Nagata rings

Let A be a commutative ring. For any set p of prime ideals of A, we define a new ring Na(A, p): the Nagata ring. This new ring has the particularity that we may transform certain properties relative to p to properties on the whole ring Na(A, p); some of these properties are: ascending chain condition, Krull dimension, Cohen-Macaulay, Gorenstein. Our main aim is to show that most of the above properties relative to a set of prime ideals p(i.e., local properties) determine and are determined by the same properties on the Nagata ring (i.e., global properties). In order to look for new applications, we show that this construction is functorial, and exhibits a functorial embedding from the localized category (A, p)-Mod into the module category Na(A, p)-Mod.     

Bicomplex Hermitian Clifford analysis

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∞(R4n,W4n)→C∞(R4n,W4n), where W4n is the tensor product of three algebras, i.e., the hyperbolic quaternion B^, the bicomplex number B, and the Clifford algebra Rn. The operator D is a square root of the Laplacian in R4n, introduced by the formula D|=∑j=03Kj?Zj with Kjbeing the basis of B^, and ?Zj denoting the twisted Hermitian Dirac operators in the bicomplex Clifford algebra B?R0,4n 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.     

Wintgen ideal submanifolds with a low-dimensional integrable distribution

Submanifolds in space forms satisfy the well-known DDVV inequality. A submanifold attaining equality in this inequality pointwise is called a Wintgen ideal submanifold. As conformal invariant objects, Wintgen ideal submanifolds are investigated in this paper using the framework of M?bius geometry. We classify Wintgen ideal submanfiolds of dimension m≥3 and arbitrary codimension when a canonically defined 2-dimensional distribution ?2 is integrable. Such examples come from cones, cylinders, or rotational submanifolds over super-minimal surfaces in spheres, Euclidean spaces, or hyperbolic spaces, respectively. We conjecture that if ?2 generates a k-dimensional integrable distribution ?kand k<m, then similar reduction theorem holds true. This generalization when k = 3 has been proved in this paper.     

Generalized area operators on Lp(?n)

We introduce the generalized area operators by using nonnegative measures defined on upper half-spaces ?+n+1. The characterization of the boundedness and compactness of the generalized area operator from L^p(?n) to L^q(?n) is investigated in terms of s-Carleson measures with 1<p, q<+∞. In the case of p = q = 1, the weak type estimate is also obtained.     

Unit groups of quotient rings of complex quadratic rings

For a square-free integer d other than 0 and 1, let K=?(d), where ? is the set of rational numbers. Then K is called a quadratic field and it has degree 2 over ?. For several quadratic fields K=?(d), the ring R_dof integers of K is not a unique-factorization domain. For d<0, there exist only a finite number of complex quadratic fields, whose ring Rd of integers, called complex quadratic ring, is a unique-factorization domain, i.e., d = −1,−2,−3,−7,−11,−19,−43,−67,−163. Let ϑ denote a prime element of Rd, and let n be an arbitrary positive integer. The unit groups of Rd/〈vn〉 was determined by Cross in 1983 for the case d = −1. This paper completely determined the unit groups of Rd/〈vn〉 for the cases d = −2,−3.     

Largest adjacency,signless Laplacian,and Laplacian H-eigenvalues of loose paths

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)=31/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.     

Upper bound of Kähler angles on the β-symplectic critical surfaces

Let (M,g) be a Kähler surface and \mathrm{\Sigma } be a \beta-symplectic critical surface in M. If L_q\left(\mathrm{\Sigma }\right) is bounded for some q>3, then we give a uniform upper bound for the Kähler angle on \mathrm{\Sigma }. This bound only depends on M,q,\beta and the L_q functional of \mathrm{\Sigma }. For q>4, this estimate is known and we extend the scope of q.     

Perpetual cutoff method and CDE′(K,N) condition on graphs

By using the perpetual cutoff method, we prove two discrete versions of gradient estimates for bounded Laplacian on locally finite graphs with exception sets under the condition of CD{E}^{\prime }(K,N). This generalizes a main result of F. Münch who considers the case of CD(K, \infty) curvature. Hence, we answer a question raised by Münch. For that purpose, we characterize some basic properties of radical form of the perpetual cutoff semigroup and give a weak commutation relation between bounded Laplacian \text{Δ} and perpetual cutoff semigroup P_t^W in our setting.     

Exponential sums involving Maass forms

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.     

Maximal function characterizations of Musielak-Orlicz-Hardy spaces associated with magnetic Schr?dinger operators

Let φ be a growth function, and let A:=-(?-ia)?(?-ia)+V be a magnetic Schr?dinger operator on L2(?n),n≥2, where α:=(α1,α2,?,αn)∈Lloc2(?n,?n) and 0≤V∈Lloc1(?n). We establish the equivalent characterizations of the Musielak-Orlicz-Hardy space HA,φ(?n), defined by the Lusin area function associated with {e-t2A}t>0, in terms of the Lusin area function associated with {e-tA}t>0, the radial maximal functions and the nontangential maximal functions associated with {e-t2A}t>0 and {e-tA}t>0, respectively. The boundedness of the Riesz transforms LkA-1/2,k∈{1,2,?,n}, from HA,φ(?n) to Lφ(?n) is also presented, where L_k is the closure of ??xk-iαk in L2(?n). These results are new even when φ(x,t):=ω(x)tp for all x∈?nand t ∈(0,+∞) with p ∈(0, 1] and ω∈A∞(?n) (the class of Muckenhoupt weights on ?n).     

Values of binary linear forms at prime arguments

value of a given binary linear form at prime arguments. Let λ₁ and λ₂ be positive real numbers such that λ₁/λ₂ is irrational and algebraic. For any (C, c) well-spaced sequence V and δ>0, let E(V, X, δ) denote the number of υ∈V with υ≤X for which the inequality |λ1p1+λ2ρ2−υ|<υ−δ has no solution in primes p₁, p₂. It is shown that for any ε>0,we have E(V, X, δ) «max(X35+2δ+ε,X23+43δ+ε).     

Functional inequalities for time-changed symmetric α-stable processes

We establish sharp functional inequalities for time-changed symmetric \alpha-stable processes on {\mathrm{ℝ}}^d with d\ge \mathrm{1} and \alpha \in (\mathrm{0},\mathrm{2}), which yield explicit criteria for the compactness of the associated semigroups. Furthermore, when the time change is defined via the special function W(x)={(\mathrm{1}+\left|x\right|)}^{\beta } with \beta >\alpha we obtain optimal Nash-type inequalities, which in turn give us optimal upper bounds for the density function of the associated semigroups.     

Boundedness of some multipliers on stratified Lie groups

Let ￡ be the sub-Laplacian on a stratified Lie group G, and let m be a function defined on [0,+∞). We give the boundedness of the multiplier operators m(￡) on Herz-type Hardy spaces on G.     

Certain categories of modules for twisted affine Lie algebras

In this paper, using generating functions, we study two categories ? and ? of modules for twisted affine Lie algebras g^[σ], which were firstly introduced and studied for untwisted affine Lie algebras by H. -S. Li [Math Z, 2004, 248: 635-664]. We classify integrable irreducible g^[σ]-modules in categories ? and ?, where ? is proved to contain the well-known evaluation modules and ? to unify highest weight modules, evaluation modules and their tensor product modules. We determine also the isomorphism classes of those irreducible modules.     

Well-posedness of degenerate differential equations in H?lder continuous function spaces

Using known operator-valued Fourier multiplier results on vectorvalued H?lder continuous function spaces, we completely characterize the wellposedness of the degenerate differential equations (Mu)'(t)=Au(t)+f(t) for t∈R in H?lder continuous function spaces Ca(R;X) by the boundedness of the M-resolvent of A, where A and M are closed operators on a Banach space X satisfying D(A)?D(M).     

Branching random walks with random environments in time

We consider a branching random walk on N with a random environment in time （denoted by ξ）. Let Zn be the counting measure of particles of generation n, and let Zn（t） be its Laplace transform. We show the convergence of the free energy n-llog Zn（t）, large deviation principles, and central limit theorems for the sequence of measures {Zn}, and a necessary and sufficient condition for the existence of moments of the limit of the martingale Zn（t）/E[Zn（t）ξ].     

OD-Characterization of certain four dimensional linear groups with related results concerning degree patterns

The prime graph of a finite group G, which is denoted by GK(G), is a simple graph whose vertex set is comprised of the prime divisors of |G| and two distinct prime divisors p and q are joined by an edge if and only if there exists an element of order pq in G. Let p₁2<?<p_k be all prime divisors of |G|. Then the degree pattern of G is defined as D(G) = (degG(p₁), degG(p₂), ? , degG(p_k)), where degG(p) signifies the degree of the vertex p in GK(G). A finite group H is said to be OD-characterizable if G? H for every finite group G such that |G| = |H| and D(G) = D(H). The purpose of this article is threefold. First, it finds sharp upper and lower bounds on ?(G), the sum of degrees of all vertices in GK(G), for any finite group G (Theorem 2.1). Second, it provides the degree of vertices 2 and the characteristic p of the base field of any finite simple group of Lie type in their prime graphs (Propositions 3.1-3.7). Third, it proves the linear groups L₄(q), q = 19, 23, 27, 29, 31, 32, and 37, are OD-characterizable (Theorem 4.2).     

Termination of algorithm for computing relative Gr?bner bases and difference differential dimension polynomials

We introduce the concept of difference-differential degree compatibility on generalized term orders. Then we prove that in the process of the algorithm the polynomials with higher and higher degree would not be produced, if the term orders ‘?’ and ‘?′’ are difference-differential degree compatibility. So we present a condition on the generalized orders and prove that under the condition the algorithm for computing relative Gr?bner bases will terminate. Also the relative Gr?bner bases exist under the condition. Finally, we prove the algorithm for computation of the bivariate dimension polynomials in difference-differential modules terminates.