On the product of semi-groups of operators
01 Apr 1959-Vol. 10, Iss: 4, pp 545-551
About: The article was published on 1959-04-01 and is currently open access. It has received 1761 citations till now. The article focuses on the topics: Product (mathematics).
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TL;DR: It is shown how the new RESPA methods are related to predictor–corrector integrators and how these methods can be used to accelerate the integration of the equations of motion of systems with Nose thermostats.
Abstract: The Trotter factorization of the Liouville propagator is used to generate new reversible molecular dynamics integrators This strategy is applied to derive reversible reference system propagator algorithms (RESPA) that greatly accelerate simulations of systems with a separation of time scales or with long range forces The new algorithms have all of the advantages of previous RESPA integrators but are reversible, and more stable than those methods These methods are applied to a set of paradigmatic systems and are shown to be superior to earlier methods It is shown how the new RESPA methods are related to predictor–corrector integrators Finally, we show how these methods can be used to accelerate the integration of the equations of motion of systems with Nose thermostats
3,155 citations
TL;DR: In this paper, the authors introduce a picture of a boson superfluid and show how superfluidity and Bose condensation manifest themselves, showing the excellent agreement between simulations and experimental measurements on liquid and solid helium for such quantities as pair correlations, the superfluid density, the energy, and the momentum distribution.
Abstract: One of Feynman's early applications of path integrals was to superfluid $^{4}\mathrm{He}$. He showed that the thermodynamic properties of Bose systems are exactly equivalent to those of a peculiar type of interacting classical "ring polymer." Using this mapping, one can generalize Monte Carlo simulation techniques commonly used for classical systems to simulate boson systems. In this review, the author introduces this picture of a boson superfluid and shows how superfluidity and Bose condensation manifest themselves. He shows the excellent agreement between simulations and experimental measurements on liquid and solid helium for such quantities as pair correlations, the superfluid density, the energy, and the momentum distribution. Major aspects of computational techniques developed for a boson superfluid are discussed: the construction of more accurate approximate density matrices to reduce the number of points on the path integral, sampling techniques to move through the space of exchanges and paths quickly, and the construction of estimators for various properties such as the energy, the momentum distribution, the superfluid density, and the exchange frequency in a quantum crystal. Finally the path-integral Monte Carlo method is compared to other quantum Monte Carlo methods.
1,908 citations
TL;DR: Explicit reversible integrators, suitable for use in large-scale computer simulations, are derived for extended systems generating the canonical and isothermal-isobaric ensembles.
Abstract: Explicit reversible integrators, suitable for use in large-scale computer simulations, are derived for extended systems generating the canonical and isothermal-isobaric ensembles. The new methods are compared with the standard implicit (iterative) integrators on some illustrative example problems. In addition, modification of the proposed algorithms for multiple time step integration is outlined.
1,564 citations
TL;DR: Peruzzo et al. as mentioned in this paper developed a variational adiabatic ansatz and explored unitary coupled cluster where they established a connection from second order unitary cluster to universal gate sets through a relaxation of exponential operator splitting.
Abstract: Many quantum algorithms have daunting resource requirements when compared to what is available today. To address this discrepancy, a quantum-classical hybrid optimization scheme known as 'the quantum variational eigensolver' was developed (Peruzzo et al 2014 Nat. Commun. 5 4213) with the philosophy that even minimal quantum resources could be made useful when used in conjunction with classical routines. In this work we extend the general theory of this algorithm and suggest algorithmic improvements for practical implementations. Specifically, we develop a variational adiabatic ansatz and explore unitary coupled cluster where we establish a connection from second order unitary coupled cluster to universal gate sets through a relaxation of exponential operator splitting. We introduce the concept of quantum variational error suppression that allows some errors to be suppressed naturally in this algorithm on a pre-threshold quantum device. Additionally, we analyze truncation and correlated sampling in Hamiltonian averaging as ways to reduce the cost of this procedure. Finally, we show how the use of modern derivative free optimization techniques can offer dramatic computational savings of up to three orders of magnitude over previously used optimization techniques.
1,490 citations
TL;DR: In this article, the authors studied the manifold structure of certain groups of diffeomorphisms, and used this structure to obtain sharp existence and uniqueness theorems for the classical equations for an incompressible viscous and non-viscous fluid on a compact C^∞ riemannian, oriented n-manifold, possibly with boundary.
Abstract: In this paper we are concerned with the manifold structure of certain groups of diffeomorphisms, and with the use of this structure to obtain sharp existence and uniqueness theorems for the classical equations for an incompressible
fluid (both viscous and non-viscous) on a compact C^∞ riemannian, oriented n-manifold M, possibly with boundary.
1,362 citations
References
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Book•
01 Jan 1948
TL;DR: The theory of semi-groups has been studied extensively in the literature, see as discussed by the authors for a survey of some of the main applications of semi groups in the context of functional analysis.
Abstract: Part One. Functional Analysis: Abstract spaces Linear transformations Vector-valued functions Banach algebras General properties Analysis in a Banach algebra Laplace integrals and binomial series Part Two. Basic Properties of Semi-Groups: Subadditive functions Semi-modules Addition theorem in a Banach algebra Semi-groups in the strong topology Generator and resolvent Generation of semi-groups Part Three. Advanced Analytical Theory of Semi-Groups: Perturbation theory Adjoint theory Operational calculus Spectral theory Holomorphic semi-groups Applications to ergodic theory Part Four. Special Semi-groups and Applications: Translations and powers Trigonometric semi-groups Semi-groups in $L_p(-\infty,\infty)$ Semi-groups in Hilbert space Miscellaneous applications Part Five. Extensions of the theory: Notes on Banach algebras Lie semi-groups Functions on vectors to vectors Bibliography Index.
3,462 citations
TL;DR: Hille's seminal work on functional analysis and semi-groups as discussed by the authors is one of the most important mathematical books of the century, and should be read seriously by every mathematician, especially in the context of semi-group analysis, where abstract spaces, linear operations, vector-valued functions, and general analysis in Banach algebras are used.
Abstract: THIS work of Prof. Einar Hille is one of the important mathematical books of the century ; it should be read seriously by every mathematician. The subject of semi-groups is one of great and growing importance, and the type of analysis on which its treatment is based—abstract spaces, linear operations, vector-valued functions, and general analysis in 'Banach algebras'—is scarcely less important. Functional Analysis and Semi-Groups By Prof. Einar Hille. (American Mathematical Society: Colloquium Publications, Vol. 31.) Pp. xi + 528. New York: American Mathematical Society, 1948.) 7.50 dollars.
633 citations
"On the product of semi-groups of op..." refers background or result in this paper
...The results are rather different from those of Phillips [3], in that the set of perturbing operators is not linear, nor even (in the noncommutative case) a positive cone....
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...Theorems 1 and 2 may be regarded as perturbation theorems in the sense of Phillips, since they assert that under certain conditions, when an infinitesimal generator ft is modified by adding another operator aft', the modified operator (or its closure) is again an infinitesimal generator....
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...If the closure of ft+oft' is the infinitesimal generator of a semi-group of class (A) (for the notation, see [3]) theni?(X —ft —aft') is dense in X for all sufficiently large X [3, p....
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...This improves on an example of Dye and Phillips [l] and answers a question raised in [3, p. 417]....
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...We consider semi-groups of operators on a Banach space X, which are of class (Co) in the terminology of [3]....
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417 citations
"On the product of semi-groups of op..." refers background in this paper
...3 of [4] could still be applied to give the desired result....
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...3 of [4], (8) and the remark following (7) imply that if D(il+ail') and i?(X —fi —afi') are dense in X for some X>co+oco', then fi0, the closure of fia, generates a semi-group...
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52 citations
"On the product of semi-groups of op..." refers background in this paper
...norm (by defining ||/||' as sup(>o e~w!\\ Tif\\) so that Tt satisfies the norm condition [2]....
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TL;DR: In this paper, a semi-group of bounded positive operators on certain function spaces is introduced to the theory of stochastic processes of the diffusion type in an essential way, and it is shown that these semi-groups cannot be imbedded in groups of positive operators, or, in more special terms, that the solution of a diffusion equation does not define a one-parameter group of positive operator on the natural function space.
Abstract: 1. Introduction. Semi-groups of bounded positive operators on certain function spaces enter the theory of stochastic processes of the diffusion type in an essential way. It is a matter of experience that these semi-groups cannot be imbedded in groups of positive operators, or, in more special terms, that the solution of a diffusion equation does not define a one-parameter group of positive operators on the natural function space.
4 citations