Related papers: Lectures on the Routh-Hurwitz problem
These are notes from a jointly taught class at the University of Chicago and lectures by the first author in Santa Cruz. Topics covered include: construction of moduli spaces of stable maps, Gromov-Witten invariants, quantum cohomology, and…
In this paper we revisit several recent results on monotone and strictly monotone Hurwitz numbers, providing new proofs. In particular, we use various versions of these numbers to discuss methods of derivation of quantum spectral curves…
We survey classical and recent developments in numerical linear algebra, focusing on two issues: computational complexity, or arithmetic costs, and numerical stability, or performance under roundoff error. We present a brief account of the…
These informal notes deal with Fourier series in one or more variables, Fourier transforms in one variable, and related matters.
This note deals with the mono-dimensional equation: $\d_t u -\d_x L(u_x) =f$ with $L(\cdot)$ merely monotone. The goal is to examine the features of facets -- flat regions of graphs of solutions appearing as $L(\cdot)$ suffers jumps. We…
In a Hilbert space setting, we study the stability properties of the regularized continuous Newton method with two potentials, which aims at solving inclusions governed by structured monotone operators. The Levenberg-Marquardt…
The purpose of these expository notes is to give a quick and elementary, yet rigorous, presentation of the rudiments of the theory of Viscosity Solutions for fully nonlinear 2nd order PDE, with applications to Calculus of Variations in the…
In this paper, we associate a class of Hurwitz matrix polynomials with Stieltjes positive definite matrix sequences. This connection leads to an extension of two classical criteria of Hurwitz stability for real polynomials to matrix…
In this note, we show that there exist solutions of the Muskat problem that shift stability regimes: they start unstable, then become stable, and finally return to the unstable regime. We also exhibit numerical evidence of solutions with…
These are notes from a basic course in Several Complex Variables
In this note, we have shown special case on Routh stability criterion, which is not discussed, in previous literature. This idea can be useful in computer science applications.
This text is a slightly edited version of lecture notes for a course I gave at ETH, during the Summer term 2001, to undergraduate Mathematics and Physics students. It covers a few selected topics from perturbation theory at an introductory…
We discuss technical results on learning function approximations using piecewise-linear basis functions, and analyze their stability and convergence using nonlinear contraction theory.
We introduce the notion of a stable instance for a discrete optimization problem, and argue that in many practical situations only sufficiently stable instances are of interest. The question then arises whether stable instances of NP--hard…
This article considers a Cauchy problem of Helmholtz equations whose solution is well known to be exponentially unstable with respect to the inputs. In the framework of variational quasi-reversibility method, a Fourier truncation is applied…
Hurwitz theory provides a large variety of enumerative problems related to algebraic geometry, mathematical physics, and combinatorics. We give a general framework to approach the large genus asymptotics of Hurwitz theory using only…
We establish various certifying determinantal representation results for a polynomial that contains as a factor a prescribed multivariable polynomials that is strictly stable on a tube domain. The proofs use a Cayley transform in…
These lecture notes are devoted to selected topics related to the uncertainty principle in harmonic analysis. Rather than attempting a systematic treatment, we emphasize only a number of both classical and deep manifestations of this…
Consider polynomial sequences that satisfy a first-order differential recurrence. We prove that if the recurrence is of a special form, then the Tur\'an expressions for the sequence are weakly Hurwitz stable (non-zero in the open right…
Here I share a few notes I used in various course lectures, talks, etc. Some may be just calculations that in the textbooks are more complicated, scattered, or less specific; others may be simple observations I found useful or curious.