The heat flow for the minimal surface under Plateau boundary condition is defined to be a parabolic variational inequality, and then the existence, uniqueness, regularity, continuous dependence on the initial data and the asymptotics are studied. It is applied as a deformation of the level sets in the critical point theory.

In this paper, we shall mainly study the p-solvable finite group in terms of p-local rank, and a group theoretic characterization will be given of finite p-solvable groups with p-local rank two.
Theorem A Let G be a finite p-solvable group with p-local rank plr(G) = 2 and O_p (G) = 1. If P is a Sylow p-subgroup of G, then P has a normal subgroup Q such that P/Q is cyclic or a generalized quaternion 2-group and the p-rank of Q is at most two.
Theorem B Let G be a finite p-solvable group with O_p (G) = 1. Then the p-length l_p (G) < plr(G); if in addition plr(G) = 1_p(G) and p > 5 is odd, then plr(G) = 0 or 1.

In this paper, a necessary and sufficient condition concerning Huygens' principle for a family of hyperbolic equations is given, and thus Stellmacher's result is extended.

Several new results on the non-existence of some generalized bent functions are proved by using properties of the decomposition law of primes in cyclotomic fields and properties of the solutions of some special Diophantine equations.

For a class of multilinear singular integral operators T_A , < where R_m(A; x, y) denotes the m-th Taylor series remainder of A at x expanded about y, A has derivatives of order m-1 in A_β(0<β<1),Ω(x)∈L^s(S^n-1)(s≥n/n-β), is homogeneous of degree zero, the authors prove that T_A is bounded from L^p(Rⁿ) to L^q(Rⁿ)((1/p)-(1/q)=β/n,1<p<n/β) and from L¹(Rⁿ) to L^n/(n-β),∞(Rⁿ) with the bound <. And if Ω has vanishing moments of order m-1 and satisfies some kinds of Dini regularity otherwise, then T_A is also bounded from L^p(Rⁿ) to F_p^β,∞(Rⁿ)(1<s'<p<∞) with the bound <.

In this paper, we derive an elementary identity for smooth solutions of the following equation:
Δu(x)+K(x)e^2u(x)=0 in R²
and use it to get some global properties of the solutions.

Concepts of g-supersolution, g-manrtingale, g-supermartingale are introduced, which are related to BSDE with Brownian motion and Poisson point process. A strict comparison theorem, monotonic limit theorem related to this type of BSDE are also discussed. As an application of these results, a nonlinear Doob-Meyer decomposition theorem is obtained.

In this paper we study the coefficient multipliers, pointwise multipliers and cyclic vectors in the Bloch type spaces B^α and little Bloch type spaces B₀^α for 0 < α < ∞. We give several full characterizations of the coefficient multipliers (B^α ,B^β) and (B₀^α,B₀^β) for 0 < α, β < ∞ and pointwise multipliers M(B^α ,B^β) and M(B₀^α,B₀^β) for 1 ≠ α, β ∈ (0,∞). We also obtain some properties of cyclic vectors for Bloch type spaces.

Let E/K be an elliptic curve with K-rational p-torsion points. The p-Selmer group of E is described by the image of a map λ_K and hence an upper bound of its order is given in terms of the class numbers of the S-ideal class group of K and the p-division field of E.

We consider the solutions of refinement equations written in the form
<
where the vector of functions φ = (φ₁,…, φ_r)^T is unknown, g is a given vector of compactly supported functions on R^s, a is a finitely supported sequence of r × r matrices called the refinement mask, and M is an s × s dilation matrix with m = |detM|. Inhomogeneous refinement equations appear in the construction of multiwavelets and the constructions of wavelets on a finite interval. The cascade algorithm with mask a, g, and dilation M generates a sequence φ_n, n = 1, 2,…, by the iterative process <
from a starting vector of function φ₀. We characterize the L_p-convergence (0 < p < 1) of the cascade algorithm in terms of the p-norm joint spectral radius of a collection of linear operators associated with the refinement mask. We also obtain a smoothness property of the solutions of the refinement equations associated with the homogeneous refinement equation.

We single out a class of translation quivers and prove combinatorially that the translation quivers in this class are coils. These coils form a class of special coils. They are easier to visualize, but still show all the strange behaviour of general coils, and contain quasi-stable tubes as special examples.

The uniform L² stability and convergence properties for the time discretization of an evolution equation with a memory term are studied. The methods are based on the second-order backward difference methods. The memory term is approximated by the second-order convolution quadrature and interpolant quadrature.

Let n be an integer with |n| > 1. If p is the smallest prime factor of |n|, we prove that a minimal non-commutative n-insertive ring contains p⁴ elements and these rings have five (2p+4) isomorphic classes for p = 2 (p ≠ 2).

Linear symmetries of a free Bose field are exploited in the framework of Hida's white noise functionals triple. General symplectic automorphisms on the single particle space are implemented by generalized operators. The intertwining operators are constructed in a physically intuitive way, characterized analytically in terms of symbols, and factorized into three fundamental parts according to Wick ordering procedure. In particular, the classical Shale's theorem is rederived.

Consider the time-periodic perturbations of n-dimensional autonomous systems with nonhyperbolic but non-critical closed orbits in the phase space. The elementary bifurcations, such as the saddle-node, transcritical, pitchfork bifurcation to a non-hyperbolic but non-critical invariant torus of the unperturbed systems in the extended phase space (x, t), are studied. Some conditions which depend only on the original systems and can be used to determine the bifurcation structures of these problems are obtained. The theory is applied to two concrete examples.

In the paper, we study a class of standard ideals which are more general than the m-primary standard ideals discussed in [2]. We will prove an important equality concerning I-weak sequences; thus a generalization of the equality of [2] is established.

We give an Edgeworth expansion for densities of order statistics with fixed rank k. The Edgeworth expansion for densities of extreme values is then obtained as a special case k = 1.

In 1993, Ahern, Flores and Rudin showed that, if f is integrable over the unit ball B_Cⁿ of Cⁿ and satisfies
<
for every ψ∈Aut(B_Cⁿ), then f is M-harmonic if and only if n ≤ 11. The present paper is about an analogous question in the context of the unit ball B_n of Rⁿ as well as in the weighted setting.

In this paper, a characterization of continuous module homomorphisms on random semi-normed modules is first given; then the characterization is further used to show that the Hahn-Banach type of extension theorem is still true for continuous module homomorphisms on random semi-normed modules.

We derive some quadratic recursion relations for some Hodge integrals by virtual localization and obtain many closed formulas. We apply our formulas to the local geometry of toric Fano surfaces in a Calabi-Yau threefold and compute some of the numbers n_β^g in Gopakumar-Vafa's formula for all g in this case.

In this paper we count the number of SL ₂(F_2⁸) representations of torus knot groups up to a conjugacy. For the finite field F_2⁸ with character 2, the counting method is similar to that in our previous work[1]. Explicit formulae of the effective counting are given in this paper. Twisted Alexander polynomials related to those representations are discussed.

This paper gives a detailed construction of Seiberg-Witten-Floer homology for a closed oriented 3-manifold with a non-torsion Spin^c structure. Gluing formulae for certain 4-dimensional manifolds splitting along an embedded 3-manifold are obtained.

We consider the boundedness of Calderón-Zygmund operators from H^p,Ψ (Rⁿ)(the predual of a Morrey space) to h^p,Ψ (Rⁿ)(the local version of H^p,Ψ). We show that Calderón's commutator is bounded from H^p,Ψ to h^p,Ψ.

In this note, we prove that the Schrödinger flow of maps from a closed Rieman surface into a compact irreducible Hermitian symmetric space admits a global weak solution. Also, we show the existence of weak solutions to the initial value problem of Heisenberg model wit Lie algebra values, which is closely related to the Schr?dinger flow on compact Hermitian symmetric spaces.

A class of pencils (operator-valued functions of a complex argument) in a separable Hilbert space is considered. Bounds for the spectra are derived. Applications to differential operators, integral operators with delay and infinite matrix pencils are discussed.

We obtain formulae and estimates for character sums of the type S{χ,f,p^m) = Σ_x=1^pmχ(f(x)), where p^m is a prime power with m ≥ 2, χ is a multiplicative character (mod p^m), and f=f₁/f₂ is a rational function over Z. In particular, if p is odd, d=deg(f₁)+deg(f₂) and d* = max(deg(f₁), deg(f₂)) then we obtain |S(χ, f, p^m)|≤(d-1)}p^m(1-1/d*) for any non-constant f (mod p) and primitive character . For p = 2 an extra factor of 2√2 is needed.

Let n_#em/em# ,m_j >1. In[5], the sperical noncommutative torus S_p^pd was defined by twisting ×Z^l in T^pd⊕C^*(Z^l) by a totally skew multiplier p on ×Z^l for T^pd a pd-homogeneous C*-algebra over . It is shown that S_p^pd is strongly Morita equivalent to .

In this paper, we prove higher integrability results for the gradient of the solutions of some elliptic equations with degenerate coercivity whose prototype is
-div{a(x,u)Du) = f\ in D'(Ω),f∈L^r(Ω), r > 1,
where for example, a(x,u)=(1+|u|)^-θ with θ ∈ (0,1). We study the same problem for minima of functionals closely related to the previous equation.

The notion of the xst-rings was introduced by García and Marín[5] in 1999. In this paper, we consider Morita context, Morita-like equivalence and the exchange property for the xst-rings. The results of the first Morita theorem are generalized to the xst-rings. So we obtain an important Morita-like equivalence of the xst-rings, from which, as an immediate consequence, we deduce the main result of Xu-Shum-Turner[4] and the standard Morita equivalence, A ～ M_n (A), for a unital ring A. Moreover, we describe the properties of those well-known intermediate matrix rings, and show that the exchange property of a unital ring A coincides with the one for any M_n(A) as well as any intermediate matrix ring sitting between FM_Γ(A) and FC_Γ(A), which is an extension of a well-known result obtained by Nicholson[7].

In this note, we will show a new proof of the classical Calderón-Zygmund estimates established in[7] 1952. The estimates are among the fundamental estimates for elliptic equations. Parabolic equations are also considered.

The L²(Rⁿ) boundedness for the multilinear singular integral operators defined by
T_Af(x) = ∫_RⁿΩ(x-y)|x-y|ⁿ⁺¹(A(x) -A(y)-∇A(y)(x-y)) f(y)dy
is considered, where Ω is homogeneous of degree zero, integrable on the unit sphere and has vanishing moment of order one, A has derivatives of order one in BMO(Rⁿ). A sufficient condition based on the Fourier transform estimate and implying the L²(Rⁿ) boundedness for the multilinear operator T_A is given.

Some results on convergence of Newton's method in Banach spaces are established under the assumption that the derivative of the operators satisfies the radius or center Lipschitz condition with a weak L average.

The relation between the inseparable prime C^*-algebras and primitive C^*-algebras is studied, and we prove that prime AW*-algebras are all primitive C^*-algebras.

In this paper we define a very simple invariant  for a k-tuple  of unitaries in a finite factor von Neumann algebra, and we show how this invariant can replace free entropy in many of the important applications. We also introduce a notion of metric free entropy and some related concepts. We include proofs, using η, of the theorems of Liming Ge and of D. Voiculescu, respectively, on the primeness of and on the absence of Cartan subalgebras in the noncommutative free group factors.

We survey some recent results on linear maps on operator algebras that preserve invertibility. We also consider related problems such as the problem of the characterization of linear maps preserving spectrum, various parts of spectrum, spectral radius, quasinilpotents, etc. We present some results on elementary operators and additive operators preserving invertibility or related properties. In particular, we give a negative answer to a problem posed by Gao and Hou on characterizing spectrum-preserving elementary operators. Several open problems are also mentioned.

During the last ten to fifteen years, a lot of progress has been achieved in the study of complex operator spaces. In this paper, we show that a corresponding theory can be developed for real operator spaces. With some appropriate modifications, many complex results still hold for real operator spaces.

We give two applications of logarithmic Sobolev inequalities to matrix models and free probability. We also provide a new characterization of semi-circular systems through a Poincaré-type inequality.

In this note, we give some estimations of the Von Neumann-Jordan constant C_NJ(‖·‖_ψ) of Banach space (Cⁿ, ‖·‖_ψ), where ‖·‖_ψ is the absolute normalized norm on Cⁿ given by function ψ. In the case where ψ and φ are comparable, n=2 and C_NJ(‖·‖_φ)=1, we obtain a formula of computing C_NJ(‖·‖_ψ). Our results generalize some results due to Saito and others.

This paper is a brief and selective survey on classification of Hardy Submodules over the polydisk. The survey reports some progress in classification of Hardy submodules.

Several problems studied by professor R. V. Kadison are shown to be closely related. The problems were originally formulated in the contexts of homomorphisms of C^*-algebras, cohomology of von Neumann algebras and perturbations of C^*-algebras. Recent research by G. Pisier has demonstrated that all of the problems considered are related to the question of whether all C^*-algebras have finite length.