The Experts below are selected from a list of 837 Experts worldwide ranked by ideXlab platform
John Stillwell - One of the best experts on this subject based on the ideXlab platform.
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Hypothesis Testing
Statistical Inference via Convex Optimization, 2019Co-Authors: John StillwellAbstract:This chapter prepares the reader's mind for reverse mathematics. As its name suggests, reverse mathematics seeks not theorems but the right Axioms to prove theorems already known. Reverse mathematics began as a technical field of mathematical logic, but its main ideas have precedents in the ancient field of geometry and the early twentieth-century field of set theory. In geometry, the Parallel Axiom is the right Axiom to prove many theorems of Euclidean geometry, such as the Pythagorean theorem. Set theory offers a more modern example: base theory called ZF, a theorem that ZF cannot prove (the well-ordering theorem) and the “right Axiom” for proving it—the Axiom of choice. From these and similar examples one can guess at a base theory for analysis, and the “right Axioms” for proving some of its well-known theorems.
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Historical Introduction
Reverse Mathematics, 2019Co-Authors: John StillwellAbstract:This chapter prepares the reader's mind for reverse mathematics. As its name suggests, reverse mathematics seeks not theorems but the right Axioms to prove theorems already known. Reverse mathematics began as a technical field of mathematical logic, but its main ideas have precedents in the ancient field of geometry and the early twentieth-century field of set theory. In geometry, the Parallel Axiom is the right Axiom to prove many theorems of Euclidean geometry, such as the Pythagorean theorem. Set theory offers a more modern example: base theory called ZF, a theorem that ZF cannot prove (the well-ordering theorem) and the “right Axiom” for proving it—the Axiom of choice. From these and similar examples one can guess at a base theory for analysis, and the “right Axioms” for proving some of its well-known theorems.
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Sparse Recovery via ℓ1 Minimization
Statistical Inference via Convex Optimization, 2019Co-Authors: John StillwellAbstract:This chapter prepares the reader's mind for reverse mathematics. As its name suggests, reverse mathematics seeks not theorems but the right Axioms to prove theorems already known. Reverse mathematics began as a technical field of mathematical logic, but its main ideas have precedents in the ancient field of geometry and the early twentieth-century field of set theory. In geometry, the Parallel Axiom is the right Axiom to prove many theorems of Euclidean geometry, such as the Pythagorean theorem. Set theory offers a more modern example: base theory called ZF, a theorem that ZF cannot prove (the well-ordering theorem) and the “right Axiom” for proving it—the Axiom of choice. From these and similar examples one can guess at a base theory for analysis, and the “right Axioms” for proving some of its well-known theorems.
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From Hypothesis Testing to Estimating Functionals
Statistical Inference via Convex Optimization, 2019Co-Authors: John StillwellAbstract:This chapter prepares the reader's mind for reverse mathematics. As its name suggests, reverse mathematics seeks not theorems but the right Axioms to prove theorems already known. Reverse mathematics began as a technical field of mathematical logic, but its main ideas have precedents in the ancient field of geometry and the early twentieth-century field of set theory. In geometry, the Parallel Axiom is the right Axiom to prove many theorems of Euclidean geometry, such as the Pythagorean theorem. Set theory offers a more modern example: base theory called ZF, a theorem that ZF cannot prove (the well-ordering theorem) and the “right Axiom” for proving it—the Axiom of choice. From these and similar examples one can guess at a base theory for analysis, and the “right Axioms” for proving some of its well-known theorems.
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Non-Euclidean geometry
2010Co-Authors: John StillwellAbstract:Surprisingly, the geometry of curved surfaces throws light on the geometry of the plane. More than 2000 years after Euclid formulated Axioms for plane geometry, differential geometry showed that the Parallel Axiom does not follow from the other Axioms of Euclid. It had long been hoped that the Parallel Axiom followed from the others, but no proof had ever been found. In particular, no contradiction had been derived from the contrary hypothesis, P 2, that there is more than one Parallel to a given line through a given point. In the 1820s, Bolyai and Lobachevsky proposed that the consequences of P 2 be accepted as a new kind of geometry—non-Euclidean geometry. To prove that no contradiction follows from P 2, however, one needs to find a model for P 2 and the other Axioms of Euclid. One seeks a mathematical structure, containing objects called “points” and “lines,” that satisfies Euclid’s Axioms with P 2 in place of the Parallel Axiom. Such a structure was first found by Beltrami (1868a), in the form of a surface of constant negative curvature with geodesics as its “lines.” By various mappings of this surface, Beltrami found other models, including a projective model in which “lines” are line segments in the unit disk, and conformal models in which “angles” are ordinary angles. Finally, Poincare (1882) showed that Beltrami’s conformal models arise naturally in complex analysis. Papers had already been published with pictures of patterns of non-Euclidean “lines,” most notably Schwarz (1872). Thus, non-Euclidean geometry was actually a part of existing mathematics, but a part whose geometric nature had not previously been understood.
Athanase Papadopoulos - One of the best experts on this subject based on the ideXlab platform.
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Hyperbolic geometry in the work of Johann Heinrich Lambert
arXiv: Metric Geometry, 2015Co-Authors: Athanase Papadopoulos, Guillaume ThéretAbstract:The memoir Theorie der Parallellinien (1766) by Johann Heinrich Lambert is one of the founding texts of hyperbolic geometry, even though its author's aim was, like many of his predecessors', to prove that such a geometry does not exist. In fact, Lambert developed his theory with the hope of finding a contradiction in a geometry where all the Euclidean Axioms are kept except the Parallel Axiom and that the latter is replaced by its negation. In doing so, he obtained several fundamental results of hyperbolic geometry. This was sixty years before the first writings of Lobachevsky and Bolyai appeared in print. In the present paper, we present Lambert's main results and we comment on them. A French translation of the Theorie der Parallellinien, together with an extensive commentary, has just appeared in print (A. Papadopoulos and G. Theret, La theorie des lignes Paralleles de Johann Heinrich Lambert. Collection Sciences dans l'Histoire, Librairie Scientifique et Technique Albert Blanchard, Paris, 2014).
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HYPERBOLIC GEOMETRY IN THE WORK OF J. H. LAMBERT
2015Co-Authors: Athanase PapadopoulosAbstract:The memoir Theorie der Parallellinien (1766) by Jo- hann Heinrich Lambert is one of the founding texts of hyperbolic geometry, even though its author's aim was, like many of his pre- decessors', to prove that such a geometry does not exist. In fact, Lambert developed his theory with the hope of finding a contra- diction in a geometry where all the Euclidean Axioms are kept except the Parallel Axiom and that the latter is replaced by its negation. In doing so, he obtained several fundamental results of hyperbolic geometry. This was sixty years before the first writings of Lobachevsky and Bolyai appeared in print. In the present paper, we present Lambert's main results and we comment on them. A French translation of the Theorie der Parallellinien, together with an extensive commentary, has just appeared in print (19). AMS classification: 01A50 ; 53-02 ; 53-03 ; 53A05 ; 53A35.
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Hyperbolic geometry in the work of Johann Heinrich Lambert
Ganita Bharati (Indian Mathematics): Journal of the Indian Society for History of Mathematics, 2014Co-Authors: Athanase Papadopoulos, Guillaume ThéretAbstract:The memoir Theorie der Parallellinien (1766) by Johann Heinrich Lambert is one of the founding texts of hyperbolic geometry, even though its author's aim was, like many of his predecessors', to prove that such a geometry does not exist. In fact, Lambert developed his theory with the hope of finding a contradiction in a geometry where all the Euclidean Axioms are kept except the Parallel Axiom and that the latter is replaced by its negation. In doing so, he obtained several fundamental results of hyperbolic geometry. This was sixty years before the first writings of Lobachevsky and Bolyai appeared in print. In the present paper, we present Lambert's main results and we comment on them. A French translation of the Theorie der Parallellinien, together with an extensive commentary, has just appeared in print (A. Papadopoulos and G. Théret, La théorie des lignes parallèles de Johann Heinrich Lambert. Collection Sciences dans l'Histoire, Librairie Scientifique et Technique Albert Blanchard, Paris, 2014).
Guillaume Théret - One of the best experts on this subject based on the ideXlab platform.
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Hyperbolic geometry in the work of Johann Heinrich Lambert
arXiv: Metric Geometry, 2015Co-Authors: Athanase Papadopoulos, Guillaume ThéretAbstract:The memoir Theorie der Parallellinien (1766) by Johann Heinrich Lambert is one of the founding texts of hyperbolic geometry, even though its author's aim was, like many of his predecessors', to prove that such a geometry does not exist. In fact, Lambert developed his theory with the hope of finding a contradiction in a geometry where all the Euclidean Axioms are kept except the Parallel Axiom and that the latter is replaced by its negation. In doing so, he obtained several fundamental results of hyperbolic geometry. This was sixty years before the first writings of Lobachevsky and Bolyai appeared in print. In the present paper, we present Lambert's main results and we comment on them. A French translation of the Theorie der Parallellinien, together with an extensive commentary, has just appeared in print (A. Papadopoulos and G. Theret, La theorie des lignes Paralleles de Johann Heinrich Lambert. Collection Sciences dans l'Histoire, Librairie Scientifique et Technique Albert Blanchard, Paris, 2014).
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Hyperbolic geometry in the work of Johann Heinrich Lambert
Ganita Bharati (Indian Mathematics): Journal of the Indian Society for History of Mathematics, 2014Co-Authors: Athanase Papadopoulos, Guillaume ThéretAbstract:The memoir Theorie der Parallellinien (1766) by Johann Heinrich Lambert is one of the founding texts of hyperbolic geometry, even though its author's aim was, like many of his predecessors', to prove that such a geometry does not exist. In fact, Lambert developed his theory with the hope of finding a contradiction in a geometry where all the Euclidean Axioms are kept except the Parallel Axiom and that the latter is replaced by its negation. In doing so, he obtained several fundamental results of hyperbolic geometry. This was sixty years before the first writings of Lobachevsky and Bolyai appeared in print. In the present paper, we present Lambert's main results and we comment on them. A French translation of the Theorie der Parallellinien, together with an extensive commentary, has just appeared in print (A. Papadopoulos and G. Théret, La théorie des lignes parallèles de Johann Heinrich Lambert. Collection Sciences dans l'Histoire, Librairie Scientifique et Technique Albert Blanchard, Paris, 2014).
Francesco Montefoschi - One of the best experts on this subject based on the ideXlab platform.
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Axiom: A Hardware-Software Platform for Cyber Physical Systems
Proceedings - 19th Euromicro Conference on Digital System Design DSD 2016, 2016Co-Authors: Somnath Mazumdar, Nicola Bettin, Sara Ermini, Carlos Alvarez Martinez, Antonio Filgueras, Daniel Jiménez-gonzález, Xavier Martorell, Javier Bueno, Eduard Ayguadé, Francesco MontefoschiAbstract:Cyber-Physical Systems (CPSs) are widely necessary for many applications that require interactions with the humans and the physical environment. A CPS integrates a set of hardware-software components to distribute, execute and manage its operations. The Axiom project (Agile, eXtensible, fast I/O Module) aims at developing a hardware-software platform for CPS such that i) it can use an easy Parallel programming model and ii) it can easily scale-up the performance by adding multiple boards (e.g., 1 to 10 boards can run in Parallel). Axiom supports task-based programming model based on OmpSs and leverage a high-speed, inexpensive communication interface called Axiom-Link. Another key aspect is that the board provides programmable logic (FPGA) to accelerate portions of an application. We are using smart video surveillance, and smart home living applications to drive our design.
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DSD - Axiom: A Hardware-Software Platform for Cyber Physical Systems
2016 Euromicro Conference on Digital System Design (DSD), 2016Co-Authors: Somnath Mazumdar, Nicola Bettin, Sara Ermini, Carlos Alvarez Martinez, Antonio Filgueras, Daniel Jiménez-gonzález, Xavier Martorell, Javier Bueno, Eduard Ayguadé, Francesco MontefoschiAbstract:Cyber-Physical Systems (CPSs) are widely necessary for many applications that require interactions with the humans and the physical environment. A CPS integrates a set of hardware-software components to distribute, execute and manage its operations. The Axiom project (Agile, eXtensible, fast I/O Module) aims at developing a hardware-software platform for CPS such that i) it can use an easy Parallel programming model and ii) it can easily scale-up the performance by adding multiple boards (e.g., 1 to 10 boards can run in Parallel). Axiom supports task-based programming model based on OmpSs and leverage a high-speed, inexpensive communication interface called Axiom-Link. Another key aspect is that the board provides programmable logic (FPGA) to accelerate portions of an application. We are using smart video surveillance, and smart home living applications to drive our design.
Temur Z. Kalanov - One of the best experts on this subject based on the ideXlab platform.
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On the Natural-Science Foundations of Geometry
viXra, 2013Co-Authors: Temur Z. KalanovAbstract:The work is devoted to solution of an actual problem – the problem of relation between geometry and natural sciences. Methodological basis of the method of attack is the unity of formal logic and of rational dialectics. It is shown within the framework of this basis that geometry represents field of natural sciences. Definitions of the basic concepts "point", "line", "straight line", "surface", "plane surface", and “triangle” of the elementary (Euclidean) geometry are formulated. The natural-scientific proof of the Parallel Axiom (Euclid’s fifth postulate), classification of triangles on the basis of a qualitative (essential) sign, and also material interpretation of Euclid’s, Lobachevski’s, and Riemann’s geometries are proposed.
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The Solution of the Problem of Relation Between Geometry and Natural Sciences
viXra, 2013Co-Authors: Temur Z. KalanovAbstract:@@The work is devoted to solution of an actual problem – the problem of relation between geometry and natural sciences. Methodological basis of the method of attack is the unity of formal logic and of rational dialectics. It is shown within the framework of this basis that geometry represents field of natural sciences. Definitions of the basic concepts "point", "line", "straight line", "surface", "plane surface", and “triangle” of the elementary (Euclidean) geometry are formulated. The natural-scientific proof of the Parallel Axiom (Euclid’s fifth postulate), classification of triangles on the basis of a qualitative (essential) sign, and also material interpretation of Euclid’s, Lobachevski’s, and Riemann’s geometries are proposed.
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Analysis of the Problem of Relation between Geometry and Natural Sciences
Prespacetime Journal, 2011Co-Authors: Temur Z. KalanovAbstract:The work is devoted to analysis of an actual problem – the problem of relation between geometry and natural sciences. Methodological basis of the analysis is the unity of formal logic and of rational dialectics. It is shown within the framework of this basis that geometry represents field of natural sciences. Definitions of the basic concepts "point", "line", "straight line", "surface", "plane surface", and “triangle” of the elementary (Euclidean) geometry are formulated. The natural-scientific proof of the Parallel Axiom ( Euclid ’s fifth postulate), classification of triangles on the basis of a qualitative (essential) sign, and also material interpretation of Euclid ’s, Lobachevski’s, and Riemann’s geometries are proposed.
