Branching random walks with random environments in time

We consider a branching random walk on N with a random environment in time （denoted by ξ）. Let Zn be the counting measure of particles of generation n, and let Zn（t） be its Laplace transform. We show the convergence of the free energy n-llog Zn（t）, large deviation principles, and central limit theorems for the sequence of measures {Zn}, and a necessary and sufficient condition for the existence of moments of the limit of the martingale Zn（t）/E[Zn（t）ξ].     

Maximal function characterizations of Musielak-Orlicz-Hardy spaces associated with magnetic Schr?dinger operators

Let φ be a growth function, and let A:=-(?-ia)?(?-ia)+V be a magnetic Schr?dinger operator on L2(?n),n≥2, where α:=(α1,α2,?,αn)∈Lloc2(?n,?n) and 0≤V∈Lloc1(?n). We establish the equivalent characterizations of the Musielak-Orlicz-Hardy space HA,φ(?n), defined by the Lusin area function associated with {e-t2A}t>0, in terms of the Lusin area function associated with {e-tA}t>0, the radial maximal functions and the nontangential maximal functions associated with {e-t2A}t>0 and {e-tA}t>0, respectively. The boundedness of the Riesz transforms LkA-1/2,k∈{1,2,?,n}, from HA,φ(?n) to Lφ(?n) is also presented, where L_k is the closure of ??xk-iαk in L2(?n). These results are new even when φ(x,t):=ω(x)tp for all x∈?nand t ∈(0,+∞) with p ∈(0, 1] and ω∈A∞(?n) (the class of Muckenhoupt weights on ?n).     

Strong law of large numbers for supercritical superprocesses under second moment condition

Consider a supercritical superprocess X = {X_t, t≥0} on a locally compact separable metric space (E,m). Suppose that the spatial motion of X is a Hunt process satisfying certain conditions and that the branching mechanism is of the form ψ(x,λ)=-a(x)λ+b(x)λ2+∫(0,+∞)(e-λy-1+λy)n(x,dy),?x∈E,λ>0, where a∈Bb(E),b∈Bb+(E), and n is a kernel from E to (0,+∞) satisfying sup?x∈E∫0+∞y2n(x,dy)<+∞. Put Ttf(x)=Pδx?f,Xt?. Suppose that the semigroup {T_t; t≥0}is compact. Let λ₀ be the eigenvalue of the (possibly non-symmetric) generator L of {T_t}that has the largest real part among all the eigenvalues of L, which is known to be real-valued. Let ?0 and ?^0 be the eigenfunctions of L and L^(the dual of L) associated with λ₀, respectively. Assume λ₀>0. Under some conditions on the spatial motion and the ?0-transform of the semigroup {T_t}, we prove that for a large class of suitable functions f, lim?t→+∞e-λ0t?f,Xt?=W∞∫E?^0(y)f(y)m(dy),?Pμ-a.s., for any finite initial measure μ on E with compact support, where W_∞ is the martingale limit defined by W∞:=lim?t→+∞e-λ0t??0,Xt?. Moreover, the exceptional set in the above limit does not depend on the initial measure μ and the function f.     

Exponential sums involving Maass forms

We study the exponential sums involving l：burmr coeffcients ot Maass forms and exponential functions of the form e（anZ）, where 0 ≠ α∈R and 0 〈 β 〈 1. An asymptotic formula is proved for the nonlinear exponential sum ∑x〈n≤2x λg（n）e（αnβ）, when β = 1/2 and ｜α｜ is close to 2√ q C Z＋, where Ag（n） is the normalized n-th Fourier coefficient of a Maass cusp form for SL2 （Z）. The similar natures of the divisor function 7（n） and the representation function r（n） in the circle problem in nonlinear exponential sums of the above type are also studied.     

Intrinsic contractivity properties of Feynman-Kac semigroups for symmetric jump processes with infinite range jumps

Let (X_t)_t≥0 be a symmetric strong Markov process generated by non-local regular Dirichlet form (D, D(D)) as follows: D(f,g)=∫?d∫?d(f(x)-f(y))(g(x)-g(y))J(x,y)dxdy,?f,g∈D(D), where J(x, y) is a strictly positive and symmetric measurable function on ?d×?d. We study the intrinsic hypercontractivity, intrinsic supercontractivity, and intrinsic ultracontractivity for the Feynman-Kac semigroup TtV(f)(x)=Ex(exp?(-∫0tV(Xs)ds)f(Xt)),?x∈?d,f∈L2(?d;dx). In particular, we prove that for J(x,y)≈|x-y|-d-al{|x-y|≤1}+e-|x-y|l{|x-y|>1} with α ∈(0, 2) and V(x)=|x|λ with λ>0, (TtV)t≥0 is intrinsically ultracontractive if and only if λ>1; and that for symmetric α-stable process (X_t)_t≥0 with α ∈(0, 2) and V(x)=log?λ(1+|x|) with some λ>0, (TtV)t≥0 is intrinsically ultracontractive (or intrinsically supercontractive) if and only if λ>1, and (TtV)t≥0 is intrinsically hypercontractive if and only if λ≥1. Besides, we also investigate intrinsic contractivity properties of (TtV)t≥0 for the case that lim inf?|x|→+∞V(x)<+∞     

Averages of shifted convolution sums for arithmetic functions

Let f be a full-level cusp form for GLm(Z) with Fourier coefficients Af(cm-2,…, c1, n): Let λ(n) be either the von Mangoldt function Λ(n) or the k-th divisor function τk(n): We consider averages of shifted convolution sums of the type Σ|h|≤H |ΣX相似文献   

Well-posedness of degenerate differential equations in H?lder continuous function spaces

Using known operator-valued Fourier multiplier results on vectorvalued H?lder continuous function spaces, we completely characterize the wellposedness of the degenerate differential equations (Mu)'(t)=Au(t)+f(t) for t∈R in H?lder continuous function spaces Ca(R;X) by the boundedness of the M-resolvent of A, where A and M are closed operators on a Banach space X satisfying D(A)?D(M).     

Unit groups of quotient rings of complex quadratic rings

For a square-free integer d other than 0 and 1, let K=?(d), where ? is the set of rational numbers. Then K is called a quadratic field and it has degree 2 over ?. For several quadratic fields K=?(d), the ring R_dof integers of K is not a unique-factorization domain. For d<0, there exist only a finite number of complex quadratic fields, whose ring Rd of integers, called complex quadratic ring, is a unique-factorization domain, i.e., d = −1,−2,−3,−7,−11,−19,−43,−67,−163. Let ϑ denote a prime element of Rd, and let n be an arbitrary positive integer. The unit groups of Rd/〈vn〉 was determined by Cross in 1983 for the case d = −1. This paper completely determined the unit groups of Rd/〈vn〉 for the cases d = −2,−3.     

Sum-connectivity index of a graph

Let G be a simple connected graph, and let di be the degree of its i-th vertex. The sum-connectivity index of the graph G is defined as χ(G)=Σvivj∈E(G)? (di+dj)−1/2. We discuss the effect on χ(G) of inserting an edge into a graph. Moreover, we obtain the relations between sum-connectivity index and Randić index.     

Fixed points of smoothing transformation in random environment

At each time n\in \mathbb{N},\mathrm{l}\mathrm{e}\mathrm{t}{\overline{Y}}^{(n)}(\xi )=(y_1^{(n)}(\xi ),y_2^{(n)}(\xi ),\cdots ) be a random sequence of non-negative numbers that are ultimately zero in a random environment\xi \text{=}（{\xi }_n{）}_{n\in \mathbb{N}}. The existence and uniqueness of the nonnegative fixed points of the associated smoothing transformation in random environment are considered. These fixed points are solutions to the distributional equation for a.e.\xi ,Z(\xi )\stackrel{\text{d}}{=}{\sum }_{i\in \mathbb{N}+}{y}_i^{(\mathrm{0})}(\xi )Z_i^{(\mathrm{1})}(\xi ),where ｛Z_i^{(\mathrm{1})}:i\in {\mathbb{N}}_{+}｝ are random variables in random environment which satisfy that for any environment\xi; under P_{\xi }; ｛Z_i^{(\mathrm{1})}:i\in {\mathbb{N}}_{+}｝are independent of each other and {\bar{Y}}^{(\mathrm{0})}(\xi ), and have the same conditional distribution P_{\xi }(Z_i^{(1)}(\xi )\in \cdot )=P_{T\xi }(Z(T\xi )\in \cdot ) where T is the shift operator. This extends the classical results of J. D. Biggins [J. Appl. Probab., 1977, 14: 25-37] to the random environment case. As an application, the martingale convergence of the branching random walk in random environment is given as well.     

Regularity for anisotropic solutions to some nonlinear elliptic system

This paper deals with anisotropic solutions u∈W1,(pi)(Ω,?N) to the nonlinear elliptic system −Σi=1nDi(aiα(χ,Du(χ)))=−Σi=1nDiFiα(χ), α=1,2,...,N, We present a monotonicity inequality for the matrix a=(aiα)∈?N×n,whichguarantees global pointwise bounds for anisotropic solutionsu.     

Fekete and Szegö problem for a subclass of quasi-convex mappings in several complex variables

Let K be the familiar class of normalized convex functions in the unit disk. Keogh and Merkes proved the well-known result that max?f∈K|a3−λa22|≤max?{1/3,|λ−1|},λ∈?, and the estimate is sharp for each λ. We investigate the corresponding problem for a subclass of quasi-convex mappings of type B defined on the unit ball in a complex Banach space or on the unit polydisk in ?n. The proofs of these results use some restrictive assumptions, which in the case of one complex variable are automatically satisfied.     

Minimizers of anisotropic Rudin-Osher-Fatemi models

We give the explicit formulas of the minimizers of the anisotropic Rudin-Osher-Fatemi models E1φ(u)=∫Ωφo(Du)dx+λ∫Ω|u−f|dx, u∈BV(Ω),E2φ(u)=∫Ωφo(Du)dx+λ∫Ω(u−f)2dx, u∈BV(Ω), where Ω⊂?2 is a domain, φo is an anisotropic norm on ?2, and f is a solution of the anisotropic 1-Laplacian equations.     

Asymptotic properties of supercritical branching processes in random environments

We consider a supercritical branching process （Zn） in an independent and identically distributed random environment ξ, and present some recent results on the asymptotic properties of the limit variable W of the natural martingale Wn = Zn/E[Zn｜ξ], the convergence rates of W - Wn （by considering the convergence in law with a suitable norming, the almost sure convergence, the convergence in Lp, and the convergence in probability）, and limit theorems （such as central limit theorems, moderate and large deviations principles） on （log Zn）.     

Continuity of functors with respect to generalized inductive limits

Let(Ai,φi,i+1) be a generalized indue Live system of a sequeiiee (Ai) of unital separable C^*-algebras,with A =limi→∞(Ai,φi,i+1). Set φj,i=φi-1,i^0…0φj+1,j+2^0 φj,j+1 for all i>j. We prove that if φj,i are order zero completely positive contractions for all j and i>j, And L:=inf{λ|λ∈σ(φj,i(1Aj)) for all j uud i>j}>0, where σ(φj,i(1Aj)) is the speetrum of φj,i(1Aj),than limi→∞(Cu(Ai),Cu((φi,i+1))=Cu(A), where Cu(A) is a stable version of the Cuntz semigroup of C^*-algebra A. Let (An,φm,n) be a generalized inductive syfitem of C^*-algahrafl, with the ipmkn order zero completely positive contractions. We also prove that if the decomposition rank (nuclear dimension) of ,4n is no more t han some integer k for each n, then the decompostition rank (nuclear dimension) of A is also no more than k.     

Matchings extend to Hamiltonian cycles in hypercubes with faulty edges

We consider the problem of existence of a Hamiltonian cycle containing a matching and avoiding some edges in an n-cube {{\displaystyle Q}}_n, and obtain the following results. Let n\ge \mathrm{3},\hspace{0.17em}M\subset E({{\displaystyle Q}}_n), and F\subset E({{\displaystyle Q}}_n)\backslash M with \mathrm{1}\le \left|F\right|\le \mathrm{2}n-\mathrm{4}-\left|M\right|. If M is a matching and every vertex is incident with at least two edges in the graph {{\displaystyle Q}}_n-F, then all edges of M lie on a Hamiltonian cycle in {{\displaystyle Q}}_n-F. Moreover, if \left|M\right|=\mathrm{1} or \left|M\right|=\mathrm{2}, then the upper bound of number of faulty edges tolerated is sharp. Our results generalize the well-known result for \left|M\right|=\mathrm{1}.     

Bicomplex Hermitian Clifford analysis

Complex Hermitian Clifford analysis emerged recently as a refinement of the theory of several complex variables, while at the same time, the theory of bicomplex numbers motivated by the bicomplex version of quantum mechanics is also under full development. This stimulates us to combine the Hermitian Clifford analysis with the theory of bicomplex number so as to set up the theory of bicomplex Hermitian Clifford analysis. In parallel with the Euclidean Clifford analysis, the bicomplex Hermitian Clifford analysis is centered around the bicomplex Hermitian Dirac operator |D:C∞(R4n,W4n)→C∞(R4n,W4n), where W4n is the tensor product of three algebras, i.e., the hyperbolic quaternion B^, the bicomplex number B, and the Clifford algebra Rn. The operator D is a square root of the Laplacian in R4n, introduced by the formula D|=∑j=03Kj?Zj with Kjbeing the basis of B^, and ?Zj denoting the twisted Hermitian Dirac operators in the bicomplex Clifford algebra B?R0,4n whose definition involves a delicate construction of the bicomplexWitt basis. The introduction of the operator D can also overturn the prevailing opinion in the Hermitian Clifford analysis in the complex or quaternionic setting that the complex or quaternionic Hermitiean monogenic functions are described by a system of equations instead of by a single equation like classical monogenic functions which are null solutions of Dirac operator. In contrast to the Hermitian Clifford analysis in quaternionic setting, the Poisson brackets of the twisted real Clifford vectors do not vanish in general in the bicomplex setting. For the operator D, we establish the Cauchy integral formula, which generalizes the Martinelli-Bochner formula in the theory of several complex variables.     

Ground state solutions for a non-autonomous nonlinear Schrödinger-KdV system

We study the Schrödinger-KdV system\{\begin{array}{ll}-\mathrm{\Delta }u+{\lambda }_{\mathrm{1}}(x)u=u^{\mathrm{3}}+\beta uv,& u\in H^{\mathrm{1}}({ℝ}^N),\\ -\mathrm{\Delta }v+{\lambda }_{\mathrm{2}}(x)v={\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}{v}^{\mathrm{2}}+{\displaystyle \frac{\beta }{\mathrm{2}}}{u}^{\mathrm{2}},& v\in H^{\mathrm{1}}({ℝ}^N),\end{array}where N=\mathrm{1},\mathrm{2},\mathrm{3},\mathrm{}{\lambda }_i(x)\in C({ℝ}^N,ℝ),{\lim }_{\left|x\right|\to \infty }{\lambda }_i(x)={\lambda }_i(\infty ), and {\lambda }_i(x)\le {\lambda }_i(\infty ),i= 1,2,a.e. x\in {ℝ}^N.We obtain the existence of nontrivial ground state solutions for the above system by variational methods and the Nehari manifold.     

Lower deviations for supercritical branching processes with immigration

For a supercritical branching processes with immigration \{{{\displaystyle Z}}_n\}; it is known that under suitable conditions on the offspring and immigration distributions, Z_n/mⁿ converges almost surely to a finite and strictly positive limit, where m is the offspring mean. We are interested in the limiting properties of P({{\displaystyle Z}}_n={{\displaystyle k}}_n) with {{\displaystyle k}}_n=o(m^n) as n\to \infty. We give asymptotic behavior of such lower deviation probabilities in both Schröder and Böttcher cases, unifying and extending the previous results for Galton-Watson processes in literature.