Related papers: First order operators on shrinking graph-like spac…
We apply the Cartan equivalence method to the study of real analytic second order ODEs under the local real analytic diffeomorphism of $\C^2$ which are area-preserving. This enables us to give a characterization of the second order ODEs…
We consider first order linear operators commuting with the operator appearing in the linearized equation of motion of Rarita-Schwinger fields which comes directly from the action. First we consider a simplified operator giving an equation…
We consider the D1D5 CFT near the orbifold point and develop methods for computing the mixing of untwisted operators to first order by using the OPE on the covering surface. We argue that the OPE on the cover encodes both the structure…
This paper gives a unified and succinct approach to the $O(1/\sqrt{k}), O(1/k),$ and $O(1/k^2)$ convergence rates of the subgradient, gradient, and accelerated gradient methods for unconstrained convex minimization. In the three cases the…
We propose potential-based analyses for first-order algorithms applied to constrained and composite minimization problems. We first propose ``idealized'' frameworks for algorithms in the strongly and non-strongly convex cases and argue…
There have been many parallel streams of research studying order isomorphisms of some specific sets $G$ of functions from a set $X$ to $\mathbb{R}\cup\{\pm\infty\}$, such as the sets of convex or Lipschitz functions. We develop in this…
We prove well-posedness for some abstract differential equations of the first order. Our result covers the usual case of Lipschitz composition operators. It also contains the case of some integro-differential operators acting on spaces with…
A link between first-order ordinary differential equations (ODEs) and 2-dimensional Riemannian manifolds is explored. Given a first-order ODE, an associated Riemannian metric on the variable space is defined, and some properties of the…
Standard macroscopic QED is built on the second-order Green's function for the electric field and discards open-system boundary terms. Here we develop a first-order electromagnetic operator approach that retains both $\mathbf{E}$ and…
A development of an inverse first-order divided difference operator for functions of several variables is presented. Two generalized derivative-free algorithms builded up from Ostrowski's method for solving systems of nonlinear equations…
In this paper we prove the first quantitative convergence rates for the graph infinity Laplace equation for length scales at the connectivity threshold. In the graph-based semi-supervised learning community this equation is also known as…
In this work, we consider a connected network of finitely many agents working cooperatively to solve a min-max problem with convex-concave structure. We propose a decentralised first-order algorithm which can be viewed as a non-trivial…
We propose a family of optimization methods that achieve linear convergence using first-order gradient information and constant step sizes on a class of convex functions much larger than the smooth and strongly convex ones. This larger…
The main result of this paper is a sharp upper bound on the first positive eigenvalue of Dirac operators in two dimensional simply connected $C^3$-domains with infinite mass boundary conditions. This bound is given in terms of a conformal…
We study various aspects of the first-order transduction quasi-order on graph classes, which provides a way of measuring the relative complexity of graph classes based on whether one can encode the other using a formula of first-order (FO)…
Recently, there has been significant progress in the development of distributed first order methods. (At least) two different types of methods, designed from very different perspectives, have been proposed that achieve both exact and linear…
Nowadays a great attention has been focused on the discrete fractional Laplace operator as the natural counterpart of the continuous one. In this paper, we discretize the fractional Laplace operator $(-\Delta)^{s}$ for an arbitrary finite…
We establish lower bounds on the complexity of finding $\epsilon$-stationary points of smooth, non-convex high-dimensional functions using first-order methods. We prove that deterministic first-order methods, even applied to arbitrarily…
In this article, we introduce a geometric and a spectral preorder relation on the class of weighted graphs with a magnetic potential. The first preorder is expressed through the existence of a graph homomorphism respecting the magnetic…
In a recent paper, Bubeck, Lee, and Singh introduced a new first order method for minimizing smooth strongly convex functions. Their geometric descent algorithm, largely inspired by the ellipsoid method, enjoys the optimal linear rate of…