Hypersurfaces with constant anisotropic mean curvature in Riemannian manifolds

We formulate a variational notion of anisotropic mean curvature for immersed hypersurfaces of arbitrary Riemannian manifolds. Hypersurfaces with constant anisotropic mean curvature are characterized as critical points of an elliptic parametric functional subject to a volume constraint. We provide examples of such hypersurfaces in the case of rotationally invariant functionals defined in product spaces. These examples include rotationally invariant hypersurfaces and graphs.     

Regularized Riesz energies of submanifolds

Given a closed submanifold, or a compact regular domain, in Euclidean space, we consider the Riesz energy defined as the double integral of some power of the distance between pairs of points. When this integral diverges, we compare two different regularization techniques (Hadamard's finite part and analytic continuation), and show that they give essentially the same result. We prove that some of these energies are invariant under Möbius transformations, thus giving a generalization to higher dimensions of the Möbius energy of knots.     

Topological lower bounds on the distance between area preserving diffeomorphisms

Area preserving diffeomorphisms of the 2-disk which are Identity near the boundary form a group which can be equipped, using theL ²-norm on its Lie algebra, with a right invariant metric. In this paper we give a lower bound on the distance between diffeomorphisms which is invariant under area preserving changes of coordinates and which improves the lower bound induced by the Calabi invariant. In the case of renormalizable and infinitely renormalizable maps, our estimate can be improved and computed.     

The Ultrametric Space of Plane Branches

We study properties of the space of irreducible germs of plane curves (branches), seen as an ultrametric space. We provide various geometrical methods to measure the distance between two branches and to compare distances between branches, in terms of topological invariants of the singularity which comprises some of the branches. We show that, in spite of being very close to the notion of intersection multiplicity between two germs, this notion of distance behaves very differently. For instance, any value in [0, 1] ∩ ? is attained as the distance between a fixed branch and some other branch, in contrast with the fact that the semigroup of the fixed branch has gaps. We also present results that lead to interpret this distance as a sort of geometric distance between the topological equivalence or equisingularity classes of branches.     

两类距离边界条件域的拟共不变性和Mius变换的两个拟不变量

研究了两类距离边界条件域KBc和jBc,得到了它们的若干性质及它们的关系,证明了在一定条件下KBc和jBc是拟共不变的以及距离K(x1,x2)和j(x1,x2)在Mobius变换下具有拟不变性.     

Best proximity results in regular cone metric spaces

First, we define the notion of distance between two subsets in regular cone metric spaces. Then, we establish some conditions which guarantee the existence of best proximity points for cyclic contraction mappings on regular cone metric spaces.     

Bounded invariant verification for time-delayed nonlinear networked dynamical systems

We present a technique for bounded invariant verification of nonlinear networked dynamical systems with delayed interconnections. The underlying problem in precise bounded-time verification lies with computing bounds on the sensitivity of trajectories (or solutions) to changes in initial states and inputs of the system. For large networks, computing this sensitivity with precision guarantees is challenging. We introduce the notion of input-to-state (IS) discrepancy of each module or subsystem in a larger nonlinear networked dynamical system. The IS discrepancy bounds the distance between two solutions or trajectories of a module in terms of their initial states and their inputs. Given the IS discrepancy functions of the modules, we show that it is possible to effectively construct a reduced (low dimensional) time-delayed dynamical system, such that the trajectory of this reduced model precisely bounds the distance between the trajectories of the complete network with changed initial states. Using the above results we develop a sound and relatively complete algorithm for bounded invariant verification of networked dynamical systems consisting of nonlinear modules interacting through possibly delayed signals. Finally, we introduce a local version of IS discrepancy and show that it is possible to compute them using only the Lipschitz constant and the Jacobian of the dynamic function of the modules.     

An estimate for the mean square of the difference of velocities for homogeneous statistical solutions of the three-dimensional Navier-Stokes system

Stationary (with respect to t) statistical solutions of the Navier-Stokes system are constructed, in the presence of random external forces which are (statistically) stationary with respect to t. The domains considered are invariant under certain groups of motions (shifts and rotations). The external forces are supposed to be (statistically) invariant with respect to these groups and the constructed statistical solutions are also invariant in the same sense. For the constructed solutions, estimates of the mean square of the difference of velocities at different points are obtained in terms of a certain power of the distance between the points. The exponent depends on the dimension of the symmetry group.Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 11, pp. 3–11, 1986.     

赋范BCK-代数(英文)

引入了BCK-代数的范数与距离的概念,给出了赋范BCK-代数的一些基本性质,证明了赋范BCK-代数的同构(同态)像和原像仍是赋范BCK-代数,研究了BCK-代数与BCK-代数笛卡儿之间的赋范性质关系.并且引入了赋范BCK-代数的点列极限概念,研究了极限的相关性质.讨论了有界赋范BCK-代数的与模糊BCK-代数的关系.     

New results on affine invariant points

We prove a conjecture of B. Grünbaum stating that the set of affine invariant points of a convex body equals the set of points invariant under all affine linear symmetries of the convex body. As a consequence we give a short proof of the fact that the affine space of affine linear points is infinite dimensional. In particular, we show that the set of affine invariant points with no dual is of the second category. We investigate extremal cases for a class of symmetry measures. We show that the centers of the John and Löwner ellipsoids can be far apart and we give the optimal order for the extremal distance between the two centers.     

Application of invariant imbedding to linear multipoint boundary value problems with general boundary conditions

Two extensions of the usual application of invariant imbedding to the solution of linear boundary value problems are presented. The invariant imbedding formulation of a linear two point boundary value problem in which functional relationships are given between the variables at either one or both of the boundary points is presented. Also, extension of invariant imbedding to linear multipoint boundary value problems is given. Using these extensions singly or in combination, a general multipoint boundary value of linear ordinary differential equations can be solved. In addition, the problems of infinite initial conditions and / or indeterminate initial derivatives are resolved. Numerical examples demonstrate the feasibility and accuracy of the method.     

A note on the dichotomy spectrum

In many ways, exponential dichotomies are an appropriate hyperbolicity notion for nonautonomous linear differential or difference equations. The corresponding dichotomy spectrum generalizes the classical set of eigenvalues or Floquet multipliers and is therefore of eminent importance in a stability theory for explicitly time-dependent systems, as well as to establish a geometric theory of nonautonomous problems with ingredients like invariant manifolds and normal forms, or to deduce continuation and bifurcation techniques.

In this note, we derive some invariance and perturbation properties of the dichotomy spectrum for nonautonomous linear difference equations in Banach spaces. They easily follow from the observation that the dichotomy spectrum is strongly related to a weighted shift operator on an ambient sequence space.     

Non-classical laws with conserved quantities from the arbitrary functions included in general solution to elasticity and application

The general solution to static and/or dynamic linear elasticity is a transformation between the displacements and new arbitrary functions, whose conservativeness depends on some independent partial differential equations (PDEs) satisfied by the new arbitrary functions. Zhang's general solutions are mathematically appropriate since the displacements are expressed in terms of two new arbitrary functions, and the sum of the highest order derivative added together from the independent PDEs satisfied by the two new arbitrary functions is the same as that of Navier–Cauchy equations. Therefore, the following points should be emphasized: (i) the independent PDEs come from the Laplace and D'Alembert operators acting on the two new arbitrary functions in static and dynamic general solutions, respectively, and it is found that the two new arbitrary functions are related to the rotations, first strain invariant and distortion; (ii) especially, conservation laws constructed from the equations satisfied by the spatial integrals of functions hold true, although some arbitrary functions of the spatial integrals have been canceled. Based on these facts, since Noether's identity not only can be applied to a Lagrangian but also can be used to construct a functional for widespread PDEs, the functionals relating to the rotations, first strain invariant and distortion are constructed with arbitrary integer order spatial derivative or integral, and the conservation laws follow. This kind of non-classical conservation laws does not come from the Lagrangian density of an elastic body and belongs to the deep-level natures of symmetries of elastic field derived by standard techniques. Availability is shown by two examples, from which the field intensity of a vertical load applied to the surface of an elastic half-space and the path-independent integrals in a coordinate system moving with Galilean transformation are presented for comparison.     

诺依曼边条件下正弦戈登方程组的动力学

利用等价模和锥压缩的方法证明了当阻尼常数和扩散常数适当大时,Neumann边条件下周期受迫的Sine-Gordon方程组的全局吸引子是一条不变曲线,系统在不变曲线上的行为类似于圆周上的保向同胚.     

An orientation preserving fixed point free homeomorphism of the plane which admits no closed invariant line

Though fixed point free homeomorphisms of the plane would appear to exhibit the simplest dynamical behavior, we show that the minimal sets can be quite complex. Every homeomorphism which is conjugate to a translation must have a closed invariant line. However we construct an orientation preserving fixed point free homeomorphism of the plane which admits no closed invariant line. We verify that no such line exists by considering the ‘fundamental regions” of our example. Fundamental regions, studied first by Stephen Andrea, are equivalence classes of points in the plane associated with a given homeomorphism. Two points are said to be in the same equivalence class if they can be connected by an arc which diverges to infinity under both the forward and backward iterates of the homeomorphism. Our example contains no invariant fundamental regions.     

WEAKLY ALMOST PERIODIC POINT AND ERGODIC MEASURE

Let X be a compact metric space and f: X→X be continuous.This pape introduces the notion of weakly almost periodic point, which is a generalization of the notion of almost periodic point, proves that each of f-invariant ergodic measures can be generated by a weakly almost periodic point of f and gives some equivalent conditions for that f has an invariant ergodic measure whose support is X and ones for that f has no non-atomic invariant ergodic measure, the latter is a generalization of the Blokh's work on self-maps of the interval. Also two formulae for calculating the togological entropy are obtained.     

套代数上Lie同态的一个局部刻画（英文）

In this paper,linear maps preserving Lie products at zero points on nest algebras are studied.It is proved that every linear map preserving Lie products at zero points on any finite nest algebra is a Lie homomorphism.As an application,the form of a linear bijection preserving Lie products at zero points between two finite nest algebras is obtained.     

A tale of two conformally invariant metrics

The Harnack metric is a conformally invariant metric defined in quite general domains that coincides with the hyperbolic metric in the disk. We prove that the Harnack distance is never greater than the hyperbolic distance and if the two distances agree for one pair of distinct points, then either the domain is simply connected or it is conformally equivalent to the punctured disk.     

Generic super-exponential stability of elliptic equilibrium positions for symplectic vector fields

In this article, we consider linearly stable elliptic fixed points (equilibrium) for a symplectic vector field and prove generic results of super-exponential stability for nearby solutions. We will focus on the neighborhood of elliptic fixed points but the case of linearly stable isotropic reducible invariant tori in a Hamiltonian system should be similar. More specifically, Morbidelli and Giorgilli have proved a result of stability over superexponentially long times if one considers an analytic Lagrangian torus, invariant for an analytic Hamiltonian system, with a diophantine translation vector which admits a sign-definite torsion. Then, the solutions of the system move very little over times which are super-exponentially long with respect to the inverse of the distance to the invariant torus. The proof proceeds in two steps: first one constructs a high-order Birkhoff normal form, then one applies the Nekhoroshev theory. Bounemoura has shown that the second step of this construction remains valid if the Birkhoff normal form linked to the invariant torus or the elliptic fixed point belongs to a generic set among the formal series. This is not sufficient to prove this kind of super-exponential stability results in a general setting. We should also establish that the most strongly non resonant elliptic fixed point or invariant torus in a Hamiltonian system admits Birkhoff normal forms fitted for the application of the Nekhoroshev theory. Actually, the set introduced by Bounemoura is already very large but not big enough to ensure that a typical Birkhoff normal form falls into this class. We show here that this property is satisfied generically in the sense of the measure (prevalence) through infinite-dimensional probe spaces (that is, an infinite number of parameters chosen at random) with methods similar to those developed in a paper of Gorodetski, Kaloshin and Hunt in another setting.     

The data singular and the data isotropic loci for affine cones

The generic number of critical points of the Euclidean distance function from a data point to a variety is called the Euclidean distance degree (or ED degree). The two special loci of the data points where the number of critical points is smaller than the ED degree are called the Euclidean distance data singular locus and the Euclidean distance data isotropic locus. In this article, we present connections between these two special loci of an a?ne cone and its dual cone.