arXiv Math @arxiv_math@qoto.org

Planes are familiar mathematical objects which lie at the subtle boundary between continuous geometry and discrete combinatorics. A plane is geometrical, certainly, but the ways that two planes can interact break cleanly into discrete sets: the planes can intersect or not. Here we review how oriented matroids can be used to try to capture the combinatorial aspect, giving a way to encode with finite sets all the ways that $n$ planes can interact. We mention how the one-to-one correspondence breaks down in 2 dimensions for 9 lines, and in 3D for 8 planes. We include illustrations of all the types of plane arrangements using $n=4$ and 5.

Some of the most challenging problems in Timaeus' cosmology arise from the geometry of a universe without any void. On the one hand, the universe is spherical in shape; on the other hand, it must be entirely filled with the four basic particles that make up all bodies in the universe, each shaped like one of four regular polyhedra (cubes, tetrahedra, octahedra and icosahedra). The faces of all these particles are composed of right triangles. However, this leads to two mathematical impossibilities. 1. Obtaining a spherical surface from linear surfaces, as it is impossible to create a circle from straight lines. 2. Obtaining a complete tiling of a sphere using regular polyhedra, without any voids or intersections between these polyhedra. The first problem is addressed in another article slated for publication. In the present one, our focus will be on the second problem, for which we will present a solution within the framework of Timaeus' cosmology. The crux of this solution lies in a feature of Timaeus' universe that sets it apart from almost all ancient cosmologies. Instead of being composed of rigid parts, it is a dynamic living body in which all basic components are in constant motion, continuously undergoing both destruction and reconstruction. In the first part, we examine the main features of Timaeus' cosmology relevant to our issue. In the second part, we analyze the paradox in details and its consequences for Timaeus' cosmology. We then discuss the common 'solutions' and highlight their shortcomings. Finally, we propose a solution that we believe is consistent with Plato's text and independent of the choice of the major schools of interpretation of the Timaeus.

Let $\mathcal{L}$ be the closure of the set of all real numbers $α$, such that there exist infinitely many integers $n$, such that $α=\log\frac{d(n+1)}{d(n)}$, where $d$ is the number of divisors of $n$. We give improved lower bounds for the density of $\mathcal{L}$.

We prove a new bound to the exponential sum of the form $$ \sum_{h \sim H}δ_h \mathop{\sum_{m\sim M}\sum_{n\sim N}}_{mn\sim x}a_{m}b_{n}\e\big(αmn + h(mn + u)^γ\big), $$ by a new approach to the Type I sum. The sum can be applied to many problems related to Piatetski-Shapiro primes, which are primes of the form $\lfloor n^c \rfloor$. In this paper, we improve the admissible range of the Balog-Friedlander condition, which leads to an improvement to the ternary Goldbach problem with Piatetski-Shapiro primes. We also investigate the distribution of Piatetski-Shapiro primes in arithmetic progressions, Piatetski-Shapiro primes in the intersection of multiple Beatty sequences and so on.

This article contains a new result in Fourier analysis concerning jump type discontinuities.

We prove that the subgroup graph of a finite group $G$ is regular if and only if $G$ is cyclic with square-free order.

In this paper, we will give a new proof for a known result of the mean square of Riemann zeta-function.

We consider a bounded open subset $Ω$ of ${\mathbb{R}}^n$ of class $C^{1,α}$ for some $α\in]0,1[$, and we define a distributional outward unit normal derivative for $α$-Hölder continuous solutions of the Helmholtz equation in the exterior of $Ω$ that may not have a classical outward unit normal derivative at the boundary points of $Ω$ and that may have an infinite Dirichlet integral around the boundary of $Ω$. Namely for solutions that do not belong to the classical variational setting. Then we show a Schauder boundary regularity result for $α$-Hölder continuous functions that have the Laplace operator in a Schauder space of negative exponent and we prove a uniqueness theorem for $α$-Hölder continuous solutions of the exterior Dirichlet and impedance boundary value problems for the Helmholtz equation that satisfy the Sommerfeld radiation condition at infinity in the above mentioned nonvariational setting.

This monograph presents a comprehensive analysis of the technological foundations of management decision-making in the reconstruction of complex gas pipeline systems. The study addresses the challenges posed by the aging infrastructure of gas supply networks and explores advanced strategies to improve their reliability, efficiency, and automation. Particular attention is given to the reconstruction of pipelines with various configurations linear, looped, and parallel systems under non-stationary gas flow conditions. The proposed models and methodologies offer solutions for optimizing operational parameters, improving emergency valve response, and ensuring uninterrupted gas supply through advanced management systems and data-driven decision support tools. Emphasis is placed on the integration of modern technologies, system theory, and feedback mechanisms in the design and operation of reconstructed pipeline systems. This work is intended for engineers, system designers, and researchers in the fields of gas supply, systems engineering, and energy infrastructure.

In this paper, some Jensen's type inequalities between quaternionic bounded selfadjoint operators on quaternionic Hilbert spaces are proved, using a log-convex function. Also, by applying a specific log-convex function, some particular cases of operator inequalities are obtained.

This paper establishes a complete framework for infinitely nested logarithmic improvements to regularity criteria for the three-dimensional incompressible Navier-Stokes equations. Building upon our previous works on logarithmically improved and multi-level logarithmically improved criteria, we demonstrate that the limiting case of infinitely nested logarithms fully bridges the gap between subcritical and critical regularity. Specifically, we prove that if the initial data $u_0 \in L^2(\mathbb{R}^3)$ satisfies the condition $\|(-Δ)^{1/4}u_0\|_{L^q(\mathbb{R}^3)} \leq C_0Ψ(\|u_0\|_{\dot{H}^{1/2}})$, where $Ψ$ incorporates infinitely nested logarithmic factors with appropriate decay conditions, then there exists a unique global-in-time smooth solution to the Navier-Stokes equations. This result establishes global well-posedness at the critical regularity threshold $s = 1/2$. The proof relies on infinitely nested commutator estimates, precise characterization of the critical exponent function in the limiting case, and careful analysis of the energy cascade. We also derive the exact Hausdorff dimension bound for potential singular sets in this limiting case, proving that the dimension reduces to zero. Through systematic construction of the limiting function spaces and detailed analysis of the associated ODEs, we demonstrate that infinitely nested logarithmic improvements provide a pathway to resolving the global regularity question for the Navier-Stokes equations.

This work explores the rich structure of spherically symmetric time-periodic solutions of the cubic conformal wave equation on $\mathbb{S}^{3}$. We discover that the families of solutions bifurcating from the eigenmodes of the linearised equation form patterns similar to the ones observed for the cubic wave equation. Alongside the Galerkin approaches, we study them using the new method based on the Padé approximants. To do so, we provide a rigorous perturbative construction of solutions. Due to the conformal symmetry, the solutions presented in this work serve as examples of large time-periodic solutions of the conformally coupled scalar field on the anti-de Sitter background.

Let $L$ be a link in $S^3$. We consider the class of meridional presentations for $π_1(S^3\backslash L)$ in which the relations are witnessed by embedded two-spheres which are in plain sight in a fixed diagram of $L$. The Wirtinger presentation is a special case. We prove that the bridge number of $L$ equals the smallest number of generators of $π_1(S^3\backslash L)$ over all such presentations.

Random geometric graphs defined on Euclidean subspaces, also called Gilbert graphs, are widely used to model spatially embedded networks across various domains. In such graphs, nodes are located at random in Euclidean space, and any two nodes are connected by an edge if they lie within a certain distance threshold. Accurately estimating rare-event probabilities related to key properties of these graphs, such as the number of edges and the size of the largest connected component, is important in the assessment of risk associated with catastrophic incidents, for example. However, this task is computationally challenging, especially for large networks. Importance sampling offers a viable solution by concentrating computational efforts on significant regions of the graph. This paper explores the application of an importance sampling method to estimate rare-event probabilities, highlighting its advantages in reducing variance and enhancing accuracy. Through asymptotic analysis and experiments, we demonstrate the effectiveness of our methodology, contributing to improved analysis of Gilbert graphs and showcasing the broader applicability of importance sampling in complex network analysis.

The compressible Navier-Stokes system with the constant viscosity and the nonlinear heat conductivity which is proportional to a positive power of the temperature and may be degenerate is considered. Under the outer pressure boundary conditions in one-dimensional unbounded spatial domains, the global existence of the strong solutions is obtained after proving that both the specific volume and temperature are bounded from below and above independently of time and space. Moreover, the asymptotically stability of global solutions is established as time tends to infinity.

Let $G$ be a finite group and $N$ a proper, nontrivial, normal subgroup of $G$. If, for every element $x$ of $G$ not lying in $N$, the elements in the coset $xN$ all have the same order as $x$, then we say that $(G,N)$ is an {\it{equal order pair}}. This generalizes the concept of a Camina pair, that was introduced by the first author. In the present paper we study several properties of equal order pairs, showing that in many respects they resemble Camina pairs, but with some important differences.

For a smooth affine algebraic group $G$, one can attach various D-module categories to it that admit convolution monoidal structure. We consider the derived category of D-modules on $G$, the stack $G/G_{ad}$ and the category of Harish-Chandra bimodules. Combining the work of Beilinson-Drinfeld on D-modules and Hecke patterns with the recent work of the author with Dimofte and Py, we show that each of the above categories (more precisely the equivariant version) is monoidal equivalent to a localization of the DG category of modules of a graded Hopf algebra. As a consequence, we give an explicit braided monoidal structure to the derived category of D-modules on $G/G_{ad}$, which when restricted to the heart, recovers the braiding of Bezrukavnikov-Finkelberg-Ostrik.

Since the 1970s, it has been known that any open connected manifold of dimension 2, 4 or 6 admits a complex analytic structure whenever its tangent bundle admits a complex linear structure. For half a century, this has been conjectured to hold true for manifolds of any dimension. In this paper, we extend the result to manifolds of dimension 8 and 10. The result is proved by applying Gromov's h-principle in order to adapt a method of Haefliger, originally used to study foliations, to the holomorphic setting. For dimension 12 and greater, the conjecture remains open.

In this article we explain the Buium--Coleman approach to the Manin--Mumford conjecture, and outline its generalisations. As an illustration, we give a $p$-adic proof of a theorem of Bombieri, Masser and Zannier on curves in tori.

We prove subelliptic estimates for ethe complex Green operator $ K_q $ at a specific level $ q $ of the $ \bar\partial_b $-complex, defined on a not necessarily pseudoconvex CR manifold satisfying the commutator finite type condition. Additionally, we obtain maximal $ L^p $ estimates for $ K_q $ by considering closed-range estimates. Our results apply to a family of manifolds that includes a class of weak $ Y(q) $ manifolds satisfying the condition $ D(q) $. We employ a microlocal decomposition and Calderón-Zygmund theory to obtain subelliptic and maximal-$ L^p $ estimates, respectively.