Spatial Mathematics

Data from the second year (1979-80) of the Section 15 statistics are used, first to test the validity of a small set of performance indicators for fixed-route bus operations, and second to define relatively homogeneous groups of operators (peer groups) that can be compared. Agencies operating 304 bus systems are included. Rail operations were excluded, as were exclusive, demand-responsive operations. The second year data are both more complete and accurate than that reported for the inaugural year. However, data from the magnetic tape had to be reorganized before it could be used with any of the major statistical software packages. A large set of performance variables are analyzed with factor analysis to establish seven dimensions of transit performance. Seven marker indicators were chosen rather than the nine proposed in previous research. Cluster analysis is used to create a typology for transit based upon characteristics of operations that are available in the Section 15 statisti...

These articles contain many references to scientific works done elsewhere, particularly at great universities in the southeastern and southern United States. • References, including those associated with various kudzu use possibilities (as a view of the 'whole') as well as those based on science.

Is mathematical category theory a unifying tool for geography? Here we look at a few basic category theoretical ideas and interpret them in geographic example. We also offer links to indicate how category theory has been used as such in other disciplines. Finally, we announce the direction of our research program on this topic as a way to facilitate the learning, and maintenance of learning, of GIS software – and in the spirit of

The Franciscan friary in Dubrovnik (Croatia) has two old wall sundials mounted in the small and large monastery cloisters. Both indicate equinoctial hours according to the apparent solar time. It is known for one who made it and the year it was made in-Paško Baletin in 1770, and for the other, there are no data. The second sundial has two scales; on the upper scale, the direction of the shadow indicates hours counted from midnight (Ore francese), whereas on the lower scale, the hours counted from the previous sunset are indicated by the tip of the shadow i.e. its "sunlet" (Ore italiche) as it has been believed so far. In order to check this, we calculated both scales, made drafts and compared them with the existing sundial scales: hour lines on the upper scale coincide, while on the lower scale all calculated hour lines deviate from the existing lines towards the west. However, when the calculations for the lower scale are done in accordance to the hours counted from the end of the civil twilight (Ore italiche da campanile) there is complete compatibility. This was confirmed by on-site observation of the sundial. It remains to search through Dubrovnik archives to discover when the two-scale sundial was made. The last-mentioned variant could have been brought to Dubrovnik in the second half of the 18 th century, and since then there was a need to construct a two-scale sundial.

Selected Common Threads and Root Concepts Linking Old to New Sandra L. Arlinghaus* and Joseph Kerski** Introduction In the twentieth century, much of mathematics experienced a well-known paradigm shift in moving from a focus on mathematical objects to a focus on transformations of objects. Thus, one sees, for example, linear algebra courses that focus on matrix vector products as linear transformations, rather than on the algebra of the matrices, themselves. As Birkhoff and Mac Lane note in their ground-breaking mid-20 th Century classic, A Survey of Modern Algebra, “It is desirable to have at hand not merely the formal operations with matrices, but also the (often neglected) interpretation of the matrices by linear transformations” (p. vi). Indeed, midway through this work (pp. 192-194) the authors develop the Gram-Schmidt Orthogonalization Process to show how to orthogonalize any basis of a Euclidean vector space, leading to the result that all finite-dimensional abstract Euclidea...

Dragi učenici, pred vama se nalaze upute za izradu PROJEKTA iz geografije i matematike, a dobivena ocjena upisuje se u rubriku GEOGRAFSKO ISTRAŽIVANJE I VJEŠTINE i RJEŠAVANJE PROBLEMA. Ovaj projekt se odnosi na dio nastavne cjeline VODE NA ZEMLJI i DECIMALNI BROJEVI a predstavlja istraživački rad koji se sastoji od PET različitih zadataka koji donose bodove. Uz svaki zadatak nalazi se i broj maksimalnih bodova koje možete postići iz pojedinog zadatka. Najveći broj bodova koje možete ostvariti iz projekta je 17 BODOVA.

The error for Arceos x is similar except that a round-off error due to subtraction is introduced. This error does not affect the conclusion that Arceos x will have been obtained correctly to within one unit in the nineteenth significant figure. 6. Conclusions. From the standpoint of machine application the procedure given is economical and yields precise results. It uses only twenty stored constants; the calculation of Arctan x requires a maximum of only eleven multiplications and one division; the calculation of Arcsin x and Arceos x requires an additional multiplication, division, and square root.

The Jordan-Sylvester graph center theorem provides an established and well-known method for calculating the center of a graph. We visualize this theorem in the context of a historical network, the Berlin Rohrpost of 1901, and interpret that visualization using contemporary visualization tools (Google Earth). Technology serves to bridge a visualization gap over time and, through this example, suggests direction for updating other historical studies. It also serves to guide further the direction of methodological research in regard to space filling as Fractalyse software offers planners ideas of how densely packed, with network nodes, an urban environment might become over time.

The importance of system reliability within military logistics should be considered in terms of the ability to ensure the readiness of all available resources, e.g., means of transport, which are necessary during the realization of operational tasks. A special role is played by technical security, which enables the performance of all the specific tasks by the realization of the process supporting the subsystem in the area of providing the necessary assemblies, subassemblies and spare parts. The objective of the work was to define reliability in relation to technical means of transport and to illustrate an original solution leading to the determination of the expected fitness time of the available vehicle fleet, using the example of a selected military unit. The GNU Octave software—designed to conduct, among other things, advanced numerical computations—was used for the study. The daily operational mileage for a selected group of means of transport and the moments of failures were re...

Symmetric elliptic integrals, which have been used as replacements for Legendre's integrals in recent integral tables and computer codes, are homogeneous functions of three or four variables. When some of the variables are much larger than the others, asymptotic approximations with error bounds are presented. In most cases they are derived from a uniform approximation to the integrand. As an application the symmetric elliptic integrals of the first, second, and third kinds are proved to be linearly independent with respect to coefficients that are rational functions.

In this paper, we will see that the determination of direct bearing angles. As it is known, in bearing angles are often computed used formulas with arctan function. The arctan function gives an angle values between-90 o and +90 o. However, the bearing angle is by definition 0 o to 360 o. Consequently, it is inevitable to examine the process of obtaining the azimuth angle. Classic formulas only work correctly if the edge is in the 1 st quarter. If the edge is located in the other quarters, the angles of the bearing should be examined. In this work we proposed new formulas for direct bearing angles on globe (sphere). Using the formula that we propose will save execution time in codes with intensive geodesic calculations.

Fifteen years ago we published an article in Solstice: An Electronic Journal of Geography and Mathematics, noting our respective experiences as creators and editors of two of the world's earliest electronic journals. We noted the opportunities that we thought this form of publication, using then cutting-edge technology, might provide the academic and scientific culture of the time; we projected different styles of electronic journals that might come about; and we summarized a variety of content, production, and archival issues. Since that article was written in 1993, each author/editor has been led in different directions through the maze of disciplinary and interdisciplinary publication using electronic media. We once again pool our efforts to consider whether the observations from 15 years ago were valid or in some way reasonable, and also to share a vision based on our variety of experience.

This chapter presents the Planning-Horizons Study that tested a conceptual bridge from organizational research into geography. Working at different levels of managerial planning seems to have an influence on how a person makes use of his/her work-related urban neighborhood. The study is based on organizational research by Jaques (1976, 1988) and the field concept by E. Wirth (1979). Data collection took place at Humboldt-Universität (employees and students) and high-tech-companies of the Berlin-Adlershof Science and Technology Park. The results are discussed in the context of interdisciplinary urban ecology, taking particular reference to L. Wirth's study of urbanism.

Проаналізовано публікації останніх років із застосування методів оптимізації та транспортної задачі для планування потоків у логістичних системах. Побудовано математичну модель логістичної системи в оптимізаційній транспортній задачі, яку розглянуто в регіоні, що має поділ на окремі територіальні елементи. Розроблено засоби для формалізованого математико-географічного опису та вивчення геопросторової організації закритої транспортної задачі, знання про який охоплює чотири блоки. Перший блок – інформація про геопозиційність суб’єктів задачі (розташування територіальних елементів, розміщення постачальників, розміщення споживачів, відстані від постачальників до споживачів, булеві матриці належності постачальників до територіальних елементів, булеві матриці належності споживачів до територіальних елементів). Другий блок – імплікативне знання про геореляційність для логістичних потоків між постачальниками та споживачами. Третій блок – імплікативне знання про геоінтегрованість суб’єктів транспортної задачі (постачальників та споживачів) з логістичними потоками. Четвертий блок – інформація про геофункціональність суб’єктів задачі (річні обсяги відправок постачальників, річні обсяги отримання споживачів, річні обсяги перевезення за транспортними потоками). Розроблено систему та алгоритми обчислення центрів для центрографічного аналізу функціонування суб’єктів
оптимізаційної транспортної задачі і виконано змістовну інтерпретацію цих центрів. Побудовано систему звичайних диференціальних рівнянь, яка описує динаміку функціонування суб’єктів оптимізаційної транспортної задачі впродовж року і показано її зв’язок з класичною транспортною задачею. Описано та вивчено ентропійні характеристики функціонування суб’єктів оптимізаційної транспортної задачі і показана інваріантність сумарної ентропії за суб’єктами задачі. Запропоновано напрямки подальших досліджень з цієї тематики.
Ключові слова: геопросторова організація, транспортна задача, оптимізація, центрографія, динаміка, ентропія.
The recent publications of application of optimization and transport problem for planning the flows in logistic systems are analyzed. A mathematical model of logistics system for optimizing transport problem in the region, which has a territorial division into separate elements, is build. The tools for formalized mathematical and geographical description and geospatial study of closed transportation problem are developed, knowledge of which includes four blocks. First block - the information about geolocalization of subjects (the location of territorial elements, placing of suppliers, placing of consumers, distance from suppliers to consumers, Boolean matrix of belonging suppliers to the territorial elements, Boolean matrix of belonging consumers to the territorial elements). The second block - implicative knowledge about georelativity for logistics flows between suppliers and consumers. The third block - implicative knowledge about geointegrity of subjects (suppliers and consumers) with logistics flows. The fourth block - the information about geofunctioning of subjects (annual suppliers sending, annual consumers receiving, annual transportation on traffic flows). The system and algorithms for computing of centers for centrography analysis of the functioning subjects of the transport problem and meaningful interpretation of these centers is made. A system of ordinary differential equations describing the dynamics of the operation of the optimization of the transportation problem during the year is build and its relation to classical transport problem is shown. Described and studied characteristics of entropy operation optimization of the transportation problem and shows the total entropy invariance on the subjects of the problem. Proposed areas for further research on the theme.
Key words: geospatial organization, transport problem, optimization, centrography, dynamics, entropy.

Concept of modeling in human geography is stated, types of mathematical and geographical models are determined, relations of modeling and theory of systems is shown, relations of modeling with the scientific laws is opened.
Key words: modelling, territorial system, mathematical method, human geography.

Створена математико-географічна модель територіальної мобільності населення в регіональній мережі поселень. Визначені поняття однорідного та ізотропного демографічного середовища. Сформульовані закономірності динаміки населення. Модель ілюструється двома прикладами елементарних систем розселення. У кожному з них отримані розв'язки диференціальних рівнянь.
Ключові слова: мережа поселень, природний рух населення, механічний рух населення, математична модель, диференціальне рівняння.
Владимир Грицевич. МЕТОДЫ МАТЕМАТИКО-ГЕОГРАФИЧЕСКОГО МОДЕЛИРОВАНИЯ ТЕРРИТОРИАЛЬНОЙ МОБИЛЬНОСТИ НАСЕЛЕНИЯ. Создана математико-географическая модель мобильности населения в региональной сети населённых пунктов. Определены понятия однородной и изотропной демографической среды. Сформулированы закономерности динамики населения. Модель иллюстрируется двумя примерами элементарных систем расселения. В каждом из них получены решения дифференциальных уравнений.
Ключевые слова: сеть населённых пунктов, естественное движение населения, механическое движение населения, математическая модель, дифференциальное уравнение.
Volodymyr Hrytsevych. METHODS OF MATHEMATICAL AND GEOGRAPHICAL MODELLING OF POPULATIONS TERRITORIAL MOBILITY. The mathematical and geographical model of populations mobility in the regional network of settlements is created. The concepts of homogeneous and isotropic demographical environment are defined. Conformities to the law of populations dynamic are formulated. A model is illustrated by two examples of the elementary systems of settling apart. The decisions of differential equations are created for each of them.
Keywords: network of settlements, natural motion of population, mechanical motion of population, mathematical model, differential equation.

Побудована топологічна схема сусідства регіонів України. Запропоновані числові характеристики топологічного положення регіонів. Побудовані математичні моделі поведінки головних числових характеристик, а також картосхеми їх просторового розподілу.
Ключові слова: граф, топологія, числова характеристика, математична модель.
Владимир Грицевич. ТОПОЛОГО-ГЕОГРАФИЧЕСКИЙ АНАЛИЗ РЕГИОНАЛЬНОГО УСТРОЙСТВА УКРАИНЫ. Построена топологическая схемма соседства регионов Украины. Предложены численные характеристики топологического положения регионов. Построены математические модели поведения главных численных характеристих, картосхеммы их пространственного распределения.
Ключевые слова: граф, топология, численная характеристика, математическая модель.
Volodymyr Hrytsevych. THE TOPOLOGY AND GEOGRAPHY ANALYSIS OF REGIONAL STRUCTURE OF UKRAINE. The topology graph of regions neighbourhooding of Ukraine is built. The numerical characteristics of regions topology position are offered. The mathematical models of main numerical characteristics variability are built, and also maps of its spatial distributing.
Key words: graph, topology, numerical characteristic, mathematical model.

Розроблені класифікації ієрархічних форм геопросторової організації територіальних суспільних комплексів. Запропонована аналітична формалізація структури комплексів. Розглянуті в графічному та аналітичному вигляді перші 29 ієрархічних форм геопросторової організації. Запропонована числова характеристика (індекс) геопросторової організації. Теоретичні положення проілюстровані на прикладі сфери охорони здоров'я адміністративного району Львівської області.
Ключові сло ва: територіальний суспільний комплекс, форма геопросторової організації, графічна модель, аналітична формула.
Владимир Грицевич. СИСТЕМА ИЕРАРХИЧЕСКИХ ФОРМ ГЕОПРОСТРАНСТВЕННОЙ ОРГАНИЗАЦИИ ТЕРРИТОРИАЛЬНЫХ ОБЩЕСТВЕННЫХ КОМПЛЕКСОВ. Разработаны классификации иерархических форм геопространственной организации территориальных общественных комплексов. Предложена аналитическая формализация структуры комплексов. Рассмотренны в графическом и аналитическом виде первые 29 иерархических форм геопространственной организации. Предложена числовая характеристика (индекс) геопространственной организации. Теоретические положения проиллюстрированы на примере сферы здравоохранения административного района Львовской области.
Ключевые сло ва: территориальный общественный комплекс, форма геопространственной организации, графическая модель, аналитическая формула.
Volodymyr Grytsevych. THE SYSTEM OF HIERARCHICAL FORMS OF GEO-SPATIAL ORGANIZATION OF TERRITORIAL SOCIAL COMPLEXES. The classifications of hierarchical forms of geo-spatial organization of territorial social complexes are developed. Analytical formalization of structure of complexes is offered. The first 29 hierarchical forms of geo-spatial organizationin in graphic and analytical view are considered. Numerical description (index) of geo-spatial organization is offered. Theoretical positions are illustrated on an example of medical spheres of administrative district of Lviv region.
Key word s: territorial social complex, form of geo-spatial organization, graphic model, analytical formula.

Map databases are integral to many ITS (Intelligent Transportation Systems) applications in navigation, traffic forecasting, and route planning. With the increasing deployment of ITS technology demands for accurate and complete digital map databases of the nation's road network are surging. The development and maintenance of high quality digital map databases is expensive and time-consuming. Database sharing will be a sensible approach whenever possible in order to reduce cost. In the US map databses are being produced by a variety of public agencies and private vendors. Quality and levels of accuracy vary depending on data sources and production procedures. Verifying the quality and accuracy of map databases for purposes of navigation is a pragmatic and important concern. The Society of Automotive Engineers has developed a Truth-in-Labeling Standard, the goal of which is to provide a consistent method for describing and comparing map databases. While the standard requires that ...

Sandra L. Arlinghaus, John D. Nystuen and Michael J. Woldenberg
Geographical Review
Vol. 82, No. 3 (Jul., 1992), pp. 244-252
Published by: American Geographical Society
DOI: 10.2307/215349
Stable URL: http://www.jstor.org/stable/215349
Page Count: 9

Geometric models often contain elements of complex geographic processes. The shape of the curve selected to model a specific process can be crucial in determining interpretations. To illustrate this relationship we use the graphical analysis of chaos theory as a geometric model and apply it to the complex process of semidesert-soil production. This application suggests strategies for choosing when human intervention to improve soil production might be appropriate.

In this paper, a methodology to classify terrain is presented. The taxonomy is devised using a terrain template based on evidence from topographic maps, and the resulting classes are characterized as steep, intermediate, and flat terrain peer groups of transit authorities. A set of 181 transit authorities was classified according to terrain type; in borderline cases, graphic displays were used to supplement the tabular display format of the classification. The terrain template was derived from applying allometric growth and census data to topographic evidence. Sets of Section 15 bus maintenance performance indicators were examined within terrain peer groups as an example of the potential for the application of these procedures. When the indicators miles per gallon, employees per vehicle-mile, and cost per vehicle-mile were displayed by terrain peer groups, relationships were found between quality of maintenance and miles per gallon in steeper environments.

Geography of City Terrain Based on Bus Routes
Sandra L. Arlinghaus and John D. Nystuen
Geographical Review
Vol. 77, No. 2 (Apr., 1987), pp. 183-195
Published by: American Geographical Society
DOI: 10.2307/214979
Stable URL: http://www.jstor.org.proxy.lib.umich.edu/stable/214979
Page Count: 13

Abstract:
A geography of city terrain, based on bus routes, supplies information about steepness that is vital to maintenance of equipment. This article presents a technique to determine vertical profiles of bus routes to evaluate topography and service demands in terrain stress on vehicles.

Behavioral Science
Volume 30, Issue 2, pages 108–117, 1985

The structure of nonverbal communication expressed as eye-contact between two human beings is analyzed using graph-theoretic tools involving a theorem of König on bipartite graphs and various results concerning directed graphs (as in Harary). A taxonomy for possible eye-contact configurations is constructed; then a theory, formed from a sequence of theorems proved about classes of eye-contact graphs derived from the taxonomy, is interpreted to analyze possible levels of communication. This theory can apply to any living system but it is interpreted here with respect to human subsystems composed of (1) individuals with normal vision, and (2) individuals with vision disorders which lead to crossed eyes.

1985 Geografiska Annaler, 67B (1985), pp. 83-88, Journal of the Stockholm School of Economics.  Presents original linkage between fractal geometry and central place theory for all orders of central places.  Uses Diophantine equations to make the linkage.  Later proved in 1989 Geographical Analysis article, 1989.

Arlinghaus:  Spatial Synthesis:  The Evidence of Cartographic:  Hierarchy and Centrality.
Arlinghaus et al.  Kioskland:  A Strategy for Linking Hierarchical Levels of Virtual Reality Maps
Larimore, Arlinghaus, Haug:  A Methodology for Historical Geography:  Internet Implementation
Arlinghaus:  Spatial Synthesis:  Investigations in Progress

Arlinghaus, S. and Arlinghaus, W. C.: Spatial Synthesis Sampler:  Geometric Visualization of Hexagonal Hierarchies:  Animation and Virtual Reality.
Beier, K-P.:  One Optimization of an Earlier Model of Virtual Downtown Ann Arbor.
Witlox, Borgers, and Timmermans:  Modelling Locational Decision Making of Firms Using Multidimensional Fuzzy Decision Tables:  An Illustration.
Arlinghaus, Beal, Kelbaugh:  The View from the Top:  Visualizing Downtown Ann Arbor in Three Dimensions.
Schlossberg:  Visualizing Accessibility II:  Access to Food.
Martin, Peter.  Energy Flow:  Spatial and Temporal Patterns
Lach, Alma.  Winter Windows:  Ice Largo.
Han:  Spatial Analysis of Subway Zones in Boston, Massachusetts
Walton:  A Golfer's Resource:  Huron Hills Golf Course, Ann Arbor, Michigan.
Chirapiwat:  Visualization of Geographic Information with VRML.

Arlinghaus, Batty, and Nystuen:  Animated Time Lines:  Coordination of Spatial and Temporal Information.
Chaudhuri and Chattopadhyay.  Viewing the relative importance of some surface parameters associated with pre-monsoon thunderstorms through Ampliative Reasoning.
Kaur:  On L1-Convergence of Modified Sine Sums.
Arlinghaus:  Ann Arbor, Michigan:  Virtual Downtown Experiments.  Tornado Siren Location:  Ann Arbor, Michigan.

Schlossberg:  Visual Accessibility with GIS.
Austin:  Cost Proxy Models in Rural Telephone Companies
Chattopadhyay:  Predicting Pre-monsoon Thunderstorms--A Statistical View through Propositional Logic.
Austin and Childers:  Disaggregation and Targeting of Universal Service Support.
Kaur, Ram, and Bhatia:  L1-Convergence of Cosine Series with Hyper Semi Convex Coefficients
Arlinghaus, S. and Arlinghaus, W. C.:  Spatial Synthesis:  A Research Program.

Arlinghaus, Haidar, and Wilson:  Animated map Timeline:  Syria
Derudder and Witlox:  Classification Techniques in Complex Databases:  Assessment of Vagueness and Sparsity in a Network of World Cities.
Nystuen:  The Thunen Society, North American Division
Arlinghaus:  The Lights Are On, All Over the World.

Nystuen:  Set in Stone:  An Analemma in Northern Italy
Wallace:  Personal Reflections on Solar Power
Arlinghaus:  Animaps, IV:  Of Time and Place.
Nystuen:  Photo Essay.  Fifty Years of Spatial Analysis:  A Symposium in Honor of William L. Garrison, 1950-2000.
Arlinghaus and Phillips:  A Neighborhood Information System within Ann Arbor, Michigan.
Gober:  Animaps, Again.
Baird:  Animap Sequences.