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Motivated by a paper of Fang (2009), we study the Samuel multiplicity and the structure of essentially semi-regular operators on an infinite-dimensional complex Banach space. First, we generalize Fang’s results concerning Samuel multiplicity from semi-Fredholm operators to essentially semi-regular operators by elementary methods in operator theory. Second, we study the structure of essentially semi-regular operators. More precisely, we present a revised version of Fang’s 4 × 4 upper triangular model with a little modification, and prove it in detail after providing numerous preliminary results, some of which are inspired by Fang’s paper. At last, as some applications, we get the structure of semi-Fredholm operators which revised Fang’s 4 × 4 upper triangular model, from a different viewpoint, and characterize a semi-regular point λ ∈ ℂ in an essentially semi-regular domain.

This short paper is based on the talk on the conference Operator Algebras and Related Topics held on July 23–27, 2010, Beijing. The author surveys recent developments of the noncommutative gravity in joint works with Chaichian, Tureanu, Sun, Wang, Xie and Zhang.

The main purpose of this paper is to extend the result obtained by Beirao da Veiga (2000) from the whole-space case to slip boundary cases. Denote by u two components of the velocity u. To fix ideas set ū = (u1,u2, 0) (the half-space) or $${\boldsymbol{\bar u}} = {\hat u_1}{\hat e_1} + {\hat u_2}{\hat e_2}$$ (the general boundary case (see (7.1))). We show that there exists a constant K, which enjoys very simple and significant expressions such that if at some time τ ∈ (0,T) one has $$\lim {\sup _{t \to \tau - 0}}\left\| {{\boldsymbol{\bar u}}(t)} \right\|_{{L^3}(\Omega )}^3 < \left\| {{\boldsymbol{\bar u}}(\tau )} \right\|_{{L^3}(\Omega )}^3 + K$$ , then u is continuous at τ with values in L3(Ω). Roughly speaking, the above norm-discontinuity of merely two components of the velocity cannot occur for steps’ amplitudes smaller than K. In particular, if the above condition holds at each τ ∈ (0,T), the solution is smooth in (0,T) × Ω. Note that here there is no limitation on the width of the norms $$\left\| {{\boldsymbol{\bar u}}(t)} \right\|_{{L^3}(\Omega )}^3$$ . So K is independent of these quantities. Many other related results are discussed and compared among them. This is a second main aim of this paper. New results are proved in Sections 5–7.

Generalizing wavelets by adding desired redundancy and flexibility, framelets (i.e., wavelet frames) are of interest and importance in many applications such as image processing and numerical algorithms. Several key properties of framelets are high vanishing moments for sparse multiscale representation, fast framelet transforms for numerical efficiency, and redundancy for robustness. However, it is a challenging problem to study and construct multivariate nonseparable framelets, mainly due to their intrinsic connections to factorization and syzygy modules of multivariate polynomial matrices. Moreover, all the known multivariate tight framelets derived from spline refinable scalar functions have only one vanishing moment, and framelets derived from refinable vector functions are barely studied yet in the literature. In this paper, we circumvent the above difficulties through the approach of quasi-tight framelets, which behave almost identically to tight framelets. Employing the popular oblique extension principle (OEP), from an arbitrary compactly supported M-refinable vector function ϕ with multiplicity greater than one, we prove that we can always derive from ϕ a compactly supported multivariate quasi-tight framelet such that: (i) all the framelet generators have the highest possible order of vanishing moments; (ii) its associated fast framelet transform has the highest balancing order and is compact. For a refinable scalar function ϕ (i.e., its multiplicity is one), the above item (ii) often cannot be achieved intrinsically but we show that we can always construct a compactly supported OEP-based multivariate quasi-tight framelet derived from ϕ satisfying item (i). We point out that constructing OEP-based quasi-tight framelets is closely related to the generalized spectral factorization of Hermitian trigonometric polynomial matrices. Our proof is critically built on a newly developed result on the normal form of a matrix-valued filter, which is of interest and importance in itself for greatly facilitating the study of refinable vector functions and multiwavelets/multiframelets. This paper provides a comprehensive investigation on OEP-based multivariate quasi-tight multiframelets and their associated framelet transforms with high balancing orders. This deepens our theoretical understanding of multivariate quasi-tight multiframelets and their associated fast multiframelet transforms.

Ever since the famous Erdős-Ko-Rado theorem initiated the study of intersecting families of subsets, extremal problems regarding intersecting properties of families of various combinatorial objects have been extensively investigated. Among them, studies about families of subsets, vector spaces and permutations are of particular concerns. Recently, we proposed a new quantitative intersection problem for families of subsets: For $${\cal F} \subseteq \left({\matrix{{[n]} \cr k \cr}} \right)$$ , define its total intersection number as $${\cal I}({\cal F}) = \sum\nolimits_{{F_1},{F_2} \in {\cal F}} {\left| {{F_1} \cap {F_2}} \right|} $$ . Then, what is the structure of $${\cal F}$$ when it has the maximal total intersection number among all the families in $$\left({\matrix{{[n]} \cr k \cr}} \right)$$ with the same family size? In a recent paper, Kong and Ge (2020) studied this problem and characterized extremal structures of families maximizing the total intersection number of given sizes. In this paper, we consider the analogues of this problem for families of vector spaces and permutations. For certain ranges of family size, we provide structural characterizations for both families of subspaces and families of permutations having maximal total intersection numbers. To some extent, these results determine the unique structure of the optimal family for some certain values of $$\left| {\cal F} \right|$$ and characterize the relationship between having maximal total intersection number and being intersecting. Besides, we also show several upper bounds on the total intersection numbers for both families of subspaces and families of permutations of given sizes.

Using the hyper-exponential recurrence criterion, a large deviation principle for the occupation measure is derived for a class of non-linear monotone stochastic partial differential equations. The main results are applied to many concrete SPDEs such as stochastic $p$-Laplace equation, stochastic porous medium equation, stochastic fast-diffusion equation, and even stochastic real Ginzburg-Landau equation driven by $\alpha$-stable noises., Comment: This paper generalizes the idea in our NOT published paper arXiv:1510.03522. There is a substantial overlap with arXiv:1510.03522

In the study of the constant in Ahlfors' second fundamental theorem involving a set E_{q} of q points, branch values of covering surfaces outside E_{q} bring a lot of troubles. To avoid this situation, for a given surface S, it is useful to construct a new surface So such that L(So) =H(S), and all branch values of So are contained in E_{q}. The goal of this paper is to prove the existence of such So, which generalizes Lemma 9.1 and Theorem 10.1 in Zhang G.Y.: The precise bound for the area-length ratio in Ahifors' theory of covering surfaces. Invent math 191:197-253(2013), This paper is a preparation for the further research on Ahlfors' theorey of covering surfaces

The current paper is devoted to the Cauchy problem for the stochastic generalized Benjamin-Ono equation. By establishing the bilinear estimate, trilinear estimates in some Bourgain spaces, we prove that the Cauchy problem for the stochastic generalized Benjamin-Ono equation is locally well-posed for the initial data u0(x, ω) ∈ L2 (Ω; Hs (ℝ)) which is $$\mathscr{F}_{0}$$ measurable with $$s \geqslant \frac{1}{2}-\frac{\alpha}{4}$$ and $$\Phi \in L_{2}^{0, s}$$ . In particular, when α = 1, we prove that it is globally well-posed for the initial data u0(x, ω) ∈ L2(Ω; H1(ℝ)) which is $$\mathscr{F}_{0}$$ measurable and $${\rm{\Phi }} \in L_2^{0,1}$$ . The key ingredients that we use in this paper are trilinear estimates, the Ito formula and the Burkholder-Davis-Gundy (BDG) inequality as well as the stopping time technique.

In this paper, we study the topology of moment-angle manifolds and prove a conjecture of S. Gitler and S. Lopez de Medrano concerned with the behavior of the moment-angle manifold under the surgery ‘cutting off a vertex’ on a simple polytope. Let P be a simple polytope of dimension n with m facets and Pv be a polytope obtained from P by cutting off one vertex v. Let Z = Z(P) and Zv = Z(Pv) be the corresponding moment-angle manifolds. S. Gitler and S. Lopez de Medrano conjectured that: Zv is diffeomorphic to $$\partial \left[ {\left( {Z - {D^{n + m}}} \right) \times {D^2}} \right]\# \,\# _{j = 1}^{m - n}\left( {_j^{m - n}} \right)\left( {{S^{j + 2}} \times {S^{m + n - j - 1}}} \right)$$ , and they proved the conjecture in the case m < 3n. In this paper we prove the conjecture in the general case.

In this paper, a type of accurate a posteriori error estimator is proposed for the Steklov eigenvalue problem based on the complementary approach, which provides an asymptotic exact estimate for the approximate eigenpair. Besides, we design a type of cascadic adaptive finite element method for the Steklov eigenvalue problem based on the proposed a posteriori error estimator. In this new cascadic adaptive scheme, instead of solving the Steklov eigenvalue problem in each adaptive space directly, we only need to do some smoothing steps for linearized boundary value problems on a series of adaptive spaces and solve some Steklov eigenvalue problems on a low dimensional space. Furthermore, the proposed a posteriori error estimator provides the way to refine mesh and control the number of smoothing steps for the cascadic adaptive method. Some numerical examples are presented to validate the efficiency of the algorithm in this paper.

This paper deals with the pulsating waves and entire solutions for a spatially periodic nonlocal dispersal model with a quiescent stage. By the method of super- and subsolutions together with the comparison principle, we establish the existence of pulsating waves with any speed larger than the spreading speed. The construction of the super- and subsolutions depends on the principal eigenvalue theory for a periodic eigenvalue problem with partially degenerate nonlocal dispersal. The joint results of this paper and our recent work (Wang J B, Li W T, Sun J W. Global dynamics and spreading speeds for a partially degenerate system with non-local dispersal in periodic habitats. Proc Roy Soc Edinburgh Sect A, 2018, 148: 849–880) show that the spreading speed coincides with the minimal wave speed of pulsating waves for the considered system. Moreover, combining the rightward and leftward pulsating waves with different speeds and a spatially periodic solution, we prove the existence and qualitative properties of entire solutions other than pulsating waves, which provide some new spreading ways in a heterogeneous habitat.

We begin with the reference measure P0 induced by simple, symmetric nearest neighbor continuous time random walk on Zd starting at 0 with jump rate 2d and then define, for β ⩾ 0, t > 0, the Gibbs probability measure Pβ,t by specifying its density with respect to P0 as $${{d{P_{\beta, t}}} \over {d{P^0}}} = {Z_{\beta, t}}{(0)^{-1}}{{\rm{e}}^{\beta \int_0^t {{\delta _0}({x_s})ds}}},$$ (0.1) where $${Z_{\beta, t}}(0) \equiv {E^0}{\rm{[}}{{\rm{e}}^{\beta \;\int_0^t {{\delta _0}({x_s})ds}}}].$$. This Gibbs probability measure provides a simple model for a homopolymer with an attractive potential at the origin. In a previous paper (Cranston and Molchanov, 2007), we showed that for dimensions d ⩾ 3 there is a phase transition in the behavior of these paths from the diffusive behavior for β below a critical parameter to the positive recurrent behavior for β above this critical value. The critical value was determined by means of the spectral properties of the operator Δ + βδ0, where Δ is the discrete Laplacian on Zd. This corresponds to a transition from a diffusive or stretched-out phase to a globular phase for the polymer. In this paper we give a description of the polymer at the critical value where the phase transition takes place. The behavior at the critical parameter is dimension-dependent.

Nonconforming grids with hanging nodes are frequently used in adaptive finite element computations. In all earlier works on such methods, proper constraints should be enforced on degrees of freedom on edges/faces with hanging nodes to keep continuity, which yield numerical computations much complicated. In 2014, Zhao et al. (2014) presented quadrilateral constraint-free finite element methods on quadrilateral grids with hanging nodes. This paper further develops a hexahedral constraint-free finite element method on hexahedral grids with hanging nodes, which is of greater challenge than the two-dimensional case. Residual-based a posteriori error reliability and efficiency are also established in this paper.

Let $A$ be a (left and right) Noetherian ring that is semiperfect. Let $e$ be an idempotent of $A$ and consider the ring $\Gamma:=(1-e)A(1-e)$ and the semi-simple right $A$-module $S_e : = eA/e{\rm rad}A$. In this paper, we investigate the relationship between the global dimensions of $A$ and $\Gamma$, by using the homological properties of $S_e$. More precisely, we consider the Yoneda ring $Y(e):={\rm Ext}^*_A(S_e,S_e)$ of $e$. We prove that if $Y(e)$ is artinian of finite global dimension, then $A$ has finite global dimension if and only if so is $\Gamma$. We also investigate the situation where both $A,\Gamma$ have finite global dimension. When $A$ is Koszul and finite dimensional, this implies that $Y(e)$ has finite global dimension. We end the paper with a reduction technique to compute the Cartan determiant of artin algebras. We prove that if $Y(e)$ has finite global dimension, then the Cartan determinants of $A$ and $\Gamma$ coincide. This provides a new way to approach the long-standing Cartan determinant conjecture., Comment: 14 pages

In this paper, we find all positive squarefree integers d such that the Pell equation X2-dY2 = +-1 has at least two positive integer solutions (X,Y) and (X',Y') such that both X and X' have Zeckendorf representations with at most two terms. This paper has been accepted for publication in SCIENCE CHINA Mathematics.

This paper is concerned with the time-step condition of linearized implicit finite difference method for solving the Gross-Pitaevskii equation with an angular momentum rotation term. Unlike the existing studies in the literature, where the cut-off function technique was used to establish the error estimates under some conditions of the time-step size, this paper introduces an induction argument and a ‘lifting’ technique as well as some useful inequalities to build the optimal maximum error estimate without any constraints on the time-step size. The analysis method can be directly extended to the general nonlinear Schrodinger-type equations in twoand three-dimensions and other linear implicit finite difference schemes. As a by-product, this paper defines a new type of energy functional of the grid functions by using a recursive relation to prove that the proposed scheme preserves well the total mass and energy in the discrete sense. Several numerical results are reported to verify the error estimates and conservation laws.

In this paper, we obtain the necessary and sufficient condition of the pre-compact sets in the variable exponent Lebesgue spaces, which is also called the Riesz-Kolmogorov theorem. The main novelty appearing in this approach is the constructive approximation which does not rely on the boundedness of the Hardy-Littlewood maximal operator in the considered spaces such that we do not need the log-Holder continuous conditions on the variable exponent. As applications, we establish the boundedness of Riemann-Liouville integral operators and prove the compactness of truncated Riemann-Liouville integral operators in the variable exponent Lebesgue spaces. Moreover, applying the Riesz-Kolmogorov theorem established in this paper, we obtain the existence and the uniqueness of solutions to a Cauchy type problem for fractional differential equations in variable exponent Lebesgue spaces.

This paper aims to introduce some new ideas into the study of submodules in Hilbert spaces of analytic functions. The effort is laid out in the Hardy space over the bidisk H2(D2). A closed subspace M in H2(D2) is called a submodule if z i M ⊂ M (i = 1, 2). An associated integral operator (defect operator) C M captures much information about M. Using a Kreĭn space indefinite metric on the range of C M , this paper gives a representation of M. Then it studies the group (called Lorentz group) of isometric self-maps of M with respect to the indefinite metric, and in finite rank case shows that the Lorentz group is a complete invariant for congruence relation. Furthermore, the Lorentz group contains an interesting abelian subgroup (called little Lorentz group) which turns out to be a finer invariant for M.

Let A,K be positive integers and u=-2,-1,1 or 2. The main contribution of the paper is a proof that each of the D(u^2)-triples {K,A^2K+2uA,(A+1)^2K+2u(A+1)} has unique extension to a D(u^2)-quadruple., This paper has 20 pages and has been accepted for publication in SCIENCE CHINA Mathematics

This paper is devoted to the existence of the traveling waves of the equations describing a diffusive susceptible-exposed-infected-recovered (SEIR) model. The existence of traveling waves depends on the basic reproduction rate and the minimal wave speed. We obtain a more precise estimation of the minimal wave speed of the epidemic model, which is of great practical value in the control of serious epidemics. The approach in this paper is to use the Schauder fixed point theorem and the Laplace transform. We also give some numerical results on the minimal wave speed.

This paper concerns with the existence of solutions for the following fractional Kirchhoff problem with critical nonlinearity: $${\left( {\int {\int {_{{\mathbb{R}^{2N}}}\frac{{{{\left| {u\left( x \right) - u\left( y \right)} \right|}^2}}}{{{{\left| {x - y} \right|}^{N + 2s}}}}dxdy} } } \right)^{\theta - 1}}{\left( { - \Delta } \right)^s}u = \lambda h\left( x \right){u^{p - 1}} + {u^{2_s^* - 1}} in {\mathbb{R}^N},$$ where (−Δ) s is the fractional Laplacian operator with 0 < s < 1, 2 s * = 2N/(N − 2s), N > 2s, p ∈ (1, 2 s *), θ ∈ [1, 2 s */2), h is a nonnegative function and λ a real positive parameter. Using the Ekeland variational principle and the mountain pass theorem, we obtain the existence and multiplicity of solutions for the above problem for suitable parameter λ > 0. Furthermore, under some appropriate assumptions, our result can be extended to the setting of a class of nonlocal integro-differential equations. The remarkable feature of this paper is the fact that the coefficient of fractional Laplace operator could be zero at zero, which implies that the above Kirchhoff problem is degenerate. Hence our results are new even in the Laplacian case.

In this paper, we consider a nonlinear integral eigenvalue problem, which is a reformulation of the transmission eigenvalue problem arising in the inverse scattering theory. The boundary element method is employed for discretization, which leads to a generalized matrix eigenvalue problem. We propose a novel method based on the spectral projection. The method probes a given region on the complex plane using contour integrals and decides if the region contains eigenvalue(s) or not. It is particularly suitable to test if zero is an eigenvalue of the generalized eigenvalue problem, which in turn implies that the associated wavenumber is a transmission eigenvalue. Effectiveness and efficiency of the new method are demonstrated by numerical examples., Comment: The paper has been accepted for publication in SCIENCE CHINA Mathematics

A direct as well as iterative method (called the orthogonally accumulated projection method, or the OAP for short) for solving linear system of equations with symmetric coefficient matrix is introduced in this paper. With the Lanczos process the OAP creates a sequence of mutually orthogonal vectors, on the basis of which the projections of the unknown vectors are easily obtained, and thus the approximations to the unknown vectors can be simply constructed by a combination of these projections. This method is an application of the accumulated projection technique proposed recently by the authors of this paper, and can be regarded as a match of conjugate gradient method (CG) in its nature since both the CG and the OAP can be regarded as iterative methods, too. Unlike the CG method which can be only used to solve linear systems with symmetric positive definite coefficient matrices, the OAP can be used to handle systems with indefinite symmetric matrices. Unlike classical Krylov subspace methods which usually ignore the issue of loss of orthogonality, OAP uses an effective approach to detect the loss of orthogonality and a restart strategy is used to handle the loss of orthogonality. Numerical experiments are presented to demonstrate the efficiency of the OAP.

This is the second paper in a series following Tian and Xu (2015), on the construction of a mathematical theory of the gauged linear σ-model (GLSM). In this paper, assuming the existence of virtual moduli cycles and their certain properties, we define the correlation function of GLSM for a fixed smooth rigidified r-spin curve.

In this paper, we introduce the notions of differential graded (DG) Poisson algebra and DG Poisson module. Let $A$ be any DG Poisson algebra. We construct the universal enveloping algebra of $A$ explicitly, which is denoted by $A^{ue}$. We show that $A^{ue}$ has a natural DG algebra structure and it satisfies certain universal property. As a consequence of the universal property, it is proved that the category of DG Poisson modules over $A$ is isomorphic to the category of DG modules over $A^{ue}$. Furthermore, we prove that the notion of universal enveloping algebra $A^{ue}$ is well-behaved under opposite algebra and tensor product of DG Poisson algebras. Practical examples of DG Poisson algebras are given throughout the paper including those arising from differential geometry and homological algebra., Comment: Accepted by Science China Mathematics

Reflected Ornstein-Uhlenbeck process is a process that returns continuously and immediately to the interior of the state space when it attains a certain boundary. It is an extended model of the traditional Ornstein-Uhlenbeck process being extensively used in finance as a one-factor short-term interest rate model. In this paper, under certain constraints, we are concerned with the problem of estimating the unknown parameter in the reflected Ornstein-Uhlenbeck processes with the general drift coefficient. The methodology of estimation is built upon the maximum likelihood approach and the method of stochastic integration. The strong consistency and asymptotic normality of estimator are derived. As a by-product of the use, we also establish Girsanovs theorem of our model in this paper.

In the previous paper \cite{L-Z}, for a characteristic problem with not necessarily small initial data given on a complete null cone decaying like that in the work \cite{Ch-K} of the stability of Minkowski spacetime by Christodoulou and Klainerman, we proved the local existence in retarded time, which means the solution to the vacuum Einstein equations exists in a uniform future neighborhood, while the global existence in retarded time is the weak cosmic censorship conjecture. In this paper, we prove that the local existence in retarded time still holds when the data is assumed to decay slower, like that in Bieri's work \cite{Bie} on the extension to the stability of Minkowski spacetime. Such decay guarantees the existence of the limit of the Hawking mass on the initial null cone, when approaching to infinity, in an optimal way., Comment: 30 pages. arXiv admin note: text overlap with arXiv:1406.0048

This paper investigates the equality-constrained minimization of polynomial functions. Let ℝ be the field of real numbers, and ℝ[x 1, …, x n ] the ring of polynomials over ℝ in variables x 1, …, x n . For an f ∈ ℝ[x 1, …, x n ] and a finite subset H of ℝ[x 1, …, x n ], denote by $$\mathcal{V}(f:H)$$ the set { $$\left. {f\left( {\bar \alpha } \right)} \right|\bar \alpha \in \mathbb{R}^n$$ , and $$h(\bar \alpha ) = 0,\forall h \in H$$ }. We provide an effective algorithm for computing a finite set U of non-zero univariate polynomials such that the infimum $$inf \mathcal{V}(f:H)$$ of $$\mathcal{V}(f:H)$$ is a root of some polynomial in U whenever $$inf \mathcal{V}(f:H) \ne \pm \infty$$ . The strategies of this paper are decomposing a finite set of polynomials into triangular chains of polynomials and computing the so-called revised resultants. With the aid of the computer algebraic system Maple, our algorithm has been made into a general program to treat the equality-constrained minimization of polynomials with rational coefficients.

We study the N = 1 super Schrodinger algebra S in (1 + 1)-dimensional spacetime. The first part of this paper determines the necessary and sufficient conditions for highest weight supermodules over S to be simple. The paper also describes the structure of all Verma supermodules and determines all simple Harish-Chandra supermodules over S.

In this paper, we develop a novel alternating linearization method for solving convex minimization whose objective function is the sum of two separable functions. The motivation of the paper is to extend the recent work Goldfarb et al. (2013) to cope with more generic convex minimization. For the proposed method, both the separable objective functions and the auxiliary penalty terms are linearized. Provided that the separable objective functions belong to C 1,1(ℝ n ), we prove the O(1/∈) arithmetical complexity of the new method. Some preliminary numerical simulations involving image processing and compressive sensing are conducted.

This paper finishes the classification of three-generator finite p -groups G such that Φ( G )≤ Z ( G ). This paper is a part of classification of finite p -groups with a minimal non-abelian subgroup of index p , and partly solves a problem proposed by Berkovich (2008).

The orientably-regular embeddings of complete multipartite graphs have been determined by the contributions of several papers. After that, a natural question can be asked: How about the regular embeddings of the multipartite graphs with m parts, while each part contains n vertices (not necessarily complete multipartite). In this paper, we classify all the orientably-regular embeddings of these graphs when m is a prime q and n is a prime power pe.

This paper provides an estimation procedure for average treatment effect through a random coefficient dummy endogenous variable model. A leading example of the model is estimating the effect of a training program on earnings. The model is composed of two equations: an outcome equation and a decision equation. Given the linear restriction in outcome and decision equations, Chen (1999) provided a distribution-free estimation procedure under conditional symmetric error distributions. In this paper we extend Chen’s estimator by relaxing the linear index into a nonparametric function, which greatly reduces the risk of model misspecification. A two-step approach is proposed: the first step uses a nonparametric regression estimator for the decision variable, and the second step uses an instrumental variables approach to estimate average treatment effect in the outcome equation. The proposed estimator is shown to be consistent and asymptotically normally distributed. Furthermore, we investigate the finite performance of our estimator by a Monte Carlo study and also use our estimator to study the return of college education in different periods of China. The estimates seem more reasonable than those of other commonly used estimators.

This paper discusses a search problem for a Helix target motion in which any information of the target position is not available to the searchers. There exist three searchers start searching for the target from the origin. The purpose of this paper is to formulate a search model and finds the conditions under which the expected value of the first meeting time between one of the searchers and the target is finite. Also, the existence of the optimal search plan that minimizes the expected value of the first meeting time is shown. Furthermore, this optimal search plan is found. The effectiveness of this method is illustrated by using an example with numerical results.

In this paper we prove three conjectures on congruences involving central binomial coefficients or Lucas sequences. Let $p$ be an odd prime and let $a$ be a positive integer. We show that if $p\equiv 1\pmod{4}$ or $a>1$ then $$ \sum_{k=0}^{\lfloor\frac34p^a\rfloor}\binom{-1/2}k\equiv\left(\frac{2}{p^a}\right)\pmod{p^2}, $$ where $(-)$ denotes the Jacobi symbol. This confirms a conjecture of the second author. We also confirm a conjecture of R. Tauraso by showing that $$\sum_{k=1}^{p-1}\frac{L_k}{k^2}\equiv0\pmod{p}\quad {\rm provided}\ \ p>5,$$ where the Lucas numbers $L_0,L_1,L_2,\ldots$ are defined by $L_0=2,\ L_1=1$ and $L_{n+1}=L_n+L_{n-1}\ (n=1,2,3,\ldots)$. Our third theorem states that if $p\not=5$ then we can determine $F_{p^a-(\frac{p^a}5)}$ mod $p^3$ in the following way: $$\sum_{k=0}^{p^a-1}(-1)^k\binom{2k}k\equiv\left(\frac{p^a}5\right)\left(1-2F_{p^a-(\frac{p^a}5)}\right)\ \pmod{p^3},$$ which appeared as a conjecture in a paper of Sun and Tauraso in 2010., 16 pages, final published version

The paper gives a new approach to statistical simulation and resampling by the use of numbertheoretic methods and representative points. Resempling techniques take samples from an approximate population. The bootstrap suggests to use a random sample to form an approximate population. We propose to construct some approximate population distribution by the use of two kinds of representative points, and samples are taken from these approximate distributions. The statistical inference is based on those samples. The statistical inference in this paper involves estimation of mean, variance, skewness, kurtosis, quantile and density of the population distribution. Our results show that the new method can significantly improve the results by the use of Monte Carlo methods.

The purpose of this paper is to solve the problem of determining the limits of multivariate rational functions. It is essential to decide whether or not {ie1-1} for two non-zero polynomials f, g ∈ ℝ[x1, …, xn] with f(0, …, 0) = g(0, …, 0) = 0. For two such polynomials f and g, we establish two necessary and sufficient conditions for the rational function {ie1-2} to have its limit 0 at the origin. Based on these theoretic results, we present an algorithm for deciding whether or not {ie1-3}, where f, g ∈ ℝ[x1, …, xn] are two non-zero polynomials. The design of our algorithm involves two existing algorithms: one for computing the rational univariate representations of a complete chain of polynomials, another for catching strictly critical points in a real algebraic variety. The two algorithms are based on the well-known Wu’s method. With the aid of the computer algebraic system Maple, our algorithm has been made into a general program. In the final section of this paper, several examples are given to illustrate the effectiveness of our algorithm.

In this paper, explicit determination of the cyclotomic numbers of order l and 2 l , for odd prime l ≡ 3 (mod 4), over finite field F q in the index 2 case are obtained, utilizing the explicit formulas on the corresponding Gauss sums. The main results in this paper are related with the number of rational points of certain elliptic curve, called "Legendre curve", and the properties and value distribution of such number are also presented.

Linear recurring sequences over finite fields play an important role in coding theory and cryptography. It is known that subfield subcodes of linear codes yield some good codes. In this paper, we study linear recurring sequences and subfield subcodes. Let M q m ( f(x) ) denote the set of all linear recurring sequences over F q m with characteristic polynomial f(x) over F q m . Denote the restriction of M q m ( f(x) ) to sequences over F q and the set after applying trace function to each sequence in M q m ( f(x) ) by M q m ( f(x) )| F q and Tr( M q m ( f(x) )), respectively. It is shown that these two sets are both complete sets of linear recurring sequences over F q with some characteristic polynomials over F q . In this paper, we firstly determine the characteristic polynomials for these two sets. Then, using these results, we determine the generator polynomials of subfield subcodes and trace codes of cyclic codes over F q m .

In this paper, we consider the following equation u t = ( u m ) xx + ( u n ) x , with the initial condition as Dirac measure. Attention is focused on existence, nonexistence, uniqueness and the asymptotic behavior near (0, 0) of solution to the Cauchy's problem. The special feature of this equation lies in nonlinear convection effect, i.e., the equation possesses nonlinear hyperbolic character as well as degenerate parabolic one. The situation leads to a more sophisticated mathematical analysis. To our knowledge, the solvability of singular solution to the equation has not been concluded yet. Here based on the previous works by the authors, we show that there exists a critical number n 0 = m +2 such that a unique source-type solution to this equation exists if 0 ≤ n n 0 , while such a solution does not exist if n ≥ n 0 (see our working paper). Our results give the perfect answer to such topical researches.

This paper discusses conditions under which the solution of linear system with minimal Schatten-p norm, 0 < p ⩽ 1, is also the lowest-rank solution of this linear system. To study this problem, an important tool is the restricted isometry constant (RIC). Some papers provided the upper bounds of RIC to guarantee that the nuclear-norm minimization stably recovers a low-rank matrix. For example, Fazel improved the upper bounds to δ 4 < 0.558 and δ 3 < 0.4721, respectively. Recently, the upper bounds of RIC can be improved to δ 2 < 0.307. In fact, by using some methods, the upper bounds of RIC can be improved to δ 2 < 0.4931 and δ 2 < 0.309. In this paper, we focus on the lower bounds of RIC, we show that there exists linear maps A with δ 2 > 1/√2 or δ > 1/3 for which nuclear norm recovery fail on some matrix with rank at most r. These results indicate that there is only a little limited room for improving the upper bounds for δ 2 and δ . Furthermore, we also discuss the upper bound of restricted isometry constant associated with linear maps A for Schatten p (0 < p < 1) quasi norm minimization problem.

This paper is concerned with the optimal error estimates and energy conservation properties of the alternating direction implicit finite-difference time-domain (ADI-FDTD) method which is a popular scheme for solving the 3D Maxwell’s equations. Precisely, for the case with a perfectly electric conducting (PEC) boundary condition we establish the optimal second-order error estimates in both space and time in the discrete H 1-norm for the ADI-FDTD scheme, and prove the approximate divergence preserving property that if the divergence of the initial electric and magnetic fields are zero, then the discrete L 2-norm of the discrete divergence of the ADI-FDTD solution is approximately zero with the second-order accuracy in both space and time. The key ingredient is two new discrete modified energy norms which are second-order in time perturbations of two new energy conservation laws for the Maxwell’s equations introduced in this paper. Furthermore, we prove that, in addition to two known discrete modified energy identities which are second-order in time perturbations of two known energy conservation laws, the ADI-FDTD scheme also satisfies two new discrete modified energy identities which are second-order in time perturbations of the two new energy conservation laws. This means that the ADI-FDTD scheme is unconditionally stable under the four discrete modified energy norms. Experimental results which confirm the theoretical results are presented.

The aim of this paper is to develop some basic theories of stochastic functional differential equations (SFDEs) under the local Lipschitz condition in continuous functions space C. Firstly, we establish a global existence-uniqueness lemma for the SFDEs under the global Lipschitz condition in C without the linear growth condition. Then, under the local Lipschitz condition in C, we show that the non-continuable solution of SFDEs still exists if the drift coefficient and diffusion coefficient are square-integrable with respect to t when the state variable equals zero. And the solution of the considered equation must either explode at the end of the maximum existing interval or exist globally. Furthermore, some more general sufficient conditions for the global existenceuniqueness are obtained. Our conditions obtained in this paper are much weaker than some existing results. For example, we need neither the linear growth condition nor the continuous condition on the time t . Two examples are provided to show the effectiveness of the theoretical results.

This paper is a continuation of the author’s paper in 2009, where the abstract theory of fold completeness in Banach spaces has been presented. Using obtained there abstract results, we consider now very general boundary value problems for ODEs and PDEs which polynomially depend on the spectral parameter in both the equation and the boundary conditions. Moreover, equations and boundary conditions may contain abstract operators as well. So, we deal, generally, with integro-differential equations, functional-differential equations, nonlocal boundary conditions, multipoint boundary conditions, integro-differential boundary conditions. We prove n-fold completeness of a system of root functions of considered problems in the corresponding direct sum of Sobolev spaces in the Banach L q -framework, in contrast to previously known results in the Hilbert L 2-framework. Some concrete mechanical problems are also presented.

In this paper, we study the stability of solutions of the Cauchy problem for 1-D compressible Narvier-Stokes equations with general initial data. The asymptotic limit of solution is found, under some conditions. The results in this paper imply the case that the limit function of solution as t→∞ is a viscous contact wave in the sense, which approximates the contact discontinuity on any finite-time interval as the heat conduction coefficients toward zero. As a by-product, the decay rates of the solution for the fast diffusion equations are also obtained. The proofs are based on the elementary energy method and the study of asymptotic behavior of the solution to the fast diffusion equation.

In the present paper, we provide an error bound for the learning rates of the regularized Shannon sampling learning scheme when the hypothesis space is a reproducing kernel Hilbert space (RKHS) derived by a Mercer kernel and a determined net. We show that if the sample is taken according to the determined set, then, the sample error can be bounded by the Mercer matrix with respect to the samples and the determined net. The regularization error may be bounded by the approximation order of the reproducing kernel Hilbert space interpolation operator. The paper is an investigation on a remark provided by Smale and Zhou.

The paper is concerned with optimal control of backward stochastic differential equation (BSDE) driven by Teugel’s martingales and an independent multi-dimensional Brownian motion, where Teugel’s martingales are a family of pairwise strongly orthonormal martingales associated with Levy processes (see e.g., Nualart and Schoutens’ paper in 2000). We derive the necessary and sufficient conditions for the existence of the optimal control by means of convex variation methods and duality techniques. As an application, the optimal control problem of linear backward stochastic differential equation with a quadratic cost criteria (or backward linear-quadratic problem, or BLQ problem for short) is discussed and characterized by a stochastic Hamilton system.

In this article we consider a sequence of hierarchical space model of inverse problems. The underlying function is estimated from indirect observations over a variety of error distributions including those that are heavy-tailed and may not even possess variances or means. The main contribution of this paper is that we establish some oracle inequalities for the inverse problems by using quantile coupling technique that gives a tight bound for the quantile coupling between an arbitrary sample p-quantile and a normal variable, and an automatic selection principle for the nonrandom filters. This leads to the data-driven choice of weights. We also give an algorithm for its implementation. The quantile coupling inequality developed in this paper is of independent interest, because it includes the median coupling inequality in literature as a special case.

In this paper, we address the existence and asymptotic analysis of higher-dimensional contrast structure of singularly perturbed Dirichlet problem. Based on the existence, an asymptotical analysis of a steplike contrast structure (i.e., an internal transition layer solution) is studied by the boundary function method via a proposed smooth connection. In the framework of this paper, we propose a first integral condition, under which the existence of a heteroclinic orbit connecting two equilibrium points is ensured in a higher-dimensional fast phase space. Then, the step-like contrast structure is constructed, and the internal transition time is determined. Meanwhile, the uniformly valid asymptotical expansion of such an available step-like contrast structure is obtained. Finally, an example is presented to illustrate the result.

By catching the so-called strictly critical points, this paper presents an effective algorithm for computing the global infimum of a polynomial function. For a multivariate real polynomial f, the algorithm in this paper is able to decide whether or not the global infimum of f is finite. In the case of f having a finite infimum, the global infimum of f can be accurately coded in the Interval Representation. Another usage of our algorithm is to decide whether or not the infimum of f is attained when the global infimum of f is finite. In the design of our algorithm, the well-known Wu’s method plays an important role.