Geometric conditions for regularity in a time-minimum problem with constant dynamics

Continuing the earlier research on local well-posedness of a time-minimum problem associated to a closed target set C in a Hilbert space H and a convex constant dynamics F we study the Lipschitz (or, in general, Hölder) regularity of the (unique) point in C achieved from x for a minimal time. As a consequence, smoothness of the value function is proved, and an explicit formula for its derivative is given

Neighbourhood retractions of nonconvex sets in a Hilbert space via sublinear functionals

For a closed subset C of a Hilbert space and for a sublinear functional, which is equivalent to the norm, we give conditions guaranteeing existence and uniqueness of the nearest points to C in the sense of the semidistance generated by given sublinear functional. This permits us to construct a continuous retraction onto C well defined in an open neighbourhood of C. In particular, according to one of the conditions, this neighbourhood can be represented in terms of balance between the local strict convexity modulus of the Minkowski (sublinear) functional and the measure of nonconvexity of the set C at each point

On minima of a functional of the gradient: upper and lower solutions

This paper studies a scalar minimization problem with an integral functional of the gradient under affine boundary conditions. A new approach is proposed using a minimal and a maximal solution to the convexified problem. We prove a density result: any relaxed solution continuously depending on boundary data may be approximated uniformly by solutions of the nonconvex problem keeping continuity relative to data. We also consider solutions to the nonconvex problem having Lipschitz dependence on boundary data with the best Lipschitz constant

Well-posedness of minimal time problems with constant dynamics in Banach spaces

This paper concerns the study of a general minimal time problem with a convex constant dynamics and a closed target set in Banach spaces. We pay the main attention to deriving sufficient conditions for the major well-posedness properties that include the existence and uniqueness of optimal solutions as well as certain regularity of the optimal value function with respect to state variables. Most of the results obtained are new even in finite-dimensional spaces. Our approach is based on advanced tools of variational analysis and generalized differentiation

The massless hexagon integral in D = 6 dimensions

We evaluate the massless one-loop hexagon integral in six dimensions. The result is given in terms of standard polylogarithms of uniform transcendental weight three, its functional form resembling the one of the remainder function of the two-loop hexagon Wilson loop in four dimensions.Comment: 3 page

Well-Posedness of Minimal Time Problem with Constant Dynamics in Banach Spaces

This paper concerns the study of a general minimal time problem with a convex constant dynamic and a closed target set in Banach spaces. We pay the main attention to deriving efficient conditions for the major well-posedness properties that include the existence and uniqueness of optimal solutions as well as certain regularity of the optimal value function with respect to state variables. Most of the results obtained are new even in finite-dimensional spaces. Our approach is based on advanced tools of variational analysis and generalized differentiation

Theoretical analysis of quantum key distribution systems when integrated with a DWDM optical transport network

A theoretical research and numerical simulation of the noise influence caused by spontaneous Raman scattering, four-wave mixing, and linear channel crosstalk on the performance of QKD systems was conducted. Three types of QKD systems were considered: coherent one-way (COW) QKD protocol, subcarrier-wave (SCW) QKD system, and continuous-variable (CV) QKD integrated with classical DWDM channels. We calculate the secure key generation rate for the systems mentioned addressing different channel allocation schemes (i.e., configurations). A uniform DWDM grid is considered with quantum channel located in C-band and O-band (at 1310 nm) of a telecommunication window. The systems' performance is analyzed in terms of the maximal achievable distance values. Configurations for the further analysis and investigation are chosen optimally, i.e., their maximal achievable distances are the best

Підвищення балансувальної ємності кульвих чи роликових автобалансирів із зменшенням часу настання автобалансування

The study has revealed an influence of the parameters of corrective weights (balls and cylindrical rollers) in auto-balancers on the balancing capacity and the duration of the transition processes of auto-balancing in fast-rotating rotors.A compact analytical function has been obtained to determine the balancing capacity of an auto-balancer (for any quantity of corrective weights – balls or rollers), with a subsequent analysis thereof.It is shown that the process of approach of the auto-balancing can be accelerated if the auto-balancer contains at least three corrective weights.It has been proved that at a fixed radius of the corrective weights the highest balancing capacity of an auto-balancer is achieved when the corrective weights occupy nearly half of the racetrack.The study has revealed that it is technically incorrect to formulate a problem of finding a radius of the corrective weights that would maximize the balancing capacity of the auto-balancer. The statement implies that if it is a ball auto-balancer, the racetrack is a sphere, but if it is a roller-type balancer, the racetrack is a cylinder. This leads to a practically useless result, suggesting that the highest balancing capacity is achieved by auto-balancers with one corrective weight. Besides, with n≥5 for balls and n≥8 for rollers, there happens a false optimization, which consists in several corrective weights being “excess”. Their removal increases the balancing capacity of the auto-balancer.It is correct (from the engineering point of view) that the mathematical task is to optimize the balancing capacity of an auto-balancer. Herewith, it is taken into account that the racetrack of the auto-balancer is torus-shaped, which restricts the radius of the corrective weights from the top. It is shown that the balancing capacity of an automatic balancer can be maximized if in a fixed volume the corrective weights have the largest possible radius and occupy almost a half of the racetrack.The research on the duration of the transition processes for the smallest value has produced the following conclusions:– to accelerate the achieving auto-balancing, the corrective weights should occupy nearly half of the racetrack;– the shortest time of the auto-balancing is achieved with three balls or five cylindrical rollers.Исследовано влияние размера и количества корригирующих грузов (шаров или цилиндрических роликов) в автобалансире на его балансировочную емкость и на продолжительность протекания переходных процессов при автобалансировке роторных систем. Установлены размер и количество корригирующих грузов, при которых достигается наибольшая балансировочная емкость автобалансира и наименьшая продолжительность переходных процессовДосліджено вплив розміру та кількості корегувальних вантажів (куль або циліндричних роликів) в автобалансирі на його балансувальну ємність та на тривалість перебігу перехідних процесів при автобалансуванні роторних систем. Знайдені розміри та кількість корегувальних вантажів, при яких досягається найбільша балансувальна ємність автобалансира та найменша тривалість перехідних процесі