3,527 Works

Nonlinear matroid optimization and experimental design

Jon Lee, Shmuel Onn, Robert Weismantel, Yael Berstein, Hugo Maruri-Aguilar, Eva Riccomagno & Henry P. Wynn
We study the problem of optimizing nonlinear objective functions over matroids presented by oracles or explicitly. Such functions can be interpreted as the balancing of multi-criteria optimization. We provide a combinatorial polynomial time algorithm for arbitrary oracle-presented matroids, that makes repeated use of matroid intersection, and an algebraic algorithm for vectorial matroids. Our work is partly motivated by applications to minimum-aberration model-fitting in experimental design in statistics, which we discuss and demonstrate in detail.

Yet another algorithm for the symmetric eigenvalue problem

Jared L. Aurentz, Thomas Mach, Raf Vandebril & David S. Watkins
In this paper we present a new algorithm for solving the symmetric matrix eigenvalue problem that works by first using a Cayley transformation to convert the symmetric matrix into a unitary one and then uses Gragg’s implicitly shifted unitary QR algorithm to solve the resulting unitary eigenvalue problem. We prove that under reasonable assumptions on the symmetric matrix this algorithm is backward stable and also demonstrate that this algorithm is comparable with other well known...

On Vietoris-Rips Complexes of Ellipses

Michal Adamaszek, Henry Adams & Samadwara Reddy

The Berry-Keating Operator on a Lattice

Jens Bolte, Sebastian Egger & Stefan Keppeler
We construct and study a version of the Berry-Keating operator with a built-in truncation of the phase space, which we choose to be a two-dimensional torus. The operator is a Weyl quantisation of the classical Hamiltonian for an inverted harmonic oscillator, producing a difference operator on a finite, periodic lattice. We investigate the continuum and the infinite-volume limit of our model in conjunction with the semiclassical limit. Using semiclassical methods, we show that a specific...

Real group orbits on flag ind-varieties of SL (∞, C)

Mikhail V. Ignatyev, Ivan Penkov & Joseph A. Wolf
We consider the complex ind-group $G=SL (\infty, \mathbb{C})$ and its real forms $G^0=SU(\infty,\infty)$, $SU(p,\infty)$, $SL(\infty,\mathbb{R})$, $SL(\infty,\mathbb{H})$. Our main object of study are the $G^0$-orbits on an ind-variety $G/P$ for an arbitrary splitting parabolic ind-subgroup $P \subset G$, under the assumption that the subgroups $G^0 \subset G$ and $P \subset G$ are aligned in a natural way. We prove that the intersection of any $G^0$-orbit on $G/P$ with a finite-dimensional flag variety $G_n/P_n$ from a given...

The T-Graph of a Multigraded Hilbert Scheme

Milena Hering & Diane Maclagan
The $T$-graph of a multigraded Hilbert scheme records the zero and one-dimensional orbits of the $T=(K^*)^n$ action on the Hilbert scheme induced from the $T$-action on $\mathbb{A}^n$. It has vertices the $T$-fixed points, and edges the onedimensional $T$-orbits. We give a combinatorial necessary condition for the existence of an edge between two vertices in this graph. For the Hilbert scheme of points in the plane, we give an explicit combinatorial description of the equations defining...

An Explicit Formula for the Dirac Multiplicities on Lens Spaces

Sebastian Boldt & Emilio A. Lauret
We present a new description of the spectrum of the (spin-) Dirac operator $D$ on lens spaces. Viewing a spin lens space $L$ as a locally symmetric space $\Gamma \setminus Spin(2m)/Spin(2m-1)$ and exploiting the representation theory of the Spin groups, we obtain explicit formulas for the multiplicities of the eigenvalues of $D$ in terms of infinitely many integer operations. As a consequence, we present conditions for lens spaces to be Dirac isospectral. Tackling classic questions...

On periodic solutions and global dynamics in a periodic differential delay equation

Anatoli F. Ivanov & Sergei I. Trofimchuk
Several aspects of global dynamics and the existence of periodic solutions are studied for the scalar differential delay equation $x'(t) = a(t)f(x([t-K]))$, where $f(x)$ is a continuous negative feedback function, $x \cdot f(x) < 0 x \neq 0, 0\leq a(t)$ is continuous $\omega$-periodic, $[\cdot]$ is the integer part function, and the integer $K \geq 0$ is the delay. The case of integer period $\omega$ allows for a reduction to finite-dimensional difference equations. The dynamics of...

Simulation of Multibody Systems with Servo Constraints through Optimal Control

Robert Altmann & Jan Heiland
We consider mechanical systems where the dynamics are partially constrained to prescribed trajectories. An example for such a system is a building crane with a load and the requirement that the load moves on a certain path. Modelling the system using Newton's second law- "The force acting on an object is equal to the mass of that object times its acceleration."- and enforcing the servo constraints directly leads to differential-algebraic equations (DAEs) of arbitrarily high...

On the δ=const Collisions of Singularities of Complex Plane Curves

Dmitry Kerner
We study a specific class of deformations of curve singularities: the case when the singular point splits to several ones, such that the total $\delta$ invariant is preserved. These are also known as equi-normalizable or equi-generic deformations. We restrict primarily to the deformations of singularities with smooth branches. A new invariant of the singular type is introduced: the dual graph. It imposes severe restrictions on the possible collisions/deformations. And allows to prove some bounds on...

Non-stationary multivariate subdivision: joint spectral radius and asymptotic similarity

Maria Charina, Costanza Conti, Nicola Guglielmi & Vladimir Protasov
In this paper we study scalar multivariate non-stationary subdivision schemes with a general integer dilation matrix. We present a new numerically efficient method for checking convergence and Hölder regularity of such schemes. This method relies on the concepts of approximate sum rules, asymptotic similarity and the so-called joint spectral radius of a finite set of square matrices. The combination of these concepts allows us to employ recent advances in linear algebra for exact computation of...

Plethysms, replicated schur functions and series, with applications to vertex operators

Bertfried Fauser, Peter D. Jarvis & Ronald C. King
Specializations of Schur functions are exploited to define and evaluate the Schur functions $s_\lambda [\alpha X]$ and plethysms $s_\lambda [\alpha s_\nu(X))]$ for any $\alpha$-integer, real or complex. Plethysms are then used to define pairs of mutually inverse infinite series of Schur functions, $M_\pi$ and $L_\pi$, specified by arbitrary partitions $\pi$. These are used in turn to define and provide generating functions for formal characters, $s^{(\pi)}_\lambda$, of certain groups $H_\pi$, thereby extending known results for orthogonal...

Very general monomial valuations of P2 and a Nagata type conjecture

Marcin Dumnicki, Brian Harbourne, Alex Küronya, Joaquim Roé & Tomasz Szemberg

Linear Syzygies, Hyperbolic Coxeter Groups and Regularity

Alexandru Constantinescu, Thomas Kahle & Matteo Varbaro
We build a new bridge between geometric group theory and commutative algebra by showing that the virtual cohomological dimension of a Coxeter group is essentially the regularity of the Stanley–Reisner ring of its nerve. Using this connection and techniques from the theory of hyperbolic Coxeter groups, we study the behavior of the Castelnuovo–Mumford regularity of square-free quadratic monomial ideals. We construct examples of such ideals which exhibit arbitrarily high regularity after linear syzygies for arbitrarily...

Reducing sub-modules of the Bergman module $\mathbb A^{(\lambda)}(\mathbb D^n)$ under the action of the symmetric group

Shibananda Biswas, Gargi Ghosh, Gadadhar Misra & Subrata Shyam Roy
The weighted Bergman spaces on the polydisc, $\mathbb A^{(\lambda)}(\mathbb D^n)$, $\lambda>0,$ splits into orthogonal direct sum of subspaces $\mathbb P_{\boldsymbol p}\big(\mathbb A^{(\lambda)}(\mathbb D^n)\big)$ indexed by the partitions $\boldsymbol p$ of $n,$ which are in one to one correspondence with the equivalence classes of the irreducible representations of the symmetric group on $n$ symbols. In this paper, we prove that each sub-module $\mathbb P_{\boldsymbol p}\big(\mathbb A^{(\lambda)}(\mathbb D^n)\big)$ is a locally free Hilbert module of rank equal...

Looking Back on Inverse Scattering Theory

David Colton & Rainer Kress
We present an essay on the mathematical development of inverse scattering theory for time-harmonic waves during the past fifty years together with some personal memories of our participation in these events.

Exact Rate of Convergence of k-Nearest-Neighbor Classification Rule

László Györfi, Maik Döring & Harro Walk
A binary classification problem is considered. The excess error probability of the k-nearest neighbor classification rule according to the error probability of the Bayes decision is revisited by a decomposition of the excess error probability into approximation and estimation error. Under a weak margin condition and under a modified Lipschitz condition, tight upper bounds are presented such that one avoids the condition that the feature vector is bounded.

On an Effective Variation of Kronecker’s Approximation Theorem Avoiding Algebraic Sets

Lenny Fukshansky, Oleg German & Nikolay Moshchevitin
Let $\Lambda \subset \mathbb R^n$ be an algebraic lattice, coming from a projective module over the ring of integers of a number field $K$. Let $\mathcal Z \subset \mathbb R^n$ be the zero locus of a finite collection of polynomials such that $\Lambda \nsubseteq \mathcal Z$ or a finite union of proper full-rank sublattices of $\Lambda$. Let $K_1$ be the number field generated over $K$ by coordinates of vectors in $\Lambda$, and let $L_1,\dots,L_t$ be...

The Colored Jones Polynomial and Kontsevich-Zagier Series for Double Twist Knots

Jeremy Lovejoy & Robert Osburn
Using a result of Takata, we prove a formula for the colored Jones polynomial of the double twist knots $K_{(-m,-p)}$ and $K_{(-m,p)}$ where $m$ and $p$ are positive integers. In the $(-m,-p)$ case, this leads to new families of $q$-hypergeometric series generalizing the Kontsevich-Zagier series. Comparing with the cyclotomic expansion of the colored Jones polynomials of $K_{(m,p)}$ gives a generalization of a duality at roots of unity between the Kontsevich-Zagier function and the generating function...

Experimenting with Zariski Dense Subgroups

Alla Detinko, Dane Flannery & Alexander Hulpke
We give a method to describe all congruence images of a finitely generated Zariski dense group $H\leq \SL(n, \Z)$. The method is applied to obtain efficient algorithms for solving this problem in odd prime degree $n$; if $n=2$ then we compute all congruence images only modulo primes. We propose a separate method that works for all $n$ as long as $H$ contains a known transvection. The algorithms have been implemented in {\sf GAP}, enabling computer...

A few shades of interpolation

Justyna Szpond
The topic of this snapshot is interpolation. In the ordinary sense, interpolation means to insert something of a different nature into something else. In mathematics, interpolation means constructing new data points from given data points. The new points usually lie in between the already-known points. The purpose of this snapshot is to introduce a particular type of interpolation, namely, polynomial interpolation. This will be explained starting from basic ideas that go back to the ancient...

Prony’s method: an old trick for new problems

Tomas Sauer
In 1795, French mathematician Gaspard de Prony invented an ingenious trick to solve a recovery problem, aiming at reconstructing functions from their values at given points, which arose from a specific application in physical chemistry. His technique became later useful in many different areas, such as signal processing, and it relates to the concept of sparsity that gained a lot of well-deserved attention recently. Prony’s contribution, therefore, has developed into a very modern mathematical concept.

Spaces of Riemannian metrics

Mauricio Bustamante & Jan-Bernhard Kordaß
Riemannian metrics endow smooth manifolds such as surfaces with intrinsic geometric properties, for example with curvature. They also allow us to measure quantities like distances, angles and volumes. These are the notions we use to characterize the “shape” of a manifold. The space of Riemannian metrics is a mathematical object that encodes the many possible ways in which we can geometrically deform the shape of a manifold.

The codimension

Antonio Lerario
In this snapshot we discuss the notion of codimension, which is, in a sense, “dual” to the notion of dimension and is useful when studying the relative position of one object insider another one.

A Well-Posedness Result for Viscous Compressible Fluids with Only Bounded Density

Raphaël Danchin, Francesco Fanelli & Marius Paicu
We are concerned with the existence and uniqueness of solutions with only bounded density for the barotropic compressible Navier-Stokes equations. Assuming that the initial velocity has slightly sub-critical regularity and that the initial density is a small perturbation (in the $L^\infty$ norm) of a positive constant, we prove the existence of local-in-time solutions. In the case where the density takes two constant values across a smooth interface (or, more generally, has striated regularity with respect...

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