Relevant bibliographies by topics / System of Diophantine equations and calculus

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  1. Journal articles
  2. Dissertations / Theses
  3. Books
  4. Book chapters
  5. Conference papers

Academic literature on the topic 'System of Diophantine equations and calculus'

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Published: 3 June 2025
Last updated: 23 June 2025

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Journal articles on the topic "System of Diophantine equations and calculus"

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Cartiere, Carmelo R. "An Analytical Study of Diophantine Equations of Pythagorean Form: Causal Inferences on Hypothesized Relations between Quadratic and Non-quadratic Triples." Athens Journal of Education 12, no. 3 (2025): 527–46. https://doi.org/10.30958/aje.12-3-10.

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In XVII century, presumably between 1637 and 1638, with a note in the margin of Diophantus’ “Arithmetica”, Pierre de Fermat stated that Diophantine equations of the Pythagorean form, , have no integer solutions for , and . Of this statement, however, Fermat never provided a proof. Only after more than 350 years, in 1994, Prof. Andrew J. Wiles was finally successful in demonstrating it (Wiles, 1995; Taylor & Wiles, 1995; Boston, 2008). However, Wiles’ proof adopts calculus techniques far beyond Fermat’s knowledge. Our aim is to show an analytical method to attempt a proof to Fermat’s last theorem with the only use of elementary calculus techniques. Keywords: number theory, Diophantine equations, Pythagorean Theorem, Fermat’s last theorem, numerical analysis
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Pocherevin, R. V. "Multidimensional system of Diophantine equations." Moscow University Mathematics Bulletin 72, no. 1 (2017): 41–43. http://dx.doi.org/10.3103/s0027132217010089.

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V, Pandichelvi, and Vanaja R. "A Paradigm for Two Classes of Simultaneous Exponential Diophantine Equations." Indian Journal of Science and Technology 16, no. 40 (2023): 3514–21. https://doi.org/10.17485/IJST/v16i40.1643.

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Abstract <strong>Objectives:</strong>&nbsp;The goal of this article is to find integer solutions to two distinct kinds of simultaneous exponential Diophantine equations in three variables.&nbsp;<strong>Methods:</strong>&nbsp;The system of exponential Diophantine equations is translated into the eminent form of Thue equations, and then their generalised solutions satisfying certain conditions are applied.&nbsp;<strong>Findings:</strong>&nbsp;The finite set of integer solutions for two disparate categories of simultaneous exponential Diophantine equations consisting of three unknowns is scrutinized. In some circumstances, there is no solution in this analysis for both the dissimilar simultaneous Diophantine equations.&nbsp;<strong>Novelty:</strong>&nbsp;The motivation is considered to be two types of simultaneous exponential Diophantine equations are first converted into specific system of Pell equations, then into Thue equations for the possibilities of the sum of the exponents, such as or . If then, the equations are transformed into a cubic equation, which is not in the form of Pell equations. So, such cases are discarded for exploring solutions to the necessary equations. <strong>Keywords:</strong> Simultaneous Exponential Equations, Simultaneous Pell Equations, Thue Equations, Integer Solutions, Divisibility
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Acewicz, Marcin, and Karol Pąk. "Basic Diophantine Relations." Formalized Mathematics 26, no. 2 (2018): 175–81. http://dx.doi.org/10.2478/forma-2018-0015.

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Summary The main purpose of formalization is to prove that two equations ya(z)= y, y = xz are Diophantine. These equations are explored in the proof of Matiyasevich’s negative solution of Hilbert’s tenth problem. In our previous work [6], we showed that from the diophantine standpoint these equations can be obtained from lists of several basic Diophantine relations as linear equations, finite products, congruences and inequalities. In this formalization, we express these relations in terms of Diophantine set introduced in [7]. We prove that these relations are Diophantine and then we prove several second-order theorems that provide the ability to combine Diophantine relation using conjunctions and alternatives as well as to substitute the right-hand side of a given Diophantine equality as an argument in a given Diophantine relation. Finally, we investigate the possibilities of our approach to prove that the two equations, being the main purpose of this formalization, are Diophantine. The formalization by means of Mizar system [3], [2] follows Z. Adamowicz, P. Zbierski [1] as well as M. Davis [4].
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Pandichelvi, V., and S. Saranya. "Application of System Linear Diophantine Equations in Balancing Chemical Equations." International Journal for Research in Applied Science and Engineering Technology 10, no. 10 (2022): 917–20. http://dx.doi.org/10.22214/ijraset.2022.47111.

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Abstract: In this manuscript, the step by step procedure for how the system of linear Diophantine equations are applied to balance chemical equations acquired by the reactions of various chemical compounds and their products is scrutinized.
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Osipyan, V. O., K. I. Litvinov, and A. S. Zhuck. "Research and development of the mathematic models of cryptosystems based on the universal Diophantine language." SHS Web of Conferences 141 (2022): 01020. http://dx.doi.org/10.1051/shsconf/202214101020.

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This paper shows the objective necessity of improving the information security systems under the development of information and telecommunication technologies. The paper for the first time involves a new area of NP-complete problems from Diophantine analysis, namely, multi-degree systems of Diophantine equations of a given dimension and degree of Tarry-Escott type. Based on a fundamentally new number-theoretic method, a mathematical model of an alphabetic information security system (ISS) has been developed that generalizes the principle of building cryptosystems with a public key – the so called dissymmetric bigram cryptosystem. This implies to implement direct and inverse transformations according to a given algorithm based on a two-parameter solution of a multi-degree system of Diophantine equations. A formalized algorithm has been developed for the specified model of a dissymmetric bigram cryptosystem and a training example based on a normal multi-degree system of Diophantine equations of the fifth degree is presented.
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Ye, Yangbo, Ge Wang, and Jiehua Zhu. "Linear diophantine equations for discrete tomography." Journal of X-Ray Science and Technology: Clinical Applications of Diagnosis and Therapeutics 10, no. 1-2 (2001): 59–66. http://dx.doi.org/10.3233/xst-2001-00057.

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In this report, we present a number-theory-based approach for discrete tomography (DT),which is based on parallel projections of rational slopes. Using a well-controlled geometry of X-ray beams, we obtain a system of linear equations with integer coefficients. Assuming that the range of pixel values is $a(i,j)=0,1&lt;FORMULA&gt;, &amp;mldr;, &lt;FORMULA&gt;M-1&lt;FORMULA&gt;, with &lt;FORMULA&gt;M$ being a prime number, we reduce the equations modulo $M$ . To invert the linear system, each algorithmic step only needs $log^2_2 M$ bit operations. In the case of a small $M$ , we have a greatly reduced computational complexity, relative to the conventional DT algorithms, which require $log^2_2 N$ bit operations for a real number solution with a precision of $1/N$ . We also report computer simulation results to support our analytic conclusions.
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IBRAHIM, M., F. ABD RABOU, and H. ZORKTA. "APPLICATION OF DIOPHANTINE EQUATIONS IN A CIPHER SYSTEM." International Conference on Electrical Engineering 1, no. 1 (1998): 313–24. http://dx.doi.org/10.21608/iceeng.1998.61085.

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Garaev, M. Z., and V. N. Chubarikov. "Concerning the Sierpinski-Schinzel system of Diophantine equations." Mathematical Notes 66, no. 2 (1999): 142–47. http://dx.doi.org/10.1007/bf02674869.

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Chilikov, A. A., and Alexey Belov-Kanel. "Exponential diophantine equations in rings of positive characteristic." Journal of Knot Theory and Its Ramifications 29, no. 02 (2020): 2040002. http://dx.doi.org/10.1142/s0218216520400027.

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In this paper, we prove an algorithmical solvability of exponential-Diophantine equations in rings represented by matrices over fields of positive characteristic. Consider the system of exponential-Diophantine equations [Formula: see text] where [Formula: see text] are constants from matrix ring of characteristic [Formula: see text], [Formula: see text] are indeterminates. For any solution [Formula: see text] of the system we construct a word (over an alphabet containing [Formula: see text] symbols) [Formula: see text] where [Formula: see text] is a [Formula: see text]-tuple [Formula: see text], [Formula: see text] is the [Formula: see text]th digit in the [Formula: see text]-adic representation of [Formula: see text]. The main result of this paper is following: the set of words corresponding in this sense to solutions of a system of exponential-Diophantine equations is a regular language (i.e., recognizable by a finite automaton). There exists an algorithm which calculates this language. This algorithm is constructed in the paper.
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Dissertations / Theses on the topic "System of Diophantine equations and calculus"

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Beauchamp, Bradley K. McCrone Sharon Rich Beverly Susan. "Exploring calculus students' understanding of L'Hôpital's Rule." Normal, Ill. : Illinois State University, 2006. http://proquest.umi.com/pqdweb?index=0&did=1273094441&SrchMode=1&sid=3&Fmt=2&VInst=PROD&VType=PQD&RQT=309&VName=PQD&TS=1181240966&clientId=43838.

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Thesis (Ph. D.)--Illinois State University, 2006.<br>Title from title page screen, viewed on June 7, 2007. Dissertation Committee: Dissertation Committee: Sharon S. McCrone, Beverly S. Rich (co-chairs), James F. Cottrill, Lucian L. Ionescu. Includes bibliographical references (leaves 155-159) and abstract. Also available in print.
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Luo, Ye. "Random periodic solutions of stochastic functional differential equations." Thesis, Loughborough University, 2014. https://dspace.lboro.ac.uk/2134/16112.

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In this thesis, we study the existence of random periodic solutions for both nonlinear dissipative stochastic functional differential equations (SFDEs) and semilinear nondissipative SFDEs in C([-r,0],R^d). Under some sufficient conditions for the existence of global semiflows for SFDEs, by using pullback-convergence technique to SFDE, we obtain a general theorem about the existence of random periodic solutions. By applying coupled forward-backward infinite horizon integral equations method, we perform the argument of the relative compactness of Wiener-Sobolev spaces in C([0,τ],C([-r,0]L²(Ω))) and the generalized Schauder's fixed point theorem to show the existence of random periodic solutions.
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Fanelli, Francesco. "Mathematical analysis of models of non-homogeneous fluids and of hyperbolic equations with low regularity coefficients." Doctoral thesis, SISSA, 2012. http://hdl.handle.net/20.500.11767/4420.

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The present thesis is devoted to the study both of strictly hyperbolic operators with low regularity coefficients and of the density-dependent incompressible Euler system. On the one hand, we show a priori estimates for a second order strictly hyperbolic operator whose highest order coefficients satisfy a log-Zygmund continuity condition in time and a log-Lipschitz continuity condition with respect to space. Such an estimate involves a time increasing loss of derivatives...
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Maulen, Soto Rodrigo. "A dynamical system perspective οn stοchastic and iΙnertial methοds fοr optimizatiοn". Electronic Thesis or Diss., Normandie, 2024. http://www.theses.fr/2024NORMC220.

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Motivé par l'omniprésence de l'optimisation dans de nombreux domaines de la science et de l'ingénierie, en particulier dans la science des données, ce manuscrit de thèse exploite le lien étroit entre les systèmes dynamiques dissipatifs à temps continu et les algorithmes d'optimisation pour fournir une analyse systématique du comportement global et local de plusieurs systèmes du premier et du second ordre, en se concentrant sur le cadre convexe, stochastique et en dimension infinie d'une part, et le cadre non convexe, déterministe et en dimension finie d'autre part. Pour les problèmes de minimisation convexe stochastique dans des espaces de Hilbert réels séparables de dimension infinie, notre proposition clé est de les analyser à travers le prisme des équations différentielles stochastiques (EDS) et des inclusions différentielles stochastiques (IDS), ainsi que de leurs variantes inertielles. Nous considérons d'abord les problèmes convexes différentiables lisses et les EDS du premier ordre, en démontrant une convergence faible presque sûre vers les minimiseurs sous hypothèse d'intégrabilité du bruit et en fournissant une analyse globale et locale complète de la complexité. Nous étudions également des problèmes convexes non lisses composites utilisant des IDS du premier ordre et montrons que, sous des conditions d'intégrabilité du bruit, la convergence faible presque sûre des trajectoires vers les minimiseurs, et avec la régularisation de Tikhonov la convergence forte presque sûre des trajectoires vers la solution de norme minimale. Nous développons ensuite un cadre mathématique unifié pour analyser la dynamique inertielle stochastique du second ordre via la reparamétrisation temporelle et le moyennage de la dynamique stochastique du premier ordre, ce qui permet d'obtenir une convergence faible presque sûre des trajectoires vers les minimiseurs et une convergence rapide des valeurs et des gradients. Ces résultats sont étendus à des EDS plus générales du second ordre avec un amortissement visqueux et Hessien, en utilisant une analyse de Lyapunov spécifique pour prouver la convergence et établir de nouveaux taux de convergence. Enfin, nous étudions des problèmes d'optimisation déterministes non convexes et proposons plusieurs algorithmes inertiels pour les résoudre, dérivés d'équations différentielles ordinaires (EDO) du second ordre combinant à la fois un amortissement visqueux sans vanité et un amortissement géométrique piloté par le Hessien, sous des formes explicites et implicites. Nous prouvons d'abord la convergence des trajectoires en temps continu des EDO vers un point critique pour des objectives vérifiant la propriété de Kurdyka-Lojasiewicz (KL) avec des taux explicites, et génériquement vers un minimum local si l'objective est Morse. De plus, nous proposons des schémas algorithmiques par une discrétisation appropriée de ces EDO et montrons que toutes les propriétés précédentes des trajectoires en temps continu sont toujours valables dans le cadre discret sous réserve d'un choix approprié de la taille du pas<br>Motivated by the ubiquity of optimization in many areas of science and engineering, particularly in data science, this thesis exploits the close link between continuous-time dissipative dynamical systems and optimization algorithms to provide a systematic analysis of the global and local behavior of several first- and second-order systems, focusing on convex, stochastic, and infinite-dimensional settings on the one hand, and non-convex, deterministic, and finite-dimensional settings on the other hand. For stochastic convex minimization problems in infinite-dimensional separable real Hilbert spaces, our key proposal is to analyze them through the lens of stochastic differential equations (SDEs) and inclusions (SDIs), as well as their inertial variants. We first consider smooth differentiable convex problems and first-order SDEs, demonstrating almost sure weak convergence towards minimizers under integrability of the noise and providing a comprehensive global and local complexity analysis. We also study composite non-smooth convex problems using first-order SDIs, and show under integrability conditions on the noise, almost sure weak convergence of the trajectory towards a minimizer, with Tikhonov regularization almost sure strong convergence of trajectory to the minimal norm solution. We then turn to developing a unified mathematical framework for analyzing second-order stochastic inertial dynamics via time scaling and averaging of stochastic first-order dynamics, achieving almost sure weak convergence of trajectories towards minimizers and fast convergence of values and gradients. These results are extended to more general second-order SDEs with viscous and Hessian-driven damping, utilizing a dedicated Lyapunov analysis to prove convergence and establish new convergence rates. Finally, we study deterministic non-convex optimization problems and propose several inertial algorithms to solve them derived from second-order ordinary differential equations (ODEs) combining both non-vanishing viscous damping and geometric Hessian-driven damping in explicit and implicit forms. We first prove convergence of the continuous-time trajectories of the ODEs to a critical point under the Kurdyka-Lojasiewicz (KL) property with explicit rates, and generically to a local minimum under a Morse condition. Moreover, we propose algorithmic schemes by appropriate discretization of these ODEs and show that all previous properties of the continuous-time trajectories still hold in the discrete setting under a proper choice of the stepsize
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(9216107), Jordan D. F. Petty. "Modeling a Dynamic System Using Fractional Order Calculus." Thesis, 2020.

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<p>Fractional calculus is the integration and differentiation to an arbitrary or fractional order. The techniques of fractional calculus are not commonly taught in engineering curricula since physical laws are expressed in integer order notation. Dr. Richard Magin (2006) notes how engineers occasionally encounter dynamic systems in which the integer order methods do not properly model the physical characteristics and lead to numerous mathematical operations. In the following study, the application of fractional order calculus to approximate the angular position of the disk oscillating in a Newtonian fluid was experimentally validated. The proposed experimental study was conducted to model the nonlinear response of an oscillating system using fractional order calculus. The integer and fractional order mathematical models solved the differential equation of motion specific to the experiment. The experimental results were compared to the integer order and the fractional order analytical solutions. The fractional order mathematical model in this study approximated the nonlinear response of the designed system by using the Bagley and Torvik fractional derivative. The analytical results of the experiment indicate that either the integer or fractional order methods can be used to approximate the angular position of the disk oscillating in the homogeneous solution. The following research was in collaboration with Dr. Richard Mark French, Dr. Garcia Bravo, and Rajarshi Choudhuri, and the experimental design was derived from the previous experiments conducted in 2018.</p>
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Sibiya, Abram Hlophane. "Numerical methods for a four dimensional hyperchaotic system with applications." Diss., 2019. http://hdl.handle.net/10500/26398.

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This study seeks to develop a method that generalises the use of Adams-Bashforth to solve or treat partial differential equations with local and non-local differentiation by deriving a two-step Adams-Bashforth numerical scheme in Laplace space. The resulting solution is then transformed back into the real space by using the inverse Laplace transform. This is a powerful numerical algorithm for fractional order derivative. The error analysis for the method is studied and presented. The numerical simulations of the method as applied to the four-dimensional model, Caputo-Lu-Chen model and the wave equation are presented. In the analysis, the bifurcation dynamics are discussed and the periodic doubling processes that eventually caused chaotic behaviour (butterfly attractor) are shown. The related graphical simulations that show the existence of fractal structure that is characterised by chaos and usually called strange attractors are provided. For the Caputo-Lu-Chen model, graphical simulations have been realised in both integer and fractional derivative orders.<br>Mathematical Sciences<br>M. Sc. (Applied Mathematics)
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Books on the topic "System of Diophantine equations and calculus"

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Hartley, T. T. Fractional system identification: An approach using continuous order-distributions. National Aeronautics and Space Administration, Glenn Research Center, 1999.

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Orlando, Merino, ed. Discrete dynamical systems and difference equations with Mathematica. Chapman & Hall/CRC, 2002.

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Kogut, Peter I. Optimal control problems for partial differential equations on reticulated domains: Approximation and asymptotic analysis. Birkhäuser, 2011.

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Mielke, Alexander. Hamiltonian and Lagrangian flows on center manifolds: With applications to elliptic variational problems. Springer-Verlag, 1991.

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Alain, Bensoussan, ed. Representation and control of infinite dimensional systems. Birkhäuser, 1992.

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Liao, Xiaoxin. Stability of dynamical systems. Elsevier, 2007.

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International Conference on p-Adic Functional Analysis (11th 2010 Université Blaise Pascal). Advances in non-Archimedean analysis: Eleventh International Conference on p-Adic Functional Analysis, July 5-9 2010, Université Blaise Pascal, Clermont-Ferrand, France. Edited by Araujo-Gomez Jesus 1965-, Diarra B. (Bertin) 1944-, and Escassut Alain. American Mathematical Society, 2011.

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Variational Methods for Strongly Indefinite Problems (Interdisciplinary Mathematical Sciences) (Interdisciplinary Mathematical Sciences). World Scientific Publishing Company, 2007.

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Merino, Orlando, and Mustafa R. S. Kulenovic. Discrete Dynamical Systems and Difference Equations with Mathematica. Taylor & Francis Group, 2002.

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Merino, Orlando, Mustafa R. S. Kulenovic, and Kulenovic M R S (Mustafa R S ). Discrete Dynamical Systems and Difference Equations with Mathematica. Taylor & Francis Group, 2002.

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Book chapters on the topic "System of Diophantine equations and calculus"

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Colombo, Fabrizio, and Irene Sabadini. "An Invitation to the S-functional Calculus." In Spectral Theory, Mathematical System Theory, Evolution Equations, Differential and Difference Equations. Springer Basel, 2012. http://dx.doi.org/10.1007/978-3-0348-0297-0_13.

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Socolescu, Dan. "On the convergence at infinity of solutions with finite dirichlet integral to the exterior dirichlet problem for the steady plane Navier-Stokes system of equations." In Calculus of Variations and Partial Differential Equations. Springer Berlin Heidelberg, 1988. http://dx.doi.org/10.1007/bfb0082901.

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Ramachandran, Parthasarathy. "Use of Extended Euclidean Algorithm in Solving a System of Linear Diophantine Equations with Bounded Variables." In Lecture Notes in Computer Science. Springer Berlin Heidelberg, 2006. http://dx.doi.org/10.1007/11792086_14.

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Salvati, Sylvain. "Syntactic Descriptions: A Type System for Solving Matching Equations in the Linear λ-Calculus." In Lecture Notes in Computer Science. Springer Berlin Heidelberg, 2006. http://dx.doi.org/10.1007/11805618_12.

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La Camera, Giancarlo. "The Mean Field Approach for Populations of Spiking Neurons." In Advances in Experimental Medicine and Biology. Springer International Publishing, 2021. http://dx.doi.org/10.1007/978-3-030-89439-9_6.

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AbstractMean field theory is a device to analyze the collective behavior of a dynamical system comprising many interacting particles. The theory allows to reduce the behavior of the system to the properties of a handful of parameters. In neural circuits, these parameters are typically the firing rates of distinct, homogeneous subgroups of neurons. Knowledge of the firing rates under conditions of interest can reveal essential information on both the dynamics of neural circuits and the way they can subserve brain function. The goal of this chapter is to provide an elementary introduction to the mean field approach for populations of spiking neurons. We introduce the general idea in networks of binary neurons, starting from the most basic results and then generalizing to more relevant situations. This allows to derive the mean field equations in a simplified setting. We then derive the mean field equations for populations of integrate-and-fire neurons. An effort is made to derive the main equations of the theory using only elementary methods from calculus and probability theory. The chapter ends with a discussion of the assumptions of the theory and some of the consequences of violating those assumptions. This discussion includes an introduction to balanced and metastable networks and a brief catalogue of successful applications of the mean field approach to the study of neural circuits.
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La Camera, Giancarlo. "The Mean Field Approach for Populations of Spiking Neurons." In Advances in Experimental Medicine and Biology. Springer International Publishing, 2021. http://dx.doi.org/10.1007/978-3-030-89439-9_6.

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AbstractMean field theory is a device to analyze the collective behavior of a dynamical system comprising many interacting particles. The theory allows to reduce the behavior of the system to the properties of a handful of parameters. In neural circuits, these parameters are typically the firing rates of distinct, homogeneous subgroups of neurons. Knowledge of the firing rates under conditions of interest can reveal essential information on both the dynamics of neural circuits and the way they can subserve brain function. The goal of this chapter is to provide an elementary introduction to the mean field approach for populations of spiking neurons. We introduce the general idea in networks of binary neurons, starting from the most basic results and then generalizing to more relevant situations. This allows to derive the mean field equations in a simplified setting. We then derive the mean field equations for populations of integrate-and-fire neurons. An effort is made to derive the main equations of the theory using only elementary methods from calculus and probability theory. The chapter ends with a discussion of the assumptions of the theory and some of the consequences of violating those assumptions. This discussion includes an introduction to balanced and metastable networks and a brief catalogue of successful applications of the mean field approach to the study of neural circuits.
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Hemion, Geoffrey. "Diophantine inequalities." In The Classification of Knots and 3-Dimensional Spaces. Oxford University PressOxford, 1993. http://dx.doi.org/10.1093/oso/9780198596974.003.0008.

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Abstract In order to proceed, it is necessary to show that the positive solution sets of systems of linear Diophantine equations are finitely generated. One might compare this with the famous simplex algorithm, which is well known to the practitioners of economic speculation. The concern there is with a system of linear equations (describing economic facts) and linear inequalities (describing economic constraints). In addition, an overall function (representing profits) is defined, and the task is to find a solution which maximizes profits while respecting the facts and constraints. The computer quickly finds a single optimal solution.
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T, Padma, and Jayashree Nair. "Diophantine Equations for Enhanced Security in Watermarking Scheme for Image Authentication." In Advanced Image Processing Techniques and Applications. IGI Global, 2017. http://dx.doi.org/10.4018/978-1-5225-2053-5.ch010.

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Hard mathematical problems having no polynomial time algorithms to determine a solution are seemly in design of secure cryptosystems. The proposed watermarking system used number theoretic concepts of the hard higher order Diophantine equations for image content authentication scheme with three major phases such as 1) Formation of Diophantine equation; 2) Generation and embedding of dual Watermarks; and 3) Image content authentication and verification of integrity. Quality of the watermarked images, robustness to compression and security are bench-marked with two peer schemes which used dual watermarks.
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Veeresha, P., M. S. Kiran, L. Akinyemi, and Mehmet Yavuz. "A Unified Approach for the Fractional System of Equations Arising in the Biochemical Reaction without Singular Kernel." In Fractional Calculus: New Applications in Understanding Nonlinear Phenomena. BENTHAM SCIENCE PUBLISHERS, 2022. http://dx.doi.org/10.2174/9789815051933122030012.

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The pivotal aim of the present work is to find the solution for the fractional system of equations arising in the biochemical reaction using q-homotopy analysis transform method (q-HATM). The hired scheme technique unification of Laplace transform with q-homotopy analysis method, and fractional derivative defined with Caputo-Fabrizio (CF) operator. To validate and illustrate the competence of the future method, we examined the model in terms of fractional order. The fixed-point theorem hired to demonstrates the existence and uniqueness. Moreover, the physical nature of achieved solutions has been captured in terms of plots for different order. The obtained results elucidate that the considered algorithm is easy to implement, highly methodical, and very effective as well as accurate to analyse the nature of nonlinear differential equations of fractional order arising in the connected areas of science and engineering.
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Pires, Robson. "Solution Methods of Large Complex-Valued Nonlinear System of Equations." In Advances in Complex Analysis and Applications. IntechOpen, 2020. http://dx.doi.org/10.5772/intechopen.92741.

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Nonlinear systems of equations in complex plane are frequently encountered in applied mathematics, e.g., power systems, signal processing, control theory, neural networks, and biomedicine, to name a few. The solution of these problems often requires a first- or second-order approximation of nonlinear functions to generate a new step or descent direction to meet the solution iteratively. However, such methods cannot be applied to functions of complex and complex conjugate variables because they are necessarily nonanalytic. To overcome this problem, the Wirtinger calculus allows an expansion of nonlinear functions in its original complex and complex conjugate variables once they are analytic in their argument as a whole. Thus, the goal is to apply this methodology for solving nonlinear systems of equations emerged from applications in the industry. For instances, the complex-valued Jacobian matrix emerged from the power flow analysis model which is solved by Newton-Raphson method can be exactly determined. Similarly, overdetermined Jacobian matrices can be dealt, e.g., through the Gauss-Newton method in complex plane aimed to solve power system state estimation problems. Finally, the factorization method of the aforementioned Jacobian matrices is addressed through the fast Givens transformation algorithm which means the square root-free Givens rotations method in complex plane.
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Conference papers on the topic "System of Diophantine equations and calculus"

1

Hu, Cheng-Feng, and Fung-Bao Liu. "Solving System of Fuzzy Diophantine Equations." In 2008 Fifth International Conference on Fuzzy Systems and Knowledge Discovery (FSKD). IEEE, 2008. http://dx.doi.org/10.1109/fskd.2008.111.

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2

Osipyan, V. O. "Development of the mathematic model of dissymmetric bigram cryptosystem based on parametric solution family of multi-degree system of Diophantine equations." In FIT-M 2020. Знание-М, 2020. http://dx.doi.org/10.38006/907345-75-1.2020.281.288.

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Предложен новый подход разработки дисимметричной биграммной криптосистемы (ДБК) на основе многопараметрических решений многостепенных систем диофантовых уравнений (МСДУ), обобщающий принцип построения криптосистем с открытым ключом. Вводится новое понятие равносильности числовых наборов или параметров заданной размерности и порядка. Описанные математические модели СЗИ демонстрируют потенциал применения МСДУ для разработки СЗИ с высокой степенью надёжности
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3

Osipyan, Valeriy, and Kirill Litvinov. "Development of the mathematic model of disymmetric bigram cryptosystem based on a parametric solution family of multi-degree system of Diophantine equations✱." In SIN 2020: 13th International Conference on Security of Information and Networks. ACM, 2020. http://dx.doi.org/10.1145/3433174.3433596.

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Balyuba, Ivan, and Evgeniy Konopatskiy. "Point calculus. Historical background and basic definitions." In International Conference "Computing for Physics and Technology - CPT2020". ANO «Scientific and Research Center for Information in Physics and Technique», 2020. http://dx.doi.org/10.30987/conferencearticle_5fd755c0adb1d9.27038265.

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The paper describes the history of the origin and formation the mathematical apparatus «Point calculus», as one of the scientific directions of the Melitopol school of applied geometry. A brief description of point calculus as a mathematical device that operates within an arithmetic, coordinate affine space, equipped with a topological structure. The basic definitions of point calculus are presented, including point parameters and point equations, space simplex and global coordinate system, independent and current points.
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5

Stojanoski, Goran, Dimitar Ninevski, Gerhard Rath, and Matthew Harker. "Multidimensional Trajectory Tracking for Numerically Stiff Independent Metering System." In SICFP’21 The 17:th Scandinavian International Conference on Fluid Power. Linköping University Electronic Press, 2021. http://dx.doi.org/10.3384/ecp182p283.

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This paper presents a new approach for solving an optimal control problem in a hydraulic system, using a variational calculus method. It uses a path tracking method of two different states with different units and of different magnitude. To ensure the uniqueness of the solution, two regularization terms were introduced, whose influence is regulated by regularization parameters. The system of differential equations, obtained from the Euler-Lagrange equations of the variational problem, was solved by a mass matrix method and discretized with linear differential operators at the interstitial points for numerical stability. This enabled the calculation of the control variables, despite the stiffness of the numerical problem. The results obtained show an energy-efficient performance and no oscillations. Finally, a Simulink model of the hydraulic system was created in which the calculated control variables were inserted as feed-forward inputs, to verify the results.
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6

Narahari Achar, B. N., and John W. Hanneken. "Response Dynamics in the Continuum Limit of the Lattice Dynamical Theory of Viscoelasticity (Fractional Calculus Approach)." In ASME 2009 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. ASMEDC, 2009. http://dx.doi.org/10.1115/detc2009-86218.

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A fractional diffusion-wave equation is derived in the continuum limit of the lattice dynamical equations of motion of a chain of coupled fractional oscillators obtained from the integral equations of motion of a linear chain of simple harmonic oscillators by generalization of the ordinary integrals into ones involving fractional integrals. The set of integral equations of motion pertaining to the chain of coupled fractional oscillators in the continuum limit is solved by using Laplace transforms. The response of the system to impulse and sinusoidal forcing is studied. Numerical applications are discussed with particular reference to energy flow and dissipation.
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Agrawal, Om P. "Fractional Optimal Control of a Distributed System Using Eigenfunctions." In ASME 2007 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. ASMEDC, 2007. http://dx.doi.org/10.1115/detc2007-35921.

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This paper presents a formulation and a numerical scheme for Fractional Optimal Control (FOC) for a class of distributed systems. The fractional derivative is defined in the Caputo sense. The performance index of a FOCP is considered as a function of both the state and the control variables, and the dynamic constraints are expressed by a Partial Fractional Differential Equations (PFDEs). Eigenfunctions are used to eliminate the space parameter, and to define the problem in terms of a set of state and control variables. This leads to a multi FOCP in which each FOCP could be solved independently. Several other strategies are pointed out to reduce the problem to a finite dimensional space, some of which may not provide a decoupled set of equations. The Calculus of Variations, the Lagrange multiplier, and the formula for fractional integration by parts are used to obtain Euler-Lagrange equations for the problem. The numerical technique presented in [1] is used to obtain the state and the control variables. In this technique, the FOC equations are reduced to Volterra type integral equations. The time domain is descretized into several segments and a time marching scheme is used to obtain the response at discrete time points. For a linear case, the numerical technique results into a set of algebraic equations which can be solved using a direct or an iterative scheme. The problem is solved for different number of eigenfunctions and time discretizations. Numerical results show that only a few eigenfunctions are sufficient to obtain good results, and the solutions converge as the size of the time step is reduced. The formulation presented is simple and can be extended to FOC of other distributed systems.
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Yan, YiJing. "Dynamics of dissipative electronic systems and quantum transport: Hierarchical equations of motion approach." In Workshop on Entanglement and Quantum Decoherence. Optica Publishing Group, 2008. http://dx.doi.org/10.1364/weqd.2008.nmd2.

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A quantum dissipation theory is formulated in terms of hierarchically coupled equations of motion for an arbitrary fermionic system coupled with grand canonical fermionic bath ensembles.1 The theoretical construction starts with the second-quantization influence functional in path integral formalism, in which the fermionic creation and annihilation operators are represented by Grassmann variables. Temporal derivatives on influence functionals are then performed in a hierarchical manner, on the basis of calculus-on-path-integral algorithm. Both the multiple-frequency-dispersion and the non-Markovian reservoir parametrization schemes are considered for the desired hierarchy construction. The resulting formalism is in principle exact, applicable to interacting systems, with arbitrary time-dependent external fields. It renders an exact tool to evaluate various transient and stationary quantum transport properties of many-electron systems. At the second-tier truncation level the present theory recovers the real-time diagrammatic formalism developed by Schön and coworkers.2 For a single-particle system, the hierarchical formalism terminates at the second tier exactly, and the Landuer-Büttiker's transport current expression is readily recovered. Numerical studies will be presented to highlight the richness of transient current through both interacting and noninteracting model systems.
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9

Moustafa, Kamal A. F., Mohamed B. Trabia, and Mohamed I. S. Ismail. "Modeling and Control of a Variable Length Flexible Cable Overhead Crane Using the Modified Galerkin Method." In ASME 2007 International Mechanical Engineering Congress and Exposition. ASMEDC, 2007. http://dx.doi.org/10.1115/imece2007-41916.

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A mathematical model that accurately represents an overhead crane with flexible cable and load hoisting/lowering is developed. The analysis includes the transverse vibrations of the flexible cable and the trolley motion as well as the load hoisting/lowering motions. A set of highly non-linear partial differential equations and ordinary differential equations that govern the motion of the crane system within time-varying spatial domain is derived via calculus of variation and Hamilton’s principle. Variable-time modified Galerkin method has been used to discretize the non-linear system. State space transformation is then used to get a set of first order ordinary differential equation. A proportional derivative control scheme is applied to derive the underlying crane so that the cable and payload swing are damped out. Numerical simulations for the control performance of the considered system are presented for various operating conditions.
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10

Konopatskiy, E. V. "Geometric Bases of Parallel Computing in Computer Modeling and Computer-Aided Design Systems." In 32nd International Conference on Computer Graphics and Vision. Keldysh Institute of Applied Mathematics, 2022. http://dx.doi.org/10.20948/graphicon-2022-816-825.

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The concept of developing a geometric CAD kernel based on the invariants of parallel projection of geometric objects on the axes of the global coordinate system, which combines the potential of constructive geometric modeling methods that can provide paralleling of geometric constructions by tasks (message passing), and the mathematical apparatus "Point calculus" capable of implementing data paralleling by means of subordinate calculations (data parallel) is proposed. Use of subordinate calculation of point equations allows not only to parallelize calculations along coordinate axes, but also to provide coherence of computational operations by threads, which significantly reduces downtime and optimizes the performance of CPU to achieve the maximum effect of parallel computations. The greater the dimensionality of the modeled geometric object, the more it lends itself to paralleling computational flows. This leads to the fact that the computation time of a multidimensional problem becomes a value independent of the number of measurements. All calculations will run in parallel and finish simultaneously. The example of parallel computational algorithm for topographic surface modeling demonstrates the possibilities of realization of the offered concept for definition of continuous and discrete geometrical objects, the analytical description of which is carried out in point-calculus. As a result, to build a single 16-point patches, the distribution of parallel computing on 12 threads for the 4 direction lines and 3 threads for the formative line is obtained. Further, the number of simultaneously involved computational threads is a value proportional to the number of 16-point patches and can be further optimized by calculating several forming lines in parallel. In the above example, all computational threads are fully balanced in the number of calculations, which greatly minimizes the downtime of calculations and optimizes the performance of the processor. Also the proposed approach to the organization of parallel computations can be effectively used for the numerical solution of differential equations using geometric interpolants, which together with the development of models of geometric objects in the point calculus creates a closed loop digital production, which by analogy with the isogeometric method eliminates the need to coordinate geometric information in the interaction between CAD and FEA systems.
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