Points of Coincidence and Common Fixed Points for Ⅱ-Expansive Mappings on Complex Valued Metric Spaces

We use the two mappings satisfying Ⅱ-expansive conditions on complex valued metric spaces to construct the convergent sequences and prove that the unique limit of the sequences is the point of coincidence or common fixed point of the two mappings.Also,we discuss the uniqueness of points of coincidence or common fixed points and give the existence theorems of unique fixed points.The obtained results generalize and improve the corresponding conclusions in references.     

Note on Vector Translation with Respect to Thompson’s Metric

It is known that vector translations are contractive with respect to Thompson’s part metric. Here, we give a simple proof, based on a representation of Thompson’s metric through positive functionals. Moreover, we use contractivity of translations to prove a fixed point result for mappings that are Lipschitz continuous with respect to Thompson’s metric with Lipschitz constant r>1. The case r = 1 for order preserving or order reversing mappings has been recently studied by Lawson and Lim. We apply our result to a nonlinear boundary value problem.     

Continuous Selections and Fixed Points for φ-maps with Their Applications to Section Problems

The concept of finitely continuous topological space is introduced andthe basic properties of the space are given.Several continuous selection theorems and fixed point theorems for φ-maps are establish...     

Simplest Singular Points of 1–forms Invariant with Respect to an Action of Group of Order Three

Singular points that cannot be removed by small deformations of 1–forms and invariant with respect to the action of a cyclic group of order 3 are classified. It is proved that for ?₃–invariant 1–forms the equivariant index of a singular point, as an element of the representation ring of the group, coincides with the class of the representation on the space of germs of forms of highest order factored by the subspace of forms divisible by the given 1–form.     

Propagation of Singularities for Schrödinger Equations on the Euclidean Space with a Scattering Metric

Given a scattering metric on the Euclidean space. We consider the Schrödinger equation corresponding to the metric, and study the propagation of singularities for the solution in terms of the “homogeneous wavefront set”. We also prove that the notion of the homogeneous wavefront set is essentially equivalent to that of the quadratic scattering wavefront set introduced by Wunsch (1999 Wunsch , J. ( 1999 ). Propagation of singularities and growth for Schrödinger operators . Duke Math. J. 98 : 137 – 186 . [CSA] [CROSSREF] [Crossref], [Web of Science ®] , [Google Scholar]). One of the main results in Wunsch (1999 Wunsch , J. ( 1999 ). Propagation of singularities and growth for Schrödinger operators . Duke Math. J. 98 : 137 – 186 . [CSA] [CROSSREF] [Crossref], [Web of Science ®] , [Google Scholar]) follows on the Euclidean space with a weaker, almost optimal condition on the potential.     

Strengthening of Landau’s Theorem for Holomorphic Self-Mappings of a Disk with Fixed Points

Mathematical Notes -     

Scalar and φ-Sectional Curvature of a Certain Type of Metric f-Structures

We consider a Riemannian manifold (M,g) equipped with an f-structure of constant rank with parallelizable kernel. We assume certain integrability conditions on such a manifold. We prove some inequalities involving the scalar and *-scalar curvature of g. We prove that the corresponding equalities characterize an -manifold, which is a generalization of a Sasakian manifold. We also give a method of constructing such structures on toroidal bundles. Dedicated to the memory of Professor Aldo Cossu Research supported by the Italian MIUR 60% and GNSAGA.     

Approximate Solutions of Variational Inequalities on Sets of Common Fixed Points of a One-Parameter Semigroup of Nonexpansive Mappings

Let \(\mathcal{T}\) be a one-parameter semigroup of nonexpansive mappings on a nonempty closed convex subset C of a strictly convex and reflexive Banach space X. Suppose additionally that X has a uniformly Gâteaux differentiable norm, C has normal structure, and \(\mathcal{T}\) has a common fixed point. Then it is proved that, under appropriate conditions on nonexpansive semigroups and iterative parameters, the approximate solutions obtained by the implicit and explicit viscosity iterative processes converge strongly to the same common fixed point of \(\mathcal{T}\), which is a solution of a certain variational inequality.     

Estimates of Variation with Respect to a Set and Applications to Optimization Problems

A variational norm that plays a role in functional optimization and learning from data is investigated. For sets of functions obtained by varying some parameters in fixed-structure computational units (e.g., Gaussians with variable centers and widths), upper bounds on the variational norms associated with such units are derived. The results are applied to functional optimization problems arising in nonlinear approximation by variable-basis functions and in learning from data. They are also applied to the construction of minimizing sequences by an extension of the Ritz method.     

The Cross-ratio Compactification of the Configuration Space of Ordered Points on 

A natural compactification of the virtual configuration space of N points on the Riemann sphere  is constructed by using cross-ratios. We show that this compactification is homeomorphic to the Bers' compactification of the virtual moduli space of a punctured Riemann sphere of type N . In particular, the system of global and explicit coordinates of this standard compactification is given by cross-ratios.     

Asymptotics of Solutions of the Sturm–Liouville Equation with Respect to a Parameter

On a finite segment [0, l], we consider the differential equation

Some Notes on Submanifolds of an Euclidean Space with Conformal Gauss Map

Let (M,g) be an m-dimensional Riemannian manifold and i:M→E~n an isometricimmersion of (M,g) into an n-dimensional Euclidean space E~n. Let VM be anopen set in which the immersion i:M→E~n is given by x~h=x~h(y~a), (h=1,…,n; α=1,…,m). Here and in the sequel x~h(h=1,…,n) are rectangular coordinates of E~nand y~α(α=1,…,m) are local coordinates of a generic point in V. The tangent planeiM_(ip)=i(M_p), P∈V, of iM can be considered after a suitable parallel displacementas a point Γ_i(P) of the Grassmann manifold G(m,n-m).The mapping Γ:iM→G(m,n-m), ipΓ(iP) =Γ_i(P) is called the Gauss map. The mapping Γ_i:M→G(m,n-m),p_i(P) is called the Gauss map associated with the immersion i, and _i(M)=F(iM) the Gauss image of M.     

The Least Regular Order with Respect to a Regular Congruence on Ordered Γ-Semigroups

The motivation mainly comes from the conditions of congruences to be regular that are of importance and interest in ordered semigroups. In 1981, Sen has introduced the concept of the Γ-semigroups. We can see that any semigroup can be considered as a Γ-semigroup. In this paper, we introduce and characterize the concept of the regular congruences on ordered Γ-semigroups and prove the following statements on an ordered Γ-semigroup M : (1) Every ordered semilattice congruences is a regular congruence. (2) There exists the least regular order on the Γ-semigroup M/ρ with respect to a regular congru- ence ρ on M . (3) The regular congruences are not ordered semilattice congruences in general.     

Optimal Mutual Location of Compact Sets in Spaces Endowed with Euclidean Invariant Gromov–Hausdorif Metric

Non–empty compact subsets of the Euclidean space located optimally (i.e., the Hausdorff distance between them cannot be decreased) are studied. It is shown that if one of them is a single point, then it is located at the Chebyshev center of the other one. Many other particular cases are considered too. As an application, it is proved that each three–point metric space cari be isometrically embedded into the orbit space of the group of proper motions acting on the compact subsets of the Euclidean space. In addition, it is proved that for each pair of optimally located compact subsets all intermediate compact sets in the sense of Hausdorff metric are also intermediate in the sense of Euclidean Gromov–Hausdorff metric.     

Automorphisms of Maps with a Given Underlying Graph and Their Application to Enumeration

A graph is called a semi-regular graph if its automorphism group action on its ordered pair of adjacent vertices is semi-regular. In this paper, a necessary and sufficient condition for an automorphism of the graph F to be an automorphism of a map with the underlying graph F is obtained. Using this result, all orientation-preserving automorphisms of maps on surfaces (orientable and non-orientable) or just orientable surfaces with a given underlying semi-regular graph F are determined. Formulas for the numbers of non-equivalent embeddings of this kind of graphs on surfaces (orientable, non-orientable or both) are established, and especially, the non-equivalent embeddings of circulant graphs of a prime order on orientable, non-orientable and general surfaces are enumerated.