Generalized Vandermonde tensors

We extend Vandermonde matrices to generalized Vandermonde tensors. We call an mth order n-dimensional real tensor A=(Ai1i2...im) a type-1 generalized Vandermonde (GV) tensor, or GV1 tensor, if there exists a vector v=(v1,v2...vn)T such that Ai1i2...im=vi1i2+i3+...+im-m+1, and call A a type-2 (mth order ndimensional) GV tensor, or GV2 tensor, if there exists an (m-1)th order tensor B=(Bi1i2...im-1) such that Ai1i2...im=Bi1i2...im-1im-1. In this paper, we mainly investigate the type-1 GV tensors including their products, their spectra, and their positivities. Applications of GV tensors are also introduced.     

Values of binary linear forms at prime arguments

value of a given binary linear form at prime arguments. Let λ₁ and λ₂ be positive real numbers such that λ₁/λ₂ is irrational and algebraic. For any (C, c) well-spaced sequence V and δ>0, let E(V, X, δ) denote the number of υ∈V with υ≤X for which the inequality |λ1p1+λ2ρ2−υ|<υ−δ has no solution in primes p₁, p₂. It is shown that for any ε>0,we have E(V, X, δ) «max(X35+2δ+ε,X23+43δ+ε).     

Upper bounds for signless Laplacian Z-spectral radius of uniform hypergraphs

Let ■ be a k-uniform hypergraph on n vertices with degree sequence △= d1≥…≥ dn =δ. In this paper, in terms of degree di , we give some upper bounds for the Z-spectral radius of the signless Laplacian tensor (Q(■)) of ■. Some examples are given to show the efficiency of these bounds.     

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).     

Waring-Goldbach problems for unlike powers with almost equal variables

We prove that almost all positive even integers n can be written as n=p22+p33+p44+p55 with |pkk-N4|≤N321325+? for 2≤k≤5. Moreover, it is proved that each sufficiently large odd integer N can be represented as N=p1+p22+p33+p44+p55 with |pkk-N5|≤N321325+?for 1≤k≤5.     

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}.     

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   

An isometrical CPn-theorem

Let M^n(n\ge \mathrm{3}) be a complete Riemannian manifold with {{\displaystyle \sec ?}}_M\ge \mathrm{1}, and let {{\displaystyle M}}_i^{{{\displaystyle n}}_i}(i=\mathrm{1},\mathrm{2}) be two complete totally geodesic submanifolds in M. We prove that if n₁ + n₂ = n − 2 and if the distance \left|{{\displaystyle M}}_{\mathrm{1}}{{\displaystyle M}}_{\mathrm{2}}\right|\ge \pi /\mathrm{2}, then M_i is isometric to {\mathbb{S}}^{{{\displaystyle n}}_i}/{{\displaystyle \mathrm{?}}}_h,{\mathrm{?}\mathrm{P}}^{{{\displaystyle n}}_i/\mathrm{2}}/{{\displaystyle \mathrm{?}}}_{\mathrm{2}}, or {\mathrm{?}\mathrm{P}}^{{{\displaystyle n}}_i/\mathrm{2}}/{{\displaystyle \mathrm{?}}}_{\mathrm{2}} with the canonical metric when n_i>0, and thus, M is isometric to {\mathbb{S}}^n/{{\displaystyle \mathrm{?}}}_h,{\mathrm{?}\mathrm{P}}^{n/\mathrm{2}}, or {\mathrm{?}\mathrm{P}}^{n/\mathrm{2}}/{{\displaystyle \mathrm{?}}}_{\mathrm{2}} except possibly when n = 3 and M₁ (or M₂) \stackrel{\mathrm{i}\mathrm{s}\mathrm{o}}{{\displaystyle \cong }}{\mathbb{S}}^{\mathrm{1}}/{{\displaystyle \mathrm{?}}}_h with h\ge \mathrm{2} or n = 4 and M₁ (or M₂) \stackrel{\mathrm{i}\mathrm{s}\mathrm{o}}{{\displaystyle \cong }}\mathrm{?}{\mathrm{P}}^{\mathrm{2}}.     

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.     

Superminimal surfaces in hyperquadric Q2

We study a superminimal surface M immersed into a hyperquadric Q₂ in several cases classified by two global defined functions {{\displaystyle \tau }}_X and {{\displaystyle \tau }}_Y, which were introduced by X. X. Jiao and J. Wang to study a minimal immersion f : M\to {{\displaystyle Q}}_{\mathrm{2}}. In case both {{\displaystyle \tau }}_X and {{\displaystyle \tau }}_Y are not identically zero, it is proved that f is superminimal if and only if f is totally real or i\circ f:M\to {ℂ\mathrm{P}}^{\mathrm{3}} is also minimal, where i:{{\displaystyle Q}}_{\mathrm{2}}\to {ℂ\mathrm{P}}^{\mathrm{3}} is the standard inclusion map. In the rest case that {{\displaystyle \tau }}_X\equiv \mathrm{0} or {{\displaystyle \tau }}_Y\equiv \mathrm{0}, the minimal immersion f is automatically superminimal. As a consequence, all the superminimal two-spheres in Q₂ are completely described.     

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.     

Oscillation and variation for Riesz transform in setting of Bessel operators on H1 and BMO

Let \lambda >\mathrm{0} and let the Bessel operator \mathrm{\Delta }\lambda ：=-{\displaystyle \frac{{\mathrm{d}}^{\mathrm{2}}}{{\mathrm{d}x}^{\mathrm{2}}}}-{\displaystyle \frac{\mathrm{2}\lambda }{x}}{\displaystyle \frac{\mathrm{d}}{\mathrm{d}x}} defined on {{\displaystyle ℝ}}_{+}:=(\mathrm{0},\infty ). We show that the oscillation and \rho-variation operators of the Riesz transform {{\displaystyle R}}_{{\mathrm{\Delta }}_{\lambda }} associated with {\mathrm{\Delta }}_{\lambda } are bounded on BMO({{\displaystyle ℝ}}_{+},{{\displaystyle dm}}_{\lambda }), where \rho >\mathrm{2} and {{\displaystyle \mathrm{d}m}}_{\lambda }=x^{\mathrm{2}\lambda }\mathrm{d}x. Moreover, we construct a {{\displaystyle (\mathrm{1},\infty )}}_{{\mathrm{\Delta }}_{\lambda }}-atom as a counterexample to show that the oscillation and \rho-variation operators of {{\displaystyle R}}_{{\mathrm{\Delta }}_{\lambda }} are not bounded from H^{\mathrm{1}}({ℝ}^{+},{{\displaystyle \mathrm{d}m}}_{\lambda }) to L^{\mathrm{1}}({ℝ}^{+},{{\displaystyle \mathrm{d}m}}_{\lambda }). Finally, we prove that the oscillation and the {{\displaystyle (\mathrm{1},\infty )}}_{{\mathrm{\Delta }}_{\lambda }}-variation operators for the smooth truncations associated with Bessel operators {{\displaystyle \tilde{R}}}_{{\mathrm{\Delta }}_{\lambda }} are bounded from H^{\mathrm{1}}({ℝ}^{+},{{\displaystyle \mathrm{d}m}}_{\lambda }) to L^{\mathrm{1}}({ℝ}^{+},{{\displaystyle \mathrm{d}m}}_{\lambda }).     

Extremum of a time-inhomogeneous branching random walk

Consider a time-inhomogeneous branching random walk, generated by the point process L_n which composed by two independent parts: ‘branching’offspring X_n with the mean \mathrm{1}+{B(\mathrm{1}+n)}^{-\beta } for \beta \in (\mathrm{0},\mathrm{1}) and ‘displacement’ {{\displaystyle \xi }}_n with a drift {A(\mathrm{1}+n)}^{-\mathrm{2}\alpha } for \alpha \in (\mathrm{0},\mathrm{1}/\mathrm{2}), where the ‘branching’ process is supercritical for B>0 but ‘asymptotically critical’ and the drift of the ‘displacement’ {{\displaystyle \xi }}_n is strictly positive or negative for \left|A\right|\rangle \mathrm{0} but ‘asymptotically’ goes to zero as time goes to infinity. We find that the limit behavior of the minimal (or maximal) position of the branching random walk is sensitive to the ‘asymptotical’ parameter \beta and \alpha.     

Slim exceptional set for sums of two squares,two cubes,and two biquadrates of primes

We prove that, with at most O\left(N^{\frac{\mathrm{17}}{\mathrm{192}}+\epsilon }\right) exceptions, all even positive integers up to Nare expressible in the form p_{\mathrm{1}}^{\mathrm{2}}+p_{\mathrm{2}}^{\mathrm{2}}+p_{\mathrm{3}}^{\mathrm{3}}+p_{\mathrm{4}}^{\mathrm{3}}+p_{\mathrm{5}}^{\mathrm{4}}+p_{\mathrm{6}}^{\mathrm{4}},where p_{\mathrm{1}},\mathrm{}{p}_{\mathrm{2}},.\mathrm{}.\mathrm{}.\mathrm{},\mathrm{}{p}_{\mathrm{6}} are prime numbers. This gives large improvement of a recent result O\left(N^{\frac{\mathrm{13}}{\mathrm{16}}+\epsilon }\right) due to M. Zhang and J. J. Li.     

Weighted product Hardy space associated with operators

Assuming that the operators L₁, L₂ are self-adjoint and {\mathrm{e}}^{-{{\displaystyle tL}}_i}(i=\mathrm{1},\mathrm{2}) satisfy the generalized Davies-Gaffney estimates, we shall prove that the weighted Hardy space {{\displaystyle H}}_{{{\displaystyle L}}_{\mathrm{1}},{{\displaystyle L}}_{\mathrm{2}},\omega }^{\mathrm{1}}({ℝ}^{{{\displaystyle n}}_{\mathrm{1}}}\times {ℝ}^{{{\displaystyle n}}_{\mathrm{2}}}) associated to operators L₁, L₂ on product domain, which is defined in terms of area function, has an atomic decomposition for some weight \omega.     

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.     

Functional inequalities for time-changed symmetric α-stable processes

We establish sharp functional inequalities for time-changed symmetric \alpha-stable processes on {\mathrm{ℝ}}^d with d\ge \mathrm{1} and \alpha \in (\mathrm{0},\mathrm{2}), which yield explicit criteria for the compactness of the associated semigroups. Furthermore, when the time change is defined via the special function W(x)={(\mathrm{1}+\left|x\right|)}^{\beta } with \beta >\alpha we obtain optimal Nash-type inequalities, which in turn give us optimal upper bounds for the density function of the associated semigroups.     

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.