CyberSec Research

The $n$-th Christoffel function for a point $z_0\in\mathbb C$ and a finite measure $\mu$ supported on a Jordan arc $\Gamma$ is \[ \lambda_n(\mu,z_0)=\inf\left\{\int_\Gamma |P|^2d\mu\mid P\text{ is a polynomial of degree at most }n\text{ and } P(z_0)=1\right\}. \] It is natural to extend this notion to $z_0=\infty$ and define $\lambda_n(\mu,\infty)$ to be the infimum of the squared $L^2(\mu)$-norm over monic polynomials of degree $n$. The classical Szeg\H{o} theorem provides an asymptotic description of $\lambda_n(\mu,z_0)$ for $|z_0|>1$ and $z_0=\infty$ and arbitrary finite measures supported on the unit circle. Widom has proved a version of Szeg\H{o}'s theorem for measures supported on $C^{2+}$-Jordan arcs for the point $z_0=\infty$ and purely absolutely continuous measures belonging to the Szeg\H{o} class. We extend this result in two directions. We prove explicit asymptotics of $\lambda_n(\mu,z_0)$ for any finite measure $\mu$ supported on a $C^{1+}$-Jordan arc $\Gamma$, and for all points $z_0\in\mathbb{C}\cup\{\infty\}\setminus\Gamma$. Moreover, if the measure is in the Szeg\H{o} class, we provide explicit asymptotics for the extremal and orthogonal polynomials.

In this article, we continue the development of the Riemann-Hilbert formalism for studying the asymptotics of Toeplitz+Hankel determinants with non-identical symbols, which we initiated in \cite{GI}. In \cite{GI}, we showed that the Riemann-Hilbert problem we formulated admits the Deift-Zhou nonlinear steepest descent analysis, but with a special restriction on the winding numbers of the associated symbols. In particular, the most natural case, namely zero winding numbers, is not allowed. A principal goal of this paper is to develop a framework that extends the asymptotic analysis of Toeplitz+Hankel determinants to a broader range of winding-number configurations. As an application, we consider the case in which the winding numbers of the Szeg\H{o}-type Toeplitz and Hankel symbols are zero and one, respectively, and compute the asymptotics of the norms of the corresponding system of orthogonal polynomials.

We study an energy minimization problem $\sum_{i \neq j} W(z_i - z_j)$ for $N$ points $\left\{z_1, \dots, z_N\right\}$ with applications in dislocation theory. The $N$ points lie in the two-dimensional domain $\mathbb{R} \times [-\pi, \pi]$, %who are trying to minimize their interaction energy where where the kernel $W$ is derived from the Volterra potential $V(x,y) = \frac{x^2}{x^2+y^2}-\frac12\log(x^2+y^2)$. We prove that the minimum energy is given by $- N \log{N} +\mathcal{O}(N)$. This lower bound recovers the leading order term of the Read-Shockley law characterizing the energy of small angle grain boundaries in polycrystals.

Consider a Hermitian operator $A$ acting on a complex Hilbert space of dimension $2^n$. We show that when $A$ has small degree in the Pauli expansion, or in other words, $A$ is a local $n$-qubit Hamiltonian, its operator norm can be approximated independently of $n$ by maximizing $|\braket{\psi|A|\psi}|$ over a small collection $\mathbf{X}_n$ of product states $\ket{\psi}\in (\mathbf{C}^{2})^{\otimes n}$. More precisely, we show that whenever $A$ is $d$-local, \textit{i.e.,} $\deg(A)\le d$, we have the following discretization-type inequality: \[ \|A\|\le C(d)\max_{\psi\in \mathbf{X}_n}|\braket{\psi|A|\psi}|. \] The constant $C(d)$ depends only on $d$. This collection $\mathbf{X}_n$ of $\psi$'s, termed a \emph{quantum norm design}, is independent of $A$, and consists of product states, and can have cardinality as small as $(1+\eps)^n$, which is essentially tight. Previously, norm designs were known only for homogeneous $d$-localHamiltonians $A$ \cite{L,BGKT,ACKK}, and for non-homogeneous $2$-local traceless $A$ \cite{BGKT}. Several other results, such as boundedness of Rademacher projections for all levels and estimates of operator norms of random Hamiltonians, are also given.

We show that the frame measure function of a frame in certain reproducing kernel Hilbert spaces on metric measure spaces is given by the reciprocal of the Beurling density of its index set. In addition, we show that each such frame with Beurling density greater than one contains a subframe with Beurling density arbitrary close to one. This confirms that the concept of frame measure function as introduced by Balan and Landau is a meaningful quantitative definition for the redundancy of a large class of infinite frames. In addition, it shows that the necessary density conditions for sampling in reproducing kernel Hilbert spaces obtained by F\"uhr, Gr\"ochenig, Haimi, Klotz and Romero are optimal. As an application, we also settle the open questions of the existence of frames near the critical density for exponential frames on unbounded sets and for nonlocalized Gabor frames. The techniques used in this paper combine a selector form of Weaver's conjecture and various methods for quantifying the overcompleteness of frames.

We study a family of fractional integral operators defined on Heisenberg groups. The kernels of these operators satisfy Zygmund dilations. We obtain a Hardy-Littlewood-Sobolev type inequality.

We establish dimensional thresholds for dot product sets associated with compact subsets of translated paraboloids. Specifically, we prove that when the dimension of such a subset exceeds $ \frac{5}{4} = \frac{3}{2} - \frac{1}{4} $ in $\mathbb{R}^3$, and $ \frac{d}{2} - \frac{1}{4} - \frac{1}{8d - 4} $ in $\mathbb{R}^d$ for $d\geq 4$, its dot product set has positive Lebesgue measure. This result demonstrates that if a compact set in $ \mathbb{R}^d $ exhibits a paraboloidal structure, then the usual dimensional barrier of $ \frac{d}{2} $ for dot product sets can be lowered for $ d \geq 3 $. Our work serves as the continuous counterpart of a paper by Che-Jui Chang, Ali Mohammadi, Thang Pham, and Chun-Yen Shen, which examines the finite field setting with partial reliance on the extension conjecture. The key idea, closely following their paper, is to reformulate the dot product set on the paraboloid as a variant of a distance set. This reformulation allows us to leverage state-of-the-art results from the pinned distance problem, as established by Larry Guth, Alex Iosevich, Yumeng Ou, and Hong Wang for $ d = 2 $, and by Xiumin Du, Yumeng Ou, Kevin Ren, and Ruixiang Zhang for higher dimensions. Finally, we present explicit constructions and existence proofs that highlight the sharpness of our results.

Let $\{e^{-tL^{\alpha}}\}_{t>0}$ be the fractional Schr\"{o}dinger semigroup associated with $L=-\Delta+V$, where $V$ is a non-negatvie potential belonging to the reverse H\"{o}lder class. In this paper, we establish weighted boundedness properties of the variation operator related to $\{e^{-tL^{\alpha}}\}_{t>0}$, including weighted $L^{p}-L^{q}$ quantitative inequalities and mixed weak-type inequalities.

In this article, the authors survey and review the studies of boundary value problems for regular functions in Clifford analysis, which include theoretical foundations and useful methods. Its theoretical bases consist of the generalized Cauchy theorem, the generalized Cauchy integral formula, the Painlev\'{e} theorem and boundary behaviors of the Cauchy type integrals, as well as various integral representations. Certain boundary value problems in the Clifford algebra setting and singular integral equations are introduced.

Given an integer $a\ge 1$, a function $f: \mathbb{R}\to \mathbb{R}$ is said to be $a$-subadditive if $$ f(ax+y) \le af(x)+f(y) \,\,\,\text{ for all }x,y \in \mathbb{R}. $$ Of course, $1$-subadditive functions (which correspond to ordinary subadditive functions) are $2$-subadditive. % and $3$-subadditive. Answering a question of Matkowski, we show that there exists a continuous function $f$ satisfying $f(0)=0$ which is $2$-subadditive but not $1$-subadditive. In addition, the same example is not $3$-subadditive, which shows that the sequence of families of continuous $a$-subadditive functions passing through the origin is not increasing with respect to $a$. The construction relies on a perturbation of a given subadditive function with an even Gaussian ring, which will destroy the original subadditivity while keeping the weaker property. Lastly, given a positive rational cone $H\subseteq (0,\infty)$ which is not finitely generated, we prove that there exists a subadditive bijection $f:H\to H$ such that $\liminf_{x\to 0}f(x)=0$ and $\limsup_{x\to 0}f(x)=1$. This is related an open question of Matkowski and {\'S}wi{\k a}tkowski in [Proc. Amer. Math. Soc. 119 (1993), 187--197].

In [Jalowy, Kabluchko, Marynych, arXiv:2504.11593v1, 2025], the authors discuss a user-friendly approach to determine the limiting empirical zero distribution of a sequence of real-rooted polynomials, as the degree goes to $\infty$. In this note, we aim to apply it to a vast range of examples of polynomials providing a unifying source for limiting empirical zero distributions. We cover Touchard, Fubini, Eulerian, Narayana and little $q$-Laguerre polynomials as well as hypergeometric polynomials including the classical Hermite, Laguerre and Jacobi polynomials. We construct polynomials whose empirical zero distributions converge to the free multiplicative normal and Poisson distributions. Furthermore, we study polynomials generated by some differential operators. As one inverse result, we derive coefficient asymptotics of the characteristic polynomial of random covariance matrices.

We study those measures whose doubling constant is the least possible among doubling measures on a given metric space. It is shown that such measures exist on every metric space supporting at least one doubling measure. In addition, a connection between minimizers for the doubling constant and superharmonic functions is exhibited. This allows us to show that for the particular case of the euclidean space $\mathbb R^d$, Lebesgue measure is the only minimizer for the doubling constant (up to constant multiples) precisely when $d=1$ or $d=2$, while for $d\geq3$ there are infinitely many independent minimizers. Analogously, in the discrete setting, we can show uniqueness of the counting measure as a minimizer for regular graphs where the standard random walk is a recurrent Markov chain. The counting measure is also shown to be a minimizer in every infinite graph where the cardinality of balls depends solely on their radii.

In the present paper, we study the asymptotic properties of the semi-exponential Post-Widder operator. It is connected with $p(x) = x^2$. The main result is a pointwise complete asymptotic expansion valid for locally smooth functions of exponential growth. All coefficients are derived and explicitly given. As a special case we recover the complete asymptotic expansion for the classical Post-Widder operator.

The Tomas-Stein inequality for a compact subset $\Gamma$ of the sphere $S^d$ states that the mapping $f\mapsto \widehat{f\sigma}$ is bounded from $L^2(\Gamma,\sigma)$ to $L^{2+4/d}(\R^{d+1})$. Then conditional on a strict comparison between the best constants for the sphere and for the Strichartz inequality for the Schr\"odinger equations, we prove that there exist functions which extremize this inequality, and any extremising sequence has a subsequence which converges to an extremizer. The method is based on the refined Tomas-Stein inequality for the sphere and the profile decompositions. The key ingredient to establish orthogonality in profile decompositions is that we use Tao's sharp bilinear restriction theorem for the paraboloids beyond the Tomas-Stein range. Similar results have been previously established by Frank, Lieb and Sabin \cite{Frank-Lieb-Sabin:2007:maxi-sphere-2d}, where they used the method of the missing mass.

We study orthogonal polynomials on a fully symmetric planar domain $\Omega$ that is generated by a certain triangle in the first quadrant. For a family of weight functions on $\Omega$, we show that orthogonal polynomials that are even in the second variable on $\Omega$ can be identified with orthogonal polynomials on the unit disk composed with a quadratic map, and the same phenomenon can be extended to the domain generated by the rotation of $\Omega$ in higher dimensions. The connection allows an immediate deduction of results for approximation and Fourier orthogonal expansions on these fully symmetric domains. It applies, for example, to analysis on a double cone or a double hyperboloid.

We study the dynamics of a delayed predator-prey system with Holling type II functional response, focusing on the interplay between time delay and carrying capacity. Using local and global Hopf bifurcation theory, we establish the existence of sequences of bifurcations as the delay parameter varies, and prove that the connected components of global Hopf branches are nested under suitable conditions. A novel contribution is to show that the classical limit cycle of the non-delayed system belongs to a connected component of the global Hopf bifurcation in Fuller's space. Our analysis combines rigorous functional differential equation theory with continuation methods to characterize the structure and boundedness of bifurcation branches. We further demonstrate that delays can induce oscillatory coexistence at lower carrying capacities than in the corresponding ODE model, yielding counterintuitive biological insights. The results contribute to the broader theory of global bifurcations in delay differential equations while providing new perspectives on nonlinear population dynamics.

Systems of N = 1, 2, . . . first-order hyperbolic conservation laws feature N undamped waves propagating at finite speeds. On their own hand, multi-step Finite Difference and lattice Boltzmann schemes with q = N + 1, N + 2, . . . unknowns involve N ''physical'' waves, which are aimed at being as closely-looking as possible to the ones of the PDEs, and q-N ''numerical-spurious-parasitic'' waves, which are subject to their own speed of propagation, and either damped or undamped. The whole picture is even more complicated in the discrete setting-as numerical schemes act as dispersive media, thus propagate different harmonics at different phase (and group) velocities. For compelling practical reasons, simulations must always be conducted on bounded domains, even when the target problem is unbounded in space. The importance of transparent boundary conditions, preventing artificial boundaries from acting as mirrors producing polluting ricochets, naturally follows. This work presents, building on Besse, Coulombel, and Noble [ESAIM: M2AN, 55 (2021)], a systematic way of developing perfectly transparent boundary conditions for lattice Boltzmann schemes tackling linear problems in one and two space dimensions. Our boundary conditions are ''perfectly'' transparent, at least for 1D problems, as they absorb both physical and spurious waves regardless of their frequency. After presenting, in a simple framework, several approaches to handle the fact that q > N , we elect the so-called ''scalar'' approach (which despite its name, also works when N > 1) as method of choice for more involved problems. This method solely relies on computing the coefficients of the Laurent series at infinity of the roots of the dispersion relation of the bulk scheme. We insist on asymptotics for these coefficients in the spirit of analytic combinatorics. The reason is two-fold: asymptotics guide truncation of boundary conditions to make them depending on a fixed number of past time-steps, and make it clearduring the process of computing coefficients-whether intermediate quantities can be safely stored using floating-point arithmetic or not. Numerous numerical investigations in 1D and 2D with N = 1 and 2 are carried out, and show the effectiveness of the proposed boundary conditions.

We study families of spherical metrics on the flat torus $E_{\tau}$ $=$ $\mathbb{C}/\Lambda_{\tau}$ with blow-up behavior at prescribed conical singularities at $0$ and $\pm p$, where the cone angle at $0$ is $6\pi$, and at $\pm p$ is $4\pi$. We prove that the existence of such a necessarily unique, even family of spherical metrics is completely determined by the geometry of the torus: such a family exists if and only if\textbf{ }the Green function $G(z;\tau)$ admits a pair of nontrivial critical points $\pm a$. In this case, the cone point $p$ must equal $a$, and the corresponding monodromy data is $\left( 2r,2s\right) $, where $a=r+s\tau.$ An explicit transformation relating this family to the one with a single conical singularity of angle $6\pi$ at the origin is established in Theorem 1.4. A rigidity result for rhombic tori is proved in Theorem 1.5.