This paper studies the Hardy-type inequalities on the intervals (may be infinite) with two weights, either vanishing at two endpoints of the interval or having mean zero. For the first type of inequalities, in terms of new isoperimetric constants, the factor of upper and lower bounds becomes smaller than the known ones. The second type of the inequalities is motivated from probability theory and is new in the analytic context. The proofs are now rather elementary. Similar improvements are made for Nash inequality, Sobolev-type inequality, and the logarithmic Sobolev inequality on the intervals.

In this paper, we prove the interior regularity for the solution to the Abreu equation in any dimension assuming the existence of the C⁰ estimate.

Though the theory of Triebel-Lizorkin and Besov spaces in one-parameter has been developed satisfactorily, not so much has been done for the multiparameter counterpart of such a theory. In this paper, we introduce the weighted Triebel-Lizorkin and Besov spaces with an arbitrary number of parameters and prove the boundedness of singular integral operators on these spaces using discrete Littlewood-Paley theory and Calderón’s identity. This is inspired by the work of discrete Littlewood- Paley analysis with two parameters of implicit dilations associated with the flag singular integrals recently developed by Han and Lu [12]. Our approach of derivation of the boundedness of singular integrals on these spaces is substantially different from those used in the literature where atomic decomposition on the one-parameter Triebel-Lizorkin and Besov spaces played a crucial role. The discrete Littlewood-Paley analysis allows us to avoid using the atomic decomposition or deep Journe’s covering lemma in multiparameter setting.

For a simple digraph G, let β(G) be the size of the smallest subset X  E(G) such that G?X has no directed cycles, and let γ(G) be the number of unordered pairs of nonadjacent vertices in G. A digraph G is called k-free if G has no directed cycles of length at most k. This paper proves that β(G) ≤ 0.3819γ(G) if G is a 4-free digraph, and β(G) ≤ 0.2679γ(G) if G is a 5-free digraph. These improve the results of Sullivan in 2008.

In this paper, we consider the real interpolation with a function parameter between martingale Hardy and BMO spaces. An interpolation theorem for martingale Hardy and BMO spaces is formulated. As an application, real interpolation between martingale Lorentz and BMO spaces is given.

R-duals of certain sequences in Hilbert spaces were introduced by Casazza, Kutyniok and Lammers in 2004 and later generalized to Banach spaces by Xiao and Zhu. In this paper we provide some characterizations of R-dual sequences in Banach spaces.

Let M be a closed smooth manifold M, and let f:M→M be a diffeomorphism. In this paper, we consider a nontrivial transitive set Λ of f. We show that if f has the C¹-stably average shadowing property on Λ, then Λ admits a dominated splitting.

This contribution gives results on the action of the Laplace-Beltrami derivative on sufficiently smooth kernels on the sphere, those defined by absolutely and uniformly expansions generated by a family of at least continuous functions. Among other things, the results show that convenient Laplace-Beltrami derivatives of positive definite kernels on the sphere are positive definite too. We also include similar results on the action of the Laplace-Beltrami derivative on condensed spherical harmonic expansions.

We introduce the quasi-Poisson enveloping algebra and Poisson enveloping algebra for a non-commutative Poisson algebra. We prove that for a non-commutative Poisson algebra, the category of quasi-Poisson modules is equivalent to the category of left modules over its quasi-Poisson enveloping algebra, and the category of Poisson modules is equivalent to the category of left modules over its Poisson enveloping algebra.

In this paper, we present a decomposition method of multivariate functions. This method shows that any multivariate function f on [0, 1]^d is a finite sum of the form Σ_jΦ_jΨ_j, where each Φ_j can be extended to a smooth periodic function, each Ψ_j is an algebraic polynomial, and each Φ_jΨ_j is a product of separated variable type and its smoothness is same as f. Since any smooth periodic function can be approximated well by trigonometric polynomials, using our decomposition method, we find that any smooth multivariate function on [0, 1]^d can be approximated well by a combination of algebraic polynomials and trigonometric polynomials. Meanwhile, we give a precise estimate of the approximation error.

In this paper we get the sharp estimates of the p-adic Hardy and Hardy-Littlewood-Pólya operators on L^q(|x|_p^α dx). Also, we prove that the commutators generated by the p-adic Hardy operators (Hardy-Littlewood-Pólya operators) and the central BMO functions are bounded on L^q(|x|_p^α dx), more generally, on Herz spaces.

In this paper, we mainly discuss the normality of two families of functions concerning shared values and proved: Let F and G be two families of functions meromorphic on a domain D ⊂ C, a₁, a₂, a₃, a₄ be four distinct finite complex numbers. If G is normal, and for every f ∈ F, there exists g ∈ G such that f(z) and g(z) share the values a₁, a₂, a₃, a₄, then F is normal on D.

We study a new trust region affine scaling method for general bound constrained optimization problems. At each iteration, we compute two trial steps. We compute one along some direction obtained by solving an appropriate quadratic model in an ellipsoidal region. This region is defined by an affine scaling technique. It depends on both the distances of current iterate to boundaries and the trust region radius. For convergence and avoiding iterations trapped around nonstationary points, an auxiliary step is defined along some newly defined approximate projected gradient. By choosing the one which achieves more reduction of the quadratic model from the two above steps as the trial step to generate next iterate, we prove that the iterates generated by the new algorithm are not bounded away from stationary points. And also assuming that the second-order sufficient condition holds at some nondegenerate stationary point, we prove the Q-linear convergence of the objective function values. Preliminary numerical experience for problems with bound constraints from the CUTEr collection is also reported.

The homogeneous approximation property (HAP) states that the number of building blocks involved in a reconstruction of a function up to some error is essentially invariant under time-scale shifts. In this paper, we show that every wavelet frame with nice wavelet function and arbitrary expansive dilation matrix possesses the HAP. Our results improve some known ones.

In this paper, we prove that some volume-preserving almost Anosov systems are ergodic if they are essentially accessible. The key idea is that there are stable and unstable manifolds with uniform size on the orbits of the hyperbolic points for these systems.

The small-world network, proposed by Watts and Strogatz, has been extensively studied for the past over ten years. In this paper, a generalized small-world network is proposed, which extends several small-world network models. Furthermore, some properties of a special type of generalized small-world network with given expectation of edge numbers have been investigated, such as the degree distribution and the isoperimetric number. These results are used to present a lower and an upper bounds for the clustering coefficient and the diameter of the given edge number expectation generalized small-world network, respectively. In other words, we prove mathematically that the given edge number expectation generalized small-world network possesses large clustering coefficient and small diameter.

Let U be a quantized enveloping algebra and   its modified form. Lusztig gives some symmetries on U and . In view of the realization of U by the reduced Drinfeld double of the Ringel-Hall algebra, one can apply the BGP-reflection functors to the double Ringel-Hall algebra to obtain Lusztig's symmetries on U and their important properties, for instance, the braid relations. In this paper, we define a modified form  of the Ringel-Hall algebra and realize the Lusztig's symmetries on  by applying the BGP-reflection functors to .

In this paper, we establish the boundedness of parameterized Littlewood-Paley operator μ_λ^*,ρ and parameterized area integral μ_Ω,S^ρ with kernel satisfying the logarithmic type Lipschitz condition on the weak Hardy space.

Let X, Y be Banach spaces, R:X → Y and S:Y → X be bounded linear operators. When λ≠0, we investigate common properties of λI-SR and λI-RS. This work should be viewed as a continuation of researches of Barnes and Lin et al.. We also apply these results obtained to B-Fredholm theory, extensions and Aluthge transforms.

This paper aims at solving a multidimensional backward stochastic differential equation (BSDE) whose generator g satisfies a weak monotonicity condition and a general growth condition in y. We first establish an existence and uniqueness result of solutions for this kind of BSDEs by using systematically the technique of the priori estimation, the convolution approach, the iteration, the truncation and the Bihari inequality. Then, we overview some assumptions related closely to the monotonicity condition in the literature and compare them in an effective way, which yields that our existence and uniqueness result really and truly unifies the Mao condition in y and the monotonicity condition with the general growth condition in y, and it generalizes some known results. Finally, we prove a stability theorem and a comparison theorem for this kind of BSDEs, which also improves some known results.

We consider a Markov chain X={X_i, i=1, 2,...} with the state space {0, 1}, and define W=∑_i=1ⁿX_iX_i+1, which is the number of 2-runs in X before time n+1. In this paper, we prove that the negative binomial distribution is an appropriate approximation for LW when VarW is greater than EW. The error estimate obtained herein improves the corresponding result in previous literatures.

Let 0 ≤ α < n, Ω be a rough kernel, and let A have derivatives of order m-1 in CMO^q,μ₂ with m ≥ 2. We consider a class of generalized commutators T_Ω,α^A of Cohen-Gosselin type, and obtain the boundedness of T_Ω,α^A from the central Morrey spaces E^p,μ₁ to E^r,λ for λ=μ₁+μ₂+α/n and 1/r=1/p+1/q-α/n.

For a non-Gaussian Lévy model, it is shown that if the model exists a trivial arbitrage-free interval, option pricing by mean correcting method is always arbitrage-free, and if the arbitrage-free interval is non-trivial, this pricing method may lead to arbitrage in some cases. In the latter case, some necessary and sufficient conditions under which option price is arbitrage-free are obtained.

This paper discusses the asymptotic behaviors of the longest run on a countable state Markov chain. Let {X_a}_a∈Z+ be a stationary strongly ergodic reversible Markov chain on countablestate space S={1, 2,...}. Let TS be an arbitrary finite subset of S. Denote by Ln the length of the longest run of consecutive i's for i ∈ T, that occurs in the sequence X₁,...,X_n. In this paper, we obtain a limit law and a week version of an Erdös-Rényi type law for L_n. A large deviation result of L_n is also discussed.

Let P be a transition matrix which is symmetric with respect to a measure π. The spectral gap of P in L²(π)-space, denoted by gap(P), is defined as the distance between 1 and the rest of the spectrum of P. In this paper, we study the relationship between gap(P) and the convergence rate of Pⁿ. When P is transient, the convergence rate of Pⁿ is equal to 1-gap(P). When P is ergodic, we give the explicit upper and lower bounds for the convergence rate of Pⁿ in terms of gap(P). These results are extended to L^∞(π)-space.

Let λ_f (n) be the n-th normalized Fourier coefficient of a holomorphic Hecke eigenform f(z) ∈S_k(Γ). In this paper, we established nontrivial estimates for  where 1 ≤ i < j ≤ 4.

In this paper, we generalize the construction of the inverse transgression map done by Adem, A., Ruan, Y. and Zhang, B. in [A stringy product on twisted orbifold K-theory. Morfismos, 11, 33-64 (2007)] and give a different proof to the statement that the image of the inverse transgression map for a gerbe with connection over an orbifold is an inner local system on its inertia orbifold. Keywords Gerbe, orbifold, Lie groupoid, inner local system, holonomy

The purpose of this paper is to investigate the growth of meromorphic functions concerning Picard values with a radially distributed value. This generalizes a classic result due to Hayman [Hayman, W. K.: Picard values of meromorphic functions and their derivatives. Ann. of Math., 70, 9-42 (1959)].

It is proved that the Auslander-Reiten conjecture is true for local Artin algebras with radical cube zero.

Let f_k(n) be the characteristic function of n with Ω(n) = k, and  The main purpose of this paper is to establish a Bombieri-type mean-value theorem for T_k(x, α), via using the modified Huxley-Hooley contour and the large-sieve type zero-density estimate for Dirichlet L-functions. The result plays an important role in handling the enlarge major arcs when we try to solve the Waring-Goldbach type problem by the circle method.

In this paper, we introduce a type of approximation operators of neural networks with sigmodal functions on compact intervals, and obtain the pointwise and uniform estimates of the approximation. To improve the approximation rate, we further introduce a type of combinations of neural networks. Moreover, we show that the derivatives of functions can also be simultaneously approximated by the derivatives of the combinations. We also apply our method to construct approximation operators of neural networks with sigmodal functions on infinite intervals.

Let (Ω, Σ, μ) be a complete probability space and let X be a Banach space. We introduce the notion of scalar equi-convergence in measure which being applied to sequences of Pettis integrable functions generates a new convergence theorem. We also obtain a Vitali type I-convergence theorem for Pettis integrals where I is an ideal on N.

Let X, Y be two real Banach spaces and ε ≥ 0. A map f: X→Y is said to be a standard ε-isometry if |||f(x)-f(y) - x - y||| ≤ ε for all x, y ∈ X and with f(0) = 0. We say that a pair of Banach spaces (X, Y) is stable if there exists γ > 0 such that, for every such ε and every standard ε-isometry f : X → Y , there is a bounded linear operator T : L(f) ≡ span f(X) → X so that Tf(x)-x ≤ γε for all x ∈ X. X(Y) is said to be universally left-stable if (X, Y) is always stable for every Y (X). In this paper, we show that if a dual Banach space X is universally left-stable, then it is isometric to a complemented w^*-closed subspace of l_∞(Γ) for some set Γ, hence, an injective space; and that a Banach space is universally left-stable if and only if it is a cardinality injective space; and universally left-stability spaces are invariant.

We prove that for any bounded type irrational number 0 < θ < 1, the boundary of the Siegel disk of f_α(z) = e^2πiθ sin(z) + α sin³(z), α ∈ C, which centered at the origin, is a quasicircle passing through 2, 4 or 6 critical points of f_α counted with multiplicity.

Let Λ be a row-finite k-graph without sources. We investigate the relationship between the complex Kumjian-Pask algebra KPC(Λ) and graph algebra C^*(Λ). We identify situations in which the Kumjian-Pask algebra is equal to the graph algebra, and the conditions in which the Kumjian-Pask algebra is finite-dimensional.

In this paper, we prove that 2-degenerate graphs and some planar graphs without adjacent short cycles are group (Δ(G)+1)-edge-choosable, and some planar graphs with large girth and maximum degree are group Δ(G)-edge-choosable.

In this paper, the problem of the global L² stability for large solutions to the nonhomogeneous incompressible Navier-Stokes equations in 3D bounded or unbounded domains is studied. By delicate energy estimates and under the suitable condition of the large solutions, it shows that if the initial data are small perturbation on those of the known strong solutions, the large solutions are stable.

In this paper, we classified the finite p-groups with exactly one A₁-subgroup of given structure of order p³.

In this paper, we give some decompositions of triples of Z_pⁿ or Z_3pⁿ into cyclic triple systems. New constructions of difference families are given. Some infinite classes of simple cyclic triple systems are obtained from these decompositions.

An edge coloring total k-labeling is a labeling of the vertices and the edges of a graph G with labels {1, 2, ..., k} such that the weights of the edges define a proper edge coloring of G. Here the weight of an edge is the sum of its label and the labels of its two end vertices. This concept was introduce by Brandt et al. They defined χ_t'(G) to be the smallest integer k for which G has an edge coloring total k-labeling and proposed a question: Is there a constant K with χ_t'(G) ≤ [Δ(G)+1]/2 +K for all graphs G of maximum degree Δ(G)? In this paper, we give a positive answer for outerplanar graphs by showing that χ_t'(G) ≤ [(Δ(G)+1)/2] + 1 for each outerplanar graph G with maximum degree Δ(G).