uniformization of analytic spaces in this article we give an explicit elementary proof of a local version of resolution of singularities in characteristic zero local means that the centres of blowing up are chosen locally so that a finite number of finite sequences of local blowings up may be required to cover a neighbourhood of a given point hence we use the term uniformization rather than local desingularization although uniformization in our sense is considerably stronger than the original idea of zariski 16 in the last two decades several mathematicians notably abhyankar 1 hironaka and spivakovsky 12 and youssin 14 15 have proposed simpler or more explicit versions of the inductive procedure in hironaka s proof of his great theorem 8 these approaches would seem to lead to some form of uniformization though full details of none of them have yet appeared the idea of seeking an explicit local procedure to determine the centres of blowing up is that a sufficiently good local choice should globalize automatically our method for example gives global resolution of singularities of surfaces in any codimension throughout this paper k denotes either the field of real numbers r or the field of complex numbers c we will work in the category of analytic spaces over k although our techniques apply as well to algebraic spaces over a field

hidden dynamics in models of discontinuity and switching a method of solving discontinuous differential equations is analyzed the central result generalizes filippov s method to nonlinear sliding modes exponentially small terms are shown to create novel switching dynamics the results are shown to account naturally for stiction static friction models random effects are shown to eliminate nonlinear terms under limited conditions a b s t r a c t sharp switches in behaviour like impacts stick slip motion or electrical relays can be modelled by differential equations with discontinuities a discontinuity approximates fine details of a switching process that lie beyond a bulk empirical model the theory of piecewise smooth dynamics describes what happens assuming we can solve the system of equations across its discontinuity what this typically neglects is that effects which are vanishingly small outside the discontinuity can have an arbitrarily large effect at the discontinuity itself here we show that such behaviour can be incorporated within the standard theory through nonlinear terms and these introduce multiple sliding modes we show that the nonlinear terms persist in more precise models for example when the discontinuity is smoothed out the nonlinear sliding can be eliminated however if the model contains an irremovable level of unknown error which provides a criterion for systems to obey the standard filippov laws for sliding dynamics at a discontinuity

consensus of high order multi agent systems with large input and communication delays we study in this paper the consensus problem for multi agent systems with agents characterized by high order linear systems with time delays in both the communication network and inputs provided that the open loop dynamics of the agents is not exponentially unstable but may be polynomially unstable and the communication topology contains a directed spanning tree a truncated predictor feedback approach is established to solve the consensus problem it is shown that if the delays are constant and exactly known the consensus problems can be solved by both full state feedback and observer based output feedback protocols for arbitrarily large yet bounded delays if it is further assumed that the open loop dynamics of the agents only contains zero eigenvalues the delays are allowed to be time varying and unknown numerical examples are worked out to illustrate the effectiveness of the proposed approaches

distributed lms estimation over networks with quantised communications this article investigates the problem of distributed least mean square d lms estimation over a network with quantised communication each node in the network has a quantiser consisting of a first order dynamical encoder decoder and can only communicate with its neighbours then a new d lms estimator is proposed by employing a weighted sum of the internal state differences between each node s quantiser and those of its neighbours performance analysis of the proposed quantised d lms algorithm is studied in terms of mean square transient and steady state measurements we show that for gaussian data and sufficiently small step sizes the proposed cooperative d lms with quantisation is mean square stable in all quantisation levels including the 1 bit case and its performance approaches the cooperative d lms without quantisation when the quantisation step size is fairly small furthermore although in unquantised case infinite precision the cooperative d lms always outperforms the non cooperative d lms s

a lower bound on the minimum euclidean distance of trellis coded modulation schemes a lower bound on the minimum free euclidean distance of trellis coded modulation tcm is derived that guarantees the existence of good tcm codes of any complexity the bound is used to compare trellis codes combined with phase shift keying pulse amplitude modulation and quadratic amplitude shift keying modulation this random coding bound is the first lower bound on the free distance of trellis codes is tighter than any upper bound for large constraint lengths and predicts the asymptotic performance of tcm when the complexity of the code becomes large the bound can be used with any code rate and any modulation scheme and shows that the free distance increases linearly with the constraint length for large values of the constraint length

adaptive coordinated tracking of multi agent systems with quantized information abstract in this paper we study the adaptive coordinated tracking problem for continuous time first order integrator systems with quantized information under switching undirected and fixed directed communication graphs respectively the combined effect of quantized relative information error and quantized absolute information error on the tracking result is investigated both the logarithmic quantizers and uniform quantizers are considered it is shown that when logarithmic quantizers are used exact coordinated tracking can still be achieved by properly choosing the design parameters in the controller while when uniform quantizers are used practical coordinated tracking can be achieved with tracking error bounds proportional to the quantizer parameter simulation examples are provided to illustrate the results

a principalization algorithm for locally monomial ideal sheaves on 3 folds with an application to toroidalization abstract suppose that i is an ideal sheaf on a nonsingular variety x a principalization of i is a proper birational morphism x x such that x is nonsingular and i o x is locally principal we provide a fast and simple algorithm to construct a principalization of a locally monomial ideal sheaf on a 3 fold as an application we prove the existence of toroidalization of locally toroidal morphisms of 3 folds over an algebraically closed field of characteristic zero

equivariant poincare polynomials and counting points over finite fields abstract suppose a finite group acts as a group of automorphisms of a smooth complex algebraic variety which is defined over a number field we show how in certain circumstances an equivariant comparison theorem in l adic cohomology may be used to convert the computation of the graded character of the induced action on cohomology into questions about numbers of rational points of varieties over finite fields this is carried through in three applications first for the symmetric group acting on the moduli space of n points on a genus zero curve second for a unitary reflection group acting on the complement of its reflecting hyperplanes and third for the symmetric group action on the space of configurations of points in any smooth variety which satisfies certain strong purity conditions

regularization of discontinuous foliations blowing up and sliding conditions via fenichel theory abstract we study the regularization of an oriented 1 foliation f on m where m is a smooth manifold and m is a closed subset which can be interpreted as the discontinuity locus of f in the spirit of filippov s work we define a sliding and sewing dynamics on the discontinuity locus as some sort of limit of the dynamics of a nearby smooth 1 foliation and obtain conditions to identify whether a point belongs to the sliding or sewing regions

pausing for arbitrarily long times in dynamical systems it is well known that continuity in dynamical systems is not sufficient to guarantee uniqueness of solutions but less obvious is that non uniqueness can carry internal structure useful to characte

elliptic genera of singular varieties the notions of orbifold elliptic genus and elliptic genus of singular varieties are introduced and the relation between them is studied the elliptic genus of singular varieties is given in terms of a resolution of singularities and extends the elliptic genus of calabi yau hypersurfaces in fano gorenstein toric varieties introduced earlier the orbifold elliptic genus is given in terms of the fixed point sets of the action we show that the generating function for the orbifold elliptic genus ellorb x 6n p for symmetric groups 6n acting on n fold products coincides with the one proposed by r dijkgraaf g moore e verlinde and h verlinde the two notions of elliptic genera are conjectured to coincide

interval type 2 fuzzy model predictive control of nonlinear networked control systems in this paper the problem of fuzzy predictive control of nonlinear networked control systems subject to parameter uncertainties and defective communication links is studied stochastic variables with bernoulli random binary distribution are used to represent the defective communication links with packets loss occurring intermittently between the controller and the physical plant an interval type 2 it2 takagi sugeno t s fuzzy model is employed to describe the nonlinear plant subject to parameter uncertainties which can be captured with the lower and upper membership functions the it2 fuzzy model and it2 fuzzy controller are not required to share the same lower and upper membership functions in order to design the state feedback fuzzy model predictive controller an optimization problem which minimizes the upper bound of a quadratic objective function subject to input constraints and packets dropout is formulated and solved at every sampling instant in the finite time horizon by introducing some slack matrices less conservative conditions are developed for system stability analysis two examples are given to demonstrate the effectiveness and merits of the proposed new design techniques

regularization of hidden dynamics in piecewise smooth flows abstract this paper studies the equivalence between differentiable and non differentiable dynamics in r n filippov s theory of discontinuous differential equations allows us to find flow solutions of dynamical systems whose vector fields undergo switches at thresholds in phase space the canonical convex combination at the discontinuity is only the linear part of a nonlinear combination that more fully explores filippov s most general problem the differential inclusion here we show how recent work relating discontinuous systems to singular limits of continuous or regularized systems extends to nonlinear combinations we show that if sliding occurs in a discontinuous systems there exists a differentiable slow fast system with equivalent slow invariant dynamics we also show the corresponding result for the pinching method a converse to regularization which approximates a smooth system by a discontinuous one

distributed consensus over digital networks with limited bandwidth and time varying topologies in this paper we consider discrete time distributed average consensus with limited communication data rate and time varying communication topologies we design a distributed encoding decoding scheme based on quantization of scaled innovations and a control protocol based on a symmetric compensation method we develop an adaptive scheme to select the numbers of quantization levels according to whether the associated channel is active or not we prove that if the network is jointly connected then under the protocol designed average consensus can be asymptotically achieved and the convergence rate is quantified especially if the duration of any link failure in the network is bounded then the control gain and the scaling function can be selected properly such that 5 level quantizers suffice for asymptotic average consensus with an exponential convergence rate

channel coding with multilevel phase signals a coding technique is described which improves error performance of synchronous data links without sacrificing data rate or requiring more bandwidth this is achieved by channel coding with expanded sets of multilevel phase signals in a manner which increases free euclidean distance soft maximum likelihood ml decoding using the viterbi algorithm is assumed following a discussion of channel capacity simple hand designed trellis codes are presented for 8 phase shift keying psk and 16 quadrature amplitude shift keying qask modulation these simple codes achieve coding gains in the order of 3 4 db it is then shown that the codes can be interpreted as binary convolutional codes with a mapping of coded bits into channel signals which we call mapping by set partitioning based on a new distance measure between binary code sequences which efficiently lower bounds the euclidean distance between the corresponding channel signal sequences a search procedure for more powerful codes is developed codes with coding gains up to 6 db are obtained for a variety of multilevel phase modulation schemes simulation results are presented and an example of carrier phase tracking is discussed

the aastrom wittenmark self tuning regulator revisited and els based adaptive trackers although considerable progress has been made in stochastic adaptive control the problem concerning the convergence of the original self tuning regulator proposed by k j aastrom and b wittenmark 1973 is still open since it is attractive in theory and important in applications this problem is examined and a rigorous proof of its stability and optimality is given related problems such as convergence of the extended least squares els based adaptive tracker are also considered

quantum k theory of grassmannians we show that equivariant k theoretic 3 point gromov witten invariants of genus zero on a grassmann variety are equal to triple intersec tions computed in the equivariant k theory of a two step flag manifold thus generalizing an earlier result of buch kresch and tamvakis in the process we show that the gromov witten variety of curves passing through 3 general points is irreducible and rational our applications include pieri and giambelli formulas for the quantum k theory ring of a grassmannian which determine the multiplication in this ring we also compute the dual schubert basis for this ring and show that its structure constants satisfy s3 symmetry our for mula for gromov witten invariants can be partially generalized to cominuscule homogeneous spaces by using a construction of chaput manivel and perrin

coset codes i introduction and geometrical classification practically all known good constructive coding techniques for bandlimited channels including lattice codes and various trellis coded modulation schemes can be characterized as coset codes a coset code is defined by a lattice partition lambda lambda and by a binary encoder c that selects a sequence of cosets of the lattice lambda the fundamental coding gain of a coset code as well as other important parameters such as the error coefficient the decoding complexity and the constellation expansion factor are purely geometric parameters determined by c lambda lambda the known types of coset codes as well as a number of new classes that systematize and generalize known codes are classified and compared in terms of these parameters

distributed coordination of multi agent systems with quantized observer based encoding decoding integrative design of communication mechanism and coordinated control law is an interesting and important problem for multi agent networks in this paper we consider distributed coordination of discrete time second order multi agent systems with partially measurable state and a limited communication data rate a quantized observer based encoding decoding scheme is designed which integrates the state observation with encoding decoding a distributed coordinated control law is proposed for each agent which is given in terms of the states of its encoder and decoders it is shown that for a connected network 2 bit quantizers suffice for the exponential asymptotic synchronization of the states of the agents the selection of controller parameters and the performance limit are discussed it is shown that the algebraic connectivity and the spectral radius of the laplacian matrix of the communication graph play key roles in the closed loop performance the spectral radius of the laplacian matrix is related to the selection of control gains while the algebraic connectivity is related to the spectral radius of the closed loop state matrix furthermore it is shown that as the number of agents increases the asymptotic convergence rate can be approximated as a function of the number of agents the number of quantization levels communication data rate and the ratio of the algebraic connectivity to the spectral radius of the laplacian matrix of the communication graph

stability of recursive stochastic tracking algorithms first the paper gives a stability study for the random linear equation x n 1 i a n x n it is shown that for a quite general class of random matrices a n of interest the stability of such a vector equation can be guaranteed by that of a corresponding scalar linear equation for which various results are given without requiring stationary or mixing conditions then these results are applied to the main topic of the paper i e to the estimation of time varying parameters in linear stochastic systems giving a unified stability condition for various tracking algorithms including the standard kalman filter least mean squares and least squares with forgetting factor

generic bifurcation of sliding vector fields abstract the main aim of this paper is to classify the singularities which generically occur in one parameter families of sliding vector fields

equimultiplicity algebraic elimination and blowing up abstract given a variety x over a perfect field we study the partition defined on x by the multiplicity into equimultiple points and the effect of blowing up at smooth equimultiple centers over fields of characteristic zero we prove resolution of singularities by using the multiplicity as an invariant instead of the hilbert samuel function

a proof of kurdyka s conjecture on arc analytic functions we prove a conjecture of kurdyka stating that every arc symmetric semialgebraic set is precisely the zero locus of an arc analytic semialgebraic function this implies in particular that arc symmetric semialgebraic sets are in one to one correspondence with radical ideals of the ring of arc analytic semialgebraic functions

the ghosts of departed quantities in switches and transitions transitions between steady dynamical regimes in diverse applications are often modeled using discontinuities but doing so introduces problems of uniqueness no matter how quickly a transition occurs its inner workings can affect the dynamics of the system significantly here we discuss the way transitions can be reduced to discontinuities without trivializing them by preserving so called hidden terms we review the fundamental methodology its motivations and where their study seems to be heading we derive a prototype for piecewise smooth models from the asymptotics of systems with rapid transitions sharpening filippov s convex combinations by encoding the tails of asymptotic series into nonlinear dependence on a switching parameter we present a few examples that illustrate the impact of these on our standard picture of smooth or only piecewise smooth dynamics

performance analysis of averaging based distributed estimation algorithm with additive quantization model abstract in this paper we consider the distributed sensor fusion problem over sensor networks under directed communication links and bandwidth constraint we investigate the impact of the additive quantization model on the proposed two stage averaging based algorithm existing works on the effect of the additive model show that convergence can be guaranteed only if the quantization error variances form a convergent series we show that the proposed algorithm achieves the performance of the optimal centralized estimate even if the quantization error variances are not vanishing this is guaranteed by establishing a law of the iterated logarithm for weighted sums of independent random vectors moreover an explicit bound of the convergence rate of the proposed algorithm is given to quantify its almost sure performance

distributed consensus algorithms in sensor networks with imperfect communication link failures and channel noise the paper studies average consensus with random topologies intermittent links and noisy channels consensus with noise in the network links leads to the bias variance dilemma running consensus for long reduces the bias of the final average estimate but increases its variance we present two different compromises to this tradeoff the a nd algorithm modifies conventional consensus by forcing the weights to satisfy a persistence condition slowly decaying to zero and the a nc algorithm where the weights are constant but consensus is run for a fixed number of iterations iota then it is restarted and rerun for a total of p runs and at the end averages the final states of the p runs monte carlo averaging we use controlled markov processes and stochastic approximation arguments to prove almost sure convergence of a nd to a finite consensus limit and compute explicitly the mean square error mse variance of the consensus limit we show that a nd represents the best of both worlds zero bias and low variance at the cost of a slow convergence rate rescaling the weights balances the variance versus the rate of bias reduction convergence rate in contrast a nc because of its constant weights converges fast but presents a different bias variance tradeoff for the same number of iterations iota p shorter runs smaller iota lead to high bias but smaller variance larger number p of runs to average over for a static nonrandom network with gaussian noise we compute the optimal gain for a nc to reach in the shortest number of iterations iota p with high probability 1 delta epsiv delta consensus epsiv residual bias our results hold under fairly general assumptions on the random link failures and communication noise

multistage decoding of multilevel block m psk modulation codes and its performance analysis multistage decoding of multilevel block multilevel phase shift keying m psk modulation codes for the additive white gaussian noise awgn channel is investigated several types of multistage decoding including a suboptimum soft decision decoding scheme are devised and analyzed upper bounds on the probability of an incorrect decoding of a code are derived for the proposed multistage decoding schemes error probabilities of some specific multilevel block 8 psk modulation codes are evaluated and simulated the computation and simulation results for these codes show that with multistage decoding significant coding gains can be achieved with large reduction in decoding complexity in one example it is shown that the difference in performance between the proposed suboptimum multistage soft decision decoding and the single stage optimum decoding is small only a fraction of a db loss in snr at the block error probability of 10 sup 6

functorial desingularization of quasi excellent schemes in characteristic zero the nonembedded case we prove that any reduced noetherian quasi excellent scheme of characteristic zero admits a strong desingularization which is functorial with respect to all regular morphisms as a main application we deduce that any reduced formal variety of characteristic zero admits a strong functorial desin gularization also we show that as an easy formal consequence of our main result one obtains strong functorial desingularization for many other spaces of characteristic zero including quasi excellent stacks and formal schemes and complex or non archimedean analytic spaces moreover these functors easily generalize to non compact setting by use of generalized convergent blow up sequences with regular centers

canonical desingularization in characteristic zero by blowing up the maximum strata of a local invariant this article contains an elementary constructive proof of resolution of singularities in characteristic zero our proof applies in particular to schemes of finite type and to analytic spaces so we recover the great theorems of hironaka we introduce a discrete local invariant rm inv x a whose maximum locus determines a smooth centre of blowing up leading to desingularization to define rm inv x we need only to work with a category of local ringed spaces x x cal o x satisfying certain natural conditions if a in x then rm inv x a depends only on widehat cal o x a more generally rm inv x is defined inductively after any sequence of blowings up whose centres have only normal crossings with respect to the exceptional divisors and lie in the constant loci of rm inv x cdot the paper is self contained and includes detailed examples one of our goals is that the reader understand the desingularization theorem rather than simply know it is true

a new description of trellis codes a trellis code is a sliding window method of encoding a binary data stream as a sequence of real or complex numbers that are input to a noisy transmission channel ungerboeck has constructed simple trellis codes that provide the same noise immunity as is given by increasing the power of uncoded transmission by factors ranging from two to four his method is to specify an underlying convolutional code and a rule mapping by set partitioning that maps the output of this code onto a fixed signal constallation a new description of a trellis code is given that combines these two steps into one the new description is analytic rather than graphical many practical codes can be described very simply and strict bounds on performance can be obtained a method for differential encoding trellis codes is presented that was suggested by the authors representation

trellis coded modulation with multidimensional constellations trellis coded modulation schemes using four eight or 16 dimensional constellations have a number of potential advantages over the usual two dimensional schemes a smaller constituent two dimensional constellation easier tolerance to phase ambiguities and a better trade off between complexity and coding gain a number of such schemes are presented and evaluated starting with a variety of multidimensional lattices we show how to select multidimensional constellations how to partition them into subsets how to construct trellis codes using those subsets and how to map hits to constellation points simplifications of the viterbi decoding algorithm are presented we conclude that there are multidimensional trellis coded modulation schemes that perform better for the same complexity than do two dimensional schemes

an efficient algorithm to compute the euclidean distance spectrum of a general intersymbol interference channel and its applications we present an efficient algorithm to compute the distance spectrum of a general finite intersymbol interference isi channel whose complexity is lower than those of existing methods closed form expressions are derived for both input output euclidean distance enumerators and asymptotic distance spectrum shapes for 2 tap and 3 tap isi channels coded and or precoded isi channels are also discussed

a generalization of manifolds with corners abstract in conventional differential geometry one studies manifolds locally modelled on r n manifolds with boundary locally modelled on 0 r n 1 and manifolds with corners locally modelled on 0 k r n k they form categories man m a n b m a n c manifolds with corners x have boundaries x also manifolds with corners with dim x dim x 1 we introduce a new notion of manifolds with generalized corners or manifolds with g corners extending manifolds with corners which form a category m a n gc with man m a n b m a n c m a n gc manifolds with g corners are locally modelled on x p hom mon p 0 for p a weakly toric monoid where x p 0 k r n k for p n k z n k most differential geometry of manifolds with corners extends nicely to manifolds with g corners including well behaved boundaries x in some ways manifolds with g corners have better properties than manifolds with corners in particular transverse fibre products in m a n gc exist under much weaker conditions than in m a n c this paper was motivated by future applications in symplectic geometry in which some moduli spaces of j holomorphic curves can be manifolds or kuranishi spaces with g corners rather than ordinary corners our manifolds with g corners are related to the interior binomial varieties of kottke and melrose 20 and the positive log differentiable spaces of gillam and molcho 6

continuous time and sampled data based average consensus with logarithmic quantizers this paper considers the average consensus problem for multi agent systems with continuous time first order dynamics logarithmic quantization is considered in the communication channels and continuous time and sampled data based protocols are proposed for the continuous time protocol we give an explicit upper bound of the consensus error in terms of the initial states the quantization density and the parameters of the network graph it is shown that in contrast with the case with uniform quantization the consensus error in the logarithmic quantization case is always uniformly bounded independent of the quantization density and the b asymptotic average consensus is ensured under the proposed protocol i e the asymptotic consensus error converges to zero as the sector bound b of the logarithmic quantizer approaches zero for the sampled data based protocol we give sufficient conditions on the sampling interval to ensure the b asymptotic average consensus numerical examples are given to demonstrate the effectiveness of the protocols

a game for the resolution of singularities we propose a combinatorial game on finite graphs called stratify t hat is played by two protagonists a and b the game captures the logical structure of a proof of the resolution of singularities in each round the graph of the game is modified by the moves of the players when it assumes a final configuration a has won otherwise the game goes on forever and nobody wins in particular b cannot win the game but can only prevent a from winning we show that a always possesses a winning strategy regardless of the initial shape of the graph and of the moves of b this implies translating the game back to algebraic geometry that there is a choice of centres for the blowup of singular varieties in characteristic zero which eventually leads to their resolution the algebra needed for this implication is elementary the transcription from varieties to graphs and from blowups to modifications of the graph thus axiomatizes the proof of the resolution of singularities in principle the same logic could also work in positive characteristic once an appropriate descent in dimension is settled

sphere packing upper bounds on the free distance of trellis codes sphere packing arguments are used to develop upper bounds on the free distance of trellis codes a general bounding procedure is presented sphere packing bounds including bounds on the density of infinite regions packings for hypercubes and packings on the surface of a unit sphere are then used to produce bounds for a wide variety of trellis codes among the applications are convolutional codes ungerboeck codes for phase shift keying and quadrature amplitude modulation coset codes and continuous phase modulation codes the new bounds are significantly tighter than existing bounds in many cases of practical interest

finite level quantized synchronization of discrete time linear multiagent systems with switching topologies in this paper the synchronization of discrete time linear multiagent systems is studied with finite communication data rate and switching topology flows a class of quantized observer based communication schemes and a class of certainty equivalence principle based cooperative control laws are proposed with adaptive encoders and decoders it is shown that if the pairs of agents state matrices and control matrices multiplied by laplacian eigenvalues of the weakly connected components are simultaneously stabilizable and the communication topology flow is frequently connected then there exist such protocols leading to synchronization exponentially fast furthermore only finite bits of information exchange per step are required to guarantee the synchronization if the communication channels are frequently active for first order dynamics the dwell time and the number of bits are both related to the unstable mode of agent dynamics the number of agents the frequency of connectivity and the laplacian eige

new trellis codes based on lattices and cosets a new technique is proposed for constructing trellis codes which provides an alternative to ungerboeck s method of set partitioning the new codes use a signal constellation consisting of points from an n dimensional lattice lambda with an equal number of points from each coset of a sublattice lambda one part of the input stream drives a generalized convolutional code whose outputs are cosets of lambda while the other part selects points from these cosets several of the new codes are better than those previously known

asymptotic upper bounds on the minimum distance of trellis codes a trellis code is a sliding window method of encoding a binary data stream as a sequence of signal points in rn when a trellis code is used to encode data at the rate of k bits channel symbol each channel input depends not only on the most recent block of k bits to enter the encoder but will also depend on a set of bits preceding this block the bits determine the state of the encoder and the most recent block of k bits generates the channel symbol conditional on the encoder state the performance of a trellis code depends on a suitably defined minimum distance property of that code this paper obtains upper bounds on this minimum distance that are better than any previously known

quantized consensus over directed networks with switching topologies abstract this paper studies the quantized consensus problem for a group of agents over directed networks with switching topologies we propose an effective distributed protocol with an adaptive finite level uniform quantized strategy under which consensus among agents is guaranteed with weaker communication conditions in particular we analytically prove that each agent sending 5 level quantized information to each of its neighbors together with 3 level quantized information to itself at each time step which suffices for attaining consensus with an exponential convergence rate as long as the duration of all link failures in the directed network is bounded by dropping the typical common left eigenvector requirement for the existence of common quadratic lyapunov function we conduct the convergence analysis based on the notion of input to output stability the proposed quantized protocol has favorable merits of requiring little communication overhead and increasing robustness to link unreliability and it fits well into the digital network framework

virtual poincare polynomial of the link of a real algebraic variety the link of an affine real algebraic variety at a point is defined to be the intersection of the variety with a small sphere centred at the point the euler characteristic of the link leads to local topological characterisation of real algebraic varieties in law dimensions we prove in the paper that not only the euler characteristic but also the stronger virtual poincare polynomial is well defined for the link at a point of an affine algebraic variety

a singular approach to discontinuous vector fields on the plane in this paper we deal with discontinuous vector fields on r 2 and we prove that the analysis of their local behavior around a typical singularity can be treated via singular perturbation the regularization process developed by sotomayor and teixeira is crucial for the development of this work

a canonical bundle formula a higher dimensional analogue of kodaira s canonical bundle formula is obtained as applications we prove that the log canonical ring of a klt pair with 3is finitely generated and that there exists an effectively computable natural number m such that mk x induces the iitaka fibering for every algebraic threefold x with kodaira dimension 1

on continuous approximation of discontinuous systems abstract by using tichonov theorem for singularly perturbed differential equations we study the relationship between dynamics of discontinuous differential equations and their continuous approximations along periodic solutions

unit memory codes for continuous phase modulation in an awgn channel the combination of unit memory codes and continuous phase modulation in an additive white gaussian noise channel is considered our motivation is based on the fact that when rouanne and costello s 1988 free distance lower bound is applied to unit memory codes an improved lower bound is obtained specific constructions are considered for 4 level modulation schemes with full response and weak partial response heuristic design rules are used for defining the subsets of codes within which an exhaustive search is conducted the coding gains of the best codes within these subsets are better than those of best previously known multi memory codes

trellis coded multidimensional phase modulation a 2l dimensional multiple phase shift keyed l mpsk signal set is obtained by forming the cartesian product of l two dimensional mpsk signal sets a systematic approach to partitioning l mpsk signal sets that is based on block coding is used an encoder system approach is developed it incorporates the design of a differential precoder a systematic convolutional encoder and a signal set mapper trellis coded l 4psk l 8psk and l 16psk modulation schemes are found for 1

the weierstrass approximation theorem for maps between real algebraic varieties p n r for some n endowed with the zariski topology and the sheaf of rvalued regular functions morphisms between real algebraic varieties will be called regular maps every real algebraic variety carries also the euclidean topology that is the topology induced by the usual metric topology on r unless explicitly stated otherwise all topological notions related to real algebraic varieties will refer to the euclidean topology given two real algebraic varieties x and y we regard the set r x y of all regular maps from x into y as a subset of the space c x y of all continuous maps fromx into y endowed with the compact open topology if x and y are nonsingular then we also regard r x y as a subset of the space c x y of all c maps from x into y endowed with the c compact open topology the weak c topology in the terminology used in 6