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Stability of regular vortex polygons in Bose-Einstein condensate 玻色-爱因斯坦凝聚中规则涡旋多边形的稳定性
IF 0.4 Q3 Mathematics Pub Date : 2020-11-01 DOI: 10.35634/2226-3594-2020-56-02
A. Kilin, E. Artemova
We consider the problem of the stability of rotating regular vortex N-gons (Thomson configurations) in a Bose-Einstein condensate in a harmonic trap. The dependence of the rotation velocity ω of the Thomson configuration around the center of the trap is obtained as a function of the number of vortices N and the radius of the configuration R. The analysis of the stability of motion of such configurations in the linear approximation is carried out. For N⩽6, regions of orbital stability of configurations in the parameter space are constructed. It is shown that vortex N-gons for N > 6 are unstable for any parameters of the system.
研究了谐波阱中玻色-爱因斯坦凝聚中旋转规则涡旋n -子(汤姆逊组态)的稳定性问题。得到了汤姆逊构型在陷阱中心周围的旋转速度ω与涡数N和构型半径r的关系,并在线性近似下分析了这类构型的运动稳定性。对于N≤6,构造了构型在参数空间中的轨道稳定性区域。结果表明,当N > 6时,对系统的任何参数都是不稳定的。
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引用次数: 3
Sufficient conditions of a nonlocal solvability for a system of two quasilinear equations of the first order with constant terms 一类一阶常项拟线性方程组非局部可解的充分条件
IF 0.4 Q3 Mathematics Pub Date : 2020-05-01 DOI: 10.35634/2226-3594-2020-55-05
M. Dontsova
We consider a Cauchy problem for a system of two quasilinear equations of the first order with constant terms. The study of the solvability of the Cauchy problem for a system of two quasilinear equations of the first order with constant terms in the original coordinates is based on the method of an additional argument. Theorems on the local and nonlocal existence and uniqueness of solutions to the Cauchy problem are formulated and proved. We prove the existence and uniqueness of the local solution of the Cauchy problem for a system of two quasilinear equations of the first order with constant terms, which has the same smoothness with respect to $x$ as the initial functions of the Cauchy problem. Sufficient conditions for the existence and uniqueness of a nonlocal solution of the Cauchy problem for a system of two quasilinear equations of the first order with constant terms are found; this solution is continued by a finite number of steps from the local solution. The proof of the nonlocal solvability of the Cauchy problem for a system of two quasilinear equations of the first order with constant terms relies on global estimates.
考虑一类一类一阶常项拟线性方程组的柯西问题。基于附加参数法,研究了两个一阶常项拟线性方程组在原坐标下的可解性问题。给出并证明了柯西问题解的局部和非局部存在唯一性定理。证明了一类一阶常项拟线性方程组的局部解的存在唯一性,该方程组对x具有与柯西问题初值函数相同的光滑性。给出了一类一阶常项拟线性方程组Cauchy问题非局部解存在唯一性的充分条件;该解从局部解开始连续有限步。对于两个一阶常项拟线性方程组的Cauchy问题的非局部可解性的证明依赖于全局估计。
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引用次数: 1
To the question of optimization of the starting point in the routing problem with restrictions 对带约束的路由问题中起点的优化问题进行了研究
IF 0.4 Q3 Mathematics Pub Date : 2020-05-01 DOI: 10.35634/2226-3594-2020-55-09
A. Chentsov, P. Chentsov
We consider the extreme routing problem with an additive criterion, the terminal component of which depends on the starting point. This dependence may be associated, in particular, with the requirement to return to the starting point region after completing the final system of tasks that need to be ordered. The work assumes that the tasks to be performed are related with visiting non-empty finite sets called megacities. In turn, the mentioned visits are associated with the performance of works, the costs of which are involved in the formation of the criterion. Finally, the costs of external movements (between megacities) supplement the formation of an additive criterion to be minimized. It is required to find a global extremum and a solution that includes a starting point, the order of visits to megacities and a specific trajectory of the process. The solution uses widely understood dynamic programming (DP). It is significant that procedures based on DP use starting point. Therefore, enumeration of mentioned points is required. The article proposes an approach to solving the problem of reducing this enumeration through the use of auxiliary DP that are universal with respect to the choice of a starting point. The optimal algorithm was built and implemented on a PC using the aforementioned approach.
考虑了具有加性准则的极端路由问题,该准则的终端分量依赖于起点。这种依赖关系可能特别与在完成需要排序的最终任务系统后返回起始点区域的需求相关联。该工作假设要执行的任务与访问称为特大城市的非空有限集有关。反过来,上述访问与作品的表现有关,其成本涉及标准的形成。最后,外部移动(特大城市之间)的成本补充了要最小化的附加标准的形成。需要找到一个全局极值和解决方案,其中包括起点、访问大城市的顺序和过程的具体轨迹。该解决方案使用广泛理解的动态规划(DP)。基于DP的程序使用起点具有重要意义。因此,需要列举上述要点。本文提出了一种通过使用辅助DP来解决减少这种枚举问题的方法,这些辅助DP在起点的选择方面是普遍的。利用上述方法构建了最优算法并在PC机上实现。
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引用次数: 2
On certain analogues of linkedness and supercompactness 关于连接和超紧性的某些类似物
IF 0.4 Q3 Mathematics Pub Date : 2020-05-01 DOI: 10.35634/2226-3594-2020-55-08
A. Chentsov
Natural generalizations of properties of the family linkedness and the topological space supercompactness are considered. We keep in mind reinforced linkedness when nonemptyness of intersection of preassigned number of sets from a family is postulated. Analogously, supercompactness is modified: it is postulated the existence of an open subbasis for which, from every covering (by sets of the subbasis), it is possible to extract a subcovering with a given number of sets (more precisely, not more than a given number). It is clear that among all families having the reinforced linkedness, one can distinguish families that are maximal in ordering by inclusion. Under natural and (essentially) “minimal”' conditions imposed on the original measurable structure, among the mentioned maximal families with reinforced linkedness, ultrafilters are certainly contained. These ultrafilters form subspaces in the sense of natural topologies corresponding conceptually to schemes of Wallman and Stone. In addition, maximal families with reinforced linkedness, when applying topology of the Wallman type, have the above-mentioned property generalizing supercompactness. Thus, an analogue of the superextension of the $T_1$-space is realized. The comparability of “Wallman”' and “Stone”' topologies is established. As a result, bitopological spaces (BTS) are realized; for these BTS, under equipping with analogous topologies, ultrafilter sets are subspaces. It is indicated that some cases exist when the above-mentioned BTS is nondegenerate in the sense of the distinction for forming topologies. At that time, in the case of “usual” linkedness (this is a particular case of reinforced linkedness), very general classes of spaces are known for which the mentioned BTS are degenerate (the cases when origial set, i.e., “unit”' is equipped with an algebra of sets or a topology).
考虑了族连性和拓扑空间超紧性性质的自然推广。当假定一个族中预先指定的集合数的交点不空时,我们注意到强化联系。类似地,对超紧性进行了修改:假设存在一个开子基,对于这个开子基,从每个覆盖(由子基的集合)中,可以提取出具有给定数量的集合(更准确地说,不超过给定数量)的子覆盖。显然,在所有具有加强联系的家庭中,人们可以通过包含来区分排序最大的家庭。在施加于原始可测结构的自然和(本质上)“最小”条件下,在所提到的具有增强连接的极大族中,肯定包含超过滤器。这些超滤波器形成了自然拓扑意义上的子空间,在概念上与Wallman和Stone的方案相对应。此外,当应用Wallman型拓扑时,具有增强连接的极大族具有上述推广超紧性的性质。这样,就实现了$T_1$-空间的超扩展。建立了“Wallman”拓扑和“Stone”拓扑的可比性。从而实现了双拓扑空间(BTS);对于这些BTS,在具有类似拓扑的情况下,超滤集合是子空间。结果表明,在某些情况下,上述BTS在形成拓扑的区别意义上是非简并的。此时,在“通常”链接的情况下(这是增强链接的特殊情况),已知非常一般的空间类,其中提到的BTS是退化的(当原始集合,即“单元”'配备了集合代数或拓扑的情况下)。
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引用次数: 1
Asymptotic expansion of a solution of a singularly perturbed optimal control problem with a convex integral quality index, whose terminal part additively depends on slow and fast variables 具有凸积分质量指标的奇摄动最优控制问题解的渐近展开式,该问题的终端部分加性地依赖于慢变量和快变量
IF 0.4 Q3 Mathematics Pub Date : 2020-05-01 DOI: 10.35634/2226-3594-2020-55-03
A. R. Danilin, A. A. Shaburov
The paper deals with the problem of optimal control with a Boltz-type quality index over a finite time interval for a linear steady-state control system in the class of piecewise continuous controls with smooth control constraints. In particular, we study the problem of controlling the motion of a system of small mass points under the action of a bounded force. The terminal part of the convex integral quality index additively depends on slow and fast variables, and the integral term is a strictly convex function of control variable. If the system is completely controllable, then the Pontryagin maximum principle is a necessary and sufficient condition for optimality. The main difference between this study and previous works is that the equation contains the zero matrix of fast variables and, thus, the results of A.B. Vasilieva on the asymptotic of the fundamental matrix of a control system are not valid. However, the linear steady-state system satisfies the condition of complete controllability. The article shows that problems of optimal control with a convex integral quality index are more regular than time-optimal problems.
研究一类具有光滑控制约束的分段连续控制的线性稳态控制系统在有限时间区间内具有玻尔兹型质量指标的最优控制问题。特别地,我们研究了在有界力作用下小质量点系统的运动控制问题。凸积分质量指标的末端部分加性地依赖于慢变量和快变量,积分项是控制变量的严格凸函数。如果系统是完全可控的,则庞特里亚金极大值原理是最优性的充分必要条件。本研究与以往工作的主要区别在于,该方程包含快速变量的零矩阵,因此A.B. Vasilieva关于控制系统基本矩阵渐近的结果不成立。而线性稳态系统满足完全可控的条件。本文证明了带凸积分质量指标的最优控制问题比时间最优问题更具有规律性。
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引用次数: 2
Construction of scattering curves in one class of time-optimal control problems with leaps of a target set boundary curvature 一类具有目标集边界曲率跳跃的时间最优控制问题散射曲线的构造
IF 0.4 Q3 Mathematics Pub Date : 2020-05-01 DOI: 10.35634/2226-3594-2020-55-07
P. Lebedev, A. A. Uspenskii
We consider a time-optimal control problem on the plane with a circular vectogram of velocities and a non-convex target set with a boundary having a finite number of points of discontinuity of curvature. We study the problem of identifying and constructing scattering curves that form a singular set of the optimal result function in the case when the points of discontinuity of curvature have one-sided curvatures of different signs. It is shown that these points belong to pseudo-vertices that are characteristic points of the boundary of the target set, which are responsible for the generation of branches of a singular set. The structure of scattering curves and the optimal trajectories starting from them, which fall in the neighborhood of the pseudo-vertex, is investigated. A characteristic feature of the case under study is revealed, consisting in the fact that one pseudo-vertex can generate two different branches of a singular set. The equation of the tangent to the smoothness points of the scattering curve is derived. A scheme is proposed for constructing a singular set, based on the construction of integral curves for first-order differential equations in normal form, the right-hand sides of which are determined by the geometry of the boundary of the target in neighborhoods of the pseudo-vertices. The results obtained are illustrated by the example of solving the control problem when the target set is a one-dimensional manifold.
考虑一个平面上的时间最优控制问题,该平面具有圆形速度矢量图和具有有限个曲率不连续点的边界的非凸目标集。研究了当曲率不连续点具有不同符号的单侧曲率时,如何识别和构造构成最优结果函数奇异集的散射曲线问题。结果表明,这些点属于伪顶点,它们是目标集边界的特征点,负责生成奇异集的分支。研究了伪顶点附近散射曲线的结构及其出发的最优轨迹。揭示了所研究案例的一个特征,即一个伪顶点可以生成一个奇异集的两个不同分支。导出了散射曲线光滑点的切线方程。在构造一阶正规微分方程的积分曲线的基础上,提出了一种构造奇异集的方法,该方法的右手边由伪顶点邻域内目标边界的几何形状决定。通过求解目标集为一维流形时的控制问题的实例说明了所得结果。
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引用次数: 2
The routing problems with optimization of the starting point: dynamic programming 带优化起点的路径问题:动态规划
IF 0.4 Q3 Mathematics Pub Date : 2019-12-09 DOI: 10.20537/2226-3594-2019-54-08
Chentsov Aleksandr Georgievich, Chentsov Pavel Aleksandrovich
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引用次数: 2
Persecution of rigidly coordinated evaders in a linear problem with fractional derivatives and a simple matrix 具有分数阶导数和简单矩阵的线性问题中刚性协调逃避者的迫害
IF 0.4 Q3 Mathematics Pub Date : 2019-11-01 DOI: 10.20537/2226-3594-2019-54-04
A. I. Machtakova
In the finite-dimensional Euclidean space, the problem of pursuit of a group of evaders by a group of pursuers is considered, which is described by a system of the form $$D^{(alpha)} z_{ij} = a z_{ij} + u_i - v,$$ where $D^{(alpha)} f$ is the Caputo derivative of the order $alpha in (0,1)$ of the function $f$. It is assumed that all evaders use the same control. The goal of the pursuers is to catch at least one of the evaders. The evaders use piecewise-program strategies, and the pursuers use piecewise-program counterstrategies. Every pursuer catches not more than one evader. The set of admissible controls is a ball of unit radius with the center at the origin, the target sets are the origin. In terms of initial positions and game parameters, a sufficient conditions for the capture are obtained.
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引用次数: 0
On the supercompactness of ultrafilter space with the topology of Wallman type Wallman型拓扑下超滤空间的超紧性
IF 0.4 Q3 Mathematics Pub Date : 2019-11-01 DOI: 10.20537/2226-3594-2019-54-07
A. Chentsov
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引用次数: 2
On the spectrum of a relativistic Landau Hamiltonian with a periodic electric potential 具有周期电势的相对论性朗道哈密顿谱
IF 0.4 Q3 Mathematics Pub Date : 2019-11-01 DOI: 10.20537/2226-3594-2019-54-01
L. I. Danilov
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引用次数: 0
期刊
Izvestiya Instituta Matematiki i Informatiki-Udmurtskogo Gosudarstvennogo Universiteta
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