Convergence and rate of convergence of a non-autonomous gradient system on Hadamard manifoldsAhmadi, P.; Khatibzadeh, H.
doi: 10.1134/S1995080214030032pmid: N/A
In this paper we consider the following nonhomogeneous gradient system on a Hadamard manifold M
$\left\{ \begin{gathered}
- x'(t) = grad\phi (x(t)) + e(t), \hfill \\
x(0) = x_0 \hfill \\
\end{gathered} \right.
$
where φ: M → ℝ is a geodesically convex function of class C
2 with argmin φ ≠ Ø. We prove global convergence of solutions of the gradient systemto a minimum point of φ. We also discuss on the rate of convergence of φ(x(t)) to the minimum value of φ as well as the rate of convergence of ‖x′(t)‖ and d(x(t), p) to zero, where p is the minimum point of φ. Finally, we present some problems and future directions to study.
On the almost sure convergence of some ergodic meansBoukhari, F.
doi: 10.1134/S1995080214030068pmid: N/A
We study the asymptotic behavior of ergodic averages of expectation of some randomly selected group of unitary operators, showing the mean convergence when the sequence of selectors is a ℤ
d
-valued random walk. We make use of the spectral decomposition of the unitary group to investigate the more difficult problem of almost sure convergence, and provide sufficient spectral conditions which carry out the almost everywhere convergence of these means when the sequence of selectors is a ℤ
d
-valued random walk satisfying some integrability conditions. We also show that this condition is optimal for d = 1, and deduce a speed of convergence for these averages using a Rademacher-Menchoff theorem on orthogonal series.
Dibaric and evolution algebras in biologyLadra, M.; Omirov, B.; Rozikov, U.
doi: 10.1134/S199508021403007Xpmid: N/A
We find conditions on ideals of an algebra under which the algebra is dibaric. Dibaric algebras have not non-zero homomorphisms to the set of the real numbers. We introduce a concept of bq-homomorphism (which is given by two linear maps f, g of the algebra to the set of the real numbers) and show that an algebra is dibaric if and only if it admits a non-zero bq-homomorphism. Using the pair (f, g) we define conservative algebras and establish criteria for a dibaric algebra to be conservative. Moreover, the notions of a Bernstein algebra and an algebra induced by a linear operator are introduced and relations between these algebras are studied. For dibaric algebras we describe a dibaric algebra homomorphism and study their properties by bq-homomorphisms of the dibaric algebras. We apply the results to the (dibaric) evolution algebra of a bisexual population. For this dibaric algebra we describe all possible bq-homomorphisms and find conditions under which the algebra of a bisexual population is induced by a linear operator. Moreover, some properties of dibaric algebra homomorphisms of such algebras are studied.
Iterative solution methods for mesh approximation of control and state constrained optimal control problem with observation in a part of the domainLapin, A.; Khasanov, M.
doi: 10.1134/S1995080214030081pmid: N/A
Iterative solution methods for finite-dimensional constrained saddle point problems are investigated theoretically and numerically. These saddle point problems arise when approximating differential optimal control problems with point-wise state and control constraints by finite element or finite difference schemes with further using Lagrange multipliers technique. The linear elliptic boundary value problems with distributed control and the observation in a part of the domain are considered. Equivalent transformations of the constructed finite-dimensional saddle point problem are executed to apply effectively Uzawa-type iterative methods. Numerical comparison of these methods with gradient method for a regularized problem and interior point method is done.
Connections in the second order tangent bundle with extended structure groupShurygin, V.; Vashurina, L.
doi: 10.1134/S199508021403010Xpmid: N/A
The second order tangent bundle T
2
M of a smooth manifold M carries a natural structure of a smooth manifold over the algebra D
2 of truncated polynomials of degree two in one variable, which gives rise to an extended structure group of T
2
M and the corresponding extended second order frame bundle
$\hat P^2 M$
associated to T
2
M. Two connections in
$\hat P^2 M$
are said to be equivalent if one of them can be mapped into the other by a fiber preserving D
2-diffeomorphism of T
2
M to itself. We establish necessary and sufficient conditions under which two connections in
$\hat P^2 M$
are equivalent and in particular the conditions under which a connection in
$\hat P^2 M$
is equivalent to a second order differential connection onM.
Tripotents in algebras: Invertibility and hyponormalityBikchentaev, A.
doi: 10.1134/S1995080214030056pmid: N/A
Let A be a unital algebra over complex field ℂ, I be the unit of A. An element A ∈ A is called tripotent if A
3 = A. Let A
tri = {A ∈ A: A
3 = A}. We show that A ∈ A
tri if and only if I ± A − A
2 ∈ A
tri. We study invertibility properties of elements I + λA with A ∈ A
tri and λ ∈ ℂ \ {−1,1}. Let X be a Banach space with the approximation property and A, B ∈ B(X)tri. If A − B is a nuclear operator then tr(A − B) ∈ ℂ. We show that if H is a Hilbert space and an operator A ∈ B(H)tri is hyponormal or cohyponormal then A = A*. We also prove that every A ∈ B(H)tri similar to a Hermitian tripotent.