Showing papers in "Acta Mathematica Sinica in 2015"

Functionals for multilinear fractional embedding

Abstract: 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.

The Fractional Metric Dimension of Permutation Graphs

Abstract: Let G = (V (G),E(G)) be a graph with vertex set V (G) and edge set E(G). For two distinct vertices x and y of a graph G, let R G {x, y} denote the set of vertices z such that the distance from x to z is not equal to the distance from y to z in G. For a function g defined on V (G) and for U ⊆ V (G), let g(U) = Σ s∈ U g(s). A real-valued function g: V (G) → [0, 1] is a resolving function of G if g(R G {x, y}) ≥ 1 for any two distinct vertices x, y ∈ V (G). The fractional metric dimension dim f (G) of a graph G is min{g(V (G)): g is a resolving function of G}. Let G 1 and G 2 be disjoint copies of a graph G, and let σ: V (G 1) → V (G 2) be a bijection. Then, a permutation graph G σ = (V, E) has the vertex set V = V (G 1) ∪ V (G 2) and the edge set E = E(G 1) ∪ E(G 2) ∪ {uv | v = σ(u)}. First, we determine dimf (T) for any tree T. We show that $1 < \dim _f (G_\sigma ) \leqslant \tfrac{1} {2}(|V(G)| + |S(G)|) $ for any connected graph G of order at least 3, where S(G) denotes the set of support vertices of G. We also show that, for any ɛ > 0, there exists a permutation graph G σ such that dim f (G σ) - 1 < e. We give examples showing that neither is there a function h 1 such that dim f (G) < h 1(dim f (G σ)) for all pairs (G, σ), nor is there a function h 2 such that h 2(dim f (G)) > dim f (G σ)) for all pairs (G, σ). Furthermore, we investigate dim f (G σ)) when G is a complete k-partite graph or a cycle.

Common properties of the operator products in local spectral theory

Abstract: Let X, Y be Banach spaces, A,D: X → Y and B,C: Y → X be the bounded linear operators satisfying operator equation set $$\left\{ \begin{gathered} ACD = DBD, \hfill \\ DBA = ACA. \hfill \\ \end{gathered} \right. $$ . 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).

On generalized Douglas–Weyl ( α , β )-metrics

Abstract: 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.

Optimal D-RIP bounds in compressed sensing

Abstract: This paper establishes new bounds on the restricted isometry constants with coherent tight frames in compressed sensing. It is shown that if the sensing matrix A satisfies the D-RIP condition δk < 1/3 or \(\delta _{2k} < \sqrt 2 /2\), then all signals f with D*f are k-sparse can be recovered exactly via the constrained l1 minimization based on y = Af, where D* is the conjugate transpose of a tight frame D. These bounds are sharp when D is an identity matrix, see Cai and Zhang’s work. These bounds are greatly improved comparing to the condition δk < 0.307 or δ2k < 0.4931. Besides, if δk < 1/3 or \(\delta _{2k} < \sqrt 2 /2\), the signals can also be stably reconstructed in the noisy cases.

Inferences in linear mixed models with skew-normal random effects

Abstract: For the linear mixed model with skew-normal random effects, this paper gives the density function, moment generating function and independence conditions. The noncentral skew chi-square distribution is defined and its density function is shown. The necessary and sufficient conditions under which a quadratic form is distributed as noncentral skew chi-square distribution are obtained. Also, a version of Cochran’s theorem is given, which modifies the result of Wang et al. (2009) and is used to set up exact tests for fixed effects and variance components of the proposed model. For illustration, our main results are applied to a real data problem.

On support τ-tilting modules over endomorphism algebras of rigid objects

Abstract: We consider a Krull–Schmidt, Hom-finite, 2-Calabi–Yau triangulated category with a basic rigid object T, and show a bijection between the set of isomorphism classes of basic rigid objects in the finite presented category pr T of T and the set of isomorphism classes of basic τ-rigid pairs in the module category of the endomorphism algebra Endc(T) op . As a consequence, basic maximal objects in pr T are one-to-one correspondence to basic support τ-tilting modules over End c (T) op . This is a generalization of correspondences established by Adachi–Iyama–Reiten.

On the structure of split Leibniz triple systems

Abstract: 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.

Entropy and renormalized solutions for nonlinear elliptic problem involving variable exponent and measure data

Abstract: 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: $$- div(a(x,u, abla u) + \varphi (u)) + g(x,u, abla u) = \mu ,$$ where the right-hand side belongs to L 1(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 ∈ C 0(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.

Busemann-Petty problems for general L p -intersection bodies

Abstract: For 0 < p < 1, Haberl and Ludwig defined the notions of symmetric Lp-intersection body and nonsymmetric Lp-intersection body. In this paper, we introduce the general Lp-intersection bodies. Furthermore, the Busemann-Petty problems for the general Lp-intersection bodies are shown.

Multiple weighted estimates for commutators of multilinear maximal function

Abstract: Let M be the multilinear maximal function and $\vec b$ = (b 1,, b m ) be a collection of locally integrable functions Denote by M $\vec b$ and $\vec b$ , M] the maximal commutator and the commutator of M with $\vec b$ , respectively In this paper, the multiple weighted strong and weak type estimates for operators M $\vec b$ and [ $\vec b$ , M] are studied Some characterizations of the class of functions b j are given, for which these operators satisfy some strong or weak type estimates

Neighbor sum distinguishing total colorings of triangle free planar graphs

Abstract: A total k-coloring c of a graph G is a proper total coloring c of G using colors of the set [k] = {1, 2, ..., k}. Let f(u) denote the sum of the color on a vertex u and colors on all the edges incident to u. A k-neighbor sum distinguishing total coloring of G is a total k-coloring of G such that for each edge uv ∈ E(G), f(u) ≠ f(v). By χ″nsd(G), we denote the smallest value k in such a coloring of G. Pilśniak and Woźniak conjectured that χ″nsd(G) ≤ Δ(G)+3 for any simple graph with maximum degree Δ(G). In this paper, by using the famous Combinatorial Nullstellensatz, we prove that the conjecture holds for any triangle free planar graph with maximum degree at least 7.

Multi-dimensional central limit theorems and laws of large numbers under sublinear expectations

Abstract: In this paper, we present some multi-dimensional central limit theorems and laws of large numbers under sublinear expectations, which extend some previous results.

Hausdorff operators on the Heisenberg group

Abstract: This paper is devoted to the high-dimensional and multilinear Hausdorff operators on the Heisenberg group H n . The sharp bounds for the strong type (p, p) (1 ≤ p ≤ ∞) estimates of n-dimensional Hausdorff operators on H n 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 H n . The weak type (p, p) (1 ≤ p≤∞) estimates are also obtained. Keywords Hausdorff operator, Heisenberg group, multilinear, sharp estimate

On the deformation of Lie-Yamaguti algebras

Abstract: The deformation theory of Lie-Yamaguti algebras is developed by choosing a suitable cohomology. The relationship between the deformation and the obstruction of Lie-Yamaguti algebras is obtained.

On gradient solitons of the Ricci-Harmonic flow

Abstract: 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.

Nonsolvable D 2 -groups

Abstract: 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.

The optimal constant in Hardy-type inequalities

Abstract: To estimate the optimal constant in Hardy-type inequalities, some variational formulas and approximating procedures are introduced. The known basic estimates are improved considerably. The results are illustrated by typical examples. It is shown that the sharp factor is meaningful for each finite interval and a classical sharp model is re-examined.

Variable selection for fixed effects varying coefficient models

Abstract: 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.

A remark on Jeśmanowicz’ conjecture for the non-coprimality case

Abstract: Let a, b, c be relatively prime positive integers such that a 2 + b 2 = c 2. Jeśmanowicz’ conjecture on Pythagorean numbers states that for any positive integer N, the Diophantine equation (aN) x + (bN) y = (cN) z has no positive solution (x, y, z) other than x = y = z = 2. In this paper, we prove this conjecture for the case that a or b is a power of 2.

A note on quasi-weakly almost periodic point

Abstract: Recently, He et al. [On quasi-weakly almost periodic points. Sci. China Math., 56, 597–606 (2013)] constructed two binary sub-shifts to solve an open problem posed by Zhou and Feng in [Twelve open problems on the exact value of the Hausdorff measure and on topological entropy: A brief survey of recent results. Nonlinearity, 17, 493–502 (2004)]. In this paper, we study more dynamical properties of those two binary sub-shifts. We show that the first one has zero topological entropy and is transitive but not weakly mixing, while the second one has positive topological entropy and is strongly mixing.

Gaussian fluctuations of eigenvalues in log-gas ensemble: Bulk case I

Abstract: We study the central limit theorem of the k-th eigenvalue of a random matrix in the log-gas ensemble with an external potential V = q2mx2m. More precisely, let Pn(dH) = Cne-nTrV(H)dH be the distribution of n × n Hermitian random matrices, ρV (x)dx the equilibrium measure, where Cn is a normalization constant, V (x) = q2mx2m with \(q2m = \frac{{\Gamma \left( m \right)\Gamma \left( {\frac{1}{2}} \right)}}{{\Gamma \left( {\frac{{2m + 1}}{2}} \right)}}\), and m ≥ 1. Let x1 ≤... ≤ xn be the eigenvalues of H. Let k:= k(n) be such that \(\frac{{k\left( n \right)}}{n} \in \left[ {a,1 - a} \right]\) for n large enough, where a ∈ (0, 1/2). Define \(G\left( s \right): = \int_{ - 1}^s {\rho v\left( x \right)dx, - 1 \leqslant s \leqslant 1} ,\) and set t:= G−1(k/n). We prove that, as n → ∞, \(\frac{{xk - t}}{{\frac{{\left( {\sqrt {\log n} } \right)}}{{\sqrt {2{\pi ^2}} n\rho v\left( t \right)}}}} \to N\left( {0,1} \right)\) in distribution. Multi-dimensional central limit theorem is also proved. Our results can be viewed as natural extensions of the bulk central limit theorems for GUE ensemble established by J. Gustavsson in 2005.

Isomorphisms of finite semi-Cayley graphs

Abstract: Let G be a finite group. A Cayley graph over G is a simple graph whose automorphism group has a regular subgroup isomorphic to G. A Cayley graph is called a CI-graph (Cayley isomorphism) if its isomorphic images are induced by automorphisms of G. A well-known result of Babai states that a Cayley graph Γ of G is a CI-graph if and only if all regular subgroups of Aut(Γ) isomorphic to G are conjugate in Aut(Γ). A semi-Cayley graph (also called bi-Cayley graph by some authors) over G is a simple graph whose automorphism group has a semiregular subgroup isomorphic to G with two orbits (of equal size). In this paper, we introduce the concept of SCI-graph (semi-Cayley isomorphism) and prove a Babai type theorem for semi-Cayley graphs. We prove that every semi-Cayley graph of a finite group G is an SCI-graph if and only if G is cyclic of order 3. Also, we study the isomorphism problem of a special class of semi-Cayley graphs.

The isometric extension problem between unit spheres of two separable Banach spaces

Abstract: 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.

Polynomials with palindromic and unimodal coefficients

Abstract: Let f(q) = a r q r + ⋯ + a s q s , with a r ≠ 0 and a s ≠ 0, be a real polynomial. It is a palindromic polynomial of darga n if r + s = n and a r+i = a s−i for all i. Polynomials of darga n form a linear subspace $$\mathcal{P}_n (q)$$ of ℝ(q) n+1 of dimension $$\left\lfloor {\tfrac{n} {2}} \right\rfloor + 1$$ . We give transition matrices between two bases {q j (1 + q + ⋯ + q n−2j)}, {q j (1 + q) n−2j } and the standard basis {q j (1 + q n−2j)} of P n (q). We present some characterizations and sufficient conditions for palindromic polynomials that can be expressed in terms of these two bases with nonnegative coefficients. We also point out the link between such polynomials and rank-generating functions of posets.

Commuting structure jacobi operator for hopf hypersurfaces in complex two-plane grassmannians

Abstract: 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.

Measure of noncompactness and semilinear nonlocal functional differential equations in Banach spaces

Abstract: 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.

Nonpositively Curved Almost Hermitian Metrics on Product of Compact Almost Complex Manifolds

Abstract: 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)].

The study of minimal period estimates for brake orbits of autonomous subquadratic Hamiltonian systems

Abstract: 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 ∈ C 2(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.

Poincaré and Sobolev inequalities for vector fields satisfying Hörmander’s condition in variable exponent Sobolev spaces

Abstract: 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 Hormander’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 Hormander’s condition, but they also hold for Grushin vector fields as well with obvious modifications.