Weighted L 2-contractivity of Langevin dynamics with singular potentials

Convergence to equilibrium of underdamped Langevin dynamics is studied under general assumptions on the potential U allowing for singularities. By modifying the direct approach to convergence in L 2 pioneered by Hérau and developed by Dolbeault et al, we show that the dynamics converges exponentially fast to equilibrium in the topologies L 2(dμ) and L 2(W* dμ), where μ denotes the invariant probability measure and W* is a suitable Lyapunov weight. In both norms, we make precise how the exponential convergence rate depends on the friction parameter γ in Langevin dynamics, by providing a lower bound scaling as min(γ, γ −1). The results hold for usual polynomial-type potentials as well as potentials with singularities such as those arising from pairwise Lennard-Jones interactions between particles.

Solution for elliptic inclusion in an infinite plate with remote loading and Eshelby’s eigenstrain of polynomial type

In this paper, a particular inhomogeneous inclusion problem is studied. In the problem, Eshelby’s eigenstrain takes the type [Formula: see text], where m+ n = 2, and the remote loadings [Formula: see text], [Formula: see text] are applied. In the solution, the complex variable method is used. The continuity conditions along the interface of the matrix and the inclusion are formulated exactly. Because the stress field is no longer uniform in inclusion in this case, the studied problem has an inherent difficulty. After some manipulation, the final result for stress components [Formula: see text], [Formula: see text] and [Formula: see text] in inclusion are obtainable. In the present study, [Formula: see text], [Formula: see text] and [Formula: see text] are no longer uniform.

On the Use of Homogeneous Polynomial Yield Functions in Sheet Metal Forming Analysis

Sheet metal forming techniques are a major class of stamping and manufacturing processes of numerous parts such as doors, hoods, and fenders in the automotive and related supplier industries. Due to series of rolling processes employed in the sheet production phase, automotive sheet metals, typically, exhibit a significant variation in the mechanical properties especially in strength and an accurate description of their so-called plastic anisotropy and deformation behaviors are essential in the stamping process and methods engineering studies. One key gradient of any engineering plasticity modeling is to use an anisotropic yield criterion to be employed in an industrial content. In literature, several orthotropic yield functions have been proposed for these objectives and usually contain complex and nonlinear formulations leading to several difficulties in obtaining positive and convex functions. In recent years, homogenous polynomial type yield functions have taken a special attention due to their simple, flexible, and generalizable structure. Furthermore, the calculation of their first and second derivatives are quite straightforward, and this provides an important advantage in the implementation of these models into a finite element (FE) software. Therefore, this study focuses on the plasticity descriptions of homogeneous second, fourth and sixth order polynomials and the FE implementation of these yield functions. Finally, their performance in FE simulation of sheet metal cup drawing processes are presented in detail.

DYNAMICS OF THE LEASING MARKET IN THE RUSSIAN FEDERATION AND FORECAST OF ITS DEVELOPMENT

На основании проведенного исследования сложившегося уровня рынка лизинговых услуг в России установлены основные тенденции его развития. При проведении сравнительного анализа за период с 2014 по 2020 гг. рассчитаны темпы роста по показателям совокупного портфеля лизинговых услуг, определены топ-10 компаний по величине стоимости капитала, а также установлены основные направления отраслевой специфики имущества лизингодателей. Научная новизна заключается в определении причин и факторов, вызвавших подобную динамику развития рынка лизинга в России. Практическая значимость состоит в составлении прогноза развития рынка лизинговых услуг на период 2021-2022 гг. при помощи трендовой модели полиномиального вида 6 порядка, выбор которой был обусловлен возможностями применяемой программной системы MS Excel. В результате проведенного исследования сформулированы рекомендации по развитию рынка лизинговых услуг в РФ. Based on the study of the current level of the leasing services market in Russia, the main trends of its development are established. When conducting a comparative analysis for the period from 2014 to 2020, the growth rates for the indicators of the total portfolio of leasing services were calculated, the top 10 companies were determined by the value of capital, and the main directions of the industry specifics of the property of lessors were established. According to the results of the study, the reasons and factors that caused such dynamics of the development of the leasing market in Russia were identified. The practical significance consists in making a forecast of the development of the leasing services market for the period 2021-2022 using a trend model of the 6-order polynomial type. the choice of which was determined by the capabilities of the MS Excel software system used. As a result of the study, recommendations for the development of the leasing services market in the Russian Federation are formulated.

SUMS OF POLYNOMIAL-TYPE EXCEPTIONAL UNITS MODULO

Let $f(x)\in \mathbb {Z}[x]$ be a nonconstant polynomial. Let $n\ge 1, k\ge 2$ and c be integers. An integer a is called an f-exunit in the ring $\mathbb {Z}_n$ of residue classes modulo n if $\gcd (f(a),n)=1$ . We use the principle of cross-classification to derive an explicit formula for the number ${\mathcal N}_{k,f,c}(n)$ of solutions $(x_1,\ldots ,x_k)$ of the congruence $x_1+\cdots +x_k\equiv c\pmod n$ with all $x_i$ being f-exunits in the ring $\mathbb {Z}_n$ . This extends a recent result of Anand et al. [‘On a question of f-exunits in $\mathbb {Z}/{n\mathbb {Z}}$ ’, Arch. Math. (Basel)116 (2021), 403–409]. We derive a more explicit formula for ${\mathcal N}_{k,f,c}(n)$ when $f(x)$ is linear or quadratic.

Permutable quasiregular maps

Let f and g be two quasiregular maps in $\mathbb{R}^d$ that are of transcendental type and also satisfy $f\circ g =g \circ f$ . We show that if the fast escaping sets of those functions are contained in their respective Julia sets then those two functions must have the same Julia set. We also obtain the same conclusion about commuting quasimeromorphic functions with infinite backward orbit of infinity. Furthermore we show that permutable quasiregular functions of the form f and $g = \phi \circ f$ , where $\phi$ is a quasiconformal map, have the same Julia sets and that polynomial type quasiregular maps cannot commute with transcendental type ones unless their degree is less than or equal to their dilatation.

A Generalised Solution to the Point to Target Problem Using the Minimum Curvature Method

An explicit solution to the general 3D point to target problem based on the minimum curvature method has been sought for more than four decades. The general case involves the trajectory's start and target points connected by two circular arcs joined by a straight line with the position and direction defined at both ends. It is known that the solutions are multi-valued and efficient iterative schemes to find the principal root have been established. This construction is an essential component of all major trajectory construction packages. However, convergence issues have been reported in cases where the intermediate tangent section is either small or vanishes and rigorous mathematical conditions under which solutions are both possible and are guaranteed to converge have not been published. An implicit expression has now been determined that enables all the roots to be identified and permits either exact, or polynomial type solution methods to be employed. Most historical attempts at solving the problem have been purely algebraic, but a geometric interpretation of related problems has been attempted, showing that a single circular arc and a tangent section can be encapsulated in the surface of a horn torus. These ideas have now been extended, revealing that the solution to the general 3D point to target problem can be represented as a 10th order self-intersecting geometric surface, characterised by the trajectory's start and end points, the radii of the two arcs and the length of the tangent section. An outline of the solution's derivation is provided in the paper together with complete details of the general expression and its various degenerate forms so that readers can implement the algorithms for practical application. Most of the degenerate conditions reduce the order of the governing equation. Full details of the critical and degenerate conditions are also provided and together these indicate the most convenient solution method for each case. In the presence of a tangent section the principal root is still most easily obtained using an iterative scheme, but the mathematical constraints are now known. It is also shown that all other cases degenerate to quadratic forms that can be solved using conventional methods. It is shown how the general expression for the general point to target problem can be modified to give the known solutions to the 3D landing problem and how the example in the published works on this subject is much simplified by the geometric, rather than algebraic treatment.

A Generalized Solution to the Point-to-Target Problem Using the Minimum Curvature Method

Summary An explicit solution to the general 3D point-to-target problem based on the minimum curvature method has been sought for more than four decades. The general case involves the trajectory's start and target points connected by two circular arcs joined by a straight line with the position and direction defined at both ends. It is known that the solutions are multivalued, and efficient iterative schemes to find the principal root have been established. This construction is an essential component of all major trajectory construction packages. However, convergence issues have been reported in cases where the intermediate tangent section is either small or vanishes and rigorous mathematical conditions under which solutions are both possible and are guaranteed to converge have not been published. An implicit expression has now been determined that enables all the roots to be identified and permits either exact or polynomial-type solution methods to be used. Most historical attempts at solving the problem have been purely algebraic, but a geometric interpretation of related problems has been attempted, showing that a single circular arc and a tangent section can be encapsulated in the surface of a horn torus. These ideas have now been extended, revealing that the solution to the general 3D point-to-target problem can be represented as a 10th-orderself-intersecting geometric surface, characterized by the trajectory's start and end points, the radii of the two arcs, and the length of the tangent section. An outline of the solution's derivation is provided in the paper together with complete details of the general expression and its various degenerate forms so that readers can implement the algorithms for practical application. Most of the degenerate conditions reduce the order of the governing equation. Full details of the critical and degenerate conditions are also provided, and together these indicate the most convenient solution method for each case. In the presence of a tangent section, the principal root is still most easily obtained using an iterative scheme, but the mathematical constraints are now known. It is also shown that all other cases degenerate to quadratic forms that can be solved using conventional methods. It is shown how the general expression for the general point-to-target problem can be modified to give the known solutions to the 3D landing problem and how the example in the published works on this subject is much simplified by the geometric, rather than algebraic treatment.