CyberSec Research

For two systems of functions that are p-ary analogues of Rademacher chaos, their q-lacunarity is proved and their uniqueness sets are studied.

We show how to prove by means of the Lagrange inversion the limit of Arnol'd that $$ \lim_{x\to0}\frac{\sin(\tan x)-\tan(\sin x)}{\arcsin(\arctan x)-\arctan(\arcsin x)}=1\,. $$ In fact, we obtain a more general result in terms of formal power series.

In this paper we compute the Assouad and lower spectra of Bedford--McMullen sponges in $\mathbb{R}^3$ explicitly. According to the formulae established, we discover that the spectra are not determined by the ratio set, and the box, lower and Assouad dimensions of the sponge anymore, which is unlike the situation in a planar carpet. As a by-product, we construct two Bedford--McMullen sponges on the same grid, both of which have non-uniform fibres. Particularly, they share the same box, lower and Assouad dimensions. However, their Assouad type spectra are different and therefore they are not bi-Lipschitz equivalent. For Bedford--McMullen sponges in higher dimensions, we also determine the dimension spectra when $\theta$ is smaller than the minimal ratio or it is bigger than the maximal ratio.

In this paper, we obtain sharp $L^p$ improving estimates for maximal averages over nondegenerate surfaces of codimension $2$ in $\mathbb R^4$. We also establish local smoothing type estimates for the averages, which are accomplished by making use of multilinear restriction estimates and decoupling inequalities for two dimensional conic extension of two dimensional nondegenerate surfaces.

We study the number of real zeros of finite combinations of $K+1$ consecutive normalized Laguerre polynomials of the form $$ q_n(x)=\sum_{j=0}^K\gamma_j\tilde L^\alpha_{n-j}(x),\quad n\ge K, $$ where $\gamma_j$, $j=0,\cdots ,K$, are real numbers with $\gamma_0=1$, $\gamma_K\not =0$. We consider four different normalizations of Laguerre polynomials: the monic Laguerre polynomials $\hat L_n^\alpha$, the polynomials $\mathcal L_n^\alpha=n!L_n^\alpha/(1+\alpha)_n$ (so that $\mathcal L_n^\alpha(0)=1$), the standard Laguerre polynomials $(L_n^\alpha)_n$ and the Brenke normalization $L_n^\alpha/(1+\alpha)_n$. We show the key role played by the polynomials $Q(x)=\sum_{j=0}^K(-1)^j\gamma_j(x)_{K-j}$ and $P(x)=\sum_{j=0}^K\gamma_jx^{K-j}$ to solve this problem: $Q$ in the first case and $P$ in the second, third and forth cases. In particular, in the first case, if all the zeros of the polynomial $Q$ are real and less than $\alpha+1$, then all the zeros of $q_n$, $n\ge K$, are positive. In the other cases, if all the zeros of $P$ are real then all the zeros of $q_n$, $n\ge K$, are also real. If $P$ has $m>1$ non-real zeros, there are important differences between the four cases. For instance in the first case, $q_n$ has still only real zeros for $n$ big enough, but in the fourth case $q_n$ has exactly $m$ non-real zeros for $n$ big enough.

The exactly solvable Schr\"{o}dinger equations with the conventional shape-invariant potentials are known to be related with each other through point cannonical transformations. In this paper, we extend the idea to integral formulae called the SWKB integrals. By virtue of this, we derive extended forms of the SWKB quantization condition for certain classes of Natanzon potentials. We further demonstrate that the same idea can also be applied to obtain an exact quantization rule for a subclass of quantum systems with position-dependent effective masses, provided their solutions involve the classical orthogonal polynomials. Based on the findings, we conjecture about the implication of the exactness of the SWKB formula in relation to the classical orthogonal polynomials.

There are two equivalent descriptions of George Wilson's adelic Grassmannian $Gr^{ad}$, one in terms of differential ``conditions'' and another in terms of Calogero-Moser Pairs. The former approach was used in the 2020 paper by Kasman-Milson which found that each family of Exceptional Hermite Polynomials has a generating function which lives in $Gr^{ad}$. This suggests that Calogero-Moser Pairs should also be useful in the study of Exceptional Hermite Polynomials, but no researchers have pursued that line of inquiry prior to the first author's thesis. The purpose of this note is to summarize highlights from that thesis, including a novel formula for Exceptional Hermite Polynomials in terms of Calogero-Moser Pairs and a theorem utilizing this correspondence to produce explicit finitely-supported distributions which annihilate them.

In this paper we consider the Moreau's sweeping processes driven by a time dependent prox-regular set $C(t)$ which is continuous in time with respect to the asymmetric distance $e$ called the excess, defined by $e(A,B) := \sup_{x \in A} d(x,B)$ for every pair of sets $A$, $B$ in a Hilbert space. As observed by J.J. Moreau in his pioneering works, the excess provides the natural topological framework for sweeping process. Assuming a uniform interior cone condition for $C(t)$, we prove that the associated sweeping process has a unique solution, thereby improving the existing result on continuous prox-regular sweeping processes in two directions: indeed, in the previous literature $C(t)$ was supposed to be continuous in time with respect to the symmetric Hausdorff distance instead of the excess and also its boundary $\partial C(t)$ was required to be continuous in time, an assumption which we completely drop. Therefore our result allows to consider a much wider class of continuously moving constraints.

We establish a geometric characterization of the sets $E \subset \mathbb{R}^n$ so that the corresponding log-distance function $\log \text{dist}(\cdot, E)$ is in $BMO$ (or equivalently $\text{dist}(\cdot, E)^{-\alpha} \in A_\infty$ for some $\alpha > 0$), generalizing the $A_1$ characterization in the work by Anderson, Lehrb\"ack, Mudarra, and V\"ah\"akangas. The proof relies on a new median-value characterization of $BMO$: For a real-valued measurable function on $\mathbb{R}^n$ and constants $0 < s < t < 1$, \[\|f\|_{BMO} \approx_{s, t, n} \sup_{Q}[M_t(f, Q) - M_s(f, Q)]\] where $M_s(f, Q)$ denotes the $s$-median value of $f$ on $Q$. We apply our characterization to the study of Hardy-Sobolev inequalities.

Symmetry is a key property of numerical methods. The geometric properties of symmetric schemes make them an attractive option for integrating Hamiltonian systems, whilst their ability to exactly recover the initial condition without the need to store the entire solution trajectory makes them ideal for the efficient implementation of Neural ODEs. In this work, we present a Hopf algebraic approach to the study of symmetric B-series methods. We show that every B-series method can be written as the composition of a symmetric and "antisymmetric" component, and explore the structure of this decomposition for Runge-Kutta schemes. A major bottleneck of symmetric Runge-Kutta schemes is their implicit nature, which requires solving a nonlinear system at each step. By introducing a new set of order conditions which minimise the antisymmetric component of a scheme, we derive what we call Explicit and Effectively Symmetric (EES) schemes -- a new class of explicit Runge-Kutta schemes with near-symmetric properties. We present examples of second-order EES schemes and demonstrate that, despite their low order, these schemes readily outperform higher-order explicit schemes such as RK4 and RK5, and achieve results comparable to implicit symmetric schemes at a significantly lower computational cost.

Let $G=(V,E)$ be a finite, connected graph. We investigate a notion of boundary $\partial G \subseteq V$ and argue that it is well behaved from the point of view of potential theory. This is done by proving a number of discrete analogous of classical results for compact domains $\Omega \subset \mathbb{R}^d$. These include (1) an analogue of P\'olya's result that a random walk in $\Omega$ typically hits the boundary $\partial \Omega$ within $\lesssim \mbox{diam}(\Omega)^2$ units of time, (2) an analogue of the Faber-Krahn inequality, (3) an analogue of the Hardy inequality, (4) an analogue of the Alexandrov-Bakelman-Pucci estimate, (5) a stability estimate for hot spots and (6) a Theorem of Bj\"orck stating that probability measures $\mu$ that maximize $\int_{\Omega \times \Omega} \|x-y\|^{\alpha} d\mu(x) d\mu(y)$ are fully supported in the boundary.

We show that it is impossible to quantify the decay rate of a semi-uniformly stable operator semigroup based on sole knowledge of its spectrum. More precisely, given an arbitrary positive function $r$ vanishing at $\infty$, we construct a Banach space $X$ and a bounded semigroup $ (T(t))_{t \geq 0}$ of operators on it whose infinitesimal generator $A$ has empty spectrum $\sigma(A)=\varnothing$, but for which, for some $x \in X$, $$\limsup_{t\to\infty} \frac{\|T(t)A^{-1}x\|_{X}}{r(t)}=\infty.$$

We provide quantitative improvements to the Berezin-Li-Yau inequality and the Kr\"oger inequality, in $\mathbb{R}^n$, $n\ge 2$. The improvement on Kr\"oger's inequality resolves an open question raised by Weidl from 2006. The improvements allow us to show that, for any open bounded domains, there are infinite many Dirichlet eigenvalues satisfying P\'olya's conjecture if $n\ge 3$, and infinite many Neumann eigenvalues satisfying P\'olya's conjecture if $n\ge 5$ and the Neumann spectrum is discrete.

In recent work, Etayo introduces a new Bombieri-type inequality for monic polynomials. Here we reinterpret this new inequality as a more general integral inequality involving the Green function for the sphere. This rather geometric interpretation allows for generalizations of the basic inequality, involving fractional zeros while also opening up the possibility to extend the setting to general compact Riemann surfaces. We derive a sharp form of these generalized Bombieri-type inequalities for the case of the sphere and the torus. These inequalities involve a quantity we call the packing number, which in turn is inspired by the geometric zero packing problems considered by Hedenmalm in the context of the asymptotic variance of the Bergman projection of a bounded function. As for the torus, we introduce analogs of polynomials (pseudopolynomials) based on the classical Weierstrass $\sigma$ function, and we explain how such pseudopolynomials fit in with the extended geometric Bombieri-type inequality. The sharpness of the packing number bound on the torus involves the construction of a lattice configuration on the torus for any given integer number of points. The corresponding bound for the sphere instead relies on the existence of well-conditioned polynomials in the sense of Shub and Smale.

We explore a class of meromorphic functions on elliptic curves, termed \emph{elliptic orthogonal a-polynomials} ($a$-EOPs), which extend the classical notion of orthogonal polynomials to compact Riemann surfaces of genus one. Building on Bertola's construction of orthogonal sections, we study these functions via non-Hermitian orthogonality on the torus, establish their recurrence properties, and derive an analogue of the Christoffel--Darboux formula. We demonstrate that, under real-valued orthogonality conditions, $a$-EOPs exhibit interlacing and simplicity of zeros similar to orthogonal polynomials on the real line (OPRL). Furthermore, we construct a general correspondence between families of OPRL and elliptic orthogonal functions, including a decomposition into multiple orthogonality relations, and identify new interlacing phenomena induced by rational deformations of the orthogonality weight.

We give a definition of Radon hypergeometric function (Radon HGF) of confluent and nonconfluent type, which is a function on the Grassmannian Gr(m,nr) obtained as a Radon transform of a character of the universal covering group of H_{\lambda}\subset GL(nr) specified by a partition \lambda of n, where H_{(1,\dots,1)}\simeq(GL(r))^{n}. When r=1, the Radon HGF reduces to the Gelfand HGF on the Grassmannian. We give a system of differential equations satisfied by the Radon HGF and show that the Hermitian matrix integral analogues of Gauss HGF and its confluent family: Kummer, Bessel, Hermite-Weber and Airy function, are obtained in a unified manner as the Radon HGF on Gr(2r,4r) corresponding to the partitions (1,1,1,1), (2,1,1), (2,2), (3,1) and (4), respectively.

We characterize the collection of sets $E \subset \mathbb{R}^n$ for which there exists $\theta \in \mathbb{R}\setminus\{0\}$ such that the distance weight $w(x) = \operatorname{dist}(x, E)^\theta$ belongs to the Muckenhoupt class $A_p$, where $p > 1$. These sets exhibit a certain balance between the small-scale and large-scale pores that constitute their complement$-$a property we show to be more general than the so-called weak porosity condition, which in turn, and according to recent results, characterizes the sets with associated distance weights in the $A_1$ case. Furthermore, we verify the agreement between this new characterization and the properties of known examples of distance weights, that are either $A_p$ weights or merely doubling weights, by means of a probabilistic approach that may be of interest by itself.

We study the initial-value problem for the nonlinear Schr\"odinger equation. Application of the inverse scattering transform method involves solving direct and inverse scattering problems for the Zakharov-Shabat system with complex potentials. We solve these problems by using new series representations for the Jost solutions of the Zakharov-Shabat system. The representations have the form of power series with respect to a transformed spectral parameter. In terms of the representations, solution of the direct scattering problem reduces to computing the series coefficients following a simple recurrent integration procedure, computation of the scattering coefficients by multiplying corresponding pairs of polynomials (partial sums of the series representations) and locating zeros of a polynomial inside the unit disk. Solution of the inverse scattering problem reduces to the solution of a system of linear algebraic equations for the power series coefficients, while the potential is recovered from the first coefficients. The system is obtained directly from the scattering relations. Thus, unlike other existing techniques, the method does not involve solving the Gelfand-Levitan-Marchenko equation or the matrix Riemann-Hilbert problem. The overall approach leads to a simple and efficient algorithm for the numerical solution of the initial-value problem for the nonlinear Schr\"odinger equation, which is illustrated by numerical examples.