modelling quasicrystals at positive temperature we consider a two dimensional lattice model of equilibrium statistical mechanics using nearest neighbor interactions based on the matching conditions for an aperiodic set of 16 wang tiles this model has uncountably many ground state configurations all of which are nonperiodic the question addressed in this paper is whether nonperiodicity persists at low but positive temperature we present arguments mostly numerical that this is indeed the case in particular we define an appropriate order parameter prove that it is identically zero at high temperatures and show by monte carlo simulation that it is nonzero at low temperatures

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

on convergence rates of simultaneously triangularizable switched linear systems in this note we introduce the concepts of best case and worst case convergence rates for switched linear systems their connections to the problems of stability and stabilization are briefly discussed for simultaneously triangularizable switched systems explicit formulas are presented for the convergence rates in terms of the system eigenvalues

numeric and mesh algorithms for the coxeter spectral study of positive edge bipartite graphs and their isotropy groups we develop algorithmic techniques for the coxeter spectral analysis of the class ubigr n of connected loop free positive edge bipartite graphs d with n 2 vertices i e signed graphs in particular we present numerical and graphical algorithms allowing us a computer search in the study of such graphs d by means of their gram matrix email protected d the complex spectrum specc email protected c of the coxeter matrix cox d email protected email protected email protected d t r and the geometry of weyl orbits in the set mor d d of matrix morsifications email protected m n z of a simply laced dynkin diagram email protected a n d n e 6 e 7 e 8 associated with d and mesh root systems of type email protected our algorithms construct the coxeter gram polynomials cox d t z t and mesh geometries of root orbits of small connected loop free positive edge bipartite graphs d we apply them to the study of the following coxeter spectral analysis problem does email protected email protected hold i e the email protected email protected d arez congruent for any pair of connected positive loop free edge bipartite graphs d d inubigr nsuch thatspecc d specc d the problem if any square integer matrix email protected m n z is z congruent with its transpose a t r is also discussed we present a solution for graphs in ubigr n with email protected 6

integral bilinear forms coxeter transformations and coxeter polynomials of finite posets abstract linear algebra technique in the study of linear representations of finite posets is developed in the paper a concept of a quadratic wandering on a class of posets i is introduced and finite posets i are studied by means of the four integral bilinear forms b i b i b i b i z i z i z 1 1 the associated coxeter transformations and the coxeter polynomials in connection with bilinear forms of dynkin diagrams extended dynkin diagrams and irreducible root systems are also studied bilinear equivalences between some of the forms are established and equivalences with the bilinear forms of dynkin diagrams and extended dynkin diagrams are discussed a homological interpretation of the bilinear forms 1 1 is given and z bilinear equivalences between them are discussed by applying well known results of bongartz loupias and zavadskij shkabara we give several characterisations of posets i with the euler form q i x b i x x weakly positive resp with the reduced euler form q i x b i x x weakly positive and posets i with the tits form q i x b i x x weakly positive

wavevector dependent susceptibility in z invariant pentagrid ising model we study the q dependent susceptibility q of a z invariant ferromagnetic ising model on a penrose tiling as first introduced by korepin using de bruijn s pentagrid for the rapidity lines the pair correlation function for this model can be calculated exactly using the quadratic difference equations from our previous papers its fourier transform q is studied using a novel way to calculate the joint probability for the pentagrid neighborhoods of the two spins reducing this calculation to linear programming since the lattice is quasiperiodic we find that q is aperiodic and has everywhere dense peaks which are not all visible at very low or high temperatures more and more peaks become visible as the correlation length increases that is as the temperature approaches the critical temperature

bifurcations in a system of interacting fronts we show that the bifurcation scenario in a high dimensional system with interacting moving fronts can be related to the universal u sequence which is known from the symbolic analysis of iterated one dimensional maps this connection is corroborated for a model of a semiconductor superlattice which describes the complex dynamics of electron accumulation and depletion fronts by a suitable poincare section we reduce the dynamics to a low dimensional iterated map for which in the most elementary case the bifurcation points can be determined analytically

self avoiding walks on a penrose lattice direct real space renormalisation group procedures are constructed to calculate the self avoiding walk exponent nu on a two dimensional penrose lattice this lattice is neither translationally invariant nor hierarchical but still self similar and therefore calls for a renormalisation group treatment within the accuracy of a small cell calculation the obtained values of nu are consistent with the universal value of nu in two dimensions which is believed to be close to nu 0 75

self similarity of quasilattices in two dimensions ii the non bravais type n gonal quasilattice for pt i see ibid vol 22 p 193 1989 presents a systematic method of dividing an n gonal lattice into identical sublattices which are also n gonal lattices a non bravais type n gonal quasilattice in two dimensions is constructed with the projection method by assigning windows with different shapes sizes and or orientations to the sublattices the symmetries and the orientations of the windows are determined by the relative point symmetries of the sublattices to the n gonal lattice and windows assigned to equivalent sublattices differ from one another only in their orientations the sublattices are transformed among themselves by a volume conserving linear transformation induced by a complex pv unit in the n cyclotomic field a necessary and sufficient condition for a pv unit to be a complex self similarity ratio of a non bravais type quasilattice is presented it is shown that every non bravais type n gonal quasilattice has a self similarity characterised by pv unit

on algorithmic coxeter spectral analysis of positive posets abstract following a general framework of coxeter spectral analysis of signed graphs and finite posets i introduced by simson siam j discrete math 27 827 854 2013 we present efficient numerical algorithms for the coxeter spectral study of finite posets i 1 n i that are positive in the sense that the symmetric gram matrix g i 1 2 c i c i t r m n q is positive definite where c i m n z is the incidence matrix of i encoding the relation i in the framework of scientific computing we present a complete coxeter spectral classification of finite positive posets i of size n i 20 it extends one of the main results obtained in gasiorek et al eur j comb 48 127 142 2015 for posets of size n 10 we also show that the connectivity of such posets i is determined by the complex coxeter spectrum specc i c equivalently by the coxeter polynomial cox i t z t od i one of the main results of the paper is a new technique to compute in a polynomial time a z invertible matrix b i m n z such that b i t r c i b i g d i where g d i m n z is a non symmetric gram matrix of a simply laced dynkin diagram d i a n d n e 6 e 7 e 8 associated with a finite positive poset i

tame wild dichotomy of birkhoff type problems for nilpotent linear operators abstract inspired by recent results of ringel schmidmeier kussin lenzing meltzer xiao wu chen and pu zhang given a field k m 1 and a finite poset i i with a unique maximal element we study the category m o n i f m of mono representations of i over the frobenius k algebra f m k t t m of k dimension m viewed as k vector spaces u with an m nilpotent k linear operator t u u together with t invariant subspaces u i u j u for all i j in i the problem of when the krull schmidt k category m o n i f m is of wild resp tame representation type is called a wild resp tame birkhoff type problem for m nilpotent operators in case when k is algebraically closed we give a complete solution of the problem by describing all minimal pairs i m resp all pairs with m 1 such that category m o n i f m is of wild resp tame representation type we reduce the problem to a birkhoff type problem for the category fspr i f m m o n i f m of subprojective representations over f m of a larger poset i i the tame wild dichotomy for the category m o n i f m is also proved surprisingly in case when i i a b is the union of two incomparable chains i and i of length i a 1 1 and i b 1 1 with i i the problem is equivalent with the wildness resp tameness of the category coh x p of coherent sheaves over the weighted projective line x p for the weight triple p a b m with a b m 2 studied by kussin lenzing and meltzer 19 in relation with the hypersurface singularity f x 1 a x 2 b x 3 m

self affine delone sets and deviation phenomena we study the growth of norms of ergodic integrals for the translation action on spaces coming from expansive self affine delone sets the linear map giving the self affinity induces a renormalization map on the pattern space and we show that the rate of growth of ergodic integrals is controlled by the induced action of the renormalizing map on the cohomology of the pattern space up to boundary errors we explore the consequences for the diffraction of such delone sets and explore in detail what the picture is for substitution tilings as well as for cut and project sets which are self affine we also explicitly compute some examples

universal groups for point sets and tilings abstract we study the universal groups of inverse semigroups associated with point sets and with tilings we focus our attention on two classes of examples the first class consists of point sets which are obtained by a cut and projection scheme so called model sets here we introduce another inverse semigroup which is given in terms of the defining data of the projection scheme and related to the model set by the empire congruence the second class is given by one dimensional tilings

quantization noise in single loop sigma delta modulation with sinusoidal inputs an exact nonlinear difference equation is derived and solved for a simple sigma delta modulator consisting of a discrete time integrator and a binary quantizer inside a single feedback loop and an arbitrary input signal it is shown that the system can be represented as an affine operation discrete time integration of a biased input followed by a memoryless nonlinearity an extension of the transform method for the analysis of nonlinear systems is applied to obtain formulas for first and second order time average moments of the binary quantization noise including the sample mean energy and autocorrelation the results are applied to the special case of a sinusoidal input signal to evaluate these time averages and the power spectrum in the limit of large oversampling ratios the marginal moments behave as if the quantization noise had a uniform distribution the spectrum is neither white nor continuous however even in the limit of large oversampling ratios

quasiperiodic tilings with tenfold symmetry and equivalence with respect to local derivability two 2d quasiperiodic tilings with generalized tenfold symmetry are derived from the lattice a4r the reciprocal of the root lattice a4 both tilings are built from four tiles triangles in one case rhombi and hexagons in the other after a brief description of the tilings and their structures the authors introduce the equivalence concept of mutual local derivability they present its key properties and its application to several tenfold tilings and discuss some implications on a future classification of tilings in position space

on the structure of loop free non negative edge bipartite graphs abstract we continue the study of a class of signed graphs called finite connected loop free edge bipartite graphs bigraphs for short started in simson 2013 33 and continued in g siorek et al 2016 14 and simson zaj c 2017 38 in this paper we present an algorithmic approach to the study of non negative bigraphs with n r 2 vertices of corank r 1 that is bigraphs with the symmetric gram matrix g 1 2 g g t r m n r 1 2 z positive semi definite of rank n 1 where g m n r z is the non symmetric gram matrix of one of the main results of this paper are theorem 2 1 theorem 2 17 asserting that every such a bigraph with n r 2 vertices can be obtained from a corank zero i e positive connected loop free bigraph with n 1 vertices by an r point extension procedure u 1 u r u 1 u r along the r 1 roots u 1 u r z n of the positive definite gram form q z n z v v g v t r it is also shown that the extension procedure yields a combinatorial algorithm allowing us to obtain inductively all connected loop free non negative corank r 1 bigraphs with n r 2 vertices from the corank zero bigraphs with n 1 vertices

a coxeter type classification of one peak principal posets abstract we present a complete coxeter spectral classification of finite one peak posets i t that are principal in the sense that their symmetric gram matrix g i 1 2 c i c i t r m i q is positive semi definite of rank i 1 where c i m i z is the incidence matrix of i equivalently posets t with the quadratic tits form q t z t 1 z in the sense of drozd non negative and the kernel ker q t v z t 1 q t v 0 z t 1 infinite cyclic we continue the study started in m g siorek d simson 2012 11 we show that up to the strong gram z congruence and the weak gram z congruence every one peak principal poset is uniquely determined by its complex coxeter spectrum specc i c moreover we calculate the exact number of all such posets of a given size up to isomorphism and present their hasse quivers we finish the paper with conjecture on the exact number of principal positive posets according to our knowledge no formula is known for the number of non isomorphic posets in a general case

technical communique existence and stability of periodic trajectories in switched server systems the paper analyzes an example of the discrete control of a continuous variable system we prove that the switched server system with one server and an arbitrary number of buffers has a set of periodic trajectories that attract all other trajectories of the system

stability analysis of switched systems using variational principles an introduction many natural and artificial systems and processes encompass several modes of operation with a different dynamical behavior in each mode switched systems provide a suitable mathematical model for such processes and their stability analysis is important for both theoretical and practical reasons we review a specific approach for stability analysis based on using variational principles to characterize the most unstable solution of the switched system we also discuss a link between the variational approach and the stability analysis of switched systems using lie algebraic considerations both approaches require the use of sophisticated tools from many different fields of applied mathematics the purpose of this paper is to provide an accessible and self contained review of these topics emphasizing the intuitive and geometric underlying ideas

algebraic construction of one dimensional quasiperiodic tilings for an arbitrary one dimensional quasiperiodic tiling constructed via the grid method with periodically spaced grids an algebraic equation is derived for the positions of the vertices dependent on a single variable the cardinal position of the vertices

special points of 2 1 reducible quasilattices in three dimensions presents a complete classification of special points of five dimensional 5d bravais lattices which yield with the cut and projection method 2 1 reducible quasilattices in 3d the quasilattices are periodic along the c axis but quasiperiodic only along the plane perpendicular to it there exist five bravais classes of 5d lattices associated with the 2 1 reducible quasilattices namely primitive octagonal decagonal and dodecagonal lattices the centred octagonal lattice and the pentagonal lattice the author also discusses the special points of the reciprocal lattices of these 5d lattices

overlapping unit cells in 3d quasicrystal structure a three dimensional quasiperiodic lattice with overlapping unit cells and periodic in one direction is constructed using grid and projection methods pioneered by de bruijn each unit cell consists of 26 points of which 22 are the vertices of a convex polytope and 4 are interior points also shared with other neighbouring unit cells using kronecker s theorem the frequencies of all possible types of overlapping are found

the equivalence of two face centred icosahedral tilings with respect to local derivability the author demonstrates show two tilings with face centred icosahedral symmetry can be derived from one another by using only local information the tiling under consideration are the zonohedral tiling proposed by socolar and steinhardt which is closely related to the original three dimensional rhombohedral tiling and the tetrahedral tiling of danzer both tilings have matching rules and his proof is based on this property

a reduction functor tameness and tits form for a class of orders abstract let d be a complete discrete valuation domain which is an algebra over an algebraically closed field k we study in the paper a class of suborders of tiled d orders by means of a rational quadratic form q associated to and a finite poset i having exactly two maximal elements and criteria for finite lattice type and for tame lattice type of are given in terms of the form q and of the category i spr of socle projective k linear representations of the poset i the shape of auslander reiten quiver latt is described in corollary 3 2 a reduction functor h latt i spr preserving representation types is constructed it is shown in corollary 2 10 that for any poset i having exactly two maximal elements there exists a d order in our class an additive functor latt i spr preserving representation types and a poset isomorphism i i

scd patterns have singular diffraction among the many families of nonperiodic tilings known so far scd tilings are still a bit mysterious here we determine the diffraction spectra of point sets derived from scd tilings and show that they have no absolutely continuous part that they have a uniformly discrete pure point part on the axis re3 and that they are otherwise supported on a set of concentric cylinder surfaces around this axis for scd tilings with additional properties more detailed results are given

theory of self similarity of periodic approximants to a quasilattice ii the case of the dodecagonal quasilattice for pt i see ibid vol 24 p 4873 87 1991 the space groups of the periodic approximants pas to the dodecagonal quasilattice in two dimensions are investigated the bravais lattices of the pas are tetragonal p4mm rectangular pmm hexagonal p6mm and rhombic cmm the author presents several series of pas each of which is derived from a prototype pa by a successive application of the deflation and rescaling dar and the space group is common among the members each series is composed of two subseries the consecutive members in each subseries are related by the proper dar and the two subseries are related to each other by the improper dar

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

theory of matching rules for the 3 dimensional penrose tilings we consider packings of the two ammann rhombohedra used for tiling the three dimensional space we define decorations for the facets of the rhombohedra using elementary algebraic topology we prove that any tiling by these rhombohedra with matching decorations is a quasiperiodic penrose tiling the proof does not involve any reference to self similarity

a classification of special points of quasilattices in two dimensions a complete classification of the special points of octagonal decagonal and dodecagonal quasilattices in two dimensions 2d is presented they are obtained by projecting the special points of the starting n gonal lattice n 8 10 or 12 in 4d onto a 2d subspace a set of equivalent special points of a quasilattice are located on the centres of a single kind of tile in the quasiperiodic tiling associated with the quasilattice the point symmetry of the special points is identical to that of the tiles they form a quasilattice with the same point symmetry as that of the original quasilattice the new quasilattice is of a bravais type or non bravais type according to whether the point symmetry group of the special points is identical to the full point symmetry group of the quasilattice or its true subgroup respectively

projection of the danzer tiling we derive the icosahedrally symmetric octahedral tiling of danzer 1991 denoted by t d locally from the tiling t 2f of kramer et al 1989 obtained by icosahedral projection from the root lattice d6 moreover we determine all windows such that the tiling t d can be obtained by projection from the 6d root lattice d6 we reconstruct all vertex configurations of the tiling t d using the tools of the projection method

symbolic computation of strong gram congruences for cox regular positive edge bipartite graphs with loops abstract the paper can be viewed as a second part of the author s paper simson 2018 43 our main aim is to construct symbolic algorithms for the coxeter spectral classification of a class of signed graphs called edge bipartite graphs with n 2 vertices we started in simson 2013 37 and bocian et al 2014 6 more precisely we construct algorithms for the classification up to the strong gram z congruence z of all finite connected cox regular edge bipartite graphs with at least one loop bigraphs for short that are positive in the sense that the associated symmetric gram matrix g 1 2 g g t r m n 1 2 z is positive definite where g m n z is the non symmetric gram matrix of defined in section 1 and z means that there is a z invertible matrix b m n z such that g b t r g b we recall from 43 that every such a bigraph with a loop is strongly gram z congruent with a bigraph d that is one of the positive cox regular bigraphs b n n 2 c n n 3 f 4 m 4 g 2 presented in section 1 here by applying the geometry of mesh root system technique we construct symbolic algorithms that compute the correspondence d and the set gl n z d of all z invertible matrices b m n z defining the strong gram z congruence z d for any connected positive bigraph with n 2 vertices and at least one loop

electrons in deterministic quasicrystalline potentials and hidden conserved quantities we propose an ansatz for the wave function of a non interacting quantum particle in a deterministic quasicrystalline potential it is applicable to both continuous and discrete models and includes sutherland s hierarchical wave function as a special case the ansatz is parameterized by a first cohomology class of the hull of the structure the structure of the ansatz and the values of its parameters are preserved by the time evolution numerical results suggest that the ground states of the standard vertex models on ammann beenker and penrose tilings belong to this class of functions this property remains valid for the models perturbed within their mutual local derivability class e g by adding links along diagonals of rhombi the convergence of the numerical simulations in a finite patch of the tiling critically depends on the boundary conditions and can be significantly improved when the choice of the latter respects the structure of the ansatz

simple example of digital control systems with chaotic rounding errors chaotic behaviour due to round off characteristics in digital compensators is called a chaotic rounding error first we show sufficient conditions for the existence of chaotic rounding errors in digital control systems with first order plants and first order finite wordlength compensators next it is shown that a chaotic rounding error which appears as a chaotic attractor in the state space can be described by a markov chain with an appropriate partition of the attractor finally by analysing the markov chain we calculate several statistical properties of the chaotic rounding error

schurian sp representation finite right peak pi rings and their indecomposable socle projective modules such that soc r is an essential right ideal in r and it is isomorphic to a direct sum of finitely many copies of p called a right peak of r here f f are local rings f f is a division ring imj is an fi fj bimodule and the multiplication in r is given by fi f bimodule maps cii imj im jm satisfying the natural associativity conditions we denote by p p p the right indecomposable row ideals of r

spaces of tilings finite telescopic approximations and gap labeling the continuous hull of a repetitive tiling t in d with the finite pattern condition fpc inherits a minimal d lamination structure with flat leaves and a transversal open image in new window which is a cantor set this class of tiling includes the penrose the amman benkker ones in 2d as well as the icosahedral tilings in 3d we show that the continuous hull with its canonical d action can be seen as the projective limit of a suitable sequence of branched oriented and flat compact d manifolds as a consequence the longitudinal cohomology and the k theory of the corresponding c algebra open image in new window are obtained as direct limits of cohomology and k theory of ordinary manifolds moreover the space of invariant finite positive measures can be identified with a cone in the d th homology group canonically associated with the orientation of d at last the gap labeling theorem holds given an invariant ergodic probability measure on the hull the corresponding integrated density of states ids of any selfadjoint operators affiliated to open image in new window takes on values on spectral gaps in the module generated by the occurrence probabilities of finite patches in the tiling

self dual tilings with respect to star duality the concept of star duality is described for self similar cut and project tilings in arbitrary dimensions this generalises thurston s concept of a galois dual tiling the dual tilings of the penrose tilings as well as the ammann beenker tilings are calculated conditions for a tiling to be self dual are obtained

quasiperiodic tiling in two and three dimensions algebraic criteria which allow the vertices of two and three dimensional penrose tiling patterns to be specified are presented the application of these criteria to the interpretation of electron microscope observations from quasicrystalline structures is discussed

quasiperiodic icosahedral tilings from the six dimensional bcc lattice the cell geometry of the six dimensional bcc lattice is investigated via klotz construction two different classes of icosahedrally projected quasiperiodic tilings are defined for both cases we determine the acceptance domains of tiles and give a detailed description of the geometry of all tiles

theory of self similarity of periodic approximants to a quasilattice iii the case of a nonbravais type quasilattice for pt ii see ibid vol 25 p 2147 1992 the mother lattice l of a non bravais type quasilattice nbtql is a non bravais type periodic lattice with higher dimensionality and has an associated bravais lattice l0 the main problems on periodic approximants pas to the nbtql are reduced to similar problems in the case of the relevant bravais type quasilattice derived from l0 and the previous theories on the space groups and self similarity of the pas apply to the nbtql the present theory includes a general prescription of obtaining pas to a nbtql the author applies the theory to several nbtqls with an octagonal decagonal or dodecagonal point symmetry the second important bravais lattice lh called the host lattice is associated with l and he has clarified the difference in roles between l0 and lh in the theory of pas to the nbtql

root systems and inflations of non negative quasi cartan matrices abstract cartan matrices quasi cartan matrices and associated integral quadratic forms and root systems play an important role in such areas like lie theory representation theory and algebraic graph theory we study quasi cartan matrices by means of the inflation algorithm an idea used in ovsienko s classical proof of the classification of positive definite integral quadratic forms and recently applied in several other classification results we prove that it is enough to perform linear number of inflations to reduce positive or non negative principal quasi cartan matrix to its canonical form i e to the cartan matrix of a dynkin or euclidean diagram respectively moreover we show that in the positive case the length of every sequence of inflations has quadratic bound and that the only other natural non negative class having such universal bound is the class of so called pos sincerely principal quasi cartan matrices and in this case the bound is linear we obtain such low bounds by applying among others some new observations on the properties of the reduced root systems in the sense of bourbaki associated with positive quasi cartan matrices

cohomology of quasiperiodic patterns and matching rules quasiperiodic patterns described by polyhedral atomic surfaces are considered it is shown that under certain rationality conditions which coincide with the necessary conditions for the existence of matching rules the cohomology ring of the continuous hull of such patterns is isomorphic to that of the complement of a torus tn to an arrangement a of thickened affine tori of codimension 2 explicit computation of betti numbers for several two dimensional tilings and for the icosahedral ammann kramer tiling confirms in most cases the results obtained previously by different methods the cohomology groups of tn a have a natural structure of a right module over the group ring of the space symmetry group of the pattern and can be decomposed in a direct sum of its irreducible representations an example of such decomposition is shown for the ammann kramer tiling

coxeter spectral classification of almost tp critical one peak posets using symbolic and numeric computations following an idea of spectral graph theory we give several characterizations of finite one peak posets i with almost p critical quadratic tits form q i zi z by applying linear algebra methods combinatorial algorithms an old idea of s a ovsienko and recent results by v m bondarenko and m v stepochkina our study is mainly motivated by an important application of the quadratic tits form of a poset in constructing linear algebra invariants that measure a geometric complexity of nazarova roiter matrix problems over a field k in particular our study is inspired by a well known result of ju a drozd explained in introduction one of our main results asserts that given a finite poset i 1 n t such that is its unique maximal element and q i is almost p critical we have i the form q i is non negative and the subgroup kerq i v zn 1 q i v 0 of zi zn 1 is generated by a vector h h1 hn h with h1 0 hn 0 ii 5 i 9 and the coxeter polynomial coxi t z t coincides with the coxeter polynomial f i t of a uniquely determined euclidean diagram i d 4 e 6 e 7 e 8 iii the z bilinear tits form b i zn 1 zn 1 z of i is z equivalent to the gram z bilinear form b i zn 1 zn 1 z of the euclidean diagram i iv there is a z invertible matrix c such that c itr ctr c i c and c2 e where c i mn 1 z is the tits matrix of i and v i is one of the posets l1 l132 listed in the paper in particular we get in an alternative way the list of p critical posets t in the form t i presented by v m bondarenko and m v stepochkina 6 and we determine their coxeter euclidean types there are 115 one peak posets i l1 l115 with q i p critical and 17 exceptional posets i l116 l132 with q i almost p critical but not p critical our lists are obtained by using computer algebra tools mainly symbolic computation in maple and numeric computation in c

topological equivalence of tilings we introduce a notion of equivalence on tilings which is formulated in terms of their local structure we compare it with the known concept of locally deriving one tiling from another and show that two tilings of finite type are topologically equivalent whenever their associated groupoids are isomorphic

statistical analysis and spectral estimation techniques for one dimensional chaotic signals signals arising out of nonlinear dynamics are compelling models for a wide range of both natural and man made phenomena in contrast to signals arising out of linear dynamics extremely rich behavior is obtained even when we restrict our attention to one dimensional 1 d chaotic systems with certain smoothness constraints an important class of such systems are the so called markov maps we develop several properties of signals obtained from markov maps and present analytical techniques for computing a broad class of their statistics in closed form these statistics include for example correlations of arbitrary order and all moments of such signals among several results we demonstrate that all markov maps produce signals with rational spectra and we can therefore view the associated signals as chaotic arma processes with chaotic white noise as a special case finally we also demonstrate how markov maps can be used to approximate to arbitrary accuracy the statistics any of a broad class of non markov chaotic maps

periodicity and chaos from switched flow systems contrasting examples of discretely controlled continuous systems two examples of the discrete control of a continuous variable system are analyzed these examples exhibit what may be regarded as the two extremes of complexity of the closed loop behavior one is eventually periodic the other is chaotic the examples are derived from sampled deterministic flow models these are of interest in their own right but have also been used as models for certain aspects of manufacturing systems in each case the authors give a precise characterization of the closed loop behavior

an n gonal quasilattice in two dimensions as a dual lattice to a multiple periodic grid an n gonal quasilattice in two dimensions is constructed for the case of n 6m m or 2 as a dual to an m tuple periodic grid gm g0 union rg0 union union rm 1g0 where g0 is a periodic grid with hexagonal point symmetry and r the rotation operation of the plane by angle 2 pi n the author investigates the case where g0 is a honeycomb triangular kagome or diced grid the resulting quasilattice is not of bravais type unless g0 is the honeycomb grid it is shown also that the same quasilattice is obtained with the projection method from the 2m dimensional lattice l l0 l0 l0 where l0 is a triangular honeycomb diced or kagome lattice being dual to g0 in the projection method the 2m dimensional euclidean space is divided into cells with a 2m dimensional grid g which is dual to l it is shown that gm is identical to a two dimensional section of g

coxeter transformation and inverses of cartan matrices for coalgebras abstract let c be a coalgebra and let z i c z i c z i c be the grothendieck groups of the category c o p inj and c inj of the socle finite injective right and left c comodules respectively one of the main aims of the paper is to study the coxeter transformation c z i c z i c and its dual c z i c z i c of a pointed sharp euler coalgebra c and to relate the action of c and c on a class of indecomposable finitely cogenerated c comodules n with the ends of almost split sequences starting with n or ending at n by applying chin kleiner and quinn 2002 5 we also show that if c is a pointed k coalgebra such that the every vertex of the left gabriel quiver q c of c has only finitely many neighbours then for any indecomposable non projective left c comodule n of finite k dimension there exists a unique almost split sequence 0 c n n n 0 in the category c comod f c of finitely cogenerated left c comodules with an indecomposable comodule c n we show that dim c n c dim n if c is hereditary or more generally if inj dim d n 1 and hom c c d n 0

a classification of two dimensional quasiperiodic tilings obtained with the grid method two dimensional quasiperiodic tilings qpt obtained from n grids are classified into local isomorphism li classes by using the invariants of the grids the number of the invariants is given by n phi n if n is odd or n phi 2n if n is even where phi is euler s function in number theory all the qpt obtained from n grids with n 2k k or 2 belong to a single li class whose point symmetry is d2n when n is not a power of 2 the qpt are classified into several li classes of the li classes two have the highest point symmetry d2n one of the two classes is associated with n grids in which all the invariants vanish and the other with n grids in which all of them take 1 2 in the case of an odd n there is also one continuous series of qpt with point symmetry dn the author also presents a general formula of the tile statistics of the quasiperiodic tilings obtained with the grid method

superquasicrystals self similar ordered structures with non crystallographic point symmetries we present a systematic method of constructing limit quasiperiodic structures with non crystallographic point symmetries such structures are different aperiodic ordered structures from quasicrystals and we call them superquasicrystals they are sections of higher dimensional limit periodic structures constructed on super bravais lattices we enumerate important super bravais lattices superquasicrystals with strong self similarities form an important subclass the simplest example is a two dimensional octagonal superquasicrystal

quasicrystalline ground states without matching rules a simple local cluster interaction is presented which has as only ground states perfectly quasicrystalline tilings from a single local isomorphism class since these tilings do not allow for any perfect matching rules it is thereby shown that the class of structures which are the ground state of some finite range interaction is considerably larger than previously anticipated cluster interactions having a quasicrystalline ground state turn out to be simple and robust and therefore provide an attractive explanation for the existence of quasicrystals a simplified version of our cluster interaction is found to have super tile random tiling ground states due to the large size of the super tiles these random tilings still look perfect on a local scale

crystallography and the penrose pattern abstract the penrose pattern is a tiling of two dimensional and of three dimensional space by identical tiles of two kinds acute and obtuse rhombi with 72 and 144 in two dimensions and acute and obtuse rhombohedra with 63 43 and 116 57 in three dimensions the two dimensional pattern is a section through that in three dimensions when joining or recursion rules are prescribed the pattern is unique and non periodic it has local five fold axes and thus represents a structure outside the formalism of classical crystallography and might be designated a quasi lattice

symmetries of quasicrystals we consider tiling models of round quasicrystals which would have diffraction patterns which are fully rotation invariant rings instead of bragg peaks they can be distinguished from glasses by self similarity of the pattern of radii of the rings

self similarity of quasilattices in two dimensions iii inflation by a non unit pv number for pt ii see ibid vol 22 205 1989 it is shown that every n gonal quasilattice in two dimensions has self similarity with a complex self similarity ratio which is equal to a non unit pv number in the n cyclotomic field in this self similarity the original quasilattice can be inflated to a number of different quasilattices which are locally isomorphous to each other the number is equal to the algebraic norm of the relevant pv number in the n cyclotomic field several examples are presented for the cases of octagonal decagonal and dodecagonal quasilattices the difference between the present self similarity and that associated with a pv unit is clarified

discontinuities of the integrated density of states for random operators on delone sets despite all the analogies with usual random models tight binding operators for quasicrystals exhibit a feature that clearly distinguishes them from the former the integrated density of states may be discontinuous this phenomenon is identified as a local effect due to the occurrence of eigenfunctions with bounded support

tiling spaces are inverse limits let m be an arbitrary riemannian homogeneous space and let be a space of tilings of m with finite local complexity relative to some symmetry group and closed in the natural topology then is the inverse limit of a sequence of compact finite dimensional branched manifolds the branched manifolds are finite unions of cells constructed from the tiles themselves and the group this result extends previous results of anderson and putnam of ormes radin and sadun of bellissard benedetti and gambaudo and of gahler in particular the construction in this paper is a natural generalization of gahler s

structure and a coxeter dynkin type classification of corank two non negative posets inspired by famous results of drozd 1974 13 nazarova ro ter 1975 42 and bondarenko stepochkina 2005 8 on the importance of finite posets i with the non negative tits quadratic form q i zi z in the representation theory we study the finite non negative posets i of corank two that is such posets i that the symmetric gram matrix gi 12 ci citr mi q of i is positive semi definite of corank two where ci mi z is the incidence matrix of i a structure of such posets is described in the paper it is shown that every such a poset i can be obtained in a natural way from a corank zero poset j and a pair of roots of j following main ideas of the coxeter spectral analysis of posets we associate with any connected corank two poset i the incidence quadratic form qi zi z the coxeter matrix coxi ci ci tr the coxeter polynomial coxi t det t e coxi z t its complex coxeter spectrum specci the coxeter number ci n the reduced coxeter number c i n the incidence defect homomorphism i zi kerqi zi and a simply laced dynkin diagram dyni in particular we study the question when the coxeter spectrum specci of i together with dyni determines the incidence matrix ci of i hence the poset i uniquely up to a z congruence by applying linear algebra techniques and a combinatorial computer aided analysis of posets we prove that the question has an affirmative answer for connected corank two posets i with at most 15 elements a complete list of such posets is presented and their coxeter type invariants are computed it follows that i 6 there is only one such a poset i of cardinality 6 and 8 such posets of cardinality 7 up to the poset duality i iop the existence of a i mesh translation quiver ri kerqi i on the set ri kerqi is discussed where ri v zi qi v 1 is the set of roots of qi

application of stochastic control techniques to chaotic nonlinear systems a control algorithm based on stochastic control techniques is devised for chaotic nonlinear systems the algorithm uses a state estimator based on the kalman filter and yields performance improvements in at least some regions of state space with respect to that obtainable by use of a controller utilizing only the conditional mean of the system state vector the method is applied to two typical chaotic nonlinear systems the henon heiles system and the lorenz system and their behavior with control is explored numerically

structural transformations in quasicrystals induced by higher dimensional lattice transitions we study the structural transformations induced via the cut and project method in quasicrystals and tilings by lattice transitions in higher dimensions with a focus on transition paths preserving at least some symmetry in intermediate lattices we discuss the effect of such transformations on planar aperiodic penrose tilings and on three dimensional aperiodic ammann tilings with icosahedral symmetry we find that locally the transformations in the aperiodic structures occur through the mechanisms of tile splitting tile flipping and tile merger and we investigate the origin of these local transformation mechanisms within the projection framework

generalised 2d penrose tilings structural properties usual penrose tilings are defined by a set of five phases gamma j which add up to an integer the authors study the general case where the phases add up to any number and show that these generalised tilings can be obtained with four tiles two thick rhombi and two thin rhombi with different colourings of the edges the consideration of generalised tilings is relevant for the study of phase distortions local structural transitions and therefore various defects of the usual things

one peak posets with positive quadratic tits form their mesh translation quivers of roots and programming in maple and python by applying linear algebra and computer algebra tools we study finite posets with positive quadratic tits form our study is motivated by applications of matrix representations of posets in representation theory where a matrix representation of a partially ordered set t p1 pn with a partial order means a block matrix m m1 m2 mn over a field k of size d d1 dn determined up to all elementary row transformations elementary column transformations within each of the substrips m1 m2 mn and additions of linear combinations of columns of mi to columns of mj if pi pj drozd 10 proves that t has only a finite number of direct sum indecomposable representations if and only if its quadratic tits form q x1 xn x x12 xn2 x 2 pi pjxixj x x1 xn is weakly positive i e q a1 an a 0 for all non zero vectors a1 an a with integral non negative coefficients in this case there exists an indecomposable representation m of size d d1 dn if and only if d1 dn d is a root of q i e q d1 dn d 1 bondarenko and stepochkina 8 give a list of posets t with positive tits form consisting of four infinite series and 108 posets up to duality in the paper we construct this list in an alternative way by applying computational algorithms implemented in maple and python moreover given any poset t of the list we show that a the coxeter polynomial coxi t of the poset i t obtained from t by adding a unique maximal element called a peak is a coxeter polynomial f i t of a simply laced dynkin diagram i an dn e6 e7 e8 and b the set of orbits of the set rq of integral roots of q admits a mesh translation quiver structure rq of a cylinder shape where is the coxeter transformation of i t in the sense of drozd 10

self similarity of quasilattices in two dimensions i the n gonal quasilattice the self similarity of a two dimensional quasilattice with an n gonal point symmetry has been investigated for even n n or 8 it is shown that for any n an n gonal quasilattice has a self similarity characterised by a complex number tau on inflation the quasilattice is scaled by mod tau mod and subsequently rotated by arg tau tau is a pv unit of the n cyclotomic field q zeta zeta exp 2 pi i n tau satisfies i tau and tau 1 are both algebraic integers in q zeta ii mod tau mod 1 and iii mod tau mod 1 for any conjugate tau but tau the complex conjugate of tau in q zeta pv units are calculated for every n gonal quasilattice whose multiplicity m phi n 2 with phi being the eulerian function is less than 5 m 2 for n 8 10 12 m 3 for n 14 18 and m 4 for n 16 20 24 30 it is found that a quasilattice has two or more independent scales of self similarity if its multiplicity is larger than 2

space group theory for a nonperiodic icosahedral quasilattice the hypercubic lattice y is considered in euclidean space e6 embedding of the icosahedral group a 5 determines a well known noncrystallographic action of a 5 on an irreducible subspace e3 from boundaries of y and its metrical dual y new convex polytopes called klotze are constructed in e6 the klotze tile e6 periodically but not face to face and provide a new fundamental domain under the hypercubic translation group all boundaries of a klotz are parallel or perpendicular to the subspace e3 the intersection of the lattice y with e3 determine a nonperiodic face to face rhombohedral tiling of e3 a quasicrystal model is introduced in e6 and e3 the model has stable density supported on the rhombohedral tiles its fourier transform yields sharp diffraction maxima and reduces to integrals in e3 over the representative tiles and their boundaries

equivalence of the generalised grid and projection methods for the construction of quasiperiodic tilings the two main techniques for the generation of quasiperiodic tilings de bruijn s grid method 1981 and the projection formalism are generalised a very broad class of quasiperiodic tilings is obtained in this way the two generalised methods are shown to be equivalent the standard calculation of fourier spectra is extended to the whole general class of tilings various applications are discussed

pattern equivariant functions and cohomology the cohomology of a tiling or a point pattern has originally been defined via the construction of the hull or the groupoid associated with the tiling or the pattern here we present a construction which is more direct and therefore more easily accessible it is based on generalizing the notion of equivariance from lattices to point patterns of finite local complexity

systems of subspaces of a unitary space abstract for a finite poset p p 1 p t we study systems u 1 u t u of subspaces of a unitary space u such that u i u j if p i p j two systems u 1 u t u and v 1 v t v are said to be isometric if there exists an isometry u v such that u i v i we classify such systems up to isometry if p is a semichain we prove that the problem of their classification is unitarily wild if p is not a semichain a classification problem is called unitarily wild if it contains the problem of classifying linear operators on a unitary space which is hopeless in a certain sense

a self similar dodecagonal quasiperiodic tiling of the plane in terms of squares regular hexagons and thin rhombi the author presents a dodecagonal quasiperiodic tiling of the plane in terms of three kinds of tiles a square a regular hexagon and a thin rhombus whose acute inner angles are 30 degrees the dodecagonal tiling is self similar with respect to a dilatation by 2 square root 3 it is associated with a dodecagonal quasiperiodic lattice which is obtained by a projection from a hyperhoneycomb lattice in four dimensions the hyperhoneycomb lattice is a non bravais lattice composed of four bravais sublattices and the author must assign windows with different orientations to different sublattices the author discusses in detail the relationships of the present method of obtaining a dodecagonal tiling to other methods

on the periodicity of symbolic observations of piecewise smooth discrete time systems a study is made of the behavior of discrete time systems composed of a set of smooth transition maps coupled by a quantized feedback function the feedback function partitions the state space into disjoint regions and assigns a smooth transition function to each region the main result is that under a constraint on the norm of the derivative of the transition maps a bounded state trajectory with limit points in the interior of the switching regions leads to a region index sequence that is eventually periodic under these assumptions it is shown that eventually the feedback function is determined by a finite state automaton a similar result is proved in the case of finite state dynamic feedback

a topological analysis of a family of dynamical systems with non standard chaotic and periodic behaviour we examine a family of maps which are motivated by a specific flow model for a manufacturing system that act on polygons inscribed within an equilateral triangle the family of maps is interesting because it provides an example of chaotic behaviour in one extreme and periodic behaviour in the other with a non standard transition between these extremes we characterize the behaviour of this family of maps and describe the interesting phenomena which occur between the two extremes of chaos and periodicity we also use symbolic dynamics to relate one of the maps to a particular subshift of finite type

self similar quasilattices with windows having fractal boundaries the point inflation rule is presented as a powerful scheme to produce a self similar quasilattice with a non crystallographic point symmetry alternatively the conjugate point inflation rule which is a set map acting on the internal space has the unique fixed point which is a compact set w and the quasilattice ql is obtained with the projection method by taking w as the window the fixed point w is the attractor of the set map if w is topologically a disc a ball for a three dimensional case its boundary is usually fractal we investigate conditions for the boundary of the window to be a von koch curve a practical method of constructing windows and the relevant qls is presented a lot of windows with fractal boundaries are determined on the basis of this method we introduce isomerism which has priority of rank to the mutual local derivability as a classification scheme for self similar qls it is argued that the pir and isomerism can be the key points of classification of self similar qls

locking to incommensurate structures a model with three competing lengths a classical pseudo spin model in one dimension is considered representing a variation on the frenkel kontorova model to include non convex interactions resulting in three competing length scales an exact algorithm is used numerically to determine the classical ground state as a function of a chemical potential approximate arguments suggest that only a first order transition should occur however when the lengths are not rationally related it is found that the mean lattice spacing appears to vary as a devil s staircase the plateaux of the staircase correspond to locking to incommensurate structures while there is no locking to the commensurate ones most of the locking values observed numerically belong to a series which can be analytically calculated on the basis of simple topological hypotheses which are also consistent with numerical observations

matching rules and growth rules for pentagonal quasicrystal tilings preuve de l existence de ces regles pour un ensemble infini de pavages pentagonaux 2d autres que le pavage de peurose

bridging genetic networks and queueing theory one of the main challenges facing biology today is the understanding of the joint action of genes proteins and rna molecules interwoven in intricate interdependencies commonly known as genetic networks to this end several mathematical approaches have been introduced to date in addition to developing the analytical tools required for this task anew one can utilize knowledge found in existing disciplines specializing in the representation and analysis of systems featuring similar aspects we suggest queueing theory as a possible source of such knowledge this discipline which focuses on the study of workloads forming in a variety of scenarios offers an assortment of tools allowing for the derivation of the statistical properties of the inspected systems we argue that a proper adaptation of modeling techniques and analytical methods used in queueing theory can contribute to the study of genetic regulatory networks this is demonstrated by presenting a queueing inspired model of a genetic network of arbitrary size and structure for which the probability distribution function is derived this model is further applied to the description of the lac operon regulation mechanism in addition we discuss the possible benefits stemming for queueing theory from the interdisciplinary dialogue with molecular biology in particular the incorporation of various dynamical behaviours into queueing networks

conjugacies for tiling dynamical systems we consider tiling dynamical systems and topological conjugacies between them we prove that the criterion of being of finite type is invariant under topological conjugacy for substitution tiling systems under rather general conditions including the penrose and pinwheel systems we show that substitutions are invertible and that conjugacies are generalized sliding block codes

dualisation of voronoi domains and klotz construction a general method for the generation of proper space fillings the authors present a general method for the generation of proper space fillings of euclidean spaces with convex polytopes by projection from higher dimensional spaces as a special case quasiperiodic tilings are obtained by the projection of periodic structures the concept is general enough for the treatment of defects in quasiperiodic tilings by placing defects in high dimensional translation lattices

local rules for quasicrystals the relationship of local ordering and long range order is studied for quasicrystalline tilings of plane and space two versions of the concept of local rules are introduced strong and weak necessary conditions of the existence of strong local rules are found they are mainly reduced to the constraints for irrational numbers related to incommensurabilities of the quasicrystals for planar quasicrystals the quadratic irrationalities a b sqrt d left a b in mathbb q d in mathbb z right play an important role for three dimensional quasicrystals not only quadratic but also cubic irrationalities a b 3 sqrt d c 3 sqrt d 2 left a b c in mathbb q d in mathbb z right are allowed the existence of weak local rules is established for almost all two dimensional quasicrystals based on quadratic irrationalities and for the three dimensional quasicrystal having icosahedral symmetry