CyberSec Research

A Young subgroup of the symmetric group $\mathcal{S}_{N}$ with three factors, is realized as the stabilizer $G_{n}$ of a monomial $x^{\lambda}$ ( $=x_{1}^{\lambda_{1}}x_{2}^{\lambda_{2}}\cdots x_{N}^{\lambda_{N}}$) with $\lambda=\left( d_{1}^{n_{1}},d_{2}^{n_{2}},d_{3}^{n_{3}}\right) $ (meaning $d_{j}$ is repeated $n_{j}$ times, $1\leq j\leq3$), thus is isomorphic to the direct product $\mathcal{S}_{n_{1}}\times\mathcal{S}_{n_{2}}\times \mathcal{S}_{n_{3}}$. The orbit of $x^{\lambda}$ under the action of $\mathcal{S}_{N}$ (by permutation of coordinates) spans a module $V_{\lambda}% $, the representation induced from the identity representation of $G_{n}$. The space $V_{\lambda}$ decomposes into a direct sum of irreducible $\mathcal{S}% _{N}$-modules. The spherical function is defined for each of these, it is the character of the module averaged over the group $G_{n}$. This paper concerns the value of certain spherical functions evaluated at a cycle which has no more than one entry in each of the three intervals $I_{j}=\left\{ i:\lambda_{i}=d_{j}\right\} ,1\leq j\leq3$. These values appear in the study of eigenvalues of the Heckman-Polychronakos operators in the paper by V. Gorin and the author (arXiv:2412:01938v1). The present paper determines the spherical function values for $\mathcal{S}_{N}$-modules $V$ of two-row tableau type, corresponding to Young tableaux of shape $\left[ N-k,k\right] $. The method is based on analyzing the effect of a cycle on $G_{n}$-invariant elements of $V$. These are constructed in terms of Hahn polynomials in two variables.

In a recent short note the first author \cite{tem} gave the first positive result on the higher order regularity of the discrete noncentered Hardy-Littlewood maximal function. In this article we conduct a thorough investigation of possible similar results for higher order derivatives. We uncover that such results are indeed a consequence of a stronger phenomenon regarding the growth of $l^p(\Z)$ norms of the derivatives of characteristic functions of finite subsets of $\Z$. Along the way we discover very interesting connections to Prouhot-Tarry-Escott (PTE) problem, and to zeros of complex polynomials with restricted coefficients (Littlewood-type polynomials).

We investigate properties of holomorphic extensions in the one-variable case of Whitney's Approximation Theorem on intervals. Improving a result of Gauthier-Kienzle, we construct tangentially approximating functions which extend holomorphically to domains of optimal size. For approximands on unbounded closed intervals, we also bound the growth of holomorphic extensions, in the spirit of Arakelyan, Bernstein, Keldych, and Kober.

In this paper, we study renormalization, that is, the procedure for eliminating singularities, for a special model using both combinatorial techniques in the framework of working with formal series, and using a limit transition in a standard multidimensional integral, taking into account the removal of the singular components. Special attention is paid to the comparative analysis of the two views on the problem. It is remarkably that the divergences, which have the same form in one approach, acquire a different nature in another approach and lead to interesting consequences. A special deformation of the spectrum is used as regularization.

In this paper, we investigate a spline frame generated by oversampling against the well-known Battle-Lemari\'e wavelet system of nonnegative integer order, $n$. We establish a characterization of the Besov and Triebel-Lizorkin (quasi-) norms for the smoothness parameter up to $s < n+1$, which includes values of $s$ where the Battle-Lemari\'e system no longer provides an unconditional basis; we, additionally, prove a result for the endpoint case $s=n+1$. This builds off of earlier work by G. Garrig\'os, A. Seeger, and T. Ullrich, where they proved the case $n=0$, i.e. that of the Haar wavelet, and work of R. Srivastava, where she gave a necessary range for the Battle-Lemari\'e system to give an unconditional basis of the Triebel-Lizorkin spaces.

In this paper twistor methods are used to construct a family of multivalued harmonic functions on ${\bf R}^{3}$ which were obtained by Dashen Yan using different methods. The branching sets for the solutions are ellipses and the functions have quadratic growth at infinity.

We provide a unified framework to proving pointwise convergence of sparse sequences, deterministic and random, at the $L^1(X)$ endpoint. Specifically, suppose that \[ a_n \in \{ \lfloor n^c \rfloor, \min\{ k : \sum_{j \leq k} X_j = n\} \} \] where $X_j$ are Bernoulli random variables with expectations $\mathbb{E} X_j = n^{-\alpha}$, and we restrict to $1 < c < 8/7, \ 0 < \alpha < 1/2$. Then (almost surely) for any measure-preserving system, $(X,\mu,T)$, and any $f \in L^1(X)$, the ergodic averages \[ \frac{1}{N} \sum_{n \leq N} T^{a_n} f \] converge $\mu$-a.e. Moreover, our proof gives new quantitative estimates on the rate of convergence, using jump-counting/variation/oscillation technology, pioneered by Bourgain. This improves on previous work of Urban-Zienkiewicz, and Mirek, who established the above with $c = \frac{1001}{1000}, \ \frac{30}{29}$, respectively, and LaVictoire, who established the random result, all in a non-quantitative setting.

In this paper we extend the theory of Rubio de Francia extrapolation for matrix weights, recently introduced by Bownik and the first author, to off-diagonal extrapolation. We also show that the theory of matrix weighted extrapolation can be extended to matrix $\mathcal{A}_p$ classes defined with respect to a general basis, provided that a version of the Christ-Goldberg maximal operator is assumed to be bounded. Finally, we extend a recent result by Vuorinen and show that all of the multiparameter bases have this property.

In this paper, we present a general method for obtaining addition theorems of the Weierstrass elliptic function $\wp(z)$ in terms of given parameters. We obtain the classical addition theorem for the Weierstrass elliptic function as a special case. Furthermore, we give novel two-term addition, three-term addition, duplication and triplication formulas. New identities for elliptic invariants are also proven.

We establish weighted norm inequalities for a class of multilinear singular integral operators with rough kernels. Specifically, we consider the multilinear singular integral operator $\mathcal{L}_\Omega$ associated with an integrable function $\Omega$ on the unit sphere $\mathbb{S}^{mn-1}$ satisfying the vanishing mean condition. Extending the classical results of Watson and Duoandikoetxea to the multilinear setting, we prove that $\mathcal{L}_\Omega$ is bounded from $L^{p_1}(w_1)\times\cdots\times L^{p_m}(w_m)$ to $L^p(v_{\vec{\boldsymbol{w}}})$ under the assumption that $\Omega\in L^q(\mathbb{S}^{mn-1})$ and that the $m$ tuple of weights $\vec{\boldsymbol{w}}= (w_1,\ldots,w_m)$ lies in the multiple weight class $A_{\vec{\boldsymbol{p}}/q'}((\mathbb{R}^n)^m)$. Here, $q'$ denotes the H\"older conjugate of $q$, and we assume $q'\le p_1,\dots,p_m<\infty$ with $1/p = 1/p_1 + \cdots + 1/p_m$.

In this paper, we derive sharp two side heat kernel estimate on exterior $C^{1,1}$ domains in the plane, and sharp upper heat kernel bound on exterior $C^{1,\mathrm{Dini}}$ domains in $\mathbb{R}^n$, $n\ge 2$. Estimates for Green's function and Riesz potentials on exterior domains in the plane are also presented. Based on the heat kernel estimates, we show the boundedness of the fractional Riesz transforms on exterior $C^{1,\mathrm{Dini}}$ domains in $\mathbb{R}^n$, $n\ge 2$. Some further applications to product and chain rules and nonlinear Schr\"odinger equation are also presented.

Let $\nu = (\nu_1, \ldots, \nu_n) \in (-1/2, \infty)^n$, with $n \ge 1$, and let $\Delta_\nu$ be the multivariate Bessel operator defined by \[ \Delta_{\nu} = -\sum_{j=1}^n\left( \frac{\partial^2}{\partial x_j^2} - \frac{\nu_j^2 - 1/4}{x_j^2} \right). \] In this paper, we develop the theory of Hardy spaces and BMO-type spaces associated with the Bessel operator $\Delta_\nu$. We then study the higher-order Riesz transforms associated with $\Delta_\nu$. First, we show that these transforms are Calder\'on-Zygmund operators. We further prove that they are bounded on the Hardy spaces and BMO-type spaces associated with $\Delta_\nu$.

We prove that, for every polyhedral or $C^1$ norm on $\mathbb{R}^d$ and every set $E \subseteq \mathbb{R}^d$ of packing dimension $s$, the packing dimension of the distance set of $E$ with respect to that norm is at least $\tfrac{s}{d}$. One of the main tools is a nonlinear projection theorem extending a result of M. J\"{a}rvenp\"{a}\"{a}. An explicit construction follows, demonstrating that these distance sets bounds are sharp for a large class of polyhedral norms.

Given a sequence of polynomials $(P_n)_{n \in \mathbb{N}}$ with only nonpositive zeros, the aim of this article is to present a user-friendly approach for determining the limiting zero distribution of $P_n$ as $\mathrm{deg}\, P_n \to \infty$. The method is based on establishing an equivalence between the existence of a limiting empirical zero distribution $\mu$ and the existence of an exponential profile $g$ associated with the coefficients of the polynomials $(P_n)_{n \in \mathbb{N}}$. The exponential profile $g$, which can be roughly described by $[z^k]P_n(z) \approx \exp(n g(k/n))$, offers a direct route to computing the Cauchy transform $G$ of $\mu$: the functions $t \mapsto tG(t)$ and $\alpha \mapsto \exp(-g'(\alpha))$ are mutual inverses. This relationship, in various forms, has previously appeared in the literature, most notably in the paper [Van Assche, Fano and Ortolani, SIAM J. Math. Anal., 1987]. As a first contribution, we present a self-contained probabilistic proof of this equivalence by representing the polynomials as generating functions of sums of independent Bernoulli random variables. This probabilistic framework naturally lends itself to tools from large deviation theory, such as the exponential change of measure. The resulting theorems generalize and unify a range of previously known results, which were traditionally established through analytic or combinatorial methods. Secondly, using the profile-based approach, we investigate how the exponential profile and the limiting zero distribution behave under certain operations on polynomials, including finite free convolutions, Hadamard products, and repeated differentiation. In particular, our approach yields new proofs of the convergence results `$\boxplus_n \to \boxplus$' and `$\boxtimes_n \to \boxtimes$', extending them to cases where the distributions are not necessarily compactly supported.

The goal of this paper is to study the boundedness and compactness of the Bergman projection commutators in two weighted settings via the weighted BMO and VMO spaces, respectively. The novelty of our work lies in the distinct treatment of the symbol b in the commutator, depending on whether it is analytic or not, which turns out to be quite different. In particular, we show that an additional weight condition due to Aleman, Pott, and Reguera is necessary to study the commutators when b is not analytic, while it can be relaxed when b is analytic. In the analytic setting, we completely characterize boundedness and compactness, while in the non-analytic setting, we provide a sufficient condition which generalizes the Euclidean case and is also necessary in many cases of interest. Our work initiates a study of the commutators acting on complex function spaces with different symbols.

The Hausdorff dimension of a set can be detected using the Riesz energy. Here, we consider situations where a sequence of points, $\{x_n\}$, ``fills in'' a set $E \subset \mathbb{R}^d$ in an appropriate sense and investigate the degree to which the discrete analog to the Riesz energy of these sets can be used to bound the Hausdorff dimension of $E$. We also discuss applications to data science and Erd\H{o}s/Falconer type problems.

Let $\nu=(\nu_1,\ldots,\nu_n)\in (-1,\vc)^n$, $n\ge 1$, and let $\mathcal{L}_\nu$ be a self-adjoint extension of the differential operator \[ L_\nu := \sum_{i=1}^n \left[-\frac{\partial^2}{\partial x_i^2} + x_i^2 + \frac{1}{x_i^2}(\nu_i^2 - \frac{1}{4})\right] \] on $C_c^\infty(\mathbb{R}_+^n)$ as the natural domain. The $j$-th partial derivative associated with $L_{\nu}$ is given by \[ \delta_{\nu_j} = \frac{\partial}{\partial x_j} + x_j-\frac{1}{x_j}\Big(\nu_j + \f{1}{2}\Big), \ \ \ \ j=1,\ldots, n. \] In this paper, we investigate the weighted estimates of the higher-order Riesz transforms $\delta_\nu^k\mathcal L^{-|k|/2}_\nu, k\in \mathbb N^n$, where $\delta_\nu^k=\delta_{\nu_n}^{k_n}\ldots \delta_{\nu_1}^{k_1}$. This completes the description of the boundedness of the higher-order Riesz transforms with the full range $\nu \in (-1,\vc)^n$.

We study the global (asymptotic) stability of the Lengyel-Epstein differential systems, sometimes called Belousov-Zhabotinsky differential systems. Such systems are topologically equivalent to a two-parameter family of cubic systems in the plane. We show that for each pair of admissible parameters the unique equilibrium point of the corresponding system is not globally (asymptotically) stable. On the other hand, we provide explicit conditions for this unique equilibrium point to be asymptotically stable and we study its basin of attraction. We also study the generic and degenerate Hopf bifurcations and highlight a subset of the set of admissible parameters for which the phase portraits of the systems have two limit cycles.