Building upon the Foundations of Branching Space-Times

This chapter introduces a variety of events that are definable in BST and discusses in which histories these events occur. This gives rise to the concept of the occurrence proposition for events of various kinds. Of particular interest are transitions, defined as pairs of events, one of which is appropriately below the other. Transitions play a crucial role in later chapters. The chapter then discusses the topological aspects of BST, which are picked up again in Chapter 9. It defines a natural topology for BST: the diamond topology, and describes some important facts about it, focusing on the Hausdorff property and local Euclidicity. The chapter also gives an overview of how BST structures may be used to build semantic models for languages with temporal and modal operators.

Continuous Functions in Hashimoto Topologies and Their Algebraic Properties

In the paper we consider the Hashimoto topologies on the interval $$[0,1]$$ [ 0 , 1 ] as well as on $$\mathbb {R}$$ R , which are connected with the natural topology on $$\mathbb {R}$$ R and with some important and well known $$\sigma $$ σ -ideals in $$\mathcal {P}(\mathbb {R})$$ P ( R ) . We study the families of continuous functions $$f:[0,1]\rightarrow \mathbb {R}$$ f : [ 0 , 1 ] → R with respect to the same Hashimoto topology $$\mathcal {H}(\mathcal {I})$$ H ( I ) (connected with the $$\sigma $$ σ -ideal $$\mathcal {I}$$ I ) on the domain and on the range of the considered functions. We show that inside common parts and differences of some such families we can find large ($$\mathfrak {c}$$ c -generated) free algebras. Some of constructed algebras appear dense in the algebra of the functions which are continuous in the usual sense.

Description of unitary representations of the group of infinite 𝑝-adic integer matrices

We classify irreducible unitary representations of the group of all infinite matrices over a p p -adic field ( p ≠ 2 p\ne 2 ) with integer elements equipped with a natural topology. Any irreducible representation passes through a group G L GL of infinite matrices over a residue ring modulo p k p^k . Irreducible representations of the latter group are induced from finite-dimensional representations of certain open subgroups.

A dichotomy for bounded displacement equivalence of Delone sets

We prove that in every compact space of Delone sets in ${\mathbb {R}}^d$ , which is minimal with respect to the action by translations, either all Delone sets are uniformly spread or continuously many distinct bounded displacement equivalence classes are represented, none of which contains a lattice. The implied limits are taken with respect to the Chabauty–Fell topology, which is the natural topology on the space of closed subsets of ${\mathbb {R}}^d$ . This topology coincides with the standard local topology in the finite local complexity setting, and it follows that the dichotomy holds for all minimal spaces of Delone sets associated with well-studied constructions such as cut-and-project sets and substitution tilings, whether or not finite local complexity is assumed.

COINCIDENCE AND COMMON FIXED POINT THEOREMS IN Ƒ-METRIC SPACES

Recently, the concept of Ƒ metric space has been introduced and have been defined a natural topology in this spaces by Jleli and Samet[6]. Furthermore, a new style of Banach contraction principle has been given in the Ƒ metric spaces. In this paper, we prove some coincidence and common fixed point theorems in Ƒ metric spaces.

Some questions of differential game theory with phase constraints

Differential game (DG) of guidance-evasion is considered; moreover, its relaxations constructed with due account for priority considerations in the implementation of target set (TS) guidance and phase constraints (PC) validity are considered. We suppose that TS is closed in a natural topology of position space. With respect to the set that defines PC, it is postulated that the sections corresponding to time fixing are closed. For this setting, with the use of program iteration method (PIM), a variant of alternative for some natural (asymmetric) classes of strategies is established. A scheme of relaxation for the game guidance problem with nonclosed (in general case) set defining PC is considered. Under relaxation construction, reasons connected with priority in the implementation of guidance to TS and PC validity are taken into account (the case of asymmetric weakening of conditions of game ending is investigated). A position function is introduced, values of which (with priority correction) play the role of an analogue of least size for neighborhoods of TS and set defining PC under which it is possible to get a guaranteed solution of a relaxed problem of a player interested in approaching with TS while observing PC. It is demonstrated that the value of given function (when fixing the position of the game) is a price of DG for minimax-maximin quality functional which characterizes both the “degree” of approaching with TS and the “degree” of observance of initial PC.

On (ϕ, ψ)-Metric Spaces with Applications

The aim of this article is to introduce the notion of a ϕ,ψ-metric space, which extends the metric space concept. In these spaces, the symmetry property is preserved. We present a natural topology τϕ,ψ in such spaces and discuss their topological properties. We also establish the Banach contraction principle in the context of ϕ,ψ-metric spaces and we illustrate the significance of our main theorem by examples. Ultimately, as applications, the existence of a unique solution of Fredholm type integral equations in one and two dimensions is ensured and an example in support is given.

MOVING OBJECTS AWARE SENSOR MESH FUSION FOR INDOOR RECONSTRUCTION FROM A COUPLE OF 2D LIDAR SCANS

Indoor mapping attracts more attention with the development of 2D and 3D camera and Lidar sensor. Lidar systems can provide a very high resolution and accurate point cloud. When aiming to reconstruct the static part of the scene, moving objects should be detected and removed which can prove challenging. This paper proposes a generic method to merge meshes produced from Lidar data that allows to tackle the issues of moving objects removal and static scene reconstruction at once. The method is adapted to a platform collecting point cloud from two Lidar sensors with different scan direction, which will result in different quality. Firstly, a mesh is efficiently produced from each sensor by exploiting its natural topology. Secondly, a visibility analysis is performed to handle occlusions (due to varying viewpoints) and remove moving objects. Then, a boolean optimization allows to select which triangles should be removed from each mesh. Finally, a stitching method is used to connect the selected mesh pieces. Our method is demonstrated on a Navvis M3 (2D laser ranger system) dataset and compared with Poisson and Delaunay based reconstruction methods.

Porosity of 𝓢-approximately continuous functions

Considering the natural topology or 𝓢-density topology on the domain and on the range we obtain different families of continuous functions f : ℝ → ℝ. In this paper we compare these families in porosity terms. In particular, we obtain strengthening of some recent results by J. Hejduk, A. Loranty, R. Wiertelak.

Existence and Density of General Components of the Noether–Lefschetz Locus on Normal Threefolds

We consider the Noether–Lefschetz problem for surfaces in ${\mathbb Q}$-factorial normal 3-folds with rational singularities. We show the existence of components of the Noether–Lefschetz locus of maximal codimension, and that there are indeed infinitely many of them. Moreover, we show that their union is dense in the natural topology.