controlled invariance for discrete time nonlinear systems with an application to the disturbance decoupling problems invariant distributions are defined for discrete time nonlinear control systems and necessary and sufficient conditions are given for their controlled invariance this extends to discrete time systems the basic tool which has been so important in solving the various synthesis problems for continuous time systems to indicate their utility in the discrete time setting they are used to locally solve the disturbance decoupling problem

differential geometric methods in variable structure control this article presents a differential geometric approach for the design of sliding modes in non linear variable structure feedback systems coordinate free characterizations of local existence conditions for sliding regimes and a geometric reformulation of some of its most salient features are presented the approach uses basic notions from differential geometry involving vector fields distributions and 1 forms both single and multiple input cases are treated with some illustrative examples

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

shift in the velocity of a front due to a cutoff we consider the effect of a small cutoff ensuremath varepsilon on the velocity of a traveling wave in one dimension simulations done over more than ten orders of magnitude as well as a simple theoretical argument indicate that the effect of the cutoff ensuremath varepsilon is to select a single velocity that converges when ensuremath varepsilon ensuremath rightarrow 0 to the one predicted by the marginal stability argument for small ensuremath varepsilon the shift in velocity has the form k mathrm ln ensuremath varepsilon ensuremath 2 and our prediction for the constant k agrees very well with the results of our simulations a very similar logarithmic shift appears in more complicated situations in particular in finite size effects of some microscopic stochastic systems our theoretical approach can also be extended to give a simple way of deriving the shift in position due to initial conditions in the fisher kolmogorov or similar equations

mutually unbiased product bases for multiple qudits we investigate the interplay between mutual unbiasedness and product bases for multiple qudits of possibly different dimensions a product state of such a system is shown to be mutually unbiased to a product basis only if each of its factors is mutually unbiased to all the states which occur in the corresponding factors of the product basis this result implies both a tight limit on the number of mutually unbiased product bases which the system can support and a complete classification of mutually unbiased product bases for multiple qubits or qutrits in addition only maximally entangled states can be mutually unbiased to a maximal set of mutually unbiased product bases

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

a generalized pauli problem and an infinite family of mub triplets in dimension 6 we exhibit an infinite family of triplets of mutually unbiased bases mubs in dimension 6 these triplets involve the fourier family of hadamard matrices f a b however in the main result of this paper we also prove that for any values of the parameters a b the standard basis and f a b cannot be extended to a mub quartet the main novelty lies in the method of proof which may successfully be applied in the future to prove that the maximal number of mubs in dimension 6 is three

the structure of nonlinear control systems possessing symmetries a concept of symmetry is defined for general nonlinear control systems it is shown under various technical conditions that nonlinear control systems with symmetries admit local and or global decompositions in terms of lower dimensional subsystems and feedback loops the structure of the individual subsystems is dependent on the structure of the symmetry group for example if the symmetry group is abelian one of the subsystems is a quadrature an additional feature of the decomposition is that the state space dimensions of the subsystems sum to the state space dimension of the original system

phenomenological theory giving the full statistics of the position of fluctuating pulled fronts we propose a phenomenological description for the effect of a weak noise on the position of a front described by the fisher kolmogorov petrovsky piscounov equation or any other traveling wave equation in the same class our scenario is based on four hypotheses on the relevant mechanism for the diffusion of the front our parameter free analytical predictions for the velocity of the front its diffusion constant and higher cumulants of its position agree with numerical simulations

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

on orthogonal systems of matrix algebras abstract in this work it is shown that certain interesting types of orthogonal system of subalgebras whose existence cannot be ruled out by the trivial necessary conditions cannot exist in particular it is proved that there is no orthogonal decomposition of m n c m n c m n 2 c into a number of maximal abelian subalgebras and factors isomorphic to m n c in which the number of factors would be 1 or 3 in addition some new tools are introduced too for example a quantity c a b which measures how close the subalgebras a b m n c are to being orthogonal it is shown that in the main cases of interest c a b where a and b are the commutants of a and b respectively can be determined by c a b and the dimensions of a and b the corresponding formula is used to find some further obstructions regarding orthogonal systems

affine constellations without mutually unbiased counterparts it has been conjectured that a complete set of mutually unbiased bases in a space of dimension d exists if and only if there is an affine plane of order d we introduce affine constellations and compare their existence properties with those of mutually unbiased constellations the observed discrepancies make a deeper relation between the two existence problems unlikely

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

geometrical description of quantal state determination under the assumption that every quantal measurement may give data about the post measurement state of the inspected ensemble the problem of the state determination is reconsidered it is shown that orthogonal decomposition of the set of complex n n hermitian matrices into the commutative subsets allows operators to be found such that post measurement information on these observables allows a partial in some cases total determination of the pre measurement state to be effected

nonlinear variable structure systems in sliding mode the general case the problem of inducing local sliding regimes on smooth state space surfaces of nonlinear single input single output controlled systems is addressed in full generality the notion of relative degree in its more general form is used to establish the most salient features of nonlinear controlled systems undergoing sliding motions including their characteristic disturbance rejection properties it is shown that the simplest possible structure at infinity must be exhibited by nonlinear systems undergoing sliding motions on the zero level set of the output feedback function

on tracking control for affine nonlinear systems by sliding mode abstract the stability problem of output tracking of a bounded time varying reference by decouplable affine nonlinear systems using sliding mode control is investigated it is shown that when the error dynamics on the sliding surface is chosen to be linear time invariant closed loop stability of systems under the presented sliding mode control can be guaranteed only if the systems are minimum phase

sliding mode control of retarded nonlinear systems via finite spectrum assignment approach in the present study a sliding mode control design method based on the finite spectrum assignment procedure is proposed the finite spectrum assignment for retarded nonlinear systems can transform retarded nonlinear systems into delay free linear systems via a variable transformation and a feedback which contain the past values of the state this method can be considered to be an extension of both the finite spectrum assignment for retarded linear systems with controllability over polynomial rings of the delay operator and the exact linearization for finite dimensional nonlinear systems the proposed method is to design a sliding surface via the variable transformation used in the finite spectrum assignment and to derive a switching feedback law the obtained surface contains not only the current values of the state variables but also the past values of the state variables in the original coordinates the effectiveness of the proposed method is tested by an illustrative example

a collective coordinate framework to study the dynamics of travelling waves in stochastic partial differential equations abstract we propose a formal framework based on collective coordinates to reduce infinite dimensional stochastic partial differential equations spdes with symmetry to a set of finite dimensional stochastic differential equations which describe the shape of the solution and the dynamics along the symmetry group we study spdes arising in population dynamics with multiplicative noise and additive symmetry breaking noise the collective coordinate approach provides a remarkably good quantitative description of the shape of the travelling front as well as its diffusive behaviour which would otherwise only be available through costly computational experiments we corroborate our analytical results with numerical simulations of the full spde

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

effect of selection on ancestry an exactly soluble case and its phenomenological generalization we consider a family of models describing the evolution under selection of a population whose dynamics can be related to the propagation of noisy traveling waves for one particular model that we shall call the exponential model the properties of the traveling wave front can be calculated exactly as well as the statistics of the genealogy of the population one striking result is that for this particular model the genealogical trees have the same statistics as the trees of replicas in the parisi mean field theory of spin glasses we also find that in the exponential model the coalescence times along these trees grow like the logarithm of the population size a phenomenological picture of the propagation of wave fronts that we introduced in a previous work as well as our numerical data suggest that these statistics remain valid for a larger class of models while the coalescence times grow like the cube of the logarithm of the population size

distributed sliding mode control in systems described by linear and quasilinear partial differential equations abstract the theory of variable structure systems and their associated sliding regimes is extended to controlled dynamical systems described by first order quasilinear partial differential equations

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

connections between mutually unbiased bases and quantum random access codes we present a new quantum communication complexity protocol the promise quantum random access code which allows us to introduce a new measure of unbiasedness for bases of hilbert spaces the proposed measure possesses a clear operational meaning and can be used to investigate whether a specific number of mutually unbiased bases exist in a given dimension by employing semidefinite programming techniques

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

analysis and design of variable structure systems using a geometric approach multivariable variable structure systems in the sliding mode are studies using a geometric approach the properties of system order reduction and disturbance rejection are proved using projector theory new design methods for the choice of switching hyperplanes are derived for the closed loop eigenvalue eigenvector assignment problem

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

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

robust adaptive sliding mode control for fuzzy systems with mismatched uncertainties this paper is devoted to design adaptive sliding mode controllers for the takagi sugeno t s fuzzy system with mismatched uncertainties and exogenous disturbances the uncertainties in state matrices are mismatched and norm bounded while the exogenous disturbances are assumed to be bounded with an unknown bound which is estimated by a simple and effective adaptive approach both state and static output feedback sliding mode control problems are considered in terms of linear matrix inequalities lmis both sliding surfaces and sliding mode controllers can be easily obtained via a convex optimization technique finally two simulation examples and a real experiment are utilized to illustrate the applicability and effectiveness of the design procedures proposed in this paper

non linear discrete variable structure systems in quasi sliding mode abstract the problem of inducing convergent quasi sliding regimes on smooth state space surfaces of non linear single input single output discrete time controlled systems is addressed in full generality a suitable extension of the notion of relative degree is used in establishing the most salient features of quasi sliding motions in non linearly controlled systems several examples are given

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

existence of carathe spl acute odory solutions in nonlinear systems with discontinuous switching feedback controllers in this note we consider the existence of a carathe spl acute odory solution in a single input single output nonlinear system with a discontinuous switching feedback controller the main contribution is to show that if the nonlinear system can be transformed into a global normal form then we can specify the value of the discontinuous switching feedback controller on the switching hypersurface so that the closed loop system has a carathe spl acute odory solution

nonlinear feedback regulator design for the cuk converter the method of extended linearization is used for designing stabilizing nonlinear proportional integral pi feedback controllers which regulate to a constant set point value either the average output inductor current the average input inductor current or the average transfer capacitor voltage of a pulse width modulation pwm controlled cuk converter the design is carried out on the basis of the frequency domain ziegler nichols method applied to a family of transfer function models of the linearized average pwm controlled circuit parametrized by constant operating equilibrium points

fisher waves an individual based stochastic model the propagation of a beneficial mutation in a spatially extended population is usually studied using the phenomenological stochastic fisher kolmogorov petrovsky piscounov sfkpp equation we derive here an individual based stochastic model founded on the spatial moran process where fluctuations are treated exactly the mean field approximation of this model leads to an equation that is different from the phenomenological fkpp equation at small selection pressure the front behavior can be mapped into a brownian motion with drift the properties of which can be derived from the microscopic parameters of the moran model finally we generalize the model to take into account dispersal kernels beyond migration to nearest neighbors we show how the effective population size which controls the noise amplitude and the diffusion coefficient can both be computed from the dispersal kernel

an example of a nonlinear regulator the purpose of this note is to use the concept of feedback equivalence to linear controllable systems in order to design a nonlinear regulator for a synchronous generator connected to an infinite bus the design of the regulator will show how recently developed theorems in the area of geometric nonlinear control theory approach a power system problem offering an interesting solution

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 sliding mode control of multivariable nonlinear systems sliding mode control of multivariable nonlinear systems is addressed from the perspective of linear differential algebra for a special but large class of linearizable systems known as differentially flat systems essential orders and differential flatness are shown to be a relevant concept and a sensible requirement respectively associated with the possibilities of designing static or dynamical sliding mode multivariable feedback regulators for nonlinear systems

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

sliding mode scheme for adaptive specific growth rate control in biotechnological fed batch processes this paper addresses the control of biomass growth rate in fed batch bioreactors the main difficulty when designing controllers for these processes is the lack of accurate on line knowledge of the controlled variable as well as the strong parameter and model uncertainties a completely novel approach to the control design is introduced in this paper which allows us to overcome these problems in fact the proposed controller which is applicable to a large class of fermentation processes requires minimal knowledge of the process parameters and only uses on line measurement of volume and biomass concentration first a reference model is proposed and a goal manifold in the state space is derived where the control objective is satisfied a partial state feedback law is then proved to be an invariant control for the goal manifold then the feedback gain is dynamically adjusted via a discontinuous action that enforces a sliding regime such that all state trajectories are steered towards the goal manifold

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

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

exact nonlinear control of large angle rotational maneuvers the rigid body attitude control problem with external torques is transformed into an equivalent linear form implementable by three double integrators the linearizing transformations themselves are formulated in vector algebra requiring no integrators for implementation it is thereby shown that optimal command generation for fast slewing maneuvers can be carried out exactly in the transformed system together with regulator design without gain scheduling for correction of unmodeled disturbances in particular a general rest to rest maneuver is computed in closed form and coupled with a previously obtained exact detumbling maneuver so that arbitrary initial conditions can be accommodated an illustrative simulation is appended

dynamical discontinuous feedback control of nonlinear systems the problem of analyzing dynamical discontinuous feedback nonlinear controlled plants is addressed the continuous feedback scheme is assumed to be constituted by a dynamical feedback plant of nonlinear nature and a controlled switch obeying a pulse width modulation pwm type of switching strategy with sufficiently high sampling rate it is found that the actual closed loop controlled responses of the system exhibit chattering motions constituting nonideal sliding regimes which converge continuously toward certain average manifolds as the sampling frequency is increased these manifolds are immersed in the regions of the composite state space where the duty ratio function is not acting under saturation conditions the average system is simply obtained by an infinite sampling frequency assumption on the pwm process a pwm feedback interconnection scheme with general duty ratio function is shown to be easily analyzable in terms of an average model which captures the essential features of the discontinuously feedback controlled system

constructing discontinuity surfaces for variable structure systems a lyapunov approach the problem of constructing discontinuity surfaces in variable structure systems is studied from a lyapunov point of view a switching surface determined by the control coefficient matrix and the associated lyapunov function is able to ensure asymptotic stability for the system in sliding mode the proposed method may also be used for systems with nonlinear dynamics and for linear systems with delays

global transformations of nonlinear systems recent results have established necessary and sufficient conditions for a nonlinear system of the form dot x t f x t u t g x t with f 0 0 to be locally equivalent in a neighborhood of the origin in rnto a controllable linear system we combine these results with several versions of the global inverse function theorem to prove sufficient conditions for the transformation of a nonlinear system to a linear system in doing so we introduce a technique for constructing a transformation under the assumptions that g ldot f dotg ad n 1 f ldotg span an n dimensional space and that g ldot f ldot g ad n 2 f ldotg is an involutive set

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 limited role of mutually unbiased product bases in dimension 6 we show that a complete set of seven mutually unbiased bases in dimension six if it exists cannot contain more than one product basis

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

orbits of mutually unbiased bases we express alltop s construction of mutually unbiased bases as orbits under the weyl heisenberg group in prime dimensions and find a related construction in dimensions 2 and 4 we reproduce alltop s mutually unbiased bases using abelian subgroups of the clifford group in prime dimensions in direct analogy to the well known construction of mutually unbiased bases using abelian subgroups of the weyl heisenberg group finally we prove three theorems relating to the distances and linear dependencies among different sets of mutually unbiased bases

output tracking control of mimo fuzzy nonlinear systems using variable structure control approach the output tracking control problem for nonlinear systems in the presence of both parameter perturbations and external disturbances is studied our approach is based on the takagi sugeno t s fuzzy modeling method and a variable structure control vsc technique therefore the systems considered are not and need not be in the triangular and parametric strict feedback form which are prevalent among adaptive model following control for nonlinear systems or in the normal form which pervades almost all existing results in neuro fuzzy model following control approach we first study the problem of stabilization of t s fuzzy systems by using a vsc technique a method for the design of a switching surface based on linear matrix inequalities is developed and a stabilizing controller based on a reaching law concept in the presence of both parameter perturbations and external disturbances is proposed then the method is extended to design controllers for output tracking of t s fuzzy nonlinear systems in two cases i e systems which possess the so called strong passive subsystems and strong stable zero dynamics respectively finally illustrative examples are presented to demonstrate the whole design procedure from the original nonlinear systems to their fuzzification and finally to the realization of the desired controllers simulation results show that the goal of output tracking can be achieved by the proposed controllers

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

h 2 reducible complex hadamard matrices of order 6 complex hadamard matrices h of order 6 are characterized in a novel manner according to the presence absence of order 2 hadamard submatrices it is shown that if there exists one such submatrix h is equivalent to a hadamard matrix where all the nine submatrices are hadamard the ensuing subset of h 2 reducible complex hadamard matrices is more general than might be thought and significantly includes all the up till now described one and two parameter families of order 6 a known isolated matrix and most numerically generated matrices fall outside the subset

the noisy edge of traveling waves traveling waves are ubiquitous in nature and control the speed of many important dynamical processes including chemical reactions epidemic outbreaks and biological evolution despite their fundamental role in complex systems traveling waves remain elusive because they are often dominated by rare fluctuations in the wave tip which have defied any rigorous analysis so far here we show that by adjusting nonlinear model details noisy traveling waves can be solved exactly the moment equations of these tuned models are closed and have a simple analytical structure resembling the deterministic approximation supplemented by a nonlocal cutoff term the peculiar form of the cutoff shapes the noisy edge of traveling waves and is critical for the correct prediction of the wave speed and its fluctuations our approach is illustrated and benchmarked using the example of fitness waves arising in simple models of microbial evolution which are highly sensitive to number fluctuations we demonstrate explicitly how these models can be tuned to account for finite population sizes and determine how quickly populations adapt as a function of population size and mutation rates more generally our method is shown to apply to a broad class of models in which number fluctuations are generated by branching processes because of this versatility the method of model tuning may serve as a promising route toward unraveling universal properties of complex discrete particle systems

the speed of evolution in large asexual populations we consider an asexual biological population of constant size n evolving in discrete time under the influence of selection and mutation beneficial mutations appear at rate u and their selective effects s are drawn from a distribution g s after introducing the required models and concepts of mathematical population genetics we review different approaches to computing the speed of logarithmic fitness increase as a function of n u and g s we present an exact solution of the infinite population size limit and provide an estimate of the population size beyond which it is valid we then discuss approximate approaches to the finite population problem distinguishing between the case of a single selection coefficient g s s s b and a continuous distribution of selection coefficients analytic estimates for the speed are compared to numerical simulations up to population sizes of order 10300

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

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

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

sliding mode control for singularly perturbed systems this paper addresses the design of sliding mode controllers for singularly perturbed systems the full order system is separated into slow and fast subsystems and a sliding mode controller is designed for each subsystem a composite of these controls is then implemented on the full order system it is shown that if specific reaching conditions are satisfied for the reduced order models and an additional condition is met then the reaching conditions are also satisfied for the full order model furthermore sufficient conditions are derived which guarantee that the approximation error between the controlled reduced order response and the controlled full order model response is small a numerical example is given which successfully demonstrates the techniques developed in this paper

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

output tracking design of affine nonlinear plant via variable structure system an output tracking design for affine nonlinear systems via a variable structure system is proposed the structure algorithm is used to determine if the considered system is left invertible it is shown that if such a condition is satisfied then a sliding mode controller can be designed to achieve output tracking the decouplability condition in conventional approaches is now replaced by the weaker left invertibility condition stable tracking is guaranteed if the internal mode is bounded input bounded state stable simulation results are included to demonstrate the design procedure