on the complexity of algebraic numbers i expansions in integer bases let b 2 be an integer we prove that the b ary expansion of every irrational algebraic number cannot have low complexity furthermore we establish that irrational morphic numbers are transcendental for a wide class of morphisms in particular irrational automatic numbers are transcendental our main tool is a new combinatorial transcendence criterion

on the input output distinguishability of single output continuous linear time invariant systems this note addresses the distinguishability problem for continuous linear time invariant systems using their input and output data the proposed approach is based on the characterization of the class of initial state vectors and control input signals that make the outputs of different continuous linear time invariant systems indistinguishable this class is defined here as the indistinguishability zone from this characterization a simple necessary and sufficient condition for the distinguishability of single input single output systems is established furthermore it is shown that multiple input single output systems are never distinguishable

a method for partial fraction expansion of transfer matrices a general method for partial fraction expansion of transfer matrices is presented and a general formula is given the main ideas of the procedure include 1 no required derivatives for the case of repeated eigenvalues 2 all the matrix residues are found at the same time and 3 there is no need for knowledge of the minimal polynomial the procedure can be easily applied to digital computer application and fills a void in the computer method for finding the partial fraction expansion in the case of repeated eigenvalues the results are presented in structured form using kronecker products in a way that facilitates easy implementation using a variety of standard software packages

the determinants of matrices with recursive entries abstract we prove a result concerning the evaluation of determinant of a matrix whose entries are given by certain linear inhomogeneous recurrence relations this result generalizes previous determinants evaluations due to krattenthaler neuwirth and the authors in c krattenthaler evaluations of some determinants of matrices related to the pascal triangle sem lothar combin 47 2002 19 article b47g e neuwirth treeway galton arrays and generalized pascal like determinants technical report university of vienna 2004 pp 1 16 a r moghaddamfar s m h pooya s navid salehy s nima salehy evaluating and generalizing certain determinants submitted for publication the main tool for proving our result is lu factorization of the recursive arrays introduced in c krattenthaler advanced determinant calculus a complement linear algebra appl 411 2005 68 166

the model matching problem for periodic discrete time systems in this paper the model matching problem via state feedback is studied for discrete time periodic systems it is shown that under some zero matching conditions it is possible to assign a target input output periodic system by means of a periodic state feedback control law furthermore this entails that a periodic system can be converted into a closed loop time invariant system via state feedback

on the rapid computation of various polylogarithmic constants we give algorithms for the computation of the d th digit of certain transcendental numbers in various bases these algorithms can be easily implemented multiple precision arithmetic is not needed require virtually no memory and feature run times that scale nearly linearly with the order of the digit desired they make it feasible to compute for example the billionth binary digit of log 2 or on a modest work station in a few hours run time

on the automaticity of the hankel determinants of a family of automatic sequences abstract hankel determinants and automatic sequences are two classical subjects widely studied in mathematics and theoretical computer science however these two topics were considered totally independently until in 1998 when allouche peyriere wen and wen proved that all the hankel determinants of the thue morse sequence are nonzero this property allowed bugeaud to prove that the irrationality exponents of the thue morse mahler numbers are exactly 2 since then the hankel determinants of several other automatic sequences in particular the paperfolding sequence the stern sequence the period doubling sequence are studied by coons vrbik guo wu wen bugeaud fu han fokkink kraaikamp and shallit on the other hand it is known that the hankel determinants of a rational power series are ultimately zero and the hankel determinants of a quadratic power series over finite fields are ultimately periodic it is therefore natural to ask if we can obtain similar results about the hankel determinants of algebraic series in the present paper we provide a partial answer to this question by establishing the automaticity of the reduced hankel determinants modulo 2 of a family of automatic sequences as an application of our result we give upper bounds for the irrationality exponent of a family of automatic numbers

conditions for strict distinguishability of single output nonlinear control affine systems abstract strict distinguishability is the property of two systems to generate identical output signals only for the zero control input and the zero initial state vector the objective of this paper is to characterize this property for single output nonlinear control affine systems in this context we first introduce the indistinguishability zone of the systems this zone represents the set of control inputs and initial state vectors that make the output signals of the systems indistinguishable the explicit identification of the elements of this set is proposed in the paper from this result necessary and sufficient conditions for strict distinguishability are then established under suitable assumptions

elliptic hypergeometric series on root systems abstract we derive a number of summation and transformation formulas for elliptic hypergeometric series on the root systems a n c n and d n in the special cases of classical and q series our approach leads to new elementary proofs of the corresponding identities

a generalized ordered bell polynomial abstract in this paper we consider a generalized ordered bell polynomial p n q defined by the following exponential generating function n 0 p n q n t n e t q q e t 1 using the method of exponential riordan arrays and orthogonal polynomials we give the jacobi continued fraction expansion of n 0 p n q t n then we get the q stieltjes moment property strong q log convexity 3 q log convexity and hankel determinants of p n q n 0

hankel transform of a series reversion of a certain rational function abstract starting from a sequence v n n n 0 defined by the series reversion of a certain rational function b x we use the method based on orthogonal polynomials to give closed form evaluation of the hankel transform of sequence v n n n 0 and corresponding shifted sequences that gives a proof the conjecture by barry 4 concerning the ratio equalities connecting those terms moreover in a special case we compute lu decomposition of the hankel matrix of a shifted sequence v n n n 0 which reproves the expression for the hankel transform of that sequence

riordan arrays generalized narayana triangles and series reversion abstract using elements of the group of riordan arrays we define a family of generalized narayana triangles and their associated generalized catalan numbers and study their links to series reversion in particular we use lagrange inversion techniques to determine the generating functions for these generalized catalan numbers

decomposition of some hankel matrices generated by the generalized rencontres polynomials abstract using the algebraic techniques of formal series we obtain a combinatorial decomposition of some matrices generated by the generalized rencontres polynomials specifically for the complete matrix and the hankel matrix we obtain an ldu decomposition r d s t where d is a diagonal matrix and r and s are two sheffer matrices then for the shifted matrices we obtain an ltu decomposition r t s t where t is a tridiagonal matrix and r and s are two sheffer matrices furthermore we give an algebraic characterization of the generalized rencontres numbers and polynomials in terms of hankel determinants finally we determine a relation between the complete matrices and the hankel matrices

binomial determinants paths and hook length formulae abstract we give a combinatorial interpretation for any minor or binomial determinant of the matrix of binomial coefficients this interpretation involves configurations of nonintersecting paths and is related to young tableaux and hook length formulae

fault detection of linear discrete time periodic systems in this note an approach to the design of optimal fault detection systems for linear discrete time periodic systems is proposed which leads to an optimized compromise between robustness to unknown disturbances and sensitivity to faults the needed computation mainly consists in solving a difference periodic riccati system the proposed approach is finally illustrated by a numerical example

model matching with strong stability in switched linear systems abstract this work deals with model matching by output dynamic feedback in switched linear systems the plant and the model are assumed to be subject to different switching signals a necessary and sufficient condition for achieving model matching with asymptotic stability of both the closed loop compensated system and the compensator for a sufficiently large dwell time is proven such condition is obtained by specializing the structural condition with a suitable redefinition of the robust controlled invariant subspace involved capable of capturing not only the structural aspect of the problem but also the stability aspects the effect of the combined action of nonzero initial states and nonzero inputs is dealt with the solution to a more demanding problem formulation where the dwell time is assumed to be fixed and given is also discussed

hankel hyperdeterminants and selberg integrals we investigate the simplest class of hyperdeterminants defined by cayley in the case of hankel hypermatrices tensors of the form ai1i2 ik f i1 i2 ik it is found that many classical properties of hankel determinants can be generalized and a connection with selberg type integrals is established in particular selberg s original formula amounts to the evaluation of all hankel hyperdeterminants built from the moments of the jacobi polynomials many higher dimensional analogues of classical hankel determinants are evaluated in closed form the toeplitz case is also briefly discussed in physical terms both cases are related to the partition functions of one dimensional coulomb systems with logarithmic potential

diophantine approximation and cantor sets we provide an explicit construction of elements of the middle third cantor set with any prescribed irrationality exponent this answers a question posed by kurt mahler

power series expansion of a hankel determinant abstract in this article we find a power series expansion for the hankel determinant whose entries are the moments of a linear functional related to discrete semiclassical orthogonal polynomials we provide explicit formulas that allow the computation of coefficients to arbitrary order and give examples for the first few terms in the series we reinterpret these results in the context of the theory of young tableaux and schur functions and find closed form expressions for the cases corresponding to the charlier and meixner polynomials we also discuss further research directions

survey invariant representations of discrete time periodic systems the correspondence between periodic and time invariant systems can take a variety of forms the purpose of this survey paper is to present a complete and clear picture on time invariant representations for discrete time periodic systems the core of the treatment evolves around the concept of periodic transfer function

an algorithmic proof theory for hypergeometric ordinary and q multisum integral identities it is shown that every proper hypergeometric multisum integral identity orq identity with a fixed number of summations and or integration signs possesses a short computer constructible proof we give a fast algorithm for finding such proofs most of the identities that involve the classical special functions of mathematical physics are readily reducible to the kind of identities treated here we give many examples of the method including computer generated proofs of identities of mehta dyson selberg hille hardy q saalschutz and others the prospect of using the method for proving multivariate identities that involve an arbitrary number of summations integrations is discussed

automaticity of the hankel determinants of difference sequences of the thue morse sequence extending the result of allouche et al we prove that for any k the two dimensional sequence modulo 2 k t np n p 0 is 2 automatic moreover we prove that the three dimensional sequence modulo 2 k t np n p k 0 is 2 automatic

fixed zeros in the model matching problem for systems over semirings in this paper solution existence conditions for the model matching problem are studied for systems over semirings which are used in many applications such as queueing systems communication networks and manufacturing systems the main contribution is the discovery of fixed pole structure in solutions to the model matching problem this fixed pole structure provides essential information contained in all the solutions to the model matching problem for a discrete event dynamic system example a common petri net component in the solutions of the model matching problem can be discovered from the fixed pole structure

multiply refined enumeration of alternating sign matrices four natural boundary statistics and two natural bulk statistics are considered for alternating sign matrices asms specifically these statistics are the positions of the 1 s in the first and last row and column of an asm and the numbers of generalized inversions and 1 s in an asm previously known results for the exact enumeration of asms with prescribed values of some of these statistics are reviewed in detail a quadratic relation which fully determines the generating function associated with all six statistics is then obtained the derivation of the relation involves combining the desnanot jacobi determinant identity with the izergin korepin formula for the partition function of the six vertex model with domain wall boundary conditions

the realization problem for linear periodic systems abstract the realization problem for discrete time linear periodic systems is considered deep insight on the various complicated aspects of minimal quasi minimal and uniform realization is provided based on the knowledge of the structural properties of discrete time periodic systems the results hinge on the lifting isomorphism between such systems and time invariant ones

factoring polynomials with rational coefficients in this paper we present a polynomial time algorithm to solve the following problem given a non zero polynomial fe q x in one variable with rational coefficients find the decomposition of f into irreducible factors in q x it is well known that this is equivalent to factoring primitive polynomials fez x into irreducible factors in z x here we call f z x primitive if the greatest common divisor of its coefficients the content of f is 1 our algorithm performs well in practice cf 8 its running time measured in bit operations is o nl2 n9 log fd3

complete integrability of relativistic calogero moser systems and elliptic function identities poincare invariant generalizations of the galilei invariant calogero moser a particle systems are studied a quantization of the classical integrals s1 sn is presented such that the operators si9 sn mutually commute as a corollary it follows that si9 9sn poisson commute these results hinge on functional equations satisfied by the weierstrass and 0 functions a generalized cauchy identity involving the function leads to an n x n matrix l whose symmetric functions are proportional to sl5 9sn

some composition determinants abstract we compute several parametric determinants in which rows and columns are indexed by compositions where the entries are either products of binomial coefficients or products of powers these results generalize previous determinant evaluations due to the first and fourth author siam j matrix anal appl 23 2001 459 471 and a polynomial generalization of the power compositions determinant linear multilinear algebra in press and they prove two conjectures of the second author advanced determinant calculus a complement preliminary version

symmetry classes of alternating sign matrices under one roof in a previous article math co 9712207 we derived the alternating sign matrix asm theorem from the izergin korepin determinant for a partition function for square ice with domain wall boundary here we show that the same argument enumerates three other symmetry classes of alternating sign matrices vsasms vertically symmetric asms even htsasms half turn symmetric asms and even qtsasms quarter turn symmetric asms the vsasm enumeration was conjectured by mills the others by robbins math co 0008045 we introduce several new types of asms uasms asms with a u turn side uuasms two u turn sides osasms off diagonally symmetric asms oosasms off diagonally off antidiagonally symmetric and uosasms off diagonally symmetric with u turn sides uasms generalize vsasms while uuasms generalize vhsasms vertically and horizontally symmetric asms and another new class vhpasms vertically and horizontally perverse osasms oosasms and uosasms are related to the remaining symmetry classes of asms namely dsasms diagonally symmetric dasasms diagonally anti diagonally symmetric and tsasms totally symmetric asms we enumerate several of these new classes and we provide several 2 enumerations and 3 enumerations our main technical tool is a set of multi parameter determinant and pfaffian formulas generalizing the izergin korepin determinant for asms and the tsuchiya determinant for uasms solv int 9804010 we evaluate specializations of the determinants and pfaffians using the factor exhaustion method

zeros of discrete time linear periodic systems in this note the concept of zero for linear discrete time siso periodic systems is discussed the definition is based on a time invariant mimo equivalent representation an input output characterization of the zeros is provided which extends a well known property for time invariant systems

a new shift invariant representation for periodic linear systems abstract this paper presents a new family of shift invariant representations for periodic linear discrete time systems this representation has the advantages over the standard representation 5 of preserving the time step of the original system and of preserving causality under arbitrary feedback the disadvantage is that the realization is not minimal an application in attitude determination is presented

nonintersecting paths pfaffians and plane partitions gessel and viennot have developed a powerful technique for enumerating various classes of plane partitions the purpose of this article is to show by similar means that one may use pfaffiants to enumerate configurations of nonintersecting paths in which the initial and or terminal vertices of the paths are allowed to vary over specified regions of the digraph this leads to the possibility of enumerating classes of plane partitions in which the shape is allowed to vary whereas the previous applications of gessel and viennot were largely confined to plane partitions of a given shape

exact solution of the lock step model of vicious walkers the lock step model of vicious walkers on a one dimensional lattice allows each walker at the tick of a clock to move either one lattice site to the left or one lattice site to the right the only restriction is that no two walkers may arrive at the same lattice site or pass one another in periodic boundary conditions the partition function and correlation function for this model are calculated exactly taking the continuum limit gives an exactly solvable model of vicious walkers undergoing brownian motion

a direct approach to h sub 2 optimal deconvolution of periodic digital channels this paper studies the h sub 2 optimal deconvolution problem for periodic finite impulse response fir and infinite impulse response iir channels it shows that the h sub 2 norm of a periodic filter can be directly quantified in terms of periodic system matrices and linear matrix inequalities lmis without resorting to the commonly used lifting technique the optimal signal reconstruction problem is then formulated as an optimization problem subject to a set of matrix inequality constraints under this framework the optimization of both the fir and iir periodic deconvolution filters can be made convex solved using the interior point method and computed by using the matlab lmi toolbox the robust deconvolution problem for periodic fir and iir channels with polytopic uncertainties are further formulated and solved also by convex optimization and the lmis compared with the lifting approach to the design of periodic filters the proposed approach is simpler yet more powerful in dealing with multiobjective deconvolution problems and channel uncertainties especially for iir deconvolution filter design the obtained solutions are applied to the design of an optimal filterbank yielding satisfactory performance

some extensions of brioschi s double alternant abstract we evaluate some determinants related to brioschi s extension of the classical double alternant of cauchy and scott s extension of brioschi s

on the expansion of some exponential periods in an integer base we derive a lower bound for the subword complexity of the base b expansion b 2 of all real numbers whose irrationality exponent is equal to 2 this provides a generalization of a theorem due to ferenczi and mauduit as a consequence we obtain the first lower bound for the subword complexity of the number e and of some other transcendental exponential periods

on the irrationality exponent of the regular paperfolding numbers abstract in this paper improving the method of allouche et al we calculate the hankel determinant of the regular paperfolding sequence and prove that the hankel determinant sequence modulo 2 is periodic with period 10 which answers coon s conjecture then we extend bugeaud s method to obatin the exact value of the irrationality exponent for some general transcendental numbers using the results above we prove that the irrationality exponents of the regular paperfolding numbers are exactly 2

confluent q extensions of some classical determinants abstract we evaluate two determinants the first is a q h extension of the classical confluent extension of the vandermonde determinant the second is a similar extension of cauchy s double alternant

advanced determinant calculus a complement abstract this is a complement to my previous article advanced determinant calculus c krattenthaler advanced determinant calculus seminaire lotharingien combin 42 1999 the andrews festschrift article b42q 67 pp in the present article i share with the reader my experience of applying the methods described in the previous article in order to solve a particular problem from number theory g almkvist c krattenthaler j petersson some new formulas for experiment math 12 2003 441 456 moreover i add a list of determinant evaluations which i consider as interesting which have been found since the appearance of the previous article or which i failed to mention there including several conjectures and open problems

ruijsenaars commuting difference operators as commuting transfer matrices for belavin s elliptic quantum r matrix we construct an l operator as a set of difference operators acting on functions on the type a weight space according to the fundamental relation rll llr taking the trace of the l operator gives a set of commuting difference operators we show that for the above mentioned l operator this approach gives macdonald type operators with elliptic theta function coefficient actually equivalent to ruijsenaars operators the relationship between the difference l operator and krichever s lax matrix is given and an explicit formula for elliptic commuting differential operators is derived we also study the invariant subspace for the system which is spanned by symmetric theta functions on the weight space

matrix methods for determinants of pascal like matrices abstract using the ldu like factorization method the evaluations of the determinants of some pascal like matrices are derived the entries of these matrices satisfy the recurrence relation a i j a i 1 j a i j 1 a i 1 j 1 i j 1 and the values of the first row a 0 j and the column a i 0 have eight cases part of the results is due to c krattenthaler 2002 3

on t extensions of the hankel determinants of certain automatic sequences in 1998 allouche peyriere wen and wen considered the thue morse sequence and proved that all the hankel determinants of the period doubling sequence are odd integers we speak of t extension when the entries along the diagonal in the hankel determinant are all multiplied by t we prove that the t extension of each hankel determinant of the period doubling sequence is a polynomial in t whose leading coefficient is the only one to be an odd integer our proof makes use of the combinatorial set up developed by bugeaud and han which appears to be very suitable for this study as the parameter t counts the number of fixed points of a permutation finally we prove that all the t extensions of the hankel determinants of the regular paperfolding sequence are polynomials in t of degree less than or equal to 3

properties of blocked linear systems this paper presents a systematic study on the properties of blocked linear systems that have resulted from blocking discrete time linear time invariant systems the main idea is to explore the relationship between the blocked and the unblocked systems existing results are reviewed and a number of important new results are derived focus is given particularly on the zero properties of the blocked system as no such study has been found in the literature

hankel continued fraction and its applications abstract the hankel determinants of a given power series f can be evaluated by using the jacobi continued fraction expansion of f however the existence of the jacobi continued fraction needs that all hankel determinants of f are nonzero we introduce hankel continued fraction whose existence and uniqueness are guaranteed without any condition for the power series f the hankel determinants can also be evaluated by using the hankel continued fraction it is well known that the continued fraction expansion of a quadratic irrational number is ultimately periodic we prove a similar result for power series if a power series f over a finite field satisfies a quadratic equation then the hankel continued fraction is ultimately periodic as an application we derive the hankel determinants of several automatic sequences in particular the regular paperfolding sequence thus we provide an automatic proof of a result obtained by guo wu and wen which was conjectured by coons vrbik

invariant algebras and major indices for classical weyl groups given a classical weyl group w that is a weyl group of type a b or d one can associate with it a polynomial with integral coefficients z w given by the ratio of the hilbert series of the invariant algebras of the natural action of w and w t on the ring of polynomials bf c x 1 ldots x n otimes t we introduce and study several statistics on the classical weyl groups of type b and d and show that they can be used to give an explicit formula for z d n more precisely we define two mahonian statistics that is statistics having the same distribution as the length function dmaj and ned on d n the statistic dmaj defined in a combinatorial way has an analogous algebraic meaning to the major index for the symmetric group and the flag major index of adin and roichman for b n namely it allows us to find an explicit formula for z d n our proof is based on the theory of t partite partitions introduced by gordon and further studied by garsia and gessel using similar ideas we define the mahonian statistic ned also on b n and we find a new and simpler proof of the adin roichman formula for z b n finally we define a new descent number ddes on d n so that the pair ddes dmaj gives a generalization to d n of the carlitz identity on the eulerian mahonian distribution of descent number and major index on the symmetric group

descent representations and multivariate statistics combinatorial identities on weyl groups of types a and b are derived from special bases of the corresponding coinvariant algebras using the garsia stanton descent basis of the coinvariant algebra of type a we give a new construction of the solomon descent representations an extension of the descent basis to type b using new multivariate statistics on the group yields a refinement of the descent representations these constructions are then applied to refine well known decomposition rules of the coinvariant algebra and to generalize various identities

model matching for discrete time periodic systems with time varying relative degree and order abstract this paper extends the well known solution for the linear time invariant model matching problem to discrete time periodic systems with time varying relative degree and order it is shown that a key step to the design of a periodic output feedback controller is to compute the stable inverse of the periodic system using input output equations this problem is solved and model matching is achieved with system internal stability