arXiv Math @arxiv_math@qoto.org

Certain notions of convergence of sequences of functions such as convergence pointwise, uniform and parts (compact sets or bounded sets) come from suitable topological functional spaces [1]. Under certain conditions these topologies involved are metrizable, which in an advantage since there is an extensive theory on convergence in metric spaces. However, the case of pointwise convergence is delicate, since it is shown that under certain hypothesis this form of convergence of sequences of functions is not equivalent to convergence in metric.

In this article, we give an explicit and uniform lower bound of the arithmetic Hilbert-Samuel function of projective hypersurfaces, which has the optimal dominant term. As an application, we apply this lower bound in the determinant method.

We survey the combinatorics of the Adinkra, a graphical device for solving differential equations in supersymmetry. These graphs represent an exceptional class of 1-factorizations with further augmentations. As a new feature, we characterize Adinkras using Latin rectangles.

Assume that $\,I\subseteq\mathbb{R}\,$ is an interval and $\,β:\,I\rightarrow\,I\,$ a strictly increasing and continuous function with a single fixed point $\,s_0\in I\,$, satisfying $\,(s_0-t)(β(t)-t)\leq 0\,$ for all $\,t\in I$, where the equality occurs only when $\,t=s_0$. Hamza et al. considered the general quantum operator, $\,D_β[f](t):=\displaystyle\frac{f\big(β(t)\big)-f(t)}{β(t)-t}\,$ when $\,t\neq s_0\,$ and $\,D_β[f](s_0):=f^{\prime}(s_0)\,$ when $\,t=s_0\,$. It generalizes the Jackson $\,q$-derivative operator $\,D_{q}\,$ as well as the Hahn (quantum derivative) operator, $\,D_{q,ω}$. We obtained Grüss type inequalities for its inverse operator, the $β$-integral. Furthermore, we introduced the concept of $\,β$-Riemann-Stieltjes integral and obtained Grüss type inequalities associated with it.

We revisit the method of $a$-contraction with shifts used for long-time behavior of barotropic Navier-Stokes flows perturbed from a Riemann shock. For the usage of the method of $a$-contraction with shifts, we do not employ the effective velocity $h$ variable even for higher order estimates. This approach would be important when handling the barotropic Navier-Stokes system with other effects, for example, such as capillary effect and boundary effect.

We prove that the ring of Weyl invariant $E_8$ weak Jacobi forms is isomorphic to that of joint covariants of a binary sextic and a binary quartic form. The ring is therefore finitely generated. A minimal basis of generators is obtained from that already known for the ring of covariants.

We construct a measure on the moduli space of super Riemann surfaces with Ramond punctures using the super Mumford isomorphism and a super period map.

Biunivalent holomorphic functions form an interesting class in geometric function theory and are connected with special functions and solutions of complex differential equations. The paper reveals a deep connection between biunivalence and geometry of Teichmuller balls and provides some sufficient conditions for biunivalence of holomorphic functions on the disk. Among the consequences, one obtains new sharp distortion theorems for univalent functions with quasiconformal extension.

Following the work of Mestre, we use Weil's explicit formulas to compute explicit lower bounds on the conductors of elliptic curves and abelian varieties over number fields. Moreover, we obtain bounds for the conductor of elliptic curves and abelian varieties over $\mathbb{Q}$ with specified bad reduction and over number fields. As an application, for specific fields, we prove the non-existence of abelian varieties with everywhere good reduction.

On a compact Riemann surface $Σ$ of genus $g>1$, equipped with a complex vector bundle $E$ of rank $2$ and degree zero let $M_H$ be the moduli space of Higgs bundles. $M_H$ admits a $\mathbb C^{\star}$-action and to each stable $\mathbb C^{\star}$-fixed point $[(\bar\partial_0,Φ_0)]$ is associated a holomorphic Lagrangian submanifold $W^1(\bar\partial_0,Φ_0)$ inside the de Rham moduli space $M_{dR}$ of complex flat connections. In this note we prove a conjecture of Simpson stating that $W^1(\bar\partial_0,Φ_0)$ is closed inside $M_{dR}$.

Starting from the well-known relationship $|{\mathrm{e}}^z| = {\mathrm{e}}^{{\mathrm Re}(z)}$, we consider the question whether $|E_{α,β}(z)|$ and $E_{α,β}({\mathrm Re}(z))$ are comparable, as functions of the complex variable $z$, where $E_{α,β}$ denotes the two-parameter Mittag-Leffler function, a generalization of the exponential function. For some ranges of the parameters $α$ and $β$ we prove inequalities between $|E_{α,β}(z)|$ and $E_{α,β}({\mathrm Re}(z))$ holding globally for all $z\in \mathbb{C}$. In some other ranges of $α$ and $β$ the same inequalities are proved to hold asymptotically, i.e. for sufficiently small or large $z$. There are moreover some values of $α$ and $β$ for which the situation is less clear, and some conjectures, motivated by numerical observations, are proposed.

This paper concerns the initial boundary value problem of three-dimensional inhomogeneous incompressible liquid crystal flows with density-dependent viscosity. When the viscosity coefficient $μ(ρ)$ is a power function of the density with the power larger than $1$, that is $μ(ρ)=μρ^α$ with $α>1$, it is proved that the system exists a unique global strong solution as long as the initial density is sufficiently large and $L^3$-norm of the derivative of the initial director is sufficiently small. This is the first result concerning the global strong solution for three-dimensional inhomogeneous liquid crystal flows without smallness of velocity.

Systems with predetermined Lyapunov functions play an important role in many areas of applied mathematics, physics and engineering: dynamic optimization methods (objective functions and their modifications), machine learning (loss functions), thermodynamics and kinetics (free energy and other thermodynamic potentials), adaptive control (various objective functions, stabilization quality criteria and other Lyapunov functions). Dimensionality reduction is one of the main challenges in the modern era of big data and big models. Dimensionality reduction for systems with Lyapunov functions requires it preserving dissipativity: the reduced system must also have a Lyapunov function, which is expected to be a restriction of the original Lyapunov function on the manifold of the reduced motion. An additional complexity of the problem is that the equations of motion themselves are often unknown in detail in advance and must be determined in the course of the study, while the Lyapunov function could be determined based on incomplete data. Therefore, the projection problem arises: for a given Lyapunov function, find a field of projectors such that the reduction of it any dissipative system is again a dissipative system. In this paper, we present an explicit construction of such projectors and prove their uniqueness. We have also taken the first step beyond the approximation by manifolds. This is required in many applications. For this purpose, we introduce the concept of monotone trees and find a projection of dissipative systems onto monotone trees that preserves dissipativity.

Differentially positive systems are the nonlinear systems whose linearization along trajectories preserves a cone field on a smooth Riemannian manifold. One of the embryonic forms for cone fields in reality is originated from the general relativity. Out of a set of measure zero, we show that almost every orbits will converge to equilibrium. This solved a reduced version (from a measure-theoretic perspective) of Forni-Sepulchre's conjecture in 2016 for globally orderable manifolds.

In this paper, we introduce the quantitative coarse Baum-Connes conjecture with coefficients (or QCBC, for short) for proper metric spaces which refines the coarse Baum-Connes conjecture. And we prove that QCBC is derived by the coarse Baum-Connes conjecture with coefficients which provides many examples satisfying QCBC. In the end, we show QCBC can be reduced to the uniformly quantitative coarse Baum-Connes conjecture with coefficients of a sequence of bounded metric spaces.

A $d$-subgroup of a lattice-ordered group ($\ell$-group) $G$ is a subgroup that contains the principal polars generated by each of its elements. We call $G$ a Martinez $\ell$-group if every convex $\ell$-subgroup of $G$ is a $d$-subgroup (equivalently: $G(a)=a^{\perp \perp}$ for every $a \in G$); known examples include all hyperarchimedean $\ell$-groups and all existentially closed abelian $\ell$-groups. This paper gives new characterizations of the Martinez $\ell$-groups, shows that the abelian Martinez $\ell$-groups form a radical class, investigates a related type of archimedean $\ell$-group called a Yosida $\ell$-group, and uses an analogue of a construction from ring theory, and other methods, to produce new examples of Martinez $\ell$-groups with special properties.

We propose and analyse a model predictive control (MPC) strategy tailored for networks of underwater agents tasked with maintaining formation while following a shared path and using acoustic communication channels. The strategy accommodates both time-division and frequency-division medium access schemes, and addresses the inherent challenges of lossy and broadcast communication over acoustic media. Our approach extends an existing distributed control algorithm originally assuming standard double precision in exchanged data, and designed for synchronous, bidirectional, and reliable communication. Here we introduce adaptations for handling broadcast asynchronous communication, for mitigating packet losses, and for quantising exchanged data. These modifications are general and intended to be applicable to other distributed control schemes that were developed under idealised assumptions. Our goal is thus to help facilitating deployment also of other control schemes in practical field conditions. We provide simulation results that quantify the impact of these adaptations on the performance of the original controller, along with sensitivity analyses on how performance losses are influenced by key hyperparameters. Additionally, we characterise the data rate savings vs. control performance losses that may be achieved through tuning such hyperparameters, showcasing the feasibility of implementing the proposed strategy for practical purposes using commercially available full-duplex or half-duplex modems.

We prove a lower bound for the exponent of the relative class group $\mathrm{Pic}^0 X_1/ϕ^* \mathrm{Pic}^0 X_2$ for a covering of curves $X_1 \to X_2$ over a finite field $\mathbb{F}_q$. The results improve on the existing best bounds (due to Stichtenoth) in the case $X_2=\mathbb{P}^1$, when the relative class group equals the class group of the function field $\mathbb{F}_q(X_1)$, and are completely new for the genuinely relative situation.

We comprehensively study weighted projective Reed-Muller (WPRM) codes on weighted projective planes $\mathbb{P}(1,a,b)$. We provide the universal Gröbner basis for the vanishing ideal of the set $Y$ of $\mathbb{F}_q$--rational points of $\mathbb{P}(1,a,b)$ to get the dimension of the code. We determine the regularity set of $Y$ using a novel combinatorial approach. We employ footprint techniques to compute the minimum distance.

We put into light the Killing vector fields on $\mathbb R^2$ endowed with a family of diagonal Riemannian metrics. According to certain restrictions on the Lamé coefficients, we concretely describe the symmetries of the metric.