This paper employs the Green's function method to investigate the stability of high-Reynolds-number Couette flow in the whole space. The core of the analysis relies on obtaining sharp estimates for the linearized Green's function. Based on these estimates, the study establishes a quantitative relationship between the transition threshold and the decay behavior of perturbations. Specifically, it proves that for any 0 < theta <= 1/2, if the initial vorticity perturbation satisfies parallel to omega(0)parallel to(L2 boolean AND L1) <= c(0)nu(1/2(1+theta)) for some sufficiently small constant c(0) independent of the viscosity nu, then the vorticity perturbation exhibits an enhanced decay rate of (1 + t)(-(1/2+theta)). Furthermore, the perturbation remains bounded within O (nu(1/2(1+theta))) of the underlying Couette flow.
Shih-Hsien Yu has done important works in the field of partial differential equations. Shih-Hsien has his unique way of doing mathematics, and often does not follow the established routes. The present article illustrates through explicit examples some of his novel creations and his pioneering ways of doing mathematics.
The present paper delineates the conditions that the projective change between two (alpha, beta)-metrics analyzable. The F = alpha(alpha + beta) is considered as a Square root (alpha, beta)-metric and F = alpha 2 beta is considered as a Kropina metric on a manifold of dimension n >= 2, where alpha and alpha are Riemannian metrics while beta and beta are two non-zero 1-forms. The primary objective of this study is to identify the necessary and sufficient conditions for a projective change between Square root (alpha, beta)-metric and Kropina metric. Furthermore we will discuss some curvature properties on a manifold.
In this paper, a general null-field boundary integral formulation for boundary value problems is proposed using degenerate kernels in orthogonal curvilinear coordinates. By introducing separable kernels, all singular integrals are handled rigorously, even when the collocation point lies on the real boundary. The polar, elliptical, and bipolar coordinates are employed, and four Jacobians are derived to establish their interrelations. The closed-form fundamental solution ln(r) is represented in terms of degenerate kernels with harmonic bases in these coordinates. The proposed formulation effectively solves boundary value problems governed by the Laplace operator. Additionally, the unit logarithmic capacity and degenerate scale in boundary integral equations are investigated for circular, elliptical, and infinite plane domains containing two circles.
The fluid-dynamic limit of the Enskog equation with a slight modification is discussed on the basis of the Chapman-Enskog method. This modified version of the Enskog equation has been shown recently by the present authors to ensure the H-theorem. In the present paper, it is shown that the modified version recovers the same fluid-dynamic description of the dense gas as the original Enskog equation, at least up to the level of the Navier-Stokes-Fourier set of equations inclusive. Since the original Enskog equation is known to recover the fluid-dynamical transport properties well, this result implies that the modified version of the Enskog equation provides consistent descriptions both thermodynamically and fluid-dynamically.
We study quantitative estimates for the flocking and uniform-time classical limit to the relativistic Cucker-Smale (in short RCS) model introduced in [15]. Different from previous works, we do not neglect the relativistic effect on the presence of the pressure in momentum equation. For the RCS model, we provide a quantitative estimate on the uniform-time classical limit with an optimal convergence rate which is the same as in finite-time classical limit under a relaxed initial condition. We also allow corresponding initial data for the RCS and Cucker-Smale (CS) model to be different in the classical limit. This removes earlier constraints employed in the previous classical limit. As a direct application of this optimal convergence rate in the classical limit of the RCS model, we derive an optimal convergence rate for the corresponding uniform-time classical limit for the kinetic RCS model.
We present a systematic algebraic approach for the weak coupling of Cauchy problems to multiple Lohe tensor models. For this, we identify an admissible Cauchy problem to the Lohe tensor (LT) model with a characteristic symbol consisting of four tuples in terms of a size vector, a natural frequency tensor, a coupling strength tensor and admissible initial configuration. In this way, the collection of all admissible Cauchy problems to the LT models is equivalent to the space of characteristic symbols. On the other hand, we introduce a binary operation, namely "fusion operation" as a binary operation between the characteristic symbols. It turns out that the fusion operation satisfies the associativity and admits the identity element in the space of characteristic symbols which naturally forms a monoid. By virtue of the fusion operation, the weakly coupled system of multiple LT models can be obtained by applying the fusion operation of multiple characteristic symbols corresponding to the LT models. As a concrete example, we consider a weak coupling of the swarm sphere model and the Lohe matrix model, and provide a sufficient framework leading to emergent dynamics to the proposed weakly coupled model.
We make some comments on our previous work for Cauchy-Riemann manifolds with S1-action. Among others, we provide the missing terms in the local index theorem. We would like to make corrections for the results in [2]. Due to the effect of pullbacks, the local index formula should include the contribution from the singular strata. We provide a complete formula in Section 1 (see (1.1) below). In Section 2, we discuss how to deal with the trace integral when a suitable pullback is inserted. For Gaussian type integrals in [2, Section 7], we give factors in full generality for some formulas in Section 3. We refer to [4] for explicit computations in some cases.
This paper provides a comprehensive derivation of geodesic equations in a twodimensional Finsler space characterized by a Matsumoto-type metric. It further explores the geometric applications of these equations on various surfaces, including cylinders, spheres, pseudo-spheres, and catenoids. Using illustrative examples and graphs, we analyze the geometric properties of geodesics for different parametric forms of these surfaces. The results enhance our understanding of geodesic behavior in Finsler geometry and shed light on curvature and geometric structures in diverse contexts.
Consider independent pairs of random variables without a finite first moment. We show that an Exact Strong Law exists for the product of these two random variables. The continuous case isn't difficult, but the discrete case is not easy at all. The example we use for the discrete distribution case is the famous St. Petersburg Game.
The aim of this paper is to study several new L-p-boundedness properties for the index F-2(1)-transform over the spaces L-p(R+, e(gamma x)dx), 1 <= p < infinity, gamma is an element of R, and L-infinity(R+). We also obtain a Parseval-type relation over the space L-1(R+, e(gamma x)dx).
In this paper, we consider a non-trivial compact hyperbolic Ricci soliton (Nn, g, X, lambda, mu). Letting the vector field X to be a 2-conformal field, we find two integral equations for compact oriented hyperbolic Ricci solitons with 2-conformal potential vector field. We show that such a manifold with constant scalar curvature is isometric to the Euclidean sphere Sn. As a consequence, our results indicate that X is gradient. Moreover, X can be decomposed (according to the Hodge-de Rham theorem) into the sum of a Killing vector field and the gradient of a suitable smooth function on N.
We introduce peaceful colourings, a variant ofhconflict free colourings. We call a colouring with no monochromatic edges p-peaceful if for each vertex v, there are at most p neighbours of v coloured with a colour appearing on another neighbour of v. An hconflict-free colouring of a graph is a (vertex)-colouring with no monochromatic edges so that for every vertex v, the number of neighbours of v which are coloured with a colour appearing on no other neighbour of v is at least the minimum of h and the degree of v. If G is Delta-regular then it has anhconflict free colouring precisely if it has a (Delta-h)-peaceful colouring. We focus on the minimum p(Delta) of those p for which every graph of maximum degree Delta has a p-peaceful colouring with Delta+1 colours. We show that p(Delta) > (1-1/e-o(1))Delta and that for graphs of bounded codegree, p(Delta) <= (1-1/e +o(1))Delta. We ask if the latter result can be improved by dropping the bound on the codegree. As a partial result, we show that p(Delta) <= 8000/8001 Delta for sufficiently large Delta.
Diffusion limits provide a framework for the asymptotic analysis of stochastic gradient descent (SGD) schemes used in machine learning. We consider an alternative framework, the Riemannian Langevin equation (RLE), that generalizes the classical paradigm of equilibration in R^n to a Riemannian manifold (M^n, g). The most subtle part of this equation is the description of Brownian motion on (M^n, g). Explicit formulas are presented for some fundamental cones.
The Thompson sporadic group admits special relationships to modular forms of two kinds. On the one hand, last century's generalized moonshine for the monster equipped the Thompson group with a module for which the associated McKay-Thompson series are distinguished weight zero modular functions. On the other hand, Griffin and Mertens verified the existence of a module for which the McKay-Thompson series are distinguished modular forms of weight one-half, that were assigned to the Thompson group in this century by the last two authors of this work. In this paper we round out this picture by proving the existence of two new avatars of Thompson moonshine: a new module giving rise to weight zero modular functions, and a new module giving rise to forms of weight one-half. We explain how the newer modules are related to the older ones by Borcherds products and traces of singular moduli. In so doing we clarify the relationship between the previously known modules, and expose a new arithmetic aspect to moonshine for the Thompson group. We also present evidence that this phenomenon extends to a correspondence between other cases of generalized monstrous moonshine and penumbral moonshine, and thereby enriches these phenomena with counterparts in weight one-half and weight zero, respectively.
For an antiferromagnetic spin-1 Bose-Einstein condensate under an applied uniform magnetic field, its ground state (psi 1, psi 0, psi-1) undergoes a phase transition from a two-component state (psi 0 equivalent to 0) to a three-component state (psi j =6 0 for all j) at a critical value of the magnetic field. This phenomenon has been observed in numerical simulations as well as in experiments. In this paper, we provide a mathematical proof based on a simple principle found by the authors: a redistribution of the mass densities between different components will decrease the kinetic energy.
This paper investigates boundedness properties, Parseval-type relations and asymptotic behaviours for operators with complex Gaussian kernels over Lebesgue spaces. Additionally, this paper explores the Gauss-Weierstrass semigroup as a specific example within the scope of our analysis.
