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Abstract
A number of ways in which mathematics is used in geography are mentioned. Plane Euclidean geometry is used in surveying small areas in the field, while spherical geometry and trigonometry are required in the construction of map projections, both traditional elements of mathematical geography. In the newer applications of mathematics to geography, topology is being used increasingly in the spatial analysis of networks. Graph theory provides indices to describe various types of network, such as drainage patterns. Differential equations are needed to study dynamic processes in geomorphology. Statistical techniques, such as trend surface analysis, factor analysis, cluster analysis and multiple discriminant analysis, can be applied to the description and analysis of the data of regional geography. Mathematical models are used in various forms to simplify the problems in geography. Examples of analogue models, such as the gravity model, are mentioned. Simulation models and Markov chain stochastic models are of value in studying certain geographical processes. Game theory is mentioned briefly. In the final section planning and prediction are very briefly referred to. In the former linear programming is a useful method, and in the latter trend fitting and extrapolation are applicable. Geography has gained a great deal in quantitative value and precision in adopting mathematical techniques.
... As aplicações matemáticas possuem imenso potencial em uma diversidade de domínios da Geografi a (KING, 1970), porém o uso do Cálculo é simultaneamente um desafi o e um campo de oportunidades -que se estende ao desenvolvimento da Geomorfologia. Por isso, este artigo é essencialmente metodológico, sem a premissa de apresentar teoremas ou corolários, apesar de serem apresentados exemplos, tanto hipotéticos quanto com uso de dados reais. ...
A declividade de canais possui um importante papel na morfologia, estrutura, dinâmica e evolução da paisagem fluvial, sendo uma variável chave para a interpretação de processos e formas de um sistema fluvial. A declividade pode ser obtida a partir de uma solução linear aplicada aos dados de cotas altimétricas, o que permite a obtenção de valores que correspondem aos trechos compreendidos entre as estações. Porém, frequentemente nota-se uma escassez de dados altimétricos ao longo do perfil longitudinal de um rio, o que limita as análises geomorfológicas detalhadas. Com base nesse panorama, o objetivo deste artigo foi discutir o uso de conceitos de cálculo diferencial para análise da declividade em perfis longitudinais. São apresentados os principais tipos de funções elementares, bem como são apresentadas soluções para obtenção da declividade utilizando-se algumas dessas funções. Observou-se que as funções polinomial e exponencial são úteis para representação de perfis longitudinais, assim como para obtenção da declividade a partir da derivada de primeira ordem. Adicionalmente, demonstrou-se que as derivadas podem ser calculadas para pontos específicos de interesse ou para um grande conjunto de pontos ao longo do perfil longitudinal. No segundo caso, tem-se como resultado que a declividade se torna uma variável contínua, podendo ser avaliada quantitativamente ao longo do perfil longitudinal. Esta pesquisa permite as seguintes conclusões: i) a partir de dados de altitude e distância horizontal de um perfil, é possível estabelecer funções que representam a forma de um perfil longitudinal; ii) a primeira derivada sobre essas funções equivale à declividade de canais fluviais; iii) a segunda derivada pode ser utilizada para obter informação sobre a concavidade do perfil e para encontrar seus pontos de inflexão; iv) finalmente, conclui-se que o Cálculo pode ser aplicado para o tratamento da variável declividade, tornando-o uma ferramenta a ser explorada em geomorfologia fluvial.
Metodika obsahuje postupy a kritéria pro posuzování účinnosti protierozních a vodohospodářských zařízení realizovaných v pozemkových úpravách, ale také pro posouzení jejich vlivu na ekologickou stabilitu, průchodnost a estetiku zemědělské krajiny. Byla vypracována jako výstup výzkumného projektu NAZV QI92A012 „Hodnocení realizací protierozních a vodohospodářských zařízení v KPÚ z pohledu ochrany a tvorby zemědělské krajiny“. Metodika je určena především Státnímu pozemkovému úřadu, pracovníkům pozemkových úřadů a státní správy pro zkvalitnění jejich rozhodovací, plánovací, řídící a kontrolní činnosti, ale má potenciál širšího využití odbornou veřejností. Publikované postupy a kritéria lze aplikovat i mimo obor pozemkových úprav, zejména pro hodnocení dopadů činnosti člověka na některé aspekty zemědělské krajiny. Její uplatnění v praxi vyžaduje základní odborné znalosti v oboru vodní a větrné eroze, hydrologie a krajinného plánování.
In recent years there has been a considerable development of emphasis on the interplay between mathematics and biology, espeically at the tertiaral levels of education. These can be seen in the established areas of Biostatistics, Catastrophe theory, Population Biology, Mathematical Genetics and Cybernetics. Prominant journals produce research works of high standard on these interplays. These recent developments have forced some curriculum developers to contemplate the introduction of the subject of “Biomathematics” into the school curriculum. Such a move, however, may be open for a debate.
A variety of geometrical forms are fitted to geomorphological features, often by means of regression analysis. Glaciated slopes approximate closely to parabolic forms. Other curves that steepen at an accelerating rate upslope also fit well. A curve‐fitting computer program was used to determine the best fit curve in relation to a surveyed profile of a slope that has probably been glaciated. River longitudinal profiles are shown to decrease logarithmically in gradient down valley. This characteristic can be used to extrapolate unrejuvenated sections downstream to ascertain the fall of sea level that initiated the rejuvenation. Lemniscate and rose curves are fitted to drumlins, which are formed beneath flowing ice, and as a result have a streamline form. The degree of drumlin elongation is related to constancy of ice flow and ice pressure in the Eden Valley‐Solway area of northwest England. The plan form of beaches that are free to become adapted to the approach of refracted waves is fitted to the logarithmic spiral form, which is shown to be an equilibrium form. The conclusion is reached that many natural forms that are in equilibrium approximate to the logarithmic or exponential form rather than a linear form in order to achieve the minimum and most uniform use of energy.
Individual channel networks ordinarily are portions of far larger, essentially infinite, networks. The overall network is by definition at infinite topologically random network if the populations of subnetworks within it are topologically random. From such a network, the probability of randomly drawing a link, a subnetwork, or a basin with Strahler order ; and that of randomly drawing a stream of order . The probability of drawing a link of magnitude , that is, one having sources ultimately tributary to it, is equal to the probability of a first passage through the origin at step in a symmetric random walk, a fact which suggests a useful mathematical analogy between random walks and infinite topologically random networks. Assuming uniform link length equal to the constant of channel maintenance, which in turn is the reciprocal of drainage density, the probability distributions for links and streams of various orders may be interpreted as crude geomorphological "laws" analogous to Horton's laws of drainage composition. These distributions predict geometric-series "laws" in which, using Strahler orders, the bifurcation ratio is 1/4, the link-number ratio is 1/2, the length ratio is 2, the cumulative-length ratio is 4, and the basin-area ratio is 4, all in good agreement with the observed ratios. They also predict values of 4/3 and 2/3, respectively, for the dimensionless ratios of total number of Strahler streams to network magnitude and of Strahler stream frequency to the square of the drainage density, in agreement with the values of 1.34 and 0.694 found empirically.
The Economics of Location
- A Lösch
A. Lösch, The Economics of Location, Yale University Press, New Haven, Connecticut, 1954.