We discuss the integrability of orthogonal almost complex structures on Riemannian products of even-dimensional round spheres and give a partial answer to the question raised by E. Calabi concerning the existence of complex structures on a product manifold of a round 2-sphere and of a round 4-sphere.

An automorphism group of a graph is said to be -regular if it acts regularly on the set of -arcs in the graph. A graph is -regular if its full automorphism group is -regular. In the present paper, all -regular cubic graphs of order are classified for each and each prime .

In this paper, we consider the Fourier-type functionals on Wiener space. We then establish the analytic Feynman integrals involving the -convolutions. Further, we give an approach to solution of the Schrdinger equation via Fourier-type functionals. Finally, we use this approach to obtain solutions of the Schrdinger equations for harmonic oscillator and double-well potential. The Schrdinger equations for harmonic oscillator and double-well potential are meaningful subjects in quantum mechanics.

For two positive integers and , let be the open convex cone in consisting of positive definite real symmetric matrices and let be the set of all real matrices. In this paper, we investigate differential operators on the non-reductive homogeneous space that are invariant under the natural action of the semidirect product group on the Minkowski-Euclid space . These invariant differential operators play an important role in the theory of automorphic forms on generalizing that of automorphic forms on .

If and are non-negative integers, then three new classes of abelian -groups are defined and studied: the , -simply presented groups, the , -balanced projective groups and the , -totally projective groups. These notions combine and generalize both the theories of simply presented groups and -projective groups. If , , these all agree with the class of totally projective groups, but when , they also include the -projective groups. These classes are related to the (strongly) n-simply presented and (strongly) -balanced projective groups considered in [15] and the n-summable groups considered in [2]. The groups in these classes whose lengths are less than are characterized, and if in addition we have , they are determined by isometries of their -socles.

Let denote the sum of the sth powers of the positive divisors of a positive integer N and let with , N, and s positive integers. Hahn [12] proved that $$16\sum_{k for .

In this work we deal with the problem of how to squeeze multiple ciphertexts without losing original message information. To do so, we formalize the notion of decomposability for public-key encryption and investigate why adding decomposability is challenging. We construct an ElGamal encryption scheme over extension fields, and show that it supports the efficient decomposition. We then analyze security of our scheme under the standard DDH assumption, and evaluate the performance of our construction.

A general result for the complete convergence of arrays of rowwise extended negatively dependent random variables is derived. As its applications eight corollaries for complete convergence of weighted sums for arrays of rowwise extended negatively dependent random variables are given, which extend the corresponding known results for independent case.

Let be an extension of integral domains, X be an indeterminate over T, and R[X] and T[X] be polynomial rings. Then is said to be LCM-stable if for all . Let be the so-called -operation on an integral domain A. In this paper, we introduce the notions of - and -LCM-stable extensions: (i) is -LCM-stable if for all and (ii) is -LCM-stable if for all . We prove that LCM-stable extensions are both -LCM-stable and -LCM-stable. We also generalize some results on LCM-stable extensions. Among other things, we show that if R is a Krull domain (resp., ), then is -LCM-stable (resp., -LCM-stable) if and only if is -LCM-stable (resp., -LCM-stable).

For any and arbitrary with compact support, it is proved that there exists a pair (L, ), with L second order uniformly elliptic operator and such that a.e. in .

We construct some hyperbolic hyperelliptic space forms whose fundamental groups are generated by only two or three isometries. Each occurring group is obtained from a supergroup, which is an extended Coxeter group generated by plane re ections and half-turns. Then we describe covering properties and determine the isometry groups of the constructed manifolds. Furthermore, we give an explicit construction of space form of the second smallest volume nonorientable hyperbolic 3-manifold with one cusp.

We describe the one-dimensional and zero-dimensional cusps of the Satake compactification for the paramodular groups in degree two for arbitrary levels. We determine the crossings of the one-dimensional cusps. Applications to computing the dimensions of Siegel modular forms are given.