Feinberg, M.: `Chemical reaction network structure and the stabiliy of complex isothermal reactors - I. The deficiency zero and deficiency one theorems', Review Article 25, Chemical Engr. Sci., 1987, 42, pp. 2229-2268.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....is discussed. A definition of parameter robustness is proposed and an explicit observer for zero deficiency chemical networks is presented, which is robust in this sense. 1 Introduction Consider the following model for chemical reaction networks of Feinberg Horn Jackson zero deficiency type ([3, 4, 5]) with mass action kinetics: x = f A (x) n (b i b j ) 1.1) together with a set of measurements y = h(x) 1.2) The vector x represents the concentration of each species involved in the reactions and thus we will be interested only in those trajectories that evolve in ....

M. Feinberg, "Chemical reaction network structure and the stabiliy of complex isothermal reactors - I. The deficiency zero and deficiency one theorems", Review Article 25, Chemical Engr. Sci., 42, 1987, 2229--2268.

....arise in the modeling of chemical reaction systems by the so called mass action kinetics. The polynomials in the system are de ned by two graphs, a weighted directed graph for the chemical reactions and a weighted bipartite graph for the involved chemicals. In the non algebraic chemical literature [4, 11, 12, 13, 14, 17, 23, 28] there have been several attempts to study the number of real positive solutions depending on the structure of the graphs while dynamic phenomena have been studied a lot by numerical mathematicians, see e.g. 7] The investigation of real positive solutions of the sparse polynomial system f(x) 0 ....

....contains precisely one positive real solution. Proof: The conditions on the ranks and components assure that Y A (x) 0 is equivalent to a binomial system. Using the Smith normal form the argumentation is analogous to the de ciency zero theorem. 2 8. Using toric varieties Feinberg formulates in [11, 12, 13] a theorem which he calls the de ciency one theorem. The assumptions in [13] are more restrictive then in [11] We give here our version. Theorem 8.1: De ciency One Theorem, Feinberg [12] p. 2259, proof in [15] Let the graph be weakly reversible and rank(Y A ) jL j 2 for each connected ....

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M. Feinberg. Chemical reaction network structure and the stability of complex isothermal reactors-II. multiple steady states for networks of deciency one. Chemical Engineering Science, 43:1-25, 1988. Review Article Number 26.

....arise in the modeling of chemical reaction systems by the so called mass action kinetics. The polynomials in the system are de ned by two graphs, a weighted directed graph for the chemical reactions and a weighted bipartite graph for the involved chemicals. In the non algebraic chemical literature [4, 11, 12, 13, 14, 17, 23, 28] there have been several attempts to study the number of real positive solutions depending on the structure of the graphs while dynamic phenomena have been studied a lot by numerical mathematicians, see e.g. 7] The investigation of real positive solutions of the sparse polynomial system f(x) 0 ....

....constants c i and so are the coecients j . 2 A trajectory in the positive orthant (R 0 ) m staying in the cone x 0 P j j B j stays in particular in (x 0 im(B) R 0 ) m . Since in our context two special cases of B are interesting we have the following de nitions. De nition: In [12] the vector space im(Y A) is called kinetic space and S = im(Y I a ) span(fy i y j j (C j C i ) 2 Rg) is called stoichiometric space. Obviously, im(Y A) S. In Section 3 we will give a sucient condition for im(Y A) S. Choosing di erent sets of constants fk ij j (C j C i ) 2 Rg in general ....

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M. Feinberg. Chemical reaction network structure and the stability of complex isothermal reactors-I. the deciency zero and deciency one theorems. Chemical Engineering Science, 42:2229-2268, 1987. Review Article Number 25.

....the theorems given here will be possible. The class of systems which we consider is, basically, that of deficiency zero chemical reaction networks with mass action kinetics (and one linkage class) in the language of the beautiful and powerful theory developed by Feinberg, Horn, and Jackson, cf. [2, 3, 4, 5, 7]. As a matter of fact, our stability (but not the robustness nor the feedback) results are basically contained in that previous work. The existence and uniqueness of equilibria, and local asymptotic stability, are already proved in the papers by Feinberg, Horn, and Jackson, and we claim no ....

....Horn, and Jackson, and we claim no originality whatsoever in that regard. And although the global stability results may not be readily apparent from a casual reading of that literature, our proofs of them consist basically of a careful repackaging of the discussion found in Feinberg s paper [2]. One should point out that an alternative approach to global stability, which would apply to a somewhat smaller class of systems, could use ideas from [10] which, in turn, was based on [12] The reliance upon the Feinberg Horn Jackson theory notwithstanding, we provide here a totally ....

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Feinberg, M., "Chemical reaction network structure and the stabiliy of complex isothermal reactors - I. The deficiency zero and deficiency one theorems," Review Article 25, Chemical Engr. Sci. 42(1987): 2229-2268.

....arise in the modelling of chemical reaction systems by the so called mass action kinetics. The polynomials in the system are de ned by two graphs, a weighted directed graph for the chemical reactions and a weighted bipartite graph for the involved chemicals. In the chemical engineering literature [3, 8, 9, 10, 11, 14, 16, 23] there have been several attempts to study the number of real positive solutions depending on the structure of the graphs. The investigation of real positive solutions of the sparse polynomial system f(x) 0 is very important for the application since this are the steady state solutions of a ....

....z 1 ; u d . For each solution z of the rst system there may be several families of solutions of the second system. The remaing statements of the theorem follow analogous to the argumentation in the proof of the de ciency zero theorem. 2 6 More binomial cases Feinberg formulates in [8, 9, 10] a theorem which he calls the de ciency one theorem. The assumptions in [10] are more restrictive then in [8] We give here our version. Theorem 6.1 Consider the system Y A (x) 0 of polynomial equations de ned by a directed graph with weighted adjacency matrix K and a bipartite graph with ....

[Article contains additional citation context not shown here]

M. Feinberg. Chemical reaction network structure and the stability of complex isothermal reactors-II. multiple steady states for networks of deciency one. Chemical Engineering Science, 43:1-25, 1988. Review Article Number 26.

....arise in the modelling of chemical reaction systems by the so called mass action kinetics. The polynomials in the system are de ned by two graphs, a weighted directed graph for the chemical reactions and a weighted bipartite graph for the involved chemicals. In the chemical engineering literature [3, 8, 9, 10, 11, 14, 16, 23] there have been several attempts to study the number of real positive solutions depending on the structure of the graphs. The investigation of real positive solutions of the sparse polynomial system f(x) 0 is very important for the application since this are the steady state solutions of a ....

.... j = R t 2 t 1 j (x(t) dt = R t 2 t 1 x y j (t)dt. If x(t) is non negative then the monomials x y j are non negative and so are the coe cients j . 2 A trajectory staying in (R 0 ) m and in the cone x 0 P j j Y a j stays in particular in (x 0 im(Y A) R 0 ) m . In [9] im(Y A) is called kinetic subspace. Since the columns of A are linear combinations of i j where C j C i are elements of the directed graph R Feinberg [8] p. 2 15 de nes the so called stoichiometric space S = span(fy i y j j (C j C i ) 2 Rg) In [8] the space S is a natural choice since ....

[Article contains additional citation context not shown here]

M. Feinberg. Chemical reaction network structure and the stability of complex isothermal reactors-I. the deciency zero and deciency one theorems. Chemical Engineering Science, 42:2229-2268, 1987. Review Article Number 25.

....exist for the standard CSTR model and other perturbations of this model. Naturally, it is important to consider other extensions to these prior results, for example the case of two consecutive reactions A B C. We refer the interested reader to Farr and Aris (1986) and Moiola et al. 1990) Feinberg (1987, 1988) has developed the Deficiency Zero and the Deficiency One Theorems and used these to analyze the multiplicity behavior of a much broader class of isothermal systems. The previous multiplicity and control studies dealing with the standard CSTR model have assumed that the inner loop of the cascade ....

Feinberg, M. (1988). Chemical reaction network structure and the stability of complex isothermal reactors - II. Multiple steady-states for networks of deficiency one. Chem. Eng. Sci., 43, 1-25.

....exist for the standard CSTR model and other perturbations of this model. Naturally, it is important to consider other extensions to these prior results, for example the case of two consecutive reactions A B C. We refer the interested reader to Farr and Aris (1986) and Moiola et al. 1990) Feinberg (1987, 1988) has developed the Deficiency Zero and the Deficiency One Theorems and used these to analyze the multiplicity behavior of a much broader class of isothermal systems. The previous multiplicity and control studies dealing with the standard CSTR model have assumed that the inner loop of the ....

Feinberg, M. (1987). Chemical reaction network structure and the stability of complex isothermal reactors - I. The deficiency zero and deficiency one theorems. Chem. Eng. Sci., 42, 2229-2268.

....exist for the standard CSTR model and other perturbations of this model. Naturally, it is important to consider other extensions to these prior results, for example the case of two consecutive reactions A B C. We refer the interested reader to Farr and Aris (1986) and Moiola et al. 1990) Feinberg (1987, 1988) has developed the Deficiency Zero and the Deficiency One Theorems and used these to analyze the multiplicity behavior of a much broader class of isothermal systems. 1.3 Current Work The previous multiplicity and control studies dealing with the standard CSTR model have assumed that the ....

Feinberg, M. (1988). Chemical reaction network structure and the stability of complex isothermal reactors - II. Multiple steady-states for networks of deficiency one. Chem. Eng. Sci., 43, 1-25.

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Feinberg, M., "Chemical reaction network structure and the stabiliy of complex isothermal reactors - I. The deficiency zero and deficiency one theorems," Review Article 25, Chemical Engr. Sci. 42(1987): 2229-2268.

....of molecules) will lead to a better understanding of cellular dynamical processes. The main objective of this paper is to construct (when and if possible) state observers for the class of systems which describe chemical reaction networks of the type introduced by Feinberg, Horn, and Jackson in [6, 7, 8, 9, 11, 12]. As outputs, we take a subset of state variables or, more generally, monomials in state variables (which could, in practice, be associated to measured reaction rates) We provide here a complete solution to the observer problem for this class. The results given in [20] provide a convenient ....

Feinberg, M., "Chemical reaction network structure and the stabiliy of complex isothermal reactors - I. The deficiency zero and deficiency one theorems," Review Article 25, Chemical Engr. Sci. 42(1987): 2229-2268.

No context found.

Feinberg, M.: `Chemical reaction network structure and the stabiliy of complex isothermal reactors - I. The deficiency zero and deficiency one theorems', Review Article 25, Chemical Engr. Sci., 1987, 42, pp. 2229-2268.

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M. Feinberg, "Chemical reaction network structure and the stabiliy of complex isothermal reactors - I. The deficiency zero and deficiency one theorems," Review Article 25, Chemical Eng. Science (1987)42:2229-2268.

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M. Feinberg. Chemical reaction network structure and the stability of complex isothermal reactors I. Chemical Engineering Science, 42:22292268, 1987.

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M. Feinberg. Chemical reaction network structure and the stability of complex isothermal reactors I. Chemical Engineering Science, 42:22292268, 1987.

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