tight frames for cyclotomic fields and other rational vector spaces abstract here we consider the construction of tight frames for rational vector spaces this is a subtle question because the inner products on q d are not all isomorphic we show that a tight frame for c d can be arbitrarily approximated by a tight frame with vectors in q i q d and hence there are many tight frames for rational inner product spaces we investigate the minimal field for which there is a tight frame with a given gramian we then consider the rational vector space given the cyclotomic field q with a primitive n th root of unity we give a simple formula for the unique inner product which makes the n th roots 1 2 n 1 into a tight frame for q from this we conclude that the associated canonical coordinates have many nice properties e g multiplication in q corresponds to convolution which makes them well suited to computation along the way we give a detailed description of the space of q linear dependencies between the n th roots which includes a cyclically invariant tight frame

local entropy averages and projections of fractal measures we show that for families of measures on euclidean space which satisfy an ergodic theoretic form of self similarity under the operation of re scaling the dimension of linear images of the measure behaves in a semi continuous way we apply this to prove the following conjecture of furstenberg let m n be integers which are not powers of the same integer and let x y be closed subsets of the unit interval which are invariant respectively under times m mod 1 and times n mod 1 then for any non zero t dim x ty min 1 dim x dim y a similar result holds for invariant measures and gives a simple proof of the rudolph johnson theorem our methods also apply to many other classes of conformal fractals and measures as another application we extend and unify results of peres shmerkin and nazarov and of moreira concerning projections of products self similar measures and gibbs measures on regular cantor sets we show that under natural irreducibility assumptions on the maps in the ifs the image measure has the maximal possible dimension under any linear projection other than the coordinate projections we also present applications to bernoulli convolutions and to the images of fractal measures under differentiable maps

the defect of generalized fourier matrices the n n complex hadamard matrices form a real algebraic manifold cn we have cn mn t nun and following tadej and yczkowski we investigate here the computation of the enveloping tangent space t hcn thmn t thnun and notably of its dimension d h dim t hcn called undephased defect of h our main result is an explicit formula for the defect of the fourier matrix fg associated to an arbitrary finite abelian group g zn1 znr we also comment on the general question does the associated quantum permutation group see the defect with a probabilistic speculation involving diaconis shahshahani type variables

integral geometric properties of capacities let m and n be positive integers 0 i for ix y we consider relations between the capacities ck e and ch e n a when e c rn and a varies over the m dimensional affine subspaces of r for example we prove that if e is compact ck e c f ch e n a d ma where x m is a rigidly invariant measure and c is a positive constant depending only on n and m

an algorithmic introduction to numerical simulation of stochastic differential equations a practical and accessible introduction to numerical methods for stochastic differential equations is given the reader is assumed to be familiar with euler s method for deterministic differential equations and to have at least an intuitive feel for the concept of a random variable however no knowledge of advanced probability theory or stochastic processes is assumed the article is built around 10 matlab programs and the topics covered include stochastic integration the euler maruyama method milstein s method strong and weak convergence linear stability and the stochastic chain rule

design of a quantum projection filter this article develops a quantum projection filtering approach to approximating the quantum filter equation based on it hat rm o stochastic taylor expansions and quantum information geometric techniques the proposed approximation scheme is designed so that the truncated taylor expansion of the difference between the true quantum trajectory and its approximation on a lower dimensional differential submanifold is minimized through an orthogonal projection operation in the quantum fisher metric in addition a convenient design for a special class of open quantum systems is formulated simulation results from a numerical example demonstrate the approximation capability of the proposed quantum projection filter

analysis of orthogonality and of orbits in affine iterated function systems we introduce a duality for affine iterated function systems aifs which is naturally motivated by group duality in the context of traditional harmonic analysis our affine systems yield fractals defined by iteration of contractive affine mappings we build a duality for such systems by scaling in two directions fractals in the small by contractive iterations and fractals in the large by recursion involving iteration of an expansive matrix by a fractal in the small we mean a compact attractor x supporting hutchinson s canonical measure and we ask when is a spectral measure i e when the hilbert space l 2 mu has an orthonormal basis onb of exponentials e lambda lambda in lambda we further introduce a fourier duality using a matched pair of such affine systems using next certain extreme cycles and positive powers of the expansive matrix we build fractals in the large which are modeled on lacunary fourier series and which serve as spectra for x our two main results offer simple geometric conditions allowing us to decide when the fractal in the large is a spectrum for x our results in turn are illustrated with concrete sierpinski like fractals in dimensions 2 and 3

the green s function for the huckel tight binding model applications of the huckel tight binding model are ubiquitous in quantum chemistry and solid state physics the matrix representation of this model is isomorphic to an unoriented vertex adjacency matrix of a bipartite graph which is also the laplacian matrix plus twice the identity in this paper we analytically calculate the determinant and when it exists the inverse of this matrix in connection with the green s function g of the n n huckel matrix a corollary is a closed form expression for a harmonic sum eq 12 we then extend the results to d dimensional lattices whose linear size is n the existence of the inverse becomes a question of number theory we prove a new theorem in number theory pertaining to vanishing sums of cosines and use it to prove that the inverse exists if and only if n 1 and d are odd and d is smaller than the smallest divisor of n 1 we corroborate our results by demonstrating the entry patterns of the green s function and discuss applications related to transpo

asymptotic resonance interaction of modes and subharmonic bifurcation we study the existence of small amplitude oscillations near elliptic equilibria of autonomous systems which mix different normal modes the reference problem is the fermi pasta ulam model a chain of nonlinear oscillators with nearest neighborhood interaction we develop a new bifurcation approach that locates secondary bifurcations from the unimodal primary branches two sufficient conditions for bifurcation are given one involves only the arithmetic properties of the eigenvalues of the linearized system asymptotic resonance while the other takes into account the nonlinear character of the interaction between normal modes nonlinear coupling both conditions are checked for the fermi pasta ulam problem

solving the kpz equation we introduce a new concept of solution to the kpz equation which is shown to extend the classical cole hopf solution this notion provides a factorisation of the cole hopf solution map into a universal measurable map from the probability space into an explicitly described auxiliary metric space composed with a new solution map that has very good continuity properties the advantage of such a formulation is that it essentially provides a pathwise notion of a solution together with a very detailed approximation theory in particular our construction completely bypasses the cole hopf transform thus laying the groundwork for proving that the kpz equation describes the uctuations of systems in the kpz universality class as a corollary of our construction we obtain very detailed new regularity results about the solution as well as its derivative with respect to the initial condition other byproducts of the proof include an explicit approximation to the stationary solution of the kpz equation a well posedness result for the fokker planck equation associated to a particle diusing in a rough space time dependent potential and a new periodic homogenisation result for the heat equation with a space time periodic potential one ingredient in our construction is an example of a non gaussian rough path such that the area process of its natural approximations needs to be renormalised by a diverging term for the approximations to converge

on prime root of unity sequences with perfect periodic correlation in this paper perfect root of unity sequences prus with entries in alpha p x in complexc x p 1 where p is a prime are studied a lower bound on the number of distinct phase

a new look at proper orthogonal decomposition we investigate some basic properties of the proper orthogonal decomposition pod method as it is applied to data compression and model reduction of finite dimensional nonlinear systems first we provide an analysis of the errors involved in solving a nonlinear ode initial value problem using a pod reduced order model then we study the effects of small perturbations in the ensemble of data from which the pod reduced order model is constructed on the reduced order model we explain why in some applications this sensitivity is a concern while in others it is not we also provide an analysis of computational complexity of solving an ode initial value problem and study the computational savings obtained by using a pod reduced order model we provide several examples to illustrate our theoretical results

equiangular tight frames and unistochastic matrices we demonstrate that a complex equiangular tight frame composed of n vectors in dimension d denoted etf d n exists if and only if a certain bistochastic matrix univocally determined by n and d belongs to a special class of unistochastic matrices this connection allows us to find new complex etfs in infinitely many dimensions and to derive a method to introduce non trivial free parameters in etfs we present an explicit six parametric family of complex etf 6 16 which defines a family of symmetric povms minimal and maximal possible average entanglement of the vectors within this qubit qutrit family are described furthermore we propose an efficient numerical procedure to compute the unitary matrix underlying a unistochastic matrix which we apply to find all existing classes of complex etfs containing up to 20 vectors

buffon s needle estimates for rational product cantor sets let s infty a infty times b infty be a self similar product cantor set in the complex plane defined via s infty bigcup j 1 l t j s infty where t j bbb c to bbb c have the form t j z 1 over l z z j and z 1 ldots z l a ib for some a b subset bbb r with a b 1 and a b l let s n be the l n neighborhood of s infty or equivalently up to constants its n th cantor iteration we are interested in the asymptotic behavior as n to infty of the it favard length of s n defined as the average with respect to direction length of its 1 dimensional projections if the sets a and b are rational and have cardinalities at most 6 then the favard length of s n is bounded from above by cn p log log n for some p0 the same result holds with no restrictions on the size of a and b under certain implicit conditions concerning the generating functions of these sets this generalizes the earlier results of nazarov perez volberg l aba zhai and bond volberg

preserving symmetries in the proper orthogonal decomposition the proper orthogonal decomposition pod also called karhunen loeve expansion has been recently used in turbulence to derive optimally fast converging bases of spatial functions leading to efficient finite truncations whether a finite number of these modes can be used in numerical simulations to derive an accurate finite set of ordinary differential equations over a certain range of bifurcation parameter values still remains an open question it is shown here that a necessary condition for achieving this goal is that the truncated system inherit the symmetry properties of the original infinite dimensional system in most cases this leads to a systematic involvement of the symmetry group in deriving a new expansion basis called the symmetric pod basis the kuramoto sivashinsky equation with periodic boundary conditions is used as a paradigm to illustrate this point of view however the conclusion is general and can be applied to other equations such as the navier stokes equations the complex g

representations of quantum permutation algebras we develop a combinatorial approach to the quantum permutation algebras as hopf images of representations of type pi a s n to b h we discuss several general problems including the commutativity and cocommutativity ones the existence of tensor product or free wreath product decompositions and the tannakian aspects of the construction the main motivation comes from the quantum invariants of the complex hadamard matrices we show here that under suitable regularity assumptions the computations can be performed up to n 6

μm d orthogonality and compatible pair abstract let m d be a self affine measure associated with an expanding integer matrix m m n z and a finite subset d z n in the present paper we study the m d orthogonality and compatible pair conditions as well as relations between them the research here is based on the structure of vanishing sums of roots of unity and is closely related to the problem of spectral self affine measure

evaluation of proper orthogonal decomposition based decomposition techniques applied to parameter dependent nonturbulent flows the proper orthogonal decomposition pod method as a systematic technique to analyze parameter dependent problems may be inappropriate as a practical tool for generating low dimensional models we propose a weighted pod w pod as an alternative to give higher priority to low energetic or important modes by simply weighting a predefined pod p pod is suggested where modes are selected not only on the basis of energy but also on some a priori knowledge of the system the techniques are tested on a flow problem undergoing steady or unsteady transition but may in general be applied to nonlinear systems of discretized pdes closed by additional linear algebraic equations various residual functions to measure the quality of the reduced models are introduced they provide lower limits of residual energy to ensure an adequate physical representation for a given modeling it is shown that modes extracted locally may generally not contain information regarding the global dynamics and the dynamics may not be correctly predicted over a longer range of model parameters it is demonstrated that by weighting and predefining base vectors it is possible to improve the pod technique s capability to generate low dimensional models

constants of cyclotomic derivations let k x k x0 xn 1 and k y k y0 yn 1 be the polynomial rings in n 3 variables over a field k of characteristic zero containing the n th roots of unity let d be the cyclotomic derivation of k x and let be the factorisable derivation of k y associated with d that is d xj xj 1 and yj yj yj 1 yj for all j zn we describe polynomial constants and rational constants of these derivations we prove among others that the field of constants of d is a field of rational functions over k in n n variables and that the ring of constants of d is a polynomial ring if and only if n is a power of a prime moreover we show that the ring of constants of is always equal to k v where v is the product y0 yn 1 and we describe the field of constants of in two cases when n is power of a prime and when n pq

non spectrality of self affine measures abstract under some supportive evidences of known results about non spectral self affine measures li proposed a conjecture in 18 j funct anal 255 2008 which is so called non spectral conjecture in this paper we prove that li s non spectral conjecture is true under some suitable condition

tiling the line with translates of one tile a region t is a closed subset of the real line of positive finite lebesgue measure which has a boundary of measure zero call a region t a tile if bbb r can be tiled by measure disjoint translates of t for a bounded tile all tilings of bbb r with its translates are periodic and there are finitely many translation equivalence classes of such tilings the main result of the paper is that for any tiling of bbb r by a bounded tile any two tiles in the tiling differ by a rational multiple of the minimal period of the tiling from it we deduce a structure theorem characterizing such tiles in terms of complementing sets for finite cyclic groups

on the well posedness of the stochastic allen cahn equation in two dimensions white noise driven nonlinear stochastic partial differential equations spdes of parabolic type are frequently used to model physical systems in space dimensions d 1 2 3 whereas existence and uniqueness of weak solutions to these equations are well established in one dimension the situation is different for d 2 despite their popularity in the applied sciences higher dimensional versions of these spde models are generally assumed to be ill posed by the mathematics community we study this discrepancy on the specific example of the two dimensional allen cahn equation driven by additive white noise since it is unclear how to define the notion of a weak solution to this equation we regularize the noise and introduce a family of approximations based on heuristic arguments and numerical experiments we conjecture that these approximations exhibit divergent behavior in the continuum limit the results strongly suggest that shrinking the mesh size in simulations of the two dimensional white noise driven allen cahn equation does not lead to the recovery of a physically meaningful limit

the uncertainty principle and a generalization of a theorem of tao abstract let g be a finite abelian group if f g c is a nonzero function with fourier transform f the classical uncertainty principle states that supp f supp f g recently tao showed that if g is cyclic of prime order p then in fact a stronger inequality supp f supp f p 1 holds in this paper we use representation theory of the unitary group and weyl s character formula to derive a generalization of tao s result for arbitrary finite cyclic groups

frames for vector spaces and affine spaces a finite frame for a finite dimensional hilbert space is simply a spanning sequence we show that the linear functionals given by the dual frame vectors do not depend on the inner product and thus it is possible to extend the frame expansion and other elements of frame theory to any finite spanning sequence for a vector space the corresponding coordinate functionals generalise the dual basis the case when the vectors are linearly independent and are characterised by the fact that the associated gramian matrix is an orthogonal projection existing generalisations of the frame expansion to banach spaces involve an analogue of the frame bounds and frame operator the potential applications of our results are considerable whenever there is a natural spanning set for a vector space computations can be done directly with it in an efficient and stable way we illustrate this with a diverse range of examples including multivariate spline spaces generalised barycentric coordinates and vector spaces over the rationals such as the cyclotomic fields

a differential geometric approach to nonlinear filtering the projection filter this paper presents a new and systematic method of approximating exact nonlinear filters with finite dimensional filters using the differential geometric approach to statistics the projection filter is defined rigorously in the case of exponential families a convenient exponential family is proposed which allows one to simplify the projection filter equation and to define an a posteriori measure of the local error of the projection filter approximation finally simulation results are discussed for the cubic sensor problem

a theory of regularity structures we introduce a new notion of regularity structure that provides an algebraic framework allowing to describe functions and or distributions via a kind of jet or local taylor expansion around each point the main novel idea is to replace the classical polynomial model which is suitable for describing smooth functions by arbitrary models that are purpose built for the problem at hand in particular this allows to describe the local behaviour not only of functions but also of large classes of distributions we then build a calculus allowing to perform the various operations multiplication composition with smooth functions integration against singular kernels necessary to formulate fixed point equations for a very large class of semilinear pdes driven by some very singular typically random input this allows for the first time to give a mathematically rigorous meaning to many interesting stochastic pdes arising in physics the theory comes with convergence results that allow to interpret the solutions obtained in this way as limits of classical solutions to regularised problems possibly modified by the addition of diverging counterterms these counterterms arise naturally through the action of a renormalisation group which is defined canonically in terms of the regularity structure associated to the given class of pdes our theory also allows to easily recover many existing results on singular stochastic pdes kpz equation stochastic quantisation equations burgers type equations and to understand them as particular instances of a unified framework one surprising insight is that in all of these instances local solutions are actually smooth in the sense that they can be approximated locally to arbitrarily high degree as linear combinations of a fixed family of random functions distributions that play the role of polynomials in the theory as an example of a novel application we solve the long standing problem of building a natural markov process that is symmetric with respect to the finite volume measure describing the phi 4 3 euclidean quantum field theory it is natural to conjecture that the markov process built in this way describes the glauber dynamic of 3 dimensional ferromagnets near their critical temperature

convolution idempotents with a given zero set we investigate the structure of n length discrete signals h satisfying h h h that vanish on a given set of indices we motivate this problem from examples in sampling fuglede s conjecture and orthogonal interpolation of bandlimited signals when n p m is a prime power we characterize all such h with a prescribed zero set in terms of base p expansions of nonzero indices in mathcal f 1 h

energy solutions of kpz are unique the kardar parisi zhang kpz equation is conjectured to universally describe the fluctuations of weakly asymmetric interface growth here we provide the first intrinsic well posedness result for the kpz equation on the real line by showing that its energy solutions as introduced by goncalves and jara in gj14 and refined in gj13 are unique together with the convergence results of gj14 and many follow up papers this establishes the weak kpz universality conjecture for a wide class of models our proof builds on an observation of funaki and quastel fq15 a remarkable consequence is that the energy solution to the kpz equation is not equal to the cole hopf solution but it involves an additional drift t 12

probability distribution of the free energy of the continuum directed random polymer in 1 1 dimensions we consider the solution of the stochastic heat equation tz d 1 2 x zz p w with delta function initial condition z t d0 x dixd0 whose logarithm with appropriate normalization is the free energy of the con tinuum directed polymer or the hopf cole solution of the kardar parisi zhang equation with narrow wedge initial conditions we obtain explicit formulas for the one dimensional marginal distributions the crossover distributions which interpolate between a standard gaussian dis tribution small time and the gue tracy widom distribution large time the proof is via a rigorous steepest descent analysis of the tracy widom formula for the asymmetric simple exclusion process with antishock initial data which is shown to converge to the continuum equations in an appropriate weakly asymmetric limit the limit also describes the crossover behavior between the symmetric and asymmetric exclusion processes 2010 wiley periodicals inc

frame theoretic analysis of dft codes with erasures discrete fourier transform dft codes have been used as joint source channel codes in order to provide robustness against channel errors and erasures this paper presents a frame theoretic analysis of lowpass dft codes with erasures first it is shown that such codes make a special class of frames then the message reconstructions by the frame theoretic signal space projection and coding theoretic syndrome decoding of erasures are analyzed and their equivalence is proved the error performance of lowpass dft codes is analyzed using the frame theory the analysis helps in classifying the subframes of a dft frame into several classes this classification leads to the development of packetization criteria in order to guarantee minimum mean square reconstruction error and thus to minimize the effect of the quantization error simulation results substantiate the presented theory and the proposed packetization schemes

minimal vanishing sums of roots of unity with large coefficients a vanishing sum where n is a primitive nth root of unity and the a i s are non negative integers is called minimal if the coefficient vector a 0 a n 1 does not properly dominate the coefficient vector of any other such non zero sum we show that for every c there is a minimal vanishing sum of nth roots of unity with its greatest coefficient equal to c where n is of the form 3pq for odd primes p q this solves an open problem posed by lenstra jr

on self similar sets with overlaps and inverse theorems for entropy we study the dimension of self similar sets and measures on the line we show that if the dimension is less than the generic bound of minf1 sg where s is the similarity dimension then there are superexponentially close cylinders at all small enough scales this is a step towards the conjecture that such a dimension drop implies exact overlaps and conrms it when the generating similarities have algebraic coecients as applications we prove furstenberg s conjecture on projections of the one dimensional sierpinski gasket and achieve some progress on the bernoulli convolutions problem and more generally on problems about parametric families of self similar measures the key tool is an inverse theorem on the structure of pairs of probability measures whose mean entropy at scale 2 n has only a small amount of growth under convolution

scaling limits of wick ordered kpz equation consider the kpz equation x d where w t x is a space time white noise this paper investigates the question of whether for some exponents and z k u k z t kx converges in some sense as k to infty and if so what are the values of these exponents the non linear term in the kpz equation is interpreted as a wick product and the equation is solved in a suitable space of stochastic distributions the main tools for establishing the scaling properties of the solution are those of white noise analysis in particular the wiener chaos expansion a notion of convergence in law in the sense of wiener chaos is formulated and convergence in this sense of k u k z t kx as k is established for various values of and z depending on the dimension d

on vanishing sums of roots of unity an unsolved problem in number theory asked the following for a given natural number m what are the possible integers n for which there exist mth roots of unity 1 n c such that 1 n 0 we show in this paper that the set of all possible n s is exactly the collection of n combinations of the prime divisors of m where n denotes the set of all non negative integers the proof is long and involves a subtle analysis of minimal vanishing sums of mth roots of unity couched in the setting of integral group rings of finite cyclic groups our techniques also recovered with ease some of the classical results on vanishing sums of roots of unity such as those of redei de bruijn and schoenberg

stochastic burgers and kpz equations from particle systems we consider two strictly related models a solid on solid interface growth model and the weakly asymmetric exclusion process both on the one dimensional lattice it has been proven that in the diffusive scaling limit the density field of the weakly asymmetric exclusion process evolves according to the burgers equation and the fluctuation field converges to a generalized ornstein uhlenbeck process we analyze instead the density fluctuations beyond the hydrodynamical scale and prove that their limiting distribution solves the non linear burgers equation with a random noise on the density current for the solid on solid model we prove that the fluctuation field of the interface profile if suitably rescaled converges to the kardar parisi zhang equation this provides a microscopic justification of the so called kinetic roughening i e the non gaussian fluctuations in some non equilibrium processes our main tool is the cole hopf transformation and its microscopic version we also develop a mathematical theory for the macroscopic equations

rough burgers like equations with multiplicative noise we construct solutions to vector valued burgers type equations perturbed by a multiplicative space time white noise in one space dimension due to the roughness of the driving noise solutions are not regular enough to be amenable to classical methods we use the theory of controlled rough paths to give a meaning to the spatial integrals involved in the definition of a weak solution subject to the choice of the correct reference rough path we prove unique solvability for the equation and we show that our solutions are stable under smooth approximations of the driving noise

rough stochastic pdes in this article we show how the theory of rough paths can be used to provide a notion of solution to a class of nonlinear stochastic pdes of burgers type that exhibit too high spatial roughness for classical analytical methods to apply in fact the class of spdes that we consider is genuinely ill posed in the sense that different approximations to the nonlinearity may converge to different limits using rough path theory a pathwise notion of solution to these spdes is formulated and we show that this yields a well posed problem that is stable under a large class of perturbations including the approximation of the rough driving noise by a mollified version and the addition of hyperviscosity we also show that under certain structural assumptions on the coefficients the spdes under consideration generate a reversible markov semigroup with respect to a diffusion measure that can be given explicitly

nonlinear fluctuations of weakly asymmetric interacting particle systems we introduce what we call the second order boltzmann gibbs principle which allows one to replace local functionals of a conservative one dimensional stochastic process by a possibly nonlinear function of the conserved quantity this replacement opens the way to obtain nonlinear stochastic evolutions as the limit of the fluctuations of the conserved quantity around stationary states as an application of this second order boltzmann gibbs principle we introduce the notion of energy solutions of the kpz and stochastic burgers equations under minimal assumptions we prove that the density fluctuations of one dimensional stationary weakly asymmetric conservative particle systems are sequentially compact and that any limit point is given by energy solutions of the stochastic burgers equation we also show that the fluctuations of the height function associated to these models are given by energy solutions of the kpz equation in this sense unfortunately we lack a uniqueness result for these energy solutions we conjecture these solutions to be unique and we show some regularity results for energy solutions of the kpz burgers equation supporting this conjecture

singular perturbations to semilinear stochastic heat equations we consider a class of singular perturbations to the stochastic heat equation or semilinear variations thereof the interesting feature of these perturbations is that as the small parameter e tends to zero their solutions converge to the wrong limit i e they do not converge to the solution obtained by simply setting e 0 a similar effect is also observed for some formally small stochastic perturbations of a deterministic semilinear parabolic pde our proofs are based on a detailed analysis of the spatially rough component of the equations combined with a judicious use of gaussian concentration inequalities

quantitative visibility estimates for unrectifiable sets in the plane the visibility of a planar set s from a point a is defined as the normalized size of the radial projection of s from a to the unit circle centered at a simon and solomyak 27 proved that unrectifiable self similar one sets are invisible from every point in the plane we quantify this by giving an upper bound on the visibility of neighbourhoods of such sets we also prove lower bounds on the visibility of neighborhoods of more general sets based in part on bourgain s discretized sum product estimates in 7

sufficient conditions for a conjecture of ryser about hadamard circulant matrices let h be a hadamard circulant matrix of order n 4h2 where h 1 is an odd positive integer with at least two prime divisors such that the exponents of the prime numbers that divide h are big enough and such that the nonzero coefficients of the cyclotomic polynomial n t are bounded by a constant independent of n then for all the n n th primitive roots w of 1 p w n is not an algebraic integer in the cyclotomic field k q w where p t is the representer polynomial of h and is the euler function this implies that p w is not a real number