on the stability and instability of n dimensional discrete systems n 1 the applicability of sufficient conditions for stability and for instability of multidimensional discrete systems is extended the approach is based on interpreting in the transform domain an extension which has been previously proposed only for one dimensional systems and had been based on time domain considerations

the dynamical properties of penrose tilings the set of penrose tilings when provided with a natural compact metric topology becomes a strictly ergodic dynamical system under the action of r2 by translation we show that this action is an almost 1 1 extension of a minimal r2 action by rotations on t4 i e it is an r2 generalization of a sturmian dynamical system we also show that the inflation mapping is an almost 1 1 extension of a hyperbolic automorphism on t4 the local topological structure of the set of penrose tilings is described and some generalizations are discussed

symmetrized β integers this article deals with b numeration systems which are numeration systems with a non integral base b 1 in this framework there exist elements which naturally play the role of integers which are called b integers the set of non negative b integers denoted by z b has various equivalent definitions which arise from different points of view nevertheless these definitions may be generalized on negative real numbers in a non unique way depending on the chosen framework we focus in this article on confluent parry units they are the positive roots of the polynomials x d i 1 d 1kx i 1 where the integers d and k satisfy d 2 and k 1 for any of these numbers we prove that there exists a discrete subset of r that we denote by z b s which is locally isomorphic to z b and such that z b s z b s moreover z b s is a model set and satisfies the inflation property there exists l 1 such that lz b s z b s l is then called the inflation factor for z b s finally we compute inflation factors for z b s of the form b i i being a positive integer

n dimensional stability margins computation and a variable transformation stability margins for n dimensional n geq 2 linear discrete causal systems are defined following a definition suggested recently in the literature for n 2 these can serve as a measure in some sense for the tendency of a stable n d system to become unstable also presented is a new procedure for the computation of these stability margins which is explicitly formulated the computational complexity is equivalent to that of solving for one indeterminate a set of n 1 real equations in n 1 real indeterminates also an n dimensional expansion of a known variable transformation which preserves the core condition of stability is presented and used to considerably simplify n dimensional stability tests and stability margins computations for a class of special cases numerical examples are also provided

stepped surfaces and rauzy fractals induced from automorphisms on the free group of rank 2 from irreducible unimodular pisot substitutions of rank 2 we can construct the discrete approximations of the lines the so called stepped surfaces and moreover we obtain rauzy fractals as geometric realizations of the symbolic dynamical systems in this paper we study a particular infinite family of automorphisms on the free group of rank 2 we see that for every hyperbolic quadratic unit integer l there is a corresponding automorphism in the infinite family whose incidence matrix has eigenvalue l and the automorphism is conjugate to a substitution or an alternating substitution and we construct stepped surfaces and rauzy fractals induced from automorphisms in the family by using tiling substitutions we also discuss the domain exchange transformations on rauzy fractals

on the boundary of the set of schur polynomials and applications to the stability of 1 d and 2 d digital recursive filters the authors show that implementation of stability test for 2 d digital quarter plane or nonsymmetric half plane recursive filters requires the testing of whether a particular resultant vanishes on the unit circle the authors prove that this cannot be avoided whatever the nature of implementation may be for the stability test this result is established by studying the set of all the schur polynomials whose coefficients belong to the space of univariate complex polynomials of degree not greater than n the authors first prove that this set of schur polynomials is connected next they give the equation which is obtained by equating a particular resultant to zero of the smallest hypersurface containing the boundary of this set finally it is shown that this equation is irreducible

a note on line spectral frequencies speech coding the main properties of line spectral frequencies lsfs are analyzed by means of the root locus technique this leads to a better insight into the nature of this efficient predictor filter parametrization the effects of quantization of the lsfs are illustrated by a numerical example

a markov partition that reflects the geometry of a hyperbolic toral automorphism we show how to construct a markov partition that reflects the geometrical action of a hyperbolic automorphism of the n torus the transition matrix is the transpose of the matrix induced by the automorphism in u dimensional homology provided this is non negative here u denotes the expanding dimension that condition is satisfied at least for some power of the original automorphism under a certain non degeneracy condition on the galois group of the characteristic polynomial the nu rectangles are constructed by an iterated function system and they resemble the product of the projection of a u dimensional face of the unit cube onto the unstable subspace and the projection of minus the orthogonal n u dimensional face onto the stable subspace

the split levinson algorithm the classical levinson algorithm for computing the predictor polynomial relative to a real positive definite toeplitz matrix is shown to be redundant in complexity it can be broken down into two simpler algorithms either of which needs only to be processed this result can be interpreted in the framework of the theory of orthogonal polynomials on the real line as follows the symmetric and antisymmetric parts of the predictors relative to the sequence of toeplitz matrices constitute two families of polynomials orthogonal on the interval 1 1 with respect to some even weight functions it turns out that the recurrence relations for these orthogonal polynomials can be used efficiently to compute the desired predictor the resulting split levinson algorithm requires roughly one half the number of multiplications and the same number of additions as the classical levinson algorithm a simple derivation of cybenko s method for computing the pisarenko frequencies is obtained from the recurrence relations underlying the split levinson algorithm

tilings induced by a class of cubic rauzy fractals we study aperiodic and periodic tilings induced by the rauzy fractal and its subtiles associated with beta substitutions related to the polynomial x 3 ax 2 bx 1 for a b 1 in particular we compute the corresponding boundary graphs describing the adjacencies in the tilings these graphs are a valuable tool for more advanced studies of the topological properties of the rauzy fractals as an example we show that the rauzy fractals are not homeomorphic to a closed disc as soon as a 2b 4 the methods presented in this paper may be used to obtain similar results for other classes of substitutions

a class of cubic rauzy fractals in this paper we study arithmetical and topological properties for a class of rauzy fractals r a given by the polynomial x 3 a x 2 x 1 where a 2 is an integer in particular we prove the number of neighbors of r a in the periodic tiling is equal to 8 we also give explicitly an automaton that generates the boundary of r a as a consequence we prove that r 2 is homeomorphic to a topological disk

a note on a classical bound for the moduli of all zeros of a polynomial considers the monic polynomial f z z sup n a sub n 1 z sup n 1 a sub 0 in the complex variable z with complex coefficients under the assumption that the nonleading coefficients of f lie in the disk mod z mod

algebraic numbers free group automorphisms and substitutions on the plane there has been much recent work on the geometric representation of pisot substitutions and the explicit construction of markov partitions for pisot toral automorphims we give a construction that extends this to the general hyperbolic case for the sake of simplicity we consider a simple example of an automorphism of the free group on 4 generators whose associated matrix has 4 distinct complex eigenvalues two of them of modulus larger than 1 and the other 2 of modulus smaller than 1 non pisot case using this generator we build substitution polygonal tilings of the contracting plane and the expanding plane of the matrix we prove that these substitution tilings can be lifted in a unique way to stepped surfaces approximating each of these planes the vertices of each of these stepped surfaces can be projected to an atomic surface a compact set with fractal boundary contained in the other plane we prove that both tilings can be refined to exact self similar tilings whose tiles have fractal boundaries and can be obtained by iteration or by a cut and project method by using the atomic surface as the window using the self similar tiling one can build a numeration system associated to a complex lambda expansion the natural extension of the lambda expansion associated with this number system is the linear map obtained by abelianization of the free group automorphism this gives an explicit markov partition of this hyperbolic toral automorphism the fractal domains can be used to define a pseudo group of translations which gives transversal dynamics in the sense of vershik 1994 or numeration systems in the sense of kamae 2005 the construction can be extended to a larger class of free group automorphisms each of which can be used to build substitution rules and dynamical systems

a circular stability test for general polynomials abstract we extend a new stability test proposed recently for discrete system polynomials 1 to polynomials with complex coefficients the method is based on a three term recursion of a conjugate symmetric sequence of polynomials the complex version has the same relative improved efficiency as the real version in comparison to the classical schur cohn formulation for counting the number of zeros of a polynomial with respect to the unit circle furthermore if desirable the complex text can be carried out using only real polynomials and arithmetic

cohomology of canonical projection tilings we define the cohomology of a tiling as the cocycle cohomology of its associated groupoid and consider this cohomology for the class of tilings which are obtained from a higher dimensional lattice by the canonical projection method in schlottmann s formulation we prove the cohomology to be equivalent to a certain cohomology of the lattice we discuss one of its qualitative features namely that it provides a topological obstruction for a generic tiling to be substitutional we develop and demonstrate techniques for the computation of cohomology for tilings of codimension smaller than or equal to 2 presenting explicit formulae these in turn give computations for the k theory of certain associated non commutative c algebras

direct bilinear routh stability criteria for discrete systems the bilinear transformation is applied to routh conditions for hurwitz polynomials to obtain a variety of equivalent direct z plane continued fraction cf expansions and stability conditions for discrete system polynomials efficient computational algorithms are provided they can be used to test the stability of a discrete system and to determine the number of zeros of unstable system polynomials outside the unit circle the many new cf forms may be also applicable to the design of bilinear transformed sampled data ladder filters

geometric theory of unimodular pisot substitutions we are concerned with the tiling flow t associated to a substitution over a finite alphabet our focus is on substitutions that are unimodular pisot i e their matrix is unimodular and has all eigenvalues strictly inside the unit circle with the exception of the perron eigenvalue 1 the motivation is provided by the still open conjecture asserting that t has pure discrete spectrum for any such we develop a number of necessary and sufficient conditions for pure discrete spectrum including injectivity of the canonical torus map the geometric realization geometric coincidence condition partial commutation of t and the dual r d 1 action measure and tiling properties of rauzy fractals and concrete algorithms some of these are original and some have already appeared in the literature as sufficient conditions only but they all emerge from a unified approach based on the new device the strand space f of the proof of the necessity hinges on determination of the discrete spectrum of t as that of the associated kronecker toral flow

stability invariance of discrete and continuous multidimensional systems under some variable tranformations a variable transformation for stability tests of multidimensional discrete systems which was previously shown to retain a nonvanishing property of a multivariable polynomial on the distinguished boundary of the multidimensional unit ball is shown here to retain this property in the entire multidimensional unit ball also a new variable transformation is presented for stability tests of multidimensional continuous systems and is shown to retain the nonvanishing property on the finite part of the distinguished boundary of the right half hyperplane the multidimensional finite imaginary axis applying this new transformation some necessary conditions are derived for the nonvanishing of multivariable quadratic form polynomials on the finite multidimensional imaginary axis also explicit necessary and sufficient conditions are derived for this case where the dimensionality is two

on robust hurwitz and schur polynomials based on results of a v lipatov and n i sokolov 1979 simple sufficient conditions for the robust stability of both strictly hurwitz and schur interval polynomials are obtained furthermore sufficient conditions are derived for robustness of strictly hurwitz and schur polynomials whose coefficients of linear functions of a single parameter several examples are discussed illustrating the derived results

the computation of line spectral frequencies using chebyshev polynomials line spectral frequencies provide an alternate parameterization of the analysis and synthesis filters used in linear predictive coding lpc of speech in this paper a new method of converting between the direct form predictor coefficients and line spectral frequencies is presented the system polynomial for the analysis filter is converted to two even order symmetric polynomial with interlacing roots on the unit circle the line spectral frequencies are given by the positions of the roots of these two auxiliary polynomials the response of each of these polynomials on the unit circle is expressed as a series expansion in chebyshev polynomials the line spectral frequencies are found using an iterative root finding algorithm which searches for real roots of a real function the algorithm developed is simple in structure and is designed to constrain the maximum number of evaluations of the series expansions the method is highly accurate and can be used in a form that avoids the storage of trigonometric tables or the computation of trigonometric functions the reconversion of line spectral frequencies to predictor coefficients uses an efficient algorithm derived by expressing the root factors as an expansion in chebyshev polynomials

a note on the generalized test for discrete system stability the purpose of this paper is to prove that the algorithms proposed in 1 6 to obtain the root distribution of the polynomial with real coefficients with respect to the unit circle are equivalent to one proposed in 14 15 and they have the same degree of complexity

dual systems of algebraic iterated function systems abstract we study a class of graph directed iterated function systems on r with algebraic parameters which we call algebraic gifs we construct a dual ifs of an algebraic gifs and study the relations between the two systems we determine when a dual system satisfies the open set condition which is fundamental for feasible pisot systems we construct the left and right rauzy thurston tilings and study their multiplicities and decompositions we also investigate their relation with codings space domain exchange transformation and the pisot spectrum conjecture the dual ifs provides a unified and simple framework for rauzy fractals tilings and related studies and allows us gain better understanding

a simple criterion for stability of linear discrete systems abstract a simple sufficient condition is presented for confining the poles of a linear discrete system on an arbitrarily specified circular region in the z plane

algorithm for determining pure pointedness of self affine tilings overlap coincidence in a self affine tiling in rd is equivalent to pure point dynamical spectrum of the tiling dynamical system we interpret the overlap coincidence in the setting of substitution delone set in rd and find an efficient algorithm to check the pure point dynamical spectrum this algorithm is easy to implement into a computer program we give the program and apply it to several examples in the course of the proof of the algorithm we show a variant of the conjecture of urbanski solomyak 2006 40 on the hausdorff dimension of the boundaries of fractal tiles

a note on the new 1 d and 2 d stability theorems for discrete systems an interesting theorem on the stability of 1 d discrete systems has been presented in 1 and it has been extended to the 2 d case in 2 in this correspondence after showing that the conditions stated by the theorems of 1 and 2 are only necessary we introduce another condition inclusion of which makes the above stability theorems necessary and sufficient

balances for fixed points of primitive substitutions an infinite word defined over a finite alphabet a is balanced if for any pair of factors of the same length and for any letter a in the alphabet a a 1 where a denotes the number of occurrences of the letter a in the word in this paper we generalize this notion and introduce a measure of balance for an infinite sequence in the case of fixed points of primitive substitutions we show that the asymptotic behaviour of this measure is in part ruled by the spectrum of the incidence matrix associated with the substitution connections with frequencies of letters and other balance properties are also discussed

comparison between some stability criteria of discrete time filters routh s algorithm is written in the form of polynomial recursion and transposed for testing stability in the wide sense of discrete time linear systems the algorithm obtained is a three term recursion of symmetric polynomials this recursion easily gives both the coefficients appearing in the continued fraction expansion associated to a given polynomial in the z domain and a computation scheme for the bilinear transformation of polynomials finally this recursion is compared to other algorithms already known for testing stability

a generalised test for discrete system stability in this correspondence we show that by using the continued fraction expansion 8 twice we can determine whether the discrete system is stable or not and if yes the number of real and complex roots appearing inside the unit circle

two dimensional iterated morphisms and discrete planes iterated morphisms of the free monoid are very simple combinatorial objects which produce infinite sequences by replacing iteratively letters by words the aim of this paper is to introduce a formalism for a notion of two dimensional morphisms we show that they can be iterated by using local rules and that they generate two dimensional patterns related to discrete approximations of irrational planes with algebraic parameters we associate such a two dimensional morphism with any usual pisot unimodular one dimensional iterated morphism over a three letter alphabet

canonical substitutions tilings of ammann beenker type we present a large class of parallelogram tilings that have two important features in common with the ammann beenker tiling they can be constructed by canonical projection from z4 r4 to a plane and they admit substitution rules