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Czachor, Marek

doi:10.1023/a:1026670215803

Cited by 8 publications

(1 citation statement)

References 42 publications

(41 reference statements)

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“…If the operator H is Hermitian it is convenient to work in the basis of eigenvectors of H since then the diagonal elements Ψ jj are time independent. Hamiltonian, Lie-Poisson, or Lie-Nambu structures, as well as the Casimirs associated with (16) were discussed in detail in [11] and generalized in [12]. In particular, for any natural n the quantities Tr(H Ψ n ) are constants of motion, and Tr( Ψ n ) are Casimir fuctions.…”

Section: Soliton Dynamics Of the Two Strandsmentioning

confidence: 99%

Two-State Dynamics for Replicating Two-Strand Systems

Aerts

Czachor

2007

Open Syst. Inf. Dyn.

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We propose a formalism for describing two-strand systems of a DNA type by means of soliton von Neumann equations, and illustrate how it works on a simple example exactly solvably by a Darboux transformation. The main idea behind the construction is the link between solutions of von Neumann equations and entangled states of systems consisting of two subsystems evolving in time in opposite directions. Such a time evolution has analogies in realistic DNA where the polymerazes move on leading and lagging strands in opposite directions. Two-Strand Systems and Mutually Time-Reflected Turing MachinesAccording to Adleman [1] the process of DNA replication may be analyzed in terms of Turing machines: One strand plays a role of an instruction tape, a polymeraze is the read/write head, and the second strand contains the results of instructions. At a molecular level each strand is a sequence of molecules. In simple models one can represent sequences of molecules in a strand as chains of two-level systems (bits) in a state |ψ(t) = B1...Bn ψ(t) B1...Bn |B 1 . . . B n , B j = 0, 1. Thinking of the motion of the head in terms of a dynamics, one can write |ψ(t) = U (t, 0)|ψ(0) , where 0 ≤ t ≤ T . The final time T is the time of arrival of the head at the end of the strand, and U (t 1 , t 2 ) is a unitary operator which, in principle, may be different for different initial states of the system (this type of evolution occurs for nonlinear Schrödinger equations). A two-strand system can be represented by an entangled state

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Section: Soliton Dynamics Of the Two Strandsmentioning

confidence: 99%

Two-State Dynamics for Replicating Two-Strand Systems

Aerts

Czachor

2007

Open Syst. Inf. Dyn.

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Nonlinear quantum mechanics: a conflict with the Ptolomean structure?

Mielnik

2001

Physics Letters A

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Density matrix interpretation of solutions of Lie-Nambu equations

Czachor

Marciniak

1998

Physics Letters A

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The spectrum of a density matrix $\rho(t)$ is conserved by a Lie-Nambu dynamics if $\rho(t)$ is a self-adjoint and Hilbert-Schmidt solution of a nonlinear triple-bracket equation. This generalizes to arbitrary separable (positive- and indefinite-metric) Hilbert spaces the previous result which was valid for finite-dimensional Hilbert spaces.Comment: final version, to be published in Phys. Lett. A 239 No. 6, pp. 353-358 (16 March 1998

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Abstract: Linear quantum mechanics can be regarded as a particular example of a nonlinear Nambu-type theory. Some elements of this approach are presented.

Cited by 8 publications

References 42 publications

Two-State Dynamics for Replicating Two-Strand Systems

Two-State Dynamics for Replicating Two-Strand Systems

Nonlinear quantum mechanics: a conflict with the Ptolomean structure?

Density matrix interpretation of solutions of Lie-Nambu equations

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