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https://en.wikipedia.org/wiki/Pierre%20Laffitte
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Pierre Laffitte (21 February 1823 – 4 January 1903) was a French positivist philosopher.
Laffitte was born at Béguey, Gironde. Residing at Paris as a teacher of mathematics, he became a disciple of Auguste Comte, who appointed him his literary executor. On the schism of the Positivist body which followed Comte's death, he was recognized as head of the section which accepted the full Comtian doctrine; the other section adhered to Émile Littré, who rejected the religion of humanity as inconsistent with the philosophy of science of Comte's earlier period. From 1853 Laffitte delivered Positivist lectures in the room formerly occupied by Comte in the rue Monsieur le Prince. He published Les Grands Types de l'humanité (1875) and Cours de philosophie première (1889). In 1893 he was appointed to the new chair founded at the Collège de France for the exposition of the general history of science, and it was largely due to his inspiration that a statue to Comte was erected in the Place de la Sorbonne in 1902. Laffitte died in Paris.
Lafitte with a delegation of positivists visited Constantinople in 1877 visited Midhat Pasha to advocate positivist principles as a non-Christian, modern system.
References
M. S. Ozervarli: Positivism in the Late Ottoman Empire, The Young Turks as Mediators, Johannes Feichtinger et al,: The Worlds of Positivism: a Global Intellectual History, 1770-1930 Palgrave MacMillan, 2018
1823 births
1903 deaths
People from Gironde
19th-century French philosophers
19th-century French historians
Academic staff of the Collège de France
Comtism
Members of the Ligue de la patrie française
Burials at Père Lachaise Cemetery
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https://en.wikipedia.org/wiki/Winged%20edge
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In computer graphics, the winged edge data structure is a way to represent polygon meshes in computer memory. It is a type of boundary representation and describes both the geometry and topology of a model. Three types of records are used: vertex records, edge records, and face records. Given a reference to an edge record, one can answer several types of adjacency queries (queries about neighboring edges, vertices and faces) in constant time. This kind of adjacency information is useful for algorithms such as subdivision surface.
Features
The winged edge data structure explicitly describes the geometry and topology of faces, edges, and vertices when three or more surfaces come together and meet at a common edge. The ordering is such that the surfaces are ordered counter-clockwise with respect to the innate orientation of the intersection edge. Moreover the representation allows numerically unstable situations like that depicted below.
The winged edge data structure allows for quick traversal between faces, edges, and vertices due to the explicitly linked structure of the network. It serves adjacency queries in constant time with little storage overhead. This rich form of specifying an unstructured grid is in contrast to simpler specifications of polygon meshes such as a node and element list, or the implied connectivity of a regular grid. An alternative to the winged edge data structure is the Half-edge data structure.
Structure and pseudocode
The face and vertex records are relatively simple, while the edge record is more complex.
For each vertex, its record stores only the vertex's position (e.g. coordinates) and a reference to one incident edge. The other edges can be found by following further references in the edge.
Similarly each face record only stores a reference to one of the edges surrounding the face. There is no need to store the direction of the edge relative to the face (CCW or CW) as the face can be trivially compared to the edge's own left and right faces to obtain this information.
Finally, the structure of the edge record is as follows. An edge is assumed to be directed. The edge record contains two references to the vertices that make up the endpoints of the edge, two references to the faces on either side of the edge, and four references to the previous and next edges surrounding the left and right face.
In short, the edge record has references to all its adjacent records, both when traversing around an adjacent vertex or around an adjacent face.
class Edge
{
Vertex *vert_origin, *vert_destination;
Face *face_left, *face_right;
Edge *edge_left_cw,
*edge_left_ccw,
*edge_right_cw,
*edge_right_ccw;
}
class Vertex
{
float x, y, z;
Edge *edge;
}
class Face
{
Edge *edge;
}
See also
Quad-edge data structure
Combinatorial maps
Doubly connected edge list
Doubly linked face list
Half-edge data structure
External links
Computer-aided design
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https://en.wikipedia.org/wiki/Borel%20summation
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In mathematics, Borel summation is a summation method for divergent series, introduced by . It is particularly useful for summing divergent asymptotic series, and in some sense gives the best possible sum for such series. There are several variations of this method that are also called Borel summation, and a generalization of it called Mittag-Leffler summation.
Definition
There are (at least) three slightly different methods called Borel summation. They differ in which series they can sum, but are consistent, meaning that if two of the methods sum the same series they give the same answer.
Throughout let denote a formal power series
and define the Borel transform of to be its equivalent exponential series
Borel's exponential summation method
Let denote the partial sum
A weak form of Borel's summation method defines the Borel sum of to be
If this converges at to some function , we say that the weak Borel sum of converges at , and write .
Borel's integral summation method
Suppose that the Borel transform converges for all positive real numbers to a function growing sufficiently slowly that the following integral is well defined (as an improper integral), the Borel sum of is given by
If the integral converges at to some , we say that the Borel sum of converges at , and write .
Borel's integral summation method with analytic continuation
This is similar to Borel's integral summation method, except that the Borel transform need not converge for all , but converges to an analytic function of near 0 that can be analytically continued along the positive real axis.
Basic properties
Regularity
The methods and are both regular summation methods, meaning that whenever converges (in the standard sense), then the Borel sum and weak Borel sum also converge, and do so to the same value. i.e.
Regularity of is easily seen by a change in order of integration, which is valid due to absolute convergence: if is convergent at , then
where the rightmost expression is exactly the Borel sum at .
Regularity of and imply that these methods provide analytic extensions to .
Nonequivalence of Borel and weak Borel summation
Any series that is weak Borel summable at is also Borel summable at . However, one can construct examples of series which are divergent under weak Borel summation, but which are Borel summable. The following theorem characterises the equivalence of the two methods.
Theorem ().
Let be a formal power series, and fix , then:
If , then .
If , and then .
Relationship to other summation methods
is the special case of Mittag-Leffler summation with .
can be seen as the limiting case of generalized Euler summation method in the sense that as the domain of convergence of the method converges up to the domain of convergence for .
Uniqueness theorems
There are always many different functions with any given asymptotic expansion. However, there is sometimes a best possible function, in the sense that the errors in the
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https://en.wikipedia.org/wiki/Mathematical%20methods%20in%20electronics
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Mathematical methods are integral to the study of electronics.
Mathematics in electronics
Electronics engineering careers usually include courses in calculus (single and multivariable), complex analysis, differential equations (both ordinary and partial), linear algebra and probability. Fourier analysis and Z-transforms are also subjects which are usually included in electrical engineering programs. Laplace transform can simplify computing RLC circuit behaviour.
Basic applications
A number of electrical laws apply to all electrical networks. These include
Faraday's law of induction: Any change in the magnetic environment of a coil of wire will cause a voltage (emf) to be "induced" in the coil.
Gauss's Law: The total of the electric flux out of a closed surface is equal to the charge enclosed divided by the permittivity.
Kirchhoff's current law: the sum of all currents entering a node is equal to the sum of all currents leaving the node or the sum of total current at a junction is zero
Kirchhoff's voltage law: the directed sum of the electrical potential differences around a circuit must be zero.
Ohm's law: the voltage across a resistor is the product of its resistance and the current flowing through it.at constant temperature.
Norton's theorem: any two-terminal collection of voltage sources and resistors is electrically equivalent to an ideal current source in parallel with a single resistor.
Thévenin's theorem: any two-terminal combination of voltage sources and resistors is electrically equivalent to a single voltage source in series with a single resistor.
Millman's theorem: the voltage on the ends of branches in parallel is equal to the sum of the currents flowing in every branch divided by the total equivalent conductance.
See also Analysis of resistive circuits.
Circuit analysis is the study of methods to solve linear systems for an unknown variable.
Circuit analysis
Components
There are many electronic components currently used and they all have their own uses and particular rules and methods for use.
Electronic components
Complex numbers and Complex Analysis
If you apply a voltage across a capacitor, it 'charges up' by storing the electrical charge as an electrical field inside the device. This means that while the voltage across the capacitor remains initially small, a large current flows. Later, the current flow is smaller because the capacity is filled, and the voltage raises across the device. Complex Analysis methods are also important in electrical engineering in fields such as signal processing, power electronics, control systems, and others
A similar though opposite situation occurs in an inductor; the applied voltage remains high with low current as a magnetic field is generated, and later becomes small with high current when the magnetic field is at maximum.
The voltage and current of these two types of devices are therefore out of phase, they do not rise and fall together as simple resistor networks do. The mathema
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https://en.wikipedia.org/wiki/Staghorn
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Staghorn may refer to:
The Horn (anatomy) of a stag
Calocera viscosa, a fungus commonly called Yellow Stagshorn or Stagshorn Fungus
Staghorn calculus, a type of kidney stone
Staghorn coral, a branching coral
Rhus typhina, a shrub commonly called Staghorn sumac
Lycopodium clavatum, a moss commonly called Staghorn moss
Platycerium, a fern commonly called Staghorn fern
Pacific staghorn sculpin, a type of fish
Staghorn (He-Man), an action figure from the Mattel
Struvite, a type of kidney stone, also referred to as Staghorn calculus
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https://en.wikipedia.org/wiki/List%20of%20insurance%20companies%20in%20Hong%20Kong
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This is a list of insurance companies in Hong Kong.
See also
List of banks in Hong Kong
References
External links
Statistics, Office of the Commissioner of Insurance
Insurance Companies in Hong Kong
Insurance
Hong Kong
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https://en.wikipedia.org/wiki/Algebra%20of%20physical%20space
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In physics, the algebra of physical space (APS) is the use of the Clifford or geometric algebra Cl3,0(R) of the three-dimensional Euclidean space as a model for (3+1)-dimensional spacetime, representing a point in spacetime via a paravector (3-dimensional vector plus a 1-dimensional scalar).
The Clifford algebra Cl3,0(R) has a faithful representation, generated by Pauli matrices, on the spin representation C2; further, Cl3,0(R) is isomorphic to the even subalgebra Cl(R) of the Clifford algebra Cl3,1(R).
APS can be used to construct a compact, unified and geometrical formalism for both classical and quantum mechanics.
APS should not be confused with spacetime algebra (STA), which concerns the Clifford algebra Cl1,3(R) of the four-dimensional Minkowski spacetime.
Special relativity
Spacetime position paravector
In APS, the spacetime position is represented as the paravector
where the time is given by the scalar part , and e1, e2, e3 are the standard basis for position space. Throughout, units such that are used, called natural units. In the Pauli matrix representation, the unit basis vectors are replaced by the Pauli matrices and the scalar part by the identity matrix. This means that the Pauli matrix representation of the space-time position is
Lorentz transformations and rotors
The restricted Lorentz transformations that preserve the direction of time and include rotations and boosts can be performed by an exponentiation of the spacetime rotation biparavector W
In the matrix representation, the Lorentz rotor is seen to form an instance of the SL(2,C) group (special linear group of degree 2 over the complex numbers), which is the double cover of the Lorentz group. The unimodularity of the Lorentz rotor is translated in the following condition in terms of the product of the Lorentz rotor with its Clifford conjugation
This Lorentz rotor can be always decomposed in two factors, one Hermitian , and the other unitary , such that
The unitary element R is called a rotor because this encodes rotations, and the Hermitian element B encodes boosts.
Four-velocity paravector
The four-velocity, also called proper velocity, is defined as the derivative of the spacetime position paravector with respect to proper time τ:
This expression can be brought to a more compact form by defining the ordinary velocity as
and recalling the definition of the gamma factor:
so that the proper velocity is more compactly:
The proper velocity is a positive unimodular paravector, which implies the following condition in terms of the Clifford conjugation
The proper velocity transforms under the action of the Lorentz rotor L as
Four-momentum paravector
The four-momentum in APS can be obtained by multiplying the proper velocity with the mass as
with the mass shell condition translated into
Classical electrodynamics
The electromagnetic field, potential, and current
The electromagnetic field is represented as a bi-paravector F:
with the Hermitian part repre
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https://en.wikipedia.org/wiki/Velidhoo%20%28Noonu%20Atoll%29
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N.Velidhoo (Dhivehi: ވެލިދޫ) is one of the inhabited islands of Noonu Atoll in the Maldives. Information from Maldives bureau of statistics
History
There exists little to no information, because of the poor understanding of the history behind the island. However it has been said that the first colonisers of this island were indians in the 5th century. This island has little to no historic landmarks, but there is one. It brings the island together, it is a tree called (Bodu Nikagas). The tree is over 300 years old and the oldest and the tallest tree in the island. This tree is located besides the island preschool (HINZ Preschool) and in the island's Volleyball court.
Demography
The islands population has started to grow rapidly.
Governance
Government Offices
This island has most government offices just like the capital island.
The Secretariat of the Velidhoo Council
Velidhoo Police Station
Noonu Velidhoo Magistrate Court
Noonu Atoll Education Centre
Velidhoo Health Centre
FENAKA Corporation - Velidhoo Branch
Maldives Post - Noonu Velidhoo Branch
Bank of Maldives - Velidhoo Branch
Velidhoo Waste Management Centre
Island chiefs
Below is a list of people who have filled the post of island chief (dates are provided where known).
Katheebu Kaleyge – Muththoshige Gasim Fulhu (late) – Muththosige
Ibrahim Moosa Kaleyfaanu (Naibu dhonkokko) (late) – Seesange
Hassan Kaleyfaanu (late) – Muththosige
Easa Gasim (late) – Daylight
Mohamed Gasim (late) – Ochidmaage
Abdul Samed Adam Kaleyfanu – Lubomage (3 September 1977 – 7 April 1982)
Mohamed Hassan (Magoodhu Mohamed Fulhu) (late) – Dorensyvila (31 March 1970)
Mohamed Hassan (late)– Muththoshige (3 June 1964 – 9 July 1977)
Mohamed Abubakur (late) – Hilihilage (19 June 1982 – 31 July 1989)
Ahmed Waheed (late) (Tholhendhu Waheed) (29 August 1970 – 14 June 1975)
Abdul Azeez Moosa Kaleyfaanu – Oasanvilla (late) (6 July 1978 – 19 July 1979)
Ali Moosa Kaleyfanu (late) – Elpaso (6 September 1976 – 16 May 1978)
Ahmed Wasif (late) – Rediyamge (2 November 1982 – 1 December 1983)
Moosa Hassanfulhu – Dheyliyaage (17 November 1965 – 1992)
Abdul Rahman Hussain – Kethi (retired)
Ibrahim Haleem Ali (late) – Bina (late) (died March 10, 2021)
Ibrahim Naeem - Waves (retired)
First Elected Velidhoo Council - 26 February 2011
Mohamed Adil - Iramaage (President)
Ahmed Saeed - Asdhoo (Vice President)
Ahmed Ziyaad - Maadhuni
Ahmed Siraaj - France
Mohamed Faiz - Athiriaage
Second Elected Velidhoo Council - 13 March 2013
President of council: Ali Ahmeen Boashimaage
Vice President: Ibrahim Ali Boashimaage
Councilor: Ali Mujthaba - Kethi
Councilor: Asma Hussain - Aahama
Councilor: Ibrahim Qasim - Fashuvimaage
Third Elected Velidhoo Council - 3 June 2017
Asma Hussain - Aahama (President)
Ahmed Jaufaru - Dearhouse (Vice President)
Adam Ali - Hasthee (Council Member)
Fourth Elected Velidhoo Council - 17 May 2021
Athif Hussain - Aahama (President)
Jeehan Mohamed - Aliha (Vice President)
Ha
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https://en.wikipedia.org/wiki/210%20%28number%29
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210 (two hundred [and] ten) is the natural number following 209 and preceding 211.
In mathematics
210 is a composite number, an abundant number, Harshad number, and the product of the first four prime numbers (2, 3, 5, and 7), and thus a primorial. It is also the least common multiple of these four prime numbers. It is the sum of eight consecutive prime numbers (13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 = 210).
It is a triangular number (following 190 and preceding 231), a pentagonal number (following 176 and preceding 247), and the second smallest to be both triangular and pentagonal (the third is 40755).
It is also an idoneal number, a pentatope number, a pronic number, and an untouchable number. 210 is also the third 71-gonal number, preceding 418. It is the first primorial number greater than 2 which is not adjacent to 2 primes (211 is prime, but 209 is not).
It is the largest number n such that all primes between n/2 and n yield a representation as a sum of two primes.
Integers between 211 and 219
211
212
213
214
215
216
217
218
219
See also
210 BC
AD 210
North American telephone area code area code 210
References
Integers
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https://en.wikipedia.org/wiki/Surface%20of%20general%20type
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In algebraic geometry, a surface of general type is an algebraic surface with Kodaira dimension 2. Because of Chow's theorem any compact complex manifold of dimension 2 and with Kodaira dimension 2 will actually be an algebraic surface, and in some sense most surfaces are in this class.
Classification
Gieseker showed that there is a coarse moduli scheme for surfaces of general type; this means that for any fixed values of the Chern numbers there is a quasi-projective scheme classifying the surfaces of general type with those Chern numbers. It remains a very difficult problem to describe these schemes explicitly, and there are few pairs of Chern numbers for which this has been done (except when the scheme is empty). There are some indications that these schemes are in general too complicated to write down explicitly: the known upper bounds for the number of components are very large, some components can be non-reduced everywhere, components may have many different dimensions, and the few pieces that have been studied explicitly tend to look rather complicated.
The study of which pairs of Chern numbers can occur for a surface of general type is known as "" and there is an almost complete answer to this question. There are several conditions that the Chern numbers of a minimal complex surface of general type must satisfy:
(as it is equal to 12χ)
(the Bogomolov-Miyaoka-Yau inequality)
where q is the irregularity of a surface (the Noether inequality).
Many (and possibly all) pairs of integers satisfying these conditions are the Chern numbers for some complex surface of general type.
By contrast, for almost complex surfaces, the only constraint is:
and this can always be realized.
Examples
This is only a small selection of the rather large number of examples of surfaces of general type that have been found. Many of the surfaces of general type that have been investigated lie on (or near) the edges of the region of possible Chern numbers. In particular Horikawa surfaces lie on or near the "Noether line", many of the surfaces listed below lie on the line the minimum possible value for general type, and surfaces on the line are all quotients of the unit ball in C2 (and are particularly hard to find).
Surfaces with χ=1
These surface which are located in the "lower left" boundary in the diagram have been studied in detail. For these surfaces with second Chern class can be any integer from 3 to 11. Surfaces with all these values are known; a few of the many examples that have been studied are:
c2 = 3: Fake projective plane (Mumford surface). The first example was found by Mumford using p-adic geometry, and there are 50 examples altogether. They have the same Betti numbers as the projective plane, but are not homeomorphic to it as their fundamental groups are infinite.
c2 = 4: Beauville surfaces are named for Arnaud Beauville and have infinite fundamental group.
c2 ≥ 4: Burniat surfaces
c2 = 10: Campedelli surfaces. Surfaces with the same
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https://en.wikipedia.org/wiki/Seven-dimensional%20cross%20product
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In mathematics, the seven-dimensional cross product is a bilinear operation on vectors in seven-dimensional Euclidean space. It assigns to any two vectors a, b in a vector also in . Like the cross product in three dimensions, the seven-dimensional product is anticommutative and is orthogonal both to a and to b. Unlike in three dimensions, it does not satisfy the Jacobi identity, and while the three-dimensional cross product is unique up to a sign, there are many seven-dimensional cross products. The seven-dimensional cross product has the same relationship to the octonions as the three-dimensional product does to the quaternions.
The seven-dimensional cross product is one way of generalizing the cross product to other than three dimensions, and it is the only other bilinear product of two vectors that is vector-valued, orthogonal, and has the same magnitude as in the 3D case. In other dimensions there are vector-valued products of three or more vectors that satisfy these conditions, and binary products with bivector results.
Multiplication table
The product can be given by a multiplication table, such as the one here. This table, due to Cayley, gives the product of orthonormal basis vectors ei and ej for each i, j from 1 to 7. For example, from the table
The table can be used to calculate the product of any two vectors. For example, to calculate the e1 component of x × y the basis vectors that multiply to produce e1 can be picked out to give
This can be repeated for the other six components.
There are 480 such tables, one for each of the products satisfying the definition. This table can be summarized by the relation
where is a completely antisymmetric tensor with a positive value +1 when ijk = 123, 145, 176, 246, 257, 347, 365.
The top left 3 × 3 corner of this table gives the cross product in three dimensions.
Definition
The cross product on a Euclidean space V is a bilinear map from V × V to V, mapping vectors x and y in V to another vector x × y also in V, where x × y has the properties
orthogonality:
magnitude:
where (x·y) is the Euclidean dot product and |x| is the Euclidean norm. The first property states that the product is perpendicular to its arguments, while the second property gives the magnitude of the product. An equivalent expression in terms of the angle θ between the vectors is
which is the area of the parallelogram in the plane of x and y with the two vectors as sides. A third statement of the magnitude condition is
if x × x = 0 is assumed as a separate axiom.
Consequences of the defining properties
Given the properties of bilinearity, orthogonality and magnitude, a nonzero cross product exists only in three and seven dimensions. This can be shown by postulating the properties required for the cross product, then deducing an equation which is only satisfied when the dimension is 0, 1, 3 or 7. In zero dimensions there is only the zero vector, while in one dimension all vectors are parallel, so in both thes
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https://en.wikipedia.org/wiki/Hadamard%20factorization%20theorem
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In mathematics, and particularly in the field of complex analysis, the Hadamard factorization theorem asserts that every entire function with finite order can be represented as a product involving its zeroes and an exponential of a polynomial. It is named for Jacques Hadamard.
The theorem may be viewed as an extension of the fundamental theorem of algebra, which asserts that every polynomial may be factored into linear factors, one for each root. It is closely related to Weierstrass factorization theorem, which does not restrict to entire functions with finite orders.
Formal statement
Define the Hadamard canonical factors Entire functions of finite order have Hadamard's canonical representation:where are those roots of that are not zero (), is the order of the zero of at (the case being taken to mean ), a polynomial (whose degree we shall call ), and is the smallest non-negative integer such that the seriesconverges. The non-negative integer is called the genus of the entire function . In this notation,In other words: If the order is not an integer, then is the integer part of . If the order is a positive integer, then there are two possibilities: or .
Furthermore, Jensen's inequality implies that its roots are distributed sparsely, with critical exponent .
For example, , and are entire functions of genus .
Critical exponent
Define the critical exponent of the roots of as the following:where is the number of roots with modulus . In other words, we have an asymptotic bound on the growth behavior of the number of roots of the function:It's clear that .
Theorem: If is an entire function with infinitely many roots, thenNote: These two equalities are purely about the limit behaviors of a real number sequence that diverges to infinity. It does not involve complex analysis.
Proposition: , by Jensen's formula.
Proof
Since is also an entire function with the same order and genus, we can wlog assume .
If has only finitely many roots, then with the function of order . Thus by an application of the Borel–Carathéodory theorem, is a polynomial of degree , and so we have .
Otherwise, has infinitely many roots. This is the tricky part and requires splitting into two cases. First show that , then show that .
Define the function where . We will study the behavior of .
Bounds on the behaviour of
In the proof, we need four bounds on :
For any , when .
For any , there exists such that when .
For any , there exists such that when .
for all , and as .
These are essentially proved in the similar way. As an example, we prove the fourth one.where is an entire function. Since it is entire, for any , it is bounded in . So inside .
Outside , we have
is well-defined
For any , we show that the sum converges uniformly over .
Since only finitely many , we can split the sum to a finite bulk and an infinite tail:The bulk term is a finite sum, so it converges uniformly. It remains to bound the tail term.
By bound (1
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https://en.wikipedia.org/wiki/Bochner%27s%20theorem
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In mathematics, Bochner's theorem (named for Salomon Bochner) characterizes the Fourier transform of a positive finite Borel measure on the real line. More generally in harmonic analysis, Bochner's theorem asserts that under Fourier transform a continuous positive-definite function on a locally compact abelian group corresponds to a finite positive measure on the Pontryagin dual group. The case of sequences was first established by Gustav Herglotz (see also the related Herglotz representation theorem.)
The theorem for locally compact abelian groups
Bochner's theorem for a locally compact abelian group G, with dual group , says the following:
Theorem For any normalized continuous positive-definite function f on G (normalization here means that f is 1 at the unit of G), there exists a unique probability measure μ on such that
i.e. f is the Fourier transform of a unique probability measure μ on . Conversely, the Fourier transform of a probability measure on is necessarily a normalized continuous positive-definite function f on G. This is in fact a one-to-one correspondence.
The Gelfand–Fourier transform is an isomorphism between the group C*-algebra C*(G) and C0(Ĝ). The theorem is essentially the dual statement for states of the two abelian C*-algebras.
The proof of the theorem passes through vector states on strongly continuous unitary representations of G (the proof in fact shows that every normalized continuous positive-definite function must be of this form).
Given a normalized continuous positive-definite function f on G, one can construct a strongly continuous unitary representation of G in a natural way: Let F0(G) be the family of complex-valued functions on G with finite support, i.e. h(g) = 0 for all but finitely many g. The positive-definite kernel K(g1, g2) = f(g1 − g2) induces a (possibly degenerate) inner product on F0(G). Quotiening out degeneracy and taking the completion gives a Hilbert space
whose typical element is an equivalence class [h]. For a fixed g in G, the "shift operator" Ug defined by (Ug)(h) (g') = h(g − g), for a representative of [h], is unitary. So the map
is a unitary representations of G on . By continuity of f, it is weakly continuous, therefore strongly continuous. By construction, we have
where [e] is the class of the function that is 1 on the identity of G and zero elsewhere. But by Gelfand–Fourier isomorphism, the vector state on C*(G) is the pull-back of a state on , which is necessarily integration against a probability measure μ. Chasing through the isomorphisms then gives
On the other hand, given a probability measure μ on , the function
is a normalized continuous positive-definite function. Continuity of f follows from the dominated convergence theorem. For positive-definiteness, take a nondegenerate representation of . This extends uniquely to a representation of its multiplier algebra and therefore a strongly continuous unitary representation Ug. As above we have f g
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https://en.wikipedia.org/wiki/Normal%20bundle
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In differential geometry, a field of mathematics, a normal bundle is a particular kind of vector bundle, complementary to the tangent bundle, and coming from an embedding (or immersion).
Definition
Riemannian manifold
Let be a Riemannian manifold, and a Riemannian submanifold. Define, for a given , a vector to be normal to whenever for all (so that is orthogonal to ). The set of all such is then called the normal space to at .
Just as the total space of the tangent bundle to a manifold is constructed from all tangent spaces to the manifold, the total space of the normal bundle to is defined as
.
The conormal bundle is defined as the dual bundle to the normal bundle. It can be realised naturally as a sub-bundle of the cotangent bundle.
General definition
More abstractly, given an immersion (for instance an embedding), one can define a normal bundle of N in M, by at each point of N, taking the quotient space of the tangent space on M by the tangent space on N. For a Riemannian manifold one can identify this quotient with the orthogonal complement, but in general one cannot (such a choice is equivalent to a section of the projection ).
Thus the normal bundle is in general a quotient of the tangent bundle of the ambient space restricted to the subspace.
Formally, the normal bundle to N in M is a quotient bundle of the tangent bundle on M: one has the short exact sequence of vector bundles on N:
where is the restriction of the tangent bundle on M to N (properly, the pullback of the tangent bundle on M to a vector bundle on N via the map ). The fiber of the normal bundle in is referred to as the normal space at (of in ).
Conormal bundle
If is a smooth submanifold of a manifold , we can pick local coordinates around such that is locally defined by ; then with this choice of coordinates
and the ideal sheaf is locally generated by . Therefore we can define a non-degenerate pairing
that induces an isomorphism of sheaves . We can rephrase this fact by introducing the conormal bundle defined via the conormal exact sequence
,
then , viz. the sections of the conormal bundle are the cotangent vectors to vanishing on .
When is a point, then the ideal sheaf is the sheaf of smooth germs vanishing at and the isomorphism reduces to the definition of the tangent space in terms of germs of smooth functions on
.
Stable normal bundle
Abstract manifolds have a canonical tangent bundle, but do not have a normal bundle: only an embedding (or immersion) of a manifold in another yields a normal bundle.
However, since every manifold can be embedded in , by the Whitney embedding theorem, every manifold admits a normal bundle, given such an embedding.
There is in general no natural choice of embedding, but for a given M, any two embeddings in for sufficiently large N are regular homotopic, and hence induce the same normal bundle. The resulting class of normal bundles (it is a class of bundles and not a specific bundle because N cou
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https://en.wikipedia.org/wiki/Kjartan%20Poskitt
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Kjartan Poskitt (born 15 May 1956 in York) is a British writer and TV presenter who is best known for writing the Murderous Maths children's series of books.
Early life and education
Poskitt was born in York, England, grew up in Selby, Yorkshire and was educated at the Selby Abbey School, at Terrington Hall, North Yorkshire, and at Bootham School, York, before studying engineering at Collingwood College, Durham University.
Career
Poskitt was a member of a comedy group while at Durham University, and toured with it to the US in 1979, playing the piano and singing, performing skits and doing a ventriloquist act with a dummy which wouldn't speak. He participated in the National Student Drama Festival from 1976-1980, and also performed solo at the Edinburgh Fringe Festival from 1979 to 1981. He was a member of the pop group Candlewick Green in the early 1980s.
He has written and directed a number of pantomimes performed by the National Student Theatre Company, including Jack and the Beanstalk (1979), The Sleeping Beauty (1980), Cinderella (1981), Jack and the Beanstalk (1983), as well as a nativity play, The Road to Bethlehem (1980), and a "musical ghost pantomime", Sammy's Magic Garden (1985).
Early children's television appearances included Swap Shop during its Edinburgh Festival broadcasts, and ITV Yorkshire's Behind the Bike Sheds.
Writing
In addition to his science and maths books, which include books on Isaac Newton (Dead Famous: Isaac Newton and his Apple), and various galaxy puzzles, practical jokes and secret codes, Poskitt has written a book of magical tricks (Magic Tricks with Underpants, Scholastic 2004), a GCSE Maths support book and four Rosie and Jim annuals. In 2007, Poskitt published the first in a series of children's novels called Urgum the Axeman. Since then he has written the "Borgon the Axeboy" series and the award winning "Agatha Parrot" series.
He also wrote the theme tune for the children's art program SMart, as well as the title theme and music for the first two series of Brum. He is also the creator of a logic puzzle, Kjarposko.
He has been a presenter for a number of (mostly BBC) educational children's TV shows.
See also
List of children's non-fiction writers
Murderous Maths
References
External links
Official website
Murderous Maths official website
Interview with Poskitt on Scholastic website
British children's writers
British non-fiction writers
Living people
People from Aylesbury
1956 births
English people of Hungarian descent
British male writers
Alumni of Collingwood College, Durham
Male non-fiction writers
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https://en.wikipedia.org/wiki/Different%20ideal
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In algebraic number theory, the different ideal (sometimes simply the different) is defined to measure the (possible) lack of duality in the ring of integers of an algebraic number field K, with respect to the field trace. It then encodes the ramification data for prime ideals of the ring of integers. It was introduced by Richard Dedekind in 1882.
Definition
If OK is the ring of integers of K, and tr denotes the field trace from K to the rational number field Q, then
is an integral quadratic form on OK. Its discriminant as quadratic form need not be +1 (in fact this happens only for the case K = Q). Define the inverse different or codifferent or Dedekind's complementary module as the set I of x ∈ K such that tr(xy) is an integer for all y in OK, then I is a fractional ideal of K containing OK. By definition, the different ideal δK is the inverse fractional ideal I−1: it is an ideal of OK.
The ideal norm of δK is equal to the ideal of Z generated by the field discriminant DK of K.
The different of an element α of K with minimal polynomial f is defined to be δ(α) = f′(α) if α generates the field K (and zero otherwise): we may write
where the α(i) run over all the roots of the characteristic polynomial of α other than α itself. The different ideal is generated by the differents of all integers α in OK. This is Dedekind's original definition.
The different is also defined for a finite degree extension of local fields. It plays a basic role in Pontryagin duality for p-adic fields.
Relative different
The relative different δL / K is defined in a similar manner for an extension of number fields L / K. The relative norm of the relative different is then equal to the relative discriminant ΔL / K. In a tower of fields L / K / F the relative differents are related by δL / F = δL / KδK / F.
The relative different equals the annihilator of the relative Kähler differential module :
The ideal class of the relative different δL / K is always a square in the class group of OL, the ring of integers of L. Since the relative discriminant is the norm of the relative different it is the square of a class in the class group of OK: indeed, it is the square of the Steinitz class for OL as a OK-module.
Ramification
The relative different encodes the ramification data of the field extension L / K. A prime ideal p of K ramifies in L if the factorisation of p in L contains a prime of L to a power higher than 1: this occurs if and only if p divides the relative discriminant ΔL / K. More precisely, if
p = P1e(1) ... Pke(k)
is the factorisation of p into prime ideals of L then Pi divides the relative different δL / K if and only if Pi is ramified, that is, if and only if the ramification index e(i) is greater than 1. The precise exponent to which a ramified prime P divides δ is termed the differential exponent of P and is equal to e − 1 if P is tamely ramified: that is, when P does not divide e. In the case when P is wildly ramified the differential expo
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https://en.wikipedia.org/wiki/Takeuti%27s%20conjecture
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In mathematics, Takeuti's conjecture is the conjecture of Gaisi Takeuti that a sequent formalisation of second-order logic has cut-elimination (Takeuti 1953). It was settled positively:
By Tait, using a semantic technique for proving cut-elimination, based on work by Schütte (Tait 1966);
Independently by Prawitz (Prawitz 1968) and Takahashi (Takahashi 1967) by a similar technique (Takahashi 1967) - although Prawitz's and Takahashi's proofs are not limited to second-order logic, but concern higher-order logics in general;
It is a corollary of Jean-Yves Girard's syntactic proof of strong normalization for System F.
Takeuti's conjecture is equivalent to the 1-consistency of second-order arithmetic in the sense that each of the statements can be derived from each other in the weak system PRA. It is also equivalent to the strong normalization of the Girard/Reynold's System F.
See also
Hilbert's second problem
References
Dag Prawitz, 1968. Hauptsatz for higher order logic. J. Symb. Log., 33:452–457, 1968.
William W. Tait, 1966. A nonconstructive proof of Gentzen's Hauptsatz for second order predicate logic. In Bulletin of the American Mathematical Society, 72:980–983.
Gaisi Takeuti, 1953. On a generalized logic calculus. In Japanese Journal of Mathematics, 23:39–96. An errata to this article was published in the same journal, 24:149–156, 1954.
Moto-o Takahashi, 1967. A proof of cut-elimination in simple type theory. In Japanese Mathematical Society, 10:44–45.
Proof theory
Conjectures that have been proved
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https://en.wikipedia.org/wiki/Roy%20Kerr
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Roy Patrick Kerr (; born 16 May 1934) is a New Zealand mathematician who discovered the Kerr geometry, an exact solution to the Einstein field equation of general relativity. His solution models the gravitational field outside an uncharged rotating massive object, including a rotating black hole. His solution to Einstein's equations predicted spinning black holes before they were discovered.
Early life and education
Kerr was born in 1934 in Kurow, New Zealand. He was born into a dysfunctional family, and his mother was forced to leave when he was three. When his father went to war, he was sent to a farm. After his father's return from war, they moved to Christchurch. He was accepted to St Andrew's College, a private school, as his father had served under a former headmaster. Kerr's mathematical talent was first recognised while he was still a student at St Andrew's College. Although there was no mathematics teacher there at the time, he was able in 1951 to go straight into the third year of mathematics at Canterbury University College, a constituent of the University of New Zealand and the precursor to the University of Canterbury. Their regulations did not permit him to graduate until 1954 and so it was not until September 1955 that he moved to the University of Cambridge, where he earned his PhD in 1959. His dissertation concerned the equations of motion in general relativity.
Career and research
After a postdoctoral fellowship at Syracuse University, where Einstein's collaborator Peter Bergmann was a professor, he spent some time working for the United States Air Force at Wright-Patterson Air Force Base. Kerr speculated that the "main reason why the US Air Force had created a General Relativity section was probably to show the U.S. Navy that they could also do pure research."
Work at Texas and Canterbury
In 1962, Kerr joined Alfred Schild and his Relativity Group at the University of Texas at Austin. As Kerr wrote in 2009:
By the summer of 1963, Maarten Schmidt at Caltech had shown that certain starlike objects (now called quasars) were actually distant objects emitting enormous amounts of energy. Nobody understood how they could be so bright. In an effort to unravel this mystery, several hundred astronomers, astrophysicists, and general relativists gathered for a conference in Dallas, held in early December that year. This would be the First (of what since then has become the biennial) Texas Symposium on Relativistic Astrophysics.
Kerr presented to the Symposium his solution to the Einstein field equations. S. Chandrasekhar (Nobel laureate, 1983) is quoted as having said :
"In my entire scientific life, extending over forty-five years, the most shattering experience has been the realization that an exact solution of Einstein's equations of general relativity, discovered by the New Zealand mathematician, Roy Kerr, provides the absolutely exact representation of untold numbers of massive black holes that populate the universe"
In 1965,
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https://en.wikipedia.org/wiki/List%20of%20things%20named%20after%20Leonhard%20Euler
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In mathematics and physics, many topics are named in honor of Swiss mathematician Leonhard Euler (1707–1783), who made many important discoveries and innovations. Many of these items named after Euler include their own unique function, equation, formula, identity, number (single or sequence), or other mathematical entity. Many of these entities have been given simple and ambiguous names such as Euler's function, Euler's equation, and Euler's formula.
Euler's work touched upon so many fields that he is often the earliest written reference on a given matter. In an effort to avoid naming everything after Euler, some discoveries and theorems are attributed to the first person to have proved them after Euler.
Conjectures
Euler's conjecture (Waring's problem)
Euler's sum of powers conjecture
Euler's Graeco-Latin square conjecture
Equations
Usually, Euler's equation refers to one of (or a set of) differential equations (DEs). It is customary to classify them into ODEs and PDEs.
Otherwise, Euler's equation may refer to a non-differential equation, as in these three cases:
Euler–Lotka equation, a characteristic equation employed in mathematical demography
Euler's pump and turbine equation
Euler transform used to accelerate the convergence of an alternating series and is also frequently applied to the hypergeometric series
Ordinary differential equations
Euler rotation equations, a set of first-order ODEs concerning the rotations of a rigid body.
Euler–Cauchy equation, a linear equidimensional second-order ODE with variable coefficients. Its second-order version can emerge from Laplace equation in polar coordinates.
Euler–Bernoulli beam equation, a fourth-order ODE concerning the elasticity of structural beams.
Euler's differential equation, a first order nonlinear ordinary differential equation
Partial differential equations
Euler conservation equations, a set of quasilinear first-order hyperbolic equations used in fluid dynamics for inviscid flows. In the (Froude) limit of no external field, they are conservation equations.
Euler–Tricomi equation – a second-order PDE emerging from Euler conservation equations.
Euler–Poisson–Darboux equation, a second-order PDE playing important role in solving the wave equation.
Euler–Lagrange equation, a second-order PDE emerging from minimization problems in calculus of variations.
Formulas
Functions
The Euler function, a modular form that is a prototypical q-series.
Euler's totient function (or Euler phi (φ) function) in number theory, counting the number of coprime integers less than an integer.
Euler hypergeometric integral
Euler–Riemann zeta function
Identities
Euler's identity .
Euler's four-square identity, which shows that the product of two sums of four squares can itself be expressed as the sum of four squares.
Euler's identity may also refer to the pentagonal number theorem.
Numbers
Euler's number, , the base of the natural logarithm
Euler's idoneal numbers, a set of 65 or possibly 66 or 67 integ
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https://en.wikipedia.org/wiki/Tangle
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Tangle may refer to:
Science, Technology, Engineering & Mathematics
The Tangle is the name of the ledger, a directed acyclic graph, used for the cryptocurrency IOTA
Tangle (mathematics), a topological object
Natural sciences & medicine
Sea tangle, another name for kelp
Neurofibrillary tangles, which occur in Alzheimer's disease
Music
Tangle (album), a 1989 album by Thinking Fellers Union Local 282
Tangle (EP), a 2016 extended play by Trash Talk
Tangles (album), a 2005 album by S. J. Tucker
Social media
tangle.com, a Christian social networking site
Fiction
Tangle (TV series), an Australian television series
Tangle, a character in The Golden Key by George MacDonald
The Tangle is a 2019 sci-fi film by Christopher Soren Kelly.
Tangle the Lemur, a character from IDW Publishing comic series Sonic the Hedgehog
"Tangles", a Hugo Award-nominated story by Seanan McGuire
See also
Tangled (disambiguation)
Knot
Rectangle
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https://en.wikipedia.org/wiki/Dihedral%20group%20of%20order%206
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In mathematics, D3 (sometimes alternatively denoted by D6) is the dihedral group of degree 3 and order 6. It equals the symmetric group S3. It is also the smallest non-abelian group.
This page illustrates many group concepts using this group as example.
Symmetry groups
The dihedral group D3 is the symmetry group of an equilateral triangle, that is, it is the set of all transformations such as reflection, rotation, and combinations of these, that leave the shape and position of this triangle fixed. In the case of D3, every possible permutation of the triangle's vertices constitutes such a transformation, so that the group of these symmetries is isomorphic to the symmetric group S3 of all permutations of three distinct elements. This is not the case for dihedral groups of higher orders.
The dihedral group D3 is isomorphic to two other symmetry groups in three dimensions:
one with a 3-fold rotation axis and a perpendicular 2-fold rotation axis (hence three of these): D3
one with a 3-fold rotation axis in a plane of reflection (and hence also in two other planes of reflection): C3v
Permutations of a set of three objects
Consider three colored blocks (red, green, and blue), initially placed in the order RGB. The symmetric group S3 is then the group of all possible rearrangements of these blocks.
If we denote by a the action "swap the first two blocks", and by b the action "swap the last two blocks", we can write all possible permutations in terms of these two actions.
In multiplicative form, we traditionally write xy for the combined action "first do y, then do x"; so that ab is the action , i.e., "take the last block and move it to the front".
If we write e for "leave the blocks as they are" (the identity action), then we can write the six permutations of the set of three blocks as the following actions:
e : RGB ↦ RGB or ()
a : RGB ↦ GRB or (RG)
b : RGB ↦ RBG or (GB)
ab : RGB ↦ BRG or (RGB)
ba : RGB ↦ GBR or (RBG)
aba : RGB ↦ BGR or (RB)
The notation in brackets is the cycle notation.
Note that the action aa has the effect , leaving the blocks as they were; so we can write .
Similarly,
bb = e,
(aba)(aba) = e, and
(ab)(ba) = (ba)(ab) = e;
so each of the above actions has an inverse.
By inspection, we can also determine associativity and closure (two of the necessary group axioms); note for example that
(ab)a = a(ba) = aba, and
(ba)b = b(ab) = bab.
The group is non-abelian since, for example, . Since it is built up from the basic actions a and b, we say that the set generates it.
The group has presentation
, also written
or
, also written
where a and b are swaps and is a cyclic permutation. Note that the second presentation means that the group is a Coxeter group. (In fact, all dihedral and symmetry groups are Coxeter groups.)
Summary of group operations
With the generators a and b, we define the additional shorthands , and , so that a, b, c, d, e, and f are all the elements of this group. We can then summarize the group op
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https://en.wikipedia.org/wiki/John%20Henry%20Michell
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John Henry Michell, FRS (26 October 1863 – 3 February 1940) was an Australian mathematician and Professor of Mathematics at the University of Melbourne.
Early life
Michell was the son of John Michell (pronounced Mitchell), a miner, and his wife Grace, née Rowse, and was born in Maldon, Victoria. His parents had migrated from Devonshire in 1854. Educated first at Maldon, he went to Wesley College, Melbourne, in 1877, where he won the Draper and Walter Powell scholarships. In 1881 he began the arts course at the University of Melbourne, and qualified for the B.A. degree at the end of 1883. He had an outstanding course, heading the list with first-class honours each year, and winning the final honour scholarship in mathematics and physics.
Michell then went to the University of Cambridge, obtained a major scholarship at Trinity College, and was bracketed senior wrangler with three others in the first part of the mathematical tripos in 1887. In the second part of the tripos in 1888, Michell was placed in division one of the first class.
University of Melbourne
Michell was elected a fellow of Trinity in 1890, but returned to Melbourne later the same year, and was appointed lecturer in mathematics at Melbourne University. He held this position for over 30 years. His academic work occupied so much of his time that it was difficult to do original research. The first of his papers, "On the theory of free streamlines", which appeared in Transactions of the Royal Society in 1890, had drawn attention to his ability as a mathematician, and during the following 12 years, about 15 papers were contributed to English mathematical journals. It was recognized that these were important contributions to the knowledge of hydrodynamics and elasticity, and in June 1902 he was elected as a Fellow of the Royal Society (FRS), London. The number of his students at the University steadily increased, but there was no corresponding staff increase for a long while. Michell continued his research work but none of it was published. In 1923 he became a professor of mathematics and, obtaining some increase in staff, established practice classes and tutorials, thus considerably improving the efficiency of his department. Michell resigned the chair at the end of 1928 and was given the title of honorary research professor. He died after a short illness on 3 February 1940 at Camberwell, and was buried at Boroondara General Cemetery. Michell remained a bachelor till his death in 1940. Michell published The Elements of Mathematical Analysis (1937), a substantial work in two volumes written in collaboration with Maurice Belz.
Legacy
Michell was regarded as a shy man and was one of the earliest graduates of an Australian university to be elected to the Royal Society. He was a good teacher, good-natured and patient with students, but his heart was really in his research work. His assistance was freely given to his engineering friends in clearing up their problems, and he did a good dea
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https://en.wikipedia.org/wiki/Mathematics%20education%20in%20New%20York
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Mathematics education in New York in regard to both content and teaching method can vary depending on the type of school a person attends. Private school math education varies between schools whereas New York has statewide public school requirements where standardized tests are used to determine if the teaching method and educator are effective in transmitting content to the students. While an individual private school can choose the content and educational method to use, New York State mandates content and methods statewide. Some public schools have and continue to use established methods, such as Montessori for teaching such required content. New York State has used various foci of content and methods of teaching math including New Math (1960s), 'back to the basics' (1970s), Whole Math (1990s), Integrated Math, and Everyday Mathematics.
How to teach math, what to teach, and its effectiveness has been a topic of debate in New York State and nationally since the "Math Wars" started in the 1960s. Often, current political events influence how and what is taught. The politics in turn influence state legislation. California, New York, and several other states have influenced textbook content produced by publishers.
The state of New York has implemented a novel curriculum for high school mathematics.
The courses Algebra I, Geometry, and Algebra II/Trigonometry are required courses mandated by the New York State Department of Education for high school graduation.
2007-present
Algebra
This is the first course in the new three-year curriculum. It was originally "Math A," but was replaced with "Integrated Algebra." In 2009 when Common Core was adopted, "Algebra I" replaced "Integrated Algebra" and is still in use today.
Students learn to how write, solve, and graph equations and inequalities. They will also learn how to solve systems of equations, as well as how to simplify exponents, quadratic equations, exponential functions, polynomials, radicals, and rational expressions. Other topics included are probability and statistics.
Some schools divided Algebra 1 into a two-year sequence. The students who receive it begin with Algebra 1A, and will cover the rest of the Algebra 1 topics in Algebra 1B in the next school year. Typically, most students are placed in Algebra 1 which covers all lessons rather than splitting them.
At the conclusion of the one-year course, students take the New York State Regents Exam. The first administration of this exam was in June 2008.
Geometry
This is the second course in the new three year curriculum. It replaced part of "Math A" and part of "Math B"
Geometric concepts such as right triangles are introduced. The course also covers topics including perpendicular and parallel lines, triangles, quadrilaterals, and transformations.
At the conclusion of this one-year course, students take a New York State Regents exam in Geometry. The first administration of this exam was in June 2009.
Algebra II
This is the third and
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https://en.wikipedia.org/wiki/Rational%20normal%20curve
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In mathematics, the rational normal curve is a smooth, rational curve of degree in projective n-space . It is a simple example of a projective variety; formally, it is the Veronese variety when the domain is the projective line. For it is the plane conic and for it is the twisted cubic. The term "normal" refers to projective normality, not normal schemes. The intersection of the rational normal curve with an affine space is called the moment curve.
Definition
The rational normal curve may be given parametrically as the image of the map
which assigns to the homogeneous coordinates the value
In the affine coordinates of the chart the map is simply
That is, the rational normal curve is the closure by a single point at infinity of the affine curve
Equivalently, rational normal curve may be understood to be a projective variety, defined as the common zero locus of the homogeneous polynomials
where are the homogeneous coordinates on . The full set of these polynomials is not needed; it is sufficient to pick of these to specify the curve.
Alternate parameterization
Let be distinct points in . Then the polynomial
is a homogeneous polynomial of degree with distinct roots. The polynomials
are then a basis for the space of homogeneous polynomials of degree . The map
or, equivalently, dividing by
is a rational normal curve. That this is a rational normal curve may be understood by noting that the monomials
are just one possible basis for the space of degree homogeneous polynomials. In fact, any basis will do. This is just an application of the statement that any two projective varieties are projectively equivalent if they are congruent modulo the projective linear group (with the field over which the projective space is defined).
This rational curve sends the zeros of to each of the coordinate points of ; that is, all but one of the vanish for a zero of . Conversely, any rational normal curve passing through the coordinate points may be written parametrically in this way.
Properties
The rational normal curve has an assortment of nice properties:
Any points on are linearly independent, and span . This property distinguishes the rational normal curve from all other curves.
Given points in in linear general position (that is, with no lying in a hyperplane), there is a unique rational normal curve passing through them. The curve may be explicitly specified using the parametric representation, by arranging of the points to lie on the coordinate axes, and then mapping the other two points to and .
The tangent and secant lines of a rational normal curve are pairwise disjoint, except at points of the curve itself. This is a property shared by sufficiently positive embeddings of any projective variety.
There are
independent quadrics that generate the ideal of the curve.
The curve is not a complete intersection, for . That is, it cannot be defined (as a subscheme of projective space) by only equations, that being the c
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https://en.wikipedia.org/wiki/Rational%20surface
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In algebraic geometry, a branch of mathematics, a rational surface is a surface birationally equivalent to the projective plane, or in other words a rational variety of dimension two. Rational surfaces are the simplest of the 10 or so classes of surface in the Enriques–Kodaira classification of complex surfaces,
and were the first surfaces to be investigated.
Structure
Every non-singular rational surface can be obtained by repeatedly blowing up a minimal rational surface. The minimal rational surfaces are the projective plane and the Hirzebruch surfaces Σr for r = 0 or r ≥ 2.
Invariants: The plurigenera are all 0 and the fundamental group is trivial.
Hodge diamond:
where n is 0 for the projective plane, and 1 for Hirzebruch surfaces
and greater than 1 for other rational surfaces.
The Picard group is the odd unimodular lattice I1,n, except for the Hirzebruch surfaces Σ2m when it is the even unimodular lattice II1,1.
Castelnuovo's theorem
Guido Castelnuovo proved that any complex surface such that q and P2 (the irregularity and second plurigenus) both vanish is rational. This is used in the Enriques–Kodaira classification to identify the rational surfaces. proved that Castelnuovo's theorem also holds over fields of positive characteristic.
Castelnuovo's theorem also implies that any unirational complex surface is rational, because if a complex surface is unirational then its irregularity and plurigenera are bounded by those of a rational surface and are therefore all 0, so the surface is rational. Most unirational complex varieties of dimension 3 or larger are not rational.
In characteristic p > 0 found examples of unirational surfaces (Zariski surfaces) that are not rational.
At one time it was unclear whether a complex surface such that q and P1 both vanish
is rational, but a counterexample (an Enriques surface) was found by Federigo Enriques.
Examples of rational surfaces
Bordiga surfaces: A degree 6 embedding of the projective plane into P4 defined by the quartics through 10 points in general position.
Châtelet surfaces
Coble surfaces
Cubic surfaces Nonsingular cubic surfaces are isomorphic to the projective plane blown up in 6 points, and are Fano surfaces. Named examples include the Fermat cubic, the Cayley cubic surface, and the Clebsch diagonal surface.
del Pezzo surfaces (Fano surfaces)
Enneper surface
Hirzebruch surfaces Σn
P1×P1 The product of two projective lines is the Hirzebruch surface Σ0. It is the only surface with two different rulings.
The projective plane
Segre surface An intersection of two quadrics, isomorphic to the projective plane blown up in 5 points.
Steiner surface A surface in P4 with singularities which is birational to the projective plane.
White surfaces, a generalization of Bordiga surfaces.
Veronese surface An embedding of the projective plane into P5.
See also
List of algebraic surfaces
References
External links
Le Superficie Algebriche: A tool to visually study the geograp
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https://en.wikipedia.org/wiki/Matrix%20congruence
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In mathematics, two square matrices A and B over a field are called congruent if there exists an invertible matrix P over the same field such that
PTAP = B
where "T" denotes the matrix transpose. Matrix congruence is an equivalence relation.
Matrix congruence arises when considering the effect of change of basis on the Gram matrix attached to a bilinear form or quadratic form on a finite-dimensional vector space: two matrices are congruent if and only if they represent the same bilinear form with respect to different bases.
Note that Halmos defines congruence in terms of conjugate transpose (with respect to a complex inner product space) rather than transpose, but this definition has not been adopted by most other authors.
Congruence over the reals
Sylvester's law of inertia states that two congruent symmetric matrices with real entries have the same numbers of positive, negative, and zero eigenvalues. That is, the number of eigenvalues of each sign is an invariant of the associated quadratic form.
See also
Congruence relation
Matrix similarity
Matrix equivalence
References
Linear algebra
Matrices
Equivalence (mathematics)
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https://en.wikipedia.org/wiki/Gabriel%20Mouton
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Gabriel Mouton (1618 – 28 September 1694) was a French abbot and scientist. He was a doctor of theology from Lyon, but was also interested in mathematics and astronomy. His 1670 book, the Observationes diametrorum solis et lunae apparentium, proposed a natural standard of length based on the circumference of the Earth, divided decimally. It was influential in the adoption of the metric system in 1799.
The milliare
Based on the measurements of the size of the Earth conducted by Riccioli of Bologna (at 321,815 Bologna feet to the degree), Mouton proposed a decimal system of measurement based on the circumference of the Earth, explaining the advantages of a system based on nature.
His suggestion was a unit, the milliare, that was defined as a minute of arc along a meridian arc, and a system of sub-units, dividing successively by factors of ten into the centuria, decuria, virga, virgula, decima, centesima, and millesima. The virga, 1/1000 of a minute of arc, corresponding to 64.4 Bologna inches, or ~2.04 m, was reasonably close to the then current unit of length, the Parisian toise (~1.95 m) – a feature which was meant to make acceptance of the new unit easier.
As a practical implementation, Mouton suggested that the actual standard be based on pendulum movement, so that a pendulum located in Lyon of length one virgula (1/10 virga) would change direction 3959.2 times in half an hour. The resulting pendulum would have a length of ~20.54 cm.
His ideas attracted interest at the time, and were supported by Jean Picard as well as Huygens in 1673, and also studied at Royal Society in London. In 1673, Leibniz independently made proposals similar to those of Mouton.
It would be over a century later, however, that the French Academy of Sciences weights and measures committee suggested the decimal metric system that defined the Metre as, at least initially, a division of the circumference of the Earth. The first official adoption of this system occurred in France in 1791.
By today's measures, his milliare corresponds directly to a nautical mile, and his virga would by definition have been 1.852 m.
See also
Introduction to the metric system
List of Roman Catholic scientist-clerics
Notes
References
G. Bigourdan: Le systeme metrique des poids et mesures, 1901, chapter Les precurseurs de la reforme des poids et mesures
Ferdinand Hoefer: Historie de l'astronomie, Paris 1873
External links
Metrication - Genesis
Un historique du METRE
1618 births
1694 deaths
17th-century French mathematicians
17th-century French Roman Catholic priests
Catholic clergy scientists
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https://en.wikipedia.org/wiki/Christos%20V.%20Massalas
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Christos V. Massalas is a Greek academic working in the field of mathematics and materials science. He is widely published and has held senior positions at the University of Ioannina and the University of Western Macedonia.
Biography
Massalas was born in Ioannina, Greece. After graduating as a civil engineer, Massalas continued his education with a diploma degree in mathematics (MSc, PhD, habilitation in mechanics). During his education he obtained scholarships from the Polytechnic Institute of Brooklyn, the Fulbright Foundation and UNESCO. He worked as a professor in the Department of Mathematics until 2000 when he was appointed professor of mechanics of materials at the Department of Materials Science in the University of Ioannina. Massalas has worked as a visiting professor at Trinity College, Dublin (1989–1990). He is a vice-director of the Institute B.R.I. and is a member of the scientific committee of Onassis Foundation Science Lectures. He is the author of several books, monographs and more than 100 research papers. His administrative course started in 1992 as chairman of the Department of Mathematics and was followed as vice-rector (1994–1997) at the University of Ioannina, rector (1997–2003) and the following three years (2003–2006) vice-rector of the university as well as the president of the University of Western Macedonia, Greece (2003-). In addition, he is the chairman of the Board of Higher Education (SAPE).
References
Year of birth missing (living people)
Living people
Polytechnic Institute of New York University alumni
People from Ioannina
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https://en.wikipedia.org/wiki/Runcinated%205-cell
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In four-dimensional geometry, a runcinated 5-cell is a convex uniform 4-polytope, being a runcination (a 3rd order truncation, up to face-planing) of the regular 5-cell.
There are 3 unique degrees of runcinations of the 5-cell, including with permutations, truncations, and cantellations.
Runcinated 5-cell
The runcinated 5-cell or small prismatodecachoron is constructed by expanding the cells of a 5-cell radially and filling in the gaps with triangular prisms (which are the face prisms and edge figures) and tetrahedra (cells of the dual 5-cell). It consists of 10 tetrahedra and 20 triangular prisms. The 10 tetrahedra correspond with the cells of a 5-cell and its dual.
Topologically, under its highest symmetry, [[3,3,3]], there is only one geometrical form, containing 10 tetrahedra and 20 uniform triangular prisms. The rectangles are always squares because the two pairs of edges correspond to the edges of the two sets of 5 regular tetrahedra each in dual orientation, which are made equal under extended symmetry.
E. L. Elte identified it in 1912 as a semiregular polytope.
Alternative names
Runcinated 5-cell (Norman Johnson)
Runcinated pentachoron
Runcinated 4-simplex
Expanded 5-cell/4-simplex/pentachoron
Small prismatodecachoron (Acronym: Spid) (Jonathan Bowers)
Structure
Two of the ten tetrahedral cells meet at each vertex. The triangular prisms lie between them, joined to them by their triangular faces and to each other by their square faces. Each triangular prism is joined to its neighbouring triangular prisms in anti orientation (i.e., if edges A and B in the shared square face are joined to the triangular faces of one prism, then it is the other two edges that are joined to the triangular faces of the other prism); thus each pair of adjacent prisms, if rotated into the same hyperplane, would form a gyrobifastigium.
Configuration
Seen in a configuration matrix, all incidence counts between elements are shown. The diagonal f-vector numbers are derived through the Wythoff construction, dividing the full group order of a subgroup order by removing one mirror at a time.
Dissection
The runcinated 5-cell can be dissected by a central cuboctahedron into two tetrahedral cupola. This dissection is analogous to the 3D cuboctahedron being dissected by a central hexagon into two triangular cupola.
Images
Coordinates
The Cartesian coordinates of the vertices of an origin-centered runcinated 5-cell with edge length 2 are:
An alternate simpler set of coordinates can be made in 5-space, as 20 permutations of:
(0,1,1,1,2)
This construction exists as one of 32 orthant facets of the runcinated 5-orthoplex.
A second construction in 5-space, from the center of a rectified 5-orthoplex is given by coordinate permutations of:
(1,-1,0,0,0)
Root vectors
Its 20 vertices represent the root vectors of the simple Lie group A4. It is also the vertex figure for the 5-cell honeycomb in 4-space.
Cross-sections
The maximal cross-section of the runci
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https://en.wikipedia.org/wiki/Runcinated%20tesseracts
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In four-dimensional geometry, a runcinated tesseract (or runcinated 16-cell) is a convex uniform 4-polytope, being a runcination (a 3rd order truncation) of the regular tesseract.
There are 4 variations of runcinations of the tesseract including with permutations truncations and cantellations.
Runcinated tesseract
The runcinated tesseract or (small) disprismatotesseractihexadecachoron has 16 tetrahedra, 32 cubes, and 32 triangular prisms. Each vertex is shared by 4 cubes, 3 triangular prisms and one tetrahedron.
Construction
The runcinated tesseract may be constructed by expanding the cells of a tesseract radially, and filling in the gaps with tetrahedra (vertex figures), cubes (face prisms), and triangular prisms (edge figure prisms). The same process applied to a 16-cell also yields the same figure.
Cartesian coordinates
The Cartesian coordinates of the vertices of the runcinated tesseract with edge length 2 are all permutations of:
Images
Structure
Eight of the cubical cells are connected to the other 24 cubical cells via all 6 square faces. The other 24 cubical cells are connected to the former 8 cells via only two opposite square faces; the remaining 4 faces are connected to the triangular prisms. The triangular prisms are connected to the tetrahedra via their triangular faces.
The runcinated tesseract can be dissected into 2 cubic cupolae and a rhombicuboctahedral prism between them. This dissection can be seen analogous to the 3D rhombicuboctahedron being dissected into two square cupola and a central octagonal prism.
Projections
The cube-first orthographic projection of the runcinated tesseract into 3-dimensional space has a (small) rhombicuboctahedral envelope. The images of its cells are laid out within this envelope as follows:
The nearest and farthest cube from the 4d viewpoint projects to a cubical volume in the center of the envelope.
Six cuboidal volumes connect this central cube to the 6 axial square faces of the rhombicuboctahedron. These are the images of 12 of the cubical cells (each pair of cubes share an image).
The 18 square faces of the envelope are the images of the other cubical cells.
The 12 wedge-shaped volumes connecting the edges of the central cube to the non-axial square faces of the envelope are the images of 24 of the triangular prisms (a pair of cells per image).
The 8 triangular faces of the envelope are the images of the remaining 8 triangular prisms.
Finally, the 8 tetrahedral volumes connecting the vertices of the central cube to the triangular faces of the envelope are the images of the 16 tetrahedra (again, a pair of cells per image).
This layout of cells in projection is analogous to the layout of the faces of the (small) rhombicuboctahedron under projection to 2 dimensions. The rhombicuboctahedron is also constructed from the cube or the octahedron in an analogous way to the runcinated tesseract. Hence, the runcinated tesseract may be thought of as the 4-dimensional analogue of the rho
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https://en.wikipedia.org/wiki/Pierre%20Dusart
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Pierre Dusart is a French mathematician at the Université de Limoges who specializes in number theory.
He has published in several countries, specially in South Korea, with his colleague Damien Sauveron who is associate professor in Computer Sciences at the Université de Limoges.
External links
Résumé and thesis: (French)
"The kth prime is greater than k(ln k + ln ln k-1) for k>=2". Mathematics of Computation 68 (1999), pp. 411–415.
"ESTIMATES OF SOME FUNCTIONS OVER PRIMES".
Notes and references
French mathematicians
Living people
Year of birth missing (living people)
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https://en.wikipedia.org/wiki/Thomas%20Callister%20Hales
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Thomas Callister Hales (born June 4, 1958) is an American mathematician working in the areas of representation theory, discrete geometry, and formal verification. In representation theory he is known for his work on the Langlands program and the proof of the fundamental lemma over the group Sp(4) (many of his ideas were incorporated into the final proof of the fundamental lemma, due to Ngô Bảo Châu). In discrete geometry, he settled the Kepler conjecture on the density of sphere packings and the honeycomb conjecture. In 2014, he announced the completion of the Flyspeck Project, which formally verified the correctness of his proof of the Kepler conjecture.
Biography
He received his Ph.D. from Princeton University in 1986 with a dissertation titled The Subregular Germ of Orbital Integrals. Hales taught at Harvard University and the University of Chicago, and from 1993 and 2002 he worked at the University of Michigan.
In 1998, Hales submitted his paper on the computer-aided proof of the Kepler conjecture, a centuries-old problem in discrete geometry which states that the most space-efficient way to pack spheres is in a tetrahedron shape. He was aided by graduate student Samuel Ferguson. In 1999, Hales proved the honeycomb conjecture, and also stated that the conjecture may have been in the minds of mathematicians before Marcus Terentius Varro.
After 2002, Hales became the University of Pittsburgh's Mellon Professor of Mathematics. In 2003, Hales started work on Flyspeck to vindicate his proof of the Kepler conjecture. His proof relied on computer calculation to verify conjectures. The project used two proof assistants, HOL Light and Isabelle. Annals of Mathematics accepted the proof in 2005; but was only 99% sure of the proof. In August 2014, the Flyspeck team's software finally verified the proof to be correct.
In 2017, he initiated the Formal Abstracts project which aims to provide formalised statements of the main results of each mathematical research paper in the language of an interactive theorem prover. The goal of this project is to benefit from the increased precision and interoperability that computer formalisation provides while circumventing the effort that a full-scale formalisation of all published proofs currently entails. In the long term, the project hopes to build a corpus of mathematical facts which would allow for the application of machine learning techniques in interactive and automated theorem proving.
Awards and memberships
Hales won the Chauvenet Prize in 2003 and a Lester R. Ford Award in 2008. In 2012 he became a fellow of the American Mathematical Society. He was invited to give the Tarski Lectures in 2019. His three lectures were titled "A formal proof of the Kepler conjecture", "Formalizing mathematics", and "Integrating with Logic".
Publications
Notes
External links
20th-century American mathematicians
21st-century American mathematicians
1958 births
Living people
University of Michigan faculty
Princeton Uni
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https://en.wikipedia.org/wiki/L%C3%A1szl%C3%B3%20Fejes%20T%C3%B3th
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László Fejes Tóth (, 12 March 1915 – 17 March 2005) was a Hungarian mathematician who specialized in geometry. He proved that a lattice pattern is the most efficient way to pack centrally symmetric convex sets on the Euclidean plane (a generalization of Thue's theorem, a 2-dimensional analog of the Kepler conjecture). He also investigated the sphere packing problem. He was the first to show, in 1953, that proof of the Kepler conjecture can be reduced to a finite case analysis and, later, that the problem might be solved using a computer.
He was a member of the Hungarian Academy of Sciences (from 1962) and a director of the Alfréd Rényi Institute of Mathematics (1970-1983). He received both the Kossuth Prize (1957) and State Award (1973).
Together with H.S.M. Coxeter and Paul Erdős, he laid the foundations of discrete geometry.
Early life and career
As described in a 1999 interview with István Hargittai, Fejes Tóth's father was a railway worker, who advanced in his career within the railway organization ultimately to earn a doctorate in law. Fejes Tóth's mother taught Hungarian and German literature in a high school. The family moved to Budapest, when Fejes Tóth was five; there he attended elementary school and high school—the Széchenyi István Reálgimnázium—where his interest in mathematics began.
Fejes Tóth attended Pázmány Péter University, now the Eötvös Loránd University. As a freshman, he developed a generalized solution regarding Cauchy exponential series, which he published in the proceedings of the French Academy of Sciences—1935. He then received his doctorate at Pázmány Péter University, under the direction of Lipót Fejér.
After university, he served as a soldier for two years, but received a medical exemption. In 1941 he joined the University of Kolozsvár (Cluj). It was here that he became interested in packing problems. In 1944, he returned to Budapest to teach mathematics at Árpád High School. Between 1946 and 1949 he lectured at Pázmány Péter University and starting in 1949 became a professor at the University of Veszprém (now University of Pannonia) for 15 years, where he was the primary developer of the "geometric patterns" theory "of the plane, the sphere and the surface space" and where he "had studied non grid-like structures and quasicrystals" which later became an independent discipline, as reported by János Pach.
The editors of a book dedicated to Fejes Tóth described some highlights of his early work; e.g. having shown that the maximum density of a packing of repeated symmetric convex bodies occurs with a lattice pattern of packing. He also showed that, of all convex polytopes of given surface area that are equivalent to a given Platonic solid (e.g. a tetrahedron or an octahedron), a regular polytope always has the largest possible volume. He developed a technique that proved Steiner's conjecture for the cube and for the dodecahedron. By 1953, Fejes Tóth had written dozens of papers devoted to these types of fundam
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https://en.wikipedia.org/wiki/San%20Bartolom%C3%A9%20Perulap%C3%ADa
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San Bartolomé Perulapía is a municipality in the Cuscatlán department of El Salvador. It is located on the highway between San Martín and Suchitoto.
The following statistics are for a city of the same name within the municipality:
Municipality statistics
Population: 12,000 (according to mayorship) or 6909 (according to SIBASI 2001)
One Hospital
Two Schools
Ten Churches
Water, electricity, phone services
Two Alcoholics Anonymous groups
A police force
A court
History
When the Spanish conquistadors came, the location was part of three native towns called pupulapan. The towns were then called, by the Europeans, San Martín, San Pedro, y San Bartolomé Perulapán, also called pupulapía, y transformed into perulapía. In 1770 Pedro Cortés y Larraz estimated that the population was 421 natives and 6 Latinos in population.
It was part of the department of San Salvador from 1824 until 1835, at which time it was turned over to Cuscatlán. Because of an earthquake in 1872, the town was moved a kilometer from its original location. Its population was 960 inhabitants in 1890.
External links
http://www.gobernacion.gob.sv/observatorio/Iniciativas%20Locales/WEB/Cuscatl%C3%A1n/snbartolomeperulapia.htm Information
Municipalities of the Cuscatlán Department
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https://en.wikipedia.org/wiki/William%20L.%20Burke
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William Lionel Burke (July 1941 – July 1996) was an astronomy, astrophysics, and physics professor at UC Santa Cruz. He is also the author of Spacetime, Geometry, Cosmology (), and of Applied differential geometry (), a text expounding the virtues of differential forms over vector calculus for theoretical physics.
Born in Bennington, Vermont, Burke obtained his Bachelor of Science degree from Caltech in 1963. His 1969 doctoral thesis, also at Caltech and supervised by Kip Thorne, Richard Feynman, and John Wheeler, was entitled The Coupling of Gravitational Radiation to Nonrelativistic Sources. His discovery of the Burke Potential, an aspect of gravitation overlooked by Einstein himself, dates from this period. He became a full professor at UCSC in 1988.
Burke is also known as the godfather of the Santa Cruz "Chaos Cabal" also known as the dynamical systems collective, that nurtured the seminal work of MacArthur Fellow Robert Shaw, Norman Packard, Doyne Farmer and James P. Crutchfield. In Tom Bass' book The Eudaemonic Pie, Burke prided himself for his Rubik's Cube costume at the end of the book which kept his identity concealed from his students.
An avid hiker, climber, skier, sailor, wind surfer, and Go player, Bill Burke died from complications due to a cervical fracture sustained in an automobile accident. Bill's understanding of science is paraphrased by his thinking: "Never descend the Grand Canyon with less than two geologists."
Bill was married and then divorced from his wife Pat (Patricia).
See also
Tom Bass, The Eudaemonic Pie
James Gleick, Chaos: Making a New Science
External links
Burke's home page at UCSC
Photograph of William L. Burke from the UC Santa Cruz Library's Digital Collections
William L. Burke obituary
1941 births
1996 deaths
20th-century American physicists
Complex systems scientists
University of California, Santa Cruz faculty
University of California, Santa Cruz alumni
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https://en.wikipedia.org/wiki/Artin%20approximation%20theorem
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In mathematics, the Artin approximation theorem is a fundamental result of in deformation theory which implies that formal power series with coefficients in a field k are well-approximated by the algebraic functions on k.
More precisely, Artin proved two such theorems: one, in 1968, on approximation of complex analytic solutions by formal solutions (in the case ); and an algebraic version of this theorem in 1969.
Statement of the theorem
Let denote a collection of n indeterminates, the ring of formal power series with indeterminates over a field k, and a different set of indeterminates. Let
be a system of polynomial equations in , and c a positive integer. Then given a formal power series solution , there is an algebraic solution consisting of algebraic functions (more precisely, algebraic power series) such that
Discussion
Given any desired positive integer c, this theorem shows that one can find an algebraic solution approximating a formal power series solution up to the degree specified by c. This leads to theorems that deduce the existence of certain formal moduli spaces of deformations as schemes. See also: Artin's criterion.
Alternative statement
The following alternative statement is given in Theorem 1.12 of .
Let be a field or an excellent discrete valuation ring, let be the henselization at a prime ideal of an -algebra of finite type, let m be a proper ideal of , let be the m-adic completion of , and let
be a functor sending filtered colimits to filtered colimits (Artin calls such a functor locally of finite presentation). Then for any integer c and any , there is a such that
.
See also
Ring with the approximation property
Popescu's theorem
Artin's criterion
References
Moduli theory
Commutative algebra
Theorems about algebras
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https://en.wikipedia.org/wiki/Luther%20P.%20Eisenhart
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Luther Pfahler Eisenhart (13 January 1876 – 28 October 1965) was an American mathematician, best known today for his contributions to semi-Riemannian geometry.
Life
Eisenhart was born in York, Pennsylvania, and graduated from Gettysburg College in 1896. He earned his doctorate in 1900 at Johns Hopkins University, where he was influenced (at long range) by the work of Gaston Darboux and at shorter range by that of Thomas Craig. During the next two decades, Eisenhart's research focused on moving frames after the French school, but around 1921 took a different turn when he became enamored of the mathematical challenges and entrancing beauty of a new theory of gravitation, Albert Einstein's general theory of relativity.
Eisenhart played a central role in American mathematics in the early twentieth century. He served as chairman of the mathematics department at Princeton University and later as Dean of the Graduate School there from 1933 to 1945. He is widely credited with guiding the development in America of the mathematical background needed for the further development of general relativity, through his influential textbooks and his personal interaction with Albert Einstein, Oswald Veblen, and John von Neumann at the nearby Institute for Advanced Study, as well as with gifted students such as Abraham Haskel Taub.
In the early 40s he chaired the "Reference Committee", formed in June 1940 for editors of scientific journals to send the papers submitted to them, in order to check that the papers did not contain results (especially regarding nuclear physics) whose public knowledge could be detrimental to the US war efforts.
Publications
Notes
External links
National Academy of Sciences Biographical Memoir
1876 births
1965 deaths
People from York, Pennsylvania
Differential geometers
20th-century American mathematicians
Gettysburg College alumni
Princeton University faculty
Presidents of the American Mathematical Society
Johns Hopkins University alumni
Mathematicians from Pennsylvania
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https://en.wikipedia.org/wiki/Potential%20%28disambiguation%29
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Potential generally refers to a currently unrealized ability, in a wide variety of fields from physics to the social sciences.
Mathematics and physics
Scalar potential, a scalar field whose gradient is a given vector field
Vector potential, a vector field whose curl is a given vector field
Potential function (disambiguation)
Potential variable (Boolean differential calculus)
Potential energy, the energy possessed by an object because of its position relative to other objects, stresses within itself, its electric charge, or other factors
Magnetic vector potential
Magnetic scalar potential (ψ)
Electric potential, the amount of work needed to move a unit positive charge from a reference point to a specific point inside the field without producing any acceleration
Electromagnetic four-potential, a relativistic vector function from which the electromagnetic field can be derived
Coulomb potential
Van der Waals force, distance-dependent interactions between atoms or molecules
Lennard-Jones potential, a mathematical model that approximates the interaction between a pair of neutral atoms or molecules.
Yukawa potential, a potential in particle physics which may arise from the exchange of a massive scalar field
Gravitational potential
Biology
Action potential, occurs when the membrane potential of a specific axon location rapidly rises and falls: this depolarisation then causes adjacent locations to similarly depolarise
Membrane potential, the difference in electric potential between the interior and the exterior of a biological cell. With respect to the exterior of the cell, typical values of membrane potential range from –40 mV to –80 mV
Water potential, the potential energy of water per unit volume relative to pure water in reference conditions
Linguistics
Potential mood
Popular culture
"Potential" (Buffy the Vampire Slayer), an episode of a television series
Potential and new Slayers, characters in Buffy the Vampire Slayer
Potential (song), a song by Danielle Bradbery on her album, I Don't Believe We've Met
Philosophy and society
Potentiality and actuality, a "possibility" that a thing can be said to have
Human Potential Movement, a social movement which asserts that all people have extraordinary untapped potential capacities
See also
Ability (disambiguation)
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https://en.wikipedia.org/wiki/Glossary%20of%20category%20theory
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This is a glossary of properties and concepts in category theory in mathematics. (see also Outline of category theory.)
Notes on foundations: In many expositions (e.g., Vistoli), the set-theoretic issues are ignored; this means, for instance, that one does not distinguish between small and large categories and that one can arbitrarily form a localization of a category. Like those expositions, this glossary also generally ignores the set-theoretic issues, except when they are relevant (e.g., the discussion on accessibility.)
Especially for higher categories, the concepts from algebraic topology are also used in the category theory. For that see also glossary of algebraic topology.
The notations and the conventions used throughout the article are:
[n] = {0, 1, 2, …, n}, which is viewed as a category (by writing .)
Cat, the category of (small) categories, where the objects are categories (which are small with respect to some universe) and the morphisms functors.
Fct(C, D), the functor category: the category of functors from a category C to a category D.
Set, the category of (small) sets.
sSet, the category of simplicial sets.
"weak" instead of "strict" is given the default status; e.g., "n-category" means "weak n-category", not the strict one, by default.
By an ∞-category, we mean a quasi-category, the most popular model, unless other models are being discussed.
The number zero 0 is a natural number.
A
B
C
D
E
F
G
H
I
K
L
M
N
O
P
Q
R
S
T
U
W
Y
Z
Notes
References
A. Joyal, The theory of quasi-categories II (Volume I is missing??)
Lurie, J., Higher Algebra
Lurie, J., Higher Topos Theory
Further reading
Groth, M., A Short Course on ∞-categories
Cisinski's notes
History of topos theory
http://plato.stanford.edu/entries/category-theory/
Emily Riehl, A leisurely introduction to simplicial sets
Categorical Logic lecture notes by Steve Awodey
(a detailed discussion of a 2-category)
Category theory
Wikipedia glossaries using description lists
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https://en.wikipedia.org/wiki/Vec
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Vec may mean:
Mathematics:
vec(A), the vectorization of a matrix A.
Vec denotes the category of vector spaces over the reals.
Other:
Venetian language (Vèneto), language code.
Vecuronium, a muscle relaxant.
vec, a sentient moravec robot from the Orion's Arm Universe Project (see also Moravec_(robot))
See also
VEC (disambiguation)
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https://en.wikipedia.org/wiki/The%20Association%20of%20Cricket%20Statisticians%20and%20Historians
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The Association of Cricket Statisticians and Historians (ACS) was founded in England in 1973 for the purpose of researching and collating information about the history and statistics of cricket. Originally called the Association of Cricket Statisticians, the words "and Historians" were added in 1992 but it has continued to use the initialism ACS.
The ACS headquarters were formerly in Nottingham, opposite Trent Bridge Cricket Ground, but relocated to Cardiff in 2006.
Although constituted in England, the ACS has a worldwide membership and is open to anyone with a relevant interest.
Origin
Following the formal definition of first-class cricket by the then Imperial Cricket Conference (ICC) in May 1947, and particularly given ICC's statement that the definition does not have retrospective effect, a number of cricket statisticians became interested in developing an agreed list of matches played before 1947 from which to compile accurate first-class records.
Roy Webber published his Playfair Book of Cricket Records in 1951 and stated his view that first-class cricket records (i.e., for statistical purposes) should not include matches played before 1864. In this first edition, Webber accepted the records used by Wisden Cricketers' Almanack, including those that summarised the career of W G Grace. In his second edition (1961), Webber challenged many existing views about match status and produced, inter alia, an alternative career record for Grace that did not include some of the matches Wisden recognised as first-class.
There was some support among cricket statisticians, including Bill Frindall, for Webber's basic arguments but there were (and remain) differences of opinion about his commencement date and about the details of his recommended matchlist. The controversial cricket historian Rowland Bowen wrote a lengthy critique of Webber's sources in 1961. Bowen then started Cricket Quarterly (1963–70), devoted to cricket statistics, which included among its contributors some of the original ACS members.
The ACS itself was founded by Robert Brooke and Dennis Lambert, two of Bowen's contributors, by means of advertisements in the October 1972 issues of The Cricketer and Playfair Cricket Monthly. These attracted a nucleus of some 50 members who formed the association in 1973.
Scope of activities
The ACS has sought to compile details of all known historically significant matches, and has widened its scope to include details of other competitions such as the Minor Counties Cricket Championship and the Second XI Championship in England. The bulk of its research concerns those matches that are officially or unofficially recognised as first-class or List A limited overs cricket. The findings have been published in-house in various guides (see list below) and in the Association's quarterly journal The Cricket Statistician. The Who's Who of first-class Cricketers is one of the few commercially published works.
Like Webber and Frindall, the ACS has no offi
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https://en.wikipedia.org/wiki/Daniel%20Goldston
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Daniel Alan Goldston (born January 4, 1954, in Oakland, California) is an American mathematician who specializes in number theory. He is currently a professor of mathematics at San Jose State University.
Early life and education
Daniel Alan Goldston was born on January 4, 1954, in Oakland, California. In 1972, he matriculated to the University of California, Berkeley, where he earned his bachelor's degree and, in 1981, a Ph.D. in mathematics. His doctoral advisor at Berkeley was Russell Sherman Lehman; his dissertation was entitled "Large Differences between Consecutive Prime Numbers".
Career
After earning his doctorate, Goldston worked at the University of Minnesota Duluth and then spent the next academic year (1982–83) at the Institute for Advanced Study (IAS) in Princeton. He has worked at San Jose State University since 1983, save for stints at the IAS (1990), the University of Toronto (1994), and the Mathematical Sciences Research Institute in Berkeley (1999).
Research
In 2009, Goldston, János Pintz, and Cem Yıldırım proved:
where denotes the nth prime number. In other words, for every , there exist infinitely many pairs of consecutive primes and which are closer to each other than the average distance between consecutive primes by a factor of , i.e., . This result was originally reported in 2003 by Goldston and Yıldırım but was later retracted. Then Pintz joined the team and they completed the proof with the GPY sieve.
Recognition
In 2014, Goldston won the Cole Prize, shared with Yitang Zhang and colleagues Cem Yildirim and János Pintz, for his contributions to number theory. Also,
Goldston was named to the 2021 class of fellows of the American Mathematical Society "for contributions to analytic number theory".
See also
Landau's problems
Yitang Zhang
James Maynard (mathematician)
References
External links
Dan Goldston's Homepage
Institute for Advanced Study visiting scholars
University of California, Berkeley alumni
San Jose State University faculty
20th-century American mathematicians
21st-century American mathematicians
Number theorists
1954 births
Living people
Fellows of the American Mathematical Society
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https://en.wikipedia.org/wiki/Cem%20Y%C4%B1ld%C4%B1r%C4%B1m
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Cem Yalçın Yıldırım (born 8 July 1961) is a Turkish mathematician who specializes in number theory.
Education
Yıldırım obtained his B.Sc from Middle East Technical University in Ankara, Turkey and his PhD from the University of Toronto in 1990. His advisor was John Friedlander. He is currently a faculty member at Boğaziçi University in Istanbul, Turkey.
Research
In 2009, Dan Goldston, János Pintz, and Yıldırım proved that for any positive number ε there exist primes p and p′ such that the difference between p and p′ is smaller than ε log p. This result was originally reported in 2003 by Goldston and Yıldırım but was later retracted. Then Janos Pintz joined the team and they completed the proof in 2005 and developed the so called GPY sieve.
See also
Landau's problems
References
External links
20th-century Turkish mathematicians
21st-century Turkish mathematicians
1961 births
Living people
Number theorists
Institute for Advanced Study visiting scholars
University of Toronto alumni
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https://en.wikipedia.org/wiki/Yang%E2%80%93Mills%20existence%20and%20mass%20gap
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The Yang–Mills existence and mass gap problem is an unsolved problem in mathematical physics and mathematics, and one of the seven Millennium Prize Problems defined by the Clay Mathematics Institute, which has offered a prize of US$1,000,000 for its solution.
The problem is phrased as follows:
Yang–Mills Existence and Mass Gap. Prove that for any compact simple gauge group G, a non-trivial quantum Yang–Mills theory exists on and has a mass gap Δ > 0. Existence includes establishing axiomatic properties at least as strong as those cited in , and .
In this statement, a quantum Yang–Mills theory is a non-abelian quantum field theory similar to that underlying the Standard Model of particle physics; is Euclidean 4-space; the mass gap Δ is the mass of the least massive particle predicted by the theory.
Therefore, the winner must prove that:
Yang–Mills theory exists and satisfies the standard of rigor that characterizes contemporary mathematical physics, in particular constructive quantum field theory, and
The mass of all particles of the force field predicted by the theory are strictly positive.
For example, in the case of G=SU(3)—the strong nuclear interaction—the winner must prove that glueballs have a lower mass bound, and thus cannot be arbitrarily light.
The general problem of determining the presence of a spectral gap in a system is known to be undecidable.
Background
The problem requires the construction of a QFT satisfying the Wightman axioms and showing the existence of a mass gap. Both of these topics are described in sections below.
The Wightman axioms
The Millennium problem requires the proposed Yang–Mills theory to satisfy the Wightman axioms or similarly stringent axioms. There are four axioms:
W0 (assumptions of relativistic quantum mechanics)
Quantum mechanics is described according to von Neumann; in particular, the pure states are given by the rays, i.e. the one-dimensional subspaces, of some separable complex Hilbert space.
The Wightman axioms require that the Poincaré group acts unitarily on the Hilbert space. In other words, they have position dependent operators called quantum fields which form covariant representations of the Poincaré group.
The group of space-time translations is commutative, and so the operators can be simultaneously diagonalised. The generators of these groups give us four self-adjoint operators, , which transform under the homogeneous group as a four-vector, called the energy-momentum four-vector.
The second part of the zeroth axiom of Wightman is that the representation U(a, A) fulfills the spectral condition—that the simultaneous spectrum of energy-momentum is contained in the forward cone:
The third part of the axiom is that there is a unique state, represented by a ray in the Hilbert space, which is invariant under the action of the Poincaré group. It is called a vacuum.
W1 (assumptions on the domain and continuity of the field)
For each test function f, there exists a set of opera
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https://en.wikipedia.org/wiki/Algebraic%20space
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In mathematics, algebraic spaces form a generalization of the schemes of algebraic geometry, introduced by Michael Artin for use in deformation theory. Intuitively,
schemes are given by gluing together affine schemes using the Zariski topology, while algebraic spaces are given by gluing together affine schemes using the finer étale topology. Alternatively one can think of schemes as being locally isomorphic to affine schemes in the Zariski topology, while algebraic spaces are locally isomorphic to affine schemes in the étale topology.
The resulting category of algebraic spaces extends the category of schemes and allows one to carry out several natural constructions that are used in the construction of moduli spaces but are not always possible in the smaller category of schemes, such as taking the quotient of a free action by a finite group (cf. the Keel–Mori theorem).
Definition
There are two common ways to define algebraic spaces: they can be defined as either quotients of schemes by etale equivalence relations, or as sheaves on a big etale site that are locally isomorphic to schemes. These two definitions are essentially equivalent.
Algebraic spaces as quotients of schemes
An algebraic space X comprises a scheme U and a closed subscheme R ⊂ U × U satisfying the following two conditions:
1. R is an equivalence relation as a subset of U × U
2. The projections pi: R → U onto each factor are étale maps.
Some authors, such as Knutson, add an extra condition that an algebraic space has to be quasi-separated, meaning that the diagonal map is quasi-compact.
One can always assume that R and U are affine schemes. Doing so means that the theory of algebraic spaces is not dependent on the full theory of schemes, and can indeed be used as a (more general) replacement of that theory.
If R is the trivial equivalence relation over each connected component of U (i.e. for all x, y belonging to the same connected component of U, we have xRy if and only if x=y), then the algebraic space will be a scheme in the usual sense. Since a general algebraic space X does not satisfy this requirement, it allows a single connected component of U to cover X with many "sheets". The point set underlying the algebraic space X is then given by |U| / |R| as a set of equivalence classes.
Let Y be an algebraic space defined by an equivalence relation S ⊂ V × V. The set Hom(Y, X) of morphisms of algebraic spaces is then defined by the condition that it makes the descent sequence
exact (this definition is motivated by a descent theorem of Grothendieck for surjective étale maps of affine schemes). With these definitions, the algebraic spaces form a category.
Let U be an affine scheme over a field k defined by a system of polynomials g(x), x = (x1, ..., xn), let
denote the ring of algebraic functions in x over k, and let X = {R ⊂ U × U} be an algebraic space.
The appropriate stalks ÕX, x on X are then defined to be the local rings of algebraic functions defined by ÕU, u,
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https://en.wikipedia.org/wiki/Inseparable
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Inseparable may refer to:
Mathematics
Inseparable differential equation, an ordinary differential equation that cannot be solved by using separation of variables
Inseparable extension, a field extension by elements that do not all satisfy a separable polynomial
Inseparable polynomial, a polynomial that does not have distinct roots in a splitting field
Music
Inseparable (album), by Natalie Cole, 1975
"Inseparable" (song), the title song
Inseparable (EP), by Veridia, 2014
Les inséparables, an album by Corneille, 2011
"Inseperable", a song by Jonas Brothers from Jonas Brothers, 2007
"Inseperable", a song by Mariah Carey from Memoirs of an Imperfect Angel, 2009
Other uses
Inseparable (book), a 2019 sports autobiography by Shaquem Griffin and Shaquill Griffin, with Mark Schlabach
Inseparable (film), a 2011 Chinese film by Dayyan Eng
The Inseparables, a 1929 British film by Adelqui Migliar and John Stafford
Inseparability, in marketing, a quality of services as distinct from goods
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https://en.wikipedia.org/wiki/Mex%20%28mathematics%29
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In mathematics, the mex ("minimum excluded value") of a subset of a well-ordered set is the smallest value from the whole set that does not belong to the subset. That is, it is the minimum value of the complement set.
Beyond sets, subclasses of well-ordered classes have minimum excluded values. Minimum excluded values of subclasses of the ordinal numbers are used in combinatorial game theory to assign nim-values to impartial games.
According to the Sprague–Grundy theorem, the nim-value of a game position is the minimum excluded value of the class of values of the positions that can be reached in a single move from the given position.
Minimum excluded values are also used in graph theory, in greedy coloring algorithms. These algorithms typically choose an ordering of the vertices of a graph and choose a numbering of the available vertex colors. They then consider the vertices in order, for each vertex choosing its color to be the minimum excluded value of the set of colors already assigned to its neighbors.
Examples
The following examples all assume that the given set is a subset of the class of ordinal numbers:
where is the limit ordinal for the natural numbers.
Game theory
In the Sprague–Grundy theory the minimum excluded ordinal is used to determine the nimber of a normal-play impartial game. In such a game, either player has the same moves in each position and the last player to move wins. The nimber is equal to 0 for a game that is lost immediately by the first player, and is equal to the mex of the nimbers of all possible next positions for any other game.
For example, in a one-pile version of Nim, the game starts with a pile of stones, and the player to move may take any positive number of stones. If is zero stones, the nimber is 0 because the mex of the empty set of legal moves is the nimber 0. If is 1 stone, the player to move will leave 0 stones, and , gives the nimber for this case. If is 2 stones, the player to move can leave 0 or 1 stones, giving the nimber 2 as the mex of the nimbers In general, the player to move with a pile of stones can leave anywhere from 0 to stones; the mex of the nimbers is always the nimber . The first player wins in Nim if and only if the nimber is not zero, so from this analysis we can conclude that the first player wins if and only if the starting number of stones in a one-pile game of Nim is not zero; the winning move is to take all the stones.
If we change the game so that the player to move can take up to 3 stones only, then with stones, the successor states have nimbers giving a mex of 0. Since the nimber for 4 stones is 0, the first player loses. The second player's strategy is to respond to whatever move the first player makes by taking the rest of the stones. For stones, the nimbers of the successor states of 2, 3, and 4 stones are the nimbers 2, 3, and 0 (as we just calculated); the mex of the set of nimbers is the nimber 1, so starting with 5 stones in this game is
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https://en.wikipedia.org/wiki/Beal%20conjecture
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The Beal conjecture is the following conjecture in number theory:
If
where A, B, C, x, y, and z are positive integers with x, y, z ≥ 3, then A, B, and C have a common prime factor.
Equivalently,
The equation has no solutions in positive integers and pairwise coprime integers A, B, C if x, y, z ≥ 3.
The conjecture was formulated in 1993 by Andrew Beal, a banker and amateur mathematician, while investigating generalizations of Fermat's Last Theorem. Since 1997, Beal has offered a monetary prize for a peer-reviewed proof of this conjecture or a counterexample. The value of the prize has increased several times and is currently $1 million.
In some publications, this conjecture has occasionally been referred to as a generalized Fermat equation, the Mauldin conjecture, and the Tijdeman-Zagier conjecture.
Related examples
To illustrate, the solution has bases with a common factor of 3, the solution has bases with a common factor of 7, and has bases with a common factor of 2. Indeed the equation has infinitely many solutions where the bases share a common factor, including generalizations of the above three examples, respectively
and
Furthermore, for each solution (with or without coprime bases), there are infinitely many solutions with the same set of exponents and an increasing set of non-coprime bases. That is, for solution
we additionally have
where
Any solutions to the Beal conjecture will necessarily involve three terms all of which are 3-powerful numbers, i.e. numbers where the exponent of every prime factor is at least three. It is known that there are an infinite number of such sums involving coprime 3-powerful numbers; however, such sums are rare. The smallest two examples are:
What distinguishes Beal's conjecture is that it requires each of the three terms to be expressible as a single power.
Relation to other conjectures
Fermat's Last Theorem established that has no solutions for n > 2 for positive integers A, B, and C. If any solutions had existed to Fermat's Last Theorem, then by dividing out every common factor, there would also exist solutions with A, B, and C coprime. Hence, Fermat's Last Theorem can be seen as a special case of the Beal conjecture restricted to x = y = z.
The Fermat–Catalan conjecture is that has only finitely many solutions with A, B, and C being positive integers with no common prime factor and x, y, and z being positive integers satisfying Beal's conjecture can be restated as "All Fermat–Catalan conjecture solutions will use 2 as an exponent".
The abc conjecture would imply that there are at most finitely many counterexamples to Beal's conjecture.
Partial results
In the cases below where n is an exponent, multiples of n are also proven, since a kn-th power is also an n-th power. Where solutions involving a second power are alluded to below, they can be found specifically at Fermat–Catalan conjecture#Known solutions. All cases of the form (2, 3, n) or (2, n, 3) have the solution 23 + 1n =
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https://en.wikipedia.org/wiki/Spence%27s%20function
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In mathematics, Spence's function, or dilogarithm, denoted as , is a particular case of the polylogarithm. Two related special functions are referred to as Spence's function, the dilogarithm itself:
and its reflection.
For , an infinite series also applies (the integral definition constitutes its analytical extension to the complex plane):
Alternatively, the dilogarithm function is sometimes defined as
In hyperbolic geometry the dilogarithm can be used to compute the volume of an ideal simplex. Specifically, a simplex whose vertices have cross ratio has hyperbolic volume
The function is sometimes called the Bloch-Wigner function. Lobachevsky's function and Clausen's function are closely related functions.
William Spence, after whom the function was named by early writers in the field, was a Scottish mathematician working in the early nineteenth century. He was at school with John Galt, who later wrote a biographical essay on Spence.
Analytic structure
Using the former definition above, the dilogarithm function is analytic everywhere on the complex plane except at , where it has a logarithmic branch point. The standard choice of branch cut is along the positive real axis . However, the function is continuous at the branch point and takes on the value .
Identities
Particular value identities
Special values
where is the Riemann zeta function.
In particle physics
Spence's Function is commonly encountered in particle physics while calculating radiative corrections. In this context, the function is often defined with an absolute value inside the logarithm:
See also
Markstein number
Notes
References
Further reading
External links
NIST Digital Library of Mathematical Functions: Dilogarithm
Special functions
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https://en.wikipedia.org/wiki/Virtual%20group
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Virtual group may refer to:
Virtual band in music
Groupoid in category theory (an area of mathematics)
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https://en.wikipedia.org/wiki/List%20of%20Eliteserien%20top%20scorers
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List of top goal scorers in the top flight of Norwegian football, currently known as Eliteserien. The statistics begin with the 1948–49 season. The League of Norway, played from 1937–38 to 1947–48, was divided in eleven conferences with different numbers of game weeks and is therefore not included in this statistics.
Top scorers
By season
The following is a list of top scorers in the top football league of Norway by season.
By player
By club
By nationality
All-time topscorers with over 50 goals
The following is an all-time top-scorer list. Players who have scored 50 or more goals are included.
References
top scorers
Norway
Norway
Association football player non-biographical articles
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https://en.wikipedia.org/wiki/Unit%20root
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In probability theory and statistics, a unit root is a feature of some stochastic processes (such as random walks) that can cause problems in statistical inference involving time series models. A linear stochastic process has a unit root if 1 is a root of the process's characteristic equation. Such a process is non-stationary but does not always have a trend.
If the other roots of the characteristic equation lie inside the unit circle—that is, have a modulus (absolute value) less than one—then the first difference of the process will be stationary; otherwise, the process will need to be differenced multiple times to become stationary. If there are d unit roots, the process will have to be differenced d times in order to make it stationary. Due to this characteristic, unit root processes are also called difference stationary.
Unit root processes may sometimes be confused with trend-stationary processes; while they share many properties, they are different in many aspects. It is possible for a time series to be non-stationary, yet have no unit root and be trend-stationary. In both unit root and trend-stationary processes, the mean can be growing or decreasing over time; however, in the presence of a shock, trend-stationary processes are mean-reverting (i.e. transitory, the time series will converge again towards the growing mean, which was not affected by the shock) while unit-root processes have a permanent impact on the mean (i.e. no convergence over time).
If a root of the process's characteristic equation is larger than 1, then it is called an explosive process, even though such processes are sometimes inaccurately called unit roots processes.
The presence of a unit root can be tested using a unit root test.
Definition
Consider a discrete-time stochastic process , and suppose that it can be written as an autoregressive process of order p:
Here, is a serially uncorrelated, zero-mean stochastic process with constant variance . For convenience, assume . If is a root of the characteristic equation, of multiplicity 1:
then the stochastic process has a unit root or, alternatively, is integrated of order one, denoted . If m = 1 is a root of multiplicity r, then the stochastic process is integrated of order r, denoted I(r).
Example
The first order autoregressive model, , has a unit root when . In this example, the characteristic equation is . The root of the equation is .
If the process has a unit root, then it is a non-stationary time series. That is, the moments of the stochastic process depend on . To illustrate the effect of a unit root, we can consider the first order case, starting from y0 = 0:
By repeated substitution, we can write . Then the variance of is given by:
The variance depends on t since , while . Note that the variance of the series is diverging to infinity with t.
There are various tests to check for the existence of a unit root, some of them are given by:
The Dickey–Fuller test (DF) or augmented Dickey–Fuller (
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https://en.wikipedia.org/wiki/Univariate
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In mathematics, a univariate object is an expression, equation, function or polynomial involving only one variable. Objects involving more than one variable are multivariate. In some cases the distinction between the univariate and multivariate cases is fundamental; for example, the fundamental theorem of algebra and Euclid's algorithm for polynomials are fundamental properties of univariate polynomials that cannot be generalized to multivariate polynomials.
In statistics, a univariate distribution characterizes one variable, although it can be applied in other ways as well. For example, univariate data are composed of a single scalar component. In time series analysis, the whole time series is the "variable": a univariate time series is the series of values over time of a single quantity. Correspondingly, a "multivariate time series" characterizes the changing values over time of several quantities. In some cases, the terminology is ambiguous, since the values within a univariate time series may be treated using certain types of multivariate statistical analyses and may be represented using multivariate distributions.
In addition to the question of scaling, a criterion (variable) in univariate statistics can be described by two important measures (also key figures or parameters): Location & Variation.
Measures of Location Scales (e.g. mode, median, arithmetic mean) describe in which area the data is arranged centrally.
Measures of Variation (e.g. span, interquartile distance, standard deviation) describe how similar or different the data are scattered.
See also
Arity
Bivariate (disambiguation)
Multivariate (disambiguation)
Univariate analysis
Univariate binary model
Univariate distribution
References
Mathematical terminology
Theory of probability distributions
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https://en.wikipedia.org/wiki/Greg%20Whitten
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Greg Whitten is an American computer engineer, investor and car collector.
Whitten graduated from the University of Virginia with a B.A. in mathematics in 1973, and from Harvard University with a Ph.D. in applied mathematics in 1978.
He worked for Compucolor, a company in Georgia established in 1977 that made the home computer Compucolor II (an early PC) but went out of business in 1983. While there, he reputedly optimized an unlicensed copy of Microsoft Basic so effectively that Microsoft later forgave Compucolor for their infringement in exchange for the rights to the enhancements.
Microsoft 1979–1998
He then worked for Microsoft from 1979 to 1998. He developed the standards for the company's BASIC compiler line.
"GW" in the name of the GW-BASIC dialect (first released 1983) of BASIC developed by Microsoft may have come from Greg Whitten's initials:
"The GW-BASIC name stands for Gee-Whiz BASIC. The GW- name was picked by Bill Gates. He is the one who knows whether it was Gee-Whiz or after me because it has been used both ways. I did set the directions for the BASIC language features after joining the company in 1979."
- Greg Whitten, 13 Apr 2005
As a chief software architect, he also oversaw the development of the enterprise support systems required in Windows for the Microsoft Office.
Numerix 1998–2013
In 1998 he became member of the board of Numerix, a financial software company established 1996, where he had made a major investment and in 2001 was elected Boardman after another major investment. After being its CEO 2003–13, he was succeeded by Steve O'Hanlon.
Car collector
He is also a vintage car enthusiast and has a famous car collection that contains various Ferraris including his first, a Ferrari F40, a Ferrari Enzo (#131632), a Ferrari 250 GTO (#3413GT, 2000–2018), a Ferrari 250 LM (#5907, 1994–2005), a Ferrari 250 GT TdF (#0703GT, 1997–), a Ferrari F50 (#104163, 1995–2004), and two LaFerrari, of which one is the unique in blue electrics. His Ferrari 250 GTO sold for $48.8 million at a public auction in August 2018.
References
University of Virginia alumni
Harvard University alumni
American computer businesspeople
Microsoft employees
Living people
American computer scientists
American car collectors
Year of birth missing (living people)
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https://en.wikipedia.org/wiki/Elliott%20H.%20Lieb
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Elliott Hershel Lieb (born July 31, 1932) is an American mathematical physicist and professor of mathematics and physics at Princeton University who specializes in statistical mechanics, condensed matter theory, and functional analysis.
Lieb is a prolific author, with over 400 publications both in physics and mathematics.
In particular, his scientific works pertain to quantum and classical many-body problem, atomic structure, the stability of matter, functional inequalities, the theory of magnetism, and the Hubbard model.
Biography
He received his B.S. in physics from the Massachusetts Institute of Technology in 1953 and his PhD in mathematical physics from the University of Birmingham in England in 1956. Lieb was a Fulbright Fellow at Kyoto University, Japan (1956–1957), and worked as the Staff Theoretical Physicist for IBM from 1960 to 1963. In 1961–1962, Lieb was on leave as professor of applied mathematics at Fourah Bay College, the University of Sierra Leone. He has been a professor at Princeton since 1975, following a leave from his professorship at MIT.
He is married to fellow Princeton professor Christiane Fellbaum.
For years, he has rejected the standard practice of transferring copyright of his research articles to academic publishers. Instead, he would only give publishers his consent to publish.
Awards
Lieb has been awarded several prizes in mathematics and physics, including the Heineman Prize for Mathematical Physics of the American Physical Society and the American Institute of Physics (1978), the Max Planck Medal of the German Physical Society (1992), the Boltzmann medal of the International Union of Pure and Applied Physics (1998), the Schock Prize (2001), the Henri Poincaré Prize of the International Association of Mathematical Physics (2003), and the Medal of the Erwin Schrödinger Institute for Mathematics and Physics (2021).
In 2022 he was awarded the Medal for Exceptional Achievement in Research from the American Physical Society for ″major contributions to theoretical physics through obtaining exact solutions to important physical problems, which have impacted condensed matter physics, quantum information, statistical mechanics, and atomic physics″
and the Carl Friedrich Gauss Prize at the International Congress of Mathematicians ″for deep mathematical contributions of exceptional breadth which have shaped the fields of quantum mechanics, statistical mechanics, computational chemistry, and quantum information theory.″ Also in 2022 he received the Dirac Medal of the ICTP jointly with Joel Lebowitz and David Ruelle.
Lieb is a member of the U.S. National Academy of Sciences and has twice served (1982–1984 and 1997–1999) as the president of the International Association of Mathematical Physics. Lieb was awarded the Austrian Decoration for Science and Art in 2002. In 2012 he became a fellow of the American Mathematical Society and in 2013 a Foreign Member of the Royal Society.
In 2023 he received Kyoto Prize in Basic Sc
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https://en.wikipedia.org/wiki/Angle%20bisector%20theorem
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In geometry, the angle bisector theorem is concerned with the relative lengths of the two segments that a triangle's side is divided into by a line that bisects the opposite angle. It equates their relative lengths to the relative lengths of the other two sides of the triangle.
Theorem
Consider a triangle . Let the angle bisector of angle intersect side at a point between and . The angle bisector theorem states that the ratio of the length of the line segment to the length of segment is equal to the ratio of the length of side to the length of side :
and conversely, if a point on the side of divides in the same ratio as the sides and , then is the angle bisector of angle .
The generalized angle bisector theorem states that if lies on the line , then
This reduces to the previous version if is the bisector of . When is external to the segment , directed line segments and directed angles must be used in the calculation.
The angle bisector theorem is commonly used when the angle bisectors and side lengths are known. It can be used in a calculation or in a proof.
An immediate consequence of the theorem is that the angle bisector of the vertex angle of an isosceles triangle will also bisect the opposite side.
Proofs
There exist many different ways of proving the angle bisector theorem. A few of them are shown below.
Proof using similar triangles
As shown in the accompanying animation, the theorem can be proved using similar triangles. In the version illustrated here, the triangle gets reflected across a line that is perpendicular to the angle bisector , resulting in the triangle with bisector . The fact that the bisection-produced angles and are equal means that and are straight lines. This allows the construction of triangle that is similar to . Because the ratios between corresponding sides of similar triangles are all equal, it follows that . However, was constructed as a reflection of the line , and so those two lines are of equal length. Therefore, , yielding the result stated by the theorem.
Proof using Law of Sines
In the above diagram, use the law of sines on triangles and :
Angles and form a linear pair, that is, they are adjacent supplementary angles. Since supplementary angles have equal sines,
Angles and are equal. Therefore, the right hand sides of equations () and () are equal, so their left hand sides must also be equal.
which is the angle bisector theorem.
If angles are unequal, equations () and () can be re-written as:
Angles are still supplementary, so the right hand sides of these equations are still equal, so we obtain:
which rearranges to the "generalized" version of the theorem.
Proof using triangle altitudes
Let be a point on the line , not equal to or and such that is not an altitude of triangle .
Let be the base (foot) of the altitude in the triangle through and let be the base of the altitude in the triangle through . Then, if is strictly between and , one and on
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https://en.wikipedia.org/wiki/Richard%20Schoen
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Richard Melvin Schoen (born October 23, 1950) is an American mathematician known for his work in differential geometry and geometric analysis. He is best known for the resolution of the Yamabe problem in 1984.
Career
Born in Celina, Ohio, and a 1968 graduate of Fort Recovery High School, he received his B.S. from the University of Dayton in mathematics. He then received his PhD in 1977 from Stanford University. After faculty positions at the Courant Institute, NYU, University of California, Berkeley, and University of California, San Diego, he was Professor at Stanford University from 1987–2014, as Bass Professor of Humanities and Sciences since 1992. He is currently Distinguished Professor and Excellence in Teaching Chair at the University of California, Irvine. His surname is pronounced "Shane."
Schoen received an NSF Graduate Research Fellowship in 1972 and a Sloan Research Fellowship in 1979. Schoen is a 1983 MacArthur Fellow. He has been invited to speak at the International Congress of Mathematicians (ICM) three times, including twice as a Plenary Speaker. In 1983 he was an Invited Speaker at the ICM in Warsaw, in 1986 he was a Plenary Speaker at the ICM in Berkeley, and in 2010 he was a Plenary Speaker at the ICM in Hyderabad. For his work on the Yamabe problem, Schoen was awarded the Bôcher Memorial Prize in 1989. He was elected to the American Academy of Arts and Sciences in 1988 and to the National Academy of Sciences in 1991, became Fellow of the American Association for the Advancement of Science in 1995, and won a Guggenheim Fellowship in 1996. In 2012 he became a Fellow of the American Mathematical Society. He received the 2014–15 Dean’s Award for Lifetime Achievements in Teaching from Stanford University. In 2015, he was elected Vice President of the American Mathematical Society. He was awarded an Honorary Doctor of Science from the University of Warwick in 2015. He received the Wolf Prize in Mathematics for 2017, shared with Charles Fefferman. In the same year, he was awarded the Heinz Hopf Prize, the Lobachevsky Medal and Prize by Kazan Federal University, and the Rolf Schock Prize.
He has had over 44 doctoral students, including Hubert Bray, José F. Escobar, Ailana Fraser, Chikako Mese, William Minicozzi, and André Neves.
Mathematical work
Schoen has investigated the use of analytic techniques in global differential geometry, with a number of fundamental contributions to the regularity theory of minimal surfaces and harmonic maps.
Harmonic maps
In 1976, Schoen and Shing-Tung Yau used Yau's earlier Liouville theorems to extend the rigidity phenomena found earlier by James Eells and Joseph Sampson to noncompact settings. By identifying a certain interplay of the Bochner identity for harmonic maps together with the second variation of area formula for minimal hypersurfaces, they also identified some novel conditions on the domain leading to the same conclusion. These rigidity theorems are complemented by their existence the
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https://en.wikipedia.org/wiki/Franz%20Kamin
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Franz Kamin (May 25, 1941 – April 11, 2010) was an American author, composer, poet, performance-installation artist, and pianist whose works explore structural principles derived from topology, general systems theory, prosody, and meditational processes in unusual combinations of genre and technique. He made use of conventional instruments and children's toys; sound poetry and puppet theater; choreography and speaking chorus; systematic chance operations and both programmed and otherwise uncommon improvisation; performance scribbling and the live reading of narrative texts.
Born in Milwaukee, Kamin studied composition at the University of Oklahoma with Spencer Norton, and at Indiana University with Roque Cordero, where he also studied piano with Alfonso Montecino. While at IU, Kamin, together with fellow composer James Brody organized FIASCO, an experimental collective which meet weekly in Bloomington from 1966 to 1972. Among his compositions from this period was "The Concert of Doors", a synaesthetic work in which a number of doors, each of vastly differing design, some found, some constructed, ranging from comical to mysterious, were set on a path through a woods to be traversed by the audience-participants. Kamin moved to New York City in the 1970s and collaborated with cellist Charlotte Moorman, a fixture in the downtown avant-garde scene at that time. He eventually settled in Saint Paul, Minnesota, where he remained for the rest of his life.
Kamin published two books full books of trans-genre work: Ann-Margret Loves You & Other Psychotopological Diversions (1980) and Scribble Death (Station Hill Press: Barrytown, New York). Several of his shorter works, musical scores and an LP recording of Behavioral Drift II and Rugugmool have also been published.
He was killed in a car crash in Roseville, Minnesota when a car driven by James Brody, who also died, left the road, jumped a curb, and hit a tree. The two composers were returning from a SEAMUS festival in St. Cloud, Minnesota. Their names were released on April 12, 2010, by the Ramsey County medical examiner's office.
References
1941 births
2010 deaths
American male composers
American composers
Road incident deaths in Minnesota
University of Oklahoma alumni
Indiana University alumni
Writers from Milwaukee
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https://en.wikipedia.org/wiki/Poincar%C3%A9%20recurrence%20theorem
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In mathematics and physics, the Poincaré recurrence theorem states that certain dynamical systems will, after a sufficiently long but finite time, return to a state arbitrarily close to (for continuous state systems), or exactly the same as (for discrete state systems), their initial state.
The Poincaré recurrence time is the length of time elapsed until the recurrence. This time may vary greatly depending on the exact initial state and required degree of closeness. The result applies to isolated mechanical systems subject to some constraints, e.g., all particles must be bound to a finite volume. The theorem is commonly discussed in the context of ergodic theory, dynamical systems and statistical mechanics. Systems to which the Poincaré recurrence theorem applies are called conservative systems.
The theorem is named after Henri Poincaré, who discussed it in 1890 and proved by Constantin Carathéodory using measure theory in 1919.
Precise formulation
Any dynamical system defined by an ordinary differential equation determines a flow map f t mapping phase space on itself. The system is said to be volume-preserving if the volume of a set in phase space is invariant under the flow. For instance, all Hamiltonian systems are volume-preserving because of Liouville's theorem. The theorem is then: If a flow preserves volume and has only bounded orbits, then, for each open set, any orbit that intersects this open set intersects it infinitely often.
Discussion of proof
The proof, speaking qualitatively, hinges on two premises:
A finite upper bound can be set on the total potentially accessible phase space volume. For a mechanical system, this bound can be provided by requiring that the system is contained in a bounded physical region of space (so that it cannot, for example, eject particles that never return) – combined with the conservation of energy, this locks the system into a finite region in phase space.
The phase volume of a finite element under dynamics is conserved (for a mechanical system, this is ensured by Liouville's theorem).
Imagine any finite starting volume of the phase space and to follow its path under the dynamics of the system. The volume evolves through a "phase tube" in the phase space, keeping its size constant. Assuming a finite phase space, after some number of steps the phase tube must intersect itself. This means that at least a finite fraction of the starting volume is recurring.
Now, consider the size of the non-returning portion of the starting phase volume – that portion that never returns to the starting volume. Using the principle just discussed in the last paragraph, we know that if the non-returning portion is finite, then a finite part of it must return after steps. But that would be a contradiction, since in a number lcm of step, both and would be returning, against the hypothesis that only was. Thus, the non-returning portion of the starting volume cannot be the empty set, i.e. all is recurring after
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https://en.wikipedia.org/wiki/Parallel%20education
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Parallel Education is a system in which boys and girls in Australia attend the same school, but are split into single sex classes for core subjects such as English, Maths, science, LOTE, and humanities. However, students will come together for drama, music and other social and cultural activities although a strict 30 cm personal space rule applies. Theoretically, this means that students will receive the best education while still being able to interact with the opposite sex.
Haileybury College, Melbourne currently operates Parallel Education at their five campuses with many other leading schools set to follow.
References
Gender and education
Educational administration
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https://en.wikipedia.org/wiki/Donald%20G.%20Saari
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Donald Gene Saari (born March 1940) is an American mathematician, a Distinguished Professor of Mathematics and Economics and former director of the Institute for Mathematical Behavioral Sciences at the University of California, Irvine.
His research interests include the -body problem, the Borda count voting system, and application of mathematics to the social sciences.
Contributions
Saari has been widely quoted as an expert in voting methods and lottery odds. He is opposed to the use of the Condorcet criterion in evaluating voting systems, and among positional voting schemes he favors using the Borda count over plurality voting, because it reduces the frequency of paradoxical outcomes (which however cannot be avoided entirely in ranking systems because of Arrow's impossibility theorem). For instance, as he has pointed out, plurality voting can lead to situations where the election outcome would remain unchanged if all voters' preferences were reversed; this cannot happen with the Borda count. Saari has defined, as a measure of the inconsistency of a voting method, the number of different combinations of outcomes that would be possible for all subsets of a field of candidates. According to this measure, the Borda count is the least inconsistent possible positional voting scheme, while plurality voting is the most inconsistent. However, other voting theorists such as Steven Brams, while agreeing with Saari that plurality voting is a bad system, disagree with his advocacy of the Borda count, because it is too easily manipulated by tactical voting. Saari also applies similar methods to a different problem in political science, the apportionment of seats to electoral districts in proportion to their populations. He has written several books on the mathematics of voting.
In economics, Saari has shown that natural price mechanisms that set the rate of change of the price of a commodity proportional to its excess demand can lead to chaotic behavior rather than converging to an economic equilibrium, and has exhibited alternative price mechanisms that can be guaranteed to converge. However, as he also showed, such mechanisms require that the change in price be determined as a function of the whole system of prices and demands, rather than being reducible to a computation over pairs of commodities.
In celestial mechanics, Saari's work on the -body problem "revived the singularity theory" of Henri Poincaré and Paul Painlevé, and proved Littlewood's conjecture that the initial conditions leading to collisions have measure zero. He also formulated the "Saari conjecture", that when a solution to the Newtonian -body problem has an unchanging moment of inertia relative to its center of mass, its bodies must be in relative equilibrium. More controversially, Saari has taken the position that anomalies in the rotation speeds of galaxies, discovered by Vera Rubin, can be explained by considering more carefully the pairwise gravitational interactions of individual
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https://en.wikipedia.org/wiki/Arnold%20Ross
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Arnold Ephraim Ross (August 24, 1906 – September 25, 2002) was a mathematician and educator who founded the Ross Mathematics Program, a number theory summer program for gifted high school students. He was born in Chicago, but spent his youth in Odesa, Ukraine, where he studied with Samuil Shatunovsky. Ross returned to Chicago and enrolled in University of Chicago graduate coursework under E. H. Moore, despite his lack of formal academic training. He received his Ph.D. and married his wife, Bee, in 1931.
Ross taught at several institutions including St. Louis University before becoming chair of University of Notre Dame's mathematics department in 1946. He started a teacher training program in mathematics that evolved into the Ross Mathematics Program in 1957 with the addition of high school students. The program moved with him to Ohio State University when he became their department chair in 1963. Though forced to retire in 1976, Ross ran the summer program until 2000. He had worked with over 2,000 students during more than forty summers.
The program is known as Ross's most significant work. Its attendees have since continued on to prominent research positions across the sciences. His program inspired several offshoots and was recognized by mathematicians as highly influential. Ross has received an honorary doctorate and several professional association awards for his instruction and service.
Early life and career
Ross was born Arnold Ephraim Chaimovich on August 24, 1906, in Chicago to Ukrainian-Jewish immigrants. He was an only child. His mother supported the family as a physical therapist. Ross returned to Odesa, Ukraine with his mother in 1909 for assistance from her extended family, and stayed once World War I and the Russian Revolution broke out. The two events led to widespread famine and economic woe in the region. Ross learned Russian at the behest of his mother, and developed a love of the theater and language. Ross's mother encouraged him to read, which he did often, and subscribed to a private library since Odesa had no public library. He credited his favorite uncle, an X-ray diagnostician, with introducing him to mathematics. The uncle had hired Samuil Shatunovsky to tutor his talented son, and Ross asked to join in. As money meant little due to inflation, Shatunovsky was paid to tutor the two boys with a pound of French hard candy. During this time, Ross was not taught with textbooks or lectured on geometric proofs. His geometry teacher would ask the class to prove and justify ideas on the blackboard per trial and error. Many universities were closed due to the famine, but Odesa University reopened and let a small group of adolescents attend, including Ross.
Ross left Odesa—now part of Ukraine—in 1922 with the intention of returning to Chicago and studying topology with E. H. Moore at the University of Chicago. After negotiating his way home, he worked at a family friend's bookbinding shop and continued to learn English at th
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https://en.wikipedia.org/wiki/Bisector%20%28music%29
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In diatonic set theory, a bisector divides the octave approximately in half (the equal tempered tritone is exactly half the octave) and may be used in place of a generator to derive collections for which structure implies multiplicity is not true such as the ascending melodic minor, harmonic minor, and octatonic scales. Well formed generated collections generators and bisectors coincide, such as the perfect fifth (circle of fifths) in the diatonic collection. The term was introduced by Jay Rahn (1977), who considers any division between one and two thirds as approximately half (major third to minor sixth or 400 to 800 cents) and who applied the term only the equally spaced collections. Clough and Johnson both adapt the term to apply to generic scale steps. Rahn also uses aliquant bisector for bisectors which may be used to generate every note in a collection, in which case the bisector and the number of notes must be coprime. Bisectors may be used to produce the diatonic, harmonic minor, and ascending melodic minor collections. (Johnson 2003, p.97, 101, 158n10-12)
The diatonic scale may be derived from a chain of perfect fifths:
P5 P5 P5 P5 P5 P5
F C G D A E B = C D E F G A B C.
5, 0, 7, 2, 9, 4, e = 0, 2, 4, 5, 7, 9, e, 0.
+7 +7 +7 +7 +7 +7 (mod 12)
For example, the octatonic scale may be derived similarly to derivations of the diatonic scale by a chain of perfect fifths (a generator), by using a bisector of 5 scale steps (3 may also be used). However, five steps in the octatonic scale alternates between 7 and 8 semitones, so it is a bisector and not a generator:
A5 P5 A5 P5 A5 P5 A5 P5
C A♭ E♭ B G♭ D A F C = C D E♭ F G♭ A♭ A B C.
0, 8, 3, e, 6, 2, 9, 5, 0 = 0, 2, 3, 5, 6, 8, 9, e, 0.
+8 +7 +8 +7 +8 +7 +8 +7
References
Johnson, Timothy (2003). Foundations of Diatonic Theory: A Mathematically Based Approach to Music Fundamentals. Key College Publishing. .
Rahn, Jay (1977). "Some Recurrent Features of Scales", In Theory Only 2, no. 11-12: 43-52
Diatonic set theory
Musicology
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https://en.wikipedia.org/wiki/Poncelet%27s%20closure%20theorem
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In geometry, Poncelet's closure theorem, also known as Poncelet's porism, states that whenever a polygon is inscribed in one conic section and circumscribes another one, the polygon must be part of an infinite family of polygons that are all inscribed in and circumscribe the same two conics. It is named after French engineer and mathematician Jean-Victor Poncelet, who wrote about it in 1822; however, the triangular case was discovered significantly earlier, in 1746 by William Chapple.
Poncelet's porism can be proved by an argument using an elliptic curve, whose points represent a combination of a line tangent to one conic and a crossing point of that line with the other conic.
Statement
Let C and D be two plane conics. If it is possible to find, for a given n > 2, one n-sided polygon that is simultaneously inscribed in C (meaning that all of its vertices lie on C) and circumscribed around D (meaning that all of its edges are tangent to D), then it is possible to find infinitely many of them. Each point of C or D is a vertex or tangency (respectively) of one such polygon.
If the conics are circles, the polygons that are inscribed in one circle and circumscribed about the other are called bicentric polygons, so this special case of Poncelet's porism can be expressed more concisely by saying that every bicentric polygon is part of an infinite family of bicentric polygons with respect to the same two circles.
Proof sketch
View C and D as curves in the complex projective plane P2. For simplicity, assume that C and D meet transversely (meaning that each intersection point of the two is a simple crossing). Then by Bézout's theorem, the intersection C ∩ D of the two curves consists of four complex points. For an arbitrary point d in D, let ℓd be the tangent line to D at d. Let X be the subvariety of C × D consisting of (c,d) such that ℓd passes through c. Given c, the number of d with (c,d) ∈ X is 1 if c ∈ C ∩ D and 2 otherwise. Thus the projection X → C ≃ P1 presents X as a degree 2 cover ramified above 4 points, so X is an elliptic curve (once we fix a base point on X). Let be the involution of X sending a general (c,d) to the other point (c,d′) with the same first coordinate. Any involution of an elliptic curve with a fixed point, when expressed in the group law, has the form x → p − x for some p, so has this form. Similarly, the projection X → D is a degree 2 morphism ramified over the contact points on D of the four lines tangent to both C and D, and the corresponding involution has the form x → q − x for some q. Thus the composition is a translation on X. If a power of has a fixed point, that power must be the identity. Translated back into the language of C and D, this means that if one point c ∈ C (equipped with a corresponding d) gives rise to an orbit that closes up (i.e., gives an n-gon), then so does every point. The degenerate cases in which C and D are not transverse follow from a limit argument.
See also
Finding E
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https://en.wikipedia.org/wiki/Ramanujan%E2%80%93Nagell%20equation
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In mathematics, in the field of number theory, the Ramanujan–Nagell equation is an equation between a square number and a number that is seven less than a power of two. It is an example of an exponential Diophantine equation, an equation to be solved in integers where one of the variables appears as an exponent.
The equation is named after Srinivasa Ramanujan, who conjectured that it has only five integer solutions, and after Trygve Nagell, who proved the conjecture. It implies non-existence of perfect binary codes with the minimum Hamming distance 5 or 6.
Equation and solution
The equation is
and solutions in natural numbers n and x exist just when n = 3, 4, 5, 7 and 15 .
This was conjectured in 1913 by Indian mathematician Srinivasa Ramanujan, proposed independently in 1943 by the Norwegian mathematician Wilhelm Ljunggren, and proved in 1948 by the Norwegian mathematician Trygve Nagell. The values of n correspond to the values of x as:-
x = 1, 3, 5, 11 and 181 .
Triangular Mersenne numbers
The problem of finding all numbers of the form 2b − 1 (Mersenne numbers) which are triangular is equivalent:
The values of b are just those of n − 3, and the corresponding triangular Mersenne numbers (also known as Ramanujan–Nagell numbers) are:
for x = 1, 3, 5, 11 and 181, giving 0, 1, 3, 15, 4095 and no more .
Equations of Ramanujan–Nagell type
An equation of the form
for fixed D, A , B and variable x, n is said to be of Ramanujan–Nagell type. The result of Siegel implies that the number of solutions in each case is finite. By representing with and with , the equation of Ramanujan–Nagell type is reduced to three Mordell curves (indexed by ), each of which has a finite number of integer solutions:
,
,
.
The equation with has at most two solutions, except in the case corresponding to the Ramanujan–Nagell equation. There are infinitely many values of D for which there are two solutions, including .
Equations of Lebesgue–Nagell type
An equation of the form
for fixed D, A and variable x, y, n is said to be of Lebesgue–Nagell type. This is named after Victor-Amédée Lebesgue, who proved that the equation
has no nontrivial solutions.
Results of Shorey and Tijdeman imply that the number of solutions in each case is finite. Bugeaud, Mignotte and Siksek solved equations of this type with A = 1 and 1 ≤ D ≤ 100. In particular, the following generalization of the Ramanujan-Nagell equation:
has positive integer solutions only when x = 1, 3, 5, 11, or 181.
See also
Pillai's conjecture
Scientific equations named after people
Notes
References
External links
Can N2 + N + 2 Be A Power Of 2?, Math Forum discussion
Diophantine equations
Srinivasa Ramanujan
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https://en.wikipedia.org/wiki/Pappus%20configuration
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In geometry, the Pappus configuration is a configuration of nine points and nine lines in the Euclidean plane, with three points per line and three lines through each point.
History and construction
This configuration is named after Pappus of Alexandria. Pappus's hexagon theorem states that every two triples of collinear points and (none of which lie on the intersection of the two lines) can be completed to form a Pappus configuration, by adding the six lines , , , , , and , and their three intersection points , , and . These three points are the intersection points of the "opposite" sides of the hexagon . According to Pappus' theorem, the resulting system of nine points and eight lines always has a ninth line containing the three intersection points , , and , called the Pappus line.
The Pappus configuration can also be derived from two triangles and that are in perspective with each other (the three lines through corresponding pairs of points meet at a single crossing point) in three different ways, together with their three centers of perspectivity , , and . The points of the configuration are the points of the triangles and centers of perspectivity, and the lines of the configuration are the lines through corresponding pairs of points.
Related constructions
The Levi graph of the Pappus configuration is known as the Pappus graph. It is a bipartite symmetric cubic graph with 18 vertices and 27 edges.
The Desargues configuration can also be defined in terms of perspective triangles, and the Reye configuration can be defined analogously from two tetrahedra that are in perspective with each other in four different ways, forming a desmic system of tetrahedra.
For any nonsingular cubic plane curve in the Euclidean plane, three real inflection points of the curve, and a fourth point on the curve, there is a unique way of completing these four points to form a Pappus configuration in such a way that all nine points lie on the curve.
Applications
A variant of the Pappus configuration provides a solution to the orchard-planting problem, the problem of finding sets of points that have the largest possible number of lines through three points. The nine points of the Pappus configuration form only nine three-point lines. However, they can be arranged so that there is another three-point line, making a total of ten. This is the maximum possible number of three-point lines through nine points.
References
External links
Configurations (geometry)
Dot patterns
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https://en.wikipedia.org/wiki/Monopole%20%28mathematics%29
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In mathematics, a monopole is a connection over a principal bundle G with a section of the associated adjoint bundle.
Physical interpretation
Physically, the section can be interpreted as a Higgs field, where the connection and Higgs field should satisfy the Bogomolny equations and be of finite action.
See also
Nahm equations
Instanton
Magnetic monopole
Yang–Mills theory
References
Differential geometry
Mathematical physics
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https://en.wikipedia.org/wiki/Hand%20geometry
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Hand geometry is a biometric that identifies users from the shape of their hands. Hand geometry readers measure a user's palm and fingers along many dimensions including length, width, deviation, and angle and compare those measurements to measurements stored in a file.
History
Viable hand geometry devices have been manufactured since the early 1970s, making hand geometry the first biometric to find widespread computerized use. Robert Miller realized the distinctive features of hand sizes and shapes could be used for identification and patented the first automated hand geometry device at the Stanford Research Institute in 1971. The device would measure the hand, and the numbers needed to match the punched holes of a user ID card to activate the circuit to be identified. David Sidlauskas was also a major player in the hand geometry device production, and he patented Handkey ID3D, the first hand scanner that worked in 3D that involved an optical measuring plate, camera, and numeric keypad to enter a personal PIN.
As an add-on
Hand geometry is not thought to be as unique as fingerprints, palm veins or irises. Fingerprinting and iris recognition remain the preferred technology for high-security applications. In large populations, hand geometry is not suitable for so-called one-to-many applications, in which a user is identified from his biometric without any other identification. However, hand geometry is very reliable when combined with other forms of identification, such as identification cards or personal identification numbers. There have also been proposed methods to include hand-geometry with palm print-based verification for better accuracy and performance.
Commercial use
There have been many patents issued for devices that measure hand geometry from the U.S. patent office.
A hand-geometry system‚ Identimat, was used at Shearson Hamil on Wall Street to track attendance, marking the beginning of biometric technology usage. Based on Robert Miller's patent, Identimat utilized light sensing cells to measure finger length and a magnetic strip card reader to verify identification cards and compared the information given to determine the authorization of the person. Although production ceased in 1987, the idea remains popular; common applications include access control and time-and-attendance operations.
Advantages
Although hand geometry is not considered to be the most secure compared to other biometric points, there are some advantages to this method. This includes:
Medium cost
Fast results due to low-computational cost algorithms
Reduced template size so takes up less storage
Easy to use
Disadvantages
Although the performance of these systems is not shown to be influenced by factors such as dry skin, large rings and swelling in the fingers may pose problems.
They are not universally accessible, as they cannot be used by those with paralysis or Parkinson's disease, and they can be deceived using high-quality bone structure models.
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https://en.wikipedia.org/wiki/Differential%20graded%20algebra
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In mathematics, in particular in homological algebra, a differential graded algebra is a graded associative algebra with an added chain complex structure that respects the algebra structure.
Definition
A differential graded algebra (or DG-algebra for short) A is a graded algebra equipped with a map which has either degree 1 (cochain complex convention) or degree −1 (chain complex convention) that satisfies two conditions:
A more succinct way to state the same definition is to say that a DG-algebra is a monoid object in the monoidal category of chain complexes.
A DG morphism between DG-algebras is a graded algebra homomorphism which respects the differential d.
A differential graded augmented algebra (also called a DGA-algebra,
an augmented DG-algebra or simply a DGA) is a DG-algebra equipped with a DG morphism to the ground ring (the terminology is due to Henri Cartan).
Warning: some sources use the term DGA for a DG-algebra.
Examples of DG-algebras
Tensor algebra
The tensor algebra is a DG-algebra with differential similar to that of the Koszul complex. For a vector space over a field there is a graded vector space defined as
where .
If is a basis for there is a differential on the tensor algebra defined component-wise
sending basis elements to
In particular we have and so
Koszul complex
One of the foundational examples of a differential graded algebra, widely used in commutative algebra and algebraic geometry, is the Koszul complex. This is because of its wide array of applications, including constructing flat resolutions of complete intersections, and from a derived perspective, they give the derived algebra representing a derived critical locus.
De-Rham algebra
Differential forms on a manifold, together with the exterior derivation and the exterior product form a DG-algebra. These have wide applications, including in derived deformation theory. See also de Rham cohomology.
Singular cohomology
The singular cohomology of a topological space with coefficients in is a DG-algebra: the differential is given by the Bockstein homomorphism associated to the short exact sequence , and the product is given by the cup product. This differential graded algebra was used to help compute the cohomology of Eilenberg–MacLane spaces in the Cartan seminar.
Other facts about DG-algebras
The homology of a DG-algebra is a graded algebra. The homology of a DGA-algebra is an augmented algebra.
See also
Homotopy associative algebra
Differential graded category
Differential graded Lie algebra
Differential graded scheme
Differential graded module
References
, see sections V.3 and V.5.6
Algebras
Homological algebra
commutative algebra
Differential algebra
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https://en.wikipedia.org/wiki/Omaha%20North%20High%20School
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Omaha North High Magnet School is a public high school located at 4410 North 36th Street in the city of Omaha, Nebraska. The school is a science, technology, engineering and mathematics (STEM) magnet school in the Omaha Public Schools district. North has won several awards, including being named a 2007 Magnet Schools of America "Magnet School of Excellence".
History
After a start to construction in 1922 at North 31st and Ames Avenues was hampered by unexpected groundwater, the present North High School was completed in 1924 at North 36th Street and Ames Avenue. Located on four acres, North opened as an eighth through twelfth grade school in September 1924, and had 650 students its first year. The building's first principal was Edward E. McMillan, who served until 1942. The junior high students were moved to other schools by 1929. The school began winning district and state awards in academic and athletic competitions in 1926, and in 1932, North High earned the top place at a national debate competition.
During World War II, 1,711 North graduates served in the United States military; 77 did not make it home. By 1945 the school's attendance doubled. In the late 1940s, the building was expanded to accommodate huge growth in the population of North Omaha. A music wing, cafeteria, gymnasium and more classrooms were added. The swimming pool was converted into a freshman study hall and library. In the late 1950s, the late E. E. McMillan was honored as the namesake of the nearby middle school, McMillan Magnet Center.
Overcrowding began to become a problem in the 1960s, and by the 1970s, more than 2,400 students were enrolled. The Omaha Public Schools initiated a desegregation plan in 1976, and in the 1980s, the temporary elimination of 9th grade at North led to decreased enrollment. North was designated a magnet school focused on math, science, and technology during that decade as well. In 1993, the school completed a $20 million renovation. Several nearby properties were bought and several additions expanded North's campus to its present twelve acres. The new student center and cafeteria, called the Viking Center, was heralded as a major development, along with a renovation of the library and new facilities that included a multipurpose physical education and athletics facility.
In the early 2000s, North shifted its magnet focus to specialize in the fields of science, engineering and technology. To complement this focus, in 2010 a new four-story, 32,092-square foot addition was dedicated at North. It consists of science, media technology, and engineering classrooms and lab space, as well as a wrestling practice room in the basement. Students taking classes in North's engineering program helped design the new building, working with RDG Planning & Design to present and finish the plans. With this building, North became the first school in Nebraska to be certified under the Leadership in Energy and Environmental Design (LEED) for Schools, earning a S
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https://en.wikipedia.org/wiki/Congruent%20transformation
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In mathematics, a congruent transformation (or congruence transformation) is:
Another term for an isometry; see congruence (geometry).
A transformation of the form A → PTAP, where A and P are square matrices, P is invertible, and PT denotes the transpose of P; see Matrix Congruence and congruence in linear algebra.
Mathematics disambiguation pages
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https://en.wikipedia.org/wiki/Universal%20probability%20bound
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A universal probability bound is a probabilistic threshold whose existence is asserted by William A. Dembski and is used by him in his works promoting intelligent design. It is defined as
Dembski asserts that one can effectively estimate a positive value which is a universal probability bound. The existence of such a bound would imply that certain kinds of random events whose probability lies below this value can be assumed not to have occurred in the observable universe, given the resources available in the entire history of the observable universe. Contrapositively, Dembski uses the threshold to argue that the occurrence of certain events cannot be attributed to chance alone. Universal probability bound is then used to argue against random evolution. However evolution is not based on random events only (genetic drift), but also on natural selection.
The idea that events with fantastically small, but positive probabilities, are effectively negligible was discussed by the French mathematician Émile Borel primarily in the context of cosmology and statistical mechanics. However, there is no widely accepted scientific basis for claiming that certain positive values are universal cutoff points for effective negligibility of events. Borel, in particular, was careful to point out that negligibility was relative to a model of probability for a specific physical system.
Dembski appeals to cryptographic practice in support of the concept of the universal probability bound, noting that cryptographers have sometimes compared the security of encryption algorithms against brute force attacks by the likelihood of success of an adversary utilizing computational resources bounded by very large physical constraints. An example of such a constraint might be obtained for example, by assuming that every atom in the observable universe is a computer of a certain type and these computers are running through and testing every possible key. Although universal measures of security are used much less frequently than asymptotic ones and the fact that a keyspace is very large may be less relevant if the cryptographic algorithm used has vulnerabilities which make it susceptible to other kinds of attacks, asymptotic approaches and directed attacks would, by definition, be unavailable under chance-based scenarios such as those relevant to Dembski's universal probability bound. As a result, Dembski's appeal to cryptography is best understood as referring to brute force attacks, rather than directed attacks.
Dembski's estimate
Dembski's original value for the universal probability bound is 1 in 10150, derived as the inverse of the product of the following approximate quantities:
1080, the number of elementary particles in the observable universe.
1045, the maximum rate per second at which transitions in physical states can occur (i.e., the inverse of the Planck time).
1025, a billion times longer than the typical estimated age of the universe in seconds.
Thus, 101
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https://en.wikipedia.org/wiki/Reflection%20group
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In group theory and geometry, a reflection group is a discrete group which is generated by a set of reflections of a finite-dimensional Euclidean space. The symmetry group of a regular polytope or of a tiling of the Euclidean space by congruent copies of a regular polytope is necessarily a reflection group. Reflection groups also include Weyl groups and crystallographic Coxeter groups. While the orthogonal group is generated by reflections (by the Cartan–Dieudonné theorem), it is a continuous group (indeed, Lie group), not a discrete group, and is generally considered separately.
Definition
Let E be a finite-dimensional Euclidean space. A finite reflection group is a subgroup of the general linear group of E which is generated by a set of orthogonal reflections across hyperplanes passing through the origin. An affine reflection group is a discrete subgroup of the affine group of E that is generated by a set of affine reflections of E (without the requirement that the reflection hyperplanes pass through the origin).
The corresponding notions can be defined over other fields, leading to complex reflection groups and analogues of reflection groups over a finite field.
Examples
Plane
In two dimensions, the finite reflection groups are the dihedral groups, which are generated by reflection in two lines that form an angle of and correspond to the Coxeter diagram Conversely, the cyclic point groups in two dimensions are not generated by reflections, nor contain any – they are subgroups of index 2 of a dihedral group.
Infinite reflection groups include the frieze groups and and the wallpaper groups , , , and . If the angle between two lines is an irrational multiple of pi, the group generated by reflections in these lines is infinite and non-discrete, hence, it is not a reflection group.
Space
Finite reflection groups are the point groups Cnv, Dnh, and the symmetry groups of the five Platonic solids. Dual regular polyhedra (cube and octahedron, as well as dodecahedron and icosahedron) give rise to isomorphic symmetry groups. The classification of finite reflection groups of R3 is an instance of the ADE classification.
Relation with Coxeter groups
A reflection group W admits a presentation of a special kind discovered and studied by H. S. M. Coxeter. The reflections in the faces of a fixed fundamental "chamber" are generators ri of W of order 2. All relations between them formally follow from the relations
expressing the fact that the product of the reflections ri and rj in two hyperplanes Hi and Hj meeting at an angle is a rotation by the angle fixing the subspace Hi ∩ Hj of codimension 2. Thus, viewed as an abstract group, every reflection group is a Coxeter group.
Finite fields
When working over finite fields, one defines a "reflection" as a map that fixes a hyperplane (otherwise for example there would be no reflections in characteristic 2, as so reflections are the identity). Geometrically, this amounts to including she
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https://en.wikipedia.org/wiki/Corrado%20Segre
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Corrado Segre (20 August 1863 – 18 May 1924) was an Italian mathematician who is remembered today as a major contributor to the early development of algebraic geometry.
Early life
Corrado's parents were Abramo Segre and Estella De Benedetti.
Career
Segre developed his entire career at the University of Turin, first as a student of Enrico D'Ovidio. In 1883 he published a dissertation on quadrics in projective space and was named an assistant to professors in algebra and analytic geometry. In 1885 he also assisted in descriptive geometry. He began to instruct in projective geometry, as a stand-in for Giuseppe Bruno, from 1885 to 1888. Then for 36 years, he had the chair in higher geometry following D'Ovidio. Segre and Giuseppe Peano made Turin known in geometry, and their complementary instruction has been noted as follows:
The Erlangen program of Felix Klein appealed early on to Segre, and he became a promulgator. First, in 1885 he published an article on conics in the plane where he demonstrated how group theory facilitated the study. As Hawkins says (page 252) "the totality of all conics in the plane is identified with P5(C)". The group of its projectivities is then the group that permutes conics. About Segre, Hawkins writes
The inspiring Geometrie der Lage (1847) of Karl Georg Christian von Staudt provided Segre with another project. He encouraged Mario Pieri to make a translation, Geometria di Posizione (1889), while Segre composed a biographical sketch of von Staudt that was included in the publication.
Segre also expanded algebraic geometry by consideration of multicomplex numbers, in particular the bicomplex numbers. Segre's 1892 contribution to Mathematische Annalen shows him extending the work of William Rowan Hamilton and William Kingdon Clifford on biquaternions. But Segre was unaware of an earlier study of tessarines that had anticipated his bicomplex numbers.
In English, the best-known work of Segre is an inspirational essay meant for Italian students, translated by J.W. Young in 1904. It provides guidance and encouragement to young people studying mathematics.
In a 1926 memorial article, H.F. Baker called Segre the "father of" the Italian school of algebraic geometry.
The 1912 article "Higher-dimensional Spaces" (Mehrdimensionale Räume) for Enzyklopädie der mathematischen Wissenschaften spanned 200 pages. In admiration, Baker (1926) wrote and Coolidge (1927) reiterated: For completeness of detail, breadth of view, and generous recognition of the work of a host of other writers, this must remain for many years a monument of the comprehensiveness of the man.
Notes
References
.
.
. (see especially pages 455–67)
Pierre Speziale (1975) "Corrado Segre", Dictionary of Scientific Biography, auspices of American Council of Learned Societies.
Livia Giacardi (2001) "The Corrado Segre Archive", Historia Mathematica 28: 296–301.
Livia Giacardi, 2002, (Ed.) I Quaderni di Corrado Segre, CD-ROM, Dipartimento di matematica, Un
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https://en.wikipedia.org/wiki/Bond%20length
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In molecular geometry, bond length or bond distance is defined as the average distance between nuclei of two bonded atoms in a molecule. It is a transferable property of a bond between atoms of fixed types, relatively independent of the rest of the molecule.
Explanation
Bond length is related to bond order: when more electrons participate in bond formation the bond is shorter. Bond length is also inversely related to bond strength and the bond dissociation energy: all other factors being equal, a stronger bond will be shorter. In a bond between two identical atoms, half the bond distance is equal to the covalent radius.
Bond lengths are measured in the solid phase by means of X-ray diffraction, or approximated in the gas phase by microwave spectroscopy. A bond between a given pair of atoms may vary between different molecules. For example, the carbon to hydrogen bonds in methane are different from those in methyl chloride. It is however possible to make generalizations when the general structure is the same.
Bond lengths of carbon with other elements
A table with experimental single bonds for carbon to other elements is given below. Bond lengths are given in picometers. By approximation the bond distance between two different atoms is the sum of the individual covalent radii (these are given in the chemical element articles for each element). As a general trend, bond distances decrease across the row in the periodic table and increase down a group. This trend is identical to that of the atomic radius.
Bond lengths in organic compounds
The bond length between two atoms in a molecule depends not only on the atoms but also on such factors as the orbital hybridization and the electronic and steric nature of the substituents. The carbon–carbon (C–C) bond length in diamond is 154 pm. It is generally considered the average length for a carbon–carbon single bond, but is also the largest bond length that exists for ordinary carbon covalent bonds. Since one atomic unit of length(i.e., a Bohr radius) is 52.9177 pm, the C–C bond length is 2.91 atomic units, or approximately three Bohr radii long.
Unusually long bond lengths do exist. Current record holder for the longest C-C bond with a length of 186.2 pm is 1,8-Bis(5-hydroxydibenzo[a,d]cycloheptatrien-5-yl)naphthalene, one of many molecules within a category of hexaaryl ethanes, which are derivatives based on hexaphenylethane skeleton. Bond is located between carbons C1 and C2 as depicted in a picture below.
Another notable compound with an extraordinary C-C bond length is tricyclobutabenzene, in which a bond length of 160 pm is reported. Longest C-C bond within the cyclobutabenzene category is 174 pm based on X-ray crystallography. In this type of compound the cyclobutane ring would force 90° angles on the carbon atoms connected to the benzene ring where they ordinarily have angles of 120°.
The existence of a very long C–C bond length of up to 290 pm is claimed in a dimer of two tetracyanoethylen
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https://en.wikipedia.org/wiki/IB%20Group%205%20subjects
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The Group 5: Mathematics subjects of the IB Diploma Programme consist of two different mathematics courses, both of which can be taken at Standard Level (SL) or Higher Level (HL). To earn an IB Diploma, a candidate must take either Mathematics Applications and Interpretation (SL/HL) or Mathematics Analysis and Approaches (SL/HL), as well as satisfying all CAS, TOK and EE requirements.
Examination structure
At the standard level (SL), there are 2 external examinations and 1 internal examination for both of the IB math courses. At the higher level (HL), there are 3 external examinations and 1 internal examination for both of the IB math courses.
Footnotes
References
International Baccalaureate
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https://en.wikipedia.org/wiki/George%20Yuri%20Rainich
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George Yuri Rainich (Rabinovich) (March 25, 1886 in Odessa – October 10, 1968) was a leading mathematical physicist in the early twentieth century.
Career
Rainich studied mathematics from 1904 to 1908 in Odessa, in Göttingen (1905–1906), and in Munich (1906–1907), eventually obtaining his doctorate (Magister of Pure Mathematics) in 1913 from the University of Kazan. After teaching at the University of Kazan, in 1922 (via Istanbul), he emigrated with his wife to the United States. After three years at Johns Hopkins University, he joined the faculty of the University of Michigan, where he remained until his retirement in 1956. After his retirement as professor emeritus, he was in 1957 at Brown University as a member of the editorial staff of Mathematical Reviews and he was for several years a visiting professor at the University of Notre Dame. After the death of his wife in 1963, he returned to the University of Michigan at Ann Arbor and organized there a seminar on general relativity theory for physicists and mathematicians.
Rainich's research centered on general relativity and early work toward a unified field theory. In 1924, Rainich found a set of equivalent conditions for a Lorentzian manifold to admit an interpretation as an exact non-null electrovacuum solution in general relativity; these are now known as the Rainich conditions.
According to some sources, Peter Gabriel Bergmann brought Rainich's suggestion that algebraic topology (and knot theory in particular) should play a role in physics to the attention of John Archibald Wheeler, which shortly led to the Ph.D. thesis of Charles W. Misner. Another version of this tale replaces Bergmann with Hugh Everett, who was a fellow student of Misner at the time.
According to the Editor of The American Mathematical Monthly, Rainich is the inventor of the Rabinowitsch trick, a clever argument to deduce the Hilbert Nullstellensatz from an easier special case. It is later explained that Rainich was born Rabinowitsch, hence the Pseudonym. This claim is, however, disputed.
Rainich, as Georg Rabinowitsch, proved new results on Euler's prime-generating polynomial.
Rainich was an Invited Speaker at the ICM in 1928 at Bologna (with talk On a Spacetime possessing the symmetry properties of radiation) and in 1932 at Zürich. He also gave a talk (concerning his results on the Euler prime-generating polynomial) at the ICM in 1912 at Cambridge, England. From 1933 to 1936 he was a member of the council (advising the board of trustees) of the American Mathematical Society.
Rainich's private papers are held at the University of Texas.
Personal life
In 1917 he married Sophie Kramkowsky. In 1930 he brought his mother from the Soviet Union to Ann Arbor, where she remained until her death in 1953. Upon his death he was survived by a daughter, two grandchildren, and a brother, Michael Rabinovich of Moscow.
Students
Several of Rainich's Ph.D. students became famous:
Ruel Vance Churchill (12 December 1899 - 31 O
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https://en.wikipedia.org/wiki/Bellos
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Bellos is a surname. Notable people with the surname include:
Alex Bellos, author of books on mathematics and football
David Bellos, English translator and biographer, father of Alex Bellos
Linda Bellos (born 1950), British activist and London politician.
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https://en.wikipedia.org/wiki/Preah%20Pithu
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Preah Pithu (, ) is a group of five temples at Angkor, Cambodia.
In fact they were in all probability not designed as a group. Despite their ruined state, the remains have good decorative carving and their semi-wooded setting is attractive and peaceful.
The site
The temples are located in Angkor Thom, north-east of the Bayon, in front of Tep Pranam. The temples are near but they weren't built in the same period, except for two of them, so there is no apparent order. They're identified by letters: T, U, V, W and X. "X" is a Buddhist temple, it remained unfinished and is probably the latest. The others are Hindu. The five temples are in bad conditions, upper levels are ruined, but their carvings are interesting and the site is rather peaceful, wooded and scarcely crowded. A moat, often dry, surrounds some of the temples. They were cleaned first by Jean Commaille in 1908, then by Henri Marchal from 1918 to 1920.
Temples "T" and "U"
They were built together quite for sure, as they are on the same W-E axis and are surrounded by a single moat. They're oriented to the west, having a cruciform terrace on two levels with nāgas balustrades as entrance. The temple "T" has a sandstone enclosure, which measures 45 by 40 meters, and has two gopuras on the main W-E axis. The sanctuary, on a 3-levels ornated platform 6 meters high, has a chamber which sheltered a large linga on its pedestal. The fragments of the lintel of western door, on the ground, show a stylized depiction of the "Churning of the Sea of Milk", while the walls are decorated with devatas and floral motifs, Bayon style.
The temple "U" is similar to "T", but smaller and simpler. Its enclosure is 35 by 28 meters. It has no gopura. The lintel of west door shows the Trimurti, with Shiva dancing on a kala-head between Vishnu and Brahma. The internal walls are sculpted with dvarapalas and the blind arches at the base of pillars are typical of Angkor Wat period. On the northern lintel there is another depiction of the "Churning of the Sea of Milk".
Temple "X"
Temple "X" stands on a large terrace, 4 meters high, about 100 meters east of temple "U". It is straight reachable from there if the moat is dry. It is similar in design to the former temples but is larger and incomplete in its decorations. "X" is a buddhist temple: on the walls of the sanctuary chamber there is a double row of sculpted Buddhas. Even the lintels show buddhist motifs. However the most interesting remains, depicting famous Siddharta's cutting of the hair, are not in place. Towards the east, the laterite terrace is surrounded by semas, delimiting the sacred area.
Temple "V"
Outside the residual moat, north of temple "U", there is temple "V". It is oriented to the East, where it opens with a double vestibule, and has no enclosure. On the West it has a causeway which ends with a 40 meter long cruciform terrace, leading into NE corner of the Royal Square. Its sanctuary chamber, standing on two-tier sculpted basement, is the la
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https://en.wikipedia.org/wiki/Ricci%20decomposition
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In the mathematical fields of Riemannian and pseudo-Riemannian geometry, the Ricci decomposition is a way of breaking up the Riemann curvature tensor of a Riemannian or pseudo-Riemannian manifold into pieces with special algebraic properties. This decomposition is of fundamental importance in Riemannian and pseudo-Riemannian geometry.
Definition of the decomposition
Let (M,g) be a Riemannian or pseudo-Riemannian n-manifold. Consider its Riemann curvature, as a (0,4)-tensor field. This article will follow the sign convention
written multilinearly, this is the convention
With this convention, the Ricci tensor is a (0,2)-tensor field defined by Rjk=gilRijkl and the scalar curvature is defined by R=gjkRjk. (Note that this is the less common sign convention for the Ricci tensor; it is more standard to define it by contracting either the first and third or the second and fourth indices, which yields a Ricci tensor with the opposite sign. Under that more common convention, the signs of the Ricci tensor and scalar must be changed in the equations below.) Define the traceless Ricci tensor
and then define three (0,4)-tensor fields S, E, and W by
The "Ricci decomposition" is the statement
As stated, this is vacuous since it is just a reorganization of the definition of W. The importance of the decomposition is in the properties of the three new tensors S, E, and W.
Terminological note. The tensor W is called the Weyl tensor. The notation W is standard in mathematics literature, while C is more common in physics literature. The notation R is standard in both, while there is no standardized notation for S, Z, and E.
Basic properties
Properties of the pieces
Each of the tensors S, E, and W has the same algebraic symmetries as the Riemann tensor. That is:
together with
The Weyl tensor has the additional symmetry that it is completely traceless:
Hermann Weyl showed that W has the remarkable property of measuring the deviation of a Riemannian or pseudo-Riemannian manifold from local conformal flatness; if it is zero, then M can be covered by charts relative to which g has the form gij=efδij for some function f defined chart by chart.
Properties of the decomposition
One may check that the Ricci decomposition is orthogonal in the sense that
recalling the general definition This has the consequence, which could be proved directly, that
Terminological note. It would be symbolically clean to present this orthogonality as saying
together with
However, there is an unavoidable ambiguity with such notation depending on whether one views as multilinear maps or as linear maps in which case the corresponding norms and inner products would differ by a constant factor. Although this would not lead to any inconsistencies in the above equations, since all terms would be changed by the same factor, it can lead to confusion in more involved contexts. For this reason, the index notation can often be easier to understand.
Related formulas
One can compute the "
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https://en.wikipedia.org/wiki/False%20nearest%20neighbor%20algorithm
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Within abstract algebra, the false nearest neighbor algorithm is an algorithm for estimating the embedding dimension. The concept was proposed by Kennel et al. (1992). The main idea is to examine how the number of neighbors of a point along a signal trajectory change with increasing embedding dimension. In too low an embedding dimension, many of the neighbors will be false, but in an appropriate embedding dimension or higher, the neighbors are real. With increasing dimension, the false neighbors will no longer be neighbors. Therefore, by examining how the number of neighbors change as a function of dimension, an appropriate embedding can be determined.
See also
Commutative ring
Local ring
Nearest neighbor
Time series
References
Statistical algorithms
Dynamical systems
Nonlinear time series analysis
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https://en.wikipedia.org/wiki/NCES
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NCES may refer to:
National Center for Education Statistics, part of the U.S. Department of Education
Net-Centric Enterprise Services, a United States Department of Defense program
Normal curve equivalents, a type of scale score based on the normal curve
See also
NCE (disambiguation)
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https://en.wikipedia.org/wiki/Erd%C5%91s%E2%80%93Straus%20conjecture
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The Erdős–Straus conjecture is an unproven statement in number theory. The conjecture is that, for every integer that is 2 or more, there exist positive integers , , and for which
In other words, the number can be written as a sum of three positive unit fractions.
The conjecture is named after Paul Erdős and Ernst G. Straus, who formulated it in 1948, but it is connected to much more ancient mathematics; sums of unit fractions, like the one in this problem, are known as Egyptian fractions, because of their use in ancient Egyptian mathematics. The Erdős–Straus conjecture is one of many conjectures by Erdős, and one of many unsolved problems in mathematics concerning Diophantine equations.
Although a solution is not known for all values of , infinitely many values in certain infinite arithmetic progressions have simple formulas for their solution, and skipping these known values can speed up searches for counterexamples. Additionally, these searches need only consider values of that are prime numbers, because any composite counterexample would have a smaller counterexample among its prime factors. Computer searches have verified the truth of the conjecture up to .
If the conjecture is reframed to allow negative unit fractions, then it is known to be true. Generalizations of the conjecture to fractions with numerator 5 or larger have also been studied.
Background and history
When a rational number is expanded into a sum of unit fractions, the expansion is called an Egyptian fraction. This way of writing fractions dates to the mathematics of ancient Egypt, in which fractions were written this way instead of in the more modern vulgar fraction form with a numerator and denominator . The Egyptians produced tables of Egyptian fractions for unit fractions multiplied by two, the numbers that in modern notation would be written , such as the Rhind Mathematical Papyrus table; in these tables, most of these expansions use either two or three terms. These tables were needed, because the obvious expansion was not allowed: the Egyptians required all of the fractions in an Egyptian fraction to be different from each other. This same requirement, that all fractions be different, is sometimes imposed in the Erdős–Straus conjecture, but it makes no significant difference to the problem, because for any solution to where the unit fractions are not distinct can be converted into a solution where they are all distinct; see below.
Although the Egyptians did not always find expansions using as few terms as possible, later mathematicians have been interested in the question of how few terms are needed. Every fraction has an expansion of at most terms, so in particular needs at most two terms, needs at most three terms, and needs at most four terms. For , two terms are always needed, and for , three terms are sometimes needed, so for both of these numerators, the maximum number of terms that might be needed is known. However, for , it is unknown whethe
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https://en.wikipedia.org/wiki/Theorem%20on%20friends%20and%20strangers
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The theorem on friends and strangers is a mathematical theorem in an area of mathematics called Ramsey theory.
Statement
Suppose a party has six people. Consider any two of them. They might be meeting for the first time—in which case we will call them mutual strangers; or they might have met before—in which case we will call them mutual acquaintances. The theorem says:
In any party of six people, at least three of them are (pairwise) mutual strangers or mutual acquaintances.
Conversion to a graph-theoretic setting
A proof of the theorem requires nothing but a three-step logic. It is convenient to phrase the problem in graph-theoretic language.
Suppose a graph has 6 vertices and every pair of (distinct) vertices is joined by an edge. Such a graph is called a complete graph (because there cannot be any more edges). A complete graph on vertices is denoted by the symbol .
Now take a . It has 15 edges in all. Let the 6 vertices stand for the 6 people in our party. Let the edges be coloured red or blue depending on whether the two people represented by the vertices connected by the edge are mutual strangers or mutual acquaintances, respectively. The theorem now asserts:
No matter how you colour the 15 edges of a with red and blue, you cannot avoid having either a red triangle—that is, a triangle all of whose three sides are red, representing three pairs of mutual strangers—or a blue triangle, representing three pairs of mutual acquaintances. In other words, whatever colours you use, there will always be at least one monochromatic triangle ( that is, a triangle all of whose edges have the same color ).
Proof
Choose any one vertex; call it P. There are five edges leaving P. They are each coloured red or blue. The pigeonhole principle says that at least three of them must be of the same colour; for if there are less than three of one colour, say red, then there are at least three that are blue.
Let A, B, C be the other ends of these three edges, all of the same colour, say blue. If any one of AB, BC, CA is blue, then that edge together with the two edges from P to the edge's endpoints forms a blue triangle. If none of AB, BC, CA is blue, then all three edges are red and we have a red triangle, namely, ABC.
Ramsey's paper
The utter simplicity of this argument, which so powerfully produces a very interesting conclusion, is what makes the theorem appealing. In 1930, in a paper entitled 'On a Problem of Formal Logic,' Frank P. Ramsey proved a very general theorem (now known as Ramsey's theorem) of which this theorem is a simple case. This theorem of Ramsey forms the foundation of the area known as Ramsey theory in combinatorics.
Boundaries to the theorem
The conclusion to the theorem does not hold if we replace the party of six people by a party of less than six. To show this, we give a coloring of K5 with red and blue that does not contain a triangle with all edges the same color. We draw K5 as a pentagon surrounding a star (a pentagram).
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https://en.wikipedia.org/wiki/Primary%20group
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A primary group may refer to:
In mathematics, a special kind of group:
a p-primary group, also called simply p-group; or
a primary cyclic group, which is a p-primary cyclic group.
In sociology, a primary group as opposed to secondary group.
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https://en.wikipedia.org/wiki/Damodara
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Vatasseri Damodara Nambudiri was an astronomer-mathematician of the Kerala school of astronomy and mathematics who flourished during the fifteenth century CE. He was a son of Paramesvara (1360–1425) who developed the drigganita system of astronomical computations. The family home of Paramesvara was Vatasseri (sometimes called Vatasreni) in the village of Alathiyur, Tirur in Kerala.
Damodara was a teacher of Nilakantha Somayaji. As a teacher he initiated Nilakantha into the science of astronomy and taught him the basic principles in mathematical computations.
See also
List of astronomers and mathematicians of the Kerala school
References
Kerala school of astronomy and mathematics
15th-century Indian mathematicians
People from Malappuram district
Scientists from Kerala
15th-century Indian astronomers
Scholars from Kerala
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https://en.wikipedia.org/wiki/Anania%20Shirakatsi
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Anania Shirakatsi (, Anania Širakac’i, anglicized: Ananias of Shirak) was a 7th-century Armenian polymath and natural philosopher, author of extant works covering mathematics, astronomy, geography, chronology, and other fields. Little is known for certain of his life outside of his own writings, but he is considered the father of the exact and natural sciences in Armenia—the first Armenian mathematician, astronomer, and cosmographer.
Seen as a part of the Armenian Hellenizing School, the last lay scholar in Christian Armenia until the 11th century, Anania was educated primarily by Tychicus, in Trebizond. He composed science textbooks and the first known geographic work in classical Armenian (Ashkharhatsuyts), which provides detailed information about Greater Armenia, Persia and the Caucasus (Georgia and Caucasian Albania).
In mathematics, his accomplishments include the earliest known table of results of the four basic operations, the earliest known collection of recreational math puzzles and problems, and the earliest book of math problems in Armenian. He also devised a system of mathematical notation based on the Armenian alphabet, although he was the only writer known to have used it.
Name
His name is usually anglicized as Ananias of Shirak (Širak). Anania is the Armenian variant of the biblical name Ananias, itself the Greek version of the Hebrew Hananiah. The second part of his name denotes his place of origin, the region of Shirak (Širak), though it may have become a sort of surname. In some manuscripts, he is called Shirakuni (Շիրակունի) and Shirakavantsi (Շիրակաւանցի).
Life
Background
Anania Shirakatsi lived in the 7th century. The dates of his birth and death have not been definitively established. Robert H. Hewsen noted in 1968 that Anania is widely believed to have been born between 595 and 600; a quarter-century later he settled on c. 610 as a birthdate and 685 as the year he died. Agop Jack Hacikyan et al. place his birth in early 600s but agrees on 685. James R. Russell, Edward G. Mathews, and Theo van Lint also concur with 610–685, while Greenwood suggests c. 600–670. Vardanyan places his death in the early 690s.
Anania is the only classical Armenian scholar to have written an autobiography. It is a brief text, characterized as "somewhat self-congratulatory" and "more a statement of academic pedigree" than autobiography. It was probably written as the preface to one of his scholarly works, possibly the K'nnikon. He was the son of Hovhannes/Yovhannes and was born in the village of Anania/Aneank' (Անեանք) or in the town of Shirakavan (Yerazgavors), in the canton of Shirak (Širak), in the central Armenian province of Ayrarat. Aneank' may be connected to the later city of Ani, the Bagratid Armenian capital.
Anania probably came from a noble family. Since his name is sometimes spelled as "Shirakuni" (Շիրակունի), Hewsen argued that he may have belonged to the house of the Kamsarakan or Arsharuni princes of Shirak and Aršarunik’, r
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https://en.wikipedia.org/wiki/Erie%20County%20Fair
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The Erie County Fair is a fair held in Hamburg in Erie County, New York, every August. Based on 2018 attendance statistics, The Erie County Fair is the second largest fair in New York and the fourth largest county fair in North America, often drawing over one million in attendance.
History
1820 to 1867
The Erie County Agricultural Society is a private, not for profit membership corporation established in 1819, then called the Niagara County Horticultural Society. It held its first fair in 1820 on what is now the site of the Donovan Office Building in Buffalo. One year later, Niagara County split into Erie and Niagara Counties, and so did the agricultural society. The Erie County Agricultural Society is the oldest civic, community member organization in Erie County. The only time in the history of the fair where the event was not held was 1943 during World War II due to rationing of supplies for the war effort. The fair was briefly renamed America's Fair during the early to mid-2000s in an effort to expand the fair beyond Erie County; it has since been renamed the Erie County Fair.
The region's first agricultural society was founded in 1819 with the goal of sponsoring a county fair to promote education and competition among farmers. This goal was realized when the first fairs were held on the Buffalo waterfront in 1820 and 1821. Dr. Cyrenius Chapin, one of Buffalo's most active energetic pioneers was elected President for the fledgling organization. The location was near Terrace and Main Streets, the current site of One Canalside (2014). Enthusiasm for the annual "Farmer’s Holiday" was shortcut when local farmers lost interest due to poor travel conditions, a downturn in the economy and the community's collective focus on the building of the Erie Canal. With the opening of the Canal in 1825, the Village of Buffalo quickly grew in size and economic stature becoming a city in 1832. After not holding a fair from 1822 through 1840, The Erie County Agricultural Society was re-activated in 1841 and sponsored a Fair held at Lafayette Square on the grounds of the Erie County Court House. So earnest were their endeavors that only once since 1841 has a year passed without a Fair. In 1943 the Fair was postponed due to World War II rationing of gasoline and other vital commodities.
The Fair would be held within the Buffalo city-limits until 1849 at which time urban expansion facilitated a move to a country location for the Fair. 1850 marked the first year that the Fair was held outside of the City of Buffalo making its debut in the Village of Aurora. Lancaster hosted the Fair September 9 and 10, 1851. Fairgoers were able to take specially scheduled trains for a fare of .25 cents from Buffalo at 9:30a & 12:30a and returning from Lancaster at 1p and 5p. 1852 saw the Fair held in East Hamburgh, now Orchard Park. The 1853 Fair was held on the enclosed grounds at Cold Spring October 7 & 8. The Fair returned to Aurora in 1854 and 1855.
The year 1855 was
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https://en.wikipedia.org/wiki/Tournesol
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Tournesol may refer to:
Sunflower
Chrozophora
Professor Calculus (French: Professeur Tryphon Tournesol), a fictional character of The Adventures of Tintin
Tournesol (satellite), a satellite launched in 1971
"Tournesol" (magazine), a French comic book published since October 1960 by Ligue pour la Lecture de la Bible in Valence
See also
Turnsole
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https://en.wikipedia.org/wiki/Formulario%20mathematico
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Formulario Mathematico (Latino sine flexione: Formulary for Mathematics) is a book by Giuseppe Peano which expresses fundamental theorems of mathematics in a symbolic language developed by Peano. The author was assisted by Giovanni Vailati, Mario Pieri, Alessandro Padoa, Giovanni Vacca, Vincenzo Vivanti, Gino Fano and Cesare Burali-Forti.
The Formulario was first published in 1894. The fifth and last edition was published in 1908.
Hubert Kennedy wrote "the development and use of mathematical logic is the guiding motif of the project". He also explains the variety of Peano's publication under the title:
the five editions of the Formulario [are not] editions in the usual sense of the word. Each is essentially a new elaboration, although much material is repeated. Moreover, the title and language varied: the first three, titled Formulaire de Mathématiques, and the fourth, titled, Formulaire Mathématique, were written in French, while Latino sine flexione, Peano's own invention, was used for the fifth edition, titled Formulario Mathematico. ... Ugo Cassina lists no less than twenty separately published items as being parts of the 'complete' Formulario!
Peano believed that students needed only precise statement of their lessons. He wrote:
Each professor will be able to adopt this Formulario as a textbook, for it ought to contain all theorems and all methods. His teaching will be reduced to showing how to read the formulas, and to indicating to the students the theorems that he wishes to explain in his course.
Such a dismissal of the oral tradition in lectures at universities was the undoing of Peano's own teaching career.
Notes
References
Ivor Grattan-Guinness (2000) The Search for Mathematical Roots 1870-1940. Princeton University Press.
1895 non-fiction books
1908 non-fiction books
Mathematics books
Mathematical terminology
Mathematical logic
Mathematical symbols
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https://en.wikipedia.org/wiki/Simplicial%20manifold
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In physics, the term simplicial manifold commonly refers to one of several loosely defined objects, commonly appearing in the study of Regge calculus. These objects combine attributes of a simplex with those of a manifold. There is no standard usage of this term in mathematics, and so the concept can refer to a triangulation in topology, or a piecewise linear manifold, or one of several different functors from either the category of sets or the category of simplicial sets to the category of manifolds.
A manifold made out of simplices
A simplicial manifold is a simplicial complex for which the geometric realization is homeomorphic to a topological manifold. This is essentially the concept of a triangulation in topology. This can mean simply that a neighborhood of each vertex (i.e. the set of simplices that contain that point as a vertex) is homeomorphic to a n-dimensional ball.
A simplicial object built from manifolds
A simplicial manifold is also a simplicial object in the category of manifolds. This is a special case of a simplicial space in which, for each n, the space of n-simplices is a manifold.
For example, if G is a Lie group, then the simplicial nerve of G has the manifold as its space of n-simplices. More generally, G can be a Lie groupoid.
Structures on manifolds
Simplicial sets
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https://en.wikipedia.org/wiki/E12
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E12 or E-12 may refer to:
Science, technology and mathematics
the E12 series of preferred numbers
E12 screw, a type of Edison screw
the code name for Microsoft Exchange Server 2007
Siding Spring Survey code
Transport
Roads and trails
European route E12
E12 European long distance path
Ampang–Kuala Lumpur Elevated Highway (AKLEH), a fully elevated expressway in Kuala Lumpur, Malaysia
Vehicles
BMW E12
HMS E12, a United Kingdom Royal Navy submarine which saw service during World War I
Spyker E12, a Dutch Spyker car
Other uses
Queen's Indian Defense, Encyclopaedia of Chess Openings code
E12, a postcode district in the E postcode area
E-12 equal temperament consisting of twelve equal semitones per octave
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https://en.wikipedia.org/wiki/Indicatrix
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Indicatrix may refer to:
Differential geometry
Dupin indicatrix, a conic section which describes the local shape of a surface
Tissot's indicatrix, which describes and visualizes the distortion of a map
Tangent indicatrix, an object in differential geometry related to a closed space curve
Optics
Indicatrix, a special case of the index ellipsoid in the study of crystals and refractive indices
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https://en.wikipedia.org/wiki/Guillermo%20Owen
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Guillermo Owen (born 1938) is a Colombian mathematician, and professor of applied mathematics at the Naval Postgraduate School in Monterey, California, known for his work in game theory. He is also the son of the Mexican Poet and Diplomat Gilberto Owen.
Biography
Guillermo Owen was born May 4, 1938, in Bogotá, Colombia, and obtained a B.S. degree from Fordham University in 1958, and a Ph.D. degree from Princeton University under the guidance of Dr. Harold Kuhn in 1962.
Owen has taught at Fordham University (1961–1969), Rice University (1969–1977) and Los Andes University in Colombia (1978–1982, 2008), apart from having given lectures in many universities in Europe and Latin America. He is currently holding the position of Distinguished Professor of applied mathematics at the Naval Postgraduate School in Monterey, California.
Owen is member of the Colombian Academy of Sciences, The Royal Academy of Arts and Sciences of Barcelona, and the Third World Academy of Sciences. He is associate editor of the International Journal of Game Theory, and fellow of the International Game Theory Society.
Honors and awards
The Escuela Naval Almirante Padilla of Cartagena gave him an honorary degree of Naval Science Professional in June 2004.
Owen was named Honorary President of the XIV Latin Ibero American Congress on Operations Research - CLAIO 2008. Cartagena, Colombia, September 2008.
The university of Lower Normandy, in Caen, France, gave him an honorary doctorate in October 2017.
Publications
Owen has authored, translated and/or edited thirteen books, and approximately one hundred and forty papers published in journals such as Management Science, Operations Research, International Journal of Game Theory, American political Science Review, and Mathematical Programming, among others. Owen's books include:
1968. Game theory. Academic Press
1970. Finite mathematics and calculus; mathematics for the social and management sciences. With M. Evans Munroe.
1983. Information pooling and group decision making : proceedings of the Second University of California, Irvine, Conference on Political Economy. Edited with Bernard Grofman.
1999. Discrete mathematics and game theory.
2001. Power indices and coalition formation. Edited with Manfred J. Holler.
References
External links
Biography. Naval Postgraduate School in Monterey, California.
1938 births
20th-century Colombian mathematicians
Colombian expatriates in the United States
Game theorists
Living people
Fordham University alumni
Princeton University alumni
Fordham University faculty
Rice University faculty
Naval Postgraduate School faculty
Operations researchers
21st-century Colombian mathematicians
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https://en.wikipedia.org/wiki/Richard%20Peto
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Sir Richard Peto (born 14 May 1943) is an English statistician and epidemiologist who is Professor of Medical Statistics and Epidemiology at the University of Oxford, England.
Education
He attended Taunton's School in Southampton and subsequently studied the Natural Sciences Tripos at Trinity College, Cambridge followed by a Master of Science degree in Statistics at Imperial College London.
Career and research
His career has included collaborations with Richard Doll beginning at the Medical Research Council Statistical Research Unit in London. He set up the Clinical Trial Service Unit (CTSU) in Oxford in 1975 and is currently co-director. Peto's paradox is named after him.
Awards and honours
He was elected a Fellow of the Royal Society in 1989 for his contributions to the development of meta-analysis. He is a leading expert on deaths related to tobacco use. "When Sir Richard Peto began work with the late Richard Doll fifty years ago, the UK had the worst death rates from smoking in the world. Smoking was the cause of more than half of all premature deaths of British men." He was knighted for his services to epidemiology and to cancer prevention in 1999, and he received an honorary Doctor of Medical Sciences degree from Yale University in 2011.
Personal life
His brother Julian Peto, with whom he has published work in mathematical statistics (e.g. on the logrank test), is also a distinguished epidemiologist. His family runs a Thai restaurant in the Covered Market, Oxford, of whose parent company he is a director.
References
Selected publications
Peto R, Schneiderman M, eds. Quantification of occupational cancer. Cold Spring Harbor, NY: Cold Spring Harbor Laboratory, 1981. .
Peto R, zur Hausen H, eds. Viral etiology of cervical cancer. Cold Spring Harbor, NY: Cold Spring Harbor Laboratory, 1986. .
Peto R, Imperial Cancer Research Fund (Great Britain), World Health Organization, et al. Mortality from smoking in developed countries, 1950–2000: indirect estimates from national vital statistics. Oxford and New York: Oxford University Press, 1994. .
1943 births
Living people
Alumni of Queens' College, Cambridge
British epidemiologists
English statisticians
Fellows of Green Templeton College, Oxford
Members of the French Academy of Sciences
Donald Reid Medalists
Royal Medal winners
Winners of the Heineken Prize
Fellows of the Royal Society
Knights Bachelor
Fellows of the AACR Academy
Tobacco researchers
Members of the National Academy of Medicine
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https://en.wikipedia.org/wiki/Giovanni%20Vacca%20%28mathematician%29
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Giovanni Enrico Eugenio Vacca (18 November 1872 – 6 January 1953) was an Italian mathematician, Sinologist and historian of science.
Vacca studied mathematics and graduated from the University of Genoa in 1897 under the guidance of G. B. Negri. He was a politically active student and was banished for that from Genoa in 1897. He moved to Turin and became an assistant to Giuseppe Peano. In 1899 he studied, at Hanover, unpublished manuscripts of Gottfried Wilhelm Leibniz, which he published in 1903. Around 1898 Vacca became interested in Chinese language and culture after attending a Chinese exhibition in Turin. He took private lessons of Chinese and continued to study it at the University of Florence. Vacca then traveled to China in 1907–8. Originally he had planned to study the history of Chinese mathematics and ancient and modern science in more detail, but he returned to Europe without truly linking his mathematical interests with the study of Chinese civilization. He pursued the latter professionally, first, as a lecturer of Chinese language and literature at the Istituto di studi superiori di Firenze in 1910, then, in charge of teaching language and literature of the Far East at the University of Rome from 1911 until 1921, transferred to Florence in succession of Carlo Puini as ordinario for History and Geography of East Asia, and finally at the University of Rome, where he held the chair for History and Geography of East Asia from 1923 to his retirement in 1948 when he was in his 76th year. Besides actively contributing to the field of East Asian Studies, Vacca continued in parallel to work on mathematical questions inspired by his intellectual proximity to Peano’s School and his extensive readings of historical sources by Euclid, Archimedes, Euler, Fermat, Napier, and others.
The interests of Vacca were almost equally split between mathematics, Sinology and history of science, with a corresponding number of papers being 38, 47 and 45. In 1910, Vacca developed a complex number iteration for pi:
The calculation efficiency of these formulas is significantly worse than of the modern Borwein's algorithm – they converge by only about half a decimal point with each iteration.
Vacca published his two major contributions to mathematics in 1910 and 1926, on series expansion (later named Vacca series) of the Euler constant. They are, respectively
Vacca noted in 1910 that:
There is some hope that this series can be of some use in the proof of the irrationality of , a very difficult problem, proposed, but not resolved, in the Correspondence, recently published, between Hermite und Stieltjes.
References
1872 births
1953 deaths
19th-century Italian mathematicians
20th-century Italian mathematicians
Italian historians of mathematics
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https://en.wikipedia.org/wiki/Mario%20Pieri
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Mario Pieri (22 June 1860 – 1 March 1913) was an Italian mathematician who is known for his work on foundations of geometry.
Biography
Pieri was born in Lucca, Italy, the son of Pellegrino Pieri and Ermina Luporini. Pellegrino was a lawyer. Pieri began his higher education at University of Bologna where he drew the attention of Salvatore Pincherle. Obtaining a scholarship, Pieri transferred to Scuola Normale Superiore in Pisa. There he took his degree in 1884 and worked first at a technical secondary school in Pisa.
When an opportunity arose at the military academy in Turin to teach projective geometry, Pieri moved there and, by 1888, he was also an assistant instructor in the same subject at the University of Turin. By 1891, he had become libero docente at the university, teaching elective courses. Pieri continued to teach in Turin until 1900 when, through competition, he was awarded the position of extraordinary professor at University of Catania on the island of Sicily.
Von Staudt's Geometrie der Lage (1847) was a much admired text on projective geometry. In 1889 Pieri translated it as Geometria di Posizione, a publication that included a study of the life and work of von Staudt written by Corrado Segre, the initiator of the project.
Pieri also came under the influence of Giuseppe Peano at Turin. He contributed to the Formulario mathematico, and Peano placed nine of Pieri's papers for publication with the Academy of Sciences of Turin between 1895 and 1912. They shared a passion for reducing geometric ideas to their logical form and expressing these ideas symbolically.
In 1898 Pieri wrote I principii della geometria di posizione composti in un sistema logico-deduttivo. It progressively introduced independent axioms:
based on nineteen sequentially independent axioms – each independent of the preceding ones – which are introduced one by one as they are needed in the development, thus allowing the reader to determine on which axioms a given theorem depends.
Pieri was invited to address the International Congress of Philosophy in 1900 in Paris. Since this was also the year he moved from Turin to Sicily, he declined to attend but sent a paper "Sur la Géométrie envisagée comme un système purement logique", which was delivered by Louis Couturat. The ideas were also advanced by Alessandro Padoa at both that congress and the International Congress of Mathematicians also held in Paris that year.
In 1900 Pieri wrote Monographia del punto e del moto, which Smith calls the Point and Motion memoire. It is noteworthy as using only two primitive notions, point and motion to develop axioms for geometry. Alessandro Padoa shared in this expression of Peano's logico-geometrical program that reduced the number of primitive notions from the four used by Moritz Pasch.
The research into the foundations of geometry led to another formulation in 1908 in a Point and Sphere memoire. Smith (2010) describes it as
a full axiomatization of Euclidean geometry based sol
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https://en.wikipedia.org/wiki/Gino%20Fano
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Gino Fano (5 January 18718 November 1952) was an Italian mathematician, best known as the founder of finite geometry. He was born to a wealthy Jewish family in Mantua, in Italy and died in Verona, also in Italy.
Fano made various contributions on projective and algebraic geometry. His work in the foundations of geometry predates the similar, but more popular, work of David Hilbert by about a decade.
He was the father of physicist Ugo Fano and electrical engineer Robert Fano and uncle to physicist and mathematician Giulio Racah.
Mathematical work
Fano was an early writer in the area of finite projective spaces. In his article on proving the independence of his set of axioms for projective n-space, among other things, he considered the consequences of having a fourth harmonic point be equal to its conjugate. This leads to a configuration of seven points and seven lines contained in a finite three-dimensional space with 15 points, 35 lines and 15 planes, in which each line contained only three points.
All the planes in this space consist of seven points and seven lines and are now known as Fano planes:
Fano went on to describe finite projective spaces of arbitrary dimension and prime orders.
In 1907 Gino Fano contributed two articles to Part III of Klein's encyclopedia. The first (SS. 221–88) was a comparison of analytic geometry and synthetic geometry through their historic development in the 19th century. The second (SS. 282–388) was on continuous groups in geometry and group theory as a unifying principle in geometry.
Notes
References
External links
1871 births
1952 deaths
Scientists from Mantua
19th-century Italian mathematicians
20th-century Italian mathematicians
Algebraic geometers
Italian algebraic geometers
20th-century Italian Jews
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https://en.wikipedia.org/wiki/Conserved%20quantity
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A conserved quantity is a property or value that remains constant over time in a system even when changes occur in the system. In mathematics, a conserved quantity of a dynamical system is formally defined as a function of the dependent variables, the value of which remains constant along each trajectory of the system.
Not all systems have conserved quantities, and conserved quantities are not unique, since one can always produce another such quantity by applying a suitable function, such as adding a constant, to a conserved quantity.
Since many laws of physics express some kind of conservation, conserved quantities commonly exist in mathematical models of physical systems. For example, any classical mechanics model will have mechanical energy as a conserved quantity as long as the forces involved are conservative.
Differential equations
For a first order system of differential equations
where bold indicates vector quantities, a scalar-valued function H(r) is a conserved quantity of the system if, for all time and initial conditions in some specific domain,
Note that by using the multivariate chain rule,
so that the definition may be written as
which contains information specific to the system and can be helpful in finding conserved quantities, or establishing whether or not a conserved quantity exists.
Hamiltonian mechanics
For a system defined by the Hamiltonian , a function f of the generalized coordinates q and generalized momenta p has time evolution
and hence is conserved if and only if . Here denotes the Poisson bracket.
Lagrangian mechanics
Suppose a system is defined by the Lagrangian L with generalized coordinates q. If L has no explicit time dependence (so ), then the energy E defined by
is conserved.
Furthermore, if , then q is said to be a cyclic coordinate and the generalized momentum p defined by
is conserved. This may be derived by using the Euler–Lagrange equations.
See also
Conservative system
Lyapunov function
Hamiltonian system
Conservation law
Noether's theorem
Charge (physics)
Invariant (physics)
References
Differential equations
Dynamical systems
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https://en.wikipedia.org/wiki/Pneumonia%20severity%20index
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The pneumonia severity index (PSI) or PORT Score is a clinical prediction rule that medical practitioners can use to calculate the probability of morbidity and mortality among patients with community acquired pneumonia.
The PSI/PORT score is often used to predict the need for hospitalization in people with pneumonia. This is consistent with the conclusions stated in the original report that published the PSI/PORT score: "The prediction rule we describe accurately identifies the patients with community-acquired pneumonia who are at low risk for death and other adverse outcomes. This prediction rule may help physicians make more rational decisions about hospitalization for patients with pneumonia."
Mortality prediction is similar to that when using CURB-65.
Development
The rule uses demographics (whether someone is older, and is male or female), the coexistence of co-morbid illnesses, findings on physical examination and vital signs, and essential laboratory findings. This study demonstrated that patients could be stratified into five risk categories, Risk Classes I-V, and that these classes could be used to predict 30-day survival.
Usage
The purpose of the PSI is to classify the severity of a patient's pneumonia to determine the amount of resources to be allocated for care. Most commonly, the PSI scoring system has been used to decide whether patients with pneumonia can be treated as outpatients or as (hospitalized) inpatients.
A Risk Class I or Risk Class II pneumonia patient can be sent home on oral antibiotics.
A Risk Class III patient, after evaluation of other factors including home environment and follow-up, may either:
be sent home with oral antibiotics
be admitted for a short hospital stay with antibiotics and monitoring.
Patients with Risk Class IV-V pneumonia patient should be hospitalized for treatment.
Algorithm
The PSI Algorithm is detailed below. An online, automated PSI calculator was once available on the US AHRQ website for Personal Digital Assistants that are no longer sold. In 2018 AHRQ presented a new toolkit on the basis of CURB-65, an older counterpart to the PSI. In the 2019 ATS/IDSA Guidelines for the Diagnosis and Treatment of Adults with Community-acquired
Pneumonia, PSI was recommended over CURB-65 because of lack of evidence supporting the safety and effectiveness of the latter.
Data source for derivation and validation
The rule was derived then validated with data from 38,000 patients from the MedisGroup Cohort Study for 1989, comprising 1 year of data from 257 hospitals across the US who used the MedisGroup patient outcome tracking software built and serviced by Mediqual Systems (Cardinal Health). One significant caveat to the data source was that patients who were discharged home or transferred from the MedisGroup hospitals could not be followed at the 30-day mark, and were therefore assumed to be "alive" at that time. Further validation was performed with the Pneumonia Patient Outcomes Research Team [PO
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