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Despite plethora of research that attends to the convincing power of different types of proofs, research related to the convincing power of counterexamples is rather slim. In this paper we examine how prospective and practicing secondary school mathematics teachers respond to different types of counterexamples. The counterexamples were presented as products of students' arguments, and the participants were asked to evaluate their correctness and comment on them. The counterexamples varied according to mathematical topic: algebra or geometry, and their explicitness. However, as we analyzed the data, we discovered that these distinctions were insufficient to explain why teachers accepted some counterexamples, but rejected others, with seemingly similar features. As we analyze the participants' perceived transparency of different counterexamples, we employ various theoretical approaches that can advance our understanding of teachers' conceptions of conviction with respect to counterexamples. [ABSTRACT FROM AUTHOR]

The object of this paper is to solve the generalized radical cubic functional equation, and discuss the stability problem in quasi-β-Banach spaces and then the stability by using subadditive and subquadratic functions in (β; p)-Banach spaces for the generalized radical cubic functional equation. Also certain non-stability results are investigated via specific counterexamples. Our results are generalization of the main results which are established by Z. Alizadeh and A. G. Ghazanfari in 2016. [ABSTRACT FROM AUTHOR]

Providing compact and understandable counterexamples for violated system properties is an essential task in model checking. Existing works on counterexamples for probabilistic systems so far computed either a large set of system runs or a subset of the system's states, both of which are of limited use in manual debugging. Many probabilistic systems are described in a guarded command language like the one used by the popular model checker PRISM. In this paper we describe how a smallest possible subset of the commands can be identified which together make the system erroneous. We additionally show how the selected commands can be further simplified to obtain a well-understandable counterexample. [ABSTRACT FROM AUTHOR]

The Cantor-Schroeder-Bernstein theorem [Can97][Sch98][Ber05] states that any two sets that have injections into each other have the same cardinality, i.e. there is a bijection between them. Another way to phrase this is if two sets A;B have monomorphisms from A to B and B to A, then they are isomorphic in the setting of sets. One naturally wonders if this may be extended to other commonly studied systems of sets with structure and functions which preserve that structure. Given two objects with injective structure preserving maps between them are the structures of these objects the same? In other words, would these two objects be isomorphic in their respective setting? We see that in vector spaces, which are determined completely by their bases sets, this is true. However, when the objects are graphs, groups, rings or topological spaces, one may find counterexamples to such an extension. This is interesting, as it contradicts the naive intuition that two objects which are "subobjects" of each other must be the same. In this paper we provide some of these counterexamples. [ABSTRACT FROM AUTHOR]

Current stochastic model checkers do not make counterexamples for property violations readily available. In this paper, we apply directed explicit state-space search to discrete and continuous-time Markov chains in order to compute counterexamples for the violation of PCTL or CSL properties. Directed explicit state-space search algorithms explore the state space on-the-fly, which makes our method very efficient and highly scalable. They can also be guided using heuristics which usually improve the performance of the method. Counterexamples provided by our method have two important properties. First, they include those traces which contribute the greatest amount of probability to the property violation. Hence, they show the most probable offending execution scenarios of the system. Second, the obtained counterexamples tend to be small. Hence, they can be effectively analyzed by a human user. Both properties make the counterexamples obtained by our method very useful for debugging purposes. We implemented our method based on the stochastic model checker PRISM and applied it to a number of case studies in order to illustrate its applicability. [ABSTRACT FROM AUTHOR]

In his 1976 book, Proofs and Refutations, Lakatos presents a collection of case studies to illustrate methods of mathematical discovery in the history of mathematics. In this paper, we reframe these methods in ways that we have found make them more amenable for use as a framework for research on learning and teaching mathematics. We present an episode from an undergraduate abstract algebra classroom to illustrate the guided reinvention of mathematics through processes that strongly parallel those described by Lakatos. Our analysis suggests that the constructs described by Lakatos can provide a useful framework for making sense of the mathematical activity in classrooms where students are actively engaged in the development of mathematical ideas and provide design heuristics for instructional approaches that support the learning of mathematics through the process of guided reinvention. [ABSTRACT FROM AUTHOR]

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In this paper, we present counterexamples showing that for any p ∈ ( 1 , ∞ ) , p ≠ 2 , there is a non-divergence form uniformly elliptic operator with piecewise constant coefficients in R 2 (constant on each quadrant in R 2 ) for which there is no W p 2 estimate. The corresponding examples in the divergence case are also discussed. One implication of these examples is that the ranges of p are sharp in the recent results obtained in [4,5] for non-divergence type elliptic and parabolic equations in a half space with the Dirichlet or Neumann boundary condition when the coefficients do not have any regularity in a tangential direction. [ABSTRACT FROM AUTHOR]

Abstract: Probabilistic annotations generalise standard Hoare Logic to quantitative properties of probabilistic programs. They can be used to express critical expected values over program variables that must be maintained during program execution. As for standard program development, probabilistic assertions can be checked mechanically relative to an appropriate program semantics. In the case that a mechanical prover is unable to complete such validity checks then a counterexample to show that the annotation is incorrect can provide useful diagnostic information. In this paper, we provide a definition of counterexamples as failure traces for probabilistic assertions within the context of the pB language , an extension of the standard B method to cope with probabilistic programs. In addition, we propose algorithmic techniques to find counterexamples where they exist, and suggest a ranking mechanism to return ‘the most useful diagnostic information’ to the pB developer to aid the resolution of the problem. [Copyright &y& Elsevier]

In this paper, we give some counterexamples to the article “Stability properties of nonlinear difference equations and conditions for boundedness”, Computers Math. Applic., 38 (2) 29–35 (1999). [Copyright &y& Elsevier]

In this paper, we show that Theorems 4.2, 4.3 and 4.6 in [Youness, J. Optim. Theory Appl. 102 (1999) 439–450] are incorrect by giving some counterexamples. We introduce a new class of semi-E-convex function and discuss some its basic properties. [Copyright &y& Elsevier]

Following several papers in the prior literature, we study the relationship between order bounded operators, topologically bounded operators and topologically continuous operators. Our main contribution is two-folded: (i) we provide a set of counterexamples to illustrate several extant results in the literature; (ii) we give conditions for the space of order bounded operators to coincide with the space of topologically bounded operators as well as conditions for these two spaces to coincide with the space of topologically continuous operators.

Generalization is an important operation for programs that learn. Anti-unification guarantees the existence of a term which is the most specific generalization of a collection of terms. This provides a formal basis for learning from examples. However, such generalization may be "too" general and therefore counterexamples are required to restrict them. In this case, the learner has to check whether the resulting formula provides a concept to learn, i.e. whether a formula of the form t/{t1,...,tn} is a generalization, where t is viewed as a generalization of a set of examples and t1,...,tn are counterexample. The formula t/{t1,...,tn} is often called an implicit representation. The implicit representation t/{t1,...,tn} is a generalization iff there exist ground (variable-free) instances of t which are not instances of t1,...,tn. This is a non-trivial task even when there is no background knowledge (see [15]). We present here a method to check whether an implicit representation t/{t1,...,tn} is a generalization with respect to a finite set of equations which describes the background knowledge problem, i.e. whether there exists a ground instance of t which is not equivalent to any ground instance of t1,...,tn with respect to a set E of equations. Whereas this problem is in general undecidable since the equality is so, we show in this paper that in this case where the set E of equations is compiled into a ground convergent term rewriting system, we can discover concepts in Universe of Discourse with background knowledge described by a finite set of equations. We show how the method applies to induce concepts in theories of elementary arithmetic. [ABSTRACT FROM AUTHOR]