An Optimal Estimate for the Anisotropic Logarithmic Potential

This paper introduces the new annulus body to establish the optimal lower bound for the anisotropic logarithmic potential as the complement to the theory of its upper bound estimate which has already been investigated. The connections with convex geometry analysis and some metric properties are also established. For the application, a polynomial dual log-mixed volume difference law is deduced from the optimal estimate.

Theory space of one unitary matrix model and its critical behavior associated with Argyres–Douglas theory

The lowest critical point of one unitary matrix model with cosine plus logarithmic potential is known to correspond with the [Formula: see text] Argyres–Douglas (AD) theory and its double scaling limit derives the Painlevé II equation with parameter. Here, we consider the critical points associated with all cosine potentials and determine the scaling operators, their vacuum expectation values (vevs) and their scaling dimensions from perturbed string equations at planar level. These dimensions agree with those of [Formula: see text] AD theory.

Oracle-Based Primal-Dual Algorithms for Packing and Covering Semidefinite Programs

Packing and covering semidefinite programs (SDPs) appear in natural relaxations of many combinatorial optimization problems as well as a number of other applications. Recently, several techniques have been proposed that utilize the particular structure of this class of problems in order to obtain more efficient algorithms than those offered by general SDP solvers. For certain applications, it may be necessary to deal with SDPs with a very large number of (e.g., exponentially or even infinitely many) constraints. In this chapter, we give an overview of some of the techniques that can be used to solve this class of problems, focusing on multiplicative weight updates and logarithmic-potential methods.

Type-based analysis of logarithmic amortised complexity

We introduce a novel amortised resource analysis couched in a type-and-effect system. Our analysis is formulated in terms of the physicist’s method of amortised analysis and is potentialbased. The type system makes use of logarithmic potential functions and is the first such system to exhibit logarithmic amortised complexity. With our approach, we target the automated analysis of self-adjusting data structures, like splay trees, which so far have only manually been analysed in the literature. In particular, we have implemented a semi-automated prototype, which successfully analyses the zig-zig case of splaying, once the type annotations are fixed.

Cahn–Hilliard equations on an evolving surface

We describe a functional framework suitable to the analysis of the Cahn–Hilliard equation on an evolving surface whose evolution is assumed to be given a priori. The model is derived from balance laws for an order parameter with an associated Cahn–Hilliard energy functional and we establish well-posedness for general regular potentials, satisfying some prescribed growth conditions, and for two singular non-linearities – the thermodynamically relevant logarithmic potential and a double-obstacle potential. We identify, for the singular potentials, necessary conditions on the initial data and the evolution of the surfaces for global-in-time existence of solutions, which arise from the fact that integrals of solutions are preserved over time, and prove well-posedness for initial data on a suitable set of admissible initial conditions. We then briefly describe an alternative derivation leading to a model that instead preserves a weighted integral of the solution and explain how our arguments can be adapted in order to obtain global-in-time existence without restrictions on the initial conditions. Some illustrative examples and further research directions are given in the final sections.

On the Viscous Cahn-Hilliard-Oono System with Chemotaxis and Singular Potential

We analyze a diffuse interface model that couples a viscous Cahn-Hilliard equation for the phase variable with a diffusion-reaction equation for the nutrient concentration. The system under consideration also takes into account some important mechanisms like chemotaxis, active transport as well as nonlocal interaction of Oono’s type. When the spatial dimension is three, we prove the existence and uniqueness of global weak solutions to the model with singular potentials including the physically relevant logarithmic potential. Then we obtain some regularity properties of the weak solutions when t>0. In particular, with the aid of the viscous term, we prove the so-called instantaneous separation property of the phase variable such that it stays away from the pure states ±1 as long as t>0. Furthermore, we study long-time behavior of the system, by proving the existence of a global attractor and characterizing its ω-limit set.

Method of variable separation for investigating exact solutions and dynamical properties of the time-fractional Fokker-Planck equation

Based on the idea of variable separation, the time-fractional Fokker-Planck equation with external force field is studied by using the property of Mittag-Leffler function and some special algorithm skills. In the cases of various external potential functions such as linear potential, harmonic potential, logarithmic potential, exponential potential, and quartic potential, exact solutions and dynamical properties of the above mentioned equation is investigated. The some interesting dynamical behaviors and phenomena are discovered. The profiles of some representative exact solutions are illustrated by 3D-graphs.

Solution for the degenerate scale for a rigid curve in antiplane elasticity by using a weakly singular integral equation

This paper provides a numerical solution for the degenerate scale for a rigid curve in antiplane elasticity. The degenerate scale problem for the rigid curve is formulated on the usage of the logarithmic potential. After assuming the displacement to be a vanishing value along the rigid curve, the boundary integral equation (BIE) is formulated. The problem can be first formulated in the degenerate scale. After making a coordinate transform, we can obtain the relevant BIE in the ordinary scale. Finally, a numerical solution is achieved. Several numerical examples are provided. In addition, the degenerate scale problem for the multiple rigid curves is also solved.