We present a partial first-order affine-scaling method for solving smooth optimization with linear inequality constraints. At each iteration, the algorithm considers a subset of the constraints to reduce the complexity. We prove the global convergence of the algorithm for general smooth objective functions, and show it converges at sublinear rate when the objective function is quadratic. Numerical experiments indicate that our algorithm is efficient.

The behavior of the Kozachenko-Leonenko estimates for the (differential) Shannon entropy is studied when the number of i.i.d. vector-valued observations tends to infinity. The asymptotic unbiasedness and L²-consistency of the estimates are established. The conditions employed involve the analogues of the Hardy-Littlewood maximal function. It is shown that the results are valid in particular for the entropy estimation of any nondegenerate Gaussian vector.

We consider the vanishing ideal of a projective space over a finite field. An explicit set of generators for this ideal has been given by Mercier and Rolland. We show that these generators form a universal Gröbner basis of the ideal. Further we give a projective analogue for the so-called footprint bound, and a version of it that is suitable for estimating the number of rational points of projective algebraic varieties over finite fields. An application to Serre's inequality for the number of points of projective hypersurfaces over finite fields is included.

The typical problem in the mechanics of deformable solids comprises a mathematical model in the form of systems of partial differential equations or inequalities. Subsequent investigations are then concerned with analysis of the model to determine its well-posedness, followed by the development and implementation of algorithms to obtain approximate solutions to problems that are generally intractable in closed form. These processes of modelling, analysis, and computation are discussed with a focus on the behaviour of elastic-plastic bodies; these are materials which exhibit path-dependence and irreversibility in their behaviour. The resulting variational problem is an inequality that is not of standard elliptic or parabolic type. Properties of this formulation are reviewed, as are the conditions under which fully discrete approximations converge. A solution algorithm, motivated by the predictor-corrector algorithms that are common in elastoplastic problems, is presented and its convergence properties summarized. An important extension of the conventional theory is that of straingradient plasticity, in which gradients of the plastic strain appear in the formulation, and which includes a length scale not present in the conventional theory. Some recent results for strain-gradient plasticity are presented, and the work concludes with a brief description of current investigations.

In the paper we present a survey of recent results obtained by the author that concern discrete fragmentation-coagulation models with growth. Models like that are particularly important in mathematical biology and ecology where they describe the aggregation of living organisms. The main results discussed in the paper are the existence of classical semigroup solutions to the fragmentation-coagulation equations.

In this paper, we study the well-posedness of the third-order differential equation with finite delay (P₃):αu"'(t) + u"(t)=Au(t) + Bu' (t) + Fu_t + f(t)(t ∈ T:=[0, 2π]) with periodic boundary conditions u(0)=u(2π), u' (0)=u' (2π), u"(0)=u" (2π), in periodic Lebesgue-Bochner spaces L^p(T; X) and periodic Besov spaces B_p,q^s (T; X), where A and B are closed linear operators on a Banach space X satisfying D(A) ∩ D(B)≠ {0}, α≠ 0 is a fixed constant and F is a bounded linear operator from L^p([-2π, 0]; X) (resp. B_p,q^s ([-2π, 0]; X)) into X, u_t is given by u_t(s)=u(t + s) when s ∈[-2π, 0]. Necessary and sufficient conditions for the L^p-well-posedness (resp. B_p,q^s -well-posedness) of (P₃) are given in the above two function spaces. We also give concrete examples that our abstract results may be applied.

Suppose that X, Y are two real Banach Spaces. We know that for a standard ∈-isometry f:X → Y, the weak stability formula holds and by applying the formula we can induce a closed subspace N of Y^*. In this paper, by using again the weak stability formula, we further show a sufficient and necessary condition for a standard ∈-isometry to be stable in assuming that N is ω^*-closed in Y^*. Making use of this result, we improve several known results including Figiel's theorem in reflexive spaces. We also prove that if, in addition, the space Y is quasi-reflexive and hereditarily indecomposable, then L(f) ≡ span[f(X)] contains a complemented linear isometric copy of X; Moreover, if X=Y, then for every ∈-isometry f:X → X, there exists a surjective linear isometry S:X → X such that f -S is uniformly bounded by 2∈ on X.

Let T ⊂ R be an interval. By studying an admissible family of branching mechanisms {ψ_t, t ∈ T} introduced in Li[Ann. Probab., 42, 41-79 (2014)], we construct a decreasing Lévy-CRTvalued process {T_t, t ∈ T} by pruning Lévy trees accordingly such that for each t ∈ T, T_t is a ψ_t-Lévy tree. We also obtain an analogous process {T_t^*, t ∈ T} by pruning a critical Lévy tree conditioned to be infinite. Under a regular condition on the admissible family of branching mechanisms, we show that the law of {T_t, t ∈ T} at the ascension time A:=inf{t ∈ T; T_t is finite} can be represented by {T_t^*, t ∈ T}. The results generalize those studied in Abraham and Delmas[Ann. Probab., 40, 1167-1211 (2012)].

The authors recently defined a new graph invariant denoted by Ω(G) only in terms of a given degree sequence which is also related to the Euler characteristic. It has many important combinatorial applications in graph theory and gives direct information compared to the better known Euler characteristic on the realizability, connectedness, cyclicness, components, chords, loops etc. Many similar classification problems can be solved by means of Ω. All graphsGso that Ω(G) ≤ -4 are shown to be disconnected, and if Ω(G) ≥ -2, then the graph is potentially connected. It is also shown that if the realization is a connected graph and Ω(G)=-2, then certainly the graph should be a tree. Similarly, it is shown that if the realization is a connected graph G and Ω(G) ≥ 0, then certainly the graph should be cyclic. Also, when Ω(G) ≤ -4, the components of the disconnected graph could not all be cyclic and if all the components of Gare cyclic, then Ω(G) ≥ 0. In this paper, we study an extremal problem regarding graphs. We find the maximum number of loops for three possible classes of graphs. We also state a result giving the maximum number of components amongst all possible realizations of a given degree sequence.

In this paper, motived by the notion of independent and identically distributed random variables under the sub-linear expectation initiated by Peng, we establish a three series theorem of independent random variables under the sub-linear expectations. As an application, we obtain the Marcinkiewicz's strong law of large numbers for independent and identically distributed random variables under the sub-linear expectations. The technical details are different from those for classical theorems because the sub-linear expectation and its related capacity are not additive.

Professor Lo Yang is a world famous mathematician of our country. He made a lot of outstanding achievements in the value distribution theory of function theory, which are highly rated and widely quoted by domestic and foreign scholars. He also did a lot of work to develop Chinese mathematics. It can be said that Professor Yang is one of the mathematicians who made main influences on the mathematical development in modern China. This paper briefly introduces Professor Yang's life, mainly discusses his academic achievement and influence, and briefly describes his contributions to the Chinese mathematics community.

This paper is concerned with small quadratic perturbations to one parameter family of generic reversible quadratic vector fields with a simple center. The first objective is to show that this system exhibits two small amplitude limit cycles emerging from a Hopf bifurcation. The second one we prove that the system has no limit cycle around the weak focus of order two. The results may be viewed as a contribution to proving the conjecture on cyclicity proposed by Iliev (1998).

This paper is devoted to investigating the existence and uniqueness of mild solution to the 3D incompressible micropolar fluid system in the Fourier-Herz framework. By taking advantage of microlocal analysis and the mutual effect in the same frequency range of convection term, we give a special initial data (u₀, ω₀) whose norm of  (q > 2) is arbitrarily small, however, the couple (u₀, ω₀) produces a solution which is arbitrarily large in  after an arbitrarily short time. This implies the system is ill-posed in the sense of "norm inflation" as q > 2.

On the Dirichlet space of the unit ball, we study some algebraic properties of Toeplitz operators. We give a relation between Toeplitz operators on the Dirichlet space and their analogues defined on the Hardy space. Based on this, we characterize when finite sums of products of Toeplitz operators are of finite rank. Also, we give a necessary and sufficient condition for the commutator and semi-commutator of two Toeplitz operators being zero.

This paper is concerned with a popular form of Cahn-Hilliard equation which plays an important role in understanding the evolution of phase separation. We get the existence and regularity of a weak solution to nonlinear parabolic, fourth order Cahn-Hilliard equation with degenerate mobility M(u)=u^m(1-u)^m which is allowed to vanish at 0 and 1. The existence and regularity of weak solutions to the degenerate Cahn-Hilliard equation are obtained by getting the limits of Cahn-Hilliard equation with non-degenerate mobility. We explore the initial value problem with compact support and obtain the local non-negative result. Further, the above derivation process is also suitable for the viscous Cahn-Hilliard equation with degenerate mobility.

Usually, the condition that T is bounded on L²(Rⁿ) is assumed to prove the boundedness of an operator T on a Hardy space. With this assumption, one only needs to prove the uniformly boundness of T on atoms, since T(f)=Σ_iλ_iT(a_i), provided that f=Σ_iλ_ia_i in L²(Rⁿ), where a_i is an L² atom of this Hardy space. So far, the L² atomic decomposition of local Hardy spaces h^p(Rⁿ), 0 < p ≤ 1, hasn't been established. In this paper, we will solve this problem, and also show that h^p(Rⁿ) can also be characterized by discrete Littlewood-Paley functions.

In the present paper, we investigate the dual Lie coalgebras of the centerless W (2, 2) algebra by studying the maximal good subspaces. Based on this, we construct the dual Lie bialgebra structures of the centerless W (2, 2) Lie bialgebra. As by-products, four new infinite dimensional Lie algebras are obtained.

Numerical range has an important applications on spectrum distribution of operators. In this paper, we devoted to characterizing operators whose numerical range contains the origin. Some necessary and sufficient conditions are given by operator decomposition technique and constructive methods. Furthermore, the closeness of the numerical range of a given operator is also investigated.

We show that if a bounded domain Ω is exhausted by a bounded strictly pseudoconvex domain D with C² boundary, then Ω is holomorphically equivalent to D or the unit ball, and show that a bounded domain has to be holomorphically equivalent to the unit ball if its Fridman's invariant has certain growth condition near the boundary.

In this paper, we generalize the concept of asymptotic Hankel operators on H²(D) to the Hardy space H²(Dⁿ) (over polydisk) in terms of asymptotic Hankel and partial asymptotic Hankel operators and investigate some properties in case of its weak and strong convergence. Meanwhile, we introduce i^th-partial Hankel operators on H²(Dⁿ) and obtain a characterization of its compactness for n > 1. Our main results include the containment of Toeplitz algebra in the collection of all strong partial asymptotic Hankel operators on H²(Dⁿ). It is also shown that a Toeplitz operator with symbol φ is asymptotic Hankel if and only if φ is holomorphic function in L^∞(Tⁿ).

In this paper, the weak (1, 1) boundedness of oscillatory singular integral with variable phase P (x)γ(y) for any x, y ∈ R,
Tf(x):=p. v.∫_-∞^∞ e^iP(x)γ(y)f(x-y) dy/y
is studied, where P is a real monic polynomial on R.

This paper studies the M₀-shadowing property for the dynamics of diffeomorphisms defined on closed manifolds. The C¹ interior of the set of all two dimensional diffeomorphisms with the M₀-shadowing property is described by the set of all Anosov diffeomorphisms. The C¹-stably M₀-shadowing property on a non-trivial transitive set implies the diffeomorphism has a dominated splitting.

This paper mainly focuses on the entire solutions of a nonlocal dispersal equation with asymmetric kernel and bistable nonlinearity. Compared with symmetric case, the asymmetry of the dispersal kernel function makes more diverse types of entire solutions since it can affect the sign of the wave speeds and the symmetry of the corresponding nonincreasing and nondecreasing traveling waves. We divide the bistable case into two monostable cases by restricting the range of the variable, and obtain some merging-front entire solutions which behave as the coupling of monostable and bistable waves. Before this, we characterize the classification of the wave speeds so that the entire solutions can be constructed more clearly. Especially, we investigate the influence of the asymmetry of the kernel on the minimal and maximal wave speeds.

In this note, we study a rich operator class denoted by PB_n(Ω) which includes all homogeneous operators and quasi-homogeneous operators in the Cowen-Douglas class. A complete unitarily classification theorem is given. Furthermore, we also concern the curvature and similarity of operators in PB_n(Ω).

We study property T for an action α of a discrete group Γ on a unital C^*-algebra A. Our main results improve some well-known results about property T for groups. Moreover, we introduce Hilbert A -module property T and show that the action α has property T if and only if the reduced crossed product A×_α,r Γ has Hilbert A -module property T.

Let G be a graph with vertex set V (G), edge set E(G) and maximum degree Δ respectively. G is called degree-magic if it admits a labelling of the edges by integers {1, 2, …,|E(G)|} such that for any vertex v the sum of the labels of the edges incident with v is equal to (1+|E(G)|)/2·d(v), where d(v) is the degree of v. Let f be a proper edge coloring of G such that for each vertex v ∈ V (G),|{e:e ∈ E_v, f(e) ≤ Δ/2}|=|{e:e ∈ E_v, f(e) > Δ/2}|, and such an f is called a balanced edge coloring of G. In this paper, we show that if G is a supermagic even graph with a balanced edge coloring and m ≥ 1, then (2m + 1)G is a supermagic graph. If G is a d-magic even graph with a balanced edge coloring and n ≥ 2, then nG is a d-magic graph. Results in this paper generalise some known results.

In this paper, we give an operator parameterization for the set of dilations of a given pair of dual g-frames and the set of dilations of pairs of dual g-frames of a given g-frame. In particular, for the dilations of a given pair of dual g-frames, we introduce the concept of joint complementary g-frames and prove that the joint complementary g-frames of a pair of dual g-frames are unique in the sense of joint similarity, which then helps to obtain a sufficient condition such that the complementary g-frames of a g-frame are unique in the sense of similarity and show that the set of dilations of a given dual g-frame pair are parameterized by a set of invertible diagonal operators. For the dilations of pairs of dual g-frames, we prove that the set of dilations of pairs of dual g-frames are parameterized by a set of invertible upper triangular operators.

The main purpose of this paper is to establish, using the Littlewood-Paley-Stein theory (in particular, the Littlewood-Paley-Stein square functions), a Calderón-Torchinsky type theorem for the following Fourier multipliers on anisotropic Hardy spaces H^p(Rⁿ; A) associated with expensive dilation A:
???20191109
Our main Theorem is the following:Assume that m(ξ) is a function on Rⁿ satisfying
???20191109-1
with s > ζ_-^-1(1/p-1/2). Then T_m is bounded from H^p(Rⁿ; A) to H^p(Rⁿ; A) for all 0 < p ≤ 1 and
???20191109-2
where A^* denotes the transpose of A. Here we have used the notations m_j(ξ)=m(A^*jξ)φ(ξ) and φ(ξ) is a suitable cut-off function on Rⁿ, and W^s(A^*) is an anisotropic Sobolev space associated with expansive dilation A^* on Rⁿ.

The Gelfand-Kirillov dimension is an invariant which can measure the size of infinitedimensional algebraic structures. In this article, we show that it can also measure the reducibility of scalar generalized Verma modules. In particular, we use it to determine the reducibility of scalar generalized Verma modules associated with maximal parabolic subalgebras in the Hermitian symmetric case.

Let D be a finite and simple digraph with vertex set V (D). The minimum degree δ of a digraph D is defined as the minimum value of its out-degrees and its in-degrees. If D is a digraph with minimum degree δ and edge-connectivity λ, then λ ≤ δ. A digraph is maximally edge-connected if λ=δ. A digraph is called super-edge-connected if every minimum edge-cut consists of edges incident to or from a vertex of minimum degree. In this note we show that a digraph is maximally edge-connected or super-edge-connected if the number of arcs is large enough.

In this paper we give a classification of special endomorphisms of nil-manifolds:Let f:N/Γ → N/Γ be a covering map of a nil-manifold and denote by A:N/Γ → N/Γ the nil-endomorphism which is homotopic to f. If f is a special T A-map, then A is a hyperbolic nil-endomorphism and f is topologically conjugate to A.

In this paper, we construct random two-faced families of matrices with non-Gaussian entries to approximate a bi-free central limit distribution with a positive definite covariance matrix. We prove that, under modest conditions weaker than independence, a family of random two-faced families of matrices with non-Gaussian entries is asymptotically bi-free from a two-faced family of constant diagonal matrices.

We consider a fourth order nonlinear PDE involving the critical Sobolev exponent on a bounded domain of Rⁿ, n ≥ 5 with Navier condition on the boundary. We study the lack of compactness of the problem and we provide an existence theorem through a new index formula.

In this paper, we study the complete f-moment convergence for widely orthant dependent (WOD, for short) random variables. A general result on complete f-moment convergence for arrays of rowwise WOD random variables is obtained. As applications, we present some new results on complete f-moment convergence for WOD random variables. We also give an application to nonparametric regression models based on WOD errors by using the complete convergence that we established. Finally, the choice of the fixed design points and the weight functions for the nearest neighbor estimator are proposed, and a numerical simulation is provided to verify the validity of the theoretical result.

Basing upon the recent development of the Patterson-Sullivan measures with a Hölder continuous nonzero potential function, we use tools of both dynamics of geodesic flows and geometric properties of negatively curved manifolds to present a new formula illustrating the relation between the exponential decay rate of Patterson-Sullivan measures with a Hölder continuous potential function and the corresponding critical exponent.

In this paper, we prove quasi-modularity property for the twisted Gromov-Witten theory of O(3) over P². Meanwhile, we derive its holomorphic anomaly equation.

This paper constructs several families of self-complementary Cayley graphs of extraspecial p-groups, where p is a prime and congruent to 1 modulo 4.