463 results
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2. On the Smoothing Property of Multigrid Methods in the Non-symmetric Case.
- Author
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Ecker, Alois and Zulehner, Walter
- Subjects
MULTIGRID methods (Numerical analysis) ,NUMERICAL analysis ,LINEAR systems ,SYSTEMS theory ,EQUATIONS ,MATHEMATICAL analysis - Abstract
In this paper we present an extension of Reusken's Lemma about the smoothing property of a multigrid method for solving non-symmetric linear systems of equations. One of the consequences of this extended lemma is the verification of the smoothing property for all damping factors ω ∈ (0,1). Additionally, a semi-iterative smoother is constructed which gives, in some sense, optimal smoothing rate estimates. [ABSTRACT FROM AUTHOR]
- Published
- 1996
- Full Text
- View/download PDF
3. ON SOLUTIONS OF THE RECURSIVE EQUATIONS xn+1 = xp n+1=xp n (p > 0) VIA FIBONACCI-TYPE SEQUENCES.
- Author
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ÖCALAN, ÖZKAN and DUMAN, OKTAY
- Subjects
INTEGERS ,REAL analysis (Mathematics) ,EQUATIONS ,FIBONACCI sequence ,NUMERICAL analysis ,MATHEMATICAL analysis - Abstract
In this paper, by using the classical Fibonacci sequence and the golden ratio, we first give the exact solution of the nonlinear recursive equation x
n +1 = xn -1/xn with respect to certain powers of the initial values x-1 and x0: Then we obtain a necessary and sufficient condition on the initial values for which the equation has a non-oscillatory solution. Later we extend our all results to the recursive equations xn+1 = xp -1=xp n (p > 0) in a similar manner. We also get a characterization for unbounded positive solutions. At the end of the paper we analyze all possible positive solutions and display some graphical illustrations verifying our results. [ABSTRACT FROM AUTHOR]n -1=xp n (p > 0) in a similar manner. We also get a characterization for unbounded positive solutions. At the end of the paper we analyze all possible positive solutions and display some graphical illustrations verifying our results. [ABSTRACT FROM AUTHOR]- Published
- 2019
4. Mutual estimates of L p -norms and the Bellman function.
- Author
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Vasyunin, V.
- Subjects
NUMERICAL analysis ,ARITHMETIC ,EQUATIONS ,INTERVAL analysis ,MATHEMATICAL analysis - Abstract
In this paper, we describe the range of the L
p -norm of a function under fixed Lp -norms with two other different exponents p and under a natural multiplicative restriction of the type of the Muckenhoupt condition. Particular cases of such results are simple inequalities as the interpolation inequality between two Lp -norms as well as such nontrivial inequalities as the Gehring inequality or the reverse Hölder inequality for Mackenhoupt weights. The basic method of our paper is the search for the exact Bellman function of the corresponding extremal problem. Bibliography: 5 [ABSTRACT FROM AUTHOR]- Published
- 2009
- Full Text
- View/download PDF
5. On Genocchi Operational Matrix of Fractional Integration for Solving Fractional Differential Equations.
- Author
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Abdulnasir Isah and Chang Phang
- Subjects
FRACTIONAL integrals ,MATHEMATICS ,POLYNOMIALS ,MATHEMATICAL analysis ,NUMERICAL analysis ,EQUATIONS ,ALGEBRA - Abstract
In this paper we present a new numerical method for solving fractional differential equations (FDEs) based on Genocchi polynomials operational matrix through collocation method. The operational matrix of fractional integration in Riemann-Liouville sense is derived. The upper bound for the error of the operational matrix of fractional integration is also shown. The properties of Genocchi polynomials are utilized to reduce the given problems to a system of algebraic equations. Illustrative examples are finally given to show the simplicity, accuracy and applicability of the method. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
6. Point-wise Integrated-RBF-based Discretisation of Differential Equations.
- Author
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Mai-Duy, Nam and Tran-Cong, Thanh
- Subjects
ELLIPTIC differential equations ,PARTIAL differential equations ,NUMERICAL analysis ,MATHEMATICAL analysis ,EQUATIONS - Abstract
This paper discusses a discretisation scheme which is based on point collocation and integrated radial basis function networks (IRBFNs) for the solution of elliptic differential equations (DEs). The use of IRBFNs to represent the field variable in a given DE gives several advantages over the case of using conventional RBFNs and polynomials. Some numerical examples are included for demonstration purposes. [ABSTRACT FROM AUTHOR]
- Published
- 2010
- Full Text
- View/download PDF
7. Generalized solutions for the H1 model in ABS List of lattice equations.
- Author
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Da-jun Zhang and Hietarinta, Jarmo
- Subjects
SOLITONS ,NONLINEAR theories ,EQUATIONS ,NUMERICAL analysis ,MATHEMATICAL analysis - Abstract
In this paper we discuss solutions in Casoratian form for H1, which is the simplest member in ABS list of lattice equations. By investigating the condition satisfied by the Casoratian basic column we propose a generalization, which yields solutions which are different form solitons. These solutions can be considered as limit solutions of solitons. Similar generalizations can apply to other lattice equations in ABS list, such as H2, H3 and Q1. [ABSTRACT FROM AUTHOR]
- Published
- 2010
- Full Text
- View/download PDF
8. ASYMPTOTIC BEHAVIOR OF NONOSCILLATORY SOLUTIONS OF HIGHER-ORDER INTEGRO-DYNAMIC EQUATIONS.
- Author
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Bohner, Martin, Grace, Said, and Sultana, Nasrin
- Subjects
EQUATIONS ,NUMERICAL analysis ,MATHEMATICAL analysis ,ALGEBRA ,MATHEMATICAL statistics - Abstract
In this paper, we establish some new criteria on the asymptotic behavior of nonoscillatory solutions of higher-order integro-dynamic equations on time scales. [ABSTRACT FROM AUTHOR]
- Published
- 2014
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9. Complicated asymptotic behavior exponents for solutions of the evolution p-Laplacian equation with absorption.
- Author
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Wang, Liangwei, Yin, Jngxue, and Wu, Yuqiu
- Subjects
LAPLACIAN operator ,LAPLACIAN matrices ,EQUATIONS ,NUMERICAL analysis ,MATHEMATICAL analysis - Abstract
In this paper, we investigate how the initial value belonging to spaces $W_{\sigma}(\mathbb{R}^{N})$ ( $0<\sigma
- Published
- 2017
- Full Text
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10. Geometrically Derived Ray-Theory Results and Direct Verification of the Pekeris Solution for Unbounded Constant-Gradient Media.
- Author
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Barnard, Thomas E.
- Subjects
HELMHOLTZ equation ,CALCULUS ,EQUATIONS ,SURFACE waves (Fluids) ,GEOMETRY ,MATHEMATICAL analysis ,NUMERICAL analysis - Abstract
This paper directly verifies the Pekeris solution to the point-source Helmholtz equation for an unbounded constant-gradient medium using only elementary vector calculus. Self-contained geometrical derivations of ray-theory results for such a medium are presented: 1) ray-path location and travel time as a function of source location, ray start angle, and ray angle; 2) the wavefront equation as a function of source location and travel time; 3) the wavefront location and ray angle along a ray as a function of source location, ray start angle, and travel time; and 4) source angle and receiver angle as a function of source location and receiver location. A short mathematical derivation gives the travel time between two points for a given source location and a given receiver location. In some cases, the form of the results seems to be simpler than that of the equivalent results previously given in the literature. [ABSTRACT FROM PUBLISHER]
- Published
- 2012
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11. Maxwellian Circuits of Conducting Circular Loops.
- Author
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Shen, Wenhui, Xue, Changwei, Mei, Kenneth K., and Lin, Jiahong
- Subjects
ELECTRONIC circuits ,ELECTRIC conductivity ,ANTENNAS (Electronics) ,DIFFERENTIAL equations ,MOMENTS method (Statistics) ,ELECTRIC lines ,SCATTERING (Physics) ,WAVE equation ,MATHEMATICAL analysis ,NUMERICAL analysis - Abstract
Circular loops are useful fundamental devices, subjects of many investigations. Before the era of the method of moments (MoM), it was difficult to analyze a circular loop because it was hard to find the current distribution on the loop. The classical investigations on circular loops involve fairly complicated mathematical analyses. In this paper, based on the theory of Maxwellian circuits (MC), it is discovered that the current of a circular loop is the solution of a classical wave equation with complex constant parameters. Once the parameters are determined, the current may be solved analytically or by simple numerical methods. Very broad band computations of circular loops up to 40 GHz have been performed. It shows that the Maxwellian circuits formulations can indeed simplify the analyses of conducting circular loops and provide valuable insights to RF engineers. [ABSTRACT FROM AUTHOR]
- Published
- 2011
- Full Text
- View/download PDF
12. On the optimal boundary control of vibrations of a spherical layer.
- Author
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Sergeev, S. A.
- Subjects
SPHERICAL functions ,NUMERICAL analysis ,MATHEMATICAL analysis ,EQUATIONS ,SPHERICAL harmonics ,TRANSCENDENTAL functions - Abstract
We seek the optimal boundary control of vibrations of a spherical layer in the spherically symmetric case. This paper continues the series of papers by V.A. Il’in and his students and, unlike the previous papers, uses a more general control optimality criterion. We obtain closed formulas for the controls; these formulas are consistent with Il’in’s earlier results. [ABSTRACT FROM AUTHOR]
- Published
- 2009
- Full Text
- View/download PDF
13. Raster cellular neural network simulator for image processing applications with numerical integration algorithms.
- Author
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Murugesh, V.
- Subjects
EVOLUTIONARY computation ,NUMERICAL analysis ,MATHEMATICAL analysis ,EQUATIONS ,IMAGE processing ,NUMERICAL integration ,ARTIFICIAL neural networks ,ARTIFICIAL intelligence ,ALGORITHMS ,ALGEBRA ,FOUNDATIONS of arithmetic - Abstract
In this paper, a universal simulator for cellular neural network (CNN) is presented. This simulator is capable of performing Raster simulation for any size of input image, and thus is a powerful tool for researchers investigating potential applications of CNN. This paper reports the latency properties of CNNs along with popular numerical integration algorithms; results and comparisons are also presented. [ABSTRACT FROM AUTHOR]
- Published
- 2009
- Full Text
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14. A class of approximate inverse preconditioners for solving linear systems.
- Author
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Zhang, Yong, Huang, Ting-Zhu, Liu, Xing-Ping, and Gu, Tong-Xiang
- Subjects
MATRICES (Mathematics) ,LINEAR systems ,MATHEMATICS ,MATHEMATICAL ability ,NUMERICAL analysis ,MATHEMATICAL analysis ,EQUATIONS ,ALGEBRA ,MATHEMATICAL combinations ,LINEAR differential equations - Abstract
Some preconditioners for accelerating the classical iterative methods are given in Zhang et al. [Y. Zhang and T.Z. Huang, A class of optimal preconditioners and their applications, Proceedings of the Seventh International Conference on Matrix Theory and Its Applications in China, 2006. Y. Zhang, T.Z. Huang, and X.P. Liu, Modified iterative methods for nonnegative matrices and M-matrices linear systems, Comput. Math. Appl. 50 (2005), pp. 1587-1602. Y. Zhang, T.Z. Huang, X.P. Liu, A class of preconditioners based on the (I+S(α))-type preconditioning matrices for solving linear systems, Appl. Math. Comp. 189 (2007), pp. 1737-1748]. Another kind of preconditioners approximating the inverse of a symmetric positive definite matrix was given in Simons and Yao [G. Simons, Y. Yao, Approximating the inverse of a symmetric positive definite matrix, Linear Algebra Appl. 281 (1998), pp. 97-103]. Zhang et al. 's preconditioners and Simons and Yao's are generalized in this paper. These preconditioners are all of low construction cost, which all could be taken as approximate inverse of M-matrices. Numerical experiments of these preconditioners applied with Krylov subspace methods show the effectiveness and performance, which also show that the preconditioners proposed in this paper are better approximate inverse for M-matrices than Simons'. [ABSTRACT FROM AUTHOR]
- Published
- 2009
- Full Text
- View/download PDF
15. Splitting cubes: a fast and robust technique for virtual cutting.
- Author
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Nico Pietroni, Fabio Ganovelli, Paolo Cignoni, and Roberto Scopigno
- Subjects
NUMERICAL analysis ,MATHEMATICAL analysis ,EQUATIONS ,INTERVAL analysis - Abstract
Abstract This paper presents the splitting cubes, a fast and robust technique for performing interactive virtual cutting on deformable objects. The technique relies on two ideas. The first one is to embed the deformable object in a regular grid, to apply the deformation function to the grid nodes and to interpolate the deformation inside each cell from its 8 nodes. The second idea is to produce a tessellation for the boundary of the object on the base of the intersections of such boundary with the edges of the grid. Please note that the boundary can be expressed in any way; for example it can be a triangle mesh, an implicit or a parametric surface. The only requirement is that the intersection between the boundary and the grid edges can be computed. This paper shows how the interpolation of the deformation inside the cells can be used to produce discontinuities in the deformation function, and the intersections of the cut surface can be used to visually show the cuts on the object. The splitting cubes is essentially a tessellation algorithm for growing, deformable surface, and it can be applied to any method for animating deformable objects. In this paper the case of the mesh-free methods (MMs) is considered: in this context, we described a practical GPU friendly method, that we named the extended visibility criterion, to introduce discontinuities of the deformation. [ABSTRACT FROM AUTHOR]
- Published
- 2009
- Full Text
- View/download PDF
16. ON THE FINE SPECTRUM OF THE GENERALIZED DIFFERENCE OPERATOR Δv OVER THE SEQUENCE SPACE C0.
- Author
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Srivastava, P. D. and Kumar, Sudhanshu
- Subjects
MATHEMATICAL analysis ,EQUATIONS ,REAL numbers ,MATHEMATICS ,NUMERICAL analysis - Abstract
The purpose of the paper is to determine fine spectrum of newly introduced operator Δ
ν on the sequence space c0 . The operator Δν on c0 is defined by Δν χ = (νn χn - νn -1χn -1)n=0 ∞ with χ-1 = 0, where ν = (νk ) is either constant or strictly decreasing sequence of positive real numbers such that lim νk = L > 0 and sup νk ≤ 2L. In this paper, it is shown that spectrum (These equations cannot be represented into ASCII text), the point spectrum σp (Δν ,c0 ) = ϕ if ν is a constant and σp (Δν ,c0 ) = {νn } if ν is a strictly decreasing sequence. We have also obtained the results on continuous spectrum σc (Δν ,c0 ), residual spectrum σr (Δν ,c0 ) and fine spectrum of the operator Δν on c0 . [ABSTRACT FROM AUTHOR]- Published
- 2009
17. A REPRESENTATION THEOREM FOR THE ERROR OF RECURSIVE ESTIMATORS.
- Author
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Gerencsér, László
- Subjects
STOCHASTIC systems ,ADAPTIVE control systems ,RECURSIVE functions ,NUMERICAL analysis ,MATHEMATICAL analysis ,EQUATIONS - Abstract
The ultimate objective of this paper is to develop new techniques that can be used for the analysis of performance degradation due to statistical uncertainty for a wide class of linear stochastic systems. For this we need new technical tools similar to those used in [L. Gerencsér, Statist. Plann. Inference, 41 (1994), pp. 303–325]. The immediate technical objective is to extend the previous technical results to the Djereveckii-Fradkov-Ljung scheme with enforced boundedness. Our starting point is a standard approximation of the estimation error used in the asymptotic theory of recursive estimation. Tight control of the difference between the estimation error and its standard approximation, referred to as residuals, is a crucial point in our applications. The main technical advance of the present paper is a set of strong approximation theorems for three closely related recursive estimation algorithms in which, for any q ≥ 1, the L
q -norms of the residual terms are shown to tend to zero with rate N-1/2-ϵ with some ϵ > 0. This is a significant extension of previous results for the recursive prediction error or RPE estimator of ARMA processes given in [L. Gerencsér, Systems Control Lett., 21 (1993), pp. 347–351]. Two useful corollaries will be derived. In the first a standard transform of the estimation-error process for the basic recursive estimation method, Algorithm CR, will be shown to be L-mixing, while in the second the asymptotic covariance matrix of the estimator for the same method will be given. Applications to multivariable adaptive prediction and the minimum-variance self-tuning regulator for ARMAX systems will be described. [ABSTRACT FROM AUTHOR]- Published
- 2006
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18. Safe and tight linear estimators for global optimization.
- Author
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Borradaile, Glencora and Van Hentenryck, Pascal
- Subjects
EQUATIONS ,ALGORITHMS ,ALGEBRA ,NUMERICAL analysis ,MATHEMATICAL analysis - Abstract
Global optimization problems are often approached by branch and bound algorithms which use linear relaxations of the nonlinear constraints computed from the current variable bounds. This paper studies how to derive safe linear relaxations to account for numerical errors arising when computing the linear coefficients. It first proposes two classes of safe linear estimators for univariate functions. Class-1 estimators generalize previously suggested estimators from quadratic to arbitrary functions, while class-2 estimators are novel. When they apply, class-2 estimators are shown to be tighter theoretically (in a certain sense) and almost always tighter numerically. The paper then generalizes these results to multivariate functions. It shows how to derive estimators for multivariate functions by combining univariate estimators derived for each variable independently. Moreover, the combination of tight class-1 safe univariate estimators is shown to be a tight class-1 safe multivariate estimator. Finally, multivariate class-2 estimators are shown to be theoretically tighter (in a certain sense) than multivariate class-1 estimators. [ABSTRACT FROM AUTHOR]
- Published
- 2005
- Full Text
- View/download PDF
19. Error Estimates for a Variable Time-Step Discretization of a Phase Transition Model with Hyperbolic Momentum.
- Author
-
Segatti, Antonio
- Subjects
NUMERICAL analysis ,EQUATIONS ,MATHEMATICAL optimization ,MATHEMATICAL analysis ,MATHEMATICS - Abstract
This paper deals with a fully implicit time discretization scheme with variable time-step for a nonlinear system modelling phase transition and mechanical deformations in shape memory alloys. The model is studied in the non-stationary case and accounts for local microscopic interactions between the phases introducing the gradients of the phase parameters. The resulting initial-boundary value problem has already been studied by the author who proved existence, uniqueness and continuous dependence on data for a suitable weak solution along some regularity results. A careful and detailed investigation of the variable time-step discretization is the goal of this paper. Thus, we deduce some estimates for the discretization error. These estimates depend only on data, impose no constraints between consecutive time-steps and show an optimal order of convergence. Finally, we prove another regularity result for the solution under stronger regularity assumptions on data. [ABSTRACT FROM AUTHOR]
- Published
- 2004
- Full Text
- View/download PDF
20. CONVERGENCE OF SUBDIVISION SCHEMES ASSOCIATED WITH NONNEGATIVE MASKS.
- Author
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Jia, Rong-Qing and Zhou, Ding-Xuan
- Subjects
STOCHASTIC matrices ,EQUATIONS ,STOCHASTIC processes ,MATHEMATICAL analysis ,NUMERICAL analysis ,MATHEMATICS - Abstract
This paper is concerned with refinement equations of the type [This symbol cannot be presented in ASCII format] where f is the unknown function defined on the s-dimensional Euclidean space R
s , a is a finitely supported sequence on Zs , and M is an s × s dilation matrix with m := | det M|. The solution of a refinement equation can be obtained by using the subdivision scheme associated with the mask. In this paper we give a characterization for the convergence of the subdivision scheme when the mask is nonnegative. Our method is to relate the problem of convergence to m column-stochastic matrices induced by the mask. In this way, the convergence of the subdivision scheme can be determined in a finite number of steps by checking whether each finite product of those column-stochastic matrices has a positive row. As a consequence of our characterization, we show that the convergence of the subdivision scheme with a nonnegative mask depends only on the location of its positive coefficients. Several examples are provided to demonstrate the power and applicability of our approach. [ABSTRACT FROM AUTHOR]- Published
- 1999
21. The generalized Landau-Raychaudhuri equation and its applications.
- Author
-
Stepanov, Sergey E. and Mikeš, Josef
- Subjects
EQUATIONS ,LANDAU theory ,NUMERICAL analysis ,MATHEMATICAL analysis ,ASYMPTOTIC expansions - Abstract
More than twenty years ago, the first of the two authors of this paper has deduced the generalized Landau-Raychaudhuri equation and demonstrated its numerous applications. Now we present some new interesting applications of the generalized Landau-Raychaudhuri equation. [ABSTRACT FROM AUTHOR]
- Published
- 2015
- Full Text
- View/download PDF
22. Outcome of special vibration controller techniques linked to a cracked beam.
- Author
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EL-Sayed, A.T. and Bauomy, H.S.
- Subjects
- *
VIBRATION (Mechanics) , *CLASSICAL mechanics , *EQUATIONS , *NUMERICAL analysis , *MATHEMATICAL analysis - Abstract
This paper presents a comparison between three different controller methods added to a cracked beam under the action of a harmonic excitation. Those three controllers are Positive Position Feedback (PPF), Integral Resonant Control (IRC) and Nonlinear Integral Positive Position Feedback (NIPPF) which be added to the measured system. The multiple scales method (MSM) is applied for getting the approximate solution on behalf of measured design. This method is effective to solve the major equations of measured system. Stability and effect of different coefficients of the system are demonstrated. The approximate solution response is established via numerical simulation outcome. NIPPF controller is the best one gives better results compared to the other two controllers in decreasing the high amplitude of the system. Comparison between mathematical solution and numerical simulation are considered. Relationship of formerly available papers is considered. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
23. Thermal Analysis of Thin Plates Using the Finite Element Method.
- Author
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Er, G. K., Iu, V. P., and Liu, X. L.
- Subjects
FINITE element method ,NUMERICAL analysis ,THERMAL analysis ,MATHEMATICAL analysis ,EQUATIONS - Abstract
The isotropic thermal plate is analyzed with finite element method. The solution procedure is presented. The elementary stiffness matrix and loading vector are derived rigorously with variation principle and the principle of minimum potential energy. Numerical results are obtained based on the derived equations and tested with available exact solutions. The problems in the finite element analysis are figured out. It is found that the finite element solutions can not converge as the number of elements increases around the corners of the plate. The derived equations presented in this paper are fundamental for our further study on more complicated thermal plate analysis. [ABSTRACT FROM AUTHOR]
- Published
- 2010
- Full Text
- View/download PDF
24. Finite Element Analysis of Polycrystalline Deformation with the Rate-dependent Crystal Plasticity.
- Author
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Yoon, J. H., Huh, H., and Lee, Y. S.
- Subjects
FINITE element method ,MATHEMATICAL continuum ,EQUATIONS ,MATERIAL plasticity ,ALGORITHMS ,NUMERICAL analysis ,MATHEMATICAL analysis - Abstract
Constitutive models for the crystal plasticity have the common objective which relates the behavior of microscopic single crystals in the crystallographic texture to the macroscopic continuum response. This paper presents the texture analysis of polycrystalline materials using the rate-dependent single crystal plasticity to develop a multi-scale description of the mechanism at the grain and aggregate levels. The texture analysis requires a numerical algorithm for integrating the constitutive equations. The implicit deformation gradient approach is employed to update the stresses and texture orientations as an integration algorithm. It considers elastic or plastic deformation gradient as the primary unknown variables and constructs the residual of the elastic and plastic velocity gradients as the governing equations. This algorithm is shown to be an efficient and robust algorithm in rather large time steps. The texture analysis of the asymmetric rolling process is also presented to show investigation of the effect of texture evolution based on the finite element analysis as a numerical example. The analysis result for texture evolution is investigated by comparing the pole figure before and after the asymmetric rolling process. © 2007 American Institute of Physics [ABSTRACT FROM AUTHOR]
- Published
- 2007
- Full Text
- View/download PDF
25. Mathematical analysis and numerical methods for Caputo-Hadamard fractional diffusion-wave equations.
- Author
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Ou, Caixia, Cen, Dakang, Vong, Seakweng, and Wang, Zhibo
- Subjects
- *
NUMERICAL analysis , *MATHEMATICAL analysis , *EQUATIONS , *ANALYTICAL solutions , *FRACTIONAL integrals - Abstract
In this paper, mathematical analysis and numerical methods for Caputo-Hadamard fractional diffusion-wave equations with initial singularity are investigated. By adopting the modified Laplace transform and the well-known finite Fourier sine transform, we obtain the analytical solution. Furthermore, the regularity and logarithmic decay of its solution are researched. Under the singularity hypothesis, the numerical methods for the problems are then studied. The model is first transformed into its equivalent form and then the technic of exponential type meshes is utilized. The fact that the discrete coefficients of Hadamard fractional integral have several graceful properties is crucial in convergence and stability analysis. For the sake of reducing storage and computational cost, the SOE technology is exploited to the new variable t = log t n s. On this basis, a fast compact difference scheme and a fast compact ADI method are constructed for the one- and two-dimensional problems, respectively. To illustrate the significance of studying singularity, a Crank-Nicolson difference scheme is proposed. The final result that the error in numerically approximating the solution depends on the parameter γ shows obviously how the regularity of the solution and the exponential type meshes affect the convergence order of the derived scheme. Ultimately, the numerical examples are provided to verify that the fast algorithm effectively reduces the computational cost compared to the direct method. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
26. Global Existence and Decay Rates of the Solutions Near Maxwellian for Non-linear Fokker-Planck Equations.
- Author
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Liao, Jie, Wang, Qianrong, and Yang, Xiongfeng
- Subjects
EQUATIONS ,NONLINEAR theories ,NUMERICAL analysis ,MATHEMATICAL analysis ,MATHEMATICAL models - Abstract
In this paper, we study the global existence and decay rates of the solutions near Maxwellian for non-linear Fokker-Planck equations in the whole space. The global existence is proved by combining uniform-in-time energy estimates with local solution constructed by Picard type iteration sequence. The decay rates of the nonlinear model is obtained by using the precise spectral analysis of the linearized Fokker-Planck operator as well as the energy method. The nonlinearity in the model brings new difficulty to the energy estimates, which is resolved by additional tailored weighted-in-v energy estimates suitable for Fokker-Planck operators. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
27. Number of solutions to kax + lby = cz.
- Author
-
Deng, Naijuan, Yuan, Pingzhi, and Luo, Wenyu
- Subjects
- *
INTEGERS , *EQUATIONS , *ALGEBRA , *NUMERICAL analysis , *MATHEMATICAL analysis - Abstract
Text Let k , l , a , b , c be positive integers such that gcd ( k a , l b ) = 1 , min { a , b , c } > 1 , a ≠ 3 , b ≠ 3 and 2 ∤ c . In this paper, we prove that there are at most four solutions in positive integers ( x , y , z ) to the equation k a x + l b y = c z and at most two solutions when 2 ∤ ( u ( l / k ) ) , where u ( m ) is the least positive integer t with m t ≡ 1 ( mod c ) . Video For a video summary of this paper, please visit https://youtu.be/Dt3Y7TDMxlg . [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
28. Numerical solutions of FitzHugh-Nagumo equation by exact finite-difference and NSFD schemes.
- Author
-
Namjoo, Mehran and Zibaei, Sadegh
- Subjects
NUMERICAL analysis ,FINITE difference method ,EQUATIONS ,MATHEMATICAL models ,MATHEMATICAL analysis - Abstract
In this paper, numerical solution of the FitzHugh-Nagumo (FN) equation is presented based on the nonstandard finite-difference (NSFD) scheme. At first, two exact finite-difference schemes for FN equation are obtained. Moreover, two NSFD schemes are presented for this equation. The positivity, boundedness, and local truncation error of the schemes are discussed. The numerical results obtained by the NSFD schemes are compared with the exact solution and some available methods, to verify the accuracy and efficiency of the NSFD schemes. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
29. The equivariant Tamagawa number conjecture and the extended abelian Stark conjecture.
- Author
-
Vallières, Daniel
- Subjects
STARK'S conjectures ,L-functions ,EQUATIONS ,NUMERICAL analysis ,MATHEMATICAL analysis - Abstract
The goal of this paper is to show that the equivariant Tamagawa number conjecture implies the extended abelian Stark conjecture contained in [12] and [11]. In particular, this gives the first proof of the extended abelian Stark conjecture for the base field Q, since the equivariant Tamagawa number conjecture away from 2 was proved in this context by Burns and Greither in [8] and Flach completed their results at 2 in [13] and [14]. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
30. Partial regularity for mass-minimizing currents in Hilbert spaces.
- Author
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Ambrosio, Luigi, De Lellis, Camillo, and Schmidt, Thomas
- Subjects
HILBERT space ,MANIFOLDS (Mathematics) ,EQUATIONS ,NUMERICAL analysis ,MATHEMATICAL analysis - Abstract
Recently, the theory of currents and the existence theory for Plateau's problem have been extended to the case of finite-dimensional currents in infinite-dimensional manifolds or even metric spaces; see [5] (and also [7, 39] for the most recent developments). In this paper, in the case when the ambient space is Hilbert, we provide the first partial regularity result, in a dense open set of the support, for n-dimensional integral currents which locally minimize the mass. Our proof follows with minor variants [34], implementing Lipschitz approximation and harmonic approximation without indirect arguments and with estimates which depend only on the dimension n and not on codimension or dimension of the target space. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
31. Which abelian tensor categories are geometric?
- Author
-
Schäppi, Daniel
- Subjects
TENSOR algebra ,LINEAR algebra ,EQUATIONS ,NUMERICAL analysis ,MATHEMATICAL analysis - Abstract
For a large class of geometric objects, the passage to categories of quasicoherent sheaves provides an embedding in the 2-category of abelian tensor categories. The notion of weakly Tannakian categories introduced by the author gives a characterization of tensor categories in the image of this embedding. However, this notion requires additional structure to be present, namely a fiber functor. For the case of classical Tannakian categories in characteristic zero, Deligne has found intrinsic properties-expressible entirely within the language of tensor categories-which are necessary and sufficient for the existence of a fiber functor. In this paper we generalize Deligne's result to weakly Tannakian categories in characteristic zero. The class of geometric objects whose tensor categories of quasi-coherent sheaves can be recognized in this manner includes both the gerbes arising in classical Tannaka duality and more classical geometric objects such as projective varieties over a field of characteristic zero. Our proof uses a different perspective on fiber functors, which we formalize through the notion of geometric tensor categories. A second application of this perspective allows us to describe categories of quasi-coherent sheaves on fiber products. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
32. Langlands program for p-adic coefficients and the petits camarades conjecture.
- Author
-
Tomoyuki Abe
- Subjects
COEFFICIENTS (Statistics) ,EQUATIONS ,MATHEMATICAL functions ,NUMERICAL analysis ,MATHEMATICAL analysis - Abstract
In this paper, we prove that, if Deligne's “petits camarades conjecture” holds, then a Langlands type correspondence holds also for p-adic coefficients on a smooth curve over a finite field. As an application, we prove that any overconvergent F -isocrystal of rank less than or equal to 2 on a smooth curve is ı-mixed. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
33. New Bounds on the Size of Optimal Meshes.
- Author
-
Sheehy, Donald R.
- Subjects
NUMERICAL analysis ,MATHEMATICAL analysis ,EQUATIONS ,DEFECT correction methods (Numerical analysis) ,ANALYTIC geometry ,NONLINEAR theories - Abstract
The theory of optimal size meshes gives a method for analyzing the output size (number of simplices) of a Delaunay refinement mesh in terms of the integral of a sizing function over the input domain. The input points define a maximal such sizing function called the feature size. This paper presents a way to bound the feature size integral in terms of an easy to compute property of a suitable ordering of the point set. The key idea is to consider the pacing of an ordered point set, a measure of the rate of change in the feature size as points are added one at a time. In previous work, Miller et al. showed that if an ordered point set has pacing ϕ, then the number of vertices in an optimal mesh will be O(ϕ
d n), where d is the input dimension. We give a new analysis of this integral showing that the output size is only θ( n+ nlogϕ). The new analysis tightens bounds from several previous results and provides matching lower bounds. Moreover, it precisely characterizes inputs that yield outputs of size O( n). [ABSTRACT FROM AUTHOR]- Published
- 2012
- Full Text
- View/download PDF
34. Novel fitted operator finite difference methods for singularly perturbed elliptic convection–diffusion problems in two dimensions.
- Author
-
Munyakazi, Justin B. and Patidar, Kailash C.
- Subjects
FINITE differences ,NUMERICAL analysis ,FLUID dynamics ,MATHEMATICAL analysis ,EQUATIONS - Abstract
We consider a class of singularly perturbed elliptic problems posed on a unit square. These problems are solved by using fitted mesh methods by many researchers but no attempts are made to solve them using fitted operator methods, except our recent work on reaction–diffusion problems [J.B. Munyakazi and K.C. Patidar, Higher order numerical methods for singularly perturbed elliptic problems, Neural Parallel Sci. Comput. 18(1) (2010), pp. 75–88]. In this paper, we design two fitted operator finite difference methods (FOFDMs) for singularly perturbed convection–diffusion problems which possess solutions with exponential and parabolic boundary layers, respectively. We observe that both of these FOFDMs are ϵ-uniformly convergent. This fact contradicts the claim about singularly perturbed convection–diffusion problems [Miller et al. Fitted Numerical Methods for Singular Perturbation Problems, World Scientific, Singapore, 1996] that ‘when parabolic boundary layers are present, …, it is not possible to design an ϵ-uniform FOFDM if the mesh is restricted to being a uniform mesh’. We confirm our theoretical findings through computational investigations and also found that we obtain better results than those of Linß and Stynes [Appl. Numer. Math. 31 (1999), pp. 255–270]. [ABSTRACT FROM AUTHOR]
- Published
- 2012
- Full Text
- View/download PDF
35. A NEW KIND OF FUZZY RELATIONAL EQUATIONS.
- Author
-
QIN, FENG and FANG, PING
- Subjects
FUZZY logic ,FUZZY systems ,EQUATIONS ,NUMERICAL analysis ,MATHEMATICAL analysis - Abstract
In this paper, a new kind of fuzzy relational equations (FREs for short) A ∘
R x = b is first introduced, and then the problem of solving solution to the FREs is discussed, where A is an m × n matrix, x and b are an n and an m dimensional column vectors, respectively. More specifically, their solvability and unique solvability are investigated, the corresponding necessary and sufficient conditions are presented, the complete solution set is obtained. It is worth noting the method to construct the complete solution set. [ABSTRACT FROM AUTHOR]* - Published
- 2010
- Full Text
- View/download PDF
36. A SUFFICIENT CONDITION FOR A GRAPH TO BE A FRACTIONAL (f, n)-CRITICAL GRAPH.
- Author
-
Sizhong Zhou
- Subjects
FRACTIONAL integrals ,GRAPH theory ,EQUATIONS ,NUMERICAL analysis ,MATHEMATICAL analysis ,NUMERICAL functions - Abstract
Let a, b and n be non-negative integers such that 1 ⩽ a ⩽ b, and let G be a graph of order p with p ⩾ (Multiple line equations cannot be presented in ASII text) and f be an integer-valued function defined on V(G) such that a ⩽ f (x) ⩽ (Multiple line equations cannot be presented in ASII text) for all x ∈ V(G). Let h : E(G) → [0, 1] be a function. If Σ
ℓ϶x h(e) = f (x) holds for any x ∈ V(G), then we call G[Fh ] a fractional f -factor of G with indicator function h, where Fh = {e ∈ E(G) : h(e) > 0}. A graph G is called a fractional (f, n)-critical graph if after deleting any n vertices of G the remaining graph of G has a fractional f -factor. In this paper, it is proved thatGis a fractional (f, n)- critical graph if ∣NG (X)∣ > (Multiple line equations cannot be presented in ASII text) for every non-empty independent subset X of V(G), and δ(G) > (Multiple line equations cannot be presented in ASII text). Furthermore, it is shown that the result in this paper is best possible in some sense. [ABSTRACT FROM AUTHOR]- Published
- 2010
- Full Text
- View/download PDF
37. Generalized Flow Rating Equations at Prototype Gated Spillways.
- Author
-
Ansar, Matahel and Chen, Zhiming
- Subjects
SPILLWAYS ,DIVERSION structures (Hydraulic engineering) ,CHANNELS (Hydraulic engineering) ,NUMERICAL analysis ,WATER currents ,HYDRODYNAMICS ,EQUATIONS ,FLOW meters ,MATHEMATICAL analysis - Abstract
Gated spillways are used to control flow in canals, rivers, and estuaries. Despite their widespread use, available flow ratings at these gated structures are largely derived from reduced-scale laboratory models. In this paper, generalized flow rating equations are developed based on field flow measurements collected mostly with an Acoustic Doppler Current Profiler at about 90 prototype gated spillways in South Florida. The proposed ratings, developed using dimensional analysis, agree well with field measurements. A single generalized equation that is applicable to all flow conditions was also developed. This generalized equation addresses the limitation of continuity of the calculations as flow transitions from one condition to another. [ABSTRACT FROM AUTHOR]
- Published
- 2009
- Full Text
- View/download PDF
38. Lie symmetry and Hojman conserved quantity of Nambu system.
- Author
-
Lin Peng, Fang Jian, Hui and, and Pang Ting
- Subjects
SYMMETRY (Physics) ,EQUATIONS ,MATHEMATICAL physics ,MATHEMATICAL analysis ,NUMERICAL analysis ,LIE algebras - Abstract
This paper studies the Lie symmetry and Hojman conserved quantity of the Nambu system. The determining equations of Lie symmetry for the system are given. The conditions for existence and the form of the Hojman conserved quantity led by the Lie symmetry for the system are obtained. Finally, an example is given to illustrate the application of the results. [ABSTRACT FROM AUTHOR]
- Published
- 2008
- Full Text
- View/download PDF
39. Routh method of reduction for Birkhoffian systems in the event space.
- Subjects
EQUATIONS ,COORDINATES ,INTEGRALS ,NUMERICAL analysis ,MATHEMATICAL analysis ,MATHEMATICAL physics - Abstract
For a Birkhoffian system in the event space, this paper presents the Routh method of reduction. The parametric equations of the Birkhoffian system in the event space are established, and the definition of cyclic coordinates for the system is given and the corresponding cyclic integral is obtained. Through the cyclic integral, the order of the system can be reduced. The Routh functions for the Birkhoffian system in the event space are constructed, and the Routh method of reduction is successfully generalized to the Birkhoffian system in the event space. The results show that if the system has a cyclic integral, then the parametric equations of the system can be reduced at least by two degrees and the form of the equations holds. An example is given to illustrate the application of the results. [ABSTRACT FROM AUTHOR]
- Published
- 2008
- Full Text
- View/download PDF
40. The Behavior of Positive Solutions of a Nonlinear Second-Order Difference Equation.
- Author
-
Stević, Stevo and Berenhaut, Kenneth S.
- Subjects
MATHEMATICS ,MATHEMATICAL analysis ,NONLINEAR theories ,PERIODIC functions ,EQUATIONS ,NUMERICAL analysis - Abstract
This paper studies the boundedness, global asymptotic stability, and periodicity of positive solutions of the equation x
n = f(xn-2 )/g(xn-1 ), n ϵ ℕ0 , where f, g ϵ C[(0,∞), (0,∞)]. It is shown that if f and g are nondecreasing, then for every solution of the equation the subsequences {x2n } and {x2n-1 } are eventually monotone. For the case when f(x) = a + βx and g satisfies the conditions g(0) = 1, g is nondecreasing, and x/g(x) is increasing, we prove that every prime periodic solution of the equation has period equal to one or two. We also investigate the global periodicity of the equation, showing that if all solutions of the equation are periodic with period three, then f(x) = c1/x and g(x) = c2 x, for some positive c1 and c2 . [ABSTRACT FROM AUTHOR]- Published
- 2008
- Full Text
- View/download PDF
41. Extended LQP Method for Monotone Nonlinear Complementarity Problems.
- Author
-
Bnouhachem, A. and Yuan, X. M.
- Subjects
EQUATIONS ,ITERATIVE methods (Mathematics) ,LOGARITHMIC functions ,VECTOR analysis ,CONJUGATE gradient methods ,NUMERICAL analysis ,MATHEMATICAL analysis - Abstract
To solve nonlinear complementarity problems (NCP), the logarithmicquadratic proximal (LQP) method solves a system of nonlinear equations at each iteration. In this paper, the iterates generated by the original LQP method are extended by explicit formulas and thus an extended LQP method is presented. It is proved theoretically that the lower bound of the progress obtained by the extended LQP method is greater than that by the original LQP method. Preliminary numerical results are provided to verify the theoretical assertions and the effectiveness of both the original and the extended LQP method. [ABSTRACT FROM AUTHOR]
- Published
- 2007
- Full Text
- View/download PDF
42. Random walks on the Comb model and its generalizations.
- Author
-
Arkhincheev, V. E.
- Subjects
DIFFUSION ,PROPERTIES of matter ,SOLUTION (Chemistry) ,SEPARATION (Technology) ,EQUATIONS ,NUMERICAL analysis ,MATHEMATICAL analysis ,RANDOM walks ,STOCHASTIC processes ,MATHEMATICAL physics - Abstract
Microscopic models with anomalous diffusion, which include the Comb model and its generalization for the finite width of the backbone, have been considered in this paper. The physical mechanisms of the subdiffusion random walks have been established. The first comes from the permanent return of the diffusing particle to the initial point of the diffusion due to “effective reducing” of the dimensionality of the considered system to the quasi-one-dimensional system. This physical mechanism has been obtained in the Comb model and in the model with a strip. The second mechanism of the subdiffusion is connected with random capture on the traps of diffusing particles and their ensuing random release from the traps. It has been shown that these different mechanisms of subdiffusion have been described by the different generalized diffusion equations of fractional order. The solutions of these different equations have been obtained, and the physical sense of the fractional order generalized equations has been discussed. [ABSTRACT FROM AUTHOR]
- Published
- 2007
- Full Text
- View/download PDF
43. A New Attack on the Filter Generator.
- Author
-
Rønjom, Sondre and Helleseth, Tor
- Subjects
MATHEMATICS ,NONLINEAR systems ,SYSTEMS theory ,EQUATIONS ,ALGEBRA ,MATHEMATICAL analysis ,ALGORITHMS ,NUMERICAL analysis - Abstract
The filter generator is an important building block in many stream ciphers. The generator consists of a linear feedback shift register of length n that generates an m-sequence of period 2′ - 1 filtered through a Boolean function of degree d that combines bits from the shift register and creates an output bit z
t at any time t. The previous best attacks aimed at reconstructing the initial state from an observed keystream, have essentially reduced the problem to solving a nonlinear system of D = (Multiple line equation(s) cannot be represented in ASCII text) (i) equations in n unknowns using techniques based on linear algebra. This attack needs about D bits of keystream and the system can be solved in complexity O (Dω ), where ω can be taken to be Strassen's reduction exponent ω = log2 (7) ≈ 2.807. This paper describes a new algorithm that recovers the initial state of most filter generators after observing O(D) keystream bits with complexity O((D - n)/2) ≈ O(D), after a pre-computation with complexity O(D(log2 D)³). [ABSTRACT FROM AUTHOR]- Published
- 2007
- Full Text
- View/download PDF
44. A STRENGTHENED CARLEMAN'S INEQUALITY.
- Author
-
Hu Yue
- Subjects
EQUATIONS ,MATHEMATICAL formulas ,MATHEMATICS ,MATHEMATICAL analysis ,NUMERICAL analysis - Abstract
In this paper, it is proved that "Multiple line equation(s) cannot be represented in ASCII text." Where "Multiple line equation(s) cannot be represented in ASCII text." [ABSTRACT FROM AUTHOR]
- Published
- 2006
45. Supersymmetric fifth order evolution equations.
- Author
-
Tian, K. and Liu, Q. P.
- Subjects
SUPERSYMMETRY ,DIFFERENTIAL equations ,EQUATIONS ,NUMERICAL analysis ,MATHEMATICAL analysis - Abstract
This paper considers supersymmetric fifth order evolution equations. Within the framework of symmetry approach, we give a list containing six equations, which are (potentially) integrable systems. Among these equations, the most interesting ones include a supersymmetric Sawada-Kotera equation and a novel supersymmetric fifth order KdV equation. For the latter, we supply some properties such as a Hamiltonian structures and a possible recursion operator. [ABSTRACT FROM AUTHOR]
- Published
- 2010
- Full Text
- View/download PDF
46. Design-Oriented Analysis of Circuits With Equality Constraints.
- Author
-
Vytyaz, Igor, Hanumolu, Pavan Kumar, Moon, Un-Ku, and Mayaram, Kartikeya
- Subjects
ELECTRONIC circuit design ,LOGIC design ,NUMERICAL analysis ,MATHEMATICAL analysis ,FINITE differences ,MATHEMATICS - Abstract
This paper presents a design-oriented circuit analysis that is augmented with design constraints. This analysis computes the circuit response and also finds the values of circuit parameters (equal to the number of design specifications) that result in a specified circuit performance. An application of this approach is demonstrated for the periodic steady-state analysis with shooting and finite difference formulations. The new analysis with design equality constraints is several times faster than search-based techniques that employ conventional analysis methods. [ABSTRACT FROM PUBLISHER]
- Published
- 2011
- Full Text
- View/download PDF
47. Stability Analysis of Autonomous Ratio-Memory Cellular Nonlinear Networks for Pattern Recognition.
- Author
-
Tsai, Su-Yung, Wang, Chi-Hsu, and Wu, Chung-Yu
- Subjects
LYAPUNOV functions ,NUMERICAL analysis ,ELECTRONIC feedback ,NEURONS ,MATHEMATICAL analysis - Abstract
The stability analysis via the Lyapunov theorem for Autonomous Ratio-Memory Cellular Nonlinear Networks (ARMCNNs) is proposed. A conservative domain of attraction (DOA) is found from the stability analysis through a graphical method without complicated numerical analysis. The stability analysis shows that ARMCNNs can tolerate large ratio weight variations. This paper also presents the ARMCNN with self-feedback (SARMCNN) to overcome the problem of isolated neurons due to low correlation between neighboring neurons. The SARMCNN recognition rate (RR) is compared with other CNN constructed via the singular value decomposition technique (SVD-CNN). [ABSTRACT FROM PUBLISHER]
- Published
- 2010
- Full Text
- View/download PDF
48. Uniqueness of Meromorphic Functions Concerning the Difference Polynomials.
- Author
-
FANGHONG LIU and HONGXUN YI
- Subjects
- *
MEROMORPHIC functions , *NEVANLINNA theory , *MATHEMATICAL analysis , *NUMERICAL analysis , *EQUATIONS - Abstract
In this article, we main study the uniqueness problem of meromorphic function which difference polynomials sharing common values. We consider the entire function (fn(fm -1) Πjs=1 f(z+cj)μj)(k) and the meromorphic function fn(fm-1) Πjs=1 f(z+cj)μjto get the main results which extend Theorem 1.1 in paper[5] and theorem 1.4 in paper[6]. [ABSTRACT FROM AUTHOR]
- Published
- 2015
- Full Text
- View/download PDF
49. Bondage number of strong product of two paths.
- Author
-
Zhao, Weisheng and Zhang, Heping
- Subjects
GRAPHIC methods ,INTEGERS ,NUMERICAL analysis ,MATHEMATICAL analysis ,EQUATIONS - Abstract
The bondage number b( G) of a graph G is the cardinality of a minimum set of edges whose removal from G results in a graph with a domination number greater than that of G. In this paper, we obtain the exact value of the bondage number of the strong product of two paths. That is, for any two positive integers m ⩾ 2 and n ⩾ 2, b( P ⊠ P) = 7 − r( m) − r( n) if ( r( m), r( n)) = (1, 1) or (3, 3), 6 − r( m) − r( n) otherwise, where r( t) is a function of positive integer t, defined as r( t) = 1 if t ≡ 1 (mod 3), r( t) = 2 if t ≡ 2 (mod 3), and r( t) = 3 if t ≡ 0 (mod 3). [ABSTRACT FROM AUTHOR]
- Published
- 2015
- Full Text
- View/download PDF
50. On the Braid Index of Kanenobu Knots.
- Author
-
HIDEO TAKIOKA
- Subjects
- *
POLYNOMIALS , *DIFFERENTIAL geometry , *MATHEMATICAL analysis , *NUMERICAL analysis , *EQUATIONS - Abstract
We study the braid indices of the Kanenobu knots. It is known that the Kanenobu knots have the same HOMFLYPT polynomial and the same Khovanov-Rozansky homology. The MFW inequality is known for giving a lower bound of the braid index of a link by applying the HOMFLYPT polynomial. Therefore, it is not easy to determine the braid indices of the Kanenobu knots. In our previous paper, we gave upper bounds and sharper lower bounds of the braid indices of the Kanenobu knots by applying the 2-cable version of the zeroth coefficient HOMFLYPT polynomial. In this paper, we give sharper upper bounds of the braid indices of the Kanenobu knots. [ABSTRACT FROM AUTHOR]
- Published
- 2015
- Full Text
- View/download PDF
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