Suppose that A and B are nonempty subsets of a complete metric space $$(\mathcal {M},d)$$ $$\phi ,\psi :A\rightarrow B$$ mappings. The aim this work is to investigate some conditions on $$ $$\psi such the two functions, one assigns each $$x\in A$$ exactly $$d(x,\phi x)$$ other $$d(x,\psi , attain global minimum value at same point in A. We have introduced notion proximally F-weakly dominated pa...

In this paper, we study the existence and uniqueness of fixed points for mappings with respect to a $wt$-distance in $b$-metric spaces endowed with a graph. Our results are significant, since we replace the condition of continuity of mapping with the condition of orbitally $G$-continuity of mapping and we consider $b$-metric spaces with graph instead of $b$-metric spaces, under which can be gen...

Let M be a complete, connected Riemannian surface and suppose that $${\mathcal {S}}\subset M$$ is discrete subset. What can we learn about from the knowledge of all distances between pairs points {S}}$$ ? We prove if in correspond to 2-dimensional lattice, or more generally an arbitrary net $${\mathbb {R}}^2$$ , then isometric Euclidean plane. thus find embeddings certain metric spaces are rath...

This paper continues research of Toda (The Complexity of Finding Medians, 31 st Symposium on Foundations of Computer Science (1990), pp. 778{787) on problems complete for function classes like FP # P and Mid P under Krentel's metric reductions. We rst show that metric reductions wipe out the diierence between Mid P and other related classes of functions which are probably diierent from Mid P. I...

We prove that under reasonable assumptions, every cat (compact abstract theory) is metric, and develop some of the theory of metric cats. We generalise Morley’s theorem: if a countable Hausdorff cat T has a unique complete model of density character λ ≥ ω1, then it has a unique complete model of density character λ for every λ ≥ ω1.

In this paper, we define a kind of lattice-valued convergence spaces based on the notion of $top$-filters, namely $top$-convergence spaces, and show the category of $top$-convergence spaces is Cartesian-closed. Further, in the lattice valued context of a complete $MV$-algebra, a close relation between the category of$top$-convergence spaces and that of strong $L$-topological spaces is establish...

‎this contribution mainly focuses on some aspects of lipschitz groups‎, ‎i.e.‎, ‎metrizable groups with lipschitz multiplication and inversion map‎. ‎in the main result it is proved that metric groups‎, ‎with a translation-invariant metric‎, ‎may be characterized as particular group objects in the category of metric spaces and lipschitz maps‎. ‎moreover‎, ‎up to an adjustment of the metric‎, ‎a...

It is known that folding a protein chain into a cubic lattice is an NP-complete problem. We consider a seemingly easier problem: given a three-dimensional (3D) fold of a protein chain (coordinates of its C(alpha) atoms), we want to find the closest lattice approximation of this fold. This problem has been studied under names such as "lattice approximation of a protein chain", "the protein chain...