source
stringlengths 31
168
| text
stringlengths 51
3k
|
|---|---|
https://en.wikipedia.org/wiki/Zero-product%20property
|
In algebra, the zero-product property states that the product of two nonzero elements is nonzero. In other words,
This property is also known as the rule of zero product, the null factor law, the multiplication property of zero, the nonexistence of nontrivial zero divisors, or one of the two zero-factor properties. All of the number systems studied in elementary mathematics — the integers , the rational numbers , the real numbers , and the complex numbers — satisfy the zero-product property. In general, a ring which satisfies the zero-product property is called a domain.
Algebraic context
Suppose is an algebraic structure. We might ask, does have the zero-product property? In order for this question to have meaning, must have both additive structure and multiplicative structure. Usually one assumes that is a ring, though it could be something else, e.g. the set of nonnegative integers with ordinary addition and multiplication, which is only a (commutative) semiring.
Note that if satisfies the zero-product property, and if is a subset of , then also satisfies the zero product property: if and are elements of such that , then either or because and can also be considered as elements of .
Examples
A ring in which the zero-product property holds is called a domain. A commutative domain with a multiplicative identity element is called an integral domain. Any field is an integral domain; in fact, any subring of a field is an integral domain (as long as it contains 1). Similarly, any subring of a skew field is a domain. Thus, the zero-product property holds for any subring of a skew field.
If is a prime number, then the ring of integers modulo has the zero-product property (in fact, it is a field).
The Gaussian integers are an integral domain because they are a subring of the complex numbers.
In the strictly skew field of quaternions, the zero-product property holds. This ring is not an integral domain, because the multiplication is not commutative.
The set of nonnegative integers is not a ring (being instead a semiring), but it does satisfy the zero-product property.
Non-examples
Let denote the ring of integers modulo . Then does not satisfy the zero product property: 2 and 3 are nonzero elements, yet .
In general, if is a composite number, then does not satisfy the zero-product property. Namely, if where , then and are nonzero modulo , yet .
The ring of 2×2 matrices with integer entries does not satisfy the zero-product property: if and then yet neither nor is zero.
The ring of all functions , from the unit interval to the real numbers, has nontrivial zero divisors: there are pairs of functions which are not identically equal to zero yet whose product is the zero function. In fact, it is not hard to construct, for any n ≥ 2, functions , none of which is identically zero, such that is identically zero whenever .
The same is true even if we consider only continuous functions, or only even infi
|
https://en.wikipedia.org/wiki/CountrySTAT
|
CountrySTAT is a Web-based information technology system for food and agriculture statistics at the national and subnational levels. It provides decision-makers access to statistics across thematic areas such as production, prices, trade and consumption. This supports analysis, informed policy-making and monitoring with the goal of eradicating extreme poverty and hunger.
Since 2005, the Statistics Division of the United Nations Food and Agriculture Organization (FAO) has introduced CountrySTAT in over 20 countries in Latin America, sub-Saharan Africa and Asia.
Overview
The CountrySTAT web system is a browser oriented statistical framework to organise, harmonise and synchronise data collections.
CountrySTAT aims are to facilitate data use by policy makers and researchers. It provides statistical standards, data exchange tools and related methods without using external data sources such as databases. The data source is a text file in a specific format, called px-file. The application supports many languages. The layout can be easily changed to match the needs of users.
Features
The CountrySTAT web system is easy to install and to operate on a standard Windows XP professional machine. It is programmed in ASP with visual basic using internet information service and suitable windows software for graphical and statistical output for the intranet and internet environment.
Criticisms
The programming with VB scripts, customised DLLs and additional windows software (PC-Axis family) makes it to a platform dependently software only run with the internet information server on a Windows server machine. To use it with the internet requires an own dedicated windows server.
See also
FAO
CountrySTAT technical documentation
External links
FAO Programme Committee (87th Session): Modernization of FAOSTAT – An update. Rome, 6-10 May 2002.
Website of FAO
CountrySTAT Web site
FAOSTAT Web site
FAO Statistics Division Web site
National CountrySTAT Web sites
CountrySTAT Philippines
CountrySTAT Bhutan
CountrySTAT Mali
CountrySTAT Niger
CountrySTAT Togo
RegionSTAT UEMOA
CountrySTAT Angola
CountrySTAT Benin
CountrySTAT Burkina Faso
CountrySTAT Ivory Coast
CountrySTAT Cameroon
CountrySTAT Ghana
CountrySTAT Kenya
CountrySTAT Senegal
CountrySTAT Uganda
CountrySTAT United Republic of Tanzania
Agricultural databases
Organizations established in 1945
Food and Agriculture Organization
Statistical data sets
cs:Organizace pro výživu a zemědělství
da:FAO
de:Food and Agriculture Organization
es:Organización para la Alimentación y la Agricultura
eo:Organizaĵo pri Nutrado kaj Agrikulturo
fr:Organisation des Nations unies pour l'alimentation et l'agriculture
id:Organisasi Pangan dan Pertanian
it:FAO
nl:Voedsel- en Landbouworganisatie
ja:国際連合食糧農業機関
nn:FAO
pt:Organização das Nações Unidas para a Agricultura e a Alimentação
ru:Продовольственная и сельскохозяйственная организация ООН
tr:Gıda ve Tarım Teşkilatı
zh:联合国粮食及农业组织
|
https://en.wikipedia.org/wiki/Hong%20Kong%20Association%20of%20Science%20and%20Mathematics%20Education
|
Hong Kong Association of Science and Mathematics Education is a society to promote and improve the teaching methodology of the science and mathematics in Hong Kong. Founded in 1964, current members are secondary school teachers, professors and lecturers in the universities and government officials in education.
External links
Official website
Education in Hong Kong
|
https://en.wikipedia.org/wiki/David%20R.%20Cheriton%20School%20of%20Computer%20Science
|
The David R. Cheriton School of Computer Science is a professional school within the Faculty of Mathematics at the University of Waterloo. QS World University Rankings ranked the David R. Cheriton School of Computer Science 24th in the world, 10th in North America and 2nd in Canada in Computer Science in 2014. U.S. News & World Report ranked the David R. Cheriton School of Computer Science 42nd in world and second in Canada.
History
In 1965, when Mathematics was still a department within the Faculty of Arts, four third-year mathematics students (Richard Shirley, Angus German, James G. Mitchell, and Bob Zarnke) wrote the WATFOR compiler for the FORTRAN programming language, under the direction of lecturer Peter Shantz. "Within a year it would be adopted by computing centres in over eight countries, and the number of student users at UW increased to over 2500." Later on in 1966, two mathematics lecturers (Paul Dirksen and Paul H. Cress) led a team that developed WATFOR 360, for which they received the 1972 Grace Murray Hopper Award from the Association for Computing Machinery.
UW's Faculty of Mathematics was later established in 1967. As a result, the Department of Applied Analysis and Computer Science (AA&CS) was created. By 1969, AA&CS had become the largest department in the faculty. At that point, the first two PhD degrees in computer science were awarded, to Byron L. Ehle, for a thesis on numerical analysis, and to Hugh Williams, for a thesis on computational number theory. In 1975 the department dropped the words "Applied Analysis" and became simply the Department of Computer Science.
In 1982, the Institute for Computer Research (ICR) was established. Its goals were "to foster computer research..., facilitate interaction with industry, and encourage advanced education in computer science and engineering." Also that year, the Ontario government announced plans to build the Davis Centre, current home of the School of Computer Science. The groundbreaking was in April 1985 and the Davis Centre was formally dedicated on November 10, 1988.
On May 1, 2002, the department officially became the School of Computer Science. On November 18, 2005, it was renamed again to the David R. Cheriton School of Computer Science, in recognition of the establishment of the David R. Cheriton Endowment for Excellence in Computer Science. Cheriton had recently donated $25 million to the university.
Support for computing within the School of Computer Science had been historically provided by the Computer Science Computing Facility (CSCF) and ICR.
Programs
The David R. Cheriton School of Computer Science offers several diverse undergraduate programs including:
Bachelor of Computer Science
Honours Computer Science
Business Option
Bioinformatics Option
Digital Hardware Option
Software Engineering Option
Honours Data Science
Bachelor of Mathematics
Honours Computer Science
Business Option
Digital Hardware Option
Bachelor of Software Engineering
Bache
|
https://en.wikipedia.org/wiki/Carlson%27s%20theorem
|
In mathematics, in the area of complex analysis, Carlson's theorem is a uniqueness theorem which was discovered by Fritz David Carlson. Informally, it states that two different analytic functions which do not grow very fast at infinity can not coincide at the integers. The theorem may be obtained from the Phragmén–Lindelöf theorem, which is itself an extension of the maximum-modulus theorem.
Carlson's theorem is typically invoked to defend the uniqueness of a Newton series expansion. Carlson's theorem has generalized analogues for other expansions.
Statement
Assume that satisfies the following three conditions. The first two conditions bound the growth of at infinity, whereas the third one states that vanishes on the non-negative integers.
is an entire function of exponential type, meaning that for some real values , .
There exists such that
for every non-negative integer .
Then is identically zero.
Sharpness
First condition
The first condition may be relaxed: it is enough to assume that is analytic in , continuous in , and satisfies
for some real values , .
Second condition
To see that the second condition is sharp, consider the function . It vanishes on the integers; however, it grows exponentially on the imaginary axis with a growth rate of , and indeed it is not identically zero.
Third condition
A result, due to , relaxes the condition that vanish on the integers. Namely, Rubel showed that the conclusion of the theorem remains valid if vanishes on a subset of upper density 1, meaning that
This condition is sharp, meaning that the theorem fails for sets of upper density smaller than 1.
Applications
Suppose is a function that possesses all finite forward differences . Consider then the Newton series
with is the binomial coefficient and is the -th forward difference. By construction, one then has that for all non-negative integers , so that the difference . This is one of the conditions of Carlson's theorem; if obeys the others, then is identically zero, and the finite differences for uniquely determine its Newton series. That is, if a Newton series for exists, and the difference satisfies the Carlson conditions, then is unique.
See also
Newton series
Mahler's theorem
Table of Newtonian series
References
F. Carlson, Sur une classe de séries de Taylor, (1914) Dissertation, Uppsala, Sweden, 1914.
, cor 21(1921) p. 6.
E.C. Titchmarsh, The Theory of Functions (2nd Ed) (1939) Oxford University Press (See section 5.81)
R. P. Boas, Jr., Entire functions, (1954) Academic Press, New York.
Factorial and binomial topics
Finite differences
Theorems in complex analysis
|
https://en.wikipedia.org/wiki/Vertical%20and%20horizontal%20bundles
|
In mathematics, the vertical bundle and the horizontal bundle are vector bundles associated to a smooth fiber bundle. More precisely, given a smooth fiber bundle , the vertical bundle and horizontal bundle are subbundles of the tangent bundle of whose Whitney sum satisfies . This means that, over each point , the fibers and form complementary subspaces of the tangent space . The vertical bundle consists of all vectors that are tangent to the fibers, while the horizontal bundle requires some choice of complementary subbundle.
To make this precise, define the vertical space at to be . That is, the differential (where ) is a linear surjection whose kernel has the same dimension as the fibers of . If we write , then consists of exactly the vectors in which are also tangent to . The name is motivated by low-dimensional examples like the trivial line bundle over a circle, which is sometimes depicted as a vertical cylinder projecting to a horizontal circle. A subspace of is called a horizontal space if is the direct sum of and .
The disjoint union of the vertical spaces VeE for each e in E is the subbundle VE of TE; this is the vertical bundle of E. Likewise, provided the horizontal spaces vary smoothly with e, their disjoint union is a horizontal bundle. The use of the words "the" and "a" here is intentional: each vertical subspace is unique, defined explicitly by . Excluding trivial cases, there are an infinite number of horizontal subspaces at each point. Also note that arbitrary choices of horizontal space at each point will not, in general, form a smooth vector bundle; they must also vary in an appropriately smooth way.
The horizontal bundle is one way to formulate the notion of an Ehresmann connection on a fiber bundle. Thus, for example, if E is a principal G-bundle, then the horizontal bundle is usually required to be G-invariant: such a choice is equivalent to a connection on the principal bundle. This notably occurs when E is the frame bundle associated to some vector bundle, which is a principal bundle.
Formal definition
Let π:E→B be a smooth fiber bundle over a smooth manifold B. The vertical bundle is the kernel VE := ker(dπ) of the tangent map dπ : TE → TB.
Since dπe is surjective at each point e, it yields a regular subbundle of TE. Furthermore, the vertical bundle VE is also integrable.
An Ehresmann connection on E is a choice of a complementary subbundle HE to VE in TE, called the horizontal bundle of the connection. At each point e in E, the two subspaces form a direct sum, such that
TeE = VeE ⊕ HeE.
Example
A simple example of a smooth fiber bundle is a Cartesian product of two manifolds. Consider the bundle B1 := (M × N, pr1) with bundle projection pr1 : M × N → M : (x, y) → x. Applying the definition in the paragraph above to find the vertical bundle, we consider first a point (m,n) in M × N. Then the image of this point under pr1 is m. The preimage of m under this same pr1 is {m} × N, so that T(m,n
|
https://en.wikipedia.org/wiki/Tridiagonal%20matrix%20algorithm
|
In numerical linear algebra, the tridiagonal matrix algorithm, also known as the Thomas algorithm (named after Llewellyn Thomas), is a simplified form of Gaussian elimination that can be used to solve tridiagonal systems of equations. A tridiagonal system for n unknowns may be written as
where and .
For such systems, the solution can be obtained in operations instead of required by Gaussian elimination. A first sweep eliminates the 's, and then an (abbreviated) backward substitution produces the solution. Examples of such matrices commonly arise from the discretization of 1D Poisson equation and natural cubic spline interpolation.
Thomas' algorithm is not stable in general, but is so in several special cases, such as when the matrix is diagonally dominant (either by rows or columns) or symmetric positive definite; for a more precise characterization of stability of Thomas' algorithm, see Higham Theorem 9.12. If stability is required in the general case, Gaussian elimination with partial pivoting (GEPP) is recommended instead.
Method
The forward sweep consists of the computation of new coefficients as follows, denoting the new coefficients with primes:
and
The solution is then obtained by back substitution:
The method above does not modify the original coefficient vectors, but must also keep track of the new coefficients. If the coefficient vectors may be modified, then an algorithm with less bookkeeping is:
For do
followed by the back substitution
The implementation in a VBA subroutine without preserving the coefficient vectors:
Sub TriDiagonal_Matrix_Algorithm(N%, A#(), B#(), C#(), D#(), X#())
Dim i%, W#
For i = 2 To N
W = A(i) / B(i - 1)
B(i) = B(i) - W * C(i - 1)
D(i) = D(i) - W * D(i - 1)
Next i
X(N) = D(N) / B(N)
For i = N - 1 To 1 Step -1
X(i) = (D(i) - C(i) * X(i + 1)) / B(i)
Next i
End Sub
Derivation
The derivation of the tridiagonal matrix algorithm is a special case of Gaussian elimination.
Suppose that the unknowns are , and that the equations to be solved are:
Consider modifying the second () equation with the first equation as follows:
which would give:
Note that has been eliminated from the second equation. Using a similar tactic with the modified second equation on the third equation yields:
This time was eliminated. If this procedure is repeated until the row; the (modified) equation will involve only one unknown, . This may be solved for and then used to solve the equation, and so on until all of the unknowns are solved for.
Clearly, the coefficients on the modified equations get more and more complicated if stated explicitly. By examining the procedure, the modified coefficients (notated with tildes) may instead be defined recursively:
To further hasten the solution process, may be divided out (if there's no division by zero risk), the newer modified coefficients, each notated with a prime, will be:
This gives the following
|
https://en.wikipedia.org/wiki/Successive%20over-relaxation
|
In numerical linear algebra, the method of successive over-relaxation (SOR) is a variant of the Gauss–Seidel method for solving a linear system of equations, resulting in faster convergence. A similar method can be used for any slowly converging iterative process.
It was devised simultaneously by David M. Young Jr. and by Stanley P. Frankel in 1950 for the purpose of automatically solving linear systems on digital computers. Over-relaxation methods had been used before the work of Young and Frankel. An example is the method of Lewis Fry Richardson, and the methods developed by R. V. Southwell. However, these methods were designed for computation by human calculators, requiring some expertise to ensure convergence to the solution which made them inapplicable for programming on digital computers. These aspects are discussed in the thesis of David M. Young Jr.
Formulation
Given a square system of n linear equations with unknown x:
where:
Then A can be decomposed into a diagonal component D, and strictly lower and upper triangular components L and U:
where
The system of linear equations may be rewritten as:
for a constant ω > 1, called the relaxation factor.
The method of successive over-relaxation is an iterative technique that solves the left hand side of this expression for x, using the previous value for x on the right hand side. Analytically, this may be written as:
where is the kth approximation or iteration of and is the next or k + 1 iteration of .
However, by taking advantage of the triangular form of (D+ωL), the elements of x(k+1) can be computed sequentially using forward substitution:
Convergence
The choice of relaxation factor ω is not necessarily easy, and depends upon the properties of the coefficient matrix.
In 1947, Ostrowski proved that if is symmetric and positive-definite then for .
Thus, convergence of the iteration process follows, but we are generally interested in faster convergence rather than just convergence.
Convergence Rate
The convergence rate for the SOR method can be analytically derived.
One needs to assume the following
the relaxation parameter is appropriate:
Jacobi's iteration matrix has only real eigenvalues
Jacobi's method is convergent:
the matrix decomposition satisfies the property that for any and .
Then the convergence rate can be expressed as
where the optimal relaxation parameter is given by
In particular, for (Gauss-Seidel) it holds that .
For the optimal we get , which shows SOR is roughly four times more efficient than Gauss–Seidel.
The last assumption is satisfied for tridiagonal matrices since for diagonal with entries and .
Algorithm
Since elements can be overwritten as they are computed in this algorithm, only one storage vector is needed, and vector indexing is omitted. The algorithm goes as follows:
Inputs: , ,
Output:
Choose an initial guess to the solution
repeat until convergence
for from 1 until do
set to 0
for
|
https://en.wikipedia.org/wiki/Projective%20cone
|
A projective cone (or just cone) in projective geometry is the union of all lines that intersect a projective subspace R (the apex of the cone) and an arbitrary subset A (the basis) of some other subspace S, disjoint from R.
In the special case that R is a single point, S is a plane, and A is a conic section on S, the projective cone is a conical surface; hence the name.
Definition
Let X be a projective space over some field K, and R, S be disjoint subspaces of X. Let A be an arbitrary subset of S. Then we define RA, the cone with top R and basis A, as follows :
When A is empty, RA = A.
When A is not empty, RA consists of all those points on a line connecting a point on R and a point on A.
Properties
As R and S are disjoint, one may deduce from linear algebra and the definition of a projective space that every point on RA not in R or A is on exactly one line connecting a point in R and a point in A.
(RA) S = A
When K is the finite field of order q, then = + , where r = dim(R).
See also
Cone (geometry)
Cone (algebraic geometry)
Cone (topology)
Cone (linear algebra)
Conic section
Ruled surface
Hyperboloid
Projective geometry
|
https://en.wikipedia.org/wiki/Difference%20polynomials
|
In mathematics, in the area of complex analysis, the general difference polynomials are a polynomial sequence, a certain subclass of the Sheffer polynomials, which include the Newton polynomials, Selberg's polynomials, and the Stirling interpolation polynomials as special cases.
Definition
The general difference polynomial sequence is given by
where is the binomial coefficient. For , the generated polynomials are the Newton polynomials
The case of generates Selberg's polynomials, and the case of generates Stirling's interpolation polynomials.
Moving differences
Given an analytic function , define the moving difference of f as
where is the forward difference operator. Then, provided that f obeys certain summability conditions, then it may be represented in terms of these polynomials as
The conditions for summability (that is, convergence) for this sequence is a fairly complex topic; in general, one may say that a necessary condition is that the analytic function be of less than exponential type. Summability conditions are discussed in detail in Boas & Buck.
Generating function
The generating function for the general difference polynomials is given by
This generating function can be brought into the form of the generalized Appell representation
by setting , , and .
See also
Carlson's theorem
Bernoulli polynomials of the second kind
References
Ralph P. Boas, Jr. and R. Creighton Buck, Polynomial Expansions of Analytic Functions (Second Printing Corrected), (1964) Academic Press Inc., Publishers New York, Springer-Verlag, Berlin. Library of Congress Card Number 63-23263.
Polynomials
Finite differences
Factorial and binomial topics
|
https://en.wikipedia.org/wiki/Stirling%20polynomials
|
In mathematics, the Stirling polynomials are a family of polynomials that generalize important sequences of numbers appearing in combinatorics and analysis, which are closely related to the Stirling numbers, the Bernoulli numbers, and the generalized Bernoulli polynomials. There are multiple variants of the Stirling polynomial sequence considered below most notably including the Sheffer sequence form of the sequence, , defined characteristically through the special form of its exponential generating function, and the Stirling (convolution) polynomials, , which also satisfy a characteristic ordinary generating function and that are of use in generalizing the Stirling numbers (of both kinds) to arbitrary complex-valued inputs. We consider the "convolution polynomial" variant of this sequence and its properties second in the last subsection of the article. Still other variants of the Stirling polynomials are studied in the supplementary links to the articles given in the references.
Definition and examples
For nonnegative integers k, the Stirling polynomials, Sk(x), are a Sheffer sequence for defined by the exponential generating function
The Stirling polynomials are a special case of the Nørlund polynomials (or generalized Bernoulli polynomials) each with exponential generating function
given by the relation .
The first 10 Stirling polynomials are given in the following table:
{| class="wikitable"
!k !! Sk(x)
|-
| 0 ||
|-
| 1 ||
|-
| 2 ||
|-
| 3 ||
|-
| 4 ||
|-
| 5 ||
|-
| 6 ||
|-
| 7 ||
|-
| 8 ||
|-
| 9 ||
|}
Yet another variant of the Stirling polynomials is considered in (see also the subsection on Stirling convolution polynomials below). In particular, the article by I. Gessel and R. P. Stanley defines the modified Stirling polynomial sequences, and where are the unsigned Stirling numbers of the first kind, in terms of the two Stirling number triangles for non-negative integers . For fixed , both and are polynomials of the input each of degree and with leading coefficient given by the double factorial term .
Properties
Below denote the Bernoulli polynomials and the Bernoulli numbers under the convention denotes a Stirling number of the first kind; and denotes Stirling numbers of the second kind.
Special values:
If and then:
If and then: and:
The sequence is of binomial type, since Moreover, this basic recursion holds:
Explicit representations involving Stirling numbers can be deduced with Lagrange's interpolation formula: Here, are Laguerre polynomials.
The following relations hold as well:
By differentiating the generating function it readily follows that
Stirling convolution polynomials
Definition and examples
Another variant of the Stirling polynomial sequence corresponds to a special case of the convolution polynomials studied by Knuth's article
and in the Concrete Mathematics reference. We first define these polynomials through the Stirling numbers of the first kind as
It follows th
|
https://en.wikipedia.org/wiki/National%20Institute%20of%20Statistics%20and%20Geography
|
The National Institute of Statistics and Geography (INEGI from its former name in ) is an autonomous agency of the Mexican Government dedicated to coordinate the National System of Statistical and Geographical Information of the country. It was created on January 25, 1983, by presidential decree of Miguel de la Madrid.
It is the institution responsible for conducting the Censo General de Población y Vivienda every ten years; as well as the economic census every five years and the agricultural, livestock and forestry census of the country. The job of gathering statistical information of the Institute includes the monthly gross domestic product, consumer trust surveys and proportion of commercial samples; employment and occupation statistics, domestic and couple violence; as well as many other jobs that are the basis of studies and projections to other governmental institutions.
The Institute headquarters are in the city of Aguascalientes in central Mexico.
Functions
With the enactment of the National System of Statistical and Geographical Information Law, (LSNIEG by its name in Spanish, Ley del Sistema Nacional de Información Estadística y Geográfica) on April 16, 2008, INEGI changed its legal personality, acquiring technical and management autonomy. Its new denomination is National Institute of Statistic and Geography (INEGI by its name in Spanish, Instituto Nacional de Estadística y Geografía), but it preserves the acronym of its former name (INEGI).
INEGI's main objective is to achieve that the National System of Statistical and Geographical Information (SNIEG, by its name in Spanish), bring to the society and to the government, quality information, pertinent, truthful and relevant, to contribute to the national development, under accessibility, transparency, objectivity and independence principles.
To this goal, its attributions are:
Regulate and coordinate the SNIEG's development.
Regulate the statistical and geographical activities.
Produce statistical and geographical information.
Provide the Public Service of Information.
Promote the knowledge and use of the information.
Maintain information up to date.
The INEGI is governed by a government board, which oversees its operations. It is integrated by the institute president, and four vice presidents, who are designated by the president of Mexico with Chamber of Senators approval.
INEGI collaborates with American and Canadian government scientists, along with the Commission for Environmental Cooperation, to produce the North American Environmental Atlas, which is used to depict and track environmental issues for a continental perspective.
See also
Sociedad Mexicana de Geografía y Estadística
Survey of Occupation and Employment
References
External links
National Institute of Statistic and Geography official website (INEGI)
Cuéntame – INEGI's educational section
National System of Statistical and Geographical Information (SNIEG)
Digital Map of Mexico
Interactive Nat
|
https://en.wikipedia.org/wiki/Tom%20Tango
|
Tom Tango and "TangoTiger" are aliases used online by a baseball sabermetrics and ice hockey statistics analyst. He runs the Tango on Baseball sabermetrics website and is also a contributor to ESPN's baseball blog TMI (The Max Info). Tango is currently the Senior Database Architect of Stats for MLB Advanced Media.
Born in Canada in 1968, he resides in New Jersey with his family and has insisted on keeping his true name secret.
In 2006, Tango's book The Book: Playing the Percentages in Baseball, which was co-written with Mitchel Lichtman and Andrew Dolphin, was published featuring a foreword by Pete Palmer. In The Book he and his coauthors analyzed many advanced baseball questions, such as how to optimize a lineup or when to issue an intentional base on balls. They also introduced the wOBA metric to measure overall offensive contributions.
Tango maintains the "Marcel the Monkey Forecasting System," a player projection system which uses three years of weighted player statistics with statistical regression and player age adjustment.
He is best known for developing the FIP (Fielding Independent Pitching) statistic, which attempts to more accurately assess the quality of a pitcher's performance than other statistics, such as ERA. 2009 American League Cy Young Award winner Zack Greinke specifically mentioned FIP as his favorite statistic. "That's pretty much how I pitch, to try to keep my FIP as low as possible".
Tango works as a consultant for several National Hockey League teams, and has worked for Major League Baseball. Tango has worked for the Seattle Mariners and Toronto Blue Jays as a statistical analysis consultant. He worked exclusively for the Chicago Cubs in a similar role.
In 2020, he was awarded by the Society for American Baseball Research the Henry Chadwick Award. The award is given "to honor those researchers, historians, analysts, and statisticians whose work has most contributed to our understanding of the game and its history."
Books
Tom Tango, Mitchel Lichtman, and Andrew Dolphin. The Book: Playing the Percentages in Baseball. Washington, D.C.: Potomac Books, 2007. .
References
Citations
Justin Sopp, "Q&A with SaberWizard Tom Tango", Beyond the Box Score, 24 August 2011
External links
Tangotiger (official)
The Book—Playing The Percentages In Baseball (official)
Interview at Baseball Digest Daily - Part I
Interview at Baseball Digest Daily - Part II
Living people
Canadian statisticians
Baseball statisticians
Baseball people from Quebec
People from Montreal
Anglophone Quebec people
1968 births
Unidentified people
|
https://en.wikipedia.org/wiki/Collineation
|
In projective geometry, a collineation is a one-to-one and onto map (a bijection) from one projective space to another, or from a projective space to itself, such that the images of collinear points are themselves collinear. A collineation is thus an isomorphism between projective spaces, or an automorphism from a projective space to itself. Some authors restrict the definition of collineation to the case where it is an automorphism. The set of all collineations of a space to itself form a group, called the collineation group.
Definition
Simply, a collineation is a one-to-one map from one projective space to another, or from a projective space to itself, such that the images of collinear points are themselves collinear. One may formalize this using various ways of presenting a projective space. Also, the case of the projective line is special, and hence generally treated differently.
Linear algebra
For a projective space defined in terms of linear algebra (as the projectivization of a vector space), a collineation is a map between the projective spaces that is order-preserving with respect to inclusion of subspaces.
Formally, let V be a vector space over a field K and W a vector space over a field L. Consider the projective spaces PG(V) and PG(W), consisting of the vector lines of V and W.
Call D(V) and D(W) the set of subspaces of V and W respectively. A collineation from PG(V) to PG(W) is a map α : D(V) → D(W), such that:
α is a bijection.
A ⊆ B ⇔ α(A) ⊆ α(B) for all A, B in D(V).
Axiomatically
Given a projective space defined axiomatically in terms of an incidence structure (a set of points P, lines L, and an incidence relation I specifying which points lie on which lines, satisfying certain axioms), a collineation between projective spaces thus defined then being a bijective function f between the sets of points and a bijective function g between the set of lines, preserving the incidence relation.
Every projective space of dimension greater than or equal to three is isomorphic to the projectivization of a linear space over a division ring, so in these dimensions this definition is no more general than the linear-algebraic one above, but in dimension two there are other projective planes, namely the non-Desarguesian planes, and this definition allows one to define collineations in such projective planes.
For dimension one, the set of points lying on a single projective line defines a projective space, and the resulting notion of collineation is just any bijection of the set.
Collineations of the projective line
For a projective space of dimension one (a projective line; the projectivization of a vector space of dimension two), all points are collinear, so the collineation group is exactly the symmetric group of the points of the projective line. This is different from the behavior in higher dimensions, and thus one gives a more restrictive definition, specified so that the fundamental theorem of projective geometry holds.
In thi
|
https://en.wikipedia.org/wiki/Millwood%20Lake
|
Millwood Lake is a reservoir in southwestern Arkansas, United States. It is located from Ashdown and is formed from the damming of the point where Little River and Saline River meet.
Statistics
Lake statistics:
Drainage area above the dam:
Elevation above sea level of the top of flood control pool:
Elevation above sea level of the top of conservation pool:
Elevation above sea level of the top of inactive pool:
Surface area of lake at top of flood control pool:
Surface area of lake at top of conservation pool:
Shoreline length at top of conservation pool:
Dam statistics:
Length of dam:
Maximum height of dam above streambed:
Length of spillway:
Length of non-overflow section:
Spillway crest gates (13), size:
Outlet conduits (2), size:
Water supply pipe (1), diameter:
Overview
Lake Millwood is mainly recognized for its fishing and birding access. It is also known for housing the 1,380-pound alligator, which was caught in the lake in 2012. Its of submerged timber provide homes for the many varieties of fish in the lake, including the indigenous Millwood lunker largemouth bass. Other species of fauna around the lake include white-tailed deer, bobwhite quail, squirrel, dove, rabbit, raccoon, armadillo, opossum, fox, mink, American gator, and beaver. Boating is also popular on Millwood Lake, but only a small part of the whole surface area of the lake can be used as boating due to the submerged timber that takes up of the pond. Lake Millwood also has a diverse flora life, with many plants and trees such as gum, oak, birch, pine, juniper, flowering shrubs, and wildflowers.
History
The Millwood Lake project was authorized by the Flood Control Act of 1946, and modified by the Flood Control Act of 1958. The dam and lake were designed and built by the Tulsa District of the Army Corps of Engineers, which still maintains the lake's Beard's Bluff recreation center. The projects construction work began in 1961, and was finished for flood control operations in 1966 at a cost of $44,000,000. The lake and dam were dedicated on December 8, 1966. The lake is the key in the general flood reduction system for the Red River below Lake Texoma.
Water use
Benefits of the lake have been restoring wildlife, providing water to nearby areas, and preventing an estimate of $9,715,000 in flood damage. In Ashdown, Arkansas, the lake supplies Domtar's (formerly Georgia Pacific) Communications Paper Division with 50 million gallons of water each day for its operations. The lake also provides drinking water to the city of Texarkana, Arkansas, through a water treatment plant located at Ashdown.
See also
List of Arkansas dams and reservoirs
References
External links
Bathymetric Map, Area/Capacity Table, and Sediment Volume Estimate for Millwood Lake, Near Ashburn, Arkansas, 2013 United States Geological Survey
Fishing Millwood Lake Black Bass, Crappie and Catfish
Reservoirs in Arkansas
Protected areas of Little River County, Arkansas
Protected areas o
|
https://en.wikipedia.org/wiki/Applied%20Mathematics%20Panel
|
The Applied Mathematics Panel (AMP) was created at the end of 1942 as a division of the National Defense Research Committee (NDRC) within the Office of Scientific Research and Development (OSRD) in order to solve mathematical problems related to the military effort in World War II, particularly those of the other NDRC divisions.
The panel's headquarters were in Manhattan, and it was directed by Warren Weaver, formerly of NDRC Division 7, Fire Control. It contracted projects out to various research groups, notably at Princeton and Columbia Universities.
In addition to work immediately relevant to the war effort, mathematicians involved with the panel also pursued problems of interest to them without contracts from outside organizations. Most notably, Abraham Wald developed the statistical technique of sequential analysis while working for AMP.
AMP was formally disbanded in 1946.
References
MacLane, Saunders. "The Applied Mathematics Group at Columbia in World War II" in A Century of Mathematics in America, vol. 3 (ed. Peter Duren). Providence: American Mathematical Society, 1989.
Owens, Larry. "Mathematicians at War: Warren Weaver and the Applied Mathematics Panel, 1942–1945" in The History of Modern Mathematics, vol. 2 (eds. David E. Rowe and John McCleary). Boston: Academic Press, 1989.
Rees, Mina. "The Mathematical Sciences and World War II". The American Mathematical Monthly (1980), 87, 607–621.
Wallis, W. Allen. "The Statistical Research Group, 1942–1945". Journal of the American Statistical Association (1980), 75, 320–330.
Agencies of the United States government during World War II
Mathematics organizations
Government agencies established in 1942
1942 establishments in the United States
|
https://en.wikipedia.org/wiki/Nachbin%27s%20theorem
|
In mathematics, in the area of complex analysis, Nachbin's theorem (named after Leopoldo Nachbin) is commonly used to establish a bound on the growth rates for an analytic function. This article provides a brief review of growth rates, including the idea of a function of exponential type. Classification of growth rates based on type help provide a finer tool than big O or Landau notation, since a number of theorems about the analytic structure of the bounded function and its integral transforms can be stated. In particular, Nachbin's theorem may be used to give the domain of convergence of the generalized Borel transform, given below.
Exponential type
A function defined on the complex plane is said to be of exponential type if there exist constants and such that
in the limit of . Here, the complex variable was written as to emphasize that the limit must hold in all directions . Letting stand for the infimum of all such , one then says that the function is of exponential type .
For example, let . Then one says that is of exponential type , since is the smallest number that bounds the growth of along the imaginary axis. So, for this example, Carlson's theorem cannot apply, as it requires functions of exponential type less than .
Ψ type
Bounding may be defined for other functions besides the exponential function. In general, a function is a comparison function if it has a series
with for all , and
Comparison functions are necessarily entire, which follows from the ratio test. If is such a comparison function, one then says that is of -type if there exist constants and such that
as . If is the infimum of all such one says that is of -type .
Nachbin's theorem states that a function with the series
is of -type if and only if
Borel transform
Nachbin's theorem has immediate applications in Cauchy theorem-like situations, and for integral transforms. For example, the generalized Borel transform is given by
If is of -type , then the exterior of the domain of convergence of , and all of its singular points, are contained within the disk
Furthermore, one has
where the contour of integration γ encircles the disk . This generalizes the usual Borel transform for exponential type, where . The integral form for the generalized Borel transform follows as well. Let be a function whose first derivative is bounded on the interval , so that
where . Then the integral form of the generalized Borel transform is
The ordinary Borel transform is regained by setting . Note that the integral form of the Borel transform is just the Laplace transform.
Nachbin resummation
Nachbin resummation (generalized Borel transform) can be used to sum divergent series that escape to the usual Borel summation or even to solve (asymptotically) integral equations of the form:
where may or may not be of exponential growth and the kernel has a Mellin transform. The solution can be obtained as with and is the Mellin transform of . An example of t
|
https://en.wikipedia.org/wiki/Exponential%20type
|
In complex analysis, a branch of mathematics, a holomorphic function is said to be of exponential type C if its growth is bounded by the exponential function for some real-valued constant as . When a function is bounded in this way, it is then possible to express it as certain kinds of convergent summations over a series of other complex functions, as well as understanding when it is possible to apply techniques such as Borel summation, or, for example, to apply the Mellin transform, or to perform approximations using the Euler–Maclaurin formula. The general case is handled by Nachbin's theorem, which defines the analogous notion of -type for a general function as opposed to .
Basic idea
A function defined on the complex plane is said to be of exponential type if there exist real-valued constants and such that
in the limit of . Here, the complex variable was written as to emphasize that the limit must hold in all directions . Letting stand for the infimum of all such , one then says that the function is of exponential type .
For example, let . Then one says that is of exponential type , since is the smallest number that bounds the growth of along the imaginary axis. So, for this example, Carlson's theorem cannot apply, as it requires functions of exponential type less than . Similarly, the Euler–Maclaurin formula cannot be applied either, as it, too, expresses a theorem ultimately anchored in the theory of finite differences.
Formal definition
A holomorphic function is said to be of exponential type if for every there exists a real-valued constant such that
for where .
We say is of exponential type if is of exponential type for some . The number
is the exponential type of . The limit superior here means the limit of the supremum of the ratio outside a given radius as the radius goes to infinity. This is also the limit superior of the maximum of the ratio at a given radius as the radius goes to infinity. The limit superior may exist even if the maximum at radius does not have a limit as goes to infinity. For example, for the function
the value of
at is dominated by the term so we have the asymptotic expressions:
and this goes to zero as goes to infinity, but is nevertheless of exponential type 1, as can be seen by looking at the points .
Exponential type with respect to a symmetric convex body
has given a generalization of exponential type for entire functions of several complex variables.
Suppose is a convex, compact, and symmetric subset of . It is known that for every such there is an associated norm with the property that
In other words, is the unit ball in with respect to . The set
is called the polar set and is also a convex, compact, and symmetric subset of . Furthermore, we can write
We extend from to by
An entire function of -complex variables is said to be of exponential type with respect to if for every there exists a real-valued constant such that
for all .
Fréchet space
Colle
|
https://en.wikipedia.org/wiki/Graffiti%20%28program%29
|
Graffiti is a computer program which makes conjectures in various subfields of mathematics (particularly graph theory) and chemistry, but can be adapted to other fields. It was written by Siemion Fajtlowicz and Ermelinda DeLaViña at the University of Houston. Research on conjectures produced by Graffiti has led to over 60 publications by other mathematicians.
References
External links
Graffiti & Automated Conjecture-Making
Siemion Fajtlowicz
Chemistry software
Mathematical software
|
https://en.wikipedia.org/wiki/Comparison%20function
|
In applied mathematics, comparison functions are several classes of continuous functions, which are used in stability theory to characterize the stability properties of control systems as Lyapunov stability, uniform asymptotic stability etc. 1 + 1 equals 2, which can be used in comparison functions.
Let be a space of continuous functions acting from to . The most important classes of comparison functions are:
Functions of class are also called positive-definite functions.
One of the most important properties of comparison functions is given by Sontag’s -Lemma, named after Eduardo Sontag. It says that for each and any there exist :
Many further useful properties of comparison functions can be found in.
Comparison functions are primarily used to obtain quantitative restatements of stability properties as Lyapunov stability, uniform asymptotic stability, etc. These restatements are often more useful than the qualitative definitions of stability properties given in language.
As an example, consider an ordinary differential equation
where is locally Lipschitz. Then:
() is globally stable if and only if there is a so that for any initial condition and for any it holds that
() is globally asymptotically stable if and only if there is a so that for any initial condition and for any it holds that
The comparison-functions formalism is widely used in input-to-state stability theory.
References
Types of functions
Stability theory
|
https://en.wikipedia.org/wiki/Borel%20transform
|
In mathematics, Borel transform may refer to
A transform used in Borel summation
A generalization of this in Nachbin's theorem
|
https://en.wikipedia.org/wiki/Ryan%20Palmer%20%28chess%20player%29
|
Ryan Palmer (born 23 January 1974) is a chess player of Jamaican origin; he was the Jamaican National Champion in 1992. During the academic years of 2004-2007, he taught mathematics at Adams' Grammar School in Newport, Shropshire, and now has moved to the United States, to pursue further studies. In both 2006 and 2007, he and his teammates were the Shropshire Chess League Division 1 Champions. In 2007, Palmer accomplished one win, one draw, and one loss leading to an accumulative score of 50%. He later returned to the UK to teach at St Olaves Grammar School, Orpington and is now teaching maths at Richmond Park Academy
References
External links
The Chess Drum Article
Ryan Palmer 365Chess.com
1974 births
Living people
Jamaican chess players
People from Newport, Shropshire
|
https://en.wikipedia.org/wiki/Latent%20class%20model
|
In statistics, a latent class model (LCM) relates a set of observed (usually discrete) multivariate variables to a set of latent variables. It is a type of latent variable model. It is called a latent class model because the latent variable is discrete. A class is characterized by a pattern of conditional probabilities that indicate the chance that variables take on certain values.
Latent class analysis (LCA) is a subset of structural equation modeling, used to find groups or subtypes of cases in multivariate categorical data. These subtypes are called "latent classes".
Confronted with a situation as follows, a researcher might choose to use LCA to understand the data: Imagine that symptoms a-d have been measured in a range of patients with diseases X, Y, and Z, and that disease X is associated with the presence of symptoms a, b, and c, disease Y with symptoms b, c, d, and disease Z with symptoms a, c and d.
The LCA will attempt to detect the presence of latent classes (the disease entities), creating patterns of association in the symptoms. As in factor analysis, the LCA can also be used to classify case according to their maximum likelihood class membership.
Because the criterion for solving the LCA is to achieve latent classes within which there is no longer any association of one symptom with another (because the class is the disease which causes their association), and the set of diseases a patient has (or class a case is a member of) causes the symptom association, the symptoms will be "conditionally independent", i.e., conditional on class membership, they are no longer related.
Model
Within each latent class, the observed variables are statistically independent. This is an important aspect. Usually the observed variables are statistically dependent. By introducing the latent variable, independence is restored in the sense that within classes variables are independent (local independence). We then say that the association between the observed variables is explained by the classes of the latent variable (McCutcheon, 1987).
In one form, the latent class model is written as
where is the number of latent classes and are the so-called recruitment
or unconditional probabilities that should sum to one. are the
marginal or conditional probabilities.
For a two-way latent class model, the form is
This two-way model is related to probabilistic latent semantic analysis and non-negative matrix factorization.
Related methods
There are a number of methods with distinct names and uses that share a common relationship. Cluster analysis is, like LCA, used to discover taxon-like groups of cases in data. Multivariate mixture estimation (MME) is applicable to continuous data, and assumes that such data arise from a mixture of distributions: imagine a set of heights arising from a mixture of men and women. If a multivariate mixture estimation is constrained so that measures must be uncorrelated within each distribution, it is termed latent pro
|
https://en.wikipedia.org/wiki/Value%20%28mathematics%29
|
In mathematics, value may refer to several, strongly related notions.
In general, a mathematical value may be any definite mathematical object. In elementary mathematics, this is most often a number – for example, a real number such as or an integer such as 42.
The value of a variable or a constant is any number or other mathematical object assigned to it.
The value of a mathematical expression is the result of the computation described by this expression when the variables and constants in it are assigned values.
The value of a function, given the value(s) assigned to its argument(s), is the quantity assumed by the function for these argument values.
For example, if the function is defined by , then assigning the value 3 to its argument yields the function value 10, since .
If the variable, expression or function only assumes real values, it is called real-valued. Likewise, a complex-valued variable, expression or function only assumes complex values.
See also
Value function
Value (computer science)
Absolute value
Truth value
References
Elementary mathematics
nl:Reëel-waardige functie
|
https://en.wikipedia.org/wiki/Heap
|
Heap or HEAP may refer to:
Computing and mathematics
Heap (data structure), a data structure commonly used to implement a priority queue
Heap (mathematics), a generalization of a group
Heap (programming) (or free store), an area of memory for dynamic memory allocation
Heapsort, a comparison-based sorting algorithm
Heap overflow, a type of buffer overflow that occurs in the heap data area
Sorites paradox, also known as the paradox of the heap
Other uses
Heap (surname)
Heaps (surname)
Heap leaching, an industrial mining process
Heap (comics), a golden-age comic book character
Heap, Bury, a former district in England
"The Heap" (Fargo), a 2014 television episode
High Explosive, Armor-Piercing, ammunition and ordnance
Holocaust Education and Avoidance Pod, an idea in Neal Stephenson's novel Cryptonomicon
See also
Skandha, Buddhist concept describing the aggregated contents of mental activity
Beap or bi-parental heap, a data structure
Treap, a form of binary search tree data structure
Heapey, a village and civil parish of the Borough of Chorley, in Lancashire, England
Pile (disambiguation)
|
https://en.wikipedia.org/wiki/Score%20function
|
The term score function may refer to:
Scoring rule, in decision theory, measures the accuracy of probabilistic predictions
Score (statistics), the derivative of the log-likelihood function with respect to the parameter
In positional voting, a function mapping the rank of a candidate to the number of points this candidate receives.
See also
Scorer's functions
|
https://en.wikipedia.org/wiki/Trigonometric%20series
|
In mathematics, a trigonometric series is an infinite series of the form
where is the variable and and are coefficients. It is an infinite version of a trigonometric polynomial.
A trigonometric series is called the Fourier series of the integrable function if the coefficients have the form:
Examples
Every Fourier series gives an example of a trigonometric series.
Let the function on be extended periodically (see sawtooth wave). Then its Fourier coefficients are:
Which gives an example of a trigonometric series:
The converse is false however, not every trigonometric series is a Fourier series. The series
is a trigonometric series which converges for all but is not a Fourier series.
Here for and all other coefficients are zero.
Uniqueness of Trigonometric series
The uniqueness and the zeros of trigonometric series was an active area of research in 19th century Europe. First, Georg Cantor proved that if a trigonometric series is convergent to a function on the interval , which is identically zero, or more generally, is nonzero on at most finitely many points, then the coefficients of the series are all zero.
Later Cantor proved that even if the set S on which is nonzero is infinite, but the derived set S''' of S is finite, then the coefficients are all zero. In fact, he proved a more general result. Let S0 = S and let Sk+1 be the derived set of Sk. If there is a finite number n for which Sn is finite, then all the coefficients are zero. Later, Lebesgue proved that if there is a countably infinite ordinal α such that Sα is finite, then the coefficients of the series are all zero. Cantor's work on the uniqueness problem famously led him to invent transfinite ordinal numbers, which appeared as the subscripts α in S''α .
Notes
References
See also
Denjoy–Luzin theorem
Fourier series
|
https://en.wikipedia.org/wiki/Marc%20Thomas%20%28computer%20scientist%29
|
Marc Phillip Thomas (1949–2017) was a professor of computer science and mathematics, retired chair and a system administrator of Computer Science department at CSU Bakersfield.
His successful research projects include the resolution of the commutative Singer–Wermer conjecture and construction of a non-standard closed ideal in a certain radical Banach algebra of power series and their quotients.
Exposition
The Relationship between C, ANSI C, and C++
(from Encyclopedia of Information Systems)
Remarks on Network Security
Typical Hacking Attempts
Typical Buffer Overflow Hack Attempts
Moronic Hacking
Efficient Hacking
Publications
Elements in the radical of a Banach algebra obeying the unbounded Kleinecke-Shirokov conjecture
Prime-like Elements and Semi-direct Products in Commutative Banach Algebras
Principal Ideals and Semi-direct Products in Commutative Banach Algebras
Single-Element Properties in Commutative Radical Banach Algebras:a Classification Scheme
Reduction of discontinuity for derivations on Frechet algebras
Radical Banach Algebrasand Quasinilpotent Weighted Shift Operators.
The image of a derivation is contained in the radical ()
Education
Degree: Ph.D. (Mathematics), UC Berkeley, 1976
Related work
Derivations with large separating subspace
External links
CSUB Computer Science Department
California State University of Bakersfield
1950 births
2017 deaths
People from Bakersfield, California
American computer scientists
American mathematicians
|
https://en.wikipedia.org/wiki/Think%20globally%2C%20act%20locally
|
The phrase "Think globally, act locally" or "Think global, act local" has been used in various contexts, including planning, environment, education, mathematics, business and the church.
Definition
"Think globally, act locally" urges people to consider the health of the entire planet and to take action in their own communities and cities. Long before governments began enforcing environmental laws, individuals were coming together to protect habitats and the organisms that live within them. These efforts are referred to as grassroots efforts. They occur on a local level and are primarily run by volunteers and helpers.
"Think Globally, Act Locally" originally began at the grassroots level, however, it is now a global concept with high importance. It is not just volunteers who take the environment into consideration. Corporations, government officials, education system, and local communities also see the importance of taking necessary actions that can impact positively the environment.
Warren Heaps states, "It's really important to recognize that markets are different around the world, and company compensation programs should reflect a balance between global corporate philosophy and local practice and culture".
Origin in town planning
The original phrase "Think global, act local" has been attributed to Scots town planner and social activist Patrick Geddes, a Scottish biologist, sociologist, philanthropist and pioneering town planner. Although the exact phrase does not appear in Geddes' 1915 book Cities in Evolution, the idea (as applied to city planning) is clearly evident: "'Local character' is thus no mere accidental old-world quaintness, as its mimics think and say. It is attained only in course of adequate grasp and treatment of the whole environment, and in active sympathy with the essential and characteristic life of the place concerned." Geddes was also responsible for introducing the concept of "region" to architecture and planning. He has made significant contributions to the consideration of the environment. Geddes believed in working with the environment, versus working against it.
Town planning is important to understanding of the idea "think globally, act locally". Urban management and development highly impacts the surrounding environment. The ways in which this is initiated is vital to the health of the environment. Corporations need to be aware of global communities when expanding their companies to new locations. Not only do corporations need to be aware of global differences, but also Urban and rural areas who plan on expanding or changing the dynamics of their community. As stated "Addressing the complex urban environmental problems, in order to improve urban livability through Urban Environmental Strategies (UES), involves taking stock of the existing urban environmental problems, their comparative analysis and prioritization, setting out objectives and targets, and identification of various measures to meet these objectives
|
https://en.wikipedia.org/wiki/Fr%C3%A9chet%20algebra
|
In mathematics, especially functional analysis, a Fréchet algebra, named after Maurice René Fréchet, is an associative algebra over the real or complex numbers that at the same time is also a (locally convex) Fréchet space. The multiplication operation for is required to be jointly continuous.
If is an increasing family of seminorms for
the topology of , the joint continuity of multiplication is equivalent to there being a constant and integer for each such that for all . Fréchet algebras are also called B0-algebras.
A Fréchet algebra is -convex if there exists such a family of semi-norms for which . In that case, by rescaling the seminorms, we may also take for each and the seminorms are said to be submultiplicative: for all -convex Fréchet algebras may also be called Fréchet algebras.
A Fréchet algebra may or may not have an identity element . If is unital, we do not require that as is often done for Banach algebras.
Properties
Continuity of multiplication. Multiplication is separately continuous if and for every and sequence converging in the Fréchet topology of . Multiplication is jointly continuous if and imply . Joint continuity of multiplication is part of the definition of a Fréchet algebra. For a Fréchet space with an algebra structure, if the multiplication is separately continuous, then it is automatically jointly continuous.
Group of invertible elements. If is the set of invertible elements of , then the inverse map is continuous if and only if is a set. Unlike for Banach algebras, may not be an open set. If is open, then is called a -algebra. (If happens to be non-unital, then we may adjoin a unit to and work with , or the set of quasi invertibles may take the place of .)
Conditions for -convexity. A Fréchet algebra is -convex if and only if for every, if and only if for one, increasing family of seminorms which topologize , for each there exists and such that for all and . A commutative Fréchet -algebra is -convex, but there exist examples of non-commutative Fréchet -algebras which are not -convex.
Properties of -convex Fréchet algebras. A Fréchet algebra is -convex if and only if it is a countable projective limit of Banach algebras. An element of is invertible if and only if its image in each Banach algebra of the projective limit is invertible.
Examples
Zero multiplication. If is any Fréchet space, we can make a Fréchet algebra structure by setting for all .
Smooth functions on the circle. Let be the 1-sphere. This is a 1-dimensional compact differentiable manifold, with no boundary. Let be the set of infinitely differentiable complex-valued functions on . This is clearly an algebra over the complex numbers, for pointwise multiplication. (Use the product rule for differentiation.) It is commutative, and the constant function acts as an identity. Define a countable set of seminorms on by where denotes the supremum of the absolute value of the th derivative . Then, by the produc
|
https://en.wikipedia.org/wiki/Cartan%20model
|
In mathematics, the Cartan model is a differential graded algebra that computes the equivariant cohomology of a space.
References
Stefan Cordes, Gregory Moore, Sanjaye Ramgoolam, Lectures on 2D Yang-Mills Theory, Equivariant Cohomology and Topological Field Theories, , 1994.
Algebraic topology
|
https://en.wikipedia.org/wiki/P%C3%B3lya%20Prize%20%28LMS%29
|
The Pólya Prize is a prize in mathematics, awarded by the London Mathematical Society. Second only to the triennial De Morgan Medal in prestige among the society's awards, it is awarded in the years that are not divisible by three – those in which the De Morgan Medal is not awarded. First given in 1987, the prize is named after Hungarian mathematician George Pólya, who was a member of the society for over 60 years.
The prize is awarded "in recognition of outstanding creativity in, imaginative exposition of, or distinguished contribution to, mathematics within the United Kingdom".
It cannot be given to anyone who has previously received the De Morgan Medal.
List of winners
1987 John Horton Conway
1988 C. T. C. Wall
1990 Graeme B. Segal
1991 Ian G. Macdonald
1993 David Rees
1994 David Williams
1996 David Edmunds
1997 John Hammersley
1999 Simon Donaldson
2000 Terence Lyons
2002 Nigel Hitchin
2003 Angus Macintyre
2005 Michael Berry
2006 Peter Swinnerton-Dyer
2008 David Preiss
2009 Roger Heath-Brown
2011 E. Brian Davies
2012 Dan Segal
2014 Miles Reid
2015 Boris Zilber
2017 Alex Wilkie
2018 Karen Vogtmann
2020 Martin W. Liebeck
2021 Ehud Hrushovski
See also
List of mathematics awards
References
List of LMS prize winners London Mathematical Society
The Pólya Prize of the London Mathematical Society MacTutor History of Mathematics
British awards
Awards established in 1987
Awards of the London Mathematical Society
|
https://en.wikipedia.org/wiki/Initial%20algebra
|
In mathematics, an initial algebra is an initial object in the category of -algebras for a given endofunctor . This initiality provides a general framework for induction and recursion.
Examples
Functor
Consider the endofunctor sending to , where is the one-point (singleton) set, the terminal object in the category. An algebra for this endofunctor is a set (called the carrier of the algebra) together with a function . Defining such a function amounts to defining a point and a function .
Define
and
Then the set of natural numbers together with the function is an initial -algebra. The initiality (the universal property for this case) is not hard to establish; the unique homomorphism to an arbitrary -algebra , for an element of and a function on , is the function sending the natural number to , that is, , the -fold application of to .
The set of natural numbers is the carrier of an initial algebra for this functor: the point is zero and the function is the successor function.
Functor
For a second example, consider the endofunctor on the category of sets, where is the set of natural numbers. An algebra for this endofunctor is a set together with a function . To define such a function, we need a point and a function . The set of finite lists of natural numbers is an initial algebra for this functor. The point is the empty list, and the function is cons, taking a number and a finite list, and returning a new finite list with the number at the head.
In categories with binary coproducts, the definitions just given are equivalent to the usual definitions of a natural number object and a list object, respectively.
Final coalgebra
Dually, a final coalgebra is a terminal object in the category of -coalgebras. The finality provides a general framework for coinduction and corecursion.
For example, using the same functor as before, a coalgebra is defined as a set together with a function . Defining such a function amounts to defining a partial function {{math|f''': X ⇸ Y}} whose domain is formed by those for which belongs to . Such a structure can be viewed as a chain of sets, on which is not defined, which elements map into by , which elements map into by , etc., and containing the remaining elements of . With this in view, the set consisting of the set of natural numbers extended with a new element is the carrier of the final coalgebra in the category, where is the predecessor function (the inverse of the successor function) on the positive naturals, but acts like the identity on the new element : , . This set that is the carrier of the final coalgebra of is known as the set of conatural numbers.
For a second example, consider the same functor as before. In this case the carrier of the final coalgebra consists of all lists of natural numbers, finite as well as infinite. The operations are a test function testing whether a list is empty, and a deconstruction function defined on non-empty lists returning a pair co
|
https://en.wikipedia.org/wiki/Prym%20variety
|
In mathematics, the Prym variety construction (named for Friedrich Prym) is a method in algebraic geometry of making an abelian variety from a morphism of algebraic curves. In its original form, it was applied to an unramified double covering of a Riemann surface, and was used by F. Schottky and H. W. E. Jung in relation with the Schottky problem, as it is now called, of characterising Jacobian varieties among abelian varieties. It is said to have appeared first in the late work of Riemann, and was extensively studied by Wirtinger in 1895, including degenerate cases.
Given a non-constant morphism
φ: C1 → C2
of algebraic curves, write Ji for the Jacobian variety of Ci. Then from φ construct the corresponding morphism
ψ: J1 → J2,
which can be defined on a divisor class D of degree zero by applying φ to each point of the divisor. This is a well-defined morphism, often called the norm homomorphism. Then the Prym variety of φ is the kernel of ψ. To qualify that somewhat, to get an abelian variety, the connected component of the identity of the reduced scheme underlying the kernel may be intended. Or in other words take the largest abelian subvariety of J1 on which ψ is trivial.
The theory of Prym varieties was dormant for a long time, until revived by David Mumford around 1970. It now plays a substantial role in some contemporary theories, for example of the Kadomtsev–Petviashvili equation. One advantage of the method is that it allows one to apply the theory of curves to the study of a wider class of abelian varieties than Jacobians. For example, principally polarized abelian varieties (p.p.a.v.'s) of dimension > 3 are not generally Jacobians, but all p.p.a.v.'s of dimension 5 or less are Prym varieties. It is for this reason that p.p.a.v.'s are fairly well understood up to dimension 5.
References
Algebraic curves
Abelian varieties
|
https://en.wikipedia.org/wiki/Q-difference%20polynomial
|
In combinatorial mathematics, the q-difference polynomials or q-harmonic polynomials are a polynomial sequence defined in terms of the q-derivative. They are a generalized type of Brenke polynomial, and generalize the Appell polynomials. See also Sheffer sequence.
Definition
The q-difference polynomials satisfy the relation
where the derivative symbol on the left is the q-derivative. In the limit of , this becomes the definition of the Appell polynomials:
Generating function
The generalized generating function for these polynomials is of the type of generating function for Brenke polynomials, namely
where is the q-exponential:
Here, is the q-factorial and
is the q-Pochhammer symbol. The function is arbitrary but assumed to have an expansion
Any such gives a sequence of q-difference polynomials.
References
A. Sharma and A. M. Chak, "The basic analogue of a class of polynomials", Riv. Mat. Univ. Parma, 5 (1954) 325–337.
Ralph P. Boas, Jr. and R. Creighton Buck, Polynomial Expansions of Analytic Functions (Second Printing Corrected), (1964) Academic Press Inc., Publishers New York, Springer-Verlag, Berlin. Library of Congress Card Number 63-23263. (Provides a very brief discussion of convergence.)
Q-analogs
Polynomials
|
https://en.wikipedia.org/wiki/Weinstein%20conjecture
|
In mathematics, the Weinstein conjecture refers to a general existence problem for periodic orbits of Hamiltonian or Reeb vector flows. More specifically, the conjecture claims that on a compact contact manifold, its Reeb vector field should carry at least one periodic orbit.
By definition, a level set of contact type admits a contact form obtained by contracting the Hamiltonian vector field into the symplectic form. In this case, the Hamiltonian flow is a Reeb vector field on that level set. It is a fact that any contact manifold (M,α) can be embedded into a canonical symplectic manifold, called the symplectization of M, such that M is a contact type level set (of a canonically defined Hamiltonian) and the Reeb vector field is a Hamiltonian flow. That is, any contact manifold can be made to satisfy the requirements of the Weinstein conjecture. Since, as is trivial to show, any orbit of a Hamiltonian flow is contained in a level set, the Weinstein conjecture is a statement about contact manifolds.
It has been known that any contact form is isotopic to a form that admits a closed Reeb orbit; for example, for any contact manifold there is a compatible open book decomposition, whose binding is a closed Reeb orbit. This is not enough to prove the Weinstein conjecture, though, because the Weinstein conjecture states that every contact form admits a closed Reeb orbit, while an open book determines a closed Reeb orbit for a form which is only isotopic to the given form.
The conjecture was formulated in 1978 by Alan Weinstein. In several cases, the existence of a periodic orbit was known. For instance, Rabinowitz showed that on star-shaped level sets of a Hamiltonian function on a symplectic manifold, there were always periodic orbits (Weinstein independently proved the special case of convex level sets). Weinstein observed that the hypotheses of several such existence theorems could be subsumed in the condition that the level set be of contact type. (Weinstein's original conjecture included the condition that the first de Rham cohomology group of the level set is trivial; this hypothesis turned out to be unnecessary).
The Weinstein conjecture was first proved for contact hypersurfaces in in 1986 by , then extended to cotangent bundles by Hofer–Viterbo and to wider classes of aspherical manifolds by Floer–Hofer–Viterbo. The presence of holomorphic spheres was used by Hofer–Viterbo. All these cases dealt with the situation where the contact manifold is a contact submanifold of a symplectic manifold. A new approach without this assumption was discovered in dimension 3 by Hofer and is at the origin of contact homology.
The Weinstein conjecture has now been proven for all closed 3-dimensional manifolds by Clifford Taubes. The proof uses a variant of Seiberg–Witten Floer homology and pursues a strategy analogous to Taubes' proof that the Seiberg-Witten and Gromov invariants are equivalent on a symplectic four-manifold. In particular, the proof pro
|
https://en.wikipedia.org/wiki/Pieter%20van%20Musschenbroek
|
Pieter van Musschenbroek (14 March 1692 – 19 September 1761) was a Dutch scientist. He was a professor in Duisburg, Utrecht, and Leiden, where he held positions in mathematics, philosophy, medicine, and astronomy. He is credited with the invention of the first capacitor in 1746: the Leyden jar. He performed pioneering work on the buckling of compressed struts. Musschenbroek was also one of the first scientists (1729) to provide detailed descriptions of testing machines for tension, compression, and flexure testing. An early example of a problem in dynamic plasticity was described in the 1739 paper (in the form of the penetration of butter by a wooden stick subjected to impact by a wooden sphere).
Early life and studies
Pieter van Musschenbroek was born on 14 March 1692 in Leiden, Holland, Dutch Republic. His father was Johannes van Musschenbroek and his mother was Margaretha van Straaten. The van Musschenbroeks, originally from Flanders, had lived in the city of Leiden since circa 1600. His father was an instrument maker, who made scientific instruments such as air pumps, microscopes, and telescopes.
Van Musschenbroek attended Latin school until 1708, where he studied Greek, Latin, French, English, High German, Italian, and Spanish. He studied medicine at Leiden University and received his doctorate in 1715. He also attended lectures by John Theophilus Desaguliers and Isaac Newton in London. He finished his study in philosophy in 1719.
Musschenbroek belonged to the tradition of Dutch thinkers who popularised the ontological argument of God's design. He is author of Oratio de sapientia divina (Prayer of Divine Wisdom. 1744).
Academic career
Duisburg
In 1719, he became professor of mathematics and philosophy at the University of Duisburg. In 1721, he also became professor of medicine.
Utrecht
In 1723, he left his posts in Duisburg and became professor at the University of Utrecht. In 1726 he also became professor in astronomy. Musschenbroek's Elementa Physica (1726) played an important part in the transmission of Isaac Newton's ideas in physics to Europe. In November 1734 he was elected a Fellow of the Royal Society.
Leiden
In 1739, he returned to Leiden, where he succeeded Jacobus Wittichius as professor.
Already during his studies at Leiden University, van Musschenbroek became interested in electrostatics. At that time, transient electrical energy could be generated by friction machines but there was no way to store it. Musschenbroek and his student Andreas Cunaeus discovered that the energy could be stored, in work that also involved Jean-Nicolas-Sébastien Allamand as collaborator. The apparatus was a glass jar filled with water into which a brass rod had been placed; and the stored energy could be released only by completing an external circuit between the brass rod and another conductor, originally a hand, placed in contact with the outside of the jar. Van Musschenbroek communicated this discovery to René Réaumur in January 1746, and
|
https://en.wikipedia.org/wiki/Q-exponential
|
In combinatorial mathematics, a q-exponential is a q-analog of the exponential function,
namely the eigenfunction of a q-derivative. There are many q-derivatives, for example, the classical q-derivative, the Askey-Wilson operator, etc. Therefore, unlike the classical exponentials, q-exponentials are not unique. For example, is the q-exponential corresponding to the classical q-derivative while are eigenfunctions of the Askey-Wilson operators.
Definition
The q-exponential is defined as
where is the q-factorial and
is the q-Pochhammer symbol. That this is the q-analog of the exponential follows from the property
where the derivative on the left is the q-derivative. The above is easily verified by considering the q-derivative of the monomial
Here, is the q-bracket.
For other definitions of the q-exponential function, see , , and .
Properties
For real , the function is an entire function of . For , is regular in the disk .
Note the inverse, .
Addition Formula
The analogue of does not hold for real numbers and . However, if these are operators satisfying the commutation relation , then holds true.
Relations
For , a function that is closely related is It is a special case of the basic hypergeometric series,
Clearly,
Relation with Dilogarithm
has the following infinite product representation:
On the other hand, holds.
When ,
By taking the limit ,
where is the dilogarithm.
In physics
The Q-exponential function is also known as the quantum dilogarithm.
References
Q-analogs
Exponentials
|
https://en.wikipedia.org/wiki/TFAE
|
TFAE may refer to:
Mathematics
TFAE: "The Following Are Equivalent"
Chemistry
Pirkle's alcohol, or TFAE: 2,2,2-trifluoro-1-(9-anthryl)ethanol
|
https://en.wikipedia.org/wiki/Schottky%20problem
|
In mathematics, the Schottky problem, named after Friedrich Schottky, is a classical question of algebraic geometry, asking for a characterisation of Jacobian varieties amongst abelian varieties.
Geometric formulation
More precisely, one should consider algebraic curves of a given genus , and their Jacobians . There is a moduli space of such curves, and a moduli space of abelian varieties, , of dimension , which are principally polarized. There is a morphismwhich on points (geometric points, to be more accurate) takes isomorphism class to . The content of Torelli's theorem is that is injective (again, on points). The Schottky problem asks for a description of the image of , denoted .
The dimension of is , for , while the dimension of is g(g + 1)/2. This means that the dimensions are the same (0, 1, 3, 6) for g = 0, 1, 2, 3. Therefore is the first case where the dimensions change, and this was studied by F. Schottky in the 1880s. Schottky applied the theta constants, which are modular forms for the Siegel upper half-space, to define the Schottky locus in . A more precise form of the question is to determine whether the image of essentially coincides with the Schottky locus (in other words, whether it is Zariski dense there).
Dimension 1 case
All elliptic curves are the Jacobian of themselves, hence the moduli stack of elliptic curves is a model for .
Dimensions 2 and 3
In the case of Abelian surfaces, there are two types of Abelian varieties: the Jacobian of a genus 2 curve, or the product of Jacobians of elliptic curves. This means the moduli spacesembed into . There is a similar description for dimension 3 since an Abelian variety can be the product of Jacobians.
Period lattice formulation
If one describes the moduli space in intuitive terms, as the parameters on which an abelian variety depends, then the Schottky problem asks simply what condition on the parameters implies that the abelian variety comes from a curve's Jacobian. The classical case, over the complex number field, has received most of the attention, and then an abelian variety A is simply a complex torus of a particular type, arising from a lattice in Cg. In relatively concrete terms, it is being asked which lattices are the period lattices of compact Riemann surfaces.
Riemann's matrix formulation
Note that a Riemann matrix is quite different from any Riemann tensor
One of the major achievements of Bernhard Riemann was his theory of complex tori and theta functions. Using the Riemann theta function, necessary and sufficient conditions on a lattice were written down by Riemann for a lattice in Cg to have the corresponding torus embed into complex projective space. (The interpretation may have come later, with Solomon Lefschetz, but Riemann's theory was definitive.) The data is what is now called a Riemann matrix. Therefore the complex Schottky problem becomes the question of characterising the period matrices of compact Riemann surfaces of genus g, formed by in
|
https://en.wikipedia.org/wiki/GHP%20formalism
|
The GHP formalism (or Geroch–Held–Penrose formalism) is a technique used in the mathematics of general relativity that involves singling out a pair of null directions at each point of spacetime. It is a rewriting of the Newman–Penrose formalism which respects the covariance of Lorentz transformations preserving two null directions. This is desirable for Petrov Type D spacetimes, where the pair is made up of degenerate principal null directions, and spatial surfaces, where the null vectors are the natural null orthogonal vectors to the surface.
The New Covariance
The GHP formalism notices that given a spin-frame with the complex rescaling does not change normalization. The magnitude of this transformation is a boost, and the phase tells one how much to rotate. A quantity of weight is one that transforms like One then defines derivative operators which take tensors under these transformations to tensors. This simplifies many NP equations, and allows one to define scalars on 2-surfaces in a natural way.
See also
General relativity
NP formalism
References
Mathematical methods in general relativity
|
https://en.wikipedia.org/wiki/Hermitian%20symmetric%20space
|
In mathematics, a Hermitian symmetric space is a Hermitian manifold which at every point has an inversion symmetry preserving the Hermitian structure. First studied by Élie Cartan, they form a natural generalization of the notion of Riemannian symmetric space from real manifolds to complex manifolds.
Every Hermitian symmetric space is a homogeneous space for its isometry group and has a unique decomposition as a product of irreducible spaces and a Euclidean space. The irreducible spaces arise in pairs as a non-compact space that, as Borel showed, can be embedded as an open subspace of its compact dual space. Harish Chandra showed that each non-compact space can be realized as a bounded symmetric domain in a complex vector space. The simplest case involves the groups SU(2), SU(1,1) and their common complexification SL(2,C). In this case the non-compact space is the unit disk, a homogeneous space for SU(1,1). It is a bounded domain in the complex plane C. The one-point compactification of C, the Riemann sphere, is the dual space, a homogeneous space for SU(2) and SL(2,C).
Irreducible compact Hermitian symmetric spaces are exactly the homogeneous spaces of simple compact Lie groups by maximal closed connected subgroups which contain a maximal torus and have center isomorphic to the circle group. There is a complete classification of irreducible spaces, with four classical series, studied by Cartan, and two exceptional cases; the classification can be deduced from Borel–de Siebenthal theory, which classifies closed connected subgroups containing a maximal torus. Hermitian symmetric spaces appear in the theory of Jordan triple systems, several complex variables, complex geometry, automorphic forms and group representations, in particular permitting the construction of the holomorphic discrete series representations of semisimple Lie groups.
Hermitian symmetric spaces of compact type
Definition
Let H be a connected compact semisimple Lie group, σ an automorphism of H of order 2 and Hσ the fixed point subgroup of σ. Let K be a closed subgroup of H lying between Hσ and its identity component. The compact homogeneous space H / K is called a symmetric space of compact type. The Lie algebra admits a decomposition
where , the Lie algebra of K, is the +1 eigenspace of σ and the –1 eigenspace. If contains no simple summand of , the pair (, σ) is called an orthogonal symmetric Lie algebra of compact type.
Any inner product on , invariant under the adjoint representation and σ, induces a Riemannian structure on H / K, with H acting by isometries. A canonical example is given by minus the Killing form. Under such an inner product, and are orthogonal. H / K is then a Riemannian symmetric space of compact type.
The symmetric space H / K is called a Hermitian symmetric space if it has an almost complex structure preserving the Riemannian metric. This is equivalent to the existence of a linear map J with J2 = −I on which preserves the inner product and
|
https://en.wikipedia.org/wiki/Krull%27s%20theorem
|
In mathematics, and more specifically in ring theory, Krull's theorem, named after Wolfgang Krull, asserts that a nonzero ring has at least one maximal ideal. The theorem was proved in 1929 by Krull, who used transfinite induction. The theorem admits a simple proof using Zorn's lemma, and in fact is equivalent to Zorn's lemma, which in turn is equivalent to the axiom of choice.
Variants
For noncommutative rings, the analogues for maximal left ideals and maximal right ideals also hold.
For pseudo-rings, the theorem holds for regular ideals.
A slightly stronger (but equivalent) result, which can be proved in a similar fashion, is as follows:
Let R be a ring, and let I be a proper ideal of R. Then there is a maximal ideal of R containing I.
This result implies the original theorem, by taking I to be the zero ideal (0). Conversely, applying the original theorem to R/I leads to this result.
To prove the stronger result directly, consider the set S of all proper ideals of R containing I. The set S is nonempty since I ∈ S. Furthermore, for any chain T of S, the union of the ideals in T is an ideal J, and a union of ideals not containing 1 does not contain 1, so J ∈ S. By Zorn's lemma, S has a maximal element M. This M is a maximal ideal containing I.
Krull's Hauptidealsatz
Another theorem commonly referred to as Krull's theorem:
Let be a Noetherian ring and an element of which is neither a zero divisor nor a unit. Then every minimal prime ideal containing has height 1.
Notes
References
Ideals (ring theory)
|
https://en.wikipedia.org/wiki/Chief%20Statistician%20of%20Canada
|
The Chief Statistician of Canada () is the senior public servant responsible for Statistics Canada (StatCan), an agency of the Government of Canada. The office is equivalent to that of a deputy minister and as a member of the public service, the position is nonpartisan.
The chief statistician advises on matters pertaining to statistical programs of the department and agencies of the government, supervises the administration of the Statistics Act, controls the operation and staff of StatCan and reports annually on the activities of StatCan to the minister of industry.
The current chief statistician of Canada is Anil Arora, since September 19, 2016.
Dominion Statisticians and Chief Statisticians of Canada (1918 to present)
References and notes
External links
Statistics Canada web site
About Statistics Canada
Statistics Act
Innovation, Science and Economic Development Canada
|
https://en.wikipedia.org/wiki/Cyclic%20module
|
In mathematics, more specifically in ring theory, a cyclic module or monogenous module is a module over a ring that is generated by one element. The concept is a generalization of the notion of a cyclic group, that is, an Abelian group (i.e. Z-module) that is generated by one element.
Definition
A left R-module M is called cyclic if M can be generated by a single element i.e. for some x in M. Similarly, a right R-module N is cyclic if for some .
Examples
2Z as a Z-module is a cyclic module.
In fact, every cyclic group is a cyclic Z-module.
Every simple R-module M is a cyclic module since the submodule generated by any non-zero element x of M is necessarily the whole module M. In general, a module is simple if and only if it is nonzero and is generated by each of its nonzero elements.
If the ring R is considered as a left module over itself, then its cyclic submodules are exactly its left principal ideals as a ring. The same holds for R as a right R-module, mutatis mutandis.
If R is F[x], the ring of polynomials over a field F, and V is an R-module which is also a finite-dimensional vector space over F, then the Jordan blocks of x acting on V are cyclic submodules. (The Jordan blocks are all isomorphic to ; there may also be other cyclic submodules with different annihilators; see below.)
Properties
Given a cyclic R-module M that is generated by x, there exists a canonical isomorphism between M and , where denotes the annihilator of x in R.
Every module is a sum of cyclic submodules.
See also
Finitely generated module
References
Module theory
|
https://en.wikipedia.org/wiki/Homeotopy
|
In algebraic topology, an area of mathematics, a homeotopy group of a topological space is a homotopy group of the group of self-homeomorphisms of that space.
Definition
The homotopy group functors assign to each path-connected topological space the group of homotopy classes of continuous maps
Another construction on a space is the group of all self-homeomorphisms , denoted If X is a locally compact, locally connected Hausdorff space then a fundamental result of R. Arens says that will in fact be a topological group under the compact-open topology.
Under the above assumptions, the homeotopy groups for are defined to be:
Thus is the mapping class group for In other words, the mapping class group is the set of connected components of as specified by the functor
Example
According to the Dehn-Nielsen theorem, if is a closed surface then i.e., the zeroth homotopy group of the automorphisms of a space is the same as the outer automorphism group of its fundamental group.
References
Algebraic topology
Homeomorphisms
|
https://en.wikipedia.org/wiki/Darboux%27s%20theorem%20%28analysis%29
|
In mathematics, Darboux's theorem is a theorem in real analysis, named after Jean Gaston Darboux. It states that every function that results from the differentiation of another function has the intermediate value property: the image of an interval is also an interval.
When ƒ is continuously differentiable (ƒ in C1([a,b])), this is a consequence of the intermediate value theorem. But even when ƒ′ is not continuous, Darboux's theorem places a severe restriction on what it can be.
Darboux's theorem
Let be a closed interval, be a real-valued differentiable function. Then has the intermediate value property: If and are points in with , then for every between and , there exists an in such that .
Proofs
Proof 1. The first proof is based on the extreme value theorem.
If equals or , then setting equal to or , respectively, gives the desired result. Now assume that is strictly between and , and in particular that . Let such that . If it is the case that we adjust our below proof, instead asserting that has its minimum on .
Since is continuous on the closed interval , the maximum value of on is attained at some point in , according to the extreme value theorem.
Because , we know cannot attain its maximum value at . (If it did, then for all , which implies .)
Likewise, because , we know cannot attain its maximum value at .
Therefore, must attain its maximum value at some point . Hence, by Fermat's theorem, , i.e. .
Proof 2. The second proof is based on combining the mean value theorem and the intermediate value theorem.
Define .
For define and .
And for define and .
Thus, for we have .
Now, define with .
is continuous in .
Furthermore, when and when ; therefore, from the Intermediate Value Theorem, if then, there exists such that .
Let's fix .
From the Mean Value Theorem, there exists a point such that .
Hence, .
Darboux function
A Darboux function is a real-valued function ƒ which has the "intermediate value property": for any two values a and b in the domain of ƒ, and any y between ƒ(a) and ƒ(b), there is some c between a and b with ƒ(c) = y. By the intermediate value theorem, every continuous function on a real interval is a Darboux function. Darboux's contribution was to show that there are discontinuous Darboux functions.
Every discontinuity of a Darboux function is essential, that is, at any point of discontinuity, at least one of the left hand and right hand limits does not exist.
An example of a Darboux function that is discontinuous at one point is the topologist's sine curve function:
By Darboux's theorem, the derivative of any differentiable function is a Darboux function. In particular, the derivative of the function is a Darboux function even though it is not continuous at one point.
An example of a Darboux function that is nowhere continuous is the Conway base 13 function.
Darboux functions are a quite general class of functions. It turns out that any real-valued function ƒ on the re
|
https://en.wikipedia.org/wiki/Peter%20B.%20Andrews
|
Peter Bruce Andrews (born 1937) is an American mathematician and Professor of Mathematics, Emeritus at Carnegie Mellon University in Pittsburgh, Pennsylvania, and the creator of the mathematical logic Q0. He received his Ph.D. from Princeton University in 1964 under the tutelage of Alonzo Church. He received the Herbrand Award in 2003. His research group designed the TPS automated theorem prover. A subsystem ETPS (Educational Theorem Proving System) of TPS is used to help students learn logic by interactively constructing natural deduction proofs.
Publications
Andrews, Peter B. (1965). A Transfinite Type Theory with Type Variables. North Holland Publishing Company, Amsterdam.
Andrews, Peter B. (1971). "Resolution in type theory". Journal of Symbolic Logic 36, 414–432.
Andrews, Peter B. (1981). "Theorem proving via general matings". J. Assoc. Comput. March. 28, no. 2, 193–214.
Andrews, Peter B. (1986). An introduction to mathematical logic and type theory: to truth through proof. Computer Science and Applied Mathematics. . Academic Press, Inc., Orlando, FL.
Andrews, Peter B. (1989). "On connections and higher-order logic". J. Automat. Reason. 5, no. 3, 257–291.
Andrews, Peter B.; Bishop, Matthew; Issar, Sunil; Nesmith, Dan; Pfenning, Frank; Xi, Hongwei (1996). "TPS: a theorem-proving system for classical type theory". J. Automat. Reason. 16, no. 3, 321–353.
Andrews, Peter B. (2002). An introduction to mathematical logic and type theory: to truth through proof. Second edition. Applied Logic Series, 27. . Kluwer Academic Publishers, Dordrecht.
References
External links
Peter B. Andrews
1937 births
Living people
20th-century American mathematicians
21st-century American mathematicians
American logicians
Mathematical logicians
Carnegie Mellon University faculty
Princeton University alumni
|
https://en.wikipedia.org/wiki/Paul%20Tannery
|
Paul Tannery (20 December 1843 – 27 November 1904) was a French mathematician and historian of mathematics. He was the older brother of mathematician Jules Tannery, to whose Notions Mathématiques he contributed an historical chapter. Though Tannery's career was in the tobacco industry, he devoted his evenings and his life to the study of mathematicians and mathematical development.
Life and career
Tannery was born in Mantes-la-Jolie on 20 December 1843, to a deeply Catholic family. He attended private school in Mantes, followed by the Lycées in Le Mans and Caen. He then entered the École Polytechnique, on whose entrance exam he excelled. His curriculum included mathematics, the sciences, and the classics, all of which would be represented in his future academic work. Tannery's life of public service began as he then entered the École d'Applications des Tabacs as an apprentice engineer.
As an assistant engineer, Tannery spent two years in the state tobacco factory at Lille. In 1867, he moved to Paris; three years later, he served as an artillery captain in the Franco-Prussian War. Biographies of Tannery describe him as an ardent patriot and claim that he never fully accepted the humiliating Treaty of Frankfurt.
After his graduation from the École Polytechnique, Tannery had become interested in Auguste Comte and his positivist philosophy. After the war, his interest in mathematics continued, and Comte's ideas would influence his approach to the study of the history of science. Tannery moved several times with his career in the tobacco industry: to Périgord in 1872, to Bordeaux in 1874, to Le Havre in 1877, and to Paris in 1883. Bordeaux had something of an intellectual atmosphere, and though Tannery moved to Le Havre (near his parents, who lived at Caen) at his own request, he would also directly request the move to Paris, where his research and academic pursuits would be able to flourish.
It was in Paris that Tannery took on his first two major editorial works. In 1883, he began an edition of Diophantus's manuscripts, and in 1885, he and Charles Henry began an edition of one of Fermat's works. This work was made possible by access to the Bibliothèque Nationale, and so Tannery had to reduce his efforts in 1886 when he was transferred to Tonneins. Even without access to the Bibliothèque, Tannery remained hard at work, however, as he published two books composed of articles he had been writing for the Revue philosophique de la France et de l'étranger and for the Bulletin de sciences mathematiques.
In 1888, Tannery moved back to Bordeaux, where he studied Greek astronomy and directed the tobacco factory. Two years later, he was back in Paris; he would remain near Paris until his death. Despite a heavy professional workload, he continued to be productive in his work in the history of science. His editions of Diophantus and Fermat were published, along with over 250 articles. From 1890 forward, Tannery's other major work focused on
|
https://en.wikipedia.org/wiki/Li%C3%A9nard%20equation
|
In mathematics, more specifically in the study of dynamical systems and differential equations, a Liénard equation is a second order differential equation, named after the French physicist Alfred-Marie Liénard.
During the development of radio and vacuum tube technology, Liénard equations were intensely studied as they can be used to model oscillating circuits. Under certain additional assumptions Liénard's theorem guarantees the uniqueness and existence of a limit cycle for such a system. A Liénard system with piecewise-linear functions can also contain homoclinic orbits.
Definition
Let and be two continuously differentiable functions on with an even function and an odd function. Then the second order ordinary differential equation of the form
is called a Liénard equation.
Liénard system
The equation can be transformed into an equivalent two-dimensional system of ordinary differential equations. We define
then
is called a Liénard system.
Alternatively, since the Liénard equation itself is also an autonomous differential equation, the substitution leads the Liénard equation to become a first order differential equation:
which is an Abel equation of the second kind.
Example
The Van der Pol oscillator
is a Liénard equation. The solution of a Van der Pol oscillator has a limit cycle. Such cycle has a solution of a Liénard equation with negative at small and positive otherwise. The Van der Pol equation has no exact, analytic solution. Such solution for a limit cycle exists if is a constant piece-wise function.
Liénard's theorem
A Liénard system has a unique and stable limit cycle surrounding the origin if it satisfies the following additional properties:
g(x) > 0 for all x > 0;
F(x) has exactly one positive root at some value p, where F(x) < 0 for 0 < x < p and F(x) > 0 and monotonic for x > p.
See also
Autonomous differential equation
Abel equation of the second kind
Biryukov equation
Footnotes
External links
Dynamical systems
Differential equations
Theorems in dynamical systems
|
https://en.wikipedia.org/wiki/Retraction
|
Retraction or retract(ed) may refer to:
Academia
Retraction in academic publishing, withdrawals of previously published academic journal articles
Mathematics
Retraction (category theory)
Retract (group theory)
Retraction (topology)
Human physiology
Retracted (phonetics), a sound pronounced to the back of the vocal tract, in linguistics
Retracted tongue root, a position of the tongue during the pronunciation of a vowel, in phonetics
Sternal retraction, a symptom of respiratory distress in humans
Retraction (kinesiology), an anatomical term of motion
Linguistics
A process which has led to the Neo-Shtokavian accentuation, also known as "Neo-Shtokavian metatony"
See also
Retractor (disambiguation)
|
https://en.wikipedia.org/wiki/Operation%20%28mathematics%29
|
In mathematics, an operation is a function which takes zero or more input values (also called "operands" or "arguments") to a well-defined output value. The number of operands is the arity of the operation.
The most commonly studied operations are binary operations (i.e., operations of arity 2), such as addition and multiplication, and unary operations (i.e., operations of arity 1), such as additive inverse and multiplicative inverse. An operation of arity zero, or nullary operation, is a constant. The mixed product is an example of an operation of arity 3, also called ternary operation.
Generally, the arity is taken to be finite. However, infinitary operations are sometimes considered, in which case the "usual" operations of finite arity are called finitary operations.
A partial operation is defined similarly to an operation, but with a partial function in place of a function.
Types of operation
There are two common types of operations: unary and binary. Unary operations involve only one value, such as negation and trigonometric functions. Binary operations, on the other hand, take two values, and include addition, subtraction, multiplication, division, and exponentiation.
Operations can involve mathematical objects other than numbers. The logical values true and false can be combined using logic operations, such as and, or, and not. Vectors can be added and subtracted. Rotations can be combined using the function composition operation, performing the first rotation and then the second. Operations on sets include the binary operations union and intersection and the unary operation of complementation. Operations on functions include composition and convolution.
Operations may not be defined for every possible value of its domain. For example, in the real numbers one cannot divide by zero or take square roots of negative numbers. The values for which an operation is defined form a set called its domain of definition or active domain. The set which contains the values produced is called the codomain, but the set of actual values attained by the operation is its codomain of definition, active codomain, image or range. For example, in the real numbers, the squaring operation only produces non-negative numbers; the codomain is the set of real numbers, but the range is the non-negative numbers.
Operations can involve dissimilar objects: a vector can be multiplied by a scalar to form another vector (an operation known as scalar multiplication), and the inner product operation on two vectors produces a quantity that is scalar. An operation may or may not have certain properties, for example it may be associative, commutative, anticommutative, idempotent, and so on.
The values combined are called operands, arguments, or inputs, and the value produced is called the value, result, or output. Operations can have fewer or more than two inputs (including the case of zero input and infinitely many inputs).
An operator is similar to an operation in tha
|
https://en.wikipedia.org/wiki/Fermat%27s%20theorem%20%28stationary%20points%29
|
In mathematics, Fermat's theorem (also known as interior extremum theorem) is a method to find local maxima and minima of differentiable functions on open sets by showing that every local extremum of the function is a stationary point (the function's derivative is zero at that point). Fermat's theorem is a theorem in real analysis, named after Pierre de Fermat.
By using Fermat's theorem, the potential extrema of a function , with derivative , are found by solving an equation in . Fermat's theorem gives only a necessary condition for extreme function values, as some stationary points are inflection points (not a maximum or minimum). The function's second derivative, if it exists, can sometimes be used to determine whether a stationary point is a maximum or minimum.
Statement
One way to state Fermat's theorem is that, if a function has a local extremum at some point and is differentiable there, then the function's derivative at that point must be zero. In precise mathematical language:
Let be a function and suppose that is a point where has a local extremum. If is differentiable at , then .
Another way to understand the theorem is via the contrapositive statement: if the derivative of a function at any point is not zero, then there is not a local extremum at that point. Formally:
If is differentiable at , and , then is not a local extremum of .
Corollary
The global extrema of a function f on a domain A occur only at boundaries, non-differentiable points, and stationary points.
If is a global extremum of f, then one of the following is true:
boundary: is in the boundary of A
non-differentiable: f is not differentiable at
stationary point: is a stationary point of f
Extension
In higher dimensions, exactly the same statement holds; however, the proof is slightly more complicated. The complication is that in 1 dimension, one can either move left or right from a point, while in higher dimensions, one can move in many directions. Thus, if the derivative does not vanish, one must argue that there is some direction in which the function increases – and thus in the opposite direction the function decreases. This is the only change to the proof or the analysis.
The statement can also be extended to differentiable manifolds. If is a differentiable function on a manifold , then its local extrema must be critical points of , in particular points where the exterior derivative is zero.
Applications
Fermat's theorem is central to the calculus method of determining maxima and minima: in one dimension, one can find extrema by simply computing the stationary points (by computing the zeros of the derivative), the non-differentiable points, and the boundary points, and then investigating this set to determine the extrema.
One can do this either by evaluating the function at each point and taking the maximum, or by analyzing the derivatives further, using the first derivative test, the second derivative test, or the higher-order derivative tes
|
https://en.wikipedia.org/wiki/2001%20Ukrainian%20census
|
The 2001 Ukrainian census is to date the only census of the population of independent Ukraine. It was conducted by the State Statistics Committee of Ukraine on 5 December 2001, twelve years after the last Soviet Union census in 1989. The next Ukrainian census was planned to be held in 2011 but has been repeatedly postponed.
The total population recorded in 2001 was 48,457,100 persons, of which the urban population was 32,574,500 (67.2%), rural: 15,882,600 (32.8%), male: 22,441,400 (46.3%), female: 26,015,700 (53.7%). The total permanent population recorded was 48,241,000 persons.
Settlements
There were 454 cities: Nine had a population over 500,000. The census recorded over 130 nationalities.
Actual population by regions
Source: Total number of actual population. 2001 Ukrainian Population Census. State Statistics Committee of Ukraine
Urban and rural population by regions
Source: Urban and rural population. 2001 Ukrainian Population Census. State Statistics Committee of Ukraine'''
Gender structure by regionsSource: Gender structure of the population. 2001 Ukrainian Population Census. State Statistics Committee of UkraineNational structure
Source: National composition of the population. 2001 Ukrainian Population Census. State Statistics Committee of Ukraine'''
National structure by regionsNote: listed are those nationalities which comprise more than 0.1% of regional population. Numbers are given in thousands.
Autonomous Republic of Crimea - 2,024.0 (100%)
Russians - 1,180.4 (58.5%)
Ukrainians - 492.2 (24.4%)
Crimean Tatars - 243.4 (12.1%)
Belarusians - 29.2 (1.5%)
Tatars - 11.0 (0.5%)
Armenians - 8.7 (0.4%)
Jews - 4.5 (0.2%)
Poles - 3.8 (0.2%)
Moldovans - 3.7 (0.2%)
Azeris - 3.7 (0.2%)
Uzbeks - 2.9 (0.1%)
Koreans - 2.9 (0.1%)
Greeks - 2.8 (0.1%)
Germans - 2.5 (0.1%)
Mordvins - 2.2 (0.1%)
Chuvashi - 2.1 (0.1%)
Cherkasy Oblast - 1,398.3 (100%)
Ukrainians - 1,301.2 (93.1%)
Russians - 75.6 (5.4%)
Belarusians - 3.9 (0.3%)
Armenians - 1.7 (0.1%)
Moldovans - 1.6 (0.1%)
Jews - 1.5 (0.1%)
Chernihiv Oblast - 1,236.1 (100%)
Ukrainians - 1,155.4 (93.5%)
Russians - 62.2 (5.0%)
Belarusians - 7.1 (0.6%)
Chernivtsi Oblast - 919.0 (100%)
Ukrainians - 689.1 (75.0%)
Romanians - 114.6 (12.5%)
Moldovans - 67.2 (7.3%)
Russians - 37.9 (4.1%)
Poles - 3.4 (0.4%)
Belarusians - 1.5 (0.2%)
Jews - 1.4 (0.2%)
Dnipropetrovsk Oblast - 3,561.2 (100%)
Ukrainians - 2,825.8 (79.3%)
Russians - 627.5 (17.6%)
Belarusians - 29.5 (0.8%)
Jews - 13.7 (0.4%)
Armenians - 10.6 (0.3%)
Azeris - 5.6 (0.2%)
Donetsk Oblast - 4,825.6 (100%)
Ukrainians - 2,744.1 (56.9%)
Russians - 1,844.4 (38.2%)
Greeks - 77.5 (1.6%)
Belarusians - 44.5 (0.9%)
Tatars - 19.2 (0.4%)
Armenians - 15.7 (0.3%)
Jews - 8.8 (0.2%)
Azeris - 8.1 (0.2%)
Georgians - 7.2 (0.2%)
Moldovans - 7.2 (0.2%)
Ivano-Frankivsk Oblast - 1,406.1 (100%)
Ukrainians - 1,371.2 (97.5%)
Russians - 24.9 (1.8%)
Poles - 1.9 (0.2%)
Belarusians - 1.5 (0.2%)
Kharkiv Oblast - 2,895.8 (100%
|
https://en.wikipedia.org/wiki/Great%20disnub%20dirhombidodecahedron
|
In geometry, the great disnub dirhombidodecahedron, also called Skilling's figure, is a degenerate uniform star polyhedron.
It was proven in 1970 that there are only 75 uniform polyhedra other than the infinite families of prisms and antiprisms. John Skilling discovered another degenerate example, the great disnub dirhombidodecahedron, by relaxing the condition that edges must be single. More precisely, he allowed any even number of faces to meet at each edge, as long as the set of faces couldn't be separated into two connected sets (Skilling, 1975). Due to its geometric realization having some double edges where 4 faces meet, it is considered a degenerate uniform polyhedron but not strictly a uniform polyhedron.
The number of edges is ambiguous, because the underlying abstract polyhedron has 360 edges, but 120 pairs of these have the same image in the geometric realization, so that the geometric realization has 120 single edges and 120 double edges where 4 faces meet, for a total of 240 edges. The Euler characteristic of the abstract polyhedron is −96. If the pairs of coinciding edges in the geometric realization are considered to be single edges, then it has only 240 edges and Euler characteristic 24.
The vertex figure has 4 square faces passing through the center of the model.
It may be constructed as the exclusive or (blend) of the great dirhombicosidodecahedron and compound of twenty octahedra.
Related polyhedra
It shares the same edge arrangement as the great dirhombicosidodecahedron, but has a different set of triangular faces. The vertices and edges are also shared with the uniform compounds of twenty octahedra or twenty tetrahemihexahedra. 180 of the edges are shared with the great snub dodecicosidodecahedron.
Dual polyhedron
The dual of the great disnub dirhombidodecahedron is called the great disnub dirhombidodecacron. It is a nonconvex infinite isohedral polyhedron.
Like the visually identical great dirhombicosidodecacron in Magnus Wenninger's Dual Models, it is represented with intersecting infinite prisms passing through the model center, cut off at a certain point that is convenient for the maker. Wenninger suggested these figures are members of a new class of stellation polyhedra, called stellation to infinity. However, he also acknowledged that strictly speaking they are not polyhedra because their construction does not conform to the usual definitions.
Gallery
See also
List of uniform polyhedra
References
.
http://www.software3d.com/MillersMonster.php
External links
http://www.orchidpalms.com/polyhedra/uniform/skilling.htm
http://www.georgehart.com/virtual-polyhedra/great_disnub_dirhombidodecahedron.html
Uniform polyhedra
|
https://en.wikipedia.org/wiki/Unit%20of%20observation
|
In statistics, a unit of observation is the unit described by the data that one analyzes. A study may treat groups as a unit of observation with a country as the unit of analysis, drawing conclusions on group characteristics from data collected at the national level. For example, in a study of the demand for money, the unit of observation might be chosen as the individual, with different observations (data points) for a given point in time differing as to which individual they refer to; or the unit of observation might be the country, with different observations differing only in regard to the country they refer to.
Unit of observation vs unit of analysis
The unit of observation should not be confused with the unit of analysis. A study may have a differing unit of observation and unit of analysis: for example, in community research, the research design may collect data at the individual level of observation but the level of analysis might be at the neighborhood level, drawing conclusions on neighborhood characteristics from data collected from individuals. Together, the unit of observation and the level of analysis define the population of a research enterprise.
Data point
A data point or observation is a set of one or more measurements on a single member of the unit of observation. For example, in a study of the determinants of money demand with the unit of observation being the individual, a data point might be the values of income, wealth, age of individual, and number of dependents. Statistical inference about the population would be conducted using a statistical sample consisting of various such data points.
In addition, in statistical graphics, a "data point" may be an individual item with a statistical display; such points may relate to either a single member of a population or to a summary statistic calculated for a given subpopulation.
Types of data
The measurements contained in a unit of observation are formally typed, where here type is used in a way compatible with datatype in computing; so that the type of measurement can specify whether the measurement results in a Boolean value from {yes, no}, an integer or real number, the identity of some category, or some vector or array.
The implication of point is often that the data may be plotted in a graphic display, but in many cases the data are processed numerically before that is done. In the context of statistical graphics, measured values for individuals or summary statistics for different subpopulations are displayed as separate symbols within a display; since such symbols can differ by shape, size and colour, a single data point within a display can convey multiple aspects of the set of measurements for an individual or subpopulation.
See also
Observation error
Sample point
References
Statistical data types
Social research
|
https://en.wikipedia.org/wiki/Petrick%27s%20method
|
In Boolean algebra, Petrick's method (also known as Petrick function or branch-and-bound method) is a technique described by Stanley R. Petrick (1931–2006) in 1956 for determining all minimum sum-of-products solutions from a prime implicant chart. Petrick's method is very tedious for large charts, but it is easy to implement on a computer. The method was improved by Insley B. Pyne and Edward Joseph McCluskey in 1962.
Algorithm
Reduce the prime implicant chart by eliminating the essential prime implicant rows and the corresponding columns.
Label the rows of the reduced prime implicant chart , , , , etc.
Form a logical function which is true when all the columns are covered. P consists of a product of sums where each sum term has the form , where each represents a row covering column .
Apply De Morgan's Laws to expand into a sum of products and minimize by applying the absorption law .
Each term in the result represents a solution, that is, a set of rows which covers all of the minterms in the table. To determine the minimum solutions, first find those terms which contain a minimum number of prime implicants.
Next, for each of the terms found in step five, count the number of literals in each prime implicant and find the total number of literals.
Choose the term or terms composed of the minimum total number of literals, and write out the corresponding sums of prime implicants.
Example of Petrick's method
Following is the function we want to reduce:
The prime implicant chart from the Quine-McCluskey algorithm is as follows:
{| class="wikitable" style="text-align:center;"
|-
! || 0 || 1 || 2 || 5 || 6 || 7 || ⇒ || A || B || C
|-
| style="a1;" | K = m(0,1) || || || || || || || ⇒ || 0 || 0 ||
|-
| style="a1;" | L = m(0,2) || || || || || || || ⇒ || 0 || || 0
|-
| style="a1;" | M = m(1,5) || || || || || || || ⇒ || || 0 || 1
|-
| style="a1;" | N = m(2,6) || || || || || || || ⇒ || || 1 || 0
|-
| style="a1;" | P = m(5,7) || || || || || || || ⇒ || 1 || || 1
|-
| style="a1;" | Q = m(6,7) || || || || || || || ⇒ || 1 || 1 ||
|}
Based on the ✓ marks in the table above, build a product of sums of the rows where each row is added, and columns are multiplied together:
(K+L)(K+M)(L+N)(M+P)(N+Q)(P+Q)
Use the distributive law to turn that expression into a sum of products. Also use the following equivalences to simplify the final expression: X + XY = X and XX = X and X+X=X
= (K+L)(K+M)(L+N)(M+P)(N+Q)(P+Q)
= (K+LM)(N+LQ)(P+MQ)
= (KN+KLQ+LMN+LMQ)(P+MQ)
= KNP + KLPQ + LMNP + LMPQ + KMNQ + KLMQ + LMNQ + LMQ
Now use again the following equ
|
https://en.wikipedia.org/wiki/Credibility%20theory
|
Credibility theory is a branch of actuarial mathematics concerned with determining risk premiums. To achieve this, it uses mathematical models in an effort to forecast the (expected) number of insurance claims based on past observations. Technically speaking, the problem is to find the best linear approximation to the mean of the Bayesian predictive density, which is why credibility theory has many results in common with linear filtering as well as Bayesian statistics more broadly.
For example, in group health insurance an insurer is interested in calculating the risk premium, , (i.e. the theoretical expected claims amount) for a particular employer in the coming year. The insurer will likely have an estimate of historical overall claims experience, , as well as a more specific estimate for the employer in question, . Assigning a credibility factor, , to the overall claims experience (and the reciprocal to employer experience) allows the insurer to get a more accurate estimate of the risk premium in the following manner:
The credibility factor is derived by calculating the maximum likelihood estimate which would minimise the error of estimate. Assuming the variance of and are known quantities taking on the values and respectively, it can be shown that should be equal to:
Therefore, the more uncertainty the estimate has, the lower is its credibility.
Types of Credibility
In Bayesian credibility, we separate each class (B) and assign them a probability (Probability of B). Then we find how likely our experience (A) is within each class (Probability of A given B). Next, we find how likely our experience was over all classes (Probability of A). Finally, we can find the probability of our class given our experience. So going back to each class, we weight each statistic with the probability of the particular class given the experience.
Bühlmann credibility works by looking at the Variance across the population. More specifically, it looks to see how much of the Total Variance is attributed to the Variance of the Expected Values of each class (Variance of the Hypothetical Mean), and how much is attributed to the Expected Variance over all classes (Expected Value of the Process Variance). Say we have a basketball team with a high number of points per game. Sometimes they get 128 and other times they get 130 but always one of the two. Compared to all basketball teams this is a relatively low variance, meaning that they will contribute very little to the Expected Value of the Process Variance. Also, their unusually high point totals greatly increases the variance of the population, meaning that if the league booted them out, they'd have a much more predictable point total for each team (lower variance). So, this team is definitely unique (they contribute greatly to the Variance of the Hypothetical Mean). So we can rate this team's experience with a fairly high credibility. They often/always score a lot (low Expected Value of Process Vari
|
https://en.wikipedia.org/wiki/Urban%20cluster
|
Urban cluster may refer to:
Urban cluster (UC) in the US census. See List of United States urban areas
Urban cluster (France), a statistical area defined by France's national statistics office
City cluster, mainly in Chinese English, synonymous with megalopolis
|
https://en.wikipedia.org/wiki/Smoothing
|
In statistics and image processing, to smooth a data set is to create an approximating function that attempts to capture important patterns in the data, while leaving out noise or other fine-scale structures/rapid phenomena. In smoothing, the data points of a signal are modified so individual points higher than the adjacent points (presumably because of noise) are reduced, and points that are lower than the adjacent points are increased leading to a smoother signal. Smoothing may be used in two important ways that can aid in data analysis (1) by being able to extract more information from the data as long as the assumption of smoothing is reasonable and (2) by being able to provide analyses that are both flexible and robust. Many different algorithms are used in smoothing.
Smoothing may be distinguished from the related and partially overlapping concept of curve fitting in the following ways:
curve fitting often involves the use of an explicit function form for the result, whereas the immediate results from smoothing are the "smoothed" values with no later use made of a functional form if there is one;
the aim of smoothing is to give a general idea of relatively slow changes of value with little attention paid to the close matching of data values, while curve fitting concentrates on achieving as close a match as possible.
smoothing methods often have an associated tuning parameter which is used to control the extent of smoothing. Curve fitting will adjust any number of parameters of the function to obtain the 'best' fit.
Linear smoothers
In the case that the smoothed values can be written as a linear transformation of the observed values, the smoothing operation is known as a linear smoother; the matrix representing the transformation is known as a smoother matrix or hat matrix.
The operation of applying such a matrix transformation is called convolution. Thus the matrix is also called convolution matrix or a convolution kernel. In the case of simple series of data points (rather than a multi-dimensional image), the convolution kernel is a one-dimensional vector.
Algorithms
One of the most common algorithms is the "moving average", often used to try to capture important trends in repeated statistical surveys. In image processing and computer vision, smoothing ideas are used in scale space representations. The simplest smoothing algorithm is the "rectangular" or "unweighted sliding-average smooth". This method replaces each point in the signal with the average of "m" adjacent points, where "m" is a positive integer called the "smooth width". Usually m is an odd number. The triangular smooth is like the rectangular smooth except that it implements a weighted smoothing function.
Some specific smoothing and filter types, with their respective uses, pros and cons are:
See also
Convolution
Curve fitting
Discretization
Edge preserving smoothing
Filtering (signal processing)
Graph cuts in computer vision
Numerical smoothing and differentiation
|
https://en.wikipedia.org/wiki/Cubohemioctahedron
|
In geometry, the cubohemioctahedron is a nonconvex uniform polyhedron, indexed as U15. It has 10 faces (6 squares and 4 regular hexagons), 24 edges and 12 vertices. Its vertex figure is a crossed quadrilateral.
It is given Wythoff symbol 4 | 3, although that is a double-covering of this figure.
A nonconvex polyhedron has intersecting faces which do not represent new edges or faces. In the picture vertices are marked by golden spheres, and edges by silver cylinders.
It is a hemipolyhedron with 4 hexagonal faces passing through the model center. The hexagons intersect each other and so only triangular portions of each are visible.
Related polyhedra
It shares the vertex arrangement and edge arrangement with the cuboctahedron (having the square faces in common), and with the octahemioctahedron (having the hexagonal faces in common).
Tetrahexagonal tiling
The cubohemioctahedron can be seen as a net on the hyperbolic tetrahexagonal tiling with vertex figure 4.6.4.6.
Hexahemioctacron
The hexahemioctacron is the dual of the cubohemioctahedron, and is one of nine dual hemipolyhedra. It appears visually indistinct from the octahemioctacron.
Since the cubohemioctahedron has four hexagonal faces passing through the model center, thus it is degenerate, and can be seen as having four vertices at infinity.
In Magnus Wenninger's Dual Models, they are represented with intersecting infinite prisms passing through the model center, cut off at a certain point that is convenient for the maker.
See also
Hemi-cube - The four vertices at infinity correspond directionally to the four vertices of this abstract polyhedron.
References
(Page 101, Duals of the (nine) hemipolyhedra)
External links
Uniform polyhedra and duals
Uniform polyhedra
|
https://en.wikipedia.org/wiki/Great%20ditrigonal%20icosidodecahedron
|
In geometry, the great ditrigonal icosidodecahedron (or great ditrigonary icosidodecahedron) is a nonconvex uniform polyhedron, indexed as U47. It has 32 faces (20 triangles and 12 pentagons), 60 edges, and 20 vertices. It has 4 Schwarz triangle equivalent constructions, for example Wythoff symbol 3 | 3 gives Coxeter diagram = . It has extended Schläfli symbol a{,3} or c{3,}, as an altered great stellated dodecahedron or converted great icosahedron.
Its circumradius is times the length of its edge, a value it shares with the cube.
Related polyhedra
Its convex hull is a regular dodecahedron. It additionally shares its edge arrangement with the small ditrigonal icosidodecahedron (having the triangular faces in common), the ditrigonal dodecadodecahedron (having the pentagonal faces in common), and the regular compound of five cubes.
References
External links
VRML model:
MathWorld
Uniform polyhedra
|
https://en.wikipedia.org/wiki/Small%20rhombihexahedron
|
In geometry, the small rhombihexahedron (or small rhombicube) is a nonconvex uniform polyhedron, indexed as U18. It has 18 faces (12 squares and 6 octagons), 48 edges, and 24 vertices. Its vertex figure is an antiparallelogram.
Related polyhedra
This polyhedron shares the vertex arrangement with the stellated truncated hexahedron. It additionally shares its edge arrangement with the convex rhombicuboctahedron (having 12 square faces in common) and with the small cubicuboctahedron (having the octagonal faces in common).
It may be constructed as the exclusive or (blend) of three octagonal prisms.
External links
Polyhedra
|
https://en.wikipedia.org/wiki/Small%20cubicuboctahedron
|
In geometry, the small cubicuboctahedron is a uniform star polyhedron, indexed as U13. It has 20 faces (8 triangles, 6 squares, and 6 octagons), 48 edges, and 24 vertices. Its vertex figure is a crossed quadrilateral.
The small cubicuboctahedron is a faceting of the rhombicuboctahedron. Its square faces and its octagonal faces are parallel to those of a cube, while its triangular faces are parallel to those of an octahedron: hence the name cubicuboctahedron. The small suffix serves to distinguish it from the great cubicuboctahedron, which also has faces in the aforementioned directions.
Related polyhedra
It shares its vertex arrangement with the stellated truncated hexahedron. It additionally shares its edge arrangement with the rhombicuboctahedron (having the triangular faces and 6 square faces in common), and with the small rhombihexahedron (having the octagonal faces in common).
Related tilings
As the Euler characteristic suggests, the small cubicuboctahedron is a toroidal polyhedron of genus 3 (topologically it is a surface of genus 3), and thus can be interpreted as a (polyhedral) immersion of a genus 3 polyhedral surface, in the complement of its 24 vertices, into 3-space. (A neighborhood of any vertex is topologically a cone on a figure-8, which cannot occur in an immersion. Note that the Richter reference overlooks this fact.) The underlying polyhedron (ignoring self-intersections) defines a uniform tiling of this surface, and so the small cubicuboctahedron is a uniform polyhedron. In the language of abstract polytopes, the small cubicuboctahedron is a faithful realization of this abstract toroidal polyhedron, meaning that it is a nondegenerate polyhedron and that they have the same symmetry group. In fact, every automorphism of the abstract genus 3 surface with this tiling is realized by an isometry of Euclidean space.
Higher genus surfaces (genus 2 or greater) admit a metric of negative constant curvature (by the uniformization theorem), and the universal cover of the resulting Riemann surface is the hyperbolic plane. The corresponding tiling of the hyperbolic plane has vertex figure 3.8.4.8 (triangle, octagon, square, octagon). If the surface is given the appropriate metric of curvature = −1, the covering map is a local isometry and thus the abstract vertex figure is the same. This tiling may be denoted by the Wythoff symbol 3 4 | 4, and is depicted on the right.
Alternatively and more subtly, by chopping up each square face into 2 triangles and each octagonal face into 6 triangles, the small cubicuboctahedron can be interpreted as a non-regular coloring of the combinatorially regular (not just uniform) tiling of the genus 3 surface by 56 equilateral triangles, meeting at 24 vertices, each with degree 7. This regular tiling is significant as it is a tiling of the Klein quartic, the genus 3 surface with the most symmetric metric (automorphisms of this tiling equal isometries of the surface), and the orientation-preseserving aut
|
https://en.wikipedia.org/wiki/Nonconvex%20great%20rhombicuboctahedron
|
In geometry, the nonconvex great rhombicuboctahedron is a nonconvex uniform polyhedron, indexed as U17. It has 26 faces (8 triangles and 18 squares), 48 edges, and 24 vertices. It is represented by the Schläfli symbol rr{4,} and Coxeter-Dynkin diagram of . Its vertex figure is a crossed quadrilateral.
This model shares the name with the convex great rhombicuboctahedron, also called the truncated cuboctahedron.
An alternative name for this figure is quasirhombicuboctahedron. From that derives its Bowers acronym: querco.
Orthographic projections
Cartesian coordinates
Cartesian coordinates for the vertices of a nonconvex great rhombicuboctahedron centered at the origin with edge length 1 are all the permutations of
(±ξ, ±1, ±1),
where ξ = − 1.
Related polyhedra
It shares the vertex arrangement with the convex truncated cube. It additionally shares its edge arrangement with the great cubicuboctahedron (having the triangular faces and 6 square faces in common), and with the great rhombihexahedron (having 12 square faces in common). It has the same vertex figure as the pseudo great rhombicuboctahedron, which is not a uniform polyhedron.
Great deltoidal icositetrahedron
The great deltoidal icositetrahedron is the dual of the nonconvex great rhombicuboctahedron.
References
External links
Great Rhombicuboctahedron Paper model
Uniform polyhedra
|
https://en.wikipedia.org/wiki/Small%20dodecahemidodecahedron
|
In geometry, the small dodecahemidodecahedron is a nonconvex uniform polyhedron, indexed as . It has 18 faces (12 pentagons and 6 decagons), 60 edges, and 30 vertices. Its vertex figure alternates two regular pentagons and decagons as a crossed quadrilateral.
It is a hemipolyhedron with six decagonal faces passing through the model center.
Related polyhedra
It shares its edge arrangement with the icosidodecahedron (its convex hull, having the pentagonal faces in common), and with the small icosihemidodecahedron (having the decagonal faces in common).
References
External links
Uniform polyhedra and duals
Uniform polyhedra
|
https://en.wikipedia.org/wiki/Small%20icosihemidodecahedron
|
In geometry, the small icosihemidodecahedron (or small icosahemidodecahedron) is a uniform star polyhedron, indexed as . It has 26 faces (20 triangles and 6 decagons), 60 edges, and 30 vertices. Its vertex figure alternates two regular triangles and decagons as a crossed quadrilateral. It is a hemipolyhedron with its six decagonal faces passing through the model center.
It is given a Wythoff symbol, but that construction represents a double covering of this model.
Related polyhedra
It shares its edge arrangement with the icosidodecahedron (its convex hull, having the triangular faces in common), and with the small dodecahemidodecahedron (having the decagonal faces in common).
See also
Pentakis icosidodecahedron
List of uniform polyhedra
References
External links
Uniform polyhedra and duals
Polyhedra
|
https://en.wikipedia.org/wiki/Small%20dodecicosahedron
|
In geometry, the small dodecicosahedron (or small dodekicosahedron) is a nonconvex uniform polyhedron, indexed as U50. It has 32 faces (20 hexagons and 12 decagons), 120 edges, and 60 vertices. Its vertex figure is a crossed quadrilateral.
Related polyhedra
It shares its vertex arrangement with the great stellated truncated dodecahedron. It additionally shares its edges with the small icosicosidodecahedron (having the hexagonal faces in common) and the small ditrigonal dodecicosidodecahedron (having the decagonal faces in common).
References
External links
Uniform polyhedra
|
https://en.wikipedia.org/wiki/Octahemioctahedron
|
In geometry, the octahemioctahedron or allelotetratetrahedron is a nonconvex uniform polyhedron, indexed as . It has 12 faces (8 triangles and 4 hexagons), 24 edges and 12 vertices. Its vertex figure is a crossed quadrilateral.
It is one of nine hemipolyhedra, with 4 hexagonal faces passing through the model center.
Orientability
It is the only hemipolyhedron that is orientable, and the only uniform polyhedron with an Euler characteristic of zero (a topological torus).
Related polyhedra
It shares the vertex arrangement and edge arrangement with the cuboctahedron (having the triangular faces in common), and with the cubohemioctahedron (having the hexagonal faces in common).
By Wythoff construction it has tetrahedral symmetry (Td), like the rhombitetratetrahedron construction for the cuboctahedron, with alternate triangles with inverted orientations. Without alternating triangles, it has octahedral symmetry (Oh). In this respect it is akin to the Morin surface, which has fourfold symmetry if orientation is ignored and twofold symmetry otherwise. However the octahemioctahedron has a higher degree of symmetry and is genus 1 rather than 0.
Octahemioctacron
The octahemioctacron is the dual of the octahemioctahedron, and is one of nine dual hemipolyhedra. It appears visually indistinct from the hexahemioctacron.
Since the hemipolyhedra have faces passing through the center, the dual figures have corresponding vertices at infinity; properly, on the real projective plane at infinity. In Magnus Wenninger's Dual Models, they are represented with intersecting prisms, each extending in both directions to the same vertex at infinity, in order to maintain symmetry. In practice the model prisms are cut off at a certain point that is convenient for the maker. Wenninger suggested these figures are members of a new class of stellation figures, called stellation to infinity. However, he also suggested that strictly speaking they are not polyhedra because their construction does not conform to the usual definitions.
The octahemioctacron has four vertices at infinity.
See also
Compound of five octahemioctahedra
Hemi-cube - The four vertices at infinity correspond directionally to the four vertices of this abstract polyhedron.
References
(Page 101, Duals of the (nine) hemipolyhedra)
External links
Uniform polyhedra and duals
Toroidal polyhedra
|
https://en.wikipedia.org/wiki/Small%20dodecicosidodecahedron
|
In geometry, the small dodecicosidodecahedron (or small dodekicosidodecahedron) is a nonconvex uniform polyhedron, indexed as U33. It has 44 faces (20 triangles, 12 pentagons, and 12 decagons), 120 edges, and 60 vertices. Its vertex figure is a crossed quadrilateral.
Related polyhedra
It shares its vertex arrangement with the small stellated truncated dodecahedron and the uniform compounds of 6 or 12 pentagrammic prisms. It additionally shares its edge arrangement with the rhombicosidodecahedron (having the triangular and pentagonal faces in common), and with the small rhombidodecahedron (having the decagonal faces in common).
Dual
The dual polyhedron to the small dodecicosidodecahedron is the small dodecacronic hexecontahedron (or small sagittal ditriacontahedron). It is visually identical to the small rhombidodecacron. Its faces are darts. A part of each dart lies inside the solid, hence is invisible in solid models.
Proportions
Faces have two angles of , one of and one of . Its dihedral angles equal . The ratio between the lengths of the long and short edges is .
References
External links
Uniform polyhedra
|
https://en.wikipedia.org/wiki/Rhombicosahedron
|
In geometry, the rhombicosahedron is a nonconvex uniform polyhedron, indexed as U56. It has 50 faces (30 squares and 20 hexagons), 120 edges and 60 vertices. Its vertex figure is an antiparallelogram.
Related polyhedra
A rhombicosahedron shares its vertex arrangement with the uniform compounds of 10 or 20 triangular prisms. It additionally shares its edges with the rhombidodecadodecahedron (having the square faces in common) and the icosidodecadodecahedron (having the hexagonal faces in common).
Rhombicosacron
The rhombicosacron is a nonconvex isohedral polyhedron. It is the dual of the uniform rhombicosahedron, U56. It has 50 vertices, 120 edges, and 60 crossed-quadrilateral faces.
References
External links
Uniform polyhedra and duals
Uniform polyhedra
|
https://en.wikipedia.org/wiki/Great%20icosicosidodecahedron
|
In geometry, the great icosicosidodecahedron (or great icosified icosidodecahedron) is a nonconvex uniform polyhedron, indexed as U48. It has 52 faces (20 triangles, 12 pentagrams, and 20 hexagons), 120 edges, and 60 vertices. Its vertex figure is a crossed quadrilateral.
Related polyhedra
It shares its vertex arrangement with the truncated dodecahedron. It additionally shares its edge arrangement with the great ditrigonal dodecicosidodecahedron (having the triangular and pentagonal faces in common) and the great dodecicosahedron (having the hexagonal faces in common).
References
External links
Uniform polyhedra
|
https://en.wikipedia.org/wiki/Small%20rhombidodecahedron
|
In geometry, the small rhombidodecahedron is a nonconvex uniform polyhedron, indexed as U39. It has 42 faces (30 squares and 12 decagons), 120 edges, and 60 vertices. Its vertex figure is a crossed quadrilateral.
Related polyhedra
It shares its vertex arrangement with the small stellated truncated dodecahedron and the uniform compounds of 6 or 12 pentagrammic prisms. It additionally shares its edge arrangement with the rhombicosidodecahedron (having the square faces in common), and with the small dodecicosidodecahedron (having the decagonal faces in common).
Small rhombidodecacron
The small rhombidodecacron (or small dipteral ditriacontahedron) is a nonconvex isohedral polyhedron. It is the dual of the small rhombidodecahedron. It is visually identical to the Small dodecacronic hexecontahedron. It has 60 intersecting antiparallelogram faces.
References
External links
Uniform polyhedra and duals
Uniform polyhedra
|
https://en.wikipedia.org/wiki/Pentagrammic%20antiprism
|
In geometry, the pentagrammic antiprism is one in an infinite set of nonconvex antiprisms formed by triangle sides and two regular star polygon caps, in this case two pentagrams.
It has 12 faces, 20 edges and 10 vertices. This polyhedron is identified with the indexed name U79 as a uniform polyhedron.
Note that the pentagram face has an ambiguous interior because it is self-intersecting. The central pentagon region can be considered interior or exterior depending on how interior is defined. One definition of interior is the set of points that have a ray that crosses the boundary an odd number of times to escape the perimeter.
In either case, it is best to show the pentagram boundary line to distinguish it from a concave decagon.
Gallery
Net
Net (fold the dotted line in the centre in the opposite direction to all the other lines):
See also
Prismatic uniform polyhedron
Pentagrammic prism
Pentagrammic crossed-antiprism
References
External links
http://www.mathconsult.ch/showroom/unipoly/04.html
https://web.archive.org/web/20050313233653/http://www.math.technion.ac.il/~rl/kaleido/data/04.html
Prismatoid polyhedra
|
https://en.wikipedia.org/wiki/Pentagrammic%20crossed-antiprism
|
In geometry, the pentagrammic crossed-antiprism is one in an infinite set of nonconvex antiprisms formed by triangle sides and two regular star polygon caps, in this case two pentagrams.
It differs from the pentagrammic antiprism by having opposite orientations on the two pentagrams.
This polyhedron is identified with the indexed name U80 as a uniform polyhedron.
The pentagrammic crossed-antiprism may be inscribed within an icosahedron, and has ten triangular faces in common with the great icosahedron. It has the same vertex arrangement as the pentagonal antiprism. In fact, it may be considered as a parabidiminished great icosahedron.
See also
Prismatic uniform polyhedron
External links
http://www.mathconsult.ch/showroom/unipoly/80.html
http://bulatov.org/polyhedra/uniform/u05.html
https://web.archive.org/web/20050313234519/http://www.math.technion.ac.il/~rl/kaleido/data/05.html
Prismatoid polyhedra
|
https://en.wikipedia.org/wiki/Small%20ditrigonal%20icosidodecahedron
|
In geometry, the small ditrigonal icosidodecahedron (or small ditrigonary icosidodecahedron) is a nonconvex uniform polyhedron, indexed as U30. It has 32 faces (20 triangles and 12 pentagrams), 60 edges, and 20 vertices. It has extended Schläfli symbol a{5,3}, as an altered dodecahedron, and Coxeter diagram or .
It is constructed from Schwarz triangle (3 3 ) with Wythoff symbol 3 | 3. Its hexagonal vertex figure alternates equilateral triangle and pentagram faces.
Related polyhedra
Its convex hull is a regular dodecahedron. It additionally shares its edge arrangement with the great ditrigonal icosidodecahedron (having the triangular faces in common), the ditrigonal dodecadodecahedron (having the pentagrammic faces in common), and the regular compound of five cubes. As a simple polyhedron, it is also a hexakis truncated icosahedron where the triangles touching the pentagons are made coplanar, making the others concave.
See also
List of uniform polyhedra
References
External links
Uniform polyhedra
|
https://en.wikipedia.org/wiki/Stellated%20truncated%20hexahedron
|
In geometry, the stellated truncated hexahedron (or quasitruncated hexahedron, and stellatruncated cube) is a uniform star polyhedron, indexed as U19. It has 14 faces (8 triangles and 6 octagrams), 36 edges, and 24 vertices. It is represented by Schläfli symbol t'{4,3} or t{4/3,3}, and Coxeter-Dynkin diagram, . It is sometimes called quasitruncated hexahedron because it is related to the truncated cube, , except that the square faces become inverted into {8/3} octagrams.
Even though the stellated truncated hexahedron is a stellation of the truncated hexahedron, its core is a regular octahedron.
Orthographic projections
Related polyhedra
It shares the vertex arrangement with three other uniform polyhedra: the convex rhombicuboctahedron, the small rhombihexahedron, and the small cubicuboctahedron.
See also
List of uniform polyhedra
References
External links
Uniform polyhedra
|
https://en.wikipedia.org/wiki/Great%20cubicuboctahedron
|
In geometry, the great cubicuboctahedron is a nonconvex uniform polyhedron, indexed as U14. It has 20 faces (8 triangles, 6 squares and 6 octagrams), 48 edges, and 24 vertices. Its square faces and its octagrammic faces are parallel to those of a cube, while its triangular faces are parallel to those of an octahedron: hence the name cubicuboctahedron. The prefix great serves to distinguish it from the small cubicuboctahedron, which also has faces in the aforementioned directions.
Orthographic projections
Related polyhedra
It shares the vertex arrangement with the convex truncated cube and two other nonconvex uniform polyhedra. It additionally shares its edge arrangement with the nonconvex great rhombicuboctahedron (having the triangular faces and 6 square faces in common), and with the great rhombihexahedron (having the octagrammic faces in common).
Great hexacronic icositetrahedron
The great hexacronic icositetrahedron (or great lanceal disdodecahedron) is the dual of the great cubicuboctahedron.
See also
List of uniform polyhedra
References
External links
Polyhedra
|
https://en.wikipedia.org/wiki/Dodecadodecahedron
|
In geometry, the dodecadodecahedron is a nonconvex uniform polyhedron, indexed as U36. It is the rectification of the great dodecahedron (and that of its dual, the small stellated dodecahedron). It was discovered independently by , and .
The edges of this model form 10 central hexagons, and these, projected onto a sphere, become 10 great circles. These 10, along with the great circles from projections of two other polyhedra, form the 31 great circles of the spherical icosahedron used in construction of geodesic domes.
Wythoff constructions
It has four Wythoff constructions between four Schwarz triangle families: 2 | 5 5/2, 2 | 5 5/3, 2 | 5/2 5/4, 2 | 5/3 5/4, but represent identical results. Similarly it can be given four extended Schläfli symbols: r{5/2,5}, r{5/3,5}, r{5/2,5/4}, and r{5/3,5/4} or as Coxeter-Dynkin diagrams: , , , and .
Net
A shape with the same exterior appearance as the dodecadodecahedron can be constructed by folding up these nets:
12 pentagrams and 20 rhombic clusters are necessary. However, this construction replaces the crossing pentagonal faces of the dodecadodecahedron with non-crossing sets of rhombi, so it does not produce the same internal structure.
Related polyhedra
Its convex hull is the icosidodecahedron. It also shares its edge arrangement with the small dodecahemicosahedron (having the pentagrammic faces in common), and with the great dodecahemicosahedron (having the pentagonal faces in common).
This polyhedron can be considered a rectified great dodecahedron. It is center of a truncation sequence between a small stellated dodecahedron and great dodecahedron:
The truncated small stellated dodecahedron looks like a dodecahedron on the surface, but it has 24 faces: 12 pentagons from the truncated vertices and 12 overlapping as (truncated pentagrams). The truncation of the dodecadodecahedron itself is not uniform and attempting to make it uniform results in a degenerate polyhedron (that looks like a small rhombidodecahedron with {10/2} polygons filling up the dodecahedral set of holes), but it has a uniform quasitruncation, the truncated dodecadodecahedron.
It is topologically equivalent to a quotient space of the hyperbolic order-4 pentagonal tiling, by distorting the pentagrams back into regular pentagons. As such, it is topologically a regular polyhedron of index two:
The colours in the above image correspond to the red pentagrams and yellow pentagons of the dodecadodecahedron at the top of this article.
Medial rhombic triacontahedron
The medial rhombic triacontahedron is a nonconvex isohedral polyhedron. It is the dual of the dodecadodecahedron. It has 30 intersecting rhombic faces.
It can also be called the small stellated triacontahedron.
Stellation
The medial rhombic triacontahedron is a stellation of the rhombic triacontahedron, which is the dual of the icosidodecahedron, the convex hull of the dodecadodecahedron (dual to the original medial rhombic triacontahedron).
Related hyperbolic ti
|
https://en.wikipedia.org/wiki/Great%20icosidodecahedron
|
In geometry, the great icosidodecahedron is a nonconvex uniform polyhedron, indexed as U54. It has 32 faces (20 triangles and 12 pentagrams), 60 edges, and 30 vertices. It is given a Schläfli symbol r{3,}. It is the rectification of the great stellated dodecahedron and the great icosahedron. It was discovered independently by , and .
Related polyhedra
The figure is a rectification of the great icosahedron or the great stellated dodecahedron, much as the (small) icosidodecahedron is related to the (small) icosahedron and (small) dodecahedron, and the cuboctahedron to the cube and octahedron.
It shares its vertex arrangement with the icosidodecahedron, which is its convex hull. Unlike the great icosahedron and great dodecahedron, the great icosidodecahedron is not a stellation of the icosidodecahedron, but a faceting of it instead.
It also shares its edge arrangement with the great icosihemidodecahedron (having the triangle faces in common), and with the great dodecahemidodecahedron (having the pentagram faces in common).
The truncated great stellated dodecahedron is a degenerate polyhedron, with 20 triangular faces from the truncated vertices, and 12 (hidden) pentagonal faces as truncations of the original pentagram faces, the latter forming a great dodecahedron inscribed within and sharing the edges of the icosahedron.
Great rhombic triacontahedron
The dual of the great icosidodecahedron is the great rhombic triacontahedron; it is nonconvex, isohedral and isotoxal. It has 30 intersecting rhombic faces. It can also be called the great stellated triacontahedron.
The great rhombic triacontahedron can be constructed by expanding the size of the faces of a rhombic triacontahedron by a factor of τ3 = 1+2τ = 2+√5, where τ is the golden ratio.
See also
List of uniform polyhedra
Rhombic hexecontahedron
Notes
References
External links
Uniform polyhedra and duals
Uniform polyhedra
|
https://en.wikipedia.org/wiki/Cubitruncated%20cuboctahedron
|
In geometry, the cubitruncated cuboctahedron or cuboctatruncated cuboctahedron is a nonconvex uniform polyhedron, indexed as U16. It has 20 faces (8 hexagons, 6 octagons, and 6 octagrams), 72 edges, and 48 vertices, and has a shäfli symbol of tr{4,3/2}
Convex hull
Its convex hull is a nonuniform truncated cuboctahedron.
Orthogonal projection
Cartesian coordinates
Cartesian coordinates for the vertices of a cubitruncated cuboctahedron are all the permutations of
(±(−1), ±1, ±(+1))
Related polyhedra
Tetradyakis hexahedron
The tetradyakis hexahedron (or great disdyakis dodecahedron) is a nonconvex isohedral polyhedron. It has 48 intersecting scalene triangle faces, 72 edges, and 20 vertices.
Proportions
The triangles have one angle of , one of and one of . The dihedral angle equals . Part of each triangle lies within the solid, hence is invisible in solid models.
It is the dual of the uniform cubitruncated cuboctahedron.
See also
List of uniform polyhedra
References
p. 92
External links
http://gratrix.net Uniform polyhedra and duals
Uniform polyhedra
|
https://en.wikipedia.org/wiki/Great%20truncated%20cuboctahedron
|
In geometry, the great truncated cuboctahedron (or quasitruncated cuboctahedron or stellatruncated cuboctahedron) is a nonconvex uniform polyhedron, indexed as U20. It has 26 faces (12 squares, 8 hexagons and 6 octagrams), 72 edges, and 48 vertices. It is represented by the Schläfli symbol tr{4/3,3}, and Coxeter-Dynkin diagram . It is sometimes called the quasitruncated cuboctahedron because it is related to the truncated cuboctahedron, , except that the octagonal faces are replaced by {8/3} octagrams.
Convex hull
Its convex hull is a nonuniform truncated cuboctahedron. The truncated cuboctahedron and the great truncated cuboctahedron form isomorphic graphs despite their different geometric structure.
Orthographic projections
Cartesian coordinates
Cartesian coordinates for the vertices of a great truncated cuboctahedron centered at the origin are all permutations of
(±1, ±(1−), ±(1−2)).
See also
List of uniform polyhedra
References
External links
Uniform polyhedra
|
https://en.wikipedia.org/wiki/Truncated%20great%20dodecahedron
|
In geometry, the truncated great dodecahedron is a nonconvex uniform polyhedron, indexed as U37. It has 24 faces (12 pentagrams and 12 decagons), 90 edges, and 60 vertices. It is given a Schläfli symbol t{5,5/2}.
Related polyhedra
It shares its vertex arrangement with three other uniform polyhedra: the nonconvex great rhombicosidodecahedron, the great dodecicosidodecahedron, and the great rhombidodecahedron; and with the uniform compounds of 6 or 12 pentagonal prisms.
This polyhedron is the truncation of the great dodecahedron:
The truncated small stellated dodecahedron looks like a dodecahedron on the surface, but it has 24 faces, 12 pentagons from the truncated vertices and 12 overlapping as (truncated pentagrams).
Small stellapentakis dodecahedron
The small stellapentakis dodecahedron (or small astropentakis dodecahedron) is a nonconvex isohedral polyhedron. It is the dual of the truncated great dodecahedron. It has 60 intersecting triangular faces.
See also
List of uniform polyhedra
References
External links
Uniform polyhedra and duals
Nonconvex polyhedra
Uniform polyhedra
|
https://en.wikipedia.org/wiki/Small%20stellated%20truncated%20dodecahedron
|
In geometry, the small stellated truncated dodecahedron (or quasitruncated small stellated dodecahedron or small stellatruncated dodecahedron) is a nonconvex uniform polyhedron, indexed as U58. It has 24 faces (12 pentagons and 12 decagrams), 90 edges, and 60 vertices. It is given a Schläfli symbol t{,5}, and Coxeter diagram .
Related polyhedra
It shares its vertex arrangement with three other uniform polyhedra: the convex rhombicosidodecahedron, the small dodecicosidodecahedron and the small rhombidodecahedron.
It also has the same vertex arrangement as the uniform compounds of 6 or 12 pentagrammic prisms.
See also
List of uniform polyhedra
References
External links
Uniform polyhedra
|
https://en.wikipedia.org/wiki/Great%20stellated%20truncated%20dodecahedron
|
In geometry, the great stellated truncated dodecahedron (or quasitruncated great stellated dodecahedron or great stellatruncated dodecahedron) is a nonconvex uniform polyhedron, indexed as U66. It has 32 faces (20 triangles and 12 decagrams), 90 edges, and 60 vertices. It is given a Schläfli symbol t0,1{5/3,3}.
Related polyhedra
It shares its vertex arrangement with three other uniform polyhedra: the small icosicosidodecahedron, the small ditrigonal dodecicosidodecahedron, and the small dodecicosahedron:
Cartesian coordinates
Cartesian coordinates for the vertices of a great stellated truncated dodecahedron are all the even permutations of
(0, ±τ, ±(2−1/τ))
(±τ, ±1/τ, ±2/τ)
(±1/τ2, ±1/τ, ±2)
where τ = (1+)/2 is the golden ratio (sometimes written φ).
See also
List of uniform polyhedra
References
External links
Uniform polyhedra
|
https://en.wikipedia.org/wiki/Truncated%20great%20icosahedron
|
In geometry, the truncated great icosahedron (or great truncated icosahedron) is a nonconvex uniform polyhedron, indexed as U55. It has 32 faces (12 pentagrams and 20 hexagons), 90 edges, and 60 vertices. It is given a Schläfli symbol t{3,} or t0,1{3,} as a truncated great icosahedron.
Cartesian coordinates
Cartesian coordinates for the vertices of a truncated great icosahedron centered at the origin are all the even permutations of
(±1, 0, ±3/τ)
(±2, ±1/τ, ±1/τ3)
(±(1+1/τ2), ±1, ±2/τ)
where τ = (1+√5)/2 is the golden ratio (sometimes written φ). Using 1/τ2 = 1 − 1/τ one verifies that all vertices are on a sphere, centered at the origin, with the radius squared equal to 10−9/τ. The edges have length 2.
Related polyhedra
This polyhedron is the truncation of the great icosahedron:
The truncated great stellated dodecahedron is a degenerate polyhedron, with 20 triangular faces from the truncated vertices, and 12 (hidden) pentagonal faces as truncations of the original pentagram faces, the latter forming a great dodecahedron inscribed within and sharing the edges of the icosahedron.
Great stellapentakis dodecahedron
The great stellapentakis dodecahedron is a nonconvex isohedral polyhedron. It is the dual of the truncated great icosahedron. It has 60 intersecting triangular faces.
See also
List of uniform polyhedra
References
External links
Uniform polyhedra and duals
Uniform polyhedra
|
https://en.wikipedia.org/wiki/Great%20ditrigonal%20dodecicosidodecahedron
|
In geometry, the great ditrigonal dodecicosidodecahedron (or great dodekified icosidodecahedron) is a nonconvex uniform polyhedron, indexed as U42. It has 44 faces (20 triangles, 12 pentagons, and 12 decagrams), 120 edges, and 60 vertices. Its vertex figure is an isosceles trapezoid.
Related polyhedra
It shares its vertex arrangement with the truncated dodecahedron. It additionally shares its edge arrangement with the great icosicosidodecahedron (having the triangular and pentagonal faces in common) and the great dodecicosahedron (having the decagrammic faces in common).
See also
List of uniform polyhedra
References
External links
Uniform polyhedra
|
https://en.wikipedia.org/wiki/Great%20dodecicosidodecahedron
|
In geometry, the great dodecicosidodecahedron (or great dodekicosidodecahedron) is a nonconvex uniform polyhedron, indexed as U61. It has 44 faces (20 triangles, 12 pentagrams and 12 decagrams), 120 edges and 60 vertices.
Related polyhedra
It shares its vertex arrangement with the truncated great dodecahedron and the uniform compounds of 6 or 12 pentagonal prisms. It additionally shares its edge arrangement with the nonconvex great rhombicosidodecahedron (having the triangular and pentagrammic faces in common), and with the great rhombidodecahedron (having the decagrammic faces in common).
See also
List of uniform polyhedra
References
External links
Uniform polyhedra
|
https://en.wikipedia.org/wiki/Small%20icosicosidodecahedron
|
In geometry, the small icosicosidodecahedron (or small icosified icosidodecahedron) is a nonconvex uniform polyhedron, indexed as U31. It has 52 faces (20 triangles, 12 pentagrams, and 20 hexagons), 120 edges, and 60 vertices.
Related polyhedra
It shares its vertex arrangement with the great stellated truncated dodecahedron. It additionally shares its edges with the small ditrigonal dodecicosidodecahedron (having the triangular and pentagrammic faces in common) and the small dodecicosahedron (having the hexagonal faces in common).
See also
List of uniform polyhedra
References
External links
Uniform polyhedra
|
https://en.wikipedia.org/wiki/Rhombidodecadodecahedron
|
In geometry, the rhombidodecadodecahedron is a nonconvex uniform polyhedron, indexed as U38. It has 54 faces (30 squares, 12 pentagons and 12 pentagrams), 120 edges and 60 vertices. It is given a Schläfli symbol t0,2, and by the Wythoff construction this polyhedron can also be named a cantellated great dodecahedron.
Cartesian coordinates
Cartesian coordinates for the vertices of a uniform great rhombicosidodecahedron are all the even permutations of
(±1/τ2, 0, ±τ2)
(±1, ±1, ±)
(±2, ±1/τ, ±τ)
where τ = (1+)/2 is the golden ratio (sometimes written φ).
Related polyhedra
It shares its vertex arrangement with the uniform compounds of 10 or 20 triangular prisms. It additionally shares its edges with the icosidodecadodecahedron (having the pentagonal and pentagrammic faces in common) and the rhombicosahedron (having the square faces in common).
Medial deltoidal hexecontahedron
The medial deltoidal hexecontahedron (or midly lanceal ditriacontahedron) is a nonconvex isohedral polyhedron. It is the dual of the rhombidodecadodecahedron. It has 60 intersecting quadrilateral faces.
See also
List of uniform polyhedra
References
External links
Uniform polyhedra and duals
Uniform polyhedra
|
https://en.wikipedia.org/wiki/Icositruncated%20dodecadodecahedron
|
In geometry, the icositruncated dodecadodecahedron or icosidodecatruncated icosidodecahedron is a nonconvex uniform polyhedron, indexed as U45.
Convex hull
Its convex hull is a nonuniform truncated icosidodecahedron.
Cartesian coordinates
Cartesian coordinates for the vertices of an icositruncated dodecadodecahedron are all the even permutations of
(±(2−1/τ), ±1, ±(2+τ))
(±1, ±1/τ2, ±(3τ−1))
(±2, ±2/τ, ±2τ)
(±3, ±1/τ2, ±τ2)
(±τ2, ±1, ±(3τ−2))
where τ = (1+)/2 is the golden ratio (sometimes written φ).
Related polyhedra
Tridyakis icosahedron
The tridyakis icosahedron is the dual polyhedron of the icositruncated dodecadodecahedron. It has 44 vertices, 180 edges, and 120 scalene triangular faces.
See also
Catalan solid Duals to convex uniform polyhedra
Uniform polyhedra
List of uniform polyhedra
References
Photo on page 96, Dorman Luke construction and stellation pattern on page 97.
External links
Uniform polyhedra
|
https://en.wikipedia.org/wiki/Truncated%20dodecadodecahedron
|
In geometry, the truncated dodecadodecahedron (or stellatruncated dodecadodecahedron) is a nonconvex uniform polyhedron, indexed as U59. It is given a Schläfli symbol t0,1,2{,5}. It has 54 faces (30 squares, 12 decagons, and 12 decagrams), 180 edges, and 120 vertices. The central region of the polyhedron is connected to the exterior via 20 small triangular holes.
The name truncated dodecadodecahedron is somewhat misleading: truncation of the dodecadodecahedron would produce rectangular faces rather than squares, and the pentagram faces of the dodecadodecahedron would turn into truncated pentagrams rather than decagrams. However, it is the quasitruncation of the dodecadodecahedron, as defined by . For this reason, it is also known as the quasitruncated dodecadodecahedron. Coxeter et al. credit its discovery to a paper published in 1881 by Austrian mathematician Johann Pitsch.
Cartesian coordinates
Cartesian coordinates for the vertices of a truncated dodecadodecahedron are all the triples of numbers obtained by circular shifts and sign changes from the following points (where is the golden ratio):
Each of these five points has eight possible sign patterns and three possible circular shifts, giving a total of 120 different points.
As a Cayley graph
The truncated dodecadodecahedron forms a Cayley graph for the symmetric group on five elements, as generated by two group members: one that swaps the first two elements of a five-tuple, and one that performs a circular shift operation on the last four elements. That is, the 120 vertices of the polyhedron may be placed in one-to-one correspondence with the 5! permutations on five elements, in such a way that the three neighbors of each vertex are the three permutations formed from it by swapping the first two elements or circularly shifting (in either direction) the last four elements.
Related polyhedra
Medial disdyakis triacontahedron
The medial disdyakis triacontahedron is a nonconvex isohedral polyhedron. It is the dual of the uniform truncated dodecadodecahedron.
See also
List of uniform polyhedra
References
External links
Uniform polyhedra
|
https://en.wikipedia.org/wiki/Great%20truncated%20icosidodecahedron
|
In geometry, the great truncated icosidodecahedron (or great quasitruncated icosidodecahedron or stellatruncated icosidodecahedron) is a nonconvex uniform polyhedron, indexed as U68. It has 62 faces (30 squares, 20 hexagons, and 12 decagrams), 180 edges, and 120 vertices. It is given a Schläfli symbol t0,1,2, and Coxeter-Dynkin diagram, .
Cartesian coordinates
Cartesian coordinates for the vertices of a great truncated icosidodecahedron centered at the origin are all the even permutations of
(±τ, ±τ, ±(3−1/τ)),
(±2τ, ±1/τ, ±τ−3),
(±τ, ±1/τ2, ±(1+3/τ)),
(±, ±2, ±/τ) and
(±1/τ, ±3, ±2/τ),
where τ = (1+)/2 is the golden ratio.
Related polyhedra
Great disdyakis triacontahedron
The great disdyakis triacontahedron (or trisdyakis icosahedron) is a nonconvex isohedral polyhedron. It is the dual of the great truncated icosidodecahedron. Its faces are triangles.
Proportions
The triangles have one angle of , one of and one of . The dihedral angle equals . Part of each triangle lies within the solid, hence is invisible in solid models.
See also
List of uniform polyhedra
References
p. 96
External links
Uniform polyhedra
|
https://en.wikipedia.org/wiki/Great%20snub%20icosidodecahedron
|
In geometry, the great snub icosidodecahedron is a nonconvex uniform polyhedron, indexed as U57. It has 92 faces (80 triangles and 12 pentagrams), 150 edges, and 60 vertices. It can be represented by a Schläfli symbol sr{,3}, and Coxeter-Dynkin diagram .
This polyhedron is the snub member of a family that includes the great icosahedron, the great stellated dodecahedron and the great icosidodecahedron.
In the book Polyhedron Models by Magnus Wenninger, the polyhedron is misnamed great inverted snub icosidodecahedron, and vice versa.
Cartesian coordinates
Cartesian coordinates for the vertices of a great snub icosidodecahedron are all the even permutations of
(±2α, ±2, ±2β),
(±(α−βτ−1/τ), ±(α/τ+β−τ), ±(−ατ−β/τ−1)),
(±(ατ−β/τ+1), ±(−α−βτ+1/τ), ±(−α/τ+β+τ)),
(±(ατ−β/τ−1), ±(α+βτ+1/τ), ±(−α/τ+β−τ)) and
(±(α−βτ+1/τ), ±(−α/τ−β−τ), ±(−ατ−β/τ+1)),
with an even number of plus signs, where
α = ξ−1/ξ
and
β = −ξ/τ+1/τ2−1/(ξτ),
where τ = (1+)/2 is the golden mean and
ξ is the negative real root of ξ3−2ξ=−1/τ, or approximately −1.5488772.
Taking the odd permutations of the above coordinates with an odd number of plus signs gives another form, the enantiomorph of the other one.
The circumradius for unit edge length is
where is the second largest real root of the polynomial .
The four positive real roots of the sextic in
are, in order, the circumradii of the great retrosnub icosidodecahedron (U74), great snub icosidodecahedron (U57), great inverted snub icosidodecahedron (U69) and snub dodecahedron (U29).
Related polyhedra
Great pentagonal hexecontahedron
The great pentagonal hexecontahedron (or great petaloid ditriacontahedron) is a nonconvex isohedral polyhedron and dual to the uniform great snub icosidodecahedron. It has 60 intersecting irregular pentagonal faces, 120 edges, and 92 vertices.
Proportions
Denote the golden ratio by . Let be the negative zero of the polynomial . Then each pentagonal face has four equal angles of and one angle of . Each face has three long and two short edges. The ratio between the lengths of the long and the short edges is given by
.
The dihedral angle equals . Part of each face lies inside the solid, hence is invisible in solid models. The other two zeroes of the polynomial play a similar role in the description of the great inverted pentagonal hexecontahedron and the great pentagrammic hexecontahedron.
See also
List of uniform polyhedra
Great inverted snub icosidodecahedron
Great retrosnub icosidodecahedron
References
External links
Uniform polyhedra
|
https://en.wikipedia.org/wiki/Small%20snub%20icosicosidodecahedron
|
In geometry, the small snub icosicosidodecahedron or snub disicosidodecahedron is a uniform star polyhedron, indexed as U32. It has 112 faces (100 triangles and 12 pentagrams), 180 edges, and 60 vertices. Its stellation core is a truncated pentakis dodecahedron. It also called a holosnub icosahedron, ß{3,5}.
The 40 non-snub triangular faces form 20 coplanar pairs, forming star hexagons that are not quite regular. Unlike most snub polyhedra, it has reflection symmetries.
Convex hull
Its convex hull is a nonuniform truncated icosahedron.
Cartesian coordinates
Cartesian coordinates for the vertices of a small snub icosicosidodecahedron are all the even permutations of
(±(1-ϕ+α), 0, ±(3+ϕα))
(±(ϕ-1+α), ±2, ±(2ϕ-1+ϕα))
(±(ϕ+1+α), ±2(ϕ-1), ±(1+ϕα))
where ϕ = (1+)/2 is the golden ratio and α = .
See also
List of uniform polyhedra
Small retrosnub icosicosidodecahedron
External links
Uniform polyhedra
|
https://en.wikipedia.org/wiki/Snub%20dodecadodecahedron
|
In geometry, the snub dodecadodecahedron is a nonconvex uniform polyhedron, indexed as . It has 84 faces (60 triangles, 12 pentagons, and 12 pentagrams), 150 edges, and 60 vertices. It is given a Schläfli symbol as a snub great dodecahedron.
Cartesian coordinates
Cartesian coordinates for the vertices of a snub dodecadodecahedron are all the even permutations of
(±2α, ±2, ±2β),
(±(α+β/τ+τ), ±(-ατ+β+1/τ), ±(α/τ+βτ-1)),
(±(-α/τ+βτ+1), ±(-α+β/τ-τ), ±(ατ+β-1/τ)),
(±(-α/τ+βτ-1), ±(α-β/τ-τ), ±(ατ+β+1/τ)) and
(±(α+β/τ-τ), ±(ατ-β+1/τ), ±(α/τ+βτ+1)),
with an even number of plus signs, where
β = (α2/τ+τ)/(ατ−1/τ),
where τ = (1+)/2 is the golden mean and
α is the positive real root of τα4−α3+2α2−α−1/τ, or approximately 0.7964421.
Taking the odd permutations of the above coordinates with an odd number of plus signs gives another form, the enantiomorph of the other one.
Related polyhedra
Medial pentagonal hexecontahedron
The medial pentagonal hexecontahedron is a nonconvex isohedral polyhedron. It is the dual of the snub dodecadodecahedron. It has 60 intersecting irregular pentagonal faces.
See also
List of uniform polyhedra
Inverted snub dodecadodecahedron
References
External links
Uniform polyhedra
|
https://en.wikipedia.org/wiki/Ditrigonal%20dodecadodecahedron
|
In geometry, the ditrigonal dodecadodecahedron (or ditrigonary dodecadodecahedron) is a nonconvex uniform polyhedron, indexed as U41. It has 24 faces (12 pentagons and 12 pentagrams), 60 edges, and 20 vertices. It has extended Schläfli symbol b{5,}, as a blended great dodecahedron, and Coxeter diagram . It has 4 Schwarz triangle equivalent constructions, for example Wythoff symbol 3 | 5, and Coxeter diagram .
Related polyhedra
Its convex hull is a regular dodecahedron. It additionally shares its edge arrangement with the small ditrigonal icosidodecahedron (having the pentagrammic faces in common), the great ditrigonal icosidodecahedron (having the pentagonal faces in common), and the regular compound of five cubes.
Furthermore, it may be viewed as a facetted dodecahedron: the pentagrammic faces are inscribed in the dodecahedron's pentagons. Its dual, the medial triambic icosahedron, is a stellation of the icosahedron.
It is topologically equivalent to a quotient space of the hyperbolic order-6 pentagonal tiling, by distorting the pentagrams back into regular pentagons. As such, it is a regular polyhedron of index two:
See also
List of uniform polyhedra
References
External links
Uniform polyhedra
|
https://en.wikipedia.org/wiki/Great%20dodecahemidodecahedron
|
In geometry, the great dodecahemidodecahedron is a nonconvex uniform polyhedron, indexed as U70. It has 18 faces (12 pentagrams and 6 decagrams), 60 edges, and 30 vertices. Its vertex figure is a crossed quadrilateral.
Aside from the regular small stellated dodecahedron {5/2,5} and great stellated dodecahedron {5/2,3}, it is the only nonconvex uniform polyhedron whose faces are all non-convex regular polygons (star polygons), namely the star polygons {5/2} and {10/3}.
It is a hemipolyhedron with 6 decagrammic faces passing through the model center.
Related polyhedra
Its convex hull is the icosidodecahedron. It also shares its edge arrangement with the great icosidodecahedron (having the pentagrammic faces in common) and the great icosihemidodecahedron (having the decagrammic faces in common).
Gallery
See also
List of uniform polyhedra
References
External links
Uniform polyhedra and duals
Uniform polyhedra
|
https://en.wikipedia.org/wiki/Small%20dodecahemicosahedron
|
In geometry, the small dodecahemicosahedron (or great dodecahemiicosahedron) is a nonconvex uniform polyhedron, indexed as U62. It has 22 faces (12 pentagrams and 10 hexagons), 60 edges, and 30 vertices. Its vertex figure is a crossed quadrilateral.
It is a hemipolyhedron with ten hexagonal faces passing through the model center.
Related polyhedra
Its convex hull is the icosidodecahedron. It also shares its edge arrangement with the dodecadodecahedron (having the pentagrammic faces in common), and with the great dodecahemicosahedron (having the hexagonal faces in common).
Gallery
See also
List of uniform polyhedra
References
External links
Uniform polyhedra and duals
Uniform polyhedra
|
https://en.wikipedia.org/wiki/Great%20dodecahemicosahedron
|
In geometry, the great dodecahemicosahedron (or great dodecahemiicosahedron) is a nonconvex uniform polyhedron, indexed as U65. It has 22 faces (12 pentagons and 10 hexagons), 60 edges, and 30 vertices. Its vertex figure is a crossed quadrilateral.
It is a hemipolyhedron with ten hexagonal faces passing through the model center.
Related polyhedra
Its convex hull is the icosidodecahedron. It also shares its edge arrangement with the dodecadodecahedron (having the pentagonal faces in common), and with the small dodecahemicosahedron (having the hexagonal faces in common).
Great dodecahemicosacron
The great dodecahemicosacron is the dual of the great dodecahemicosahedron, and is one of nine dual hemipolyhedra. It appears visually indistinct from the small dodecahemicosacron.
Since the hemipolyhedra have faces passing through the center, the dual figures have corresponding vertices at infinity; properly, on the real projective plane at infinity. In Magnus Wenninger's Dual Models, they are represented with intersecting prisms, each extending in both directions to the same vertex at infinity, in order to maintain symmetry. In practice, the model prisms are cut off at a certain point that is convenient for the maker. Wenninger suggested these figures are members of a new class of stellation figures, called stellation to infinity. However, he also suggested that strictly speaking, they are not polyhedra because their construction does not conform to the usual definitions.
The great dodecahemicosahedron can be seen as having ten vertices at infinity.
See also
List of uniform polyhedra
Hemi-icosahedron - The ten vertices at infinity correspond directionally to the 10 vertices of this abstract polyhedron.
References
(Page 101, Duals of the (nine) hemipolyhedra)
External links
Uniform polyhedra and duals
Uniform polyhedra
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.