arXiv Math @arxiv_math@qoto.org

In this paper, we study the following critical fractional Schrödinger equation: \begin{equation} (-Δ)^s u+V(|y'|,y'')u=K(|y'|,y'')u^{\frac{n+2s}{n-2s}},\quad u>0,\quad y =(y',y'') \in \mathbb{R}^3\times\mathbb{R}^{n-3}, \qquad(0.1)\end{equation} where $n\geq 3$, $s\in(0,1)$, $V(|y'|,y'')$ and $K(|y'|,y'')$ are two bounded nonnegative potential functions. Under the conditions that $K(r,y'')$ has a stable critical point $(r_0,y_0'')$ with $r_0>0$, $K(r_0,y_0'')>0$ and $V(r_0,y_0'')>0$, we prove that equation (0.1) has a new type of infinitely many solutions that concentrate at points lying on the top and the bottom of a cylinder. In particular, the bubble solutions can concentrate at a pair of symmetric points with respect to the origin. Our proofs make use of a modified finite-dimensional reduction method and local Pohozaev identities.

In this paper, we study the Cauchy problem of the classical incompressible Navier-Stokes equations and the parabolic-elliptic Keller-Segel system in the framework of the Fourier-Besov spaces with variable regularity and integrability indices. By fully using some basic properties of these variable function spaces, we establish the linear estimates in variable Fourier-Besov spaces for the heat equation. Such estimates are fundamental for solving certain PDE's of parabolic type. As an applications, we prove global well-posedness in variable Fourier-Besov spaces for the 3D classical incompressible Navier-Stokes equations and the 3D parabolic-elliptic Keller-Segel system.

Nonlinear hydroelastic waves along a compressed ice sheet lying on top of a two-dimensional fluid of infinite depth are investigated. Based on a Hamiltonian formulation of this problem and by applying techniques from Hamiltonian perturbation theory, a Hamiltonian Dysthe equation is derived for the slowly varying envelope of modulated wavetrains. This derivation is further complicated here by the presence of cubic resonances for which a detailed analysis is given. A Birkhoff normal form transformation is introduced to eliminate non-resonant triads while accommodating resonant ones. It also provides a non-perturbative scheme to reconstruct the ice-sheet deformation from the wave envelope. Linear predictions on the modulational instability of Stokes waves in sea ice are established, and implications for the existence of solitary wavepackets are discussed for a range of values of ice compression relative to ice bending. This Dysthe equation is solved numerically to test these predictions. Its numerical solutions are compared to direct simulations of the full Euler system, and very good agreement is observed.

The aim of this manuscript is to address non-linear differential equations of the Lane Emden equation of second order using the shifted Legendre neural network (SLNN) method. Here all the equations are classified as singular initial value problems. To manage the singularity challenge, we employ an artificial neural network method. The approach utilizes a neural network of a single layer, where the hidden layer is omitted by enlarge the input using shifted Legendre polynomials. We apply a feed forward neural network model along with the principle of error back propagation. The effectiveness of the Legendre Neural Network model is demonstrated through LaneEmden equations.

This paper deals with the problem of robust matrix completion -- retrieving a low-rank matrix and a sparse matrix from the compressed counterpart of their superposition. Though seemingly not an unresolved issue, we point out that the compressed matrix in our case is sampled in a deterministic pattern instead of those random ones on which existing studies depend. In fact, deterministic sampling is much more hardware-friendly than random ones. The limited resources on many platforms leave deterministic sampling the only choice to sense a matrix, resulting in the significance of investigating robust matrix completion with deterministic pattern. In such spirit, this paper proposes \textit{restricted approximate $\infty$-isometry property} and proves that, if a \textit{low-rank} and \textit{incoherent} square matrix and certain deterministic sampling pattern satisfy such property and two existing conditions called \textit{isomerism} and \textit{relative well-conditionedness}, the exact recovery from its sampled counterpart grossly corrupted by a small fraction of outliers via convex optimization happens with very high probability.

We show that the independence number of $ G_{n,m}$ is concentrated on two values for $ n^{5/4+ ε} < m \le \binom{n}{2}$. This result establishes a distinction between $G_{n,m}$ and $G_{n,p}$ with $p = m/ \binom{n}{2}$ in the regime $ n^{5/4 + ε} < m< n^{4/3}$. In this regime the independence number of $ G_{n,m}$ is concentrated on two values while the independence number of $ G_{n,p}$ is not; indeed, for $p$ in this regime variations in $ α( G_{n,p})$ are determined by variations in the number of edges in $ G_{n,p}$.

A graph is said to be neighborhood 3-balanced if there exists a vertex labeling with three colors so that each vertex has an equal number of neighbors of each color. We give order constraints on 3-balanced graphs, determine which generalized Petersen and Pappus graphs are 3-balanced, discuss when being 3-balanced is preserved under various graph constructions, give two general characterizations of cubic 3-balanced graphs, and classify cubic 3-balanced graphs of small order.

In this work, we look for a spacial SEIAR-type epidemic model consediring quarantined population $(Q)$, namely SQEIAR model. The dynamic of the SQEIAR model involves six partial differential equations that decribe the changes in susceptible, quarantined, exposed, asymptomatic, infected and recovered population. Our goal is to reduce the number of exposed, asymptomatic, and infected population while taking into account the environment, which plays a critical role in the spread of epidemics. Then, we implement a new strategy based on two control actions: regional quarantine for susceptible population and treatment for infected population. To demonstrate the practical utility of the obtained results, a numerical example centered on COVID-19 is presented.

Spherical and polar geometries arise in many important areas of computational science, including weather and climate forecasting, optics, and astrophysics. In these applications, tensor-product grids are often used to represent unknowns. However, interpolation schemes that exploit the tensor-product structure can introduce artificial boundaries at the poles in spherical coordinates and at the origin in polar coordinates, leading to numerical challenges, especially for high-order methods. In this paper, we present new bivariate trigonometric barycentric interpolation formulas for spheres and bivariate trigonometric/polynomial barycentric formulas for disks, designed to overcome these issues. These formulas are also efficient, as they only rely on a set of (precomputed) weights that depend on the grid structure and not the data itself. The formulas are based on the Double Fourier Sphere (DFS) method, which transforms the sphere into a doubly periodic domain and the disk into a domain without an artificial boundary at the origin. For standard tensor-product grids, the proposed formulas exhibit exponential convergence when approximating smooth functions. We provide numerical results to demonstrate these convergence rates and showcase an application of the spherical barycentric formulas in a semi-Lagrangian advection scheme for solving the tracer transport equation on the sphere.

Consider a group $G$ of order $M$ acting unitarily on a real inner product space $V$. We show that the sorting based embedding obtained by applying a general linear map $α: \mathbb{R}^{M \times N} \to \mathbb{R}^D$ to the invariant map $β_Φ: V \to \mathbb{R}^{M \times N}$ given by sorting the coorbits $(\langle v, g ϕ_i \rangle_V)_{g \in G}$, where $(ϕ_i)_{i=1}^N \in V$, satisfies a bi-Lipschitz condition if and only if it separates orbits. Additionally, we note that any invariant Lipschitz continuous map (into a Hilbert space) factors through the sorting based embedding, and that any invariant continuous map (into a locally convex space) factors through the sorting based embedding as well.

A convex body $R$ in the hyperbolic plane is reduced if any convex body $K\subset R$ has a smaller minimal width than $R$. We answer a few of Lassak's questions about ordinary reduced polygons regarding its perimeter, diameter and circumradius, and we also obtain a hyperbolic extension of a result of Fabińska.

The paper studies spectral representation as well as sampling and interpolation problems for non-vanishing continuous time signals. The notions of transfer functions, the spectrum gaps, and bandlimitness are introduced for these signals. As an application, an analog Sampling Theorem and a modification of Whittaker-Shannon interpolation formula for unbounded signals is obtained.

A new proof following a suggestion of Kaplansky to use a result of Dixmier, and in this way avoid unbounded nets of operators, is given of the Kaplansky density theorem.

This paper introduces and investigates the concept of the $q$-numerical range for tuples of bounded linear operators in Hilbert spaces. We establish various inequalities concerning the $q$-numerical radius associated with these operator tuples. Furthermore, we extend our study to define the $q$-numerical range in semi-Hilbert spaces and provide a proof of its convexity. Additionally, we explore several related results in this context.

In 1967, Peetre proposed to give a precise description of the real interpolation space for Besov hierarchical spaces $l^{s,q}(A)$. In 1974, Cwikel proved that the Lions-Peetre formula for $(l^{q_0}(A_0), l^{q_1}(A_1))_{θ,r}$ have no reasonable generalization for any $r\neq q$. In this paper, we apply wavelets to transform the study of real interpolation space into {\bf the study of nonlinear functional structure and nonlinear topological structure}. We solve completely Peetre's longstanding open problem.

In this paper, Peetre's conjecture about the real interpolation space of Besov space {\bf is solved completely } by using the classification of vertices of cuboids defined by {\bf wavelet coefficients and wavelet's grid structure}. Littlewood-Paley analysis provides only a decomposition of the function on the ring. We extend Lorentz's rearrangement function and Hunt's Marcinkiewicz interpolation theorem to more general cases. We use the method of calculating the topological quantity of the grid to replace the traditional methods of data classification such as gradient descent method and distributed algorithm. We developed a series of new techniques to solve this longstanding open problem. These skills make up for the deficiency of Lions-Peetre iterative theorem in dealing with strong nonlinearity. Using the properties of wavelet basis, a series of {\bf functional nonlinearities} are studied. Using the lattice property of wavelet, we study the lattice topology. By three kinds of {\bf topology nonlinearities}, we give the specific wavelet expression of K functional.

We visualize the identity p(n) = sum s(k) p(n-k)/n for the integer partition function p(n) involving the divisor function s, add comments on the history of visualizations of numbers, illustrate how different mathematical fields play together when proving lim p(n)^(1/n)=1 and introduce finite or infinite rings of partitions.

The Goormaghtigh conjecture states that the only two numbers which have two non-trivial representations as repunits are $31$ and $8191$. We call such a prime number a {\it Goormaghtigh prime}. We show that there are no other Goormaghtigh primes less than $10^{700}$.

This note determines an effective asymptotic formula for the number of squarefree totients $p-1$ with a fixed primitive root $u\ne \pm 1, v^2$.

This paper focuses on the stability of both local and global error bounds for a proper lower semicontinuous convex function defined on a Banach space. Without relying on any dual space information, we first provide precise estimates of error bound moduli using directional derivatives. For a given proper lower semicontinuous convex function on a Banach space, we prove that the stability of local error bounds under small perturbations is equivalent to the directional derivative at a reference point having a non-zero minimum over the unit sphere. Additionally, the stability of global error bounds is shown to be equivalent to the infimum of the directional derivatives, at all points on the boundary of the solution set, being bounded away from zero over some neighborhood of the unit sphere.