axisymmetric shape of gas bubbles in a uniform flow this paper investigates axisymmetric shapes of an incompressible inviscid fluid bubble in a uniform flow approximate solutions are constructed for the governing nonlinear boundary value problem using the domain perturbation method bifurcation and turning points are located by applying a special type of hermite pade approximation

duo rings and ore extensions abstract we show that in an ore polynomial ring every left ideal is two sided only in the trivial commutative case it is unknown whether the analogous statement about right ideals is true we establish a variety of restrictions on any putative counterexample

an intrinsic algebraic setting for poles and zeros of linear time varying systems in this paper poles and zeros are defined for linear time varying systems using suitable ground field extensions the definitions of the system poles transmission poles invariant zeros hidden modes etc are given in an intrinsic module based framework and are consistent in the sense that the poles are connected to the stability of the system and the zeros to the zeroing of the output for non zero inputs in particular it is proved that the necessary and sufficient condition for a continuous time system to be exponentially stable is similar to the well known condition in the time invariant case

skew and linearized reed solomon codes and maximum sum rank distance codes over any division ring abstract reed solomon codes and gabidulin codes have maximum hamming distance and maximum rank distance respectively a general construction using skew polynomials called skew reed solomon codes has already been introduced in the literature in this work we introduce a linearized version of such codes called linearized reed solomon codes we prove that they have maximum sum rank distance such distance is of interest in multishot network coding or in singleshot multi network coding to prove our result we introduce new metrics defined by skew polynomials which we call skew metrics we prove that skew reed solomon codes have maximum skew distance and then we translate this scenario to linearized reed solomon codes and the sum rank metric the theories of reed solomon codes and gabidulin codes are particular cases of our theory and the sum rank metric extends both the hamming and rank metrics we develop our theory over any division ring commutative or non commutative field we also consider non zero derivations which give new maximum rank distance codes over infinite fields not considered before

fixed structure synthesis of induced norm controllers this paper proposes a fixed structure technique for synthesizing controllers that are optimal with respect to various operator norms an optimal control problem is developed for each of these operator norms and necessary conditions for sub optimal performance are derived mixed norm optimal control problems are also formulated a continuation algorithm using quasi newton corrections is used to compute approximate solutions of the necessary conditions for a sequence of problems whose solutions approach an optimal controller optimal controllers with respect to each of the operator norms are synthesized for a 4th order mass spring dashpot system

l sub 2 optimal filter reduction a closed loop approach this paper is concerned with the problem of order reduction of a full order kalman filter for a stable linear signal model so that the steady state filtering error variance associated with the reduced order filter is minimized by an orthogonal parameterization the above problem is formulated to minimize the filtering error variance over a set of orthogonal matrices both continuous and iterative algorithms are derived to compute an optimal reduced order filter the algorithms are shown to possess good properties including the desirable convergence property the proposed algorithms are simple and effective numerical examples are presented to demonstrate the effectiveness and the significant advantages of the proposed algorithms over the existing open loop methods such as the well known balanced truncation method

on metrics for error correction in network coding the problem of error correction in both coherent and noncoherent network coding is considered under an adversarial model for coherent network coding where knowledge of the network topology and network code is assumed at the source and destination nodes the error correction capability of an outer code is succinctly described by the rank metric as a consequence it is shown that universal network error correcting codes achieving the singleton bound can be easily constructed and efficiently decoded for noncoherent network coding where knowledge of the network topology and network code is not assumed the error correction capability of a subspace code is given exactly by a new metric called the injection metric which is closely related to but different than the subspace metric of koumltter and kschischang in particular in the case of a non constant dimension code the decoder associated with the injection metric is shown to correct more errors then a minimum subspace distance decoder all of these results are based on a general approach to adversarial error correction which could be useful for other adversarial channels beyond network coding

all optimal hankel norm approximations of linear multivariable systems and their l error bounds the problem of approximating a multivariable transfer function g s of mcmillan degree n by s of mcmillan degree k is considered a complete characterization of all approximations that minimize the hankel norm is derived the solution involves a characterization of all rational functions s f s that minimize where s has mcmillan degree k and f s is anticavisal the solution to the latter problem is via results on balanced realizations all pass functions and the inertia of matrices all in terms of the solutions to lyapunov equations it is then shown that where k 1 g s is the k l st hankel singular value of g s and for one class of optimal hankel norm approximations the method is not computationally demanding and is applied to a 12 state model

universal and dynamic locally repairable codes with maximal recoverability via sum rank codes locally repairable codes lrcs are considered with equal or unequal localities local distances and local field sizes an explicit two layer architecture with a sum rank outer code is obtained having disjoint local groups and achieving maximal recoverability mr for all families of local linear codes mds or not simultaneously up to a specified maximum locality r furthermore the local linear codes thus the localities local distances and local fields can be efficiently and dynamically modified without global recoding or changes in architecture or outer code while preserving the mr property easily adapting to new configurations in storage or new hot and cold data in addition local groups and file components can be added removed or updated without global recoding the construction requires global fields of size roughly g r for g local groups and maximum or specified locality r for equal localities these global fields are smaller than those of previous mr lrcs when r leq h global parities for unequal localities they provide an exponential field size reduction on all previous best known mr lrcs for bounded localities and a large number of local groups the global erasure correction complexity of the given construction is comparable to that of tamo barg codes or reed solomon codes with local replication while local repair is as efficient as for the cartesian product of the local codes reed solomon codes with local replication and cartesian products are recovered from the given construction when r 1 and h 0 respectively the given construction can also be adapted to provide hierarchical mr lrcs for all types of hierarchies and parameters finally subextension subcodes and sum rank alternant codes are introduced to obtain further exponential field size reductions at the expense of lower information rates

on the similarities between generalized rank and hamming weights and their applications to network coding rank weights and generalized rank weights have been proved to characterize error and erasure correction and information leakage in linear network coding in the same way as hamming weights and generalized hamming weights describe classical error and erasure correction and information leakage in wire tap channels of type ii and code based secret sharing although many similarities between both the cases have been established and proved in the literature many other known results in the hamming case such as bounds or characterizations of weight preserving maps have not been translated to the rank case yet or in some cases have been proved after developing a different machinery the aim of this paper is to further relate both weights and generalized weights show that the results and proofs in both cases are usually essentially the same and see the significance of these similarities in network coding some of the new results in the rank case also have new consequences in the hamming case

principal component analysis in linear systems controllability observability and model reduction kalman s minimal realization theory involves geometric objects controllable unobservable subspaces which are subject to structural instability specifically arbitrarily small perturbations in a model may cause a change in the dimensions of the associated subspaces this situation is manifested in computational difficulties which arise in attempts to apply textbook algorithms for computing a minimal realization structural instability associated with geometric theories is not unique to control it arises in the theory of linear equations as well in this setting the computational problems have been studied for decades and excellent tools have been developed for coping with the situation one of the main goals of this paper is to call attention to principal component analysis hotelling 1933 and an algorithm golub and reinsch 1970 for computing the singular value decompositon of a matrix together they form a powerful tool for coping with structural instability in dynamic systems as developed in this paper principal component analysis is a technique for analyzing signals singular value decomposition provides the computational machinery for this reason kalman s minimal realization theory is recast in terms of responses to injected signals application of the signal analysis to controllability and observability leads to a coordinate system in which the internally balanced model has special properties for asymptotically stable systems this yields working approximations of x c x bar o the controllable and unobservable subspaces it is proposed that a natural first step in model reduction is to apply the mechanics of minimal realization using these working subspaces

brief paper positivity preserving h model reduction for positive systems this paper is concerned with the model reduction of positive systems for a given stable positive system our attention is focused on the construction of a reduced order model in such a way that the positivity of the original system is preserved and the error system is stable with a prescribed h performance based upon a system augmentation approach a novel characterization on the stability with h performance of the error system is first obtained in terms of linear matrix inequality lmi then a necessary and sufficient condition for the existence of a desired reduced order model is derived accordingly furthermore iterative lmi approaches with primal and dual forms are developed to solve the positivity preserving h model reduction problem finally a compartmental network is provided to show the effectiveness of the proposed techniques

multiobjective h sub 2 h sub spl infin control for a linear time invariant system with several disturbance inputs and controlled outputs we show how to minimize the nominal h sub 2 norm performance in one channel while keeping bounds on the h sub 2 norm or h sub spl infin norm performance implying robust stability in the other channels this multiobjective h sub 2 h sub spl infin problem in an infinite dimensional space is reduced to sequences of finite dimensional convex optimization problems we show how to compute the optimal value and how to numerically detect the existence of a rational optimal controller if it exists we reveal how the novel trick of optimizing the trace norm of the youla parameter over certain convex constraints allows one to design a nearly optimal controller whose youla parameter is of the same order as the optimal one

codes from unit groups of division algebras over number fields lenstra and guruswami described number field analogues of the algebraic geometry codes of goppa recently the first author and oggier generalised these constructions to other arithmetic groups unit groups in number fields and orders in division algebras they suggested to use unit groups in quaternion algebras but could not completely analyse the resulting codes we prove that the noncommutative unit group construction yields asymptotically good families of codes for the sum rank metric from division algebras of any degree and we estimate the size of the alphabet in terms of the degree

convolutional codes with maximum column sum rank for network streaming the column hamming distance of a convolutional code determines the error correction capability when streaming over a class of packet erasure channels we introduce a metric known as the column sum rank that parallels the column hamming distance when streaming over a network with link failures we prove the rank analogues of several known column hamming distance properties and introduce a new family of convolutional codes that maximize the column sum rank up to the code memory our construction involves finding a class of super regular matrices that preserve this property after multiplication with non singular block diagonal matrices in the ground field

evaluation and interpolation over multivariate skew polynomial rings abstract the concepts of evaluation and interpolation are extended from univariate skew polynomials to multivariate skew polynomials with coefficients over division rings iterated skew polynomial rings are in general not suitable for this purpose instead multivariate skew polynomial rings are constructed in this work as follows first free multivariate skew polynomial rings are defined where multiplication is additive on degrees and restricts to concatenation for monomials this allows to define the evaluation of any skew polynomial at any point by unique remainder division multivariate skew polynomial rings are then defined as the quotient of the free ring by two sided ideals that vanish at every point the main objectives and results of this work are descriptions of the sets of zeros of these multivariate skew polynomials the families of functions that such skew polynomials define and how to perform lagrange interpolation with them to obtain these descriptions the existing concepts of p closed sets p independence p bases which are shown to form a matroid and skew vandermonde matrices are extended from the univariate case to the multivariate one

bilinear forms on frobenius algebras abstract we analyze the homothety types of associative bilinear forms that can occur on a hopf algebra or on a local frobenius k algebra r with residue field k if r is symmetric then there exists a unique form on r up to homothety iff r is commutative if r is frobenius then we introduce a norm based on the nakayama automorphism of r we show that if two forms on r are homothetic then the norm of the unit separating them is central and we conjecture the converse we show that if the dimension of r is even then the determinant of a form on r taken in k k 2 is an invariant for r

writing representations over proper division subrings abstract let e be a division ring and g a finite group of automorphisms of e whose elements are distinct modulo inner automorphisms of e let f e g be the division subring of elements of e fixed by g given a representation a e d d of an f algebra a we give necessary and sufficient conditions for to be writable over f here e d d denotes the algebra of d d matrices over e and a matrix a writes over f if a 1 a a f d d we give an algorithm for constructing an a or proving that no a exists the case of particular interest to us is when e is a field and is absolutely irreducible the algorithm relies on an explicit formula for a and a generalization of hilbert s theorem 90 that arises in galois cohomology the algorithm has applications to the construction of absolutely irreducible group representations especially for solvable groups and to the recognition of class c 5 in aschbacher s matrix group classification scheme m aschbacher on the maximal subgroups of the finite classical groups invent math 76 1984 469 514 mr0746539 shangzhi li on the subgroup structure of classical groups in group theory in china in math appl china ser vol 365 kluwer acad publ dordrecht 1996 pp 70 90 mr1447199 1 13

formulation and solution of structured total least norm problems for parameter estimation the total least squares tls method is a generalization of the least squares ls method for solving overdetermined sets of linear equations ax spl ap b the tls method minimizes spl par e r spl par sub f where r b a e x so that b r spl isin range a e given a spl isin c sup m spl times n with m spl ges n and b spl isin c sup m spl times 1 the most common tls algorithm is based on the singular value decomposition svd of a b however the svd based methods may not be appropriate when the matrix a has a special structure since they do not preserve the structure previously a new problem formulation known as structured total least norm stln and the algorithm for computing the stln solution have been developed the stln method preserves the special structure of a or a b and can minimize the error in the discrete l sub p norm where p 1 2 or spl infin in this paper the stln problem formulation is generalized for computing the solution of stln problems with multiple right hand sides ax spl ap b it is shown that these problems can be converted to ordinary stln problems with one right hand side in addition the method is shown to converge to the optimal solution in certain model reduction problems furthermore the application of the stln method to various parameter estimation problems is studied in which the computed correction matrix applied to a or a b keeps the same toeplitz structure as the data matrix a of a b respectively in particular the l sub 2 norm stln method is compared with the ls and tls methods in deconvolution transfer function modeling and linear prediction problems

vector orthogonal polynomials abstract the following well known framework will be applied to hermite pade approximants first the hermite pade type approximants will be defined from the error of approximation some supplementary conditions will be fixed in order to obtain hermite pade forms vector orthogonal polynomials will be deduced from these conditions their existence and uniqueness are studied in the normal case these results are ensured then some properties of vector orthogonal polynomials will be proved

approximants de pade hermite 2eme partie programmation in this second paper we propose an algorithm for the construction of the tables and of the pade hermite tables 3 then we present some numerical applications

model reduction via balanced state space representations a model reduction procedure based on balanced state space representations is studied in this paper the reduced order model is examined from the point of view of stability controllability and observability both continuous time and discrete time systems are considered

approximants de pade hermite 1ere partie theorie we present in this first paper a generalization of pade approximants which gives us as particular cases shafer s and baker sd log approximants first we define these approximants following an old idea of hermite then we prove some fundamental properties for their constructions

brief paper on the stability of a reduced order filter based on dominant singular value decomposition of the system dynamics the stability of a reduced order filter rof the gain of which is constructed on the basis of a subspace of dominant singular vectors of the system dynamics is examined a definition of s detectability is introduced it is found that the observability of all unstable and neutral singular vectors s detectability is a sufficient condition for the existence of a stable filter

hankel norm approximation for well posed linear systems the sub optimal hankel norm approximation problem is solved for a well posed linear system with generating operators a b c and transfer function g satisfying some mild assumptions in the special case of the sub optimal nehari problem an explicit parameterization of all solutions is obtained in terms of the system parameters a b c and g 0

model reduction for state space symmetric systems in this paper the model reduction problem for state space symmetric systems is investigated first it is shown that several model reduction methods such as balanced truncation balanced truncation which preserves the dc gain optimal and suboptimal hankel norm approximations inherit the state space symmetric property furthermore for single input and single output siso state space symmetric systems we prove that the h norm of its transfer functions can be calculated via two simple formulas moreover the siso state space symmetric systems are equivalent to systems with zeros interlacing the poles zip under mild conditions

equivalence and characterizations of linear rank metric codes based on invariants abstract we show that the sequence of dimensions of the linear spaces generated by a given rank metric code together with itself under several applications of a field automorphism is an invariant for the whole equivalence class of the code the same property is proven for the sequence of dimensions of the intersections of itself under several applications of a field automorphism these invariants give rise to easily computable criteria to check if two codes are inequivalent we derive some concrete values and bounds for these dimension sequences for some known families of rank metric codes namely gabidulin and generalized twisted gabidulin codes we then derive conditions on the length of the codes with respect to the field extension degree such that codes from different families cannot be equivalent furthermore we derive upper and lower bounds on the number of equivalence classes of gabidulin codes and twisted gabidulin codes improving a result of schmidt and zhou for a wider range of parameters in the end we use the aforementioned sequences to determine a characterization result for gabidulin codes

directional interpolation approach to h optimization and robust stabilization it is shown that under a mild condition h infty optimization and robust stabilization are equivalent to a directional interpolation problem which is a matrix extension of the classical pick nevanlinna interpolation problem a classical iterative method which is an extension of the schur nevanlinna algorithm is given for solving the problem this method does not require the inner outer factorization nor the balanced realization of the original plant a circuit theoretical parameterization of all solutions is derived that is expected to enhance the physical insight to the h infty optimal control and robust stabilization this parameterization has the degree much less than the one obtained previously

vandermonde and wronskian matrices over division rings in l the first author initiated the study of vandermonde matrices with entries over a division ring k even in the case when k is a field the definition of a vandermonde matrix in l is already more general than that in the classical sense as it takes into account a given endomorphism s of k under the epilogue of l it was further pointed out that the results of that paper remain valid if in the definition of a vandermonde matrix one allows for a given s derivation d and uses the appropriate definition of the power functions in the s d setting such a generalization is worthwhile because in this setting the study of vandermonde matrices over k also encompasses the study of wronskian matrices with respect to d so results on vandermonde matrices may be used to study linear differential equations arising from the operator d more generally vandermonde matrices seem to be an important tool in studying the skew polynomial ring k t s d so a deeper understanding of vandermonde matrices should be important to the study of the arithmetic of a division ring the purpose of this paper is threefold first we shall develop under sec tion 2 the basic facts on skew polynomials in the s d setting define the evaluation of such polynomials and prove the all important product theorem 2 7 second we shall give a general computation of the rank of a vandermonde matrix with respect to s d as in l the computation is first reduced to the case when the elements a a used to build the vandermonde matrix are pairwise s d conjugate in this case the rank of the vandermonde matrix is computed by the dimension of a vector space over a certain division subring of k cf theorem 4 5 last under

convolution and hankel operator norms for linear systems some norms are derived for convolution and hankel operators associated with linear time invariant systems in certain cases these norms are shown to be identical the tightest possible bound has been obtained for the absolute magnitude of the euclidean 2 or infinity norm of the time domain response of a multioutput system to certain classes of input disturbance

universal secure network coding via rank metric codes the problem of securing a network coding communication system against an eavesdropper is considered the network implements linear network coding to deliver n packets from source to each receiver and the adversary can eavesdrop on arbitrarily chosen links the objective is to provide reliable communication to all receivers while guaranteeing that the source information remains information theoretically secure from the adversary a coding scheme is proposed that can achieve the maximum possible rate of n packets the scheme which is based on rank metric codes has the distinctive property of being universal it can be applied on top of any communication network without requiring knowledge of or any modifications on the underlying linear network code the only requirement of the scheme is that the packet length be at least n which is shown to be strictly necessary for universal communication at the maximum rate a further scenario is considered where the adversary is allowed not only to eavesdrop but also to inject up to t erroneous packets into the network and the network may suffer from a rank deficiency of at most in this case the proposed scheme can be extended to achieve the rate of n 2t packets this rate is shown to be optimal under the assumption of zero error communication

on the theory of space time codes for psk modulation the design of space time codes to achieve full spatial diversity over fading channels has largely been addressed by handcrafting example codes using computer search methods and only for small numbers of antennas the lack of more general designs is in part due to the fact that the diversity advantage of a code is the minimum rank among the complex baseband differences between modulated codewords which is difficult to relate to traditional code designs over finite fields and rings we present general binary design criteria for psk modulated space time codes for linear bpsk qpsk codes the rank of binary projections of the unmodulated codewords as binary matrices over the binary field is a sufficient design criterion full binary rank guarantees full spatial diversity this criterion accounts for much of what is currently known about psk modulated space time codes we develop new fundamental code constructions for both quasi static and time varying channels these are perhaps the first general constructions other than delay diversity schemes that guarantee full spatial diversity for an arbitrary number of transmit antennas

hilbert 90 theorems over division rings hubert s satz 90 is well known for cyclic extensions of fields but attempts at generalizations to the case of division rings have only been partly successful jacobson s criterion for logarithmic derivatives for fields equipped with derivations is formally an analogue of satz 90 but the exact relationship between the two was apparently not known in this paper we study triples k s d where s is an endomorphism of the division ring k and d is an s derivation using the technique of ore extensions k t s d we char acterize the notion of 5 d algebraicity for elements a e k and give an effective criterion for two elements a b k to be s conjugate in the case when the s z conjugacy class of a is algebraic this criterion amounts to a general hubert 90 theorem for division rings in the k s z setting sub suming and extending all known forms of hubert 90 in the literature including the aforementioned jacobson criterion two of the working tools used in the paper the conjugation theorem 2 2 and the composite function theorem 2 3 are of independent interest in the theory of ore extensions

convolutional codes in rank metric with application to random network coding random network coding recently attracts attention as a technique to disseminate information in a network this paper considers a noncoherent multishot network where the unknown and time variant network is used several times in order to create dependence between the different shots particular convolutional codes in rank metric are used these codes are so called partial unit memory p um codes i e convolutional codes with memory one first distance measures for convolutional codes in rank metric are shown and two constructions of p um codes in rank metric based on the generator matrices of maximum rank distance codes are presented second an efficient error erasure decoding algorithm for these codes is presented its guaranteed decoding radius is derived and its complexity is bounded finally it is shown how to apply these codes for error correction in random linear and affine network coding

a new perspective of cyclicity in convolutional codes in this paper we propose a new way of providing cyclic structures to convolutional codes we define the skew cyclic convolutional codes as left ideals of a quotient ring of a suitable non commutative polynomial ring in contrast to the previous approaches to cyclicity for convolutional codes we use ore polynomials with coefficients in a field the rational function field over a finite field so their arithmetic is very well known and we may proceed similarly to cyclic block codes in particular we show how to obtain easily skew cyclic convolutional codes of a given dimension and we compute an idempotent generator of the code and its dual