Vol 87, No 5 (2023)

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.

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.

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.

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.

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.

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.