The Experts below are selected from a list of 219 Experts worldwide ranked by ideXlab platform
Paul S Pedersen - One of the best experts on this subject based on the ideXlab platform.
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cauchy s integral theorem on a finitely generated real commutative and associative algebra
Advances in Mathematics, 1997Co-Authors: Paul S PedersenAbstract:LetR[α]=R[α1, α2, …, αn] (whereα1=1) be a real, unitary, finitely generated, commutative, and associative algebra. We consider functionsf(z)=f(∑ni=1 xiαi) which mapR[α]n={∑ni=1 aiαi | 1⩽i⩽n,ai∈R} intoR[α]=finite dimensional subspaces of {∑k∈Nn0 bkαk | bk∈R,k=(k1, …, kn)∈Nn0} whereN0={0, 1, 2, …}. We impose a total order on an algorithmically defined BasisBforR[α]. The resulting algebra and Ordered Basis will be written as (R[α], <). We then use this Basis to define a norm ‖·‖ on (R[α], <). Continuous functions, differentiable functions, and the concept of Riemann integration will then be defined and discussed in this new setting. We then show that ∫γ f(z) dz=0 whenf(z) is a continuous and differentiable function defined in a simply connected regionG⊂R[α]n⊂(R[α], <) containing the closed pathγ.
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Cauchy's Integral Theorem on a Finitely Generated, Real, Commutative, and Associative Algebra
Advances in Mathematics, 1997Co-Authors: Paul S PedersenAbstract:LetR[α]=R[α1, α2, …, αn] (whereα1=1) be a real, unitary, finitely generated, commutative, and associative algebra. We consider functionsf(z)=f(∑ni=1 xiαi) which mapR[α]n={∑ni=1 aiαi | 1⩽i⩽n,ai∈R} intoR[α]=finite dimensional subspaces of {∑k∈Nn0 bkαk | bk∈R,k=(k1, …, kn)∈Nn0} whereN0={0, 1, 2, …}. We impose a total order on an algorithmically defined BasisBforR[α]. The resulting algebra and Ordered Basis will be written as (R[α],
M. Razavy - One of the best experts on this subject based on the ideXlab platform.
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Integration of the Heisenberg equations for inverse power-law potentials
Physical review. A Atomic molecular and optical physics, 1996Co-Authors: M. Hron, M. RazavyAbstract:The direct method of integration of the operator Heisenberg equations of motion is extended to the solution of quantum tunneling when the central potential is a sum of inverse powers of the radial distance. By obtaining the equation of motion for the Weyl-Ordered Basis set {l_brace}{ital S}{sub {ital m},{ital n}}({ital t}){r_brace} formed from {ital r}({ital t}) and {ital p}{sub {ital r}}({ital t}), one can express the time evolution of any member of the set as an infinite sum involving the operators {l_brace}{ital S}{sub {ital m},{ital n}}(0){r_brace}. The direct integration enables one to find the expectation values of the radial position operator and its conjugate momentum and higher moments of these operators as functions of time. {copyright} {ital 1996 The American Physical Society.}
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Integration of the Heisenberg equations of motion for quartic potentials.
Physical review. A Atomic molecular and optical physics, 1995Co-Authors: M. Hron, M. RazavyAbstract:A direct integration of the Heisenberg equations of motion yields expansions of the position and momentum operators x(t) and p(t), each as an infinite series in terms of the initial Weyl-Ordered Basis set {${\mathit{S}}_{\mathit{m},}$n} formed from x(0) and p(0), with c-number time-dependent coefficients. This method is applied to the problem of tunneling in a symmetric and an asymmetric quartic potential. The expectation values of the position and momentum operators with minimum uncertainty wave packet 〈0\ensuremath{\Vert}x(t)\ensuremath{\Vert}0〉 and 〈0\ensuremath{\Vert}p(t)\ensuremath{\Vert}0〉 can be calculated accurately for a maximum time that is short compared to the period of oscillation of the wave packet. From the result of this calculation, with the help of Prony's method one can determine the level spacings for the low-lying states. In addition, in this formulation the wave packet retains its shape; therefore, one can study the trajectory 〈0\ensuremath{\Vert}x(t)\ensuremath{\Vert}0〉 and 〈0\ensuremath{\Vert}p(t)\ensuremath{\Vert}0〉 as the quantal analogue of the motion of the system in phase space.
Giulio Landolfi - One of the best experts on this subject based on the ideXlab platform.
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Nonautonomous Hamiltonian quantum systems, operator equations, and representations of the Bender–Dunne Weyl-Ordered Basis under time-dependent canonical transformations
Theoretical and Mathematical Physics, 2017Co-Authors: Mariagiovanna Gianfreda, Giulio LandolfiAbstract:We solve the problem of integrating operator equations for the dynamics of nonautonomous quantum systems by using time-dependent canonical transformations. The studied operator equations essentially reproduce the classical integrability conditions at the quantum level in the basic cases of one-dimensional nonautonomous dynamical systems. We seek solutions in the form of operator series in the Bender–Dunne Basis of pseudodifferential operators. Together with this problem, we consider quantum canonical transformations. The minimal solution of the operator equation in the representation of the Basis at a fixed time corresponds to the lowest-order contribution of the solution obtained as a result of applying a canonical linear transformation to the Basis elements.
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nonautonomous hamiltonian quantum systems operator equations and representations of the bender dunne weyl Ordered Basis under time dependent canonical transformations
Theoretical and Mathematical Physics, 2017Co-Authors: Mariagiovanna Gianfreda, Giulio LandolfiAbstract:We solve the problem of integrating operator equations for the dynamics of nonautonomous quantum systems by using time-dependent canonical transformations. The studied operator equations essentially reproduce the classical integrability conditions at the quantum level in the basic cases of one-dimensional nonautonomous dynamical systems. We seek solutions in the form of operator series in the Bender–Dunne Basis of pseudodifferential operators. Together with this problem, we consider quantum canonical transformations. The minimal solution of the operator equation in the representation of the Basis at a fixed time corresponds to the lowest-order contribution of the solution obtained as a result of applying a canonical linear transformation to the Basis elements.
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non autonomous hamiltonian quantum systems operator equations and representations of bender dunne weyl Ordered Basis under time dependent canonical transformations
arXiv: Quantum Physics, 2015Co-Authors: Mariagiovanna Gianfreda, Giulio LandolfiAbstract:We address the problem of integrating operator equations concomitant with the dynamics of non autonomous quantum systems by taking advantage of the use of time-dependent canonical transformations. In particular, we proceed to a discussion in regard to basic examples of one-dimensional non-autonomous dynamical systems enjoying the property that their Hamiltonian can be mapped through a time-dependent linear canonical transformation into an autonomous form, up to a time-dependent multiplicative factor. The operator equations we process essentially reproduce at the quantum level the classical integrability condition for these systems. Operator series form solutions in the Bender-Dunne Basis of pseudo-differential operators for one dimensional quantum system are sought for such equations. The derivation of generating functions for the coefficients involved in the \emph{minimal} representation of the series solutions to the operator equations under consideration is particularized. We also provide explicit form of operators that implement arbitrary linear transformations on the Bender-Dunne Basis by expressing them in terms of the initial Weyl Ordered Basis elements. We then remark that the matching of the minimal solutions obtained independently in the two Basis, i.e. the Basis prior and subsequent the action of canonical linear transformation, is perfectly achieved by retaining only the lowest order contribution in the expression of the transformed Bender-Dunne Basis elements.
Mariagiovanna Gianfreda - One of the best experts on this subject based on the ideXlab platform.
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Nonautonomous Hamiltonian quantum systems, operator equations, and representations of the Bender–Dunne Weyl-Ordered Basis under time-dependent canonical transformations
Theoretical and Mathematical Physics, 2017Co-Authors: Mariagiovanna Gianfreda, Giulio LandolfiAbstract:We solve the problem of integrating operator equations for the dynamics of nonautonomous quantum systems by using time-dependent canonical transformations. The studied operator equations essentially reproduce the classical integrability conditions at the quantum level in the basic cases of one-dimensional nonautonomous dynamical systems. We seek solutions in the form of operator series in the Bender–Dunne Basis of pseudodifferential operators. Together with this problem, we consider quantum canonical transformations. The minimal solution of the operator equation in the representation of the Basis at a fixed time corresponds to the lowest-order contribution of the solution obtained as a result of applying a canonical linear transformation to the Basis elements.
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nonautonomous hamiltonian quantum systems operator equations and representations of the bender dunne weyl Ordered Basis under time dependent canonical transformations
Theoretical and Mathematical Physics, 2017Co-Authors: Mariagiovanna Gianfreda, Giulio LandolfiAbstract:We solve the problem of integrating operator equations for the dynamics of nonautonomous quantum systems by using time-dependent canonical transformations. The studied operator equations essentially reproduce the classical integrability conditions at the quantum level in the basic cases of one-dimensional nonautonomous dynamical systems. We seek solutions in the form of operator series in the Bender–Dunne Basis of pseudodifferential operators. Together with this problem, we consider quantum canonical transformations. The minimal solution of the operator equation in the representation of the Basis at a fixed time corresponds to the lowest-order contribution of the solution obtained as a result of applying a canonical linear transformation to the Basis elements.
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non autonomous hamiltonian quantum systems operator equations and representations of bender dunne weyl Ordered Basis under time dependent canonical transformations
arXiv: Quantum Physics, 2015Co-Authors: Mariagiovanna Gianfreda, Giulio LandolfiAbstract:We address the problem of integrating operator equations concomitant with the dynamics of non autonomous quantum systems by taking advantage of the use of time-dependent canonical transformations. In particular, we proceed to a discussion in regard to basic examples of one-dimensional non-autonomous dynamical systems enjoying the property that their Hamiltonian can be mapped through a time-dependent linear canonical transformation into an autonomous form, up to a time-dependent multiplicative factor. The operator equations we process essentially reproduce at the quantum level the classical integrability condition for these systems. Operator series form solutions in the Bender-Dunne Basis of pseudo-differential operators for one dimensional quantum system are sought for such equations. The derivation of generating functions for the coefficients involved in the \emph{minimal} representation of the series solutions to the operator equations under consideration is particularized. We also provide explicit form of operators that implement arbitrary linear transformations on the Bender-Dunne Basis by expressing them in terms of the initial Weyl Ordered Basis elements. We then remark that the matching of the minimal solutions obtained independently in the two Basis, i.e. the Basis prior and subsequent the action of canonical linear transformation, is perfectly achieved by retaining only the lowest order contribution in the expression of the transformed Bender-Dunne Basis elements.
Michael Manry - One of the best experts on this subject based on the ideXlab platform.
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A Functional Link Network With Ordered Basis Functions
2007 International Joint Conference on Neural Networks, 2007Co-Authors: Saurabh Sureka, Michael ManryAbstract:A procedure is presented for selecting and ordering the polynomial Basis functions in the functional link net (FLN). This procedure, based upon a modified Gram Schmidt orthonormalization, eliminates linearly dependent and less useful Basis functions at an early stage, reducing the possibility of combinatorial explosion. The number of passes through the training data is minimized through the use of correlations. A one-pass method is used for validation and network sizing. Function approximation and learning examples are presented. Results for the Ordered FLN are compared with those for the FLN, group method of data handling, and multi-layer perceptron.
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IJCNN - A Functional Link Network With Ordered Basis Functions
2007 International Joint Conference on Neural Networks, 2007Co-Authors: Saurabh Sureka, Michael ManryAbstract:A procedure is presented for selecting and ordering the polynomial Basis functions in the functional link net (FLN). This procedure, based upon a modified Gram Schmidt orthonormalization, eliminates linearly dependent and less useful Basis functions at an early stage, reducing the possibility of combinatorial explosion. The number of passes through the training data is minimized through the use of correlations. A one-pass method is used for validation and network sizing. Function approximation and learning examples are presented. Results for the Ordered FLN are compared with those for the FLN, group method of data handling, and multi-layer perceptron.
