arXiv Math @arxiv_math@qoto.org

Reaping the benefits of multi-antenna communication systems in frequency division duplex (FDD) requires channel state information (CSI) reporting from mobile users to the base station (BS). Over the last decades, the amount of CSI to be collected has become very challenging owing to the dramatic increase of the number of antennas at BSs. To mitigate the overhead associated with CSI reporting, compressed CSI techniques have been proposed with the idea of recovering the original CSI at the BS from its compressed version sent by the mobile users. Channel charting is an unsupervised dimensionality reduction method that consists in building a radio-environment map from CSIs. Such a method can be considered in the context of the CSI compression problem, since a chart location is, by definition, a low-dimensional representation of the CSI. In this paper, the performance of channel charting for a task-based CSI compression application is studied. A comparison of the proposed method against baselines on realistic synthetic data is proposed, showing promising results.

An irreducible quintic equation is solvable by radicals if and only if its Galois group is solvable. In this work, we provide necessary and sufficient conditions for solvability, expressed in terms of invariants of the quintic.

We discuss a specific type of pseudospherical surfaces defined by a class of third order differential equations, of the form $u_t - u_{xxt} = λu^2 u_{xxx} + G(u, u_x, u_{xx})$, and poses a question about the dependence of the triples $\{a,b,c\}$ of the second fundamental form in the context of local isometric immersion in $\mathbb{E}^3$. It is demonstrated that the triples $\{a,b,c\}$ of the second fundamental form are not influenced by a jet of finite order of $u$. Instead, they are shown to rely on a jet of order zero, making them universal and not reliant on the specific solution chosen for $u$.

We establish optimal error bounds on an exponential wave integrator (EWI) for the space fractional nonlinear Schrödinger equation (SFNLSE) with low regularity potential and/or nonlinearity. For the semi-discretization in time, under the assumption of $L^\infty$-potential, $C^1$-nonlinearity, and $H^α$-solution with $1<α\leq 2$ being the fractional index of $(-Δ)^\fracα{2}$, we prove an optimal first-order $L^2$-norm error bound $O(τ)$ and a uniform $H^α$-norm bound of the semi-discrete numerical solution, where $τ$ is the time step size. We further discretize the EWI in space by the Fourier spectral method and obtain an optimal error bound without introducing any CFL-type time step size restrictions. In particular, the spatial convergence is optimal with respect to the regularity of the exact solution. Moreover, under slightly stronger regularity assumptions, we obtain optimal error bounds in $H^\fracα{2}$-norm, which is the norm associated to the energy. Extensive numerical examples are provided to validate the optimal error bounds and show their sharpness. We also find distinct evolving patterns between the SFNLSE and the classical nonlinear Schrödinger equation.

We consider the class of ring $Q$-homeomorphisms with respect to $p$-modulus in $\mathbb{R}^{n}$ with $p > n$, and obtain lower bounds for limsups of the distance distortions under such mappings. These estimates can be treated as Hölder's continuity of the inverses near the origin. The sharpness is illustrated by example

This paper is devoted to order-one explicit approximations of random periodic solutions to multiplicative noise driven stochastic differential equations (SDEs) with non-globally Lipschitz coefficients. The existence of the random periodic solution is demonstrated as the limit of the pull-back of the discretized SDE. A novel approach is introduced to analyze mean-square error bounds of the proposed scheme that does not depend on a prior high-order moment bounds of the numerical approximations. Under mild assumptions, the proposed scheme is proved to achieve an expected order-one mean square convergence in the infinite time horizon. Numerical examples are finally provided to verify the theoretical results.

A highly cited and inspiring article by Bates et al (2024) demonstrates that the prediction errors estimated through cross-validation, Bootstrap or Mallow's $C_P$ can all be independent of the actual prediction errors. This essay hypothesizes that these occurrences signify a broader, Heisenberg-like uncertainty principle for learning: optimizing learning and assessing actual errors using the same data are fundamentally at odds. Only suboptimal learning preserves untapped information for actual error assessments, and vice versa, reinforcing the `no free lunch' principle. To substantiate this intuition, a Cramer-Rao-style lower bound is established under the squared loss, which shows that the relative regret in learning is bounded below by the square of the correlation between any unbiased error assessor and the actual learning error. Readers are invited to explore generalizations, develop variations, or even uncover genuine `free lunches.' The connection with the Heisenberg uncertainty principle is more than metaphorical, because both share an essence of the Cramer-Rao inequality: marginal variations cannot manifest individually to arbitrary degrees when their underlying co-variation is constrained, whether the co-variation is about individual states or their generating mechanisms, as in the quantum realm. A practical takeaway of such a learning principle is that it may be prudent to reserve some information specifically for error assessment rather than pursue full optimization in learning, particularly when intentional randomness is introduced to mitigate overfitting.

This expository article is devoted to the notion of quasianalytic classes and the Borel mapping. Although quasianalytic classes are well known in analysis since several decades. We are interested in certain properties of Denjoy-Carleman's quasianalytic classes, such as the non-surjectivity of the Borel mapping, the property of monotonicity. We try to see if it remains true for other quasianalytic classes, such as for example, the classes of indefinitely differentiable functions definable in a polynomially bounded o-minimal structures. What motivated this is the fact of having shown in a previous article the existence of quasianalytic classes where Borel mapping is surjective.

This paper focuses on the derivations and automorphism groups of certain finite-dimensional associative algebras over the field of complex numbers. Using classification results for algebras of dimensions two, three, and four, along with computational tools like Mathematica and Maple, we offer detailed descriptions of the derivations and automorphism groups for these algebras. Our analysis of these groups helps to uncover important structural features and symmetries in low-dimensional associative algebras.

Codes in the generalized quaternion group algebra $\mathbb{F}_q[Q_{4n}]$ are considered. Restricting to char$\mathbb{F}_q \nmid 4n$ the structure of an arbitrary code $C \subseteq \mathbb{F}_q[Q_{4n}]$ is described via the Wedderburn decomposition. Moreover it is known that in this case every code $C \subseteq \mathbb{F}_q[Q_{4n}]$ has a generating idempotent $λ\in \mathbb{F}_q[Q_{4n}]$. Given the generating idempotent of a code $C$ we determine the different components in its decomposition $C \cong \bigoplus_{j=1}^{r+s}C_j \oplus \bigoplus_{i=1}^{k+t}C'_{i}.$ Afterwards we apply this result to describe the blocks of codes induced by cyclic group codes.

In 1946, S. Ulam invented Monte Carlo method, which has since become the standard numerical technique for making statistical predictions for long-term behaviour of dynamical systems. We show that this, or in fact any other numerical approach can fail for the simplest non-linear discrete dynamical systems given by the logistic maps $f_{a}(x)=ax(1-x)$ of the unit interval. We show that there exist computable real parameters $a\in (0,4)$ for which almost every orbit of $f_a$ has the same asymptotical statistical distribution in $[0,1]$, but this limiting distribution is not Turing computable.

The ever-changing world of disease study heavily relies on mathematical models. They are key in finding and controlling infectious diseases. We aim to explore these mathematical tools used for studying disease spread in biology. The SEIR model holds our focus. It is a super important tool known for being flexible and useful. We look at the modified SEIR models' design and analysis. We dive right into vital parts like the equations that make the modified SEIR model work, setting parameter identities, and then checking its solutions' positivity and limits. The study begins with a detailed examination of the design and analysis of a modified SEIR model, demonstrating its angularity. We delve into the model's heart, dealing with critical issues such as the equations that drive the modified SEIR model, establishing parameter identities, and ensuring the positivity and boundlessness of its solutions. Basic Reproduction Number marks a significant milestone. We investigate the local stability, DFE, and EE. Global stability, a paramount consideration in understanding the long-term behaviors of the systems, is scrutinized by employing the Lyapunov stability theorem. The bifurcation analysis classifies and elucidates the fundamental concepts therein. One-dimensional bifurcation and forward and backward bifurcation analyses are intricately examined, providing a comprehensive understanding of the dynamical behavior and basic concepts. In summary, we offer a thorough description and analysis of the SEIR model but also lay the groundwork for advancing mathematical modeling in epidemiology. By bridging theoretical insights with practical implications, this study strives to empower researchers and policymakers with a deep understanding of infectious disease dynamics, thereby contributing to targeted public health strategies.

This work investigates the long time asymptotic behavior of some inhomogeneous non-linear Schrödinger type equations. We give sharp a threshold of scattering versus non-scattering of mass solutions, depending on the source term. This work complements the literature to the inhomogeneous regime.

In this paper we present a brief study of the $σ$-set-$σ$-antiset duality that occurs in $σ$-set theory and we also present the development of the integer space $3^{A}=\left\langle 2^{A}, 2^{A^{-}} \right\rangle$ for the cardinals $|A|=2,3$ together with its algebraic properties. In this article, we also develop a presentation of some of the properties of fusion of $σ$-sets and finally we present the development and definition of a type of equations of one $σ$-set variable.

We study the Einstein-Yang-Mills system in both the Lorenz and harmonic gauges, where the Yang-Mills fields are valued in any arbitrary Lie algebra $\cal G$, associated to any compact Lie group $G$. This gives a system of hyperbolic partial partial differential that does not satisfy the null condition and that has new complications that are not present for the Einstein vacuum equations nor for the Einstein-Maxwell system. We prove the exterior stability of the Minkowski space-time, $\mathbb{R}^{1+3}$, governed by the fully coupled Einstein-Yang-Mills system in the Lorenz gauge, valued in any arbitrary Lie algebra $\cal G$, without any assumption of spherical symmetry. We start with an arbitrary sufficiently small initial data, defined in a suitable energy norm for the perturbations of the Yang-Mills potential and of the Minkowski space-time, and we show the well-posedness of the Cauchy development in the exterior, and we prove that this leads to solutions converging in the Lorenz gauge and in wave coordinates to the zero Yang-Mills fields and to the Minkowski space-time. This provides a first detailed proof of the exterior stability of Minkowski governed by the fully non-linear Einstein-Yang-Mills equations in the Lorenz gauge, by using a null frame decomposition that was first used by H. Lindblad and I. Rodnianski for the case of the Einstein vacuum equations. We note that in contrast to the much simpler case of the Einstein-Maxwell equations where one can omit the potential, in fact in the non-abelian case of the Einstein-Yang-Mills equations, the question of stability, or non-stability, is a purely gauge dependent statement and the partial differential equations depend on the gauge on the Yang-Mills potential that is needed to write up the equations.

Community detection, also known as graph partitioning, is a well-known NP-hard combinatorial optimization problem with applications in diverse fields such as complex network theory, transportation, and smart power grids. The problem's solution space grows drastically with the number of vertices and subgroups, making efficient algorithms crucial. In recent years, quantum computing has emerged as a promising approach to tackling NP-hard problems. This study explores the use of a quantum-inspired algorithm, Simulated Bifurcation (SB), for community detection. Modularity is employed as both the objective function and a metric to evaluate the solutions. The community detection problem is formulated as a Quadratic Unconstrained Binary Optimization (QUBO) problem, enabling seamless integration with the SB algorithm. Experimental results demonstrate that SB effectively identifies community structures in benchmark networks such as Zachary's Karate Club and the IEEE 33-bus system. Remarkably, SB achieved the highest modularity, matching the performance of Fujitsu's Digital Annealer, while surpassing results obtained from two quantum machines, D-Wave and IBM. These findings highlight the potential of Simulated Bifurcation as a powerful tool for solving community detection problems.

This paper proposes risk-averse and risk-agnostic formulations to robust design in which solutions that satisfy the system requirements for a set of scenarios are pursued. These scenarios, which correspond to realizations of uncertain parameters or varying operating conditions, can be obtained either experimentally or synthetically. The proposed designs are made robust to variations in the training data by considering perturbed scenarios. This practice allows accounting for error and uncertainty in the measurements, thereby preventing data overfitting. Furthermore, we use relaxation to trade-off a lower optimal objective value against lesser robustness to uncertainty. This is attained by eliminating a given number of optimally chosen outliers from the dataset, and by allowing for the perturbed scenarios to violate the requirements with an acceptably small probability. For instance, we can pursue a riskier design that attains a lower objective value in exchange for a few scenarios violating the requirements, or we might seek a more conservative design that satisfies the requirements for as many perturbed scenarios as possible. The design of a flexible wing subject to aeroelastic constraints is used for illustration.

We prove that meromorphic differentials $ω^{(0)}_n(z_1,...,z_n)$ which are recursively generated by an involution identity are symmetric in all their arguments $z_1,...,z_n$ -- provided that an intriguing combinatorial identity between integer partitions into given number of parts is true.

We prove the existence of infinitely many \v Soltés' digraphs, the digraph analogue of \v Soltés' graphs. We also give an example of a \v Soltés' digraph with trivial automorphism group.

We prove that the space of algebraic maps between two smooth projective varieties, under certain conditions, admit a configuration space model, thereby obtaining an algebro-geometric analogue of Bendersky-Gitler's result on topological function spaces. Our result is a natural higher dimensional counterpart of \cite[Theorem 3]{Ban24}.