arXiv Math @arxiv_math@qoto.org

This paper discusses a generalization of virtual knot theory that we call multi-virtual knot theory. Multi-virtual knot theory uses a multiplicity of types of virtual crossings. As we will explain, this multiplicity is motivated by the way it arises first in a graph-theoretic setting in relation to generalizing the Penrose evaluation for colorings of planar trivalent graphs to all trivalent graphs, and later by its uses in a virtual knot theory. As a consequence, the paper begins with the graph theory as a basis for our constructions, and then proceeds to the topology of multi-virtual knots and links. The second section of the paper is a review of our previous work (See arXiv:1511.06844). The reader interested in seeing our generalizations of the original Penrose evaluation, can begin this paper at the beginning and see the graph theory. A reader primarily interested in multi-virtual knots and links can begin reading in section 4 with references to the earlier part of the paper.

Given a graph $G$, its $2$-color Turán number $\mathrm{ex}^{(2)}(n,G)$ is the largest number of edges in an $n$-vertex graph whose edges can be colored with two colors avoiding a monochromatic copy of $G$. Let $π^{(2)}(G)=\lim_{n\to\infty}\mathrm{ex}^{(2)}(n,G)/\binom{n}{2}$ be the $2$-color Turán density of $G$. What real numbers in the interval $(0,1)$ are realized as the $2$-color Turán density of some graph? It is known that $π^{(2)}(G)=1-(R_χ(G)-1)^{-1}$, where $R_χ(G)$ is the chromatic Ramsey number of $G$. However, determining specific values of $R_χ(G)$ is challenging. Burr, Erdős, and Lovász showed that $(k-1)^2+1\leqslant{R_χ(G)}\leqslant{R(k)}$, for any $k$-chromatic graph $G$, where $R(k)$ is the classical Ramsey number. The upper bound here can be attained by a clique and the lower bound is achieved by a graph constructed by Zhu. To the best of our knowledge, there are no other, besides these two, known values of $R_χ(G)$ among $k$-chromatic graphs $G$ for general $k$. In this paper we prove that there are $Ω(k)$ different values of $R_χ(G)$ among $k$-chromatic graphs $G$. In addition, we determine a new value for the chromatic Ramsey numbers of $4$-chromatic graphs. This sheds more light into the possible $2$-color Turán densities of graphs.

Motivated by numerical continuation studies of coupled mechanical oscillators, we investigate branches of localized time-periodic solutions of one-dimensional chains of coupled oscillators. We focus on Ginzburg-Landau equations with nonlinearities of lambda-omega type and establish the existence of on-site and off-site solutions in the case of weak coupling and weak-amplitude dependence of the oscillator periods. Utilizing a core-far-field decomposition, we demonstrate that in-phase solutions lie on branches that exhibit snaking behavior.

Given a matrix model, by combining the Schwinger-Dyson equations with positivity constraints on its solutions, in the large $N$ limit one is able to obtain explicit and numerical bounds on its moments. This technique is known as bootstrapping with positivity. In this paper we use this technique to estimate the critical points and exponents of several matrix multi-models. As a proof of concept, we first show it can be used to find the well-studied quartic single matrix model's critical phenomena. We then apply the method to several similar ``unsolved" 2-matrix models with various quartic interactions. We conjecture and present strong evidence for the string susceptibility exponent for some of these models to be $γ= 1/2$, which heuristically indicates that the continuum limit will likely be the Continuum Random Tree. For the other 2-matrix models, we find estimates of new string susceptibility exponents that may indicate a new continuum limit. We then study an unsolved 3-matrix model that generalizes the 3-colour model with cubic interactions. Additionally, for all of these models, we are able to derive explicitly the first several terms of the free energy in the large $N$ limit as a power series expansion in the coupling constants at zero by exploiting the structure of the Schwinger-Dyson equations.

Let $p\ge 5$ be a prime and let $P$ be a Sylow $p$-subgroup of a finite symmetric group. To every irreducible character of $P$ we associate a collection of labelled, complete $p$-ary trees. The main results of this article describe Sylow branching coefficients for symmetric groups for all irreducible characters of $P$ in terms of some combinatorial properties of these trees, extending previous work on the linear characters of $P$.

In this paper we study monomial cycles in Koszul homology over a monomial ring. The main result is that a monomial cycle is a boundary precisely when the monomial representing that cycle is contained in an ideal we introduce called the boundary ideal. As a consequence, we obtain necessary ideal-theoretic conditions for a monomial ideal to be Golod. We classify Golod monomial ideals in four variables in terms of these conditions. We further apply these conditions to symmetric monomial ideals, allowing us to classify Golod ideals generated by the permutations of one monomial. Lastly, we show that a class of ideals with linear quotients admit a basis for Koszul homology consisting of monomial cycles. This class includes the famous case of stable monomial ideals as well as new cases, such as symmetric shifted ideals.

We develop a theory of derived rigid spaces and quasi-coherent sheaves and analytic crystals on them. Amongst other things, we obtain a six-functor formalism for these quasi-coherent sheaves and analytic crystals. We provide evidence that the category of analytic crystals is related to the theory of D-cap-modules introduced by Ardakov--Wadsley.

We compare the multi-dimensional generalisation of the Aw-Rascle model with the pressureless Euler-alignment system, in which the communication weight is matrix-valued. Our generalisation includes the velocity offset in the form of a gradient of a non-local density function, given by the convolution with a kernel $K$. We investigate connections between these models at the macroscopic, mesoscopic and macroscopic (hydrodynamic) level, and overview the results on the mean-field limit for various assumptions on $K$.

We present natural conjectural generalizations of the `positivity and integrality of mirror maps' phenomenon, encompassing the mirror maps appearing in the Batyrev--Borisov construction of mirror Calabi--Yau complete intersections in Fano toric varieties as a special case. We find that, given the combinatorial data from which one constructs a mirror pair of Calabi--Yau complete intersections, there are two ways of writing down an associated `mirror map': one which is the `true mirror map', meaning the one which appears in mirror symmetry theorems; and one which is the `naive mirror map'. The two are equal under a certain combinatorial criterion which holds e.g. for the quintic threefold, but not in general. We conjecture (based on substantial computer checks, together with proofs under extra hypotheses) that the naive mirror map always has positive integer coefficients, while the true mirror map always has integer (but not necessarily positive) coefficients. Almost all previous works on the integrality of mirror maps concern the naive mirror map, and in particular, only apply to the true mirror map under the combinatorial criterion mentioned above.

We give a new short proof of the classification of rank at least two invariant subvarieties in genus three, which is due to Aulicino, Nguyen, and Wright. The proof uses techniques developed in recent work of Apisa and Wright.

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).