the vinnicombe metric for nonlinear operators describes an extension of the vinnicombe metric on linear operators to a pseudometric on nonlinear operators a metric for finite dimensional time varying operators is shown to be capable of guaranteeing stability and performance robustness and reduces to the standard vinnicombe metric for the time invariant operator case which is known to be less conservative than the gap metric the analysis exploits the time varying operator equivalents of unstable poles and normalized coprime fractional descriptions in addition a time varying operator equivalent of the winding number is defined

the integral 3 harmonic graphs abstract let g be a graph on n vertices v1 vn and let d vi be the degree of the vertex vi if d g d v1 d vn is an eigenvector of the 0 1 adjacency matrix a of g i e a g d g d g then g is said to be harmonic in this paper all connected integral 3 harmonic graphs are determined there are exactly 26 such graphs

integral graphs and k τ regular sets abstract a subset of the vertex set of a graph g w v g is a k regular set if it induces a k regular subgraph of g and every vertex not in the subset has neighbors in it in this paper we deal with the existence of k regular sets associated with all distinct eigenvalues we show some families that have this property and we give some results concerning the existence of such sets considering restrictions on the symbol of circulant graphs

on spectral integral variations of mixed graphs in this paper we characterize the mixed graphs with exactly one laplacian eigenvalue moving up by an integer and other laplacian eigenvalues remaining invariant when an edge is added the results extend those of fan linear and multilinear algebra 50 2002 133 for general graphs and so linear and multilinear algebra 46 1999 193 for simple graphs

regular graphs with girth at least 5 and small second largest eigenvalue abstract in this paper we present an upper bound on the degree of a regular graph with girth at least 5 in terms of its second largest eigenvalue usually denoted by 2 next we consider bipartite r regular graphs with girth at least 6 whose second largest eigenvalue satisfies 2 r and prove that such graphs are the incidence graphs of two class partially balanced incomplete block designs and also balanced incomplete block designs upper bound on the order of such graphs is also given we also consider the relations between the incidence graphs of two class partially balanced incomplete block designs and distance regular graphs in particular we determine all regular reflexive graphs of degree 3 with girth at least 5 all regular graphs with girth at least 5 satisfying 2 3 and all bipartite regular reflexive graphs with girth at least 6

gap metric on the subgraphs of systems and the robustness problem the gap metric between the shift invariant subspaces of the graphs or subgraphs of systems is investigated under certain index conditions it is shown that the gap metric on the subgraphs shares the same fundamental property of robust stability as those well known metrics such as the gap metric and the spl nu gap metric it is also shown that the spl nu gap metric between two systems is the distance measured by the gap metric between their respective sets of all subgraphs under certain index conditions

distance spectra and distance energy of integral circulant graphs the distance energy of a graph g is a recently developed energy type invariant defined as the sum of absolute values of the eigenvalues of the distance matrix of g there was a vast research for the pairs and families of non cospectral graphs having equal distance energy and most of these constructions were based on the join of graphs a graph is called circulant if it is cayley graph on the circulant group i e its adjacency matrix is circulant a graph is called integral if all eigenvalues of its adjacency matrix are integers integral circulant graphs play an important role in modeling quantum spin networks supporting the perfect state transfer in this paper we characterize the distance spectra of integral circulant graphs and prove that these graphs have integral eigenvalues of distance matrix d furthermore we calculate the distance spectra and distance energy of unitary cayley graphs in conclusion we present two families of pairs g1 g2 of integral circulant graphs with equal distance energy in the first family g1 is subgraph of g2 while in the second family the diameter of both graphs is three

regular graphs with four eigenvalues we study the connected regular graphs with four distinct eigenvalues properties and feasibility conditions of the eigenvalues are found several examples constructions and characterizations are given as well as some uniqueness and nonexistence results

capacity bounds and concatenated codes over segmented deletion channels we develop an information theoretic characterization and a practical coding approach for segmented deletion channels compared to channels with independent and identically distributed i i d deletions where each bit is independently deleted with an equal probability the segmentation assumption imposes certain constraints i e in a block of bits of a certain length only a limited number of deletions are allowed to occur this channel model has recently been proposed and motivated by the fact that for practical systems when a deletion error occurs it is more likely that the next one will not appear very soon we first argue that such channels are information stable hence their channel capacity exists then we introduce several upper and lower bounds with two different methods in an attempt to understand the channel capacity behavior the first scheme utilizes certain information provided to the transmitter and or receiver while the second one explores the asymptotic behavior of the bounds when the average bit deletion rate is small in the second part of the paper we consider a practical channel coding approach over a segmented deletion channel specifically we utilize outer ldpc codes concatenated with inner marker codes and develop suitable channel detection algorithms for this scenario different maximum a posteriori map based channel synchronization algorithms operating at the bit and symbol levels are introduced and specific ldpc code designs are explored simulation results clearly indicate the advantages of the proposed approach in particular for the entire range of deletion probabilities less than unity our scheme offers a significantly larger transmission rate compared to the other existing solutions in the literature

pointwise gap metrics on transfer matrices a family of metrics pointwise gap metrics in the space of real rational matrices of fixed size is developed and used to study open loop and closed loop stability robustness of linear time invariant finite dimensional continuous time systems it is shown that pointwise gap metrics have the desired qualitative properties for the study of stability robustness necessary and sufficient conditions on the robustness are obtained in terms of the radii of the pointwise gap metric balls centered at the nominal plant and or the nominal controller comparison with the gap metric and the graph metric is made all of these metrics induce the same topology many of the quantitative properties of pointwise gap metrics are the same as those of the gap metric although they differ in value pointwise gap metrics in the scalar case have a very simple expression which is potentially useful for accessing the relationship between the uncertainty of physical parameters and uncertainty measured by pointwise gap metrics

stability analysis for time varying systems via quadratic constraints abstract in this paper we introduce the concept of quadratic constraint qc to deal with various stability problems for infinite dimensional linear discrete time varying ltv systems first we derive a necessary and sufficient condition for the close loop stability based on quadratic constraints qcs an equivalent condition of this stability criterion presents the relationship between the stabilization of each finite dimensional truncation system and that of the whole system moreover applying this stability criterion and the complete finiteness of the nest algebra we show that the plant is stabilizable if and only if it has a single strong representation finally we use qcs to give a necessary and sufficient condition for the robust stabilization of connected uncertainty set defined by gap metric

further results on the perfect state transfer in integral circulant graphs for a given graph g denote by a its adjacency matrix and f t exp iat we say that there exist a perfect state transfer pst in g if f t a b 1 for some vertices a b and a positive real number t such a property is very important for the modeling of quantum spin networks with nearest neighbor couplings we consider the existence of the perfect state transfer in integral circulant graphs circulant graphs with integer eigenvalues some results on this topic have already been obtained by saxena et al 2007 5 basic et al 2009 6 and basic petkovic 2009 7 in this paper we show that there exists an integral circulant graph with n vertices having a perfect state transfer if and only if 4 n several classes of integral circulant graphs have been found that have a perfect state transfer for the values of n divisible by 4 moreover we prove the nonexistence of a pst for several other classes of integral circulant graphs whose order is divisible by 4 these classes cover the class of graphs where the divisor set contains exactly two elements the obtained results partially answer the main question of which integral circulant graphs have a perfect state transfer

robust stabilization bibo stability distance notions and robustness optimization abstract this paper studies robust stabilization of both linear shift invariant causal systems in an lp setting and linear time invariant causal continuous time systems in an lp p 1 or setting two key technical results in the paper establish the existence of lp and lp stable normalized coprime factorizations for discrete time and continuous time systems respectively which have coprime factorizations as lp and lp stable operators several distance measures for systems are then introduced including the graph metric the function the gap between the graphs of the systems and the projection gap it is shown that these distance measures lead to the weakest convergence notions for systems for which closed loop stability is a robust property the function can be computed using the dahleh pearson theory for l1 l1 optimal control robustness optimization in a directed p function is shown to be closely related to robustness optimization for bibo stable normalized coprime factor perturbations this result connects the stability margin of dahleh for coprime factor perturbations to the function these considerations are further supported by a robustness result in terms of the projection gap

adaptive control and robustness in the gap metric we consider the construction of adaptive controllers for minimum phase linear systems that achieve nonzero robustness margins in the sense of the gap metric the gap perturbation margin may be more constrained for larger disturbances and for larger parametric uncertainties working in both and settings and within the framework of the nonlinear gap metric universal adaptive controllers are first given achieving stabilization for first order nominal plants and the results are then generalized to relative degree one nominal plants a notion of a semiuniversal control design is introduced which is the property that a bound on performance exists that is independent of the a priori known uncertainty level and a characterization is given for when semiuniversal designs outperform the class of memoryless controllers and the class of linear time invariant controllers robust semiuniversal adaptive control designs are given for nominal plants under the classical assumptions of adaptive control in both the and settings the results are applied throughout to explicit classes of unmodeled dynamics including the rohrs example

graphs with integral spectrum abstract it is shown that only a fraction of 2 n of the graphs on n vertices have an integral spectrum although there are several explicit constructions of such graphs no upper bound for their number has been known graphs of this type play an important role in quantum networks supporting the so called perfect state transfer

sufficient conditions for robust bibo stabilization given by the gap metric a relation between coprime fractions and the gap metric is presented using this result we provide some sufficient conditions for robust bibo stabilization for a wide class of systems these conditions allow the plant and the compensator to be disturbed simultaneously

on the existence of nominal plants abstract in this paper we consider single input single output siso linear time invariant systems represented by their meromorphic transfer functions k s we assume that k s is representative of a given set of uncertainty containing k and we would like to introduce conditions for determining if a nominal plant exists for k which would then be used for both analysis and design it is shown that uncertainty sets representing marginally stable or unstable dipoles do not possess nominal plants some concluding remarks and suggestions for future research are provided

mutual information bounds via adjacency events the mutual information between two jointly distributed random variables x and y is a functional of the joint distribution p xy which is sometimes difficult to handle or estimate a coarser description of the statistical behavior of x y is given by the marginal distributions p x p y and the adjacency relation induced by the joint distribution where x and y are adjacent if p x y 0 we derive a lower bound on the mutual information in terms of these entities the bound is obtained by viewing the channel from x to y as a probability distribution on a set of possible actions where an action determines the output for any possible input and is independently drawn we also provide an alternative proof based on convex optimization that yields a generally tighter bound finally we derive an upper bound on the mutual information in terms of adjacency events between the action and the pair x y where in this case an action a and a pair x y are adjacent if y a x as an example we apply our bounds to the binary deletion channel and show that for the special case of an independent identically distributed input distribution and a range of deletion probabilities our lower and upper bounds both outperform the best known bounds for the mutual information

bundle reduction and the alignment distance on spaces of state space lti systems this paper introduces a large class of differential geometric distances between finite dimensional linear dynamical systems collectively called the alignment distance contrary to the existing distances the alignment distance is based on the state space description of dynamical systems and is defined on the manifolds of systems of fixed order and fixed input output dimension under a matrix rank constraint e g minimality controllability or observability while the quotient topology and principal fiber bundle structure associated with such manifolds have been known since the early days of modern control theory distances natural to this structure have not been studied the starting point for defining such a distance is to identify a linear system of order n with its equivalence class of state space realizations all related by the so called similarity action i e state space change of basis under gl n the lie group of nonsingular n times n matrices the main idea of the alignment distance is to first find the best state space change of basis that brings a realization of a system as close as possible to a realization of another system the alignment step and then compare the aligned realizations a direct implementation of this idea due to noncompactness of gl n is complicated however using the notion of reduction of the structure group of a principal bundle we show that the change of basis can be restricted to an orthogonal change of basis provided one uses realizations in a reduced subbundle this key observation brings about significant computational benefits as a technical contribution possibly of independent interest we show that several forms of realization balancing available in the control literature have differential geometric significance and are indeed examples of reducing the structure group from gl n to its subgroup of orthogonal matrices o n the alignment distance can be defined for stable and unstable systems discrete or continuous time and stochastic systems

the complement of the path is determined by its spectrum abstract it is proved that a graph whose 0 1 adjacency matrix has the spectrum of p n the complement of the path on n vertices must be p n

laplacian integral graphs in s a b abstract let g n m be the family of graphs with n vertices and m edges when n and m are previously given it is well known that there is a subset of g n m constituted by graphs g such that the vertex connectivity the edge connectivity and the minimum degree are all equal in this paper s a b classes of connected a b linear graphs with n vertices and m edges are described where m is given as a function of a b n 2 some of them have extremal graphs for which the equalities above are extended to algebraic connectivity these graphs are laplacian integral although they are not threshold graphs however we do build threshold graphs in s a b

identification of stabilizable systems closed loop approximation approximate modelling and identification of linear shift invariant possibly unstable discrete time systems or plants is studied in a framework compatible with the so called robust stability concept for feedback systems this unified framework is based on approximate modelling of the plant in the gap and graph metrics which is achieved here through approximation of certain closed loop transfer functions by finite dimensional systems properties of this approximate inverse modelling approach are studied and concrete rate of approximation results are given furthermore a consistency result in the gap and graph metrics is given for a certain experimental estimate of the plant constructed from closed loop input output data in a stationary noise set up under mild conditions on the unknown plant

conservatism of the gap metric it is shown by way of simple examples that the gap metric may be inappropriate for quantifying the uncertainty for lightly damped systems e g flexible structures such as spacecraft and aircraft two plants having otherwise identical transfer functions except that their zeros on the imaginary axis are located slightly apart due to perturbations are usually considered very close i e their closed loop characteristics under most stabilizing feedbacks are very similar however it is demonstrated that when the gap is used to quantify the plant perturbations the computed gap may be large consequently the stability of the perturbed plant with a stabilizing controller designed for the nominal plant cannot be guaranteed via the existing stability theorems using the gap metric

directly lower bounding the information capacity for channels with i i d deletions and duplications in this paper we directly lower bound the information capacity for channels with independent identically distributed i i d deletions and duplications our approach differs from previous work in that we focus on the information capacity using ideas from renewal theory rather than focusing on the transmission capacity by analyzing the error probability of some randomly generated code using a combinatorial argument of course the transmission and information capacities are equal but our change of perspective allows for a much simpler analysis that gives more general theoretical results we then apply these results to the binary deletion channel to improve existing lower bounds on its capacity

bounds on the capacity of discrete memoryless channels corrupted by synchronization and substitution errors we study the capacity of discrete memoryless channels with synchronization errors and additive noise we first show that with very large alphabets their capacity can be achieved by independent and identically distributed input sources and establish proven tight lower and upper capacity bounds we also derive tight numerical capacity bounds for channels where the synchronization between the input and output is partly preserved for instance using incorruptible synchronization markers such channels include channels with duplication errors channels that only insert or delete zeros and channels with bitshift errors studied in magnetic recording channels with small alphabets and corrupted by synchronization errors have an infinite memory revisiting the theoretical work of dobrushin and adapting techniques used to compute capacity bounds for finite state source channel models we compute improved numerical capacity lower bounds for discrete memoryless channels with small alphabets synchronization errors and memoryless noise an interesting and some what surprising result is that as long as the input sequences are not completely deleted the capacity of channels corrupted by discrete timing errors is always nonzero even if all the symbols are corrupted

perfect state transfer in integral circulant graphs of non square free order abstract this paper provides further results on the perfect state transfer in integral circulant graphs icg graphs the non existence of pst is proved for several classes of icg graphs containing an isolated divisor d 0 i e the divisor which is relatively prime to all other divisors from d d d 0 the same result is obtained for classes of integral circulant graphs having the nsf property i e each n d is square free for every d d a direct corollary of these results is the characterization of icg graphs with two divisors which have pst a similar characterization is obtained for icg graphs where each two divisors are relatively prime finally it is shown that icg graphs with the number of vertices n 2 p 2 do not have pst

graph topology and gap topology for unstable systems a reformation is provided of the graph topology and the gap topology for a general setting including lumped linear time invariant systems and distributed linear time invariant systems in the frequency domain some essential properties and their comparisons are clearly presented in the reformulation it is shown that the gap topology is suitable for general systems rather than square systems with unity feedback it is shown that whenever an unstable plant can be stabilized by feedback it is a closed operator mapping a subspace of the input space to the output space hence the gap topology can always be applied whenever the unstable plant can be stabilized the graph topology and the gap topology are suitable for different subsets of systems and have many similar characteristics if one confines them to the same subset they will be identical the definitions of the graph metric and the gap metric are discussed

improved lower bounds for the capacity of i i d deletion and duplication channels this paper considers the capacity of binary deletion channels where bits are deleted independently with probability d it improves significantly upon the best previous framework used to obtain provable lower bounds on this capacity by utilizing a stronger definition of a typical output from the channel the new results give the best known provable bounds on the capacity for all values of d moreover the techniques presented here extend to yield lower bounds for channels with certain types of random insertions namely duplications or combinations of duplications and deletions to demonstrate these techniques in this context two binary channels are analyzed a channel where each transmitted bit is copied with probability nu and a channel where each transmitted bit is copied a geometrically distributed number of times

bounds on the capacity of random insertion and deletion additive noise channels we develop several analytical lower bounds on the capacity of binary insertion and deletion channels by considering independent uniformly distributed i u d inputs and computing lower bounds on the mutual information between the input and output sequences for the deletion channel we consider two different models i i d deletion substitution channel and i i d deletion channel with additive white gaussian noise awgn these two models are considered to incorporate effects of the channel noise along with the synchronization errors for the insertion channel case we consider gallager s model in which the transmitted bits are replaced with two random bits and uniform over the four possibilities independently of any other insertion events the general approach taken is similar in all cases however the specific computations differ furthermore the approach yields a useful lower bound on the capacity for a wide range of deletion probabilities of the deletion channels while it provides a beneficial bound only for small insertion probabilities less than 0 25 of the insertion model adopted we emphasize the importance of these results by noting that 1 our results are the first analytical bounds on the capacity of deletion awgn channels 2 the results developed are the best available analytical lower bounds on the deletion substitution case 3 for the gallager insertion channel model the new lower bound improves the existing results for small insertion probabilities

on the capacity of duplication channels the i i d duplication channel which duplicates each symbol independently with a certain probability is studied the contribution is twofold first a tight lower bound on the capacity of such channels is introduced second the capacity is computed for the small values of the duplication probability using a series expansion representation

optimal robustness in the gap metric the application of the gap metric to robust stabilization of feedback systems is considered in particular a solution to the problem of robustness optimization in the gap metric is presented the problem of robust stabilization under simultaneous plant controller perturbations is addressed and the least amount of combined plant controller uncertainty measured by the gap metric that can cause instability of a nominally stable feedback system is determined included are a detailed summary of the main properties of the gap metric and the introduction of a dual metric called the t gap metric a key contribution of this study is to show that the problem of robustness optimization in the gap metric is equivalent to robustness optimization for normalized coprime factor perturbations this settles the question as to whether maximizing allowable coprime factor uncertainty corresponds to tolerating the largest ball of uncertainty in a well defined metric

robust identification of strongly stabilizable systems for strongly stabilizable systems for which a strongly stabilizing controller is known approximately the authors consider system identification in the graph gap and chordal metrics using robust h sub infinity identification of the closed loop transfer function in the framework proposed by a j helmicki et al 1990 error bounds are derived showing that robust convergence is guaranteed and that the identification can be satisfactorily combined with a model reduction step two notions of robust identification of stable systems are compared and an alternative robust identification technique based on smoothing which can be used to yield polynomial models directly is developed

the gap metric for time varying systems abstract a formula is given relating the gap metric to a 2 block uniform optimization problem georgiou s formula for the gap between time invariant systems is obtained as a special case

on the gap metric and coprime factor perturbations abstract new conditions are derived for when the distance between two linear systems in the gap metric is less than one by including a coprimeness assumption in the coprime factor uncertainty description it is shown that an open ball in the gap metric is equivalent to an open ball phrased in terms of coprime factor perturbations a new criterion is given for robust stabilization

polynomial time decodable codes for the binary deletion channel in the random deletion channel each bit is deleted independently with probability p for the random deletion channel the existence of codes of rate 1 p 9 and thus bounded away from 0 for any p has been known we give an explicit construction with polynomial time encoding and deletion correction algorithms with rate c 0 1 p for an absolute constant c 0 0

on the computation of the gap metric abstract this paper deals with the computation of the gap metric introduced by zames and el sakkary 17 it is shown that the gap between two systems p1 p2 is precisely the maximum of the two expressions inf q h d i n i d j n j q for i j equal to 1 2 and 2 1 and ni di being normalized right coprine factorizations of pi i 1 2 in the sense of vidyasagar 12 this expression is computable using well known techniques from interpolation theory

estimating the robust dead time for closed loop stability a method based on the small gain theorem is provided for estimating the robust dead time for closed loop stability the method lends itself to generalization for multivariable systems and also to graphical implementation some examples are provided to demonstrate the use of the method and the results are compared to the exact critical values of dead time

singular perturbation margin and generalised gain margin for linear time invariant systems in this paper we propose a singular perturbation margin spm and a generalised gain margin ggm as stability metrics for single input single output siso linear time invariant lti systems from the view of singular perturbations and regular perturbations which have bijective correspondences with the classical phase margin pm and the gain margin gm respectively both of the numerical and analytical time domain spm and ggm assessment methods are provided and relationships between the singular perturbation parameter pm of the perturbed system pm and spm of the nominal system and the monotonically increasing phase of the fast system are also revealed these results make it possible to assess the pm of the nominal system in the time domain for siso lti systems using the spm with a standardised testing system called pm gauge as demonstrated by examples the concepts of spm and ggm can be used as metrics of stability margins for linear time varying systems and nonlinear systems

optimal asymptotic identification under bounded disturbances the intrinsic limitation of worst case identification of linear time invariant systems using data corrupted by bounded disturbances when the unknown plant is known to belong to a given model set is studied this is done by analyzing the optimal worst case asymptotic error achievable by performing experiments using any bounded input and estimating the plant using any identification algorithm it is shown that under some topological conditions on the model set there is an identification algorithm which is asymptotically optimal for any input and the optimal asymptotic error is characterized as a function of the inputs these results which hold for any error metric and disturbance norm are applied to three specific identification problems identification of stable systems in the l sub 1 norm identification of stable rational systems in the h sub infinity norm and identification of unstable rational systems in the gap metric for each of these problems the general characterization of optimal asymptotic error is used to find near optimal inputs to minimize the error

a robust control framework for linear time invariant spatially distributed systems a robust control framework for linear time invariant lti spatially distributed systems is outlined in this paper we adopt an input output approach which takes account of the spatially distributed nature of the input and output signals for such systems the approach is a generalization of h infty xspace control in the sense that the 2 norm in both time and space is used to quantify the size of signals it is shown that a frequency domain representation in the form of a graph symbol exists for every lti spatially distributed system under very mild assumptions the graph symbol gives rise to left and right coprime representations if the system is also stabilizable we investigate fundamental issues of feedback control such as feedback stability and robust stability to plant and or controller uncertainty quantified in the gap metric this includes a generalization of the sefton ober gap formula to the infinite dimensional operator case a design example in which an electrostatically destabilized membrane is feedback stabilized concludes the paper

stability of control systems and graphs of linear systems new conditions for internal stability of a closed loop control system are given in terms of the graphs of the multiplication operators induced by the transfer functions of the plant and the controller these conditions can be given a geometrical interpretation this relates closed loop stability to the minimal angle between the graph space associated with the system and the graph space associated with the controller the maximally stabilizing controller is defined as the controller that maximizes the minimum angle between the graph space associated with the system and the graph space associated with the controller it is shown that this controller can be calculated as a nehari extension of the coprime factors of the system

regular bipartite graphs with three distinct non negative eigenvalues abstract we derive some structural and spectral properties of regular bipartite graphs with three distinct non negative eigenvalues next we consider the relations between these graphs and two class partially balanced incomplete block designs and we present a number of situations when the graphs we consider are in fact the incidence graphs of those designs as a consequence we give a several constructions of connected regular bipartite graphs with six distinct eigenvalues and we also determine all such graphs with degree 3 and all such graphs on at most 20 vertices

remarks on computing coprime factors for distributed parameter systems presents an indirect method for computing coprime factors for a class of linear time invariant distributed parameter systems the method overcomes the numerical instability of an existing direct decomposition method it is shown that coprime factorization can be reduced to constructing simultaneously stabilizing finite dimensional controllers for which an efficient iterative procedure is developed

a simple lower bound for the capacity of the deletion channel we present a simple proof that the capacity of the binary independent and identically distributed i i d deletion channel where each bit is deleted independently with probability d is at least 1 d 9 by developing a correspondence between the deletion channel and an insertion deletion channel that we call a poisson repeat channel

upper bounds on the capacity of deletion channels using channel fragmentation we study memoryless channels with synchronization errors as defined by a stochastic channel matrix allowing for symbol drop outs or symbol insertions with particular emphasis on the binary and non binary deletion channels we offer a different look at these channels by considering equivalent models by fragmenting the input sequence where different subsequences travel through different channels the resulting output symbols are combined appropriately to come up with an equivalent input output representation of the original channel which allows for derivation of new upper bounds on the channel capacity we consider both random and deterministic types of fragmentation processes applied to binary and nonbinary deletion channels with two specific applications of this idea a random fragmentation applied to a binary deletion channel and a deterministic fragmentation process applied to a nonbinary deletion channel we prove certain inequality relations among the capacities of the original channels and those of the introduced subchannels the resulting inequalities prove useful in deriving tighter capacity upper bounds for 1 independent identically distributed i i d deletion channels when the deletion probability exceeds 0 65 and 2 nonbinary deletion channels some extensions of these results for instance to the case of deletion substitution channels are also explored

computation of the singular values of toeplitz operators and the gap metric it is widely recognized that the computation of gap metric is equivalent to a certain two block h problem i e the gap is equal to the norm of a certain two block operator however it can also be characterized as the smallest singular value of a certain toeplitz operator this paper derives a simple computational method for finding such singular values and the gap between two plants by using a state space approach

on the chromatic number of integral circulant graphs integral circulant graphs are a generalization of unitary cayley graphs recently studied by klotz and sander the integral circulant graph x n d has vertices 0 1 n 1 and two vertices a and b are adjacent iff gcd x y n d where d d d n 1 d 3 we construct a family of counterexamples using exhaustive search algorithm for graph coloring and disprove the conjecture in this case

geometric structures of stable state feedback systems a parameterization of the set of all stabilizing state feedback gains is given and the geometric structure of the set of stable state feedback systems in the set of stable matrices is exploited some spectral properties related to this structure are derived the results give not only fundamental guidelines for designing state feedback gains using this parameterization but also a new approach to analyzing structures of linear stable systems

on algebraic connectivity and spectral integral variations of graphs abstract let g be a simple connected graph and l g be the laplacian matrix of g let a g be the second smallest eigenvalue of l g an eigenvector of l g corresponding to the eigenvalue a g is called a fiedler vector of g let y be a fiedler vector of g a characteristic vertex is a vertex u of g such that y u 0 and such that there is a vertex w adjacent to u satisfying y w 0 a characteristic edge is an edge u v such that y u y v

the integral graphs with index 3 and exactly two main eigenvalues a graph is called integral if the spectrum of its adjacency matrix has only integral eigenvalues an eigenvalue of a graph is called main eigenvalue if it has an eigenvector such that the sum of whose entries is not equal to zero in this paper we show that there are exactly 25 connected integral graphs with exactly two main eigenvalues and index 3

the gap metric robustness of stabilization of feedback systems in this paper we introduce the gap metric to study the robustness of the stability of feedback systems which may employ not necessarily stable open loop systems we elaborate on the computational aspects of the gap metric and provide upper bounds to the gap in cases where the exact formulas do not apply by admissible uncertainties we mean those which preserve closed loop stability and a specified small tolerance on the i o behavior of a feedback system we show that admissible uncertainties are precisely those which are constrained in the gap finally we conclude that any metric which preserves a continuous relationship between open loop systems and the corresponding stable feedback interconnections must have the topology of the gap metric

a comparative study of input output stability results at present many similar but disparate input output i o stability criteria exist without means for comparison it is unclear which result is best used in any given application this paper proposes a means for comparison between i o stability results involving norms and inner products of inputs and outputs the extended conic sector theorem provides a framework for determining which results are least conservative and most broadly applicable in so doing numerous existing stability results are unified and revealed as more powerful than previously thought

walks and regular integral graphs we establish a useful correspondence between the closed walks in regular graphs and the walks in infinite regular trees which after counting the walks of a given length between vertices at a given distance in an infinite regular tree provides a lower bound on the number of closed walks in regular graphs this lower bound is then applied to reduce the number of the feasible spectra of the 4 regular bipartite integral graphs by more than a half next we give the details of the exhaustive computer search on all 4 regular bipartite graphs with up to 24 vertices which yields a total of 47 integral graphs

on lower bounds for the capacity of deletion channels this correspondence considers binary deletion channels where bits are deleted independently with probability d it improves upon the framework used to analyze the capacity of binary deletion channels established by diggavi and grossglauser improving on their lower bounds diggavi and grossglauser considered codebooks with codewords generated by a first order markov chain they only consider typical outputs where an output is typical if an n bit input gives an n 1 d 1 epsi bit output the improvements in this correspondence arise from two considerations first a stronger notion of a typical output from the channel is used which yields better bounds even for the codebooks studied by diggavi and grossglauser second codewords generated by more general processes than first order markov chains are considered

optimal angle reduction a behavioral approach to linear system approximation we investigate the problem of optimal state reduction under minimization of the angle between system behaviors the angle is defined in a worst case sense as the largest angle that can occur between a system trajectory and its optimal approximation in the reduced order model this problem is analyzed for linear time invariant finite dimensional systems in a behavioral l2 setting without reference to input output decompositions and stability considerations the notion of a weakest past future link is introduced and it is shown how this concept is applied for the purpose of model reduction a method that reduces the state dimension by one is presented and shown to be optimal specific algorithms are provided for the numerical implementation of the approximation method the concepts and results are explicitly translated to an input output setting and related to balancing hankel norm reduction and normalized doubly coprime factorizations

capacity bounds for sticky channels the capacity of sticky channels a subclass of insertion channels where each symbol may be duplicated multiple times is considered the primary result is to provide nearly tight numerical upper and lower bounds for the independent and identically distributed i i d duplication channel

indecomposable laplacian integral graphs abstract a graph that can be constructed from isolated vertices by the operations of union and complement is decomposable every decomposable graph is laplacian integral i e its laplacian spectrum consists entirely of integers an indecomposable graph is not decomposable the main purpose of this note is to demonstrate the existence of infinitely many indecomposable laplacian integral graphs

worst case analysis of identification bibo robustness for closed loop data this paper deals with the worst case analysis of identification of linear shift invariant possibly infinite dimensional systems a necessary and sufficient input richness condition for the existence of robustly convergent identification algorithms in l sup 1 is given a closed loop identification setting is studied to cover both stable and unstable but bibo stabilizable systems identification or modeling error is then measured by distance functions which lead to the weakest convergence notions for systems such that closed loop stability in the sense of bibo stability is a robust property worst case modeling error bounds in several distance functions are included

the parallel projection operators of a nonlinear feedback system abstract this paper defines and studies a pair of nonlinear parallel projection operators associated with a nonlinear feedback system these operators have been seen to play an important role in the robustness and design of linear systems especially in the theory of the gap metric the use of weighted gaps in control system design and glover mcfarlane loop shaping we show that the input output l 2 stability of a feedback system amounts to a coordinatization of the input and output spaces which is also equivalent to the existence of a pair of nonlinear parallel projection operators onto the graph of the plant and the inverse graph of the controller respectively these projections are shown to have equal norms whenever one of the feedback elements is linear a bound on this norm is given in the case of passive systems with unity negative feedback

reliable communication over non binary insertion deletion channels we consider the problem of reliable communication over non binary insertion deletion channels where symbols are randomly deleted from or inserted in the received sequence and all symbols are corrupted by additive white gaussian noise to this end we utilize the inherent redundancy achievable in non binary symbol sets by first expanding the symbol set and then allocating part of the bits associated with each symbol to watermark symbols the watermark sequence known at the receiver is then used by a forward backward algorithm to provide soft information for an outer code which decodes the transmitted sequence through numerical results and discussions we evaluate the performance of the proposed solution and show that it leads to significant system ability to detect and correct insertions deletions we also provide estimates of the maximum achievable information rates of the system compare them with the available bounds and construct practical codes capable of approaching these limits

infinite families of q integral graphs abstract in this article we characterize all signless laplacian integral graphs here called q integral graphs in the following classes complete split graphs multiple complete split like graphs extended complete split like graphs multiple extended split like graphs all these graphs were defined by hansen et al p hansen h melot d stevanovic integral complete split graphs univ beograd publ elektrotehn fak ser mat 13 2002 89 95 also we characterize all q integral graphs of the form k n 1 k n 2 k n 3 where k n i i 1 2 3 is a complete graph on n i vertices these characterizations allow us to exhibit many infinite families of q integral graphs

optimal hankel norm identification of dynamical systems abstract the problem of optimal approximate system identification is addressed with a newly defined measure of misfit between observed time series and linear time invariant models the behavioral framework is used as a suitable axiomatic setting for a non parametric introduction of system complexity and allows for a notion of misfit of dynamical systems that is independent of system representations the misfit function introduced here is characterized in terms of the induced norm of a hankel operator associated with the data and a co inner kernel representation of a model sets of pareto optimal models are defined as feasible trade offs between complexity and misfit of models and it is shown how all pareto optimal models are characterized as exact models of compressed data sets obtained from hankel norm approximations of data matrices this leads to new conceptual algorithms for optimal approximate identification of time series 1997 elsevier science ltd

two spectral characterizations of regular bipartite graphs with five eigenvalues graphs with a few distinct eigenvalues usually possess an interesting combinatorial structure we show that regular bipartite graphs with at most six distinct eigenvalues have the property that each vertex belongs to the constant number of quadrangles this enables to determine from the spectrum alone the feasible families of numbers of common neighbors for each vertex with other vertices in its part for particular spectra such as 6 29 06 29 6 where exponents denote eigenvalue multiplicities there is a unique such family which makes it possible to characterize all graphs with this spectrum using this lemma we also to show that for r 2 a graph has spectrum r rr r 1 02 r 1 rr r 1 r if and only if it is a graph of a 1 resolvable transversal design td r r i e if it corresponds to the complete set of mutually orthogonal latin squares of size r in a well defined manner

measure of nonlinearity for estimation nonlinearity among other factors is often the root cause of difficulties in nonlinear problems it is important to quantify a problem s degree of nonlinearity to decide a proper solution for example a full blown nonlinear filter is needed in general if the estimation problem is highly nonlinear but a quasi linear filter e g an extended kalman filter is sufficient for a weakly nonlinear case this paper first surveys various measures of nonlinearity mons for different applications for nonlinear estimation we conclude that these mons are not suitable and a better measure is needed in view of this we propose a general mon for estimation it measures the mean square closeness between a point and a subspace in a functional space properties and computation of this measure are studied numerical examples of static models for parameter estimation and dynamic models for process estimation are given to illustrate our measure

uncertainty in the weighted gap metric a geometric approach the stability of control systems is studied in the context of weighted input output signal spaces necessary and sufficient conditions for a controller to stabilize a plant are given in terms of geometric notions these geometric quantities can be calculated by solving h optimization problems maximally stabilizing controllers in a weighted signal space are introduced and characterized in terms of nehari extensions the robustness properties of maximally stabilizing controllers are analysed in terms of weighted coprime factor uncertainty necessary and sufficient conditions are established for a controller of a given plant to be the maximally stabilizing controller of the plant with respect to a weight an upper bound for the mixed sensitivity of a control system is given where the controller is the maximally stabilizing controller of the plant

estimating robustness on the riemann sphere abstract a simple presentation and derivation of the robustness properties of single input single output feedback systems with rational transfer functions are given frequency response plots in the complex plane are stereographically projected onto the riemann sphere here we adopt the chordal distance between two such projections as the distance between the two systems a metric is thus obtained for approximating stable and unstable systems this chordal metric is the basis of the proposed study of robustness several results in parallel with those obtained by zames and el sakkary 1980 are developed using elementary geometric properties of the plane and the sphere

distance measures for uncertain linear systems a general theory in this paper we propose a generic notion of distance between systems that can be used to measure discrepancy between open loop systems in a feedback sense under several uncertainty structures when the uncertainty structure is chosen to be four block or equivalently normalized coprime factor uncertainty then this generic distance measure reduces to the well known nu gap metric associated with this generic distance notion we also define a generic stability margin notion that allows us to give the distance measure a feedback interpretation by deriving generic robust stability and robust performance results the proposed distance notion and the corresponding results exploit a powerful generalization of the small gain theorem which handles perturbations in rfrl infin rather than only in rfrh infin when the uncertainty structure is fixed to one of the standard structures e g additive multiplicative inverse multiplicative coprime factor four block or any mixtures of the above we give a step by step procedure based on model validation ideas that shows how the generic notion of distance and the correspondingly generic winding number conditions can be reduced to simple formulae this work provides a unified framework that captures and embeds previous results in this area and also completes the picture by showing how other results of a similar nature can be obtained from the same framework the techniques used involve only basic linear algebra so they also provide a simplification of previous advanced proofs furthermore the various distance measures so created can be used for non conservative model embedding into the smallest uncertain family an illustrative example is also given that demonstrates the superior qualities above the nu gap metric of a particular distance measure obtained from this work in situations where the plant is lightly damped all systems considered in this paper are linear time invariant

linear feedback systems and the graph topology it is established for general linear systems that the gap metric induces the coarsest topology with respect to which both closed loop stability and closed loop performance are robust properties in earlier works similar topological results were obtained by exploiting the existence of particular coprime factor system representations not known to exist in general by contrast the development here does not rely on any specific system representations systems are simply characterized as subspaces of norm bounded input output pairs and the analysis hinges on the underlying geometric structure of the feedback stabilization problem unlike other work developed within such a framework fundamental issues concerning the causality of feedback interconnections are discussed explicitly the key result of this paper concerns the difference between linear feedback interconnections with identical controllers in terms of a performance robustness related closed loop mapping upper and lower bounds on the induced norm of this difference are derived allowing for possibly infinite dimensional input output spaces and time varying behavior the bounds are both proportional to the gap metric distance between the plants which clearly demonstrates the gap to be an appropriate measure of the difference between open loop systems from the perspective of closed loop behavior to conclude an example is presented to show that bounds of the form derived here for linear systems do not hold in a general nonlinear setting

which graphs are determined by their spectrum for almost all graphs the answer to the question in the title is still unknown here we survey the cases for which the answer is known not only the adjacency matrix but also other types of matrices such as the laplacian matrix are considered

combinatorial designs with two singular values ii partial geometric designs in this and an earlier paper e r van dam e spence combinatorial designs with two singular values i uniform multiplicative designs j combin theory a 107 2004 127 142 we study combinatorial designs whose incidence matrix has two distinct singular values these generalize v k designs and include uniform multiplicative designs and partial geometric designs here we study the latter which are precisely the designs with constant replication and block size we collect most known results give new characterization results and we enumerate partly by computer all small ones

relating the gap and graph metrics via the triangle inequality a bound relating a special class of constrained optimization problems to their unconstrained counterparts is presented an application of the results is a bound on the graph metric in terms of the computable gap metric

polynomial time low density parity check codes with rates very close to the capacity of the q ary random deletion channel for large q we demonstrate polynomial time deletion codes based on the verification based decoding paradigm that come arbitrarily close to capacity the random deletion channel takes n symbols from a q ary alphabet and each symbol is deleted independently with probability p taking advantage of recent improvements on the results of luby and mitzenmacher for verification based decoding by shokrollahi and wang we show how to design for any epsi 0 and sufficiently large n and q deletion codes with the following properties the rate is 1 p 1 epsi the failure probability is no 1 epsi2 q and the computational complexity for encoding and decoding is no 1 epsi2 log q we also extend these schemes to obtain the same results even if the undeleted symbols are also transposed arbitrarily and if a sufficiently small number of random symbols are inserted

gap metric robustness analysis of linear periodically time varying feedback systems quantitative robust stability results are established in this paper for feedback systems that evolve in continuous time and exhibit linear periodically time varying lptv behavior the results presented are analogous to results known to hold for linear time invariant lti systems although they do not follow directly from these system uncertainty is measured using the gap metric which quantifies the distance between systems in terms of the aperture between their graphs the subspaces corresponding to all input output pairs for each system the main robustness result characterizes the largest gap ball of lptv plants stabilized by a nominal lptv feedback controller known to stabilize a nominal lptv plant at the center of this ball a key step in the proof of this result makes use of a formula derived for the directed gap between lptv systems this formula is essentially a generalization of georgiou s for lti systems importantly all of the results presented apply to a class of sampled data control systems as a special case

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

guess check codes for deletions insertions and synchronization we consider the problem of constructing codes that can correct delta deletions occurring in an arbitrary binary string of length n bits varshamov tenengolts vt codes dating back to 1965 are zero error single deletion delta 1 correcting codes and have an asymptotically optimal redundancy finding similar codes for delta geq 2 deletions remains an open problem in this paper we relax the standard zero error i e worst case decoding requirement by assuming that the positions of the delta deletions or insertions are independent of the code word our contribution is a new family of explicit codes that we call guess check gc codes that can correct with high probability up to a constant number of delta deletions or insertions gc codes are systematic and have deterministic polynomial time encoding and decoding algorithms we also describe the application of gc codes to file synchronization

constructably laplacian integral graphs a graph is laplacian integral if the spectrum of its laplacian matrix consists entirely of integers we consider the class of constructably laplacian integral graphs those graphs that be constructed from an empty graph by adding a sequence of edges in such a way that each time a new edge is added the resulting graph is laplacian integral we characterize the constructably laplacian integral graphs in terms of certain forbidden vertex induced subgraphs and consider the number of nonisomorphic laplacian integral graphs that can be constructed by adding a suitable edge to a constructably laplacian integral graph we also discuss the eigenvalues of constructably laplacian integral graphs and identify families of isospectral nonisomorphic graphs within the class