randomized oblivious integral routing for minimizing power cost given an undirected network g v e and a set of traffic requests r the minimum power cost routing problem requires that each r k r be routed along a single path to minimize e e l e where l e is the traffic load on edge e and is a constant greater than 1 typically 1 3 this problem is important in optimizing the energy consumption of networks to address this problem we propose a randomized oblivious routing algorithm an oblivious routing algorithm makes decisions independently of the current traffic in the network this feature enables the efficient implementation of our algorithm in a distributed manner which is desirable for large scale high capacity networks an important feature of our work is that our algorithm can satisfy the integral constraint which requires that each traffic request r k should follow a single path we prove that given this constraint no randomized oblivious routing algorithm can guarantee a competitive ratio bounded by o e 1 1 by contrast our approach provides a competitive ratio of o e 1 1 log 2 1 v log 1 d where d is the maximum demand of traffic requests furthermore our results also hold for a more general case where the objective is to minimize e l e p where p 1 is an arbitrary unknown parameter with a given upper bound 1 the theoretical results established in proving these bounds can be further generalized to a framework of designing and analyzing oblivious integral routing algorithms which is significant for research on minimizing e l e in specific scenarios with simplified problem settings for instance we prove that this framework can generate an oblivious integral routing algorithm whose competitive ratio can be bounded by o log v log 1 d and o log 3 v log 1 d on expanders and hypercubes respectively original study on energy saving in oblivious integral routing an e 1 1 lower bound on competitive ratio a random oblivious integral routing algorithm with polylog tight competitive ratio a general framework to design and analyze oblivious integral routing algorithms polylog bound on competitive ratio for expanders and hypercubes

expander graphs and their applications a major consideration we had in writing this survey was to make it accessible to mathematicians as well as to computer scientists since expander graphs the protagonists of our story come up in numerous and often surprising contexts in both fields but perhaps we should start with a few words about graphs in general they are of course one of the prime objects of study in discrete mathematics however graphs are among the most ubiquitous models of both natural and human made structures in the natural and social sciences they model relations among species societies companies etc in computer science they represent networks of communication data organization computational devices as well as the flow of computation and more in mathematics cayley graphs are useful in group theory graphs carry a natural metric and are therefore useful in geometry and though they are just one dimensional complexes they are useful in certain parts of topology e g knot theory in statistical physics graphs can represent local connections between interacting parts of a system as well as the dynamics of a physical process on such systems the study of these models calls then for the comprehension of the significant structural properties of the relevant graphs but are there nontrivial structural properties which are universally important expansion of a graph requires that it is simultaneously sparse and highly connected expander graphs were first defined by bassalygo and pinsker and their existence first proved by pinsker in the early 70s the property of being an expander seems significant in many of these mathematical computational and physical contexts it is not surprising that expanders are useful in the design and analysis of communication networks what is less obvious is that expanders have surprising utility in other computational settings such as in the theory of error correcting codes and the theory of pseudorandomness in mathematics we will encounter e g their role in the study of metric embeddings and in particular in work around the baum connes conjecture expansion is closely related to the convergence rates of markov chains and so they play a key role in the study of monte carlo algorithms in statistical mechanics and in a host of practical computational applications the list of such interesting and fruitful connections goes on and on with so many applications we will not even

eigenvalues and expansion of regular graphs the spectral method is the best currently known technique to prove lower bounds on expansion ramanujan graphs which have asymptotically optimal second eigenvalue are the best known explicit expanders the spectral method yielded a lower bound of k 4 on the expansion of linear sized subsets of k regular ramanujan graphs we improve the lower bound on the expansion of ramanujan graphs to approximately k 2 moreover we construct a family of k regular graphs with asymptotically optimal second eigenvalue and linear expansion equal to k 2 this shows that k 2 is the best bound one can obtain using the second eigenvalue method we also show an upper bound of roughly 1 k 1 on the average degree of linear sized induced subgraphs of ramanujan graphs this compares positively with the classical bound 2 k 1 as a byproduct we obtain improved results on random walks on expanders and construct selection networks respectively extrovert graphs of smaller size respectively degree than was previously known

are bitvectors optimal we study the it static membership problem given a set s of at most n keys drawn from a universe u of size m store it so that queries of the form is u in s can be answered by making few accesses to the memory we study schemes for this problem that use space close to the information theoretic lower bound of omega n log frac m n bits and yet answer queries by reading a small number of bits of the memory we show that for epsilon 0 there is a scheme that stores o frac n epsilon 2 log m bits and answers membership queries using a randomized algorithm that reads just one bit of memory and errs with probability at most epsilon we consider schemes that make no error for queries in s but are allowed to err with probability at most epsilon for queries not in s we show that there exist such schemes that store o frac n epsilon 2 log m bits and answer queries using just one bitprobe if multiple probes are allowed then the number of bits stored can be reduced to o n 1 delta log m for any delta 0 the schemes mentioned above are based on probabilistic constructions of set systems with small intersections we show lower bounds that come close to our upper bounds for a large range of n and epsilon schemes that answer queries with just one bitprobe and error probability epsilon must use omega frac n epsilon log 1 epsilon log m bits of storage if the error is restricted to queries not in s then the scheme must use omega frac n 2 epsilon 2 log n epsilon log m bits of storage we also consider deterministic schemes for the static membership problem and show tradeoffs between space and the number of probes

output feedback robust mpc of uncertain norm bounded linear systems with disturbance the output feedback robust model predictive control mpc for the linear systems with norm bounded uncertainty and disturbance is studied a new state estimator is formulated where both estimator d

probabilistic checking of proofs a new characterization of np we give a new characterization of np the class np contains exactly those languages l for which membership proofs a proof that an input x is in l can be verified probabilistically in polynomial time using logarithmic number of random bits and by reading sublogarithmic number of bits from the proof we discuss implications of this characterization specifically we show that approximating clique and independent set even in a very weak sense is np hard

entropy waves the zig zag graph product and new constant degree expanders the main contribution of this work is a new type of graph product which we call the zig zag product taking a product of a large graph with a small graph the resulting graph inherits roughly its size from the large one its degree from the small one and its expansion properties from both iteration yields simple explicit constructions of constant degree expanders of arbitrary size starting from one constant size expander crucial to our intuition and simple analysis of the properties of this graph product is the view of expanders as functions which act as entropy wave propagators they transform probability distributions in which entropy is concentrated in one area to distributions where that concentration is dissipated in these terms the graph product affords the constructive interference of two such waves

finite simple groups as expanders abstract we prove that there exist k and 0 e such that every non abelian finite simple group g which is not a suzuki group has a set of k generators for which the cayley graph cay g s is an e expander expander graphs ramanujan complexes

the spectra of infinite hypertrees a model of regular infinite hypertrees is developed to mimic for hypergraphs what infinite trees do for graphs two notions of spectra or first eigenvalue are then examined for the infinite tree obtaining a precise value for the first notion and obtaining some estimates for the second the results indicate agreement of the first eigenvalue of the infinite hypertree with the second eigenvalue of a random hypergraph of the same degree to within logarithmic factors at least for the first notion of first eigenvalue author s abstract

robust constrained model predictive control using linear matrix inequalities abstract the primary disadvantage of current design techniques for model predictive control mpc is their inability to deal explicitly with plant model uncertainty in this paper we present a new approach for robust mpc synthesis that allows explicit incorporation of the description of plant uncertainty in the problem formulation the uncertainty is expressed in both the time and frequency domains the goal is to design at each time step a state feedback control law that minimizes a worst case infinite horizon objective function subject to constraints on the control input and plant output using standard techniques the problem of minimizing an upper bound on the worst case objective function subject to input and output constraints is reduced to a convex optimization involving linear matrix inequalities lmis it is shown that the feasible receding horizon state feedback control design robustly stabilizes the set of uncertain plants several extensions such as application to systems with time delays problems involving constant set point tracking trajectory tracking and disturbance rejection which follow naturally from our formulation are discussed the controller design is illustrated with two examples

electric routing and concurrent flow cutting we investigate an oblivious routing scheme amenable to distributed computation and resilient to graph changes based on electrical flow our main technical contribution is a new rounding method which we use to obtain a bound on the 1 1 operator norm of the inverse graph laplacian we show how this norm reflects both latency and congestion of electric routing

tube based model predictive control for linear parameter varying systems with bounded rate of parameter variation abstract this paper introduces a tube based model predictive control mpc for linear parameter varying lpv systems which exploits knowledge about bounds on the parameters rate of change to extrapolate its admissible values over the prediction horizon this information is used to construct state tubes to which the future trajectories of the state are confined the tubes are consequently used for constraint tightening then an mpc optimization problem subject to tightened sets for the state and control constraints is solved for only a nominal system corresponding to a nominal trajectory of the scheduling parameter starting from its current value recursive feasibility and asymptotic stability are proven and two numerical examples are given to demonstrate the effectiveness of the proposed approach

the κ μ extreme distribution a number of realistic wireless applications can be found for which the propagated signal may experience extreme fading conditions in particular a recent work has reported on an experiment conducted within an enclosed environment where it has been shown that the fading conditions are indeed harsh drastically departing from those predicted by traditional fading models this paper investigates the extreme fading distribution a member of the fading model and obtained from it when its parameters reach their extreme values and 0 for a fixed amount of fading field data reported elsewhere and collected in enclosed environment along with other fading data collected by the authors team are used to compare the curve fit performance of the extreme model and the recently proposed two ray distribution the results show that the extreme distribution is flexible and suitable to model these scenarios the performance of this new model is then explored in the three basic combining techniques namely selection maximum ratio and equal gain and a number of exact closed form formulas are obtained interestingly it is shown that in extreme fading conditions the selection combiner may outperform the equal gain combiner

on the sum of kappa stochastic variates and applications to equal gain combining in this paper we study the statistics of the sum of not necessarily identically distributed kappa that is k random variables rv s assuming half integer values for the shaping parameters novel closed form expressions for the probability density function pdf of the sum of independent k rvs are obtained while for arbitrary values of the shaping parameters a corresponding pdf expression is derived in terms of fast converging infinite series furthermore an infinite series representation for the pdf of the sum of two arbitrarily correlated k rvs is derived the proposed analysis is employed to the performance analysis of equal gain combining egc receivers operating over composite fading shadowing channels modeled by the k distribution more specifically the outage and the average bit error probabilities as well as the average channel capacity of egc receivers operating over such composite environment are studied considering different channel fading shadowing conditions and correlation effects various numerical performance evaluation results are presented these results are complemented by equivalent computer simulated ones that validate the accuracy of the proposed analysis

difference equations isoperimetric inequality and transience of certain random walks the difference laplacian on a square lattice in rn has been stud ied by many authors in this paper an analogous difference operator is studied for an arbitrary graph it is shown that many properties of the laplacian in the continuous setting e g the maximum principle the harnack inequality and cheeger s bound for the lowest eigenvalue hold for this difference oper ator the difference laplacian governs the random walk on a graph just as the laplace operator governs the brownian motion as an application of the theory of the difference laplacian it is shown that the random walk on a class of graphs is transient the random walks we consider are defined as follows let k be a connected graph i e a one dimensional simplicial complex for a vertex x e k let m x denote the number of edges emanating from x the probability that a particle moves from x to another vertex y e k is l m x if x and y are connected by an edge and it is zero otherwise as observed by courant friedrichs and lewy cfl for the case of a square lattice in the plane this random walk is intimately related to the difference analog of the laplacian

stabilizing tube based model predictive control terminal set and cost construction for lpv systems abstract this paper presents a stabilizing tube based mpc synthesis for lpv systems we employ terminal constraint sets which are required to be controlled periodically contractive periodically or finite step contractive sets are easier to compute and can be of lower complexity than true contractive ones lowering the required computational effort both off line and on line under certain assumptions on the tube parameterization recursive feasibility of the scheme is proven subsequently asymptotic stability of the origin is guaranteed through the construction of a suitable terminal cost based on a novel lyapunov like metric for compact convex sets containing the origin a periodic variant on the well known homothetic tube parameterization that satisfies the necessary assumptions and yields a tractable lpv mpc algorithm is derived the resulting mpc algorithm requires the on line solution of a single linear program with linear complexity in the prediction horizon the properties of the approach are demonstrated by a numerical example

robust tube mpc for linear systems with multiplicative uncertainty recently a technical note 4 ensured recursive feasibility in robust model predictive control with multiplicative uncertainty by applying terminal constraints to the degrees of freedom available to the controller while constructing polytopic tube cross sections the current note extends this technique by bounding the tube parameters in norm and constructing a terminal set based on this bound this reduces the online computational load and so allows longer prediction horizons to be used

how to compute the volume in high dimension in some areas of theoretical computer science we feel that randomized algorithms are better and in some others we can prove that they are more efficient than the deterministic ones

accurate approximations to the sum of generalized random variables and applications in the performance analysis of diversity systems accurate closed form approximations to the sum of independent identically distributed eta mu and kappa mu random variables are provided the proposed approximations turn out to be simple precise and useful for obtaining important performance metrics of communications systems where sums of variates arise in particular average bit error rate and level crossing rate of multibranch equal gain combining receivers are attained to illustrate the applicability of the approximations some sample examples show that the intricate exact solution and the simple approximate expressions yield results that are almost indistinguishable from each other

robust synergetic control design under inputs and states constraints abstractin this paper a novel robust constrained control methodology for discrete time linear parameter varying dt lpv systems is proposed based on a synergetic control theory sct approach it is shown that in dt lpv systems without uncertainty and for any unmeasured bounded additive disturbance the proposed controller accomplishes the goal of stabilizing the system by asymptotically driving the error of the controlled variable to a bounded set containing the origin and then maintaining it there moreover given an uncertain dt lpv system jointly subject to unmeasured and constrained additive disturbances and constraints in states input commands and reference signals setpoints then invariant set theory is used to find an appropriate polyhedral robust invariant region in which the proposed control framework is guaranteed to robustly stabilize the closed loop system furthermore this is achieved even for the case of varying non zero control setpoints in such uncertain dt lpv systems the contr

homothetic tube model predictive control the robust model predictive control for constrained linear discrete time systems is solved through the development of a homothetic tube model predictive control synthesis method the method employs several novel features including a more general parameterization of the state and control tubes based on homothety and invariance a more flexible form of the terminal constraint set and a relaxation of the controlled dynamics of the sets that define the state and control tubes under natural assumptions the proposed method is computationally efficient and it induces strong system theoretic properties