on the representation of operators in bases of compactly supported wavelets this paper describes exact and explicit representations of the differential operators d n dx n n 1 2 cdots in orthonormal bases of compactly supported wavelets as well as the representations of the hilbert transform and fractional derivatives the method of computing these representations is directly applicable to multidimensional convolution operators also sparse representations of shift operators in orthonormal bases of compactly supported wavelets are discussed and a fast algorithm requiring o n log n operations for computing the wavelet coefficients of all n circulant shifts of a vector of the length n 2 n is constructed as an example of an application of this algorithm it is shown that the storage requirements of the fast algorithm for applying the standard form of a pseudodifferential operator to a vector see g beylkin r r coifman and v rokhlin comm pure appl math 44 1991 pp 141 183 may be reduced from o n to o log 2 n significant entries

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

a chebyshev collocation method for solving two phase flow stability problems this paper describes a chebyshev collocation method for solving the eigenvalue problem that governs the stability of parallel two phase flow the method is based on the expansion of the eigenfunctions in terms of chebyshev polynomials point collocation and the subsequent solution of the resulting generalized eigenvalue problem with the qz algorithm we concentrate on the question how to handle difficulties that arise when these standard techniques are applied to the stability problem of a thin film of liquid that is sheared by a gas after discussing this specific problem in detail it is argued that the method of solution can readily be applied to other two phase flow configurations as well

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

asymptotic error expansion of wavelet approximations of smooth functions ii we generalize earlier results concerning an asymptotic error expansion of wavelet approximations the properties of the monowavelets which are the building blocks for the error expansion are studied in more detail and connections between spline wavelets and euler and bernoulli polynomials are pointed out the expansion is used to compare the error for different wavelet families we prove that the leading terms of the expansion only depend on the multiresolution subspaces v j and not on how the complementary subspaces w j are chosen consequently for a fixed set of subspaces v j the leading terms do not depend on the fact whether the wavelets are orthogonal or not we also show that daubechies orthogonal wavelets need in general one level more than spline wavelets to obtain an approximation with a prescribed accuracy these results are illustrated with numerical examples

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

approximation accuracy analysis of fuzzy systems as function approximators this paper establishes the approximation error bounds for various classes of fuzzy systems i e fuzzy systems generated by different inferential and defuzzification methods based on these bounds the approximation accuracy of various classes of fuzzy systems is analyzed and compared it is seen that the class of fuzzy systems generated by the product inference and the center average defuzzifier has better approximation accuracy and properties than the class of fuzzy systems generated by the min inference and the center average defuzzifier and the class of fuzzy systems defuzzified by the mom defuzzifier in addition it is proved that fuzzy systems can represent any linear and multilinear function and explicit expressions of fuzzy systems generated by the mom defuzzified method are given

the compound matrix method for ordinary differential systems abstract a generalization of the compound matrix method is presented to deal with eigenvalue and boundary value problems involving unstable systems of ordinary differential equations details are given for fourth and sixth order problems next is shown that a simple equivalence relation exists between the compounds of the solution matrices of the eigenvalue problems and their adjoints and how this relation can be exploited to simplify the calculation of the adjoint eigenfunctions is discussed using the orr sommerfeld problem as an example it is also shown how the techniques used in the derivation of certain auxiliary systems which play a crucial role in the generalization of the compound matrix method can also provide an alternate method for the direct computation of certain quantities defined in terms of the eigenfunctions finally there is a brief discussion of the relationship between the compound matrix method anf the riccati method

a discontinuous galerkin conservative level set scheme for interface capturing in multiphase flows the accurate conservative level set acls method of desjardins et al o desjardins v moureau h pitsch an accurate conservative level set ghost fluid method for simulating turbulent atomization j comput phys 227 18 2008 8395 8416 is extended by using a discontinuous galerkin dg discretization dg allows for the scheme to have an arbitrarily high order of accuracy with the smallest possible computational stencil resulting in an accurate method with good parallel scaling this work includes a dg implementation of the level set transport equation which moves the level set with the flow field velocity and a dg implementation of the reinitialization equation which is used to maintain the shape of the level set profile to promote good mass conservation a near second order converging interface curvature is obtained by following a height function methodology common amongst volume of fluid schemes in the context of the conservative level set various numerical experiments are conducted to test the properties of the method and show excellent results even on coarse meshes the tests include zalesak s disk two dimensional deformation of a circle time evolution of a standing wave and a study of the kelvin helmholtz instability finally this novel methodology is employed to simulate the break up of a turbulent liquid jet

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

pseudospectra of the orr sommerfeld operator this paper investigates the pseudospectra and the numerical range of the orr sommerfeld operator for plane poiseuille flow a number z in bf c is in the epsilon pseudospectrum of a matrix or operator a if zi a 1 geq epsilon 1 or equivalently if z is in the spectrum of a e for some perturbation e satisfying e leq epsilon the numerical range of a is the set of numbers of the form au u where cdot cdot is the inner product and u is a vector or function with huh u 1 the spectrum of the orr sommerfeld operator consists of three branches it is shown that the eigenvalues at the intersection of the branches are highly sensitive to perturbations and that the sensitivity increases dramatically with the reynolds number the associated eigenfunctions are nearly linearly dependent even though they form a complete set to understand the high sensitivity of the eigenvalues a model operator is considered related to the airy equation tha

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

on the solution of differential equations with fuzzy spline wavelets fuzzy systems built with spline wavelets can approximate any function in a general hilbert space since wavelets are used extensively for the efficient solution of various types of differential equations it is demonstrated that fuzzy spline wavelets can be used for the solution of the same type of problems the advantage in using fuzzy spline wavelets for the solution of such problems is that the solution would enjoy the excellent numerical and computational characteristics of the fast wavelet transform while retaining the explanatory power of fuzzy system the method is demonstrated with the feedforward control of a flexible robotic arm

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

fuzzy rule based networks for control the authors present a method for learning fuzzy logic membership functions and rules to approximate a numerical function from a set of examples of the function s independent variables and the resulting function value this method uses a three step approach to building a complete function approximation system first learning the membership functions and creating a cell based rule representation second simplifying the cell based rules using an information theoretic approach for induction of rules from discrete valued data and finally constructing a computational neural network to compute the function value given its independent variables this function approximation system is demonstrated with a simple control example learning the truck and trailer backer upper control system

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

adaptive fuzzy wavelet network control design for nonlinear systems this paper presents a new adaptive fuzzy wavelet network controller a fwnc for control of nonlinear affine systems inspired by the theory of multiresolution analysis mra of wavelet transforms and fuzzy concepts the proposed adaptive gain controller which results from the direct adaptive approach has the ability to tune the adaptation parameter in the then part of each fuzzy rule during real time operation each fuzzy rule corresponds to a sub wavelet neural network sub wnn and one adaptation parameter each sub wnn consists of wavelets with a specified dilation value the degree of contribution of each sub wnn can be controlled flexibly orthogonal least square ols method is used to determine the number of fuzzy rules and to purify the wavelets for each sub wnn since the efficient procedure of selecting wavelets used in the ols method is not very sensitive to the input dimension the dimension of the approximated function does not cause the bottleneck for constructing fwn fwn is constructed based on the training data set of the nominal system and the constructed fuzzy rules can be adjusted by learning the translation parameters of the selected wavelets and also determining the shape of membership functions then the constructed adaptive fwn controller is employed such that the feedback linearization control input can be best approximated and the closed loop stability is guaranteed the performance of the proposed a fwnc is illustrated by applying a second order nonlinear inverted pendulum system and compared with previously published methods simulation results indicate the remarkable capabilities of the proposed control algorithm it is worth noting that the proposed controller significantly improves the transient response characteristics and the number of fuzzy rules and on line adjustable parameters are reduced

approximation power of biorthogonal wavelet expansions this paper looks at the effect of the number of vanishing moments on the approximation power of wavelet expansions the strang fix conditions imply that the error for an orthogonal wavelet approximation at scale a 2 sup i globally decays as a sup n where n is the order of the transform this is why for a given number of scales higher order wavelet transforms usually result in better signal approximations we prove that this result carries over for the general biorthogonal case and that the rate of decay of the error is determined by the order properties of the synthesis scaling function alone we also derive asymptotic error formulas and show that biorthogonal wavelet transforms are equivalent to their corresponding orthogonal projector as the scale goes to zero these results strengthen sweldens earlier analysis and confirm that the approximation power of biorthogonal and semi orthogonal wavelet expansions is essentially the same finally we compare the asymptotic performance of various wavelet transforms and briefly discuss the advantages of splines we also indicate how the smoothness of the basis functions is beneficial in reducing the approximation error

fuzzy solutions to partial differential equations adaptive approach a new technique using an adaptive fuzzy algorithm to obtain the solutions to a class of partial differential equations pdes is presented the design objective is to find a fuzzy solution to satisfy precisely the pdes with boundary conditions according to the adaptive scheme of fuzzy logic systems a fuzzy solution with adjustable parameters for the pde is first described then a set of adaptive laws for tuning the free parameters in the consequent part is derived from minimizing an appropriate error function in addition an elegant approximated error bound between the exact solution and the proposed fuzzy solution with respect to the number of fuzzy rules and solution errors has also been derived furthermore the convergence of error equations in mesh points is also discussed from the energy perspective in this paper we show that the proposed method can solve a variety of pdes encountered in engineering comparisons are also made with solutions obtained by the finite element method

numerical solution of eigenvalue problems using spectral techniques abstract two algorithms based on spectral chebyshev and pseudospectral chebyshev methods are presented for solving difficult eigenvalue problems that are valid over connected domains coupled through interfacial conditions to demonstrate the applicability of these methods we have examined the eigenvalue problems that describe the linear stability of two superposed newtonian and inelastic power law fluids in plane poiseuille flow for a selected range of parameters both algorithms provide accurate results and the pseudospectral code appears to be more efficient in handling linear stability problems

the wavelet transform time frequency localization and signal analysis two different procedures for effecting a frequency analysis of a time dependent signal locally in time are studied the first procedure is the short time or windowed fourier transform the second is the wavelet transform in which high frequency components are studied with sharper time resolution than low frequency components the similarities and the differences between these two methods are discussed for both schemes a detailed study is made of the reconstruction method and its stability as a function of the chosen time frequency density finally the notion of time frequency localization is made precise within this framework by two localization theorems

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

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

numerical solution of eigenvalue problem using the compound matrix method abstract an algorithm based on the compound matrix method is presented for solving difficult eigen value problems details are given for systems of ordinary differential equations of fourth order that are valid over connected domains coupled through interfacial conditions as an example we examine the linear stability of two superposed fluids in plane poiseuille flow and consider both interfacial and shear modes the orr sommerfeld system that describes linear stability is solved for a selected range of parameters

fuzzy spline interpolation with optimal property in parametric form in this paper we propose a new fuzzy interpolation procedure that uses cubic splines with an optimal property obtained in parametric form this fuzzy interpolation operator is also a linear operator of uniform approximation that preserves the monotonicity moreover by integrating this fuzzy cubic spline we obtain the trapezoidal quadrature rule for fuzzy henstock integrable functions

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