A complete metric in the set of mixing transformations

Tikhonov, S. V.

doi:10.1070/sm2007v198n04abeh003850

Cited by 34 publications

(40 citation statements)

References 20 publications

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“…For monostable networks, the rate equations admit a single steady state which is the same for the full and the reduced REs, i.e., Eqs. (15) and (16). It is therefore clear that all the solutions will tend to this state with time, quicker for fast variables and slower for the slow ones.…”

Section: Derivation Of the Slow-scale Linear Noise Approximationmentioning

confidence: 99%

“…(15a) and (15b) for the slow and fast variables are typically referred to as the degenerate and adjoined systems, respectively [11]. Tikhonov's first theorem [16,17] states that a simplification of the above equations under timescale separation conditions is possible whenever certain requirements are met: (1) the solutions of both the degenerate and adjoined systems [Eqs. (15)] are unique and their right-hand sides are continuous functions; (2) the root x f = h( x s ,τ ) is the stable solution of the adjoined system; and (3) the initial values x f (τ = 0) are in the domain of influence of the solution as in (2).…”

Section: Derivation Of the Slow-scale Linear Noise Approximationmentioning

confidence: 99%

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Rigorous elimination of fast stochastic variables from the linear noise approximation using projection operators

2012

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Rigorous elimination of fast stochastic variables from the linear noise approximation using projection operators Citation for published version: Thomas, P, Grima, R & Straube, AV 2012, 'Rigorous elimination of fast stochastic variables from the linear noise approximation using projection operators ' Physical Review E -Statistical, Nonlinear and Soft Matter Physics, vol. 86, no. 4, 041110, pp. - General rightsCopyright for the publications made accessible via the Edinburgh Research Explorer is retained by the author(s) and / or other copyright owners and it is a condition of accessing these publications that users recognise and abide by the legal requirements associated with these rights. Take down policyThe University of Edinburgh has made every reasonable effort to ensure that Edinburgh Research Explorer content complies with UK legislation. If you believe that the public display of this file breaches copyright please contact openaccess@ed.ac.uk providing details, and we will remove access to the work immediately and investigate your claim. The linear noise approximation (LNA) offers a simple means by which one can study intrinsic noise in monostable biochemical networks. Using simple physical arguments, we have recently introduced the slow-scale LNA (ssLNA), which is a reduced version of the LNA under conditions of timescale separation. In this paper we present the first rigorous derivation of the ssLNA using the projection operator technique and show that the ssLNA follows uniquely from the standard LNA under the same conditions of timescale separation as those required for the deterministic quasi-steady-state approximation. We also show that the large molecule number limit of several common stochastic model reduction techniques under timescale separation conditions constitutes a special case of the ssLNA.

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Section: Derivation Of the Slow-scale Linear Noise Approximationmentioning

confidence: 99%

Section: Derivation Of the Slow-scale Linear Noise Approximationmentioning

confidence: 99%

Rigorous elimination of fast stochastic variables from the linear noise approximation using projection operators

2012

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“…It is noteworthy that the chemical scheme consists of elementary reactions, that is monomolecular and bi-molecular reactions without autocatalytic steps. We assume that the total concentrations of catalysts (enzymes) E and E' are much smaller than the concentrations of the reactant S and the product P. On the basis of the Tikhonov theorem [33], the concentrations of both catalysts (enzymes) and their complexes may be eliminated as fast variables, and the dynamics of the system may be described by the two kinetic equations for the reactant S and the product P only.…”

Section: Model and Resultsmentioning

confidence: 99%

New type of the source of travelling impulses in two-variable model of reaction–diffusion system

Kawczyński

Nowakowski

2016

Reac Kinet Mech Cat

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A two-variable model of a one-dimensional, open, excitable, finite reaction-diffusion system describing time-space evolution of traveling impulses is investigated. It is shown that depending on the size of the system, the traveling impulse after reflection can generate either a source of decaying traveling impulses or a stationary periodical structure. A continuous increase of the size of the system causes periodical repetitions of these patterns. The chemical model is realistic and can become a stimulus for seeking the described effect in experiments.

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“…According to Tikhonov [30], Takens [29], Jones [20] and Kaper [21] singularly perturbed systems may be defined such as:…”

Section: Singularly Perturbed Systemsmentioning

confidence: 99%

“…The classical geometric theory of differential equations developed originally by Andronov [1], Tikhonov [30] and Levinson [23] stated that singularly perturbed systems possess invariant manifolds on which trajectories evolve slowly, and toward which nearby orbits contract exponentially in time (either forward or backward) in the normal directions. These manifolds have been called asymptotically stable (or unstable) slow invariant manifolds 1 .…”

Section: Introductionmentioning

confidence: 99%