Generalization of the separation of variables in the Jacobi identities for finite-dimensional Poisson systems

We investigate anN-state spin model called quantum relativistic Toda chain and based on the unitary finite-dimensional representations of the Weyl algebra withqbeingNth primitive root of unity. Parameters of the finite-dimensional representation of the local Weyl algebra form the classical discrete integrable system. Nontrivial dynamics of the classical counterpart corresponds to isospectral transformations of the spin system. Similarity operators are constructed with the help of modified Baxter'sQ-operators. The classical counterpart of the modifiedQ-operator for the initial homogeneous spin chain is a Bäcklund transformation. This transformation creates an extra Hirota-type soliton in a parameterization of the chain structure. Special choice of values of solitonic amplitudes yields a degeneration of spin eigenstates, leading to the quantum separation of variables, or the functional Bethe ansatz. A projector to the separated eigenstates is constructed explicitly as a product of modifiedQ-operators.

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Conditional Lie-Bäcklund Symmetry Reductions and Exact Solutions of a Class of Reaction-Diffusion Equations

Advances in Mathematical Physics ◽

10.1155/2018/3916814 ◽

2018 ◽

Vol 2018 ◽

pp. 1-6

Author(s):

Xinyang Wang ◽

Junquan Song

Keyword(s):

Dynamical Systems ◽

Exact Solutions ◽

Separation Of Variables ◽

Reaction Diffusion ◽

Reaction Diffusion Equations ◽

Diffusion Equations ◽

Financial Mathematics ◽

Symmetry Reductions ◽

Finite Dimensional ◽

Wide Range

The method of conditional Lie-Bäcklund symmetry is applied to solve a class of reaction-diffusion equations ut+uxx+Qxux2+Pxu+Rx=0, which have wide range of applications in physics, engineering, chemistry, biology, and financial mathematics theory. The resulting equations are either solved exactly or reduced to some finite-dimensional dynamical systems. The exact solutions obtained in concrete examples possess the extended forms of the separation of variables.

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Non-Linear Dynamics of Cardiovascular Variability Signals

Methods of Information in Medicine ◽

10.1055/s-0038-1634981 ◽

1994 ◽

Vol 33 (01) ◽

pp. 81-84 ◽

Cited By ~ 14

Author(s):

S. Cerutti ◽

S. Guzzetti ◽

R. Parola ◽

M.G. Signorini

Keyword(s):

Dimensional Space ◽

Severe Heart Failure ◽

Parametric Models ◽

Deterministic Systems ◽

Finite Dimensional ◽

Return Maps ◽

Pathological Conditions ◽

Non Linear ◽

Space State

Abstract:Long-term regulation of beat-to-beat variability involves several different kinds of controls. A linear approach performed by parametric models enhances the short-term regulation of the autonomic nervous system. Some non-linear long-term regulation can be assessed by the chaotic deterministic approach applied to the beat-to-beat variability of the discrete RR-interval series, extracted from the ECG. For chaotic deterministic systems, trajectories of the state vector describe a strange attractor characterized by a fractal of dimension D. Signals are supposed to be generated by a deterministic and finite dimensional but non-linear dynamic system with trajectories in a multi-dimensional space-state. We estimated the fractal dimension through the Grassberger and Procaccia algorithm and Self-Similarity approaches of the 24-h heart-rate variability (HRV) signal in different physiological and pathological conditions such as severe heart failure, or after heart transplantation. State-space representations through Return Maps are also obtained. Differences between physiological and pathological cases have been assessed and generally a decrease in the system complexity is correlated to pathological conditions.

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Center of Mass Theorem and the Separation of Variables for Two-Body System on a Three-Dimensional Sphere

Nonlinear Phenomena in Complex Systems ◽

10.33581/1561-4085-2020-23-3-306-311 ◽

2020 ◽

Vol 23 (3) ◽

pp. 306-311

Author(s):

Yu. Kurochkin ◽

Dz. Shoukavy ◽

I. Boyarina

Keyword(s):

Constant Curvature ◽

Jacobi Equation ◽

Separation Of Variables ◽

Center Of Mass ◽

Three Dimensional ◽

Relative Distance ◽

Dimensional Sphere ◽

Body System ◽

Spaces Of Constant Curvature ◽

Hamilton Jacobi Equation

The immobility of the center of mass in spaces of constant curvature is postulated based on its definition obtained in [1]. The system of two particles which interact through a potential depending only on the distance between particles on a three-dimensional sphere is considered. The Hamilton-Jacobi equation is formulated and its solutions and trajectory equations are found. It was established that the reduced mass of the system depends on the relative distance.

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Finite-Dimensional Modeling of Network-Induced Delays for Real-Time Control Systems

10.23919/acc.1988.4789881 ◽

1988 ◽

Cited By ~ 1

Author(s):

Asok Ray ◽

Yoram Halevi

Keyword(s):

Control Systems ◽

Real Time ◽

Real Time Control ◽

Time Control ◽

Finite Dimensional ◽

Dimensional Modeling

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Approaches to mixed Dirichlet-Neumann boundary condition in the method of separation of variables

2019 Applications of Electromagnetics in Modern Engineering and Medicine (PTZE) ◽

10.23919/ptze.2019.8781727 ◽

2019 ◽

Author(s):

Pawel Jablonski

Keyword(s):

Boundary Condition ◽

Neumann Boundary Condition ◽

Separation Of Variables ◽

Neumann Boundary

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A closer look at the stable completion

10.23943/princeton/9780691161686.003.0005 ◽

2017 ◽

Author(s):

Ehud Hrushovski ◽

François Loeser

Keyword(s):

Inverse Limit ◽

Unit Vector ◽

Vector Spaces ◽

Dimensional Vector ◽

Compact Sets ◽

Linear Topology ◽

Topological Embedding ◽

Definable Set ◽

Finite Dimensional ◽

The One

This chapter introduces the concept of stable completion and provides a concrete representation of unit vector Mathematical Double-Struck Capital A superscript n in terms of spaces of semi-lattices, with particular emphasis on the frontier between the definable and the topological categories. It begins by constructing a topological embedding of unit vector Mathematical Double-Struck Capital A superscript n into the inverse limit of a system of spaces of semi-lattices L(Hsubscript d) endowed with the linear topology, where Hsubscript d are finite-dimensional vector spaces. The description is extended to the projective setting. The linear topology is then related to the one induced by the finite level morphism L(Hsubscript d). The chapter also considers the condition that if a definable set in L(Hsubscript d) is an intersection of relatively compact sets, then it is itself relatively compact.

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Preface to the special issue, CMAM 2011, no. 3.

Computational Methods in Applied Mathematics ◽

10.2478/cmam-2011-0014 ◽

2011 ◽

Vol 11 (3) ◽

pp. 272

Author(s):

Ivan Gavrilyuk ◽

Boris Khoromskij ◽

Eugene Tyrtyshnikov

Keyword(s):

Separation Of Variables ◽

Numerical Algorithms ◽

Multilevel Methods ◽

High Dimensional ◽

Special Issue ◽

High Dimensions ◽

Guiding Principle ◽

Tensor Methods ◽

Closed World ◽

The One

Abstract In the recent years, multidimensional numerical simulations with tensor-structured data formats have been recognized as the basic concept for breaking the "curse of dimensionality". Modern applications of tensor methods include the challenging high-dimensional problems of material sciences, bio-science, stochastic modeling, signal processing, machine learning, and data mining, financial mathematics, etc. The guiding principle of the tensor methods is an approximation of multivariate functions and operators with some separation of variables to keep the computational process in a low parametric tensor-structured manifold. Tensors structures had been wildly used as models of data and discussed in the contexts of differential geometry, mechanics, algebraic geometry, data analysis etc. before tensor methods recently have penetrated into numerical computations. On the one hand, the existing tensor representation formats remained to be of a limited use in many high-dimensional problems because of lack of sufficiently reliable and fast software. On the other hand, for moderate dimensional problems (e.g. in "ab-initio" quantum chemistry) as well as for selected model problems of very high dimensions, the application of traditional canonical and Tucker formats in combination with the ideas of multilevel methods has led to the new efficient algorithms. The recent progress in tensor numerical methods is achieved with new representation formats now known as "tensor-train representations" and "hierarchical Tucker representations". Note that the formats themselves could have been picked up earlier in the literature on the modeling of quantum systems. Until 2009 they lived in a closed world of those quantum theory publications and never trespassed the territory of numerical analysis. The tremendous progress during the very recent years shows the new tensor tools in various applications and in the development of these tools and study of their approximation and algebraic properties. This special issue treats tensors as a base for efficient numerical algorithms in various modern applications and with special emphases on the new representation formats.

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