Interval Topology Research Articles

The mean square displacement of a ballistic quantum particle

A commonly used quantum mechanical formulation of the mean square displacement δ x 2 ( t ) is based on the quantum correlation function of the position operator. While this quantity yields the classically expected result δ x 2 ( t ) ∝ t 2 for a ballistic particle in the interval topology, it is found here to diverge at essentially all times, when evaluated on infinitely large rings. A somewhat different formulation of the mean square displacement in quantum mechanics was proposed in a previous work (R. Marquardt, Mol. Phys. 119, e1971315 (2021)) and yielded the result δ x 2 ( t ) ∝ t for the ballistic particle on an infinitely large ring. Here, it is shown analytically that that formulation yields δ x 2 ( t ) ≡ 0 for the ballistic particle on an infinitely long interval. The two formulations define two different, topology dependent quantities that have the dimension of an area and that could in principle be determined from the same experiment, following a measurement idea proposed in the aforementioned work. That idea is critically reviewed here.

Stability of multi-link delayed impulsive stochastic complex networks with Markovian switching

This article presents an analysis of p-th moment exponential stability for a class of stochastic functional differential systems with Markovian switching on complex multi-link networks using multi-delayed impulsive control. The multi-delayed impulsive control scheme implies that the impulsive jumps are related to multiple past states of the systems, which increases the complexity of the analysis. Additionally, single-link networks are extended to complex multi-link networks, which are more extensive. By combining graph theory and the Razumikhin method, several criteria for achieving p-th moment exponential stability for systems are derived. The criteria are related to the maximum upper bound of delays, impulsive interval, and network topology. Subsequently, the stability of multi-link stochastic complex networks with time delays and Markovian switching is studied using delayed impulsive control. Finally, the exponential stability of a class of stochastic coupled multi-link oscillators with Markovian switching is analyzed, and several numerical simulations are provided to verify the validity of the theoretical results.

Interval reliability-based topology optimization of piezoelectric structures under single-loop sequential strategy and negative feedback control theory

Interval reliability-based topology optimization of piezoelectric structures under single-loop sequential strategy and negative feedback control theory

Ordered Field Valued Continuous Functions with Countable Range

For a Hausdorff zero-dimensional topological space X and a totally ordered field F with interval topology, let $$C_c(X,F)$$ be the ring of all F-valued continuous functions on X with countable range. It is proved that if F is either an uncountable field or countable subfield of $${\mathbb {R}}$$ , then the structure space of $$C_c(X,F)$$ is $$\beta _0X$$ , the Banaschewski Compactification of X. The ideals $$\{O^{p,F}_c:p\in \beta _0X\}$$ in $$C_c(X,F)$$ are introduced as modified countable analogue of the ideals $$\{O^p:p\in \beta X\}$$ in C(X). It is realized that $$C_c(X,F)\cap C_K(X,F)=\bigcap _{p\in \beta _0X{\setminus } X} O^{p,F}_c$$ , and this may be called a countable analogue of the well-known formula $$C_K(X)=\bigcap _{p\in \beta X{\setminus } X}O^p$$ in C(X). Furthermore, it is shown that the hypothesis $$C_c(X,F)$$ is a Von-Neumann regular ring is equivalent to amongst others the condition that X is a P-space.

Using the Interval Metric for Modeling Entities Geometrics in ℝ2 - Case Study Interval Circumference

ABSTRACT The study of some distances provide science a way to separate two entities. It has applications in various fields such as remote sensing, data mining, pattern recognition and multivariate data analysis and others. If the distance is a Hausdorff metric, the guarantee is that all individuals are available. With the use of the distance of Trindade et al, we intend to extend the real topology to an interval topology, since the interval distance preserves the uncertainties and exits noise in the data. The present work proposes an interval circumference using an interval distance of a point to the center (pixel), like a set of pixels obeying certain distances to the center. With the interval circumference we intend to extend the notion of open ball and the concepts of neighborhood for the construction of the interval topology. A circumference separates a space into three regions, inner region, border region and outer region, where we construct our notion of neighborhood. In this work we will explore only the geometric properties of the interval circumference, we will extrapolate the notion from point to pixel by providing a differentiated frontier region for the clustering area.

Using the Interval Metric for Modeling Entities Geometrics in R2 – Case Study Interval Circumference

The study of some distances provide science away to separate two entities.It has applications in various fields such as remote sensing, datamining, pattern recognition and multivariate data analysis and others. If the distance is a Hausdorff metric, the guarantee is that all individuals are available. With the use of the distance of Trindade et al, we intend to extend the real topology to an interval topology, since the interval distance preserves the uncertainties and exits noise in the data. The present work proposes an interval circumference using an interval distance of a point to the center (pixel), like a set of pixels obeying certain distances to the center. With the interval circumference we intend to extend the notion of open ball and the concepts of neighborhood for the construction of the interval topology. A circumference separates a space into three regions, inner region, border region and outer region, where we construct our notion of neighborhood. In this work we will explore only the geometric properties of the intervalcircumference, we will extrapolate the notion from point to pixel by providing a differentiated frontier region for the clustering area.

Split Hausdorff internal topologies on posets

Abstract In this paper, the concepts of weak quasi-hypercontinuous posets and weak generalized finitely regular relations are introduced. The main results are: (1) when a binary relation ρ : X ⇀ Y satisfies a certain condition, ρ is weak generalized finitely regular if and only if (φρ(X, Y), ⊆) is a weak quasi-hypercontinuous poset if and only if the interval topology on (φρ(X, Y), ⊆) is split T2; (2) the relation ≰ on a poset P is weak generalized finitely regular if and only if P is a weak quasi-hypercontinuous poset if and only if the interval topology on P is split T2.

Globally Hyperbolic Spacetimes as Posets

It is well-known that a spacetime with its causal relation is a partially ordered set (poset for short). If it is globally hyperbolic, then it is a bicontinuous poset whose the interval topology is the manifold topology. In this work, we will state a new condition on a poset, which is called DS-FI cluster point condition and we show that when a causally simple spacetime ${\mathscr{M}}$ as a poset with the manifold topology satisfies this condition, then ≪= I+ (where by ≪, we mean the way-below relation on arbitrary poset and by I+, we mean the chronological relation on spacetime), and ${\mathscr{M}}$ is a bicontinuous poset whose the interval topology is the manifold topology. Furthermore, we show that, on a causally simple spacetime ${\mathscr{M}}$ as a poset, the global hyperbolicity condition is strictly stronger than the DS-FI cluster point condition.

Interval topology in contact geometry

A topology is introduced on spaces of Legendrian submanifolds and groups of contactomorphisms. The definition is motivated by the Alexandrov topology in Lorentz geometry.

Derived topologies on ordinals and stationary reflection

We study the transfinite sequence of topologies on the ordinal numbers that is obtained through successive closure under Cantor’s derivative operator on sets of ordinals, starting from the usual interval topology. We characterize the non-isolated points in the $\xi$th topology as those ordinals that satisfy a strong iterated form of stationary reflection, which we call $\xi$-simultaneous-reflection. We prove some properties of the ideals of non-$\xi$-simultaneous-stationary sets and identify their tight connection with indescribable cardinals. We then introduce a new natural notion of $\Pi ^1_\xi$-indescribability, for any ordinal $\xi$, which extends to the transfinite the usual notion of $\Pi ^1_n$-indescribability, and prove that in the constructible universe $L$, a regular cardinal is $(\xi +1)$-simultaneously-reflecting if and only if it is $\Pi ^1_\xi$-indescribable, a result that generalizes to all ordinals $\xi$ previous results of Jensen [Ann. Math. Logic 4 (1972), pp. 229–308] in the case $\xi =2$, and Bagaria–Magidor–Sakai [Israel J. Math. 208 (2015), pp. 1–11] in the case $\xi =n$. This yields a complete characterization in $L$ of the non-discreteness of the $\xi$-topologies, both in terms of iterated stationary reflection and in terms of indescribability.

Representing topologies using partially ordered semigroups

Two common ways of using the partially ordered semigroup structure of the reals to model topological spaces are:Defining a distance into IR and using the balls of positive radius about a point as its basic neighborhoods, andSeeing if the space is a subspace of IR with its interval topology or its upper topology.We show that many po-semigroups can be used in place of IR and in fact:Every topological space is induced by a quasimetric into and set of positives in some po-semigroup, and is also a subspace of a po-semigroup with its upper topology.Also, the following are equivalent for any topological space:It is completely regular.It is induced by a pseudometric into and set of positives in some po-semigroup.It is a subspace of some po-semigroup with set of positives in their induced interval topology.

Interval neutrosophic sets and topology

In 2005, Smarandache generalized the Atanassov’s intuitionistic fuzzy sets (IFSs) to neutrosophic sets (NS), and other researchers introduced the notion of interval neutrosophic set (INSs),which is an instance of NS, and studied various properties. The notion of neutrosophic topology on the non-standard interval is also due to Smarandache. The purpose of this paper is to study relations between INSs and topology.

The upper topology and interval topology on quasi-hypercontinuous posets

In this paper, we consider a common generalization of both hypercontinuous posets and s2-quasicontinuous posets, and introduce a new concept of quasi-hypercontinuous posets. The main results are: (1) A poset is a quasi-hypercontinuous poset iff the upper topology is a hypercontinuous lattice iff the S⁎-convergence is topological with respect to the upper topology; (2) For a quasi-hypercontinuous poset, the quasi-liminf convergence is topological with respect to the interval topology; (3) The interval topology on a quasi-hypercontinuous poset is Tychonoff; (4) A lattice is quasi-hypercontinuous as a poset iff its normal completion is a quasi-hypercontinuous lattice.

Leader–follower H∞ consensus of linear multi-agent systems with aperiodic sampling and switching connected topologies

This paper is concerned with the distributed H∞ consensus of leader–follower multi-agent systems with aperiodic sampling interval and switching topologies. Under the assumption that the sampling period takes values from a given set, a new discrete-time model is proposed for the tracking error system. For the multi-agent systems with time-varying sampling period, switching topologies and external disturbance, the considered tracking problem is converted to a robust H∞ control problem. With help of the Lyapunov stability theory, a sufficient condition for the existence of mode-dependent controller is established and it guarantees the exponential stability of tracking error system and a prescribed H∞ disturbance attenuation level. The influence of sampling period on the overall control performance is also discussed. Two simulation examples are given to show the effectiveness of the proposed control algorithm.

Stone MV-algebras and strongly complete MV-algebras

Characterizations of compact Hausdorff topological MV-algebras, Stone MV-algebras, and MV-algebras that are isomorphic to their profinite completions are established. It is proved that compact Hausdorff topological MV-algebras are products (both topological and algebraic) of copies [0, 1] with the interval topology and finite Łukasiewicz chains with the discrete topology. Going one step further, we also prove that Stone MV-algebras are products (both topological and algebraic) of finite Łukasiewicz chains with the discrete topology. Finally, it is proved that an MV-algebra is isomorphic to its profinite completion if and only if it is profinite and each of its maximal ideals of finite rank is principal.

Continuity of the sequential product of sequential quantum effect algebras

In order to study quantum measurement theory, sequential product defined by A∘B = A1/2BA1/2 for any two quantum effects A, B has been introduced. Physically motivated conditions ask the sequential product to be continuous with respect to the strong operator topology. In this paper, we study the continuity problems of the sequential product A∘B = A1/2BA1/2 with respect to other important topologies, such as norm topology, weak operator topology, order topology, and interval topology.

Time ontology extended for non-Gregorian calendar applications

We have extended OWL-Time to support the encoding of temporal position in a range of reference systems, in addition to the Gregorian calendar and conventional clock. Two alternative implementations are provided: as a pure extension or OWL- Time, or as a replacement, both of which preserve the same representation for the cases originally supported by OWL-Time. The combination of the generalized temporal position encoding and the temporal interval topology from OWL-Time support a range of applications in a variety of cultural and technical settings. These are illustrated with examples involving non-Gregorian calendars, Unix-time, and geologic time using both chronometric and stratigraphic timescales.

Idempotent generated algebras and Boolean powers of commutative rings

A Boolean power S of a commutative ring R has the structure of a commutative R-algebra, and with respect to this structure, each element of S can be written uniquely as an R-linear combination of orthogonal idempotents so that the sum of the idempotents is 1 and their coefficients are distinct. In order to formalize this decomposition property, we introduce the concept of a Specker R-algebra, and we prove that the Boolean powers of R are up to isomorphism precisely the Specker Ralgebras. We also show that these algebras are characterized in terms of a functorial construction having roots in the work of Bergman and Rota. When R is indecomposable, we prove that S is a Specker R-algebra iff S is a projective R-module, thus strengthening a theorem of Bergman, and when R is a domain, we show that S is a Specker R-algebra iff S is a torsion-free R-module. For indecomposable R, we prove that the category of Specker R-algebras is equivalent to the category of Boolean algebras, and hence is dually equivalent to the category of Stone spaces. In addition, when R is a domain, we show that the category of Baer Specker R-algebras is equivalent to the category of complete Boolean algebras, and hence is dually equivalent to the category of extremally disconnected compact Hausdorff spaces. For totally ordered R, we prove that there is a unique partial order on a Specker R-algebra S for which it is an f-algebra over R, and show that S is isomorphic to the R-algebra of piecewise constant continuous functions from a Stone space X to R equipped with the interval topology.

Domain Theory in General Relativity

It is proved that k- causal spacetimes, with inner continuous Int(K±(.)), are jointly bicontinuous posets whose interval topology is the manifold topology. In particular, using the relation K +, causally continuous space-times can be reconstructed

Insertion and extension theorems for completely distributive lattice valued functions

Insertion and extension theorems are presented for lattice-valued functions where the lattice is an appropriately based completely distributive lattice endowed with its interval topology.