CyberSec Research

We analyze the joint numerical range $W$ of three complex hermitian matrices of order four. In the generic case this $3D$ convex set has a smooth boundary. We analyze non-generic structures and investigate non-elliptic faces in the boundary $\partial W$. Fifteen possible classes regarding the numbers of non-elliptic faces are identified and an explicit example is presented for each class. Secondly, it is shown that a nonempty intersection of three mutually distinct one-dimensional faces is a corner point. Thirdly, introducing a tensor product structure into $\mathbb C^4=\mathbb C^2\otimes\mathbb C^2$, one defines the separable joint numerical range -- a subset of $W$ useful in studies of quantum entanglement. The boundary of the separable joint numerical range is compared with that of $W$.

We show that when $m>n$, the space of $m\times n$-matrix-valued rational inner functions in the disk is path connected.

We extend G\'erard's results on orthogonality of ${\rm L}^2_{\rm loc}$ sequences as a consequence of mutual singularity of corresponding H-measures (microlocal defect measures) to ${\rm L}^p$/${\rm L}^q$ sequences and newly introduced notion of orhogonality for H-distributions. We apply the result to a homogenisation problem for the heterogeneous Boltzmann equation with space-dependent drift and periodic opacity.

For a Borel probability measure $\mu$ on $\mathbb{R}^{n}$, it is called a spectral measure if the Hilbert space $L^{2}(\mu)$ admits an orthogonal basis of exponential functions. In this paper, we study the spectrality of fractal measures generated by an iterated function system (IFS) with $m$-periodic alternating contraction ratios. Specifically, for fixed $m,N\in\mathbb{N}^{+}$ and $\rho\in(0,1)$, we define the IFS as follows: $$\{\tau_d(\cdot)=(-1)^{\lfloor\frac{d}{m}\rfloor}\rho(\cdot+d)\}_{d\in D_{2Nm}},$$ where $D_k=\{0,1,\cdots,k-1\}$ and $\lfloor x\rfloor$ denotes the floor function. We prove that the associated self-similar measure $\nu_{\rho,D_{2Nm}}$ is a spectral measure if and only if $\rho^{-1}=p\in\mathbb{N}$ and $2Nm\mid p$. Furthermore, for any positive integers $p,s\geq2$, if $m=1$ and $\gcd(p,s)=1$ we show that $\nu_{p^{-1},D_{s}}$ is not a spectral measure and $L^2(\nu_{p^{-1},D_{s}})$ contains at most $s$ mutually orthogonal exponential functions. These results generalize recent work of Wu [25] [H.H. Wu, Spectral self-similar measures with alternate contraction ratios and consecutive digits, Adv. Math., 443 (2024), 109585].

This work develops, from a functional analytic perspective, the construction of random variables in Lebesgue spaces L^p. It extends classical notions of measurability, integrability, and expectation to L^p valued functions, using Pettis's theorem and the Riesz representation theorem to define the Bochner integral as a natural generalization of classical expectation.

In this paper, we calculate the covering constants for single-valued mappings in Euclidean space by using Mordukhovich derivatives (or coderivatives). At first, we prove the guideline for calculating the Frechet derivatives of single-valued mappings by their partial derivatives. Then, by using the connections between Frechet derivatives and Mordukhovich derivatives (or coderivatives) of single-valued mappings in Banach spaces, we derive the useful rules for calculating the Mordukhovich derivatives of single-valued mappings in Euclidean spaces. For practicing these rules, we find the precise solutions of the Frechet derivatives and Mordukhovich derivatives for some single-valued mappings between Euclidean spaces. By using these solutions, we find or estimate the covering constants for the considered mappings. As applications of the results about the covering constants involved in the Arutyunov Mordukhovich and Zhukovskiy Parameterized Coincidence Point Theorem, we solve some parameterized equations

A quasi-projection pair consists of two operators $P$ and $Q$ acting on a Hilbert $C^*$-module $H$, where $P$ is a projection and $Q$ is an idempotent satisfying $Q^*=(2P-I)Q(2P-I)$, in which $Q^*$ denotes the adjoint operator of $Q$, and $I$ is the identity operator on $H$. Such a pair is said to be harmonious if both $P(I-Q)$ and $(I-P)Q$ admit polar decompositions. The primary goal of this paper is to present the block matrix representations for a harmonious quasi-projection pair $(P,Q)$ on a Hilbert $C^*$-module, and additionally to derive new block matrix representations for the matched projection, the range projection, and the null space projection of $Q$. Several applications of these newly obtained block matrix representations are also explored.

This paper studies strict fixed point and stability results for multivalued operators which does not satisfy a \'Ciri\'c type contraction condition, but their admissible perturbation does. We focus on the conditions imposed on the admissible perturbation $T_G$ of a Picard operator $T:X\rightarrow P(X)$ such that the strict fixed point and stability results still hold for T. The results obtained are reformulated in terms of admissible perturbations in the sense of Takahashi and illustrated with some examples.

We prove existence and finite degeneracy of ground states of energy forms satisfying logarithmic Sobolev inequalities with respect to non vacuum states of Clifford algebras. We then derive the stability of the ground state with respect to certain unbounded perturbations of the energy form. Finally, we show how this provides an infinitesimal approach to existence and uniqueness of the ground state of Hamiltonians considered by L. Gross in QFT, describing spin $1/2$ Dirac particles subject to interactions with an external scalar field.

In the paper, we represent a comparison analysis of the methods of the topological alignment and extract the main mathematical principles forming the base of the concept. The main narrative is devoted to the so-called coupled methods dealing with the data sets of various nature. As a main theoretical result, we obtain harmonious generalizations of the graph Laplacian and kernel based methods with the central idea to find a natural structure coupling data sets of various nature. Finally, we discuss prospective applications and consider far reaching generalizations related to the hypercomplex numbers and Clifford algebras.

A $7$-tuple of commuting bounded operators $\mathbf{T} = (T_1, \dots, T_7)$ defined on a Hilbert space $\mathcal{H}$ is said to be a \textit{$\Gamma_{E(3; 3; 1, 1, 1)}$-contraction} if $\Gamma_{E(3; 3; 1, 1, 1)}$ is a spectral set for $\mathbf{T}$. Let $(S_1, S_2, S_3)$ and $(\tilde{S}_1, \tilde{S}_2)$ be tuples of commuting bounded operators on $\mathcal{H}$ satisfying $S_i \tilde{S}_j = \tilde{S}_j S_i$ for $1 \leq i \leq 3$ and $1 \leq j \leq 2$. The tuple $\mathbf{S} = (S_1, S_2, S_3, \tilde{S}_1, \tilde{S}_2)$ is called a \textit{$\Gamma_{E(3; 2; 1, 2)}$-contraction} if $\Gamma_{E(3; 2; 1, 2)}$ is a spectral set for $\mathbf{S}$. In this paper, we establish the existence and uniqueness of the fundamental operators associated with $\Gamma_{E(3; 3; 1, 1, 1)}$-contractions and $\Gamma_{E(3; 2; 1, 2)}$-contractions. Furthermore, we obtain a Beurling-Lax-Halmos type representation for invariant subspaces corresponding to a pure $\Gamma_{E(3; 3; 1, 1, 1)}$-isometry and a pure $\Gamma_{E(3; 2; 1, 2)}$-isometry. We also construct a conditional dilation for a $\Gamma_{E(3; 3; 1, 1, 1)}$-contraction and a $\Gamma_{E(3; 2; 1, 2)}$-contraction and develop an explicit functional model for a certain subclass of these operator tuples. Finally, we demonstrate that every $\Gamma_{E(3; 3; 1, 1, 1)}$-contraction (respectively, $\Gamma_{E(3; 2; 1, 2)}$-contraction) admits a unique decomposition as a direct sum of a $\Gamma_{E(3; 3; 1, 1, 1)}$-unitary (respectively, $\Gamma_{E(3; 2; 1, 2)}$-unitary) and a completely non-unitary $\Gamma_{E(3; 3; 1, 1, 1)}$-contraction (respectively, $\Gamma_{E(3; 2; 1, 2)}$-contraction).

We show that the main homological dimensions of the algebra of analytic functionals on a connected complex Lie group, as well as some of its completions, coincide with the dimension of the simply connected solvable factor in the canonical decomposition of the linearization of this group. Thus, the possible nontriviality of a linearly complex reductive factor does not affect the homological properties of the algebras under consideration.

This work proves an analogue of Fej\'er's Theorem for higher-order weighted Dirichlet spaces. For a holomorphic function $f \in \widehat{\mathcal{H}}_{\mu, m}$, the partial sums $S_{m,n}f$ do not generally converge unless their coefficients are strategically modified. We introduce a sequence of polynomials $p_n(z) = \sum\limits_{k=m+1}^{n+m+1} w_{n,k}a_kz^k$ with a weight array $\{w_{n,k}\}$ satisfying specific oscillation and decay properties. By leveraging the local Douglas formula and a coefficient correspondence theorem, we prove the convergence $\lim\limits_{n \to \infty} \|f - p_n\|_{\mu,m} = 0$. Notably, for a measure $\mu$ composed of Dirac masses, the adjusted coefficients admit an explicit closed-form expression.

Let $g \in L^2(\mathbb{R})$ be a rational function of degree $M$, i.e. there exist polynomials $P, Q$ such that $g = {{P} \over {Q}}$ and $deg(P) < deg(Q) \leq M$. We prove that for any $\varepsilon>0$ and any $M \in \mathbb{N}$ there exists universal set $\Lambda \subset \mathbb{R}$ of density less than $1+\varepsilon$ such that the system $$\left\{ e^{2\pi i \lambda t } g(t-n) \colon (\lambda, n) \in \Lambda \times \mathbb{Z} \right\}$$ is a frame in $L^2(\mathbb{R})$ for any well-behaved rational function $g$.

We show that the heat kernel measures based at the north pole of the spheres $S^{N-1}(\sqrt N)$, with properly scaled radius $\sqrt N$ and adjusted center, converges to a Gaussian measure in $\R^\infty$, and find an explicit formula for this measure.

In a recent preprint, Frank and the second author proved that if a metric on the sphere of dimension $d>4$ almost minimizes the total $\sigma_2$-curvature in the conformal class of the standard metric, then it is almost the standard metric (up to M\"obius transformations). This is achieved quantitatively in terms of a two-term distance to the set of minimizing conformal factors. We extend this result to the case $d=3$. While the standard metric still minimizes the total scalar curvature for $d=3$, it maximizes the total $\sigma_2$-curvature, which turns the related functional inequality into a reverse Sobolev-type inequality. As a corollary of our result, we obtain quantitative versions for a family of interpolation inequalities including the Andrews--De Lellis--Topping inequality on the $3$-sphere. The latter is itself a stability result for the well-known Schur lemma and is therefore called almost-Schur lemma. This makes our stability result an almost-almost-Schur lemma.

We construct a family of purely PI unrectifiable Lipschitz differentiability spaces and investigate the possible of Banach spaces targets for which Lipschitz differentiability holds. We provide a general investigation into the geometry of \emph{shortcut} metric spaces and characterise when such spaces are PI rectifiable, and when they are $Y$-LDS, for a given $Y$. The family of spaces arises as an example of our characterisations. Indeed, we show that Laakso spaces satisfy the required hypotheses.

A $7$-tuple of commuting bounded operators $\textbf{T} = (T_1, \dots, T_7)$ on a Hilbert space $\mathcal{H}$ is called a \textit{$\Gamma_{E(3; 3; 1, 1, 1)} $-contraction} if $\Gamma_{E(3; 3; 1, 1, 1)}$ is a spectral set for $\textbf{T}. $ Let $(S_1, S_2, S_3)$ and $(\tilde{S}_1, \tilde{S}_2)$ be tuples of commuting bounded operators defined on a Hilbert space $\mathcal{H}$ with $S_i\tilde{S}_j = \tilde{S}_jS_i$ for $1 \leqslant i \leqslant 3$ and $1 \leqslant j \leqslant 2$. We say that $\textbf{S} = (S_1, S_2, S_3, \tilde{S}_1, \tilde{S}_2)$ is a $\Gamma_{E(3; 2; 1, 2)} $-contraction if $ \Gamma_{E(3; 2; 1, 2)}$ is a spectral set for $\textbf{S}$. We derive various properties of $\Gamma_{E(3; 3; 1, 1, 1)}$-contractions and $\Gamma_{E(3; 2; 1, 2)}$-contractions and establish a relationship between them. We discuss the fundamental equations for $\Gamma_{E(3; 3; 1, 1,1 )}$-contractions and $\Gamma_{E(3; 2; 1, 2)}$-contractions. We explore the structure of $\Gamma_{E(3; 3; 1, 1, 1)}$-unitaries and $\Gamma_{E(3; 2; 1, 2)}$-unitaries and elaborate on the relationship between them. We also study various properties of $\Gamma_{E(3; 3; 1, 1, 1)}$-isometries and $\Gamma_{E(3; 2; 1, 2)}$-isometries. We discuss the Wold Decomposition for a $\Gamma_{E(3; 3; 1, 1, 1)}$-isometry and a $\Gamma_{E(3; 2; 1, 2)}$-isometry. We further outline the structure theorem for a pure $\Gamma_{E(3; 3; 1, 1, 1)}$-isometry and a pure $\Gamma_{E(3; 2; 1, 2)}$-isometry.