arXiv Math @arxiv_math@qoto.org

Concatenated bit-interleaved and multilevel coded modulation with outer Reed--Solomon codes, inner Chase-algorithm-based soft-decision-decoded Bose--Ray-Chaudhuri--Hocquenghem codes, and four-level pulse amplitude modulation is considered. A semi-analytical formula is derived for estimating the decoded frame error rate (FER) at the output the additive white Gaussian noise channel, obviating the need for time-consuming Monte Carlo simulations. The formula is used to search a large space of codes (including the KP4 code) to find those achieving good trade-offs among performance (measured by the gap to the constrained Shannon limit at $10^{-13}$ FER), complexity (measured by the number of elementary decoder operations), and latency (measured by overall block length).

Orthogonal Frequency Division Multiplexing (OFDM) is a popular modulation technique for transmitting digital data over wireless radio channels, including medical images due to its high transmission capacity, low interference, bandwidth efficiency, and scalability. However, the security of medical images is a major concern, and combining OFDM with encryption techniques such as chaos-based image encryption can enhance security measures. This study proposes a secure medical image transmission system that combines OFDM, 6D hyperchaotic system, and Fibonacci Q-matrix and analyzes its impact on image transmission quality using simulation results obtained through MATLAB. The study examines the Q-PSK constellation diagram, fast Fourier transform (IFFT) signal, cyclic prefix (CP) techniques, NIST, signal noise ratio (SNR), and bit error rate (BER). The results provide insights into the effectiveness of OFDM in securely transmitting high-quality medical images.

This paper studies a new latency optimization problem in unmanned aerial vehicles (UAVs)-enabled federated learning (FL) with integrated sensing and communication. In this setup, distributed UAVs participate in model training using sensed data and collaborate with a base station (BS) serving as FL aggregator to build a global model. The objective is to minimize the FL system latency over UAV networks by jointly optimizing UAVs' trajectory and resource allocation of both UAVs and the BS. The formulated optimization problem is troublesome to solve due to its non-convexity. Hence, we develop a simple yet efficient iterative algorithm to find a high-quality approximate solution, by leveraging block coordinate descent and successive convex approximation techniques. Simulation results demonstrate the effectiveness of our proposed joint optimization strategy under practical parameter settings, saving the system latency up to 68.54\% compared to benchmark schemes.

This paper is devoted to studying the maximal-in-time estimates and Strichartz estimates for orthonormal functions and convergence problem of density functions related to Boussinesq operator on manifolds. Firstly, we present the pointwise convergence of density function related to Boussinesq operator with $γ_{0}\in\mathfrak{S}^β(\dot{H}^{\frac{1}{4}}(\mathbf{R}))(β<2)$ with the aid of the maximal-in-time estimate related to Boussinesq operator with orthonormal function on $\R$. Secondly, we present the pointwise convergence of density function related to Boussinesq operator with $γ_{0}\in\mathfrak{S}^β(\dot{H}^{s})(\frac{d}{4}\leq s<\frac{d}{2},\, 0<α\leq d, 1\leqβ<\fracα{d-2s})$ with the aid of the maximal-in-time estimates related to Boussinesq operator with orthonormal function on the unit ball $\mathbf{B}^{d}(d\geq1)$ established in this paper; we also present the Hausdorff dimension of the divergence set of density function related to Boussinesq operator $dim_{H}D(γ_{0})\leq (d-2s)β$. Thirdly, we show the Strichartz estimates for orthonormal functions and Schatten bound with space-time norms related to Boussinesq operator on $\mathbf{T}$ with the aid of the noncommutative-commutative interpolation theorems established in this paper, which are just Lemmas 3.1-3.4 in this paper; we also prove that Theorems 1.5, 1.6 are optimal. Finally, by using full randomization, we present the probabilistic convergence of density function related to Boussinesq operator on $\R$, $\mathbf{T}$ and $Θ=\{x\in\R^{3}:|x|<1\}$ with $γ_{0}\in\mathfrak{S}^{2}$.

In this paper, we study the inverse problem of finding a time-dependent multiplier of the right-hand side of a time-fractional one-dimensional diffusion equation with variables coefficients in the case where the usual Cauchy, homogeneous Dirichlet boundary, and an integral overdetermination conditions are given. The overdetermination condition has the form of an integral with a weight over a spatial segment from the solution of the direct problem, in which the weight function is a spatially dependent known factor of the right-hand side of the equation. This made it possible to construct a solution to the inverse problem in explicit form and prove its correctness in the class of regular solutions

Effective and reliable data retrieval is critical for the feasibility of DNA storage, and the development of random access efficiency plays a key role in its practicality and reliability. In this paper, we study the Random Access Problem, which asks to compute the expected number of samples one needs in order to recover an information strand. Unlike previous work, we took a geometric approach to the problem, aiming to understand which geometric structures lead to codes that perform well in terms of reducing the random access expectation (Balanced Quasi-Arcs). As a consequence, two main results are obtained. The first is a construction for $k=3$ that outperforms previous constructions aiming to reduce the random access expectation. The second, exploiting a result from [1], is the proof of a conjecture from [2] for rate $1/2$ codes in any dimension.

Subwavelength resonance is a vital acoustic phenomenon in contrasting media. The narrow bandgap width of single-layer resonator has prompted the exploration of multi-layer metamaterials as an effective alternative, which consist of alternating nests of high-contrast materials, called ``resonators'', and a background media. In this paper, we develop a general mathematical framework for studying acoustics within multi-layer high-contrast structures. Firstly, by using layer potential techniques, we establish the representation formula in terms of a matrix type operator with a block tridiagonal form for multi-layer structures within general geometry. Then we prove the existence of subwavelength resonances via Gohberg-Sigal theory, which generalizes the celebrated Minnaert resonances in single-layer structures. Intriguingly, we find that the primary contribution to mode splitting lies in the fact that as the number of nested resonators increases, the degree of the corresponding characteristic polynomial also increases, while the type of resonance (consists solely of monopolar resonances) remains unchanged. Furthermore, we derive original formulas for the subwavelength resonance frequencies of concentric dual-resonator. Numerical results associated with different nested resonators are presented to corroborate the theoretical findings.

In this article, we apply the techniques developed in our previous article ``Local generation of tilings'', in which we introduced two definitions capturing the intuitive idea that some subshifts admit a procedure that can generate any tiling and working in a local way. We classify all the Wang tilesets with two colors in which each tile has an even number of each color.

Here we give a survey of consequences from the theory of the Beltrami equation in the complex plane $\mathbb C$ to generalized Cauchy-Riemann equation in the real plane $\mathbb R^2$ and clarify the relationships of the latter to the $A-$harmonic equation ${\rm div} A(z)\,{\rm grad}\, u(z) = 0$, $z=x+iy$, with matrix valued coefficients $A$ that is one of the main equations of the potential theory, namely, of the hydro\-mechanics (fluid mechanics) in anisotropic and inhomogeneous media. The survey includes various types of results as theorems on existence, representation and regularity of its solutions, in particular, for the main boundary value problems of Hilbert, Dirichlet, Neumann, Poincare and Riemann.

Given a fibred hyperbolic 3-manifold with boundary, we coarsely relate the Euclidean geometry of its cusps to the classical fractional Dehn twist coefficient of its monodromy. This result fits into the broader programme of coarsely describing the geometry of a hyperbolic 3-manifold via combinatorial data. We are thus able to study the hyperbolic geometry of certain fibred 3-manifolds under Dehn filling. For example, we find coarse volume estimates for sufficiently twisted braid closures in terms of their braid words. We also prove that for any open book decomposition of a fixed manifold (that is not a lens space or solid torus) with fibre of fixed Euler characteristic the fractional Dehn twist coefficient in some boundary component is uniformly bounded. Finally, we obtain applications to contact topology. We give a geometric criterion on the binding of an open book decomposition for the corresponding contact structure in a hyperbolic 3-manifold to be tight.

The concept of input-to-state stability (ISS) proposed in the late 1980s is one of the central notions in robust nonlinear control. ISS has become indispensable for various branches of nonlinear systems theory, such as robust stabilization of nonlinear systems, design of nonlinear observers, analysis of large-scale networks, etc. The success of the ISS theory of ODEs and the need for robust stability analysis of partial differential equations (PDEs) motivated the development of ISS theory in the infinite-dimensional setting. For instance, the Lyapunov method for analysis of iISS of nonlinear parabolic equations. For an overview of the ISS theory for distributed parameter systems. ISS of control systems with application to robust global stabilization of the chemostat has been studied with the help of vector Lyapunov functions.

Recently, general fractional calculus was introduced by Kochubei (2011) and Luchko (2021) as a further generalisation of fractional calculus, where the derivative and integral operator admits arbitrary kernel. Such a formalism will have many applications in physics and engineering, since the kernel is no longer restricted. We first extend the work of Al-Refai and Luchko (2023) on finite interval to arbitrary orders. Followed by, developing an efficient Petrov-Galerkin scheme by introducing Jacobi convolution polynomials as basis functions. A notable property of this basis function, the general fractional derivative of Jacobi convolution polynomial is a shifted Jacobi polynomial. Thus, with a suitable test function it results in diagonal stiffness matrix, hence, the efficiency in implementation. Furthermore, our method is constructed for any arbitrary kernel including that of fractional operator, since, its a special case of general fractional operator.

In this paper, we formulate the Collatz conjecture (or the $3n{+}1$-problem) as some operator theoretic problems and prove that each of those statements is equivalent to the conjecture.

We characterise absolutely dilatable completely positive maps on the space of all bounded operators on a Hilbert space that are also bimodular over a given von Neumann algebra as rotations by a suitable unitary on a larger Hilbert space followed by slicing along the trace of an additional ancilla. We define the local, quantum and approximately quantum types of absolutely dilatable maps, according to the type of the admissible ancilla. We show that the local absolutely dilatable maps admit an exact factorisation through an abelian ancilla and show that they are limits in the point weak* topology of conjugations by unitaries in the commutant of the given von Neumann algebra. We show that the Connes Embedding Problem is equivalent to deciding if all absolutely dilatable maps are approximately quantum.

The properties of a hypergraph explored through the spectrum of its unified matrix was made by the authors in [26]. In this paper, we introduce three different hypergraph matrices: unified Laplacian matrix, unified signless Laplacian matrix, and unified normalized Laplacian matrix, all defined using the unified matrix. We show that these three matrices of a hypergraph are respectively identical to the Laplacian matrix, signless Laplacian matrix, and normalized Laplacian matrix of the associated graph. This allows us to use the spectra of these hypergraph matrices as a means to connect the structural properties of the hypergraph with those of the associated graph. Additionally, we introduce certain hypergraph structures and invariants during this process, and relate them to the eigenvalues of these three matrices.

Let $SPP(n)$ be the set $\{\big(|A+A|,|A\cdot A|\big) : A\subseteq {\mathbb N}, |A|=n \}$, where $A+A$ is the sumset $\{a+b : a,b\in A\}$ and $AA$ is the product set $\{ab : a,b\in A\}$. We prove the value of $SPP(n)$ exactly for $n\le 6$, and compute a large subset of $SPP(n)$ for $n\le 32$. This is a dataset of 1,158,717 sets of positive integers that are addtiively and multiplicatively diverse. We do {\bf not} see evidence in favor of Erdős's Sum-Product Conjecture in our dataset. We include a number of conjectures, open problems, and observations motivated by this dataset, and a large number of color visualizations.

Accurate estimates of causal effects play a key role in decision-making across applications such as healthcare, economics, and operations. In the absence of randomized experiments, a common approach to estimating causal effects uses \textit{covariate adjustment}. In this paper, we study covariate adjustment for discrete distributions from the PAC learning perspective, assuming knowledge of a valid adjustment set $\bZ$, which might be high-dimensional. Our first main result PAC-bounds the estimation error of covariate adjustment by a term that is exponential in the size of the adjustment set; it is known that such a dependency is unavoidable even if one only aims to minimize the mean squared error. Motivated by this result, we introduce the notion of an \emph{$\eps$-Markov blanket}, give bounds on the misspecification error of using such a set for covariate adjustment, and provide an algorithm for $\eps$-Markov blanket discovery; our second main result upper bounds the sample complexity of this algorithm. Furthermore, we provide a misspecification error bound and a constraint-based algorithm that allow us to go beyond $\eps$-Markov blankets to even smaller adjustment sets. Our third main result upper bounds the sample complexity of this algorithm, and our final result combines the first three into an overall PAC bound. Altogether, our results highlight that one does not need to perfectly recover causal structure in order to ensure accurate estimates of causal effects.

Bourgain used the Rudin-Shapiro sequences to construct a basis of uniformly bounded holomorphic functions on the unit sphere in $\mathbb{C}^2$. They are also spherical harmonics (i.e., Laplacian eigenfunctions) on $\mathbb{S}^3 \subset \mathbb{R}^4$. In this paper, we prove that these functions tend to be equidistributed on $\mathbb{S}^3$, based on an estimate of the auto-correlation of the Rudin-Shapiro sequences. Moreover, we identify the semiclassical measure associated to these spherical harmonics by the singular measure supported on the family of Clifford tori in $\mathbb{S}^3$. In particular, this demonstrates a new localization pattern in the study of Laplacian eigenfunctions.

In this paper, we present a telegraph diffusion model with variable exponents for image despeckling. Moving beyond the traditional assumption of a constant exponent in the telegraph diffusion framework, we explore three distinct variable exponents for edge detection. All of these depend on the gray level of the image or its gradient. We rigorously prove the existence and uniqueness of weak solutions of our model in a functional setting and perform numerical experiments to assess how well it can despeckle noisy gray-level images. We consider both a range of natural images contaminated by varying degrees of artificial speckle noise and synthetic aperture radar (SAR) images. We finally compare our method with the nonlocal speckle removal technique and find that our model outperforms the latter at speckle elimination and edge preservation.

Quantum low-density parity-check (LDPC) codes are a promising family of quantum error-correcting codes for fault tolerant quantum computing with low overhead. Decoding quantum LDPC codes on quantum erasure channels has received more attention recently due to advances in erasure conversion for various types of qubits including neutral atoms, trapped ions, and superconducting qubits. Belief propagation with guided decimation (BPGD) decoding of quantum LDPC codes has demonstrated good performance in bit-flip and depolarizing noise. In this work, we apply BPGD decoding to quantum erasure channels. Using a natural modification, we show that BPGD offers competitive performance on quantum erasure channels for multiple families of quantum LDPC codes. Furthermore, we show that the performance of BPGD decoding on erasure channels can sometimes be improved significantly by either adding damping or adjusting the initial channel log-likelihood ratio for bits that are not erased. More generally, our results demonstrate BPGD is an effective general-purpose solution for erasure decoding across the quantum LDPC landscape.