CyberSec Research

We provide a foundation for an approach to the open problem of bilipschitz extendability of mappings defined on a Euclidean separated net. In particular, this allows for the complete positive solution of the problem in dimension two. Along the way, we develop a set of tools for bilipschitz extensions of mappings between subsets of Euclidean spaces.

We study expansions of Hilbert spaces with a bounded normal operator $T$. We axiomatize this theory in a natural language and identify all of its completions. We prove the definability of the adjoint $T^*$ and prove quantifier elimination for every completion after adding $T^*$ to the language. We identify types with measures on the spectrum of the operator and show that the logic topology on the type space corresponds to the weak*-topology on the space of measures. We also give a precise formula for the metric on the space of $1$-types. We prove all completions are stable and characterize the stability spectrum of the theory in terms of the spectrum of the operator. We also show all completions, regardless of their spectrum, are $\omega$-stable up to perturbations.

We review recent work connected with the invariant subspace problem for operators, in particular new developments in the last 15 years. In particular, we include discussions of almost-invariant subspaces, universal operators, specific classes of operators and new results in the framework of Banach spaces.

We characterize invariant subspaces of Brownian shifts on vector-valued Hardy spaces. We also solve the unitary equivalence problem for the invariant subspaces of these shifts.

We investigate isometric and algebraic isomorphism problems for multiplier algebras associated with Hilbert spaces of Dirichlet series whose kernels possess the complete Nevanlinna-Pick (CNP) property. We begin by providing a complete characterization of all normalized CNP Dirichlet series kernels in terms of weight and frequency data. A central aspect of our work is the explicit determination of the multiplier varieties associated with CNP Dirichlet series kernels. We show that these varieties are defined by explicit polynomial equations derived from the arithmetic structure of the weight and frequency data associated with the kernel. This explicit description of multiplier varieties enables us to classify when the multiplier algebras of certain CNP Dirichlet series kernels are (isometrically) isomorphic. As an application, we resolve an open question posed by McCarthy and Shalit ([18]) in the negative.

Using the Witten deformation and localization algebra techniques, we compute the $G$-equivariant $K$-homology class of the de Rham operator on a proper cocompact $G$-spin manifold, where $G$ is an almost connected Lie group. By applying a $G$-invariant Morse-Bott perturbation, this class is localized near the zero set of the perturbation and can be identified explicitly with an element in the representation rings associated to some isotropy subgroups. The result yields an equivariant Poincar\'e-Hopf formula and supplies concise tools for equivariant index computations.

We prove a sharp bound between sampling numbers and entropy numbers in the uniform norm for general convex sets of bounded functions.

The main purpose of this article is to explore the possibility of extending the notion of peripheral Poisson boundary of unital completely positive (UCP) maps to contractive completely positive (CCP) maps and to unital and non-unital contractive quantum dynamical semigroups on von Neumann algebras. We observe that the theory extends easily in the setting of von Neumann algebras and normal maps. Surprisingly, the peripheral Poisson boundary is unital, whenever it is nontrivial, even for contractive semigroups. The strong operator limit formula for computing the extended Choi-Effros product remains intact. However, there are serious obstacles in the framework of $C^*$-algebras, and we are unable to define the extended Choi-Effros product in such generality. We provide several intriguing examples to illustrate this.

We show that every real-valued Lipschitz function on a subset of a metric space can be extended to the whole space while preserving the slope and, up to a small error, the global Lipschitz constant. This answers a question posed by Di Marino, Gigli, and Pratelli, who established the analogous property for the asymptotic Lipschitz constant. We also prove the same result for the ascending slope and for the descending slope.

We characterize the d x d matrices whose numerical ranges are invariant by rotations of angle 2$\pi$/d.

In this work, we study the minimization of nonlinear functionals in dimension $d\geq 1$ that depend on a degenerate radial weight $w$. Our goal is to prove the existence of minimizers in a suitable functional class here introduced and to establish that the minimizers of such functionals, which exhibit $p$-growth with $1 < p < +\infty$, are radially symmetric. In our analysis, we adopt the approach developed in [Chiad\`o Piat, De Cicco and Melchor Hernandez, NoDEA $2025$, De Cicco and Serra Cassano, ESAIM:COCV $2024$], where $w$ does not satisfy classical assumptions such as doubling or Muckenhoupt conditions. The core of our method relies on proving the validity of a weighted Poincar\'e inequality involving a suitably constructed auxiliary weight.

We study the $L^p$-mean distortion functionals, \[{\cal E}_p[f] = \int_\mathbb Y K^p_f(z) \; dz, \] for Sobolev homeomorphisms $f: \overline{\mathbb Y}\xrightarrow{\rm onto} \overline{\mathbb X}$ where $\mathbb X$ and $\mathbb Y$ are bounded simply connected Lipschitz domains, and $f$ coincides with a given boundary map $f_0 \colon \partial \mathbb Y \to \partial \mathbb X$. Here, $K_f(z)$ denotes the pointwise distortion function of $f$. It is conjectured that for every $1 < p < \infty$, the functional $\mathcal{E}_p$ admits a minimizer that is a diffeomorphism. We prove that if such a diffeomorphic minimizer exists, then it is unique.

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 $x \in X$, $$\limsup_{t\to\infty} \frac{\|T(t)A^{-1}x\|_{X}}{r(t)}=\infty.$$

In [9], authors studied spectrally optimal dual frames for 1-erasure and 2-erasures of frames generated by graph. In this paper, we study spectrally optimal dual frames for r-erasures. We show that the spectral radius of the error operator of unitary equivalent frames is same with respect to their respective canonical dual frames. We prove that if a frame is generated by a connected graph, then its canonical dual frame is the unique spectrally optimal dual frame for r-erasures. Further, we show that the canonical dual of frames generated by disconnected graphs are non-unique spectrally optimal dual frames for r-erasures.

We show that the free spectrahedron determined by universal anticommuting self-adjoint unitaries is not equal to the minimal matrix convex set over the ball in dimension three or higher. This example, as well as other matrix convex sets over the ball, then provides context for structure results on the extreme points of coordinate projections. In particular, we show that the free polar dual of a real free spectrahedron is rarely the projection of a real free spectrahedron, contrasting a prior result of Helton, Klep, and McCullough over the complexes. We use this to show that spanning results for free spectrahedra that are closed under complex conjugation do not extend to free spectrahedrops that meet the same assumption. These results further clarify the role of the coefficient field.

A classical result in approximation theory states that for any continuous function \( \varphi: \mathbb{R} \to \mathbb{R} \), the set \( \operatorname{span}\{\varphi \circ g : g \in \operatorname{Aff}(\mathbb{R})\} \) is dense in \( \mathcal{C}(\mathbb{R}) \) if and only if \( \varphi \) is not a polynomial. In this note, we present infinite dimensional variants of this result. These extensions apply to neural network architectures and improves the main density result obtained in \cite{BDG23}. We also discuss applications and related approximation results in vector lattices, improving and complementing results from \cite{AT:17, bhp,BT:24}.

We introduce and study the Rhaly operator on K\"othe spaces, with a primary focus on understanding its well-definedness, continuity, and compactness. We especially examine operators acting on power series spaces of both infinite and finite type. In the sequel, we provide integral representations for the Rhaly operator on the space of entire functions $H(\mathbb{C})$ and the space of holomorphic functions on the unit disc $H(\mathbb{D})$. We also investigate the topologizability and power boundedness of the Rhaly operators, which leads to findings about their mean ergodicity, uniform mean ergodicity, and Ces\`aro boundedness.

We present a novel microlocal analysis of a non-linear ray transform, $\mathcal{R}$, arising in Compton Scattering Tomography (CST). Due to attenuation effects in CST, the integral weights depend on the reconstruction target, $f$, which has singularities. Thus, standard linear Fourier Integral Operator (FIO) theory does not apply as the weights are non-smooth. The V-line (or broken ray) transform, $\mathcal{V}$, can be used to model the attenuation of incoming and outgoing rays. Through novel analysis of $\mathcal{V}$, we characterize the location and strength of the singularities of the ray transform weights. In conjunction, we provide new results which quantify the strength of the singularities of distributional products based on the Sobolev order of the individual components. By combining this new theory, our analysis of $\mathcal{V}$, and classical linear FIO theory, we determine the Sobolev order of the singularities of $\mathcal{R}f$. The strongest (lowest Sobolev order) singularities of $\mathcal{R}f$ are shown to correspond to the wavefront set elements of the classical Radon transform applied to $f$, and we use this idea and known results on the Radon transform to prove injectivity results for $\mathcal{R}$. In addition, we present novel reconstruction methods based on our theory, and we validate our results using simulated image reconstructions.