Vertices of FFLV polytopes

FFLV polytopes describe monomial bases in irreducible representations of \(\mathfrak {sl}_n\) and \(\mathfrak {sp}_{2n}\). We study various sets of vertices of FFLV polytopes. First, we consider the special linear case. We prove the locality of the set of vertices with respect to the type A Dynkin diagram. Then we describe all the permutation vertices and after that we describe all the simple vertices and prove that their number is equal to the large Schröder number. Finally, we derive analogous results for symplectic Lie algebras.     

On Some Electroconvection Models

We consider a model of electroconvection motivated by studies of the motion of a two-dimensional annular suspended smectic film under the influence of an electric potential maintained at the boundary by two electrodes. We prove that this electroconvection model has global in time unique smooth solutions.     

High-Energy Waves in Superpolynomial FPU-Type Chains

We consider periodic traveling waves in FPU-type chains with superpolynomial interaction forces and derive explicit asymptotic formulas for the high-energy limit as well as bounds for the corresponding approximation error. In the proof we adapt twoscale techniques that have recently been developed by Herrmann and Matthies for chains with singular potential and provide an asymptotic ODE for the scaled distance profile.     

Numerical Estimation of Balanced and Falling States for Constrained Legged Systems

Instability and risk of fall during standing and walking are common challenges for biped robots. While existing criteria from state-space dynamical systems approach or ground reference points are useful in some applications, complete system models and constraints have not been taken into account for prediction and indication of fall for general legged robots. In this study, a general numerical framework that estimates the balanced and falling states of legged systems is introduced. The overall approach is based on the integration of joint-space and Cartesian-space dynamics of a legged system model. The full-body constrained joint-space dynamics includes the contact forces and moments term due to current foot (or feet) support and another term due to altered contact configuration. According to the refined notions of balanced, falling, and fallen, the system parameters, physical constraints, and initial/final/boundary conditions for balancing are incorporated into constrained nonlinear optimization problems to solve for the velocity extrema (representing the maximum perturbation allowed to maintain balance without changing contacts) in the Cartesian space at each center-of-mass (COM) position within its workspace. The iterative algorithm constructs the stability boundary as a COM state-space partition between balanced and falling states. Inclusion in the resulting six-dimensional manifold is a necessary condition for a state of the given system to be balanced under the given contact configuration, while exclusion is a sufficient condition for falling. The framework is used to analyze the balance stability of example systems with various degrees of complexities. The manifold for a 1-degree-of-freedom (DOF) legged system is consistent with the experimental and simulation results in the existing studies for specific controller designs. The results for a 2-DOF system demonstrate the dependency of the COM state-space partition upon joint-space configuration (elbow-up vs. elbow-down). For both 1- and 2-DOF systems, the results are validated in simulation environments. Finally, the manifold for a biped walking robot is constructed and illustrated against its single-support walking trajectories. The manifold identified by the proposed framework for any given legged system can be evaluated beforehand as a system property and serves as a map for either a specified state or a specific controller’s performance.     

Phase Separation Patterns from Directional Quenching

We study the effect of directional quenching on patterns formed in simple bistable systems such as the Allen–Cahn and the Cahn–Hilliard equation on the plane. We model directional quenching as an externally triggered change in system parameters, changing the system from monostable to bistable across a trigger line. We are then interested in patterns forming in the bistable region, in particular as the trigger progresses with small speed and increases this bistable region. We find existence and nonexistence results of single interfaces and striped patterns. For zero speed, we find stripes parallel or perpendicular to the trigger line and exclude stripes with an oblique orientation. Single interfaces are always perpendicular to the trigger line. For small positive speed, striped patterns can align perpendicularly. Other orientations are excluded in Allen–Cahn for all nonnegative speeds. Single interfaces for positive trigger speeds are excluded for Cahn–Hilliard and align perpendicularly in Allen–Cahn.     

Convergence of Asymptotic Systems of Non-autonomous Neural Network Models with Infinite Distributed Delays

In this paper, we investigate the global convergence of solutions of non-autonomous Hopfield neural network models with discrete time-varying delays, infinite distributed delays, and possible unbounded coefficient functions. Instead of using Lyapunov functionals, we explore intrinsic features between the non-autonomous systems and their asymptotic systems to ensure the boundedness and global convergence of the solutions of the studied models. Our results are new and complement known results in the literature. The theoretical analysis is illustrated with some examples and numerical simulations.     

Fregean Facts

Two important Fregean ideas appear to conflict. The first is that a thought can be decomposed in different ways, and the second is that a thought is constituted by the senses of its constituents. This paper is a defense of Dummett’s suggestion of a way to reconcile between those two theses through the claim that although the same thought can be structured in different ways by different sentences; one of the structures is privileged. My defense focuses on the charge raised by Levine (Ratio (new series), XIX, 43–63, 2006) that Dummett’s claim about the privileged structure of a thought conflicts with the Slingshot argument. I show that this charge is misconceived; a careful examination of the Slingshot argument’s methodology reveals that Dummett’s claim does not conflict with it.     

Non-Realist Cognitivism,Truth and Objectivity

In On What Matters, Derek Parfit defends a new metaethical theory, which he calls non-realist cognitivism. It claims that (i) normative judgments are beliefs; (ii) that some normative beliefs are true; (iii) that the normative concepts that are a part of the propositions that are the contents of normative beliefs are irreducible, unanalysable and of their own unique kind; and (iv) that neither the natural features of the reality nor any additional normative features of the reality make the relevant normative beliefs true. The aim of this article is to argue that Parfit’s theory is problematic because its defenders have no resources to make sense of the nature of normative truth, which is an essential element of their view. I do this by showing how the traditional theories of truth are not available for the non-realist cognitivists.     

Necessarily Incompatible Consistent Wants

This paper argues that the wants or desires of a person can be consistent with each other and still necessarily incompatible with each other and for interesting reasons. It is argued here that this problem is not rare and that there is no solution in sight.