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A. A. Azamov 《Mathematical Notes》2011,90(1-2):157-161
We describe an example of a three-dimensional linear differential game with convex compact sets of control. In this example, the integrand in Pontryagin’s first direct method is discontinuous on a set of positive measure. 相似文献
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The time-optimal problem for a controlled system with evolution-type distributed parameters is considered. An upper estimate is obtained for the optimal transition time into the zero state. 相似文献
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A. A. Azamov O. S. Akhmedov 《Computational Mathematics and Mathematical Physics》2011,51(8):1353-1359
The DN-tracking method is used to prove the existence of a closed trajectory in a quadratic system of ordinary differential equations in three dimensions. 相似文献
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Rahmatullaev M. M. Rafikov F. K. Azamov Sh. Kh. 《Ukrainian Mathematical Journal》2021,73(7):1092-1106
Ukrainian Mathematical Journal - We consider the Potts model on a Cayley tree and prove the existence of Gibbs measures constructed by the method proposed in [H. Akin, U. A. Rozikov, and S. Temir,... 相似文献
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At the 1974 International Congress, I. M. Singer proposed that eta invariants and hence spectral flow should be thought of
as the integral of a one form. In the intervening years this idea has lead to many interesting developments in the study of
both eta invariants and spectral flow. Using ideas of [24] Singer’s proposal was brought to an advanced level in [16] where
a very general formula for spectral flow as the integral of a one form was produced in the framework of noncommutative geometry.
This formula can be used for computing spectral flow in a general semifinite von Neumann algebra as described and reviewed
in [5]. In the present paper we take the analytic approach to spectral flow much further by giving a large family of formulae
for spectral flow between a pair of unbounded self-adjoint operators D and D + V with D having compact resolvent belonging to a general semifinite von Neumann algebra and the perturbation . In noncommutative geometry terms we remove summability hypotheses. This level of generality is made possible by introducing
a new idea from [3]. There it was observed that M. G. Krein’s spectral shift function (in certain restricted cases with V trace class) computes spectral flow. The present paper extends Krein’s theory to the setting of semifinite spectral triples
where D has compact resolvent belonging to and V is any bounded self-adjoint operator in . We give a definition of the spectral shift function under these hypotheses and show that it computes spectral flow. This
is made possible by the understanding discovered in the present paper of the interplay between spectral shift function theory
and the analytic theory of spectral flow. It is this interplay that enables us to take Singer’s idea much further to create
a large class of one forms whose integrals calculate spectral flow. These advances depend critically on a new approach to
the calculus of functions of non-commuting operators discovered in [3] which generalizes the double operator integral formalism
of [8–10]. One surprising conclusion that follows from our results is that the Krein spectral shift function is computed,
in certain circumstances, by the Atiyah-Patodi-Singer index theorem [2]. 相似文献
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Let be a Schrödinger operator on , , or 3, where is a bounded measurable real‐valued function on . Let V be an operator of multiplication by a bounded integrable real‐valued function and put for real r. We show that the associated spectral shift function (SSF) ξ admits a natural decomposition into the sum of absolutely continuous and singular SSFs. In particular, the singular SSF is integer‐valued almost everywhere, even within the absolutely continuous spectrum where the same cannot be said of the SSF itself. This is a special case of an analogous result for resolvent comparable pairs of self‐adjoint operators, which generalises the case of a trace class perturbation appearing in [2] while also simplifying its proof. We present two proofs which demonstrate the equality of the singular SSF with two a priori different and intrinsically integer‐valued functions which can be associated with the pair H0, V: the total resonance index [3] and the singular μ‐invariant [2]. 相似文献
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