arXiv Math @arxiv_math@qoto.org

I give a short introduction to data ethics. My focal audience is mathematicians, but I hope that my discussion will also be useful to others. I am not an expert about data ethics, and my article is only a starting point. I encourage readers to examine the resources that I discuss and to continue to reflect carefully on data ethics and on the societal implications of data and data analysis throughout their lives.

Immersive video, such as virtual reality (VR) and multi-view videos, is growing in popularity. Its wireless streaming is an instance of general multicast, extending conventional unicast and multicast, whose effective design is still open. This paper investigates the optimization of general rate splitting with linear beamforming for general multicast. Specifically, we consider a multi-carrier single-cell wireless network where a multi-antenna base station (BS) communicates to multiple single-antenna users via general multicast. Linear beamforming is adopted at the BS, and joint decoding is adopted at each user. We consider the maximization of the weighted sum rate, which is a challenging nonconvex problem. Then, we propose an iterative algorithm for the problem to obtain a KKT point using the concave-convex procedure (CCCP). The proposed optimization framework generalizes the existing ones for rate splitting for various types of services. Finally, we numerically show substantial gains of the proposed solutions over existing schemes and reveal the design insights of general rate splitting for general multicast.

We consider Aichinger's equation $$f(x_1+\cdots+x_{m+1})=\sum_{i=1}^{m+1}g_i(x_1,x_2,\cdots, \widehat{x_i},\cdots, x_{m+1})$$ for functions defined on commutative semigroups which take values on commutative groups. The solutions of this equation are, under very mild hypotheses, generalized polynomials. We use the canonical form of generalized polynomials to prove that compositions and products of generalized polynomials are again generalized polynomials and that the bounds for the degrees are, in this new context, the natural ones. In some cases, we also show that a polynomial function defined on a semigroup can uniquely be extended to a polynomial function defined on a larger group. For example, if $f$ solves Aichinger's equation under the additional restriction that $x_1,\cdots,x_{m+1}\in \mathbb{R}_+^p$, then there exists a unique polynomial function $F$ defined on $\mathbb{R}^p$ such that $F_{|\mathbb{R}_+^p}=f$. In particular, if $f$ is also bounded on a set $A\subseteq \mathbb{R}_+^p$ with positive Lebesgue measure then its unique polynomial extension $F$ is an ordinary polynomial of $p$ variables with total degree $\leq m$, and the functions $g_i$ are also restrictions to $\mathbb{R}_+^{pm}$ of ordinary polynomials of total degree $\leq m$ defined on $\mathbb{R}^{pm}$.

In a chemical compound, it is very important for chemists to find the size of the smallest set of atoms so that they can identify other atoms relative to that the smallest set of atoms, so chemists require mathematical forms for a set of chemical compound to give distinct representations to distinct compound structures and its corresponding in graph theory is to find the minimal resolving set. The study of the resolving set and its related parameters such as doubly resolving set and strong resolving set are significant, and it is well known that these problems are NP hard. In this article, we define the structure of the crystal cubic carbon CCC(n) by a new method, and we provide a much simpler formula for obtaining its vertices than the formula for calculating its vertices in (Baig et al., 2017). Moreover, we focus on some resolving parameters of two graphs the crystal cubic carbon CCC(n) and the layer cycle graph will be denoted by LCG(n, k).

The interpretation of non-Markovian effects as due to the information exchange between an open quantum system and its environment has been recently formulated in terms of properly regularized entropic quantities, as their revivals in time can be upper bounded by means of quantities describing the storage of information outside the open system [Phys. Rev. Lett. 127, 030401 (2021)]. Here, we elaborate on the wider mathematical framework of the theory, specifying the key properties that allow us to associate distinguishability quantifiers with the information flow from and towards the open system. We point to the Holevo quantity as a distinguished quantum divergence to which the formalism can be applied, and we show how several distinct quantifiers of non-Markovianity can be related to each other within this general framework. Finally, we present an application of our analysis to a non-trivial physical model in which an exact evaluation of all quantities can be performed.

The present paper concerns a space-time homogenization problem for nonlinear diffusion equations with periodically oscillating (in space and time) coefficients. Main results consist of corrector results (i.e., strong convergences of solutions with corrector terms) for gradients, diffusion fluxes and time-derivatives without assumptions for smoothness of coefficients. Proofs of the main results are based on the space-time version of the unfolding method, which is deeply concerned with the strong two-scale convergence theory.

Let $K$ be a number field, and $g \geq 2$ a positive integer. We define $c_K(g)$ as the smallest integer $n$ such that there exist infinitely many $\overline{K}$-isomorphism classes of genus $g$ hyperelliptic curves $C/K$ with all Weierstrass points in $K$ having potentially good reduction outside $n$ primes in $K$. We show that $c_K(g) > π_{K, \textrm{odd}}(2g) + 1$, where $π_{K, \textrm{odd}}(n)$ denotes the number of odd primes in $K$ with norm no greater than $n$, as well as present a summary of various conditional and unconditional results on upper bounds for $c_K(g)$.

In this work, we consider the fractional Stefan-type problem in a Lipschitz bounded domain $Ω\subset\mathbb{R}^d$ with time-dependent Dirichlet boundary condition for the temperature $\vartheta=\vartheta(x,t)$, $\vartheta=g$ on $Ω^c\times]0,T[$, and initial condition $η_0$ for the enthalpy $η=η(x,t)$, given in $Ω\times]0,T[$ by \[\frac{\partial η}{\partial t} +\mathcal{L}_A^s \vartheta= f\quad\text{ with }η\in β(\vartheta),\] where $\mathcal{L}_A^s$ is an anisotropic fractional operator defined in the distributional sense by \[\langle\mathcal{L}_A^su,v\rangle=\int_{\mathbb{R}^d}AD^su\cdot D^sv\,dx,\] $β$ is a maximal monotone graph, $A(x)$ is a symmetric, strictly elliptic and uniformly bounded matrix, and $D^s$ is the distributional Riesz fractional gradient for $0<s<1$. We show the existence of a unique weak solution with its corresponding weak regularity. We also consider the convergence as $s\nearrow 1$ towards the classical local problem, the asymptotic behaviour as $t\to\infty$, and the convergence of the two-phase Stefan-type problem to the one-phase Stefan-type problem by varying the maximal monotone graph $β$.

Let $\overline{p}(n)$ denote the overpartition function. In this paper, we obtain an inequality for the sequence $Δ^{2}\log \ \sqrt[n-1]{\overline{p}(n-1)/(n-1)^α}$ which states that \begin{equation*} \log \biggl(1+\frac{3π}{4n^{5/2}}-\frac{11+5α}{n^{11/4}}\biggr) < Δ^{2} \log \ \sqrt[n-1]{\overline{p}(n-1)/(n-1)^α} < \log \biggl(1+\frac{3π}{4n^{5/2}}\biggr) \ \ \text{for}\ n \geq N(α), \end{equation*} where $α$ is a non-negative real number, $N(α)$ is a positive integer depending on $α$ and $Δ$ is the difference operator with respect to $n$. This inequality consequently implies $\log$-convexity of $\bigl\{\sqrt[n]{\overline{p}(n)/n}\bigr\}_{n \geq 19}$ and $\bigl\{\sqrt[n]{\overline{p}(n)}\bigr\}_{n \geq 4}$. Moreover, it also establishes the asymptotic growth of $Δ^{2} \log \ \sqrt[n-1]{\overline{p}(n-1)/(n-1)^α}$ by showing $\underset{n \rightarrow \infty}{\lim} Δ^{2} \log \ \sqrt[n]{\overline{p}(n)/n^α} = \dfrac{3 π}{4 n^{5/2}}.$

Isbell and Semadeni [Trans. Amer. Math. Soc. 107 (1963)] proved that every infinite-dimensional $1$-injective Banach space contains a hyperplane that is $(2+\varepsilon)$-injective for every $\varepsilon > 0$, yet is is \emph{not} $2$-injective and remarked in a footnote that Pełczyński had proved for every $λ> 1$ the existence of a $(λ+ \varepsilon)$-injective ($\varepsilon > 0$) that is not $λ$-injective. Unfortunately, no trace of the proof of Pełczyński's result has been preserved. In the present paper, we establish the said theorem for $λ\in (1,2]$ by constructing an appropriate renorming of $\ell_\infty$. This contrasts (at least for real scalars) with the case $λ= 1$ for which Lindenstrauss [Mem. Amer. Math. Soc. 48 (1964)] proved the contrary statement.

In this paper, we introduce the notion of conditional $h$-convex functions and we prove an operator version of the Jensen inequality for conditional $h$-convex functions. Using this type of functions, we give some refinements for Ky-Fan's inequality, arithmetic-geometric mean inequality, Chrystal inequality, and Hölder-McCarthy inequality. Many of the other inequalities can be refined by applying this new notion.

Let $F(y):=\displaystyle\int_t^TL(s, y(s), y'(s))\,ds$ be a positive functional (the "energy"), unnecessarily autonomous, defined on the space of Sobolev functions $W^{1,p}([t,T]; \mathbb R^n)$ ($p\ge 1$). We consider the problem of minimizing $F$ among the functions $y$ that possibly satisfy one, or both, end point conditions. In many applications, where the lack of regularity or convexity or growth conditions does not ensure the existence of a minimizer of $F$, it is important to be able to approximate the value of the infimum of $F$ via a sequence of Lipschitz functions satisfying the given boundary conditions. Sometimes, even with some polynomial, coercive and convex Lagrangians in the velocity variable, thus ensuring the existence of a minimizer in the given Sobolev space, this is not achievable: this fact is know as the Lavrentiev phenomenon. The paper deals on the avoidance of the Lavrentiev phenomenon under the validity of a further given state constraint of the form $y(s)\in\mathcal S\subset\mathbb R^n$ for all $s\in [t,T]$. Given $y\in W^{1,p}([t,T];\mathbb R^n)$ with $F(y)<+\infty$ we give a constructive recipe for building a sequence $(y_h)_h$ of Lipschitz reparametrizations of $y$, sharing with $y$ the same boundary condition(s), that converge in energy to $F(y)$. With respect to previous literature on the subject, we distinguish the case of (just) one end point condition from that of both, enlarge the class of Lagrangians that satisfy the sufficient conditions and show that $(y_h)_h$ converge also in $W^{1,p}$ to $y$. Moreover, the results apply also to extended valued Lagrangians whose effective domain is bounded. The results gives new clues even when the Lagrangian is autonomous, i.e., of the form $L(s,y,y')=Λ(y,y')$. The paper follows two recent papers \[23, 24] of the author on the subject.

By making a seminal use of the maximum modulus principle of holomorphic functions we prove existence of $n$-best kernel approximation for a wide class of reproducing kernel Hilbert spaces of holomorphic functions in the unit disc, and for the corresponding class of Bochner type spaces of stochastic processes. This study thus generalizes the classical result of $n$-best rational approximation for the Hardy space and a recent result of $n$-best kernel approximation for the weighted Bergman spaces of the unit disc. The type of approximations have significant applications to signal and image processing and system identification, as well as to numerical solutions of the classical and the stochastic type integral and differential equations.

Let $G$ be a locally compact group and $(Φ, Ψ)$ be a complementary pair of $N$-functions. In this paper, using the powerful tool of porosity, it is proved that when $G$ is an amenable group, then the Figà-Talamanca-Herz-Orlicz algebra ${\A}_Φ(G)$ is a Banach algebra under convolution product if and only if $G$ is compact. Then it is shown that ${\A}_Φ(G)$ is a Segal algebra, and as a consequence, the amenability of ${\A}_Φ(G)$ and the existence of a bounded approximate identity for ${\A}_Φ(G)$ under the convolution product is discussed. Furthermore, it is shown that for a compact abelian group $G$, the character space of ${\A}_Φ(G)$ under convolution product can be identified with $\widehat{G}$, the dual of $G$.

We present new soliton equations related to the $A$-type spin Calogero-Moser (CM) systems introduced by Gibbons and Hermsen. These equations are spin generalizations of the Benjamin-Ono (BO) equation and the recently introduced non-chiral intermediate long-wave (ncILW) equation. We obtain multi-soliton solutions of these spin generalizations of the BO equation and the ncILW equation via a spin-pole ansatz where the spin-pole dynamics is governed by the spin CM system in the rational and hyperbolic cases, respectively. We also propose physics applications of the new equations, and we introduce a spin generalization of the standard intermediate long-wave equation which interpolates between the matrix Korteweg-de Vries equation, the Heisenberg ferromagnet equation, and the spin BO equation.

We consider random walks on the nonnegative integers in a space-time dependent random environment. We assume that transition probabilities are given by independent $\mathrm{Beta}(μ,μ)$ distributed random variables, with a specific behaviour at the boundary, controlled by an extra parameter $η$. We show that this model is exactly solvable and prove a formula for the mixed moments of the random heat kernel. We then provide two formulas that allow us to study the large-scale behaviour. The first involves a Fredholm Pfaffian, which we use to prove a local limit theorem describing how the boundary parameter $η$ affects the return probabilities. The second is an explicit series of integrals, and we show that non-rigorous critical point asymptotics suggest that the large deviation behaviour of this half-space random walk in random environment is the same as for the analogous random walk on $\mathbb{Z}$.

We prove a large deviation principle for the point process associated to $k$-element connected components in $\mathbb R^d$ with respect to the connectivity radii $r_n\to\infty$. The random points are generated from a homogeneous Poisson point process, so that $(r_n)_{n\ge1}$ satisfies $n^kr_n^{d(k-1)}\to\infty$ and $nr_n^d\to0$ as $n\to\infty$ (i.e., sparse regime). The rate function for the obtained large deviation principle can be represented as relative entropy. As an application, we deduce large deviation principles for various functionals and point processes appearing in stochastic geometry and topology. As concrete examples of topological invariants, we consider persistent Betti numbers of geometric complexes and the number of Morse critical points of the min-type distance function.

Let $f$ be a function in the inhomogeneous analytic Besov space $B_{\infty,1}^1$. For a pair $(L,M)$ of not necessarily commuting maximal dissipative operators, we define the function $f(L,M)$ of $L$ and $M$ as a densely defined linear operator. We prove for $p\in[1,2]$ that if $(L_1,M_1)$ and $(L_2,M_2)$ are pairs of not necessarily commuting maximal dissipative operators such that both differences $L_1-L_2$ and $M_1-M_2$ belong to the Schatten--von Neumann class $\boldsymbol{S}_p$ than for an arbitrary function $f$ in the inhomogeneous analytic Besov space $B_{\infty,1}^1$, the operator difference $f(L_1,M_1)-f(L_2,M_2)$ belongs to $\boldsymbol{S}_p$ and the following Lipschitz type estimate holds: $$ \|f(L_1,M_1)-f(L_2,M_2)\|_{\boldsymbol{S}_p} \le\operatorname{const}\|f\|_{B_{\infty,1}^1}\max\big\{\|L_1-L_2\|_{\boldsymbol{S}_p},\|M_1-M_2\|_{\boldsymbol{S}_p}\big\}. $$

Lights nodes are commonly used in blockchain systems to combat the storage burden. However, light nodes are known to be vulnerable to data availability (DA) attacks where they accept an invalid block with unavailable portions. Previous works have used LDPC codes with Merkle Trees to mitigate DA attacks. However, LDPC codes have issues in the finite length due to the NP-hardness of ascertaining the minimum stopping set size, and in the asymptotic regime due to probabilistic guarantees on code performance. We circumvent both issues by proposing the novel Polar Coded Merkle Tree (PCMT) which is a Merkle Tree built from the encoding graphs of polar codes. We provide a specialized polar code construction called Sampling-Efficient Freezing that efficiently calculates the minimum stopping set size, thus simplifying design. PCMT performs well in detecting DA attacks for large transaction block sizes.

We introduce a new generalization of the Pseudo-Lindley distribution by applying alpha power transformation. The obtained distribution is referred as the Pseudo-Lindley alpha power transformed distribution (\textit{PL-APT}). Some tractable mathematical properties of the \textit{PL-APT} distribution as reliability, hazard rate, order statistics and entropies are provided. The maximum likelihood method is used to obtain the parameters' estimation of the \textit{PL-APT} distribution. The asymptotic properties of the proposed distribution are discussed. Also, a simulation study is performed to compare the modeling capability and flexibility of \textit{PL-APT} with Lindley and Pseudo-Lindley distributions. The \textit{PL-APT} provides a good fit as the Lindley and the Pseudo-Lindley distribution. The extremal domain of attraction of \textit{PL-APT} is found and its quantile and extremal quantile functions studied. Finally, the extremal value index is estimated by the double-indexed Hill's estimator (Ngom and Lo, 2016) and related asymptotic statistical tests are provided and characterized.