In this paper, we explore the virtual technique that is very useful in studying the moduli problem from a differential geometric point of view. We introduce a class of new objects “virtual manifolds/orbifolds”, on which we develop the integration theory. In particular, the virtual localizationformula is obtained.

This note studies the Chern–Simons invariant of a closed oriented Riemannian 3-manifold M. The first achievement is to establish the formula CS(e) ? CS(e) = degA, where e and e are two (global) frames of M, and A : M → SO(3) is the “difference” map. An interesting phenomenon is that the “jumps” of the Chern–Simons integrals for various frames of many 3-manifolds are at least two, instead of one. The second purpose is to give an explicit representation of CS(e₊) and CS(e_?), where e₊ and e_? are the “left” and “right” quaternionic frames on M³ induced from an immersion M³ → E⁴, respectively. Consequently we find many metrics on S³ (Berger spheres) so that they can not be conformally embedded in E⁴.

By using the Ringel–Hall algebra approach,we investigate the structure of the Lie alge-bra L(Λ)generated by indecomposable constructible sets in the varieties of modules for any finite-dimensional C-algebraΛ.We obtain a geometric realization of the universal enveloping algebra R(Λ)of L(Λ),this generalizes the main result of Riedtmann.We also obtain Green’s formula in a geometric form for any finite-dimensional C-algebraΛand use it to give the comultiplication formula in R(Λ).

Block introduced certain analogues of the Zassenhaus algebras over a field of character-istic 0.The nongraded infinite-dimensional simple Lie algebras of Block type constructed by Xu can be viewed as generalizations of the Block algebras.In this paper,we construct a family of irreducible modules in terms of multiplication and di?erentiation operators on“polynomials”for four-devivation nongraded Lie algebras of Block type based on the finite-dimensional irreducible weight modules with multiplicity one of general linear Lie algebras.We also find a new series of submodules from which some irreducible quotient modules are obtained.

Comparing to the Chen–Ruan cohomology theory for the almost complex orbifolds,we study the orbifold cohomology theory for almost contact orbifolds.We define the Chen–Ruan cohomol-ogy group of any almost contact orbifold.Using the methods for almost complex orbifolds,we define the obstruction bundle for any 3-multisector of the almost contact orbifolds and the Chen–Ruan cup product for the Chen–Ruan cohomology.We also prove that under this cup product the direct sum of all dimensional orbifold cohomology groups constitutes a cohomological ring.Finally we calculate two examples.

In this paper,we shall study a fourth-order stochastic heat equation driven by a fractionalnoise,which is fractional in time and white in space.We will discuss the existence and uniqueness of the solution to the equation.Furthermore,the regularity of the solution will be obtained.On the other hand,the large deviation principle for the equation with a small perturbation will be established through developing a classical method.

Let{X_n,n≥1}be a strictly stationary LNQD(LPQD)sequence of positive random variables with EX₁=μ>0,and VarX₁=σ²<∞.Denote by S_n=−ⁿ_i=1 X_i andγ=σ/μthe coeffcients of variation.In this paper,under some suitable conditions,we show that a general law of precise asymptotics for products of sums holds.It can describe the relations among the boundary function,weighted function,convergence rate and limit value in the study of complete nvergence.

In this paper,the authors establish the regularity in generalized Morrey spaces of solutions to parabolic equations with VMO coeffcients by means of the theory of singular integrals and linear commutators.

In this paper,by discovering a new fact that the Lebesgue boundedness of a class of pseudo-differential operators implies the Sobolev boundedness of another related class of pseudo-differential operators,the authors establish the boundedness of pseudo-differential operators with symbols in ^m_ρ,δ on Sobolev spaces,where m∈R,ρ≤1 andδ≤1.As its applications,the boundedness of commutators generated by pseudo-differential operators on Sobolev and Bessel potential spaces is deduced.Moreover,the boundedness of pseudo-differential operators on Lipschitz spaces is also obtained.

Let G be the complexification of the real Lie algebra so(3)and A=C[t^±1₁,t^±1₂] be the Lau-rent polynomial algebra with commuting variables.Let L(t₁,t₂,1)=G _c A be the twisted multi-loop Lie algebra.Recently we have studied the universal central extension,derivations and its vertex opera-tor representations.In the present paper we study the automorphism group and bosonic representations of L(t₁,t₂,1).

Consider all the arithmetic progressions of odd numbers,no term of which is of the form 2^k+p,where K is a positive integer and p is an odd prime.Erdös ever asked whether all these progressions can be obtained from covering congruences.In this paper,we characterize all arithmetic progressions in which there are positive proportion natural numbers that can be expressed in the form 2^k+p,and give a quantitative form of Romanoff’s theorem on arithmetic progressions.As a corollary,we prove that the answer to the above Erdös problem is affirmative.

Let G be a finite group with order|G|=p^α1₁p^α2₂···p^αk_k,where p₁<p₂<···<p_k are prime numbers.One of the well-known simple graphs associated with G is the prime graph(or Gruenberg–Kegel graph)denoted byΓ(G)(or GK(G)).This graph is constructed as follows:The vertex set of it isπ(G)={p₁,p₂,...,p_k}and two vertices p_i,p_j with i≠j are adjacent by an edge(and we write p_i～p_j)if and only if G contains an element of order p_ip_j.The degree deg(p_i)of a vertex p_i∈π(G)is the number of edges incident on pi.We define D(G):=(deg(p₁),deg(p₂),...,deg(p_k)),which is called the degree pattern of G.A group G is called k-fold OD-characterizable if there exist exactly k non-isomorphic groups H such that|H|=|G|and D(H)=D(G).Moreover,a 1-fold OD-characterizable group is simply called OD-characterizable.Let L:=U₃(5) be the projective special unitary group.In this paper,we classify groups with the same order and degree pattern as an almost simple group related to L.In fact,we obtain that L and L.2 are OD-characterizable;L.3 is 3-fold OD-characterizable;L.S₃ is 6-fold OD-characterizable.

We prove general reduction theorems for gauge natural operators transforming principal connections and classical linear connections on the base manifold into sections of an arbitrary gauge natural bundle.Then we apply our results to the principal prolongation of connections.Finally we describe all such gauge natural operators for some special cases of a Lie group G.

We obtain upper and lower bounds of the exit times from balls of a jump-type symmetric Markov process.The proofs are delivered separately.The upper bounds are obtained by using the Lévy system corresponding to the process,while the precise expression of the(L²-)generator of the Dirichlet form associated with the process is used to obtain the lower bounds.

Let E be a real reflexive Banach space which admits a weakly sequentially continuous duality mapping from E to E^*,and C be a nonempty closed convex subset of E.Let{T(t):t≥0}be a nonexpansive semigroup on C such that F:=∩_t≥0 Fix(T(t))≠Ø,and f:C→C be a fixed contractive mapping.If{α_n},{β_n},{a_n},{b_n},{t_n}satisfy certain appropriate conditions,then we suggest and analyze the two modified iterative processes as: y_n=α_nx_n+(1-αn)T(tn)xn,xn=β_nf(x_n)+(1-β _n)y_n. u₀∈C,v_n=a_nu_n+(1-a_n)T(t_n)u_n,u_n+1=b_nf(u_n)+(1-b_n)v_n.We prove that the approximate solutions obtained from these methods converge strongly to q∈∩_t≥0Fix(TT(t)),which is a unique solution in F to the following variational inequality:(I-f)q,j(q-u)≤0∀u∈F.Our results extend and improve the corresponding ones of Suzuki[Proc.Amer.Math.Soc.,131,2133–2136(2002)],and Kim and XU[Nonlear Analysis,61,51–60(2005)]and Chen and He[Appl.Math.Lett.,20,751–757(2007)].

In this paper,we prove that an into isometry form S(l^∞(_(n)))to S(E),which under someconditions,can be extended to be a linear isometry defined on the whole space.Therefore we improve the results of[Ding,G.G.:The isometric extension of an into mapping from the unit sphere S(l^∞(₍₂₎)) to S(L¹(_μ)).Acta Mathematica Sinica,English Series,22(6),1721–1724(2006)].

In this paper we prove the behaviour in weighted L^p spaces of the oscillation and variation of the Hilbert transform and the Riesz transform associated with the Hermite operator of dimension 1. We prove that this operator maps L^p(R,W(x)dx) into itself when w is a weight in the Ap class for 1 < p < ∞. For p = 1 we get weak type for the A₁ class. Weighted estimated are also obtained in the extreme case p = ∞.

We study the space of positively expansive differentiable maps of a compact connected C^∞ Riemannian manifold without boundary. It is proved that (i) the C¹-interior of the set of positively expansive differentiable maps coincides with the set of expanding maps, and (ii) C¹-generically, a differentiable map is positively expansive if and only if it is expanding.

In this paper, the authors study the L^p-mapping properties of certain maximal operators with non-isotropic dilation on product domains. As an application, the L^p-boundedness of the corresponding non-isotropic multiple singular integral operator is also obtained. Here the integral kernelfunctions Ω belong to the spaces L(logL)^α(Σ₁ × Σ₂) for some α > 0, which is optimal.

For a strongly connected digraph D the restricted arc-connectivity λ' (D) is defined as the minimum cardinality of an arc-cut over all arc-cuts S satisfying that D ? S has a non-trivial strong component D1 such that D?V(D₁) contains an arc. Let S be a subset of vertices of D. We denote by ω⁺(S) the set of arcs uv with u ∈ S and v∉S, and by ω^?(S) the set of arcs uv with u∉S and v ∈ S. A digraph D = (V,A) is said to be λ' -optimal if λ' (D) = ξ' (D), where ξ' (D) is the minimum arc-degree of D defined as ξ (D) = min{em>ξ (xy) : xy∈ A}, and ξ (xy) = min{|ω +({x, y})|, |ω-({x, y|, |ω+(x) ∪ ω-(y)|, |ω-(x)∪ω+(y)|}. In this paper a sufficient condition for a s-geodetic strongly connected digraph D to be λ'-optimal is given in terms of its diameter. Furthermore we see that the h-iterated line digraph L^h(D) of a s-geodetic digraph is λ'-optimal for certain iteration h.

We consider the properties on solutions of some q-difference equations of the form (?)where a₀(z), . . . , a_n+1(z) are meromorphic functions, a₀(z)a_n(z) ≡ 0 and q ∈ C such that 0 < |q| ≤ 1.We give estimates on the upper bound for the length of the gap in the power series of entire solutions of (?) when the coefficients a₀(z), . . . , a_n+1(z) are polynomials and 0 < |q| < 1. For some special cases,we give estimates of growth of f(z). And we also show that the case 0 < |q| < 1 is different from the case |q| = 1.

As a natural generalization of a restricted Lie algebra, a restricted Lie triple system was defined by Hodge. In this paper, we develop initially the Frattini theory for restricted Lie triple systems, generalize some results of Frattini p-subalgebra for restricted Lie algebras, obtain some properties of the Frattini p_{-subsystem and give the relationship between φp(T) and φ(T) for solvable Lie triple systems.}

A directed triple system of order v with index λ, briefly by DTS(v, λ), is a pair (X, B)where X is a v-set and B is a collection of transitive triples (blocks) on X such that every ordered pair of X belongs to λ blocks of B. A simple DTS(v, λ) is a DTS(v, λ) without repeated blocks. A simple DTS(v, λ) is called pure and denoted by PDTS(v, λ) if (x, y, z) ∈ B implies z, y, x), (z, x, y), (y, x, z),(y, z, x), (x, z, y) ∈ B. A large set of disjoint PDTS(v, λ), denoted by LPDTS(v, λ), is a collection of 3(v ? 2)/λ disjoint pure directed triple systems on X. In this paper, some results about the existence for LPDTS(v, λ) are presented. Especially, we determine the spectrum of LPDTS(v, 2).

In this paper we study the cohomogeneity one de Sitter space Sⁿ_{1 .} We consider the actions in both proper and non-proper cases. In the first case we characterize the acting groups and orbits and we prove that the orbit space is homeomorphic to R. In the latter case we determine the groups and consequently the orbits in some different cases and prove that the orbit space is not Hausdorff.

Estimations of parametric functions under a system of linear regression equations with correlated errors across equations involve many complicated operations of matrices and their generalized inverses. In the past several years, a useful tool — the matrix rank method was utilized to simplifyvarious complicated operations of matrices and their generalized inverses. In this paper, we use the matrix rank method to derive a variety of new algebraic and statistical properties for the best linear unbiased estimators (BLUEs) of parametric functions under the system. In particular, we give thenecessary and sufficient conditions for some equalities, additive and block decompositions of BLUEs of parametric functions under the system to hold.

The existence and uniqueness of the solutions are proved for a class of fourth-order stochastic heat equations driven by multi-parameter fractional noises. Furthermore the regularity of the solutions is studied for the stochastic equations and the existence of the density of the law of the solutionis obtained.

In this paper, using the integral method observed by Mai Jiehua recently, we show that no dendrite admits a sensitive commutative group action.

As a discrete analogue of Aleksandrov’s projection theorem, it is natural to ask the following question: Can an o-symmetric convex lattice set of the integer lattice Zⁿ be uniquely determined by its lattice projection counts? In 2005, Gardner, Gronchi and Zong discovered a counterexample with cardinality 11 in the plane. In this paper, we will show that it is the only counterexample in Z² up tounimodular transformations and with cardinality not larger than 17.

We consider the problem u_xx(x, t) = u_t(x, t), 0 ≤ x < 1, t≥ 0, where the Cauchy data g(t) is given at x = 1. This is an ill-posed problem in the sense that a small disturbance on the boundary g(t) can produce a big alteration on its solution (if it exists). We shall define a wavelet solution to obtain the well-posed approximating problem in the scaling space V_j . In the previous papers, the theoretical results concerning the error estimate are L²-norm and the solutions aren’t stable at x = 0. However, in practice, the solution is usually required to be stable at the boundary. In this paper we shall give uniform convergence on interval x ∈ [0, 1].

We introduce the concept of α times C-second resolvent families and present the relationship between α times C-resolvent families and α times C-second resolvent families. Moreover,the perturbation and square root for α times C-resolvent families are considered in this paper which generalize the counterparts of C-Cosine operator functions.

Some of the variation formulas of a metric were derived in the literatures by using the local coordinates system. In this paper, We give the first and the second variation formulas of the Riemannian curvature tensor, Ricci curvature tensor and scalar curvature of a metric by using the moving frame method. We establish a relation between the variation of the volume of a metric and that of a submanifold. We find that the latter is a consequence of the former. Finally we give an application of these formulas to the variations of heat invariants. We prove that a conformally flat metric g is a ritical point of the third heat invariant functional for a compact 4-dimensional manifold M, then (M,g) is either scalar flat or a space form.

The Navier–Stokes system for one-dimensional compressible fluids with density-dependent viscosities when the initial density connects to vacuum continuously is considered in the present paper.When the viscosity coefficient μ is proportional to ρ^θ with 0 < θ < 1, the global existence and the uniqueness of weak solutions are proved which improves the previous results in [Vong, S. W., Yang,T., Zhu, C. J.: Compressible Navier–Stokes equations with degenerate viscosity coefficient and vacuum II. J. Differential Equations, 192(2), 475–501 (2003)]. Here ρ is the density. Moreover, a stabilization rate estimate for the density as t → +∞ for any θ > 0 is also given.

In this article we prove that some of the su?cient and necessary optimality conditions ob-tained by Ginchev,Guerraggio,Luc[Appl.Math.,51,5–36(2006)]generalize(strictly)those presentedby Guerraggio,Luc[J.Optim.Theory Appl.,109,615–629(2001)].While the former paper showsexamples for which the conditions given there are e?ective but the ones from the latter paper fail,itdoes not prove that generally the conditions it proposes are stronger.In the present note we completethis comparison with the lacking proof.

We introduce oriented tree diagram Lie algebras which are generalized from Xu’s bothupward and downward tree diagram Lie algebras,and study certain numerical invariants of thesealgebras related to abelian ideals.

Let I be an interval of positive rational numbers.Then the set S(I)=T∩N,where T isthe submonoid of( Q⁺₀,+)generated by T,is a numerical semigroup.These numerical semigroups arecalled proportionally modular and can be characterized as the set of integer solutions of a Diophantineinequality of the form ax mod b≤cx.In this paper we are interested in the study of the maximal in-tervals I subject to the condition that S(I)has a given multiplicity.We also characterize the numericalsemigroups associated with these maximal intervals.

In this paper,we study Triebel–Lizorkin space estimates for an oscillating multiplierm_Ω,α,β.This operator was initially studied by Wainger and by Fefferman–Stein in the Lebesgue spaces.We obtain the boundedness results on the Triebel–Lizorkin spaceF^α,q_p(Rⁿ)for different p,q.

We prove a global version of the implicit function theorem under a special condition andapply this result to the proof of a modified Hyers–Ulam–Rassias stability of exact differential equationsof the form,g(x,y)+h(x,y)y=0.

The author establishesλ-central BMO estimates for commutators of multilinear fractionalintegral operators on central Morrey spaces.

Let T be a bounded linear operator on a complex Hilbert space H.In this paper weintroduce a new class denoted by l-*-A,of operators satisfying T^*|T²|T≥T^*|T^*|²T,and we prove thebasic properties of these operators.Using these results,we also prove that if T or T^*∈l-*-A,thenw(f(T))=f(w(T)),σ_ea(f(T))=f(σ_ea(T))for every f∈H(σ(T)),where H(σ(T))denotes the set ofall analytic functions on an open neighborhood ofσ(T).

In this paper,we introduce the concept of a g-Besselian frame in a Hilbert space and discussthe relations between a g-Besselian frame and a Besselian frame.We also give some characterizationsof g-Besselian frames.In the end of this paper,we discuss the stability of g-Besselian frames.Ourresults show that the relations and the characterizations between a g-Besselian frame and a Besselianframe are different from the corresponding results of g-frames and frames.