Multiple Capture in a Group Pursuit Problem with Fractional Derivatives and Phase Restrictions

The problem of conflict interaction between a group of pursuers and an evader in a finite-dimensional Euclidean space is considered. All participants have equal opportunities. The dynamics of all players are described by a system of differential equations with fractional derivatives in the form D(α)zi=azi+ui−v,ui,v∈V, where D(α)f is a Caputo derivative of order α of the function f. Additionally, it is assumed that in the process of the game the evader does not move out of a convex polyhedral cone. The set of admissible controls V is a strictly convex compact and a is a real number. The goal of the group of pursuers is to capture of the evader by no less than m different pursuers (the instants of capture may or may not coincide). The target sets are the origin. For such a conflict-controlled process, we derive conditions on its parameters and initial state, which are sufficient for the trajectories of the players to meet at a certain instant of time for any counteractions of the evader. The method of resolving functions is used to solve the problem, which is used in differential games of pursuit by a group of pursuers of one evader.

Sensitivity Analysis for the Stationary Distribution of Reflected Brownian Motion in a Convex Polyhedral Cone

Reflected Brownian motion (RBM) in a convex polyhedral cone arises in a variety of applications ranging from the theory of stochastic networks to mathematical finance, and under general stability conditions, it has a unique stationary distribution. In such applications, to implement a stochastic optimization algorithm or quantify robustness of a model, it is useful to characterize the dependence of stationary performance measures on model parameters. In this paper, we characterize parametric sensitivities of the stationary distribution of an RBM in a simple convex polyhedral cone, that is, sensitivities to perturbations of the parameters that define the RBM—namely the covariance matrix, drift vector, and directions of reflection along the boundary of the polyhedral cone. In order to characterize these sensitivities, we study the long-time behavior of the joint process consisting of an RBM along with its so-called derivative process, which characterizes pathwise derivatives of RBMs on finite time intervals. We show that the joint process is positive recurrent and has a unique stationary distribution and that parametric sensitivities of the stationary distribution of an RBM can be expressed in terms of the stationary distribution of the joint process. This can be thought of as establishing an interchange of the differential operator and the limit in time. The analysis of ergodicity of the joint process is significantly more complicated than that of the RBM because of its degeneracy and the fact that the derivative process exhibits jumps that are modulated by the RBM. The proofs of our results rely on path properties of coupled RBMs and contraction properties related to the geometry of the polyhedral cone and directions of reflection along the boundary. Our results are potentially useful for developing efficient numerical algorithms for computing sensitivities of functionals of stationary RBMs.

Unstable Loci in Flag Varieties and Variation of Quotients

We consider the action of a semisimple subgroup $\hat{G}$ of a semisimple complex group $G$ on the flag variety $X=G/B$ and the linearizations of this action by line bundles $\mathcal L$ on $X$. We give an explicit description of the associated unstable locus in dependence of $\mathcal L$, as well as a formula for its (co)dimension. We observe that the codimension is equal to 1 on the regular boundary of the $\hat{G}$-ample cone and grows towards the interior in steps by 1, in a way that the line bundles with unstable locus of codimension at least $q$ form a convex polyhedral cone. We also give a description and a recursive algorithm for determining all GIT-classes in the $\hat{G}$-ample cone of $X$. As an application, we give conditions ensuring the existence of GIT-classes $C$ with an unstable locus of codimension at least two and which moreover yield geometric GIT quotients. Such quotients $Y_C$ reflect global information on $\hat{G}$-invariants. They are always Mori dream spaces, and the Mori chambers of the pseudoeffective cone $\overline{\textrm{Eff}}(Y_C)$ correspond to the GIT chambers of the $\hat{G}$-ample cone of $X$. Moreover, all rational contractions $f: Y_{C} \ \scriptsize{-}\scriptsize{-}{\scriptsize{-}\kern-5pt\scriptsize{>}}\ Y^{\prime}$ to normal projective varieties $Y^{\prime}$ are induced by GIT from linearizations of the action of $\hat{G}$ on $X$. In particular, this is shown to hold for a diagonal embedding $\hat{G} \hookrightarrow (\hat{G})^k$, with sufficiently large $k$.

A polynomial algorithm for the quadratic programming problem

An approach based on projection of a vector onto a pointed convex polyhedral cone is proposed for solving the quadratic programming problem with a positive definite matrix of the quadratic form. It is proved that this method has polynomial complexity. A method is said to be of polynomial computational complexity if the solution to the problem can be obtained in N

Necessary and sufficient conditions for stability of a bin-packing system

Objects of various integer sizes, o1, · ··, on, are to be packed together into bins of size N as they arrive at a service facility. The number of objects of size oi which arrive by time t is , where the components of are independent renewal processes, with At/t → λ as t → ∞. The empty space in those bins which are neither empty nor full at time t is called the wasted space and the system is declared stabilizable if for some finite B there exists a bin-packing algorithm whose use guarantees the expected wasted space is less than B for all t. We show that the system is stabilizable if the arrival processes are Poisson and λ lies in the interior of a certain convex polyhedral cone Λ. In this case there exists a bin-packing algorithm which stabilizes the system without needing to know λ. However, if λ lies on the boundary of Λ the wasted space grows as and if λ is exterior to Λ it grows as O(t); these conclusions hold even if objects may be repacked as often as desired.

Necessary and sufficient conditions for stability of a bin-packing system

Objects of various integer sizes, o 1, · ··, on, are to be packed together into bins of size N as they arrive at a service facility. The number of objects of size oi which arrive by time t is , where the components of are independent renewal processes, with At /t → λ as t → ∞. The empty space in those bins which are neither empty nor full at time t is called the wasted space and the system is declared stabilizable if for some finite B there exists a bin-packing algorithm whose use guarantees the expected wasted space is less than B for all t. We show that the system is stabilizable if the arrival processes are Poisson and λ lies in the interior of a certain convex polyhedral cone Λ. In this case there exists a bin-packing algorithm which stabilizes the system without needing to know λ. However, if λ lies on the boundary of Λ the wasted space grows as and if λ is exterior to Λ it grows as O(t); these conclusions hold even if objects may be repacked as often as desired.

Necessary and sufficient conditions for stability of a bin-packing system

Objects of various integer sizes,o1, · ··,on,are to be packed together into bins of sizeNas they arrive at a service facility. The number of objects of sizeoiwhich arrive by timetis, where the components ofare independent renewal processes, withAt/t → λast → ∞. The empty space in those bins which are neither empty nor full at timetis called thewasted spaceand the system is declaredstabilizableif for some finiteBthere exists a bin-packing algorithm whose use guarantees the expected wasted space is less thanBfor allt.We show that the system is stabilizable if the arrival processes are Poisson and λ lies in the interior of a certain convex polyhedral cone Λ. In this case there exists a bin-packing algorithm which stabilizes the system without needing to know λ. However, if λ lies on the boundary of Λ the wasted space grows asand if λ is exterior to Λ it grows asO(t); these conclusions hold even if objects may be repacked as often as desired.