Shallow elastic shells with a given circular boundary are considered. An axisymmetric shell shape is sought that maximizes the fundamental oscillation frequency for a given shell weight. The choice of functionals considered in optimal design is a part of the optimization problem formulation. This choice is influenced by many factors, including the primary purpose of the structure, operating conditions, and model properties. Oscillation frequency is one of the key characteristics of a structure. The most typical in the theory of optimal design of compressed structures are problems of maximizing the critical value of $\omega_0$ ($\omega_0$ is the minimum of the natural frequencies) for a given structural weight and problems of minimizing the weight subject to the constraint $\omega_0\geq\mu,$ where $\mu$ is a given number. Unlike dynamic optimal design problems, in which constraints are imposed not only on the fundamental frequency but also on higher frequencies, the consideration of stability constraints in optimal design problems is based on considering only the minimum eigenvalues. Based on the obtained multiplicity of the minimum eigenvalue and the Frechet differentiability of the same functional, the necessary optimality conditions are obtained in this paper.

The authors of this article prove the Riemann hypothesis for a special case when the real $\sigma$ and imaginary $T=1/(2a)$ parts of the zeta function argument belong to a certain region $\Theta_R$ of the complex plane. It is based on V. Blinovsky's method: 1) an expression is derived for the function $L(\sigma,a),$ the square of the modulus of the fundamental part of the zeta function, which affects its zeros, 2) the proof of the Riemann hypothesis is reduced to proving the U-convexity of the function $L$ by $\sigma,$ i. e. the positivity of $L_{\sigma\sigma}^{(2)}(\sigma,a).$ The authors provide estimates that allow us to describe the region $\Theta_R,$ and present the data of a computational experiment conducted in the Maxima mathematical package by constructing the region $\Theta_R.$ A literature review on the current state of hypothesis proof is provided. Further development of the method is proposed both in terms of expanding the region $\Theta_R$ and for increasing computational efficiency.

In this work, we investigate a spectral problem for a system of second-order ordinary differential equations with a complex spectral parameter subject to weighted integral boundary conditions involving the first derivatives of the unknown functions. For sufficiently large values of the spectral parameter, we derive a priori estimates for solutions of the system and establish the Fredholm solvability of the corresponding operator. The obtained results extend known spectral and solvability properties for ordinary differential operators with nonlocal integral conditions to coupled systems.

This paper is devoted to techniques intended for acceleration of convergence of the variants of the Newton method to singular solutions of systems of nonlinear equations. For equations with smooth mappings, there exists a rich literature on these issues. In particular, it is known that under natural assumptions, the rate of convergence of the basic Newton method to singular solutions is linear, with the asymptotic common ratio equal to 1/2, from wide domains of starting points. Making use of a special pattern of such convergence, it can be in some senses accelerated by introducing the corresponding modifications of the basic Newton method, namely, the extrapolation and overrelaxation procedures. Besides providing a survey of the related local convergence theories, the main results of the paper consist of extensions of the specified algorithms and their local convergence and rate-of-convergence theorems to the piecewise Newton method for equations with piecewise smooth mappings. The important role in this is played by previously developed constructions allowing to characterize smooth local selection mappings that can be active along the iterations of the method. The paper also contains the results of numerical comparison of the considered acceleration techniques for the piecewise Newton method for piecewise-smooth reformulations of nonlinear complementarity problems, including some illustrative examples.

This paper investigates stochastic approximation methods based on finite differences for the minimization of quasiconvex functions. Traditional approaches to convex optimization problems primarily rely on exact or stochastic subgradients. However, in many practical situations, only noisy information about function values is available. In such cases, stochastic quasi-gradient schemes constructed using randomized finite-difference estimators are considered. In particular, we study batch two-point schemes that provide unbiased approximations of the gradient of a smoothed objective function. Variance bounds are established for these estimators, which enable a rigorous convergence analysis of projected stochastic descent methods.

The paper considers the minimization problem of functions of the form $f(x) = \max_{i \in I} f_i(x),$ where each function $f_i(x),$ $i \in I,$ is quasiconvex and has a Lipschitz-continuous gradient on a convex compact set. The main result of the paper is the derivation of convergence rate estimates for stochastic finite-difference methods in the quasiconvex setting. It is shown that the expected suboptimality decreases at the rate $O\!\left(1/\sqrt{k}\right).$ The obtained results significantly broaden the applicability of stochastic finite-difference methods to nonsmooth quasiconvex optimization problems and provide a rigorous theoretical justification of the algorithm in black-box settings where only noisy function evaluations are available.

We introduce a new nonlinear Volterra integro-differential equation incorporating a weakly singular temporal peridynamic operator. Under suitable assumptions on the kernel and the nonlinear terms, we establish the existence and uniqueness of the exact solution in an appropriate Banach space via a fixed-point approach. Furthermore, a consistent discretization scheme is proposed, and the well-posedness of the associated discrete problem is proved. The convergence of the numerical approximation toward the exact solution is rigorously analyzed and illustrated by numerical experiments.

We consider a linear two-component system of reaction–diffusion partial differential equations on the real axis, modeling a reversible chemical reaction. The main focus is on the case of distinct diffusion coefficients, where the diffusion and reaction operators do not commute, but the law of total mass conservation holds. This precludes solution factorization and leads to non-trivial spatiotemporal dynamics. Using the Fourier transform, the problem is reduced to a family of linear systems of ordinary differential equations. For comparison, the classical factorized solution is presented for the case of equal diffusion coefficients, where the operators commute. The solution is constructed using the Picard successive approximation method for the Volterra integral equation, which exhibits a high convergence rate, and its series terms admit a transparent physical interpretation. The expressions for concentration are represented as a sum of the direct diffusion contribution of molecules that have not undergone reactions and that of molecules that have undergone a certain sequence of transitions between states. The obtained formulas provide an effective tool both for qualitative analysis of spatiotemporal dynamics and for quantitative calculations with controlled accuracy in applied problems of chemical kinetics and mass transport.