Fuzzy inequational logic

We present a logic for reasoning about graded inequalities which generalizes the ordinary inequational logic used in universal algebra. The logic deals with atomic predicate formulas of the form of inequalities between terms and formalizes their semantic entailment and provability in graded setting which allows to draw partially true conclusions from partially true assumptions. We follow the Pavelka approach and define general degrees of semantic entailment and provability using complete residuated lattices as structures of truth degrees. We prove the logic is Pavelka-style complete. Furthermore, we present a logic for reasoning about graded if–then rules which is obtained as particular case of the general result.     

Cut-Elimination Theorem for the Logic of Constant Domains

The logic CD is an intermediate logic (stronger than intuitionistic logic and weaker than classical logic) which exactly corresponds to the Kripke models with constant domains. It is known that the logic CD has a Gentzen-type formulation called LD (which is same as LK except that (→) and (?–) rules are replaced by the corresponding intuitionistic rules) and that the cut-elimination theorem does not hold for LD . In this paper we present a modification of LD and prove the cut-elimination theorem for it. Moreover we prove a “weak” version of cut-elimination theorem for LD , saying that all “cuts” except some special forms can be eliminated from a proof in LD . From these cut-elimination theorems we obtain some corollaries on syntactical properties of CD : fragments collapsing into intuitionistic logic. Harrop disjunction and existence properties, and a fact on the number of logical symbols in the axiom of CD . Mathematics Subject Classification : 03B55. 03F05.     

Analysis and Applications of Extended Kantorovich–Krylov Method

In our prior work, the two-dimensional bending and in-plane mode shape functions of isotropic rectangular plates were solved based on the extended Kantorovich–Krylov method. These plate modes were then applied to sandwich plate analysis using the assumed modes method. Numerical results has shown these two-dimensional plate modes improved our sandwich plate analysis. However, the rigorous mathematical convergence proof of the extended Kantorovich–Krylov method is lacking. In this article, we provide a rigorous mathematical convergence proof of the extended Kantorovich–Krylov method using the example of rectangular plate bending vibration, in which the governing equation is a biharmonic equation. The predictions of natural frequency and mode shape functions based on the extended Kantorovich–Krylov method were calculated and the results were numerically validated by other analyses. A similar convergence proof can be applied to other types of partial differential equations (PDEs) that govern vibration problems in engineering applications. Based on these results, the extended Kantorovich–Krylov method was proven to be a powerful tooi for the boundary value problems of partial differential equations in the structural vibrations.     

Some General Results About Proof Normalization

In this paper, we provide a general setting under which results of normalization of proof trees such as, for instance, the logicality result in equational reasoning and the cut-elimination property in sequent or natural deduction calculi, can be unified and generalized. This is achieved by giving simple conditions which are sufficient to ensure that such normalization results hold, and which can be automatically checked since they are syntactical. These conditions are based on basic properties of elementary combinations of inference rules which ensure that the induced global proof tree transformation processes do terminate.     

Classroom note: Sawtooth functions

Using MAPLE enables students to consider many examples which would be very tedious to work out by hand. This applies to graph plotting as well as to algebraic manipulation. The challenge is to use these observations to develop the students’ understanding of mathematical concepts. In this note an interesting relationship arising from inverse trigonometric functions is analysed. To understand what is going on students have to develop an understanding of how to deal with inverses where a function is not 1–1, by restricting the domain. The piece of work developed here also provides some interesting exercises in proof by induction.     

Valuations and Dedekind's Prague Theorem

To any field K we associate an entailment relation in the sense of Scott (Proceedings of the Tarski Symposium, 1974, pp. 411–435). In this way we can interpret an abstract propositional theory representing a generic valuation ring of a field, and obtain a simple effective proof of Dedekind's Prague theorem (Edwards, Arch. Hist. Ex. Sci (1980) 321–378; Divisor Theory, Birkhäuser, Boston, 1990).     

A framework for reasoning under uncertainty based on non-deterministic distance semantics

In this paper, we introduce a general and modular framework for formalizing reasoning with incomplete and inconsistent information. Our framework is composed of non-deterministic semantic structures and distance-based considerations. This combination leads to a variety of entailment relations that can be used for reasoning about non-deterministic phenomena and are inconsistency-tolerant. We investigate the basic properties of these entailments, as well as some of their computational aspects, and demonstrate their usefulness in the context of model-based diagnostic systems.     

Truth versus validity in mathematical proof

In mathematics education, it is often said that mathematical statements are necessarily either true or false. It is also well known that this idea presents a great deal of difficulty for many students. Many authors as well as researchers in psychology and mathematics education emphasize the difference between common sense and mathematical logic. In this paper, we provide both epistemological and didactic arguments to reconsider this point of view, taking into account the distinction made in logic between truth and validity on one hand, and syntax and semantics on the other. In the first part, we provide epistemological arguments showing that a central concern for logicians working with a semantic approach has been finding an appropriate distance between common sense and their formal systems. In the second part, we turn from these epistemological considerations to a didactic analysis. Supported by empirical results, we argue for the relevance of the distinction and the relationship between truth and validity in mathematical proof for mathematics education.     

The role of the researcher’s epistemology in mathematics education: an essay on the case of proof

Is there a shared meaning of “mathematical proof” among researchers in mathematics education? Almost all researchers may agree on a formal definition of mathematical proof. But beyond this minimal agreement, what is the state of our field? After three decades of activity in this area, being familiar with the most influential pieces of work, I realize that the sharing of keywords hides important differences in the understanding. These differences could be obstacles to scientific progress in this area, if they are not made explicit and addressed as such. In this essay I take a sample of research projects which have impacted the teaching and learning of mathematical proof, in order to describe where the gaps are. Then I suggest a possible scientific programme which aspires to strengthen the research practice in this domain. Eventually, I make the additional claim that this programme could hold for other areas of research in mathematics education.     

On a New Idiom in the Study of Entailment

This paper is an experiment in Leibnizian analysis. The reader will recall that Leibniz considered all true sentences to be analytically so. The difference, on his account, between necessary and contingent truths is that sentences reporting the former are finitely analytic; those reporting the latter require infinite analysis of which God alone is capable. On such a view at least two competing conceptions of entailment emerge. According to one, a sentence entails another when the set of atomic requirements for the first is included in the corresponding set for the other; according to the other conception, every atomic requirement of the entailed sentence is underwritten by an atomic constituent of the entailing one. The former conception is classical on the twentieth century understanding of the term; the latter is the one we explore here. Now if we restrict ourselves to the formal language of the propositional calculus, every sentence has a finite analysis into its conjunctive normal form. Semantically, then, every sentence of that language can be represented as a simple hypergraph, H, on the powerset of a universe of states. Entailment of the sort we wish to study can be represented as a known relation, subsumption between hypergraphs. Since the lattice of hypergraphs thus ordered is a DeMorgan lattice, the logic of entailment thus understood is the familiar system, FDE of first-degree entailment. We observe that, extensionalized, the relation of subsumption is itself a DeMorgan Lattice ordered by higher-order subsumption. Thus the semantic idiom that hypergraph-theory affords reveals a hierarchy of lattices capable of representing entailments of every finite degree.     

Automated prover for attribute dependencies in data with grades

We present a new axiomatization of logic for dependencies in data with grades, which includes ordinal data and data over domains with similarity relations, and an efficient reasoning method that is based on the axiomatization. The logic has its ordinary-style completeness characterizing the ordinary, bivalent entailment as well as the graded-style completeness characterizing the general, possibly intermediate degrees of entailment. A core of the method is a new inference rule, called the rule of simplification, from which we derive convenient equivalences that allow us to simplify sets of dependencies while retaining semantic closure. The method makes it possible to compute a closure of a given collection of attributes with respect to a collection of dependencies, decide whether a given dependency is entailed by a given collection of dependencies, and more generally, compute the degree to which the dependency is entailed by a collection of dependencies. We also present an experimental evaluation of the presented method.     

Conservation laws for Lotka–Volterra models

We derive necessary and sufficient conditions on a Lotka–Volterra model to admit a conservation law of Volterra's type. The result and the proof for the corresponding linear algebra problem are given in graph‐theoretical terms; they refer to the directed graph which is defined by the coefficients of the differential equation system. Copyright © 2003 John Wiley & Sons, Ltd.     

Normal forms,linearity, and prime algebraicity over nonflat domains

Using representations of nonflat Scott domains to model type systems, it is natural to wish that they be “linear”, in which case the complexity of the fundamental test for entailment of information drops from exponential to linear, the corresponding mathematical theory becomes much simpler, and moreover has ties to models of computation arising in the study of sequentiality, concurrency, and linear logic. Earlier attempts to develop a fully nonflat semantics based on linear domain representations for a rich enough type system allowing inductive types, were designed in a way that felt rather artificial, as it featured certain awkward and counter‐intuitive properties; eventually, the focus turned on general, nonlinear representations. Here we try to turn this situation around, by showing that we can work linearly in a systematic way within the nonlinear model, and that we may even restrict to a fully linear model whose objects are in a bijective correspondence with the ones of the nonlinear and are easily seen to form a prime algebraic domain. To obtain our results we study mappings of finite approximations of objects that can be used to turn approximations into normal and linear forms.     

Unifying hidden-variable problems from quantum mechanics by logics of dependence and independence

We study hidden-variable models from quantum mechanics and their abstractions in purely probabilistic and relational frameworks by means of logics of dependence and independence, which are based on team semantics. We show that common desirable properties of hidden-variable models can be defined in an elegant and concise way in dependence and independence logic. The relationship between different properties and their simultaneous realisability can thus be formulated and proven on a purely logical level, as problems of entailment and satisfiability of logical formulae. Connections between probabilistic and relational entailment in dependence and independence logic allow us to simplify proofs. In many cases, we can establish results on both probabilistic and relational hidden-variable models by a single proof, because one case implies the other, depending on purely syntactic criteria. We also discuss the ‘no-go’ theorems by Bell and Kochen-Specker and provide a purely logical variant of the latter, introducing non-contextual choice as a team-semantical property.     

Sharp Threshold of Global Existence for a Nonlocal Nonlinear Schrödinger System in ℝ3

In this paper, the authors investigate the sharp threshold of a three-dimensional nonlocal nonlinear Schr\"{o}dinger system. It is a coupled system which provides the mathematical modeling of the spontaneous generation of a magnetic field in a cold plasma under the subsonic limit. The main difficulty of the proof lies in exploring the inner structure of the system due to the fact that the nonlocal effect may bring some hinderance for establishing the conservation quantities of the mass and of the energy, constructing the corresponding variational structure, and deriving the key estimates to gain the expected result. To overcome this, the authors must establish local well-posedness theory, and set up suitable variational structure depending crucially on the inner structure of the system under study, which leads to define proper functionals and a constrained variational problem. By building up two invariant manifolds and then making a priori estimates for these nonlocal terms, the authors figure out a sharp threshold of global existence for the system under consideration.     

A note on the extension of the null-additive set function on the algebra of subsets

In this work, we point out that the proof of Theorem 2 in [E. Pap, Extension of null-additive set functions on algebra of subsets, Novi Sad J. Math. 31 (2) (2001) 9–13] is incorrect and give a correct proof. Moreover, we also get a corresponding theorem on extension of the weakly null-additive set function.     

A linear programming proof of the second order conditions of non-linear programming

In this note we give a new, simple proof of the standard first and second order necessary conditions, under the Mangasarian–Fromovitz constraint qualification (MFCQ), for non-linear programming problems. We work under a mild constraint qualification, which is implied by MFCQ. This makes it possible to reduce the proof to the relatively easy case of inequality constraints only under MFCQ. This reduction makes use of relaxation of inequality constraints and it makes use of a penalty function. The new proof is based on the duality theorem for linear programming; the proofs in the literature are based on results of mathematical analysis. This paper completes the work in a recent note of Birbil et al. where a linear programming proof of the first order necessary conditions has been given, using relaxation of equality constraints.     

Exact and heuristic approaches for the set cover with pairs problem

This work deals with the set cover with pairs problem (SCPP) which is a generalization of the set cover problem (SCP). In the SCPP the elements have to be covered by specific pairs of objects, instead of a single object. We propose a new mathematical formulation using extended variables that is capable of consistently solve instances with up to 500 elements and 500 objects. We also developed an ILS heuristic which was capable of finding better solutions for several tested instances in less computational time.     

Computerized proof techniques for undergraduates

The use of computer algebra systems such as Maple and Mathematica is becoming increasingly important and widespread in mathematics learning, teaching and research. In this article, we present computerized proof techniques of Gosper, Wilf–Zeilberger and Zeilberger that can be used for enhancing the teaching and learning of topics in discrete mathematics. We demonstrate by examples how one can use these computerized proof techniques to raise students' interests in the discovery and proof of mathematical identities and enhance their problem-solving skills.     

Boundary value problems for a quasilinear parabolic equation with an unknown coefficient

The work is connected with the mathematical modeling of physical–chemical processes in which inner characteristics of materials are subjected to changes. The considered nonlinear parabolic models consist of a boundary value problem for a quasilinear parabolic equation with an unknown coefficient multiplying the derivative with respect to time and, moreover, involve an additional relationship for a time dependence of this coefficient. For such a system, conditions of unique solvability in a class of smooth functions are studied on the basis of the Rothe method. The proposed approach involves the proof of a priori estimates in the difference-continuous Hölder spaces for the corresponding differential-difference nonlinear system that approximates the original system by the Rothe method. These estimates allow one to establish the existence of the smooth solutions and to obtain the error estimates of the approximate solutions.As examples of applications of the considered nonlinear boundary value problems, the models of destruction of heat-protective composite under the influence of high temperature heating are discussed.