arXiv Math @arxiv_math@qoto.org

Intelligent reflecting surface (IRS) has recently received much attention from the research community due to its potential to achieve high spectral and power efficiency cost-effectively. In addition to traditional cellular networks, the use of IRS in vehicular networks is also considered. Prior works on IRS-aided vehicle-to-everything communications focus on deploying reflection surfaces on the facades of buildings along the road for sidelink performance enhancement. This paper goes beyond the state of the art by presenting a novel paradigm coined Intelligent Reflecting Vehicle Surface (IRVS). It embeds a massive number of reflection elements on vehicles' surfaces to aid moving vehicular networks in military and emergency communications. We propose an alternative optimization method to optimize jointly active beamforming at the base station and passive reflection across multiple randomly-distributed vehicle surfaces. Performance evaluation in terms of sum spectral efficiency under continuous, discrete, and random phase shifts is conducted. Numerical results reveal that IRVS can substantially improve the capacity of a moving vehicular network.

Intelligent reflecting surface (IRS) is envisioned to become a key technology for the upcoming six-generation (6G) wireless system due to its potential of reaping high performance in a power-efficient and cost-efficient way. With its disruptive capability and hardware constraint, the integration of IRS imposes some fundamental particularities on the coordination of multi-user signal transmission. Consequently, the conventional orthogonal and non-orthogonal multiple-access schemes are hard to directly apply because of the joint optimization of active beamforming at the base station and passive reflection at the IRS. Relying on an alternating optimization method, we develop novel schemes for efficient multiple access in IRS-aided multi-user multi-antenna systems in this paper. Achievable performance in terms of the sum spectral efficiency is theoretically analyzed. A comprehensive comparison of different schemes and configurations is conducted through Monte-Carlo simulations to clarify which scheme is favorable for this emerging 6G paradigm.

We propose a new type of quantum statistics, which we call inclusion statistics, in which particles tend to coalesce more than ordinary bosons. Inclusion statistics is defined in analogy with exclusion statistics, in which statistical exclusion is stronger than in Fermi statistics, but now extrapolating beyond Bose statistics, resulting in statistical inclusion. A consequence of inclusion statistics is that the lowest space dimension in which particles can condense in the absence of potentials is $d=2$, unlike $d=3$ for the usual Bose-Einstein condensation. This reduction in the dimension happens for any inclusion stronger than bosons, and the critical temperature increases with stronger inclusion. Possible physical realizations of inclusion statistics involving attractive interactions between bosons may be experimentally achievable.

We present IMPACT, a flexible toolchain for nonlinear model predictive control (NMPC) specification with automatic code generation capabilities. The toolchain reduces the engineering complexity of NMPC implementations by providing the user with an easy-to-use application programming interface, and with the flexibility of using multiple state-of-the-art tools and numerical optimization solvers for rapid prototyping of NMPC solutions. IMPACT is written in Python, users can call it from Python and MATLAB, and the generated NMPC solvers can be directly executed from C, Python, MATLAB and Simulink. An application example is presented involving problem specification and deployment on embedded hardware using Simulink, showing the effectiveness and applicability of IMPACT for NMPC-based solutions.

The answers for Feynman diagrams satisfy various kinds of differential equations -- which is not a surprise, because they are defined as Gaussian correlators, possessing a vast variety of Ward identities and superintegrability properties. We study these equations in the simplest example of banana diagrams. They contain any number of loops, but can be efficiently handled in position rather than momentum representation, where loop integrals do not show up. We derive equations for the case of scalar fields, explain their origins and drastic simplification at coincident masses. To further simplify the story we do not consider coincident points, i.e. ignore delta-function contributions and ultraviolet divergences for the most part. The equations in this case reduce to homogeneous and have as many solutions as there are different Green functions -- $2^n$ for $n$ loops in quadratic theory, what reduces to just $n+1$ for coincident masses, i.e. for a single field. We comment on the recovery of the delta-functions directly from the homogeneous equations and also compare our result with momentum space formulas known in the literature.

A finite transformation method is introduced. This method is equivalent to the $Z$ transform method to a certain extent but generalizes it. By applying the presented method to the Bessel functions, it is possible to solve related ordinary differential equations of order higher than two with given initial conditions.

In this paper, we first obtain a refined version of the Bohr inequality of norm-type for holomorphic mappings with lacunary series on the polydisk in $\mathbb{C}^n$ under some restricted conditions. Next, we determine the refined version of the Bohr inequality for holomorphic functions defined on a balanced domain $ G $ of a complex Banach space $ X $ and take values from the unit disk $ \mathbb{D} $. Furthermore, as a consequence of one of this results, we obtain a refined version of the Bohr-type inequalities for harmonic functions $ f=h+\bar{g} $ defined on a balanced domain $ G\subset X $. All the results are proved to be sharp.

In this paper, a non-intrusive reduced-order model (ROM) for parametric reactor kinetics simulations is presented. Time-dependent ROMs are notoriously data intensive and difficult to implement when nonlinear multiphysics phenomena are considered. These challenges are exacerbated when parametric dependencies are included. The proper orthogonal decomposition mode coefficient interpolation (POD-MCI) ROM presented in this work can be constructed directly from lower-dimensional quantities of interest (QoIs) and is independent of the underlying model. This greatly alleviates the data requirement of many existing ROMs and can be used without modification on arbitrarily complex models or experimental data. The POD-MCI ROM is demonstrated on a number of examples and yields accurate characterizations of the QoIs within the selected parameter spaces at extremely attractive computational speed-up factors relative to the full-order models (FOMs).

An example of two $\mathbb R$-factorizable groups whose product is not $\mathbb R$-factorizable is constructed. One of these groups is second-countable.

This paper presents a parallel preconditioning approach based on incomplete LU (ILU) factorizations in the framework of Domain Decomposition (DD) for general sparse linear systems. We focus on distributed memory parallel architectures, specifically, those that are equipped with graphic processing units (GPUs). In addition to block Jacobi, we present general purpose two-level ILU Schur complement-based approaches, where different strategies are presented to solve the coarse-level reduced system. These strategies are combined with modified ILU methods in the construction of the coarse-level operator, in order to effectively remove smooth errors. We leverage available GPU-based sparse matrix kernels to accelerate the setup and the solve phases of the proposed ILU preconditioner. We evaluate the efficiency of the proposed methods as a smoother for algebraic multigrid (AMG) and as a preconditioner for Krylov subspace methods, on challenging anisotropic diffusion problems and a collection of general sparse matrices.

The generalized inverse Gaussian-Poisson (GIGP) distribution proposed by Sichel in the 1970s has proved to be a flexible fitting tool for diverse frequency data, collectively described using the item production model. In this paper, we identify the limit shape (specified as an incomplete gamma function) of the properly scaled diagrammatic representations of random samples from the GIGP distribution (known as Young diagrams). We also show that fluctuations are asymptotically normal and, moreover, the corresponding empirical random process is approximated via a rescaled Brownian motion in inverted time, with the inhomogeneous time scale determined by the limit shape. Here, the limit is taken as the number of production sources is growing to infinity, coupled with an intrinsic parameter regime ensuring that the mean number of items per source is large. More precisely, for convergence to the limit shape to be valid, this combined growth should be fast enough. In the opposite regime referred to as "chaotic", the empirical random process is approximated by means of an inhomogeneous Poisson process in inverted time. These results are illustrated using both computer simulations and some classic data sets in informetrics.

Consider the recursive relation generating a new positive integer $n_{\ell +1}$ from the positive integer $n_{\ell }$ according to the following simple rules: if the integer $n_{\ell }$ is odd, $n_{\ell +1}=3n_{\ell }+1$; if the integer $n_{\ell }$ is even, $n_{\ell +1}=n_{\ell }/2$. The so-called Collatz conjecture states that, starting from any positive integer $N$, the recursion characterized by the continued application of these rules ends up in the cycle $4,$ $2,1$. This conjecture is generally believed to be true (on the basis of extensive numerical checks), but it is as yet unproven. In this paper -- based on the assumption that the Collatz conjecture is indeed true -- we present a quite simple extension of it, which entails the possibility to divide all natural numbers into $3$ disjoint classes, to each of which we conjecture -- on the basis of (not very extensive) numerical checks -- $1/3$ of all natural numbers belong; or, somewhat equivalently, to $2$ disjoint classes, to which we conjecture that respectively $1/3$ and $2/3$ of all natural numbers belong.

The study of vector-like spectra in 4-dimensional F-theory compactifications involves root bundles, which are important for understanding the Quadrillion F-theory Standard Models (F-theory QSMs) and their potential implications in physics. Recent studies focused on a superset of physical root bundles whose cohomologies encode the vector-like spectra for certain matter representations. It was found that more than 99.995\% of the roots in this superset for the family $B_3( Δ_4^\circ )$ of $\mathcal{O}(10^{11})$ different F-theory QSM geometries had no vector-like exotics, indicating that this scenario is highly likely. To study the vector-like spectra, the matter curves in the F-theory QSMs were analyzed. It was found that each of them can be deformed to nodal curve that is identical across all spaces in $B_3( Δ^\circ )$. Therefore, from studying a few nodal curves, one can probe the vector-like spectra of a large fraction of F-theory QSMs. To this end, the cohomologies of all limit roots were determined, with line bundle cohomology on rational nodal curves playing a major role. A computer algorithm was used to enumerate all limit roots and analyze the global sections of all tree-like limit roots. For the remaining circuit-like limit roots, the global sections were manually determined. These results were organized into tables, which represent -- to the best knowledge of the author -- the first arithmetic steps towards Brill-Noether theory of limit roots.

This work outlines a fast, high-precision time-domain solver for scalar, electromagnetic and gravitational perturbations on hyperboloidal foliations of Kerr space-times. Time-domain Teukolsky equation solvers have typically used explicit methods, which numerically violate Noether symmetries and are Courant-limited. These restrictions can limit the performance of explicit schemes when simulating long-time extreme mass ratio inspirals, expected to appear in LISA band for 2-5 years. We thus explore symmetric (exponential, Padé or Hermite) integrators, which are unconditionally stable and known to preserve certain Noether symmetries and phase-space volume. For linear hyperbolic equations, these implicit integrators can be cast in explicit form, making them well-suited for long-time evolution of black hole perturbations. The 1+1 modal Teukolsky equation is discretized in space using polynomial collocation methods and reduced to a linear system of ordinary differential equations, coupled via mode-coupling arrays and discretized (matrix) differential operators. We use a matricization technique to cast the mode-coupled system in a form amenable to a method-of-lines framework, which simplifies numerical implementation and enables efficient parallelization on CPU and GPU architectures. We test our numerical code by studying late-time tails of Kerr spacetime perturbations in the sub-extremal and extremal cases.

Inspired by the definition of $\bf{t}$-spread monomial ideals, in this paper, we introduce $\bf{t}$-spread $d$-partite hypergraph $K^{\bf t}_V$ and study its edge ideal $I(K^{\bf t}_V)$. We prove that $I(K^{\bf t}_V)$ has linear quotients, all powers of $I(K^{\bf t}_V)$ have linear resolution and the Rees algebra of $I(K^{\bf t}_V)$ is a normal Cohen-Macaulay domain. It is also shown that $I(K^{\bf t}_V)$ is normally torsion-free and a complete characterization of Cohen-Macaulay $S/I(K^{\bf t}_V)$ is given.

We address the problem of controlling the reactive power setpoints of a set of distributed energy resources (DERs) in a power distribution network so as to mitigate the impact of variability in uncontrolled power injections associated with, e.g., renewable-based generation. We formulate the control design problem as a stochastic optimization problem, which we solve online using a modified version of a projected stochastic gradient descent (PSGD) algorithm. The proposed PSGD-based algorithm utilizes sensitivities of changes in bus voltage magnitudes to changes in DER reactive power setpoints; such sensitivities are learned online via a recursive least squares estimator (rLSE). To ensure proper operation of the rLSE, the sequence of incremental changes in DER reactive power setpoints needs to be persistently exciting, which is guaranteed by a mechanism built into the controller. We analyze the stability of the closed-loop system and showcase controller performance via numerical simulations on the IEEE 123-bus distribution test feeder.

Let $k * G$ be a crossed product of a division ring $k$ and a torsion-free virtually compact special group $G$. We embed $k * G$ into a division ring $\mathcal D$ and use this to confirm a recent conjecture of Kielak and Linton. In the case where $G$ is locally indicable, we prove that $\mathcal D$ is Hughes-free. If $H$ is a torsion-free one-relator group, let $\overline{kH}$ be the division ring containing $kH$ constructed by Lewin and Lewin. We prove that $\overline{kH}$ is Hughes-free whenever a Hughes-free $kH$-division ring exists. This is always the case when $k$ is of characteristic zero; in positive characteristic, our previous result implies this happens when $H$ is virtually compact special.

The determinant lower bound of Lovasz, Spencer, and Vesztergombi [European Journal of Combinatorics, 1986] is a powerful general way to prove lower bounds on the hereditary discrepancy of a set system. In their paper, Lovasz, Spencer, and Vesztergombi asked if hereditary discrepancy can also be bounded from above by a function of the hereditary discrepancy. This was answered in the negative by Hoffman, and the largest known multiplicative gap between the two quantities for a set system of $m$ substes of a universe of size $n$ is on the order of $\max\{\log n, \sqrt{\log m}\}$. On the other hand, building on work of Matoušek [Proceedings of the AMS, 2013], recently Jiang and Reis [SOSA, 2022] showed that this gap is always bounded up to constants by $\sqrt{\log(m)\log(n)}$. This is tight when $m$ is polynomial in $n$, but leaves open what happens for large $m$. We show that the bound of Jiang and Reis is tight for nearly the entire range of $m$. Our proof relies on a technique of amplifying discrepancy via taking Kronecker products, and on discrepancy lower bounds for a set system derived from the discrete Haar basis.

In this paper we overview the Poisson gauge theory focusing on the most recent developments. We discuss the general construction and its symplectic-geometric interpretation. We consider explicit realisations of the formalism for all non-commutativities of the Lie algebraic type. We discuss Seiberg-Witten maps between Poisson gauge field-theoretical models.

We prove that the cut-and-paste K-theory of polyhedra, also called scissors congruence K-theory, is a homotopy orbit spectrum. More generally, we show that a broad class of algebraic K-theory constructions, based on covering families, all commute with the formation of homotopy orbits. In other words, "K-theory of coverings" always has a Farrell-Jones type isomorphism. Finally, we introduce a trace map for K-theory of coverings, and use it to prove the existence of some classes in the higher K-theory of polyhedra.