Proof Of Existence Research Articles

Combinatorial Analysis of a Classic Card Trick

Summary We demonstrate how the analysis of Miraskill, a classic card trick by the legendary Canadian magician Stewart James (1908–1996), leads to a connection with the Legendre polynomials and provides a constructive proof of combinatorial identities that occur in quantum mechanics.

Numerical analysis of the stress‐based formulation of linear elasticity

AbstractThe Beltrami–Michell equations of linear elasticity differ from the Navier–Cauchy equations, in that the primary field in former equations is the stress tensor rather than the displacement vector. Consequently, the equations can be used for circumstances where the displacement field is not of interest, for example in design, or when increased smoothness of the solution of the stress tensor is desired. In this work, we explore the stress‐based Beltrami–Michell equations for isotropic linear elastic materials. We introduce the equations in modern tensor notation and investigate their limitations. Further, we demonstrate how to symmetrise and stabilise their weak formulation, complemented by existence and uniqueness proofs. With latter at hand, we construct a conforming finite element discretisation of the equations, avoiding the need for intermediate stress functions. Finally, we present some numerical examples.

Language is a “quite useless” tool: A rejoinder to Fedorenko, Piantadosi, and Gibson’s “Language is primarily a tool for communication rather than thought”

Contrary to the prevailing assumption that language is “primarily a tool for communication rather than thought”, I argue that language is, to invoke Oscar Wilde, “quite useless”. Arguing from aesthetic philosophy and the minimalist program for linguistic theory, I conject that language, like art, is not “for” anything—it simply is, conforming to aesthetic rather than utilitarian principles. Of course, like art, language can be a powerful instrument of communication, but its function is not that of expressing thought; it creates thoughts, “primarily” for communicating with oneself, engaging in Popperian critical rationalism, making thoughts (e.g., sentences, constructive proofs) to match Platonic objects (e.g., propositions, classical proofs).

The Mean Value Theorem and It’s Application

The Mean Value Theorem (MVT) is the central theorem of differential Calculus. It’s the major tool to research on function, also bridging the gap between function and derivative. There are lot of researchers focus on it from ancient times until now. This article introduces details about two mean theorems include the Rolle’s Theorem and Lagrange Theorem. This paper uses the sample question and refutation to explain the conditions of the theorems, thus addressing the questions that many students will ask when learning definitions of these theorems. The article later examines the application of the MVT. The proof of the unique existence of roots and the inequality is included. The Mean Value Theorem reveals the link between the macroscopic, overall properties for a function on an interval and the microscopic, localized properties of the function at a point. The significance of the application for the mean value theorem is profound, not only promoting the development of mathematical analysis theory, but also playing an important role in multiple fields

Extracting efficient exact real number computation from proofs in constructive type theory

Abstract Exact real computation is an alternative to floating-point arithmetic where operations on real numbers are performed exactly, without the introduction of rounding errors. When proving the correctness of an implementation, one can focus solely on the mathematical properties of the problem without thinking about the subtleties of representing real numbers. We propose a new axiomatization of the real numbers in a dependent type theory with the goal of extracting certified exact real computation programs from constructive proofs. Our formalization differs from similar approaches, in that we formalize the reals in a conceptually similar way as some mature implementations of exact real computation. Primitive operations on reals can be extracted directly to the corresponding operations in such an implementation, producing more efficient programs. We particularly focus on the formalization of partial and nondeterministic computation, which is essential in exact real computation. We prove the soundness of our formalization with regards to the standard realizability interpretation from computable analysis and show how to relate our theory to a classical formalization of the reals. We demonstrate the feasibility of our theory by implementing it in the Coq proof assistant and present several natural examples. From the examples we have automatically extracted Haskell programs that use the exact real computation framework AERN for efficiently performing exact operations on real numbers. In experiments, the extracted programs behave similarly to native implementations in AERN in terms of running time and memory usage.

Noncompact self-shrinkers for mean curvature flow with arbitrary genus

Abstract In his lecture notes on mean curvature flow, Ilmanen conjectured the existence of noncompact self-shrinkers with arbitrary genus. Here, we employ min-max techniques to give a rigorous existence proof for these surfaces. Conjecturally, the self-shrinkers that we obtain have precisely one (asymptotically conical) end. We confirm this for large genus via a precise analysis of the limiting object of sequences of such self-shrinkers for which the genus tends to infinity. Finally, we provide numerical evidence for a further family of noncompact self-shrinkers with odd genus and two asymptotically conical ends.

On Existence Proofs, Mathematical Norms, and Professional Obligations

On Existence Proofs, Mathematical Norms, and Professional Obligations

SOME APPLICATIONS OF WEYL CALCULUS TO BURCHNALL-CHAUNDY THEORY, III

We show how Weyl calculus can be applied to the following problem. Suppose that are differential operators with polynomial coefficients and they commute. Then there is a polynomial such that . We use Weyl calculus, but Kohn-Nirenberg calculus also can be used. In this paper, we give the proof of existence of in general case, prove that the ideal of all such is the principal ideal, and give an algorithm for finding generator of in some particular cases.

Partitioning the projective plane into two incidence‐rich parts

AbstractAn internal or friendly partition of a vertex set of a graph is a partition to two nonempty sets such that every vertex has at least as many neighbours in its own class as in the other one. Motivated by Diwan's existence proof on internal partitions of graphs with high girth, we give constructive proofs for the existence of internal partitions in the incidence graph of projective planes and discuss its geometric properties. In addition, we determine exactly the maximum possible difference between the sizes of the neighbour set in its own class and the neighbour set of the other class that can be attained for all vertices at the same time for the incidence graphs of Desarguesian planes of square order.

Analysis of Caputo fractional variable order multi-point initial value problems: existence, uniqueness, and stability

In this paper, we examine the existence, uniqueness, and stability of solutions for a Caputo variable order ϑ-initial value problem (ϑ-IVP) with multi-point initial conditions. The proofs for uniqueness and existence leverage Sadovski’s and Banach’s fixed point theorems, along with the Kuratowski measure of noncompactness. Furthermore, we explore the Ulam–Hyers–Rassias (UHR) stability of the solution. To validate our findings, we present a numerical example.

Numerical simulation of an effective transform mechanism with convergence analysis of the fractional diffusion-wave equations

In the current study, we solve two very important mathematical models, such as the time fractional-order space-fractional telegraph and diffusion-wave equations using a reliable technique called the Adomian decomposition natural method (ADNM), which combines Adomian decomposition and natural transform. The diffusion wave equation describes the flood wave propagation, which is used in solving overland and open channel flow problems. For this reason, it is critical to fully understand and effectively solve the diffusion wave equations. Because telegraph equations are crucial for modeling and developing voltage or frequency transmission, they are widely used in physics and engineering. Furthermore, the designing process is greatly impacted by the uncertainty in the system parameters. For nonlinear ordinary differential equations based on the theorem of Banach fixed point, we provide existence and uniqueness theorem proofs. The present approach has been successfully used to explore exact solutions for time fractional-order and space fractional-order applications. The results show how effective and valuable the ADNM. This paper presents a methodology that will be used in future work to address similar nonlinear problems related to fractional calculus.

Invariant idempotent ⁎-measures generated by iterated function systems

It is known that every continuous t-norm ⁎ generates a functor of the so-called ⁎-measures in the category of compact Hausdorff spaces. Similarly to the case of the hyperspace functor and the probability measure functors one can define the notion of invariant ⁎-measure for iterated function systems of contractions on compact metric spaces.We provide a simple proof of existence and uniqueness of invariant ⁎-measures. Some examples of invariant ⁎-measures, for different t-norms ⁎, are presented.

An Alternate Form of Merino-Mǐcka-Mütze’s Solution to a Knuth’s Combinatorial Generation Problem

A modification of Merino-Mǐcka-Mütze’s solution to a combinatorial generation problem of Knuth is proposed in this survey. The resulting alternate form to such solution is compatible with a reinterpretation by the author of a proof of existence of Hamilton cycles in the middle-levels graphs. Such reinterpretation is given in terms of a dihedral quotient graph associated to each middle-levels graph. The vertices of such quotient graph represent Dyck words and their associated ordered trees. Those Dyck words are linearly ordered via a rooted tree that covers all their tight, or irreducible, forms, offering an universal reference point of view to express and integrate the periodic paths, or blocks, whose concatenation leads to Hamilton cycles resulting from the said solution.

Super Metric Space and Fixed Point Results

Banach contraction principle is the beginning of the Metric fixed point theory. This principle gives existence and uniqueness of fixed points and methods for obtaining approximate fixed points. It is the basic tool of finding fixed points of all contraction type maps. It has a constructive proof which makes the theorem worthy because it yields an algorithm for computing a fixed point. Banach fixed point result has been extended by various authors in many directions either by weakening the conditions of contraction mapping or by changing the abstract structure. Several generalizations and extensions of metric spaces have been introduced. Among these, the prominent extensions are b-metric space, fuzzy metric space, partial metric space and a lot more of their combinations. In particular, a new structure namely Super metric space is introduced. In the present paper, we generalize and extend the fixed point results of fixed point theory in literature in the framework of super metric space.

Radiation of the energy-critical wave equation with compact support

We prove exterior energy lower bounds for (nonradial) solutions to the energy-critical nonlinear wave equation in space dimensions 3≤d≤5, with compactly supported initial data. In particular, it is shown that nontrivial global solutions with compact spatial support must be radiative in a sharp sense. In space dimensions 3 and 4, a nontrivial soliton background is also considered. As an application, we obtain partial results on the rigidity conjecture concerning solutions with the compactness property, including a new proof for the global existence of such solutions.

Construction Theorems and Constructive Proofs in Geometry

Construction Theorems and Constructive Proofs in Geometry

Elastic Localization With Particular Reference to Tape-Springs

Abstract Many elastic systems localize under applied displacement, precipitating into regions of lower and higher strain; further displacement is accommodated by growth of the high strain region at a constant load. Such systems can be studied as propagating instabilities, focusing on the work required to propagate the high strain region, or as two-phase energy minimization problems. It is shown that the Maxwell “equal-areas” construction, and the related common tangent construction, provide the solution to either approach. A new, graphical, proof of the Maxwell equal-areas construction using total strain energy diagrams is presented. Tape-springs are investigated as a case study, with localization presenting as the formation of elastic folds—developable regions with high curvature. One notable property of tape-spring folds is that the fold radius is approximately equal to the initial transverse radius. This result was first proven by Rimrott, and later improved by Calladine and Seffen. A further improvement is obtained here by application of the common tangent construction, and all solutions are shown to be approximations to the Maxwell equal-areas construction in the limit of zero thickness.

Self-organization in computation and chemistry: Return to AlChemy.

How do complex adaptive systems, such as life, emerge from simple constituent parts? In the 1990s, Walter Fontana and Leo Buss proposed a novel modeling approach to this question, based on a formal model of computation known as the λ calculus. The model demonstrated how simple rules, embedded in a combinatorially large space of possibilities, could yield complex, dynamically stable organizations, reminiscent of biochemical reaction networks. Here, we revisit this classic model, called AlChemy, which has been understudied over the past 30 years. We reproduce the original results and study the robustness of those results using the greater computing resources available today. Our analysis reveals several unanticipated features of the system, demonstrating a surprising mix of dynamical robustness and fragility. Specifically, we find that complex, stable organizations emerge more frequently than previously expected, that these organizations are robust against collapse into trivial fixed points, but that these stable organizations cannot be easily combined into higher order entities. We also study the role played by the random generators used in the model, characterizing the initial distribution of objects produced by two random expression generators, and their consequences on the results. Finally, we provide a constructive proof that shows how an extension of the model, based on the typed λ calculus, could simulate transitions between arbitrary states in any possible chemical reaction network, thus indicating a concrete connection between AlChemy and chemical reaction networks. We conclude with a discussion of possible applications of AlChemy to self-organization in modern programming languages and quantitative approaches to the origin of life.

Weighted Hardy factorizations in terms of Multilinear Calderón-Zygmund and fractional integral operators

The foundational paper of Coifman-Rochberg-Weiss provided a constructive proof of the weak factorizations of the classical Hardy space in terms of Riesz transforms. In this paper, we establish the factorization theorems for the weighted Hardy spaces. The results are also extended to the multilinear setting. Furthermore, we show that the weighted BMO space and weighted Lipschitz spaces are necessary for the weighted boundedness of commutators of the multilinear Calderón-Zygmund and fractional integral operators, respectively.

D-algebraic functions

Differentially-algebraic (D-algebraic) functions are solutions of polynomial equations in the function, its derivatives, and the independent variables. We revisit closure properties of these functions by providing constructive proofs. We present algorithms to compute algebraic differential equations for compositions and arithmetic manipulations of univariate D-algebraic functions and derive bounds for the order of the resulting differential equations. We apply our methods to examples in the sciences.