A novel representation is developed as a measure for multilinear fractional embedding. Corresponding extensions are given for the Bourgain-Brezis-Mironescu theorem and Pitt's inequality. New results are obtained for diagonal trace restriction on submanifolds as an application of the Hardy-Littlewood-Sobolev inequality. Smoothing estimates are used to provide new structural understanding for density functional theory, the Coulomb interaction energy and quantum mechanics of phase space. Intriguing connections are drawn that illustrate interplay among classical inequalities in Fourier analysis.

We announce a result on quantum McKay correspondence for disc invariants of outer legs in toric Calabi-Yau 3-orbifolds, and illustrate our method in a special example [C³/Z₅(1, 1, 3)].

In this paper, we completely characterize the compactness of Toeplitz operators with continuous symbol on the weighted Dirichlet space.

In the present paper, we prove the existence of global solutions for the Navier-Stokes equations in Rⁿ when the initial velocity belongs to the weighted weak Lorentz space Λ^n,∞(u) with a sufficiently small norm under certain restriction on the weight u. At the same time, self-similar solutions are induced if the initial velocity is, besides, a homogeneous function of degree -1. Also the uniqueness is discussed.

In this paper, we give a classification of almost Hermitian metrics with nonpositive holomorphic bisectional curvature on a product of compact almost complex manifolds. This generalizes previous results of Zheng [Ann. of Math. (2), 137(3), 671-673 (1993)] and the author [Proc. Amer. Math. Soc., 139(4), 1469-1472 (2011)].

A conditional log-Laplace functional (CLLF) for a class of branching processes in random environments is derived. The basic idea is the decomposition of a dependent branching dynamic into a no-interacting branching and an interacting dynamic generated by the random environments. CLLF will play an important role in the investigation of branching processes and superprocesses with interaction.

We consider the problem of variable selection for the fixed effects varying coefficient models. A variable selection procedure is developed using basis function approximations and group nonconcave penalized functions, and the fixed effects are removed using the proper weight matrices. The proposed procedure simultaneously removes the fixed individual effects, selects the significant variables and estimates the nonzero coefficient functions. With appropriate selection of the tuning parameters, an asymptotic theory for the resulting estimates is established under suitable conditions. Simulation studies are carried out to assess the performance of our proposed method, and a real data set is analyzed for further illustration.

We classify real hypersurfaces in complex two-plane Grassmannians whose structure Jacobi operator commutes either with any other Jacobi operator or with the normal Jacobi operator.

In this paper, we are concerned with the following three types of nonlinear degenerate parabolic equations with time-dependent singular potentials:
∂u^q/∂t=▽_α·(‖z‖^-pγ|▽_αu|^p-2▽_αu)+V(z, t)u^p-1,
∂u^q/∂t=▽_α·(‖z‖^-2γ▽_αu^m)+V(z, t)u^m,
∂u^q/∂t=u^μ▽_α·(u^τ|▽_αu|^p-2▽_αu)+V(z, t)u^p-1+μ+τ
in a cylinder Ω×(0, T) with initial condition u(z, 0)=u₀(z) ≥ 0 and vanishing on the boundary ∂Ω×(0, T), where Ω is a Carnot-Carathéodory metric ball in R^d+k and the time-dependent singular potential function is V(z, t) ∈ L_loc¹ (Ω×(0, T)). We investigate the nonexistence of positive solutions of these three problems and present our results on nonexistence.

This paper is concerned with the measure of noncompactness in the spaces of continuous functions and semilinear functional differential equations with nonlocal conditions in Banach spaces. The relationship between the Hausdorff measure of noncompactness of intersections and the modulus of equicontinuity is studied for some subsets related to the semigroup of linear operators in Banach spaces. The existence of mild solutions is obtained for a class of nonlocal semilinear functional differential equations without the assumption of compactness or equicontinuity on the associated semigroups of linear operators.

We give an existence result of entropy and renormalized solutions for strongly nonlinear elliptic equations in the framework of Sobolev spaces with variable exponents of the type:
-div (a(x, u,▽u)+φ(u))+g(x, u,▽u)=μ,
where the right-hand side belongs to L¹(Ω)+W^-1,p'(x)(Ω), -div(a(x, u,▽u)) is a Leray-Lions operator defined from W^-1,p'(x)(Ω) into its dual and φ∈C⁰(R,R^N). The function g(x, u,▽u) is a non linear lower order term with natural growth with respect to |▽u| satisfying the sign condition, that is, g(x, u,▽u)u ≥ 0.

In this paper, the cycle structures for directed graphs on surfaces are studied. If G is a strongly connected graph, C is a ∏-contractible directed cycle of G, then both of Int(C,∏) and Ext(C,∏) are strongly connected graph; the dimension of cycles space of G is identified. If G is a strongly connected graph, then the structure of MCB in G is unique. Let G be a strongly connected graph, if G has been embedded in orientable surface S_g with f_w(G) ≥ 2(f_w(G) is the face-width of G), then any cycle base of G must contain at least 2g noncontractible directed cycles; if G has been embedded in non-orientable surface N_g, then any cycle base of G must contain at least g noncontractible directed cycles.

In this paper, we study the structure theory of a class of not-finitely graded Lie algebras related to generalized Heisenberg-Virasoro algebras. In particular, the derivation algebras, the automorphism groups and the second cohomology groups of these Lie algebras are determined.

In this paper, continuing with Hu-Li-Vrancken and the recent work of Anti?-Dillen-Schoels-Vrancken, we obtain a decomposition theorem which settled the problem of how to determine whether a given locally strongly convex affine hypersurface can be decomposed as a generalized Calabi composition of two affine hyperspheres, based on the properties of its difference tensor K and its affine shape operator S.

A weighted graph is one in which every edge e is assigned a nonnegative number, called the weight of e. The sum of the weights of the edges incident with a vertex v is called the weighted degree of v, denoted by d^w(v). The weight of a cycle is defined as the sum of the weights of its edges. Fujisawa proved that if G is a 2-connected triangle-free weighted graph such that the minimum weighted degree of G is at least d, then G contains a cycle of weight at least 2d. In this paper, we proved that if G is a 2-connected triangle-free weighted graph of even size such that d^w(u) + d^w(v) ≥ 2d holds for any pair of nonadjacent vertices u, v ∈ V (G), then G contains a cycle of weight at least 2d.

A proper edge coloring of a graph G is acyclic if there is no 2-colored cycle in G. The acyclic chromatic index of G, denoted by χ'_a(G), is the least number of colors such that G has an acyclic edge coloring. A graph is 1-planar if it can be drawn on the plane so that each edge is crossed by at most one other edge. In this paper, it is proved that χ'_a(G) ≤ Δ(G)+22, if G is a triangle-free 1-planar graph.

We consider Leray's problem on stationary Navier-Stokes flows with arbitrary large fluxes in an unbounded cylinder with several exits to infinity. For a stationary Navier-Stokes flow with large fluxes in the unbounded cylinder, we prove that, if the difference between the pressure of the main flow and the pressure of the Poiseuille flow with the same flux in a branch of the cylinder remains bounded at |x| → ∞, then the flow behaves at infinity of the branch like the Poiseuille flow.

We study the long time behavior of J-flows on toric manifolds. By introducing the transition maps between moment maps, we get a quasilinear parabolic system for J-flows. Some basic estimates for transition maps are obtained.

Let G be a finite centerless group, let π(G) be the set of prime divisors of the order of G, and let n_p(G) be the number of Sylow p-subgroups of G, that is, n_p(G) = |Sylp(G)|. Set NS(G) := {n_p(G)|p ∈ π(G)}. In this paper, we are investigating whether L₂(r) is determined up to isomorphism by NS(L₂(r)) when r is prime.

In this paper, we study the existence of positive periodic solutions for singular second order equations x" + n²/4x+h(x) = p(t), where h has a singularity at the origin and n is a positive integer. We give an explicit condition to ensure the existence of positive periodic solutions when h is an unbounded perturbation at infinity by using qualitative analysis and topological degree theory.

In this paper, we study generalized Douglas-Weyl (α, β)-metrics. Suppose that a regular (α, β)-metric F is not of Randers type. We prove that F is a generalized Douglas-Weyl metric with vanishing S-curvature if and only if it is a Berwald metric. Moreover, by ignoring the regularity, if F is not a Berwald metric, then we find a family of almost regular Finsler metrics which is not Douglas nor Weyl. As its application, we show that generalized Douglas-Weyl square metric or Matsumoto metric with isotropic mean Berwald curvature are Berwald metrics.

Let F and G be closed subspaces of the complex Hilbert space H, and U and V be closed subspaces of F and G, respectively. In this paper, using the technique of operator block, we present the necessary and sufficient conditions under which (U, V) is a pair of (strictly, non-degenerate) principal invariant subspaces for (F, G).

We study the structure of arbitrary split Leibniz triple systems with a coherent 0-root space. By developing techniques of connections of roots for this kind of triple systems, under certain conditions, in the case of T being of maximal length, the simplicity of the Leibniz triple systems is characterized.

In this paper, we consider the minimal period estimates for brake orbits of autonomous subquadratic Hamiltonian systems. We prove that if the Hamiltonian function H ∈ C²(R²ⁿ,R) is unbounded and not uniformly coercive, there exists at least one nonconstant T-periodic brake orbit (z, T) with minimal period T or T/2 for every number T > 0.

We classify the stable homotopy types of (n-1)-connected, (n+k)-dimensional polyhedra with 2 and 3-torsion free homologies for k ≤ 6. The technique used is matrix problem (bimodule categories) which is given by Drozd.

Let G be a finite group. Let Irr₁(G) be the set of nonlinear irreducible characters of G and cd₁(G) the set of degrees of the characters in Irr₁(G). A group G is said to be a D₂-group if |cd₁(G)| = |Irr₁(G)| - 2. The main purpose of this paper is to classify nonsolvable D₂-groups.

This paper is devoted to the high-dimensional and multilinear Hausdorff operators on the Heisenberg group Hⁿ. The sharp bounds for the strong type (p, p) (1 ≤ p ≤ ∞) estimates of n-dimensional Hausdorff operators on Hⁿ are obtained. The sharp bounds for strong (p, p) estimates are further extended to multilinear cases. As an application, we derive the sharp constant for the multilinear Hardy operator on Hⁿ. The weak type (p, p) (1 ≤ p ≤ ∞) estimates are also obtained.

Let X, Y be Banach spaces, A,D : X → Y and B,C : Y → X be the bounded linear operators satisfying operator equation set

In this paper, we show that AC and BD share some basic operator properties such as the injectivity and the invertibility. Moreover, we show that AC and BD share many common local spectral properties including SVEP, Bishop property (β) and Dunford property (C).

In this paper, we study the nonperiodic first-order Hamiltonian system ?= JL(t)u + JH' (t, u), where H ∈ C¹(R × R²ⁿ). With some assumptions on L, the corresponding Hamiltonian operator has only discrete spectrum. By using the index theory for self-adjoint operator equation, we establish the existence of multiple homoclinic orbits for the asymptotically quadratic nonlinearty satisfying some twist conditions between infinity and origin.

It is a well-known fact that characters of a finite group can give important information about the group's structure. Also it was proved by the third author of this article that a finite simple group can be uniquely determined by its character table. Here the authors attempt to investigate how to characterize a finite almost simple group by using less information of its character table, and successfully characterize the almost simple K₃-groups by their orders and at most three irreducible character degrees of their character tables.

Let x : M → Q₁ⁿ⁺¹ be a regular hypersurface in the conformal space Q₁ⁿ⁺¹. We classify all the space-like Blaschke isoparametric hypersurfaces with two distinct Blaschke eigenvalues in the conformal space up to the conformal equivalence.

In this paper, we study fractional square functions associated with the Poisson semigroup for Schrödinger operators. We characterize the potential spaces in the Schrödinger setting by using vertical, area and g_λ^* fractional square functions.

In this article, we put forward the concept of the (i, j)-type L^p-mixed affine surface area, such that the notion of L^p-affine surface area which be shown by Lutwak is its special cases. Furthermore, applying this concept, the Minkowski inequality for the (i,-p)-type L^p-mixed affine surface area and the extensions of the well-known L^p-Petty affine projection inequality are established, respectively. Besides, we give an affirmative answer for the generalized L^p-Winterniz monotonicity problem.

This paper characterizes a class of normal weighted composition operators and their spectrum on the Fock space of Cⁿ.

In this paper, we study gradient solitons to the Ricci flow coupled with harmonic map heat flow. We derive new identities on solitons similar to those on gradient solitons of the Ricci flow. When the soliton is compact, we get a classification result. We also discuss the relation with quasi-Einstein manifolds.

In this paper, we study the number of limit cycles of a near-Hamiltonian system having Z₄- equivariant quintic perturbations. Using the methods of Hopf and heteroclinic bifurcation theory, we find that the perturbed system can have 28 limit cycles, and its location is also given. The main result can be used to improve the lower bound of the maximal number of limit cycles for some polynomial systems in a previous work, which is the main motivation of the present paper.

Suppose A is a unital C^*-algebra and r >1. In this paper, we define a unital C^*-algebra C_cb^*(A, r) and a completely bounded unital homomorphism α_r: A → C_cb^* (A, r) with the property that C_cb^*(A, r) = C^*(α_r(A)) and, for every unital C^*-algebra B and every unital completely bounded homomorphism φ: A → B, there is a (unique) unital *-homomorphism π: C_cb^* (A, r) → B such that φ = π ○ α_r. We prove that, if A is generated by a normal set {t_λ: λ ∈ Λ}, then C_cb^* (A, r) is generated by the set {α_r(t_λ): λ ∈ Λ}. By proving an equation of the norms of elements in a dense subset of C_cb^*(A, r) we obtain that, if B is a unital C^*-algebra that can be embedded into A, then C_cb^*(B, r) can be naturally embedded into C_cb^*(A, r). We give characterizations of C_cb^*(A, r) for some special situations and we conclude that C_cb^*(A, r) will be "nice" when dim(A) ≤ 2 and "quite complicated" when dim(A) ≥ 3. We give a characterization of the relation between K-groups of A and K-groups of C_cb^*(A, r). We also define and study some analogous of C_cb^* (A, r).

In this paper, we obtain the existence, uniqueness and asymptotic behavior of steady states to a class of Schrödinger-Poisson-Slater System.

This paper studies the problem of split convex feasibility and a strong convergent alternating algorithm is established. According to this algorithm, some strong convergent theorems are obtained and an affirmative answer to the question raised by Moudafi is given. At the same time, this paper also generalizes the problem of split convex feasibility.

In this article, we use some analytic and geometric characters of the smooth points in a sphere to study the isometric extension problem in the separable or reflexive real Banach spaces. We obtain that under some condition the answer to this problem is affirmative.