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1.
Let be a complete Riemannian manifold with , and let be two complete totally geodesic submanifolds in M. We prove that if n1 + n2 = n − 2 and if the distance , then Mi is isometric to , or with the canonical metric when ni>0, and thus, M is isometric to , or except possibly when n = 3 and M1 (or M2) with or n = 4 and M1 (or M2) . 相似文献
2.
Consider a supercritical superprocess X = {Xt, 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 ) λ 2 + ∫ ( 0 , + ∞ ) ( e - λ y - 1 + λ y ) n ( x , d y ) , ? x ∈ E , λ > 0 , where a ∈ B b ( E ) , b ∈ B b + ( E ) , and n is a kernel from E to (0,+∞) satisfying sup ? x ∈ E ∫ 0 + ∞ y 2 n ( x , d y ) < + ∞ . Put T t f ( x ) = P δ x ? f , X t ? . Suppose that the semigroup {Tt; t≥0}is compact. Let λ0 be the eigenvalue of the (possibly non-symmetric) generator L of {Tt}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 {Tt}, we prove that for a large class of suitable functions f, lim ? t → + ∞ e - λ 0 t ? f , X t ? = W ∞ ∫ E ? ^ 0 ( y ) f ( y ) m ( d y ) , ? P μ - 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 - λ 0 t ? ? 0 , X t ? . Moreover, the exceptional set in the above limit does not depend on the initial measure μ and the function f. 相似文献
3.
Miao LOU 《Frontiers of Mathematics in China》2019,14(1):123-134
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
4.
Xie-Bin CHEN 《Frontiers of Mathematics in China》2019,14(6):1117
We consider the problem of existence of a Hamiltonian cycle containing a matching and avoiding some edges in an n-cube , and obtain the following results. Let , and with . If M is a matching and every vertex is incident with at least two edges in the graph , then all edges of M lie on a Hamiltonian cycle in . Moreover, if or , then the upper bound of number of faulty edges tolerated is sharp. Our results generalize the well-known result for . 相似文献
5.
Let φ be a growth function, and let A : = - ( ? - i a ) ? ( ? - i a ) + V be a magnetic Schr?dinger operator on L 2 ( ? n ) , n ≥ 2 , where α : = ( α 1 , α 2 , ? , α n ) ∈ L l o c 2 ( ? n , ? n ) and 0 ≤ V ∈ L l o c 1 ( ? n ) . We establish the equivalent characterizations of the Musielak-Orlicz-Hardy space H A , φ ( ? n ) , defined by the Lusin area function associated with { e - t 2 A } t > 0 , in terms of the Lusin area function associated with { e - t A } t > 0 , the radial maximal functions and the nontangential maximal functions associated with { e - t 2 A } t > 0 and { e - t A } t > 0 , respectively. The boundedness of the Riesz transforms L k A - 1 / 2 , k ∈ { 1 , 2 , ? , n } , from H A , φ ( ? n ) to L φ ( ? n ) is also presented, where Lk is the closure of ? ? x k - i α k in L 2 ( ? n ) . These results are new even when φ ( x , t ) : = ω ( x ) t p for all x ∈ ? n and t ∈(0,+∞) with p ∈(0, 1] and ω ∈ A ∞ ( ? n ) (the class of Muckenhoupt weights on ? n ). 相似文献
6.
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. 相似文献
7.
Rui ZHANG 《Frontiers of Mathematics in China》2019,14(5):1017
We prove that, with at most exceptions, all even positive integers up to Nare expressible in the form ,where are prime numbers. This gives large improvement of a recent result due to M. Zhang and J. J. Li. 相似文献
8.
9.
We extend Vandermonde matrices to generalized Vandermonde tensors. We call an mth order n-dimensional real tensor A = ( A i 1 i 2 ... i m ) a type-1 generalized Vandermonde (GV) tensor, or GV1 tensor, if there exists a vector v = ( v 1 , v 2 ... v n ) T such that A i 1 i 2 ... i m = v i 1 i 2 + i 3 + ... + i m - 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 = ( B i 1 i 2 ... i m - 1 ) such that A i 1 i 2 ... i m = B i 1 i 2 ... i m - 1 i m - 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. 相似文献
10.
We study a superminimal surface M immersed into a hyperquadric Q2 in several cases classified by two global defined functions and , which were introduced by X. X. Jiao and J. Wang to study a minimal immersion f : . In case both and are not identically zero, it is proved that f is superminimal if and only if f is totally real or is also minimal, where is the standard inclusion map. In the rest case that or , the minimal immersion f is automatically superminimal. As a consequence, all the superminimal two-spheres in Q2 are completely described. 相似文献
11.
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 ∞ ( R 4 n , W 4 n ) → C ∞ ( R 4 n , W 4 n ) , where W 4 n is the tensor product of three algebras, i.e., the hyperbolic quaternion B ^ , the bicomplex number B , and the Clifford algebra R n . The operator D is a square root of the Laplacian in R 4 n , introduced by the formula D | = ∑ j = 0 3 K j ? Z j with K j being the basis of B ^ , and ? Z j denoting the twisted Hermitian Dirac operators in the bicomplex Clifford algebra B ? R 0,4 n 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. 相似文献
12.
We study the Schrödinger-KdV system where , , and ,i= 1,2,a.e. .We obtain the existence of nontrivial ground state solutions for the above system by variational methods and the Nehari manifold. 相似文献
13.
Let (Xt)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 ) d x d y , ? 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 T t V ( f ) ( x ) = E x ( exp ? ( - ∫ 0 t V ( X s ) d s ) f ( X t ) ) , ? x ∈ ? d , f ∈ L 2 ( ? d ; d x ) . In particular, we prove that for J ( x , y ) ≈ | x - y | - d - a l { | x - y | ≤ 1 } + e - | x - y | l { | x - y | > 1 } with α ∈(0, 2) and V ( x ) = | x | λ with λ>0, ( T t V ) t ≥ 0 is intrinsically ultracontractive if and only if λ>1; and that for symmetric α-stable process (Xt)t≥0 with α ∈(0, 2) and V ( x ) = log ? λ ( 1 + | x | ) with some λ>0, ( T t V ) t ≥ 0 is intrinsically ultracontractive (or intrinsically supercontractive) if and only if λ>1, and ( T t V ) t ≥ 0 is intrinsically hypercontractive if and only if λ ≥ 1 . Besides, we also investigate intrinsic contractivity properties of ( T t V ) t ≥ 0 for the case that lim inf ? | x | → + ∞ V ( x ) < + ∞ 相似文献
14.
We investigate k-uniform loose paths. We show that the largest Heigenvalues of their adjacency tensors, Laplacian tensors, and signless Laplacian tensors are computable. For a k-uniform loose path with length l ≥ 3 , we show that the largest H-eigenvalue of its adjacency tensor is ( ( 1 + 5 ) / 2 ) 2 / k when l = 3 and λ ( A ) = 3 1 / k when l = 4 , respectively. For the case of l ≥ 5 , we tighten the existing upper bound 2. We also show that the largest H-eigenvalue of its signless Laplacian tensor lies in the interval (2, 3) when l ≥ 5 . Finally, we investigate the largest H-eigenvalue of its Laplacian tensor when k is even and we tighten the upper bound 4. 相似文献
15.
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. 相似文献
16.
Let c>1 and : We study the solubility of the Diophantine inequality in Piatetski-Shapiro primes p1,p2, .., ps of the form for some , and improve the previous results in the cases s = 2, 3, 4. 相似文献
17.
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 | a 3 − λ a 2 2 | ≤ 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. 相似文献
18.
Assuming that the operators L1, L2 are self-adjoint and satisfy the generalized Davies-Gaffney estimates, we shall prove that the weighted Hardy space associated to operators L1, L2 on product domain, which is defined in terms of area function, has an atomic decomposition for some weight . 相似文献
19.
Let be a supercritical branching process in an independent and identically distributed random environment. We prove Cramér moderate deviations and Berry-Esseen bounds for log( ) uniformly in ,which extend the corresponding results by I. Grama, Q. Liu, and M. Miqueu [Stochastic Process. Appl., 2017, 127: 1255–1281] established for . The extension is interesting in theory, and is motivated by applications. A new method is developed for the proofs; some conditions of Grama et al. are relaxed in our present setting. An example of application is given in constructing confidence intervals to estimate the criticality parameter in terms of log( ) and n. 相似文献
20.
We give the explicit formulas of the minimizers of the anisotropic Rudin-Osher-Fatemi models E 1 φ ( u ) = ∫ Ω φ o ( D u ) d x + λ ∫ Ω | u − f | d x , u ∈ B V ( Ω ) , E 2 φ ( u ) = ∫ Ω φ o ( D u ) d x + λ ∫ Ω ( u − f ) 2 d x , u ∈ B V ( Ω ) , where Ω ⊂ ? 2 is a domain, φ o is an anisotropic norm on ? 2 , and f is a solution of the anisotropic 1-Laplacian equations. 相似文献