Mathematical Structures in Group Decision-Making on Resource Allocation Distributions.

Optimal decisions on the distribution of finite resources are explicitly structured by mathematical models that specify relevant variables, constraints, and objectives. Here we report analysis and evidence that implicit mathematical structures are also involved in group decision-making on resource allocation distributions under conditions of uncertainty that disallow formal optimization. A group's array of initial distribution preferences automatically sets up a geometric decision space of alternative resource distributions. Weighted averaging mechanisms of interpersonal influence reduce the heterogeneity of the group's initial preferences on a suitable distribution. A model of opinion formation based on weighted averaging predicts a distribution that is a feasible point in the group's implicit initial decision space

Non-Euclidean Contractivity of Recurrent Neural Networks

Critical questions in dynamical neuroscience and machine learning are related to the study of recurrent neural networks and their stability, robustness, and computational efficiency. These properties can be simultaneously established via a contraction analysis.This paper develops a comprehensive contraction theory for recurrent neural networks. First, for non-Euclidean ℓ 1 /ℓ ∞ logarithmic norms, we establish quasiconvexity with respect to positive diagonal weights and closed-form worst-case expressions over certain matrix polytopes. Second, for locally Lipschitz maps (e.g., arising as activation functions), we show that their one-sided Lipschitz constant equals the essential supremum of the logarithmic norm of their Jacobian. Third and final, we apply these general results to classes of recurrent neural circuits, including Hopfield, firing rate, Persidskii, Lur’e and other models. For each model, we compute the optimal contraction rate and corresponding weighted non-Euclidean norm via a linear program or, in some special cases, via a Hurwitz condition on the Metzler majorant of the synaptic matrix. Our non-Euclidean analysis establishes also absolute, connective, and total contraction properties

Non-Euclidean Contraction Analysis of Continuous-Time Neural Networks

Critical questions in dynamical neuroscience and machine learning are related to the study of continuous-time neural networks and their stability, robustness, and computational efficiency. These properties can be simultaneously established via a contraction analysis. This paper develops a comprehensive non-Euclidean contraction theory for continuous-time neural networks. First, for non-Euclidean $\ell_{1}/\ell_{\infty}$ logarithmic norms, we establish quasiconvexity with respect to positive diagonal weights and closed-form worst-case expressions over certain matrix polytopes. Second, for locally Lipschitz maps (e.g., arising as activation functions), we show that their one-sided Lipschitz constant equals the essential supremum of the logarithmic norm of their Jacobian. Third and final, we apply these general results to classes of continuous-time neural networks, including Hopfield, firing rate, Persidskii, Lur'e and other models. For each model, we compute the optimal contraction rate and corresponding weighted non-Euclidean norm via a linear program or, in some special cases, via a Hurwitz condition on the Metzler majorant of the synaptic matrix. Our non-Euclidean analysis establishes also absolute, connective, and total contraction properties

Non-Euclidean Monotone Operator Theory with Applications to Recurrent Neural Networks

We provide a novel transcription of monotone operator theory to the non-Euclidean finite-dimensional spaces $\ell_1$ and $\ell_{\infty}$. We first establish properties of mappings which are monotone with respect to the non-Euclidean norms $\ell_1$ or $\ell_{\infty}$. In analogy with their Euclidean counterparts, mappings which are monotone with respect to a non-Euclidean norm are amenable to numerous algorithms for computing their zeros. We demonstrate that several classic iterative methods for computing zeros of monotone operators are directly applicable in the non-Euclidean framework. We present a case-study in the equilibrium computation of recurrent neural networks and demonstrate that casting the computation as a suitable operator splitting problem improves convergence rates

Structural Balance via Gradient Flows over Signed Graphs

Structural balance is a classic property of signed graphs satisfying Heider's seminal axioms. Mathematical sociologists have studied balance theory since its inception in the 1940s. Recent research has focused on the development of dynamic models explaining the emergence of structural balance. In this paper, we introduce a novel class of parsimonious dynamic models for structural balance based on an interpersonal influence process. Our proposed models are gradient flows of an energy function, called the dissonance function, which captures the cognitive dissonance arising from violations of Heider's axioms. Thus, we build a new connection with the literature on energy landscape minimization. This gradient flow characterization allows us to study the transient and asymptotic behaviors of our model. We provide mathematical and numerical results describing the critical points of the dissonance function

Perturbations and chaos in quantum maps

The local density of states (LDOS) is a distribution that characterizes the effect of perturbations on quantum systems. Recently, it was proposed a semiclassical theory for the LDOS of chaotic billiards and maps. This theory predicts that the LDOS is a Breit-Wigner distribution independent of the perturbation strength and also gives a semiclassical expression for the LDOS witdth. Here, we test the validity of such an approximation in quantum maps varying the degree of chaoticity, the region in phase space where the perturbation is applying and the intensity of the perturbation. We show that for highly chaotic maps or strong perturbations the semiclassical theory of the LDOS is accurate to describe the quantum distribution. Moreover, the width of the LDOS is also well represented for its semiclassical expression in the case of mixed classical dynamics.Comment: 9 pages, 11 figures. Accepted for publication in Phys. Rev.

Moving constraints as stabilizing controls in classical mechanics

The paper analyzes a Lagrangian system which is controlled by directly assigning some of the coordinates as functions of time, by means of frictionless constraints. In a natural system of coordinates, the equations of motions contain terms which are linear or quadratic w.r.t.time derivatives of the control functions. After reviewing the basic equations, we explain the significance of the quadratic terms, related to geodesics orthogonal to a given foliation. We then study the problem of stabilization of the system to a given point, by means of oscillating controls. This problem is first reduced to the weak stability for a related convex-valued differential inclusion, then studied by Lyapunov functions methods. In the last sections, we illustrate the results by means of various mechanical examples.Comment: 52 pages, 4 figure

Exercise prescription to improve clinical practice on cancer patients suffering chemotherapy-induced peripheral neuropathy undergoing treatment: a systematic review

This document aims to summarize and analyze systematically the current body of evidence about the effects of specific exercise proto- cols on physical function, balance control and quality of life in patients with peripheral neuropathy (PNP) induced by chemotherapy. Methods: Systematic Review, Literature survey Specific terms were identified for the literature research in MEDLINE, Scopus, Bandolier, PEDro, and Web of Science. Only studies pub- lished in peer-reviewed journals written in English language were considered. Four manuscripts were classified as eligible with 88 total participants, with an average of 57.1 years old. Quality appraisal classified two studies as high quality investigations while two with low quality. Results were summarized in the following domains: \u201cCIPN symptoms\u201d, \u201cStatic balance control\u201d, \u201cDynamic balance control\u201d, \u201cQuali- ty of life and Physical function\u201d. Results Specific exercise protocols were able to counteract common symptoms of chemotherapy-induced peripheral neuropathy (CIPN) during chemotherapy treatments. Significant improvements were detected on postural control. Additionally, patients\u2019 quality of life and inde- pendence were found ameliorated after exercise sessions, together with reductions on altered sensations and in other peripheral neu- ropathy symptoms. Combined exercise protocols including endurance, strength and sensorimotor training showed larger improvements. Conclusions Exercise prescriptions for cancer patients undergoing chemotherapy with CIPN symptoms should be recommended since these exercise interventions appeared as feasible and have been demonstrated as useful tools to counteract some common side effects of chemother- apeutic agents