joint source channel coding with side information using hybrid digital analog codes we study the joint source channel coding problem of transmitting a gaussian source over a gaussian channel in two cases i the presence of interference known only to the transmitter and ii in the presence of side information about the source known only to the receiver we introduce hybrid digital analog forms of the costa and wyner ziv coding schemes we present the random coding counterpart of schemes based on lattices proposed by kochman and zamir then we discuss applications of the hybrid digital analog schemes in the case of channel signal to noise ratio mismatch and for lossy multicasting of a common source with bandwidth compression

from time series to linear system part i finite dimensional linear time invariant systems abstract in the first part of this paper the definition of a dynamical system as simply consisting of a family of time series will be developed in this context the notions of linearity time invariance and finite dimensionality will be introduced it will be shown that a given family of time series may be represented by a system of ar equations riw t l rl 1w t l 1 r0w t 0 or equivalently by a finite dimensional linear time invariant system x t 1 ax t bu t y t cx t du t w u y if and only if this family is linear shift invariant and complete or as is equivalent closed in the topology of pointwise convergence this yields a very high level and elegant set of axioms which characterize these familiar objects it is emphasized however that no a priori choice is made as to which components of w are inputs and which are outputs such a separation always exists in any specific linear time invariant model starting from these definitions the structural indices of such systems are introduced and it is shown how an ar representation of a system having a given behaviour can be constructed these results will be used in a modelling context in part ii of the paper

realization by inspection we investigate which first order representations can be obtained from high order representations of linear systems by inspection that is just by rearrangement of the data under quite weak conditions it is possible to obtain minimal realizations in the so called pencil form under stronger conditions one can obtain minimal realizations in standard state space form by inspection the development is based on a reformulation of the realization problem as a problem of finding a complete set of basis vectors for the nullspace of a given constant matrix since no numerical computation is needed the realization method in particular is suitable for situations in which some of the coefficients are symbolic rather than numerical

a behavioral characterization of the positive real and bounded real characteristic values in balancing in passivity preserving and bounded realness preserving model reduction by balanced truncation an important role is played by the so called positive real pr and bounded real br characteristic values both for the positive real as well as the bounded real case these values are defined in terms of the extremal solutions of the algebraic riccati associated with the system more precisely as the square roots of the eigenvalues of the product matrix obtained by multiplying the smallest solution with the inverse of the largest solution of the riccati equation in this paper we will establish a representation free characterization of these values in terms of the behavior of the system we will consider positive realness and bounded realness as special cases of half line dissipativity of the behavior we will then show that both for the pr and the br case the characteristic values coincide with the singular values of the linear operator that assigns to each past trajectory in the input output behavior its unique maximal supply extracting future continuation we will explain that the term singular values should be interpreted here in a generalized sense since in our setup the future behavior is only an indefinite inner product space c 2010 elsevier b v all rights reserved

a proof of marton s coding theorem for the discrete memoryless broadcast channel corresp a simple proof using random partitions and typicality is given for marton s coding theorem for broadcast channels

minimality of descriptor representations under external equivalence abstract necessary and sufficient conditions are derived for the minimality of descriptor representations under external equivalence these conditions are stated completely in terms of the matrices e a b c and d use is made of the close connection between the descriptor representation and the so called pencil representation this connection is further exploited to derive the transformation group for minimal descriptor representations it is shown that the transformations coincide with the operations of strong equivalence as introduced by verghese et al ieee trans aut control ac 26 811 831 1981

system structure and singular control the general continuous time linear quadratic control problem is considered it is shown that recently developed linear system theoretic properties and algorithms play an important role in solving this singular control problem

a behavioral approach to linear exact modeling the behavioral approach to system theory provides a parameter free framework for the study of the general problem of linear exact modeling and recursive modeling the authors present the solution of the continuous time polynomial exponential time series modeling problem both recursive and nonrecursive solutions are provided and classified according to properties like complexity and controllability it is shown in particular that recursive modeling corresponds to updating by means of a cascade inter connection of systems as a special case the solution of several other problems such as rational interpolation realization and modeling of arbitrary discrete time time series is obtained

input output structure of linear differential algebraic systems the problem of describing the input output structure of general linear differential algebraic systems is addressed explicit formulas are given in terms of the original parameters for systems with an arbitrary amount of redundancy these formulas allow one to establish whether the system determines an input output relation at all and if so they describe the rank of the transfer matrix and its pole zero structure at infinity the formulas may be seen as generalizations of a number of classical results on the input output structure of standard state space systems and descriptor systems satisfying certain constraints for the derivation the pencil representation rather than the descriptor representation is used

input output and state space representations of finite dimensional linear time invariant systems in this paper we develop various representations for systems described by aset of high order differential equations of the form r0w r1w rsw s 0 with r0 r1 rs not necessarily square matrices the variables w are the external variables particular attention is paid to the problem of obtaining minimal state space realizations and input output or input state output representations of such systems

practical dirty paper coding with sum codes in this paper we present a practical method to construct dirty paper coding dpc schemes using sum codes unlike the commonly used approach to dpc where the coding scheme involves concatenation of a channel code and a quantization code the proposed method embodies a unified approach that emulates the binning method used in the proof of the dpc result auxiliary bits are used to create the desired number of code vectors in each bin sum codes are obtained when information sequences augmented with auxiliary bits are encoded using linear block codes sum code based dpc schemes can be implemented using any linear block code and entail a relatively small increase in decoder complexity when compared to standard communication schemes they can also lead to significant reduction in transmit power in comparison to standard schemes

transformations of linear systems under external equivalence abstract we consider systems of linear differential and algebraic equations in which some of the variables are distinguished as external variables two systems are called equivalent if the set of solutions for the external variables is the same for both systems we give an operational form for this definition of equivalence i e we describe a set of system transformations having the property that two systems are equivalent if and only if they can be taken into each other by transformations from that set next an algorithm is described to transform a given system in general form to a system in minimal state space form this algorithm differs from existing methods in that it first takes the equations to first order form so that each subsequent step can be formulated and interpreted in state space terms we also compute the structure indices in terms of a state space description in nonminimal form and use this to prove the minimality of the end result of the algorithm finally an application is shown to the problem of ill posedness of feedback connections

nested linear lattice codes for structured multiterminal binning network information theory promises high gains over simple point to point communication techniques at the cost of higher complexity however lack of structured coding schemes limited the practical application of these concepts so far one of the basic elements of a network code is the binning scheme wyner 1974 1978 and other researchers proposed various forms of coset codes for efficient binning yet these schemes were applicable only for lossless source or noiseless channel network coding to extend the algebraic binning approach to lossy source or noisy channel network coding previous work proposed the idea of nested codes or more specifically nested parity check codes for the binary case and nested lattices in the continuous case these ideas connect network information theory with the rich areas of linear codes and lattice codes and have strong potential for practical applications we review these developments and explore their tight relation to concepts such as combined shaping and precoding coding for memories with defects and digital watermarking we also propose a few novel applications adhering to a unified approach

bounded real and positive real balanced truncation using σ normalised coprime factors in this article we will extend the method of balanced truncation using normalised right coprime factors of the system transfer matrix by meyer 1990 3 to balanced truncation with preservation of half line dissipativity special cases are preservation of positive realness and bounded realness we consider a half line dissipative input output system with quadratic supply rate given by a nonsingular symmetric matrix sigma with the property that its positive signature is equal to the number of input components of the system the transfer matrix of such system allows a sigma normalised right coprime factorisation we associate with such factorisation two lyapunov equations one of which is nonstandard involving the indefinite matrix sigma balancing will be based on making the unique solutions of these two lyapunov equations equal and diagonal the diagonal elements will be called the hankel sigma singular values because their squares are the nonzero eigenvalues of the composition of the graph hankel operator multiplication by sigma and the adjoint graph hankel operator this method of balanced truncation will be shown to preserve stability minimality and half line dissipativeness we will characterize the classical positive real and bounded real characteristic values in terms of the new hankel sigma singular values finally we will derive one step error bounds for the special case of balanced truncation of bounded real systems c 2009 elsevier b v all rights reserved

a test for behavioral equivalence external equivalence of behavioral representations is investigated within an impulsive smooth framework as an attendant problem initial value consistency is also considered explicit necessary and sufficient conditions are derived applying the external equivalence result to conventional system representations reveals some interesting interpretations

near capacity dirty paper code design a source channel coding approach this paper examines near capacity dirty paper code designs based on source channel coding we first point out that the performance loss in signal to noise ratio snr in our code designs can be broken into the sum of the packing loss from channel coding and a modulo loss which is a function of the granular loss from source coding and the target dirty paper coding rate or snr we then examine practical designs by combining trellis coded quantization tcq with both systematic and nonsystematic irregular repeat accumulate ira codes like previous approaches we exploit the extrinsic information transfer exit chart technique for capacity approaching ira code design but unlike previous approaches we emphasize the role of strong source coding to achieve as much granular gain as possible using tcq instead of systematic doping we employ two relatively shifted tcq codebooks where the shift is optimized via tuning the exit charts to facilitate the ira code design our designs synergistically combine tcq with ira codes so that they work together as well as they do individually by bringing together tcq the best quantizer from the source coding community and exit chart based ira code designs the best from the channel coding community we are able to approach the theoretical limit of dirty paper coding for example at 0 25 bit per symbol b s our best code design with 2048 state tcq performs only 0 630 db away from the shannon capacity

achievable rates of multi antenna downlink channels with peak power constraints this paper considers a gaussian multiple input single output miso broadcast channel with a per antenna peak power constraint or simply peak power constraint it is more realistic to consider the peak power constraint on each transmit antenna because in many practical implementations each antenna is equipped with its own power amplifier assuming the perfect knowledge of the channel state information csi at the transmitter we propose an achievable scheme using dirty tape coding dtc the ideal dirty paper coding dpc which is a capacity achieving scheme for the gaussian multiple input multiple output mimo broadcast channel cannot be used for our model because its optimal input distribution is gaussian and thus the peak power constraint is violated on the other hand the channel input of dtc is uniformly distributed in a fixed range which helps to control the peak power of the transmit signal easily we also present an algorithm that finds capacity achieving beamforming vectors and power allocation factors under a per antenna average power constraint and use the optimized parameters in the proposed scheme simulation results show that the proposed scheme provides gains of 2 7db over a non dtc scheme based on minimum mean square error mmse beamforming at high signal to noise ratio snr when there are three receivers and the transmitter has three antennas

reed solomon list decoding from a system theoretic perspective in this paper the sudan guruswami approach to list decoding of reed solomon rs codes is cast in a system theoretic framework with the data a set of trajectories or time series is associated which is then modeled as a so called behavior in this way a connection is made with the behavioral approach to system theory it is shown how a polynomial representation of the modeling behavior gives rise to the bivariate interpolating polynomials of the sudan guruswami approach the concept of weighted row reduced is introduced and used to achieve minimality two decoding methods are derived and a parametrization of all bivariate interpolating polynomials is given

technical communique duality in the behavioral systems theory in this note willems behavioral framework is slightly generalized in order to define the dual of a linear system and make the concepts of controllability and observability dual to each other

representing a nonlinear state space system as a set of higher order differential equations in the inputs and outputs an algorithm is presented for transforming a nonlinear state space system into a threefold set of equations the first subset describing the dynamics of the unobservable part of the system the second subset representing the remaining state variables as functions of inputs and outputs and their derivatives and the last subset defining the external behaviour of the system

quantization based methods additive attacks performance analysis the main goal of this study consists in the development of the worst case additive attack wcaa for ary quantization based data hiding methods using as design criteria the error probability and the maximum achievable rate of reliable communications our analysis focuses on the practical scheme known as distortion compensated dither modulation dc dm from the mathematical point of view the problem of the worst case attack wca design using probability of error as a cost function is formulated as the maximization of the average probability of error subject to the introduced distortion for a given decoding rule when mutual information is selected as a cost function a solution of the minimization problem should provide such an attacking noise probability density function pdf that will maximally decrease the rate of reliable communications for an arbitrary decoder structure the obtained results demonstrate that within the class of additive attacks the developed attack leads to a stronger performance decrease for the considered class of embedding techniques than the additive white gaussian or uniform noise attacks

a graph theoretic method for computing a formal representation of the largest controlled invariant subspace of a structured system the paper discusses a method to compute a kernel representation of the largest controlled invariant subspace in the kernel of the output mapping of a structured system the graph of the structured system is used to make the relevant equations and then a formula manipulation package can be used to compute a condensed kernel representation or even an image representation of the largest controlled invariant subspace

superposition coding for side information channels we present simple practical codes designed for the binary and gaussian dirty paper channels we show that the dirty paper decoding problem can be transformed into an equivalent multiple access decoding problem for which we apply superposition coding our concept is a generalization of the nested lattices approach of zamir shamai and erez in a theoretical setting our constructions are capable of achieving capacity using random component codes and maximum likelihood decoding we also present practical implementations of the constructions and simulation results for both dirty paper channels our results for the gaussian dirty paper channel are on par with the best known results for nested lattices we discuss the binary dirty tape channel for which we present a simple effective coding technique finally we propose a framework for extending our approach to general gel fand pinsker channels

capacity and lattice strategies for canceling known interference we consider the generalized dirty paper channel y x s n e x sup 2 spl les p sub x where n is not necessarily gaussian and the interference s is known causally or noncausally to the transmitter we derive worst case capacity formulas and strategies for strong or arbitrarily varying interference in the causal side information si case we develop a capacity formula based on minimum noise entropy strategies we then show that strategies associated with entropy constrained quantizers provide lower and upper bounds on the capacity at high signal to noise ratio snr conditions i e if n is weak relative to the power constraint p sub x these bounds coincide the optimum strategies take the form of scalar lattice quantizers and the capacity loss due to not having s at the receiver is shown to be exactly the shaping gain 1 2log 2 spl pi e 12 spl ap 0 254 bit we extend the schemes to obtain achievable rates at any snr and to noncausal si by incorporating minimum mean squared error mmse scaling and by using k dimensional lattices for gaussian n the capacity loss of this scheme is upper bounded by 1 2log2 spl pi eg spl lambda where g spl lambda is the normalized second moment of the lattice with a proper choice of lattice the loss goes to zero as the dimension k goes to infinity in agreement with the results of costa these results provide an information theoretic framework for the study of common communication problems such as precoding for intersymbol interference isi channels and broadcast channels

self adjoint elliptic operators and manifold decompositions part iii determinant line bundles and lagrangian intersection the theory of spectral flows developed in the series 10 11 12 and the present paper has a wide range of applications to important geometric operators on compact manifolds to present our results on spectral flow and manifold decomposition the present paper develops a theory of determinant line bundles and infinitedimensional lagrangians associated to self adjoint elliptic operators on compact manifolds the trace class properties of these infinite lagrangians established here and the precise uniform estimates relating them to finite lagrangians are crucial for such a determinant line bundle approach to analytical questions as an application we elucidate the walker s and other generalizations of casson s su 2 representation theoretic invariant of 3 manifolds in terms of the invariant of certain dirac operators this is carried out by introducing the technique of canonical perturbations of singular lagrangian subvarieties in symplectic geometry at the end of part ii of this series we obtained a formulation of the spectral flow of a family of self adjoint elliptic operators d u l2 e l2 e in terms

hybrid mu miso scheduling with limited feedback using hierarchical codebooks the multiuser mimo mu mimo systems tend to have inter user interference in practical systems with limited feedback practical mu mimo systems with limited feedback can be divided into two classes in general one is zero forcing beamforming zfbf and the other is per user unitary rate control pu2rc which is an extended version of random beamforming rbf to improve performance zfbf needs more feedback bits and pu2rc needs a larger number of users we propose a new hybrid mu miso system which has the advantages of the two mu mimo schemes simultaneously and analyze the sum rate performance and the quantization error of the hybrid scheme the analysis shows how the number of feedback bits the number of users and the signal power affect the sum rate the throughput scaling laws are also derived in the high and the medium snr regimes simulation results show that the proposed hybrid mu miso system performs better than either of the two mu mimo systems with the same number of feedback bits another advantage is that it has lower complexity for codeword search since it has a hierarchical codebook structure

a complement about the structure of linear systems the aim of this paper is to establish new relationships between the steps of the well known basic algorithms i s a and c i s a it is suggested that this new structural information will provide a better understanding of the structural modifications and more particularly the increase of the infinite zero structure due to the use of non invertible input transformations

generalized linear systems with lumped or distributed parameters and differential vector spaces abstract by employing differential vector spaces we obtain results for linear systems with constant or varying coefficients and with lumped or distributed parameters which were only known for constant linear systems described by rational transfer matrices questions such as the characterization of input and output input output equivalence invertibility feedback decoupling and realization are investigated

on behavior homomorphisms and system equivalence in this paper we introduce the concept of a behavior homomorphisms and isomorphisms and use it to present a uni ed approach to the study of equivalence of di erent behavior representations concentrating on a special representation we call normalized arma narma representation c 2001 elsevier science b v all rights reserved

an instanton invariant for 3 manifolds to an oriented closed 3 dimensional manifoldm withh1 m 0 we assign a 8 graded homology groupi m whose euler characteristic is twice casson s invariant the definition uses a construction on the space of instantons onm

converting high order linear pdes to first order abstract a straightforward way requiring no calculations is offered to convert a system of high order constant coefficient linear pdes into a system of first order pdes

equivalence of rational representations of behaviors this article deals with the equivalence of representations of behaviors of linear differential systems in general the behavior of a given linear differential system has many different representations in this paper we restrict ourselves to kernel and image representations two kernel representations are called equivalent if they represent one and the same behavior for kernel representations defined by polynomial matrices necessary and sufficient conditions for equivalence are well known in this paper we deal with the equivalence of rational representations i e kernel and image representations that are defined in terms of rational matrices as the first main result of this paper we will derive a new condition for the equivalence of rational kernel representations of possibly noncontrollable behaviors secondly we will derive conditions for the equivalence of rational representations of a given behavior in terms of the polynomial modules generated by the rows of the rational matrices we will also establish conditions for the equivalence of rational image representations finally we will derive conditions under which a given rational kernel representation is equivalent to a given rational image representation

driving variable realizations and the non negative realization problem for controllable behaviours non negative linear systems traditionally investigated within the state space framework have been recently analysed within the behavioural setting in a couple of recent papers j w nieuwenhuis when to call a linear system nonnegative linear algebra its appl 281 1998 pp 43 58 m e valcher nonnegative linear systems in the behavioural approach the autonomous case linear algebra and its appl 2000 319 pp 147 162 several definitions and results about non negative behaviours and in particular about non negative autonomous behaviours have been derived moreover the non negative realization problem for autonomous behaviours has been fully explored in m e valcher non negative realization of autonomous systems in the behavioral approach siam j on control and opt 2001 40 pp 540 556 thus deriving an extended set of necessary and sufficient conditions for an autonomous behaviour to admit a non negative realization here we focus our attention on the non negative re

design and analysis of nonbinary ldpc codes for arbitrary discrete memoryless channels we present an analysis under the iterative decoding of coset low density parity check ldpc codes over gf q designed for use over arbitrary discrete memoryless channels particularly nonbinary and asymmetric channels we use a random coset analysis to produce an effect that is similar to output symmetry with binary channels we show that the random selection of the nonzero elements of the gf q parity check matrix induces a permutation invariance property on the densities of the decoder messages which simplifies their analysis and approximation we generalize several properties including symmetry and stability from the analysis of binary ldpc codes we show that under a gaussian approximation the entire q 1 dimensional distribution of the vector messages is described by a single scalar parameter like the distributions of binary ldpc messages we apply this property to develop extrinsic information transfer exit charts for our codes we use appropriately designed signal constellations to obtain substantial shaping gains simulation results indicate that our codes outperform multilevel codes at short block lengths we also present simulation results for the additive white gaussian noise awgn channel including results within 0 56 db of the unrestricted shannon limit i e not restricted to any signal constellation at a spectral efficiency of 6 bits s hz

smooth impulsive linear systems axiomatic description abstract paper concerns with linear systems having arbitrary singularity at infinity these systems deserve to study since they are limits of familiar classical linear systems an axiomatic description of such systems is obtained

geometric minimization under external equivalence for implicit descriptions we consider minimality under external equivalence for linear time invariant implicit e a b c descriptions of the type ex t ax t bu t y t cx t necessary and sufficient conditions for external minimality have been previously given from an algebraic point of view by kuijper we propose here alternative geometric characterizations as well as a geometric reduction procedure that extracts from a given e a b c description a minimal model of the same type we also geometrically characterize in terms of the geometry of e a b c the corresponding minimal dimensions for the state space and for the state equation space

dynamic pole assignment and schubert calculus the output feedback pole assignment problem is a classical problem in linear systems theory in this paper we calculate the number of complex dynamic compensators of order q assigning a given set of poles for a q nondegenerate m input p output system of mcmillan degree n q m p 1 mp as a corollary it follows that when this number is odd the generic system can be arbitrarily pole assigned by output feedback with a real dynamic compensator of order at most q if and only if q m p 1 mp geq n

achievable rates for multiple descriptions consider a sequence of independent identically distributed i i d random variables x l x 2 cdots x n and a distortion measure d x i x i on the estimates x i of x i two descriptions i x in 1 2 cdots 2 nr 1 and j x in 1 2 cdots 2 nr 2 are given of the sequence x x 1 x 2 cdots x n from these two descriptions three estimates i x x2 j x and hat x o i x j x are formed with resulting expected distortions e frac 1 n sum n k 1 d x k hat x mk d m m 0 1 2 we find that the distortion constraints d 0 d 1 d 2 are achievable if there exists a probability mass distribution p x p hat x 1 hat x 2 hat x 0 x with ed x hat x m leq d m such that r 1 i x hat x 1 r 2 i x hat x 2 where i cdot denotes shannon mutual information these rates are shown to be optimal for deterministic distortion measures

structural matrix minimization algorithm for implicit descriptions in this paper we propose a matrix minimization algorithm for linear time invariant implicit e a b c descriptions of the type ex t ax t bu t y t cx t given the initial matrices e a b c we obtain an externally equivalent minimal realization em am bm cm after applying a matricial procedure based on maximal row and column compressions

a study of behaviors abstract the paper presents the results of the study of behavior theory as developed by j c willems from the point of view of polynomial and rational models considering behaviors in the discrete time case to be generalizations of rational models a natural focal point becomes the concept of a behavior homomorphism we give a characterization of behavior homomorphisms and analyze their invertibility properties in terms of embeddings in unimodular polynomial matrices these results which are of intrinsic interest are then applied to the uniform derivation of a large number of results for equivalence in different classes of behavior representations to a certain extent these are generalizations of the strict system equivalence concept for the class of polynomial matrix description of systems in the style of rosenbrock a study of behavioral controllability is undertaken and gives some new insights into connections with geometric control theory

a cell structure for the set of autoregressive systems the set of autoregressive systems generalizes the set of transfer functionsin a natural way we describe a topology for the set of all autoregressive systems of fixed size and bounded mcmillan degree we show that this topological space has the structure of a finite cw complex

a close to capacity dirty paper coding scheme the writing on dirty paper channel model offers an information theoretic framework for precoding techniques for canceling arbitrary interference known at the transmitter it indicates that lossless precoding is theoretically possible at any signal to noise ratio snr and thus dirty paper coding may serve as a basic building block in both single user and multiuser communication systems we design an end to end coding realization of a system materializing a significant portion of the promised gains we employ multidimensional quantization based on trellis shaping at the transmitter coset decoding is implemented at the receiver using virtual bits combined with iterative decoding of capacity approaching codes we achieve an improvement of 2db over the best scalar quantization scheme code design is done using the exit chart technique

writing on dirty paper corresp a channel with output y x s z is examined the state s sim n 0 qi and the noise z sim n 0 ni are multivariate gaussian random variables i is the identity matrix the input x in r n satisfies the power constraint l n sum i 1 n x i 2 leq p if s is unknown to both transmitter and receiver then the capacity is frac 1 2 ln 1 p n q nats per channel use however if the state s is known to the encoder the capacity is shown to be c ast frac 1 2 ln 1 p n independent of q this is also the capacity of a standard gaussian channel with signal to noise power ratio p n therefore the state s does not affect the capacity of the channel even though s is unknown to the receiver it is shown that the optimal transmitter adapts its signal to the state s rather than attempting to cancel it

rational differential operators and their kernels abstract a linear differential integral operator is associated to a rational matrix in a natural way and the behavior is defined to be the kernel of this operator conditions under which two rational matrices have the same behavior are derived

a new and simple proof of the equivalence theorem for behaviors two full row rank representations of the same behavior are related through a left unimodular transformation we present a new and extremely simple and insightful proof for this well established fact

floer homology and splittings of manifolds in f floer defined a mod8 graded homology group i m for an oriented integral homology 3 sphere m it is an invariant of the differentiable structure of m roughly speaking i m is a homology group coming from the morse theory of the chern simons functional f on the infinite dimensional manifold of all the gauge equivalence classes of irreducible connections on the principal su 2 bundle m x su 2 the critical point set of f is the set of the gauge equivalence classes of irreducible flat connections in general these critical sets may be degenerate and f not be a genuine morse functional a suitable perturbation of f is needed to define i m in this paper we consider only non degenerate critical points of f this is partly because we want to avoid some inessential technical complications and partly because it is sufficient for our computations of i nk for homology 3 spheres nk obtained by dehn surgery along the figure eight knot in s for a smooth connection a on m x su 2 a self adjoint fredholm operator da is defined section 2 the gauge equivalence class a of an irreducible flat connection a is a non degenerate critical point of f if and only if ker da 0 in this case a determines a generator of the mod 8 graded chain group of l m the mod 8 degree d a is related to the spectral flow invariant section 2 as follows let ao and a be two irreducible flat connections on m x su 2 such that ker da 0 ker da let at be a smooth path of smooth connections on m x su 2 connecting ao and a1 then d ai d a 0 is the mod 8 reduction of the spectral flow of the path of the self adjoint fredholm operators da the object of this paper is to give a practical method of calculation of the above spectral flow when m is split as m m1 u m2 where m1 n m2 am dm2 x and x is an orientable surface of genus g 2 such a decomposition in 3 dimensional gauge theory was first considered by c h

when are linear differentiation invariant spaces differential it is shown that a linear differentiation invariant subspace of a c trajectory space is differential i e can be represented as the kernel of a linear constant coefficient differential operator if and only if its mcmillan degree is finite

impulsive smooth behavior in multimode systems part ii minimality and equivalence this is the second part of a two part study of linear multimode systems in the first part it was argued that the behavior of such a system on an interval between switches should be described in a framework that allows for impulses at the switching instant and both first order and polynomial representations were introduced that satisfy this requirement here we determine the conditions under which first order representations are minimal we also show how two minimal representations of the same behavior are related this leads in particular to an appropriate state space isomorphism theorem the minimality conditions are given a dynamic interpretation

state space approach to behavioral systems theory the dirac bergmann algorithm this paper develops an approach to behavioral systems theory in which a state space representation of behaviors is utilised this representation is a first order hybrid representation of behaviors called pencil representation an algorithm well known after dirac and bergmann db is shown to be central in obtaining a constraint free and observable cfo state space representation of a behavior results and criteria for asymptotic stability controllability inclusions and markovianity of behaviors are derived in terms of the matrices of this representation which involve linear algebraic processes in their computation

the equivalence structure of descriptor representations of systems with possibly inconsistent initial conditions abstract necessary and sufficient conditions for minimality of descriptor representations of impulsive smooth behaviors are derived we obtain a complete set of transformations by which minimal descriptor representations that give rise to the same behavior can be transformed into each other in particular this leads to a jump behavioral interpretation of the notion of strong equivalence of descriptor representations

causal state communication the problem of state communication over a discrete memoryless channel with discrete memoryless state is studied when the state information is available strictly causally at the encoder it is shown that block markov encoding in which the encoder communicates a description of the state sequence in the previous block by incorporating side information about the state sequence at the decoder yields the minimum state estimation error when the same channel is used to send additional independent information at the expense of a higher channel state estimation error the optimal tradeoff between the rate of the independent information and the state estimation error is characterized via the capacity distortion function it is shown that any optimal tradeoff pair can be achieved via rate splitting these coding theorems are then extended optimally to the case of causal channel state information at the encoder using the shannon strategy

behaviors and symbols of rational matrices abstract recently we have defined rational differential equations and derived conditions when two such equations have the same solution set in this article we study the same question but present a different approach based on the new notion of symbols also we try to describe the relationship with the work behaviors defined by rational functions of willems and yamamoto

a pointwise criterion for controller robustness abstract we present a pointwise criterion for controller robustness with respect to stability the term point here refers to complex frequency in the right half plane the proposed test is based on the concept of the minimal angle between subspaces determined by the plant and the compensator the test leads to separate balls of uncertainty at each frequency and may therefore help to reduce conservativeness in the analysis of robustness

some inequalities for the hadamard product and the fan product of matrices abstract if a and b are nonsingular m matrices a sharp lower bound on the smallest eigenvalue a b for the fan product of a and b is given and a sharp lower bound on a b 1 for the hadamard product of a and b 1 is derived in addition we also give a sharp upper bound on the spectral radius a b for nonnegative matrices a and b

on the gluing problem for the eta invariant we solve the gluing problem for the invariant consider a generalized dirac operator d over a compact riemannian manifold m that is partitioned by a compact hypersurface n such that m m un m2 we assume that the riemannian metric of m and d have a product structure near n i e d i d d dn with some dirac operator dn on n using boundary conditions of atiyah patodi singer type parametrized by lagrangian subspaces l of kerd we define selfadjoint extensions d i 1 2 over m we express the invariant of d in terms of the invariants of di an invariant m l l2 of the pair of the lagrangian subspaces l l2 which is related to the maslov index and an integer valued term j in the adiabatic limit i e if a tubular neighborhood of n is long enough the vanishing of j is shown under certain regularity conditions we apply this result in order to prove cutting and pasting formulas for the invariant a wall nonadditivity result for the index of atiyah patodi singer boundary value problems and a splitting formula for the spectral flow 1 the gluing problem and applications 1 1 introduction we solve the gluing problem for the invariant of generalized dirac operators consider a closed compact riemannian manifold carrying a dirac bundle with a generalized dirac operator assume that this manifold is separated into two pieces by a compact hypersurface the gluing problem for the invariant consists in expressing the invariant of the original dirac operator in terms of the invariants of the dirac operators living on the pieces the selfadjoint operators on the two components with boundary depend on a boundary condition given by lagrangian subspaces lx l2 of a certain symplectic vector space the gluing formula also contains an additional real valued term m l l2 which is nicely related to the symplectic geometry and the maslov index