Network Configuration Control Via Connectivity Graph Processes
|
|
- Rosalind Baker
- 5 years ago
- Views:
Transcription
1 Network Configuration Control Via Connectivity Grah Processes Abubakr Muhammad Deartment of Electrical and Systems Engineering University of Pennsylvania Philadelhia, PA 90 Magnus Egerstedt School of Electrical and Comuter Engineering Georgia Institute of Technology Atlanta, GA 0 magnus@ece.gatech.edu Abstract In this aer, we discuss how to generate trajectories, modeled as connectivity grah rocesses, on the sace of grahs induced by the network toology. We discuss the role of feasibility, reachability, and otimality in this context. In articular, we study in detail the role of reachability and the comutation of reachable sets by using the cylindrical algebraic decomosition (CAD) algorithm. I. INTRODUCTION The roblem of coordinating multile autonomous agents has attracted significant attention in recent years. Due to advances in many enabling technologies such as advanced communication systems, novel sensing latforms, and chea comutation devices, the realization of large scale networks of cooerating mobile agents has become ossible. An imortant theme in the study of such systems is the use of grah-theoretic models for describing the local interactions in the network. Notable results have been resented in [], [], [], [], []. The conclusion to be drawn from these research efforts is that a number of uestions can be answered in a natural way by abstracting away the continuous dynamics of the individual agents. In [6], [7], the authors have resented a detailed study of grahs that arise due to the limited sensory ercetion or communication of individual agents in a formation. We have shown that the grahs that can reresent formations do in fact corresond to a roer subset of all grahs, denoted by the set of connectivity grahs. We have resented several examles of grahs that fail to exist as connectivity grahs. The idea of infeasibility in the configuration sace has been exlored further in [8], where we have used techniues from semidefinite rogramming to obtain the reuired infeasibility certificates. Based on these results, we have resented in [9] a comutational framework in the context of formation switching for achieving a global objective. At the heart of this framework lies the concet of a connectivity grah rocess, that is made ossible by understanding issues concerning feasibility, reachability and otimality. Several alications of this framework have been outlined in [9], that include the roduction of low-comlexity formations and collaborative beamforming in sensor networks. In this aer, we focus on one articular asect of this framework, namely the uestion of reachability in connectivity grah rocesses. In articular, This work was sonsored by the US Army Research Office through the grant #9988. we give details on the use of the cylindrical algebraic decomosition (CAD) algorithm from real algebraic geometry for obtaining connectivity grah rocesses. This aer is organized as follows. We first summarize our revious work on connectivity grahs (Section II). We introduce the concet of connectivity grah rocesses and use semi-definite rogramming techniues for determining feasible switchings (Section III). We then discuss the comutation of reachable sets from a given initial grah and rovide details on the use of the CAD algorithm (Section IV). We set u the framework for obtaining otimal trajectories as grah rocesses (Section V). We then rovide some simulation results to show that our method is comutable, followed by our conclusions (Section VI). II. FORMATIONS AND CONNECTIVITY GRAPHS Grahs can model local interactions between agents, when individual agents are constrained by limited knowledge of other agents. In this section we summarize some revious results [6] of a grah theoretic formalism for describing formations in which the rimary limitation of ercetion for each agent is the limited range of its sensor. Suose we have N such agents with identical dynamics evolving on R. Each agent is euied with a range limited sensor by which it can sense the osition of other agents. All agents have identical sensor ranges δ. Let the osition of each agent be x n R, and its dynamics be given by ẋ n = f(x n, u n ), () where u n R m is the control for agent n and f : R R m R is a smooth vector field. The configuration sace C N (R ) of the agent formation is made u of all ordered N-tules in R, with the roerty that no two oints coincide, i.e. C N (R ) = (R R... R ), () where = {(x, x,..., x N ) : x i = x j for some i j}. The evolution of the formation can be reresented as a trajectory F : R + C N (R ), usually written as F(t)(x (t), x (t),... x N (t)) to signify time evolution. The satial relationshi between agents can be reresented as a grah in which the vertices of the grah reresent the agents, and the air of vertices on each edge tells us that the corresonding agents are within sensor range δ of each other.
2 Let G N denote the sace of all ossible grahs that can be formed on N vertices V = {v, v,..., v N }. Then we can define a function Φ N : C N (R ) G N, with Φ N (F(t)) = G(t), where G(t) = (V, E(t)) G N is the connectivity grah of the formation F(t). v i V reresents agent i at osition x i, and E(t) denotes the edges of the grah. e ij (t) = e ji (t) E(t) if and only if x i (t) x j (t) δ, i j. The grahs are always undirected because the sensor ranges are identical. The motion of agents in a formation may result in the removal or addition of edges in the grah. Therefore G(t) is a dynamic structure. Lastly and most imortantly, every grah in G N is not a connectivity grah. The last observation is not as obvious as the others, and it has been analyzed in detail in [6]. A realization of a grah G G N is a formation F C N (R ), such that Φ N (F) = G. An arbitrary grah G G N can therefore be realized as a connectivity grah in C N (R ) if Φ N (G) is nonemty. We denote by G N,δ G N, the sace of all ossible grahs on N agents with sensor range δ, that can be realized in C N (R ). Formations can roduce a wide variety of grahs for N vertices. This includes grahs that have disconnected subgrahs or totally disconnected grahs with no edges. However the roblem of switching between different formations or of finding interesting structures within a formation of sensor range limited agents can only be tackled if no subformation of agents is totally isolated from the rest of the formation. This means that the connectivity grah G(t) of the formation F(t) should always remain connected (in the sense of connected grahs) for all time t. III. FEASIBLE CONNECTIVITY GRAPH TRANSITIONS Connectivity grah rocesses are generated through the movement of individual nodes. For a connectivity grah G j = (V j, E j ) = Φ N (x(t j )) let the nodes be artitioned as V j = Vj 0 Vm m j, where the movement of the nodes in Vj facilitates the transition from G j to the next grah G j+ and Vj 0 is the set of nodes that are stationary. With the ositions x 0 j = {x m(t i )} m V 0 j being fixed, let Feas(G j, Vj m, x0 j ) G N,δ be the set of all connected connectivity grahs that are feasible by an unconstrained lacement of ositions corresonding to Vj m in R. (We will often denote this set as Feas(G j, Vj m), when the the ositions x0 j are understood from context.) It will be aroriate to exlain the reason for keeing track of mobile and stationary nodes at each transition. In rincile, it is ossible to comute this entire set of feasible transitions by an enumeration rocedure. However, in order to manage the combinatorial growth in the number of ossible grahs, it is desirable to let the transitions be generated by the movements of a small subset of nodes only. In fact, we will investigate the situation where only one node is allowed to move at a time. Hence, we let Vj 0 = {,..., k, k +,..., N} and Vm j = {k}. It should be noted that the movement of node k can only result in the addition or deletion of edges that have node k as one of its vertices. Therefore the enumeration of the ossible resulting grahs should count all ossible combinations of G 0 Set of feasible transitions Feas(G 0, {}). Infeasible transitions. Fig.. Feasible and infeasible grahs by movement of node. such deletions and additions. This number can be easily seen to be N for N nodes. Since we are also reuired to kee the grah connected at all times, this number is actually N, obtained after removing the grah in which node k has no edge with any other node. Now, we can use the S-rocedure to evaluate whether each of the new grahs resulting from this enumeration is feasible. Since all nodes are fixed excet for x k = (x, y), the semialgebraic set we need to check for non-feasibility is defined by N olynomial ineualities over R[x, y]. Each of these ineualities has either of the following two forms, [ ] x y 0 x i 0 y i x i y i δ x i y i x y 0, if e ik E, or [ ] 0 x i x y 0 y i x y > 0, if e ik E. x i y i x i + y i δ where i N, E is the edge set of the new grah and we denote x i (t j ) by (x i, y i ) for i k. This comutation can be reeated for all N nodes so that we have a choice of N( N ) grahs. If we let ρ i = δ x i y i then by denoting A i = 0 x i, B j = 0 y i x i y i ρ i 0 x j 0 y j x j y j ρ j and ignoring the lossy asect of the S-rocedure [0], we need to solve the LMI, A α λ αi A αi λ αj B αj 0. i,e αi k E j,e αj k E An examle of such a calculation is given in Figure, where V 0 = {,,, } and V m = {}. The LMI control toolbox [] for MATLAB has been used to solve the LMI for each of these grahs in order to get the aroriate certificates.,
3 We now give a detailed study of reachability in the context of connectivity grah rocesses as the main contribution of this aer. IV. REACHABILITY AND CONNECTIVITY GRAPH PROCESSES Note that the set Feas(G 0, V0 m ) does not deend on the actual movement of the individual nodes. In fact, even if G Feas(G 0, V0 m ), it does not necessarily mean that there exists a trajectory by which G 0 G or even that G 0 G G... G. The geometrical configuration of nodes may create an obstruction in obtaining a grah rocess that takes G 0 to G. There are two ways by which this obstruction is created: The reuirement to maintain connectivity and conformity to a fixed set of mobile nodes. We therefore need some notion of reachability on the sace G N,δ. We say that a connectivity grah G f is reachable from an initial grah G 0 if there exists a connectivity grah rocess of finite length G 0 G... G f and a seuence of vertex-sets {Vk m} such that each G k+ Feas(G k, Vk m). If Vm k = Vm at each transition, then every G k Feas(G 0, V m ). (In articular, G f Feas(G 0, V m ).) Consider all such G s that are reachable from G 0 with a fixed V m. We will denote this set by Reach(G 0, V m ). It is easy to see that Reach(G 0, V m ) Feas(G 0, V m ). In the revious sub-section, it was shown how to determine the membershi for the set Feas(G 0, V m ). For all such grahs, determining whether they also belong Reach(G 0, V m ) is not very straightforward, secially under the restriction that the intermediate grahs have to be connected. For the secial case of a single mobile node, the situation is manageable as discussed below. We first describe what is called a cylindrical algebraic decomosition or CAD of a semi-algebraic set [], []. Cylindrical Algebraic Decomosition: We first give some definitions. A Nash manifold M R n is an analytic submanifold which is a semialgebraic set. Let U R n. A Nash function f : U R is a smooth function for which there exists a olynomial (x, t) = (x,... x n, t) such that (x, f(x)) = 0 for all x U. A Nash cell in R n is a Nash manifold which is diffeomorhic to an oen box (, ) d of dimension d. Every semialgebraic set can be decomosed into a disjoint union of Nash cells. More recisely []; Theorem.: Let A,, A be semialgebraic subsets of R n. Then there exists a finite semialgebraic artition of R n into Nash cells such that each A j is a union of some of these cell. The existence of such a decomosition is given by a techniue in real algebraic geometry known as the cylindrical algebraic decomosition (CAD) []. A CAD of R n is a artition into finitely many semialgebraic subsets. The CAD of R n is given by induction on n as follows. ) A CAD of R is a subdivision by finitely many oints a <... < a l. The cells are the singletons {a i } and the oen intervals delimited by these oints. ) For n >, a CAD of R n is given by a CAD of R n and for each cell C of R n, the Nash functions ζ C, < < ζ C,lC : C R. The cells of the CAD of R n are the grahs of ζ C,j and the bands in the cylinders C R determined by the grahs. Observe that the cells generated by CAD are Nash, thus roviding a artition satisfying Theorem.. For our need, we only need a CAD of R. For notational convenience, let us denote the collection of Nash cells corresonding to a semialgebraic set S by CAD(S). With V m = {k} and V 0 = {,, k, k +,, N}, consider the semialgebraic set X(V 0 ) = {(x, y) : (x x j ) + (y y j ) δ } R. j V 0 () The first thing to note about this set is that it is comact. This relieves us of any comlicated comactification rocedures that are needed to get the CAD of non-comact sets. Let C be a cell in CAD(X(V 0 )), then we call r C a configuration oint in C if r x j for all j V 0. We then have the following result. Proosition.: Let C CAD(X(V 0 )). If, are any two configuration oints in C then Φ N ((x,..., x k,, x k+,..., x N )) = Φ N ((x,..., x k,, x k+,..., x N )). Proof: If C is a 0-cell we have the result trivially. We therefore assume that dim(c) > 0. For notational convenience denote (x,..., x k,, x k+,..., x N ) by x and the grah Φ N ((x,..., x k,, x k+,..., x N )) by G for a oint C. C B δ (x m ) δ B δ (x m ) dim(c) = dim(c) = Fig.. Toological obstruction used in the roof of Proosition.. First note that any air of ositions x i, x j, where i, j V 0 are always different. This is because they are the roduct of a valid configuration on the configuration sace C N (R ) described in Euation. If is a configuration oint then the N-tule x = (x,, x k,, x k+,, x N ) is a valid configuration on C N (R ), hence the name configuration oint. The configuration oint would corresond to the k- th node in G. This shows that the maing Φ N (x ) is well defined for all configuration oints in C. We now rove the roosition by contradiction. Assume that there exist oints, C such that G G. The grah G would differ from G in at least one edge incident on node k. Without loss of generality, assume that e km is an edge between a node m V 0 and the k-th node corresonding to in G, and that there is no edge between nodes k and m in G. The existence of this edge in G means that is inside the closed ball B δ (x m ), while / B δ (x m ) as shown in Figure. From the definition of δ C
4 a Nash cell, C is diffeomorhic to (, ) dim(c) which is simly connected. This means that C, in addition to being an oen set, is also simly connected. Therefore the ball B δ (x m ) induces a artition C = (C \ B δ (x m )) (C B δ (x m )), such that B δ (x m )) C. From Figure, it is clear that each connected comonent of B δ (x m )) C induces dim(c) Nash cells that are not already resent in the CAD of CAD(X(V 0 )). Moreover, this means that there exist lower dimensional Nash cells that are roerly contained in C. Both imlications are contrary to the CAD construction described above. Therefore, the connectedness of C creates a toological obstruction in getting G G. This roves the Proosition. x(t0) x x(t) x(t) x x(t) x x(t) x(t) In articular, the -skeleton CAD () (S) consists of all - dimensional Nash cells in CAD(S). Recall that the set X(V 0 ) described by Euation is a union of closed disks in R. Moreover the boundary of of this set X(V 0 ) CAD (0) (X(V 0 )) CAD () (X(V 0 )). If C CAD () (S) and C denotes its closure in R, then by construction of the CAD, C CAD (0) (X(V 0 )) CAD () (X(V 0 )). Therefore, any connected comonent of X(V 0 ) cannot be made u of cells in CAD () (S) alone. In fact, we get the following useful lemma. Lemma.: The set CAD(X(V 0 )) is connected if and only if CAD (0) (X(V 0 )) CAD () (X(V 0 )) is connected. The above lemma suggests the existence of a ath lanning algorithm for moving the mobile node between any two oints of a connected comonent of X(V 0 ). Given, in the same connected comonent of X(V 0 ), we construct a ath γ : [0, T ] X(V 0 ) such that γ(0) = and γ(t ) =. The lacement of both oints, in the same connected comonent ensures that T <. Also, each of the oints, lie in their uniue cells of CAD(X(V 0 )), which we denote by C, C resectively. We build our algorithm from the following observations. Fig.. nodes. Trajectory of a mobile node in the CAD generated by the fixed From this we get the following useful result. Corollary.: Let x(t) = (x,..., x k, γ(t), x k+,..., x N ) be a trajectory on C N (R ). If γ(t) C for all t [t 0, t f ) then the connectivity grah rocess has no transitions while t (t 0, t f ). For a trajectory x(t) = (x,..., x k, γ(t), x k+,..., x N ) on C N (R ) that is not confined to a single Nash cell, the grah may change as the trajectories goes from one cell to another. Moreover, since the connectivity grah remains unchanged inside one cell, the transition must take lace at the boundary of Nash cells. We can now begin to areciate the connection between the CAD decomosition and the geometric origin of transitions in a connectivity grah rocess. We further observe that the trajectory is artitioned into a finite number of ieces, where each iece is confined to a single Nash cell. In more recise terms, for each grah rocess G 0 G... G N with V m i = {k} for 0 i N and transitions at t < t <... < t N, there exist a finite number of Nash cells C x = {C 0, C,..., C M } C, where M N, such that the trajectory x(t) intersects with a finite sub-collection C x,i C x for t (t i, t i+ ). One such trajectory is deicted in Figure. We will construct an exlicit lanning algorithm to go from one oint in the CAD to another. Let us define the k-th skeleton of the CAD of a set S as CAD (k) (S) = {C CAD(S) dim(c) = k}. Toologically, a simly connected set means ath-wise connectedness and the absence of any holes in the set. Fig.. φ s(t) C φ s(t) φ 0 (,) (,) Path between two oints, in a -cell of the CAD. Case : When C = C = C, i.e. when both oints lie in the same cell, consider the diffeomorhism φ : C (, ) dim(c). Then the oints φ() and φ() in the oen cube can be joined by a straight line s(t) = tφ() + ( t)φ(). Now ma this line back to C by γ(t) = φ (s(t)), t [0, ]. This gives us the desired ath. Such an exlicit construction in terms of the diffeomorhism may only be needed when dim(c) =, as deicted in Figure. For lower dimensions the construction of γ(t) is rather obvious. Case : When C C. We study various situations in this case. (a). When dim(c ), dim(c ) <. In this case one can construct the ath γ exlicitly by building a grah G P = (V P, E P ) in the following way. Each v i V corresonds to a 0-cell Ci 0 in the CAD(0) (X(V 0 )) and an edge e ij E if and only if there exists a -cell C CAD () (X(V 0 )) such that Ci 0, C0 j C. Note that by construction of the CAD, if such a -cell exists, it will be uniue. Moreover, each -cell will be maed to a uniue edge in G P. If dim(c ) = dim(c ) = 0 we are done. If not, we modify as follows. We add one vertex for each of the oints G P
5 γ(t) v C 0 i C 0 j v i w ij v j v x k(t 0) x x x CAD(X(V 0 )) G P x k(t f) Fig.. Path lanning via lanning grah., that belong to a -cell. Suose dim(c ) =. Then C corresonds to an edge e jk = (v j, v k ) induced by 0-cells Cj 0 and Ck 0. We add a vertex v corresonding to. Now modify the grah G P by removing the edge e jk and inserting two edges e j = (v j, v ) and e k = (v, v k ). A similar rocedure modifies G P if dim(c ) =. We call this modified grah our lanning grah G P. An examle of this construction is shown in Figure. We can now convert this grah into a weighted grah, by assigning each edge e ij a real number w ij. One choice of weights can be a constant weight on all edges. We will show later how to assign a more natural set of weights. Once we have that, we can use a standard discrete lanning algorithm such as the Dijkstra s algorithm [] to get a ath on G P that connects v to v by a seuence of edges and vertices. This grahical ath can now be maed back to R to get an exlicit ath γ(t) that connects to. We now give this maing. Note first that a -cell Cj has two 0-cells say C0 j 0 and Cj 0 on its boundary. If this -cell is also a subset of B δ (x k ) for some k V 0 then by the roerties of CAD construction, Cj has a natural arametrization as a curve in R given by c j : s (s, sgn(c j ) δ (s y k ) +x k ), s (x j0, x j ), where sgn(cj ) can be determined uniuely for each such - cell. If w j0j is the weight on the corresonding edge e j0j E P, then we can re-arameterize by a linear maing T j : R R that sends [0, w j0 j ) [x j0, x j ).We can therefore define a arameterized curve γ j0 j (t) = c j (T j (t)), where t [0, w j0j ). In case the -cell Ck is a vertical line segment, the arametrization is more simle. Simly, let γ k0k (t) = T k (t) where t [0, w k0k ) and T k mas [0, w k0k ) [y k0, y k ). In this way we have a method to ma back an edge in the lanning grah to R. We have omitted here details for edges that have either v or v as one of the vertices. It is easy to see that the maing for these edges is very similar. Now, given a ath in G P as a seuence of vertices and edges, one can build exlicitly a ath in R as a concatenation of the curves described above. More recisely let the ath be given by a seuence v, e k, v k, e k k, v k,..., v km, e km, v, then we define a ath that connects to by γ(t) = γ k γ k k... γ km (t), t [0, W ) Fig. 6. Path between two oints that belong to their resective -cells in the CAD. where W = w k + w k k +... w km and is the ath concatenation oeration defined by γ i γ j (t) = { γi (t), t [0, w i ); γ j (t), t [w i, w i + w j ). (b). General Case: We now discuss the most general case. We resent a strategy for the case when either or both of and belong to a -cell. Suose, are inside C, C CAD () (X(V 0 )) resectively. Now consider the line segment γ 0 (s) = s + ( s), where s [0, ]. By the construction of CAD, this line segment cuts at least cells of dimension less than. Let s be the minimum value of s at which the line cuts a lower dimensional cell. Also let s be the corresonding maximum value. Now let = γ 0 (s ) and = γ 0 (s ). Using the rocedure described in the revious case, a lanning grah can be constructed to find a ath that connects to. Concatenating the resulting curve with γ 0 (t) for t [0, s ) in the beginning and γ 0 (t) for t [s, ] at the end we get the resulting ath. One such construction is shown in Figure 6. The above discussion makes it clear that it ossible to find a ath that transorts the k-th node at time t 0 from any initial osition x(t 0 ) to a desired osition inside the same connected comonent of CAD(X(V 0 )) in a finite time. In order to give a meaning to this transortation from the oint of view of control, it enough to assume that the dynamics of the nodes described by Euation are globally controllable. Combining these observations, we get our main result. Theorem.: Assume that the individual nodes are globally controllable. Let Vi m = {k} be fixed. Given two connectivity grahs Φ N (x(0)) and G f, there exists a finite connectivity grah rocess Φ N (x(0)) G... G f, and a corresonding trajectory x(t) C N (R ), t [t 0, t f ] such that x(t f ) Φ N (G f ) if and only if there exist a finite collection of Nash cells C x = {C 0, C,..., C M } CAD(X(V 0 )) such that x(0) C 0, x(t f ) C M and x(t) C j for some C j C x for all t [t 0, t f ] This gives us a way to realize trajectories on the sace of connectivity grahs and a method to characterize the reachable set with one mobile node. In fact, by construction this node only needs two different tyed of motions, namely along constant vector fields and rotations about fixed oints.
6 V. GLOBAL OBJECTIVES, DESIRABLE TRANSITIONS AND OPTIMALITY The urose of a coordinated control strategy in a multiagent system is to evolve towards the fulfilment of a global objective. This tyically reuires the minimization (or maximization) of a cost associated with each global configuration. Viewed in this way, a lanning strategy should basically be a search rocess over the configuration sace, evolving towards this otimum. If the global objective is fundamentally a function of the grahical abstraction of the formation, then it is better to erform this search over the sace of grahs instead of the full configuration sace of the system. By introducing various grahical abstractions in the context of connectivity grahs, we have the right machinery to erform this kind of lanning. In other words, we will associate a cost or score with each connectivity grah and then work towards minimizing it. Given Reach(G 0, V m ), a decision need to be taken regarding what G f Reach(G 0, V m ) the system should switch to. For this we define a cost function Ψ : G N,δ R and we choose the transition through G f = arg min Ψ(G) G Reach(G 0,V m ) Here Ψ is analogous to a terminal cost in otimal control. If, in addition, we also take into account the cost associated with every transition in the grah rocess G 0 G... G M = G f that takes us to G f, then we would instead consider the minimization of the cost M J = Ψ(G f ) + β(i)l(g i, G i+ ), i=0 where L : G N,δ G N,δ R is the analogue of a discrete Lagrangian, β(i) are weighting constants, and G i+ Reach(G i, V m ) at each ste i. The choice of a Lagrangian lets us control the transient behavior of the system during the evolution of the grah rocess. In rincile, the framework described in this aer can be used for any alication that reuired otimization over connectivity grahs. We have resented such several such alications in a recent work [9]. Here we reroduce one such simulation result in Figure 7, where a low-comlexity formation called as a δ-chain [7] is achieved by a series of maneuvers in which different nodes take the role of the mobile node at aroriate stes. VI. CONCLUSIONS We have resented a generic framework for connectivity grah rocesses. The concets of feasibility and reachability are useful for obtaining otimal trajectories on the sace of connectivity grahs. These grahical abstractions are comutable using the techniues of semi-definite rogramming and CAD, as verified by simulation results. Fig. 7. Snashots of a connectivity grah rocess that generates a δ-chain by choosing different mobile nodes at aroriate intermediate transitions. REFERENCES [] R.Saber and R. Murray, Agreement Problems in Networks with Directed Grahs and Switching Toology, in Proc. IEEE Conference on Decision and Control, 00. [] A. Jadbabaie, J. Lin, and A. Morse, Coordination of grous of mobile autonomous agents using nearest neighbor rules, IEEE Transactions on Automatic Control, Vol. 8, No. 6, , 00. [] M. Mesbahi, On State-Deendent Dynamic Grahs and their Controllability Proerties, IEEE Conference on Decision and Control, 00 [] E. Klavins, R. Ghrist, and D. Lisky, A Grammatical Aroach to Self-Organizing Robotic Systems, IEEE Transactions on Automatic Control, 00. (To aear) [] Z. Lin, B. Francis, and M. Maggiore, Necessary and sufficient grahical conditions for formation control of unicycles, IEEE Transactions on Automatic Control, 00. [6] A. Muhammad and M. Egerstedt, Connectivity Grahs as Models of Local Interactions. Journal of Alied Mathematics and Comutation, Vol. 68, No.,. -69, Set. 00. [7] A. Muhammad and M. Egerstedt, On the Structural Comlexity of Multi-Agent Agent Formations, in Proc. American Control Conference, Boston, Massachusetts, USA, 00. [8] A. Muhammad and M. Egerstedt, Positivstellensatz Certificates for Non-Feasibility of Connectivity Grahs in Multi-agent Coordination, 6th IFAC World Congress, Prague, July -8, 00. [9] A. Muhammad and M. Egerstedt, Alications of Connectivity Grah Processes in Networked Sensing and Control, Worksho on Networked Embedded Sensing and Control, University of Notre Dame, 00. [0] S. Boyd, L. El Ghaoui, E. Feron and V. Balakrishnan, Linear Martix Ineualities in Systems and Control Theory, SIAM Studies in Alied Mathematics, 99. [] The Math- Works Inc., LMI Control Toolbox, Version.0.7, May 00. [] S. Basu, R. Pollack and M. Roy, Algorithms in Real Algebraic Geometry, Algorithms and Comutation in Mathematics Series, Vol. 0, Sringer, 00. [] J. Bochnak, M. Coste, M. Roy, Real Algebraic Geometry, Sringer- Verlag, Berlin, 998. [] E. Dijkstra, A Note on Two Problems in Connexion With Grahs, in Numerische Mathematik, vol.,. 697, 99.
arxiv:math/ v4 [math.gn] 25 Nov 2006
arxiv:math/0607751v4 [math.gn] 25 Nov 2006 On the uniqueness of the coincidence index on orientable differentiable manifolds P. Christoher Staecker October 12, 2006 Abstract The fixed oint index of toological
More informationApproximating min-max k-clustering
Aroximating min-max k-clustering Asaf Levin July 24, 2007 Abstract We consider the roblems of set artitioning into k clusters with minimum total cost and minimum of the maximum cost of a cluster. The cost
More informationTopic 7: Using identity types
Toic 7: Using identity tyes June 10, 2014 Now we would like to learn how to use identity tyes and how to do some actual mathematics with them. By now we have essentially introduced all inference rules
More informationCombinatorics of topmost discs of multi-peg Tower of Hanoi problem
Combinatorics of tomost discs of multi-eg Tower of Hanoi roblem Sandi Klavžar Deartment of Mathematics, PEF, Unversity of Maribor Koroška cesta 160, 000 Maribor, Slovenia Uroš Milutinović Deartment of
More informationDIFFERENTIAL GEOMETRY. LECTURES 9-10,
DIFFERENTIAL GEOMETRY. LECTURES 9-10, 23-26.06.08 Let us rovide some more details to the definintion of the de Rham differential. Let V, W be two vector bundles and assume we want to define an oerator
More informationElementary Analysis in Q p
Elementary Analysis in Q Hannah Hutter, May Szedlák, Phili Wirth November 17, 2011 This reort follows very closely the book of Svetlana Katok 1. 1 Sequences and Series In this section we will see some
More informationp-adic Measures and Bernoulli Numbers
-Adic Measures and Bernoulli Numbers Adam Bowers Introduction The constants B k in the Taylor series exansion t e t = t k B k k! k=0 are known as the Bernoulli numbers. The first few are,, 6, 0, 30, 0,
More informationOn the Toppling of a Sand Pile
Discrete Mathematics and Theoretical Comuter Science Proceedings AA (DM-CCG), 2001, 275 286 On the Toling of a Sand Pile Jean-Christohe Novelli 1 and Dominique Rossin 2 1 CNRS, LIFL, Bâtiment M3, Université
More informationLINEAR SYSTEMS WITH POLYNOMIAL UNCERTAINTY STRUCTURE: STABILITY MARGINS AND CONTROL
LINEAR SYSTEMS WITH POLYNOMIAL UNCERTAINTY STRUCTURE: STABILITY MARGINS AND CONTROL Mohammad Bozorg Deatment of Mechanical Engineering University of Yazd P. O. Box 89195-741 Yazd Iran Fax: +98-351-750110
More informationMath 751 Lecture Notes Week 3
Math 751 Lecture Notes Week 3 Setember 25, 2014 1 Fundamental grou of a circle Theorem 1. Let φ : Z π 1 (S 1 ) be given by n [ω n ], where ω n : I S 1 R 2 is the loo ω n (s) = (cos(2πns), sin(2πns)). Then
More informationVarious Proofs for the Decrease Monotonicity of the Schatten s Power Norm, Various Families of R n Norms and Some Open Problems
Int. J. Oen Problems Comt. Math., Vol. 3, No. 2, June 2010 ISSN 1998-6262; Coyright c ICSRS Publication, 2010 www.i-csrs.org Various Proofs for the Decrease Monotonicity of the Schatten s Power Norm, Various
More informationSECTION 5: FIBRATIONS AND HOMOTOPY FIBERS
SECTION 5: FIBRATIONS AND HOMOTOPY FIBERS In this section we will introduce two imortant classes of mas of saces, namely the Hurewicz fibrations and the more general Serre fibrations, which are both obtained
More informationSYMPLECTIC STRUCTURES: AT THE INTERFACE OF ANALYSIS, GEOMETRY, AND TOPOLOGY
SYMPLECTIC STRUCTURES: AT THE INTERFACE OF ANALYSIS, GEOMETRY, AND TOPOLOGY FEDERICA PASQUOTTO 1. Descrition of the roosed research 1.1. Introduction. Symlectic structures made their first aearance in
More informationCHAPTER 2: SMOOTH MAPS. 1. Introduction In this chapter we introduce smooth maps between manifolds, and some important
CHAPTER 2: SMOOTH MAPS DAVID GLICKENSTEIN 1. Introduction In this chater we introduce smooth mas between manifolds, and some imortant concets. De nition 1. A function f : M! R k is a smooth function if
More informationAn Inverse Problem for Two Spectra of Complex Finite Jacobi Matrices
Coyright 202 Tech Science Press CMES, vol.86, no.4,.30-39, 202 An Inverse Problem for Two Sectra of Comlex Finite Jacobi Matrices Gusein Sh. Guseinov Abstract: This aer deals with the inverse sectral roblem
More informationLinear diophantine equations for discrete tomography
Journal of X-Ray Science and Technology 10 001 59 66 59 IOS Press Linear diohantine euations for discrete tomograhy Yangbo Ye a,gewang b and Jiehua Zhu a a Deartment of Mathematics, The University of Iowa,
More informationCMSC 425: Lecture 4 Geometry and Geometric Programming
CMSC 425: Lecture 4 Geometry and Geometric Programming Geometry for Game Programming and Grahics: For the next few lectures, we will discuss some of the basic elements of geometry. There are many areas
More informationRANDOM WALKS AND PERCOLATION: AN ANALYSIS OF CURRENT RESEARCH ON MODELING NATURAL PROCESSES
RANDOM WALKS AND PERCOLATION: AN ANALYSIS OF CURRENT RESEARCH ON MODELING NATURAL PROCESSES AARON ZWIEBACH Abstract. In this aer we will analyze research that has been recently done in the field of discrete
More informationA CONCRETE EXAMPLE OF PRIME BEHAVIOR IN QUADRATIC FIELDS. 1. Abstract
A CONCRETE EXAMPLE OF PRIME BEHAVIOR IN QUADRATIC FIELDS CASEY BRUCK 1. Abstract The goal of this aer is to rovide a concise way for undergraduate mathematics students to learn about how rime numbers behave
More informationFinding Shortest Hamiltonian Path is in P. Abstract
Finding Shortest Hamiltonian Path is in P Dhananay P. Mehendale Sir Parashurambhau College, Tilak Road, Pune, India bstract The roblem of finding shortest Hamiltonian ath in a eighted comlete grah belongs
More informationMATH 6210: SOLUTIONS TO PROBLEM SET #3
MATH 6210: SOLUTIONS TO PROBLEM SET #3 Rudin, Chater 4, Problem #3. The sace L (T) is searable since the trigonometric olynomials with comlex coefficients whose real and imaginary arts are rational form
More informationThe Graph Accessibility Problem and the Universality of the Collision CRCW Conflict Resolution Rule
The Grah Accessibility Problem and the Universality of the Collision CRCW Conflict Resolution Rule STEFAN D. BRUDA Deartment of Comuter Science Bisho s University Lennoxville, Quebec J1M 1Z7 CANADA bruda@cs.ubishos.ca
More informationConvexification of Generalized Network Flow Problem with Application to Power Systems
1 Convexification of Generalized Network Flow Problem with Alication to Power Systems Somayeh Sojoudi and Javad Lavaei + Deartment of Comuting and Mathematical Sciences, California Institute of Technology
More informationBest approximation by linear combinations of characteristic functions of half-spaces
Best aroximation by linear combinations of characteristic functions of half-saces Paul C. Kainen Deartment of Mathematics Georgetown University Washington, D.C. 20057-1233, USA Věra Kůrková Institute of
More informationThe Fekete Szegő theorem with splitting conditions: Part I
ACTA ARITHMETICA XCIII.2 (2000) The Fekete Szegő theorem with slitting conditions: Part I by Robert Rumely (Athens, GA) A classical theorem of Fekete and Szegő [4] says that if E is a comact set in the
More information4. Score normalization technical details We now discuss the technical details of the score normalization method.
SMT SCORING SYSTEM This document describes the scoring system for the Stanford Math Tournament We begin by giving an overview of the changes to scoring and a non-technical descrition of the scoring rules
More informationFor q 0; 1; : : : ; `? 1, we have m 0; 1; : : : ; q? 1. The set fh j(x) : j 0; 1; ; : : : ; `? 1g forms a basis for the tness functions dened on the i
Comuting with Haar Functions Sami Khuri Deartment of Mathematics and Comuter Science San Jose State University One Washington Square San Jose, CA 9519-0103, USA khuri@juiter.sjsu.edu Fax: (40)94-500 Keywords:
More informationDUAL NUMBERS, WEIGHTED QUIVERS, AND EXTENDED SOMOS AND GALE-ROBINSON SEQUENCES. To Alexandre Alexandrovich Kirillov on his 3 4 th anniversary
DUAL NUMBERS, WEIGHTED QUIVERS, AND EXTENDED SOMOS AND GALE-ROBINSON SEQUENCES VALENTIN OVSIENKO AND SERGE TABACHNIKOV Abstract. We investigate a general method that allows one to construct new integer
More informationCorrespondence Between Fractal-Wavelet. Transforms and Iterated Function Systems. With Grey Level Maps. F. Mendivil and E.R.
1 Corresondence Between Fractal-Wavelet Transforms and Iterated Function Systems With Grey Level Mas F. Mendivil and E.R. Vrscay Deartment of Alied Mathematics Faculty of Mathematics University of Waterloo
More informationMobius Functions, Legendre Symbols, and Discriminants
Mobius Functions, Legendre Symbols, and Discriminants 1 Introduction Zev Chonoles, Erick Knight, Tim Kunisky Over the integers, there are two key number-theoretic functions that take on values of 1, 1,
More informationPositive decomposition of transfer functions with multiple poles
Positive decomosition of transfer functions with multile oles Béla Nagy 1, Máté Matolcsi 2, and Márta Szilvási 1 Deartment of Analysis, Technical University of Budaest (BME), H-1111, Budaest, Egry J. u.
More informationCR extensions with a classical Several Complex Variables point of view. August Peter Brådalen Sonne Master s Thesis, Spring 2018
CR extensions with a classical Several Comlex Variables oint of view August Peter Brådalen Sonne Master s Thesis, Sring 2018 This master s thesis is submitted under the master s rogramme Mathematics, with
More informationSTABILITY ANALYSIS TOOL FOR TUNING UNCONSTRAINED DECENTRALIZED MODEL PREDICTIVE CONTROLLERS
STABILITY ANALYSIS TOOL FOR TUNING UNCONSTRAINED DECENTRALIZED MODEL PREDICTIVE CONTROLLERS Massimo Vaccarini Sauro Longhi M. Reza Katebi D.I.I.G.A., Università Politecnica delle Marche, Ancona, Italy
More informationA Social Welfare Optimal Sequential Allocation Procedure
A Social Welfare Otimal Sequential Allocation Procedure Thomas Kalinowsi Universität Rostoc, Germany Nina Narodytsa and Toby Walsh NICTA and UNSW, Australia May 2, 201 Abstract We consider a simle sequential
More informationLilian Markenzon 1, Nair Maria Maia de Abreu 2* and Luciana Lee 3
Pesquisa Oeracional (2013) 33(1): 123-132 2013 Brazilian Oerations Research Society Printed version ISSN 0101-7438 / Online version ISSN 1678-5142 www.scielo.br/oe SOME RESULTS ABOUT THE CONNECTIVITY OF
More informationFeedback-error control
Chater 4 Feedback-error control 4.1 Introduction This chater exlains the feedback-error (FBE) control scheme originally described by Kawato [, 87, 8]. FBE is a widely used neural network based controller
More informationRESOLUTIONS OF THREE-ROWED SKEW- AND ALMOST SKEW-SHAPES IN CHARACTERISTIC ZERO
RESOLUTIONS OF THREE-ROWED SKEW- AND ALMOST SKEW-SHAPES IN CHARACTERISTIC ZERO MARIA ARTALE AND DAVID A. BUCHSBAUM Abstract. We find an exlicit descrition of the terms and boundary mas for the three-rowed
More informationMatching Transversal Edge Domination in Graphs
Available at htt://vamuedu/aam Al Al Math ISSN: 19-9466 Vol 11, Issue (December 016), 919-99 Alications and Alied Mathematics: An International Journal (AAM) Matching Transversal Edge Domination in Grahs
More informationConvex Optimization methods for Computing Channel Capacity
Convex Otimization methods for Comuting Channel Caacity Abhishek Sinha Laboratory for Information and Decision Systems (LIDS), MIT sinhaa@mit.edu May 15, 2014 We consider a classical comutational roblem
More informationMATH 2710: NOTES FOR ANALYSIS
MATH 270: NOTES FOR ANALYSIS The main ideas we will learn from analysis center around the idea of a limit. Limits occurs in several settings. We will start with finite limits of sequences, then cover infinite
More informationOn rendezvous for visually-guided agents in a nonconvex polygon
CDC 2005, To aear On rendezvous for visually-guided agents in a nonconvex olygon Anurag Ganguli Jorge Cortés Francesco Bullo Abstract This aer resents coordination algorithms for mobile autonomous agents
More informationRECIPROCITY LAWS JEREMY BOOHER
RECIPROCITY LAWS JEREMY BOOHER 1 Introduction The law of uadratic recirocity gives a beautiful descrition of which rimes are suares modulo Secial cases of this law going back to Fermat, and Euler and Legendre
More informationSystem Reliability Estimation and Confidence Regions from Subsystem and Full System Tests
009 American Control Conference Hyatt Regency Riverfront, St. Louis, MO, USA June 0-, 009 FrB4. System Reliability Estimation and Confidence Regions from Subsystem and Full System Tests James C. Sall Abstract
More informationOn generalizing happy numbers to fractional base number systems
On generalizing hay numbers to fractional base number systems Enriue Treviño, Mikita Zhylinski October 17, 018 Abstract Let n be a ositive integer and S (n) be the sum of the suares of its digits. It is
More informationGENERICITY OF INFINITE-ORDER ELEMENTS IN HYPERBOLIC GROUPS
GENERICITY OF INFINITE-ORDER ELEMENTS IN HYPERBOLIC GROUPS PALLAVI DANI 1. Introduction Let Γ be a finitely generated grou and let S be a finite set of generators for Γ. This determines a word metric on
More informationGOOD MODELS FOR CUBIC SURFACES. 1. Introduction
GOOD MODELS FOR CUBIC SURFACES ANDREAS-STEPHAN ELSENHANS Abstract. This article describes an algorithm for finding a model of a hyersurface with small coefficients. It is shown that the aroach works in
More informationNumerical Linear Algebra
Numerical Linear Algebra Numerous alications in statistics, articularly in the fitting of linear models. Notation and conventions: Elements of a matrix A are denoted by a ij, where i indexes the rows and
More informationArc spaces and some adjacency problems of plane curves.
Arc saces and some adjacency roblems of lane curves. María Pe Pereira ICMAT, Madrid 3 de junio de 05 Joint work in rogress with Javier Fernández de Bobadilla and Patrick Poescu-Pamu Arcsace of (C, 0).
More informationMODELING THE RELIABILITY OF C4ISR SYSTEMS HARDWARE/SOFTWARE COMPONENTS USING AN IMPROVED MARKOV MODEL
Technical Sciences and Alied Mathematics MODELING THE RELIABILITY OF CISR SYSTEMS HARDWARE/SOFTWARE COMPONENTS USING AN IMPROVED MARKOV MODEL Cezar VASILESCU Regional Deartment of Defense Resources Management
More informationANALYTIC NUMBER THEORY AND DIRICHLET S THEOREM
ANALYTIC NUMBER THEORY AND DIRICHLET S THEOREM JOHN BINDER Abstract. In this aer, we rove Dirichlet s theorem that, given any air h, k with h, k) =, there are infinitely many rime numbers congruent to
More informationARTICLE IN PRESS Discrete Mathematics ( )
Discrete Mathematics Contents lists available at ScienceDirect Discrete Mathematics journal homeage: wwwelseviercom/locate/disc Maximizing the number of sanning trees in K n -comlements of asteroidal grahs
More informationModel checking, verification of CTL. One must verify or expel... doubts, and convert them into the certainty of YES [Thomas Carlyle]
Chater 5 Model checking, verification of CTL One must verify or exel... doubts, and convert them into the certainty of YES or NO. [Thomas Carlyle] 5. The verification setting Page 66 We introduce linear
More informationApproximating l 2 -Betti numbers of an amenable covering by ordinary Betti numbers
Comment. Math. Helv. 74 (1999) 150 155 0010-2571/99/010150-6 $ 1.50+0.20/0 c 1999 Birkhäuser Verlag, Basel Commentarii Mathematici Helvetici Aroximating l 2 -Betti numbers of an amenable covering by ordinary
More informationIntroduction to MVC. least common denominator of all non-identical-zero minors of all order of G(s). Example: The minor of order 2: 1 2 ( s 1)
Introduction to MVC Definition---Proerness and strictly roerness A system G(s) is roer if all its elements { gij ( s)} are roer, and strictly roer if all its elements are strictly roer. Definition---Causal
More informationA Special Case Solution to the Perspective 3-Point Problem William J. Wolfe California State University Channel Islands
A Secial Case Solution to the Persective -Point Problem William J. Wolfe California State University Channel Islands william.wolfe@csuci.edu Abstract In this aer we address a secial case of the ersective
More informationOn the stability and integration of Hamilton-Poisson systems on so(3)
Rendiconti di Matematica, Serie VII Volume 37, Roma (016), 1 4 On the stability and integration of Hamilton-Poisson systems on so(3) R. M. ADAMS R. BIGGS W. HOLDERBAUM C. C. REMSING Abstract: We consider
More informationImprovement on the Decay of Crossing Numbers
Grahs and Combinatorics 2013) 29:365 371 DOI 10.1007/s00373-012-1137-3 ORIGINAL PAPER Imrovement on the Decay of Crossing Numbers Jakub Černý Jan Kynčl Géza Tóth Received: 24 Aril 2007 / Revised: 1 November
More informationCommutators on l. D. Dosev and W. B. Johnson
Submitted exclusively to the London Mathematical Society doi:10.1112/0000/000000 Commutators on l D. Dosev and W. B. Johnson Abstract The oerators on l which are commutators are those not of the form λi
More informationChapter 1 Fundamentals
Chater Fundamentals. Overview of Thermodynamics Industrial Revolution brought in large scale automation of many tedious tasks which were earlier being erformed through manual or animal labour. Inventors
More informationMATH 361: NUMBER THEORY EIGHTH LECTURE
MATH 361: NUMBER THEORY EIGHTH LECTURE 1. Quadratic Recirocity: Introduction Quadratic recirocity is the first result of modern number theory. Lagrange conjectured it in the late 1700 s, but it was first
More informationIntroduction to Banach Spaces
CHAPTER 8 Introduction to Banach Saces 1. Uniform and Absolute Convergence As a rearation we begin by reviewing some familiar roerties of Cauchy sequences and uniform limits in the setting of metric saces.
More informationHAUSDORFF MEASURE OF p-cantor SETS
Real Analysis Exchange Vol. 302), 2004/2005,. 20 C. Cabrelli, U. Molter, Deartamento de Matemática, Facultad de Cs. Exactas y Naturales, Universidad de Buenos Aires and CONICET, Pabellón I - Ciudad Universitaria,
More informationState Estimation with ARMarkov Models
Deartment of Mechanical and Aerosace Engineering Technical Reort No. 3046, October 1998. Princeton University, Princeton, NJ. State Estimation with ARMarkov Models Ryoung K. Lim 1 Columbia University,
More informationNotes on duality in second order and -order cone otimization E. D. Andersen Λ, C. Roos y, and T. Terlaky z Aril 6, 000 Abstract Recently, the so-calle
McMaster University Advanced Otimization Laboratory Title: Notes on duality in second order and -order cone otimization Author: Erling D. Andersen, Cornelis Roos and Tamás Terlaky AdvOl-Reort No. 000/8
More informationEstimation of the large covariance matrix with two-step monotone missing data
Estimation of the large covariance matrix with two-ste monotone missing data Masashi Hyodo, Nobumichi Shutoh 2, Takashi Seo, and Tatjana Pavlenko 3 Deartment of Mathematical Information Science, Tokyo
More informationOn a Markov Game with Incomplete Information
On a Markov Game with Incomlete Information Johannes Hörner, Dinah Rosenberg y, Eilon Solan z and Nicolas Vieille x{ January 24, 26 Abstract We consider an examle of a Markov game with lack of information
More informationRobust Performance Design of PID Controllers with Inverse Multiplicative Uncertainty
American Control Conference on O'Farrell Street San Francisco CA USA June 9 - July Robust Performance Design of PID Controllers with Inverse Multilicative Uncertainty Tooran Emami John M Watkins Senior
More informationf(r) = a d n) d + + a0 = 0
Math 400-00/Foundations of Algebra/Fall 07 Polynomials at the Foundations: Roots Next, we turn to the notion of a root of a olynomial in Q[x]. Definition 8.. r Q is a rational root of fx) Q[x] if fr) 0.
More informationComputer arithmetic. Intensive Computation. Annalisa Massini 2017/2018
Comuter arithmetic Intensive Comutation Annalisa Massini 7/8 Intensive Comutation - 7/8 References Comuter Architecture - A Quantitative Aroach Hennessy Patterson Aendix J Intensive Comutation - 7/8 3
More informationStable Delaunay Graphs
Stable Delaunay Grahs Pankaj K. Agarwal Jie Gao Leonidas Guibas Haim Kalan Natan Rubin Micha Sharir Aril 29, 2014 Abstract Let P be a set of n oints in R 2, and let DT(P) denote its Euclidean Delaunay
More informationBrownian Motion and Random Prime Factorization
Brownian Motion and Random Prime Factorization Kendrick Tang June 4, 202 Contents Introduction 2 2 Brownian Motion 2 2. Develoing Brownian Motion.................... 2 2.. Measure Saces and Borel Sigma-Algebras.........
More informationSolution sheet ξi ξ < ξ i+1 0 otherwise ξ ξ i N i,p 1 (ξ) + where 0 0
Advanced Finite Elements MA5337 - WS7/8 Solution sheet This exercise sheets deals with B-slines and NURBS, which are the basis of isogeometric analysis as they will later relace the olynomial ansatz-functions
More informationComputations in Quantum Tensor Networks
Comutations in Quantum Tensor Networks T Huckle a,, K Waldherr a, T Schulte-Herbrüggen b a Technische Universität München, Boltzmannstr 3, 85748 Garching, Germany b Technische Universität München, Lichtenbergstr
More informationGaps in Semigroups. Université Pierre et Marie Curie, Paris 6, Equipe Combinatoire - Case 189, 4 Place Jussieu Paris Cedex 05, France.
Gas in Semigrous J.L. Ramírez Alfonsín Université Pierre et Marie Curie, Paris 6, Equie Combinatoire - Case 189, 4 Place Jussieu Paris 755 Cedex 05, France. Abstract In this aer we investigate the behaviour
More informationDialectical Theory for Multi-Agent Assumption-based Planning
Dialectical Theory for Multi-Agent Assumtion-based Planning Damien Pellier, Humbert Fiorino Laboratoire Leibniz, 46 avenue Félix Viallet F-38000 Grenboble, France {Damien.Pellier,Humbert.Fiorino}.imag.fr
More informationFactorability in the ring Z[ 5]
University of Nebraska - Lincoln DigitalCommons@University of Nebraska - Lincoln Dissertations, Theses, and Student Research Paers in Mathematics Mathematics, Deartment of 4-2004 Factorability in the ring
More informationInformation collection on a graph
Information collection on a grah Ilya O. Ryzhov Warren Powell February 10, 2010 Abstract We derive a knowledge gradient olicy for an otimal learning roblem on a grah, in which we use sequential measurements
More information1 1 c (a) 1 (b) 1 Figure 1: (a) First ath followed by salesman in the stris method. (b) Alternative ath. 4. D = distance travelled closing the loo. Th
18.415/6.854 Advanced Algorithms ovember 7, 1996 Euclidean TSP (art I) Lecturer: Michel X. Goemans MIT These notes are based on scribe notes by Marios Paaefthymiou and Mike Klugerman. 1 Euclidean TSP Consider
More informationMATHEMATICAL MODELLING OF THE WIRELESS COMMUNICATION NETWORK
Comuter Modelling and ew Technologies, 5, Vol.9, o., 3-39 Transort and Telecommunication Institute, Lomonosov, LV-9, Riga, Latvia MATHEMATICAL MODELLIG OF THE WIRELESS COMMUICATIO ETWORK M. KOPEETSK Deartment
More informationMobility-Induced Service Migration in Mobile. Micro-Clouds
arxiv:503054v [csdc] 7 Mar 205 Mobility-Induced Service Migration in Mobile Micro-Clouds Shiiang Wang, Rahul Urgaonkar, Ting He, Murtaza Zafer, Kevin Chan, and Kin K LeungTime Oerating after ossible Deartment
More information#A64 INTEGERS 18 (2018) APPLYING MODULAR ARITHMETIC TO DIOPHANTINE EQUATIONS
#A64 INTEGERS 18 (2018) APPLYING MODULAR ARITHMETIC TO DIOPHANTINE EQUATIONS Ramy F. Taki ElDin Physics and Engineering Mathematics Deartment, Faculty of Engineering, Ain Shams University, Cairo, Egyt
More informationThe inverse Goldbach problem
1 The inverse Goldbach roblem by Christian Elsholtz Submission Setember 7, 2000 (this version includes galley corrections). Aeared in Mathematika 2001. Abstract We imrove the uer and lower bounds of the
More informationStone Duality for Skew Boolean Algebras with Intersections
Stone Duality for Skew Boolean Algebras with Intersections Andrej Bauer Faculty of Mathematics and Physics University of Ljubljana Andrej.Bauer@andrej.com Karin Cvetko-Vah Faculty of Mathematics and Physics
More informationEconometrica Supplementary Material
Econometrica Sulementary Material SUPPLEMENT TO WEAKLY BELIEF-FREE EQUILIBRIA IN REPEATED GAMES WITH PRIVATE MONITORING (Econometrica, Vol. 79, No. 3, May 2011, 877 892) BY KANDORI,MICHIHIRO IN THIS SUPPLEMENT,
More informationarxiv: v2 [math.ac] 5 Jan 2018
Random Monomial Ideals Jesús A. De Loera, Sonja Petrović, Lily Silverstein, Desina Stasi, Dane Wilburne arxiv:70.070v [math.ac] Jan 8 Abstract: Insired by the study of random grahs and simlicial comlexes,
More informationMaxisets for μ-thresholding rules
Test 008 7: 33 349 DOI 0.007/s749-006-0035-5 ORIGINAL PAPER Maxisets for μ-thresholding rules Florent Autin Received: 3 January 005 / Acceted: 8 June 006 / Published online: March 007 Sociedad de Estadística
More informationB8.1 Martingales Through Measure Theory. Concept of independence
B8.1 Martingales Through Measure Theory Concet of indeendence Motivated by the notion of indeendent events in relims robability, we have generalized the concet of indeendence to families of σ-algebras.
More informationInformation collection on a graph
Information collection on a grah Ilya O. Ryzhov Warren Powell October 25, 2009 Abstract We derive a knowledge gradient olicy for an otimal learning roblem on a grah, in which we use sequential measurements
More informationON THE NORM OF AN IDEMPOTENT SCHUR MULTIPLIER ON THE SCHATTEN CLASS
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 00, Number 0, Pages 000 000 S 000-9939XX)0000-0 ON THE NORM OF AN IDEMPOTENT SCHUR MULTIPLIER ON THE SCHATTEN CLASS WILLIAM D. BANKS AND ASMA HARCHARRAS
More informationNew Information Measures for the Generalized Normal Distribution
Information,, 3-7; doi:.339/info3 OPEN ACCESS information ISSN 75-7 www.mdi.com/journal/information Article New Information Measures for the Generalized Normal Distribution Christos P. Kitsos * and Thomas
More information1 Riesz Potential and Enbeddings Theorems
Riesz Potential and Enbeddings Theorems Given 0 < < and a function u L loc R, the Riesz otential of u is defined by u y I u x := R x y dy, x R We begin by finding an exonent such that I u L R c u L R for
More informationA Note on Guaranteed Sparse Recovery via l 1 -Minimization
A Note on Guaranteed Sarse Recovery via l -Minimization Simon Foucart, Université Pierre et Marie Curie Abstract It is roved that every s-sarse vector x C N can be recovered from the measurement vector
More informationOn the Chvatál-Complexity of Knapsack Problems
R u t c o r Research R e o r t On the Chvatál-Comlexity of Knasack Problems Gergely Kovács a Béla Vizvári b RRR 5-08, October 008 RUTCOR Rutgers Center for Oerations Research Rutgers University 640 Bartholomew
More informationAnalysis of some entrance probabilities for killed birth-death processes
Analysis of some entrance robabilities for killed birth-death rocesses Master s Thesis O.J.G. van der Velde Suervisor: Dr. F.M. Sieksma July 5, 207 Mathematical Institute, Leiden University Contents Introduction
More informationThe Knuth-Yao Quadrangle-Inequality Speedup is a Consequence of Total-Monotonicity
The Knuth-Yao Quadrangle-Ineuality Seedu is a Conseuence of Total-Monotonicity Wolfgang W. Bein Mordecai J. Golin Lawrence L. Larmore Yan Zhang Abstract There exist several general techniues in the literature
More informationSome results of convex programming complexity
2012c12 $ Ê Æ Æ 116ò 14Ï Dec., 2012 Oerations Research Transactions Vol.16 No.4 Some results of convex rogramming comlexity LOU Ye 1,2 GAO Yuetian 1 Abstract Recently a number of aers were written that
More informationTRACES OF SCHUR AND KRONECKER PRODUCTS FOR BLOCK MATRICES
Khayyam J. Math. DOI:10.22034/kjm.2019.84207 TRACES OF SCHUR AND KRONECKER PRODUCTS FOR BLOCK MATRICES ISMAEL GARCÍA-BAYONA Communicated by A.M. Peralta Abstract. In this aer, we define two new Schur and
More informationElementary theory of L p spaces
CHAPTER 3 Elementary theory of L saces 3.1 Convexity. Jensen, Hölder, Minkowski inequality. We begin with two definitions. A set A R d is said to be convex if, for any x 0, x 1 2 A x = x 0 + (x 1 x 0 )
More informationPOINTS ON CONICS MODULO p
POINTS ON CONICS MODULO TEAM 2: JONGMIN BAEK, ANAND DEOPURKAR, AND KATHERINE REDFIELD Abstract. We comute the number of integer oints on conics modulo, where is an odd rime. We extend our results to conics
More informationGENERALIZED FACTORIZATION
GENERALIZED FACTORIZATION GRANT LARSEN Abstract. Familiarly, in Z, we have unique factorization. We investigate the general ring and what conditions we can imose on it to necessitate analogs of unique
More information