The augmented Lagrangian method is a classical method for solving constrained optimization. Recently, the augmented Lagrangian method attracts much attention due to its applications to sparse optimization in compressive sensing and low rank matrix optimization problems. However, most Lagrangian methods use first order information to update the Lagrange multipliers, which lead to only linear convergence. In this paper, we study an update technique based on second order information and prove that superlinear convergence can be obtained. Theoretical properties of the update formula are given and some implementation issues regarding the new update are also discussed.

A pseudo-real Riemann surface admits anticonformal automorphisms but no anticonformal involution. We obtain the classification of actions and groups of automorphisms of pseudo-real Riemann surfaces of genera 2, 3 and 4. For instance the automorphism group of a pseudo-real Riemann surface of genus 4 is either C₄ or C₈ or the Fröbenius group of order 20, and in the case of C₄ there are exactly two possible topological actions. Let M_PR,g^K be the set of surfaces in the moduli space M_g^K corresponding to pseudo-real Riemann surfaces. We obtain the equisymmetric stratification of M_PR,g^K for genera g = 2, 3, 4, and as a consequence we have that M_PR,g^K is connected for g = 2, 3 but M_PR,4^K has three connected components.

We prove a fluctuating limit theorem of a sequence of super-stable processes over R with a single point catalyst. The weak convergence of the processes on the space of Schwartz distributions is established. The limiting process is an Ornstein-Uhlenbeck type process solving a Langevin type equation driven by a one-dimensional stable process.

A real continuous function which is defined on an interval is said to be A-convex if it is convex on the set of self-adjoint elements, with spectra in the interval, in all matrix algebras of the unital C^*-algebra A. We give a general formation of Jensen's inequality for A-convex functions.

We investigate the asymptotic behavior of the entropy numbers of Besov classes BB_p,θ^Ω (S^d-1) of generalized smoothness on the sphere in L_q(S^d-1) for 1 ≤ p, q, θ ≤ ∞, and get their asymptotic orders. We also obtain the exact orders of entropy numbers of Sobolev classes BW_pr (S^d-1) in L_q(S^d-1) when p and/or q is equal to 1 or ∞. This provides the last piece as far as exact orders of entropy numbers of BW_pr (S^d-1) in L_q(S^d-1) are concerned.

By virtue of Cauchy's integral formula in the theory of complex functions, the authors establish an integral representation for the weighted geometric mean, apply this newly established integral representation to show that the weighted geometric mean is a complete Bernstein function, and find a new proof of the well-known weighted arithmetic-geometric mean inequality.

We establish a central limit theorem for a branching Brownian motion with random immigration under the annealed law, where the immigration is determined by another branching Brownian motion. The limit is a Gaussian random measure and the normalization is t3/4 for d = 3 and t1/2 for d ≥ 4, where in the critical dimension d = 4 both the immigration and the branching Brownian motion itself make contributions to the covariance of the limit.

The study of hyperbolic graphs is an interesting topic since the hyperbolicity of a geodesic metric space is equivalent to the hyperbolicity of a graph related to it. The main result in this paper is a very simple characterization of the hyperbolicity of a large class of periodic planar graphs.

Let G be a planar graph with maximum degree Δ. In this paper, we prove that if any 4-cycle is not adjacent to an i-cycle for any i ∈ {3, 4} in G, then the list edge chromatic number χ_l'(G) = Δ and the list total chromatic number χ_l" (G) = Δ+1.

We compute the index of the real Cauchy-Riemann operator defined in FJRW theory in case of the smooth metric. For the cylindrical metric, we study the relation between the index of the linearized operator of Witten map and weights in weighted Sobolev spaces.

In the paper, we investigate the complete convergence and complete moment convergence for the maximal partial sum of martingale difference sequence. Especially, we get the Baum-Katz-type Theorem and Hsu-Robbins-type Theorem for martingale difference sequence. As an application, a strong law of large numbers for martingale difference sequence is obtained.

In this paper, we analyze the blow-up behavior of sequences {uk} satisfying the following conditions
?Δuk = |x|^2α_kV_ke^uk in Ω, (0.1)
where Ω ⊂⊂ R², V_k → V in C¹, |∇V_k| ≤ A, 0 < a ≤ V_k ≤ b, 0 ≤ α_k → α∈ (0,∞), and
∫_Ω |x|^2α_ke^ukdx ≤ Λ₁. (0.2)
Furthermore, we assume that there exists some q ∈ (1, 2) such that
r^q?2∫_Br(p) |∇uk|^{qdx ≤ Λ₂ (0.3)
for any B_r(p)} ⊆ Ω^. As a result, we give a new proof of the concentration-compactness theorem for the mean field equation^.

For α >-1, let dvα denote the weighted Lebesgue measure on the bidisk and μ a complex measure satisfying some Carleson-type conditions. In this paper, we show a sufficient and necessary condition for the Toeplitz operator T_μ^α to be bounded or compact on weighted Bergman space L_a¹(dvα).

In this paper, we consider the Brownian motion risk model with interest. The Laplace transform of the first exit time from the upper barrier before hitting the lower barrier is obtained. Using the obtained result and exploiting the limitation idea, we derive the Laplace transform of total duration of negative surplus.

An umbilical free oriented hypersurface x : M → Rⁿ with non-zero principal curvatures is called a Laguerre isoparametric hypersurface if its Laguerre form C =∑_iC_iω_i = ∑_iρ^-1(E_i(log ρ)(r -r_i)-E_i(r))ω_i vanishes and Laguerre shape operator S = ρ^-1(S^-1-rid) has constant eigenvalues. Here ρ =∑_i(r -r_i)², r = (r1+r2+…+r_n-1)/n-1 is the mean curvature radius and S is the shape operator of x. {E_i} is a local basis for Laguerre metric g = ρ²Ⅲ with dual basis {ω_i} and Ⅲ is the third fundamental form of x. In this paper, we classify all Laguerre isoparametric hypersurfaces in Rⁿ(n > 3) with two distinct non-zero principal curvatures up to Laguerre transformations.

Let G be a graph, and k ≥ 2 be a positive integer. A graph G is fractional independentset-deletable k-factor-critical (in short, fractional ID-k-factor-critical), if G-I has a fractional k-factor for every independent set I of G. The binding number bind(G) of a graph G is defined as
bind(G) = min {|NG(X)|/|X|: ø≠X ⊆ V (G),NG(X) = V (G)}.
In this paper, it is proved that a graph G is fractional ID-k-factor-critical if n ≥ 6k-9 and bind(G) >((3k-1)(n-1))/(kn-2k+2).

In this paper, we construct a category of short exact sequences of vector bundles and prove that it is equivalent to the category of double vector bundles. Moreover, operations on double vector bundles can be transferred to operations on the corresponding short exact sequences. In particular, we study the duality theory of double vector bundles in term of the corresponding short exact sequences. Examples including the jet bundle and the Atiyah algebroid are discussed.

In the paper, we further realize the higher rank quantized universal enveloping algebra U_q(sl_n+1) as certain quantum differential operators in the quantum Weyl algebra W_q(2n) defined over the quantum divided power algebra A_q(n) of rank n. We give the quantum differential operators realization for both the simple root vectors and the non-simple root vectors of U_q(sl_n+1). The nice behavior of the quantum root vectors formulas under the action of the Lusztig symmetries once again indicates that our realization model is naturally matched.

In this note, we show that a von Neumann algebra M is injective if and only if the weak^* similarity degree d_*(M) ≤ 2.

We show that the Aubry sets, the Mañé sets and the barrier functions are the same for two commuting time-periodic Tonelli Hamiltonians.

By using the Mecke identity, we study a class of birth-death type Dirichlet forms associated with the mixed Poisson measure. Both Poincaré and weak Poincaré inequalities are established, while another Poincaré type inequality is disproved under some reasonable assumptions.

In this paper, we study Lichnerowicz type estimate for eigenvalues of drifting Laplacian operator and the decay rates of L¹ and L² energy for drifting heat equation on closed Riemannian manifolds with weighted measure.

In this paper, we consider the scattering for the nonlinear Schrödinger equation with small,smooth, and localized data. In particular, we prove that the solution of the quadratic nonlinearSchrödinger equation with nonlinear term |u|² involving some derivatives in two dimension exists globallyand scatters. It is worth to note that there exist blow-up solutions of these equations withoutderivatives. Moreover, for radial data, we prove that for the equation with p-order nonlinearity withderivatives, the similar results hold for p≥(2d+3)/(2d-1) and d≥ 2, which is lower than the Strauss exponents.Keywords Schrödinger equation, global well-posedness, scatter, small data, quadratic

In this paper, we study the sharp Jackson inequality for the best approximation of f ∈L_2,κ(R^d) by a subspace E_κ²(σ) (SE_κ²(σ)), which is a subspace of entire functions of exponential type (spherical exponential type) at most σ. Here L_2,κ(R^d) denotes the space of all d-variate functions f endowed with the L₂-norm with the weight v_κ(x) = ∏_ξ∈R+ |(ξ, x)|^2κ(ξ), which is defined by a positive subsystem R₊ of a finite root system R ⊂ R^d and a function κ(ξ): R → R₊ invariant under the reflection group G(R) generated by R. In the case G(R) = Z₂^d, we get some exact results. Moreover, the deviation of best approximation by the subspace E_κ²(σ) (SE_κ²(σ)) of some class of the smooth functions in the space L_2,κ(R^d.) is obtained.

Some new characterizations of nonnegative Hamiltonian operator matrices are given. Several necessary and sufficient conditions for an unbounded nonnegative Hamiltonian operator to be invertible are obtained, so that the main results in the previously published papers are corollaries of the new theorems. Most of all we want to stress the method of proof. It is based on the connections between Pauli operator matrices and nonnegative Hamiltonian matrices.

In this paper we give a Carleson measure characterization for the compact composition operators between Dirichlet type spaces. We use this characterization to show that every compact composition operator on Dirichlet type spaces is compact on the Bloch space.

This work focuses on the second type of generalized Feigenbaum's equation

where φ(x) is C^∞-increasing function on [0, 1] and satisfies that φ(0) = 0, 0 < φ'(x) < 1 (x ∈ [0, 1]). Using constructive method, we discuss the existence of C^∞-single-valley solutions whose derivatives are not equal to 0 on origin of the above equation.

In this paper, we mainly study zeros and poles of the forward differences Δⁿf(z), where f(z) is a finite order meromorphic function with two Borel exceptional values.

In this study, a new filter algorithm is presented for solving the nonlinear semidefinite programming. This algorithm is inspired by the classical sequential quadratic programming method. Unlike the traditional filter methods, the sufficient descent is ensured by changing the step size instead of the trust region radius. Under some suitable conditions, the global convergence is obtained. In the end, some numerical experiments are given to show that the algorithm is effective.

By explicit constructions, we give direct proofs of the following results: for any distinct homotopy classes of simple closed curves α and β in a closed surface of genus g > 1, there exist a hyperbolic structure X and a holomorphic quadratic differential q on X such that l_X(α)≠ l_X(β),ext_X(α) ≠ ext_X(β) and l_q(α) ≠ l_q(β), where l_X(·), extX(·) and l_q(·) are the hyperbolic length, the extremal length and the quadratic differential length respectively. These imply that there are no equivalent simple closed curves in hyperbolic surfaces or in flat surfaces.

A non-Archimedean Banach space has the orthogonal finite-dimensional decomposition property (OFDDP) if it is the orthogonal direct sum of a sequence of finite-dimensional subspaces.This property has an influence in the non-Archimedean Grothendieck's approximation theory, where an open problem is the following: Let E be a non-Archimedean Banach space of countable type with the OFDDP and let D be a closed subspace of E. Does D have the OFDDP? In this paper we give a negative answer to this question; we construct a Banach space of countable type with the OFDDP having a one-codimensional subspace without the OFDDP. Next we prove that, however, for certain classes of Banach spaces of countable type, the OFDDP is preserved by taking finite-codimensional subspaces.

The paper solves analytically the Riemann problem for a nonstrictly hyperbolic system of conservation laws arising in geometrical optics, in which the flux contains the nonconvex function possessing an infinite number of inflection points. Firstly, the generalized Rankine-Hugoniot relations and entropy condition of delta shock waves and left(right)-contact delta shock waves are proposed and clarified. Secondly, with the help of the convex hull, seven kinds of structures of Riemann solutions are obtained. The solutions fall into three broad categories with a series of geometric structures involving simultaneously contact discontinuities, vacuums and delta shock waves. Finally, numerical experiments confirm the theoretical analysis.

Let F_p be the finite field of p elements with p prime. If A is a subset of F_p and g is an element of F_p^* with order v, then
max{|A + g · A|,|A · A|}>>(v/(v+|A|²))^1/12|A|^13/12.

A graph is called 1-planar if it admits a drawing in the plane such that each edge is crossed at most once. In this paper, we study 1-planar graph joins. We prove that the join G + H is 1-planar if and only if the pair [G,H] is subgraph-majorized by one of pairs [C₃ ∪ C₃C₃], [C₄, C₄], [C₄,C₃],[K_2,1,1, P₃] in the case when both elements of the graph join have at least three vertices. If one element has at most two vertices, then we give several necessary/sufficient conditions for the bigger element.

We determine a 2-codimensional para-CR structure on the slit tangent bundle T₀M of a Finsler manifold (M,F) by imposing a condition regarding the almost paracomplex structure P associated to F when restricted to the structural distribution of a framed para-f-structure. This condition is satisfied when (M,F) is of scalar flag curvature (particularly constant) or if the Riemannian manifold (M,g) is of constant curvature.

Sure independence screening (SIS) has been proposed to reduce the ultrahigh dimensionality down to a moderate scale and proved to enjoy the sure screening property under Gaussian linear models. However, the observed response is often skewed or heavy-tailed with extreme values in practice,which may dramatically deteriorate the performance of SIS. To this end, we propose a new robust sure independence screening (RoSIS) via considering the correlation between each predictor and the distribution function of the response. The proposed approach contributes to the literature in the following three folds: First, it is able to reduce ultrahigh dimensionality effectively. Second, it is robust to heavy tails or extreme values in the response. Third, it possesses both sure screening property and ranking consistency property under milder conditions. Furthermore, we demonstrate its excellent finite sample performance through numerical simulations and a real data example.

It is established that all even positive integers up to N but at most O(N^15/16+ε) exceptions can be expressed in the form p₁²+p₂³+p₃⁴+p₄⁵, where p₁,p₂,p₃ and p₄ are prime numbers.

The mixed volume and the measure of asymmetry for convex bodies are two important topics in convex geometry. In this paper, we first reveal a close connection between the L_p-mixed volumes proposed by E. Lutwak and the p-measures of asymmetry, which have the Minkowski measure as a special case, introduced by Q. Guo. Then, a family of measures of asymmetry is defined in terms of the Orlicz mixed volumes introduced by R. J. Gardner, D. Hug and W. Weil recently, which is an extension of the p-measures.

Let X be a space of homogenous type and φ; : X ×[0,∞) → [0,∞) be a growth function such that φ;(·,t) is a Muckenhoupt weight uniformly in t and φ;(x,·) an Orlicz function of uniformly upper type 1 and lower type p ∈ (0,1]. In this article, the authors introduce a new Musielak-Orlicz BMO-type space BMO_A^φ(X) associated with the generalized approximation to the identity, give out its basic properties and establish its two equivalent characterizations, respectively, in terms of the spaces BMO_{A, max}^φ(X) and (X). Moreover, two variants of the John-Nirenberg inequality on BMO_A^φ(X) are obtained. As an application, the authors further prove that the space BMO_√Δ^φ(Rⁿ),associated with the Poisson semigroup of the Laplace operator Δ on Rⁿ, coincides with the space BMO^φ(Rⁿ) introduced by Ky.

Based on the double penalized estimation method, a new variable selection procedure is proposed for partially linear models with longitudinal data. The proposed procedure can avoid the effects of the nonparametric estimator on the variable selection for the parameters components. Under some regularity conditions, the rate of convergence and asymptotic normality of the resulting estimators are established. In addition, to improve efficiency for regression coefficients, the estimation of the working covariance matrix is involved in the proposed iterative algorithm. Some simulation studies are carried out to demonstrate that the proposed method performs well.