Evolution equations with a free boundary condition

In this paper we consider the heat flow of harmonic maps between two compact Riemannian Manifolds M and N (without boundary) with a free boundary condition. That is, the following initial boundary value problem ∂₁,u −Δu = Γ(u)(∇u, ∇u) [tT T_{u u}N, on M × [0, ∞), u(t, x) ∈ Σ, for x ∈ ∂M, t > 0, ∂u/t6n(t, x) ⊥_u T_u(t,x) Σ, for x ∈ ∂M, t > 0, u(o,x) = u_o(x), on M, where Σ is a smooth submanifold without boundary in N and n is a unit normal vector field of M along ∂M. Due to the higher nonlinearity of the boundary condition, the estimate near the boundary poses considerable difficulties, even for the case N = ℝⁿ, in which the nonlinear equation reduces to ∂_tu-Δu = 0. We proved the local existence and the uniqueness of the regular solution by a localized reflection method and the Leray-Schauder fixed point theorem. We then established the energy monotonicity formula and small energy regularity theorem for the regular solutions. These facts are used in this paper to construct various examples to show that the regular solutions may develop singularities in a finite time. A general blow-up theorem is also proven. Moreover, various a priori estimates are discussed to obtain a lower bound of the blow-up time. We also proved a global existence theorem of regular solutions under some geometrical conditions on N and Σ which are weaker than K_N <-0 and Σ is totally geodesic in N.     

On the coincidence of maximal stable bridges in two approach game problems for stationary control systems

In the present paper we deal with fully nonlinear two-dimensional smooth control systems with scalar input $ \dot q = f(q,u),q \in M,u \in U $ \dot q = f(q,u),q \in M,u \in U , where M and U are differentiable smooth manifolds of respective dimensions two and one. For such systems, we provide two microlocal normal forms, i.e., local in the state-input space, using the fundamental necessary condition of optimality for optimal control problems: the Pontryagin maximum principle. One of these normal forms will be constructed around a regular extremal, and the other one will be constructed around an abnormal extremal. These normal forms, which in both cases are parametrized only by one scalar function of three variables, lead to a nice expression for the control curvature of the system. This expression shows that the control curvature, a priori defined for normal extremals, can be smoothly extended to abnormals.     

The canonical foliations of a locally conformal Kähler manifold

Summary Let M be a locally conformal Kähler manifold. Then the Kähler form of M satisfies d= for some closed 1 -form , called the Lee form of M. We show that M admits three canonical foliations (four if is parallel) and we prove several properties of them, improving previous results of I. Vaisman. In particular all of these foliations are totally geodesic and Riemannian, and one of them is also almost complex. If this latter foliation is regular on a compact M, then we prove that M is a locally trivial fiber bundle over a compact Kähler manifold M, and the fibers are totally geodesic flat 2-tori. Finally we study geometrical properties, the canonical class and the Godbillon-Vey class of the totally real foliation of a CR-submanifold N cM.Work done during a visit of the second author at Michigan State University; this visit was supported by C.N.R., Italy.     

On an Approximation of Functions Regular in Convex Polygons by Linear Means of Their Exponent Series

We study linear summation methods for exponent series which are expansions of functions f(z) that are regular on an open convex polygon M and continuous on . We find estimates for deviations of the arithmetic mean and the Vallée-Poussin and Rogozinskii means.     

Some properties of a regular solution of the Helmholtz equation in a two-dimensional domain

In this paper we prove that if the function u_λ is a regular solution of the equation Δ₂u + λu = 0 in an arbitrary two-dimensional domain g and if at an arbitrary point M of the domain g we introduce polar coordinates r and?, then for an arbitrary value of the polar radius r, less than the distance of the point M from the boundary of the domain g, the following formula is valid: . Simultaneously, we show that the derivative is an n-th order trigonometric polynomial.     

Elements in one-sided unit regular rings

Let R be regular. We show that the following are equivalent:(1) R is a one sided unit regular ring. (2) For every x [euro] R, there exist an idempotente and a right or left invertible u such that x [d] eu or x [d] ue. (3) For every x [euro] R,there exists a right or left invertible u such that xu or ux is an idempotent. Moreover, we give some characterizations of one-sided unit regular rings by group inverses.     

An Integral Equality on the Laplace-Beltrami Operator

Let M be an n-dimensional Riemanniaii manifold of class C^r(r> 2)> and D \subset M be a regular oriented domain on M with boundary \partial D, and W_2,0^2(D) denote the Sobolev space defined on D with the property that every u \in W_2,0^2(D) vanishes at \partial D,i. e., $u|_\partial D=0$. Let (\alpha_=\alpha \beta) be the symmetric matrix of the positive definite metric of M,and $\nabla _\alpha$ denote the operator of the covariant derivative with respect to $\alpha _\alpha \beta$. For any $u \in C^\infty(M)$, it is convenient to define $u_\alpha=\nabla_\alpha u,u_\alpha \beta=\nabla_\alpha u_\beta$ $u^\alpha=a^\alpha\beta=a^\alpha\gamma \nabla_\gamma u^\beta.(1 \leq \alpha,\beta,\gamma \leq n)$ In this paper we establish the following Theorem. Let $u \in W_2,0^2(D)$ . Then $\int_D{u^\alpha \beta u_\alpha \beta dV}=\int_D{(\Delta u)^2dV}+\int_D{Ric(du,du)dV}-\int_\partial D{(\nabla_Nu)^2\Omega ds}$ where $\Delta$ is the Laplace-Beltrami operator, Ric (du, du) is the Ricci curvature of M with respect to the vector field du, \nabla_N is the directional derivative in the direction of the exterior normal vector N at $\partial D$ is the mean curvature of \partial D in M.     

THE GLOBAL SOLUTION OF INITIALBOUNDARY VALUE PROBLEM OF HIGHER-ORDER MULTIDIMENSIONAL EQUATION OF CHANGING TYPE

In practical problems there appears the higher-order equations of changing type. But,there is only a few of papers, which studied the problems for this kind of equations. In this paper a kind of the higher-order m     

Use of a computer to find the number of regular pentagons that can simultaneously touch a given one

Around an initial regular pentagon one describes a contour L on which one introduces a measure m. One investigates the difference S(M)=1/7m(L)?m(L∩M) where M is a pentagon touching the initial one and congruent to it. The geometric part of the investigation reduces the proof of the inequality S(M)<0 for all M to the proof of the negativity of two effectively computable functions F(u,v) and G(v) in the compact domain of the variation of the arguments. By the method of demonstrative computations, one calculates on a computer the values of these functions at the nodes of a rectangular net of the domain of the variation of the arguments by taking into account the monotonicity and one estimates the computational error. The results of the computation show that we have the inequality S(M)<0, from where it follows that the desired number is equal to six.     

Modified (0, 2)-Interpolation on the Roots of Jacobi Polynomials. II (Convergence)

In [4] A. M. Chak, A. Sharma and J. Szabados characterized the Jacobi matrices P(,), (, > –1) for which the (0,2)-interpolation problem is regular. It follows from their result, that if n is odd and = , or if , are both odd integers and n > 1 + ( + )/2, then the (0,2)-interpolation problem is not regular. Recently, the author proved that for , both odd integers, the (0,2)-interpolation problem augmented with boundary (Hermite-type) conditions at the endpoints of the interval [–1,1] is regular. In this paper the convergence of this modified (0,2)-interpolation procedure is studied, if the inner nodal points are the roots of the ultraspherical polynomials with odd integer parameter.     

Regular Component Splittable Languages

Every infinite regular language contains a regular subset of the formuv+w for some words u, v, w, where v is not the empty word. The regularsubsets of the above form are called regular components. Some well-knowncontext-free languages, such as the Dyck language and the balanced language(over an alphabet X with |X| = 2), are decomposed as disjointunions of disjunctive languages. In this paper, we investigate thedecompositions of some of the regular languages and the context-freelanguages as disjoint unions of regular components. An infinite language iscalled regular component splittable if it can be expressed as a disjointunion of regular components and a finite set. We show that the Dycklanguage, the balanced language and some disjunctive context-free splittablelanguages are regular component splittable. We also present an example toshow that there is a disjunctive context-free language which is not regularcomponent splittable.     

REGULARITY RESULTS FOR SOME QUASI-LINEAR ELLIPTIC SYSTEMS INVOLVING CRITICAL EXPONENTS

The authors show the regularity of weak solutions for some typical quasi-linear elliptic systems governed by two p-Laplacian operators. The weak solutions of the following problem with lack of compactness are proved to be regular when α(x) and α,β,p, q satisfy some conditions: where Ω(?) RN (N≥3) is a smooth bounded domain.     

非周期Dirac 方程的稳态解

本文研究Dirac方程-iΣαkku＋aβu＋M（x）u=g（x,｜u｜）u的解,其中M（x）是位势函数,g（x,｜u｜）u在无穷远处关于u是超线性的.本文用变分法来研究这一问题.借助于与此方程的＂极限方程＂相关的某个辅助系统,构造了变分泛函ΦM的环绕水平,使得建立在ΦM环绕结构上的极小极大值CM满足0〈CM〈C,这里C是＂极限方程＂的最小能量.从而可以证明（C）c条件对所有c〈C成立,因此得到了方程的最小能量解.     

二阶奇异非线性微分方程边值问题的正解

分别在0≤f₀+<M₁,m₁<f_∞-≤∞和0≤f_∞+<M₁,m₁<f₀-≤∞的情形下研究了非线性奇异边值问题u″+g(t)f(u)=0,0<t<1,αu(0)-βu′(0)=0,γu(1)+δu′(1)=0正解的存在性,其中f₀+=₀f(u)/u,f_∞-=_∞f(u)/u,f₀-=₀f(u)/u,f_∞+=_∞f(u)/u,g在区间[0,1]的端点可以具有奇性。     

拟线性椭圆型方程带有混合奇异项的正整体解

在本文中,研究了方程div(｜u｜p-2u) f(x,u)=0,x∈RN,N≥3的正整体解,其中f(x,u)在u=0未假定是正则的,且f(x,u)可以同时包含超线性,亚线性项和奇异项.     

The rank of regular morse dynamical systems

Summary If x=b ⁰ × b ¹ × . is a regular Morse sequence and sup \ ⁱ<+, then x has rank 2. There are regular Morse shifts with rank one. If x is a Kakutani sequence, then x has rank one iff x is not regular. If is a nonperiodic substitution of constant length on two symbols, then is of rank 2 iff is a continuous substitution. Every Morse sequence has a simple spectrum.     

The Isoenergy Inequality for Harmonic Maps from Rotational Symmetric Manifolds

Let u be a harmonic map from a rotational symmetric manifold M and B a unit ball in M, let E(u|B) be the energy of the map u|B and E(u|∂B) the energy of the map u|∂B, then we obtain the relationship which is called the isoenergy inequality between E(u|B) and E(u|∂B):     

Regular pseudo-median graphs

A graph is pseudo-median if for every triple u, v, w of vertices there exists either a unique vertex between each pair of them (if their mutual distances sum up to an even number) or a unique triangle whose edges lie between the three pairs of u, v, w, respectively (if the distance sum is odd). We show that a finite pseudo-median graph is regular if and only if it is the Cartesian product of a hypercube with either a complete graph or a hyper-octahedron. Every self-map of a pseudo-median graph that preserves or collapses edges has an invariant regular pseudo-median subgraph. Furthermore, the set of all vertices minimizing the total distance to the vertices of a pseudo-median graph induces a regular pseudo-median subgraph.     

Moser-Trudinger Inequality on Compact Riemannian Manifolds of Dimension Two

ln this paper, we prove Moser-Trüdinger inequality in any two dimensional manifolds. Let (M,g_M,) be a two dimensional manifold without boundary and (g, g_N) with boundary, we shall prove the following three inequalities: u∈H¹(M), \sup\limits_{and ||u||_{H¹(M)}}=1∫_M^{e^{4\pi u²}<+∞} u∈H¹(M), \sup\limits_{∫_M u=0, and} ∫_M|∇u|²=1∫_M^{e^{4\pi u²}<+∞} u∈H¹_0(N), \sup\limits_{and ∫_M|∇u²|=1∫_M^{e^{4\pi u²}<+∞} Moreover, we shall show that there exist of extremal functions which at tain the above three inequalities.     

Quotient rings of polynomial rings

Let R be a commutative ring with identity. The multiplicatively closed sets U₂={fR[X]: c(f)–1=R}, (U₂)={fU₂: f is regular} and S={fR[X]: c(f)=R} are studied. By considering various equalities between these sets, many characterizations of Noetherian rings are found. In particular, a Noetherian ring R has depth 1 if and only if S=(U₂): and each maximal ideal of a Noetherian ring is regular if and only if U₂=(U₂).The theory of Prüfer v-multiplication rings (PVMR's) is developed for rings with zero divisors. Six equivalent conditions are given to the statement that an additively regular v-ring R is a PVMR.