Ашық рұқсат
Рұқсат берілді
Тек жазылушылар үшін
Том 87, № 5 (2023)
Articles
Vasilii Sergeevich Vladimirov
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2023;87(5):3-3
3-3
Fermions from classical probability and statistics defined by stochastic independence
Аннотация
The case study of fermions and the attempt to deduce their structure from classical probability opens new ways for classical and quantum probability, in particular, for the notion of stochastic coupling which, on the basis of the example of fermions, we enlarge to the notion of algebraic coupling, and for the various notions of stochastic independence. These notions are shown to be strictly correlated with algebraic and stochastic couplings. This approach allows to expand considerably the notion of open system. The above statements will be illustrated with some examples. The last section shows how, from these new stochastic couplings, new statistics emerge alongside the known Maxwell–Boltzmann, Bose–Einstein and Fermi–Dirac statistics.Bibliography: 5 titles.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2023;87(5):5-40
5-40
On weak solutions of boundary value problems for certain general differential equations
Аннотация
We study the general settings of the Dirichlet, Neiman, other boundary value problems for equations and systems of the form $\mathcal{L}^+ A\mathcal{L}u=f$ with general matrix differential operation $\mathcal L$ and some linear or non-linear operator $A$ acting in vector spaces $L^k_2(\Omega)$. Statements about the existence and uniqueness of a weak solution and the well-posedness of the formulated boundary value problems are obtained. As an operator $A$, the cases of operators Nemytsky and integral operators are considered. Also the cases of occurrence of lower derivatives are studied.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2023;87(5):41-56
41-56
On the detailed structure of quantum control landscape for fast single qubit phase-shift gate generation
Аннотация
In this work, we study the detailed structure of quantum control landscape for the problem of single-qubit phase shift gate generation on the fast time scale. In previous works, the absence of traps for this problem was proven on various time scales. A special critical point which was known to exist in quantum control landscapes was shown to be either a saddle or a global extremum, depending on the parameters of the control system. However, in the case of saddle the numbers of negative and positive eigenvalues of Hessian at this point and their magnitudes have not been studied. At the same time, these numbers and magnitudes determine the relative ease or difficulty for practical optimization in a vicinity of the critical point. In this work, we compute the numbers of negative and positive eigenvalues of Hessian at this saddle point and moreover, give estimates on magnitude of these eigenvalues. We also significantly simplify our previous proof of the theorem about this saddle point of the Hessian [Theorem 3 in B.O. Volkov, O.V. Morzhin, A.N. Pechen, J. Phys. A: Math. Theor. 54, 215303 (2021)].
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2023;87(5):57-70
57-70
Two-weight estimates for Hardy–Littlewood maximal functions and Hausdorff operators on $p$-adic Herz spaces
Аннотация
This paper is concerned with some sufficient conditions for the boundedness of Hardy–Littlewood maximal functions, rough Hausdorff and matrix Hausdorff operators on two-weighted Herz spaces on $p$-adic fields through its atomic decomposition.Bibliography: 34 titles.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2023;87(5):71-91
71-91
92-98
Operator-norm Trotter product formula on Banach spaces
Аннотация
Proof of the operator-norm convergent Trotter product formula on a Banach space is unexpectedly elaborate and a few of known results are based on assumption that at least one of the semigroups involved into this formula is holomorphic. Here we present an example of the operator-norm convergent Trotter product formula on a Banach space, where this condition is relaxed to demand that involved semigroups are contractive.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2023;87(5):99-123
99-123
Discrete symmetries of dynamics equations with polynomial integrals of higher degrees
Аннотация
In the article, systems with toric configuration space and kinetic energy in the form of a “flat” Riemannian metric on the torus are considered.The potential energy $V$ is a smooth function on the configuration torus.The dynamics of such systems are described by “natural” Hamiltonian systems of differential equations. If we replace $V$ with $\varepsilon V$, where $\varepsilon$ is a small parameter, then, according to Poincare, the study of such Hamiltonian systems at small values of $\varepsilon$ refers to the “main problem of dynamics”. The paper discusses a well-known hypothesis about unambiguous momentum-polynomial integrals of the equations of motion: if there is a momentum-polynomial integral of degree $m$, then there will necessarily be a linear or quadratic momentum first integral. It is fully proved for $m=3$ and $m=4$. The cases of “higher” degrees when $m=5$ and $m=6$ are discussed as well.Following the theory of Hamiltonian systems’ perturbations, we introduce resonant lines on the plane of impulses. If the system allows for a polynomial integral, then the number of these lines is finite. The symmetries of the set of resonant lines are found, which gives, in particular, the necessary conditions for integrability. Some new criteria for the existence of unambiguous polynomial integrals are obtained.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2023;87(5):124-139
124-139
On the positivity of direct image bundles
Аннотация
In the present paper, we obtain an equivalent relation between the log-plurisubharmonicity of the relative Bergman kernel, the Griffiths and Nakano positivity for the direct image with the natural $L^2$ metric, by finding a converse of Berndtsson’s Theorem on the direct image. A converse of Berndtsson’s generalization of Kiselman minimal principle is also obtained.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2023;87(5):140-163
140-163
Renormalization group transformation in the generalized fermionic hierarchical model
Аннотация
A two-dimensional hierarchical lattice is considered, in which an elementary cell is represented by the vertices of a square. In the generalized hierarchical model, the distance between opposite vertices of a square differs from the distance between adjacent vertices and is, in fact, a parameter of the new model. The Gaussian part of the Hamiltonian of the 4-component generalized fermionic hierarchical model is invariant under the block-spin transformation of the renormalization group. The transformation of the renormalization group in the space of coefficients that determine the Grassmann-valued density of the free measure is calculated explicitly as a homogeneous mapping of the fourth degree in a two-dimensional projective space. The properties of this mapping are described.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2023;87(5):164-176
164-176
On optimization of coherent and incoherent controls for two-level quantum systems
Аннотация
This article considers some control problems for closed and open two-level quantum systems. The closed system’s dynamics is governed by the Schrödinger equation with coherent control. The open system’s dynamics is governed by the Gorini–Kossakowski–Sudarshan–Lindblad master equation whose Hamiltonian depends on coherent control and superoperator of dissipation depends on incoherent control. For the closed system, we consider the problem for generation of the phase shift gate for some values of phases and final times for which we numerically show that zero coherent control, which is a stationary point of the objective functional, is not optimal; it gives an example of subtle point for practical solving quantum control problems. For the open system, in the two-stage method which was developed for generic $N$-level quantum systems in [A. Pechen, Phys. Rev. A., 84, 042106 (2011)] for approximate generation of target density matrices, here we consider the case of two-level systems, for which modify the first (“incoherent”) stage by numerically optimizing piecewise constant incoherent control instead of using constant incoherent control analytically computed using eigenvalues of the target density matrix. Exact analytical formulas are derived for the system’s state evolution, the objective functions and their gradients for the modified first stage. These formulas are then applied in the two-step gradient projection method. The numerical simulations show that the modified first stage’s duration can be significantly less than the unmodified first stage’s duration, but at the cost of performing optimization in the class of piecewise constant controls.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2023;87(5):177-203
177-203
Topological phases in solid state physics
Аннотация
The paper is a review devoted to topological phases – one of the actively developing directions in solid state physics. It is given an interpretation of topological phases in terms of generalized cohomology theories and K-theory.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2023;87(5):204-214
204-214
On the nontrivial solvability of a system nonlinear integral equations on the whole line
Аннотация
A system of singular integral equations with monotone and convex nonlinearity on the entire real line is considered. This system has applications in many areas of natural sciences. In particular, such systems are encountered in the theory of p-adic open-closed strings, in mathematical theory spatiotemporal spread of an epidemic in the framework of the well-known Diekmann-Kaper model, in the kinetic theory of gases, and in the theory of radiative transfer. An existence theorem for a nontrivial and bounded solution is proved. We also study the asymptotic behavior of the constructed solution on $\pm\infty$. At the end of the paper, concrete examples of applied nonlinearities and kernel functions are given.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2023;87(5):215-231
215-231
Classification of weighted dual graphs consisting of $-2$-curves and exactly one $-3$-curve
Аннотация
Let $(V, p)$ be a normal surface singularity. Let $\pi\colon (M, A)\to (V, p)$ be a minimal good resolution of $V$. The weighted dual graphs $\Gamma$ associated to $A$ completely describes the topology and differentiable structure of the embedding of $A$ in $M$. In this paper, we classify all the weighted dual graphs of $A=\bigcup_{i=1}^n A_i$ such that one of the curves $A_i$ is $-3$-curve, and the rest are all $-2$-curves. This is a natural generalization of Artin's classification of rational triple points. Moreover, we compute the fundamental cycles of maximal graphs (see Section 5) which can be used to determine whether the singularities are rational, minimally elliptic or weakly elliptic. We also give the formulas for computing arithmetic and geometric genera of star-shaped graphs.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2023;87(5):232-270
232-270