CyberSec Research

In many imaging applications it is important to assess how well the edges of the original object, $f$, are resolved in an image, $f^\text{rec}$, reconstructed from the measured data, $g$. In this paper we consider the case of image reconstruction in 2D X-ray Computed Tomography (CT). Let $f$ be a function describing the object being scanned, and $g=Rf + \eta$ be the Radon transform data in $\mathbb{R}^2$ corrupted by noise, $\eta$, and sampled with step size $\sim\epsilon$. Conventional microlocal analysis provides conditions for edge detectability based on the scanner geometry in the case of continuous, noiseless data (when $\eta = 0$), but does not account for noise and finite sampling step size. We develop a novel technique called \emph{Statistical Microlocal Analysis} (SMA), which uses a statistical hypothesis testing framework to determine if an image edge (singularity) of $f$ is detectable from $f^\text{rec}$, and we quantify edge detectability using the statistical power of the test. Our approach is based on the theory we developed in \cite{AKW2024_1}, which provides a characterization of $f^\text{rec}$ in local $O(\epsilon)$-size neighborhoods when $\eta \neq 0$. We derive a statistical test for the presence and direction of an edge microlocally given the magnitude of $\eta$ and data sampling step size. Using the properties of the null distribution of the test, we quantify the uncertainty of the edge magnitude and direction. We validate our theory using simulations, which show strong agreement between our predictions and experimental observations. Our work is not only of practical value, but of theoretical value as well. SMA is a natural extension of classical microlocal analysis theory which accounts for practical measurement imperfections, such as noise and finite step size, at the highest possible resolution compatible with the data.

The concept of mixed norm spaces has emerged as a significant interest in fields such as harmonic analysis. In addition, the problem of function approximation through sampling series has been particularly noteworthy in the realm of approximation theory. This paper aims to address both these aspects. Here we deal with the problem of function approximation in diverse mixed norm function spaces. We utilise the family of Kantorovich type sampling operators as approximator for the functions in mixed norm Lebesgue space, and mixed norm Orlicz space. The Orlicz spaces are well-known as a generalized family that encompasses many significant function spaces. We establish the boundedness of the family of generalized as well as Kantorovich type sampling operators within the framework of these mixed norm spaces.Further, we study the approximation properties of Kantorovich-type sampling operators in both mixed norm Lebesgue and Orlicz spaces. At the end, we discuss a few examples of suitable kernel involved in the discussed approximation procedure.

A bounded linear operator $T$ on a Banach space $X$ (not necessarily separable) is said to be $J$-class operator whenever the extended limit set, say $J_T(x)$ equals $X$ for some vector $x\in X$. Practically, the extended limit sets localize the dynamical behavior of operators. In this paper, using the extended limit sets we will examine the necessary and sufficient conditions for the weighted translation $T_{a,\omega}$ to be $J$-class on a locally compact group $G$, within the setting of $ L^p$-spaces for $ 1 \leq p < \infty $. Precisely, we delineate the boundary between $J$-class and hypercyclic behavior for weighted translations. Then, we will show that for torsion elements in locally compact groups, unlike the case of non-dense orbits of weighted translations, we have $J_{T_{a,\omega}}(0)=L^p(G)$. Finally, we will provide some examples on which the weighted translation $ T_{a,\omega}$ is $J$-class but it fails to be hypercyclic.

We show that the existence of a strongly convex function with a Lipschitz derivative on a Banach space already implies that the space is isomorphic to a Hilbert space. Similarly, if both a function and its convex conjugate are $C^2$ then the underlying space is also isomorphic to a Hilbert space.

This work presents a rigorous characterization of inner products on the Hilbert space $S_2$ of Hilbert--Schmidt operators. We first deal with a general setting of continuous sesquilinear forms on a Hilbert space $\mathcal H$, and provide a characterization of all inner products by means of positive operators in $\mathcal {B(H)}$. Next, we establish necessary and sufficient conditions for an operator in $\mathcal B(S_2)$ to be positive. Identifying an inner product with a positive operator enables us to rigorously describe inner products on $S_2$.

We study the properties of power-boundedness, Li-Yorke chaos, distributional chaos, absolutely Ces\`aro boundedness and mean Li-Yorke chaos for weighted composition operators on $L^p(\mu)$ spaces and on $C_0(\Omega)$ spaces. We illustrate the general results by presenting several applications to weighted shifts on the classical sequence spaces $c_0(\mathbb{N})$, $c_0(\mathbb{Z})$, $\ell^p(\mathbb{N})$ and $\ell^p(\mathbb{Z})$ ($1 \leq p < \infty$) and to weighted translation operators on the classical function spaces $C_0[1,\infty)$, $C_0(\mathbb{R})$, $L^p[1,\infty)$ and $L^p(\mathbb{R})$ ($1 \leq p < \infty$).

Given a finite-dimensional inner product space $V$ and a group $G$ of isometries, we consider the problem of embedding the orbit space $V/G$ into a Hilbert space in a way that preserves the quotient metric as well as possible. This inquiry is motivated by applications to invariant machine learning. We introduce several new theoretical tools before using them to tackle various fundamental instances of this problem.

In this paper, we find a gap between the lower bound of the Bakry-\'Emery $N$-Ricci tensor ${\rm Ric}_N$ and the Bakry-\'Emery gradient estimate ${\sf BE}$ in the space associated with the finite-particle Dyson Brownian motion (DBM) with inverse temperature $0<\beta<1$. Namely, we prove that, for the weighted space $(\mathbb R^n, w_\beta)$ with $w_\beta=\prod_{i<j}^n |x_i-x_j|^\beta$ and any $N\in[n+\frac{\beta}{2}n(n-1),+\infty]$, $\beta \ge 1 \implies {\rm Ric}_N \ge 0 \ \& \ {\sf BE}(0,N)$ hold; $0 < \beta < 1 \implies {\rm Ric}_N \ge 0$ holds while ${\sf BE}(0,N)$ does not, which shows a phase transition of the Dyson Brownian motion regarding the Bakry-\'Emery curvature bound in the small inverse temperature regime.

Let B(X) be the algebra of all bounded linear operators on a complex Banach space X of dimension at least three. For an arbitrary nonzero complex number t we determine the form of mappings f: B(X)-->B(X) with sufficiently large range such that t(AB+BA) is idempotent if and only if t(f(A)f(B)+f(B)f(A)) is idempotent, for all A, B in B(X). Note that f is not assumed to be linear or additive.

We investigate determinants of random unitary pencils (with scalar or matrix coefficients), which generalize the characteristic polynomial of a single unitary matrix. In particular we examine moments of such determinants, obtained by integrating against the Haar measure on the unitary group. We obtain an exact formula in the case of scalar coefficients, and conjecture an asymptotic formula in the general case, and prove a special case of the conjecture.

We present a noncommutative optimal transport framework for quantum channels acting on von Neumann algebras. Our central object is the Lipschitz cost measure, a transportation-inspired quantity that evaluates the minimal cost required to move between quantum states via a given channel. Accompanying this is the Lipschitz contraction coefficient, which captures how much the channel contracts the Wasserstein-type distance between states. We establish foundational properties of these quantities, including continuity, dual formulations, and behavior under composition and tensorization. Applications include recovery of several mathematical quantities including expected group word length and Carnot-Carath\'eodory distance, via transportation cost. Moreover, we show that if the Lipschitz contraction coefficient is strictly less than one, one can get entropy contraction and mixing time estimates for certain classes of non-symmetric channels.

We introduce a notion of weak convergence in arbitrary metric spaces. Metric functionals are key in our analysis: weak convergence of sequences in a given metric space is tested against all the metric functionals defined on said space. When restricted to bounded sequences in normed linear spaces, we prove that our notion of weak convergence agrees with the standard one.

In this article, we completely characterize the Berezin range of Toeplitz operators with harmonic symbols acting on weighted Bergman spaces, illustrating the necessity of the harmonicity condition through examples. We then introduce a new class of weighted composition operators on these spaces, investigating their fundamental properties and determining their Berezin range and Berezin number. Finally, we study the convexity of the Berezin range of composition operators on weighted Bergman spaces and show that the origin lies in its closure of Berezin range but not in the range itself.

In this paper, we extend the Marcinkiewicz--Zygmund inequality to the setting of Orlicz and Lorentz spaces. Furthermore, we generalize a Kadec--Pe{\l}czy\'nski-type result -- originally established by the first and third authors for $L^p$ spaces with $1 \le p < 2$ -- to a broader class of Orlicz spaces defined via Young functions $\psi$ satisfying $x \le \psi(x) \le x^2$.

We continue the analysis of random series associated to the multidimensional harmonic oscillator $-\Delta + |x|^2$ on $\mathbb{R}^d$ with d \geq 2$$. More precisely we obtain a necessary and sufficient condition to get the almost sure uniform convergence on the whole space $\mathbb{R}^d$ . It turns out that the same condition gives the almost sure uniform convergence on the sphere $\mathbb{S}^{d-1}$ (despite $\mathbb{S}^{d-1}$ is a zero Lebesgue measure of $\mathbb{R}^d$). From a probabilistic point of view, our proof adapts a strategy used by the first author for boundaryless Riemannian compact manifolds. However, our proof requires sharp off-diagonal estimates of the spectral function of $-\Delta + |x|^2$ . Such estimates are obtained using elementary tools.

In this paper, we begin by introducing some necessary and sufficient conditions for generalized $n$-strong Drazin invertibility (g$n$s-invertibility) and pseudo $n$-strong Drazin invertibility (p$n$s-invertibility) of an element in a Banach algebra for $n\in\mathbb{N}$. Subsequently, these results are utilized to prove some additive properties of g$n$s (p$n$s)-Drazin inverse in a Banach algebra. This process produces a generalization of some recent results of H Chen, M Sheibani (Linear and Multilinear Algebra \textbf{70.1} (2022): 53-65) for g$n$s and p$n$s-Drazin inverse. Furthermore, we define and characterize weighted g$n$s and weighted p$n$s-Drazin inverse in a Banach algebra.

We derive a complete left-tail asymptotic series for the density of the {\it martingale limit} of a Galton-Watson process with immigration. We show that the series converges everywhere, not only for small arguments. This is the first complete result regarding the left tails of branching processes with immigration. A good, quickly computed approximation for the density will also be derived from the series.

We introduce the CP-distance as a measure of how far a Hermitian map is from being completely positive, deriving key properties and bounds. We investigate the role of CP-distance in the structural analysis of positive Hermitian linear maps between matrix algebras, focusing on its implications for quantum information theory. In particular, we derive bounds on the detection strength of entanglement witnesses. We elucidate the interplay between CP-distance and the structural properties of positive maps, offering insights into their decompositions. We also analyze how the CP-distance influences the decompositions of positive Hermitian maps, revealing its impact on the balance between completely positive components.