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Добірка наукової літератури з теми "Polynomial potentials"

Автор: Grafiati

Опубліковано: 14 березня 2023

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Статті в журналах з теми "Polynomial potentials"

Ichinose, Wataru. "On the Feynman path integral for the magnetic Schrödinger equation with a polynomially growing electromagnetic potential." Reviews in Mathematical Physics 32, no. 01 (August 5, 2019): 2050003. http://dx.doi.org/10.1142/s0129055x20500038.

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The Feynman path integrals for the magnetic Schrödinger equations are defined mathematically, in particular, with polynomially growing potentials in the spatial direction. For example, we can handle electromagnetic potentials [Formula: see text] such that [Formula: see text] “a polynomial of degree [Formula: see text] in [Formula: see text]” [Formula: see text] and [Formula: see text] are polynomials of degree [Formula: see text] in [Formula: see text]. The Feynman path integrals are defined as [Formula: see text]-valued continuous functions with respect to the time variable.

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Lévai, Géza. "Potentials from the Polynomial Solutions of the Confluent Heun Equation." Symmetry 15, no. 2 (February 9, 2023): 461. http://dx.doi.org/10.3390/sym15020461.

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Polynomial solutions of the confluent Heun differential equation (CHE) are derived by identifying conditions under which the infinite power series expansions around the z=0 singular point can be terminated. Assuming a specific structure of the expansion coefficients, these conditions lead to four non-trivial polynomials that can be expressed as special cases of the confluent Heun function Hc(p,β,γ,δ,σ;z). One of these recovers the generalized Laguerre polynomials LN(α), and another one the rationally extended X1 type Laguerre polynomials L^N(α). The two remaining solutions represent previously unknown polynomials that do not form an orthogonal set and exhibit features characteristic of semi-classical orthogonal polynomials. A standard method of generating exactly solvable potentials in the one-dimensional Schrödinger equation is applied to the CHE, and all known potentials with solutions expressed in terms of the generalized Laguerre polynomials within, or outside the Natanzon confluent potential class, are recovered. It is also found that the potentials generated from the two new polynomial systems necessarily depend on the N quantum number. General considerations on the application of the Heun type differential differential equations within the present framework are also discussed.

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QUESNE, C. "HIGHER-ORDER SUSY, EXACTLY SOLVABLE POTENTIALS, AND EXCEPTIONAL ORTHOGONAL POLYNOMIALS." Modern Physics Letters A 26, no. 25 (August 20, 2011): 1843–52. http://dx.doi.org/10.1142/s0217732311036383.

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Exactly solvable rationally-extended radial oscillator potentials, whose wave functions can be expressed in terms of Laguerre-type exceptional orthogonal polynomials, are constructed in the framework of kth-order supersymmetric quantum mechanics, with special emphasis on k = 2. It is shown that for μ = 1, 2, and 3, there exist exactly μ distinct potentials of μth type and associated families of exceptional orthogonal polynomials, where μ denotes the degree of the polynomial gμ arising in the denominator of the potentials.

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Natanson, Gregory. "Quantization of rationally deformed Morse potentials by Wronskian transforms of Romanovski-Bessel polynomials." Acta Polytechnica 62, no. 1 (February 28, 2022): 100–117. http://dx.doi.org/10.14311/ap.2022.62.0100.

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The paper advances Odake and Sasaki’s idea to re-write eigenfunctions of rationally deformed Morse potentials in terms of Wronskians of Laguerre polynomials in the reciprocal argument. It is shown that the constructed quasi-rational seed solutions of the Schrödinger equation with the Morse potential are formed by generalized Bessel polynomials with degree-independent indexes. As a new achievement we can point to the construction of the Darboux-Crum net of isospectral rational potentials using Wronskians of generalized Bessel polynomials with no positive zeros. One can extend this isospectral family of solvable rational potentials by including ‘juxtaposed’ pairs of Romanovski-Bessel polynomials into the aforementioned polynomial Wronskians which results in deleting the corresponding pairs of bound energy states.

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Tezuka, Hirokazu. "Confinement by polynomial potentials." Zeitschrift für Physik C Particles and Fields 65, no. 1 (March 1995): 101–4. http://dx.doi.org/10.1007/bf01571309.

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Lehr, H., and C. A. Chatzidimitriou-Dreismann. "Complex scaling of polynomial potentials." Chemical Physics Letters 201, no. 1-4 (January 1993): 278–83. http://dx.doi.org/10.1016/0009-2614(93)85071-u.

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Casahorran, J. "Solitary waves and polynomial potentials." Physics Letters A 153, no. 4-5 (March 1991): 199–203. http://dx.doi.org/10.1016/0375-9601(91)90794-9.

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QUESNE, C. "RATIONALLY-EXTENDED RADIAL OSCILLATORS AND LAGUERRE EXCEPTIONAL ORTHOGONAL POLYNOMIALS IN kTH-ORDER SUSYQM." International Journal of Modern Physics A 26, no. 32 (December 30, 2011): 5337–47. http://dx.doi.org/10.1142/s0217751x11054942.

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A previous study of exactly solvable rationally-extended radial oscillator potentials and corresponding Laguerre exceptional orthogonal polynomials carried out in second-order supersymmetric quantum mechanics is extended to kth-order one. The polynomial appearing in the potential denominator and its degree are determined. The first-order differential relations allowing one to obtain the associated exceptional orthogonal polynomials from those arising in a (k-1)th-order analysis are established. Some nontrivial identities connecting products of Laguerre polynomials are derived from shape invariance.

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Brandon, David, Nasser Saad, and Shi-Hai Dong. "On some polynomial potentials ind-dimensions." Journal of Mathematical Physics 54, no. 8 (August 2013): 082106. http://dx.doi.org/10.1063/1.4817857.

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Vigo-Aguiar, M. I., M. E. Sansaturio, and J. M. Ferrándiz. "Integrability of Hamiltonians with polynomial potentials." Journal of Computational and Applied Mathematics 158, no. 1 (September 2003): 213–24. http://dx.doi.org/10.1016/s0377-0427(03)00467-9.

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Дисертації з теми "Polynomial potentials"

Bridle, Ismail Hamzaan. "Non-polynomial scalar field potentials in the local potential approximation." Thesis, University of Southampton, 2017. https://eprints.soton.ac.uk/410270/.

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We present the renormalisation group analysis of O(N) invariant scalar field theory in the local potential approximation. Linearising around the Gaussian fixed point, we find the same eigenoperators solutions exist for both the Wilsonian and the Legendre effective actions, given by solutions to Kummer’s equations. We find the usual polynomial eigenoperators and the Hilbert space they define are a natural subset of these solutions given by a specific set of quantised eigenvalues. Allowing for continuous eigenvalues, we find non-polynomial eigenoperator solutions, the so called Halpern-Huang directions, that exist outside of the polynomial Hilbert space due to the exponential field dependence. Carefully analysing the large field behaviour shows that the exponential dependence implies the Legendre effective action does not have a well defined continuum limit. In comparison, flowing towards the infrared we find that the non-polynomial eigenoperators flow into the polynomial Hilbert space. These conclusions are based off RG flow initiated at an arbitrary scale, implying non-polynomial eigenoperators are dependent upon a scale other than k. Therefore, the asymptotic field behaviour forbids self-similar scaling. These results hold when generalised from the Halpern-Huang directions around the Gaussian fixed point to a general fixed point with a general non-polynomial eigenoperator. Legendre transforming to results of the Polchinski equation, we find the flow of the Wilsonian effective action is much better regulated and always fall into the polynomial Hilbert space. For large Wilsonian effective actions, we find that the non-linear terms of the Polchinski equation forbid any non-polynomial field scaling, regardless of the fixed point. These observations lead to the conclusion that only polynomial eigenoperators show the correct, self-similar, scaling behaviour to construct a non-perturbatively renormalisable scalar QFT.

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Hyder, Asif M. "Green's operator for Hamiltonians with Coulomb plus polynomial potentials." California State University, Long Beach, 2013.

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Capraro, Patrick Leonardo [Verfasser]. "Feynman path integrals in configuration space, momentum space and phase space for perturbative and polynomial potentials / Patrick Leonardo Capraro." München : Verlag Dr. Hut, 2018. http://d-nb.info/1155058496/34.

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Capraro, Patrick [Verfasser]. "Feynman path integrals in configuration space, momentum space and phase space for perturbative and polynomial potentials / Patrick Leonardo Capraro." München : Verlag Dr. Hut, 2018. http://d-nb.info/1155058496/34.

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Hoffmann, Jan. "Types with potential: polynomial resource bounds via automatic amortized analysis." Diss., lmu, 2011. http://nbn-resolving.de/urn:nbn:de:bvb:19-139552.

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Zeriahi, Ahmed. "Fonctions plurisousharmoniques extremales, approximation et croissance des fonctions holomorphes sur des ensembles algebriques." Toulouse 3, 1986. http://www.theses.fr/1986TOU30105.

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On etudie les proprietes extremales de la "mesure capacitaire d'equilibre" associee a un "condensateur" d'un espace de stein x de dimension pure. On introduit sur un espace de stein parabolique x la notion de fonction extremale associee a un compact kcx. On generalise la theorie des fonctions extremales de siciak-zaharyuta donnant une nouvelle approche de celle-ci basee sur la theorie du potentiel complexe pour l'operateur de monge ampere complexe. On en deduit des resultats sur la theorie des fonctions a croissance controlee a l'infini. On etudie le cas d'un ensemble algebrique de c**(n). Dans ce cas, on demontre des inegalites polynomiales, une version precise d'un theoreme d'approximation de type bernstein-walsh et on en deduit le comportement asymptotique de certaines suites de polynomes orthogonaux pour la mesure d'equilibre. Pour une variete de c**(n) intersection complete, on construit des operateurs lineaires integraux pour la meilleure approximation polynomiale sur un compact avec un controle precis de l'erreur et on en deduit des resultats sur la croissance des fonctions entieres sur de telles varietes

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Alexandersson, Per. "On eigenvalues of the Schrödinger operator with a complex-valued polynomial potential." Licentiate thesis, Stockholms universitet, Matematiska institutionen, 2010. http://urn.kb.se/resolve?urn=urn:nbn:se:su:diva-52064.

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In this thesis, we generalize a recent result of A. Eremenko and A. Gabrielov on irreducibility of the spectral discriminant for the Schroedinger equation with quartic potentials. In the first paper, we consider the eigenvalue problem with a complex-valued polynomial potential of arbitrary degree d and show that the spectral determinant of this problem is connected and irreducible. In other words, every eigenvalue can be reached from any other by analytic continuation. We also prove connectedness of the parameter spaces of the potentials that admit eigenfunctions satisfying k > 2 boundary conditions, except for the case d is even and k = d/2. In the latter case, connected components of the parameter space are distinguished by the number of zeros of the eigenfunctions. In the second paper, we only consider even polynomial potentials, and show that the spectral determinant for the eigenvalue problem consists of two irreducible components. A similar result to that of paper I is proved for k boundary conditions.

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Hackl, Peter. "Optimal Design for Experiments with Potentially Failing Trials." Department of Statistics and Mathematics, WU Vienna University of Economics and Business, 1994. http://epub.wu.ac.at/68/1/document.pdf.

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We discuss the problem of optimal allocation of the design points of an experiment for the case where the trials may fail with non-zero probability. Numerical results for D-optimal designs are given for estimating the coefficients of a polynomial regression. For small sample sizes these designs may deviate substantially from the corresponding designs in the case of certain response. They can be less efficient, but are less affected by failing trials. (author's abstract)
Series: Forschungsberichte / Institut für Statistik

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Hoffmann, Jan [Verfasser], and Martin [Akademischer Betreuer] Hofmann. "Types with potential : polynomial resource bounds via automatic amortized analysis / Jan Hoffmann. Betreuer: Martin Hofmann." München : Universitätsbibliothek der Ludwig-Maximilians-Universität, 2011. http://d-nb.info/1020143665/34.

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Haese-Hill, William. "Spectral properties of integrable Schrodinger operators with singular potentials." Thesis, Loughborough University, 2015. https://dspace.lboro.ac.uk/2134/19929.

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The integrable Schrödinger operators often have a singularity on the real line, which creates problems for their spectral analysis. In several particular cases we show that all closed gaps lie on the infinite spectral arc. In the second part we develop a theory of complex exceptional orthogonal polynomials corresponding to integrable rational and trigonometric Schrödinger operators, which may have a singularity on the real line. In particular, we study the properties of the corresponding complex exceptional Hermite polynomials related to Darboux transformations of the harmonic oscillator, and exceptional Laurent orthogonal polynomials related to trigonometric monodromy-free operators.

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Книги з теми "Polynomial potentials"

Limit theorems of polynomial approximation with exponential weights. Providence, R.I: American Mathematical Society, 2008.

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Saff, E. B., Douglas Patten Hardin, Brian Z. Simanek, and D. S. Lubinsky. Modern trends in constructive function theory: Conference in honor of Ed Saff's 70th birthday : constructive functions 2014, May 26-30, 2014, Vanderbilt University, Nashville, Tennessee. Providence, Rhode Island: American Mathematical Society, 2016.

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Ibragimov, Zair. Topics in several complex variables: First USA-Uzbekistan Conference on Analysis and Mathematical Physics, May 20-23, 2014, California State University, Fullerton, California. Providence, Rhode Island: American Mathematical Society, 2016.

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Akemann, Gernot. Random matrix theory and quantum chromodynamics. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198797319.003.0005.

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This chapter was originally presented to a mixed audience of physicists and mathematicians with some basic working knowledge of random matrix theory. The first part is devoted to the solution of the chiral Gaussian unitary ensemble in the presence of characteristic polynomials, using orthogonal polynomial techniques. This includes all eigenvalue density correlation functions, smallest eigenvalue distributions, and their microscopic limit at the origin. These quantities are relevant for the description of the Dirac operator spectrum in quantum chromodynamics with three colors in four Euclidean space-time dimensions. In the second part these two theories are related based on symmetries, and the random matrix approximation is explained. In the last part recent developments are covered, including the effect of finite chemical potential and finite space-time lattice spacing, and their corresponding orthogonal polynomials. This chapter also provides some open random matrix problems.

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Olshanski, Grigori. Enumeration of maps. Edited by Gernot Akemann, Jinho Baik, and Philippe Di Francesco. Oxford University Press, 2018. http://dx.doi.org/10.1093/oxfordhb/9780198744191.013.26.

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This article discusses the relationship between random matrices and maps, i.e. graphs drawn on surfaces, with particular emphasis on the one-matrix model and how it can be used to solve a map enumeration problem. It first provides an overview of maps and related objects, recalling the basic definitions related to graphs and defining maps as graphs embedded into surfaces before considering a coding of maps by pairs of permutations. It then examines the connection between matrix integrals and maps, focusing on the Hermitian one-matrix model with a polynomial potential and how the formal expansion of its free energy around a Gaussian point (quadratic potential) can be represented by diagrams identifiable with maps. The article also illustrates how the solution of the map enumeration problem can be deduced by means of random matrix theory (RMT). Finally, it explains how the matrix model result can be translated into a bijective proof.

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Burda, Zdzislaw, and Jerzy Jurkiewicz. Phase transitions. Edited by Gernot Akemann, Jinho Baik, and Philippe Di Francesco. Oxford University Press, 2018. http://dx.doi.org/10.1093/oxfordhb/9780198744191.013.14.

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This article considers phase transitions in matrix models that are invariant under a symmetry group as well as those that occur in some matrix ensembles with preferred basis, like the Anderson transition. It first reviews the results for the simplest model with a nontrivial set of phases, the one-matrix Hermitian model with polynomial potential. It then presents a view of the several solutions of the saddle point equation. It also describes circular models and their Cayley transform to Hermitian models, along with fixed trace models. A brief overview of models with normal, chiral, Wishart, and rectangular matrices is provided. The article concludes with a discussion of the curious single-ring theorem, the successful use of multi-matrix models in describing phase transitions of classical statistical models on fluctuating two-dimensional surfaces, and the delocalization transition for the Anderson, Hatano-Nelson, and Euclidean random matrix models.

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Частини книг з теми "Polynomial potentials"

Lubinsky, Doron S., та Edward B. Saff. "Polynomial approximation of potentials". У Strong Asymptotics for Extremal Polynomials Associated with Weights on ℝ, 40–48. Berlin, Heidelberg: Springer Berlin Heidelberg, 1988. http://dx.doi.org/10.1007/bfb0082419.

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Descalzi, O., and E. Tirapegui. "Polynomial Approximations for Nonequilibrium Potentials Near Instabilities." In Instabilities and Nonequilibrium Structures II, 297–306. Dordrecht: Springer Netherlands, 1989. http://dx.doi.org/10.1007/978-94-009-2305-8_23.

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Znojil, Miloslav. "Re-construction of Polynomial Potentials with a Perturbation-Interpolation Constraint." In Lecture Notes in Physics, 458–64. Berlin, Heidelberg: Springer Berlin Heidelberg, 1994. http://dx.doi.org/10.1007/978-3-662-13969-1_29.

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Znojil, Miloslav. "Re-construction of polynomial potentials with a perturbation-interpolation constraint." In Lecture Notes in Physics, 458–64. Berlin, Heidelberg: Springer Berlin Heidelberg, 1994. http://dx.doi.org/10.1007/3-540-57576-6_29.

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Heinemann, Christian, and Christiane Kraus. "Cahn-Hilliard systems with polynomial chemical potentials coupled with damage processes and homogeneous elasticity." In Phase Separation Coupled with Damage Processes, 51–90. Wiesbaden: Springer Fachmedien Wiesbaden, 2014. http://dx.doi.org/10.1007/978-3-658-05252-2_4.

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Uwano, Yoshio. "Separability and the Birkhoff–Gustavson normalization of the perturbed harmonic oscillators with homogeneous polynomial potentials." In Superintegrability in Classical and Quantum Systems, 253–67. Providence, Rhode Island: American Mathematical Society, 2004. http://dx.doi.org/10.1090/crmp/037/22.

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Wermer, John. "Polynomial Hulls and Envelopes of Holomorphy." In Potential Theory, 339–42. Boston, MA: Springer US, 1988. http://dx.doi.org/10.1007/978-1-4613-0981-9_42.

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Mover, Sergio, Alessandro Cimatti, Alberto Griggio, Ahmed Irfan, and Stefano Tonetta. "Implicit Semi-Algebraic Abstraction for Polynomial Dynamical Systems." In Computer Aided Verification, 529–51. Cham: Springer International Publishing, 2021. http://dx.doi.org/10.1007/978-3-030-81685-8_25.

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AbstractSemi-algebraic abstraction is an approach to the safety verification problem for polynomial dynamical systems where the state space is partitioned according to the sign of a set of polynomials. Similarly to predicate abstraction for discrete systems, the number of abstract states is exponential in the number of polynomials. Hence, semi-algebraic abstraction is expensive to explicitly compute and then analyze (e.g., to prove a safety property or extract invariants).In this paper, we propose an implicit encoding of the semi-algebraic abstraction, which avoids the explicit enumeration of the abstract states: the safety verification problem for dynamical systems is reduced to a corresponding problem for infinite-state transition systems, allowing us to reuse existing model-checking tools based on Satisfiability Modulo Theory (SMT). The main challenge we solve is to express the semi-algebraic abstraction as a first-order logic formula that is linear in the number of predicates, instead of exponential, thus letting the model checker lazily explore the exponential number of abstract states with symbolic techniques. We implemented the approach and validated experimentally its potential to prove safety for polynomial dynamical systems.

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Zheng, Zhiyong, Kun Tian, and Fengxia Liu. "A Generalization of NTRUencrypt." In Financial Mathematics and Fintech, 175–88. Singapore: Springer Nature Singapore, 2022. http://dx.doi.org/10.1007/978-981-19-7644-5_7.

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AbstractNTRU cryptosystem is a new public key cryptosystem based on lattice hard problem proposed in 1996 by three digit theorists Hoffstein, Piper and Silverman of Brown University in the United States. The essence of NTRU cryptographic design is the generalization of RSA on polynomials, so it is called the cryptosystem based on polynomial rings. Its main feature is that the key generation is very simple, and the encryption and decryption algorithm is much faster than the commonly used RSA and elliptic curve cryptography. In particular, NTRU can resist quantum computing attacks and is considered to be a potential public key cryptography that can replace RSA in the post-quantum cryptography era.

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Hoffmann, Jan, and Martin Hofmann. "Amortized Resource Analysis with Polynomial Potential." In Programming Languages and Systems, 287–306. Berlin, Heidelberg: Springer Berlin Heidelberg, 2010. http://dx.doi.org/10.1007/978-3-642-11957-6_16.

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Тези доповідей конференцій з теми "Polynomial potentials"

ABENDA, S., and YU FEDOROV. "INTEGRABLE ELLIPSOIDAL BILLIARDS WITH SEPARABLE POLYNOMIAL POTENTIALS." In Proceedings of the International Conference on Differential Equations. WORLD SCIENTIFIC, 2005. http://dx.doi.org/10.1142/9789812702067_0114.

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Maiz, F., Moteb M. Alqahtani, and I. Ghnaim. "Sextic and decatic anharmonic oscillator potentials including odd power terms: Polynomial solutions." In THE SIXTH SAUDI INTERNATIONAL MEETING ON FRONTIERS OF PHYSICS 2018 (SIMFP2018). Author(s), 2018. http://dx.doi.org/10.1063/1.5042401.

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Mohankumar, K. V., and K. Kannan. "A New Approach in Kinetic Modeling Using Thermodynamic Framework for Chemically Reacting Systems and Oxidative Ageing in Polymer Composites." In ASME 2013 International Mechanical Engineering Congress and Exposition. American Society of Mechanical Engineers, 2013. http://dx.doi.org/10.1115/imece2013-64436.

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A thermodynamic framework for chemically reacting systems is put to use in kinetic modeling of any chemical system with N species undergoing M reactions. A new approach of deriving kinetic models from a Gibbs potential, of multivariate polynomial function, is demonstrated with an example of single reaction system involving three species. Also, the usual first order kinetics is deduced as a special case in the example. The distinct advantages of the new approach lies in obtaining the evolution of concentrations of species, their individual chemical potentials and the specific Gibbs potential and is demonstrated for a single reaction system as an example. Oxidation in polymer composites is studied with a coupled reaction-diffusion model obtained using first order kinetics and is solved for a boundary value problem that predicts the concentration of species over space and time. Concentration of oxidized products is correlated with modulus of aged sample and degradation effects is calculated in case of simple torsion.

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Ren, Ping, and Clément Gosselin. "Trajectory Planning of Cable-Suspended Parallel Robots Using Interval Positive-Definite Polynomials." In ASME 2012 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. American Society of Mechanical Engineers, 2012. http://dx.doi.org/10.1115/detc2012-71205.

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In this paper, the dynamic point-to-point trajectory planning of cable-suspended robots is investigated. A simple planar two-degree-of-freedom (2-dof) robot is used to demonstrate the technique. In order to maintain the cables’ positive tension, a set of algebraic inequalities is derived from the dynamic model of the 2-dof robot. The trajectories are defined using parametric polynomials with the coefficients determined by the prescribed initial and final states, and the variable time duration. With the polynomials substituted into the inequality constraints, the planning problem is then converted into an algebraic investigation on how the coefficients of the polynomials will affect the number of real roots over a given interval. An analytical approach based on a polynomial’s Discrimination Matrix and Discriminant Sequence is proposed to solve the problem. It is shown that, by adjusting the time duration within appropriate ranges, it is possible to find positive-definite polynomials such that the polynomial-based trajectories always satisfy the inequality constraints of the dynamic system. Feasible dynamic trajectories that are able to travel both beyond and within the static workspace will exploit more potential of the cable-suspended robotic platform.

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Shibata, Daisuke, and Takayuki Utsumi. "Numerical Solutions of Poisson Equation by the CIP-Basis Set Method." In ASME 2009 InterPACK Conference collocated with the ASME 2009 Summer Heat Transfer Conference and the ASME 2009 3rd International Conference on Energy Sustainability. ASMEDC, 2009. http://dx.doi.org/10.1115/interpack2009-89150.

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An accurate and reliable real space method for the ab initio calculation of electronic-structures of materials has been desired. Historically, the most popular method in this field has been the Plane Wave method. However, because the basis functions of the Plane Wave method are not local in real space, it is inefficient to represent the highly localized inner-shell electron state and it generally give rise to a large dense potential matrix which is difficult to deal with. Moreover, it is not suitable for parallel computers, because it requires Fourier transformations. These limitations of the Plane Wave method have led to the development of various real space methods including finite difference method and finite element method, and studies are still in progress. Recently, we have proposed a new numerical method, the CIP-Basis Set (CIP-BS) method [1], by generalizing the concept of the Constrained Interpolation Profile (CIP) method from the viewpoint of the basis set. This method uses a simple polynomial basis set that is easily extendable to any desired higher-order accuracy. The interpolating profile is chosen so that the sub-grid scale solution approaches the local real solution by the constraints from the spatial derivative of the original equation. Thus the solution even on the sub-grid scale becomes consistent with the master equation. By increasing the order of the polynomial, this solution quickly converges. The governing equations are unambiguously discretized into matrix form equations requiring the residuals to be orthogonal to the basis functions via the same procedure as the Galerkin method. We have already demonstrated that the method can be applied to calculations of the band structures for crystals with pseudopotentials. It has been certified that the method gives accurate solutions in the very coarse meshes and the errors converge rapidly when meshes are refined. Although, we have dealt with problems in which potentials are represented analytically, in Kohn-Sham equation the potential is obtained by solving Poisson equation, where the charge density is determined by using wave functions. In this paper, we present the CIP-BS method gives accurate solutions for Poisson equation. Therefore, we believe that the method would be a promising method for solving self-consistent eigenvalue problems in real space.

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Deshpande, Vishrut, Oliver Myers, Georges Fadel, and Suyi Li. "A New Analytical Approach for Bistable Composites." In ASME 2021 Conference on Smart Materials, Adaptive Structures and Intelligent Systems. American Society of Mechanical Engineers, 2021. http://dx.doi.org/10.1115/smasis2021-68224.

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Abstract Structures with adaptive capabilities offer many potentials to achieve future needs in efficiency, reliability, and intelligence. To this end, bistable CFRP (Carbon Fibre Reinforced Polymers) composites with asymmetric fiber layout are a promising concept that has shown shape morphing capabilities that adapt to the changes in the environment such as external forces and moments. This adaptability opens them to endless application potentials, ranging from small micro-switches to large airfoil sections in airplane wings or wind turbine blades. To harness this potential, it is essential to predict these composites’ physical shapes and behavior accurately. To this end, Hyer and Dano devised the first analytical model based on the concepts of Classical Lamination Theory, and this model has become the cornerstone of almost all subsequent studies. However, this theory uses Kirchoff’s theory of thin plates that are limited by several assumptions. As a result, Hyer’s theory can predict the overall shape of these laminates but lacks accuracy. A reason for this model’s underperformance is that it ignores the inter-laminar stresses and strains, but such stresses/strains play a vital role in the balance of the overall stress field and are found significantly higher near the free edges. To overcome these fundamental limitations, we propose a new analytical approach by combining the Reissner-Mindlin theory with concepts from the Classical Lamination Theory. This new model introduces in-plane rotations as two additional degrees of freedom. Thus, it has five independent variables compared to only three in Hyer and Dano’s model and its derivatives. Hence, we have a more complex but more accurate model. This paper outlines our new analytical approach by 1) introducing these two additional degrees of freedom; 2) selecting appropriate polynomial approximations; 3) formulating inter-laminar stresses that are functions of these added rotations; and 4) incorporating these inter-laminar stresses in the potential energy equation. By comparing this model’s prediction with the finite element simulation results, we found the new model slightly under predicts the laminate deformation, but the overall accuracy is promising, as evidenced by high R-squared correlation.

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Koelzow, Felix, Muhammad Mohsin Khan, Christian Kontermann, and Matthias Oechsner. "Application of Damage Mechanics and Polynomial Chaos Expansion for Lifetime Prediction of High-Temperature Components Under Creep-Fatigue Loading." In ASME Turbo Expo 2020: Turbomachinery Technical Conference and Exposition. American Society of Mechanical Engineers, 2020. http://dx.doi.org/10.1115/gt2020-16205.

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Abstract Several (accumulative) lifetime models were developed to assess the lifetime consumption of high-temperature components of steam and gas turbine power plants during flexible operation modes. These accumulative methods have several drawbacks, e.g. that measured loading profiles cannot be used within accumulative lifetime methods without manual corrections, and cannot be combined directly to sophisticated probabilistic methods. Although these methods are widely accepted and used for years, the accumulative lifetime prediction procedures need improvement regarding the lifetime consumption of thermal power plants during flexible operation modes. Furthermore, previous investigations show that the main influencing factor from the materials perspective, the critical damage threshold, cannot be statistically estimated from typical creep-fatigue experiments due to massive experimental effort and a low amount of available data. This paper seeks to investigate simple damage mechanics concepts applied to high-temperature components under creep-fatigue loading to demonstrate that these methods can overcome some drawbacks and use improvement potentials of traditional accumulative lifetime methods. Furthermore, damage mechanics models do not provide any reliability information, and the assessment of the resultant lifetime prediction is nearly impossible. At this point, probabilistic methods are used to quantify the missing information concerning failure probabilities and sensitivities and thus, the combination of both provides rigorous information for engineering judgment. Nearly 50 low cycle fatigue experiments of a high chromium cast steel, including dwell times and service-type cycles, are used to investigate the model properties of a simple damage evolution equation using the strain equivalence hypothesis. Furthermore, different temperatures from 300 °C to 625 °C and different strain ranges from 0.35% to 2% were applied during the experiments. The determination of the specimen stiffness allows a quantification of the damage evolution during the experiment. The model parameters are determined by Nelder-Mead optimization procedure, and the dependencies of the model parameters concerning to different temperatures and strain ranges are investigated. In this paper, polynomial chaos expansion (PCE) is used for uncertainty propagation of the model uncertainties while using non-intrusive methods (regression techniques). In a further post-processing step, the computed PCE coefficients of the damage variable are used to determine the probability of failure as a function of cycles and evolution of the probability density function (pdf). Except for the selected damage mechanics model which is considered simple, the advantages of using damage mechanics concepts combined with sophisticated probabilistic methods are presented in this paper.

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van Zutphen, Hermione J., and Joost den Haan. "Practical Implementation of the Polynomial Representation of Potential Damping in Time Domain Simulations." In ASME 2005 24th International Conference on Offshore Mechanics and Arctic Engineering. ASMEDC, 2005. http://dx.doi.org/10.1115/omae2005-67385.

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Time domain simulations are required when analyzing nonlinear vessel behaviour. The usual approach conducting time domain simulations is to transform a complex valued function of frequency dependent damping and added mass to a convolution integral in the time domain. Evaluating the integrals during time domain simulations is computational expensive and the accuracy of the calculation of the limit value of added mass in diffraction calculations is dependent on the panel size of the model. In this paper, an alternative approach based on a polynomial model for damping proposed by K.E. Kaasen et al is extended from a single degree of freedom to a 6 degrees of freedom model of a heavy lift barge. Polynomials for contributions of velocity to the damping force are constructed generically using a least square curve fitting method. The polynomials then are transformed to the time domain counterpart using a state space representation. The quality of the fits of the damping function has a large influence on the resulting damping force in time domain. Furthermore, the higher the order of the differential equation, the larger the number of variables to integrate during a time domain simulation. Consequently, the presented method is not necessarily more efficient in simulations than the traditional retardation functions.

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Lal, Mayank, Suhada Jayasuriya, and Swaminathan Sethuraman. "Motion Planning of a Group of Agents Using the Homotopy Approach." In ASME 2007 International Mechanical Engineering Congress and Exposition. ASMEDC, 2007. http://dx.doi.org/10.1115/imece2007-42677.

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In this paper motion planning of a group of agents is done to move the group from an initial configuration to a final configuration through obstacles in 2-D. Also we introduce a new homotopy approach which uses potential fields to find paths in polynomial space. We use the homotopy approach for changing the group shape of the mobile agents and at the same time treat the group as a single agent by finding a bounding disc for it to plan the motion of the group through obstacles. A time varying polynomial is constructed, the roots of which represent the current positions of the mobile agents in a frame attached to the bounding disc. The real and imaginary parts of the roots of this polynomial represent the x and y coordinates of the mobile agents in this frame. This polynomial is constructed such that it avoids the discriminant variety or the set of polynomials having multiple roots. This is equivalent to saying that the mobile agents do not collide with each other at all times. The bounding disc is then used to plan the motion of the agents through obstacles such that the group avoids the obstacles at all times.

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Maignan, Aude, and Tony Scott. "Quantum Clustering Analysis: Minima of the Potential Energy Function." In 9th International Conference on Signal, Image Processing and Pattern Recognition (SPPR 2020). AIRCC Publishing Corporation, 2020. http://dx.doi.org/10.5121/csit.2020.101914.

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Quantum clustering (QC), is a data clustering algorithm based on quantum mechanics which is accomplished by substituting each point in a given dataset with a Gaussian. The width of the Gaussian is a 𝜎 value, a hyper-parameter which can be manually defined and manipulated to suit the application. Numerical methods are used to find all the minima of the quantum potential as they correspond to cluster centers. Herein, we investigate the mathematical task of expressing and finding all the roots of the exponential polynomial corresponding to the minima of a two-dimensional quantum potential. This is an outstanding task because normally such expressions are impossible to solve analytically. However, we prove that if the points are all included in a square region of size 𝜎, there is only one minimum. This bound is not only useful in the number of solutions to look for, by numerical means, it allows to to propose a new numerical approach “per block”. This technique decreases the number of particles (or samples) by approximating some groups of particles to weighted particles. These findings are not only useful to the quantum clustering problem but also for the exponential polynomials encountered in quantum chemistry, Solid-state Physics and other applications.

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