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In this article we extend the notion of orthogonal metric space to weak orthogonal metric space. Then we establish fixed point results for a mapping satisfying a more general contraction condition. Several nontrivial examples are given in…
The theory of Morse functions and their higher dimensional versions or fold maps on manifolds and its application to geometric theory of manifolds is one of important branches of geometry and mathematics. Studies related to this was started…
We construct random Morse functions on surfaces by random walk and compute related distributions. We study the space of Morse functions through these random variables. We consider subspaces characterized by the surfaces with boundary…
As a topic of mathematics, "arrangements", systems of hyperplanes, circles, and general (regular) submanifolds, attract us strongly. We present a natural elementary study of arrangements of circles. It is also a kind of new studies. Our…
An operad structure on certain bicoloured noncrossing configurations in regular polygons is studied. Motivated by this study, a general functorial construction of enveloping operad, with input a coloured operad and output an operad, is…
Spider mechanisms are the simplest examples of arachnoid mechanisms, they are one step more complicated than polygonal linkages. Their configuration spaces have been studied intensively, but are yet not completely understood. In the paper…
We introduce and study properties of certain new harmonic function spaces on products of upper half-spaces.Norm estimates for the so-called expanded Bergman projections are obtained.Sharp theorems on multipliers acting on certain Sobolev…
Involutions of the Clifford algebra of a quadratic space induced by orthogonal symmetries are investigated.
This paper introduces a novel generalization of the classical concept of $S$-metric spaces, referred to as composed $S$-metric spaces. By incorporating a composed function, we impose more general conditions on the triangle inequality,…
A regular circle-valued Morse function on the knot complement C(K) = S^3\K is a function f from C(K) to S^1 which separates critical points and which behaves nicely in a neighborhood of the knot. Such a function induces a handle…
We construct the differential geometry of smooth manifolds equipped with an algebraic curvature map acting as an area measure. Area metric geometry provides a spacetime structure suitable for the discussion of gauge theories and strings,…
This work explores the behaviour of demagnetizing tensors for general ellipsoids under arbitrary rotations in homogeneous magnetic fields. The work is motivated by the concerns in magnetic resonance imaging safety and their practical…
In this work, we introduce a combinatorial-geometric model for the space of discrete Morse functions on any CW complex $X$. We relate this version of a space of discrete Morse functions to the space of cellular filtrations of $X$ and…
Morrey spaces can complement the boundedness properties of operators that Lebesgue spaces can not handle. Morrey spaces which we have been handling are called classical Morrey spaces. However, classical Morrey spaces are not totally enough…
The optical properties of nanoparticle clusters vary with the spatial arrangement of the constituent particles, but also the overall orientation of the cluster with respect to the incident light. This is particularly important in the…
This article arose from a series of three lectures given at the Banach Center, Warsaw, during period of 24 March to 13 April, 2003. Morse functions are useful tool in revealing the geometric formation of its domain manifolds $M$. They…
In this paper we consider planar polygons with parallel opposite sides. This type of polygons can be regarded as discretizations of closed convex planar curves by taking tangent lines at samples with pairwise parallel tangents. For this…
Plank problems concern the covering of convex bodies by planks in Euclidean space and are related to famous open problems in convex geometry. In this survey, we introduce plank problems and present surprising applications of plank theorems…
Oscillons are extremely long lived, oscillatory, spatially localized field configurations that arise from generic initial conditions in a large number of non-linear field theories. With an eye towards their cosmological implications, we…
We construct a localization for operads with respect to one-ary operations based on the Dwyer-Kan hammock localization. For an operad O and a sub-monoid of one-ary operations W we associate an operad LO and a canonical map O to LO which…