Best approximation and moduli of smoothness: Computation and equivalence theorems

In this paper we investigate the trigonometric series with the β-general monotone coefficients. First, we study the uniform convergence criterion. The estimates of best approximations and moduli of smoothness of the series in uniform metrics are obtained in terms of coefficients. These results imply several important relations between moduli of smoothness of different orders (in particular, Marchaud-type inequality) and best approximations.     

Ergodicity and asymptotic stability of Feller semigroups on Polish metric spaces

We provide some sharp criteria for studying the ergodicity and asymptotic stability of general Feller semigroups on Polish metric spaces. As an application, the 2D Navier-Stokes equations with degenerate stochastic forcing will be simply revisited.     

Exact bound on the number of limit cycles arising from a periodic annulus bounded by a symmetric heteroclinic loop

In this paper, the bound on the number of limit cycles by Poincare bifurcation in a small perturbation of some seventh-degree Hamiltonian system is concerned. The lower and upper bounds on the number of limit cycles have been obtained in two previous works, however, the sharp bound is still unknown. We will employ some new techniques to determine which is the exact bound between $3$ and $4$. The asymptotic expansions are used to determine the four vertexes of a tetrahedron, and the sharp bound can be reached when the parameters belong to this tetrahedron.     

Balls have the worst best Sobolev inequalities

Using transportation techniques in the spirit of Cordero-Erausquin, Nazaret and Villani [7], we establish an optimal non parametric trace Sobolev inequality, for arbitrary locally Lipschitz domains in ℝⁿ. We deduce a sharp variant of the Brézis-Lieb trace Sobolev inequality [4], containing both the isoperimetric inequality and the sharp Euclidean Sobolev embedding as particular cases. This inequality is optimal for a ball, and can be improved for any other bounded, Lipschitz, connected domain. We also derive a strengthening of the Brézis-Lieb inequality, suggested and left as an open problem in [4]. Many variants will be investigated in a companion article [10].     

The Congruences of Clausen-von Staudt and Kummer for Half-integral Weight Eisenstein Series

We extend sharp forms of the classical uncertainty principle to the context of commutative hypergroups. This hypergroup setting includes Gelfand pairs, Riemannian symmetric spaces, and locally compact abelain groups. For some Gelfand pairs our inequalities will be sharper than those in a recent paper by J. A. WOLF.     

Stability of reaction-fronts in porous media

The stability of reaction-fronts in porous media is studied with analytical and numerical methods. A stability criterion has been derived using linear stability analysis assuming a sharp font. The sharp front assumption is an approximation of the mathematical model in the limit of an infinite rapid reaction. The criterion shows that the stability of a sharp reaction front is dependent on the permeability that develops behind it. The sharp front is unstable for perturbations of any wave-length if the permeability increases behind the front. The criterion shows that short wave-length perturbations are more unstable than long wave-length perturbations. The sharp front is labile when the permeabilities are the same at both sides of the front. This means that the perturbed front moves unchanged forward. Finally, perturbations will die out in case the permeability decreases behind the sharp front. The stability of non-sharp fronts are simulated numerically when dissolution is by first order kinetics, the transport is by convection and diffusion and when the permeability and specific reactive surface depends on the porosity. The numerical experiments behave according to the stability criterion.     

Sign changing solutions of the Brezis-Nirenberg problem in the Hyperbolic space

In this article we will study the existence and nonexistence of sign changing solutions for the Brezis-Nirenberg type problem in the Hyperbolic space. We will also establish sharp asymptotic estimates for the solutions and the compactness properties of solutions.     

The Exact Limits and Improved Decay Estimates for All Order Derivatives of the Global Weak Solutions to a Two-Dimensional Incompressible Dissipative Quasi-Geostrophic Equation

We will accomplish the exact limits for all order derivatives of the global weak solutions to a two-dimensional incompressible dissipative quasi-geostrophic equation. We will also establish the improved decay estimates with sharp rates for all order derivatives. We will consider two cases for the initial function and the external force and prove the optimal results for both cases. We will couple together existing ideas (including the Fourier transformation and its properties, Parseval''s identity, iteration technique, Lebesgue''s dominated convergence theorem, Gagliardo-Nirenberg-Sobolev interpolation inequality, squeeze theorem, Cauchy-Schwartz''s inequality, etc) existing results (the existence of global weak solutions, the existence of local smooth solution on $(T,\infty)$ and the elementary decay estimate with a sharp rate) and a few novel ideas to obtain the main results.     

Remarks to an equivalent formulation of ekeland’s variational principle

A recent theorem Of W.Takahashi will be pointed out as an equivalent formulation of Ekelands variational principle. This gives rise to study functions having sets of weak sharp minimas in a generalized sense. Connections to the proximal point algorithm in the convex case leads to the basic and still open question: How to use Ekelands principle numerically     

Sufficient conditions for boundedness of solutions of linear differential equations

In this paper we obtain sufficient conditions for boundedness of every solution and its derivatives of a linear differential equation. We will also show that our conditions are sharp.     

On the local in time well-posedness of an elliptic–parabolic ferroelectric phase-field model

We consider a state-of-the-art ferroelectric phase-field model arising from the engineering area in recent years, which is mathematically formulated as a coupled elliptic–parabolic differential system. We utilize the maximal parabolic regularity theory to show the local in time well-posedness of the ferroelectric problem in both 2D and 3D spaces, which is sharp in the sense that the local solution is unique and a blow-up criterion is present. The well-posedness result will firstly be proved under some general assumptions. Afterwards we give sufficient geometric and regularity conditions which will guarantee the fulfillment of the imposed assumptions.     

Fourier Method for Riemann-Hilbert-Poincaré Problem of Analytic Function

Spectrum problem with Riemann-Hilbert-Poincaré boundary condition is studied. This problem will lead to inhomogeneous Fuchsian differential equations with its right hand side depending on some constants to be determined simultaneously. We find out that the multiplicities of eigenfunctions for different eigenvalues are not necessary the same, which are in sharp contrast to the known results of Riemann-Hilbert problem for analytic functions.     

On the scalar curvature of constant mean curvature hypersurfaces in space forms

In this paper we study the behavior of the scalar curvature S of a complete hypersurface immersed with constant mean curvature into a Riemannian space form of constant curvature, deriving a sharp estimate for the infimum of S. Our results will be an application of a weak Omori-Yau maximum principle due to Pigola, Rigoli, Setti (2005) [17].     

THE IMPROVED FOURIER SPLITTING METHOD AND DECAY ESTIMATES OF THE GLOBAL SOLUTIONS OF THE CAUCHY PROBLEMS FOR NONLINEAR SYSTEMS OF FLUID DYNAMICS EQUATIONS

Consider the Cauchy problems for an n-dimensional nonlinear system of fluid dynamics equations. The main purpose of this paper is to improve the Fourier splitting method to accomplish the decay estimates with sharp rates of the global weak solutions of the Cauchy problems. We will couple together the elementary uniform energy estimates of the global weak solutions and a well known Gronwall''s inequality to improve the Fourier splitting method. This method was initiated by Maria Schonbek in the 1980''s to study the optimal long time asymptotic behaviours of the global weak solutions of the nonlinear system of fluid dynamics equations. As applications, the decay estimates with sharp rates of the global weak solutions of the Cauchy problems for $n$-dimensional incompressible Navier-Stokes equations, for the $n$-dimensional magnetohydrodynamics equations and for many other very interesting nonlinear evolution equations with dissipations can be established.     

An Upper Bound on the Binomial Process Approximation to the Exceedance Process

In this paper we deal with an approximation to the point process of exceedances over a higher threshold. We compute sharp bounds on the remainder terms when actual distributions of exceedances are replaced by appropriate generalized Pareto distributions (GPD). The bound will be formulated in terms of the von Mises function. The ultimate as well as the penultimate approximations are considered; in the latter case the shape parameter of the approximating GPD depends on the threshold.     

On sharp triangle inequalities in Banach spaces

We shall present a couple of norm inequalities which will much improve the sharp triangle inequality with n elements and its reverse inequality in a Banach space shown recently by the last three authors.     

A lower bound on the number of sharp shadow-boundaries of convex polytopes

We give the lower bound on the number of sharp shadow-boundaries of convexd-polytopes (or unbounded convex polytopal sets) withn facets. The polytopes (sets) attaining these bounds are characterized. Additionally, our results will be transferred to the dual theory.The research work of the first author was (partially) supported by Hungarian National Foundation for Scientific Research, grant no. 1812.     

A New Definition of A Priori Knowledge: In Search of a Modal Basis

In this paper I will offer a novel understanding of a priori knowledge. My claim is that the sharp distinction that is usually made between a priori and a posteriori knowledge is groundless. It will be argued that a plausible understanding of a priori and a posteriori knowledge has to acknowledge that they are in a constant bootstrapping relationship. It is also crucial that we distinguish between a priori propositions that hold in the actual world and merely possible, non-actual a priori propositions, as we will see when considering cases like Euclidean geometry. Furthermore, contrary to what Kripke seems to suggest, a priori knowledge is intimately connected with metaphysical modality, indeed, grounded in it. The task of a priori reasoning, according to this account, is to delimit the space of metaphysically possible worlds in order for us to be able to determine what is actual.     

Jackson and smoothness theorems for freud weights

We prove Jackson, realization, and converse theorems for Freud weights inL _p, 0<p ≤ ∞. Even forp ≥ 1, our conditions on the weight in our Jackson theorems are far less restrictive than those previously imposed. Moreover, the method—of first approximating by a spline, and then by a polynomial—is new in this context, and of intrinsic interest, since it avoids the use of orthogonal polynomials for Freud weights. We establish some properties of the modulus of smoothness, valid inL _p for 0<p ≤ ∞. Since theK-functional is identically zero inL _p,p<1, the analysis of the modulus of continuity involves a different tool, namely, realization, which works inL _p for all 0<p ≤ ∞. We deduce Marchaud-type inequalities.     

Non‐smooth atomic decomposition for generalized Orlicz‐Morrey spaces

In the present paper, we consider the non‐smooth atomic decomposition of generalized Orlicz‐Morrey spaces. The result will be sharper than the existing results. As an application, we consider the boundedness of the bilinear operator, which is called the Olsen inequality nowadays. To obtain a sharp norm estimate, we first investigate their predual space, which is even new, and we make full advantage of the vector‐valued inequality for the Hardy‐Littlewood maximal operator.