Proof Of Theorem Research Articles

On the deformation theory of pseudo-group structure

We study the deformation theory of pseudo-group structure, and the purpose of this paper is to give a new simple proof of the existence theorem of pseudo-group structure under certain assumptions.

Unifying some classical results on Artinian rings and modules

In this note, we introduce a very crude but natural notion of measure on the class of left R-modules over a ring R. We use this notion to give short proofs of some classical theorems on (left) Artinian rings and modules, due to Akizuki, Anderson, Hopkins, and Levitzki, as well as of some new results.

Investigating insight and rigour as separate constructs in mathematical proof

ABSTRACT We investigate undergraduate mathematics students' conceptions of rigour and insight. We conducted comparative judgement experiments in which students were asked to judge different proofs of the same theorem with five separate criteria: rigour, insight, understanding, simplicity and assessment marks. We predicted, and our experiment found, that rigour is a reliable construct. We predicted that insight is also a reliable construct but asking students to judge on the basis of “which proof gives you more insight into why a theorem is true” did not result in a reliable judging scale. Our analysis suggests two distinct dimensions: rigour and marks contribute to one factor whereas simplicity and personal understanding relate to a second factor. We suggest reasons why insight was related almost equally to both factors. First, students may not have developed an aesthetic sense in which they appreciate insight in ways that are regularly discussed by mathematics educators. Second, insight may not be a coherent and well-defined construct after all.

Three-dimensional quasi-umbilical locally strongly convex homogeneous affine hypersurfaces

The study of strongly convex, locally homogeneous hypersurfaces in the affine space ℝn+1 has a rich history. In dimension 2, these hypersurfaces were first examined in Guggenheimer's 1977 book “Differential Geometry” [4], and the classification was completed by Nomizu and Sasaki in their 1991 paper “A new model of unimodular-affinely homogeneous surfaces” [6]. Apart from the work of Magid and Vrancken in their 1995 paper [5] and Ooguri’s 2013 paper [9], most studies have focused on locally strongly convex affine hypersurfaces. These are characterized by an induced affine metric that is positive definite, and have been extensively studied in Sasaki’s 1980 paper “Hyperbolic affine hyperspheres” [10]. When the hypersurface is an affine sphere, where the shape operator has all eigenvalues equal to zero, Dillen and Vrancken provided a complete classification in a series of papers. In dimension 3, their 1993 paper “The classification of 3-dimensional locally strongly convex homogeneous affine hypersurfaces” [3] offers a comprehensive classification for any dimensions of eigenvalues of the shape operator. In dimension 4, the quasi-umbilical case was explored by Dillen and Vrancken in their 1994 paper “Quasi-umbilical, locally strongly convex homogeneous affine hypersurfaces” [2]. The case where the affine shape operator has two distinct eigenvalues, each with a multiplicity of two, was investigated by Chikh-Salah and Vrancken in their 2017 paper “Four-dimensional locally strongly convex homogeneous affine hypersurfaces” [1]. In this paper, we provide a different proof of the final theorem in Dillen and Vrancken’s [3], utilizing Mathematica software to calculate all Christoffel symbol functions.

Note on Intuitionistic Fuzzy Metric-like Spaces with Application in Image Processing

Recently, the fixed-point theorem for fuzzy contractive mappings has been investigated within the framework of intuitionistic fuzzy metric-like spaces. This interesting topic was explored through the utilization of G-Cauchy sequences as defined by Grabiec. The aim of this study is to enhance the aforementioned results in a few aspects. Initially, the proof of the fixed-point theorem is simplified and condensed, allowing for potential generalization to papers focusing on similar fixed-point analyses. Furthermore, instead of G-Cauchy sequences, the classical Cauchy sequences proposed by George and Veeramani are examined, incorporating an additional condition on the fuzzy metric. Within this context, a solution to an old unresolved question posed by Gregory and Sapena is provided. The findings are reinforced by relevant examples. Finally, the introduced fuzzy metrics are applied to the field of image processing.

Proof exploration using dynamic geometry systems with integrated automated deduction capabilities

Due to its formal, logical and spatial properties, geometry is well suited for the exploration of new knowledge, new properties that arise from different constructions and new insights into already known constructions, to the exploration of proofs of new conjectures, to the exploration of new proofs of known theorems. With a Dynamic Geometry System (DGS) we get an environment where new conjectures can be easily formulated and visually verified (but not proven). With a Geometry Automated–Theorem–Prover (GATP) we get an environment where conjectures/theorems can be formally proved. We explore some environments where DGS and GATP are combined, analysing their possibilities for the exploration of new conjectures and proofs, having in mind their use in a learning situation.

Signature-based standard basis algorithm under the framework of GVW algorithm

The GVW algorithm, one of the most important so-called signature-based algorithms, is designed to eliminate a large number of useless polynomial reductions from Buchberger's algorithm. The cover theorem serves as the theoretical foundation of the GVW algorithm, and up to now, it applies only to a certain class of monomial orders, namely global orders and a special class of local orders. In this paper we extend this theorem to any semigroup order, which can be either global, local or even mixed. Building upon the pioneering idea of the Mora normal form algorithm, we propose a more comprehensive and general proof for the cover theorem while bypassing the need to choose a minimal element from an infinite set of monomials in all the existing proofs. Therefore, the algorithm for signature-based standard bases is presented for any semigroup order under the framework of the GVW algorithm, and an example is given to provide an illustration of the algorithm.

Intrinsic Geometries of the Simple Group of Order 168

It is well known that, up to isomorphism, there is a unique simple group of order 168. The projective special linear groups PSL(2, 7) and PSL(3, 2) are concrete examples of such groups, and are therefore isomorphic. This isomorphism is not obvious, but it has been clarified from several viewpoints. Here we use purely group-theoretic methods—the Sylow theorems, Poincaré’s theorem, and the orbit-stabilizer theorem—to show that any simple group of order 168 has a natural action on the projective line over F 7 , so it is isomorphic to PSL(2, 7). We discuss an analogous action on the projective plane over F 2 described by Smith and Tabachnikova. Both geometries are constructed using conjugacy classes of maximal subgroups. This gives another proof of the uniqueness theorem and may dispel some of the air of mystery that often surrounds the isomorphism between PSL(2, 7) and PSL(3, 2).

Multi-level nonstandard analysis and the axiom of choice

Model-theoretic frameworks for Nonstandard Analysis depend on the existence of nonprincipal ultrafilters, a strong form of the Axiom of Choice (AC). Hrbacek and Katz, APAL Volume 72 formulate axiomatic nonstandard set theories SPOT and SCOT that are conservative extensions of respectively ZF and ZF + ADC (the Axiom of Dependent Choice), and in which a significant part of Nonstandard Analysis can be developed.The present paper extends these theories to theories with many levels of standardness, called respectively SPOTS and SCOTS. It shows that Jin's recent nonstandard proof of Szemeredi's Theorem can be carried out in SPOTS and that SCOTS is a conservative extension of ZF + ADC.

Generalized kernel distance covariance in high dimensions: non-null CLTs and power universality

Abstract Distance covariance is a popular dependence measure for two random vectors $X$ and $Y$ of possibly different dimensions and types. Recent years have witnessed concentrated efforts in the literature to understand the distributional properties of the sample distance covariance in a high-dimensional setting, with an exclusive emphasis on the null case that $X$ and $Y$ are independent. This paper derives the first non-null central limit theorem for the sample distance covariance, and the more general sample (Hilbert–Schmidt) kernel distance covariance in high dimensions, in the distributional class of $(X,Y)$ with a separable covariance structure. The new non-null central limit theorem yields an asymptotically exact first-order power formula for the widely used generalized kernel distance correlation test of independence between $X$ and $Y$. The power formula in particular unveils an interesting universality phenomenon: the power of the generalized kernel distance correlation test is completely determined by $n\cdot \operatorname{dCor}^{2}(X,Y)/\sqrt{2}$ in the high-dimensional limit, regardless of a wide range of choices of the kernels and bandwidth parameters. Furthermore, this separation rate is also shown to be optimal in a minimax sense. The key step in the proof of the non-null central limit theorem is a precise expansion of the mean and variance of the sample distance covariance in high dimensions, which shows, among other things, that the non-null Gaussian approximation of the sample distance covariance involves a rather subtle interplay between the dimension-to-sample ratio and the dependence between $X$ and $Y$.

Embedding of Neutrosophic Graphs on Topological Surfaces

Background: A planar graph (PG) is a graph with no intersecting edges. Particular to both crisp and neutrosophic graphs (NG) is the planar graph, in contrast to crisp planar graphs. NPGs allow for the intersection of neutrosophic edge NEs, since the value of planarity in these graphs is the degree of planarity of the intersected NEs. The NPGs are often represented on a flat surface. Materials and Methods: This study discusses how to embed NGs on surfaces such as spheres and m-toruses by defining the degree of intersection of the neutrosophic edges of NGs with finding the faces on the given graph structures using Euler's theorems. Here, the proofs of Euler's theorems help us find, given the total NFV of G, the interval containing that value. Result: As result of this work obtained that for any two isomorphic planer graphs, they have the same planarity value. For any neutrosophic planer graph with f = (1,1,1) can be embedded in the plane if it can be embedded in the sphere and according to NPGs, for planar and spherical surfaces, equivalent theorems to Euler's formula are proved and shown. Conclusion: It concludes that by using neutrosophic sets and crisp graphs to construct neutrosophic graphs with the benefit of Euler’s theorem, it can provide the concept of embedding neutrosophic graphs in different topological surfaces such as a plane, sphere, and m-torus.

A short elementary proof of Beben and Theriault's theorem on homotopy fibers

Beben and Theriault proved a theorem on the homotopy fiber of an extension of a map with respect to a cone attachment, which has produced several applications. We give a short and elementary proof of this theorem.

Revealing the Correlation Between Lyapunov Exponent and Modulus of an n-Dimensional Nondegenerate Hyperchaotic Map

For their good randomness and long iteration periods, chaotic maps have been widely used in cryptography. Recently, we have revealed the correlation between Lyapunov exponent and sequence randomness of multidimensional chaotic maps based on modular operation. Since the modular operation can realize the boundedness of chaotic state points, it is important to further reveal the deterministic correlation between Lyapunov exponent and modulus. First, we constructed an [Formula: see text]-dimensional nondegenerate hyperchaotic map model with the desired Lyapunov exponents. Then, we gave the existence and uniqueness proof of quadrature rectangle decomposition theorem and revealed the correlation between Lyapunov exponent and modulus. The novelty lies in that (1) in order to realize the irreversibility of the iterative processes of chaotic maps, we constructed a chaotic map based on modular exponentiation, and its inverse function is the discrete logarithm problem; and (2) we reveal for the first time the correlation between Lyapunov exponent and modulus, and give the lower bound of the modulus of the nondegenerate chaotic map. In addition, to verify the effectiveness of the scheme, we constructed four-dimensional and five-dimensional chaotic maps, respectively, and analyzed their dynamical behaviors, and the results revealed that there exist linear or nonlinear correlation between Lyapunov exponent and modulus.

New characterization of plurisubharmonic functions and positivity of direct image sheaves

abstract: We discover a new characterization of plurisubharmonic functions in terms of $L^p$ extension from one point and Griffiths positivity of holomorphic vector bundles with singular Finsler metrics in terms of $L^p$ extensions. As applications, we give a stronger result or new proof of some well-known theorems on the Griffiths positivity of the holomorphic vector bundles and their direct image sheaves associated to certain holomorphic fibrations.

A useful formula for periodic Jacobi matrices on trees

We introduce a function of the density of states for periodic Jacobi matrices on trees and prove a useful formula for it in terms of entries of the resolvent of the matrix and its "half-tree" restrictions. This formula is closely related to the one-dimensional Thouless formula and associates a natural phase with points in the bands. This allows streamlined proofs of the gap labeling and Aomoto index theorems. We give a complete proof of gap labeling and sketch the proof of the Aomoto index theorem. We also prove a version of this formula for the Anderson model on trees.

Differentiably simple rings and ring extensions defined by p-basis

We review the concept of differentiably simple ring and we give a new proof of Harper's Theorem on the characterization of Noetherian differentiably simple rings in positive characteristic. We then study flat families of differentiably simple rings, or equivalently, finite flat extensions of rings which locally admit p-basis. These extensions are called Galois extensions of exponent one. For such an extension A⊂C, we introduce an A-scheme, called the Yuan scheme, which parametrizes subextensions A⊂B⊂C such that B⊂C is Galois of a fixed rank. So, roughly, the Yuan scheme can be thought of as a kind of Grassmannian of Galois subextensions. We finally prove that the Yuan scheme is smooth and compute the dimension of the fibers.

Erratum to "The multiple-slit version of Loewner's differential equation and pointwise Hölder continuity of driving functions"

The proof of Theorem 1.2 of [S. Schleissinger, Ann. Acad. Sci. Fenn. Math. 37:1, 2012, 191–201] contains a gap and one implication in Theorem 1.3 of the same paper is wrong.

Uniform convergence of the one-dimensional cubic Schrödinger equation

In this article, we investigate the uniform convergence problem of the one-dimensional cubic Schrödinger equation. By using the Kato smoothing estimate, the maximal function estimate and the dyadic mixed Lebesgue spaces, we establish the uniform convergence of the one-dimensional cubic Schrödinger equation in H s ( R ) ( s > 1 6 ) which is an alternative proof of Theorem 1.1 ( n = 1 , p = 3 ) of Compaan et al. [Pointwise convergence of the Schrödinger flow. Int Math Res Not. 2021;1:596–647].

Unifying the Three Algebraic Approaches to the CSP via Minimal Taylor Algebras

This paper focuses on the algebraic theory underlying the study of the complexity and the algorithms for the Constraint Satisfaction Problem (CSP). We unify, simplify, and extend parts of the three approaches that have been developed to study the CSP over finite templates -- absorption theory that was used to characterize CSPs solvable by local consistency methods (JACM'14), and Bulatov's and Zhuk's theories that were used for two independent proofs of the CSP Dichotomy Theorem (FOCS'17, JACM'20). As the first contribution we present an elementary theorem about primitive positive definability and use it to obtain the starting points of Bulatov's and Zhuk's proofs as corollaries. As the second contribution we propose and initiate a systematic study of minimal Taylor algebras. This class of algebras is broad enough that it suffices to verify the CSP Dichotomy Theorem on this class only, but still is unusually well behaved. In particular, many concepts from the three approaches coincide in this class, which is in striking contrast with the general setting. We believe that the theory initiated in this paper will eventually result in a simple and more natural proof of the Dichotomy Theorem that employs a simpler and more efficient algorithm, and will help in attacking complexity questions in other CSP-related problems.

The K-functional and reiteration theorems for Left and Right spaces, Part II

We consider the K-interpolation methods involving slowly varying functions. Let A‾θ,⁎L and A‾θ,⁎R (0≤θ≤1) be the so-called L or R limiting interpolation spaces which arise naturally in reiteration formulae for the limiting cases. We characterize the interpolation spaces (X,Y)θ,r,a (0≤θ≤1), where X=A‾θ0,⁎L or X=A‾θ0,⁎R and Y=A‾θ1,⁎L or Y=A‾θ1,⁎R, provided that 0≤θ0<θ1≤1 and at least one of the parameters θ0 or θ1 has the limiting value θ0=0 or θ1=1. This supplements the first part of the paper, where one of the operands is standard interpolation spaces involving slowly varying functions. The proofs of most reiteration theorems are based on Holmstedt-type formulae. Applications to grand and small Lorentz spaces are given.