In the papers of 2002 and 2007, Cai et al. introduced a series of congruences involving binomial coefficients under perfect moduli. This article generalizes these congruences to cubic cases leading to many new statements. For example, the congruence Πd|n（└d/e┘kd-1）μ(n/d) module n3 for e=2, 3, 4 and 6, and the following congruence
The parallel sum for adjointable operators on Hilbert C*-modules is introduced and studied. Some results known for matrices and bounded linear operators on Hilbert spaces are generalized to the case of adjointable operators on Hilbert C*-modules. It is shown that there exist a Hilbert C*-module H and two positive operators A, B ∈ L (H) such that the operator equation A1/2=(A + B)1/2X, X ∈ L (H) has no solution, where L (H) denotes the set of all adjointable operators on H.
We define the pure singularity category Dpsgb(R) as the Verdier quotient of the bounded pure derived category Dpurb(R) by the triangulated subcategory Kb(PP) of the bounded homotopy category consisting of pure projective modules, a sufficient and necessary condition under which Dpsgb(R) is equivalent to the stable category of the Gorenstein category G (PP) of pure projective modules is given. Moreover, we give a sufficient condition for the triangle-equivalence Dpsgb(R)≈Dpsgb(S), where R and S are rings.
We characterize boundedness and compactness of Hankel operators on a very general class of weighted Fock spaces over Cn in terms of a certain notion of bounded and vanishing mean oscillation. The analogous description holds for the commutators[Mf, P] where Mf denotes the multiplication operator with symbol f and P is the Toeplitz projection. We also give geometric descriptions for the spaces BMO and VMO which are defined in terms of the Berezin transform.
Wen and Wu introduced the notion of homogeneous perfect sets as a generalization of Cantor type sets and determined their exact Hausdorff dimension based on the length of their basic intervals and the gaps between them. In this paper, we considered the quasisymmetrically minimality of the homogeneous perfect sets, proved the homogeneous perfect sets with Hausdorff dimension 1 are 1-dimensional quasisymmetrically minimal under some conditions.
Inspired by the central limit theorem established by Peng, we investigate the generalized central limit theorem under sublinear expectations based on three weaker conditions with the notion of G-normal distribution. Initially, the condition E[Xn]=E[Xn]=0 is replaced by|E[Xn]|+|E[Xn]|=O(1/n). Furthermore, the original 2-nd and (2+δ)-th moments conditions are weakened through the truncation of random variables. Finally, we develop the theorem for convolutionary random variables, which can be seen as a generalization of Peng-independence.
Using large deviation and small deviation of Brownian motion in Hölder norm, local functional Chung's law of the iterated logarithm for increments of a Brownian motion in Hölder norm can be obtained.
In this paper, the Schauder basis property of the root vector systems for a class of unbounded Hamiltonian operators is studied by the induced quadratic operator pencils. First, the relationships between the root vectors of unbounded Hamiltonian operators and the corresponding quadratic operator pencils are established. Next, the eigenvalues distribution and the algebraic index of eigenvalues of unbounded Hamiltonian operators are described with the aid of the spectral properties of quadratic operator pencils. Then, a necessary and sufficient condition is given for the root vector systems of unbounded Hamiltonian operators to be a block Schauder basis of some Hilbert space. Eventually, the obtained results are validated for the bending problems of rectangular plates.
The second cohomology groups of the generalized map Schrödinger-Virasoro algebras are determined. And the universal central extensions of the generalized map Schrödinger-Virasoro algebras are given.
The paper[A new characterization of automorphism groups of Suzuki-Ree groups, Acta Math. Sin., Chin. Ser., 2013, 56(4), 545-552] proved that Aut(2F4(q)), q=2f and Aut(2G2(q)), q=3f can be characterized by their order components, where f=3s, s is a positive integer. We proved that Aut(2B2(q)), q=2f and Aut(2G2(q)), q=3f also can be characterized by their order components, where f is a odd prime. Combining both, we can obtain that the automorphism groups of the Suzuki-Ree simple groups whose prime graphs are not connected can be characterized by their order components.
Let R⊂A be a Frobenius extension of rings, where A is right coherent. Let M be any left A-module. We first show that AM is Gorenstein flat if and only if the underlying R-module RM is Gorenstein flat. Then we prove a "Gorenstein version" of Nakayama and Tsuzuku's theorem on transfer of flat dimensions along Frobenius extensions:if AM has finite Gorenstein flat dimension, then GfdA(M)=GfdR(M). Moreover, it is proved that if R ⊂ S is a separable Frobenius extension, then for any A-module (not necessarily of finite Gorenstein flat dimension), its Gorenstein flat dimension is invariant along such ring extension.
In this study, the lower semicontinuity of solution mappings to parametric generalized set-valued strong vector equilibrium problems is proved, by employing properties of the limit to set-valued mapping and the assumption of weakf-property, which method is different from the recent ones (Zhao, 2016 and Meng, 2018). Moreover, a sufficient condition of the solution sets to parametric generalized set-valued strong vector equilibrium problems is established, and examples are also given for the illustration of the correctness to the obtained results. The obtained results generalize and improve the ones in the literature (Gong, 2008, Xu, 2009, Chen, 2010, Xu, 2013 and Zhao, 2013).
In this paper, exponential time differencing schemes with Padé approximation to the Mittag-Leffler function are proposed for the time-space-fractional nonlinear Schrödinger equation. Ways of increasing the efficiency of the proposed schemes are discussed. Numerical experiments are performed on the time-space-fractional nonlinear Schrödinger equations with various parameters. The accuracy, efficiency, and reliability of the proposed method are illustrated by numerical results.
