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.     

二元语义Bonferroni集成算子及其在多属性群决策中的应用

Bonferroni平均算子的优点是在进行信息集成的过程中能够考虑到输入变量之间的相互影响。本文首先将其推广到语言环境中,提出了二元语义Bonferroni平均(2TLBA)算子、加权2TLBA(W2TLBA)算子和组合W2TLBA(C-W2TLBA)算子的概念,研究它们的相关性质。然后针对决策信息以语言变量给出的多属性群决策问题,提出一种基于W2TLBA算子和C-W2TLBA算子的群决策方法。最后应用实例说明了该方法是切实可行的。     

Assuring consistency in mixed models

Information systems security defines three properties of information: confidentiality, integrity, and availability. These characteristics remain major concerns throughout the commercial and military industry. Ordinary users have taken these features as basis for their businesses. Furthermore, users may find it necessary to combine policies in order to protect their information in a suitable way. However, inconsistencies may arise as a result of implementing multiple secrecy and privacy models; and therefore, render these services unsecure. In this paper, we propose an approach to detect and report inconsistencies when choosing mixed models for integrity and security. It is based on specifying the policies in first order logic and applying formal analysis. We demonstrate the feasibility of our proposition by applying it to the Clark Wilson and role based access control models. We use the Alloy language and analyzer to formalize the mixed model and check for any inconsistencies.     

Generalized intuitionistic fuzzy geometric aggregation operators and their application to multi-criteria decision making

In this paper, we extend the generalized weighted geometric and generalized ordered weighted geometric operators to intuitionistic fuzzy environments, that is, we develop a series of generalized intuitionistic fuzzy geometric operators to aggregate input arguments that are expressed by intuitionistic fuzzy values based on Archimedean t-conorm and t-norm. Then some desired properties of these aggregation operators are investigated. The relations between these operators and some existing intuitionistic fuzzy geometric aggregation operators are discussed in detail. Furthermore, applying these proposed operators, we develop an approach for multi-criteria decision making with intuitionistic fuzzy information. Finally, a practical example is given to verify the developed approach and to demonstrate its practicality and effectiveness.     

关于判断矩阵一致性检验与调整的一个注记

首先分析了判断矩阵不一致形成的原因,认为一个判断矩阵中的不一致是由强矛盾判断,弱矛盾判断,标度离散性,标度有限性共同作用的结果,并通过两个例子指出现有一致性检验与调整方法中存在的问题,最后在已有研究基础上给出了判断矩阵一致性调整的新步骤.     

On a class of ill-posed non-linear integral equations arising in the interpretation of indirect measurements

In this paper we consider a class of specific Urysohn integral equations for which the solutions are only determined with the exception of rearrangements of function values and associated arguments. As an alternative to Tikhonov's regularization method approximating minimum-norm solutions for this ill-posed class of inverse problems, a constrained least-squares approach is presented. This approach is aimed at finding decreasing rearrangements serving as appropriate solution representatives. It is shown that the inverses of these decresing solutions solve a Fredholm linear integral equation of the first kind.     

Another approach to proof: Arguments from physics

In the first part of the paper we will explore the use of arguments from physics in mathematical proof and give some reasons why this approach might be worthwhile. In the second part we will relate this idea to Freudenthal's concept of local organization. The third part of the paper will present the results of an empirical study conducted in Canada on the classroom use of arguments from physics in mathematical proof.     

Discretisation and duality of optimal Skorokhod embedding problems

We prove a strong duality result for a linear programming problem which has the interpretation of being a discretised optimal Skorokhod embedding problem, and we recover this continuous time problem as a limit of the discrete problems. With the discrete setup we show that for a suitably chosen objective function, the optimiser takes the form of a hitting time for a random walk. In the limiting problem we then reprove the existence of the Root, Rost, and cave embedding solutions of the Skorokhod embedding problem.The main strength of this approach is that we can derive properties of the discrete problem more easily than in continuous time, and then prove that these properties hold in the limit. For example, a consequence of the strong duality result is that dual optimisers exist, and our limiting arguments can be used to derive properties of the continuous time dual functions. These arguments are applied in Cox and Kinsley (2017), where the existence of dual solutions is required to prove characterisation results for optimal barriers in a financial application.     

犹豫模糊语言Heronian平均算子在多属性决策中的应用

针对输入变量之间的相互影响以及评价值为犹豫模糊语言信息的多属性决策问题,提出一种基于犹豫模糊语言Heronian平均算子的多属性决策方法。由于Heronian平均(HM)算子具有能够反映输入变量之间相互关联的良好特性,在犹豫模糊语言信息环境下,提出了两种新的集成算子,即犹豫模糊语言Heronian平均(HFLHM)算子和犹豫模糊语言几何Heronian平均(HFLGHM)算子,同时研究了它们的一些特性。考虑到输入变量具有不同的重要程度,还定义了犹豫模糊语言加权Heronian平均(HFLWHM)算子和犹豫模糊语言加权几何Heronian平均(HFLWGHM)算子。最后提出了基于HFLWHM算子和HFLWGHM算子的犹豫模糊语言多属性决策方法,并通过实例验证了这些算子的合理性和可行性。     

Effect of Delayed Payments (Trade Credit) on Order Quantities

We address the problem of determining the economic order quantity when the vendor permits delay in payment. Some early researchers argued that the best order quantity is invariant with respect to the trade credit. Others argued that the order quantity should increase as the delay in payment increases. We analyse this problem using the discounted cash-flow approach, and provide clarification on the inconsistencies between these approaches. First, we show that the best order quantity is an increasing function of the permitted delay in payment. We also show that an approach suggested by Chand and Ward not only yields an upper bound on the optimum, but also provides robust results. Though the classical square-root formula disregards trade-credit information, under some circumstances, it will yield better results than the formulation taking into account that information. We illustrate this anomaly with an example, and provide analytical explanation for it. We also discuss some potential conceptual pitfalls in using the average cost analysis as an approximation to the discounted cash-flow approach.     

Rich preference-based argumentation frameworks

An argumentation framework is seen as a directed graph whose nodes are arguments and arcs are attacks between the arguments. Acceptable sets of arguments, called extensions, are computed using a semantics. Existing semantics are solely based on the attacks and do not take into account other important criteria like the intrinsic strengths of arguments.The contribution of this paper is three fold. First, we study how preferences issued from differences in strengths of arguments can help in argumentation frameworks. We show that they play two distinct and complementary roles: (i) to repair the attack relation between arguments, (ii) to refine the evaluation of arguments. Despite the importance of both roles, only the first one is tackled in existing literature. In a second part of this paper, we start by showing that existing models that repair the attack relation with preferences do not perform well in certain situations and may return counter-intuitive results. We then propose a new abstract and general framework which treats properly both roles of preferences. The third part of this work is devoted to defining a bridge between the argumentation-based and the coherence-based approaches for handling inconsistency in knowledge bases, in particular when priorities between formulae are available. We focus on two well-known models, namely the preferred sub-theories introduced by Brewka and the demo-preferred sets defined by Cayrol, Royer and Saurel. For each of these models, we provide an instantiation of our abstract framework which is in full correspondence with it.     

Completeness, Sections and Selections

In this paper, we develop a general approach to set-valued semi-continuous selections which is based on order-like arguments rather than on classical approximations. The approach works nice in a number of situations demonstrating the genesis of such selection properties of set-valued mappings. In particular, it allows to generalize several known results, also to get some new results about sections of set-valued mappings.      

On Lagrangian duality in vector optimization: applications to the linear case

We recall a general scheme for vector problems based on separation arguments and alternative theorems, and then, this approach is exploited to study Lagrangian duality in vector optimization. We show that the vector linear duality theory due to Isermann can be embedded in this separation approach. The theoretical part of this paper serves the purpose of introducing two possible applications. Some well-known classical applications in economics are the minimization of costs and the maximization of profit for a firm. We extend these two examples to the multiobjective framework in the linear case, exploiting the duality theory of Isermann. For the former, we consider the minimization of costs and of pollution as two different and conflicting goals; for the latter, we introduce as second objective function the profit for a competitor firm. This allows us to study the relationships between the shadow prices referred to the two different goals and to introduce a new representation of the feasible region of the dual problem.     

Size does matter: The use of fish motion for improving human swimming simulations

In this study we evaluated the length and volume inconsistencies that occur when applying simple motion functions onto surface models of swimming fish and human, respectively. The larger the size (diameter and length) of the body, the larger are these inconsistencies. To eliminate these errors, different surface correction algorithms are introduced. Additional terms are implemented to overcome structural insufficiencies through over-stretching and overlapping in sensible body parts. For human undulatory swimming these models combine the motion of rigid extremities with the smooth surface bending at joints. The advantages of this hybrid model can be crucial for Computational Fluid Dynamics, where deviation from natural behaviour results in unrealistic surface pressures and artificial vortex structures. The structural improvements are also promising for computational graphic programming, where skin-like stretching surfaces mediate a natural impression.     

Numerical Quadrature Computation of the Macdonald Function for Complex Orders

The use of Gaussian quadrature formulae is explored for the computation of the Macdonald function (modified Bessel function) of complex orders and positive arguments. It is shown that for arguments larger than one, Gaussian quadrature applied to the integral representation of this function is a viable approach, provided the (nonclassical) weight function is suitably chosen. In combination with Gauss–Legendre quadrature the approach works also for arguments smaller than one. For very small arguments, power series can be used. A Matlab routine is provided that implements this approach. AMS subject classification (2000) 33-04, 33C10, 65D15, 65D32     

Forced oscillatory result of hyperbolic equations with continuous distributed deviating arguments

In this paper, we consider hyperbolic equations with continuous distributed deviating arguments, and sufficient conditions are presented for every solution of some boundary value problems to be oscillatory in a cylindrical domain. Our approach is to reduce multi-dimensional problems to one-dimensional problems by using some integral means of solution.     

Basic representations for Eisenstein series from their differential equations

In this paper we provide a new approach for the derivation of parameterizations for the Eisenstein series. We demonstrate that a variety of classical formulas may be derived in an elementary way, without knowledge of the inversion formulae for the corresponding Schwarzian triangle functions. In particular, we provide a new derivation for the parametric representations of the Eisenstein series in terms of the complete elliptic integral of the first kind. The proof given here is distinguished from existing elementary proofs in that we do not employ the Jacobi-Ramanujan inversion formula relating theta functions and hypergeometric series. Our alternative approach is based on a Lie symmetry group for the differential equations satisfied by certain Eisenstein series. We employ similar arguments to obtain parameterizations from Ramanujan's alternative signatures and those associated with the inversion formula for the modular J-function. Moreover, we show that these parameterizations represent the only possible signatures under a certain assumed form for the Lie group parameters.     

Forced oscillation of certain neutral hyperbolic equations with continuous distributed deviating arguments

In this paper, we consider certain hyperbolic equations with continuous distributed deviating arguments, and sufficient conditions are presented for every solution of some boundary value problems to be oscillatory in a cylindrical domain. Our approach is to reduce multi-dimensional problems to one-dimensional problems by using some integral means of solutions.     

Necessary optimality conditions for a set-valued fractional extremal programming problem under inclusion constraints

In this paper, we are concerned with a set-valued fractional extremal programming problem under inclusion constraints. Our approach consists of using the extremal principle (an approach initiated by Mordukhovich, which does not involve any convex approximations and convex separation arguments) for the study of necessary optimality conditions.