We consider the following oscillatory integral operator:

where the function f is assumed to be a Schwartz function on Rⁿ and m, k > 0. In this paper, we characterize the sufficient and necessary conditions which ensure the boundedness for T_m,k,n from L^p(Rⁿ) (1 ≤ p < ∞) to L^q(Rⁿ). In addition, the operator T_m,k,n also maps L¹(Rⁿ) into l₀(Rⁿ).

There is a close connection between Hochschild homology groups of a k-algebra and cycles of the Gabriel quiver associated to the k-algebra. In this paper,based on the minimal projective bimodule resolution of a self-injective Koszul four-point algebra constructed by Furuya, we calculate the dimensions of Hochschild homology spaces of the algebra by using combinatorial methods, and give a k-basis of every Hochschild homology space in terms of cycles. Moreover, we obtain the dimensions of cyclic homology groups of the algebra when the base field k is of zero characteristic.

In this paper, the concepts of weak medial idempotent and quasi-medial idempotent of regular semigroups are introduced. The aim of this paper is to explore the properties of the two types of idempotents. Several regular semigroups having weak(quasi-) medial idempotents are constructed to indicate the relationship between weak medial idempotents and quasi-medial idempotents, various ways are given to confirm that an idempotent is a weak medial idempotent or a quasi-medial idempotent, and some descriptions of orthodox semigroups in terms of quasi-media idempotents are hold. At last, the structure theorem of every regular semigroup with a quasi-medial idempotent is obtained. As an application of the results, such kind of construction method help us to get a way to determine whether a regular semigroup contains a multiplicative inverse transversal or not.

In the Formal Deductive System L^* of Fuzzy Propositional Calculus, the notion of closed theory is introduced and its properties are investigated. Furthermore, the completeness of Formal Deductive System L^* is proved through closed theory based on formula set F(S). At first, in the formal deductive system L^*, a concept of closed theory is introduced, and a method for extending theories to closed theories is given; at second, in the formal deductive system L^*, a concept of total closed theory is introduced, and the existence of a total closed theory satisfying relevant conditions is proved; at third, in the formal deductive system L^*, the properties of congruence relations determined by closed theories are investigated, a concept of strong congruence relations is introduced to formulas set F(S), and methods of changing each other between strong congruence relations and closed theories are revealed; at fourth, in the formal deductive system L^*, it is proved that closed theory style L^*-Lindenbaum algebras determined by closed theories are R₀-algebras, and a closed theory-L^*-Lindenbaum algebra is linear if and only if a closed theory is total; at last, the completeness of the formal deductive system L^* is accomplished by making use of total closed theory style L^*-Lindenbaum algebras, and the results obtained before have been improved.

The inclusion ideal graph of a ring R, written as Γ_I(R), is a directed graph which has all nontrivial left rings of R as vertex set and there is a directed edge from a vertex I₁ to a distinct vertex I₂ if and only if I₁ is properly contained in I₂. In addition, the ideal-relation graph of a ring R, written as Γ_i(R), is also a directed graph which has R as vertex set and there is a directed edge from a vertex A to a distinct vertex B if and only if the left ideal of R generated by A is properly contained in the left ideal generated by B. Let F_q be a finite field, the set of n×n matrices over F_q be denoted by M_n(F_q). In this paper, both the automorphisms of Γ_I(M_n(F_q)) and the automorphisms of Γ_i(M_n(F_q)) are characterized.

This article researches the batch Markov arrival process (BMAP) by using path-analysis method which is different from using conventional matrix analysis method. We are capable of calculating its jumping probability through the representation (D_k, k=0, 1, 2,...) of BMAP, demonstrating the fact that BMAP's phase process is time-homogeneous Markov chain, figuring out the transition probabilities and density matrix of the phase process. Moreover, if we give a Q process J with a finite state space and define the counting process of its jumping points as N, we can demonstrate that adjoint process X^*=(N, J) of Q process J is a MAP. The MAP's transition probabilities and representation (D₀, D₁) are exactly worked out, which are expressed by density matrix Q.

The extinction and the stationary distribution to a stochastic SEIR epidemic model with nonlinear incidence rate in a population of varying size are discussed. Under moderate conditions, the existence-and-uniqueness of the global solution, exponential stability and the stationary distribution with ergodicity are obtained. By means of linearization and Fourier transform, we prove that the solution obeys a fourdimensional normal distribution, and the mean and the variance matrix are followed. Then numerical simulations are carried out to illustrate our results.

We consider a branching process indexed by a general renewal process. Assume that the renewal distribution satisfies the Cramér condition and the branching process belongs to the Böttcher case, i.e., if always at least two offspring are born, we obtain the large deviations of such process.

The background for the introduction of Kadison-Singer algebras (KSalgebras, for short) is reviewed. Definitions and basics properties are explained. Studies on hyperfinite KS-algebras, non-hyperfinite case, KS-lattices and strong KS-algebras are described in details, as well as their connections with classical open problems such as the invariant subspace problem, Kadison's transitive algebra problem, von Neumann algebra generator problem. Operations on non-selfadjoint operator algebras are also discussed and two new operations are included in the discussion. Sixteen open problems are listed with some explanations in the end.

We are concerned with a class of Kirchhoff problem

where a, b > 0 are constants. The existence of ground state solutions, i.e., nontrivial solutions with least possible energy of this Kirchhoff problem is obtained. Moreover, when Q ≡ 1, under suitable conditions on h(x), we prove the existence of ground state solutions for the the following Kirchhoff problem
.

We give a geometric definition of sub-orbifold groupoid, and a criterion for determining a sub-orbifold groupoid. Then we prove the existence of orbifold tubular neighborhoods of compact sub-orbifold groupoids, the existence of symplectic neighborhoods of compact symplectic sub-orbifold groupoids in symplectic orbifold groupoids, and, the existence of Lagrangian neighborhoods of compact Lagrangian sub-orbifold groupoids.

We are interested in considering the following nonlocal critical problem

where Ω ⊂ R⁴ is a smooth bounded domain with 0 ∈ Ω, a ≥ 0, b, λ, μ > 0, 1 < q < 2, 0 < β < 2. By using the variational method, some existence and multiplicity results are obtained.

Computing height of piecewise monotonic functions is difficult because the value of functions may interact each other under iteration. In this paper we consider the set of continuous functions with a single non-monotonic point. We first present a sufficient and necessary condition for heights which gives a classification of those functions. Then we provide an equivalent condition and a new construction method for topological conjugacy for a nonempty subset of the mentioned continuous functions. Furthermore, we prove that topological conjugacy is a sufficient but not necessary condition for equal heights of piecewise monotonic functions. Finally, some examples are given to illustrate our results.

Convergence for identically distributed negatively associated (NA) random variables and independent and identically distributed (i.i.d.) random variables are studied under general moment conditions, the extension of the Baum-Katz Theorem and strong law of large numbers are obtained. The theorems extend the related known works in the literature.

Ye etc introduced the L_p-mixed geominimal surface areas of multiple convex bodies for any real p(p≠-n). In this paper, we define the L_p-dual mixed geominimal surface areas of multiple star bodies for any real p(p≠n), and establish some inequalities related to this concept.

Sampling in shift-invariant subspaces of L_p is commonly studied under the requirement that the generator is in a Wiener amalgam space independent of p, but such condition is too strong because it does not allow us to control p. In this paper, we mainly discuss the nonuniform average sampling and reconstruction of signals in a non-decaying shift-invariant space under the assumption that the generator is in a hybrid-norm space. The new condition is weaker than the Wiener amalgam space and depends on the parameter p. Based on some lemmas in hybrid-norm spaces, we first give the sampling stability for two kinds of average sampling functionals, and then the corresponding iterative reconstruction algorithms with exponential convergence are established.

We present the canonical forms of coupled self-adjoint boundary conditions for regular differential operators of order four by using new methods. In this new canonical forms, the four small block matrices are symmetric, and the determinant has absolute value equal to 1. This fourth order form is quite similar to the new second order one, and it provides a new way to give the canonical forms of the self-adjoint boundary condition for general high-order differential operators.

Assume T_Ω is the Calderón-Zygmund singular integral operator with rough kernel and I is the closed arc of unit circle that I ? S¹. In this paper, we prove that if Ω is supported in I and monotonous on I, then T_Ω is bounded from Hardy space H¹(R²) to L¹(R²) if and only if||Ω||_{L log L} < ∞.

The limiting behaviors for controlled branching processes in random environments are discussed. Under the normalization factor {S_n:n ∈ N}, the normalization processes {?_n:n ∈ N} of are studied, and the sufficient conditions of {?_n:n ∈ N} a.s., L¹ and L² convergence are given; A sufficient condition and a necessary condition for convergence to a non-degenerate at 0 random variable are got; Under the normalization factor {I_n:n ∈ N}, the normalization processes {W_n:n ∈ N} are discussed, and the sufficient conditions of {W_n:n ∈ N} a.s., and L¹ convergence are obtained.

We study the stochastic analysis of the measure determined by fractional diffusion process (fractional diffusion measure). We first give the pull back formula by Bismut method, then establish the integration by parts formula for fractional diffusion measure. Finally, we generalize the classic Clark-Ocone theorem to martingale representation theorem under fractional diffusion measure.

We apply the coupling method in inhomogeneous Markov processes, and generalize the fundamental coupling theorem of homogeneous Markov processes, which provides a theoretical basis for the further study of the coupling methord for inhomogeneous Markov processes.

We define the abscissa of convergence σ_c, the uniform abscissa of convergence σ_u and the absolute abscissa of convergence σ_a on Dirichlet series Σ_n=0^∞ a_ne^λ_ns,use the relationship between the index λ_n and the coefficients a_n to estimate the three convergence abscissas, and obtain that the convergence conditions of two Dirichlet series Σ_n=0^∞ a_ne^λ_ns and Σ_n=0^∞ a_ne^-λ_ns are the same.

We propose a flexible additive-multiplicative hazard model allowing the additive time-varying effects of covariate for the subdistribution in a competing risks circumstance. For inference on the model parameters, weighted estimating equation approaches under an covariates-dependent adjusted weight by fitting the Cox proportional hazard model for the censoring distribution are established. In addition, large number properties and a goodness-of-fit test procedure are presented. The finite sample behavior of the proposed estimators is evaluated through simulation studies, estimators from the proposed method perform satisfactorily on reduction of the bias, and an application to a competing risks data set from a follicular cell lymphoma study is illustrated.

The aim of this paper is to study the standing wave of the generalized Davey-Stewartson equation iu_t+Δu+a|u|^p-1u+E₁(|u|²)u=0, t ≥ 0,x∈Rⁿ, where a,b > 0, 1 < p < (n-2)/((n+2)⁺), n∈{2,3}. When 1+4/n ≤ p < (n-2)/((n+2)⁺),[Sharp threshold of global existence and instability of standing wave for a Davey-Stewartson system, Commun. Math. Phys., 2008, 283:93-125], under an assumption about the frequency of the standing wave, obtained the strongly instability of the ground state standing waves. In the present paper, We establish the same conclusion without that assumption.

We prove a large deviation principle for homeomorphism flows of stochastic differential equations (SDEs) without global Lipschitz conditions and apply it to stochastic Hamiltonian systems, which in particular covers the following nonlinear oscillator equation:
?_t=C₀?_t-Z_t³+Θ(Z_t)?_t,(Z₀,?₀)=(z,u)∈R²,
where C₀∈R, Θ∈C²(R) has a bounded first order derivative, and ?_t is a onedimensional Brownian white noise.

Let {X_n,n ≥ 1} be a sequence of strictly stationary NA random variables and set S_n=∑_i=1ⁿX_i,M_n=max_{1 ≤ i ≤ n}|S_i|. Under some proper conditions, the precise asymptotics in the law of iterated logarithm for the moment convergence of NA random variables of the partial sum and the maximum of the partial sum are obtained.

In the paper, the following Kirchhoff-type equations with critical exponent (a+b∫_R³[|∇u|²+V(x)u²]dx)·[-△u+V(x)u]=μf(x,u)+K(x)u⁵,x∈R³, is studied, in which a, b, μ > 0, the potential functions V, K satisfy some conditions and the nonlinear term f verifies sup-cubic or sup-linear growth. With the aid of the mountain pass theorem, three existence results are obtained.

We study some equivalent properties of the curvature dimension inequality CD(n,K) on locally finite graphs. These equivalences are gradient estimate, Poincaré type inequalities and reverse Poincaré inequalities. We also obtain one equivalent property of gradient estimate for a new notion of curvature dimension inequality CDE (∞,K) at the same assumption on graphs.

In this note, we investigate the partial augmentations of the normalized torsion units in integral group rings of the direct product of two finite groups. It is proved that every normalized torsion unit is conjugate to one element in the group within the rational group algebra under some conditions.

In this paper, the closed range of block operator matrices is studied. Using the perturbation theory and Hyers-Ulam stability, the sufficient conditions of the closed range of operator matrix are given. In the end, some examples are given to illustrate the effectiveness of the proposed criterion.

We defined a multi-branching process in varying environment by it's fitness. We studied the properties of its generating function and gave the recurrence relation of the generating function. We calculated the expectation and variance of the process, just like Galton-Watson process, we studied its extinction probability and constructed a nonnegative martingale, in the case that the first and second order moments of offspring are bounded, we also proved that this martingale converged in L².

We introduce stochastic volatility to Tobin's q theory, in order to find how volatility of productivity shock affects a firm's value and its investment decision. We show that volatility of productivity shock has significant effect on firm's Tobin's q, which decreases as volatility rises and the effect gets more obvious as volatility gets bigger. On the other hand, this effect can also affect firm's investment decision because the decrease of Tobin's q can reduce firm's investment, and the effect gets larger when volatility rises too. We also illustrate the effect of volatility upon firm's assets in place and growth opportunities. Our assumption that volatility of productivity shock is stochastic is more realistic than existing literature, so our conclusion is able to provide a reference for firm's valuation and investment decision.