Proof Of Theorem Research Articles

A global Morse index theorem and applications to Jacobi fields on CMC surfaces

In this paper, we establish a “global” Morse index theorem. Given a hypersurface [Formula: see text] of constant mean curvature, immersed in [Formula: see text]. Consider a continuous deformation of “generalized” Lipschitz domain [Formula: see text] enlarging in [Formula: see text]. The topological type of [Formula: see text] is permitted to change along [Formula: see text], so that [Formula: see text] has an arbitrary shape which can “reach afar” in [Formula: see text], i.e. cover any preassigned area. The proof of the global Morse index theorem is reduced to the continuity in [Formula: see text] of the Sobolev space [Formula: see text] of variation functions on [Formula: see text], as well as the continuity of eigenvalues of the stability operator. We devise a “detour” strategy by introducing a notion of “set-continuity” of [Formula: see text] in [Formula: see text] to yield the required continuities of [Formula: see text] and of eigenvalues. The global Morse index theorem thus follows and provides a structural theorem of the existence of Jacobi fields on domains in [Formula: see text].

FROM EUCLID TO ARTIFICIAL INTELLIGENCE

We give 19 proofs of the famous Angle Bisector Theorem from Euclid's Elements. The first proof is the Euclid's original proof, the remaining proofs use the methods of Euclidean Geo\-metry, Trigonometry, Analytic Geometry, Complex Numbers, and Gr\"obner Bases. The Gr\"obner Bases proof is in the area of Automatic Proving and Artificial Intelligence, so that the proofs in a way symbolise the development of mathematics from 300 BC (the Euclid's time) to modern days. We discuss what the proofs illustrate and why they are important for Math Education. All the proofs, except the first one, are original.

Twisted conjugacy in residually finite groups of finite Prüfer rank

Abstract Suppose 𝐺 is a residually finite group of finite upper rank admitting an automorphism 𝜑 with finite Reidemeister number R ⁢ ( φ ) R(\varphi) (the number of 𝜑-twisted conjugacy classes). We prove that such a 𝐺 is soluble-by-finite (in other words, any residually finite group of finite upper rank that is not soluble-by-finite has the R ∞ R_{\infty} property). This reduction is the first step in the proof of the second main theorem of the paper: suppose 𝐺 is a residually finite group of finite Prüfer rank and 𝜑 is its automorphism. Then R ⁢ ( φ ) R(\varphi) (if it is finite) is equal to the number of equivalence classes of finite-dimensional irreducible unitary representations of 𝐺, which are fixed points of the dual map φ ̂ : [ ρ ] ↦ [ ρ ∘ φ ] \hat{\varphi}\colon[\rho]\mapsto[\rho\circ\varphi] (i.e. we prove the TBFT𝑓, the finite version of the conjecture about the twisted Burnside–Frobenius theorem, for such groups).

On Rio’s proof of limit theorems for dependent random fields

This paper presents an exposition of Rio’s proof of the strong law of large numbers and extends his method to random fields. In addition to considering the rate of convergence in the Marcinkiewicz–Zygmund strong law of large numbers, we go a step further by establishing (i) the Hsu–Robbins–Erdös–Spitzer–Baum–Katz theorem, (ii) the Feller weak law of large numbers, and (iii) the Pyke–Root theorem on mean convergence for dependent random fields. These results significantly improve several particular cases in the literature. The proof is based on new maximal inequalities that hold for random fields satisfying a very general dependence structure.

A Green’s Function Proof of the Positive Mass Theorem

In this paper, a new proof of the Positive Mass Theorem is established through a newly discovered monotonicity formula, holding along the level sets of the Green’s function of an asymptotically flat 3-manifolds. In the same context and for 1<p<3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1<p<3$$\\end{document}, a Geroch-type calculation is performed along the level sets of p-harmonic functions, leading to a new proof of the Riemannian Penrose Inequality under favourable assumptions. A new characterisation of scalar curvature lower bounds in terms of the monotonicity formulas is also given.

An efficient approach to study multi-polar fuzzy ideals of semirings

The multi polar fuzzy (m-PF) set has an extensive range of implementations in real world problems related to the multi-polar information, multi-index and multi-attributes data. This paper introduces innovative extensions to algebraic structures. We present the definitions and some important results of m-polar fuzzy subsemirings (m-PFSSs), m-polar fuzzy ideals (m-PFIs), m-polar fuzzy generalized bi-ideals (m-PFGBIs), m-polar fuzzy bi-ideals (m-PFBIs) and m-polar fuzzy quasi-ideals (m-PFQIs) in semirings. The main contributions of the paper include the derivation and proof of key theorems that shed light on the algebraic interplay and computational aspects of m-polar fuzzy ideals (m-PFIs), m-polar fuzzy generalized bi-ideals (m-PFGBIs), m-polar fuzzy bi-ideals (m-PFBIs) and m-polar fuzzy quasi-ideals (m-PFQIs) in semirings along with examples. Moreover, this paper deals with several important properties of m-PFIs and characterizes regular and intra-regular semirings by the properties of these ideals.

Positive Mass Theorems for Spin Initial Data Sets With Arbitrary Ends and Dominant Energy Shields

Abstract We prove a positive mass theorem for spin initial data sets $(M,g,k)$ that contain an asymptotically flat end and a shield of dominant energy (a subset of $M$ on which the dominant energy scalar $\mu -|J|$ has a positive lower bound). In a similar vein, we show that for an asymptotically flat end $\mathcal{E}$ that violates the positive mass theorem (i.e., $\textrm{E} &amp;lt; |\textrm{P}|$), there exists a constant $R&amp;gt;0$, depending only on $\mathcal{E}$, such that any initial data set containing $\mathcal{E}$ must violate the hypotheses of Witten’s proof of the positive mass theorem in an $R$-neighborhood of $\mathcal{E}$. This implies the positive mass theorem for spin initial data sets with arbitrary ends, and we also prove a rigidity statement. Our proofs are based on a modification of Witten’s approach to the positive mass theorem involving an additional independent timelike direction in the spinor bundle.

The BPHZ Theorem for Regularity Structures via the Spectral Gap Inequality

We provide a relatively compact proof of the BPHZ theorem for regularity structures of decorated trees in the case where the driving noise satisfies a suitable spectral gap property, as in the Gaussian case. This is inspired by the recent work (Linares et al. in A diagram-free approach to the stochastic estimates in regularity structures, 2021. arXiv:2112.10739) in the multi-index setting, but our proof relies crucially on a novel version of the reconstruction theorem for a space of “pointed Besov modelled distributions”. As a consequence, the analytical core of the proof is quite short and self-contained, which should make it easier to adapt the proof to different contexts (such as the setting of discrete models).

Inverse polynomials of numerical semigroup rings

We study the defining ideal of a numerical semigroup ring k[H] using the inverse polynomial attached to the Artinian ring k[H]/(th). We give a criterion for H to be symmetric or almost symmetric using the annihilator of the inverse system. Also we give characterizations of symmetric numerical semigroups with small multiplicity and give a new proof of Bresinsky's Theorem for symmetric semigroups generated by 4 elements.

A short proof of the elliptical range theorem

A short proof of the elliptical range theorem concerning the numerical range of 2×2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$2\ imes 2$$\\end{document} complex matrices is given.

Internal and Observational Parametricity for Cubical Agda

Two approaches exist to incorporate parametricity into proof assistants based on dependent type theory. On the one hand, parametricity translations conveniently compute parametricity statements and their proofs solely based on individual well-typed polymorphic programs. But they do not offer internal parametricity: formal proofs that any polymorphic program of a certain type satisfies its parametricity statement. On the other hand, internally parametric type theories augment plain type theory with additional primitives out of which internal parametricity can be derived. But those type theories lack mature proof assistant implementations and deriving parametricity in them involves low-level intractable proofs. In this paper, we contribute Agda --bridges: the first practical internally parametric proof assistant. We provide the first mechanized proofs of crucial theorems for internal parametricity, like the relativity theorem. We identify a high-level sufficient condition for proving internal parametricity which we call the structure relatedness principle (SRP) by analogy with the structure identity principle (SIP) of HoTT/UF. We state and prove a general parametricity theorem for types that satisfy the SRP. Our parametricity theorem lets us obtain one-liner proofs of standard internal free theorems. We observe that the SRP is harder to prove than the SIP and provide in Agda --bridges a shallowly embedded type theory to compose types that satisfy the SRP. This type theory is an observational type theory of logical relations and our parametricity theorem ought to be one of its inference rules.

Moments of permutation statistics and central limit theorems

We show that if a permutation statistic can be written as a linear combination of bivincular patterns, then its moments can be expressed as a linear combination of factorials with constant coefficients. This generalizes a result of Zeilberger. We use an approach of Chern, Diaconis, Kane and Rhoades, previously applied on set partitions and matchings. In addition, we give a new proof of the central limit theorem (CLT) for the number of occurrences of classical patterns, which uses a lemma of Burstein and Hästö. We give a simple interpretation of this lemma and an analogous lemma that would imply the CLT for the number of occurrences of any vincular pattern. Furthermore, we obtain explicit formulas for the moments of the descents and the minimal descents statistics. The latter is used to give a new direct proof of the fact that we do not necessarily have asymptotic normality of the number of pattern occurrences in the case of bivincular patterns. Closed forms for some of the higher moments of several popular statistics on permutations are also obtained.

Research on the higher-order logic formalization of fractance element

Fractance element reflects the fractional order behavior of circuits, which can show the characteristics of the actual circuits. Higher-order logic theorem proving is based on the rigorous and correct mathematical theories. It becomes more and more important in the verifications of high-reliability systems. Fractance element is formalized using the proof of higher-order logic theorem in this paper. Firstly, the formalized model of fractional calculus which is based on Caputo definition is established in higher-order logic theorem proof tool. Then some properties of fractional calculus are proved, including the zero order property, the fractional differential of a constant and the consistency of fractional calculus and integer order calculus. Finally, fractance element and fractional differential circuit constituted by fractance element are formally analyzed. These formalizations demonstrate the effectiveness of the formal method in the analysis of fractance element.

[formula omitted]-isometries and their harmonious applications to Hilbert-Schmidt operators

Numerous works have been dedicated to the topic of m-isometries, including [2–6,14,18–20,27,47,48]. In this article, we introduce the concept of (m,NA)-isometry, where A is a non-zero operator and m is a positive integer, as an extension of the m-isometry class created by J. Alger and M. Stankus in the 1980s. We present some algebraic and spectral characteristics of (m,NA)-isometries. Additionally, we investigate the product of an (m,NA)-isometry by an (n,NB)-isometry, which enhances and broadens the previous work of Gu et al. on m-isometries [40]. Finally, we apply our main findings to elementary operators defined on the Hilbert-Schmidt class, which can be identified with a tensor product. This provides a new, less complicated, and non-combinatorial proof of Theorem 2.10 of [30].

Some 3D-Determinant Properties for Calculating of Cubic-Matrix of Order 2 and Order 3

In this paper, as a continuation of our work on the determinant of cubic matrices, we have studied some properties of calculation of the determinant of cubic matrices for the cubic matrix of the order 2×2, as well as the order 3×3. Like properties of square determinants, these properties are analogous to some properties for determinants of the square matrix we have proved and noted that these properties also are applicable (or not in some details) to this concept for the determinant of the cubic matrix of orders 2×2 and order 3×3. All properties that have been proved in this paper, are proved for each case for orders 2×2 and order 3×3, by including all elements of the matrix, as well as all horizontal plans, horizontal layers, and vertical layers. All results in this paper, are presented in detail during the theorem proofs.

Persistent homology for functionals

We introduce topological conditions on a broad class of functionals that ensure that the persistent homology modules of their associated sublevel set filtration admit persistence diagrams, which, in particular, implies that they satisfy generalized Morse inequalities. We illustrate the applicability of these results by recasting the original proof of the Unstable Minimal Surface Theorem given by Morse and Tompkins in a modern and rigorous framework.

G**β-Continuous and G**β-Irresolute Mappings in Topological Spaces

Topology being somehow very recent in nature but has got tremendous applications over almost all other fields. Theoretical or fundamental topology is a bit dry but the application part is what drives crazy once we get used. Topology has applications in various fields of Science and Technology, like applications to Biology, Robotics, GIS, Engineering, Computer Sciences. Topology though being a part of mathematics but it has influenced the whole world with so strong effects and incredible applications. The concept of continuity is fundamental in large parts of contemporary mathematics. In the nineteenth century, precise definitions of continuity were formulated for functions of a real or complex variable, enabling mathematicians to produce rigorous proofs of fundamental theorems of real and complex analysis, such as the Intermediate Value Theorem, Taylor’s Theorem, the Fundamental Theorem of Calculus, and Cauchy’s Theorem. In the early years of the Twentieth Century, the concept of continuity was generalized so as to be applicable to functions between metric spaces, and subsequently to functions between topological spaces. Topology is an area of mathematics concerned with the properties of space that are preserved under continuous deformations including stretching and bending but not tearing. In 2023, Dr. T. Delcia and M. S, Thillai introduced a new type of closed sets called g**β-closed sets and investigated their basic properties including their relationship with already existing concepts in Topological Spaces. In this paper, we introduce g**β-continuous function, g**β-irresolute function, g**β-open function, g**β-closed function, pre-g**β-open function, and pre-g**β-closed function, and investigate properties and characterizations of these new types of mappings in topological spaces.

The axiom of choice and its applications: An exploration of the '100 prisoners problem'

The Axiom of Choice (AC), a cardinal principle in set theory, postulates that for any assortment of disjoint non-empty sets, its possible to construct a new set by selecting one element from each set in the collection. Using choice functions, this idea suggests that every collection of nonempty sets can be associated with a choice function. In the mathematical landscape, ACs significance is accentuated by its extensive application in a myriad of mathematical deductions, marking it as a cornerstone among mathematical axioms. This study delves into the practical implications and applications of AC, employing rigorous analytical methods to investigate its vast and multifaceted influence on the broader mathematical domain. Key findings indicate that AC has facilitated the proof of various theorems, some of which, on the surface, appear unrelated. While its inception sparked considerable debates due to concerns over its intuitive validity, its impact on contemporary mathematics is profound. This research underscores ACs central role in advancing mathematical thought, highlighting its contributions to both foundational theories and intricate proofs.

Ways to introduce the concept of "vector" in geometry

In this paper, the methods of teaching the topic "vector" when studying the course of geometry at school are considered. One of the most difficult topics in geometry lessons for secondary students is considered to be the topics "vectors" and "Vector method in solving problems". In addition, the concept of a vector is one of the basic concepts in modern geometry. To increase the effectiveness of the study of the topic of vectors, historical analysis is considered, as well as teaching methods that students face when studying the topic. These problems are identified by special scientific methods aimed at ensuring the study of the topic of vectors in the geometry course of a secondary educational institution. The teacher, teaching students the vector method, develops their cognitive interest, because on the basis of the vector method, it will be possible to enter coordinates on the plane in the future. The topic of vectors is very important. This is because, first of all, it allows you to find a solution using vectors, making it easier to solve many problems of mathematics that are solved by other methods. In addition, it provides a great opportunity not only to make the proofs of many theorems understandable to students through vectors, but also to teach the search for solutions to problems and proofs of theorems. Secondly, the concept of vector is used in many applications of the mathematics course. For example, it is also used in school algebra and geometry, or in function theory and probability theory. Thirdly, the concept of a vector is an important concept in physics. Vectors also establish the interdisciplinary relationship between physics and mathematics.

Proof and Application of the Mean Value Theorem

In calculus, mean value theorem (MVT) connects a function's derivative and its rate of change over a certain interval. This paper delves into the mathematical intricacies of the MVT and its multifaceted applications. Through rigorous proofs and illustrative examples, this study establishes the MVT's fundamental role in calculus and its relevance in understanding the behavior of functions. The paper extends its exploration to encompass related theorems, including extreme value theorem, which connects function’s continuity and extrema, Intermediate Value Theorem, which states that the function value within an interval of a continuous function must be between the maximum and minimum values, local extreme value theorem, Rolle’s theorem, a specific situation of the theorem, and the integral MVT, an application in integral aspect of MVT, further enriching the comprehension of these pivotal concepts. These theorems provide powerful tools for understanding the properties of continuous functions, identifying critical points, and establishing relationships between function values and their derivatives. This paper highlights the significance of proving these theorems and solving mathematical problems as applications. Through a systematic exploration of the mathematical foundations, this paper contributes to a deeper comprehension of the core principles underlying calculus and their applied theorems in different contexts.