arXiv Math @arxiv_math@qoto.org

These are expanded notes for a short series of lectures, presented at the University of Luxembourg in 2017, giving an introduction to some of the ideas of supersymmetry and supergeometry. In particular, we start from some motivating facts in physics, pass to the theory of supermanifolds, then to spinors, ending up at super-Minkowski space-times. We examine some salient mathematical issues with understanding supersymmetry in a classical setting and make no attempt to discuss phenomenologically interesting models. Moreover, the presentation is "light" in the sense that nothing is carefully proved. The audience of the seminars ranged from Ph.D. students to more experienced postdocs in mathematics. The audience was assumed to have a working knowledge of differential geometry, elementary category theory, and basic ideas from classical & quantum mechanics. No knowledge of supersymmetry and supergeometry is assumed.

The long computational time and large memory requirements for computing Vietoris Rips persistent homology from point clouds remains a significant deterrent to its application to big data. This paper aims to reduce the memory footprint of these computations. It presents a new construction, the distilled Vietoris Rips filtration, and proves that its persistent homology is isomorphic to that of standard Vietoris Rips. The distilled complex is constructed using a discrete Morse vector field defined on the reduced Vietoris Rips complex. The algorithm for building and reducing the distilled filtration boundary matrix is highly parallelisable and memory efficient. It can be implemented for point clouds in any metric space given the pairwise distance matrix.

Kazhdan and Lusztig introduce the $W$-graphs to describe the cells and molecules corresponding to the Coxeter groups. Building on this foundation, Lusztig defines the a-funtion to classify the cells, as well as the molecules. Marberg then generalizes Kazhdan and Lusztig's $W$-graphs, using fixed-point-free involutions as their indices. The molecules of the two new $S_n$-graphs are then classified via two correspondence similar to RSK correspondence by Marberg and me. In this paper, we define an analogue of the Lusztig a-function and finish the classification of cells by proving that every molecule in the $S_n$-graphs is indeed a cell.

An arithmetical structure on a finite and connected graph G is a pair (d, r) of positive integer vectors such that r is primitive (the gcd of its entries is 1) and (diag(d) - A)r = 0, where A is the adjacency matrix of G. In this article, we investigate arithmetical structures on the wheel graphs.

We use Hodge theory to relate poles of the Archimedean zeta function Z_f of a holomorphic function f with several invariants of singularities. First, we prove that the largest nontrivial pole of Z_f is the negative of the minimal exponent of f, whose order is determined by the multiplicity of the corresponding root of the Bernstein-Sato polynomial b_f(s). This resolves in a strong sense a question of Mustaţă-Popa. On the other hand, we give an example of f where a root of b_f(s) is not a pole of Z_f, answering a question of Loeser from 1985 in the negative. In general, we determine poles of Z_f from the Hodge filtration on vanishing cycles, sharpening a result of Barlet. Finally, we obtain analytic descriptions of the V-filtration of Kashiwara and Malgrange, Hodge ideals, and higher multiplier ideals, addressing another question of Mustaţă-Popa. The proofs mainly rely on a positivity property of the polarization on the lowest piece of the Hodge filtration on a complex Hodge module in the sense of Sabbah-Schnell.

This paper addresses the finite-time non-overshooting leader-following consensus problem for multi-agent systems, whose agents are modeled by a dynamical system topologically equivalent to the integrator chain. Based on the weighted homogeneity, a nonlinear consensus control protocol is designed. A tuning scheme ensures the finite-time stability of the consensus error such that the agents do not have overshoots in the first component of the state vector. Simulations are presented to demonstrate the effectiveness of the proposed design.

In this article we study the quasi-linear equation \[ \left\{ \begin{aligned} \mathrm{div}\, \mathcal A(x,u,\nabla u)&=\mathcal B(x,u,\nabla u)&&\text{in }Ω,\\ u\in H^{1,p}_{loc}&(Ω;wdx) \end{aligned} \right. \] where $\mathcal A$ and $\mathcal B$ are functions satisfying $\mathcal A(x,u,\nabla u)\sim \mathcal B(x,u,\nabla u)\sim w(|\nabla u|^{p-2}\nabla u+|u|^{p-2}u)$ for $p>1$ and a $p$-admissible weight function $w$. We establish interior regularity results of weak solutions and use those results to obtain point-wise asymptotic estimates for solutions to \[ \left\{ \begin{aligned} -\mathrm{div}\,(w|\nabla u|^{p-2}\nabla u)&=w|u|^{q-2}u&&\text{in }Ω,\\ u\in D^{1,p}&(Ω,wdx) \end{aligned} \right. \] for a critical exponent $q>p$ in the sense of Sobolev.

The purpose of this paper is to obtain Fourier transforms of multivariate orthogonal polynomials on the cone such as Laguerre polynomials on the cone and Jacobi polynomials on the cone and to define two new families of multivariate orthogonal functions by using Parseval's identity. Also, the obtained results are expressed in terms of the continuous Hahn polynomials.

Parking functions correspond with preferences of $n$ cars which enter sequentially to park on a one-way street where (1) each car parks in the first available spot greater than or equal to its preference and (2) all cars successfully park. When a car parks in its preferred spot then the corresponding car and corresponding spot are deemed ``lucky.'' This paper looks briefly at lucky cars which have previously been studied and in simple cases can be understood by a generalization of a result due to Pollak. We also consider lucky spots where the situation is more complex and not previously studied. Probabilities and asymptotics for lucky spots are given for the first few spots on the one-way street. We close with an exploration of the special cases when cars enter the one-way street in either weakly-increasing or weakly-decreasing order of their preferences.

Let $(X_N)_{N\geq 1}$ denote a sequence of real random variables and let $\vartheta$ be the mode of the random variable of interest $X$. In this paper, we study the kernel mode estimator (say) $\vartheta_n$ when the data are widely orthant dependent (WOD) and subject to Random Left Truncation (RLT) mechanism. We establish the uniform consistency rate of the density estimator (say) $f_n$ of the underlying density $f$ as well as the almost sure convergence rate of $\vartheta_n$. The performance of the estimators are illustrated via some simulation studies and applied on a real dataset of car brake pads.

This paper explores the evolution and monotonicity of geometric constants within the framework of extended Ricci flows, incorporating variable coupling parameters. Building on Hamiltons foundational Ricci flow and subsequent extensions by List (2008), we introduce modifications to the extended Ricci flow by varying parameters that affect the interaction between the metric and scalar fields. Specifically, we modify the coefficients in the evolution equations governing geometric constants, thereby introducing new degrees of freedom in the analysis. The primary contributions include deriving evolution formulas for the modified geometric constant lambda under the extended and normalized extended Ricci flows, and proving conditions under which monotonicity is maintained.

A parameter of a mathematical model is structurally identifiable if it can be determined from noiseless experimental data. Here, we examine the identifiability properties of two important classes of linear compartmental models: directed-cycle models and catenary models (models for which the underlying graph is a directed cycle or a bidirected path, respectively). Our main result is a complete characterization of the directed-cycle models for which every parameter is (generically, locally) identifiable. Additionally, for catenary models, we give a formula for their input-output equations. Such equations are used to analyze identifiability, so we expect our formula to support future analyses into the identifiability of catenary models. Our proofs rely on prior results on input-output equations, and we also use techniques from linear algebra and graph theory.

In the work "Generalized approximation spaces generation from $\mathbb{I}_{j}$-neighborhoods and ideals with application to Chikungunya disease, published in \emph{AIMS Mathematics}, \textbf{9}(4) (2024), 10050$-$10077," Al-Shami and Hosny introduced a novel approach for generating generalized neighborhoods through $\mathbb{I}_{j}$-neighborhoods with ideals, subsequently deriving new methods for generalized approximations based on these neighborhoods. However, several errors have been identified in their results, concepts, and methods, along with significant inaccuracies in the provided examples and comparison tables. This paper aims to address and correct these errors, providing counterexamples to illustrate the inaccuracies in the proposed results. Furthermore, we correct several flawed proofs and present the revised formulations of these results and concepts. Additionally, we offer properties and clarifications to enhance the understanding of these concepts.

In this paper, we propose $λ_{g}$ conjecture for Hodge integrals with target varieties. Then we establish relations between Virasoro conjecture and $λ_{g}$ conjecture, in particular, we prove $λ_{g}$ conjecture in all genus for smooth projective varieties with semisimple quantum cohomology or smooth algebraic curves. Meanwhile, we also prove $λ_{g}$ conjecture in genus zero for any smooth projective varieties. In the end, together with DR formula for $λ_g$ class, we obtain a new type of universal constraints for descendant Gromov-Witten invariants. As an application, we prove $λ_{g}$ conjecture in genus one for any smooth projective varieties.

Due to the processes that occur during the functioning of modern electromechanical systems, these systems can be considered complex nonlinear dynamic systems from the point of view of the theory of dynamic systems. The movement of such systems is completely determined by external influences acting on the EMS, their parameters, and initial operating conditions. The above-mentioned factors complicate the study of electromechanical systems and, in the general case, make it impossible to use classical methods of analyzing the dynamics of the EMS since the latter neglect the features of nonlinear systems and describe their dynamics using ordinary linear differential equations with constant coefficients. At the same time, many methods and approaches have been developed in control theory for analyzing stability and synthesizing motion trajectories based on linear differential equations. Therefore, an important task arises to create mathematical models of nonlinear EMS that consider the peculiarities of their motion but have a form similar to linear models.

This article is devoted to the classification of anti-dendriform algebras that are associated with associativity. They are characterized as algebras with two operations whose sum is associative. In the paper all four-dimensional complex anti-dendriform algebras associated to four-dimensional associative algebras with one-dimensional center are classified

In this work, we investigate the long-time behavior of solutions to the non-cutoff Boltzmann equation on compact Riemannian manifolds with bounded Ricci curvature. The paper introduces new results on the exponential decay of hydrodynamic quantities, such as density, momentum, and energy fields, influenced by both the curvature of the manifold and singularities in the collision kernel. We demonstrate that for initial data in $H^s_x \times L^p_v$, the solutions exhibit sharp exponential decay rates in Sobolev norms, with the decay rate determined by the manifold's geometry and the regularity of the kernel. Specifically, we prove that the density $ρ(t, x)$, momentum $\mathbf{m}(t, x)$, and energy field $E(t, x)$ all decay exponentially in time, with decay rates that depend on the manifold's curvature and the nature of the collision kernel's singularity. Additionally, we address the case of angular singularities in the collision kernel, providing conditions under which the exponential decay persists. The analysis combines energy methods, Fourier analysis, and coercivity estimates for the collision operator, extended to curved geometries. These results extend the understanding of dissipation mechanisms in kinetic theory, especially in curved settings, and offer valuable insights into the behavior of rarefied gases and plasma flows in non-Euclidean environments. The findings have applications in plasma physics, astrophysics, and the study of rarefied gases, opening new directions for future research in kinetic theory and geometric analysis.

The purpose of this paper is to explore properties of the whisker topology, which is a topology endowed on the fundamental group and whose utility is to detect locally complicated phenomena in pathological topological spaces. We show that the whisker topology preserves products, resolve an open question regarding the existence of a space which makes $π_1^{wh}(X,x_0)$ a non-discrete, non-abelian, and Hausdorff topological group, and show the whisker topology is not separable on the earring group $π_1(\Er^1,x_0)$.

The longstanding Godbersen's conjecture states that for any convex body $K \subset \mathbb R^n$ of volume $1$ and any $j \in \{0, \ldots, n\}$, the mixed volume $V_j = V(K[j], -K[n - j])$ is bounded by $\binom{n}{j}$, with equality if and only if $K$ is a simplex. We demonstrate that several consequences of this conjecture are true: certain families of linear combinations of the $V_j$, arising from different geometric constructions, are bounded above by their values when one substitutes $\binom{n}{j}$ for $V_j$, with equality if and only if $K$ is a simplex. One of our results implies that for any $K$ of volume $1$ we have $\frac{1}{n + 1} \sum_{j = 0}^n \binom{n}{j}^{-1} V_j \le 1$, showing that Godbersen's conjecture holds ``on average'' for any body. Another result generalizes the well-known Rogers-Shephard inequality for the difference body.

This research paper talks about using complex mathematical tools to study and figure out the behavior of biological populations in porous media. Porous media offer a unique environment where various factors, including fluid flow and nutrient diffusion, significantly influence population dynamics. The theory of Lie symmetries is used to find inherent symmetries in the governing equation of the population model, helping to find conservation laws and invariant solutions. The derivation and analysis of the optimal system provide insights into the most influential parameters affecting population growth and distribution. Furthermore, the study explores the construction of invariant solutions, which aid in characterizing long-term population behavior. The article concludes with the non-linear self-adjointness property and conservation laws for the model.