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Dopico, Froilán M. - One of the best experts on this subject based on the ideXlab platform.
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Alan Turing and the origins of modern Gaussian elimination
Consejo Superior de Investigaciones Científicas, 2013Co-Authors: Dopico, Froilán M.Abstract:The solution of a system of linear equations is by far the most important problem in Applied Mathematics. It is important both in itself and because it is an intermediate step in many other important problems. Gaussian elimination is nowadays the standard method for solving this problem numerically on a computer and it was the first numerical algorithm to be subjected to rounding error analysis. In 1948, Alan Turing published a remarkable paper on this topic: “Rounding-off errors in matrix processes” (Quart. J. Mech. Appl. Math. 1, pp. 287-308). In this paper, Turing formulated Gaussian elimination as the matrix LU factorization and introduced the “condition number of a matrix”, both of them fundamental notions of modern Numerical Analysis. In addition, Turing presented an error analysis of Gaussian elimination for general matrices that deeply influenced the spirit of the definitive analysis developed by James Wilkinson in 1961. Alan Turing’s work on Gaussian elimination appears in a fascinating period for modern Numerical Analysis. Other giants of Mathematics, as John von Neumann, Herman Goldstine, and Harold Hotelling were also working in the mid-1940s on Gaussian elimination. The goal of these researchers was to find an efficient and reliable method for solving systems of linear equations in modern “automatic computers”. At that time, it was not clear at all whether Gaussian elimination was a right choice or not. The purpose of this paper is to revise, at an introductory level, the contributions of Alan Turing and other authors to the error analysis of Gaussian elimination, the historical context of these contributions, and their influence on modern Numerical Analysis.La resolución de sistemas de ecuaciones lineales es sin duda el problema más importante en Matemática Aplicada. Es importante en sí mismo y también porque es un paso intermedio en la resolución de muchos otros problemas de gran relevancia. La eliminación Gaussiana es hoy en día el método estándar para resolver este problema en un ordenador y, además, fue el primer algoritmo numérico para el que se realizó un análisis de errores de redondeo. En 1948, Alan Turing publicó un artículo de gran relevancia sobre este tema: “Rounding-off errors in matrix processes” (Quart. J. Mech. Appl. Math. 1, pp. 287-308). En este artículo, Turing formuló la eliminación Gaussiana en términos de la factorización LU de una matriz e introdujo la noción de número de condición de una matriz, que son dos de las nociones más fundamentales del Análisis Numérico moderno. Además, Turing presentó un análisis de errores de la eliminación Gaussiana para matrices generales que influyó profundamente en el espíritu del análisis de errores definitivo desarrollado por Wilkinson en 1961. El trabajo de Alan Turing sobre la eliminación Gaussiana aparece en un periodo fascinante del Análisis Numérico moderno. Otros gigantes de las matemáticas como John von Neumann, Herman Goldstine y Harold Hotelling también realizaron investigaciones sobre la eliminación Gaussiana en la década de 1940-50. El objetivo de estos investigadores era encontrar un método eficiente y fiable para resolver sistemas de ecuaciones lineales en los ordenadores modernos que estaban desarrollándose por entonces. En aquella época, no estaba claro en absoluto si utilizar la eliminación Gaussiana era una elección adecuada o no. El propósito de este artículo es revisar, a nivel básico, las contribuciones realizadas por Alan Turing y otros investigadores al análisis de errores de la eliminación Gaussiana, el contexto histórico de esas contribuciones y su influencia en el Análisis Numérico moderno
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Alan Turing y los orígenes de la eliminación gaussiana moderna
'Editorial CSIC', 2013Co-Authors: Dopico, Froilán M.Abstract:The solution of a system of linear equations is by far the most important problem in Applied Mathematics. It is important both in itself and because it is an intermediate step in many other important problems. Gaussian elimination is nowadays the standard method for solving this problem numerically on a computer and it was the first numerical algorithm to be subjected to rounding error analysis. In 1948, Alan Turing published a remarkable paper on this topic: “Rounding-off errors in matrix processes” (Quart. J. Mech. Appl. Math. 1, pp. 287-308). In this paper, Turing formulated Gaussian elimination as the matrix LU factorization and introduced the “condition number of a matrix”, both of them fundamental notions of modern Numerical Analysis. In addition, Turing presented an error analysis of Gaussian elimination for general matrices that deeply influenced the spirit of the definitive analysis developed by James Wilkinson in 1961. Alan Turing’s work on Gaussian elimination appears in a fascinating period for modern Numerical Analysis. Other giants of Mathematics, as John von Neumann, Herman Goldstine, and Harold Hotelling were also working in the mid-1940s on Gaussian elimination. The goal of these researchers was to find an efficient and reliable method for solving systems of linear equations in modern “automatic computers”. At that time, it was not clear at all whether Gaussian elimination was a right choice or not. The purpose of this paper is to revise, at an introductory level, the contributions of Alan Turing and other authors to the error analysis of Gaussian elimination, the historical context of these contributions, and their influence on modern Numerical Analysis.La resolución de sistemas de ecuaciones lineales es sin duda el problema más importante en Matemática Aplicada. Es importante en sí mismo y también porque es un paso intermedio en la resolución de muchos otros problemas de gran relevancia. La eliminación Gaussiana es hoy en día el método estándar para resolver este problema en un ordenador y, además, fue el primer algoritmo numérico para el que se realizó un análisis de errores de redondeo. En 1948, Alan Turing publicó un artículo de gran relevancia sobre este tema: “Rounding-off errors in matrix processes” (Quart. J. Mech. Appl. Math. 1, pp. 287-308). En este artículo, Turing formuló la eliminación Gaussiana en términos de la factorización LU de una matriz e introdujo la noción de número de condición de una matriz, que son dos de las nociones más fundamentales del Análisis Numérico moderno. Además, Turing presentó un análisis de errores de la eliminación Gaussiana para matrices generales que influyó profundamente en el espíritu del análisis de errores definitivo desarrollado por Wilkinson en 1961. El trabajo de Alan Turing sobre la eliminación Gaussiana aparece en un periodo fascinante del Análisis Numérico moderno. Otros gigantes de las matemáticas como John von Neumann, Herman Goldstine y Harold Hotelling también realizaron investigaciones sobre la eliminación Gaussiana en la década de 1940-50. El objetivo de estos investigadores era encontrar un método eficiente y fiable para resolver sistemas de ecuaciones lineales en los ordenadores modernos que estaban desarrollándose por entonces. En aquella época, no estaba claro en absoluto si utilizar la eliminación Gaussiana era una elección adecuada o no. El propósito de este artículo es revisar, a nivel básico, las contribuciones realizadas por Alan Turing y otros investigadores al análisis de errores de la eliminación Gaussiana, el contexto histórico de esas contribuciones y su influencia en el Análisis Numérico moderno
Martínez Dopico, Froilán César - One of the best experts on this subject based on the ideXlab platform.
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Alan Turing y los orígenes de la eliminación gaussiana moderna
Centro Superior de Investigaciones Científicas, 2012Co-Authors: Martínez Dopico, Froilán CésarAbstract:The proceeding at: The International Symposium "The Alan Turing Legacy" held in Madrid (Spain) in October 23-24, 2012. This symposium was organized and funded by the Real Academia de Ciencias Exactas, Físicas y Naturales of Spain and Fundación Ramón Areces.The solution of a system of linear equations is by far the most important problem in Applied Mathematics. It is important both in itself and because it is an intermediate step in many other important problems. Gaussian elimination is nowadays the standard method for solving this problem numerically on a computer and it was the first numerical algorithm to be subjected to rounding error analysis. In 1948, Alan Turing published a remarkable paper on this topic: "Rounding-off errors in matrix processes" (Quart. J. Mech. Appl. Math. 1, pp. 287-308). In this paper, Turing formulated Gaussian elimination as the matrix LU factorization and introduced the "condition number of a matrix", both of them fundamental notions of modern Numerical Analysis. In addition, Turing presented an error analysis of Gaussian elimination for general matrices that deeply influenced the spirit of the definitive analysis developed by James Wilkinson in 1961. Alan Turing's work on Gaussian elimination appears in a fascinating period for modern Numerical Analysis. Other giants of Mathematics, as John von Neumann, Herman Goldstine, and Harold Hotelling were also working in the mid-1940s on Gaussian elimination. The goal of these researchers was to find an efficient and reliable method for solving systems of linear equations in modern "automatic computers". At that time, it was not clear at all whether Gaussian elimination was a right choice or not. The purpose of this paper is to revise, at an introductory level, the contributions of Alan Turing and other authors to the error analysis of Gaussian elimination, the historical context of these contributions, and their influence on modern Numerical Analysis.La resolución de sistemas de ecuaciones lineales es sin duda el problema más importante en Matemática Aplicada. Es importante en sí mismo y también porque es un paso intermedio en la resolución de muchos otros problemas de gran relevancia. La eliminación Gaussiana es hoy en día el método estándar para resolver este problema en un ordenador y, además, fue el primer algoritmo numérico para el que se realizó un análisis de errores de redondeo. En 1948, Alan Turing publicó un artículo de gran relevancia sobre este tema: “Rounding-off errors in matrix processes” (Quart. J. Mech. Appl. Math. 1, pp. 287-308). En este artículo, Turing formuló la eliminación Gaussiana en términos de la factorización LU de una matriz e introdujo la noción de número de condición de una matriz, que son dos de las nociones más fundamentales del Análisis Numérico moderno. Además, Turing presentó un análisis de errores de la eliminación Gaussiana para matrices generales que influyó profundamente en el espíritu del análisis de errores definitivo desarrollado por Wilkinson en 1961. El trabajo de Alan Turing sobre la eliminación Gaussiana aparece en un periodo fascinante del Análisis Numérico moderno. Otros gigantes de las matemáticas como John von Neumann, Herman Goldstine y Harold Hotelling también realizaron investigaciones sobre la eliminación Gaussiana en la década de 1940-50. El objetivo de estos investigadores era encontrar un método eficiente y fiable para resolver sistemas de ecuaciones lineales en los ordenadores modernos que estaban desarrollándose por entonces. En aquella época, no estaba claro en absoluto si utilizar la eliminación Gaussiana era una elección adecuada o no. El propósito de este artículo es revisar, a nivel básico, las contribuciones realizadas por Alan Turing y otros investigadores al análisis de errores de la eliminación Gaussiana, el contexto histórico de esas contribuciones y su influencia en el Análisis Numérico moderno.This work was partially supported by the Ministerio de Economía y Competitividad of Spain through grant MTM-2009-09281.Publicad
Godoy Díaz, Carlos Sebastian - One of the best experts on this subject based on the ideXlab platform.
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Modelo de Hotelling: poner en evidencia el cumplimiento de regla de Hotelling para la explotación de cobre en Chile
Universidad Andrés Bello, 2012Co-Authors: Godoy Díaz, Carlos SebastianAbstract:Tesis (Magíster en Administración de Empresas)Los recursos naturales no renovables son aquellos para los que no existe un procesos espontaneo y natural de regeneración o repoblación. En la tierra se alberga una cantidad fija de recursos no renovables y cualquier extracción sólo puede disminuir la disponibilidad de los mismos para el futuro. De ahí que estos recursos planteen importantes cuestionamientos sobre los límites del crecimiento económico, la velocidad de explotación y precio al cual se transan. Estas cuestiones van desde preguntas básicas sobre si estamos agotando un recurso que es fundamental para el funcionamiento de la economía hasta si es posible disfrutar de un crecimiento sostenido de largo plazo. ¿Dónde está la medida justa en la explotación? ¿Es rentable extraer los recursos naturales en un tiempo finito o explotar el recurso infinitamente de manera que el remanente se aproxime a cero en el límite?, si la mina es de propiedad pública ¿cómo debería llevarse a cabo la explotación para un mayor bienestar general y cómo una política que tiene tal objetivo se compara con objetivos empresariales que buscan mayores beneficios económicos? ¿Cómo puede el estado a través de programas de tributación o regulación, inducir al propietario de una mina a establecer programas de producción más armónicos con el bienestar público? Harold Hotelling en 1931, formuló una pauta de extracción óptima, es decir, un periodo óptimo de agotamiento y la tasa de extracción óptima de un recurso natural durante toda su vida útil o el tiempo que debe ser utilizado por la economía, el cual está determinado por la demanda, tasa de interés, los avances tecnológicos y las reservas disponibles
Li Menggang - One of the best experts on this subject based on the ideXlab platform.
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An index of financial safety of China
Universitat Politècnica de Catalunya, 2015Co-Authors: Jia Xiaojun, Li MenggangAbstract:Purpose: This paper combines a synthetic index system by the variables and evaluates China’s financial safety through the change of indexes in a comprehensive way. First of all, it builds the financial industry evaluation index system composed of 25indicators in terms of the operation of the financial industry and external economic environment and particularly takes into consideration factors which might trigger liquidity risks such as off-balance-sheet business, interbank business and shadow banking; then it selects 10 indicators to conduct empirical analysis and identifies the indicator weight through principal component analysis; finally it combines the financial safety indexes through the linear weighted comprehensive evaluation model. Design/methodology/approach: Synthesis of indexes is made by constructing a proper comprehensive evaluation mathematical model, integrating a number of evaluation indexes into one comprehensive evaluation index and then obtaining corresponding comprehensive evaluation results. In this paper, it selects 10 indexes to conduct empirical analysis and identifies the index weight through principal component analysis; finally it combines the financial safety indexes through the linear weighted comprehensive evaluation model. Principal component analysis (PCA) is a statistical procedure that uses an orthogonal transformation to convert a set of observations of possibly correlated variables into a set of values of linearly uncorrelated variables called principal components. PCA was invented in 1901 and was later independently developed (and named) by Harold Hotelling in the 1930s. Findings: From 2003 to 2013 China’s financial safety indexes fluctuated. From 2003 to 2007 indexes rose, which indicates China’s financial safety status gradually improved; from 2007 to 2009 indexes declined, which indicates due to the impact of subprime crisis, China’s financial safety status took a turn for the worse; from 2009 to 2012 indexes rose, which indicates the external environment improved so did China’s financial safety status; from 2012 to 2013 indexes declined because due to the rapid development of banks’ financial products and trust products, banks’ off-balance-sheet assets and liquidity risks increased. The changes of financial safety indexes are generally identical with those of China’s financial safety status. Research limitations/implications: In the empirical analysis part, this article tries to selective 24 indicators synthetic index of China's financial security, but due to some of the indicators data acquisition is relatively difficult, can only Selective 10 of 25 indicators and gather the annual data of 10 indicators from 2003 to 2013 to synthetic index. The information of eliminated indicators cannot be reflected in the index. Index change also does not reflect of the risk from these indicators. In order to make up for the above limitations, this paper is mainly to introduce and analysis our latest financial institutions business trends associated with these eliminated indicators to get the conclusions more reliable. Originality/value: The aim of this research is to estimate financial safety of China with the application of the index of financial safety of a country using the annual data of 2003-2013. Through synthetic index of financial security measure the risks of China's financial system, provide the basis for the government macro financial policy. The Originality of the paper is mainly manifested in incorporating factors which have made important impacts on China’s financial safety in recent years, but have not been taken into consideration in the existing studies into the newly constructed financial safety index system. For example, some factors that cannot be controlled easily might have huge hidden risk hazards. To be more specific, factors such as off-balance-sheet business, interbank business and shadow banking might trigger liquidity risks. In this way, the research results will be more practical
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An index of financial safety of China
OmniaScience, 2015Co-Authors: Jia Xiaojun, Li MenggangAbstract:Purpose: This paper combines a synthetic index system by the variables and evaluates China’s financial safety through the change of indexes in a comprehensive way. First of all, it builds the financial industry evaluation index system composed of 25indicators in terms of the operation of the financial industry and external economic environment and particularly takes into consideration factors which might trigger liquidity risks such as off-balance-sheet business, interbank business and shadow banking; then it selects 10 indicators to conduct empirical analysis and identifies the indicator weight through principal component analysis; finally it combines the financial safety indexes through the linear weighted comprehensive evaluation model. Design/methodology/approach: Synthesis of indexes is made by constructing a proper comprehensive evaluation mathematical model, integrating a number of evaluation indexes into one comprehensive evaluation index and then obtaining corresponding comprehensive evaluation results. In this paper, it selects 10 indexes to conduct empirical analysis and identifies the index weight through principal component analysis; finally it combines the financial safety indexes through the linear weighted comprehensive evaluation model. Principal component analysis (PCA) is a statistical procedure that uses an orthogonal transformation to convert a set of observations of possibly correlated variables into a set of values of linearly uncorrelated variables called principal components. PCA was invented in 1901 and was later independently developed (and named) by Harold Hotelling in the 1930s. Findings: From 2003 to 2013 China’s financial safety indexes fluctuated. From 2003 to 2007 indexes rose, which indicates China’s financial safety status gradually improved; from 2007 to 2009 indexes declined, which indicates due to the impact of subprime crisis, China’s financial safety status took a turn for the worse; from 2009 to 2012 indexes rose, which indicates the external environment improved so did China’s financial safety status; from 2012 to 2013 indexes declined because due to the rapid development of banks’ financial products and trust products, banks’ off-balance-sheet assets and liquidity risks increased. The changes of financial safety indexes are generally identical with those of China’s financial safety status. Research limitations/implications: In the empirical analysis part, this article tries to selective 24 indicators synthetic index of China's financial security, but due to some of the indicators data acquisition is relatively difficult, can only Selective 10 of 25 indicators and gather the annual data of 10 indicators from 2003 to 2013 to synthetic index. The information of eliminated indicators cannot be reflected in the index. Index change also does not reflect of the risk from these indicators. In order to make up for the above limitations, this paper is mainly to introduce and analysis our latest financial institutions business trends associated with these eliminated indicators to get the conclusions more reliable. Originality/value: The aim of this research is to estimate financial safety of China with the application of the index of financial safety of a country using the annual data of 2003-2013. Through synthetic index of financial security measure the risks of China's financial system, provide the basis for the government macro financial policy. The Originality of the paper is mainly manifested in incorporating factors which have made important impacts on China’s financial safety in recent years, but have not been taken into consideration in the existing studies into the newly constructed financial safety index system. For example, some factors that cannot be controlled easily might have huge hidden risk hazards. To be more specific, factors such as off-balance-sheet business, interbank business and shadow banking might trigger liquidity risks. In this way, the research results will be more practical.Peer Reviewe
Laura CismaŞ - One of the best experts on this subject based on the ideXlab platform.
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CURRENT TRENDS OF THE REGIONAL DEVELOPMENT POLICY IN THE EUROPEAN UNION. THE DEVELOPMENT OF COMPETITIVE ECONOMIC AGGLOMERATIONS OF CLUSTER TYPE
University of Petrosani, 2010Co-Authors: Laura CismaŞ, Andra Miculescu, Maria OŢilAbstract:The study of economic agents’ behaviour, whose nowadays tendency is togroup themselves in space as clusters, has an important place in the field of localizing industrialactivities. This is due to domestic scale economies, known as agglomerations economies.According to Edgar M. Hoover (Hoover, 1948), domestic scale economies are specific tocompanies; the economies of localizing - to a certain branch, whose companies form clusters incertain geographical arias, and the urbanization economies are specific to cities, where thereare clusters of companies from different branches. The specialty literature regarding localeconomic development, based on the idea of cluster starts from well-known economic theories,such as: agglomeration theory (Alfred Marshall), the theory of spatial localizing of industrialunits (Alfred Weber), the theory of interdependence of locations (Harold Hotelling), the diamondtheory (Michael Porter), the theory of entrepreneurship (Joseph Schumpeter), the theory ofgeographical concentration. Basically, the common point which links them are the conceptswhich occur in these theories, such as: industrial district, industrial agglomeration, spatialinterdependence, concepts which lie at the basis of the cluster idea. Clusters represent animportant instrument for promoting industrial development, innovation, competitiveness andeconomic growth. If, at the beginning, the effort to develop clusters belonged to private personsand companies, nowadays, the actors involved in their development are the governments andpublic institutions of national or regional level.The objective established within the Lisbon Strategy (2000), to make the EuropeanUnion “the most competitive and dynamic knowledge-based economy”, is tightly linked to thenew approaches of the European economic policy, to competitiveness. One of the policies isfocused on developing at the European Union level clusters in the high competitiveness fields. with an innovative character. Using statistical data relatively recently by the European ClusterObservatory (2007), our paper aims at revealing the fact that clusters are linked to prosperityand that it exists a necessity to consider them as a central part of each economic strategy for theEuropean Union member states. We shall also present the initiatives of cluster type between theEuropean states, successful clusters, with a possible multiplication effect. The paper will alsopresent Romania’s trials to achieve an industrial policy based on competitive economicagglomeration
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Current Trends Of The Regional Development Policy In The European Union. The Development Of Competitive Economic Agglomerations Of Cluster Type
2024Co-Authors: Laura CismaŞ, Andra Miculescu, Maria OţilAbstract:The study of economic agents’ behaviour, whose nowadays tendency is to group themselves in space as clusters, has an important place in the field of localizing industrial activities. This is due to domestic scale economies, known as agglomerations economies. According to Edgar M. Hoover (Hoover, 1948), domestic scale economies are specific to companies; the economies of localizing - to a certain branch, whose companies form clusters in certain geographical arias, and the urbanization economies are specific to cities, where there are clusters of companies from different branches. The specialty literature regarding local economic development, based on the idea of cluster starts from well-known economic theories, such as: agglomeration theory (Alfred Marshall), the theory of spatial localizing of industrial units (Alfred Weber), the theory of interdependence of locations (Harold Hotelling), the diamond theory (Michael Porter), the theory of entrepreneurship (Joseph Schumpeter), the theory of geographical concentration. Basically, the common point which links them are the concepts which occur in these theories, such as: industrial district, industrial agglomeration, spatial interdependence, concepts which lie at the basis of the cluster idea. Clusters represent an important instrument for promoting industrial development, innovation, competitiveness and economic growth. If, at the beginning, the effort to develop clusters belonged to private persons and companies, nowadays, the actors involved in their development are the governments and public institutions of national or regional level. The objective established within the Lisbon Strategy (2000), to make the European Union “the most competitive and dynamic knowledge-based economy”, is tightly linked to the new approaches of the European economic policy, to competitiveness. One of the policies is focused on developing at the European Union level clusters in the high competitiveness fields. The efforts are concentrated at microeconomic level, by partnerships between universities, the private sector and other institutions, aiming to achieve macroeconomic results through the real growth of companies’ productivity. This is also the objective of our paper, to demonstrate the fact that for the European Union, clusters represent the economic model of development, which is suitable for organizing these efforts and, in the same time, for effectively launching initiatives with an innovative character. Using statistical data relatively recently by the European Cluster Observatory (2007), our paper aims at revealing the fact that clusters are linked to prosperity and that it exists a necessity to consider them as a central part of each economic strategy for the European Union member states. We shall also present the initiatives of cluster type between the European states, successful clusters, with a possible multiplication effect. The paper will also present Romania’s trials to achieve an industrial policy based on competitive economic agglomeration.clusters, competitive economic agglomeration, regional development policies, innovation, competitiveness
