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Journal articles on the topic 'Representations of solutions'

Author: Grafiati

Published: 4 June 2021

Last updated: 17 February 2022

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1

Guidotti, Patrick. "Semiclassical fundamental solutions." Abstract and Applied Analysis 2005, no. 1 (2005): 45–57. http://dx.doi.org/10.1155/aaa.2005.45.

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It is the aim of this paper to show how the classical theory, based on fundamental solutions and explicit representations, via special functions can be combined with the functional analytical approach to partial differential equations, to produce semiclassical representation formulae for the solution of equations in cylinder-like domains.

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Sarkar, Prabir, and Amaresh Chakrabarti. "The effect of representation of triggers on design outcomes." Artificial Intelligence for Engineering Design, Analysis and Manufacturing 22, no. 2 (2008): 101–16. http://dx.doi.org/10.1017/s0890060408000073.

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AbstractCreativity of designers can be enhanced by the application of appropriate triggers. The presence of triggers helps designers to search solution spaces. The searching of a solution space increases the possibility of finding creative solutions. Both representation and content of the triggers or stimulus to which the designers are exposed are believed to play a vital role in the representation and content of the outcome of the designers during problem solving. We studied the effect of representation of triggers on ideas generated by six design engineers while trying to solve a given probl

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3

Khutoryansky, Naum, and Horacio Sosa. "Construction of Dynamic Fundamental Solutions for Piezoelectric Solids." Applied Mechanics Reviews 48, no. 11S (1995): S222—S229. http://dx.doi.org/10.1115/1.3005076.

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Fundamental solutions are derived within the framework of transient dynamic, three-dimensional piezoelectricity. The purpose of the article is to show alternate integral representations for such solutions. Thus, a representation over the unit sphere in accordance to a methodology based on the plane wave decomposition is provided. It is shown, however, that more efficient representations from a computational point of view can be achieved through appropriate coordinate transformations. Hence, representations of the fundamental solutions over surfaces of slowness are provided as novel alternative

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BECK, J. CHRISTOPHER, TOM CARCHRAE, EUGENE C. FREUDER, and GEORG RINGWELSKI. "A SPACE-EFFICIENT BACKTRACK-FREE REPRESENTATION FOR CONSTRAINT SATISFACTION PROBLEMS." International Journal on Artificial Intelligence Tools 17, no. 04 (2008): 703–30. http://dx.doi.org/10.1142/s0218213008004114.

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In this paper we present a radical approach to obtaining a backtrack-free representation for a constraint satisfaction problem: remove values that lead to dead-ends. This technique does not require additional space but has the drawback of removing solutions. We investigate a number of variations on the basic algorithm including the use of seed solutions, consistency techniques, and a variety of pruning heuristics. Our experimental results indicate that a significant proportion of the solutions to the original problem can be retained especially when an optimization algorithm that specifically s

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5

Devert, Alexandre, Thomas Weise, and Ke Tang. "A Study on Scalable Representations for Evolutionary Optimization of Ground Structures." Evolutionary Computation 20, no. 3 (2012): 453–72. http://dx.doi.org/10.1162/evco_a_00054.

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This paper presents a comparative study of two indirect solution representations, a generative and an ontogenic one, on a set of well-known 2D truss design problems. The generative representation encodes the parameters of a trusses design as a mapping from a 2D space. The ontogenic representation encodes truss design parameters as a local truss transformation iterated several times, starting from a trivial initial truss. Both representations are tested with a naive evolution strategy based optimization scheme, as well as the state of the art HyperNEAT approach. We focus both on the best object

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6

Abian, Alexander, and Sergei Sverchkov. "Solutions and representations by iterated radicals." International Journal of Mathematical Education in Science and Technology 24, no. 3 (1993): 449–55. http://dx.doi.org/10.1080/0020739930240316.

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7

MAJID, SHAHN. "SOLUTIONS OF THE YANG-BAXTER EQUATIONS FROM BRAIDED-LIE ALGEBRAS AND BRAIDED GROUPS." Journal of Knot Theory and Its Ramifications 04, no. 04 (1995): 673–97. http://dx.doi.org/10.1142/s0218216595000284.

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We obtain an R-matrix or matrix representation of the Artin braid group acting in a canonical way on the vector space of every (super)-Lie algebra or braided-Lie algebra. The same result applies for every (super)-Hopf algebra or braided-Hopf algebra. We recover some known representations such as those associated to racks. We also obtain new representations such as a non-trivial one on the ring k[x] of polynomials in one variable, regarded as a braided-line. Representations of the extended Artin braid group for braids in the complement of S1 are also obtained by the same method.

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8

BRACKEN, A. J., M. D. GOULD, and R. B. ZHANG. "QUANTUM SUPERGROUPS AND SOLUTIONS OF THE YANG-BAXTER EQUATION." Modern Physics Letters A 05, no. 11 (1990): 831–40. http://dx.doi.org/10.1142/s0217732390000925.

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A method is developed for systematically constructing trigonometric and rational solutions of the Yang-Baxter equation using the representation theory of quantum supergroups. New quantum R-matrices are obtained by applying the method to the vector representations of quantum osp (1/2) and gl (m/n).

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9

Montgomery, James, Marcus Randall, and Tim Hendtlass. "Automated Selection of Appropriate Pheromone Representations in Ant Colony Optimization." Artificial Life 11, no. 3 (2005): 269–91. http://dx.doi.org/10.1162/1064546054407149.

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Ant colony optimization (ACO) is a constructive metaheuristic that uses an analogue of ant trail pheromones to learn about good features of solutions. Critically, the pheromone representation for a particular problem is usually chosen intuitively rather than by following any systematic process. In some representations, distinct solutions appear multiple times, increasing the effective size of the search space and potentially misleading ants as to the true learned value of those solutions. In this article, we present a novel system for automatically generating appropriate pheromone representati

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10

ZHANG, JIANMING, and SHUMING LI. "BIFURCATIONS OF TRAVELING WAVE SOLUTIONS AND EXACT SOLUTIONS FOR THE GENERALIZED SCHRÖDINGER EQUATION." International Journal of Bifurcation and Chaos 21, no. 09 (2011): 2623–28. http://dx.doi.org/10.1142/s0218127411029987.

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Using the method of dynamical systems for the generalized Schrödinger equation, the bright soliton solution, dark soliton solution, uncountably infinite many periodic wave solutions and breaking bounded wave solutions are obtained. Exact explicit parametric representations of the bounded traveling solutions are given. To guarantee the existence of the above solutions, all parameter conditions are determined.

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11

Minchenko, Andrey, and Alexey Ovchinnikov. "Extensions of differential representations of SL2 and tori." Journal of the Institute of Mathematics of Jussieu 12, no. 1 (2012): 199–224. http://dx.doi.org/10.1017/s1474748012000692.

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AbstractLinear differential algebraic groups (LDAGs) measure differential algebraic dependencies among solutions of linear differential and difference equations with parameters, for which LDAGs are Galois groups. Differential representation theory is a key to developing algorithms computing these groups. In the rational representation theory of algebraic groups, one starts with ${\mathbf{SL} }_{2} $ and tori to develop the rest of the theory. In this paper, we give an explicit description of differential representations of tori and differential extensions of irreducible representation of ${\ma

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12

Millán-Martínez, Pere, and Pedro Valero-Mora. "Automating statistical diagrammatic representations with data characterization." Information Visualization 17, no. 4 (2017): 316–34. http://dx.doi.org/10.1177/1473871617715326.

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The search for an efficient method to enhance data cognition is especially important when managing data from multidimensional databases. Open data policies have dramatically increased not only the volume of data available to the public, but also the need to automate the translation of data into efficient graphical representations. Graphic automation involves producing an algorithm that necessarily contains inputs derived from the type of data. A set of rules are then applied to combine the input variables and produce a graphical representation. Automated systems, however, fail to provide an ef

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13

Walker, Douglas C. "Schwa and /œ/ in French." Canadian Journal of Linguistics/Revue canadienne de linguistique 38, no. 1 (1993): 43–64. http://dx.doi.org/10.1017/s0008413100022295.

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Few topics in the phonological literature have inspired the amount of attention that has been lavished on the notorious “mute-e” of contemporary French. Perhaps the most perplexing difficulty for phonologists is how (or even whether) to include schwa in phonological representations. Solution of the representation problem, moreover, is necessary for a proper identification of the conditions under which schwa is deleted, maintained or inserted, and for an analysis of alternations linking schwa and other vowels. Many different solutions to the representation problem have been suggested: schwa as

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14

Haghir Chehreghani, Morteza, and Mostafa Haghir Chehreghani. "Learning representations from dendrograms." Machine Learning 109, no. 9-10 (2020): 1779–802. http://dx.doi.org/10.1007/s10994-020-05895-3.

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Abstract We propose unsupervised representation learning and feature extraction from dendrograms. The commonly used Minimax distance measures correspond to building a dendrogram with single linkage criterion, with defining specific forms of a level function and a distance function over that. Therefore, we extend this method to arbitrary dendrograms. We develop a generalized framework wherein different distance measures and representations can be inferred from different types of dendrograms, level functions and distance functions. Via an appropriate embedding, we compute a vector-based represen

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15

Marchenko, Vladimir M. "Representations of Solutions for Controlled Hybrid Systems." Journal of Automation and Information Sciences 34, no. 11 (2002): 8. http://dx.doi.org/10.1615/jautomatinfscien.v34.i11.20.

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16

Anikonov, Yu E., N. B. Ayupova, and M. V. Neshchadim. "Some representations of solutions to Blokhintsev equation." Sibirskie Elektronnye Matematicheskie Izvestiya 17 (October 21, 2020): 1697–709. http://dx.doi.org/10.33048/semi.2020.17.114.

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17

Skripko, A. A., N. V. Fyodorova, V. V. Dorofeeva, and L. N. Geller. "INSTITUTE OF MEDICAL REPRESENTATIONS: PROBLEMS AND SOLUTIONS." Pharmacy & Pharmacology 3, no. 3(10) (2015): 21. http://dx.doi.org/10.19163/2307-9266-2015-3-3(10)-21-26.

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18

Koller, Daphne, and Avi Pfeffer. "Representations and solutions for game-theoretic problems." Artificial Intelligence 94, no. 1-2 (1997): 167–215. http://dx.doi.org/10.1016/s0004-3702(97)00023-4.

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19

Sengupta, Bhaskar. "Phase-type representations for matrix-geometric solutions." Communications in Statistics. Stochastic Models 6, no. 1 (1990): 163–67. http://dx.doi.org/10.1080/15326349908807142.

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20

HOFFMANN, CHRISTOPH M., and PAMELA J. VERMEER. "ELIMINATING EXTRANEOUS SOLUTIONS IN CURVE AND SURFACE OPERATIONS." International Journal of Computational Geometry & Applications 01, no. 01 (1991): 47–66. http://dx.doi.org/10.1142/s0218195991000050.

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We study exact representations for offset curves and surfaces, for equal-distance curves and surfaces, and for fixed- and variable-radius blending surfaces. The representations are systems of nonlinear equations that define the curves and surfaces as natural projections from a higher-dimensional space into 3-space. We show that the systems derived by naively translating the geometric constraints defining the curves and surfaces can entail degeneracies that result in additional solutions that have no geometric significance. We characterize these extraneous solution points geometrically, and the

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21

Leite, Jefferson, Moiseis Cecconello, Jackellyne Leite, and R. C. Bassanezi. "On Fuzzy Solutions for Diffusion Equation." Journal of Applied Mathematics 2015 (2015): 1–10. http://dx.doi.org/10.1155/2015/874931.

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Our main goal is to define a fuzzy solution for problems involving diffusion. To this end, the solution of fuzzy diffusion-reaction-advection equation will be defined as Zadeh’s extension of deterministic solution of the associated problem. Important aspects such as unity and stability of these solutions will also be studied. Graphical representations of these solutions will be presented.

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22

Ren, Yudong. "Solving Integral Representations Problems for the Stationary Schrödinger Equation." Abstract and Applied Analysis 2013 (2013): 1–5. http://dx.doi.org/10.1155/2013/715252.

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When solutions of the stationary Schrödinger equation in a half-space belong to the weighted Lebesgue classes, we give integral representations of them, which imply known representation theorems of classical harmonic functions in a half-space.

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23

Piltner, Reinhard. "Overview about Solution Representations for Elasticity Problems and Some Selected Particular Solutions." Mathematics and Mechanics of Solids 6, no. 2 (2001): 193–220. http://dx.doi.org/10.1177/108128650100600205.

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24

Horwitz, L. P., and R. I. Arshansky. "Symmetry of the Relativistic Two-Body Bound State." Symmetry 12, no. 2 (2020): 313. http://dx.doi.org/10.3390/sym12020313.

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We show that in a relativistically covariant formulation of the two-body problem, the bound state spectrum is in agreement, up to relativistic corrections, with the non-relativistic bound-state spectrum. This solution is achieved by solving the problem with support of the wave functions in an O ( 2 , 1 ) invariant submanifold of the Minkowski spacetime. The O ( 3 , 1 ) invariance of the differential equation requires, however, that the solutions provide a representation of O ( 3 , 1 ) . Such solutions are obtained by means of the method of induced representations, providing a basic insight int

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25

GOMES, CARLA P. "Artificial intelligence and operations research: challenges and opportunities in planning and scheduling." Knowledge Engineering Review 15, no. 1 (2000): 1–10. http://dx.doi.org/10.1017/s0269888900001090.

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Both the Artificial Intelligence (AI) and the Operations Research (OR) communities are interested in developing techniques for solving hard combinatorial problems, in particular in the domain of planning and scheduling. AI approaches encompass a rich collection of knowledge representation formalisms for dealing with a wide variety of real-world problems. Some examples are constraint programming representations, logical formalisms, declarative and functional programming languages such as Prolog and Lisp, Bayesian models, rule-based formalism, etc. The downside of such rich representations is th

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26

Matsuura, Tsutomu, and Saburou Saitoh. "Integral and Direct Representations of Nonlinear Inverse Mapping." Applied Mechanics and Materials 36 (October 2010): 476–84. http://dx.doi.org/10.4028/www.scientific.net/amm.36.476.

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In this paper we shall give practical, numerical and explicit representations of inverse mappings of n-dimensional mappings (of the solutions of n-nonlinear simultaneous equations) and show their numerical experiments by using computers. We derive those concrete formulas from very general ideas for the representation of the inverse functions.

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27

LECHTENFELD, OLAF, and STUART SAMUEL. "THE SOLUTION OF THE d-DIMENSIONAL TWISTED GROUP LATTICES." International Journal of Modern Physics A 11, no. 18 (1996): 3227–46. http://dx.doi.org/10.1142/s0217751x96001541.

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The general d-dimensional twisted group lattice is solved. The irreducible representations of the corresponding group are constructed by an explicit procedure. It is proven that they are complete. All matrix representation solutions to the quantum hyperplane equations are obtained.

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28

Чепок, О. О. "Asymptotic representations of solutions with slowly varying derivatives." Researches in Mathematics and Mechanics 23, no. 2(32) (2018): 110–20. http://dx.doi.org/10.18524/2519-206x.2018.2(32).149708.

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29

Doshi, Prashant, Yifeng Zeng, and Qiongyu Chen. "Graphical models for interactive POMDPs: representations and solutions." Autonomous Agents and Multi-Agent Systems 18, no. 3 (2008): 376–416. http://dx.doi.org/10.1007/s10458-008-9064-7.

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30

Rajeev, B., and S. Thangavelu. "Probabilistic Representations of Solutions of the Forward Equations." Potential Analysis 28, no. 2 (2008): 139–62. http://dx.doi.org/10.1007/s11118-007-9074-0.

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31

Rajeev, B., and S. Thangavelu. "Probabilistic representations of solutions to the heat equation." Proceedings Mathematical Sciences 113, no. 3 (2003): 321–32. http://dx.doi.org/10.1007/bf02829609.

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32

Wu, Da-jin, and Li Cao. "Relations between different representations of stochastic adiabatic solutions." Zeitschrift f�r Physik B Condensed Matter 81, no. 3 (1990): 451–55. http://dx.doi.org/10.1007/bf01390828.

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33

Kurmanbayev, D., and K. Yesmakhanova. "SOLITON DEFORMATION OF INVERTED CATENOID." PHYSICO-MATHEMATICAL SERIES 2, no. 336 (2021): 24–32. http://dx.doi.org/10.32014/2021.2518-1726.17.

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The minimal surface (see [1]) is determined using the Weierstrass representation in three-dimensional space. The solution of the Dirac equation [2] in terms of spinors coincides with the representations of this surface with conservation of isothermal coordinates. The equation represented through the Dirac operator, which is included in the Manakov’s L, A, B triple [3] as equivalent to the modified Veselov-Novikov equation (mVN) [4]. The potential 𝑈 of the Dirac operator is the potential of representing a minimal surface. New solutions of the mVN equation are constructed using the pre-known pot

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34

Kyrchei, Ivan I. "Determinantal Representations of Solutions and Hermitian Solutions to Some System of Two-Sided Quaternion Matrix Equations." Journal of Mathematics 2018 (November 1, 2018): 1–12. http://dx.doi.org/10.1155/2018/6294672.

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Within the framework of the theory of quaternion row-column determinants previously introduced by the author, we derive determinantal representations (analogs of Cramer’s rule) of solutions and Hermitian solutions to the system of two-sided quaternion matrix equations A1XA1⁎=C1 and A2XA2⁎=C2. Since the Moore-Penrose inverse is a necessary tool to solve matrix equations, we use determinantal representations of the Moore-Penrose inverse previously obtained by the theory of row-column determinants.

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35

LEE, H. C., M. L. GE, M. COUTURE, and Y. S. WU. "STRANGE STATISTICS, BRAID GROUP REPRESENTATIONS AND MULTIPOINT FUNCTIONS IN THE N-COMPONENT MODEL." International Journal of Modern Physics A 04, no. 09 (1989): 2333–70. http://dx.doi.org/10.1142/s0217751x89000947.

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The statistics of fields in low dimensions is studied from the point of view of the braid group Bn of n strings. Explicit representations MR for the N-component model, N=2 to 5, are derived by solving the Yang-Baxter-like braid group relations for the statistical matrix R, which describes the transformation of the bilinear product of two N-component fields under the transposition of coordinates. When R2≠1 the statistics is neither Bose-Einstein nor Fermi-Dirac; it is strange. It is shown that for each N, the N+1 parameter family of solutions obtained is the most general one under a given set o

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36

Li, Jibin. "On the Exact Traveling Wave Solutions of (2 + 1)-Dimensional Higher Order Broer–Kaup Equation." International Journal of Bifurcation and Chaos 24, no. 01 (2014): 1450007. http://dx.doi.org/10.1142/s0218127414500072.

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In this paper, we study the dynamical behavior and exact parametric representations of all traveling wave solutions for (2 + 1)-dimensional higher order Broer–Kaup equation. By using the method of dynamical systems, under different parametric conditions, for the solution component U, exact monotonic and nonmonotonic kink wave solutions, two-peak wave solutions, periodic wave solutions, as well as unbounded traveling wave solutions are obtained. Exact wave profiles of traveling wave solutions for all solution components U, V, W, P are shown.

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37

Ruchti, Wendy P., and Cory A. Bennett. "Develop Reasoning through Pictorial Representations." Mathematics Teaching in the Middle School 19, no. 1 (2013): 30–36. http://dx.doi.org/10.5951/mathteacmiddscho.19.1.0030.

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38

Cutting, James E. "Representing Motion in a Static Image: Constraints and Parallels in Art, Science, and Popular Culture." Perception 31, no. 10 (2002): 1165–93. http://dx.doi.org/10.1068/p3318.

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Representing motion in a picture is a challenge to artists, scientists, and all other imagemakers. Moreover, it presents a problem that will not go away with electronic and digital media, because often the pedagogical purpose of the representation of motion is more important than the motion itself. All satisfactory solutions evoke motion—for example, dynamic balance (or broken symmetry), stroboscopic sequences, affine shear (or forward lean), and photographic blur—but they also typically sacrifice the accuracy of the motion represented, a solution often unsuitable for science. Vector represent

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39

Malinowski, Marek T. "On Existence Theorems to Symmetric Functional Set-Valued Differential Equations." Symmetry 13, no. 7 (2021): 1219. http://dx.doi.org/10.3390/sym13071219.

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In this paper, we consider functional set-valued differential equations in their integral representations that possess integrals symmetrically on both sides of the equations. The solutions have values that are the nonempty compact and convex subsets. The main results contain a Peano type theorem on the existence of the solution and a Picard type theorem on the existence and uniqueness of the solution to such equations. The proofs are based on sequences of approximations that are constructed with appropriate Hukuhara differences of sets. An estimate of the magnitude of the solution’s values is

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40

Goto, Yoshiaki, and Keiji Matsumoto. "The monodromy representation and twisted period relations for Appell’s hypergeometric function F 4." Nagoya Mathematical Journal 217 (March 2015): 61–94. http://dx.doi.org/10.1017/s0027763000026957.

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AbstractWe consider the systemF4(a, b, c)of differential equations annihilating Appell's hypergeometric seriesF4(a,b,c;x). We find the integral representations for four linearly independent solutions expressed by the hypergeometric seriesF4. By using the intersection forms of twisted (co)homology groups associated with them, we provide the monodromy representation ofF4(a, b, c)and the twisted period relations for the fundamental systems of solutions ofF4.

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Paris, R. B., and A. D. Wood. "Results old and new on the hyper-Bessel equation." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 106, no. 3-4 (1987): 259–65. http://dx.doi.org/10.1017/s0308210500018400.

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SynopsisWe consider a variety of integral representations, single and multiple, old and new, for solutions of the hyper-Bessel equation u(n) – zmu =0. In particular, we show how a very early multiple Laplace integral solution of Molins (1876) may be related to recent Mellin–Barnes integral representations given by the present authors by way of multiple integral solutions given by Saxton and the second author for an associated equation. Although both these multiple integral solutions may be found by elementary methods, it is not easy to find their asymptotic expansions for large z, and we show

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42

Kyrchei, Ivan I. "Determinantal Representations of the Core Inverse and Its Generalizations with Applications." Journal of Mathematics 2019 (October 1, 2019): 1–13. http://dx.doi.org/10.1155/2019/1631979.

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In this paper, we give the direct method to find of the core inverse and its generalizations that is based on their determinantal representations. New determinantal representations of the right and left core inverses, the right and left core-EP inverses, and the DMP, MPD, and CMP inverses are derived by using determinantal representations of the Moore-Penrose and Drazin inverses previously obtained by the author. Since the Bott-Duffin inverse has close relation with the core inverse, we give its determinantal representation and its application in finding solutions of the constrained linear equ

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43

Jauhari, Farid, Wayan Firdaus Mahmudy, and Achmad Basuki. "An Effective Chromosome Representation on Proportional Tuition Fees Assessment Using NSGA-II." Journal of Information Technology and Computer Science 4, no. 3 (2019): 291. http://dx.doi.org/10.25126/jitecs.201943161.

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Proportional tuition fees assessment is an optimization process to find a compromise point between student willingness to pay and institution income. Using a genetic algorithm to find optimal solutions requires effective chromosome representations, parameters, and operator genetic to obtain efficient search. This paper proposes a new chromosome representation and also finding efficient genetic parameters to solve the proportional tuition fees assessment problem. The results of applying the new chromosome representation are compared with another chromosome representation in the previous study.

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44

Privault, Nicolas. "Extended Mellin integral representations for the absolute value of the gamma function." Analysis 38, no. 1 (2018): 11–20. http://dx.doi.org/10.1515/anly-2017-0046.

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AbstractWe derive Mellin integral representations in terms of Macdonald functions for the squared absolute value{s\mapsto|\Gamma(a+is)|^{2}}of the gamma function and its Fourier transform when{a<0}is non-integer, generalizing known results in the case{a>0}. This representation is based on a renormalization argument using modified Bessel functions of the second kind, and it applies to the representation of the solutions of a Fokker–Planck equation.

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45

Vu, Hung, Tu Dinh Nguyen, Trung Le, Wei Luo, and Dinh Phung. "Robust Anomaly Detection in Videos Using Multilevel Representations." Proceedings of the AAAI Conference on Artificial Intelligence 33 (July 17, 2019): 5216–23. http://dx.doi.org/10.1609/aaai.v33i01.33015216.

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Detecting anomalies in surveillance videos has long been an important but unsolved problem. In particular, many existing solutions are overly sensitive to (often ephemeral) visual artifacts in the raw video data, resulting in false positives and fragmented detection regions. To overcome such sensitivity and to capture true anomalies with semantic significance, one natural idea is to seek validation from abstract representations of the videos. This paper introduces a framework of robust anomaly detection using multilevel representations of both intensity and motion data. The framework consists

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46

Fu, Y. N., and J. Laskar. "Frequency analysis and representation of slowly diffusing planetary solutions." Astronomy & Astrophysics 628 (August 2019): A84. http://dx.doi.org/10.1051/0004-6361/201730492.

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Context. Over short time-intervals, planetary ephemerides have traditionally been represented in analytical form as finite sums of periodic terms or sums of Poisson terms that are periodic terms with polynomial amplitudes. This representation is not well adapted for the evolution of planetary orbits in the solar system over million of years which present drifts in their main frequencies as a result of the chaotic nature of their dynamics. Aims. We aim to develop a numerical algorithm for slowly diffusing solutions of a perturbed integrable Hamiltonian system that will apply for the representat

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47

Villarroel, Javier. "On representations of solutions to certain stochastic differential equations." Journal of Computational Methods in Sciences and Engineering 4, no. 1-2 (2004): 97–103. http://dx.doi.org/10.3233/jcm-2004-41-212.

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Hien, Marco, and Céline Roucairol. "Integral representations for solutions of exponential Gauß-Manin systems." Bulletin de la Société mathématique de France 136, no. 4 (2008): 505–32. http://dx.doi.org/10.24033/bsmf.2564.

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Ma, Jin, and Jianfeng Zhang. "Representations and regularities for solutions to BSDEs with reflections." Stochastic Processes and their Applications 115, no. 4 (2005): 539–69. http://dx.doi.org/10.1016/j.spa.2004.05.010.

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Janušauskas, A. "Integral representations of solutions of some partial differential equations." Lithuanian Mathematical Journal 37, no. 1 (1997): 29–37. http://dx.doi.org/10.1007/bf02465437.

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