A glance into the anatomy of monotonic maps

<p>Given an autohomeomorphism on an ordered topological space or its subspace, we show that it is sometimes possible to introduce a new topology-compatible order on that space so that the same map is monotonic with respect to the new ordering. We note that the existence of such a re-ordering for a given map is equivalent to the map being conjugate (topologically equivalent) to a monotonic map on some homeomorphic ordered space. We observe that the latter cannot always be chosen to be order-isomorphic to the original space. Also, we identify other routes that may lead to similar affirmative statements for other classes of spaces and maps.</p>

Scale, Landscape and Indigenous Bedouin Land Use: Spatial Order and Agricultural Sedentarisation in the Negev Highland

By taking a small-scale perspective, Bedouin pastoral space in the Israeli Negev in the modern period has been misinterpreted as chaotic by various Israeli institutions. In critiquing this ontology we suggest that a knowledge gap with regard to an appropriate scale of understanding Bedouin settlement patterns and mechanisms of sedentarisation is at its root, and that a larger-scale analysis indicates that their space is in fact highly ordered. Field surveys and interviews with the local Bedouin showed that household cultivation plots in the Negev Highland during the period of the British Mandate were organised at a large scale through natural and man-made landscape features reflecting their structure, development and deployment in a highly ordered space. This analysis carries significant implications for understanding pastoral spaces at the local scale, particularly offering better comprehension of various sedentary forms and suggesting new approaches to sustainable planning and development for the Bedouin.

Compactness on Soft Topological Ordered Spaces and Its Application on the Information System

It is well known every soft topological space induced from soft information system is soft compact. In this study, we integrate between soft compactness and partially ordered set to introduce new types of soft compactness on the finite spaces and investigate their application on the information system. First, we initiate a notion of monotonic soft sets and establish its main properties. Second, we introduce the concepts of monotonic soft compact and ordered soft compact spaces and show the relationships between them with the help of examples. We give a complete description for each one of them by making use of the finite intersection property. Also, we study some properties associated with some soft ordered spaces and finite product spaces. Furthermore, we investigate the conditions under which these concepts are preserved between the soft topological ordered space and its parametric topological ordered spaces. In the end, we provide an algorithm for expecting the missing values of objects on the information system depending on the concept of ordered soft compact spaces.

Two-sided estimates for solutions of boundary value problems for implicit differential equations

We consider a two-point (including periodic) boundary value problem for the following system of differential equations that are not resolved with respect to the derivative of the desired function: f_i (t,x,x ̇,(x_i ) ̇ )=0,i= (1,n) ̅. Here, for any i= (1,n) ̅ the function f_i:[0,1]×R^n×R^n×R→R is measurable in the first argument, continuous in the last argument, right-continuous, and satisfies the special condition of monotonicity in each component of the second and third arguments. Assertions about the existence and two-sided estimates of solutions (of the type of Chaplygin’s theorem on differential inequality) are obtained. Conditions for the existence of the largest and the smallest (with respect to a special order) solution are also obtained. The study is based on results on abstract equations with mappings acting from a partially ordered space to an arbitrary set (see [S. Benarab, Z.T. Zhukovskaya, E.S. Zhukovskiy, S.E. Zhukovskiy. On functional and differential inequalities and their applications to control problems // Differential Equations, 2020, 56:11, 1440–1451]).

Eckland and Bishop-Phelps variational principles in partially ordered spaces

In this paper, an assertion about the minimum of the graph of a mapping acting in partially ordered spaces is obtained. The proof of this statement uses the theorem on the minimum of a mapping in a partially ordered space from [A.V. Arutyunov, E.S. Zhukovskiy, S.E. Zhukovskiy. Caristi-like condition and the existence of minima of mappings in partially ordered spaces // Journal of Optimization Theory and Applications. 2018. V. 180. Iss. 1, 48–61]. It is also shown that this statement is an analogue of the Eckland and Bishop-Phelps variational principles which are effective tools for studying extremal problems for functionals defined on metric spaces. Namely, the statement obtained in this paper and applied to a partially ordered space created from a metric space by introducing analogs of the Bishop-Phelps order relation, is equivalent to the classical Eckland and Bishop-Phelps variational principles.

Industrial Parks as Industrial Renewal

This chapter looks at the other main industrial redevelopment strategy that emerged in the 1960s. It mentions the building of industrial parks in the 1960s and 1970s, such as the Ashland Industrial Center in the deindustrialized Stockyards, that became a failure despite the allocation of significant funds by public–private partnerships. It emphasizes how the institutional fix of the industrial park could not solve the Chicago's manufacturing decline. The chapter refers to industrial areas that consist of many lots managed on a long-term basis by industrial and nonindustrial promoters. It outlines two principles that shaped the development of the industrial park in postwar Chicago: First was modern planning ideas that emerged after 1890 and second was the search for ordered space paralleled by the search for property profits.

Sum of the spaces on ordered setting

One of the divergences between topology and ordered topology is that some topological concepts such as separation axioms and continuous maps are defined using open neighborhoods or neighborhoods without any difference, however, they are distinct on the ordered topology according to the neighborhoods: Are they open neighborhoods or not? In this paper, we present the concept of sum of the ordered spaces using pairwise disjoint topological ordered spaces and study main properties. Then, we introduce the properties of ordered additive, finitely ordered additive and countably ordered additive which associate topological ordered spaces with their sum. We prove that the properties of being Ti-ordered and strong Ti-ordered spaces are ordered additive, however, the properties of monotonically compact and ordered compact spaces are finitely ordered additive. Also, we define a mapping between two sums of the ordered spaces using mappings between the ordered spaces and deduce some results related to some types of continuity and homeomorphism. We complete this work by determining the conditions under which a topological ordered space is sum of the ordered spaces.

A Lotus Blooms in the End Times: Cosmological Topography and the Tibetan State

The ruler of the central Tibetan state, the Desi Sangyé Gyatso (1653–1705), recognized its capital city of Lhasa as having the radial form of an eight-petaled flower or eight-spoked wheel. This article examines the Desi’s writings to reflect on the relationship between symbolically ordered space and cosmology. Scholars have often explained such spaces as representing a cosmological model, assigning that model the role of a static foundation and distancing it from human activity. This Tibetan case is read as evidence for another way of thinking about cosmological topography, namely as a creative process in a self-consciously critical relationship with its encompassing world. At stake is the general question of how humans both inhabit the cosmos and actively participate in ordering it.

History Channels

This chapter reads Du Fu’s commemorative poems on Yu the Great and Zhuge Liang in light of the specific communicative conditions of the Three Gorges, the vital yet perilous channel of transport and trade on the Tang empire’s remote southwest frontier. It argues that the poems for both figures are distinguished by imaginings of commemorative form—comprising both the materiality of local monuments and the means by which they extend their influence to the wider world. Picturing Yu’s creation of Qutang Gorge, Du Fu considers the founding act of “channeling and carving” as a transformation of elemental matter into ordered space that models a heroic vision of poetic composition. His poems on Zhuge Liang’s monuments dwell on the threat of material decay, while visualizing connections between Zhuge’s temples and those of his lord and patron, Liu Bei. Taken together, these poems suggest how the Three Gorges activated Du Fu’s desire for communication with history and prompted imaginative reflection on the channels that made it possible.