Dirichlet series with real frequencies which represent entire functions on the complex plane ? have been investigated by many authors. Several proeperties such as topological structures, linear continuous functionals, and bases have been considered. Le Hai Khoi drived some results with Dirichlet series having negative real frequencies which represent holomorphic functions in a half plane. In the present paper, we have obtained some properties of holomorphic Dirichlet series having positive exponents, whose coefficients belong to a Banach algebra.

The restricted parameter range set cover problem is a weak form of the NP-hard set cover problem with the restricted range of parameters. We give a polynomial time algorithm for this problem by lattices.

This paper deals with the principal eigenvalue of discrete p-Laplacian on the set of nonnegative integers. Alternatively, it is studying the optimal constant of a class of weighted Hardy inequalities. The main goal is the quantitative estimates of the eigenvalue. The paper begins with the case having reflecting boundary at origin and absorbing boundary at infinity. Several variational formulas are presented in different formulation: the difference form, the single summation form, and the double summation form. As their applications, some explicit lower and upper estimates, a criterion for positivity (which was known years ago), as well as an approximating procedure for the eigenvalue are obtained. Similarly, the dual case having absorbing boundary at origin and reflecting boundary at infinity is also studied. Two examples are presented at the end of Section 2 to illustrate the value of the investigation.

We investigate the construction of two-direction tight wavelet frames. First, a sufficient condition for a two-direction refinable function generating two-direction tight wavelet frames is derived. Second, a simple constructive method of two-direction tight wavelet frames is given. Third, based on the obtained two-direction tight wavelet frames, one can construct a symmetric multiwavelet frame easily. Finally, some examples are given to illustrate the results.

We study the strength of some combinatorial principles weaker than Ramsey theorem for pairs over RCA₀. First, we prove that Rainbow Ramsey theorem for pairs does not imply Thin Set theorem for pairs. Furthermore, we get some other related results on reverse mathematics using the same method. For instance, Rainbow Ramsey theorem for pairs is strictly weaker than Erdös-Moser theorem under RCA₀.

Similar to the property of a linear Calderón-Zygmund operator, a linear fractional type operator I_α associated with a BMO function b fails to satisfy the continuity from the Hardy space H^p into L^p for p≤1. Thus, an alternative result was given by Y. Ding, S. Lu and P. Zhang, they proved that [b, I_α] is continuous from an atomic Hardy space Hbp into L^p,where <?Pub Caret?>Hbp is a subspace of the Hardy space H^p for n/(n+1)<p≤1. In this paper, we study the commutators of multilinear fractional type operators on product of certain Hardy spaces. The endpoint (Hb1p1×?×Hbmpm, L^p) boundedness for multilinear fractional type operators is obtained. We also give the boundedness for the commutators of multilinear Calderón-Zygmund operators and multilinear fractional type operators on product of certain Hardy spaces when b∈(Lipβ)m(?n).

We study the exponential sums involving Fourier coefficients of Maass forms and exponential functions of the form e(αn^β),where 0≠α∈? and 0<β<1. An asymptotic formula is proved for the nonlinear exponential sum ∑X<n≤2Xλg(n)e(αnβ), when β = 1/2 and |α| is close to 2q, q∈?+, where λ_g(n) is the normalized n-th Fourier coefficient of a Maass cusp form for SL₂(?). The similar natures of the divisor function τ(n) and the representation function r(n) in the circle problem in nonlinear exponential sums of the above type are also studied.

A super Jaulent-Miodek hierarchy and its super Hamiltonian structures are constructed by means of a kind of Lie super algebras and super trace identity. Moreover, the self-consistent sources of the super Jaulent-Miodek hierarchy is presented based on the theory of self-consistent sources. Furthermore, the infinite conservation laws of the super Jaulent-Miodek hierarchy are also obtained. It is worth noting that as even variables are boson variables, odd variables are fermi variables in the spectral problem, the commutator is different from the ordinary one.

A semiring is an algebraic structure similar to a ring, but without the requirement that each element must have an additive inverse. A bounded semiring is a semiring equipped with a compatible bounded partial order. In this paper, properties of zero divisors and prime elements of a bounded semiring are studied. In particular, it is proved that under some mild assumption, the set Z(A) of nonzero zero divisors of A is A \ {0, 1}, and each prime element of A is a maximal element. For a bounded semiring A with Z(A) = A \ {0, 1}, it is proved that A has finitely many maximal elements if ACC holds either for elements of A or for principal annihilating ideals of A. As an application of prime elements, we show that the structure of a bounded semiring A is completely determined by the structure of integral bounded semirings if either |Z(A)| = 1 or |Z(A)| = 2 and Z(A)² ≠ 0. Applications to the ideal structure of commutative rings are also considered. In particular, when R has a finite number of ideals, it is shown that the chain complex of the poset II(R) is pure and shellable, where II(R) consists of all ideals of R.

The varietal hypercube VQn is a variant of the hypercube Qn and has better properties than Qn with the same number of edges and vertices. This paper proves that VQn is vertex-transitive. This property shows that when VQn is used to model an interconnection network, it is high symmetrical and obviously superior to other variants of the hypercube such as the crossed cube.

We give the necessary and sufficient condition for a bounded linear operator with property (ω) by means of the induced spectrum of topological uniform descent, and investigate the permanence of property (ω) under some commuting perturbations by power finite rank operators. In addition, the theory is exemplified in the case of algebraically paranormal operators.

We prove the boundedness of all solutions for the equation x″ + V′(x) = D_xG(x, t), where V(x) is of singular potential, i.e., lim_x→-1V(x) = +∞, and G(x, t) is bounded and periodic in t. We give sufficient conditions on V(x) and G(x, t) to ensure that all solutions are bounded.

We consider the spectrally negative Lévy processes and determine the joint laws for the quantities such as the first and last passage times over a fixed level, the overshoots and undershoots at first passage, the minimum, the maximum, and the duration of negative values. We apply our results to insurance risk theory to find an explicit expression for the generalized expected discounted penalty function in terms of scale functions. Furthermore, a new expression for the generalized Dickson’s formula is provided.