arXiv Math @arxiv_math@qoto.org

The conventional approach to the fronthaul design for cell-free massive MIMO system follows the compress-and-precode (CP) paradigm. Accordingly, encoded bits and precoding coefficients are shared by the distributed unit (DU) on the fronthaul links, and precoding takes place at the radio units (RUs). Previous theoretical work has shown that CP can be potentially improved by a significant margin by precode-and-compress (PC) methods, in which all baseband processing is carried out at the DU, which compresses the precoded signals for transmission on the fronthaul links. The theoretical performance gain of PC methods are particularly pronounced when the DU implements multivariate quantization (MQ), applying joint quantization across the signals for all the RUs. However, existing solutions for MQ are characterized by a computational complexity that grows exponentially with the sum-fronthaul capacity from the DU to all RUs. This work sets out to design scalable MQ strategies for PC-based cell-free massive MIMO systems. For the low-fronthaul capacity regime, we present alpha-parallel MQ (alpha-PMQ), whose complexity is exponential only in the fronthaul capacity towards an individual RU, while performing close to full MQ. alpha-PMQ tailors MQ to the topology of the network by allowing for parallel local quantization steps for RUs that do not interfere too much with each other. For the high-fronthaul capacity regime, we then introduce neural MQ, which replaces the exhaustive search in MQ with gradient-based updates for a neural-network-based decoder, attaining a complexity that grows linearly with the sum-fronthaul capacity. Numerical results demonstrate that the proposed scalable MQ strategies outperform CP for both the low and high-fronthaul capacity regimes at the cost of increased computational complexity at the DU (but not at the RUs).

Determining the crossing numbers of Cartesian products of small graphs with arbitrarily large paths has been an ongoing topic of research since the 1970s. Doing so requires the establishment of coincident upper and lower bounds; the former is usually demonstrated by providing a suitable drawing procedure, while the latter often requires substantial theoretical arguments. Many such papers have been published, which typically focus on just one or two small graphs at a time, and use ad hoc arguments specific to those graphs. We propose a general approach which, when successful, establishes the required lower bound. This approach can be applied to the Cartesian product of any graph with arbitrarily large paths, and in each case involves solving a modified version of the crossing number problem on a finite number (typically only two or three) of small graphs. We demonstrate the potency of this approach by applying it to Cartesian products involving all 133 graphs $G$ of orders five or six, and show that it is successful in 128 cases. This includes 60 cases which a recent survey listed as either undetermined, or determined only in journals without adequate peer review.

We introduce a set of special functions called multiple polyexponential integrals, defined as iterated integrals of the exponential integral $\text{Ei}(z)$. These functions arise in certain perturbative expansions of the local solutions of second-order ODEs around an irregular singularity. In particular, their recursive definition describes the asymptotic behavior of these local solutions. To complement the study of the multiple polyexponential integrals on the entire complex plane, we relate them with two other sets of special functions - the undressed and dressed multiple polyexponential functions - which are characterized by their Taylor series expansions around the origin.

Causality analysis is a powerful tool for determining cause-and-effect relationships between variables in a system by quantifying the influence of one variable on another. Despite significant advancements in the field, many existing studies are constrained by their focus on unidirectional causality or Gaussian external forcing, limiting their applicability to complex real-world problems. This study proposes a novel data-driven approach to causality analysis for complex stochastic differential systems, integrating the concepts of Liang-Kleeman information flow and linear inverse modeling. Our method models environmental noise as either memoryless Gaussian white noise or memory-retaining Ornstein-Uhlenbeck colored noise, and allows for self and mutual causality, providing a more realistic representation and interpretation of the underlying system. Moreover, this LIM-based approach can identify the individual contribution of dynamics and correlation to causality. We apply this approach to re-examine the causal relationships between the El Niño-Southern Oscillation (ENSO) and the Indian Ocean Dipole (IOD), two major climate phenomena that significantly influence global climate patterns. In general, regardless of the type of noise used, the causality between ENSO and IOD is mutual but asymmetric, with the causality map reflecting an ENSO-like pattern consistent with previous studies. Notably, in the case of colored noise, the noise memory map reveals a hotspot in the Niño 3 region, which is further related to the information flow. This suggests that our approach offers a more comprehensive framework and provides deeper insights into the causal inference of global climate systems.

We show that for any surface of genus at least 3 equipped with any choice of framing, the graph of non-separating curves with winding number 0 with respect to the framing is hierarchically hyperbolic but not Gromov hyperbolic. We also describe how to build analogues of the curve graph for marked strata of abelian differentials that capture the combinatorics of their boundaries, analogous to how the curve graph captures the combinatorics of the augmented Teichmueller space. These curve graph analogues are also shown to be hierarchically, but not Gromov, hyperbolic.

Let $\mathfrak{M}$ and $\mathfrak{J}$ be JBW$^*$-algebras admitting no central summands of type $I_1$ and $I_2,$ and let $Φ: \mathfrak{M} \rightarrow \mathfrak{J}$ be a linear bijection preserving operator commutativity in both directions, that is, $$[x,\mathfrak{M},y] = 0 \Leftrightarrow [Φ(x),\mathfrak{J},Φ(y)] = 0,$$ for all $x,y\in \mathfrak{M}$, where the associator of three elements $a,b,c$ in $\mathfrak{M}$ is defined by $[a,b,c]:=(a\circ b)\circ c - (c\circ b)\circ a$. We prove that under these conditions there exist a unique invertible central element $z_0$ in $\mathfrak{J}$, a unique Jordan isomorphism $J: \mathfrak{M} \rightarrow \mathfrak{J}$, and a unique linear mapping $β$ from $\mathfrak{M}$ to the centre of $\mathfrak{J}$ satisfying $$ Φ(x) = z_0 \circ J(x) + β(x), $$ for all $x\in \mathfrak{M}.$ Furthermore, if $Φ$ is a symmetric mapping (i.e., $Φ(x^*) = Φ(x)^*$ for all $x\in \mathfrak{M}$), the element $z_0$ is self-adjoint, $J$ is a Jordan $^*$-isomorphism, and $β$ is a symmetric mapping too. In case that $\mathfrak{J}$ is a JBW$^*$-algebra admitting no central summands of type $I_1$, we also address the problem of describing the form of all symmetric bilinear mappings $B : \mathfrak{J}\times \mathfrak{J}\to \mathfrak{J}$ whose trace is associating (i.e., $[B(a,a),b,a] = 0,$ for all $a, b \in \mathfrak{J})$ providing a complete solution to it. We also determine the form of all associating linear maps on $\mathfrak{J}$.

Two central problems in extremal combinatorics are concerned with estimating the number $ex(n,H)$, the size of the largest $H$-free hypergraph on $n$ vertices, and the number $forb(n,H)$ of $H$-free hypergraph on $n$ vertices. While it is known that $forb(n,H)=2^{(1+o(1))ex(n,H)}$ for $k$-uniform hypergraphs that are not $k$-partite, estimates for hypergraphs that are $k$-partite (or degenerate) are not nearly as tight. In a recent breakthrough, Ferber, McKinley, and Samotij proved that for many degenerate hypergraphs $H$, $forb(n, H) = 2^{O(ex(n,H))}$. However, there are few known instances of degenerate hypergraphs $H$ for which $forb(n,H)=2^{(1+o(1))ex(n,H)}$ holds. In this paper, we show that $forb(n,H)=2^{(1+o(1))ex(n,H)}$ holds for a wide class of degenerate hypergraphs known as $2$-contractible hypertrees. This is the first known infinite family of degenerate hypergraphs $H$ for which $forb(n,H)=2^{(1+o(1))ex(n,H)}$ holds. As a corollary of our main results, we obtain a surprisingly sharp estimate of $forb(n,C^{(k)}_\ell)=2^{(\lfloor\frac{\ell-1}{2}\rfloor+o(1))\binom{n}{k-1}}$ for the $k$-uniform linear $\ell$-cycle, for all pairs $k\geq 5, \ell\geq 3$, thus settling a question of Balogh, Narayanan, and Skokan affirmatively for all $k\geq 5, \ell\geq 3$. Our methods also lead to some related sharp results on the corresponding random Turan problem. As a key ingredient of our proofs, we develop a novel supersaturation variant of the delta systems method for set systems, which may be of independent interest.

We study linear versions of Reedy categories in relation with finite dimensional algebras and abelian model structures. We prove that, for a linear Reedy category $\mathcal{C}$ over a field, the category of left $\mathcal{C}$--modules admits a highest weight structure, which in case $\mathcal{C}$ is finite corresponds to a quasi-hereditary algebra with an exact Borel subalgebra. We also lift complete cotorsion pairs and abelian model structures to certain categories of additive functors indexed by linear Reedy categories, generalizing analogous results from the hereditary case.

Pendulum-driven systems have emerged as a notable modification of vibro-impact mechanisms, replacing the conventional mass-on-spring oscillator with a pendulum. Such systems exhibit intricate behavior resulting from the interplay of directional dynamics, pendulum motion, and contact forces between the designed device and the underlying surface. This paper delves into the application of a Fourier series-based greedy algorithm for control optimization in pendulum capsule drives, which hold potential for diverse scenarios, including endoscopy capsule robots, pipeline inspection, and rescue operations in confined spaces. The emphasis is placed on experimental studies involving prototype development to validate the system's efficacy with previous computational simulations. Empirical findings closely align (<2% loss) with numerical investigations, showcasing the pendulum capsule drive's ability to achieve average speeds of 2.48 cm/s and 2.58 cm/s for three and six harmonics, respectively. These results are reinforced by high-quality signal-tracking accuracy, which demonstrates resilience against potential disturbances during motion. The authors envision the Fourier series-based control optimization method as a significant step towards ensuring enhanced locomotion performance in discontinuous systems, effectively handling the non-linearities arising from dry friction.

For the action of the orthogonal group or euclidean group on k-tuples of vectors we construct a bi-Lipschitz embedding from the orbit space into euclidean space.This embedding has distortion sqrt(2).

Mobile edge generation (MEG) is an emerging technology that allows the network to meet the challenging traffic load expectations posed by the rise of generative artificial intelligence~(GAI). A novel MEG model is proposed for deploying GAI models on edge servers (ES) and user equipment~(UE) to jointly complete text-to-image generation tasks. In the generation task, the ES and UE will cooperatively generate the image according to the text prompt given by the user. To enable the MEG, a pre-trained latent diffusion model (LDM) is invoked to generate the latent feature, and an edge-inferencing MEG protocol is employed for data transmission exchange between the ES and the UE. A compression coding technique is proposed for compressing the latent features to produce seeds. Based on the above seed-enabled MEG model, an image quality optimization problem with transmit power constraint is formulated. The transmitting power of the seed is dynamically optimized by a deep reinforcement learning agent over the fading channel. The proposed MEG enabled text-to-image generation system is evaluated in terms of image quality and transmission overhead. The numerical results indicate that, compared to the conventional centralized generation-and-downloading scheme, the symbol number of the transmission of MEG is materially reduced. In addition, the proposed compression coding approach can improve the quality of generated images under low signal-to-noise ratio (SNR) conditions.

This paper introduces an auto-stabilized weak Galerkin (WG) finite element method for biharmonic equations with built-in stabilizers. Unlike existing stabilizer-free WG methods limited to convex elements in finite element partitions, our approach accommodates both convex and non-convex polytopal meshes, offering enhanced versatility. It employs bubble functions without the restrictive conditions required by existing stabilizer-free WG methods, thereby simplifying implementation and broadening application to various partial differential equations (PDEs). Additionally, our method supports flexible polynomial degrees in discretization and is applicable in any dimension, unlike existing stabilizer-free WG methods that are confined to specific polynomial degree combinations and 2D or 3D settings. We demonstrate optimal order error estimates for WG approximations in both a discrete $H^2$ norm for $k\geq 2$ and a $L^2$ norm for $k>2$, as well as a sub-optimal error estimate in $L^2$ when $k=2$, where $k\geq 2$ denotes the degree of polynomials in the approximation.

We present Russell's antinomy using three distinct deductive systems, which are then compared to deepen the logical deductions that lead to the contradiction. Some inferential paths are then presented, alternative to the commonly accepted one, that allow for the formal assertion of the antinomy without deriving the contradiction, thus preserving the coherence of the system. In light of this, the purpose of this article is to propose a review of the consequences of asserting Russell's antinomy and, by extension, the widespread belief that any attempt to resolve a paradox is doomed to failure.

We study the (generalized) one-dimensional population model developed by Anguige \& Schmeiser [1], which reflects cell-cell adhesion and volume filling under no-flux boundary condition. In this generalized model, depending on the adhesion and volume filling parameters $α,β\in[0,1],$ the resulting equation is classified into six types. Among these, we focus on the type exhibiting strong effects of both adhesion and volume filling, which results in a class of advection-diffusion equations of the forward-backward-forward type. For five distinct cases of initial maximum, minimum and average population densities, we derive the corresponding patterns for the global behavior of weak solutions to the initial and no-flux boundary value problem. Due to the presence of a negative diffusion regime, we indeed prove that the problem is ill-posed and admits infinitely many global-in-time weak solutions, with the exception of one specific case of the initial datum. This nonuniqueness is inherent in the method of convex integration that we use to solve the Dirichlet problem of a partial differential inclusion arising from the ill-posed problem.

In this short article, we will reconstruct the KP equation from Plucker relations and provide some generalizations on this topic. Additionally, in the final section, we define the discrete function $τ$ in a similar manner, leading to the construction of an integer sequence that has not yet been listed in the OEIS. Furthermore, this approach allows us to construct many other sequences that are not listed in the OEIS.

This addendum presents a relevant stronger consequence of the main theorem of the paper "Higher order stroboscopic averaged functions: a general relationship with Melnikov functions" [arXiv:2011.03663], EJQTDE No. 77 (2021).

A matching $M$ in a graph $G$ is {\em connected} if $G$ has an edge linking each pair of edges in $M$. The problem to find large connected matchings in graphs $G$ with $α(G)=2$ is closely related to Hadwiger's conjecture for graphs with independence number 2. The problem of finding a large connected matching in a general graph is NP-hard. F{ü}redi et al. in 2005 conjectured that each $(4t-1)$-vertex graph $G$ with $α(G)=2$ contains a connected matching of size at least $t$. Cambie recently showed that if this conjecture is false, then so is Hadwiger's conjecture. In this paper, we present a number of properties possessed by a counterexample to F{ü}redi et al.'s conjecture, and then using these properties, we prove that F{ü}redi et al.'s conjecture holds for $t\leq22$.

A graph $G$ is said to be Ramsey size-linear if $r(G,H) =O_G (e(H))$ for every graph $H$ with no isolated vertices. Erdős, Faudree, Rousseau, and Schelp observed that $K_4$ is not Ramsey size-linear, but each of its proper subgraphs is, and they asked whether there exist infinitely many such graphs. In this short note, we answer this question in the affirmative.

We study the property of uniform discreteness within discrete orbits of non-uniform lattices in $SL_2(\mathbb{R})$, acting on $\mathbb{R}^2$ by linear transformations. We provide a new proof of the conditions under which the orbit of a non-uniform lattice in $SL_2(\mathbb{R})$ is uniformly discrete, by using Diophantine properties. Our results include a detailed analysis of the asymptotic behavior of the error terms. Focusing on a specific group $Γ$ and a discrete orbit of it, $S$, the main result of this paper is that for any $ε>0$, three points in $S$ can be found on a horizontal line within distance $ε$ of each other. This gives a partial result toward a conjecture of Lelièvre. The set $S$ and group $Γ$ are respectively the set of long cylinder holonomy vectors, and Veech group, of the "golden L" translation surface.

We study the distribution of partial sums of Rademacher random multiplicative functions $(f(n))_n$ evaluated at polynomial arguments. We show that for a polynomial $P\in \mathbb Z[x]$ that is a product of distinct linear factors or an irreducible quadratic satisfying a natural condition, there exists a constant $κ_P>0$ such that \[ \frac{1}{\sqrt{κ_P N}}\sum_{n\leq N}f(P(n))\xrightarrow{d}\mathcal{N}(0,1), \] as $N\rightarrow\infty$, where convergence is in distribution to a standard (real) Gaussian. This confirms a conjecture of Najnudel and addresses a question of Klurman-Shkredov-Xu. We also study large fluctuations of $\sum_{n\leq N}f(n^2+1)$ and show that there almost surely exist arbitrarily large values of $N$ such that \[ \Big|\sum_{n\leq N}f(n^2+1)\Big|\gg \sqrt{N \log\log N}. \] This matches the bound one expects from the law of iterated logarithm.