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## On the index of the critical Möbius band in 𝔹4

In this paper we prove that the Morse index of the critical Möbius band in the 4−dimensional Euclidean ball 𝔹4 equals 5. It is conjectured that this is the only embedded non-orientable free boundary minimal surface of index 5 in 𝔹4. One of the ingredients in the proof is a comparison theorem between the spectral index of the Steklov problem and the energy index. The latter also enables us to give another proof of the well-known result that the index of the critical catenoid in 𝔹3 equals 4.

Lomonosov T., Journal of Mathematical Sciences 2020 Vol. 244 No. 4 P. 649-654

We obtain criteria for the L2-dissipativity of finite-difference schemes based on regularizations
of 1D barotropic and full gas dynamics systems of equations that are linearized
at a constant solution. Bibliography: 8 titles. ...

Added: February 18, 2020

Trautmann P., Vexler B., Zlotnik A., Mathematical Control and Related Fields 2018 Vol. 8 No. 2 P. 411-449

This work is concerned with the optimal control problems governed by the 1D wave equation with variable coefficients and the control spaces $\mathcal M_T$ of either measure-valued functions $L^2(I,\mathcal M(\Omega))$ or vector measures $\mathcal M(\Omega,L^2(I))$. The cost functional involves the standard quadratic terms and the regularization term $\alpha\|u\|_{\mathcal M_T}$, $\alpha>0$. We construct and study three-level ...

Added: April 8, 2017

Zlotnik A., Applied Mathematics Letters 2019 Vol. 92 P. 115-120

We deal with an explicit finite-difference scheme with a regularization for the 1D gas dynamics equations linearized at the constant solution. The sufficient condition on the Courant number for the $L^2$-dissipativity of the scheme is derived in the case of the Cauchy problem and a non-uniform spatial mesh. The energy-type technique is developed to this end, and ...

Added: January 20, 2019

Zlotnik A., Čiegis R., Applied Mathematics Letters 2021 Vol. 115 Article 106949

We study necessary conditions for stability of a Numerov-type compact higher-order finite-difference scheme for the 1D homogeneous wave equation in the case of non-uniform spatial meshes. We first show that the uniform in time stability cannot be valid in any spatial norm provided that the complex eigenvalues appear in the associated mesh eigenvalue problem. Moreover, we prove ...

Added: December 9, 2020

Zlotnik A., Čiegis R., On properties of compact 4th order finite-difference schemes for the variable coefficient wave equation / Cornell University. Series arXiv "math". 2021. No. ArXiv: 2101.10575v2[math.NA].

We consider an initial-boundary value problem for the $n$-dimensional wave equation with the variable sound speed, $n\geq 1$. We construct three-level implicit in time compact in space (three-point in each space direction) 4th order finite-difference schemes on the uniform rectangular meshes including their one-parameter (for $n=2$) and three-parameter (for $n=3$) families. They are closely connected to some ...

Added: February 2, 2021

Zlotnik A., Koltsova N., Computational Methods in Applied Mathematics 2013 Vol. 13 No. 2 P. 119-138

An initial-boundary value problem for the 1D self-adjoint parabolic equation on the half-axis is solved. We study a broad family of two-level finite-difference schemes with two parameters related to averages both in time and space. Stability in two norms is proved by the energy method. Also discrete transparent boundary conditions are rigorously derived for schemes ...

Added: April 6, 2013

Ducomet B., Zlotnik A., Zlotnik I. A., The splitting in potential Crank-Nicolson scheme with discrete transparent boundary conditions for the Schrödinger equation on a semi-infinite strip / Cornell University. Series math "arxiv.org". 2013. No. arxiv: 1303.3471.

We consider an initial-boundary value problem for a generalized 2D time-dependent Schrödinger equation on a semi-infinite strip. For the Crank-Nicolson finite-difference scheme with approximate or discrete transparent boundary conditions (TBCs), the Strang-type splitting with respect to the potential is applied. For the resulting method, the uniform in time L2-stability is proved. Due to the ...

Added: March 16, 2013

Pardalos P. M., Rassias T. undefined., Springer, 2014

This volume consists of chapters written by eminent scientists and engineers from the international community and presents significant advances in several theories, and applications of an interdisciplinary research. These contributions focus on both old and recent developments of Global Optimization Theory, Convex Analysis, Calculus of Variations, and Discrete Mathematics and Geometry, as well as several ...

Added: May 30, 2014

Zlotnik A., Romanova A. V., A Numerov-Crank-Nicolson-Strang scheme with discrete transparent boundary conditions for the Schrödinger equation on a semi-infinite strip / Cornell University. Series math "arxiv.org". 2013. No. arxiv: 1307.5398.

We consider an initial-boundary value problem for a 2D time-dependent Schrödinger equation on a semi-infinite strip. For the Numerov-Crank-Nicolson finite-difference scheme with discrete transparent boundary conditions, the Strang-type splitting with respect to the potential is applied. For the resulting method, the uniqueness of a solution and the uniform in time L_2-stability (in particular, L_2-conservativeness) are ...

Added: July 24, 2013

Bogomolov F. A., Lukzen E., Stable vector bundles on the families of curves / Cornell University. Series arXiv "math". 2020.

We offer a new approach to proving the Chen-Donaldson-Sun theorem which we demonstrate with a series of examples. We discuss the existence of a construction of a special metric on stable vector bundles over the surfaces formed by a families of curves and its relation to the one-dimensional cycles in the moduli space of stable ...

Added: October 27, 2020

Burov A. A., Герман А. Д., Косенко И. И. et al., Acta Astronautica 2018 Vol. 143 P. 126-132

Relative equilibria of a pendulum attached to the surface of a uniformly rotating celestial body are considered. The locations of the tether anchor that correspond to a given spacecraft position are defined. The domains, where the spacecraft can be held with the help of such a pendulum, are also described. Stability of the found relative ...

Added: September 10, 2018

СПб.: Издательство Санкт-Петербургского университета, 2008

В сборнике представлены результаты исследований по механике сплошной среды, в основном задач колебаний и устойчивости упругих конструкций. Характерной чертой исследований является использование разнообразных компьютерных методов: методов вычислительной механики сплошной среды, компьютерной алгебры, визуализации и др. Анализ опирается на сопоставление данных, полученных в различных подходах, причем наиболее часто сопоставляются результаты, полученные асимптотическими методами и по методу ...

Added: February 4, 2013

Ducomet B., Zlotnik A., Zlotnik I. A., ESAIM: Mathematical Modelling and Numerical Analysis 2014 Vol. 48 No. 6 P. 1681-1699

We consider an initial-boundary value problem for a generalized 2D time-dependent Schrödinger equation (with variable coefficients) on a semi-infinite strip. For the Crank-Nicolson-type finite-difference scheme with approximate or discrete transparent boundary conditions (TBCs), the Strang-type splitting with respect to the potential is applied. For the resulting method, the unconditional uniform in time $L^2$-stability is proved. ...

Added: May 23, 2014

Zlotnik A., Lomonosov T., Applied Mathematics Letters 2020 Vol. 103 Article 106198

We study an explicit in time and symmetric in space finite-difference scheme with a kinetic regularization for the 2D and 3D gas dynamics system of equations linearized at a constant solution (with any velocity). We derive both necessary and sufficient conditions for $L^2$-dissipativity of the Cauchy problem for the scheme by the spectral method. The Courant number ...

Added: December 21, 2019

М.: Издательство математико-механического факультета МГУ, 2009

В настоящий сборник вошли аннотации докладов участников XVI международной конференции «Ломоносов» по секции «Математика и механика». ...

Added: February 4, 2013

Pardalos P. M., Rassias T. undefined., Springer, 2014

The contributions in this volume have been written by eminent scientists from the international mathematical community and present significant advances in several theories, methods and problems of Mathematical Analysis, Discrete Mathematics, Geometry and their Applications. The chapters focus on both old and recent developments in Functional Analysis, Harmonic Analysis, Complex Analysis, Operator Theory, Combinatorics, Functional ...

Added: May 30, 2014

Zlotnik A., Lomonosov T., Computational Mathematics and Mathematical Physics 2019 Vol. 59 No. 3 P. 452-464

Explicit two-level in time and symmetric in space finite-difference schemes constructed by approximating the 1D barotropic quasi-gas-/quasi-hydrodynamic systems of equations are studied. The schemes are linearized about a constant solution with a nonzero velocity, and, for them, necessary and sufficient conditions for the L2-dissipativity of solutions to the Cauchy problem are derived depending on the Mach number. These conditions ...

Added: March 11, 2019

Zlotnik A., On Error Estimates of the Crank-Nicolson-Polylinear Finite Element Method with the Discrete TBC for the Generalized Schrödinger Equation in an Unbounded Parallelepiped / Cornell University. Series math "arxiv.org". 2015.

We deal with an initial-boundary value problem for the generalized time-dependent Schr\"odinger equation with variable coefficients in an unbounded $n$--dimensional parallelepiped ($n\geq 1$). To solve it, the Crank-Nicolson in time and the polylinear finite element in space method with the discrete transpa\-rent boundary conditions is considered. We present its stability properties and derive new error ...

Added: March 27, 2015

Shalimova E., Burov A. A., Technische Mechanik 2017 Vol. 37 No. 2-5 P. 129-138

Dynamics of a massive point on a rotating wire or surface under dry friction force action is considered. Existence, stability and bifurcations of non-isolated relative equilibria sets of the point located - on a sphere uniformly rotating about an inclined fixed axis; - on a thin circular hoop rotating about an inclined fixed axis; - ...

Added: December 7, 2017

Zlotnik A., Lomonosov T., Doklady Mathematics, Germany, Springer 2020 Vol. 101 No. 3 P. 198-204

We study an explicit two-level symmetric in space finite-difference scheme for the multi-dimensional barotropic gas dynamics system of equations with quasi-gasdynamic regularization linearized at a constant solution (with an arbitrary velocity). A criterion and both necessary and sufficient conditions for the $L^2$-dissipativity of the solutions to the Cauchy problem for the scheme are derived by the ...

Added: September 13, 2020

Zlotnik A., Kireeva O., On compact 4th order finite-difference schemes for the wave equation / Cornell University. Series arXiv "math". 2020. No. arXiv:2011.14104v2[math.NA].

We consider compact finite-difference schemes of the 4th approximation order for an initial-boundary value problem (IBVP) for the $n$-dimensional non-homogeneous wave equation, $n\geq 1$. Their construction is accomplished by both the classical Numerov approach and alternative technique based on averaging of the equation, together with further necessary improvements of the arising scheme for $n\geq 2$. The ...

Added: December 1, 2020

Zlotnik A., Chetverushkin B., Differential Equations 2020 Vol. 56 No. 7 P. 910-922

We consider a multidimensional hyperbolic quasi-gasdynamic system of differential equations of the second order in time and space linearized at a constant solution (with an arbitrary velocity). For the linearized system with constant coefficients, we study an implicit three-level weighted difference scheme and an implicit two-level vector difference scheme. The important domination property of the operator of ...

Added: July 16, 2020

Zlotnik A., Lomonosov T., Доклады Российской Академии наук. Математика, информатика, процессы управления (Российская Федерация) 2020 Т. 492 С. 31-37

We study an explicit two-level symmetric in space finite-difference scheme for the multi\-di\-men\-si\-onal barotropic gas dynamics system of equations with quasi-gasdynamic regulari\-za\-tion linearized at a constant solution (with arbitrary velocity). A criterion and both necessary and sufficient conditions for the $L^2$-dissipativity of the solutions to the Cauchy problem for the scheme are derived by the spectral ...

Added: March 4, 2020

Zlotnik A., Kireeva O., Mathematical Modelling and Analysis 2021 Vol. 26 No. 3 P. 479-502

We consider compact finite-difference schemes of the 4th approximation order for an initial-boundary value problem (IBVP) for the $n$-dimensional non-homogeneous wave equation, $n\geq 1$. Their construction is accomplished by both the classical Numerov approach and alternative technique based on averaging of the equation, together with further necessary improvements of the arising scheme for $n\geq 2$. The alternative ...

Added: December 9, 2020