Search

We prove that for a finite first order structure $${\mathbf {A}}$$A and a set of first order formulas $$\Phi $$Φ in its language with certain closure properties, the finitary relations on A that are definable via formulas in $$\Phi $$Φ are uniquely determined by those of arity $$|A|^{2}$$|A|2. This yields new proofs for some finiteness results from universal algebraic geometry.

The algebraic approach by E.W. Dijkstra and C.S. Scholten to formal logic is a proof calculus, where the notion of proof is a sequence of equivalences proved – mainly – by using substitution of ‘equals for equals’. This paper presents Set , a first-order logic axiomatization for set theory using the approach of Dijkstra and Scholten. What is novel about the approach presented in this paper is that symbolic manipulation of formulas is an effective tool for teaching an axiomatic set theory course to sophomore-year undergraduate students in mathematics. This paper contains many examples on how argumentative proofs can be easily expressed in Set and points out how the rigorous approach of Set can enrich the learning experience of students. The results presented in this paper are part of a larger effort to formally study and mechanize topics in mathematics and computer science with the algebraic approach of Dijkstra and Scholten., El enfoque algebraico de E.W. Dijkstra y C.S. Scholten a la lógica formal es un cálculo de prueba, donde la noción de prueba es una secuencia de equivalencias probadas, principalmente, mediante la sustitución de "iguales por iguales". Este artículo presenta Set, una axiomatización lógica de primer orden para la teoría de conjuntos utilizando el enfoque de Dijkstra y Scholten. Lo novedoso del enfoque presentado en este artículo es que la manipulación simbólica de fórmulas es una herramienta eficaz para enseñar un curso de teoría axiomática de conjuntos a estudiantes de segundo año de pregrado en matemáticas. Este artículo contiene muchos ejemplos sobre cómo las pruebas argumentativas se pueden expresar fácilmente en Set y señala cómo el enfoque riguroso de Set puede enriquecer la experiencia de aprendizaje de los estudiantes. Los resultados presentados en este artículo son parte de un esfuerzo mayor para estudiar y mecanizar formalmente temas en matemáticas e informática con el enfoque algebraico de Dijkstra y Scholten., Colombian Conference on Computing

The use of comparison functions has become standard in systems and con- trol theory, particularly for the purposes of studying stability properties. The use of these functions typically allows elegant and succinct statements of stability properties suchasasymptoticstabilityandinput-to-statestabilityanditsseveralvariants.Further- more, over the last 20years several inequalities involving these comparison functions have been developed that simplify their manipulation in the service of proving more significant results. Many of these inequalities have appeared in the body of proofs or in appendices of various papers. Our goal herein is to collect these inequalities in one place.

Skeletal signatures were introduced in [J W Anderson and A Wootton, A Lower Bound for the Number of Group Actions on a Compact Riemann Surface, Algebr. Geom. Topol. 12 (2012) 19--35.] as a tool to describe the space of all signatures with which a group can act on a surface of genus $\sigma \geq 2$. In the present paper we provide a complete description of the gaps that appear in the space of skeletal signatures, together with proofs of the conjectures posed in our earlier work., Comment: 10 pages, 1 figure

P rime numbers are the atoms of our mathematical universe. Euclid showed that there are infinitely many primes, but the subtleties of their distribution continue to fascinate mathematicians. Letting p(n) denote the number of primes p B n, Gauss conjectured in the early nineteenth century that pðnÞ n=lnðnÞ. In 1896, this conjecture was proven independently by Jacques Hadamard and Charles de la Vallee-Poussin. Their proofs both used complex analysis. The search was then on for an ‘‘elementary proof’’ of this result. G. H. Hardy was doubtful that such a proof could be found, saying if one was found ‘‘that it is time for the books to be cast aside and for the theory to be rewritten.’’ But in the Spring of 1948 such a proof was found. Almost immediately there was controversy. Was the proof attributable to Atle Selberg or was the proof attributable to Atle Selberg and Paul Erd} os? For decades there seemed to be two mathematical camps with wildly different viewpoints. In the twenty-first century the controversy has finally subsided. Among previous discussions of the controversy, we mention particularly Goldfeld [1] (from which the previously mentioned quotation by Hardy is taken) and the book [3] of Paul Hoffman. Ernst Straus was in a unique position to observe the beginnings of the controversy. He then held a position at the Institute for Advanced Study as a special assistant to Albert Einstein. (We believe Straus is the only person to have joint papers with Einstein and Erd} os.) Straus had already worked a great deal with Erd} os, and this work would continue throughout his life. Sometime in the early 1970s (we aren’t sure of the exact dates), Straus wrote the account we present here. He did not want the notes to be published while the participants were still alive. Ernst Straus was, for us and for many of his friends, a man of great wisdom. He certainly attempted in these notes to give as faithful an account of the events as he could. Whether he succeeded is a judgment for the reader to make. Certainly, he was far closer to Erd} os than to Selberg. Let us be clear that we two authors both have an Erd} os number of one, and our own associations with Erd} os were long and profound. We feel that the Straus recollections are an important historic contribution. We also believe that the controversy itself sheds considerable light on the changing nature of mathematical research. In November 2005, Atle Selberg was interviewed by Nils A. Baas and Christian F. Skau [4]. He recalled the events of 1948 with remarkable precision. We quote extensively from that account. Selberg had first shown that X

We establish different estimates for the sums of negative eigenvalues of elliptic operators. Our proofs are based on a property of the eigenvalue sums that might be viewed as a certain convexity with respect to the perturbation.

We consider an elliptic hemivariational inequality with nonlocal nonlinearities. Assuming only certain growth conditions on the data, we are able to prove existence results for the problem under consideration. In particular, no continuity assumptions are imposed on the nonlocal term. The proofs rely on a combined use of recent results due to the authors on hemivariational inequalities and operator equations in partially ordered sets.

Anderson-like ontological proofs, studied in this paper, employ contingent identity, free principles of quantification of the 1st order variables and classical principles of quantification of the 2nd order variables. All these theories are strongly complete wrt. classes of modal structures containing families of world-varying objectual domains of the 1st order and constant conceptual domains of the 2nd order. In such structures, terms of the 1st order receive only rigid extensions, which are elements of the union of all 1st order domains. Terms of the 2nd order receive extensions and intensions. Given a family of preselected world-varying objectual domains of the 2nd order, non-rigid extensions of the 2nd order terms belong always to a preselected domain connected with a given world. Rigid intensions of the 2nd order terms are chosen from among members of a conceptual domain of the 2nd order, which is the set of all functions from the set of worlds to the union of all 2nd order preselected domains such that values of these functions at a given world belong to a preselected domain connected with this world.

The paper is devoted to the study of LlogL inequalities and other related bounds for two classical operators on the real line: the truncated Hilbert transform and the segment multiplier. Using duality, these estimates are deduced from corresponding sharp exponential-type bounds, the proofs of which rest on the construction of appropriate harmonic functions on the strip \([-1,1]\times \mathbb{R }\) and transference-type arguments.

We consider optimal control problems with initial---final state equality and inequality constraints and running mixed state-control equality constraints given by smooth functions. The mixed constraints satisfy the regularity assumption of linear independence of gradients with respect to the control. We present simple proofs of second-order necessary conditions for (extended) weak minimum for extremals with discontinuous controls in these problems.

First, we describe a psychological experiment in which the participants were asked to determine whether sentences of first-order logic were true or false in finite graphs. Second, we define two proof systems for reasoning about truth and falsity in first-order logic. These proof systems feature explicit models of cognitive resources such as declarative memory, procedural memory, working memory, and sensory memory. Third, we describe a computer program that is used to find the smallest proofs in the aforementioned proof systems when capacity limits are put on the cognitive resources. Finally, we investigate the correlation between a number of mathematical complexity measures defined on graphs and sentences and some psychological complexity measures that were recorded in the experiment.

In this paper, we show that the strong NP-hardness proofs of some scheduling problems with start time dependent job processing times presented in Gawiejnowicz (Eur J Oper Res 180:472–478, 2007) and Zhao and Tang (Optim Lett 5:183–190, 2011) are incorrect. Namely, the applied transformations from 4-Product problem to the considered scheduling problems are polynomial not pseudopolynomial. Thus, the related problems are NP-hard, but their complete computational status is still an open issue: ordinary or strongly NP-hard?

We present a coinductive proof of Moessner’s theorem. This theorem describes the construction of the stream (1n,2n,3n,…) (for n≥1) out of the stream of positive natural numbers by repeatedly dropping and summing elements. Our formalisation consists of a direct translation of the operational description of Moessner’s procedure into the equivalence of—in essence—two functional programs. Our proof fully exploits the circularity that is implicitly present in Moessner’s procedure, and it is more elementary than existing proofs. As such, it serves as a non-trivial illustration of the relevance and power of coinduction.

Leo-II is an automated theorem prover for classical higher-order logic. The prover has pioneered cooperative higher-order---first-order proof automation, it has influenced the development of the TPTP THF infrastructure for higher-order logic, and it has been applied in a wide array of problems. Leo-II may also be called in proof assistants as an external aid tool to save user effort. For this it is crucial that Leo-II returns proof information in a standardised syntax, so that these proofs can eventually be transformed and verified within proof assistants. Recent progress in this direction is reported for the Isabelle/HOL system.

This paper shows how to build a formal analytical solution for a differential equation of arbitrary order and with variable coefficients. It proofs that the most known approximated solutions for such a problem can be derived from the analytical expression presented in the paper. The formalism can be easily extended to the infinite dimensional case such as the quantum time-dependent Hamiltonian problem., Comment: 16 pages, 4 figures

We provide new proofs of two key results of p-adic Hodge theory: the Fontaine-Wintenberger isomorphism between Galois groups in characteristic 0 and characteristic p, and the Cherbonnier–Colmez theorem on decompletion of \((\varphi , \Gamma )\)-modules. These proofs are derived from joint work with Liu on relative p-adic Hodge theory, and are closely related to the theory of perfectoid algebras and spaces, as in the work of Scholze.

In this paper we argue that an account of proof-theoretic harmony based on reductions and expansions delivers an inferentialist picture of meaning which should be regarded as intensional, as opposed to other approaches to harmony that will be dubbed extensional. We show how the intensional account applies to any connective whose rules obey the inversion principle first proposed by Prawitz and Schroeder-Heister. In particular, by improving previous formulations of expansions, we solve a problem with quantum-disjunction first posed by Dummett. As recently observed by Schroeder-Heister, however, the specification of an inversion principle cannot yield an exhaustive account of harmony. The reason is that there are more collections of elimination rules than just the one obtained by inversion which we are willing to acknowledge as being in harmony with a given collection of introduction rules. Several authors more or less implicitly suggest that what is common to all alternative harmonious collection of rules is their being interderivable with each other. On the basis of considerations about identity of proofs and formula isomorphism, we show that this is too weak a condition for a given collection of elimination rules to be in harmony with a collection of introduction rules, at least if the intensional picture of meaning we advocate is not to collapse on an extensional one.

In this paper, we consider, discuss, improve, and complement recent fixed points results for mappings satisfying cyclical contractive conditions established by Pacurar and Rus (Nonlinear Anal. 72:1181-1187, 2010) and Chandok and Postolache (Fixed Point Theory Appl. 2013:28, 2013). By using a new lemma we get much shorter and nicer proofs of some results with the new concept of mappings.

We discuss several contractions of integral type by using Jachymski’s approach. We give alternative proofs of recent generalizations of the Banach contraction principle due to Ri (Indag. Math. 27:85-93, 2016) and Wardowski (Fixed Point Theory Appl. 2012:94, 2012).

Let K be an algebraically closed, complete non-Archimedean field. The purpose of this paper is to carefully study the extent to which finite morphisms of algebraic K-curves are controlled by certain combinatorial objects, called skeleta. A skeleton is a metric graph embedded in the Berkovich analytification of X. A skeleton has the natural structure of a metrized complex of curves. We prove that a finite morphism of K-curves gives rise to a finite harmonic morphism of a suitable choice of skeleta. We use this to give analytic proofs of stronger "skeletonized" versions of some foundational results ofLiu-Lorenzini, Coleman, and Liu on simultaneous semistable reduction of curves. We then consider the inverse problem of lifting finite harmonic morphisms of metrized complexes to morphisms of curves over K. We prove that every tamely ramified finite harmonic morphism of \Lambda-metrized complexes of k-curves lifts to a finite morphism of K-curves. If in addition the ramification points are marked, we obtain a complete classification of all such lifts along with their automorphisms. This generalizes and provides new analytic proofs of earlier results of Sa\"idi and Wewers. As an application, we discuss the relationship between harmonic morphisms of metric graphs and induced maps between component groups of N\'eron models, providing a negative answer to a question of Ribet motivated by number theory. This article is the first in a series of two. The second article contains several applications of our lifting results to questions about lifting morphisms of tropical curves., Comment: 52 pages, 3 figures. The previous version of this arXiv post has been split into two parts; the second part is arXiv:1404.3390

We consider a system of seminlinear parabolic variational inequalities with time-dependent convex obstacles. We prove the existence and uniqueness of its solution. We also provide a stochastic representation of the solution and show that it can be approximated by the penalization method. Our proofs are based upon probabilistic methods from the theory of Markov processes and the theory of backward stochastic differential equations.

In this paper, using the properties of Schur-convex function, Schur-geometrically convex function and Schur-harmonically convex function, we provide much simpler proofs of the Schur-convexity, Schur-geometric convexity and Schur-harmonic convexity for a composite function of the complete symmetric function.

Very recently, Hussain et al. (Fixed Point Theory Appl. 2015:185, 2015) introduced the concept of JS-contraction and established some fixed point theorems for such contractions. In this paper, we introduce a new method of proofs that allows us to prove fixed point theorems for JS-contraction in complete metric spaces by removing two conditions in theorems of Hussain et al. Thus, we prove that fixed point Theorems 2.3-2.8 and Corollary 2.9 of Hussain et al. actually are consequences, and not generalizations, of the corresponding theorems of Ciric, Chatterjea, Kannan, and Reich.

One of the goals of this short note is to alert researchers as regards some mistakes that appeared in a recent paper (Chauhan, Khan and Kumar in J. Inequal. Appl. 2013:182, 2013). This entails main proofs based on a false result, which invalidates all statements. We also give a complete revision of the antecedents of this work in order to find the main reasons of the mistakes. Finally, the main aim of this note is to propose a correct, more general version of the main theorems in the paper mentioned.

In this paper we prove three power-exponential inequalities for positive real numbers. In particular, we conclude that this proofs give affirmatively answers to three, until now, open problems (conjectures~4.4, 2.1 and 2.2) posed by C{\^i}rtoaje in the following two works: "{\it J. Inequal. Pure Appl. Math.} 10, Article 21, 2009" and "{\it J. Nonlinear Sci. Appl.} 4:2:130-137, 2011". Moreover, we present a new proof of the inequality $a^{ra}+b^{rb}\ge a^{rb}+b^{ra}$ for all positive real numbers $a$ and $b$ and $r\in [0,e]$. In addition, three new conjectures are presented., Comment: 9 pages

The paper focuses on the precise asymptotics of the largest eigenvalues of β-Hermite ensemble and β-Laguerre ensemble. In particular, we obtain a general law on the precise moment convergence rates for a type of weighted series constructed by the first-order conditional moment of the largest eigenvalues of β ensembles. The results are motivated by the complete convergence for partial sums of independent random variables. The proofs depend on the small deviations for largest eigenvalue of β ensembles and non-asymptotic tail probability inequalities of the general β Tracy-Widom law.

The main aim of this paper is to advise researchers in the field of Fixed Point Theory against an extended mistake that can be found in some proofs. We illustrate our claim proving that theorems in the very recent paper (Wang in Fixed Point Theory Appl. 2014:137, 2014) are incorrect, and we provide different corrected versions of them.

In this note we provide simple and short proofs for a class of Hardy-Rellich type inequalities with the best constant, which extends some recent results. MSC:26D15, 35A23.

In this paper we give a new refinement of discrete Jensen’s inequality, which generalizes a former result. The introduced sequences depend on parameters. The strict monotonicity and the convergence are investigated. We also study the behavior of the sequences when the parameters vary. One of the proofs requires an interesting convergence theorem with probability theoretical background. This result is an extension of a former result, but its proof is simpler. The results are applied to define and study some new quasi-arithmetic means.

We establish some new Hermite-Hadamard-type inequalities involving product of two functions. Other integral inequalities for two functions are obtained as well. The analysis used in the proofs is fairly elementary and based on the use of the Minkowski, Hölder, and Young inequalities.

Nemeth introduced the notion of order weakly L-Lipschitz mapping and employed this concept to obtain nontrivial solutions of nonlinear complementarity problems. In this article, we shall extend this concept to two mappings and obtain the solution of common fixed point equations and hence coincidence point equations in the framework of vector lattices. We present some examples to show that the solution of nonlinear complementarity problems and implicit complementarity problems can be obtained using these results. We also provide an example of a mapping for which the conclusion of Banach contraction principle fails but admits one of our fixed point results. Our proofs are simple and purely order-theoretic in nature.

In this paper, we introduce and consider a new class of general variational inclusions involving the difference of operators in a Hilbert space. We establish the equivalence between the general variational inclusions and the fixed point problems as well as with a new class of resolvent equations using the resolvent operator technique. We use this alternative formulation to discuss the existence of a solution of the general variational inclusions. We again use this alternative equivalent formulation to suggest and analyze a number of iterative methods for finding a zero of the difference of operators. We also discuss the convergence of the iterative method under suitable conditions. Our methods of proofs are very simple as compared with other techniques. Several special cases of these problems are also considered. The results proved in this paper may be viewed as a refinement and an improvement of the known results in this area.

We give elementary proofs of two theorems concerning bounds on the maximum argument of the eigenvalues of a product of two unitary matrices--one by Childs et al. [J. Mod. Phys. 47, 155 (2000)] and the other one by Chau [Quant. Inf. Comp. 11, 721 (2011)]. Our proofs have the advantages that the necessary and sufficient conditions for equalities are apparent and that they can be readily generalized to the case of infinite-dimensional unitary operators.