A function model for the Teichmüller space of a closed hyperbolic Riemann surface

We introduce a function model for the Teichmüller space of a closed hyperbolic Riemann surface.Then we introduce a new metric on the Teichmüller space by using the maximum norm on the function space.We prove that the identity map from the Teichmüller space equipped with the Teichmüller metric to the Teichmüller space equipped with this new metric is uniformly continuous. Moreover, we prove that the inverse of the identity, i.e., the identity map from the Teichmüller space equipped with this new metric to the Teichmüller space equipped with the Teichmüller metric, is continuous(but not uniformly). Therefore, the topology induced by the new metric is the same as the topology induced by the Teichmüller metric on the Teichmüller space.Finally, we give a remark about the pressure metric on the function model and the Weil-Petersson metric on the Teichmüller space.     

Geometric Gibbs theory

We extend the classical Gibbs theory for smooth potentials to the geometric Gibbs theory for certain continuous potentials. We study the existence and uniqueness and the compatibility of geometric Gibbs measures associated with these continuous potentials. We introduce a complex Banach manifold structure on the space of these continuous potentials as well as on the space of all geometric Gibbs measures. We prove that with this complex Banach manifold structure, the space is complete and, moreover, is the completion of the space of all smooth potentials as well as the space of all classical Gibbs measures. There is a maximum metric on the space, which is incomplete. We prove that the topology induced by the newly introduced complex Banach manifold structure and the topology induced by the maximal metric are the same. We prove that a geometric Gibbs measure is an equilibrium state, and the infimum of the metric entropy function on the space is zero.

     

Geometric Gibbs theory In Memory of Professor Shantao Liao

We extend the classical Gibbs theory for smooth potentials to the geometric Gibbs theory for certain continuous potentials. We study the existence and uniqueness and the compatibility of geometric Gibbs measures associated with these continuous potentials. We introduce a complex Banach manifold structure on the space of these continuous potentials as well as on the space of all geometric Gibbs measures. We prove that with this complex Banach manifold structure, the space is complete and, moreover, is the completion of the space of all smooth potentials as well as the space of all classical Gibbs measures. There is a maximum metric on the space,which is incomplete. We prove that the topology induced by the newly introduced complex Banach manifold structure and the topology induced by the maximal metric are the same. We prove that a geometric Gibbs measure is an equilibrium state, and the infimum of the metric entropy function on the space is zero.     

A note on convex renorming and fragmentability

Using the game approach to fragmentability, we give new and simpler proofs of the following known results: (a) If the Banach space admits an equivalent Kadec norm, then its weak topology is fragmented by a metric which is stronger than the norm topology. (b) If the Banach space admits an equivalent rotund norm, then its weak topology is fragmented by a metric. (c) If the Banach space is weakly locally uniformly rotund, then its weak topology is fragmented by a metric which is stronger than the norm topology.     

Metric spaces in synthetic topology

We investigate the relationship between the synthetic approach to topology, in which every set is equipped with an intrinsic topology, and constructive theory of metric spaces. We relate the synthetic notion of compactness of Cantor space to Brouwer’s Fan Principle. We show that the intrinsic and metric topologies of complete separable metric spaces coincide if they do so for Baire space. In Russian Constructivism the match between synthetic and metric topology breaks down, as even a very simple complete totally bounded space fails to be compact, and its topology is strictly finer than the metric topology. In contrast, in Brouwer’s intuitionism synthetic and metric notions of topology and compactness agree.     

Wijsman topology on function spaces

LetC(X,Y) be the space of continuous functions from a metric space (X,d) to a metric space (Y, e).C(X, Y) can be thought as subset of the hyperspaceCL(X×Y) of closed and nonempty subsets ofX×Y by identifying each element ofC(X,Y) with its graph. We considerC(X,Y) with the topology inherited from the Wijsman topology induced onCL(X×Y) by the box metric ofd ande. We study the relationships between the Wijsman topology and the compact-open topology onC(X,Y) and also conditions under which the Wijsman topology coincide with the Fell topology. Sufficient conditions under which the compactopen topology onC(X,Y) is weaker than the Wijsman topology are given (IfY is totally bounded, then for every metric spaceX the compactopen topology onC(X,Y) is weaker than the Wijsman topology and the same is true forX locally connected andY rim-totally bounded). We prove that a metric spaceX is boundedly compact iff the Wijsman topology onC(X, ℝ) is weaker than the compact-open topology. We show that ifX is a σ-compact complete metric space andY a compact metric space, then the Wijsman topology onC(X,Y) is Polish.     

Extension of an abstract attainability problem with the use of the stone representation space

We consider an attainability problem in a complete metric space on values of an objective operator h. We assume that the latter admits a uniform approximation by mappings which are tier with respect to a given measurable space with an algebra of sets. Let asymptotic-type constraints be defined as a nonempty family of sets in this measurable space. We treat ultrafilters of the measurable space as generalized elements; we equip this space of ultrafilters with a topology of a zero-dimensional compact (the Stone representation space). On this base we construct a correct extension of the initial problem, realizing the set of attraction in the form of a continuous image of the compact of feasible generalized elements. Generalizing the objective operator, we use the limit with respect to ultrafilters of the measurable space. This provides the continuity of the generalized version of h understood as a mapping of the zero-dimensional compact into the topological space metrizable with a total metric.     

Metric properties of quality criteria space for multicriteria optimization problems

The problem of the construction of an object functioning in the regime of optimum performance at the design stage is reduced to the solution of the problem of multicriterion optimization, where the quality criteria are chosen to be its most essential characteristics (parameters). At the same time in all methods of multicriterion optimization the vector quality criterion is considered basically in the linear Euclidean space. Actually, in most cases, the criterion space is non-Euclidean - it is curved. Therefore, such setting cannot give results adequately reflecting the processes running in real systems.In order for the design system to really satisfy the optimality requirements the authors of the given paper offer an absolutely new approach to the solution of the problems of multicriterion optimization based on the definition of the quality criteria space and on finding an invariant corresponding to the distance between any two points of that space.The idea of the study of the metric properties of the quality criteria space and their use in solving problems of multicriterion optimization was offered in the work [1]. But that idea, due to its complexity, has not been completely realized until now. When solving such problems the quality criteria space was automatically identified with the Euclidean space with corresponding metrics. In the general case this could not give results adequately reflecting the processes occurring in real systems.In the present paper metric properties of space criteria are studied for the first time, using as the main instrument the mathematical apparatus of tensor analysis, Riemannian geometry, differential equations in partial derivatives etc. Boundary problems relative to the components of the metric tensor of the n-dimensional space of the phenomenon states enabling to determine its metric properties are posed. The knowledge of the metric tensor furthers the objective appraisal of the phenomenon state and the definition of the optimal state.     

A differential operator and weak topology for Lipschitz maps

We show that the Scott topology induces a topology for real-valued Lipschitz maps on Banach spaces which we call the L-topology. It is the weakest topology with respect to which the L-derivative operator, as a second order functional which maps the space of Lipschitz functions into the function space of non-empty weak^∗ compact and convex valued maps equipped with the Scott topology, is continuous. For finite dimensional Euclidean spaces, where the L-derivative and the Clarke gradient coincide, we provide a simple characterization of the basic open subsets of the L-topology. We use this to verify that the L-topology is strictly coarser than the well-known Lipschitz norm topology. A complete metric on Lipschitz maps is constructed that is induced by the Hausdorff distance, providing a topology that is strictly finer than the L-topology but strictly coarser than the Lipschitz norm topology. We then develop a fundamental theorem of calculus of second order in finite dimensions showing that the continuous integral operator from the continuous Scott domain of non-empty convex and compact valued functions to the continuous Scott domain of ties is inverse to the continuous operator induced by the L-derivative. We finally show that in dimension one the L-derivative operator is a computable functional.     

Mappings onto metric spaces

We give characterizations of perfect images and open and compact images of spaces that can be mapped onto metrizable spaces by a mapping with fibers having a given property $\text{P}$. We use these characterizations to obtain conditions which imply that such images can be mapped onto a metric space by a mapping with fibers satisfying $\text{P}$. Such a treatment includes the investigation of spaces with a weaker metric topology [2, Ch. 5].     

Generalized-Finslerian connection coefficients for the static spherically symmetric space

Summary The metric torsionless connection coefficients are found in an explicit way in the case of static spherically symmetric spaces defined in a generalized-Finslerian way. The connection coefficients are determined in terms of the metric tensor and its first derivatives.     

Smooth approximations

We prove, among other things, that a Lipschitz (or uniformly continuous) mapping f:X→Y can be approximated (even in a fine topology) by smooth Lipschitz (resp. uniformly continuous) mapping, if X is a separable Banach space admitting a smooth Lipschitz bump and either X or Y is a separable C(K) space (resp. super-reflexive space). Further, we show how smooth approximation of Lipschitz mappings is closely related to a smooth approximation of C¹-smooth mappings together with their first derivatives. As a corollary we obtain new results on smooth approximation of C¹-smooth mappings together with their first derivatives.     

Typical property of the topological entropy of continuous mappings of compact sets

We show that the topological entropy viewed as a functional on the space of continuous mappings of a metric compact set into itself with the uniform topology is a function of the second Baire class and is lower semicontinuous at a Baire typical point. In particular, we show that the topological entropy is zero at a Baire typical point of the space of continuous mappings of the Baire space of sequences of zeros and units.     

Generic existence of a nonempty compact set of fixed points

Let X be a complete metric space, a set of continuous mappings from X into itself, endowed with a metric topology finer than the compact-open topology. Assuming that there exists a dense subset contained in such that for every mapping T in the set {x ε X: Tx = x} is nonempty, it is proved that most mappings (in the Baire category sense) in do have a nonempty compact set of fixed points. Some applications to α-nonexpansive operators, semiaccretive operators and differential equations in Banach spaces are derived.     

Caristi’s condition and existence of a minimum of a lower bounded function in a metric space. Applications to the theory of coincidence points

We consider a lower bounded function on a complete metric space. For this function, we obtain conditions, including Caristi’s conditions, under which this function attains its infimum. These results are applied to the study of the existence of a coincidence point of two mappings acting from one metric space to another. We consider both single-valued and set-valued mappings one of which is a covering mapping and the other is Lipschitz continuous. Special attention is paid to the study of a degenerate case that includes, in particular, generalized contraction mappings.     

The space of general conservation laws and topology changes

We construct a 5D space in which conservation laws are always and absolutely valid. This general result holds also scaling in 4D space even inducing, when necessary, changes of topology. The inequality of representations of physical laws in 4D and in 5D generalized spaces is founded not on phenomenological but on mathematical results. Essentially, the mechanism is based on the fact that we take into account also transformations where the Jacobian of the change of variables can assume null values. We derive that stress-energy and metric tensors, due to Bianchi identities, are always conserved through the mechanism presented in the following based on the fact that we introduce transformations with null Jacobian in the space definition. Moreover, in the case of metric tensor, conservation is a necessary feature in order to have physically defined theories.     

Wijsman topology on function spaces

LetC(X,Y) be the space of continuous functions from a metric space (X,d) to a metric space (Y, e).C(X, Y) can be thought as subset of the hyperspaceCL(X×Y) of closed and nonempty subsets ofX×Y by identifying each element ofC(X,Y) with its graph. We considerC(X,Y) with the topology inherited from the Wijsman topology induced onCL(X×Y) by the box metric ofd ande. We study the relationships between the Wijsman topology and the compact-open topology onC(X,Y) and also conditions under which the Wijsman topology coincide with the Fell topology. Sufficient conditions under which the compactopen topology onC(X,Y) is weaker than the Wijsman topology are given (IfY is totally bounded, then for every metric spaceX the compactopen topology onC(X,Y) is weaker than the Wijsman topology and the same is true forX locally connected andY rim-totally bounded). We prove that a metric spaceX is boundedly compact iff the Wijsman topology onC(X, ?) is weaker than the compact-open topology. We show that ifX is a σ-compact complete metric space andY a compact metric space, then the Wijsman topology onC(X,Y) is Polish.     

Generic mixing transformations are rank 1

In 2007, S. V. Tikhonov introduced a complete metric on the space of mixing transformations. This metric generates a topology called the leash topology. Tikhonov posed the following problem: what conditions should be satisfied by a mixing transformation T for its conjugacy class to be dense in the space of mixing transformations equipped with the leash topology. We show the conjugacy class to be dense for every mixing transformation T. As a corollary, we find that a generic mixing transformation is rank 1.     

On the variation of a metric and its application

Some of the variation formulas of a metric were derived in the literatures by using the local coordinates system, In this paper, We give the first and the second variation formulas of the Riemannian curvature tensor, Ricci curvature tensor and scalar curvature of a metric by using the moving frame method. We establish a relation between the variation of the volume of a metric and that of a submanifold. We find that the latter is a consequence of the former. Finally we give an application of these formulas to the variations of heat invariants. We prove that a conformally flat metric g is a critical point of the third heat invariant functional for a compact 4-dimensional manifold M, then （M, g） is either scalar flat or a space form.     

INVARIANT VARIATIONAL PRINCIPLES INVOLVING VECTOR AND METRIC FIELDS IN 4-DIMENSIONS

Abstract

Variational principles in which the Lagrangian is a scalar density and a function of a metric tensor and a vector field, together with their first derivatives, are investigated in a 4-dimensional space. Associated with such Lagrangians are two expressions, the metric Euler-Lagrange expression and the vector Euler-Lagrange expression. The most general Lagrangians (of this kind) for which either of these Euler-Lagrange expressions vanishes identically, are obtained.

The most general Lagrangian (of this kind) for which the vector Euler-Lagrange equations are precisely Maxwell's equations is obtained. Although this Lagrangian is more general than the one commonly used, it still has essentially the same energy-momentum tensor.

The most general Lagrangian (of this kind) for which the metric Euler-Lagrange expression is precisely the electromagnetic energy-momentum tensor is derived. Although this Lagrangian is also more general than the one commonly used, the associated vector Euler-Lagrange equations are still Maxwell's equations.

Finally it is shown that, in contrast to the situation which obtains in the case of scalar densities which are functions of up to second derivatives of the metric and first derivatives of the vector field, there does not exist a Lagrangian, of the kind under investigation, for which the metric Euler-Lagrange expression is precisely the Einstein tensor and the vector Euler-Lagrange expression vanishes identically.