Let $\mathcal A$ be a unital operator algebra and let $\theta: \mathcal A \to B(\mathcal H)$ be a continuous unital homomorphism. We prove that for all $n \in \mathbb{N}$ and all $\beta$ in the dual space of $\mathcal{A}$, \begin{equation*} \|\theta^{(n)}\| \le \max(1, \|\theta^{(n)} + \beta^{(n)} I \|). \end{equation*} Here, $\theta^{(n)}: M_n(\mathcal A) \to B(\mathcal H^n)$ denotes the $n$-th matrix ampliation of $\theta$. This extends a result of Clou\^atre, Ostermann and Ransford (the case $n=1$), who were motivated by Crouzeix's conjecture. Our result allows us to control the completely bounded norm of $\theta$, which in turn has dilation theoretic consequences. As an application, we obtain a new proof of the similarity theorem of Okubo and Ando, which says that if $T \in B(\mathcal H)$ is an operator of class $C_\rho$, where $\rho \ge 1$, then there exists an invertible operator $S$ such that $\|S^{-1} T S \| \le 1$ and such that $\|S\| \|S^{-1}\| \le \rho$. The conclusion of the Okubo--Ando theorem implies that the inequality \[ \|f(T)\| \le \rho \|f\|_{\overline{\mathbb D}} \] holds for all polynomials $f$. A direct proof of this inequality was recently given by Clou\^atre, Ostermann and Ransford. Our main result shows that this inequality in fact holds for matrix-valued polynomials, so that the existence of the similarity $S$ follows from Paulsen's similarity theorem, which says that an operator $T$ is completely polynomially bounded with constant $\rho$ if and only if there exists an invertible operator $S$ such that $\|S^{-1} T S \| \le 1$ and such that $\|S\| \|S^{-1}\| \le \rho$. The key to proving our main result is to adapt a variational argument of Caldwell, Greenbaum and Li to matrix-valued holomorphic functions. Rather than working with biholomorphic automorphisms of the disc, we work with Potapov--M\"obius transforms $m_A$, where $A$ belongs to the unit ball of $M_n(\mathbb C)$. A slightly simplified version of our key lemma, which is shown using this technique, then reads as follows: If $R \in M_n(B(\mathcal H))$ satisfies $\|R\| > 1$ and $\|m_{A \otimes I}(R)\| \le \|R\|$ for all $A$ close to $0$ and if $x \in \mathcal H^n$ with $\|x\| = 1$ and $\|R x\| = \|R\|$, then \[ \langle R x, (A \otimes I) x \rangle = 0 \] for all $A \in M_n(\mathcal C)$.
The article is dedicated to the 125th anniversary of the birth of Naum Illich Akhiezer, an outstanding mathematician, professor of Kharkiv University, and leader of the Kharkiv school of mathematicians of the mid-twentieth century. It briefly outlines his life, scientific and pedagogical work, his role in the development of mathematics and mathematical education in Kharkiv, and the preservation of his memory.
The paper studies an implicit first-order linear difference equation $BX_{n+1}=AX_n+F_n,\quad n=0,1,2,\ldots$ over a finite commutative ring $R$ with identity, that is, an equation with a noninvertible element $B$ of the ring $R$. In contrast to the classical (explicit) linear difference equation, an implicit linear difference equation over the finite ring $R$ may have no solutions, and may also have infinitely many solutions. Since any finite commutative ring with identity is isomorphic to a finite direct sum of local commutative rings with identity, the equation decomposes into a system of equations over local finite commutative rings with identity. It is shown that the condition that the ideal $(A,B)$ generated by the elements $A,B\in R$ coincides with $R$ is necessary and sufficient for the existence of a finite number of solutions of this equation; the number of solutions in the case of their existence is counted and a formula for the general solution is provided. The condition $(A,B)\ne R$ is a necessary and sufficient condition for the existence of an infinite number of solutions of the corresponding homogeneous equation $BX_{n+1}=AX_n,\quad n=0,1,2,\ldots$. It is also established that in the case $(A,B)\ne R$ the condition $F_n\in (A,B),\quad n=0,1,2,\ldots$ is necessary for the solvability of the nonhomogeneous implicit linear difference equation, but it is not sufficient, as examples show. Under the additional restriction that $(A,B)$ is a proper principal ideal of the ring $R$, this condition is also sufficient for the existence of a solution of the considered nonhomogeneous equation; in this situation the equation has infinitely many solutions. Under the assumption that $(A,B)$ is a principal ideal, first a criterion for the existence of a solution is proved in the case of a local finite commutative ring with identity, and then in the case of an arbitrary finite commutative ring with identity. As in Fredholm theory, it is shown that if the corresponding homogeneous equation has only the trivial solution, then the studied nonhomogeneous equation has a unique solution. The work of the proved theorems is demonstrated by concrete examples.
It is well known that the Cauchy problem is ill posed for partial differential equations with constant coefficients if they do not satisfy the Petrovsky well-posedness condition. In this work, an integral condition is proposed under which the resulting boundary value problem becomes well-posed in the Schwartz space, as well as in spaces of functions with polynomial growth in spatial variables. The presented integral condition is an integral over a semi-axis with an exponential weight, which ensures the convergence of this integral. By considering this integral as the Laplace transform of the function $\exp(-t^2/2)$, one can obtain its asymptotics, which allow to estimate the resolving function. This estimate makes it possible to prove a theorem on the well-posedness of the resulting boundary value problem in the Schwartz space, as well as in a scale of functions of finite smoothness with polynomial growth. Furthermore, the problem is considered for an inhomogeneous differential equation where the right-hand side belongs to the Schwartz space in the spatial variables and is compactly supported in the time variable. For the resulting problem, a Green's function is found and estimated from above. Using this estimate, a theorem on the well-posedness of this problem is proved in the Schwartz space and its dual space. Examples of equations that are ill posed in the sense of Petrovsky are provided, and specific integral conditions are indicated under which the resulting problems will be well-posed in the Schwartz space. \\ Examples of Petrovsky-incorrect equations are given and specific integral conditions are indicated under which the resulting problems will be correct in the L. Schwartz space. Consider the equation $$\displaystyle\frac{\partial u(x_1,x_2,t)}{\partial t}=\displaystyle\frac{\partial^2 u(x_1,x_2,t)}{\partial x_1^2}-\displaystyle\frac{\partial^2 u(x_1,x_2,t)}{\partial x_2^2}. $$ For this equation, the Petrovsky conditions are not satisfied, since the polynomial $P(s_1,s_2)=-s_1^2+s_2^2$ is unbounded. But if we add the condition $$\int\limits_0^{\infty} \exp\left(-\displaystyle\frac{t^2}{2}\right) u(x_1,x_2,t)dt=\varphi(x_1,x_2),$$ then which the boundary value problem becomes well-posed in the L.\,Schwartz space, as well as in the scale of Banach spaces.
The paper addresses the stabilization problem for canonical nonlinear systems, which in the triangular case correspond to systems in p--normal form. These systems are inherently nonlinear because they are not feedback linearizable, and their linear approximation is degenerate and does not characterize the stability of the equilibrium at the origin. In the triangular case, backstepping is a standard feedback design tool due to its recursive nature. Backstepping provides a powerful framework for stabilizing a wide class of nonlinear systems in strict-feedback form. However, this recursive approach typically leads to feedback controllers of increasing complexity as the system dimension grows. This motivates the investigation of simpler control design methods. Several notable results on non-recursive stabilization have been obtained in recent years. The classes of polynomial control laws have been constructed for the case of strictly decreasing exponents of power terms in the right-hand side of the system. In particular, it has been shown that the system can be asymptotically stabilized by a linear feedback control law in the strictly decreasing case. Moreover, control coefficients can be chosen as arbitrary positive numbers. It is also established that it is possible to achieve stabilization in a certain special case of non-strictly decreasing exponents under additional conditions on the control coefficients. In this work, we solve the linear stabilization problem in a previously unexplored case of non-strictly decreasing exponents for a three-dimensional system. We propose constructive method of finding conditions for the control coefficients to guarantee asymptotic stabilization. Our approach is based on the Lyapunov function method. We also use stabilization through nonlinear approximation to generalize our result. The effectiveness of the proposed approach is demonstrated and confirmed by numerical examples.
Статтю присвячено 125-річчю від дня народження Наума Ілліча Ахієзера, видатного математика, професора Харківського університету та лідера Харкiвської школи математикiв середини минулого сторiччя. Коротко висвітлено його життєвий шлях, наукову й педагогічну діяльність, роль у розвитку математики та математичної освіти в Харкові, а також збереження пам’яті про нього.
A real scalar polynomial whose roots lie in the left half--plane of the complex plane is called a Hurwitz polynomial. This notion goes back to the works of J. C. Maxwell, E. J. Routh, and A. Hurwitz, and was later studied using techniques such as Sturm sequences, Markov parameters, and continued fractions; see, for instance, \emph{The Theory of Matrices}, Vol. II, Chapter XV, AMS Chelsea (2000), by F. Gantmacher. A $q\times q$ matrix polynomial $P(z)$ is called Hurwitz if $\det P(z)$ is a Hurwitz polynomial. Every matrix polynomial $f_n$ can be written in the form $f_n(z)=h_n(z^2)+z\,g_n(z^2)$. The matrix polynomial $f_{2m}$ is said to be of Hurwitz type if the expression $g_{2m}(z)h_{2m}^{-1}(z)$ admits a representation as a finite continued fraction with positive definite matrix coefficients. Similarly, the odd-degree matrix polynomial $f_{2m+1}$ is of Hurwitz type if $\frac{1}{z}h_{2m+1}(z)g_{2m+1}^{-1}(z)$ has the same property. The concept of Hurwitz-type matrix polynomials was introduced in \emph{On matrix Hurwitz type polynomials and their interrelations to Stieltjes positive definite sequences and orthogonal matrix polynomials}, Linear Algebra Appl. 476 (2015), by A. E. Choque Rivero. In the present work, we derive an explicit form of the Bezoutian associated with Hurwitz-type matrix polynomials. The fact that Hurwitz-type matrix polynomials are Hurwitz matrix polynomials was suggested and partially proved using Bezoutians in \emph{On generalization of classical Hurwitz stability criteria for matrix polynomials}, J. Comput. Appl. Math. 383 (2021), by X. Zhan and A. Dyachenko. In contrast to that work, we employ the decomposition of the Bezoutian form introduced in \emph{Some Questions in the Theory of Moments}, Translations of Mathematical Monographs~2, AMS, 1962, by N. I. Akhiezer and M. G. Krein, for scalar polynomials. Additionally, we propose a method to enlarge the class of Hurwitz-type matrix polynomials by adding to a given polynomial a matrix polynomial that is not of Hurwitz type, so that the resulting polynomial becomes of Hurwitz type.
Як відомо, задача Коші некоректна для диференціальних рівнянь у частинних похідних зі сталими коефіцієнтами, які не є коректними за Петровським. У цій роботі пропонується інтегральна умова, в якій отримана крайова задача стає коректною в просторі Л. Шварца, а також у просторах функцій степеневого зростання за просторовими змінними. Наведена інтегральна умова є інтегралом по півосі з експоненційною вагою, що забезпечує збіжність даного інтеграла. Якщо розглядати цей інтеграл як перетворення Лапласа від функції $\exp\left(-t^2/2\right)$, то можна отримати його асимптотику, яка дозволяє оцінити \textbf{розв'язувальну функцію}. Дана оцінка дозволяє довести теорему про коректність отриманої крайової задачі в просторі Л. Шварца, а також у шкалі функцій кінцевої гладкості степеневого зростання. Потім розглядається задача для неоднорідного диференціального рівняння, у якого права частина за просторовими змінними належить простору Л. Шварца, а за часовою змінною --- фінітна. Для отриманої задачі знаходиться функція Гріна, що оцінюється зверху. За допомогою цієї оцінки доводиться теорема про коректність даної задачі в просторі Л. Шварца й у спряженому просторі. Наведено приклади некоректних за Петровським рівнянь та вказано конкретні інтегральні умови, за яких одержувані задачі будуть коректними в просторі Л.Шварца. Розглянемо рівняння $$\displaystyle\frac{\partial u(x_1,x_2,t)}{\partial t}=\displaystyle\frac{\partial^2 u(x_1,x_2,t)}{\partial x_1^2}-\displaystyle\frac{\partial^2 u(x_1,x_2,t)}{\partial x_2^2}.$$ Для даного рівняння не виконані умови за Петровським, оскільки поліном $P(s_1,s_2)=-s_1^2+s_2^2$ необмежений. Але якщо додати умову $$\int\limits_0^{\infty} \exp\left(-\displaystyle\frac{t^2}{2}\right) u(x_1,x_2,t)dt=\varphi(x_1,x_2),$$ то така задача буде коректною в просторі Л. Шварца, а також у шкалі банахових просторів.
У статті досліджується неявне лінійне різницеве рівняння першого порядку $BX_{n+1}=AX_n+F_n,\quad n=0,1,2,\ldots$ над скінченним комутативним кільцем $R$ з одиницею, тобто рівняння з необоротним елементом $B$ кільця $R$. На відміну від класичного (явного) лінійного різницевого рівняння, неявне лінійне різницеве рівняння над скінченним кільцем $R$ може не мати розв'язків, а також може мати нескінченну кількість розв'язків. Оскільки будь-яке скінченне комутативне кільце з одиницею ізоморфне скінченній прямій сумі локальних комутативних кілець з одиницею, то рівняння розпадається на систему рівнянь над локальними скінченними комутативними кільцями з одиницею. Показано, що умова співпадіння з $R$ ідеалу $(A,B)$, який породжено елементами $A,B\in R$, є необхідною і достатньою для існування скінченного числа розв'язків цього рівняння, підраховано кількість розв’язків в разі їх існування та надано формулу для загального розв’язку. Умова $(A,B)\ne R$ є необхідною і достатньою умовою існування нескінченного числа розв'язків відповідного однорідного рівняння $BX_{n+1}=AX_n,\quad n=0,1,2,\ldots$. Також встановлено, що у випадку $(A,B)\ne R$ умова $F_n\in (A,B),\quad n=0,1,2,\ldots$ є необхідною для розв'язності неоднорідного неявного лінійного різницевого рівняння, але не є достатньою, як показують приклади. При додатковому обмеженні, що $(A,B)$ є власним головним ідеалом кільця $R$, ця умова буде також достатньою умовою для існування розв'язку розглянутого неоднорідного рівняння, в цій ситуації рівняння має нескінченно багато розв’язків. За умови, що $(A,B)$ є головним ідеалом, спочатку доводиться критерій існування розв’язку у випадку локального комутативного скінченного кільця з одиницею, а потім у випадку довільного скінченного комутативного кільця з одиницею. Як і в теорії Фредгольма, показано наступне: якщо відповідне однорідне рівняння має тільки тривіальний розв'язок, то досліджуване неоднорідне рівняння має єдиний розв'язок. На конкретних прикладах продемонстровано роботу доведених теорем.
Let D be a Dedekind domain of characteristic zero, let K be the fraction field of D, and consider the Cauchy problem ay′ = byᵐ, y(0) = c₀, in the ring D[[x]], where a, b, c₀ ∈ D, a, b ≠ 0, and m ∈ ℕ. The paper studies when the unique formal solution y ∈ K[[x]] with initial value c₀ actually has all coefficients in D, and therefore belongs to D[[x]]. The argument starts from the coefficient recursion in K[[x]] and derives an explicit formula for the coefficients of the unique solution. This reduces the existence problem in D[[x]] to an arithmetic integrality question governed by the valuations attached to nonzero prime ideals of D. The main point is that over a Dedekind domain the relevant obstruction is not ordinary divisibility by prime elements, but comparison of valuations of fractional ideals after localization at prime ideals. For the linear case m = 1, the zero initial value always gives the zero solution. For c₀ ≠ 0, the coefficients contain factorial denominators, and this produces a global obstruction unless only finitely many rational primes remain nonunits in D. In that finite-prime situation the exact criterion for a nonzero initial value is the ideal containment (b) ⊆ (a)r₁, where r₁ is an explicitly described correction ideal built from the prime ideals lying over the rational primes that are nonunits in D. For the nonlinear case m ≥ 2, writing d = m − 1, the coefficients are expressed through the product Cₖ(d) = ∏(di + 1), where i runs from 0 to k − 1. The paper shows that prime ideals over rational primes not dividing d create no additional obstruction beyond the basic condition (bc₀ᵈ) ⊆ (a), while the prime ideals over rational primes dividing d contribute a correction ideal r(d). As a result, the exact existence criterion becomes (bc₀ᵈ) ⊆ (a)r(d). In particular, for m = 2 one has r(1) = D, so the condition reduces to (bc₀) ⊆ (a). Several examples are included to illustrate the theorem over ℤ, localizations of ℤ, Gaussian integers, and the Dedekind domain ℤ[√−5], which is not a unique factorization domain.
The article explores a generalization of the concept of the derivative of a real-valued function of one variable based on filter theory. A new construction is proposed that allows the definition of a derivative of a function with respect to a filter, which reflects the manner in which the variable approaches a given point. Unlike the classical definition, where the limit is taken via a linear approach of the argument, the new definition permits a wider range of approaches to the point, thus providing a more flexible framework for analyzing the local behavior of functions. The introduced concept includes the classical definition of the derivative as a special case when an appropriate filter is chosen. The paper presents proofs of generalized versions of basic derivative properties: linearity, product rule, quotient rule, and chain rule. In particular, it is shown that the derivative with respect to a filter satisfies the same formal differentiation rules as the classical derivative while preserving greater flexibility in how the argument approaches the point. The results obtained expand the scope of differential calculus to cases where the classical approach is either inapplicable or lacks precision or interpretative convenience. It is demonstrated that, in some situations, the derivative with respect to a filter better reflects real processes of change, such as in problems with asymmetric or constrained neighborhoods of a point. The proposed approach opens new perspectives for applications in the theory of generalized functions, measure theory, and functional analysis. The article also provides examples illustrating the application of the new concept and offers a comparative analysis with the classical theory. The presented material may be of interest to researchers in the field of mathematical analysis as well as to educators seeking to extend the traditional approach to differentiation. This work holds both theoretical and methodological value, as it introduces a new tool for further research in the field of modern limit theory.
Many works are devoted to control theory but most of them are related to ordinary differential equations. Of the partial differential equations, mathematical physics equations are most often considered, for example, wave equations. In the article by Makarov O.A. "Controllability of an evolutionary system of partial differential equations. Visnyk of V. N. Karazin Kharkiv National University, Series "Mathematics, Applied Mathematics and Mechanics", 2016, Vol. 83. pp. 47-56" a system of partial differential equations has previously been considered. The complete controllability of such systems in the L. Schwartz space under certain control conditions has been investigated in this work. In particular, it was proved that if the eigenvalues of the system matrix are real, then there is a time-independent control. In addition, the case when the eigenvalues are imaginary was investigated. The purpose of this article is to study the controllability of a system of linear partial differential equations for an arbitrary matrix of the system under the constraints on the search for control in the form $u(x,t)=u(x)\cdot\exp(-\alpha t)$, where the vector of the function $u(x)$ belongs to the L. Schwartz space. Necessary and sufficient conditions for the complete controllability of this system are obtained and examples of both controllable and uncontrollable systems are given. As a consequence of this theorem, sufficient conditions for the complete controllability of the system are obtained, the real parts of the eigenvalues of the matrix $P(s)$ are bounded from above or below. In addition, it is proved that if the spatial variable belongs to the space $\mathbb{R}$, then such a system is completely controllable. Examples are given for each case. A partial differential equation of the second order in time is also considered. It has been proven that if the roots of the characteristic equation satisfy the conditions of the criterion, then this equation is completely controllable. Thus, the Helmholtz equation are completely controllable.
It is known that, up to isomorphism, there are exactly four finite commutative rings with identity, whose order is equal to $p^2$, where p is a prime number. Namely, these rings are the residue class ring modulo $p^2$, the direct sum of two residue class rings $\mathbb{Z}_p$ modulo $p$, the field of order $p^2$ and the ring $\mathcal{S}_p = \mathbb{Z}_p[t]/(t^2)$. Recently, a solvability criterion was established for the first-order linear difference equation over the residue class ring modulo $m \ge 2$. Considering this, it appears necessary to solve the solvability problem for the linear difference equation over the ring $\mathcal{S}_p$ of order $p^2$. This paper investigates first-order implicit linear difference equations over the ring $\mathcal{S}_p$. The paper presents the solvability criterion for the mentioned equation over this ring. In addition, the obtained results describe both the number of solutions and the form of the general solution of this equation. Analogous results were obtained for the initial problem over the ring $\mathcal{S}_p$. In particular, it was established that, unlike in the case of an integral domain, the initial problem over the ring $\mathcal{S}_p$ may have infinitely many solutions. Moreover, if it has a finite number of solutions, then the solution of this initial problem is unique. We obtain several corollaries of the solvability criterion for the implicit linear difference equation over the ring $\mathcal{S}_p$. In particular, as in Fredholm theory, we show that if a homogeneous equation, which corresponds to the non-homogeneous equation, has only the trivial solution, then the non-homogeneous equation, which is being investigated, has a unique solution. The article includes an example demonstrating the application of the obtained theoretical results to solving a certain equation over the ring $\mathcal{S}_p$ and the corresponding initial problem. The results may be applied to further studies of linear difference equations over finite rings, and also to the general theory of discrete dynamical systems.
In this paper, we study a categorical extension of the classical Gelfand-Naimark duality between compact Hausdorff spaces and commutative unital C*-algebras. We establish an equivalence between the category of compact Hausdorff spaces with closed equivalence relations and the category of pairs consisting of a commutative unital C*-algebra together with one of its unital C*-subalgebras. The motivation is that Gelfand duality can be enriched by additional structure: closed equivalence relations encode quotient spaces and invariance on the topological side, while subalgebras reflect restrictions and symmetries on the algebraic side. Shilov’s theorem, which identifies closed unital self-adjoint subalgebras of C(X) with algebras of functions invariant under closed equivalence relations, provides an essential link between these settings. We introduce the category EqRel, whose objects are compact Hausdorff spaces with closed equivalence relations and whose morphisms are continuous trajectory-preserving maps, and the category C*Pairs, whose objects are pairs (A,B) with A a commutative unital C*-algebra and B ⊂ A a unital C*-subalgebra, with morphisms given by unital *-homomorphisms preserving B. Contravariant functors are defined in both directions: (X,R) → (C(X),BR), where BR consists of functions constant on R-classes, and (A,B) → (Σ(A),RB), where Σ(A) is the spectrum and RB relates characters agreeing on B. Using the Kolmogorov-Gelfand theorem, the Gelfand transform, and Shilov’s theorem, we show that these functors are mutually inverse up to morphism of functors and thus prove the categorical equivalence EqRel ≃ C*Pairsop. This result demonstrates that the geometric notion of closed equivalence relations on compact spaces is in perfect correspondence with the algebraic notion of unital subalgebras of commutative C*-algebras.
У статті розглянуто узагальнення поняття похідної функції однієї дійсної змінної на основі теорії фільтрів. Запропоновано нову конструкцію, що дозволяє визначити похідну функції відносно фільтра, який відображає спосіб зближення змінної до заданої точки. На відміну від класичного означення, де границя визначається через прямолінійне зближення аргументу, нове означення дозволяє враховувати ширший спектр підходів до точки, що забезпечує гнучкіший апарат для аналізу локальної поведінки функцій. Введене поняття охоплює класичне означення похідної як частковий випадок при виборі відповідного фільтра. Наведено доведення узагальнення базових властивостей похідної: лінійності, правила добутку, частки, складеної функції. Зокрема, продемонстровано, що похідна відносно фільтра задовольняє ті самі формальні правила диференціювання, що й класична похідна, при збереженні суттєвої гнучкості у виборі характеру зближення аргументу. Отримані результати дозволяють розширити сферу застосування диференціального числення до випадків, де класичний підхід або не є застосовним, або втрачає точність чи інтерпретаційну зручність. Показано, що у деяких ситуаціях похідна за фільтром краще відображає реальні процеси зміни величин, наприклад у задачах з асиметричними або обмеженими околами точки. Запропонований підхід відкриває нові перспективи для застосування в теорії узагальнених функцій, теорії міри та функціональному аналізі. Також у статті наведено приклади застосування нового поняття та здійснено порівняльний аналіз з класичною теорією. Представлений матеріал може бути корисним для дослідників, що працюють у галузі математичного аналізу, а також для викладачів, які прагнуть розширити традиційний підхід до диференціювання. Робота має як теоретичну, так і методологічну цінність, оскільки вводить новий інструмент для подальших досліджень у галузі сучасної математичної теорії границь.
We consider oriented immersed minimal surfaces in three-dimensional sub-Riemannian manifolds which are vertical, i.e., perpendicular to the two-dimensional horizontal distribution of the sub-Riemannian structure. We showed earlier that a vertical surface is minimal in the sub-Riemannian sense if and only if it is minimal in the Riemannian sense and that its sub-Riemannian stability implies its Riemannian stability. We introduce the sub-Riemannian version of the Jacobi operator for such surfaces and prove a sufficient condition for the stability of vertical minimal surfaces similar to a theorem of Fischer-Colbrie and Schoen: if a surface allows a positive function with the vanishing Jacobi operator then it is stable. Next, we use the Jacobi operator technique to investigate vertical minimal surfaces in the Lie group $\widetilde{\mathrm{SL}(2,\mathbb{R})}$ that can be described as the universal covering of the unit tangent bundle of the hyperbolic plane with the standard left-invariant Sasaki metric (that corresponds to one of the Thurston geometries) and with two different types of sub-Riemannian structures. First, we consider a family of non-left-invariant structures defined by some parameters, find the values of parameters for which vertical minimal surfaces exist, and describe such complete connected surfaces. These are Euclidean half-planes and cylinders, and they all are stable in the sub-Riemannian sense and thus in the Riemannian sense. In particular, this gives us examples of structures that do not allow vertical minimal surfaces. Then, we describe complete connected vertical minimal surfaces for another sub-Riemannian structure that is left-invariant. These are half-planes and helicoidal surfaces that also appear to be stable in the sub-Riemannian sense and thus in the Riemannian sense.
The scalar moment problem was first introduced by T. J. Stieltjes in his work ``Recherches sur les fractions continues'' Annals of the Faculty of Sciences of Toulouse 8, 1--122, (1895). He formulated it as follows: Given the moments of order $k$ ($k=0,1,2,\dots$), find a positive mass distribution on the half-line $[0,+\infty)$. The study of matrix and operator moment problems was initiated by M. G. Krein in his seminal paper ``Fundamental aspects of the representation theory of Hermitian operators with deficiency index $(m,m)$'' Translations of the American Mathematical Society, Series II, 97, 75--143, (1949). This paper is related to the truncated Hausdorff matrix moment (THMM) problem: the truncated moment problem on a compact interval $[a,b]$ in contrast to the Stieltjes moment problem on $[0,+\infty)$ and the Hamburger moment problem on $(-\infty,+\infty)$. Our approach relies on V. P. Potapov’s method, which reformulates interpolation and moment problems as equivalent matrix inequalities and introduces auxiliary matrices that satisfy the $\widetilde{J}_q$--inner function property of the Potapov class, together with a system of column pairs. The method begins by constructing Hankel matrices from the prescribed moments. If these matrices are positive semidefinite, the THMM problem is solvable. In the strictly positive definite case, known as the non-degenerate case, we transform the associated matrix inequalities to derive the Nevanlinna (or resolvent) matrix of the THMM problem, which characterizes its solutions. This framework has been extensively applied, for instance in A. E. Choque Rivero, Yu. M. Dyukarev, B. Fritzsche, and B. Kirstein, ``A truncated matricial moment problem on a finite interval'', in Interpolation, Schur Functions and Moment Problems, Operator Theory: Advances and Applications, Birkhäuser, Basel, 165, 121--173, (2006). The main contribution of the present work is to represent the Nevanlinna matrix of the THMM problem in terms of orthogonal matrix polynomials (OMP) and their associated polynomials of the second kind at point $b$. Note that the representation at point $a$ was obtained earlier in A. E. Choque Rivero, ``From the Potapov to the Krein–Nudel’man representation of the resolvent matrix of the truncated Hausdorff matrix moment problem'' Bulletin of the Mexican Mathematical Society, 21(2), 233--259 (2015). In addition, we establish new identities involving OMP and reformulate an explicit relationship between the Nevanlinna matrices of the THMM problem at points $a$ and $b$, through OMP.
Відомо, що, з точністю до ізоморфізму, існує рівно чотири скінченні комутативні кільця з одиницею, порядок яких дорівнює p2, де p - просте число. А саме, цими кільцями є кільце лишків за модулем p2, пряма сума двох полів лишків Z_p за модулем p, поле з p2 елементів і кільце $\mathcal{S}_p = \mathbb{Z}_p[t]/(t^2)$. Нещодавно було встановлено критерій розв'язності лінійного різницевого рівняння першого порядку над кільцем лишків за модулем $m\ge 2$. З огляду на це, актуальною є задача розв'язання лінійного різницевого рівняння над кільцем $\mathcal{S}_p$ порядку p2. У статті досліджуються неявні лінійні різницеві рівняння першого порядку над кільцем $\mathcal{S}_p$. Встановлено необхідні та достатні умови розв'язності розглянутого рівняння над цим кільцем. Крім того, отримані у статті результати описують як кількість розв'язків у випадку їх існування, так і вигляд загального розв'язку цього рівняння. Аналогічні результати встановлено для відповідної початкової задачі над кільцем $\mathcal{S}_p$. Зокрема, показано, що, на відміну від випадку цілісного кільця, початкова задача над кільцем $\mathcal{S}_p$ може мати нескінченно багато розв'язків. Водночас, якщо вона має скінченну кількість розв'язків, то її розв'язок є єдиним. Також отримано наслідки з одержаного критерію розв'язності розглянутого неявного лінійного різницевого рівняння над кільцем $\mathcal{S}_p$. Зокрема, аналогічно до теорії Фредгольма, показано, що якщо відповідне однорідне рівняння має тільки тривіальний розв'язок, то досліджуване неоднорідне рівняння має єдиний розв'язок. У статтi наведено приклад, що демонструє застосування отриманих теоретичних результатiв до розв’язання конкретного рiвняння над кільцем $\mathcal{S}_p$ i вiдповiдної початкової задачi. Результати роботи можуть бути використані для подальшого вивчення лінійних різницевих рівнянь над скінченними кільцями, а також у загальній теорії дискретних динамічних систем.
This paper is devoted to the problem of null-controllability for the oscillating linear system $\dot{x}_{2i-1} = x_{2i}, \dot{x}_{2i} = - x_{2i-1} + u$, $i = \overline{1,n}$ with constraints on the control $u \in [c, 1]$ and $u \in \{c, 1\}$ in the case when the origin is not an equilibrium point. Null-controllability means that there exists a moment of time $T_0$ such that for any time $T > T_0$ we are able to reach the origin in precisely this time. The criterion of controllability into a non-equilibrium point was obtained by V. I. Korobov and a new condition called the return condition on the interval was introduced, which must be satisfied, together with the classical conditions for controllability into an equilibrium point. This condition means that there exists a time interval $I = [T, T + \alpha]$, $\alpha > 0$ so that a trajectory starting from the origin may return there at any time $T\in I$. The aim of this paper is to show that the return conditions are satisfied for the considered oscillatory system, and to obtain the analytical solution for the control that solves the return condition problem. The considered approach involves constructing a piecewise-constant control using values $u = c$ and $u = 1$ and transforming the problem into a trigonometric moment problem on an interval. This problem has a non-unique solution, and in our paper we present one involving $2n$ switching points and another with only $2$ in the case when $c \le \frac{1}{2}$. The solution with $2$ switching moments is especially interesting since it does not depend on the dimensionality of the system. We also generalize the problem to the case where the eigenvalues are of the form $\lambda_{2j}, \lambda_{2j-1} = \pm \nu_k i$, where $\nu_k$ are rational numbers. Additionally, we discuss some partial cases where $c > \frac{1}{2}$ and where the eigenvalues are irrational.
Let be $P$ the probability on a finite group $G$, i.e. the function $P\left(g\right)$ takes non-negative values and $\sum _{g}P\left(g\right) =1$ $\left(g\in G\right)$. For any two functions $F_{1} \left(g\right)$ and $F_{2} \left(g\right)$ their $G$ convolution \[\left(F_{1} *F\right)_{2} \left(t\right)=\sum _{h\in G}F_{1} \left(h\right)F_{2} \left(h^{-1} t\right), t\in G\] is also a function on $G$. In recent years, the topic of studying random walks (and not only on groups) has become very popular. From an analytical point of view, the study of random walks is only the study of their transition function, that is, the n-fold convolution of probability measures. It is well known that under simple conditions, imposed on the probability support $P$ on the group $G$, n-fold convolution $P^{(n)} =P*...*P$ ($n$ times) with $n\to \infty $ converges to uniform probability $U\left(g\right)=\frac{1}{\left|G\right|} $( \textbf{$g\in G$) }, which is obviously a constant on $G$. We study convolution of functions that may be different but are constant in or outside some subgroup. In short, we study the cases where the convolution of such functions has the same properties of constancy with respect to some subgroup. The article considers the convolution of probabilities (and, in general, real functions) constant outside (or inside) a subgroup of $H$ the finite group $G$. Let $D=G\backslash H$. For a given function $F$ on $G$ the subgroup $H$, for which $F$ is constant on $D$, is unique in the following sense: there is at most one such number $c$ and one smallest subgroup $H$ of the group $G$ such that $F\left(g\right)=c$ for all elements $g\in D$. It is proved that if the functions $F_{1} ,\ldots $, $F_{n} $ are constants on $D$, $F_{i} (x)=c_{i} $ for arbitrary $x\in D$ $\left(i=1,\ldots ,n\right)$, then their convolution $F_{1} *\ldots *F_{n} $ is also a constant on $D$. An expression for this constant is found in terms of the numbers $c_{i} ,\; \; i=1,\ldots ,n$. Functions on the group $G$ that are constant on $D$ form a semigroup with respect to the convolution. In previous statements we studied the case, when all functions $F_{1},\ldots, F_{n}$, except one, are constant on the subgroup $H$, and this one is constant on $D=G\backslash H$. Under these conditions it is proved that the convolution $F_{1} *\ldots *F_{n} $ is constant on $H$. In the above statements about a convolution, at least one of its factors is constant outside the subgroup. But if all factors are constant on some subgroup, then their convolution does not necessarily have the same property. An example is given of two functions that are constant on subgroup $H$, but their convolution is not constant on $H$. This example is not presented in the language of probabilities (or functions) on a group (as in all other parts of the article), but in the language of group algebras. The group algebra $KG$ of the group $G$ over the field $K$ appears in questions related to convolutions of functions on the group $G$ in the following way: each function $F\left(g\right)$ on $G$ with values in an arbitrary field $K$ determines an element $\sum _{g}F\left(g\right)g$ $\left(g\in G\right)$ of the algebra $KG$; the convolution of probabilities corresponds to the product of elements of the algebra $KG$. If the function $F\left(g\right)$ is a probability, then $K$ is the field of real numbers.
