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1,100 |
Simple proofing of Jordan's theorem
|
math.GM
|
Here is present short proofing of Jordan's theorem about dividing of flat on
two disjoint subsets by one closed curve.
|
math
|
1,101 |
A Concise and Direct Proof of "Fermat's Last Theorem"
|
math.GM
|
The recently developed proof of Fermat's Last Theorem is very lengthy and
difficult, so much so as to be beyond all but a small body of specialists.
While certainly of value in the developments that resulted, that proof could
not be, nor was offered as being, possibly the proof Fermat had in mind. The
present proof being brief, direct and concise is a candidate for being what
Fermat had in mind. It is also completely accessible to any one trained in
common algebra. That critical suggestions offered by significant mathematics
authorities have been unable to invalidate this concise and direct proof would
tend to be major confirmation that: The proof stands, valid and not validly
challenged.
|
math
|
1,102 |
The Theory of Ultralogics Part I
|
math.GM
|
As of the date of this version, this monograph (parts I and II) contains all
of the known technical results relative to the Robinson-styled nonstandard
modeling of natural languages and certain associated linguistic processes such
as deduction via consequence operators among other concepts. These results have
direct application to the construction of the GGU-model, the GID-model, the
GD-model, the MA-model and also apply to philosophy, psychology, properton
theory and other aspects of the Nonstandard Physical World (NSP-world).
|
math
|
1,103 |
The Theory of Ultralogics Part II
|
math.GM
|
As of this date of this version, this monograph (part I and II) contains most
of the technical results relative to the Robinson-styled nonstandard modeling
of natural languages and certain associated linguistic processes such as
deduction via consequence operators among other concepts. These results have
direct application to the construction of the GGU-model, GID-model, D-world
model, MA-model and also apply to philosophy, psychology, properton theory and
other aspects of the Nonstandard Physical World (NSP-world).
|
math
|
1,104 |
Riemann Hypothesis
|
math.GM
|
Through an equivalent condition on the Farey series set forth by Franel and
Landau, we prove Riemann Hypothesis for the Riemann zeta-function and the
Dirichlet L-function.
|
math
|
1,105 |
Sieve Method and Landau Problem
|
math.GM
|
We solve Landau's four unattackable problems, including Goldbach Conjecture
and Twin Prime Conjecture through sieve method.
|
math
|
1,106 |
Tactical games & behavioral self-organization
|
math.GM
|
The interactive game theoretical approach to tactics and behavioral
self-organization is developed. Though it uses the interactive game theoretical
formalization of dialogues as psycholinguistic phenomena, the crucial role is
played by the essentially new concept of a tactical game. Applications to the
perception processes and related subjects (memory, recollection, image
understanding, imagination) are discussed together with relations to the
computer vision and pattern recognition (the dynamical formation of patterns
and perception models during perception as a result of its self-organization)
and computer games (modelling of the tactical behavior and self-organization,
tactical RPG and elaboration of new tactical game techniques). The appendix is
devoted to the operative computer games and the user programming of operative
units in a multi-user online operative computer game.
|
math
|
1,107 |
A counting method for finding rational approximates to arbitrary order roots of integers
|
math.GM
|
It is shown that for finding rational approximates to m'th root of any
integer to any accuracy one only needs the ability to count and to distinguish
between m different classes of objects. To every integer N can be associated a
'replacement rule' that generates a word W* from another word W consisting of
symbols belonging to a finite 'alphabet' of size m. This rule applied
iteratively on almost any initial word W0, yields a sequence of words {Wi} such
that the relative frequency of different symbols in the word Wi approaches
powers of the m'th root of N as i tends to infinity
|
math
|
1,108 |
A `replacement sequence' method for finding the largest real root of an integer monic polynomial
|
math.GM
|
To every integer monic polynomial of degree m can be associated a
`replacement rule' that generates a word W* from another word W consisting of
symbols belonging to a finite `alphabet' of size 2m. This rule applied
iteratively on almost any initial word Wo, yields a sequence of words {Wi}.
From acount of different symbols in the word Wi, one can obtain a rational
approximate to the largest real root of the polynomial.
|
math
|
1,109 |
Simple Divisibility Rules for the 1st 1000 Prime Numbers
|
math.GM
|
Simple divisibility rules are given for the 1st 1000 prime numbers.
|
math
|
1,110 |
Tactics, dialectics, representation theory
|
math.GM
|
This article is devoted to the tactical game theoretical interpretation of
dialectics. Dialectical games are considered as abstractly as well as models of
the internal dialogue and reflection. The models related to the representation
theory (representative dynamics) are specially investigated in detail, they
correlate with the hypothesis on the dialectical features of human thinking in
general and mathematical thought (the constructing of a solution of
mathematical problem) in particular.
|
math
|
1,111 |
Integer Sequences associated with Integer Monic Polynomial
|
math.GM
|
To every integer monic polynomial of degree m can be associated m integer
sequences having interesting properties to the roots of the polynomial. These
sequences can be used to find the real roots of any integer monic polynomial by
using recursion relation involving integers only. This method is faster than
the conventional methods using floating point arithmetic.
|
math
|
1,112 |
Symbolic computation of the roots of any polynomial with integer coefficients
|
math.GM
|
The roots of any polynomial of degree m with integer coefficients, can be
computed by manipulation of sequences made from 2m distinct symbols and
counting the different symbols in the sequences. This method requires only
'primitive' operations like replacement of sequences and counting of symbols.
No calculations using 'advanced' operations like multiplication, division,
logarithms etc. are needed. The method can be implemented as a geometric
construction of roots of polynomials to arbitrary accuracy using only a
straight edge, a compass, and pencils of 2m different colors. In particular,
the ancient problem of the "doubling of cube" is soluble asymptotically by the
above-mentioned construction. This method, by which a cube can be doubled,
albeit, in infinite steps, is probably the closest to the original problem of
construction using only a straight edge and compass in a finite number of
steps.
Moreover, to every polynomial of degree m over the field of rationals, can be
associated an m-term recurrence relation for generating integer sequences. A
set of m such sequences, which together exhibit interesting properties related
to the roots of the polynomial, can be obtained if the m initial terms of each
of these m sequences is chosen in a special way using a matrix associated with
the polynomial. Only two of these integer sequences need to be computed to
obtain the real root having the largest absolute value. Since this method
involves only integers, it is faster than the conventional methods using
floating-point arithmetic.
|
math
|
1,113 |
Roots of any Polynomial with Complex Integer Coefficients using Replacement Sequences, Ruler and Compass
|
math.GM
|
The roots of any polynomial of degree m with complex integer coefficients can
be computed by manipulation of sequences made from distinct symbols and
counting the different symbols in the sequences. This method requires only
primitive operations like replacement of sequences and counting of symbols. No
calculations using advanced operations like multiplication, division,
logarithms etc. are needed. The method can be implemented as a geometric
construction using only a ruler and a compass.
|
math
|
1,114 |
An Abel ordinary differential equation class generalizing known integrable classes
|
math.GM
|
We present a multi-parameter non-constant-invariant class of Abel ordinary
differential equations with the following remarkable features. This one class
is shown to unify, that is, contain as particular cases, all the integrable
classes presented by Abel, Liouville and Appell, as well as all those shown in
Kamke's book and various other references. In addition, the class being
presented includes other new and fully integrable subclasses, as well as the
most general parameterized class of which we know whose members can
systematically be mapped into Riccati equations. Finally, many integrable
members of this class can be systematically mapped into an integrable member of
a different class. We thus find new integrable classes from previously known
ones.
|
math
|
1,115 |
Random triangle problem: geometrical approach
|
math.GM
|
Classical problem of random triangle in square is solved by simple and
transparent geometrical method.
|
math
|
1,116 |
Bifurcating Continued Fractions
|
math.GM
|
The notion of 'bifurcating continued fractions' is introduced. Two coupled
sequences of non-negative integers are obtained from an ordered pair of
positive real numbers in a manner that generalizes the notion of continued
fractions. These sequences enable simple representations of roots of cubic
equations. In particular, remarkably simple and elegant 'bifurcating continued
fraction' representations of Tribonacci and Moore numbers, the cubic variations
of the 'golden mean', are obtained. This is further generalized to associate m
non-negative integer sequences with a set of m given real numbers so as to
provide simple 'bifurcating continued fraction' representation of roots of
polynomial equations of degree m+1.
|
math
|
1,117 |
Experimental detection of interactive phenomena and their analysis
|
math.GM
|
The article is devoted to mathematical methods of experimental detection of
interactive phenomena in complex systems and their analysis.
|
math
|
1,118 |
Matrix exponentials
|
math.GM
|
We give a formula for matrix exponentials and partial fraction
decompositions.
|
math
|
1,119 |
Four multiplicative cohomology theorems
|
math.GM
|
I try to find natural statement and proof of the de Rham Theorem and of other
cohomology theorems.
|
math
|
1,120 |
On the Complete Solution to the Most General Fifth Degree Polynomial
|
math.GM
|
The motivation behind this note, is due to the non success in finding the
complete solution to the General Quintic Equation. The hope was to have a
solution with all the parameters precisely calculated in a straight forward
manner. This paper gives the closed form solution for the five roots of the
General Quintic Equation. They can be generated on Maple V, or on the new
version Maple VI. On the new version of maple, Maple VI, it may be possible to
insert all the substitutions calculated in this paper, into one another, and
construct one large equation for the Tschirnhausian Transformation. The
solution also uses the Generalized Hypergeometric Function which Maple V can
calculate, robustly.
|
math
|
1,121 |
A lemma on the minimal surfaces
|
math.GM
|
Stated lemma contains the assertions about isomorphism of exact m-forms and
exterior differentials of regular m-maps, of linearly harmonic m-forms and
exterior differentials of regular harmonic m-maps, of global minimal
(n-m)-surfaces and level (n-m)-surfaces of regular minimal m-maps. It hold in
n-dimensional Euclidean space.
|
math
|
1,122 |
A Proof of "Goldbach's Conjecture"
|
math.GM
|
"Goldbach's Conjecture" proven by analysis of how all combinations of the odd
primes, summed in pairs, generates all of the even numbers.
|
math
|
1,123 |
On the dynamic flows
|
math.GM
|
It is investigated a possibility of physical interpretation of vector fields
(dynamic flows) in Euclidean spaces of higher dimension. There are analyzed the
methods of measurements of dynamic flows, the characteristics of dynamic flow
and the connection between its differential and integral characteristics. It is
obtained the criterion of local minimality of (n-1)-surfaces that is not
connected with interior geometry of surface. It is analyzed some analogy
between harmonicity of dynamic flows and dynamic principle of nature.
|
math
|
1,124 |
On the quantum model of gravitational electrodynamics
|
math.GM
|
It is shown that application of dynamic flows concept in 4-dimensional
Euclidean space makes possible to form Minkowski space and to formulate the
generalized variational problem of electrodynamics and gravi- dynamics. It is
shown that 1-dimensional (cylindrical) factorization of 4-dimensional Euclidean
space provides a quantization of ths model.
|
math
|
1,125 |
Goldbach's Rule
|
math.GM
|
Goldbach`s Conjecture, "every even number greater than 2 can be expressed as
the sum of two primes" is renamed Goldbach`s Rule for it can not be otherwise.
The conjecture is proven by showing that the existence of prime pairs adding to
any even number greater than 2 is a natural by-product of the existence of the
prime sequence less than that even number. First it is shown that the remainder
of cancellations process which identifies primes less than an even number also
remainders prime pairs adding to that even number as a natural part of the
process. Then a minimum limit for the remaindered number of prime pairs adding
to an even number is expressed in terms of that even number and shown to exist
for every even number greater than 2. Furthermore, the reasonings and
formulations used in the proof are demonstrated to hold against observations.
|
math
|
1,126 |
On the generation of linear groups by combinatoric groups
|
math.GM
|
It is described the group of arrowy permutations (that is extension of
symmetric group) and the consequent process of generation of GL(n) and some its
subgroups by this combinatoric group and its subgroups.
|
math
|
1,127 |
On the rigid algebraic structure of Euclidean spaces
|
math.GM
|
It is shown that the groups of automorphisms of Euclidean spaces are
isomorphic to the groups of topologic automorphisms of respectively factored
arithmetic spaces. In particular, the geometry of Euclidean n-space with
positive signature is associated with factorization of n-dimensional arithmetic
space into n-dimensional sphere.
|
math
|
1,128 |
On some mathematical construction of many-dimensional cosmos
|
math.GM
|
It is shown that classical Clifford algebras are group algebras of cyclic
subgroups of arrowy rermutations. It is established that Euclidean 3-space,
Pauli and Dirac algebras and groups of global guage transformations are
corollary from the geometry of 8-dimensional vacuum and 9-dimensional cosmos.
|
math
|
1,129 |
On the interior structure of exterior algebra
|
math.GM
|
It is constructed the functor from category of product linear space to
category of skew-symmetric tensor space. It is defined and described the bound
bundle as analog of a symplex and as basis element of new constructive homology
theory.
|
math
|
1,130 |
New mathematical methods for psychophysical filtering of experimental data and their processing
|
math.GM
|
The article is devoted to new mathematical methods for psychophysical
filtering of experimental data and their processing.
|
math
|
1,131 |
Antinomies of Mathematical Reason: The Inconsistency of PM Arithmetic and Related Systems
|
math.GM
|
We give a proof of the inconsistency of PM arithmetic, classical set theory
and related systems, incidentally exposing an error in Goedel's own proof of
Goedel's Theorems. The inconsistency proof, that formulae of the form R and ~R
occur as theorems in the PM-isomorphic system P, proceeds from a reflexive
substitution instance of the first axiom of the propositional calculus (axiom
II.1 of P). Goedel's formalism is used throughout.
|
math
|
1,132 |
Bifurcating Continued Fractions II
|
math.GM
|
In an earlier paper we introduced the notion of 'bifurcating continued
fractions' in a heuristic manner. In this paper a formal theory is developed
for the 'bifurcating continued fractions'.
|
math
|
1,133 |
Why a Conjecture of Poincare Doesn't Work
|
math.GM
|
Poincare had conjectured that the fact that closed loops could be shrunk to
points on a surface topologically equivalent to the surface of a sphere can be
generalised to three (and more) dimensions. After nearly a century the
conjecture has remained unproven. We given arguments below to show that the
conjecture doesn't work in three dimensions.
|
math
|
1,134 |
Introductory Calculus from the Viewpoint of Non-Standard Analysis - Derivative of Sine and Cosine
|
math.GM
|
This article exemplifies a novel approach to the teaching of introductory
differential calculus using the modern notion of ``infinitesimal'' as opposed
to the traditional approach using the notion of ``limit''. I illustrate the
power of the new approach with a discussion of the derivatives of the sine and
cosine functions.
|
math
|
1,135 |
An Introduction to the Neutrosophic Probability Applied in Quantum Physics
|
math.GM
|
In this paper one generalizes the classical probability and imprecise
probability to the notion of "neutrosophic probability" in order to be able to
model Heisenberg's Uncertainty Principle of a particle's behavior,
Schr"dinger's Cat Theory, and the state of bosons which do not obey Pauli's
Exclusion Principle (in quantum physics). Neutrosophic probability is close
related to neutrosophic logic and neutrosophic set, and etymologically derived
from "neutrosophy".
|
math
|
1,136 |
On Rugina's System of Thought
|
math.GM
|
In this article one investigates Rugina's Orientation Table and one gives
particular examples for several of its seven models. Leon Walras's Economics of
Stable Equilibrium and Keynes's Economics of Disequilibrium are combined in
Rugina's Orientation Table in systems which are s% stable and 100-s% unstable,
where s may be 100, 95, 65, 50, 35, 5, and 0. The Classical Logic and Modern
Logic are united in Rugina's Integrated Logic, and then generalized in the
Neutrosophic Logic.
|
math
|
1,137 |
Neutrosophy
|
math.GM
|
Neutrosophy is a new branch of philosophy which studies the origin, nature,
and scope of neutralities, as well as their interactions with different
ideational spectra.
|
math
|
1,138 |
Special Algebraic Structures
|
math.GM
|
New notions are introduced in algebra in order to better study the
congruences in number theory. For example, the <special semigroups> makes an
important such contribution.
|
math
|
1,139 |
Mixed Non-Euclidean Geometries
|
math.GM
|
The goal of this paper is to experiment new math concepts and theories,
especially if they run counter to the classical ones. To prove that
contradiction is not a catastrophe, and to learn to handle it in an (un)usual
way. To transform the apparently unscientific ideas into scientific ones, and
to develop their study (The Theory of Imperfections). And finally, to
interconnect opposite (and not only) human fields of knowledge into
as-heterogeneous-as-possible other fields.
|
math
|
1,140 |
Linguistic Paradoxes and Tautologies
|
math.GM
|
Classes of linguistic paradoxes and linguistic tautologies are introduced
with examples and explanations. They are part of the author's work on the
Paradoxist Philosophy based on mathematical logic.
The general cases exposed below are modeled on the English language structure
in a rigid way. In order to find nice particular examples of such paradoxes and
tautologies one grammatically adjusts the sentences.
|
math
|
1,141 |
A Set of Sequences in Number Theory
|
math.GM
|
74 new integer sequences are introduced in number theory, and for each of
them is given a characterization, followed by open problems. each one a general
question: how many primes each sequence has.
|
math
|
1,142 |
Numerology
|
math.GM
|
One presents many Concatenated and Operation Sequences, P-Q Relationships,
Digital Sequences, Magic Squares, Prime Conjectures, k-Divisibility and Strong
Divisibility Sequences, Geometric Conjectures, Proposed problems.
|
math
|
1,143 |
Funny Problems!
|
math.GM
|
Thirty original and collected problems, puzzles, and paradoxes in mathematics
and physics are explained in this paper, taught by the author to the elementary
and high school teachers at the University of New Mexico - Gallup in 1997-8 and
afterwards. They have more an educational interest, because make the students
think different! For each "solution" a funny logic is invented in order to give
the problem a sense.
|
math
|
1,144 |
Integer Algorithms to Solver Diophantine Linear Equations and Systems
|
math.GM
|
The present work includes some of the author's original researches on integer
solutions of Diophantine liner equations and systems. The notion of "general
integer solution" of a Diophantine linear equation with two unknowns is
extended to Diophantine linear equations with $n$ unknowns and then to
Diophantine linear systems. The proprieties of the general integer solution are
determined (both for a Diophantine linear equation and for a Diophantine linear
system). Seven original integer algorithms (two for Diophantine linear
equations, and five for Diophantine linear systems) are exposed. The algorithms
are strictly proved and an example for each of them is given. These algorithms
can be easily implemented on the computer.
|
math
|
1,145 |
Another Set of Sequences, Sub-Sequences, and Sequences of Sequences
|
math.GM
|
In this paper 101 new integer sequences, sub-sequences, and sequences of
sequences, together with related unsolved problems and conjectures, are
presented. Also, definitions, examples, solved or open questions, and
references for each sequence are given.
|
math
|
1,146 |
Thirty-Six Unsolved Problems in Number Theory
|
math.GM
|
Partially or totally unsolved questions in number theory and geometry
especially, such as coloration problems, elementary geometric conjectures,
partitions, generalized periods of a number, length of a generalized period,
arithmetic and geometric progressions are exposed.
|
math
|
1,147 |
G Add-On, Digital, Sieve, General Periodical, and Non-Arithmetic Sequences
|
math.GM
|
In this paper a small survey is presented on fourteen sequences, such as: G
Add-on Sequences, Sieve Sequences, Digital Sequences, Non-Arithmetic
Progressions, recreational sequences (Lucky
Method/Operation/Algorithm/Differentiation/Integration etc.), General
Periodical Sequences, and numerical functions.
|
math
|
1,148 |
Considerations on New Functions in Number Theory
|
math.GM
|
In this paper a small survey is presented on eighteen new functions and four
new sequences, such as: Inferior/Superior f-Part, Fractional f-Part,
Complementary function with respect with another function, S-Multiplicative,
Primitive Function, Double Factorial Function, S-Prime and S-Coprime Functions,
Smallest Power Function.
|
math
|
1,149 |
A Generalized Numeration Base
|
math.GM
|
A Generalized Numeration Base is defined in this paper, and then particular
cases are presented, such as Prime Base, Square Base, m-Power Base, Factorial
Base, and operations in these bases. These bases are important for partitions
of integers into primes, squares, cubes, generally into m-powers, also into
factorials, and into any strictly increasing sequence.
|
math
|
1,150 |
The 42 Assessors and the Box-Kites they fly: Diagonal Axis-Pair Systems of Zero-Divisors in the Sedenions' 16 Dimensions
|
math.GM
|
G. Moreno's abstract depiction of the Sedenions' normed zero-divisors, as
homomorphic to the exceptional Lie group G2, is fleshed out by exploring
further structures the A-D-E approach of Lie algebraic taxonomy keeps hidden. A
breakdown of table equivalence among the half a trillion multiplication schemes
the Sedenions allow is found; the 168 elements of PSL(2,7), defining the finite
projective triangle on which the Octonions' 480 equivalent multiplication
tables are frequently deployed, are shown to give the exact count of primitive
unit zero-divisors in the Sedenions. (Composite zero-divisors, comprising all
points of certain hyperplanes of up to 4 dimensions, are also determined.) The
168 are arranged in point-set quartets along the 42 Assessors (pairs of
diagonals in planes spanned by pure imaginaries, each of which zero-divides
only one such diagonal of any partner Assessor). These quartets are
multiplicatively organized in systems of mutually zero-dividing trios of
Assessors, a D4-suggestive 28 in number, obeying the 6-cycle crossover logic of
trefoils or triple zigzags. 3 trefoils and 1 zigzag determine an octahedral
vertex structure we call a box-kite -- seven of which serve to partition
Sedenion space. By sequential execution of proof-driven production rules, a
complete interconnected box-kite system, or Seinfeld production (German for
field of being; American for 1990's television's Show About Nothing), can be
unfolded from an arbitrary Octonion and any (save for two) of the Sedenions.
Indications for extending the results to higher dimensions and different
dynamic contexts are given in the final pages.
|
math
|
1,151 |
Correspondence principle for idempotent calculus and some computer applications
|
math.GM
|
This paper is devoted to heuristic aspects of the so-called idempotent
calculus. There is a correspondence between important, useful and interesting
constructions and results over the field of real (or complex) numbers and
similar constructions and results over idempotent semirings in the spirit of N.
Bohr's correspondence principle in Quantum Mechanics.
Some problems nonlinear in the traditional sense (for example, the Bellman
equation and its generalizations) turn out to be linear over a suitable
semiring; this linearity considerably simplifies the explicit construction of
solutions.
The theory is well advanced and includes, in particular, new integration
theory, new linear algebra, spectral theory and functional analysis. It has a
wide range of applications.
Besides a survey of the subject, in this paper the correspondence principle
is used to develop an approach to object-oriented software and hardware design
for algorithms of idempotent calculus.
|
math
|
1,152 |
Multi-dimensional Meta-analysis for Assessment of Relationships between Asthma Rates and Particulate Air Pollution
|
math.GM
|
Multi-dimensional meta-analysis (MDMA) is an innovative technique for
investigating complex scientific problems influenced by "external" factors,
such as social, medical, economic, political or climatic trends. MDMA extends
traditional meta-analysis by identifying significant data from diverse and
independent disciplines ("orthogonal dimensions") and incorporating truth
tables and non-parametric analysis methods in the interpretation protocol. In
this paper, we outline the methodology of MDMA. We then demonstrates how to
apply the method to a specific problem: the relationship between asthma and air
particulates. The conclusions from the example show that the further reduction
of atmospheric particulate levels is not necessarily the answer to the
increasing asthma incidence. This example also demonstrates the strength of
this method of analysis for complex problems.
|
math
|
1,153 |
A Unifying Field in Logics: Neutrosophic Logic, Neutrosophic Set, Neutrosophic Probability and Statistics (fourth edition)
|
math.GM
|
In this book one makes an introduction to non-standard analysis in the first
part, needed to the next four chapters in order to study the neutrosophics:
1. Neutrosophy - a new branch of philosophy.
2. Neutrosophic Logic - a unifying field in logics.
3. Neutrosophic Set - a unifying field in sets.
4. Neutrosophic Probability - a generalization of classical and imprecise
probabilities - and Neutrosophic Statistics.
|
math
|
1,154 |
Riemann hypothesis and super-conformal invariance
|
math.GM
|
A strategy for proving (not a proof of, as was the first over-optimistic
belief) the Riemann hypothesis is suggested. The vanishing of Riemann Zeta
reduces to an orthogonality condition for the eigenfunctions of a non-Hermitian
operator D^+ having the zeros of Riemann Zeta as its eigenvalues. The
construction of D^+ is inspired by the conviction that Riemann Zeta is
associated with a physical system allowing superconformal transformations as
its symmetries and second quantization in terms of the representations of
superconformal algebra. The eigenfunctions of D^+ are analogous to the so
called coherent states and in general not orthogonal to each other. The states
orthogonal to a vacuum state (having a negative norm squared) correspond to the
zeros of Riemann Zeta. The physical states having a positive norm squared
correspond to the zeros of Riemann Zeta at the critical line. Riemann
hypothesis follows by reductio ad absurdum from the hypothesis that ordinary
superconformal algebra acts as gauge symmetries for all coherent states
orthogonal to the vacuum state, including also the non-physical coherent states
that might exist off from the critical line.
|
math
|
1,155 |
Triplets and Symmetries of Arithmetic mod p^k
|
math.GM
|
The finite ring Z_k = Z(+,.) mod p^k of residue arithmetic with odd prime
power modulus is analysed. The cyclic group of units G_k in Z_k(.) has order
(p-1)p^{k-1}, implying product structure G_k = A_k B_k. Here core A_k of order
p-1 is an extension for k >1 of Fermat's Small Theorem (FST*), where n^p == n
(mod p^k) for each core residue, while extension subgroup B_k has order
p^{k-1}. It is shown that each subgroup S >1 of core A_k has zero sum, and that
p+1 generates subgroup B_k of all n == 1 (mod p) in G_k. The p-th power
residues n^p mod p^k in G_k form an order |G_k|/p subgroup F_k, with
|F_k|/|A_k| = p^{k-2}, so F_k properly contains core A_k for k >2. By quadratic
analysis (mod p^3) rather than linear analysis (mod p^2, re Hensel's lemma
[5]), the additive structure of subgroups G_k and F_k is derived. ... Successor
function S(n)=n+1 combines with the two arithmetic symmetries -n (complement)
and 1/n (inverse) to yield the "triplet structure" of G_k : three inverse pairs
{n_i, 1/(n_i)} with (n_i)+1 = - 1/n_{i+1} (mod p^k), with indices mod 3, and
product n_0.n_1.n_2 = 1 mod p^k. In case n_0 = n_1 = n_2 = n this reduces to
the cubic root solution n+1 = -(1/n) = -(n^2) (mod p^k, p=1 mod 6). The
property "EDS" of exponent p distributing over a sum of core residues: (x+y)^p
== x+y == x^p + y^p (mod p^k), is employed to derive the known FLT inequality
for integers. In other words, to any FLT(mod p^k) equivalence for k digits
correspond p-th power integers of pk digits, and the (p-1)k "carries" make the
difference, representing the sum of mixed-terms in the binomial expansion.
|
math
|
1,156 |
On primitive roots of unity, divisors of p+/-1, Wieferich primes, and quadratic analysis mod p^3
|
math.GM
|
Primitive roots of 1 mod p^k (k>2 and odd prime p) are sought, in cyclic
units group G_k = A_k B_k mod p^k, coprime to p, of order (p-1)p^{k-1}. 'Core'
subgroup A_k has order p-1 independent of k, and p+1 generates 'extension'
subgroup B_k of all p^{k-1} residues 1 mod p. Divisors r,t of powerful
generator p-1=rs=tu of \pm B_k mod p^k, and of p+1, are investigated as
primitive root candidates. Fermat's Small Theorem: x^{p-1} \e 1 mod p for 0<x<p
is, with recursion r^{n+1}-t^{n+1}=(r^n-t^n)(r+t)-(r^{n-1}-t^{n-1})rt (divisors
r != t) extended to: all divisors r | p \pm 1 have distinct r^n mod p^3 (0<n
\leq p). So for proper divisors: r^{p-1} != 1 mod p^3, a necessary (not
sufficient) condition for a primitive root mod p^{k>2}. And for prime p: 2^p
!=2 and 3^p != 3 (mod p^3). Re: Wieferich primes [4] and FLT case_1. Conj: at
least one divisor of p \pm 1 is a semi primitive root of 1 mod p^k. -- (paper
withdrawn, re thm2.2)
|
math
|
1,157 |
Powersums representing residues mod p^k, from Fermat to Waring
|
math.GM
|
The ring Z_k(+,.) mod p^k with prime power modulus (prime p>2) is analysed.
Its cyclic group G_k of units has order (p-1)p^{k-1}, and all p-th power n^p
residues form a subgroup F_k with |F_k|=|G_k|/p. The subgroup of order p-1, the
core A_k of G_k, extends Fermat's Small Theorem (FST) to mod p^{k>1},
consisting of p-1 residues with n^p = n mod p^k. The concept of "carry", e.g.
n' in FST extension n^{p-1} = n'p+1 mod p^2, is crucial in expanding residue
arithmetic to integers, and to allow analysis of divisors of 0 mod p^k. . . . .
For large enough k \geq K_p (critical precison K_p < p depends on p), all
nonzero pairsums of core residues are shown to be distinct, upto commutation.
The known FLT case_1 is related to this, and the set F_k + F_k mod p^k of p-th
power pairsums is shown to cover half of units group G_k. -- Yielding main
result: each residue mod p^k is the sum of at most four p-th power residues.
Moreover, some results on the generative power (mod p^{k>2}) of divisors of
p^2-1 are derived. -- [Publ.: "Computers and Mathematics with Applications",
V39 N7-8 (Apr.2000) p253-261]
|
math
|
1,158 |
Additive structure of Z(.) mod m_k (squarefree) and Goldbach's Conjecture
|
math.GM
|
The product m_k of the first k primes (2..p_k) has neighbours m_k +/- 1 with
all prime divisors beyond p_k, implying there are infinitely many primes
[Euclid]. All primes between p_k and m_k are in the group G_1 of units in
semigroup Z_{m_k}(.) of mutiplication mod m_k. Due to the squarefree modulus
Z_{m_k} is a disjoint union of 2^k groups, with as many idempotents - one per
divisor of m_k, which form a Boolean lattice BL. The generators of Z_{m_k} and
the additive properties of its lattice are studied. It is shown that each
complementary pair in BL adds to 1 mod m_k and each even idempotent e in BL has
successor e+1 in G_1. It follows that G_1+G_1 \equiv E, the set of even
residues in Z_{m_k}, so each even residue is the sum of two roots of unity,
proving "Goldbach for Residues" mod m_k ("GR"). . . . Induction on k by
extending residues mod m_k with "carry" a < p_{k+1} of weight m_k, yields a
prime sieve for integers. Failure of Goldbach's Conjecture ("GC") for some 2n
contradicts GR(k) for some k. By Bertrand's Postulate (on prime i<p<2i for each
i>1) successive 2n are in overlapping intervals, while the smallest composite
unit in G_1 mod m_k is p_{k+1}^2, yielding "GC": Each 2n > 4 is the sum of two
odd primes.
|
math
|
1,159 |
Finite Semigroups of Constant Rank, and the five Basic State Machine types
|
math.GM
|
Constant Rank (CR) state machines play an important role in the general
structure theory of Finite State Machines. A machine is of constant rank if
each input and input-sequence maps the state set onto the same number of next
states. CR-machines are analysed via their sequential closure (semigroup),
which is a simple semigroup, thus: a semi- direct product (L \times R)*G of a
left- and a right-copy semigroup, and a group. . . . So in general a CR-machine
is a composition of: a branch-, a reset- and a permutation machine, which are
three of the five basic types of state machines, to be derived.
|
math
|
1,160 |
On the cardinality of the set of the real numbers
|
math.GM
|
It is shown that any denumerable list L to which Cantor's diagonal method was
applied is incomplete. However, this doesn't allow us to affirm that the
cardinality of the real numbers of the interval [0, 1] is greater than the
cardinality of the finite natural numbers. Paper withdrawn (its essential part
is included in the version 3 of math.GM/0108119).
|
math
|
1,161 |
On the Goedel's formula
|
math.GM
|
This article examines the formula G (of Goedel). We demonstrated that the
Goedel's number of the formula G is not a finite number if (i) G is
comprehended as a self-referential statement or (ii) there is an infinite set S
of well-formed formulae such that the elements of S are theorems or
antitheorems in T.
|
math
|
1,162 |
Stochastic processes on non-Archimedean spaces. I. Stochastic processes on Banach spaces
|
math.GM
|
Non-Archimedean analogs of Markov quasimeasures and stochastic processes are
investigated. Thery are used for the development of stochastic antiderivations.
The non-Archimedean analog of the It$\hat o$ formula is proved.
|
math
|
1,163 |
Stochastic processes on non-Archimedean spaces. II. Stochastic antiderivational equations
|
math.GM
|
Stochastic antiderivational equations on Banach spaces over local
non-Archimedean fields are investigated. Theorems about existence and
uniqiuness of the solutions are proved under definite conditions. In particular
Wiener processes are considered in relation with the non-Archimedean analog of
the Gaussian measure.
|
math
|
1,164 |
Symmetric Logic Synthesis with Phase Assignment
|
math.GM
|
Decomposition of any Boolean Function BF_n of n binary inputs into an optimal
inverter coupled network of Symmetric Boolean functions SF_k (k \leq n) is
described. Each SF component is implemented by Threshold Logic Cells, forming a
complete and compact T-Cell Library. Optimal phase assignment of input
polarities maximizes local symmetries. The "rank spectrum" is a new BF_n
description independent of input ordering, obtained by mapping its minterms
onto an othogonal n \times n grid of (transistor-) switched conductive paths,
minimizing crossings in the silicon plane. Using this ortho-grid structure for
the layout of SF_k cells, without mapping to T-cells, yields better area
efficiency, exploiting the maximal logic path sharing in SF's. Results obtained
with an optimization tool "Ortolog" based on these concepts, for very fast
O(n^2) detecting and enhancing local symmetries of a BF_n, are reported.
Relaxing symmetric- to planar- Boolean functions is sketched, to improve low-
symmetry BF decomposition.
|
math
|
1,165 |
On the set of natural numbers
|
math.GM
|
This paper was withdrawn by the authors.
|
math
|
1,166 |
Generalized Steiner's Problem and its Solution with the Concepts in Field Thoery
|
math.GM
|
We generalized the Steiner's shortest line problem and found its connection
with the concepts in classical field theory. We solved the generalized
Steiner's problem by introducing a conservative potential and a dissipative
force in the field and gave a computing method by using a testing point and a
corresponding iterative curve.
|
math
|
1,167 |
Le cryptosysteme non-commutatif
|
math.GM
|
It is showed a new cryptosystem based on non-commutativ calculations of
matrices, more specially nilpotent matrices. The cryptosystem seems powerful to
restsist against usual attacks.
|
math
|
1,168 |
A new Binary Number Code and a Multiplier, based on 3 as semi-primitive root of 1 mod 2^k
|
math.GM
|
The powers of 3 generate half of the odd residues mod 2^k (k>2), and a sign
change yields the other half. In other words: 3 is a semi-primitive root of 1
mod 2^k (k>2). Hence each k-bit residue is n = +/- 3^i.2^j mod 2^k, with unique
non-neg exponent pair: i<2^{k-2} and j<k. -- A new "dual base logarithmic"
binary number code (bases 2 and 3) employs this property. This (binary)
log-code [s,i,j] - where s is the corresponding sign, simplifies binary
multiplication by translating it to addition of the exponents of 2 and 3, and
XOR of the signs involved. -- Patent US-5923888 (13jul99)
|
math
|
1,169 |
The Origin of a Metric
|
math.GM
|
In the context of earlier work, we investigate the emergence of a "distance"
in the physical world. For this we consider a Cantor ternary like process, but
much more general: properties like perfectness and disconnectedness are not
invoked, but instead we deal with Borel sets. An interesting case from a
physical point of view is considered: when the process is truncated.
|
math
|
1,170 |
Stochastic processes on non-Archimedean spaces. III. Stochastic processes on totally disconnected topological groups
|
math.GM
|
Stochastic processes on totally disconnected topological groups are
investigated. In particular they are considered for diffeomorphism groups and
loop groups of manifolds on non-Archimedean Banach spaces. Theorems about a
quasi-invariance and a pseudo-differentiability of transition measures are
proved. Transition measures are used for the construction of strongly
continuous representations including irreducible of these groups. In addition
stochastic processes on general Banach-Lie groups, loop monoids, loop spaces
and path spaces of manifolds on Banach spaces over non-Archimedean local fields
also are investigated.
|
math
|
1,171 |
Quasi-invariant and pseudo-differentiable measures on a non-Archimedean Banach space.I. Real-valued measures
|
math.GM
|
Quasi-invariant and pseudo-differentiable measures on a Banach space $X$ over
a non-Archimedean locally compact infinite field with a non-trivial valuation
are defined and constructed. Measures are considered with values in $\bf R$.
Theorems and criteria are formulated and proved about quasi-invariance and
pseudo-differentiability of measures relative to linear and non-linear
operators on $X$. Characteristic functionals of measures are studied. Moreover,
the non-Archimedean analogs of the Bochner-Kolmogorov and Minlos-Sazonov
theorems are investigated. Infinite products of measures also are considered.
Convergence of quasi-invariant and pseudo-differentiable measures in the
corresponding spaces of measures is investigated.
|
math
|
1,172 |
Quasi-invariant and pseudo-differentiable measures on a non-Archimedean Banach space. II. Measures with values in non-Archimedean fields
|
math.GM
|
Quasi-invariant and pseudo-differentiable measures on a Banach space $X$ over
a non-Archimedean locally compact infinite field with a non-trivial valuation
are defined and constructed. Measures are considered with values in
non-Archimedean fields, for example, the field $\bf Q_p$ of $p$- adic numbers.
Theorems and criteria are formulated and proved about quasi-invariance and
pseudo-differentiability of measures relative to linear and non-linear
operators on $X$. Characteristic functionals of measures are studied. Moreover,
the non-Archimedean analogs of the Bochner-Kolmogorov and Minlos-Sazonov
theorems are investigated. Infinite products of measures are considered and the
analog of the Kakutani theorem is proved. Convergence of quasi-invariant and
pseudo-differentiable measures in the corresponding spaces of measures is
investigated.
|
math
|
1,173 |
The cardinality of the set of real numbers
|
math.GM
|
A proof that the set of real numbers is denumerable is given.
|
math
|
1,174 |
The Essence of Intuitive Set Theory
|
math.GM
|
Intuitive Set Theory (IST) is defined as the theory we get, when we add Axiom
of Monotonicity and Axiom of Fusion to Zermelo-Fraenkel set theory. In IST,
Continuum Hypothesis is a theorem, Axiom of Choice is a theorem, Skolem paradox
does not appear, nonLebesgue measurable sets are not possible, and the unit
interval splits into a set of infinitesimals.
|
math
|
1,175 |
White Hole, Black Whole, and The Book
|
math.GM
|
Intellectual space is defined as the set of all proofs of mathematical logic,
contained in The Book conceived by Erdos. Physical and intellectual spaces are
visualized, making use of concepts from intuitive set theory.
|
math
|
1,176 |
Testing the Existence of a Supporting Plane
|
math.GM
|
We present an algorithm testing wheather, for given four vectors in R^3,
there is a plane through the origin such that all four vectors fall into the
same open halfspace.
|
math
|
1,177 |
A further step in the proof of Riemann hypothesis
|
math.GM
|
Paper has been withdrawn due to an error in the basic argument that the
states corresponding to the zeros of Riemann Zeta with Re[s]<1/2 allow a
Fourier expansion in the basis provided by the states having Re[s]>= 1/2.
|
math
|
1,178 |
A Computational Algorithm for /pi(N)
|
math.GM
|
An algorithm for computing /pi(N) is presented.It is shown that using a
symmetry of natural numbers we can easily compute /pi(N).This method relies on
the fact that counting the number of odd composites not exceeding N suffices to
calculate /pi(N).
|
math
|
1,179 |
Introductory Topics in Distributions over Binary Test Functions
|
math.GM
|
We note with B2 the Boole algebra with two elements. We define for the R->B2
functions the limits, the derivatives, the differentiability, the test
functions, the integrals. We also define the distributions over the space of
these test functions, the regular and the singular distributions, the support
sets of the distributions. We also define for the RxR->{0,1} functions the test
functions and the distributions over them. The direct product of the
distributions is presented, as well as the convolution algebras of
distributions. Generalizations of the binary test functions and of the
distributions over them are given.
|
math
|
1,180 |
Introductory Topics in Binary Set Functions
|
math.GM
|
Let X be a non-empty set and U a ring of subsets of X. The countable additive
functions U->{0,1} are called measures. The paper gives some definitions
(derivable measures, the Lebesgue-Stieltjes measures) and properties of these
functions, its purpose being that of reconstruction of the measure theory
within this frame, by analogy with the real measure theory. We mention the
special case of the Riemann integrals.
|
math
|
1,181 |
Comments to Neutrosophy
|
math.GM
|
Any system based on axioms is incomplete because the axioms cannot be proven
from the system, just believed. But one system can be less-incomplete than
other. Neutrosophy is less-incomplete than many other systems because it
contains them. But this does not mean that it is finished, and it can always be
improved. The comments presented here are an attempt to make Neutrosophy even
less-incomplete. I argue that less-incomplete ideas are more useful, since we
cannot perceive truth or falsity or indeterminacy independently of a context,
and are therefore relative. Absolute being and relative being are defined. Also
the "silly theorem problem" is posed, and its partial solution described. The
issues arising from the incompleteness of our contexts are presented. We also
note the relativity and dependance of logic to a context. We propose
"metacontextuality" as a paradigm for containing as many contexts as we can, in
order to be less-incomplete and discuss some possible consequences.
|
math
|
1,182 |
A Strategy for Proving Riemann Hypothesis
|
math.GM
|
A strategy for proving Riemann hypothesis is suggested. The vanishing of the
Rieman Zeta reduces to an orthogonality condition for the eigenfunctions of a
non-Hermitian operator $D^+$ having the zeros of Riemann Zeta as its
eigenvalues. The construction of $D^+$ is inspired by the conviction that
Riemann Zeta is associated with a physical system allowing conformal
transformations as its symmetries. The eigenfunctions of $D^+$ are analogous to
the so called coherent states and in general not orthogonal to each other. The
states orthogonal to a vacuum state (which has a negative norm squared)
correspond to the zeros of the Riemann Zeta. The induced metric in the space
${\cal{V}}$ of states which correspond to the zeros of the Riemann Zeta at the
critical line $Re[s]=1/2$ is hermitian and both hermiticity and positive
definiteness properties imply Riemann hypothesis. Conformal invariance in the
sense of gauge invariance allows only the states belonging to ${\cal{V}}$.
Riemann hypothesis follows also from a restricted form of a dynamical conformal
invariance in ${\cal{V}}$ and one can reduce the proof to a standard analytic
argument used in Lie group theory.
|
math
|
1,183 |
On power sets
|
math.GM
|
This work presents theorems which state (i) Z is a proper subset for any
bijection f between A and Z, where Z is contained in P(A), A is a non-finite
set and |Z|=|A|, and (ii) being Z a proper subset of P(A) nothing affirms or
denies that |P(A)|>|A|. Russell's paradox is examined and it is shown that the
set of all the ordinary sets does not exist. A mistake in Cantor's proof on
cardinality of power sets is shown.
|
math
|
1,184 |
Can our number system be improved?
|
math.GM
|
Our number system is a magnificent tool. But it is far from perfect. Can it
be improved? In this paper some possibilities are discussed, including the use
of a different base or directed (negative as well as positive) numerals. We
also put forward some suggestions for further research.
|
math
|
1,185 |
Intentionally and Unintentionally. On Both, A and Non-A, in Neutrosophy
|
math.GM
|
The paper presents a fresh new start on the neutrality of neutrosophy in that
"both A and Non-A" as an alternative to describe Neuter-A in that we
conceptualize things in both intentional and unintentional background. This
unity of opposites constitutes both objective world and subjective world. The
whole induction of such argument is based on the intensive study on Buddhism
and Daoism including I-ching. In addition, a framework of contradiction
oriented learning philosophy inspired from the Later Trigrams of King Wen in
I-ching is meanwhile presented. It is shown that although A and Non-A are
logically inconsistent, but they are philosophically consistent in the sense
that Non-A can be the unintentionally instead of negation that leads to
confusion. It is also shown that Buddhism and Daoism play an important role in
neutrosophy, and should be extended in the way of neutrosophy to all sciences
according to the original intention of neutrosophy.
|
math
|
1,186 |
Beyond Goedel : Simply consistent constructive systems of first order Peano's Arithmetic that do not yield undecidable propositions by Goedel's reasoning
|
math.GM
|
In this paper, we argue that formal systems of first order Arithmetic that
admit Goedelian undecidable propositions validly are abnormally
non-constructive.
We argue that, in such systems, the strong representation of primitive
recursive predicates admits abnormally non-constructive, Platonistic, elements
into the formal system that are not reflected in the predicates which they are
intended to formalise.
We argue that the source of such abnormal Platonistic elements in these
systems is the non-constructive Generalisation rule of inference of first order
logic.
We argue that, in most simply consistent systems that faithfully formalise
intuitive Arithmetic, we cannot infer from Goedel's reasoning the Platonistic
existence of abnormally non-constructive propositions that are formally
undecidable, but true under every interpretation.
We define a constructive formal system of Peano's Arithmetic, omega2-PA,
whose axioms are identical to the axioms of standard Peano's Arithmetic PA, but
lead to significantly different logical consequences.
We thus argue that the formal undecidability of true Arithmetical
propositions is a characteristic not of relations that are Platonistically
inherent in any Arithmetic of the natural numbers, but of the particular
formalisation chosen to represent them.
|
math
|
1,187 |
Rapid growth sequences
|
math.GM
|
Studying Fermat sequence we can simply find infinitely many other rapidly
growing sequences of similar properties. On the other hand this approach allows
us simple construction of such sequences.
|
math
|
1,188 |
Infinite and natural numbers
|
math.GM
|
The infinite numbers of the set M of finite and infinite natural numbers are
defined starting from the sequence 0\Phi, where 0 is the first natural number,
\Phi is a succession of symbols S and xS is the successor of the natural number
x. The concept of limit of the natural number n, when n tends to infinite, is
examined. Definitions and theorems about operations with elements of M,
equivalence and equality of natural numbers, distance between elements of M and
the order of the elements are presented.
|
math
|
1,189 |
An Extension to Fermat's Factorisation and a simple primality test
|
math.GM
|
An extension to the factorisation principle as suggested by Fermat is
presented.We start from a symmetry of natural numbers and obtain the
factorisation principle therefrom.Later it is extended further to test the
primality of any natural number and finally used to factorise any given number.
|
math
|
1,190 |
The Banach-Tarski paradox or what mathematics and religion have in common
|
math.GM
|
We give a popular account of the Banach-Tarski paradox and its connections
with the axiom of choice.
|
math
|
1,191 |
Fuzziness and Funds Allocation in Portfolio Optimization
|
math.GM
|
Each individual investor is different, with different financial goals,
different levels of risk tolerance and different personal preferences. From the
point of view of investment management, these characteristics are often defined
as objectives and constraints. Objectives can be the type of return being
sought, while constraints include factors such as time horizon, how liquid the
investor is, any personal tax situation and how risk is handled. It's really a
balancing act between risk and return with each investor having unique
requirements, as well as a unique financial outlook - essentially a constrained
utility maximization objective. To analyze how well a customer fits into a
particular investor class, one investment house has even designed a structured
questionnaire with about two-dozen questions that each has to be answered with
values from 1 to 5. The questions range from personal background (age, marital
state, number of children, job type, education type, etc.) to what the customer
expects from an investment (capital protection, tax shelter, liquid assets,
etc.). A fuzzy logic system has been designed for the evaluation of the answers
to the above questions. We have investigated the notion of fuzziness with
respect to funds allocation.
|
math
|
1,192 |
A Family of Estimators of Population Mean Using Multiauxiliary Information in Presence of Measurement Errors
|
math.GM
|
This paper proposes a family of estimators of population mean using
information on several auxiliary variables and analyzes its properties in the
presence of measurement errors.
|
math
|
1,193 |
A Generator System of Invariant differential forms
|
math.GM
|
We obtain a generator system of the algebra of $\mathrm{GL}(V)$-invariant
differential forms on $\mathrm{End}_{\bf k} (V)$. The proof uses the Weyl-Schur
reciprocity.
|
math
|
1,194 |
Prime number logarithmic geometry on the plane
|
math.GM
|
We found a regularity of the behavior of primes that allows to represent both
prime and natural numbers as infinite matrices with a common formation rule of
their rows. This regularity determines a new class of infinite cyclic groups
that permit the proposition a plane--spiral geometric concept of the
arithmetic.
|
math
|
1,195 |
Oscillating Population Models
|
math.GM
|
Oscillating population model realistic situations in different contexts.We
examine this situation with reasonable mathematical models and come to
interesting conclusions,such as for example,that the population at most points
of the cycle approximately equals half the maximum attainable population.
|
math
|
1,196 |
Reviewing Goedel's and Rosser's meta-reasoning of undecidability
|
math.GM
|
I review the classical conclusions drawn from Goedel's meta-reasoning
establishing an undecidable proposition GUS in standard PA. I argue that, for
any given set of numerical values of its free variables, every recursive
arithmetical relation can be expressed in PA by different, but formally
equivalent, propositions. This asymmetry yields alternative Representation and
Self-reference meta-Lemmas. I argue that Goedel's meta-reasoning can thus be
expressed avoiding any appeal to the truth of propositions in the standard
interpretation IA of PA. This now establishes GUS as decidable, and PA as
omega-inconsistent. I argue further that Rosser's extension of Goedel's
meta-reasoning involves an invalid deduction.
|
math
|
1,197 |
On the polycirculant conjecture
|
math.GM
|
In the paper the foundation of the $k$-orbit theory is developed. The theory
opens a new simple way to the investigation of groups and multidimensional
symmetries.
The relations between combinatorial symmetry properties of a $k$-orbit and
its automorphism group are found. It is found the local property of a
$k$-orbit. The difference between 2-closed group and $m$-closed group for $m>2$
is discovered. It is explained the specific property of Petersen graph
automorphism group $n$-orbit. It is shown that any non-trivial primitive group
contains a transitive imprimitive subgroup and as a result it is proved that
the automorphism group of a vertex transitive graph (2-closed group) contains a
regular element (polycirculant conjecture).
Using methods of the $k$-orbit theory, it is considered different
possibilities of permutation representation of a finite group and shown that
the most informative, relative to describing of the structure of a finite
group, is the permutation representation of the lowest degree. Using this
representation it is obtained a simple proof of the W. Feit, J.G. Thompson
theorem: Solvability of groups of odd order. It is described the enough simple
structure of lowest degree representation of finite groups and found a way to
constructing of the simple full invariant of a finite group.
To the end, using methods of $k$-orbit theory, it is obtained one of possible
polynomial solutions of the graph isomorphism problem.
|
math
|
1,198 |
Heuristic algorithm for solving of the graph isomorphism problem
|
math.GM
|
We consider heuristic algorithm for solving graph isomorphism problem. The
algorithm based on a successive splitting of the eigenvalues of the matrices
which are modifications (to positive defined) of graphs' adjacency matrices.
Modification of the algorithm allows to find a solution for Frobenius problem.
Formulation of the Frobenius problem is following one. Given a pair of two
matrices with the same number of rows and columns. We must find out whether one
of the matrix can be acquired from another by permutation of it's rows and
strings or not. For example, solution of Frobenius problem can give to us
efficient way for decrypting of double permutation cyphers problem for high
dimension matrices.
|
math
|
1,199 |
Proof of Goldbach's Conjecture
|
math.GM
|
After certain subsets of Natural numbers called Range and Row are defined, we
assume (1) there is a function that can produce prime numbers and (2) each even
number greater than 2, like A, can be represented as the sum of n prime
numbers. We show this by DC(A)less than or equal to n. Each Row is similar to
each other in properties,(so is each Range). It is proven that in an arbitrary
Row for any even number greater than 2, DC(A)=2, that is to say, each prime
number greater than two is the sum of two prime numbers. So Goldbach's
conjecture is proved.
|
math
|
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