solving the algebraic riccati equation with the matrix sign function abstract this paper presents some improvements to the matrix sign function algorithm for the algebraic riccati equation a simple reorganization changes nonsymmetric matrix inversions into symmetric matrix inversions scaling accelerates convergence of the basic iteration and yields a new quadratic formula for certain 2 by 2 algebraic riccati equations numerical experience suggests the algorithm be supplemented with a refinement strategy similar to iterative refinement for systems of linear equations refinement also produces an error estimate the resulting procedure is numerically stable it compares favorably with current schur vector based algorithms

dense dirac combs in euclidean space with pure point diffraction regular model sets describing the point positions of ideal quasicrystallographic tilings are mathematical models of quasicrystals an important result in mathematical diffraction theory of regular model sets which are defined on locally compact abelian groups is the pure pointedness of the diffraction spectrum we derive an extension of this result valid for dense point sets in euclidean space which is motivated by the study of quasicrystallographic random tilings

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 the ideal theory of graphs we study algebras defined by finite sets g m1 mq of monomials of a polynomial ring r there are two basic algebras i k g k m1 mq the k subalgebra of r spanned by the mi and ii the quotient ring r i g where i g m1 mq they come together in the construction of the rees algebra r i g of the ideal i g the emphasis is almost entirely on sets of squarefree monomials of degree two and their attached graphs the main results are assertions about the cohen macaulay behaviour of the koszul homology of i g and how normality or cohen macaulayness of one of the algebras can be read off the properties of the graph or in the other algebra

solving the matrix differential riccati equation a lyapunov equation approach in this technical note we investigate a solution of the matrix differential riccati equation that plays an important role in the linear quadratic optimal control problem unlike many methods in the literature the approach that we propose employs the negative definite anti stabilizing solution of the matrix algebraic riccati equation and the solution of the matrix differential lyapunov equation an illustrative numerical example is provided to show the efficiency of our approach

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

random fields on model sets with localized dependency and their diffraction for a random field on a general discrete set we introduce a condition that the range of the correlation from each site is within a predefined compact set d for such a random field defined on the model set that satisfies a natural geometric condition we develop a method to calculate the diffraction measure of the random field the method partitions the random field into a finite number of random fields each being independent and admitting the law of large numbers the diffraction measure of consists almost surely of a pure point component and an absolutely continuous component the former is the diffraction measure of the expectation e while the inverse fourier transform of the absolutely continuous component of turns out to be a weighted dirac comb which satisfies a simple formula moreover the pure point component will be understood quantitatively in a simple exact formula if the weights are continuous over the internal space of then we provide a sufficient condition that the diffraction measure of a random field on a model set is still pure point

how should one define a weak crystal we compare two proposals for the study of positional long range order one in terms of the spectrum of the translation operator the other in terms of the fourier spectrum we point out that only the first one allows for the consideration of molecular as opposed to atomic weakly periodic structures we illustrate this point on the thue morse system

diffraction of stochastic point sets explicitly computable examples stochastic point processes relevant to the theory of long range aperiodic order are considered that display diffraction spectra of mixed type with special emphasis on explicitly computable cases together with a unified approach of reasonable generality the latter is based on the classical theory of point processes and the palm distribution several pairs of autocorrelation and diffraction measures are discussed which show a duality structure analogous to that of the poisson summation formula for lattice dirac combs

pure point diffraction and cut and project schemes for measures the smooth case we present a cut and project formalism based on measures and continuous weight functions of sufficiently fast decay the emerging measures are strongly almost periodic the corresponding dynamical systems are compact groups and homomorphic images of the underlying torus in particular they are strictly ergodic with pure point spectrum and continuous eigenfunctions their diffraction can be calculated explicitly our results cover and extend corresponding earlier results on dense dirac combs and continuous weight functions with compact support they also mark a clear difference in terms of factor maps between the case of continuous and non continuous weight functions

pointwise theorems for amenable groups in this paper we prove the pointwise ergodic theorem for general locally compact amenable groups along folner sequences that obey some restrictions these restrictions are mild enough so that such sequences exist for all amenable groups we also prove a generalization of the shannon mcmillan breiman theorem to all discrete amenable groups

diffraction theory and almost periodic distributions we introduce and study the notions of translation bounded tempered distributions and autocorrelation for a tempered distribution we further introduce the spaces of weakly strongly and null weakly almost periodic tempered distributions and show that for weakly almost periodic tempered distributions the eberlein decomposition holds for translation bounded measures all these notions coincide with the classical ones we show that tempered distributions with measure fourier transform are weakly almost periodic and that for this class the eberlein decomposition is exactly the fourier dual of the lebesgue decomposition with the fourier bohr coefficients specifying the pure point part of the fourier transform we complete the project by looking at few interesting examples

irreducible decomposition of powers of edge ideals abstract in this paper by using some tools from graph theory mainly the theorem on ear decomposition of factor critical graphs given by l lovasz and the canonical decomposition of a graph given by edmonds and gallai we can describe each irreducible component of powers of edge ideals of a graph as an application we use our results to persistence of irreducible components to bound dstab and to count the number of irreducible components of powers of edge ideals of a graph

how model sets can be determined by their two point and three point correlations we show that real model sets with real internal spaces are determined up to translation and changes of density 0 by their 2 and 3 point correlations we also show that there exist pairs of real even 1d aperiodic model sets with internal spaces that are products of real spaces and finite cyclic groups whose 2 and 3 point correlations are identical but which are not related by either translation or inversion of their windows all these examples are pure point diffractive

the covariogram determines three dimensional convex polytopes the cross covariogram gk l of two convex sets k l rn is the function which associates to each x rn the volume of the intersection of k with l x the problem of determining the sets from this function is relevant in stochastic geometry in probability and it is equivalent to a particular case of the phase retrieval problem in fourier analysis it is also relevant for the inverse problem of determining the atomic structure of a quasicrystal from its x ray diffraction image the two main results of this paper are that gk k determines three dimensional convex polytopes k and that gk l determines both k and l when k and l are convex polyhedral cones satisfying certain assumptions these results settle a conjecture of g matheron in the class of convex polytopes further results regard the known counterexamples in dimension n 4

extinctions and correlations for uniformly discrete point processes with pure point dynamical spectra the paper investigates how correlations can completely specify a uniformly discrete point process the setting is that of uniformly discrete point sets in real space for which the corresponding dynamical hull is ergodic the first result is that all of the essential physical information in such a system is derivable from its n point correlations n 2 3 if the system is pure point diffractive an upper bound on the number of correlations required can be derived from the cycle structure of a graph formed from the dynamical and bragg spectra in particular if the diffraction has no extinctions then the 2 and 3 point correlations contain all the relevant information

finite horizon optimal control of singularly perturbed systems a differential lyapunov equation approach in this note we study the finite time horizon optimal control problem for singularly perturbed systems the solution is obtained in terms of the corresponding solution of the algebraic riccati equation and the decomposition of the singularly perturbed differential lyapunov equation into reduced order differential lyapunov sylvester equations an illustrative numerical example is provided to show the efficiency of the proposed approach

the singular continuous diffraction measure of the thue morse chain the paradigm for singular continuous spectra in symbolic dynamics and in mathematical diffraction is provided by the thue morse chain in its realization as a binary sequence with values in 1 we revisit this example and derive a functional equation together with an explicit form of the corresponding singular continuous diffraction measure which is related to the known representation as a riesz product

bounds on the optimum quadratic cost of structure constrained controllers in this note lower and upper bounds are derived on the quadratic cost of optimal output feedback controllers subject to constraints on the controller structure the use of these bounds in forming a comparative assessment of alternative controller confugurations is described and an illustrative example is given the principal advantage of these bounds arises from the fact that they can save considerable computational effort

improved potter anderson moore algorithm for the differential riccati equation abstract the paper presents a method to find the solution of the constant coefficient matrix differential riccati differential in terms of solutions of algebraic riccati and lyapunov equations and the state transition matrix matrix exponential of the corresponding linear dynamic system the method presented represents an improved method of potter anderson moore since the solution is obtained under milder assumptions than the original algorithm of potter anderson moore an aircraft and satellite examples done in the paper demonstrate the advantages of the improved algorithm

renormalisation of pair correlation measures for primitive inflation rules and absence of absolutely continuous diffraction the pair correlations of primitive inflation rules are analysed via their exact renormalisation relations we introduce the inflation displacement algebra that is generated by the fourier matrix of the inflation and deduce various consequences of its structure moreover we derive a sufficient criterion for the absence of absolutely continuous diffraction components as well as a necessary criterion for its presence this is achieved via estimates for the lyapunov exponents of the fourier matrix cocycle of the inflation rule we also discuss some consequences for the spectral measures of such systems while we develop the theory first for the classic setting in one dimension we also present its extension to primitive inflation rules in higher dimensions with finitely many prototiles up to translations

an ultrametric state space with a dense discrete overlap distribution paperfolding sequences we compute the parisi overlap distribution for paperfolding sequences it turns out to be discrete and to live on the dyadic rationals hence it is a pure point measure whose support as a closed set is the full interval 1 1 the space of paperfolding sequences has an ultrametric structure our example provides an illustration of some properties which were suggested to occur for pure states in spin glass models

universal bounds on the selfaveraging of random diffraction measures we consider diffraction at random point scatterers on general discrete point sets in restricted to a finite volume we allow for random amplitudes and random dislocations of the scatterers we investigate the speed of convergence of the random scattering measures applied to an observable towards its mean when the finite volume tends to infinity we give an explicit universal large deviation upper bound that is exponential in the number of scatterers the rate is given in terms of a universal function that depends on the point set only through the minimal distance between points and on the observable only through a suitable sobolev norm our proof uses a cluster expansion and also provides a central limit theorem

stability of depths of powers of edge ideals abstract let g be a graph and let i i g be its edge ideal in this paper we provide an upper bound of n from which depth r i g n is stationary and compute this limit explicitly this bound is always achieved if g has no cycles of length 4 and every its connected component is either a tree or a unicyclic graph

on diffraction by aperiodic structures this paper gives a rigorous treatment of some aspects of diffraction by aperiodic structures such as quasicrystals it analyses diffraction in the limit of the infinite system through an appropriately defined autocorrelation the main results are a justification of the standard way of calculating the diffraction spectrum of tilings obtained by the projection method and a proof of a variation on a conjecture by bombieri and taylor

translation representations and scattering by two intervals studying unitary one parameter groups in hilbert space u t h we show that a model for obstacle scattering can be built up to unitary equivalence with the use of translation representations for l2 functions in the complement of two finite and disjoint intervals the model encompasses a family of systems u t h for each we obtain a detailed spectral representation and we compute the scattering operator and scattering matrix we illustrate our results in the lax phillips model where u t h represents an acoustic wave equation in an exterior domain and in quantum tunneling for dynamics of quantum states

the inverse problem of pure point diffraction examples and open questions this paper considers some open questions related to the inverse problem of pure point diffraction in particular what types of objects may diffract and which of these may exhibit the same diffraction some diverse objects with the same simple lattice diffraction are constructed including a tempered distribution that is not a measure and it is shown that there are uncountably many such objects in the diffraction solution class of any pure point diffraction measure with an infinite supporting set

numerical integration of the differential matrix riccati equation two new bernoulli substitution methods for solving the riccati differential equation are tested numerically against direct integration of the riccati equation the chandrasekhar algorithm and the davison maki method on a large set of problems taken from the literature the first of these new methods was developed for the time invariant case and uses the matrix analog of completing the square to transform the problem to a bisymmetric riecati equation whose solution can be given explicitly in terms of a matrix exponential of order n this method is fast and accurate when the extremal solutions of the associated algebraic riccati equation are well separated the second new method was developed as a means of eliminating the instabilities associated with the davison maki algorithm by using reinitialization at each time step the davison maki algorithm can be recast as a recursion which is over three times faster than the original method and is easily shown to be stable for both time invariant and time dependent problems from the results of our study we conclude that the modified davison maki method gives superior performance except for those problems where the number of observers and controllers is small relative to the number of states in which ease the chandrasekhar algorithm is better

limit quasi periodic point sets as quasicrystals with p adic internal spaces model sets or cut and project sets provide a familiar and commonly used method of constructing and studying nonperiodic point sets here we extend this method to situations where the internal spaces are no longer euclidean but instead spaces with p adic topologies or even with mixed euclidean p adic topologies we show that a number of well known tilings precisely fit this form including the chair tiling and the robinson square tilings thus the scope of the cut and project formalism is considerably larger than is usually supposed applying the powerful consequences of model sets we derive the diffractive nature of these tilings

close packed dimers on the line diffraction versus dynamical spectrum the translation action of d on a translation bounded measure leads to an interesting class of dynamical systems with a rather rich spectral theory in general the diffraction spectrum of which is the carrier of the diffraction measure lives on a subset of the dynamical spectrum it is known that under some mild assumptions a pure point diffraction spectrum implies a pure point dynamical spectrum the opposite implication always being true for other systems the diffraction spectrum can be a proper subset of the dynamical spectrum as was pointed out for the thue morse sequence with singular continuous diffraction by van enter and mi kisz j stat phys 66 1147 1153 1992 here we construct a random system of close packed dimers on the line that have some underlying long range periodic order as well and display the same type of phenomenon for a system with absolutely continuous spectrum an interpretation in terms of atomic versus molecular spectrum suggests a way to come to a more general correspondence between these two types of spectra

the torus parametrization of quasiperiodic li classes the torus parametrization of quasiperiodic local isomorphism classes is introduced and used to determine the number of elements in such a class with special symmetries or inflation properties the method is explained in an illustrative fashion for some widely used tiling classes with golden mean rescaling namely for the fibonacci chain one dimensional the triangle and penrose patterns two dimensional and for kramer s and danzer s icosahedral tilings three dimensional we obtain a rather complete picture of the orbit structure within these classes and also discuss various general results

spectral and topological properties of a family of generalised thue morse sequences the classic middle thirds cantor set leads to a singular continuous measure via a distribution function that is known as the devil s staircase the support of the cantor measure is a set of zero lebesgue measure here we discuss a class of singular continuous measures that emerge in mathematical diffraction theory and lead to somewhat similar distribution functions yet with significant differences various properties of these measures are derived in particular these measures have supports of full lebesgue measure and possess strictly increasing distribution functions in this sense they mark the opposite end of what is possible for singular continuous measures for each member of the family the underlying dynamical system possesses a topological factor with maximal pure point spectrum and a close relation to a solenoid which is the kronecker factor of the system the inflation action on the continuous hull is sufficiently explicit to permit the calculation of the corresponding dynamical zeta functions this is achieved as a corollary of analysing the anderson putnam complex for the determination of the cohomological invariants of the corresponding tiling spaces

overlap distributions for deterministic systems with many pure states we discuss the parisi overlap distribution function for various deterministic systems with uncountably many pure ground states we show examples of trivial countably discrete and continuous distributions

quasicrystals and almost periodicity we give in this paper topological and dynamical characterizations of mathematical quasicrystals let open image in new window denote the space of uniformly discrete subsets of the euclidean space let open image in new window denote the elements of open image in new window that admit an autocorrelation measure a patterson set is an element of open image in new window such that the fourier transform of its autocorrelation measure is discrete patterson sets are mathematical idealizations of quasicrystals we prove that s open image in new window is a patterson set if and only if s is almost periodic in open image in new window open image in new window where open image in new window denotes the besicovitch topology let be an ergodic random element of open image in new window we prove that is almost surely a patterson set if and only if the dynamical system has a discrete spectrum as an illustration we study deformed model sets

on the fourier analysis of measures with meyer set support abstract in this paper we show the existence of the generalized eberlein decomposition for fourier transformable measures with meyer set support we prove that each of the three components is also fourier transformable and has meyer set support we obtain that each of the pure point absolutely continuous and singular continuous components of the fourier transform is a strong almost periodic measure and hence is either trivial or has relatively dense support we next prove that the fourier transform of a measure with meyer set support is norm almost periodic and hence so is each of the pure point absolutely continuous and singular continuous components we show that a measure with meyer set support is fourier transformable if and only if it is a linear combination of positive definite measures which can be chosen with meyer set support solving a particular case of an open problem we complete the paper by discussing some applications to the diffraction of weighted dirac combs with meyer set support

almost periodic measures and bragg diffraction in this paper we prove that the cone of positive positive definite discrete and strong almost periodic measures over a compact locally compact abelian group g has an interesting property given any positive and positive definite measure smaller than some measure in the strong almost periodic part s of is also in we then use this result to prove that given a positive weighted dirac comb with finite local complexity and pure point diffraction any positive dirac comb less than has either a trivial bragg spectrum or a relatively dense set of bragg peaks

weighted dirac combs with pure point diffraction a class of translation bounded complex measures which have the form of weighted dirac combs on locally compact abelian groups is investigated given such a dirac comb we are interested in its diffraction spectrum which emerges as the fourier transform of the autocorrelation measure we present a sufficient set of conditions to ensure that the diffraction measure is a pure point measure simultaneously we establish a natural link to the cut and project formalism and to the theory of almost periodic measures our conditions are general enough to cover the known theory of model sets but also to include examples such as the visible lattice points

brief paper eigenvector approach for order reduction of singularly perturbed linear quadratic optimal control problems in this paper we show how to decompose the singularly perturbed algebraic riccati equation and the corresponding linear quadratic optimal control problem at steady state in terms of reduced order pure slow and pure fast problems by using the eigenvector approach the eigenvector approach should be used for decomposition of singularly perturbed control systems in the cases when the singular perturbation parameter is not very small in such cases the decomposition methods based on series expansions fixed point iterations subspace iterations and newton iterations either fail to produce solutions of the corresponding algebraic equations or display very slow convergence

on the minors of an incidence matrix and its smith normal form abstract consider the vertex edge incidence matrix of an arbitrary undirected loopless graph we completely determine the possible minors of such a matrix these depend on the maximum number of vertex disjoint odd cycles i e the odd tulgeity of the graph the problem of determining this number is shown to be np hard turning to maximal minors we determine the rank of the incidence matrix this depends on the number of components of the graph containing no odd cycle we then determine the maximum and minimum absolute values of the maximal minors of the incidence matrix as well as its smith normal form these results are used to obtain sufficient conditions for relaxing the integrality constraints in integer linear programming problems related to undirected graphs finally we give a sufficient condition for a system of equations whose coefficient matrix is an incidence matrix to admit an integer solution

concentration inequalities for functions of gibbs fields with application to diffraction and random gibbs measures we derive useful general concentration inequalities for functions of gibbs fields in the uniqueness regime we also consider expectations of random gibbs measures that depend on an additional disorder field and prove concentration w r t the disorder field both fields are assumed to be in the uniqueness regime allowing in particular for non independent disorder fields the modification of the bounds compared to the case of an independent field can be expressed in terms of constants that resemble the dobrushin contraction coefficient and are explicitly computable on the basis of these inequalities we obtain bounds on the deviation of a diffraction pattern created by random scatterers located on a general discrete point set in euclidean space restricted to a finite volume here we also allow for thermal dislocations of the scatterers around their equilibrium positions extending recent results for independent scatterers we give a universal upper bound on the probability of a deviation of the random scattering measures applied to an observable from its mean the bound is exponential in the number of scatterers with a rate that involves only the minimal distance between points in the point set

continuity of eigenfunctions of uniquely ergodic dynamical systems and intensity of bragg peaks we study uniquely ergodic dynamical systems over locally compact sigma compact abelian groups we characterize uniform convergence in wiener wintner type ergodic theorems in terms of continuity of the limit our results generalize and unify earlier results of robinson and assani respectively