Given two doubling measures μ and ν in a metric space (S, ρ) of homogeneous type, let B₀⊂S be a given ball. It has been a well-known result by now (see [1-4]) that the validity of an L¹→L¹ Poincaré inequality of the following form:

for all metric balls B⊂B₀⊂S, implies a variant of representation formula of fractional integral type: for ν-a.e. x∈B₀,

One of the main results of this paper shows that an L¹ to L^q Poincaré inequality for some 0 < q < 1, i.e.,

for all metric balls B?B₀, will suffice to imply the above representation formula. As an immediate corollary, we can show that the weak-type condition,

also implies the same formula.
Analogous theorems related to high-order Poincaré inequalities and Sobolev spaces in metric spaces are also proved.

In this paper we prove the existence of global weak solutions of the p-harmonic flow with potential between Riemannian manifolds M and N for arbitrary initial data having finite p-energy in the case when the target N is a homogeneous space with a left invariant metric.

We are concerned with the following Dirichlet problem:
(P)
where f(x, t) ∈C (Ω×R), f(x, t)/t is nondecreasing in t∈R and tends to an L^∞-function q(x) uniformly in x∈Ω as t→ + ∞ (i.e., f(x, t) is asymptotically linear in t at infinity). In this case, an Ambrosetti-Rabinowitz-type condition, that is, for some θ > 2, M > 0,
(AR)
is no longer true, where As is well known, (AR) is an important technical condition in applying Mountain Pass Theorem. In this paper, without assuming (AR) we prove, by using a variant version of Mountain Pass Theorem, that problem (P) has a positive solution under suitable conditions on f(x, t) and q(x). Our methods also work for the case where f(x, t) is superlinear in t at infinity, i.e., q(x) ≡ +∞.

This paper considers the admissibility of the estimators for finite population when the parameter space is restricted. We obtain all admissible linear estimators of an arbitrary linear function of characteristic values of a finite population in the class of linear estimators under the criterion of the expectation of mean squared error.

New kinds of strongly zero-dimensional locales are introduced and characterized, which are different from Johnstone's, and almost all the topological properties for strongly zero-dimensional spaces have the pointless localic forms. Particularly, the Stone-Čech compactification of a strongly zero-dimensional locale is strongly zero-dimensional.

In this paper it is proved that for all completely distributive lattices L, the category of L-fuzzifying topological spaces can be embedded in the category of L-topological spaces (stratified Chang-Goguen spaces) as a simultaneously bireflective and bicoreflective full subcategory.

Suppose X is a superdiffusion in R^d with general branching mechanism ψ, and denotes the total weighted occupation time of X in a bounded smooth domain D. We discuss the conditions on ψ to guarantee that has absolutely continuous states. And for particular ψ(z) = z^1+β , 0 < β≤ 1, we prove that, in the case d < 2 + 2/β, is absolutely continuous with respect to the Lebesgue measure in D, whereas in the case d > 2 + 2/β, it is singular. As we know the absolute continuity and singularity of have not been discussed before.

In this paper, we investigate the growth of solutions of the second-order linear homogeneous differential equations with entire coefficients of the same order, and obtain precise estimates of the hyper-order of their solutions.

We obtain a new substitution for modules over exchange rings satisfying related comparability. Also we investigate the structure of modules over exchange rings satisfying power comparability and provide a new class of exchange rings satisfying related comparability.

This paper constructs a new family in the stable homotopy of spheres π_t-6S represented by in the Adams spectral sequence which revisits the b_n-1g₀γ₃-elements ∈ π_t-7S constructed in [3], where t = 2pⁿ (p-1) + 6(p² + p + 1)(p-1) and p≥ 7 is a prime, n≥ 4.

In the fifties, Calderón established a formal relation between symbol and kernel distribution, but it is difficult to establish an intrinsic relation. The Calderón-Zygmund (C-Z) school studied the C-Z operators, and Hörmander , Kohn and Nirenberg, et al. studied the symbolic operators. Here we apply a refinement of the Littlewood-Paley (L-P) decomposition, analyse under new wavelet bases, to characterize both symbolic operators spaces and kernel distributions spaces with other spaces composed of some almost diagonal matrices, then get an isometric between and kernel distribution spaces.

In this paper, an existence result of entropy solutions to some parabolic problems is established. The data belongs to L¹ and no growth assumption is made on the lower-order term in divergence form.

Associated with each finite directed quiver Q is a quasi-hereditary algebra, the so-called twisted double of the path algebra kQ. Characteristic tilting modules over this class of quasi-hereditary algebras are constructed. Their endomorphism algebras are explicitly described. It turns out that this class of quasi-hereditary algebras is closed under taking the Ringel dual.

The truncation equation for the derivative nonlinear Schrödinger equation has been discussed in this paper. The existence of a special heteroclinic orbit has been found by using geometrical singular perturbation theory together with Melnikov's technique.

In this paper, we give characterizations of Nash inequalities for birth-death process and diffusion process on the line. As a by-product, we prove that for these processes, transience implies that the semigroups P(t) decay as

Sufficient conditions for general Markov chains are also obtained.

We address the question that if π₁-surjective maps between closed aspherical 3-manifolds have the same rank on π₁ they must be of non-zero degree. The positive answer is proved for Seifert manifolds, which is used in constructing the first known example of minimal Haken manifold. Another motivation is to study epimorphisms of 3-manifold groups via maps of non-zero degree between 3-manifolds. Many examples are given.

The notion of the star chromatic number of a graph is a generalization of the chromatic number. In this paper, we calculate the star chromatic numbers of three infinite families of planar graphs. The first two families are derived from a 3-or 5-wheel by subdivisions, their star chromatic numbers being 2+2/(2n + 1), 2+3/(3n + 1), and 2+3(3n-1), respectively. The third family of planar graphs are derived from n odd wheels by Hajos construction with star chromatic numbers 3 + 1/n, which is a generalization of one result of Gao et al.

Let X be a connected finite CW complex and d_X : K₀(C(X)) →Z be the dimension function. We show that, if A is a unital separable simple nuclear C^*-algebra of TR(A) = 0 with the unique tracial state and satisfying the UCT such that K₀(A) = Q⊕ kerd_x and K₁(A) = K₁(C(X)), then A is isomorphic to an inductive limit of M_n! (C(X)).

In this paper, we study the bifurcation problems of rough heteroclinic loops connecting three saddle points for a higher-dimensional system. Under some transversal conditions and the nontwisted condition, the existence, uniqueness, and incoexistence of the 1-heteroclinic loop with three or two saddle points, 1-homoclinic orbit and 1-periodic orbit near Γ are obtained. Meanwhile, the bifurcation surfaces and existence regions are also given. Moreover, the above bifurcation results are extended to the case for heteroclinic loop with l saddle points.

A surface Σ is a graph in R⁴ if there is a unit constant 2-form ω on R⁴ such that <e₁∧e₂, ω≥v₀>0 where {e₁, e₂} is an orthonormal frame on Σ. We prove that, if on the initial surface, then the mean curvature flow has a global solution and the scaled surfaces converge to a self-similar solution. A surface Σ is a graph in M₁×M₂ where M₁ and M₂ are Riemann surfaces, if <e₁∧e₂, ω₁>≥v₀>0 where ω₁ is a Köhler form on M₁. We prove that, if M is a K?hler-Einstein surface with scalar curvature R, on the initial surface, then the mean curvature flow has a global solution and it sub-converges to a minimal surface, if, in addition, R≥0 it converges to a totally geodesic surface which is holomorphic.

In this paper, the exchange rings R whose primitive factor rings are artinian are studied. The following results are proved: for any exchange ring R and any two-sided ideal I of R, K₀(π) : K₀(R)→K₀(R/I) is a group epimorphism with the kernel {[P]-[Q]|P = PI, Q = QI}; there is an isomorphism of ordered groups from K₀(R) to the gorup of all such functions f_P : X→Q(P∈p(R)), where X is the set of all primitive ideals of R and Q, the rational integers.

The aim of this paper is to investigate abundant semigroups with a multiplicative adequate transversal. Some properties and characterizations for such semigroups are obtained. In particular, we establish the structure of this class of abundant semigroups in terms of left normal bands, right normal bands and adequate semigroups with some simple compatibility conditions. Finally, we apply this structure to some special cases.

Let A, B be associative rings with identity, and (S, ≤) a strictly totally ordered monoid which is also artinian and finitely generated. For any bimodule _A M_B , we show that the bimodule defines a Morita duality if and only if _A M_B defines a Morita duality and A is left noetherian, B is right noetherian. As a corollary, it is shown that the ring[[A^S,≤]] of generalized power series over A has a Morita duality if and only if A is a left noetherian ring with a Morita duality induced by a bimodule _A M_B such that B is right noetherian.

This paper uses the method of upper and lower solutions and the monotone iterative technique to investigate the existence of maximal and minimal solutions of the periodic boundary value problem for first order impulsive functional differential equations.

In this paper, we study the existence of almost periodic solutions of neutral differential difference equations with piecewise constant arguments via difference equation methods.

In this paper, we show several arithmetic properties on the residues of binomial coefficients and their products modulo prime powers, e.g., for any distinct odd primes p and q. Meanwhile, we discuss the connections with the prime recognitions.

With an effort to investigate a unified approach to the Lagrange inverse Krattenthaler established operator method we finally found a general pair of inverse relations, called the Krattenthaler forumlas. The present paper presents a very short proof of this formula via Lagrange interpolation. Further, our method of proof declares that the Krattenthaler result is unique in the light of Lagrange interpolation.

We present an upper and a lower bound for the spectral radius of non-negative matrices. Then we give the bounds for the spectral radius of digraphs.

We establish the concept of a quotient affine Poisson group, and study the reduced Poisson action of the quotient of an affine Poisson group G on the quotient of an affine Poisson-G-variety V. The Poisson morphisms (including equivariant cases) between Poisson affine varieties are also discussed.

Let {X_n ;n≥1} be a sequence of i.i.d. random variables and let if |X_j | is the r-th maximum of |X₁|,..., |X_n |. Let S_n = X₁+…+X_n and Sufficient and necessary conditions for ^(r)S_n approximating to sums of independent normal random variables are obtained. Via approximation results, the convergence rates of the strong law of large numbers for ^(r)S_n are studied.

We show that there do not exist computable functions f₁(e, i), f₂(e, i), g₁(e, i), g₂(e, i) such that for all e, i ∈ ω, (1)

It follows that the splitting theorems of Sacks and Cooper cannot be combined uniformly.

Let N be a normal subgroup of a finite group G. Let ψ be an irreducible Brauer character of N. Assume π is a set of primes and χ(1)/φ(1) is a π'-number of any χ∈IBr_p (G/φ). If p'|G:N|, and N is p-solvable, then G/N has an abelian-by-metabelian Hall-π subgroup; If pψπ then G/N has a metabelian Hall-π subgroup.

Some equivalent characterizations for a skew group ring to be a Dubrovin valuation ring are given. Among them all the prime ideals of a Dubrovin valuation skew group ring are characterised.

In this paper, we investigate ideals of regular rings and give several characterizations for an ideal to satisfy the comparability. In addition it is shown that, if I is a minimal two-sided ideal of a regular ring R, then I satisfies the comparability if and only if I is separative. Furthermore, we prove that, for ideals with stable range one, Roth's problem has an affirmative solution. These extend the corresponding results on unit-regularity and one-sided unit-regularity.

It is proved that every proper holomorphic self-mapping of some kinds of Generalized Hartogs Triangles is an automorphism, and its explicit expression is given.

We generalize a theorem of Ky Fan about the nearest distance between a closed convex set D in a Banach space E and its image by a function f:D→E, in several directions: (1) for noncompact sets D, when f(D) precompact; (2) for compact D and upper semicontinuous multifunction f; and more generally, (3) for noncompact D and upper semicontinuous multifunction f with f(D) Hausdorff precompact.In particular, we prove a version of the fixed point theorem of Kakutani-Ky Fan for multifunctions whose values are convex closed bounded, thus not necessarily compact.

We derive the explicit fundamental solutions for a class of degenerate (or singular) one-parameter subelliptic differential operators on groups of Heisenberg (H) type. This extends the results of Kaplan of the sub-Laplacian on H-type groups, which in turn generalizes Folland's result on the Heisenberg group. As an application, we obtain a one-parameter representation formula for Sobolev functions of compact support on H-type groups. By choosing the parameter equal to the homogeneous dimension Q and using the Moser-Trudinger inequality for the convolutional type operator on stratified groups obtained in[18], we get the following theorem which gives the best constant for the Moser-Trudinger inequality for Sobolev functions in H-type groups.Let G be any group of Heisenberg type whose Lie algebra is generated by m left invariant vector fields and with a q-dimensional center.Let Q = m+ 2q, Q
= Q/Q−1 and 

Then

with A_Q as the sharp constant, where _G denotes the subelliptic gradient on G.This continues the research originated in our earlier study of the best constants in Moser-Trudinger inequalities and fundamental solutions for one-parameter subelliptic operators on the Heisenberg group[18].

In this paper, the autors study the continuity properties of higher order commutators generated by the homogeneous fractional integral and BMO functions on certain Hardy spaces, weak Hardy spaces and Herz-type Hardy spaces.

Consider a linear regression model, Y=β' X+ε where Y may be right censored and the cdf F_o of ε is unknown. We show that a modified semi-parametric MLE, denoted by , is strongly consistent under certain regularity conditions. Moreover, if F_o is discontinuous, then P(≠β i.o.)=0, which means that P( =β if the sample size is large)=1. The latter property has not been reported for the existing estimators. By contrast, most estimators, such as the Buckley-James estimator and M-estimators , satisfy that P( ≠β i.o.)=1.

Motivated from the study of logarithmic Sobolev, Nash and other functional inequalities, the variational formulas for Poincaré inequalities are extended to a large class of Banach (Orlicz) spaces of functions on the line. Explicit criteria for the inequalities to hold and explicit estimates for the optimal constants in the inequalities are presented. As a typical application, the logarithmic Sobolev constant is carefully examinated.