The teaching-learning process is analyzed in a course for a group of professors who were taught subjects on Calculus, to study the episodes of problem-solving in them, focused on the identification ...of patterns and argumentation using counterexamples. The explanation and the argument in the classroom can be used together so that the argument (issued as a counterexample) supports the explanation (conjecture). Developing the mathematics class so that the above occurs is a form of interaction and how to encourage students to move from explanation to argumentation (placing a hybrid system). Furthermore, both forms of reasoning can influence dialogue protocols and strategies. In this work, the dialogue model is described as a tool to address the problem that arises when working with students.
We study the fractal pointwise convergence for the equation iħ∂tu+P(D)u=0, where the symbol P is real, homogeneous and non-singular. We prove that for initial data f∈Hs(Rn) with s>(n−α+1)/2 the ...solution u converges to fHα-a.e., where Hα is the α-dimensional Hausdorff measure. We improve upon this result depending on the dispersive strength of P. On the other hand, for a family of polynomials P and given α, we exploit a Talbot-like effect to construct initial data whose solutions u diverge in sets of Hausdorff dimension α. To compute the dimension of the sets of divergence, we adopt the Mass Transference Principle from Diophantine approximation. We also construct counterexamples for quadratic symbols like the saddle to show that our positive results are sometimes best possible.
In this article I will review McGee's famous counterexample to Modus Ponens and I will argue that it is not a real counterexample. I will claim that the problem lies in an infelicitous assertion of ...the second premise. As result of this diagnosis, I will suggest that when dealing with indicative conditionals a pragmatic theory is needed.
This paper presents a counterexample-guided iterative algorithm to compute convex, piecewise linear (polyhedral) Lyapunov functions for continuous-time piecewise linear systems. Polyhedral Lyapunov ...functions provide an alternative to commonly used polynomial Lyapunov functions. Our approach first characterizes intrinsic properties of a polyhedral Lyapunov function including its “eccentricity” and “robustness” to perturbations. We then derive an algorithm that either computes a polyhedral Lyapunov function proving that the system is asymptotically stable, or concludes that no polyhedral Lyapunov function exists whose eccentricity and robustness parameters satisfy some user-provided limits. Significantly, our approach places no a-priori bound on the number of linear pieces that make up the desired polyhedral Lyapunov function. The algorithm alternates between a learning step and a verification step, always maintaining a finite set of witness states. The learning step solves a linear program to compute a candidate Lyapunov function compatible with a finite set of witness states. In the verification step, our approach verifies whether the candidate Lyapunov function is a valid Lyapunov function for the system. If verification fails, we obtain a new witness. We prove a theoretical bound on the maximum number of iterations needed by our algorithm. We demonstrate the applicability of the algorithm on numerical examples.
A conjecture of Mallows and Sloane conveys the dominance of Hilbert series for finding basic invariants of finite linear groups if the Hilbert series of the invariant ring is of a certain explicit ...canonical form. However, the conjecture does not hold in general by a well-known counterexample of Stanley. In this article, we give a constraint on lower bounds for the degrees of homogeneous system of parameters of rings of invariants of finite linear groups depending on the universal denominator of Hilbert series defined by Derksen. We consider the conjecture with the universal denominator on abelian groups and provide some criteria guaranteeing the existence of homogeneous system of parameters of certain degrees. In this case, Stanley’s counterexample could be avoided, and the homogeneous system of parameters is optimal.
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