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Integration of the mKdV Equation with nonstationary coefficients and additional terms in the case of moving eigenvalues 带有非平稳系数和附加项的mKdV方程在移动特征值情况下的积分
IF 0.4 Q4 MATHEMATICS Pub Date : 2023-05-01 DOI: 10.35634/2226-3594-2023-61-08
A. B. Khasanov, U. Hoitmetov, Sh.Q. Sobirov
In this paper, we consider the Cauchy problem for the non-stationary modified Korteweg–de Vries equation with an additional term and a self-consistent source in the case of moving eigenvalues. Also, the evolution of the scattering data of the Dirac operator is obtained, the potential of which is the solution of the loaded modified Korteweg–de Vries equation with a self-consistent source in the class of rapidly decreasing functions. Specific examples are given to illustrate the application of the obtained results.
本文研究了带有附加项和自洽源的非平稳修正Korteweg-de Vries方程在移动特征值情况下的Cauchy问题。同时,得到了Dirac算子散射数据的演化过程,其势为速降函数类中具有自相容源的加载修正Korteweg-de Vries方程的解。通过具体实例说明所得结果的应用。
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引用次数: 0
Asymptotic expansion of the solution to an optimal control problem for a linear autonomous system with a terminal convex quality index depending on slow and fast variables 一类终端凸质量指标依赖于慢变量和快变量的线性自治系统最优控制问题解的渐近展开
IF 0.4 Q4 MATHEMATICS Pub Date : 2023-05-01 DOI: 10.35634/2226-3594-2023-61-03
A. R. Danilin, O. O. Kovrizhnykh
In this paper, we investigate a problem of optimal control over a finite time interval for a linear autonomous system with slow and fast variables in the class of piecewise continuous controls with smooth geometric constraints in the form of a ball. We consider a terminal convex quality index that depends on slow and fast variables. We substantiate a limit relation for the vector determining the optimal control as the small parameter tends to zero. We refine the limit relation for the case of an indirect control problem with a terminal quality index, which is the sum of values of two strictly convex continuously differentiable functions, the first of which depends only on slow variables, and the second one depends only on fast variables and has a minimum at zero. In doing so, we show that the first component of the determining vector converges to the determining vector of the limit problem while the second component tends to zero. In the problem of indirect control of a system of material points in a medium with resistance, we obtain the complete asymptotics of the determining vector in powers of a small parameter.
本文研究了一类具有光滑几何约束的球形分段连续控制中具有慢变量和快变量的线性自治系统在有限时间区间的最优控制问题。我们考虑了一个依赖于慢速和快速变量的终端凸质量指标。我们证明了当小参数趋于零时确定最优控制的向量的极限关系。我们改进了具有终端质量指标的间接控制问题的极限关系,该问题是两个严格凸连续可微函数的值和,其中第一个函数只依赖于慢变量,第二个函数只依赖于快变量,并且在零处有最小值。在此过程中,我们证明了决定向量的第一个分量收敛于极限问题的决定向量,而第二个分量趋于零。在有阻力介质中质点系统的间接控制问题中,我们得到了决定向量的小参数幂的完全渐近性。
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引用次数: 0
A bottleneck routing problem with a system of priority tasks 优先级任务系统的瓶颈路由问题
IF 0.4 Q4 MATHEMATICS Pub Date : 2023-05-01 DOI: 10.35634/2226-3594-2023-61-09
A. G. Chentsov
We consider a minimax routing problem related to visiting megacities under precedence conditions and cost functions with task list dependence. It is supposed that some megacity system requiring visiting above all is selected. For solving, an approach with decomposition into a set of two minimax routing problems is proposed. A two-step widely understood dynamic programming procedure realizing an optimal composition solution is constructed. The above-mentioned optimality is established by theoretical methods. Application of the results obtained is possible under investigation of multi-stage processes connected with regular allocation of resources. Another variant of application concerns the particular case of one-element megacities (i.e., cities) and may be related to the issues of aviation logistics under organization of flights using one tool (airplane or helicopter) under system of tasks on the realization of passing cargo transportation with prioritization of visits realized above all.
考虑了一类具有任务列表依赖的代价函数和优先条件下访问特大城市的极大极小路由问题。假设选择了一些最需要参观的超大城市系统。为了求解这一问题,提出了一种分解为两个极大极小路由问题的方法。构造了一个可实现最优组合解的两步动态规划过程。通过理论方法建立了上述最优性。在与资源的定期分配有关的多阶段过程的研究中,可以应用所获得的结果。应用的另一种变体涉及单元素特大城市(即城市)的特殊情况,可能与使用一种工具(飞机或直升机)组织航班的任务系统下的航空物流问题有关,该任务系统实现了通过货物运输的优先级,首先实现了访问。
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引用次数: 0
Algorithms for constructing suboptimal coverings of plane figures with disks in the class of regular lattices 在正则格类中构造带盘平面图形次优覆盖的算法
IF 0.4 Q4 MATHEMATICS Pub Date : 2023-05-01 DOI: 10.35634/2226-3594-2023-61-05
P. Lebedev, O. Kuvshinov
The problem of covering a compact planar set $M$ with a set of congruent disks is considered. It is assumed that the centers of the circles belong to some lattice. The criterion of optimality in one case is the minimum of the number of elements of the covering, and in the other case — the minimum of the Hausdorff deviation of the union of elements of the covering from the set $M$. To solve the problems, transformations of parallel transfer and rotation with the center at the origin can be applied to the lattice. Statements concerning sufficient conditions for sets of circles that provide solutions to the problems are proved. Numerical algorithms based on minimizing the Hausdorff deviation between two flat compacts are proposed. Solutions of a number of examples are given for various figures of $M$.
研究了紧平面集合$M$被一组同盘覆盖的问题。我们假设这些圆的中心属于某个晶格。一种情况下的最优性准则是覆盖的元素数的最小值,另一种情况下的最优性准则是覆盖的元素的并集与集合$M$的Hausdorff偏差的最小值。为了解决这些问题,可以对晶格进行以原点为中心的平行转移和旋转变换。证明了关于圆集合的充分条件的陈述,这些陈述提供了问题的解。提出了基于最小化两个平面紧致之间的豪斯多夫偏差的数值算法。给出了若干实例的解法,适用于不同的$M$数值。
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引用次数: 0
On a class of Besicovitch almost periodic type selections of multivalued maps 关于一类多值映射的Besicovitch概周期型选择
IF 0.4 Q4 MATHEMATICS Pub Date : 2023-05-01 DOI: 10.35634/2226-3594-2023-61-04
L. I. Danilov
Let ${mathcal B}$ be a Banach space and let ${mathcal M}^p({mathbb R};{mathcal B})$, $pgeqslant 1$, be the Marcinkiewicz space with a seminorm $| cdot | _{{mathcal M}^p}$. By $widetilde {mathfrak B}^p_c({mathbb R};{mathcal B})$ we denote the set of functions ${mathcal F}in {mathcal M}^p({mathbb R};{mathcal B})$ that satisfy the following three conditions: (1) $| {mathcal F}(cdot )-{mathcal F}(cdot +tau )| _{{mathcal M}^p}to 0$ as $tau to 0$, (2) for every $varepsilon >0$ the set of ($varepsilon ,| cdot | _{{mathcal M}^p}$)-almost periods of the function ${mathcal F}$ is relatively dense, (3) for every $varepsilon >0$ there exists a set $X(varepsilon )subseteq {mathbb R}$ such that $| chi _{X(varepsilon )}| _{{mathcal M}^1({mathbb R};{mathbb R})}0$ there is a number $delta >0$ such that the estimate $| chi _X{mathcal F}| _{{mathcal M}^p}0$, we prove under the condition $rho (u,F(cdot ))in widetilde {mathcal M}^{p,circ }({mathbb R};{mathbb R})$ the existence of a function ${mathcal F}in widetilde {mathfrak B}^p_c({mathbb R};U)cap widetilde {mathcal M}^{p,circ }({mathbb R};U)$ such that ${mathcal F}(t)in F(t)$ and $rho (u,{mathcal F}(t))
设${mathcal B}$为Banach空间设${mathcal M}^p({mathbb R};{mathcal B})$, $pgeqslant 1$为Marcinkiewicz空间,带半形式$| cdot | _{{mathcal M}^p}$。我们用$widetilde {mathfrak B}^p_c({mathbb R};{mathcal B})$表示满足以下三个条件的函数集${mathcal F}in {mathcal M}^p({mathbb R};{mathcal B})$:(1) $| {mathcal F}(cdot )-{mathcal F}(cdot +tau )| _{{mathcal M}^p}to 0$为$tau to 0$,(2)对于每一个$varepsilon >0$,函数${mathcal F}$的($varepsilon ,| cdot | _{{mathcal M}^p}$)-几乎周期的集合相对密集,(3)对于每一个$varepsilon >0$存在一个集合$X(varepsilon )subseteq {mathbb R}$,使得$| chi _{X(varepsilon )}| _{{mathcal M}^1({mathbb R};{mathbb R})}0$存在一个数字$delta >0$,使得估计$| chi _X{mathcal F}| _{{mathcal M}^p}0$,在条件$rho (u,F(cdot ))in widetilde {mathcal M}^{p,circ }({mathbb R};{mathbb R})$下,我们证明了一个函数${mathcal F}in widetilde {mathfrak B}^p_c({mathbb R};U)cap widetilde {mathcal M}^{p,circ }({mathbb R};U)$的存在性,使得${mathcal F}(t)in F(t)$和$rho (u,{mathcal F}(t))
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引用次数: 0
Synchronous implementation of simultaneous multiple captures of evaders 同时捕获多个逃税者的同步实现
IF 0.4 Q4 MATHEMATICS Pub Date : 2023-05-01 DOI: 10.35634/2226-3594-2023-61-01
A. I. Blagodatskikh
The problem of pursuit of a group of $m$ evaders $(mgeqslant 1)$ in conflict-controlled processes with equal opportunities is considered. It is said that in the problem of chasing one evader $(m = 1)$, multiple capture occurs if a given number of pursuers catch him, and the moments of capture may not coincide. In the problem of simultaneous multiple capture of one evader, it is required that the moments of capture coincide. Simultaneous multiple capture of the whole group of evaders $(m geqslant 2)$ occurs if, as a result of pursuit, each evader is repeatedly caught simultaneously, and at the same time. In terms of the initial positions of the participants, necessary and sufficient conditions for the simultaneous multiple capture of the whole group of evaders are obtained.
研究了机会均等的冲突控制过程中$m$逃逃者$(mgeqslant 1)$群体的追捕问题。据说,在追捕一个逃逃者$(m = 1)$的问题中,如果给定数量的追捕者抓住了他,则会发生多次捕获,并且捕获的时刻可能不一致。在同时捕获多个逃兵的问题中,要求捕获的力矩一致。如果在追捕过程中,每个逃兵在同一时间同时被多次捕获,则整个逃兵群体$(m geqslant 2)$会同时被多次捕获。根据参与者的初始位置,得到了同时多次捕获整个逃逃者群的充分必要条件。
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引用次数: 0
On the exploitation of a population given by a system of linear equations with random parameters 一类具有随机参数的线性方程组给出的总体的开发
IF 0.4 Q4 MATHEMATICS Pub Date : 2023-05-01 DOI: 10.35634/2226-3594-2023-61-02
M. S. Woldeab, L. I. Rodina
We consider a population whose dynamics in the absence of exploitation is given by a system of linear homogeneous differential equations, and some random shares of the resource of each species at fixed times, are extracted from this population. We assume that the harvesting process can be controlled in such a way as to limit the amount of the extracted resource in order to increase the size of the next harvesting. A method for harvesting a resource is described, in which the largest value of the average time benefit is reached with a probability of one, provided that the initial amount of the population is constantly maintained or periodically restored. The harvesting modes are also considered in which the average time benefit is infinite. To prove the main assertions, we use the corollary of the law of large numbers proved by A.N. Kolmogorov. The results on the optimal resource extraction for systems of linear difference equations, a particular case of which are Leslie and Lefkovich population dynamics models, are given.
我们考虑一个种群,在没有开发的情况下,其动态由线性齐次微分方程系统给出,并且从该种群中抽取每个物种在固定时间的一些随机资源份额。我们假设采集过程可以通过这样一种方式进行控制,即限制提取资源的数量,以便增加下一次采集的规模。本文描述了一种获取资源的方法,只要不断保持或定期恢复种群的初始数量,以1的概率达到平均时间效益的最大值。同时考虑了平均时间效益无限的收获模式。为了证明主要的断言,我们使用柯尔莫哥洛夫证明的大数定律的推论。以Leslie和Lefkovich种群动力学模型为例,给出了线性差分方程系统的最优资源提取的结果。
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引用次数: 0
About the geometry of the Cassini oval, its non-convexity degree and $varepsilon$-offset layer 关于卡西尼椭圆的几何,它的非凸度和$varepsilon$偏移层
IF 0.4 Q4 MATHEMATICS Pub Date : 2022-11-01 DOI: 10.35634/2226-3594-2022-60-03
O. Kuvshinov
The paper studies the geometry of a closed nonconvex smooth simply connected curve on a plane — of the Cassini oval, as well as the geometry of the $varepsilon$-layer around the set whose boundary is the Cassini oval. Various analytical representations of the $varepsilon$-layer boundary are formed, and special points of this boundary are described. The measure of nonconvexity $alpha$ of a simply connected set whose boundary is the Cassini oval is determined, and the angular characteristic of nonconvexity of its $varepsilon$-neighborhood is defined.
研究了卡西尼椭圆平面上的闭非凸光滑单连通曲线的几何性质,以及以卡西尼椭圆为边界的集合周围$varepsilon$ -层的几何性质。形成了$varepsilon$层边界的各种解析表示,并描述了该边界的特殊点。确定了以卡西尼椭圆为边界的单连通集的非凸性测度$alpha$,并定义了其$varepsilon$邻域的非凸性的角特性。
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引用次数: 0
New generalized integral inequalities via $(h,m)$-convex modified functions 基于$(h,m)$-凸修正函数的新的广义积分不等式
IF 0.4 Q4 MATHEMATICS Pub Date : 2022-11-01 DOI: 10.35634/2226-3594-2022-60-01
B. Bayraktar, J. N. Nápoles Valdés
In this article, we establish several inequalities for $(h,m)$-convex maps, related to weighted integrals, used in previous works. Throughout the work, we show that our results generalize several of the integral inequalities known from the literature.
在这篇文章中,我们建立了$(h,m)$-凸映射的几个不等式,这些不等式与以前的作品中使用的加权积分有关。在整个工作中,我们证明了我们的结果推广了文献中已知的几个积分不等式。
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引用次数: 2
On the approach problem for a control system on a finite time interval 有限时间区间上控制系统的逼近问题
IF 0.4 Q4 MATHEMATICS Pub Date : 2022-11-01 DOI: 10.35634/2226-3594-2022-60-07
V. Ushakov, A. Ushakov
A conflict-controlled system in a finite-dimensional Euclidean space is considered. We study the game problem of approaching the system to the goal set in the phase space over a finite time interval. The study of the problem is based on methods developed in the theory of positional differential games. Within the framework of this theory, an approach to constructing approximate solutions to the approach problem is presented.
研究有限维欧氏空间中的冲突控制系统。研究了系统在有限时间间隔内逼近相空间目标集的对策问题。这个问题的研究是基于在位置微分对策理论中发展起来的方法。在该理论的框架内,提出了一种构造逼近问题近似解的方法。
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引用次数: 0
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Izvestiya Instituta Matematiki i Informatiki-Udmurtskogo Gosudarstvennogo Universiteta
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