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Hirschhorn, Daniel B., and Denisse R. Thompson. "Technology and Reasoning in Algebra and Geometry." Mathematics Teacher 89, no. 2 (February 1996): 138–42. http://dx.doi.org/10.5951/mt.89.2.0138.
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If one topic is likely to be stressed by algebra and geometry teachers, it is reasoning. In algebra classes, students are constantly being asked to show their work and justify their simplifications, often without formal connection to proof concepts or the proof process. In geometry classes, students are expected to learn how to write simple proofs. However, evidence shows that students are not learning these reasoning skills. In the 1985–86 National Assessment of Educational Progress, Silver and Carpenter (1989, 18) found that “many eleventhgrade students are confused about the fundamental distinctions among mathematical demonstrations, assumptions, and proofs.” Most students thought a theorem was a demonstration or an assumption. Senk (1985) found that only about 30 percent of students mastered proof wTiting in geometry, despite being enrolled in a year-long course emphasizing proof. Thompson (1992) found that roughly 60 percent of precalculus students were successful at trigonometric-identity proofs, more than 30 percent could complete number-theory proofs dealing with divisibility, and less than 20 percent could handle indirect arguments or proof by mathematical induction.
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Karunakaran, Shiv, Ben Freeburn, Nursen Konuk, and Fran Arbaugh. "Improving Preservice Secondary Mathematics Teachers' Capability With Generic Example Proofs." Mathematics Teacher Educator 2, no. 2 (March 2014): 158–70. http://dx.doi.org/10.5951/mathteaceduc.2.2.0158.
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Preservice mathematics teachers are entrusted with developing their future students' interest in and ability to do mathematics effectively. Various policy documents place an importance on being able to reason about and prove mathematical claims. However, it is not enough for these preservice teachers, and their future students, to have a narrow focus on only one type of proof (demonstration proof), as opposed to other forms of proof, such as generic example proofs or pictorial proofs. This article examines the effectiveness of a course on reasoning and proving on preservice teachers' awareness of and abilities to recognize and construct generic example proofs. The findings support assertions that such a course can and does change preservice teachers' capability with generic example proofs.
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Meo, Santolo, and Luisa Toscano. "On the Existence and Uniqueness of the ODE Solution and Its Approximation Using the Means Averaging Approach for the Class of Power Electronic Converters." Mathematics 9, no. 10 (May 19, 2021): 1146. http://dx.doi.org/10.3390/math9101146.
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Power electronic converters are mathematically represented by a system of ordinary differential equations discontinuous right-hand side that does not verify the conditions of the Cauchy-Lipschitz Theorem. More generally, for the properties that characterize their discontinuous behavior, they represent a particular class of systems on which little has been investigated over the years. The purpose of the paper is to prove the existence of at least one global solution in Filippov’s sense to the Cauchy problem related to the mathematical model of a power converter and also to calculate the error in norm between this solution and the integral of its averaged approximation. The main results are the proof of this theorem and the analytical formulation that provides to calculate the cited error. The demonstration starts by a proof of local existence provided by Filippov himself and already present in the literature for a particular class of systems and this demonstration is generalized to the class of electronic power converters, exploiting the non-chattering property of this class of systems. The obtained results are extremely useful for estimating the accuracy of the averaged model used for analysis or control of the effective system. In the paper, the goodness of the analytical proof is supported by experimental tests carried out on a converter prototype representing the class of power electronics converter.
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Hodds, Mark, Lara Alcock, and Matthew Inglis. "Self-Explanation Training Improves Proof Comprehension." Journal for Research in Mathematics Education 45, no. 1 (January 2014): 62–101. http://dx.doi.org/10.5951/jresematheduc.45.1.0062.
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In this article, the authors report 3 experiments demonstrating that a simple booklet containing self-explanation training, designed to focus students' attention on logical relationships within a mathematical proof, can significantly improve their proof comprehension.
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Barbero, M., T. Fritzsch, L. Gonella, F. Hügging, H. Krüger, M. Rothermund, and N. Wermes. "A via last TSV process applied to ATLAS pixel detector modules: proof of principle demonstration." Journal of Instrumentation 7, no. 08 (August 10, 2012): P08008. http://dx.doi.org/10.1088/1748-0221/7/08/p08008.
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Arpaia, P., M. Buzio, M. Kazazi, and S. Russenschuck. "Proof-of-principle demonstration of a translating coils-based method for measuring the magnetic field of axially-symmetric magnets." Journal of Instrumentation 10, no. 02 (February 10, 2015): P02004. http://dx.doi.org/10.1088/1748-0221/10/02/p02004.
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Anellis, Irving. "Charles Peirce and Bertrand Russell on Euclid." Revista Brasileira de História da Matemática 19, no. 37 (October 16, 2020): 79–94. http://dx.doi.org/10.47976/rbhm2019v19n3779-94.
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Both Charles Sanders Peirce (1839–1914) and Bertrand Russell (1872–1970) held that Euclid’s proofs in geometry were fundamentally flawed, and based largely on mathematical intuition rather than on sound deductive reasoning. They differed, however, as to the role which diagramming played in Euclid’s emonstrations. Specifically, whereas Russell attributed the failures on Euclid’s proofs to his reasoning from diagrams, Peirce held that diagrammatic reasoning could be rendered as logically rigorous and formal. In 1906, in his manuscript “Phaneroscopy” of 1906, he described his existential graphs, his highly iconic, graphical system of logic, as a moving picture of thought, “rendering literally visible before one’s very eyes the operation of thinking in actu”, and as a “generalized diagram of the Mind” (Peirce 1906; 1933, 4.582). More generally, Peirce personally found it more natural for him to reason diagrammatically, rather than algebraically. Rather, his concern with Euclid’s demonstrations was with its absence of explicit explanations, based upon the laws of logic, of how to proceed from one line of the “proof” to the next. This is the aspect of his criticism of Euclid that he shared with Russell; that Euclid’s demonstrations drew from mathematical intuition, rather than from strict formal deduction.
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Ivliev, Y. "DIAGNOSTICS OF MATHEMATICAL PROOF OF THE BEAL CONJECTURE IN MEDICAL PSYCHOLOGY (REMAKE OF PREVIOUS AUTHOR’S ARTICLES CONCERNING FERMAT’S LAST THEOREM)." East European Scientific Journal 1, no. 5(69) (June 15, 2021): 28–33. http://dx.doi.org/10.31618/essa.2782-1994.2021.1.69.48.
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In the given work diagnostics of mathematical proof of the Beal Conjecture (Generalized Fermat’s Last Theorem) obtained in the earlier author’s works was conducted and truthfulness of the suggested proof was established. Realizing the process of the Bill Conjecture solution, the mathematical structure defining hypothetical equality of the Fermat theorem was determined. Such a structure turned to be one of Pythagorean theorem with whole numbers. With help of Euclid’s geometrical theorem and Fermat’s method of infinite descent one can manage to set that Pythagorean equation in whole numbers representing Fermat’s Last Theorem cannot exist and then the Fermat theorem is true, that is Fermat’s equality in natural numbers does not exist. Thus mental scheme of “demonstratio mirabile”, which Pierre de Fermat mentioned on the margins of Diophantus’s “Arithmetic”, was reconstructed.
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Casesnoves, Francisco. "Inverse methods and integral-differential model demonstration for optimal mechanical operation of power plants – numerical graphical optimization for second generation of tribology models." Electrical, Control and Communication Engineering 14, no. 1 (July 1, 2018): 39–50. http://dx.doi.org/10.2478/ecce-2018-0005.
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Abstract Stepping forward from a previous conference contribution, the article focuses on extension of inverse problem algorithms to integral-differential modelling and formal/strict demonstration of graphical-optimization method. It shows evident-mathematical and 3D-imaging proofs of the graphical optimization method with L1 Norm simulations and algorithms. At present, Linear/Nonlinear Optimization mathematical methods constitute the choice of preference in getting improvements for erosion and corrosion simulations- determinations in general tribology, biotribology and tribocorrosion. The method(s) developed are classical numerical optimization settings for objective functions, programming optimization and simulations, and special software for imaging in 3D. Results are diverse and the range of their applications is wide. First, the article provides a definite formal demonstration of the nonlinear graphical optimization both in numerical results and in imaging. Then, the authors propose the development of programming optimization and mathematical proofs-algorithms of the integral-differential model for various models. Subsequently, an overview of stochastic erosion methods based on Markov Chain is presented in the article. Finally, the second generation of tribology models is defined and conceptually explained. To summarise, the article comprises new findings towards modernization of tribology, biotribology and tribocorrosion models, gathering innovative research branches for future extension of the mathematical modelling progress. The results can be applied to both general techniques and mechanical engineering. The analytical and numerical demonstration of the integral-differential model constitutes a key point and essential result of the research. Extension to electromagnetic and electronic models of these methods is also considered feasible and practical
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Di Liscia, Daniel. "The conclusio pulchra, mirabilis et bona: an ingenious demonstration attributable to Nicole Oresme." Mediaevalia Textos e estudos 37 (2018): 139–68. http://dx.doi.org/10.21747/21836884/med37a7.
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Hardly any other concept has occupied the minds of philosophers and scientists as much as the con-cept of infinity. Late medieval philosophy is not an exception. Especially within the context of the so-calledcalculatorestradition a new approach emerged which prioritised the analysis of physical, mathematical, and logical problems over the determination of the essence of infinity and its defini-tion. From the fourteenth century onward, it was not unusual in this context to discuss in detail some special cases of motion which included an augmentationin infinitumof the “degrees of velocity”. This paper focuses on a particular case, the “conclusio mirabilis”, a demonstration to which Oresme could have self-referred in this treatiseDe configurationibusas a “more subtle and more difficult” proof. Whereas this short text has until now been analysed according to only one manuscript, the present contribution involves a research regarding a text conglomerate made up of at least seven manuscripts which are somehow mutually connected. It is argued that an attribution of this demon-stration to Oresme is, with due caution, possible, even if further research is still needed to determine the original shape of the text. In addition, this paper includes a short reference to two later important authors, Biagio Pelacani da Parma and Jacques Almain, whose reception of theconclusio mirabi-lisremained unnoticed in the scholarship until now.
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Yasev, Aleksandr. "Improvement quality of preparation specialists in area of mathematical modeling." System technologies 3, no. 134 (April 5, 2021): 49–55. http://dx.doi.org/10.34185/1562-9945-3-134-2021-06.
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Improvement quality of preparation specialists in area of mathematical modeling is an actual problem in education students (especially applied) and non mathematical special-ties. For solving this problem must be used system and successive exposition of mainte-nance of the stages of building model with obligatory demonstration of there intercom-munications. Each stage of modeling makes some part of quality of model. Most difficult stage is idealization of original, because in this case the most important properties of original must be distinguished and the experimental samples of output of variables of original and calculation samples of model are got.. Main attention it must be spared to estimation of adequacy original and model with the help of famous nonparametric criteria such as criterion Wilcoxon and special criterion for small samples. This criterion is based on the statistical analysis of output variables of the model and original. At a necessity new structural or technological suggestions are developed for perfection of original. Last part of modeling usually is optimization parameters of original. The method of optimization is elected depending on the type of model and maintenance of task.Methodology of study, that provides a receipt of the deeper and complete under-standing of essence of mathematical modeling of technical objects is offered. Positive results are achieved by using method which is proposed.System exposition of material and accent during the teaching on research and proof adequacy of model allow to attain the deeper and complete understanding by students of essence of mathematical design of technical objects and promote efficiency of preparation of specialists in area of the applied mathematical modeling.
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Kuperman, Gilbert G., and William A. Perez. "A Frame-Based Mission Decomposition Model." Proceedings of the Human Factors Society Annual Meeting 32, no. 2 (October 1988): 135–39. http://dx.doi.org/10.1177/154193128803200229.
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This paper presents the results of a mathematical modeling (computer simulation) effort that applied frame-based, data processing constructs, originally developed and applied in the context of artificial intelligence, to the decomposition of a complex Air Force bomber mission. The model was written in LISP to facilitate the development of a concurrent processing environment in which to simulate the simultaneous occurrence of multiple external events/crew tasks. The model simulated a four hour segment of a strategic mission scenario. Two distinct crew complements, four-man and two-man, together with their respective levels of aircraft avionics automation, were represented during a proof-of-concept demonstration. The model provided measures of resource (crew and “black box”) utilization, presumed to correlate to “workload,” at different levels of specificity. These measures were used to identify crew task “chokepoints” (large queue sizes, task interrupts) and to evaluate the effects of automation.
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Dodge, Wally, Virginia Horak, and David Masunaga. "Message From the Editorial Panel: Welcome to Our Focus Issue on the Concept of Proof." Mathematics Teacher 91, no. 8 (November 1998): 646–47. http://dx.doi.org/10.5951/mt.91.8.0646.
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The concept of proof is one concerning which the pupil should have a growing and increasing understanding. It is a concept which not only pervades his work in mathematics but is also involved in all situations where conclusions are to be reached and decisions to be made. Mathematics has a unique contribution to make in the development of this concept, and up to the present time teachers of mathematics have, in general, assumed this contribution can best be made in the tenth year through the study of demonstrative geometry. The practice resulting from this assumption has tended to isolate the concept of proof, whereas this concept may well serve to unify the mathematical experiences of the pupil.
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Nicolau, Dan V., Mercy Lard, Till Korten, Falco C. M. J. M. van Delft, Malin Persson, Elina Bengtsson, Alf Månsson, Stefan Diez, Heiner Linke, and Dan V. Nicolau. "Parallel computation with molecular-motor-propelled agents in nanofabricated networks." Proceedings of the National Academy of Sciences 113, no. 10 (February 22, 2016): 2591–96. http://dx.doi.org/10.1073/pnas.1510825113.
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The combinatorial nature of many important mathematical problems, including nondeterministic-polynomial-time (NP)-complete problems, places a severe limitation on the problem size that can be solved with conventional, sequentially operating electronic computers. There have been significant efforts in conceiving parallel-computation approaches in the past, for example: DNA computation, quantum computation, and microfluidics-based computation. However, these approaches have not proven, so far, to be scalable and practical from a fabrication and operational perspective. Here, we report the foundations of an alternative parallel-computation system in which a given combinatorial problem is encoded into a graphical, modular network that is embedded in a nanofabricated planar device. Exploring the network in a parallel fashion using a large number of independent, molecular-motor-propelled agents then solves the mathematical problem. This approach uses orders of magnitude less energy than conventional computers, thus addressing issues related to power consumption and heat dissipation. We provide a proof-of-concept demonstration of such a device by solving, in a parallel fashion, the small instance {2, 5, 9} of the subset sum problem, which is a benchmark NP-complete problem. Finally, we discuss the technical advances necessary to make our system scalable with presently available technology.
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ROSSMANITH, PETER, and MARTIN WAGNER. "A Novel Poisson Distribution–Based Approach for Testing Boundaries of Real-Time PCR Assays for Food Pathogen Quantification." Journal of Food Protection 74, no. 9 (September 1, 2011): 1404–12. http://dx.doi.org/10.4315/0362-028x.jfp-10-458.
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The validation of quantitative real-time PCR systems and above all, proof of the detection limit of this method, is a frequently and intensively discussed topic in food pathogen detection. Among proper sample collection, assay design, careful experimental design, execution of real-time PCR, and data analysis, the validation of the method per se ensuring reliable quantification data is of prime importance. The purpose of this study was to evaluate a novel validation tool for real-time PCR assays, based on the theoretical possibility of the amplification of a single DNA target. The underlying mathematical basis for the work is Poisson distribution, which describes patterns of low particle numbers in a volume. In this context, we focused on the quantitative aspect of real-time PCR for the first time. This allowed for demonstration of the reliable amplification of a lone target DNA molecule and the demonstration of the distinct discrimination between integer molecular numbers when using low initial copy numbers. A real-time PCR assay amplifying a 274-bp fragment of the positive regulatory protein A locus of Listeria monocytogenes was used for this work. Evidence for a linear range of quantification from a single target copy to 10 ng of target DNA was experimentally demonstrated, and evidence for the significance of this novel validation approach is presented here.
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Sovrano, Elisa. "About Chaotic Dynamics in the Twisted Horseshoe Map." International Journal of Bifurcation and Chaos 26, no. 06 (June 15, 2016): 1650092. http://dx.doi.org/10.1142/s0218127416500929.
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The twisted horseshoe map was developed in order to study a class of density dependent Leslie population models with two age classes. From the beginning, scientists have tried to prove that this map presents chaotic dynamics. Some demonstrations that have appeared in mathematical literature present some difficulties or delicate issues. In this paper, we give a simple and rigorous proof based on a different approach. We also highlight the possibility of getting chaotic dynamics for a broader class of maps.
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Fleischman, Tomaž, and Paolo Dini. "Mathematical Foundations for Balancing the Payment System in the Trade Credit Market." Journal of Risk and Financial Management 14, no. 9 (September 21, 2021): 452. http://dx.doi.org/10.3390/jrfm14090452.
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The increasingly complex economic and financial environment in which we live makes the management of liquidity in payment systems and the economy in general a persistent challenge. New technologies make it possible to address this challenge through alternative solutions that complement and strengthen existing payment systems. For example, interbank balancing and clearing methods (such as real-time gross settlement) can also be applied to private payments, complementary currencies, and trade credit clearing to provide better liquidity and risk management. The paper defines the concept of a balanced payment system mathematically and demonstrates the effects of balancing on a few small examples. It then derives the construction of a balanced payment subsystem that can be settled in full and therefore that can be removed in toto to achieve debt reduction and payment gridlock resolution. Using well-known results from graph theory, the main output of the paper is the proof—for the general formulation of a payment system with an arbitrary number of liquidity sources—that the amount of liquidity saved is maximum, along with a detailed discussion of the practical steps that a lending institution can take to provide different levels of service subject to the constraints of available liquidity and its own cap on total overdraft exposure. From an applied mathematics point of view, the original contribution of the paper is two-fold: (1) the introduction of a liquidity node with a store of value function in obligation-clearing; and (2) the demonstration that the case with one or more liquidity sources can be solved with the same mathematical machinery that is used for obligation-clearing without liquidity. The clearing and balancing methods presented are based on the experience of a specific application (Tetris Core Technologies), whose wider adoption in the trade credit market could contribute to the financial stability of the whole economy and a better management of liquidity and risk overall.
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Rajabalipanah, Hamid, Ali Abdolali, Shahid Iqbal, Lei Zhang, and Tie Jun Cui. "Analog signal processing through space-time digital metasurfaces." Nanophotonics 10, no. 6 (March 29, 2021): 1753–64. http://dx.doi.org/10.1515/nanoph-2021-0006.
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Abstract In the quest to realize analog signal processing using subwavelength metasurfaces, in this paper, we present the first demonstration of programmable time-modulated metasurface processors based on the key properties of spatial Fourier transformation. Exploiting space-time coding strategy enables local, independent, and real-time engineering of not only amplitude but also phase profile of the contributing reflective digital meta-atoms at both central and harmonic frequencies. Several illustrative examples are demonstrated to show that the proposed multifunctional calculus metasurface is capable of implementing a large class of useful mathematical operators, including 1st- and 2nd-order spatial differentiation, 1st-order spatial integration, and integro-differential equation solving accompanied by frequency conversions. Unlike the recent proposals based on the Green’s function (GF) method, the designed time-modulated signal processor effectively operates for input signals containing wide spatial frequency bandwidths with an acceptable gain level. Proof-of-principle simulations are also reported to demonstrate the successful realization of image processing functions like edge detection. This time-varying wave-based computing system can set the direction for future developments of programmable metasurfaces with highly promising applications in ultrafast equation solving, real-time and continuous signal processing, and imaging.
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Coghetto, Roland. "Gauge Integral." Formalized Mathematics 25, no. 3 (October 1, 2017): 217–25. http://dx.doi.org/10.1515/forma-2017-0021.
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Summary Some authors have formalized the integral in the Mizar Mathematical Library (MML). The first article in a series on the Darboux/Riemann integral was written by Noboru Endou and Artur Korniłowicz: [6]. The Lebesgue integral was formalized a little later [13] and recently the integral of Riemann-Stieltjes was introduced in the MML by Keiko Narita, Kazuhisa Nakasho and Yasunari Shidama [12]. A presentation of definitions of integrals in other proof assistants or proof checkers (ACL2, COQ, Isabelle/HOL, HOL4, HOL Light, PVS, ProofPower) may be found in [10] and [4]. Using the Mizar system [1], we define the Gauge integral (Henstock-Kurzweil) of a real-valued function on a real interval [a, b] (see [2], [3], [15], [14], [11]). In the next section we formalize that the Henstock-Kurzweil integral is linear. In the last section, we verified that a real-valued bounded integrable (in sense Darboux/Riemann [6, 7, 8]) function over a interval a, b is Gauge integrable. Note that, in accordance with the possibilities of the MML [9], we reuse a large part of demonstrations already present in another article. Instead of rewriting the proof already contained in [7] (MML Version: 5.42.1290), we slightly modified this article in order to use directly the expected results.
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Almouloud, Saddo Ag, and Maridete Brito Ferreira. "Proposta e análise de uma organização didática para o ensino de conceitos geométricos via provas e demonstração." Revista Brasileira de Educação em Ciências e Educação Matemática 1, no. 1 (December 21, 2017): 30. http://dx.doi.org/10.33238/rebecem.2017.v.1.n.1.18556.
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Resumo: Este artigo é um recorte de uma pesquisa maior que analisa uma proposta didática, cujas tarefas articulam provas e demonstrações, como alternativa metodológica para minimizar as dificuldades relacionadas ao tópico ‘quadriláteros’, em um curso de licenciatura em Matemática. Essas tarefas envolvem construções geométricas em um ambiente de papel e lápis, em que os alunos são solicitados a construir figuras geométricas e justificar matematicamente as técnicas utilizadas. Em uma parte das tarefas propostas, solicitava-se cumpri-las por meio de um raciocínio hipotético-dedutivo. Neste texto, apresentamos uma análise a priori de algumas das tarefas que não puderam ser experimentadas, mas que faziam parte do rol de situações propostas na investigação supracitada. A análise vislumbrada apoia-se, principalmente, na Teoria Antropológica do Didático (TAD) e na concepção de prova e demonstração. Nas análises que propusemos, além da TAD, as fases exploratórias das diferentes tarefas e a articulação entre os registros de representação (tratamento e conversão) nos permitiram mostrar que não basta o aluno se basear apenas na apreensão perceptiva para fazer uma validação formal das afirmações tecidas nas diferentes tarefas. É necessário que se explore, principalmente as apreensões operatórias e discursivas da figura para vislumbrar caminhos que permitem a construção de provas intelectuais.Palavras-chave: Prova; Demonstração; Geometria; Quadriláteros; Análise a priori. Proposal and analysis of a didactic organization for the geometric concepts teaching by tests and demonstrationAbstract: This article is a clipping of a larger survey that analyzes a didactic proposal whose tasks articulate proof and demonstrations as a methodological alternative to minimizing the difficulties related to the topic ' quadrilaterals ' in a bachelor's degree course in mathematics. These tasks involve geometric constructs, in a paper and pencil environment, in which students are asked to construct geometric figures and to justify mathematically the techniques used. In a part of the proposed tasks, it was requested to fulfil them by a hypothetical-deductive reasoning. In this text, we presented a priori analysis of some of the tasks that could not be experienced, but which were part of the list of situations proposed in the aforementioned investigation. The envisaged analysis is primarily supported by the anthropological theory of the didactic (ATD) and the conception of proof and demonstration. In the analysis we have proposed, beyond the ATD, the exploratory phases of the different tasks, the articulation between semiotic register representation (treatment and conversion) allowed us to show that it is not enough that student only supports the perceptual apprehension to make formal validation of the affirmations woven in the different tasks. It is necessary to explore, mainly, the operative seizures and discursive of the figure to glimpse paths that allow the construction of intellectual evidence.Keywords: Proof; Demonstration; Geometry; Quadrilaterals; A priori analysis.
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Wallace, William A. "The Problem of Apodictic Proof in Early Seventeenth-Century Mechanics. Galileo, Guevara, and the Jesuits." Science in Context 3, no. 1 (1989): 67–87. http://dx.doi.org/10.1017/s0269889700000727.
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The ArgumentThe argument developed herein, a countertheme to the Merton thesis, is that the ideal of science pursued by Galileo and his contemporaries in Italy would be unaffected by their Catholic faith if it could achieve apodictic proof in the subject of its investigations, in which case it would attain truth – the very goal sought by that faith. Unfortunately such proof was hard to come by in early seventeenth-century mechanics. A case study is proposed to show Galileo's difficulty demonstrating the tensile strength of columns in mathematical physics on the basis of improper suppositions, contrasting these with the suppositions of a contemporary, Giovanni de Guevara. The case study casts new light on the subject of Galileo's atomism and its relation to the Eucharist, refuting Redondi's claim that this was the real motive behind Galileo's trial in 1633. A further lesson relating to the Merton thesis can also be drawn, namely, that while challenges to science from religious orthodoxy may appear as temporary obstacles, in the long run they are an aid to science in its efforts to bring mankind closer to the truth.
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Elkin, Yury, and Vitaliy Kurlin. "Isometry Invariant Shape Recognition of Projectively Perturbed Point Clouds by the Mergegram Extending 0D Persistence." Mathematics 9, no. 17 (September 1, 2021): 2121. http://dx.doi.org/10.3390/math9172121.
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Rigid shapes should be naturally compared up to rigid motion or isometry, which preserves all inter-point distances. The same rigid shape can be often represented by noisy point clouds of different sizes. Hence, the isometry shape recognition problem requires methods that are independent of a cloud size. This paper studies stable-under-noise isometry invariants for the recognition problem stated in the harder form when given clouds can be related by affine or projective transformations. The first contribution is the stability proof for the invariant mergegram, which completely determines a single-linkage dendrogram in general position. The second contribution is the experimental demonstration that the mergegram outperforms other invariants in recognizing isometry classes of point clouds extracted from perturbed shapes in images.
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D'Angio, Richard G., and Peter R. Gwilt. "Time to Steady State following a Change in Dosing Rate." DICP 23, no. 6 (June 1989): 468–72. http://dx.doi.org/10.1177/106002808902300607.
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Drugs administered at fixed intervals or by continuous infusion will accumulate in the body until steady state is achieved. The time to a given percentage of the eventual steady-state concentration has previously been considered to be dependent only on the elimination half-life. This is incorrect. Although the rate of drug accumulation in the body is dependent only on the elimination half-life, the time to a given percentage of steady state is dependent on both the elimination half-life of the drug and the initial concentration. This paper presents the mathematical proof of this concept, computer simulations demonstrating the use of these equations, and nomograms for use in clinical practice. The use of this method allows serum drug concentrations to be evaluated earlier than previously predicted after changes in the dosing rate.
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Ugaglia, Monica. "Knowing by Doing: The Role of Geometrical Practice in Aristotle’s Theory of Knowledge." Elenchos 36, no. 1 (March 1, 2015): 45–88. http://dx.doi.org/10.1515/elen-2015-360103.
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Abstract Aristotle’s way of conceiving the relationship between mathematics and other branches of scientific knowledge is completely different from the way a contemporary scientist conceives it. This is one of the causes of the fact that we look at the mathematical passages we find in Aristotle’s works with the wrong expectation. We expect to find more or less stringent proofs, while for the most part Aristotle employs mere analogies. Indeed, this is the primary function of mathematics when employed in a philosophical context: not a demonstrative tool, but a purely analogical model. In the case of the geometrical examples discussed in this paper, the diagrams are not conceived as part of a formalized proof, but as a work in progress. Aristotle is not interested in the final diagram but in the construction viewed in its process of development; namely in the figure a geometer draws, and gradually modifies, when he tries to solve a problem. The way in which the geometer makes use of the elements of his diagram, and the relation between these elements and his inner state of knowledge is the real feature which interests Aristotle. His goal is to use analogy in order to give the reader an idea of the states of mind involved in a more general process of knowing.
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Hayden, Patrick, Kevin Milner, and Mark M. Wilde. "Two-message quantum interactive proofs and the quantum separability problem." Quantum Information and Computation 14, no. 5&6 (May 2014): 384–416. http://dx.doi.org/10.26421/qic14.5-6-2.
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Suppose that a polynomial-time mixed-state quantum circuit, described as a sequence of local unitary interactions followed by a partial trace, generates a quantum state shared between two parties. One might then wonder, does this quantum circuit produce a state that is separable or entangled? Here, we give evidence that it is computationally hard to decide the answer to this question, even if one has access to the power of quantum computation. We begin by exhibiting a two-message quantum interactive proof system that can decide the answer to a promise version of the question. We then prove that the promise problem is hard for the class of promise problems with "quantum statistical zero knowledge" QSZK proof systems by demonstrating a polynomial-time Karp reduction from the QSZK-complete promise problem "quantum state distinguishability" to our quantum separability problem. By exploiting Knill's efficient encoding of a matrix description of a state into a description of a circuit to generate the state, we can show that our promise problem is NP-hard with respect to Cook reductions. Thus, the quantum separability problem (as phrased above) constitutes the first nontrivial promise problem decidable by a two-message quantum interactive proof system while being hard for both NP and QSZK. We also consider a variant of the problem, in which a given polynomial-time mixed-state quantum circuit accepts a quantum state as input, and the question is to decide if there is an input to this circuit which makes its output separable across some bipartite cut. We prove that this problem is a complete promise problem for the class QIP of problems decidable by quantum interactive proof systems. Finally, we show that a two-message quantum interactive proof system can also decide a multipartite generalization of the quantum separability problem.
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Aksteiner, Steffen, Lars Andersson, Thomas Bäckdahl, Igor Khavkine, and Bernard Whiting. "Compatibility Complex for Black Hole Spacetimes." Communications in Mathematical Physics 384, no. 3 (May 7, 2021): 1585–614. http://dx.doi.org/10.1007/s00220-021-04078-y.
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AbstractThe set of local gauge invariant quantities for linearized gravity on the Kerr spacetime presented by two of the authors (Aksteiner and Bäckdahl in Phys Rev Lett 121:051104, 2018) is shown to be complete. In particular, any gauge invariant quantity for linearized gravity on Kerr that is local and of finite order in derivatives can be expressed in terms of these gauge invariants and derivatives thereof. The proof is carried out by constructing a complete compatibility complex for the Killing operator, and demonstrating the equivalence of the gauge invariants from Aksteiner and Bäckdahl (Phys Rev Lett 121:051104, 2018) with the first compatibility operator from that complex.
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FUJIKAWA, KAZUO, C. H. OH, and CHENGJIE ZHANG. "LINEARITY OF QUANTUM PROBABILITY MEASURE AND HARDY'S MODEL." International Journal of Modern Physics A 29, no. 02 (January 20, 2014): 1450017. http://dx.doi.org/10.1142/s0217751x14500171.
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We re-examine d = 4 hidden-variables model for a system of two spin-1/2 particles in view of the concrete model of Hardy, who analyzed the criterion of entanglement without referring to inequality. The basis of our analysis is the linearity of the probability measure related to the Born probability interpretation, which excludes noncontextual hidden-variables model in d≥3. To be specific, we note the inconsistency of the noncontextual hidden-variables model in d = 4 with the linearity of the quantum mechanical probability measure in the sense 〈ψ|a⋅σ ⊗b ⋅σ|ψ〉+ 〈ψ|a ⋅σ ⊗b′ ⋅σ|ψ〉 = 〈ψ|a⋅σ⊗(b + b′)⋅σ|ψ〉 for noncollinear b and b′. It is then shown that Hardy's model in d = 4 does not lead to a unique mathematical expression in the demonstration of the discrepancy of local realism (hidden-variables model) with entanglement and thus his proof is incomplete. We identify the origin of this nonuniqueness with the nonuniqueness of translating quantum mechanical expressions into expressions in hidden-variables model, which results from the failure of the above linearity of the probability measure. In contrast, if the linearity of the probability measure is strictly imposed, which tantamounts to asking that the noncontextual hidden-variables model in d = 4 gives the Clauser–Horne–Shimony–Holt (CHSH) inequality |〈B〉|≤2 uniquely, it is shown that the hidden-variables model can describe only separable quantum mechanical states; this conclusion is in perfect agreement with the so-called Gisin's theorem which states that |〈B〉|≤2 implies separable states.
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Davis, Simon. "A rationality condition for the existence of odd perfect numbers." International Journal of Mathematics and Mathematical Sciences 2003, no. 20 (2003): 1261–93. http://dx.doi.org/10.1155/s0161171203108277.
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A rationality condition for the existence of odd perfect numbers is used to derive an upper bound for the density of odd integers such thatσ(N)could be equal to2N, whereNbelongs to a fixed interval with a lower limit greater than10300. The rationality of the square root expression consisting of a product of repunits multiplied by twice the base of one of the repunits depends on the characteristics of the prime divisors, and it is shown that the arithmetic primitive factors of the repunits with different prime bases can be equal only when the exponents are different, with possible exceptions derived from solutions of a prime equation. This equation is one example of a more general prime equation,(qjn−1)/(qin−1)=ph, and the demonstration of the nonexistence of solutions whenh≥2requires the proof of a special case of Catalan's conjecture. General theorems on the nonexistence of prime divisors satisfying the rationality condition and odd perfect numbersNsubject to a condition on the repunits in factorization ofσ(N)are proven.
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Oehring, Charles. "Singular numbers of smooth kernels." Mathematical Proceedings of the Cambridge Philosophical Society 103, no. 3 (May 1988): 511–14. http://dx.doi.org/10.1017/s0305004100065129.
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In [12] we elaborate the vague principle that the behaviour at infinity of the decreasing sequence of singular numbers sn(K) of a Hilbert–Schmidt kernel K is at least as good as that of the sequence {n−1/qω(n−1;K)}, where ωp is an Lp-modulus of continuity of K and q = p/(p − 1), where 1 ≤ p ≤ 2. Despite the author's effort to justify his study of refinements of the half-century old theorem of Smithies [13], that theorem remains the central result of the subject (viz. that for 0 < a ≤ 1, K∈Lip(a, p) implies that sn(K) = O(n−α−1/q)). For example, Cochran's omnibus theorems [5, 6] that delimit the Schatten classes to which a kernel belongs are based on the blending of ‘smoothness’ conditions and emphasize the pivotal role of the principal corollary of Smithies' theorem (viz. {sn}∈lr if r−1 < α + q−1). Cochran later offered in [7] a very simple derivation of the corollary from a Fourier series theorem of Konyushkov (see [2], vol. II, p. 197), whose proof was, however, at least as intricate as Smithies' demonstration.
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Yudenkov, Aleksey V., Aleksandr M. Volodchenkov, and Liliya P. Rimskaya. "STABILITY OF SYSTEMS OF SINGULAR INTEGRAL EQUATIONS WITH CAUCHY KERNEL." T-Comm 14, no. 9 (2020): 48–55. http://dx.doi.org/10.36724/2072-8735-2020-14-9-48-55.
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Singular Cauchy integral equations have been widely used for mathematical simulation of the actual physical and technical systems. They are considered universal at every level of simulation beginning with quantum field theory and up to strength analysis of the underground constructions. Therefore investigating system stability of such models under perturbation of their absolute terms and coefficients appears an urgent scientific task. The aim of the study is to show various aspects of stability of singular Cauchy integral sets of equations which are generalizing simulation models of the primal problems of the elasticity theory for homogeneous isotropic bodies. The methods of study are based on the properties of the Cauchy singular integral, on the general theory of Fredholm operators. When in use, systems of the singular integral equations are reduced to a set of Fredholm integral equations of the second kind and a set of the boundary value problems for analytic functions. The key results of the study are the following: development of the general determination method of the system index for singular integral equations, proof of the system stability against perturbations of the absolute terms of the set. Against perturbations of the boundary coefficients, the singular integral system is unstable. Demonstration of the stability of the singular integral Cauchy sets generalizing primal problems of the elasticity theory appears a significantly new result. The research of singular integral equations sets has been performed conducted on the space of functions satisfying the Holder condition. However the main research results prove to be true if we operate random functions converting in mean square. Stability of singular integral equations sets against perturbations of the absolute terms lays a foundation for calculus of approximations in real world tasks of defining the built-in stress of an elastic complex body.
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Hawkes, P. W. "Aberration correction past and present." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 367, no. 1903 (September 28, 2009): 3637–64. http://dx.doi.org/10.1098/rsta.2009.0004.
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Electron lenses are extremely poor: if glass lenses were as bad, we should see as well with the naked eye as with a microscope! The demonstration by Otto Scherzer in 1936 that skilful lens design could never eliminate the spherical and chromatic aberrations of rotationally symmetric electron lenses was therefore most unwelcome and the other great electron optician of those years, Walter Glaser, never ceased striving to find a loophole in Scherzer’s proof. In the wartime and early post-war years, the first proposals for correcting C s were made and in 1947, in a second milestone paper, Scherzer listed these and other ways of correcting lenses; soon after, Dennis Gabor invented holography for the same purpose. These approaches will be briefly summarized and the work that led to the successful implementation of quadupole–octopole and sextupole correctors in the 1990s will be analysed. In conclusion, the elegant role of image algebra in describing image formation and processing and, above all, in developing new methods will be mentioned.
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Ward, Andrew. "Proof and Demonstration." International Philosophical Quarterly 48, no. 1 (2008): 23–37. http://dx.doi.org/10.5840/ipq200848123.
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Hildebrandt, Mireille. "The Trial of the Expert: ÉÉpreuve and Preuve." New Criminal Law Review 10, no. 1 (January 1, 2007): 78–101. http://dx.doi.org/10.1525/nclr.2007.10.1.78.
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This article explores the complex relationship between law and (scientific) expertise. The author first discusses the difference between scientific knowledge and scientific expertise, tracing the historical roots of the concept of proof in a legal rather than a mathematical context. Second, the historical roots of the fair trial are discussed in terms of the premodern éépreuve and the modern preuve, demonstrating the relationship with coordinate and subordinate types of justice. Third, Foucault's analysis of éépreuve, enquêête, and examèèn is extended to clarify how preuve and éépreuve have been integrated into the fair trial, which is explained in relation to the formal and the substantive notions of the "Rechtsstaat." This analysis finally allows the author to discuss the difference between two approaches of scientific expertise in court: one claiming that judges should defer to science and another claiming that a court of law is one of the spaces in which such expertise can be contested. The article concludes that the integration of subordinate and coordinate justice exemplified in the fair trial, opens new perspectives for the testing of scientific expertise, taking into account whose interests are at stake.
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Hornby, Richard. "Mathematical Proof." Hudson Review 54, no. 1 (2001): 110. http://dx.doi.org/10.2307/3852829.
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Curran, W. S. "Mathematical proof." Journal of Interdisciplinary Mathematics 8, no. 1 (January 2005): 119–25. http://dx.doi.org/10.1080/09720502.2005.10700395.
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THOMAS, R. S. D. "Mathematical Proof." Philosophia Mathematica 7, no. 1 (February 1, 1999): 3–4. http://dx.doi.org/10.1093/philmat/7.1.3.
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Traversa, Fabio Lorenzo, Chiara Ramella, Fabrizio Bonani, and Massimiliano Di Ventra. "Memcomputing NP-complete problems in polynomial time using polynomial resources and collective states." Science Advances 1, no. 6 (July 2015): e1500031. http://dx.doi.org/10.1126/sciadv.1500031.
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Memcomputing is a novel non-Turing paradigm of computation that uses interacting memory cells (memprocessors for short) to store and process information on the same physical platform. It was recently proven mathematically that memcomputing machines have the same computational power of nondeterministic Turing machines. Therefore, they can solve NP-complete problems in polynomial time and, using the appropriate architecture, with resources that only grow polynomially with the input size. The reason for this computational power stems from properties inspired by the brain and shared by any universal memcomputing machine, in particular intrinsic parallelism and information overhead, namely, the capability of compressing information in the collective state of the memprocessor network. We show an experimental demonstration of an actual memcomputing architecture that solves the NP-complete version of the subset sum problem in only one step and is composed of a number of memprocessors that scales linearly with the size of the problem. We have fabricated this architecture using standard microelectronic technology so that it can be easily realized in any laboratory setting. Although the particular machine presented here is eventually limited by noise—and will thus require error-correcting codes to scale to an arbitrary number of memprocessors—it represents the first proof of concept of a machine capable of working with the collective state of interacting memory cells, unlike the present-day single-state machines built using the von Neumann architecture.
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K.V., Niedialkova. "FORMATION OF COMPONENT OF METHODICAL COMPETENCE OF FUTURE MATHEMATICS TEACHERS ABOUT TEACHING PUPILS TO PROVE MATHEMATICAL STATEMENTS." Collection of Research Papers Pedagogical sciences, no. 90 (November 4, 2020): 110–18. http://dx.doi.org/10.32999/ksu2413-1865/2020-90-18.
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Актуальність дослідження визначається положенням, що вивчення теорем та їх доведень є одним з ключових завдань шкільного курсу математики і водночас важливою методичною проблемою, оскільки останнім часом засвідчується зменшення інтересу здобувачів середньої освіти до доказових міркувань. Метою статті є з’ясування й експериментальна перевірка умов ефективного формування складника методичної компетентності майбутніх учителів математики щодо навчання учнів доводити математичні твердження. Під час педагогічного експерименту автором було використано власні мето-дичні розробки з досліджуваної теми, включаючи демонстрацію основних етапів роботи над теоремами, тестові завдання, завдання для самостійної та індивідуальної роботи, презентації, QR-словник понять теми дослідження тощо. Для реалізації експерименту та з’ясування його ефективності використано такі методи, як-от: аналіз нормативних освітніх документів, педагогічне спостереження, анкетування студентів, бесіди з учителями і викладачами, аналіз модульних контрольних робіт з фахової дисципліни «Шкільний курс математики і методика його навчання», математична обробка результатів експеримен-тальної роботи. Висновками з проведеної роботи були визначені умови щодо вдосконалення фахової підготовки майбутніх учителів математики стосовно формування такого складника їхньої методичної компетентності, як навчання учнів доводити математичні твердження, із-поміж яких: вміння здійсню-вати пропедевтичну роботу; набуття глибоких і міцних знань щодо логічних основ ШКМ, принципів і прийомів навчання школярів готових доведень та самостійного пошуку учнями доведень математич-них тверджень; здійснення творчого підходу до реалізації основних етапів роботи з теоремами; усві-домлення значущості та доречності застосування різних способів доведень теорем; урізноманітнення викладачем фахових дисциплін методів, засобів, форм організації навчальної діяльності студентів та ін.Перспективою подальших досліджень є визначення умов підвищення ефективності формування інших складників методичної компетентності майбутніх учителів математики. The relevance of the article is determined by the fact that the study of theorems and their proofs is one of the main tasks of the school course of mathematics and, at the same time, an important methodical problem, as recently there has been a decrease in pupils’ interest in evidence. The purpose of the article is to clarify and experimentally test the conditions for the effective formation of a component of the methodological competence of future mathematics teachers to teach pupils to prove mathematical statements. During the pedagogical experiment the author used his own methodical tasks on the research topic, including demonstration of the main stages of work on theorems, test tasks, tasks for independent and individual work, presentations, QR-dictionary of research topic concepts, etc. To implement the experiment and determine its effectiveness, the following methods were used: analysis of normative educational documents, pedagogical observation, questionnaires of students, conversations with teachers, analysis of modular tests in the discipline “School course of mathematics and methods of its teaching”, mathematical processing of results of experimental work. The conclusions of the research are certain conditions for improving the professional training of future teachers of mathematics, namely: the formation of such a component of their methodical competence as teaching pupils to prove mathematical statements. Among these conditions are ability to carry out propaedeutic work; acquisition of deep and solid knowledge about the logical foundations of SCM, principles and methods of teaching pupils ready-made proofs and independent search by pupils for proofs of mathematical statements; using of a creative approach to the implementation of the main stages of work with theorems; awareness of the significance and appropriateness of the application of different methods of proving theorems; diversification by the professor of methods, means, forms of the organization of educational activity of students, etc.The prospect of further research is to determine the conditions for improving the effectiveness of the formation of other components of methodical competence of future teachers of mathematics.
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Rodriguez-Padilla, Consuelo, Enrique Cuan-Urquizo, Armando Roman-Flores, José L. Gordillo, and Carlos Vázquez-Hurtado. "Algorithm for the Conformal 3D Printing on Non-Planar Tessellated Surfaces: Applicability in Patterns and Lattices." Applied Sciences 11, no. 16 (August 16, 2021): 7509. http://dx.doi.org/10.3390/app11167509.
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In contrast to the traditional 3D printing process, where material is deposited layer-by-layer on horizontal flat surfaces, conformal 3D printing enables users to create structures on non-planar surfaces for different and innovative applications. Translating a 2D pattern to any arbitrary non-planar surface, such as a tessellated one, is challenging because the available software for printing is limited to planar slicing. The present research outlines an easy-to-use mathematical algorithm to project a printing trajectory as a sequence of points through a vector-defined direction on any triangle-tessellated non-planar surface. The algorithm processes the ordered points of the 2D version of the printing trajectory, the tessellated STL files of the target surface, and the projection direction. It then generates the new trajectory lying on the target surface with the G-code instructions for the printer. As a proof of concept, several examples are presented, including a Hilbert curve and lattices printed on curved surfaces, using a conventional fused filament fabrication machine. The algorithm’s effectiveness is further demonstrated by translating a printing trajectory to an analytical surface. The surface is tessellated and fed to the algorithm as an input to compare the results, demonstrating that the error depends on the resolution of the tessellated surface rather than on the algorithm itself.
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REIS, Carlos José dos, Laerte Dias de CARVALHO, Lucas Monteiro CHAVES, and Devanil Jaques de SOUZA. "THREE SIMPLE HEURISTICS MATHEMATICAL PROOFS ON LASSO THEORY." REVISTA BRASILEIRA DE BIOMETRIA 38, no. 2 (June 29, 2020): 243. http://dx.doi.org/10.28951/rbb.v38i2.444.
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Three relevant facts about the least absolute shrinkage and selection operator (Lasso) are studied: The estimatives follows piecewise linear curves in relation to tuning parameter, the number of nonzero selected covariates is an unbiased estimator of its degrees of freedom and when the number of covariates p is greater than the numbers of observations n at most n covariates are selected. These results are well known and described in the literature, but with no simple demonstrations. We present, based on a geometrical approach, simple and intuitive heuristics proofs for these results.
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Tselishchev, Vitaliy V., and Aleksandr V. Khlebalin. "Proof Versus Understanding in Mathematical Proof." Vestnik Tomskogo gosudarstvennogo universiteta. Filosofiya, sotsiologiya, politologiya, no. 54 (April 1, 2020): 29–38. http://dx.doi.org/10.17223/1998863x/54/3.
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Burn, Bob, Rowan Garnier, John Taylor, and R. B. J. T. Allenby. "100% Mathematical Proof." Mathematical Gazette 83, no. 496 (March 1999): 169. http://dx.doi.org/10.2307/3618736.
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Nizar, Achmad, Siti Maghfirotun Amin, and Agung Lukito. "A Learning Trajectory of Indonesian 12-years Old Students Understanding of Division of Fractions." Southeast Asian Mathematics Education Journal 7, no. 2 (December 29, 2017): 41–52. http://dx.doi.org/10.46517/seamej.v7i2.52.
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The purpose of this study was to describe mathematical hands-on activities that can support students to gain better understanding of dividing fractions. This preliminary research phase was started by testing, analyzing, and refining the initial hypothetical learning trajectory (HLT), then in the pilot experimental phase the revised HLT was implemented, and ended with the teaching experimental phase by developing a learning trajectory for 12-year old students in understanding division of fractions. In developing the trajectory, a design research methodology was employed by using four contextual-based learning series (sharing biscuit, sharing remaining chocolate bar, arranging bedroom mats, and running around school yard), including providing some concrete materials or pictorial models as manipulative tools. Seven mathematics experts and twenty five 12-year old students were involved during the research. The four designed learning goals were determining the quotient of division of integer by proper fraction, proper fraction by integer, two proper fractions, and two fractions. Students succeeded in demonstrating their understanding and stated that 16:1/2 = 32; 25/36:5 = 5/36; 24/64:1/2 = 6/8; and 15/2:3/4 = 10 respectively at the end of each designed activity. The interesting results of this study are not a proof, so that a much larger study is needed to determine if the results are due to this approach or due to the teachers’ enthusiasm or what is known as the Hawthorne Effect.
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Pedruzzi, Gabriele, and Igor M. Rouzine. "An evolution-based high-fidelity method of epistasis measurement: Theory and application to influenza." PLOS Pathogens 17, no. 6 (June 21, 2021): e1009669. http://dx.doi.org/10.1371/journal.ppat.1009669.
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Linkage effects in a multi-locus population strongly influence its evolution. The models based on the traveling wave approach enable us to predict the average speed of evolution and the statistics of phylogeny. However, predicting statistically the evolution of specific sites and pairs of sites in the multi-locus context remains a mathematical challenge. In particular, the effects of epistasis, the interaction of gene regions contributing to phenotype, is difficult to predict theoretically and detect experimentally in sequence data. A large number of false-positive interactions arises from stochastic linkage effects and indirect interactions, which mask true epistatic interactions. Here we develop a proof-of-principle method to filter out false-positive interactions. We start by demonstrating that the averaging of haplotype frequencies over multiple independent populations is necessary but not sufficient for epistatic detection, because it still leaves high numbers of false-positive interactions. To compensate for the residual stochastic noise, we develop a three-way haplotype method isolating true interactions. The fidelity of the method is confirmed analytically and on simulated genetic sequences evolved with a known epistatic network. The method is then applied to a large sequence database of neurominidase protein of influenza A H1N1 obtained from various geographic locations to infer the epistatic network responsible for the difference between the pre-pandemic virus and the pandemic strain of 2009. These results present a simple and reliable technique to measure epistatic interactions of any sign from sequence data.
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D'Hondt, E., and P. Panangaden. "The computational power of the W and GHZ states." Quantum Information and Computation 6, no. 2 (March 2006): 173–83. http://dx.doi.org/10.26421/qic6.2-3.
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It is well understood that the use of quantum entanglement significantly enhances the computational power of systems. Much of the attention has focused on Bell states and their multipartite generalizations. However, in the multipartite case it is known that there are several inequivalent classes of states, such as those represented by the W-state and the GHZ-state. Our main contribution is a demonstration of the special computational power of these states in the context of paradigmatic problems from classical distributed computing. Concretely, we show that the W-state is the only pure state that can be used to exactly solve the problem of leader election in anonymous quantum networks. Similarly we show that the GHZ-state is the only one that can be used to solve the problem of distributed consensus when no classical post-processing is considered. These results generalize to a family of W- and GHZ-like states. At the heart of the proofs of these impossibility results lie symmetry arguments.
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Shiu, P., and David S. G. Stirling. "Mathematical Analysis and Proof." Mathematical Gazette 82, no. 493 (March 1998): 148. http://dx.doi.org/10.2307/3620189.
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Wright, Crispin. "Wittgenstein on Mathematical Proof." Royal Institute of Philosophy Supplement 28 (March 1990): 79–99. http://dx.doi.org/10.1017/s1358246100005257.
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To be asked to provide a short paper on Wittgenstein's views on mathematical proof is to be given a tall order (especially if little or no familiarity either with mathematics or with Wittgenstein's philosophy is to be presupposed!). Close to one half of Wittgenstein's writings after 1929 concerned mathematics, and the roots of his discussions, which contain a bewildering variety of underdeveloped and sometimes conflicting suggestions, go deep to some of the most basic and difficult ideas in his later philosophy. So my aims in what follows are forced to be modest. I shall sketch an intuitively attractive philosophy of mathematics and illustrate Wittgenstein's opposition to it. I shall explain why, contrary to what is often supposed, that opposition cannot be fully satisfactorily explained by tracing it back to the discussions of following a rule in the Philosophical Investigations and Remarks on the Foundations of Mathematics. Finally, I shall try to indicate very briefly something of the real motivation for Wittgenstein's more strikingly deflationary suggestions about mathematical proof, and canvass a reason why it may not in the end be possible to uphold them.
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Avigad, Jeremy. "Mathematical Method and Proof." Synthese 153, no. 1 (November 2006): 105–59. http://dx.doi.org/10.1007/s11229-005-4064-5.
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Nathanson, Melvyn B. "Desperately Seeking Mathematical Proof." Mathematical Intelligencer 31, no. 2 (April 2009): 8–10. http://dx.doi.org/10.1007/s00283-009-9040-3.
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Wheeler, David. "Aspects of mathematical proof." Interchange 21, no. 1 (March 1990): 1–5. http://dx.doi.org/10.1007/bf01809604.
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