In this note, we present a connection between equivariant Bott-Chern classes and Kähler-Ricci solitons. We also propose a generalized version the of the K-energy.

Kenneth R. Davidson raised ten open problems in the book Nest Algebras. One of these open problems is
Problem 7 If K∩AlgL is weak^*. dense in AlgL, where K is the set of all compact operators in B(H), is L completely distributive?
In this note, we prove that there is a reflexive subspace lattice L on some Hilbert space, which satisfies the following conditions:
(a) F(AlgL) is dense in AlgL in the ultrastrong operator topology, where F(AlgL) is the set of all finite rank operators in AlgL;
(b) L isn't a completely distributive lattice.
The subspace lattices that satisfy the above conditions form a large class of lattices. As a special case of the result, it easy to see that the answer to Problem 7 is negative.

In this paper, we give an example of a Hausdorff centered-Lindelöf space X such that St-l(X) = c⁺, which negatively answers two questions rasied by Bonanzinga and Matveev.

By using the Riccati technique and the technique, new oscillation criteria are obtained for the second order matrix differential system
(P(t)Y'(t))' + r(t)P(t)Y'(t) + Q(t)Y (t) = 0, t ≥ t₀.
These results in the present paper generalize and improve many known conclusions. Furthermore, some results are different from the most known ones in the sense that they are based on the information only on a sequence of subintervals of [t₀,∞), rather than on the whole half-line. In particular, our results complement a number of existing results and handle the case that is not covered by the known criteria.

The concept of an RS-set in a complex Banach space is introduced and the problem of best approximation from an RS-set in a complex space is investigated. Results consisting of characterizations, uniqueness and strong uniqueness are established.

By using the linear independent fundamental solutions of the linear variational equation along the heteroclinic loop to establish a suitable local coordinate system in some small tubular neighborhood of the heteroclinic loop, the Poincaré map is constructed to study the bifurcation problems of a fine 3-point loop in higher dimensional space. Under some transversal conditions and the non-twisted condition, the existence, coexistence and incoexistence of 2-point-loop, 1-homoclinic orbit, simple 1-periodic orbit and 2-fold 1-periodic orbit, and the number of 1-periodic orbits are studied. Moreover, the bifurcation surfaces and existence regions are given. Lastly, the above bifurcation results are applied to a planar system and an inside stability criterion is obtained.

In this paper, we establish sufficient conditions on weights which ensure that high-order Riesz-Bessel transformations generated by the generalized shift operator act boundedly from one weighted L_p-space into another.

In this paper we study the singularity at the origin with three-fold zero eigenvalue for symmetric vector fields with nilpotent linear part and 3-jet C^∞-equivalent to
y∂/∂x+z∂/∂y+ax²y∂/∂z
with a ≠ 0. We first obtain several subfamilies of the symmetric versal unfoldings of this singularity by using the normal form and blow-up methods under some conditions, and derive the local and global bifurcation behavior, then prove analytically the existence of the Sil'nikov homoclinic bifurcation for some subfamilies of the symmetric versal unfoldings of this singularity, by using the generalized Mel'nikov methods of a homoclinic orbit to a hyperbolic or non-hyperbolic equilibrium in a highdimensional space.

In this paper, two classes of closely related multilinear singular and fractional integrals, which include the commutators as special cases, are studied and their boundedness on Herz type spaces is discussed. In fact, it is proved that these operators are actually not bounded in certain extreme cases.

We obtain sufficient conditions for the existence of periodic solutions of the following second order nonlinear differential equation:

Our approach is based on the continuation theorem of the coincidence degree, and the priori estimate of periodic solutions.

The main purpose of this paper is, using the mean-value theorem of Dirichlet l-functions, to study the distribution properties of the hybrid mean value involving certain Hardy sums and Ramanujan sum, and give four interesting identities.

We propose a one-step smoothing Newton method for solving the non-linear complementarity problem with P₀-function (P₀-NCP) based on the smoothing symmetric perturbed Fisher function (for short, denoted as the SSPF-function). The proposed algorithm has to solve only one linear system of equations and performs only one line search per iteration. Without requiring any strict complementarity assumption at the P₀-NCP solution, we show that the proposed algorithm converges globally and superlinearly under mild conditions. Furthermore, the algorithm has local quadratic convergence under suitable conditions. The main feature of our global convergence results is that we do not assume a priori the existence of an accumulation point. Compared to the previous literatures, our algorithm has stronger convergence results under weaker conditions.

This paper contains a generalization of the well-known Palais-Smale and Cerami compactness conditions. The compactness condition introduced is used to prove some general existence theorems for critical points. Some applications are given to differential equations.

An excellent introduction to the topic of poset matroids is due to M. Barnabei, G. Nicoletti and L. Pezzoli. On the basis of their work, we have obtained the global rank axioms for poset matroids. In this paper, we study the special integral function f and obtain a new class of poset matroids from the old ones, and then we generalize this result according to the properties of f. Almost all of these results can be regarded as the application of global rank axioms for poset matroids. The main results in our paper have, indeed, investigated the restriction of the basis of the poset matroid, and we give them the corresponding geometric interpretation.

In this paper, we extend a classical result of Hua to arithmetic progressions with large moduli. The result implies the Linnik Theorem on the least prime in an arithmetic progression.

We consider the stabilization of Naghdi's model by boundary feedbacks where the model has a middle surface of any shape. First, applying the semigroup approach and the regularity of elliptic boundary value problems, we obtain the existence, the uniqueness, and the properties of solutions to Naghdi's model. Finally, we establish the exponential decay rates for Naghdi's model under some checkable geometric conditions on the middle surface.

In this work we study Lie symmetries of planar quasihomogeneous polynomial vector fields from different points of view, showing its integrability. Additionally, we show that certain perturbations of such vector fields which generalize the so-called degenerate infinity vector fields are also integrable.

Let a₀ < a₁ < … < a_n be positive integers with sums ε_ia_i(ε_i= 0,1) distinct. P. Erdös conjectured that 1/a_i≤ 1/2^#em/em#. The best known result along this line is that of Chen: Let f be any given convex decreasing function on [A, B] with α₀, α₁, …, α_n, β₀, β₁, …, β_n being real numbers in [A, B] with α₀ ≤ α₁ ≤ … ≤ α_n, a_i≥ β_i,k=0,…,n.Then f(a_i)≤ f(β_i). In this paper, we obta_in two generalizations of the above result; each is of special interest in itself. We prove:
Theorem 1 Let f and g be two given non-negative convex decreasing functions on [A, B], and α₀, α₁, …, α_n, β₀, β₁, …, β_n, α'₀, α'₁, …, α'_n, β'₀, β'₁, …, β'_n be real numbers in [A, B] with α₀ ≤ α₁ ≤ … ≤ α_n, a_i≥ β_i,t=0,…,n,α'₀ ≤ α'₁ ≤ … ≤ α'_n, α'_i≥ β'_#em/em#,t=0,…,n.Then f(a_i)g(α'_i)≤ f(β_i)g(β'_#em/em#).
Theorem 2 Let f be any given convex decreasing function on [A, B] with k₀, k₁, …, k_n being nonnegative real numbers and α₀, α₁, …, α_n, β₀, β₁, …, β_n being real numbers in [A, B] with α₀ ≤ α₁ ≤ … ≤ α_n, k_ia_i≥ k_iβ_i,t=0, …,n.Then k_if(a_i)≤ k_if(β_i).

We deal with the existence of periodic solutions for doubly degenerate quasilinear parabolic equations of higher order, which can degenerate, on a part of the boundary, on a segment in the interior of the domain and in time.

In this paper, the authors discuss a class of multilinear singular integrals and obtain that the operators are bounded from H¹ (Rⁿ) to weak L¹ (Rⁿ). Using this result, we can directly prove a main theorem in [5].

This paper provides some functional equations satisfied by the generating functions for enumerating general rooted planar maps with up to three parameters. Furthermore, the generating functions can be obtained explicitly by employing the Lagrangian inversion. This is also an answer to an open problem in 1989.

A graph is called a semi-regular graph if its automorphism group action on its ordered pair of adjacent vertices is semi-regular. In this paper, a necessary and sufficient condition for an automorphism of the graph Γ to be an automorphism of a map with the underlying graph Γ is obtained. Using this result, all orientation-preserving automorphisms of maps on surfaces (orientable and non-orientable) or just orientable surfaces with a given underlying semi-regular graph Γ are determined. Formulas for the numbers of non-equivalent embeddings of this kind of graphs on surfaces (orientable, non-orientable or both) are established, and especially, the non-equivalent embeddings of circulant graphs of a prime order on orientable, non-orientable and general surfaces are enumerated.

In this paper, we consider the relation of the Morse index of a closed geodesic with the Maslov-type index of a path in a symplectic group. More precisely, for a closed geodesic c on a Riemannian manifold M with its linear Poincaré map P (a symplectic matrix), we construct a symplectic path γ(t) starting from identity I and ending at P, such that the Morse index of the closed geodesic c equals the Maslov-type index of γ. As an application of this result, we study the parity of the Morse index of any closed geodesic.

In this paper, we prove that R is a two-sided Artinian ring and J is a right annihilator ideal if and only if (i) for any nonzero right module, there is a nonzero linear map from it to a projective module; (ii) every submodule of R _R is not a radical module for some right coherent rings. We call a ring a right X ring if Hom_R(M, R) = 0 for any right module M implies that M = 0. We can prove some left Goldie and right X rings are right Artinian rings. Moreover we characterize semisimple rings by using X rings. A famous Faith's conjecture is whether a semipimary PF ring is a QF ring. Similarly we study the relationship between X rings and QF and get many interesting results.

In this paper, we study one conjecture proposed by W. Bergweiler and show that any transcendental meromorphic functions f(z) have the form exp(αz+β) if f(z)f"(z)-a(f'(z))²≠0, where a ≠ 1, n±1/n, in n∈N. Moreover, an analogous normality criterion is obtained.

Criteria for extreme points and strongly extreme points in Musielak-Orlicz sequence spaces, equipped with both the Luxemburg norm and the Orlicz norm, are given.

The aim of this paper is to investigate higher level orderings on modules over commutative rings. On the basis of the theory of higher level orderings on fields and commutative rings, some results involving existence of higher level orderings are generalized to the category of modules over commutative rings. Moreover, a strict intersection theorem for higher level orderings on modules is established.

In this paper, for an arbitrary regular biordered set E, by using biorder-isomorphisms between the ω-ideals of E, we construct a fundamental regular semigroup W_E called NH-semigroup of E, whose idempotent biordered set is isomorphic to E. We prove further that W_E can be used to give a new representation of general regular semigroups in the sense that, for any regular semigroup S with the idempotent biordered set isomorphic to E, there exists a homomorphism from S to W_E whose kernel is the greatest idempotent-separating congruence on S and the image is a full symmetric subsemigroup of W_E. Moreover, when E is a biordered set of a semilattice E₀, W_E is isomorphic to the Munn-semigroup T_E0; and when E is the biordered set of a band B, W_E is isomorphic to the Hall-semigroup W_B.

We consider the maximum likelihood estimator of the unknown parameter in a class of nonstationary diffusion processes. We give further a precise estimate for the error of the estimator.

We prove a unified convergence theorem, which presents, in four equivalent forms, the famous Antosik-Mikusinski theorems. In particular, we show that Swartz' three uniform convergence principles are all equivalent to the Antosik-Mikusinski theorems.

We consider the system of four linear matrix equations A₁X = C₁, XB₂ = C₂, A₃XB₃ = C₃ and A₄XB₄ = C₄ over R, an arbitrary von Neumann regular ring with identity. A necessary and sufficient condition for the existence and the expression of the general solution to the system are derived. As applications, necessary and sufficient conditions are given for the system of matrix equations A₁X = C₁ and A₃X = C₃ to have a bisymmetric solution, the system of matrix equations A₁X = C₁ and A₃XB₃ = C₃ to have a perselfconjugate solution over R with an involution and char R≠2, respectively. The representations of such solutions are also presented. Moreover, some auxiliary results on other systems over R are obtained. The previous known results on some systems of matrix equations are special cases of the new results.

It is proved that with at most O(N^11/12+ε) exceptions, all positive integers n ≤ N satisfying some necessary congruence conditions are the sum of three squares of primes. This improves substantially the previous results in this direction.

Let Γ be a portion of a C ^1,α boundary of an n-dimensional domain D. Let u be a solution to a second order parabolic equation in D×(-T, T) and assume that u = 0 on Γ×(-T, T), 0∈Γ. We prove that u satis.es a three cylinder inequality near Γ×(-T, T) . As a consequence of the previous result we prove that if u (x, t) = O (|x|^k) for every t∈(-T, T) and every k∈N, then u is identically equal to zero.

By means of the continuation theorem of coincidence degree theory, some new results on the non-existence, existence and unique existence of periodic solutions for a kind of second order neutral functional differential equation are obtained.

We consider the Cauchy problem of a shallow water equation and its local wellposedness.

Let J_*,k^r denote the ideal in MO_* of cobordism classes containing a representative that admits (Z₂)^k-actions with a fixed point set of constant codimension r. In this paper we determine J_*,k^{2k + 2} and J_*,3^{23 + 1}.

Let G be a graph with vertex set V (G) and edge set E(G) and let g and f be two integervalued functions defined on V (G) such that 2k - 2 ≤ g(x) ≤ f(x) for all x ∈ V (G). Let H be a subgraph of G with mk edges. In this paper, it is proved that every (mg +m- 1,mf - m + 1)-graph G has (g, f)-factorizations randomly k-orthogonal to H under some special conditions.

In this paper, we consider a bound on a general version of the integral inequalities for functions and also study the qualitative behavior of the solutions of certain classes of the hyperbolic partial delay differential equations under the integral inequalities.

Using the relation between the set of embeddings of tori into Euclidean spaces modulo ambient isotopies and the homotopy groups of Stiefel manifolds, we prove new results on embeddings of tori into Euclidean spaces.

Let S^d-1 = {x:|x| = 1} be a unit sphere of the d-dimensional Euclidean space R^d and let H^p≡H^p(S^d-1) (0 < p ≤ 1) denote the real Hardy space on S^d-1. For 0 < p ≤ 1 and f∈H^p(S^d-1), let E_j(f,H^p) (j = 0, 1,…) be the best approximation of f by spherical polynomials of degree less than or equal to j, in the space H^p(S^d-1), its Cesàro mean of order δ > -1 is denoted by σ_k^δ(f). For 0 < p ≤ 1, it is known that δ(p):= d-1/p-d/2 is the critical index for the uniform summability of σ_k^δ in the metric H^p. In this paper, the following result is proved:   Theorem Let 0<p<1 and δ(p):= d-1/p-d/2. Then for f∈H^p(S^d-1),
,
where A_N(f)≈B_N(f) means that there's a positive constant C, independent of N and f, such that C^-1 A_N(f)≤B_N(f)≤A_N(f).
In the case d = 2, this result was proved by Belinskii in 1996.