This note discusses proofs for convergence of first-order methods based on simple potentialfunction arguments. We cover methods like gradient descent (for both smooth and non-smooth settings), mirror descent, and some accelerated variants.

‎In this work‎, ‎we consider the nonlinear first-order evolution‎ ‎equations‎: ‎$u_t=f(x,t,u,u_x,u_{xx})$ for $0 ‎to initial condition $u(x,0)=g(x)$‎, ‎where $u$ is a function of‎ ‎$x$ and $t$ and $f$ is a known analytic function‎. ‎The purpose of‎ ‎this paper is to introduce the method of RBF to existing method‎ ‎in solving nonlinear first-ord...

This paper investigates the complexity of query problem for first-order formulas on quasi-unary signatures, that is, on vocabularies made of a single unary function and any number of monadic predicates. We first prove a form of quantifier elimination result: any query defined by a quasi-unary first-order formula can be equivalently defined, up to a suitable linear-time reduction, by a quantifie...

Let s(n) denote the sum of the positive divisors of n except for n iteself. Discussed since Pythagoras, s may be the first function of mathematics. Pythagoras also suggested iterating s, so perhaps considering the first dynamical system. The historical legacy has left us with some colorful and attractive problems, mostly still unsolved. Yet the efforts have been productive in the development of...

The first measurement of F γ 2,c is presented. At low x the measurement indicates a non-zero hadron-like component to F γ 2,c. At large x the measurement constitutes a test of perturbative QCD at next-to-leading order, with only mc and αs as free parameters, with a precision of O(40%). INTRODUCTION For about 20 years measurements of photon structure functions give deep insight into the rich str...

BACKGROUND We tested the hypothesis that schizophrenia is primarily a frontostriatal disorder by examining executive function in first-episode patients. Previous studies have shown either equal decrements in many cognitive domains or specific deficits in memory. Such studies have grouped test results or have used few executive measures, thus, possibly losing information. We, therefore, measured...