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Journal articles on the topic 'Algebraic singularity'

Author: Grafiati

Published: 2 June 2025

Last updated: 31 July 2025

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1

Lafortune, S., and A. Goriely. "Singularity confinement and algebraic integrability." Journal of Mathematical Physics 45, no. 3 (2004): 1191–208. http://dx.doi.org/10.1063/1.1640797.

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2

Hamid Yasser Al-Yassery, Dr Kamal, and Zahraa Hameed Ali Al-Kafaji. "Perturbed Taylor expansion for bifurcation of solution of singularly parameterized perturbed ordinary differentia equations and differential algebraic equations." Journal of Education for Pure Science- University of Thi-Qar 10, no. 2 (2021): 219–34. http://dx.doi.org/10.32792/jeps.v10i2.80.

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In This paper deals with the study of singularity perturbed ordinary differential equation, and is considered the basis for obtaining the system of differential algebraic equations. In this study the we use implicit function theorem to solve for fast variable y to get a reduced model in terms of slow dynamics locally around x. It is well known that solving nonlinear algebraic equations analytical is quite difficult and numerical solution methods also face many uncertainties since nonlinear algebraic equations may have many solutions, especially around bifurcation points. We have used singularl

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3

Xu, Chenyang. "Finiteness of algebraic fundamental groups." Compositio Mathematica 150, no. 3 (2014): 409–14. http://dx.doi.org/10.1112/s0010437x13007562.

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4

Fan, Jian Hui, Bin Li, and Xin Hua Zhao. "Kinematics and Singularity Analyses of a 2-RPU&SPR Parallel Manipulator." Applied Mechanics and Materials 441 (December 2013): 568–71. http://dx.doi.org/10.4028/www.scientific.net/amm.441.568.

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In this paper, kinematics and singularity of a 2-RPU&SPR parallel mechanism are analyzed by algebraic method. Firstly, the inverse kinematics of the parallel mechanism is derived. Secondly, the Jacobian matrix of the parallel mechanism is obtained and the singularity of the mechanism is analyzed. Finally, the correctness of singularity analysis of the mechanism is verified by numerical simulations.

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5

Polishchuk, Alexander, and Arkady Vaintrob. "Matrix factorizations and cohomological field theories." Journal für die reine und angewandte Mathematik (Crelles Journal) 2016, no. 714 (2016): 1–122. http://dx.doi.org/10.1515/crelle-2014-0024.

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6

Amari, Shun-ichi, and Hiroyuki Nakahara. "Difficulty of Singularity in Population Coding." Neural Computation 17, no. 4 (2005): 839–58. http://dx.doi.org/10.1162/0899766053429426.

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Fisher information has been used to analyze the accuracy of neural population coding. This works well when the Fisher information does not degenerate, but when two stimuli are presented to a population of neurons, a singular structure emerges by their mutual interactions. In this case, the Fisher information matrix degenerates, and the regularity condition ensuring the Cramér-Rao paradigm of statistics is violated. An animal shows pathological behavior in such a situation. We present a novel method of statistical analysis to understand information in population coding in which algebraic singul

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7

Yasir, Kamal H., and Zahraa A. Mutar. "Bifurcation in Singularity Parameterized ODEs." JOURNAL OF ADVANCES IN MATHEMATICS 12, no. 1 (2016): 5808–16. http://dx.doi.org/10.24297/jam.v12i1.592.

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In this paper we will study differential algebraic equations (DAEs) through studying singularly perturbed ODEs. That's the ODEs will be transformed to an DAEs when the perturbed parameter approach to 0. This will permit us to apply the classicalbifurcation theory of ODEs for the new system (DAEs). So we will show by giving theorems, sufficient conditions for fold, pitchfork and transcritical bifurcation to be occurred in (DAEs). An illustrative example is given.

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8

Mourtada, Hussein. "Hilbert meets Ramanujan: Singularity theory and integer partitions." Bulletin of the American Mathematical Society 62, no. 1 (2024): 93–111. https://doi.org/10.1090/bull/1854.

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9

Raptis, Ioannis. "Finitary-Algebraic ‘Resolution’ of the Inner Schwarzschild Singularity." International Journal of Theoretical Physics 45, no. 1 (2006): 79–128. http://dx.doi.org/10.1007/s10773-005-9011-1.

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10

Cutkosky, S. Dale, and Hema Srinivasan. "The Algebraic Fundamental Group of a Curve Singularity." Journal of Algebra 230, no. 1 (2000): 101–26. http://dx.doi.org/10.1006/jabr.1999.7941.

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11

Domitrz, Wojciech, and Żaneta Trȩbska. "Symplectic T7, T8 singularities and Lagrangian tangency orders." Proceedings of the Edinburgh Mathematical Society 55, no. 3 (2012): 657–83. http://dx.doi.org/10.1017/s0013091510001124.

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AbstractWe study the local symplectic algebra of curves. We use the method of algebraic restrictions to classify symplectic T7, T8 singularities. We define discrete symplectic invariants (the Lagrangian tangency orders) and compare them with the index of isotropy. We use these invariants to distinguish symplectic singularities of classical T7 singularity. We also give the geometric description of symplectic classes of the singularity.

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12

Gazor, Majid, та Mahsa Kazemi. "Normal Form Analysis of ℤ2-Equivariant Singularities". International Journal of Bifurcation and Chaos 29, № 02 (2019): 1950015. http://dx.doi.org/10.1142/s0218127419500159.

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Singular parametric systems usually experience bifurcations when their parameters slightly vary around certain critical values, that is, surprising changes occur in their dynamics. The bifurcation analysis is important due to their applications in real world problems. Here, we provide a brief review of the mathematical concepts in the extension of our developed Maple library, Singularity, for the study of [Formula: see text]-equivariant local bifurcations. We explain how the process of this analysis is involved with algebraic objects and tools from computational algebraic geometry. Our procedu

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13

Paemurru, Erik. "Counting Divisorial Contractions with Centre a $cA_{n}$-Singularity." Publications of the Research Institute for Mathematical Sciences 60, no. 4 (2024): 699–721. http://dx.doi.org/10.4171/prims/60-4-2.

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First, we simplify the existing classification due to Kawakita and Yamamoto of 3 -dimensional divisorial contractions with centre a cA_{n} -singularity, also called a compound A_{n} singularity. Next, we describe the global algebraic divisorial contractions corresponding to a given local analytic equivalence class of divisorial contractions with centre a point. Finally, we consider divisorial contractions of discrepancy at least 2 to a fixed variety with centre a cA_{n} -singularity. We show that if there exists one such divisorial contraction, then there exist uncountably many such divisorial

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14

DU, DONGYUN, and YUN TANG. "BIFURCATION ANALYSIS OF DIFFERENTIAL-DIFFERENCE-ALGEBRAIC EQUATIONS." International Journal of Bifurcation and Chaos 14, no. 08 (2004): 2853–65. http://dx.doi.org/10.1142/s0218127404010886.

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The dynamics of differential-difference-algebraic equations is studied. The paper extends the study of the local bifurcations to transcritical and pitchfork bifurcation under certain nondegenerate conditions using Lyapunov–Schmidt reduction. Furthermore, an improved version of singularity induced bifurcation theorem is given in this paper.

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15

Mansour, Eman A. ""On Lyapunov- Schmidt Reduction to Reduce Differential Algebraic Equation "." Muthanna Journal of Pure Science 4, no. 2 (2017): 20–22. http://dx.doi.org/10.52113/2/04.02.2017/20-22.

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"In this paper , we transformation singularity differential – Algebraic equation to an ordinary differential equations by use Lyapunov– Schmidt reduction and constructing Lyapunov functions depending on Reiss and Geiss method."

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16

Stefan Carstea, Adrian. "Singularity analysis and bilinear approach to some Bogoyavlenski-type differential-difference equations and tau-functions." Journal of Physics: Conference Series 3002, no. 1 (2025): 012011. https://doi.org/10.1088/1742-6596/3002/1/012011.

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Abstract We review the results published in [6] about singularity structure and bilinear formalism for Bogoyavlesnki differential-difference systems. Also in this paper a new Bogoyavlenski system is proposed which is showed to have multi-soliton solution. The connection between strictly confined singularity patterns and different representations in terms of tau-functions are discussed. Finally comments on application of the “express method” for computing the algebraic entropy are given.

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17

Harte, Robin. "Variations on a theme of Kaplansky." Filomat, no. 16 (2002): 19–30. http://dx.doi.org/10.2298/fil0216019h.

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18

Darvas, Tamás, and Mingchen Xia. "The closures of test configurations and algebraic singularity types." Advances in Mathematics 397 (March 2022): 108198. http://dx.doi.org/10.1016/j.aim.2022.108198.

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19

Eshkuvatov, Z. K., N. M. A. Nik Long, Habshah Midi, and Abduvali Khaldjigitov. "Polynomial Spline Approach for Double Integrals with Algebraic Singularity." Journal of Physics: Conference Series 435 (April 26, 2013): 012009. http://dx.doi.org/10.1088/1742-6596/435/1/012009.

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20

Takenawa, Tomoyuki. "A geometric approach to singularity confinement and algebraic entropy." Journal of Physics A: Mathematical and General 34, no. 10 (2001): L95—L102. http://dx.doi.org/10.1088/0305-4470/34/10/103.

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21

Bruno, A. D., and A. B. Batkhin. "Resolution of an algebraic singularity by power geometry algorithms." Programming and Computer Software 38, no. 2 (2012): 57–72. http://dx.doi.org/10.1134/s036176881202003x.

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22

Schmale, Wiland. "Algebraic condition for the singularity of certain Toeplitz pencils." Linear Algebra and its Applications 533 (November 2017): 386–96. http://dx.doi.org/10.1016/j.laa.2017.07.028.

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23

Jiang, Guangfeng. "A formula for the Euler characteristic of line singularities on singular spaces." Bulletin of the Australian Mathematical Society 61, no. 2 (2000): 335–43. http://dx.doi.org/10.1017/s0004972700022346.

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We prove an algebraic formula for the Euler characteristic of the Milnor fibres of functions with critical locus a smooth curve on a space which is a weighted homogeneous complete intersection with isolated singularity.

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24

KLEINSCHMIDT, AXEL, and HERMANN NICOLAI. "ARITHMETIC QUANTUM GRAVITY." International Journal of Modern Physics D 19, no. 14 (2010): 2305–10. http://dx.doi.org/10.1142/s0218271810018311.

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The arithmetic chaos of classical (super)gravity near a spacelike singularity is elevated to the quantum level via the construction of a cosmological quantum billiard system. Its precise formulation, together with its underlying algebraic structure, allows for a general analysis of the wavefunction of the universe near the singularity. We argue that the extension of these results beyond the billiard approximation may provide a concrete mechanism for emergent space as well as new perspectives on several long-standing issues in canonical quantum gravity. The exponentially growing complexity of t

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25

Dzhalladova, Irada Agayevna, and Oleg Yevhenovych Kaminsky. "MATHEMATICAL MODELING AND ANALYSIS OF CONTROLLABILITY OF SINGULAR HYBRID SYSTEMS IN APPLIED PROBLEMS." Bulletin of the National Technical University "KhPI". Series: Mathematical modeling in engineering and technologies, no. 1(8) (July 23, 2025): 61–66. https://doi.org/10.20998/2222-0631.2025.01(8).07.

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The article investigates singular hybrid systems (SHS), which combine differential-algebraic structures with hybrid dynamics involving mode switching. A mathematical model is presented, approaches to stability and controllability analysis are substantiated, and numerical simulation is performed. Such systems naturally arise in complex engineering applications – from energy systems to biomedical devices – where the system behavior changes due to faults, mode transitions, or internal structural variations. A mathematical model of SHS is proposed based on matrices, which depend on the discrete mo

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26

Hassan Kubba, Adel Ahmed, and Nasikh Mohamed Elfaki Ahmed. "Efficient Jacobi Collocation Method for Nonlinear Singular Fractional Integro-Differential Equations." Journal of Research in Humanities and Social Science 13, no. 4 (2025): 131–39. https://doi.org/10.35629/9467-1304131139.

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This paper presents an efficient numerical approach for solving nonlinear weakly singular fractional integrodifferential equations. We develop a Jacobi collocation method that effectively handles the singularity while maintaining high accuracy. The proposed technique transforms the original problem into a system of algebraic equations through careful discretization. We provide rigorous error analysis demonstrating the method's spectral convergence. Numerical examples confirm the theoretical results and showcase the algorithm's superior performance compared to existing methods, particularly in

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27

Polishchuk, Alexander. "Homogeneity of cohomology classes associated with Koszul matrix factorizations." Compositio Mathematica 152, no. 10 (2016): 2071–112. http://dx.doi.org/10.1112/s0010437x16007557.

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In this work we prove the so-called dimension property for the cohomological field theory associated with a homogeneous polynomial $W$ with an isolated singularity, in the algebraic framework of [A. Polishchuk and A. Vaintrob, Matrix factorizations and cohomological field theories, J. Reine Angew. Math. 714 (2016), 1–122]. This amounts to showing that some cohomology classes on the Deligne–Mumford moduli spaces of stable curves, constructed using Fourier–Mukai-type functors associated with matrix factorizations, live in prescribed dimension. The proof is based on a homogeneity result establish

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28

Magalhães, Renan B., Gabriel P. Ribeiro, Haroldo C. D. Lima Junior, Gonzalo J. Olmo, and Luís C. B. Crispino. "Singular space-times with bounded algebraic curvature scalars." Journal of Cosmology and Astroparticle Physics 2024, no. 05 (2024): 114. http://dx.doi.org/10.1088/1475-7516/2024/05/114.

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Abstract We show that the absence of unbounded algebraic curvature invariants constructed from polynomials of the Riemann tensor cannot guarantee the absence of strong singularities. As a consequence, it is not sufficient to rely solely on the analysis of such scalars to assess the regularity of a given space-time. This conclusion follows from the analysis of incomplete geodesics within the internal region of asymmetric wormholes supported by scalar matter which arise in two distinct metric-affine gravity theories. These wormholes have bounded algebraic curvature scalars everywhere, which high

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29

SORET, MARC, та MARINA VILLE. "SINGULARITY KNOTS OF MINIMAL SURFACES IN ℝ4". Journal of Knot Theory and Its Ramifications 20, № 04 (2011): 513–46. http://dx.doi.org/10.1142/s0218216511009406.

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We study knots in 𝕊3 obtained by the intersection of a minimal surface in ℝ4 with a small 3-sphere centered at a branch point. We construct new examples of minimal knots. In particular we show the existence of non-fibered minimal knots. We show that simple minimal knots are either reversible or fully amphicheiral; this yields an obstruction for a given knot to be a simple minimal knot. Properties and invariants of these knots such as the algebraic crossing number of a braid representative and the Alexander polynomial are studied.

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30

Hauser, Herwig, and Gerd Müller. "Algebraic singularities have maximal reductive automorphism groups." Nagoya Mathematical Journal 113 (March 1989): 181–86. http://dx.doi.org/10.1017/s0027763000001343.

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Let X = On/i be an analytic singularity where ṫ is an ideal of the C-algebra On of germs of analytic functions on (Cn, 0). Let denote the maximal ideal of X and A = Aut X its group of automorphisms. An abstract subgroup equipped with the structure of an algebraic group is called algebraic subgroup of A if the natural representations of G on all “higher cotangent spaces” are rational. Let π be the representation of A on the first cotangent space and A1 = π(A).

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31

CHEN, MENG. "Canonical stability in terms of singularity index for algebraic threefolds." Mathematical Proceedings of the Cambridge Philosophical Society 131, no. 2 (2001): 241–64. http://dx.doi.org/10.1017/s030500410100531x.

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Throughout the ground field is always supposed to be algebraically closed of characteristic zero. Let X be a smooth projective threefold of general type, denote by ϕm the m-canonical map of X which is nothing but the rational map naturally associated with the complete linear system [mid ]mKX[mid ]. Since, once given such a 3-fold X, ϕm is birational whenever m [Gt ] 0, quite an interesting thing to find is the optimal bound for such an m. This bound is important because it is not only crucial to the classification theory, but also strongly related to other problems. For example, it can be appl

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32

Hasegawa, Takemitsu, and Hiroshi Sugiura. "Uniform approximation to fractional derivatives of functions of algebraic singularity." Journal of Computational and Applied Mathematics 228, no. 1 (2009): 247–53. http://dx.doi.org/10.1016/j.cam.2008.09.018.

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33

Zhang, Yi, Varun Jain, Artur Palha, and Marc Gerritsma. "The Discrete Steklov–Poincaré Operator Using Algebraic Dual Polynomials." Computational Methods in Applied Mathematics 19, no. 3 (2019): 645–61. http://dx.doi.org/10.1515/cmam-2018-0208.

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AbstractIn this paper, we will use algebraic dual polynomials to set up a discrete Steklov–Poincaré operator for the mixed formulation of the Poisson problem. The method will be applied in curvilinear coordinates and to a test problem which contains a singularity. Exponential convergence of the trace variable in {H^{1/2}}-norm will be shown.

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34

Zhang, Chunyan, Yu Wan, Dan Zhang, and Qihua Ma. "A New Mathematical Method to Study the Singularity of 3-RSR Multimode Mobile Parallel Mechanism." Mathematical Problems in Engineering 2019 (April 11, 2019): 1–11. http://dx.doi.org/10.1155/2019/1327167.

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In the process of parallel mechanism design, it is difficult to avoid the singularity, especially in the mobile parallel mechanism. Therefore, a new mathematical method to study the singularity of multimode mobile parallel mechanism is proposed. In this paper, the singularity of 3-RSR parallel mechanism (PM) is analyzed by using reciprocal screw methods and linear geometry theory from two aspects of fixed mode and all-attitude multiple motion modes. Specifically, the complete Jacobian matrix of the PM is obtained by using the screw theory, and the reciprocal screw of each branch is expressed w

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35

Basu, Saugata, and Ali Mohammad-Nezhad. "Semi-Algebraic Structure and Central Paths in Nonlinear Optimization." ACM Communications in Computer Algebra 58, no. 3 (2024): 97–100. https://doi.org/10.1145/3717582.3717593.

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We consider the concept of so-called central path in nonlinear optimization that lies at the heart of interior point methods. Using singularity theory and real algebraic geometry, we provide conditions for the existence, convergence, and analyticity of central paths of nonlinear optimization problems. We also prove quantitative bounds on the convergence rate and analyticity of the central path for the special case of semi-definite optimization.

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36

Yee, Tat Leung. "Algebraic construction of exact travelling wave solutions for the complex Gerdjikov-Ivanov equation." Advanced Studies in Theoretical Physics 19, no. 3 (2025): 129–43. https://doi.org/10.12988/astp.2025.92303.

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This study employs birational transformations to derive explicit exact solutions of the complex Gerdjikov--Ivanov (GI) equation. Through travelling wave reduction and singularity analysis, we establish the system's complete integrability and its algebraic-geometric structure via elliptic curves. We construct a comprehensive set of solutions, including elliptic functions, bright and dark solitons, and rational solutions, with detailed analysis of their physical relevance to nonlinear optics, Bose–Einstein condensates, and wave focusing phenomena. Our approach highlights the power of algebraic g

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37

Rana, Julie. "A boundary divisor in the moduli spaces of stable quintic surfaces." International Journal of Mathematics 28, no. 04 (2017): 1750021. http://dx.doi.org/10.1142/s0129167x17500215.

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We give a bound on which singularities may appear on Kollár–Shepherd-Barron–Alexeev stable surfaces for a wide range of topological invariants and use this result to describe all stable numerical quintic surfaces (KSBA-stable surfaces with [Formula: see text]) whose unique non-Du Val singularity is a Wahl singularity. We then extend the deformation theory of Horikawa to the log setting in order to describe the boundary divisor of the moduli space [Formula: see text] corresponding to these surfaces. Quintic surfaces are the simplest examples of surfaces of general type and the question of descr

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38

Ladopoulos, E. G., G. Tsamasphyros, and V. A. Zisis. "Finite-part singular integral approximations in Hilbert spaces." International Journal of Mathematics and Mathematical Sciences 2004, no. 52 (2004): 2787–93. http://dx.doi.org/10.1155/s016117120431135x.

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Some new approximation methods are proposed for the numerical evaluation of the finite-part singular integral equations defined on Hilbert spaces when their singularity consists of a homeomorphism of the integration interval, which is a unit circle, on itself. Therefore, some existence theorems are proved for the solutions of the finite-part singular integral equations, approximated by several systems of linear algebraic equations. The method is further extended for the proof of the existence of solutions for systems of finite-part singular integral equations defined on Hilbert spaces, when th

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39

Alves, L. S. de B., S. C. Hirata, M. Schuabb, and A. Barletta. "Identifying linear absolute instabilities from differential eigenvalue problems using sensitivity analysis." Journal of Fluid Mechanics 870 (May 15, 2019): 941–69. http://dx.doi.org/10.1017/jfm.2019.275.

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Identifying the convective/absolute instability nature of a local base flow requires an analysis of its linear impulse response. One must find the appropriate singularity in the eigenvalue problem with complex frequencies and wavenumbers and prove causality. One way to do so is to show that the appropriate integration contour of this response, a steepest decent path through the relevant singularity, exists. Due to the inherent difficulties of such a proof, one often verifies instead whether this singularity satisfies the collision criterion. In other words, one must show that the branches invo

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40

Sinuvasan, R., K. Krishnakumar, K. M. Tamizhmani, and PGL Leach. "FURTHER GENERALISATIONS OF THE KUMMER-SCHWARZ EQUATION: ALGEBRAIC AND SINGULARITY PROPERTIES." Acta Polytechnica 57, no. 6 (2017): 467. http://dx.doi.org/10.14311/ap.2017.57.0467.

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The Kummer–Schwarz Equation, 2<em>y'y'''</em>− 3(<em>y''</em>)<sup>2</sup> = 0, has a generalisation, (<em>n</em> − 1)<em>y</em><sup>(<em>n</em>−2)</sup><em>y</em><sup>(<em>n</em>)</sup> − <em>ny</em><sup>(<em>n</em>−1)<sup>2</sup></sup> = 0, which shares many properties with the parent form in terms of symmetry and singularity. All equations of the class are integrable in closed form. Here we introduce a new class, (<em>n

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41

Wang, Shida. "On the first singularity for the Upsilon invariant of algebraic knots." Bulletin of the London Mathematical Society 48, no. 2 (2016): 349–54. http://dx.doi.org/10.1112/blms/bdw007.

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42

Kang, Hongchao, and Shuhuang Xiang. "On the calculation of highly oscillatory integrals with an algebraic singularity." Applied Mathematics and Computation 217, no. 8 (2010): 3890–97. http://dx.doi.org/10.1016/j.amc.2010.09.050.

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43

Baricz, Árpád, and Tivadar Danka. "Zeros of Orthogonal Polynomials Near an Algebraic Singularity of the Measure." Constructive Approximation 47, no. 3 (2018): 407–35. http://dx.doi.org/10.1007/s00365-017-9411-5.

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44

Dash, Anjan Kumar, I.-Ming Chen, Song Huat Yeo, and Guilin Yang. "Instantaneous kinematics and singularity analysis of three-legged parallel manipulators." Robotica 22, no. 2 (2004): 189–203. http://dx.doi.org/10.1017/s0263574703005496.

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Instantaneous kinematics and singularity analysis of a class of three-legged, 6-DOF parallel manipulators are addressed in this paper. A generic method of derivation of reciprocal screw and consequently, the instantaneous kinematics model is presented. The advantage of this formulation is that the instantaneous kinematics model possesses well-defined geometric meaning and algebraic structure. Singularity analysis is performed under three categories, namely forward, inverse and combined singularities. A new concept of Passive Joint Plane is introduced to correlate the physical structure of the

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45

Ma, Haicheng. "The Singularity of the K4 Homeomorphic Graph." Axioms 14, no. 1 (2024): 17. https://doi.org/10.3390/axioms14010017.

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Let G be a finite simple graph and let A(G) be its adjacency matrix. Then, G is singular if A(G) is singular. The singularity of graphs is of certain interest in graph theory and algebraic combinatorics. For positive integers ai≥2, i=1,2,…,6. Insert a1−2, a2−2, a3−2, a4−2, a5−2 and a6−2 vertices in the six edges of the complete graph K4, respectively, then the resulting graph is called the K4 homeomorphic graph, denoted by K(a1,a2,a3,a4,a5,a6). In this paper, we give the necessary and sufficient condition for the singularity of K(a1,a2,a3,a4,a5,a6), and we also show that the probability of a K

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46

Drenovac, Vojin, and Veljko Potkonjak. "Contribution to the modelling of impact with robotic systems." Robotica 11, no. 5 (1993): 445–52. http://dx.doi.org/10.1017/s0263574700016994.

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SUMMARYThis paper discusses the problem of impact with robotic systems. The original method for the solution of impact is presented. The main idea is the replacement of impact with a singularity and hence the approach is called the IVSA (Impact-Via-Singularity-Analysis) Method. This goal is achieved by considering the obstacle as a unilateral constraint and introducing the new set of generalized coordinates so as to incorporate the constraint in the dynamic model. Using the IVSA Method the impact is not described by algebraic equations but by a reduced set of differential equations resulting d

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47

Aoyagi, Miki, and Kenji Nagata. "Learning Coefficient of Generalization Error in Bayesian Estimation and Vandermonde Matrix-Type Singularity." Neural Computation 24, no. 6 (2012): 1569–610. http://dx.doi.org/10.1162/neco_a_00271.

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The term algebraic statistics arises from the study of probabilistic models and techniques for statistical inference using methods from algebra and geometry (Sturmfels, 2009 ). The purpose of our study is to consider the generalization error and stochastic complexity in learning theory by using the log-canonical threshold in algebraic geometry. Such thresholds correspond to the main term of the generalization error in Bayesian estimation, which is called a learning coefficient (Watanabe, 2001a , 2001b ). The learning coefficient serves to measure the learning efficiencies in hierarchical learn

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48

GINÉ, JAUME, and JAUME LLIBRE. "ON THE CENTER CONDITIONS FOR ANALYTIC MONODROMIC DEGENERATE SINGULARITIES." International Journal of Bifurcation and Chaos 22, no. 12 (2012): 1250303. http://dx.doi.org/10.1142/s0218127412503038.

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In this paper, we present two methods for detecting centers of monodromic degenerate singularities of planar analytic vector fields. These methods use auxiliary symmetric vector fields and can be applied independently so that the singularity is algebraic solvable or not, or has characteristic directions or not. We remark that these are the first methods which allow to study monodromic degenerate singularities with characteristic directions.

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GAZOR, MAJID, and PEI YU. "INFINITE ORDER PARAMETRIC NORMAL FORM OF HOPF SINGULARITY." International Journal of Bifurcation and Chaos 18, no. 11 (2008): 3393–408. http://dx.doi.org/10.1142/s0218127408022445.

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In this paper, we introduce a suitable algebraic structure for efficient computation of the parametric normal form of Hopf singularity based on a notion of formal decompositions. Our parametric state and time spaces are respectively graded parametric Lie algebra and graded ring. As a consequence, the parametric state space is also a graded module. Parameter space is observed as an integral domain as well as a vector space, while the near-identity parameter map acts on the parametric state space. The method of multiple Lie bracket is used to obtain an infinite order parametric normal form of co

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Ohta, Y., K. M. Tamizhmani, B. Grammaticos, and A. Ramani. "Singularity confinement and algebraic entropy: the case of the discrete Painlevé equations." Physics Letters A 262, no. 2-3 (1999): 152–57. http://dx.doi.org/10.1016/s0375-9601(99)00670-2.

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