On Some Symmetries of Quadratic Systems Han, Maoan; Petek, Tatjana; Romanovski, Valery G.
Symmetry (Bandung),
08/2020, Letnik:
12, Številka:
8
Journal Article
Recenzirano
Odprti dostop
We provide a general method for identifying real quadratic polynomial dynamical systems that can be transformed to symmetric ones by a bijective polynomial map of degree one, the so-called affine ...map. We mainly focus on symmetry groups generated by rotations, in other words, we treat equivariant and reversible equivariant systems. The description is given in terms of affine varieties in the space of parameters of the system. A general algebraic approach to find subfamilies of systems having certain symmetries in polynomial differential families depending on many parameters is proposed and computer algebra computations for the planar case are presented.
Knowledge of the mathematical models of the fermentation processes is indispensable for their simulation and optimization and for the design and synthesis of the applicable control systems. The paper ...focuses on determining a dynamic mathematical model of the milk fermentation process taking place in a batch bioreactor. Models in the literature describe milk fermentation in batch bioreactors as an autonomous system. They do not enable the analysis of the effect of temperature changes on the metabolism during fermentation. In the presented extensive multidisciplinary study, we have developed a new mathematical model that considers the impact of temperature changes on the dynamics of the CO2 produced during fermentation in the batch bioreactor. Based on laboratory tests and theoretical analysis, the appropriate structure of the temperature-considered dynamic model was first determined. Next, the model parameters of the fermentation process in the laboratory bioreactor were identified by means of particle swarm optimization. Finally, the experiments with the laboratory batch bioreactor were compared with the simulations to verify the derived mathematical model. The developed model proved to be very suitable for simulations, and, above all, it enables the design and synthesis of a control system for batch bioreactors.
Let B(X) be the algebra of all bounded linear operators on a complex or real Banach space X with dimX≥3. In this paper, we characterize the maps from B(X) into itself which preserve the ascent of ...product of operators or, they preserve the descent of product of operators. It turns out that both problems are connected with preservers of the rank-one nilpotency of the product.
Let
X
be a real or complex Banach space of dimension at least 3. We give a complete description of surjective mappings on
B
(
X
) that preserve the ascent of Jordan triple product of operators or, ...preserve the descent of Jordan triple product of operators.
Given n≥2 let 2≤k≤n be fixed. We study mappings ϕ on Mn, the algebra of n×n complex matrices, that satisfy Nk(ϕ(A)ϕ(B)+ϕ(B)ϕ(A))=Nk(AB+BA) for all A, B∈Mn, where Nk(C) denotes the Ky-Fan k-norm, the ...sum of k greatest singular values of the matrix C. If n≥3, we additionally suppose either ϕ(μ1I)=μ2I, for some unimodular complex μ1, μ2, or, that ϕ is surjective; and when n=2, the complete description is obtained without additional assumptions.
Surjective isometries in a given unitarily invariant norm on n-by-n complex matrices and with respect to Jordan product, are classified. Moreover, the similar problem is considered for a much wider ...class of unitarily invariant functions.
We give the complete description of bijective real-linear transformations on B(H), the algebra of all bounded linear operators on an infinite-dimensional complex Hilbert space H, where every unitary ...U∈B(H) is mapped to a unitary ϕ(U). In turn, bijective real-linear maps on B(H) preserving equivalence by unitaries will be determined.
For a given family of real planar polynomial systems of ordinary differential equations depending on parameters, we consider the problem of how to find the systems in the family which become ...time-reversible after some affine transformation. We first propose a general computational approach to solve this problem, and then demonstrate its usage for the case of the family of quadratic systems.
Let S(H) be the set of all linear positive-semidefinite self-adjoint Trace-one operators (states) on H where H is an at least two-dimensional finite-dimensional real or complex Hilbert space or at ...least three-dimensional left quaternionic Hilbert space of dimension n. Given a strictly convex function f:0,1↦R, for any ρ∈S(H) we define F(ρ)=∑if(λi), where λ1,λ2,…,λn are the eigenvalues of ρ counted with multiplicities. In this note, we completely describe maps ϕ:S(H)→S(H) having the property F(tρ+(1−t)σ)=F(tϕ(ρ)+(1−t)ϕ(σ)) for all t∈0,1 and every ρ,σ∈S(H). It turns out that ϕ(ρ)=UρU⁎, ρ∈S(H), where U is a real-linear isometry of H. Note that there is no surjectivity assumption and that our result in particular improves the description of maps preserving the von Neumann entropy of convex combinations of states in the complex Hilbert space. It can as well be applied to preserving Schatten or some other strictly convex norms of convex combinations of states.
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