entropy of random quantum states the authors use a simple generating function to calculate exactly the entropy of random quantum states for finite dimensional hilbert spaces over real complex and quaternionic scalars this allows them to extend their previous formula for the quantum correlation information of a state determination apparatus to include real and quaternionic von neumann analysers

separation of unistochastic matrices from the double stochastic ones recovery of a 3 3 unitary matrix from experimental data the aim of the paper is to provide a constructive method for recovering a unitary matrix from experimental data since there is a natural immersion of unitary matrices within the set of double stochastic ones the problem to solve is to find necessary and sufficient criteria that separate the two sets a complete solution is provided for the three dimensional case accompanied by a 2 test necessary for the reconstruction of a unitary matrix from error affected data

entropy and hadamard matrices the entropy of an orthogonal matrix is defined it provides a new interpretation of hadamard matrices as those that saturate the bound for entropy it appears to be a useful morse function on the group manifold it has sharp maximum and other saddle points the matrices corresponding to the maxima for three and five dimensions are presented they are integer matrices up to a rescaling

permutation equivalence classes of kronecker products of unitary fourier matrices kronecker products of unitary fourier matrices play an important role in solving multilevel circulant systems by a multidimensional fast fourier transform they are also special cases of complex hadamard zeilinger matrices arising in many problems of mathematics and theoretical physics the main result of the paper is splitting the set of all kronecker products of unitary fourier matrices into permutation equivalence classes the choice of the permutation equivalence to relate the products is motivated by the quantum information theory problem of constructing maximally entangled bases of finite dimensional quantum systems permutation inequivalent products can be used to construct inequivalent in a certain sense maximally entangled bases

thermodynamically optimal creation of correlations correlations lie at the heart of almost all scientific predictions it is therefore of interest to ask whether there exist general limitations to the amount of correlations that can be created at a finite amount of invested energy within quantum thermodynamics such limitations can be derived from first principles in particular it can be shown that establishing correlations between initially uncorrelated systems in a thermal background has an energetic cost this cost which depends on the system dimension and the details of the energy level structure can be bounded from below but whether these bounds are achievable is an open question here we put forward a framework for studying the process of optimally correlating identical thermal quantum systems the framework is based on decompositions into subspaces that each support only states with diagonal classical marginals using methods from stochastic majorisation theory we show that the creation of correlations at minimal energy cost is possible for all pairs of three and four dimensional quantum systems for higher dimensions we provide sufficient conditions for the existence of such optimally correlating operations which we conjecture to exist in all dimensions

on entropy preserving stochastic averages abstract when an n n doubly stochastic matrix a acts on r n on the left as a linear transformation and p is an n long probability vector we refer to the new probability vector ap as the stochastic average of the pair a p let n denote the set of pairs a p whose stochastic average preserves the entropy of p h ap h p n is a subset of b n n where b n is the birkhoff polytope and n is the probability simplex we characterize n and determine its geometry topology and combinatorial structure for example we find that a p n if and only if a t ap p we show that for any n n is a connected set and is in fact piecewise linearly contractible in b n n we write n as a finite union of product subspaces in two distinct ways we derive the geometry of the fibers a and p of n 3 is worked out in detail our analysis exploits the convexity of x log x and the structure of an efficiently computable bipartite graph that we associate to each ds matrix this graph also lets us represent such a matrix in a permutation equivalent block diagonal form where each block is doubly stochastic and fully indecomposable

isolated hadamard matrices from mutually unbiased product bases a new construction of complex hadamard matrices of composite order d pq with primes p q is presented which is based on pairs of mutually unbiased bases containing only product states for product dimensions d 100 we illustrate the method by deriving many previously unknown complex hadamard matrices we obtain at least 12 new isolated matrices of butson type with orders ranging from 9 to 91

a generalization of the aubin lions simon compactness lemma for problems on moving domains abstract this work addresses an extension of the aubin lions simon compactness result to generalized bochner spaces l 2 0 t h t where h t is a family of hilbert spaces parameterized by t a compactness result of this type is needed in the study of the existence of weak solutions to nonlinear evolution problems governed by partial differential equations defined on moving domains we identify the conditions on the regularity of the domain motion in time under which our extension of the aubin lions simon compactness result holds concrete examples of the application of the compactness theorem are presented including a classical problem for the incompressible navier stokes equations defined on a given non cylindrical domain and a class of fluid structure interaction problems for the incompressible navier stokes equations coupled to the elastodynamics of a koiter shell the compactness result presented in this manuscript is crucial in obtaining constructive existence proofs to nonlinear moving boundary problems using rothe s method

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

on biunimodular vectors for unitary matrices abstract a biunimodular vector of a unitary matrix a u n is a vector v t n c n such that av t n as well over the last 30 years the sets of biunimodular vectors for fourier matrices have been the object of extensive research in various areas of mathematics and applied sciences here we broaden this basic harmonic analysis perspective and extend the search for biunimodular vectors to arbitrary unitary matrices this search can be motivated in various ways the main motivation is provided by the fact that the existence of biunimodular vectors for an arbitrary unitary matrix allows for a natural understanding of the structure of all unitary matrices

some formulas for invariant phases of unitary matrices by jarlskog we describe a calculation of jarlskog s determinant in the case of n 4 in detail next we investigate some formulas for invariant phases of unitary matrices and derive some explicit relations for them

families of complex hadamard matrices what is the dimension of a smooth family of complex hadamard matrices including the fourier matrix we address this problem with a power series expansion studying all dimensions up to 100 we find that the first order result is misleading unless the dimension is 6 or a power of a prime in general the answer depends critically on the prime number decomposition of the dimension our results suggest that a general theory is possible we discuss the case of dimension 12 in detail and argue that the solution consists of two 13 dimensional families intersecting in a previously known 9 dimensional family a precise conjecture for all dimensions equal to a prime times another prime squared is formulated

parametrisation of unitary matrices an algorithm for parametrisation of arbitrary n n unitary matrices is presented it is given with a minimum number of real parameters out of which a part take values in the positive unit cube the others being arbitrary phases the case n 4 is worked out in detail

unbiased bases hadamards for six level systems four ways from fourier in quantum mechanics some properties are maximally incompatible such as the position and momentum of a particle or the vertical and horizontal projections of a two level spin given any definite state of one property the other property is completely random or unbiased for n level systems the six level ones are the smallest for which a tomographically efficient set of n 1 mutually unbiased bases has not been found to facilitate the search we numerically extend the classification of unbiased bases or hadamards by incrementally adjusting relative phases in a standard basis we consider the nonunitarity caused by small adjustments with a second order taylor expansion and choose incremental steps within the four dimensional null space of the curvature in this way we prescribe a numerical integration of a four parameter set of hadamards of order of 6

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

three parameter complex hadamard matrices of order 6 a three parameter family of complex hadamard matrices of order 6 is presented it significantly extends the set of closed form complex hadamard matrices of this order and in particular contains all previously described one and two parameter families as subfamilies

an entropy approach to the scaling of ordinal categorical data this paper studies the problem of scaling ordinal categorical data observed over two or more sets of categories measuring a single characteristic scaling is obtained by solving a constrained entropy model which finds the most probable values of the scales given the data a kullback leibler statistic is generated which operationalizes a measure for the strength of consistency among the sets of categories a variety of data of two and three sets of categories are analyzed using the entropy approach

on the existence and uniqueness of maximum likelihood estimates in the rasch model necessary and sufficient conditions for the existence and uniqueness of a solution of the so called unconditional uml and the conditional cml maximum likelihood estimation equations in the dichotomous rasch model are given the basic critical condition is essentially the same for uml and cml estimation for complete data matricesa it is formulated both as a structural property ofa and in terms of the sufficient marginal sums in case of incomplete data the condition is equivalent to complete connectedness of a certain directed graph it is shown how to apply the results in practical uses of the rasch model

all teleportation and dense coding schemes we establish a one to one correspondence between 1 quantum teleportation schemes 2 dense coding schemes 3 orthonormal bases of maximally entangled vectors 4 orthonormal bases of unitary operators with respect to the hilbert schmidt scalar product and 5 depolarizing operations whose kraus operators can be chosen to be unitary the teleportation and dense coding schemes are assumed to be tight in the sense that all hilbert spaces involved have the same finite dimension d and the classical channel involved distinguishes d 2 signals a general construction procedure for orthonormal bases of unitaries involving latin squares and complex hadamard matrices is also presented

some results on the parametrization of complex hadamard matrices in this paper we provide an analytical procedure which leads to a system of n 2 2 polynomial equations whose solutions give the parametrization of the complex n n hadamard matrices it is shown that in general the hadamard matrices depend on a number of arbitrary phases and a lower bound for this number is given the moduli equations define interesting geometrical objects whose study will shed light on the parametrization of the hadamard matrices as well as on some interesting geometrical varieties defined by them

permutation matrices whose convex combinations are orthostochastic abstract let p 1 p m be n n permutation matrices in this note we give a simple necessary condition for all convex combinations of p 1 p m to be orthostochastic we show that for n 15 the condition is also sufficient but for n 15 whether the condition is sufficient or not is still open

mutually unbiased triplets from non affine families of complex hadamard matrices in dimension 6 we study the problem of constructing mutually unbiased bases in dimension 6 this approach is based on an efficient numerical method designed to find solutions to the quantum state reconstruction problem in finite dimensions our technique suggests the existence of previously unknown symmetries in karlsson s non affine family k 2 6 which we confirm analytically also we obtain strong evidence that not more than three mutually unbiased bases can be constructed from pairs which contain members of some non affine families of complex hadamard matrices

mutually unbiased bases and hadamard matrices of order six we report on a search for mutually unbiased bases mubs in six dimensions we find only triplets of mubs and thus do not come close to the theoretical upper bound 7 however we point out that the natural habitat for sets of mubs is the set of all complex hadamard matrices of the given order and we introduce a natural notion of distance between bases in hilbert space this allows us to draw a detailed map of where in the landscape the mub triplets are situated we use available tools such as the theory of the discrete fourier transform to organize our results finally we present some evidence for the conjecture that there exists a four dimensional family of complex hadamard matrices of order 6 if this conjecture is true the landscape in which one may search for mubs is much larger than previously thought

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

finite level systems hermitian operators isometries and a novel parametrization of stiefel and grassmann manifolds in this paper we provide an explicit parametrization of arbitrary n dimensional hermitian operators on the hilbert space operators that may be considered either as hamiltonians or density matrices for finite level quantum systems the description of which gives a complete solution to the over parametrization problem it is shown that all the spectral multiplicities are encoded in a flag unitary matrix obtained as an ordered product of special unitary matrices each one generated by a complex n k dimensional unit vector k 0 1 n 2 as a byproduct an alternative and simple parametrization of stiefel and grassmann manifolds is obtained

unitary stochastic matrix ensembles and spectral statistics we propose to study unitary matrix ensembles defined in terms of unitary stochastic transition matrices associated with markov processes on graphs we argue that the spectral statistics of such an ensemble after ensemble averaging depends crucially on the spectral gap between the leading and subleading eigenvalue of the underlying transition matrix it is conjectured that unitary stochastic ensembles follow one of the three standard ensembles of random matrix theory in the limit of infinite matrix size n if the spectral gap of the corresponding transition matrices closes slower than 1 n the hypothesis is tested by considering several model systems ranging from binary graphs to uniformly and non uniformly connected star graphs and diffusive networks in arbitrary dimensions

relative entropy convergence for depolarizing channels we study the convergence of states under continuous time depolarizing channels with full rank fixed points in terms of the relative entropy the optimal exponent of an upper bound on the relative entropy in this case is given by the log sobolev 1 constant our main result is the computation of this constant as an application we use the log sobolev 1 constant of the depolarizing channels to improve the concavity inequality of the von neumann entropy this result is compared to similar bounds obtained recently by kim and we show a version of pinsker s inequality which is optimal and tight if we fix the second argument of the relative entropy finally we consider the log sobolev 1 constant of tensor powers of the completely depolarizing channel and use a quantum version of shearer s inequality to prove a uniform lower bound

forced classification a simple application of a quantification method this study formulates a property of a quantification method the principle of equivalent partitioning pep when the pep is used together with guttman s principle of internal consistency pic in a simple way the combination offers an interesting way of analyzing categorical data in terms of the variate s chosen by the investigator a type of canonical analysis the study discusses applications of the technique to multiple choice rank order and paired comparison data

the birkhoff theorem for unitary matrices of prime power dimension abstract the unitary birkhoff theorem states that any unitary matrix with all row sums and all column sums equal unity can be decomposed as a weighted sum of permutation matrices such that both the sum of the weights and the sum of the squared moduli of the weights are equal to unity if the dimension n of the unitary matrix equals a power of a prime p i e if n p w then the birkhoff decomposition does not need all n possible permutation matrices as the epicirculant permutation matrices suffice this group of permutation matrices is isomorphic to the general affine group ga w p of order only p w p w 1 p w p p w p w 1 p w

a note on factoring unitary matrices abstract the note concerns three factorizations of unitary matrices a recent factorization allowing to obtain any unitary matrix of dimension 2n from four unitary matrices of dimension n the cosine sine decomposition and a new factorization decomposing a given unitary matrix into a product of two orthogonal matrices and a diagonal unitary matrix

minimum total squared correlation quaternary signature sets new bounds and optimal designs we derive new bounds on the total squared correlation tsc of quaternary quadriphase signature sequence sets for all lengths l and set sizes k then for all k l we design minimum tsc optimal sets that meet the new bounds with equality direct numerical comparison with the tsc value of the recently obtained optimal binary sets shows under what k l realizations gains are materialized by moving from the binary to the quaternary code division multiplexing alphabet on the other hand comparison with the welch tsc value for real complex field sets shows that arguably not much is to be gained by raising the alphabet size above four for any k l the sum capacity as well as the maximum squared correlation and total asymptotic efficiency of minimum tsc quaternary sets is also evaluated in closed form and contrasted against the sum capacity of minimum tsc optimal binary and real complex sets

on the reconstruction of a unitary matrix from its moduli we study the problem of reconstructing a unitary matrix from the knowledge of the moduli of its matrix elements first in the case of a symmetric matrix which could be thes matrix forn coupled channels second in the case of a non symmetric matrix like the cabibbo kobayashi maskawa matrix forn generations of quarks and leptons in the symmetric case we find conditions under which the problem has 2 n 2 3n 2 solutions differing in a non trivial way but also situations where one has continuous ambiguities

on the crossing rule abstract the paper gives conditions on a family of matrices which guarantee that some matrix in the family will have a multiple eigenvalue in particular the main theorem states exactly which dimensions admit k dimensional subspaces of matrices for which all nonzero elements have distinct eigenvalues this question arises naturally in the theory of first order hyperbolic systems of partial differential equations the main theorem in this context tells exactly for which integers n an n x n system in k space variables may be strictly hyperbolic author

an angular momentum approach to quadratic fourier transform hadamard matrices gauss sums mutually unbiased bases the unitary group and the pauli group the construction of unitary operator bases in a finite dimensional hilbert space is reviewed through a nonstandard approach combining angular momentum theory and representation theory of su 2 a single formula for the bases is obtained from a polar decomposition of su 2 and is analyzed in terms of cyclic groups quadratic fourier transforms hadamard matrices and generalized gauss sums weyl pairs generalized pauli operators and their application to the unitary group and the pauli group naturally arise in this approach

factorization of unitary matrices factorization of n n unitary matrices as a product of n diagonal phase matrices interlaced with n 1 orthogonal matrices each one generated by a real vector is provided as a byproduct an explicit form for the weyl factorization of unitary matrices is given the results can be used at the parametrization of complex hadamard matrices and in finding the laplace beltrami operators on unitary groups

on a spectral property of doubly stochastic matrices and its application to their inverse eigenvalue problem abstract in this note we present a useful theorem concerning the spectral properties of doubly stochastic matrices as applications we use this result together with some known results that we recall as a tool for extracting new sufficient conditions for the inverse eigenvalue problem for doubly stochastic matrices

volume of the set of unistochastic matrices of order 3 and the mean jarlskog invariant a bistochastic matrix b of size n is called unistochastic if there exists a unitary u such that bij uij 2 for i j 1 n the set u3 of all unistochastic matrices of order n 3 forms a proper subset of the birkhoff polytope which contains all bistochastic doubly stochastic matrices we compute the volume of the set u3 with respect to the flat lebesgue measure and analytically evaluate the mean entropy of an unistochastic matrix of this order we also analyze the jarlskog invariant j defined for any unitary matrix of order three and derive its probability distribution for the ensemble of matrices distributed with respect to the haar measure on u 3 and for the ensemble which generates the flat measure on the set of unistochastic matrices for both measures the probability of finding j smaller than the value observed for the cabbibo kobayashi maskawa matrix which describes the violation of the cp parity is shown to be small similar statistical reasoning may also be applied to the maki nakagawa sakata matrix which plays role in describing the neutrino oscillations some conjectures are made concerning analogous probability measures in the space of unitary matrices in higher dimensions

block circulant matrices with circulant blocks weil sums and mutually unbiased bases ii the prime power case in our previous paper combescure m circulant matrices gauss sums and the mutually unbiased bases i the prime number case cubo a mathematical journal unpublished we have shown that the theory of circulant matrices allows to recover the result that there exists p 1 mutually unbiased bases in dimension p p being an arbitrary prime number two orthonormal bases b b of cd are said mutually unbiased if b b b b one has that b b 1 d b b hermitian scalar product in cd in this paper we show that the theory of block circulant matrices with circulant blocks allows to show very simply the known result that if d pn p a prime number and n any integer there exists d 1 mutually unbiased bases in cd our result relies heavily on an idea of klimov et al geometrical approach to the discrete wigner function j phys a 39 14471 2006 as a subproduct we recover properties of quadratic weil sums for p 3 which generalizes the fact that in the prime case the quadratic gauss sum properties fo

random unistochastic matrices an ensemble of random unistochastic orthostochastic matrices is defined by taking squared moduli of elements of random unitary orthogonal matrices distributed according to the haar measure on u n or o n an ensemble of symmetric unistochastic matrices is obtained with use of unitary symmetric matrices pertaining to the circular orthogonal ensemble we study the distribution of complex eigenvalues of bistochastic unistochastic and orthostochastic matrices in the complex plane we compute averages entropy traces over the ensembles of unistochastic matrices and present inequalities concerning the entropies of products of bistochastic matrices

existence of a weak solution to the fluid structure interaction problem in 3d abstract we study a nonlinear fluid structure interaction problem in which the fluid is described by the three dimensional incompressible navier stokes equations and the elastic structure is modeled by the nonlinear plate equation which includes a generalization of kirchhoff von karman and berger plate models the fluid and the structure are fully coupled via kinematic and dynamic boundary conditions the existence of a weak solution is obtained by designing a hybrid approximation scheme that successfully deals with the nonlinearities of the system we combine time discretization and operator splitting to create two sub problems one piece wise stationary for the fluid and one in the galerkin basis for the plate to guarantee the convergence of approximate solutions to a weak solution a sufficient condition is given on the number of time discretization sub intervals in every step in a form of dependence with number of the galerkin basis functions and nonlinearity order of the plate equation

linear representations of probabilistic transformations induced by context transitions by using straightforward frequency arguments we classify transformations of probabilities which can be generated by transition from one preparation procedure context to another there are three classes of transformations corresponding to statistical deviations of different magnitudes a trigonometric b hyperbolic c hyper trigonometric it is shown that not only quantum preparation procedures can have trigonometric probabilistic behaviour we propose generalizations of c linear space probabilistic calculus to describe non quantum trigonometric and hyperbolic probabilistic transformations we also analyse the superposition principle in this framework

hadamard matrices from mutually unbiased bases an analytical method for getting new complex hadamard matrices by using mutually unbiased bases and a nonlinear doubling formula is provided the method is illustrated with the n 4 case that leads to a rich family of eight dimensional hadamard matrices that depend on five arbitrary parameters whose modulus is equal to unity

product of diagonal entries of the unitary orbit of a 3 by 3 normal matrix abstract let n be a 3 3 normal matrix we investigate the sets w k n l 1 k unu ll u u 3 where u 3 is the group of 3 3 unitary matrices and 1 k 3 geometric properties of these sets are studied namely star shapedness and simple connectedness are investigated a method for the numerical estimation of w k n is also provided for normal matrices of size 3

traces extensions and co normal derivatives for elliptic systems on lipschitz domains this is the post print version of the article the official published version can be accessed from the link below copyright 2011 elsevier

a composite parameterization of unitary groups density matrices and subspaces unitary transformations and density matrices are central objects in quantum physics and various tasks require to introduce them in a parameterized form in this paper we present a parameterization of the unitary group of arbitrary dimension d which is constructed in a composite way we show explicitly how any element of can be composed of matrix exponential functions of generalized anti symmetric matrices and one dimensional projectors the specific form makes it considerably easy to identify and discard redundant parameters in several cases in this way redundancy free density matrices of arbitrary rank k can be formulated our construction can also be used to derive an orthonormal basis of any k dimensional subspaces of with the minimal number of parameters as an example it is shown that this feature leads to a significant reduction of parameters in the case of investigating distillability of quantum states via lower bounds of an entanglement measure the m concurrence

hermitian unitary matrices with modular permutation symmetry abstract we study hermitian unitary matrices s c n n with the following property there exist r 0 and t 0 such that the entries of s satisfy s j j r and s j k t for all j k 1 n j k we derive necessary conditions on the ratio d r t and show that these conditions are very restrictive except for the case when n is even and the sum of the diagonal elements of s is zero examples of families of matrices s are constructed for d belonging to certain intervals the case of real matrices s is examined in more detail it is demonstrated that a real s can exist only for d n 2 1 or for n even and n 2 d 1 mod 2 we provide a detailed description of the structure of real s with d n 4 3 2 and derive a sufficient and necessary condition of its existence in terms of the existence of certain symmetric v k designs we prove that there exists no real s with d n 6 1 n 4 3 2 a parametrization of hermitian unitary matrices is also proposed and its generalization to general unitary matrices is given at the end of the paper the role of the studied matrices in quantum mechanics on graphs is briefly explained

composed ensembles of random unitary matrices composed ensembles of random unitary matrices are defined via products of matrices each pertaining to a given canonical circular ensemble of dyson we investigate statistical properties of spectra of some composed ensembles and demonstrate their physical relevance we also discuss the methods of generating random matrices distributed according to invariant haar measure on the orthogonal and unitary group

composite parameterization and haar measure for all unitary and special unitary groups we adopt the concept of the composite parameterization of the unitary group u d to the special unitary group su d furthermore we also consider the haar measure in terms of the introduced parameters we show that the well defined structure of the parameterization leads to a concise formula for the normalized haar measure on u d and su d with regard to possible applications of our results we consider the computation of high order integrals over unitary groups

product of two diagonal entries of a 3 by 3 normal matrix abstract the set of product of two diagonal entries of 3 by 3 normal matrices unitarily similar to a diagonal matrix is a triangle if the diagonal entries form an acute angled triangle inscribed to the unit circle

solution regularity and co normal derivatives for elliptic systems with non smooth coefficients on lipschitz domains abstract elliptic pde systems of the second order with coefficients from l or holder lipschitz spaces are considered in the paper continuity of the operators in corresponding sobolev spaces is stated and the internal local solution regularity theorems are generalized to the non smooth coefficient case for functions from the sobolev space h s 1 2 s 3 2 definitions of non unique generalized and unique canonical co normal derivatives are considered which are related to possible extensions of a partial differential operator and the pde right hand side from the domain to its boundary it is proved that the canonical co normal derivatives coincide with the classical ones when both exist a generalization of the boundary value problem settings which makes them insensitive to the co normal derivative inherent non uniqueness is given

on orthostochastic unistochastic and qustochastic matrices abstract we introduce qustochastic matrices as the bistochastic matrices arising from quaternionic unitary matrices by replacing each entry with the square of its norm this is the quaternionic analogue of the unistochastic matrices studied by physicists we also introduce quaternionic hadamard matrices and quaternionic mutually unbiased bases mub in particular we show that the number of mub in an n dimensional quaternionic hilbert space is at most 2 n 1 the bound is attained for n 2 we also determine all quaternionic hadamard matrices of size n 4

a new method to construct families of complex hadamard matrices in even dimensions we present a new method for constructing affine families of complex hadamard matrices in every even dimension this method has an intersection with di s construction and generalizes szoll si s method we extend some known families and present new ones existing in even dimensions in particular we find more than 13 millon inequivalent affine families in dimension 32 we also find analytical restrictions for any set of four mutually unbiased bases existing in dimension six and for any family of complex hadamard matrices existing in every odd dimension

the canonical coset decomposition of unitary matrices through householder transformations this paper reveals the relation between the canonical coset decomposition of unitary matrices and the corresponding decomposition via householder reflections these results can be used to parametrize unitary matrices via householder reflections

the birkhoff theorem for unitary matrices of arbitrary dimensions abstract it was shown recently that birkhoff s theorem for doubly stochastic matrices can be extended to unitary matrices with equal line sums whenever the dimension of the matrices is prime we prove a generalization of the birkhoff theorem for unitary matrices with equal line sums for arbitrary dimension

coherent states for su 3 we define coherent states for su 3 using six bosonic creation and annihilation operators these coherent states are explicitly characterized by six complex numbers with constraints for the completely symmetric representations n 0 and 0 m only three of the bosonic operators are required for mixed representations n m all six operators are required the coherent states provide a resolution of identity satisfy the continuity property and possess a variety of group theoretic properties we introduce an explicit parametrization of the group su 3 and the corresponding integration measure finally we discuss the path integral formalism for a problem in which the hamiltonian is a function of su 3 operators at each site

fourier duality for fractal measures with affine scales for a family of fractal measures wend an explicit fourier d uality the measures in the pair have compact support in r d and they both have the same matrix scaling but the two use dierent translation vectors one by a subset b in r d and the other by a related subset l among other things we show that there is then a pair of innite disc rete sets l and b in r d such that the l fourier exponentials are orthogonal in l 2 b and the b fourier exponentials are orthogonal in l 2 l these sets of orthogonal frequencies are typically lac unary and they will be obtained by scaling in the large the nature of our duality is explored below both in higher dimensions and for examples on the real line our duality pairs do not always yield orthonormal fourier bases in the respective l 2 hilbert spaces but depending on the geometry of certainnite orbit s we show that they do in some cases we further show that there are new and surprising scaling symmetries of relevance for the ergodic theory of these ane fractal measures

birkhoff s polytope and unistochastic matrices n 3 and n 4 the set of bistochastic or doubly stochastic n n matrices is a convex set called birkhoff s polytope which we describe in some detail our problem is to characterize the set of unistochastic matrices as a subset of birkhoff s polytope for n 3 we present fairly complete results for n 4 partial results are obtained an interesting difference between the two cases is that there is a ball of unistochastic matrices around the van der waerden matrix for n 3 while this is not the case for n 4

extreme points of the local differential privacy polytope abstract we study the convex polytope of n n stochastic matrices that define locally differentially private mechanisms we first present invariance properties of the polytope and results reducing the number of constraints needed to define it our main results concern the extreme points of the polytope in particular we completely characterise these for matrices with 1 2 or n non zero columns

existence of a weak solution to a fluid elastic structure interaction problem with the navier slip boundary condition abstract we study a nonlinear moving boundary fluid structure interaction fsi problem between an incompressible viscous newtonian fluid modeled by the 2d navier stokes equations and an elastic structure modeled by the shell or plate equations the fluid and structure are coupled via the navier slip boundary condition and balance of contact forces at the fluid structure interface the slip boundary condition might be more realistic than the classical no slip boundary condition in situations e g when the structure is rough and in modeling fsi dynamics near or at a contact cardiovascular tissue and cell seeded tissue constructs which consist of grooves in tissue scaffolds that are lined with cells are examples of rough elastic interfaces interacting with an incompressible viscous fluid the problem of heart valve closure is an example of a fsi problem with a contact involving elastic interfaces we prove the existence of a weak solution to this class of problems by designing a constructive proof based on the time discretization via operator splitting this is the first existence result for fluid structure interaction problems involving elastic structures satisfying the navier slip boundary condition

families of line graphs and their quantization any directed graph g with n vertices and j edges has an associated line graph l g where the j edges form the vertices of l g we show that the non zero eigenvalues of the adjacency matrices are the same for all graphs of such a family ln g we give necessary and sufficient conditions for a line graph to be quantisable and demonstrate that the spectra of associated quantum propagators follow the predictions of random matrices under very general conditions line graphs may therefore serve as models to study the semiclassical limit of large matrix size of a quantum dynamics on graphs with fixed classical behaviour

classical 1d maps quantum graphs and ensembles of unitary matrices we study a certain class of classical one dimensional piecewise linear maps for these systems we introduce an infinite family of markov partitions in equal cells the symbolic dynamics generated by these systems is described by bi stochastic doubly stochastic matrices we analyse the structure of graphs generated from the corresponding symbolic dynamics we demonstrate that the spectra of quantized graphs corresponding to the regular classical systems have locally poissonian statistics while quantized graphs derived from classically chaotic systems display statistical properties characteristic of the circular unitary ensemble even though the corresponding unitary matrices are sparse

defect of a kronecker product of unitary matrices abstract the generalized defect d u of a unitary n n matrix u with no zero entries is the dimension of the real space of directions moving into which from u we do not disturb the moduli u i j as well as the gram matrix u u in the first order then the defect d u is equal to d u 2 n 1 that is the generalized defect diminished by the dimension of the manifold d r ud c d r d c unitary diagonal calculation of d u involves calculating the dimension of the space in r n 2 spanned by a certain set of vectors associated with u we split this space into a direct sum assuming that u is a kronecker product of unitary matrices thus making it easier to perform calculations numerically basing on this we give a lower bound on d u equivalently d u supposing it is achieved for most unitaries with a fixed kronecker product structure also supermultiplicativity of d u with respect to kronecker subproducts of u is shown

binary programming and test design an algorithmic approach to test design using information functions is presented the approach uses a special branch of linear programming i e binary programming in addition results of some benchmark problems are presented within the same framework it is also possible to formulate the problem of individualized testing

defect of a unitary matrix abstract we analyze properties of a map f sending a unitary matrix u of size n into a doubly stochastic matrix b f u defined by bi j ui j 2 for any u we define its defect determined by the dimension of the image df t u u of the space t u u tangent to the manifold of unitary matrices u at u under the tangent map df corresponding to f the defect of u equal to zero for a generic unitary matrix gives an upper bound for the dimension of a smooth orbit a manifold stemming from u of inequivalent unitary matrices mapped into the same doubly stochastic matrix b f u we demonstrate several properties of the defect and prove an explicit formula for the defect of the fourier matrix fn of size n in this way we obtain an upper bound for the dimension of a smooth orbit of inequivalent unitary complex hadamard matrices stemming from fn it is equal to zero iff n is prime and coincides with the dimension of the known orbits if n is a power of a prime two constructions of these orbits are presented at the end of this work

on the parametrisation of unitary matrices by the moduli of their elements the parametrisation of ann n unitary matrix by the moduli of its elements is not a well posed problem i e there are continuous and discrete ambiguities which naturally appear we show that the continuous ambiguity is n 1 n 3 dimensional in the general case and frac n n 3 2 in the symmetric casesij sij we give also lower bounds on the number of discrete ambiguities the number of solutions being at least 2 frac n n 3 2 in the first case and 2 left frac n 2 right left frac n 1 2 right 1 for the symmetric one where r denotes the integral part ofr

existence of a solution to a fluid multi layered structure interaction problem abstract we study a nonlinear unsteady moving boundary fluid structure interaction fsi problem in which the structure is composed of two layers a thick layer and a thin layer which serves as a fluid structure interface with mass the fluid flow which is driven by the time dependent dynamic pressure data is modeled by the navier stokes equations for an incompressible viscous fluid defined on a 2d cylinder the elastodynamics of the cylinder wall is governed by the 1d linear wave equation modeling the thin structural layer and by the 2d equations of linear elasticity modeling the thick structural layer we prove existence of a weak solution to this nonlinear fsi problem as long as the cylinder radius is greater than zero the spaces of weak solutions presented in this manuscript reveal a striking new feature the presence of a thin fluid structure interface with mass regularizes solutions of the coupled problem

two parameter complex hadamard matrices for n 6 a new two parameter nonaffine family of complex hadamard matrices of order 6 is reported it interpolates between the two fourier families and contains as one parameter subfamilies the di family a symmetric family and an almost up to equivalence self adjoint family

why two qubits are special we analyze some special properties of a system of two qubits and in particular of the so called bell basis for this system and discuss the possibility of extending these properties to higher dimensional systems we give a general construction for orthonormal bases of maximally entangled vectors which works in any dimension and is based on latin squares and complex hadamard matrices however for none of these bases the special properties of the operation of complex conjugation in bell basis hold namely that maximally entangled vectors have up to a phase real coefficients and that factorizable unitaries have real matrix elements

a recursive parametrization of unitary matrices a simple recursive scheme for parametrization of n by n unitary matrices is presented the n by n matrix is expressed as a product containing the n 1 by n 1 matrix and a unitary matrix that contains the additional parameters needed to go from n 1 to n the procedure is repeated to obtain recursion formulas for n by n unitary matrices

the birkhoff theorem for unitary matrices of prime dimension abstract the birkhoff s theorem states that any doubly stochastic matrix lies inside a convex polytope with the permutation matrices at the corners we prove that any unitary matrix with equal line sums can also be written as a sum of permutation matrices with sum of weights equal 1 furthermore when the matrix dimension is prime we prove that the unitary matrix lies inside a convex complex birkhoff polytope

existence of a weak solution to a nonlinear fluid structure interaction problem modeling the flow of an incompressible viscous fluid in a cylinder with deformable walls we study a nonlinear unsteady moving boundary fluid structure interaction fsi problem arising in modeling blood flow through elastic and viscoelastic arteries the fluid flow which is driven by the time dependent pressure data is governed by two dimensional incompressible navier stokes equations while the elastodynamics of the cylindrical wall is modeled by the one dimensional cylindrical koiter shell model two cases are considered the linearly viscoelastic and the linearly elastic koiter shell the fluid and structure are fully coupled two way coupling via the kinematic and dynamic lateral boundary conditions describing continuity of velocity the no slip condition and the balance of contact forces at the fluid structure interface we prove the existence of weak solutions to the two fsi problems the viscoelastic and the elastic case as long as the cylinder radius is greater than zero the proof is based on a novel semi discrete operator splitting numerical scheme known as the kinematically coupled scheme introduced in guidoboni et al j comput phys 228 18 6916 6937 2009 to numerically solve the underlying fsi problems the backbone of the kinematically coupled scheme is the well known marchuk yanenko scheme also known as the lie splitting scheme we effectively prove convergence of that numerical scheme to a solution of the corresponding fsi problem