Abstract In this paper, for the Hamilton–Jacobi–Bellman equation with an infinite horizon and state constraints, we construct a suitably regular representation. This allows us to reduce the problem of existence and uniqueness of solutions to the Frankowska and Basco theorem from Basco and Frankowska (Nonlinear Differ Equ Appl 26:1–24, 2019). Furthermore, we demonstrate that our representations are stable. The obtained results are illustrated with examples.

Conformally weighted Einstein manifolds: The uniqueness problem

The aim of the work is to identify the main directions of artificial intelligence development in the provision of telemedicine care that require additional legal regulation. The provision of medical care is fundamentally changing under the influence of informatization, which is manifested, among other things, in the increasing role of artificial intelligence and telemedicine technologies. Taken together, these two phenomena create difficulties for legislative and law enforcement agencies. The work is aimed at identifying and developing proposals for improving the legal regulation of artificial intelligence in telemedicine. As a result of the study, the main problems of using artificial intelligence in telemedicine are identified: (1) the problem of the quality of the source data; (2) the problem of loss of generalization ability; (3) the problem of the duration of continuous learning; (4) the problem of data uniqueness; (5) the problem of responsibility; (6) The problem of personal data. The study is financially supported by the Russian Science Foundation, project № 23-78-10175, https://rscf.ru/project/23-78-10175/.

Let \( f \) be a meromorphic function in the complex plane \( \mathbb{C} \). The uniqueness problems concerning \( f(z) \) and its shifted counterpart \( f(z+c) \) under various shared-value and growth conditions have been extensively studied over the past two decades. Many researchers have explored the uniqueness of the derivatives of these functions when they share values, considering both cases: ignoring multiplicities (IM) and counting multiplicities (CM). Recently, attention has been directed toward the uniqueness of \( f^{(j)}(z) \) and \( f^{(k)}(z+c) \) in the context of three shared values—specifically, one shared value in the IM sense and two shared values in the partial CM~sense. The objective of this study is to refine and extend the existing sharing conditions. In this article, we investigate the uniqueness between \( f^{(j)}(z) \) and \( f^{(k)}(z+c) \) when sharing two values in the CM sense. We provide examples to demonstrate that the proposed conditions are optimal. Additionally, we examine how deficiency conditions affect value sharing and their influence on the unicity results. Our final result extends our investigation by proving the uniqueness between \( f^{(j)}(z) \) and \( f^{(k)}(z+c) \) for one CM shared value, subject to suitable deficiency conditions. We also show that the same result holds when these derivatives share a value in the IM sense, along with a stronger deficiency condition.

Abstract In the paper, using the idea of normal family, we investigate the uniqueness problem of entire functions that share two values partially with their k -th derivatives. The obtained results improve the results of Lü, Xu and Yi (Ann. Polon. Math. 95(1) (2009), 67–75) in a large scale. Also, as an application of our results, we have settled the conjecture posed by Li and Yang (Illinois J. Math. 44(2) (2000), 349–362).

Abstract We consider a certain uniqueness problem concerning the Fourier coefficients of normalized Cauchy transforms. These problems inherently involve proving a simultaneous approximation phenomenon and establishing the existence of cyclic inner functions in certain sequence spaces. Our results have several applications in different directions. First, we offer a new non-probabilistic proof of a classic theorem by Kahane, Katzenelson and Nestoridis on simultaneous approximation. Secondly, we demonstrate the absence of uniform admissible majorants of Fourier coefficients in de Branges–Rovnyak spaces, and deduce complementary results to the classical Ivashev-Musatov Theorem.

In connection to Brück conjecture we improve a uniqueness problem for entire functions that share a polynomial with linear differential polynomial.

Abstract We investigate the concept of Pauli pairs and a discrete counterpart to it. In particular, we make substantial progress on the question of when a discrete Pauli pair is automatically a classical Pauli pair. Effectively, if one of the functions has space and frequency Gaussian decay, and one has that and on two sets which accumulate like suitable small multiples of at infinity, then and . Furthermore, we show that if one drops either the assumption that one of the functions has space–frequency decay or that the discrete sets accumulate at a high rate, then the desired property no longer holds. Our techniques are inspired by and directly connected to several recent results in the realm of Fourier uniqueness problems [43, 47, 50], and our results may be seen as a nonlinear generalization of those. As a consequence of said techniques, we are able to prove a sharp discrete version of Hardy's uncertainty principle.

Abstract When overconstrained multibody mechanisms are modeled as rigid body systems, some or all of the reaction forces are non-unique, which may affect other simulation results, making them incorrect. Despite this global ambiguity, unambiguously determinable reactions may be present in the system. For some multibody formulations, methods for checking which reactions are unique and which are non-unique are available. This paper presents a new version of the reaction uniqueness analysis approach based on the divide-and-conquer algorithm. In this method, single- and multi-joint connections are treated in the same way, and the approach does not require additional passes along a binary tree. The uniqueness test may be performed by considering the coefficient matrix (in full or reduced form)—after the main pass of the DCA is finished—by using one of the following numerical methods: rank comparison, QR decomposition, SVD, or nullspace approaches. All of them are considered in this article. The new approach can be applied to a broader class of mechanisms than the DCA-based method developed so far. Moreover, more complex reaction sets may be considered. A spatial parallelogram mechanism with a triple pendulum is studied to illustrate the approach. Furthermore, the proposed method is compared with different approaches to uniqueness analysis in terms of computational efficiency.

The article presents the method for analyzing the stability of linear matrix differential equations with constant coefficients. One of the classical typesof such equations is the class of linear matrix differential equations, which includes the Lyapunov equation as a particular case. Matrix differentialequations arise in problems of stability theory, practical stability, optimal control theory, and state estimation of systems under uncertainty. Therefore,it is necessary to compute and analyze the qualitative properties of solutions to matrix differential equations. This involves addressing problems of existence, uniqueness, continuation, and analysis of stability conditions for various types of such mathematical equations. The method proposed in thearticle is based on algebraic properties of eigenvalues, Jordan forms of matrices, and characteristics of polynomial roots. A theorem is established regardingthe conditions for stability, asymptotic stability, and instability of solutions to linear matrix differential equations with constant coefficients.The developed approach includes the computation of the maximal real parts of eigenvalues and the analysis of the Jordan form structure of the systemmatrices. As a consequence, corresponding stability conditions for the Lyapunov matrix equation are also obtained. An algorithm is proposed for computingthe maximal real part of the roots of a polynomial, as well as for finding all roots. The approach relies on the Routh – Hurwitz theorem. The article also presents results of computational experiments.

Abstract The paper deals with the uniqueness problem of derivatives of meromorphic functions when they share certain values with their shift and difference operators. In the case of shift operators, our result improves and extends all the existing results in this direction. We establish two uniqueness theorems for difference operators that also extend and improve certain theorems in this direction. Furthermore, we exhibit some examples pointing out the sharpness of some of our conditions.

In the paper, we apply the concept of weighted sharing to study the uniqueness problems of differential polynomial of meromorphic function of zero order with its $q$-shift. The results of the paper improve and extend some recent results due to H. P. Waghamore and M. M. Manakame [Int. J. Open Problems Compt. Math., 18 (2025), 22-34]. A typical theorem obtained in the paper is as follows: Let $P$ be a polynomial, $f(z)$ be a non-constant meromorphic function of zero-order. Suppose that $q$ is a non-zero complex constant, $\eta \in \mathbb{C}$ and $n$ is an integer satisfying $n\geq m+3\tau+3\Omega +6$, where $m=\deg P$, $\tau =\sum_{j=1}^{s}\mu _{j}$ and $\Omega =\sum_{j=1}^{s}j\mu _{j}.$ If $f^{n}(z)P(f(z))\prod _{j=1}^{s}f^{(j)}(z)^{\mu _{j}}$ and $f^{n}(qz+\eta )P(f(qz+\eta ))\prod _{j=1}^{s}f^{(j)}(qz+\eta )^{\mu _{j}}$ share $(1,2)$ and $(\infty,\infty)$, then $\displaystyle f^{n}(z)P(f(z))\prod _{j=1}^{s}f^{(j)}(z)^{\mu _{j}}\equiv f^{n}(qz+\eta )P(f(qz+\eta ))\prod _{j=1}^{s}f^{(j)}(qz+\eta )^{\mu _{j}}.$ Three other similar theorems are also obtained in the paper.

In this paper, we study the problem of uniqueness of fixed points for operators acting from a Banach space X into a subspace Y with a stronger norm. Our main objective is to preserve the expected regularity of fixed points, as determined by the norm of Y, while analyzing their uniqueness without imposing the classical or generalized contraction condition on Y. The results presented here provide generalized uniqueness theorems that extend existing fixed-point theorems to a broader class of operators and function spaces. The results are used to study fractional initial value problems in generalized Hölder spaces.

In this paper, we have studied the uniqueness problems of meromorphic functions with their difference or generalized linear shift operators in the light of partial sharing in several complex variables. By relaxing one sharing condition from CM (counting multiplicities) to IM (ignoring multiplicities), one of the results of our paper improved a result of W. Wu and T.-B. Cao [Comput. Methods Funct. Theory 2022, 22 (2), 379-399]. Our other results also extend and improve some results of the same paper.

Abstract For any n ∈ N n\in\mathbb{N} , n ≥ 2 n\geq 2 , we construct a real analytic, one-parameter family of compact embedded CMC annuli with free boundary in the unit ball B 3 \mathbb{B}^{3} of R 3 \mathbb{R}^{3} with a prismatic symmetry group of order 4 ⁢ n 4n . These examples give a negative answer to the uniqueness problem by Nitsche and Wente of whether any annular solution to the partitioning problem in the ball should be rotational.

This paper extends the investigation into the uniqueness problems concerning meromorphic functions and their differences or difference polynomials and explores the conditions under which two transcendental meromorphic functions 𝑓 (𝑧) and 𝑔(𝑧), with hyper-order less than one, they share certain values or small functions. For instance, 𝑓 (𝑧)𝑛 and share common values together with 𝑔(𝑧)𝑛 and share common values. Moreover, we address scenarios where 𝑓 (𝑧)𝑛 and 𝑔(𝑘)(𝑧 + 𝑐) share values 𝐼𝑀 together with 𝑔(𝑧)𝑛 and 𝑓 (𝑘)(𝑧 + 𝑐) share values 𝐼𝑀, leading to conclusions about the relationship between the functions.

There are some uniqueness problems with meromorphic functions with difference operators that we looked into in this paper. We looked at them in the light of partial sharing. Specifically, we have obtained two uniqueness results by considering sharing and partial sharing of small functions. In the first theorem and shares CM, whereas in the second theorem and partially share CM. 2010 AMS Classification: 30D35, 30D45 Keywords and phrases: Uniqueness, Meromorphic function, Difference operator, Small function, Partial sharing.

There are algebraic and variational approaches of construction in the spline theory. In algebraic approach splines are considered as some smooth piecewise polynomial functions. In the variational approach splines are elements of Hilbert or Banach spaces minimizing certain functionals. Then we study the problems of existence, uniqueness, and convergence of splines and algorithms for constructing them based on their own properties of splines. In this paper, we study the problem of natural spline functions in a Hilbert space. Here, using the Sobolev method, an algorithm is given for solving a system of linear algebraic equations for the coefficients of the natural spline functions. For ???? = 2, explicit expressions for the optimal coefficients of the natural spline function in the Hilbert space ????(2,0) are obtained.

Abstract Inspired by the Abel–Goncharov interpolation problem, we consider a collection of uniqueness problems, with related interpolation and sampling issues. We call them deep zero problems, as they are concerned with local properties at a small number of given points.

ABSTRACTThe study of level sets of solutions to partial differential equations, particularly for the semilinear elliptic problem on a Riemannian manifold , has been a fundamental area in mathematical analysis. This type of equation appears in various physical models, including reaction‐diffusion processes and phase transition phenomena. In this survey, we provide a historical overview and discussion of the main properties and results related to the level sets of solutions under zero Dirichlet boundary conditions. Additionally, we address the related problem of uniqueness of the critical point of the solution .