Applications of Wallis Theorem
In this paper we present theorems and applications of Wallis theorem related to trigonometric integrals.
math.GM↗
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A novel operation associated with Gauss' arithmetic-geometric means
The arithmetic mean is the mean for addition and the geometric mean is that for multiplication. Then what kind of binary operation is associated with the arithmetic-geometric mean (AGM) due to C. F. Gauss? If it is possible to construct an arithmetic operation such that AGM is the mean for this operation, it can be regarded as an intermediate operation between addition and multiplication in view of the meaning of AGM. In this paper such an operation is introduced and several of its algebraic properties are proved.
math.NT↗
An Ergodic Result
A rather general ergodic type scheme is presented on arbitrary sets X, as they are generated by arbitrary mappings T : X \longrightarrow X. The structures considered on X are given by suitable subsets of the set of all of its finite partitions. Ergodicity is studied not with respect to subsets of X, but with the {\it inverse limits} of families of finite partitions.
math.GM↗
About Goldbach strong conjecture
In this work we use the number classification in families of the form 6n+1, and 6n+5 with n integer (Such families contain all odd prime numbers greater than 3 and other compound numbers related with primes). We will use this kind of classification in order to attempt an approach to Goldbach strong conjecture. By means of a geometric method of binary bands of numbers we conceive a new form of study of the stated problem.
math.GM↗
An elementary sieve
In this paper we review the properties of families of numbers of the form $6n\pm1$, with $n$ integer (in which there are all prime numbers greater than 3 and other compound numbers with particular properties) to later use them in a new sieve that allows the separation of numbers $n$ that generate primes from those that only generate compounds. In principle, this can be used to find the amount of prime numbers up to a given number $h$; this means, $π(h)$.
math.GM↗
A Fixed Point Conjecture
Inverse limits, unlike direct limits, can in general be void, [1]. The existence of fixed points for arbitrary mappings $T : X \longrightarrow X$ is conjectured to be equivalent with the fact that related direct limits of all finite partitions of X are not void.
math.GM↗
A Problem in Categories
The problem is posed to find out for arbitrary nonvoid sets $X$ which are all the mappings $T : X \longrightarrow X$ that can be defined and each separately identified through means of categories alone. As argued, this problem may have a certain foundational relevance.
math.GM↗
Painlevé scheme
In this note, we review the notion of Painlevé scheme of the sixth Painlevé equation from the viewpoint of accessible singular point and its local index in the Hirzebruch surface of degree two ${Σ_2}$. The key method is Painlevé $α$-method for each accessible singular point. Giving a Painlevé scheme in the differential system satisfying certain conditions, we can recover the Painlevé VI system with the polynomial Hamiltonian. We also consider the case of the Painlevé V,IV and III systems, respectively. Finally, we study non-linear ordinary differential systems in dimension two with only simple accessible singular $(n+2)$-points in the Hirzebruch surface of degree $n$; ${Σ_n}$. This equation has symmetry of symmetric group of degree $n+2$.
math.GM↗
Godel's theorem as a corollary of impossibility of complete axiomatization of geometry
Not any geometry can be axiomatized. The paradoxical Godel's theorem starts from the supposition that any geometry can be axiomatized and goes to the result, that not any geometry can be axiomatized. One considers example of two close geometries (Riemannian geometry and $σ$-Riemannian one), which are constructed by different methods and distinguish in some details. The Riemannian geometry reminds such a geometry, which is only a part of the full geometry. Such a possibility is covered by the Godel's theorem.
math.GM↗
Ring geometries, Two-Weight Codes and Strongly Regular Graphs
It is known that a linear two-weight code $C$ over a finite field $\F_q$ corresponds both to a multiset in a projective space over $\F_q$ that meets every hyperplane in either $a$ or $b$ points for some integers $a<b$, and to a strongly regular graph whose vertices may be identified with the codewords of $C$. Here we extend this classical result to the case of a ring-linear code with exactly two nonzero homogeneous weights and multisets of points in an associated projective ring geometry. We will show that a two-weight code over a finite Frobenius ring gives rise to a strongly regular graph, and we will give some constructions of two-weight codes using ring geometries. These examples all yield infinite families of strongly regular graphs with non-trivial parameters.
math.CO↗
Note on pre-Courant algebroid structures for parabolic geometries
This note aims to demonstrate that every parabolic geometry has a naturally defined per-Courant algebroïd structure. This structure is a Courant algebroïd if and only if the the curvature $κ$ of the Cartan connection vanishes. In all other cases, if the parabolic geometry is regular, there does not exist a natural universal expression for a Courant bracket.
math.DG↗
One page proof of the Riemann hypothesis
We give a short Wiener measure proof of the Riemann hypothesis based on a surprising, unexpected and deep relation between the Riemann zeta $ζ(s)$ and the trivial zeta $ζ_{t}(s):=Im(s)(2Re(s)-1)$.
math.GM↗
The Set of Prime Numbers
In this work we show that the prime distribution is deterministic. Indeed the set of prime numbers P can be expressed in terms of two subsets of N using three specific selection rules, acting on two sets of prime candidates. The prime candidates are obtained in terms of the first perfect number. The asymptotic behaviour is also considered. We obtain for the first time an explicit relation for generating the full set P of prime numbers smaller than n or equal to n.
math.GM↗
A general learning algorithm for functions between metric spaces
In this paper we show how to approximate ("learn") a function f, where X and Y are metric spaces.
math.FA↗
Reference Frame Fields based on Quantum Theory Representations of Real and Complex Numbers
A quantum theory representations of real (R) and complex (C) numbers is given that is based on states of single, finite strings of qukits for any base k > 1. Both unary representations and the possibility that qukits with k a prime number are elementary and the rest composite are discussed. Cauchy sequences of qukit string states are defined from the arithmetic properties. The representations of R and C, as equivalence classes of these sequences, differ from classical kit string state representations in two ways: the freedom of choice of basis states, and the fact that each quantum theory representation is part of a mathematical structure that is itself based on the real and complex numbers. These aspects enable the description of 3 dimensional frame fields labeled by different k values, different basis or gauge choices, and different iteration stages. The reference frames in the field are based on each R and C representation where each frame contains representations of all physical theories as mathematical structures based on the R and C representation. Approaches to integrating this with physics are described. It is observed that R and C values of physical quantities, matrix elements, etc. which are viewed in a frame as elementary and featureless, are seen in a parent frame as equivalence classes of Cauchy sequences of qukit string states.
quant-ph↗
Different representations of Euclidean geometry and their application to the space-time geometry
Three different representation of the proper Euclidean geometry are considered. They differ in the number of basic elements, from which the geometrical objects are constructed. In E-representation there are three basic elements (point, segment, angle) and no additional structures. V-representation contains two basic elements (point, vector) and additional structure: linear vector space. In sigma-representation there is only one basic element and additional structure: world function σ=ρ^{2}/2, where ρis the distance. The concept of distance appears in all representations. However, as a structure, determining the geometry, the distance appears only in the sigma-representation. The sigma-representation is most appropriate for modification of the proper Euclidean geometry. Practically any modification of the proper Euclidean geometry turns it into multivariant geometry, where there are many vectors Q_0Q_1, Q_0Q_1^{\prime},..., which are equal to the vector P_0P_1, but they are not equal between themselves, in general. Concept of multivariance is very important in application to the space-time geometry. The real space-time geometry is multivariant. Multivariance of the space-time geometry is responsible for quantum effects.
math.GM↗
The Infinite in Sciences and Arts
Actual infinity in its various forms is discussed, searched and not found.
math.GM↗
New families of cryptographic systems
A symmetric encryption method based on properties of quasicrystals is proposed. The advantages of the cipher are strict aperiodicity and everywhere discontinuous property as well as the speed of computation, simplicity of implementation and a straightforward possibility of extending the method to encryption of higher dimensional data.
cs.CR↗