The aim of this paper is to clarify the role played by resolvent estimates for nonlinear stability. We will give examples showing that a large resolvent may lead to a small domain of nonlinear stability. In other examples the resolvent is large, but the domain of nonlinear stability is completely unrestricted. Which case prevails depends on the details of the problem.We will also show that the size of the resolvent depends in an essential way on the norms that are used.

Let A be a noncommutative Banach algebra. Suppose there exists a continuous linear Jordan derivation D : A→A such that [D(x), x]D(x)[D(x),x] ∈ rad(A) for all x∈A. In this case, D(A) ⊆ rad (A)

Let M and N be two compact Riemannian manifolds. Let u_k (x, t) be a sequence of strong stationary weak heat flows from M×R⁺ to N with bounded energies. Assume that u_k→u weakly in H^1,2(M×R⁺, N) and that Σ^t is the blow-up set for a fixed t > 0. In this paper we first prove Σ^t is an H^m-2-rectifiable set for almost all t∈R⁺. And then we prove two blow-up formulas for the blow-up set and the limiting map. From the formulas, we can see that if the limiting map u is also a strong stationary weak heat flow, Σ^t is a distance solution of the (m-2)-dimensional mean curvature flow [1]. If a smooth heat flow blows-up at a finite time, we derive a tangent map or a weakly quasi-harmonic sphere and a blow-up set ∪_t<0Σ^t× {t}. We prove the blow-up map is stationary if and only if the blow-up locus is a Brakke motion.

Let G be a finite group. The question how the properties of its minimal subgroups influence the structure of G is of considerable interest for some scholars. In this paper we try to use c-normal condition on minimal subgroups to characterize the structure of G. Some previously known results are generalized.

In this paper, a sufficient condition is given under which the smash product A#H is a transfinite left free normalizing extension of an algebra A. Moreover, the result is applied to a skew semigroup ring, a skew group ring and the quantum group U_q(sl(2)) such that some properties are shown.

In this paper a special kind of triangulated maps on the sphere called fair triangulations is enumerated with the size of maps as parameter. Moreover, the number of several other kinds of triangulations are enumerated as well.

In this paper, we study the boundedness and compactness of composition operator C_φ on the Bloch space β(Ω), Ω being a bounded homogeneous domain. For Ω = B_n, we give the necessary and sufficient conditions for a composition operator C_φ to be compact on β(B_n) or β₀(B_n).

This note is devoted to the study of the stochastic comparability of jump processes. On the basis of [2] and [3], it is proved that two jump processes are stochatically comparable if and only if their q-pairs are comparable. Meanwhile, the result concerning the uniqueness given in [6] is also improved upon.

Given any positive integers k≥ 3 and λ, let c(k, λ) denote the smallest integer such that v∈B(k, λ) for every integer v≥c(k, λ) that satisfies the congruences λv(v-1) ≡ 0(mod k(k-1)) and λ(v-1) ≡ 0(mod k-1). In this article we make an improvement on the bound of c(k, λ) provided by Chang in [4] and prove that c(k,λ)≤exp{k^3k6}. In particular, c(k,λ)≤exp{k^3k6}.

In this paper, we use the ordinary differential equation theory of Banach spaces and minimax theory, and in particular, the relative mountain pass lemma to study semilinear elliptic boundary value problems with jumping nonlinearities at zero or infinity, and get new multiple solutions and sign-changing solutions theorems, at last we get up to six nontrivial solutions.

This paper is concerned with an optimal control problem for semilinear elliptic variational inequalities associated with bilateral constraints. Existence and optimality conditions of the optimal pair are established.

We introduce certain Calderón-Zygmund-type operators and discuss their boundedness on spaces such as weighted Lebesgue spaces, weighted weak Lebesgue spaces, weighted Hardy spaces and weighted weak Hardy spaces. The sharpness of some results is also investigated.

This paper develops asymptotic properties of singularly perturbed Markov chains with inclusion of absorbing states. It focuses on both unscaled and scaled occupation measures. Under mild conditions, a mean-square estimate is obtained. By averaging the fast components, we obtain an aggregated process. Although the aggregated process itself may be non-Markovian, its weak limit is a Markov chain with much smaller state space. Moreover, a suitably scaled sequence consisting of a component of scaled occupation measures and a component of the aggregated process is shown to converge to a pair of processes with a switching diffusion component.

Let A be a unital simple C^*-algebra of real zero, stable rank one, with weakly unperforated K₀(A) and unique normalized quasi-trace τ, and let X be a compact metric space. We show that two monomorphisms φ, ψ : C(X)→A are approximately unitarily equivalent if and only if φψ induce the same element in KL(C(X), A) and the two lineal functionals τοφ and τοψ are equal. We also show that, with an injectivity condition, an almost multiplicative morphism from C(X) into A with vanishing KK-obstacle is close to a homomorphism.

A central problem in the mathematical analysis of fluid dynamics is the asymptotic limit of the fluid flow as viscosity goes to zero. This is particularly important when boundaries are present since vorticity is typically generated at the boundary as a result of boundary layer separation. The boundary layer theory, developed by Prandtl about a hundred years ago, has become a standard tool in addressing these questions. Yet at the mathematical level, there is still a lack of fundamental understanding of these questions and the validity of the boundary layer theory. In this article, we review recent progresses on the analysis of Prandtl's equation and the related issue of the zero-viscosity limit for the solutions of the Navier-Stokes equation. We also discuss some directions where progress is expected in the near future.

In this note, by using the method of S. Zhang[1], we obtain the local version of a theorem of BGS[2] which links Faltings heights of projective varieties with the Philippon heights for the corresponding generalized Chow points. By the stable reduction theorem of S. Zhang[1], we prove that Chow semistabilities and generalized Chow semistabilities are the same.

We prove in this paper that the BSD conjecture holds for a certain kind of elliptic curves.

In this paper, we prove that for every symplectic matrix M possessing eigenvalues on the unit circle, there exists a symplectic matrix PP^-1 MP is a symplectic matrix of the normal forms defined in this paper.

In this paper, for two given transition probabilities, the existence of the optimal Markovian coupling with respect to a non-negative lower semi-continuous function is proved. As an application of this result, the well-known Strassen's theorem is generalized. Moreover, it is proved that the existence of an order-preserving Markovian coupling of two given jump processes is equivalent to their stochastical comparability.

In this paper, we prove that if two multiresolutions satisfy a relation, then they have the same accuracy in the multivariate FSI case.

In this paper we use the Calderón-Zygmund operator theory to provide a discrete Calderón-type reproducing formula. Since translation, dilation and, in particular, the Fourier transform are never used in the proofs, all results still hold on spaces of homogenous type introduced by Coifman and Weiss. As a consequence, we obtain a class of frames with the minimum regularity properties.

Let κ∈N. We prove that the multilinear operators of finite sums of products of singular integrals on R_n are bounded from into if they have vanishing moments up to a certain order dictated by the target spaces. These conditions on vanishing moments satisfied by the multilinear operators are also necessary when α_j≥ 0 and the singular integrals considered here include the Calderón-Zygmund singular integrals and the fractional integrals of any orders.

The structure of the set S of shiftable points of wavelet subspaces is researched in this paper. We prove that S = R or S = 1/qZ where q∈N. The spectral and functional characterizations for the shiftability are given. Furthermore, the nonharmonic wavelet bases are discussed.

In this paper, we consider nonlinear degenerate quasilinear parabolic initial boundary value problems of second order. Using results from the theory of pseudomonotone operators, we show that there exists at least one weak solution in a suitable weighted Sobolev space.

We introduce the quasi-homeomorphisms of generalized Dirichlet forms and prove that any quasi-regular generalized Dirichlet form is quasi-homeomorphic to a semi-regular generalized Dirichlet form. Moreover, we apply this quasi-homeomorphism method to study the measures of finite energy integrals of generalized Dirichlet forms. We show that any 1-coexcessive function which is dominated by a function in is associated with a measure of finite energy integral. Consequently, we prove that a Borel set B is ε-exceptional if and only if μ(B) = 0 for any measure μ of finite energy integral.

In this paper, we consider the bidimensional exterior unsteady Navier-Stokes equations with nonhomogeneous boundary conditions and present an Oseen coupling problem which approximates the Navier-Stokes problem, obtained by coupling the Navier-Stokes equations in the inner region and the Oseen equations in the outer region. Moreover, we prove the existence, uniqueness and the approximate accuracy of the weak solution of the Oseen coupling equations.

In this paper we prove a very general result concerning solvability of the resonant problem: Δu + λ_k u + g(x, u) = h(x); u = 0, x ∈∂Ω, which immediately gives three generalized Landesman-Lazer conditions. The most interesting application of the general result is concerned with the problem when λ_k = λ₁, in which case we prove solvability results for it under conditions which are not the standard Landesman-Lazer condition or only partly enjoy it. Furthermore, we propose a new sign condition and give a comprehensive extension of a main result of Figueiredo and Ni.

A variational formula for the lower bound of the principal eigenvalue of general Markov jump processes is presented. The result is complete in the sense that the condition is fulfilled and the resulting bound is sharp for Markov chains under some mild assumptions.

In this paper we shall characterize the large deviation principles (abbreviated to LDP) of Donsker-Varadhan of a Markov process both for the weak convergence topology and for the τ-topology, by means of a hyper-exponential recurrence property. A Lyapunov criterion for this type of recurrence property is presented. These results are applied to countable Markov chains, unidimensional diffusions, elliptic or hypoelliptic diffusions on Rienmannian manifolds. Several counter-examples are equally presented.

Let f(x) be a continued fraction with elements a_nx, where coefficients a_n are positive algebraic numbers. Using the criterion of [1] for any nonzero real algebraic numbers α₁,..., α_s with distinct absolute values the algebraic independence of the values f(α₁),...,f(α_s) is proved under certain assumption concerning only with a_n. For some transcendental numbers ξ the algebraic independence of values f(ξ^j)(j∈Z) is also established.

This paper has studied two open questions about normal functions due to Lappan, and obtained two corresponding results for α-normal functions.

This paper consists of three main parts. One of them is to develop local and global Sobolev interpolation inequalities of any higher order for the nonisotropic Sobolev spaces on stratified nilpotent Lie groups. Despite the extensive research after Jerison's work [3] on Poincaré-type inequalities for Hörmander's vector fields over the years, our results given here even in the nonweighted case appear to be new. Such interpolation inequalities have crucial applications to subelliptic or parabolic PDE's involving vector fields. The main tools to prove such inequalities are approximating the Sobolev functions by polynomials associated with the left invariant vector fields on G. Some very useful properties for polynomials associated with the functions are given here and they appear to have independent interests in their own rights. Finding the existence of such polynomials is the second main part of this paper. Main results of these two parts have been announced in the author's paper in Mathematical Research Letters [38]. The third main part of this paper contains extension theorems on anisotropic Sobolev spaces on stratified groups and their applications to proving Sobolev interpolation inequalities on (ε, δ) domains. Some results of weighted Sobolev spaces are also given here. We construct a linear extension operator which is bounded on different Sobolev spaces simultaneously. In particular, we are able to construct a bounded linear extension operator such that the derivatives of the extended function can be controlled by the same order of derivatives of the given Sobolev functions. Theorems are stated and proved for weighted anisotropic Sobolev spaces on stratified groups.

The binomial tree method is the most popular numerical approach to pricing options. However, for currency lookback options, this method is not consistent with the corresponding continuous models, which leads to slow speed of convergence. On the basis of the PDE approach, we develop a consistent numerical scheme called the modified binomial tree method. It possesses one order of accuracy and its efficiency is demonstrated by numerical experiments. The convergence proofs are also produced in terms of numerical analysis and the notion of viscosity solution.

In this paper, we introduce the concept of the G class of functions of the parabolic class, and show the Hölder continuity of the G class of functions. The introduction of this concept contributes to the proof of the regularity and existence of the solution for the first boundary problem of parabolic equation in divergence form.

In this paper, we study real Banach * algebras systematically. We present the right form of Pták's inequality [1,4] in the real case, and generalize the results of Vukman in [3] to the general case (algebras with or without an identity). Moreover, this paper is a real analogue of Pták's work [1] in the complex case.

In this paper we show that there exists a unique local smooth solution for the Cauchy problem of the inhomogeneous Schrödinger flow for maps from a compact Riemannian manifold M with dim (M)≤3 into a compact Kähler manifold (N, J) with nonpositive Riemannian sectional curvature.

This paper is concerned with the existence of periodic solutions for a nonlinear system of ordinary differential equations. We obtain a Nagumo-type a priori bound for the periodic solutions and then by using this a priori bound we prove the existence of at least one T-periodic solution under some general conditions.

In this paper, we establish the global fast dynamics for the derivative Ginzburg-Landau equation in two spatial dimensions. We show the squeezing property and the existence of finite dimensional exponential attractors for this equation.

A hierarchy of multidimensional Hénon-Heiles (M-H-H) systems are constructed via the x- and t_n-higher-order-constrained flows of KdV hierarchy. The Lax representation for the M-H-H hierarchy is determined from the adjoint representation of the auxiliary linear problem for the KdV hierarchy. By using the Lax representation the classical Poisson structure and r-matrix for the hierarchy are found and the Jacobi inversion problem for the hierarchy is constructed.

Hua's estimate is established for character sums in a number field. A relationship between liftings of a character sum in a local field is also studied.