Delay-Dependent Exponential Stability for Nonlinear Reaction-Diffusion Uncertain Cohen-Grossberg Neural Networks with Partially Known Transition Rates via Hardy-Poincaré Inequality

In this paper, stochastic global exponential stability criteria for delayed impulsive Markovian jumping reaction-diffusion Cohen-Grossberg neural networks(CGNNs for short) are obtained by using a novel Lyapunov-Krasovskii functional approach, linear matrix inequalities(LMIs for short) technique, It formula, Poincar′e inequality and Hardy-Poincaré inequality, where the CGNNs involve uncertain parameters, partially unknown Markovian transition rates, and even nonlinear p-Laplace diffusion(p 1). It is worth mentioning that ellipsoid domains in Rm(m ≥ 3) can be considered in numerical simulations for the first time owing to the synthetic applications of Poincar′e inequality and Hardy-Poincar′e inequality. Moreover, the simulation numerical results show that even the corollaries of the obtained results are more feasible and effective than the main results of some recent related literatures in view of significant improvement in the allowable upper bounds of delays.     

Delay-Dependent Exponential Stability for Nonlinear Reaction-Diffusion Uncertain Cohen-Grossberg Neural Networks with Partially Known Transition Rates via Hardy-Poincar′e Inequality

In this paper, stochastic global exponential stability criteria for delayed impulsive Markovian jumping reaction-diffusion Cohen-Grossberg neural networks (CGNNs for short) are obtained by using a novel Lyapunov-Krasovskii functional approach, linear matrix inequalities (LMIs for short) technique, It? formula, Poincare inequality and Hardy-Poincare inequality, where the CGNNs involve uncertain parameters, partially unknown Markovian transition rates, and even nonlinear p-Laplace diffusion (p > 1). It is worth mentioning that ellipsoid domains in $R^m$ (m ≥ 3) can be considered in numerical simulations for the first time owing to the synthetic applications of Poincar′e inequality and Hardy-Poincar′e inequality. Moreover, the simulation numerical results show that even the corollaries of the obtained results are more feasible and effective than the main results of some recent related literatures in view of significant improvement in the allowable upper bounds of delays.     

Translative containment measure and symmetric mixed isohomothetic inequalities

We first investigate the translative containment measure for convex domain K0 to contain, or to be contained in, the homothetic copy of another convex domain K1, i.e., given two convex domains K0, K1 of areas A0, A1, respectively, in the Euclidean plane R2, is there a translation T so that t(T K1)  K0 or t(T K1) ? K0 for t 0? Via the translative kinematic formulas of Poincar′e and Blaschke in integral geometry,we estimate the symmetric mixed isohomothetic deficit σ2(K0, K1) ≡ A201- A0A1, where A01 is the mixed area of K0 and K1. We obtain a sufficient condition for K0 to contain, or to be contained in, t(T K1). We obtain some Bonnesen-style symmetric mixed isohomothetic inequalities and reverse Bonnesen-style symmetric mixed isohomothetic inequalities. These symmetric mixed isohomothetic inequalities obtained are known as Bonnesen-style isopermetric inequalities and reverse Bonnesen-style isopermetric inequalities if one of domains is a disc. As direct consequences, we obtain some inequalities that strengthen the known Minkowski inequality for mixed areas and the Bonnesen-Blaschke-Flanders inequality.     

Nonexistence of Proper Holomorphic Maps Between Certain Classical Bounded Symmetric Domains

The author,motivated by his results on Hermitian metric rigidity,conjectured in [4] that a proper holomorphic mapping f：Ω→Ω′from an irreducible bounded symmetric domainΩof rank≥2 into a bounded symmetric domainΩ′is necessarily totally geodesic provided that r′：=rank（Ω′）≤rank（Ω）：=r.The Conjecture was resolved in the affirmative by I.-H.Tsai [8].When the hypothesis r′≤r is removed,the structure of proper holomorphic maps f：Ω→Ω′is far from being understood,and the complexity in studying such maps depends very much on the difference r′-r,which is called the rank defect.The only known nontrivial non-equidimensional structure theorems on proper holomorphic maps are due to Z.-H.Tu [10],in which a rigidity theorem was proven for certain pairs of classical domains of type I,which implies nonexistence theorems for other pairs of such domains.For both results the rank defect is equal to 1,and a generaliza- tion of the rigidity result to cases of higher rank defects along the line of arguments of [10] has so far been inaccessible. In this article, the author produces nonexistence results for infinite series of pairs of （Ω→Ω′） of irreducible bounded symmetric domains of type I in which the rank defect is an arbitrarily prescribed positive integer. Such nonexistence results are obtained by exploiting the geometry of characteristic symmetric subspaces as introduced by N. Mok and L-H Tsai [6] and more generally invariantly geodesic subspaces as formalized in [8]. Our nonexistence results motivate the formulation of questions on proper holomorphic maps in the non-equirank case.     

WEAK TYPE INEQUALITIES FOR FRACTIONAL INTEGRAL OPERATORS ON GENERALIZED NON-HOMOGENEOUS MORREY SPACES

We obtain weak type (1,q) inequalities for fractional integral operators on generalized non-homogeneous Morrey spaces.The proofs use some properties of maximal operators.Our results are closely related to the strong type inequalities in [13,14,15].     

ON THE TOPOLOGY,VOLUME,DIAMETER AND GAUSS MAP IMAGE OF SUBMANIFOLDS IN A SPHERE

In this paper, the author uses Gauss map to study the topology, volume and diameter of submanifolds in a sphere. It is proved that if there exist ε, 1≥ε > 0 and a fixed unit simple p-vector a such that the Gauss map g of an n-dimensional complete and connected submanifold M in Sn p satisfies (g, a) ≥ε, then M is diffeomorphic to Sn, and the volume and diameter of M satisfy εnvol(Sn) ≤vol(M) ≤ vol(Sn)/ε and επ ≤diam(M) ≤ π/ε, respectively. The author also characterizes the case where these inequalities become equalities. As an application, a differential sphere theorem for compact submanifolds in a sphere is obtained.     

Inequalities for Eigenvalues of a System of Equations of Elliptic Operator in Weighted Divergence Form on Metric Measure Space

Let A be a symmetric and positive definite(1, 1) tensor on a bounded domain Ω in an ndimensional metric measure space■. In this paper, we investigate the Dirichlet eigenvalue problem of a system of equations of elliptic operators in weighted divergence form■,where ■, α is a nonnegative constant and u is a vector-valued function.Some universal inequalities for eigenvalues of this problem are established. Moreover, as applications of these results, we give some estimates for the upper bound of ?k+1 and the gap of ?k+1-?k in terms of the first k eigenvalues. Our results contain some results for the Lam′e system and a system of equations of the drifting Laplacian.     

Orlicz-Hardy spaces and their duals

We establish the theory of Orlicz-Hardy spaces generated by a wide class of functions.The class will be wider than the class of all the N-functions.In particular,we consider the non-smooth atomic decomposition.The relation between Orlicz-Hardy spaces and their duals is also studied.As an application,duality of Hardy spaces with variable exponents is revisited.This work is different from earlier works about Orlicz-Hardy spaces H(Rn)in that the class of admissible functions is largely widened.We can deal with,for example,(r)≡(rp1(log(e+1/r))q1,0r 6 1,rp2(log(e+r))q2,r1,with p1,p2∈(0,∞)and q1,q2∈(.∞,∞),where we shall establish the boundedness of the Riesz transforms on H(Rn).In particular,is neither convex nor concave when 0p11p2∞,0p21p1∞or p1=p2=1 and q1,q20.If(r)≡r(log(e+r))q,then H(Rn)=H(logH)q(Rn).We shall also establish the boundedness of the fractional integral operators I of order∈(0,∞).For example,I is shown to be bounded from H(logH)1./n(Rn)to Ln/(n.)(log L)(Rn)for 0n.     

Lyapunov-type Inequalities for Fractional (p, q)-Laplacian Systems

In this paper, we establish some new Lyapunov type inequalities for fractional (p, q)-Laplacian operators in an open bounded set Ω ? RN , under homogeneous Dirichlet boundary conditions. Next, we use the obtained inequalities to derive some geometric properties of the generalized spectrum associated to the considered problem.     

Infinitely Many Solutions for Schr?dinger–Choquard–Kirchhoff Equations Involving the Fractional p-Laplacian

In this article,we show the existence of infinitely many solutions for the fractional pLaplacian equations of Schr?dinger-Kirchhoff type equation ■ ,where(-△)_p~s is the fractional p-Laplacian operator,[u]_(s,p) is the Gagliardo p-seminorm,0 s 1 q p N/s,α∈(0,N),M and V are continuous and positive functions,and k(x) is a non-negative function in an appropriate Lebesgue space.Combining the concentration-compactness principle in fractional Sobolev space and Kajikiya's new version of the symmetric mountain pass lemma,we obtain the existence of infinitely many solutions which tend to zero for suitable positive parameters λ and β.     

Modulation space estimates for the fractional integral operators

We obtain the boundedness for the fractional integral operators from the modulation Hardy space μp,q to the modulation Hardy space μr,q for all 0 < p < ∞. The result is an extension of the known result for the case 1 < p < ∞ and it contains a larger range of r than those in the classical result of the Lp → Lr boundedness in the Lebesgue spaces. We also obtain some estimates on the modulation spaces for the bilinear fractional operators.     

REGULARITY OF WEAK SOLUTIONS TO MAGNETO-MICROPOLAR FLUID EQUATIONS

In this article,we study the regularity of weak solutions and the blow-up criteria for smooth solutions to the magneto-micropolar fluid equations in R~3.We obtain the classical blow-up criteria for smooth solutions(u,ω,b),i.e.,u ∈ L q(0,T;L p(R 3)) for 2 q + 3 p ≤ 1 with 3p≤∞,u ∈ C([0,T);L 3(R 3)) or u ∈L q(0,T;L p) for 3 2p≤∞ satisfying 2 q + 3p≤2.Moreover,our results indicate that the regularity of weak solutions is dominated by the velocity u of the fluid.In the end-point case p = ∞,the blow-up criteria can be extended to more general spaces u∈ L~1(0,T;B_(∞,∞)~0(R~3)).     

The Two Hyperplane Conjecture

We introduce a conjecture that we call the Two Hyperplane Conjecture, saying that an isoperimetric surface that divides a convex body in half by volume is trapped between parallel hyperplanes. The conjecture is motivated by an approach we propose to the Hots Spots Conjecture of J. Rauch using deformation and Lipschitz bounds for level sets of eigenfunctions. We will relate this approach to quantitative connectivity properties of level sets of solutions to elliptic variational problems, including isoperimetric inequalities, Poincar′e inequalities, Harnack inequalities, and NTA(non-tangentially accessibility). This paper mostly asks questions rather than answering them, while recasting known results in a new light. Its main theme is that the level sets of least energy solutions to scalar variational problems should be as simple as possible.     

Inequalities for the Extended Best Polynomial Approximation Operator in Orlicz Spaces

In this paper we pursue the study of the best approximation operator extended from L~Φ to L~φ, where φ denotes the derivative of the function Φ. We get pointwise convergence for the coefficients of the extended best approximation polynomials for a wide class of function f, closely related to the Calder′on–Zygmund class t_m~p(x) which had been introduced in 1961. We also obtain weak and strong type inequalities for a maximal operator related to the extended best polynomial approximation and a norm convergence result for the coefficients is derived. In most of these results, we have to consider Matuszewska–Orlicz indices for the function φ.     

Sharp Observability Inequalities for the 1-D Plate Equation with a Potential

This paper deals with the problem of sharp observability inequality for the 1-D plate equation wtt + wxxxx + q(t,x)w = 0 with two types of boundary conditions w = wxx = 0 or w = wx = 0,and q(t,x) being a suitable potential.The author shows that the sharp observability constant is of order exp(C q ∞27) for q ∞≥ 1.The main tools to derive the desired observability inequalities are the global Carleman inequalities,based on a new point wise inequality for the fourth order plate operator.     

Maximal Inequalities for the Best Approximation Operator and Simonenko Indices

In an abstract set up, we get strong type inequalities in L~(p+1) by assuming weak or extra-weak inequalities in Orlicz spaces. For some classes of functions, the number p is related to Simonenko indices. We apply the results to get strong inequalities for maximal functions associated to best Φ-approximation operators in an Orlicz space L~Φ.     

Poincaré and Sobolev Inequalities for Vector Fields Satisfying Hrmander's Condition in Variable Exponent Sobolev Spaces

In this paper,we will establish Poincare inequalities in variable exponent non-isotropic Sobolev spaces.The crucial part is that we prove the boundedness of the fractional integral operator on variable exponent Lebesgue spaces on spaces of homogeneous type.We obtain the first order Poincare inequalities for vector fields satisfying Hrmander's condition in variable non-isotropic Sobolev spaces.We also set up the higher order Poincare inequalities with variable exponents on stratified Lie groups.Moreover,we get the Sobolev inequalities in variable exponent Sobolev spaces on whole stratified Lie groups.These inequalities are important and basic tools in studying nonlinear subelliptic PDEs with variable exponents such as the p(x)-subLaplacian.Our results are only stated and proved for vector fields satisfying Hrmander's condition,but they also hold for Grushin vector fields as well with obvious modifications.     

Existence results of solutions for anti-periodic fractional Langevin equation

Recently, Khalili and Yadollahzadeh [ Y. Khalili and M. Yadollahzadeh, Existence results for a new class of nonlinear Langevin equations of fractional orders, Iranian J. Sci. Tech., Trans. A: Sci., 2019, 43(5), 2335–2342] have investigated the uniqueness and existence of solution $u(t),~t\in[0,1]$ for a class of nonlocal boundary conditions to fractional Langevin equation. The authors used the boundary condition $u"(0)=0$ by incorrect method. In the current contribution, we show the correct method for using this condition and study the existence and uniqueness of solution for the same class of equation in slightly different form with anti-periodic and nonlocal integral boundary conditions as well as the boundary condition $u"(0)=0$. An exemplar is provided to illustrate our results.     

Homogenization of Elliptic Problems with Neumann Boundary Conditions in Non-smooth Domains

We consider a family of second-order elliptic operators {L_ε} in divergence form with rapidly oscillating and periodic coefficients in Lipschitz and convex domains in R~n. We are able to show that the uniform W~(1,p) estimate of second order elliptic systems holds for 2n/(n+1)-δ p 2n/(n-1)+ δ where δ 0 is independent of ε and the ranges are sharp for n = 2, 3. And for elliptic equations in Lipschitz domains, the W~(1,p) estimate is true for 3/2-δ p 3 + δ if n ≥ 4, similar estimate was extended to convex domains for 1 p ∞.     

A note on the existence of fractional f-factors in random graphs

Let G=Gn,p be a binomial random graph with n vertices and edge probability p=p(n),and f be a nonnegative integer-valued function defined on V(G) such that 0a≤f(x)≤bnp-2np ㏒n for every x ∈V(G). An fractional f-indicator function is an function h that assigns to each edge of a graph G a number h(e) in [0,1] so that for each vertex x,we have dh G(x)=f(x),where dh G(x) = x∈e h(e) is the fractional degree of x in G. Set Eh = {e:e ∈E(G) and h(e)=0}.If Gh is a spanning subgraph of G such that E(Gh)=Eh,then Gh is called an fractional f-factor of G. In this paper,we prove that for any binomial random graph Gn,p with p≥n-23,almost surely Gn,p contains an fractional f-factor.