generalized entropic uncertainty relations a new class of uncertainty relations is derived for pairs of observables in a finite dimensional hilbert space which do not have any common eigenvector this class contains an entropic uncertainty relation which improves a previous result of deutsch and confirms a recent conjecture by kraus some comments are made on the extension of these relations to the case where the hilbert space is infinite dimensional

measuring time frequency information content using the renyi entropies the generalized entropies of renyi inspire new measures for estimating signal information and complexity in the time frequency plane when applied to a time frequency representation tfr from cohen s class or the affine class the renyi entropies conform closely to the notion of complexity that we use when visually inspecting time frequency images these measures possess several additional interesting and useful properties such as accounting and cross component and transformation invariances that make them natural for time frequency analysis this paper comprises a detailed study of the properties and several potential applications of the renyi entropies with emphasis on the mathematical foundations for quadratic tfrs in particular for the wigner distribution we establish that there exist signals for which the measures are not well defined

solution of poisson and laplace equations by quadrilateral quadrature element the quadrature element method qem is a new concept in numerical methods which was introduced only recently a quadrilateral quadrature element is developed in the paper and applied to the solution of two dimensional potential problems governed by poisson or laplace equations the results of three examples are in good agreement with the available solutions it is shown that the present quadrilateral quadrature element is very efficient and effective most significantly it can be employed to solve differential equations of physical problems with irregular geometry and complex boundary conditions

asymptotic formulae for the quantum renyi entropies of position application to the infinite well general asymptotic formulae are derived by means of the wkb approximation for the continuous and discrete renyi entropies of position of one dimensional quantum systems in energy eigenstates in terms of the corresponding entropies for a microcanonical ensemble of analogous classical systems these results are checked in the simplest particular case of the infinite potential well where the asymptotic formula for continuous entropies holds as an exact identity for the discrete entropies analytical expressions are obtained from which the asymptotic formulae given for the limiting cases of large and small measurement resolution can both be verified

majorization entropic uncertainty relations entropic uncertainty relations in a finite dimensional hilbert space are investigated making use of the majorization technique we derive explicit lower bounds for the sum of r enyi entropies describing probability distributions associated with a given pure state expanded in eigenbases of two observables obtained bounds are expressed in terms of the largest singular values of submatrices of the unitary rotation matrix numerical simulations show that for a generic unitary matrix of size n 5 our bound is stronger than the well known result of maassen and uffink mu with a probability larger than 98 we also show that the bounds investigated are invariant under the dephasing and permutation operations finally we derive a classical analogue of the mu uncertainty relation which is formulated for stochastic transition matrices

on classes of non gaussian asymptotic minimizers in entropic uncertainty principles abstract in this paper we revisit the bialynicki birula and mycielski uncertainty principle and its cases of equality this shannon entropic version of the well known heisenberg uncertainty principle can be used when dealing with variables that admit no variance in this paper we extend this uncertainty principle to renyi entropies we recall that in both shannon and renyi cases and for a given dimension n the only case of equality occurs for gaussian random vectors we show that as n grows however the bound is also asymptotically attained in the cases of n dimensional student t and student r distributions a complete analytical study is performed in a special case of a student t distribution we also show numerically that this effect exists for the particular case of a n dimensional cauchy variable whatever the renyi entropy considered extending the results of abe and illustrating the analytical asymptotic study of the student t case in the student r case we show numerically that the same behavior occurs for uniformly distributed vectors these particular cases and other ones investigated in this paper are interesting since they show that this asymptotic behavior cannot be considered as a gaussianization of the vector when the dimension increases

dynamical aspects of tsallis entropy abstract the description of the temporal evolution of some physical systems is tackled from an information theory view point in which tsallis entropy is extremalized with the constraints posed by the knowledge of the generalized expectation values of a selected set of observables the concomitant tsallis density operator is shown to be a particular solution of von neumann s equation and tsallis entropy is seen to provide one with a constant of the motion the pertinent ideas are illustrated with reference to the larmor precession

characterization of time series via renyi complexity entropy curves abstract one of the most useful tools for distinguishing between chaotic and stochastic time series is the so called complexity entropy causality plane this diagram involves two complexity measures the shannon entropy and the statistical complexity recently this idea has been generalized by considering the tsallis monoparametric generalization of the shannon entropy yielding complexity entropy curves these curves have proven to enhance the discrimination among different time series related to stochastic and chaotic processes of numerical and experimental nature here we further explore these complexity entropy curves in the context of the renyi entropy which is another monoparametric generalization of the shannon entropy by combining the renyi entropy with the proper generalization of the statistical complexity we associate a parametric curve the renyi complexity entropy curve with a given time series we explore this approach in a series of numerical and experimental applications demonstrating the usefulness of this new technique for time series analysis we show that the renyi complexity entropy curves enable the differentiation among time series of chaotic stochastic and periodic nature in particular time series of stochastic nature are associated with curves displaying positive curvature in a neighborhood of their initial points whereas curves related to chaotic phenomena have a negative curvature finally periodic time series are represented by vertical straight lines

equivalence theorem of uncertainty relations we present an equivalence theorem to unify the two classes of uncertainty relations i e the variance based ones and the entropic forms showing that the entropy of an operator in a quantum system can be built from the variances of a set of commutative operators this means that an uncertainty relation in the language of entropy may be mapped onto a variance based one and vice versa employing the equivalence theorem alternative formulations of entropic uncertainty relations are obtained for the qubit system that are stronger than the existing ones in the literature and variance based uncertainty relations for spin systems are reached from the corresponding entropic uncertainty relations

simple random walk on long range percolation clusters ii scaling limits supported in part at the technion by a landau fellowship supported in part by an alfred sloan fellowship in mathematics

sub poissonian and anti bunching criteria via majorization of statistics we use majorization and confidence intervals as a convenient tool to compare the uncertainty in photon number for different quantum light states to this end majorization is formulated in terms of confidence intervals as a suitable case study we apply this tool to sub and super poissonian behavior and bunching and anti bunching effects we focus on the most significant classical and nonclassical states such as glauber coherent thermal photon number and squeezed states we show that majorization provides a more complete analysis that in some relevant situations contradicts the predictions of variance

a quantum uncertainty relation based on fisher s information we explore quantum uncertainty relations involving the fisher information functionals ix and ip evaluated respectively on a wavefunction x defined on a d dimensional configuration space and the concomitant wavefunction on the conjugate momentum space we prove that the associated fisher functionals obey the uncertainty relation ixip 4d2 when either x or is real on the other hand there is no lower bound to the above product for arbitrary complex wavefunctions we give explicit examples of complex wavefunctions not obeying the above bound in particular we provide a parametrized wavefunction for which the product ixip can be made arbitrarily small

general entropy like uncertainty relations in finite dimensions we revisit entropic formulations of the uncertainty principle up for an arbitrary pair of positive operator valued measures povm a and b acting on finite dimensional hilbert space salicr generalized entropies including r nyi and tsallis ones among others are used as uncertainty measures associated with the distribution probabilities corresponding to the outcomes of the observables we obtain a nontrivial lower bound for the sum of generalized entropies for any pair of entropic functionals which is valid for both pure and mixed states the bound depends on the overlap triplet with ca respectively cb being the overlap between the elements of the povm a respectively b and the overlap between the pair of povm our approach is inspired by that of de vicente and s nchez ruiz 2008 phys rev a 77 042110 and consists in a minimization of the entropy sum subject to the landau pollak inequality that links the maximum probabilities of both observables we solve the constrained optimization problem in a geometrical way and furthermore when dealing with r nyi or tsallis entropic formulations of the up we overcome the h lder conjugacy constraint imposed on the entropic indices by the riesz thorin theorem in the case of nondegenerate observables we show that for given the bound obtained is optimal and that for r nyi entropies our bound improves deutsch one but maassen uffink bound prevails when finally we illustrate by comparing our bound with known previous results in particular cases of r nyi and tsallis entropies

upper bounds on shannon and renyi entropies for central potentials the renyi and shannon entropies are information theoretic measures which have enabled to formulate the position momentum uncertainty principle in a much more adequate and stringent way than the variance based heisenberg like relation moreover they are closely related to various energetic density functionals of quantum systems here we derive upper bounds on these quantities in terms of the second order moment r2 for general central potentials this improves previous results of this type the proof uses the renyi maximization procedure with a covariance constraint due to costa et al in proceedings of the fourth international workshop on energy minimization methods in computer vision and pattern recognition emmcvpr edited by a rangarajan m a t figueiredo and j zerubia springer verlag lisbon 2003 lect notes comput sci 52 211 2003 the contributions to these bounds coming from the radial and angular parts of the physical wave functions are taken into account finally the appl

entropic functionals of laguerre and gegenbauer polynomials with large parameters the determination of the physical entropies renyi shannon tsallis of high dimensional quantum systems subject to a central potential requires the knowledge of the asymptotics of some power and logarithmic integral functionals of the hypergeometric orthogonal polynomials which control the wavefunctions of the stationary states for the d dimensional hydrogenic and oscillator like systems the wavefunctions of the corresponding bound states are controlled by the laguerre and gegenbauer

on the axiomatic requirement of range to measure uncertainty abstract how to measure uncertainty is still an open issue probability theory is a primary tool to express the aleatoric uncertainty the shannon s information entropy is an effective measure for the uncertainty in probability theory dempster shafer theory an extension of probability theory has the ability to express the aleatoric and epistemic uncertainty simultaneously with respect to such uncertainties in dempster shafer theory a justifiable uncertainty measure is required to satisfy five axiomatic requirements based on previous studies in this paper we show that one of the axiomatic requirements the requirement of range is questionable the correct range of uncertainty should be 0 log 2 2 x rather than 0 log 2 x according to the concept of entropy

pseudo diffusion equation and information entropy of squeezed coherent states abstract as the squeezed coherent states are labeled by 3 tuple the momentum the coordinate and the squeeze parameter a phase space probability distribution function associated with a generic wavefunction can be introduced we verify that this probability function is the solution of a partial differential equation the pseudo diffusion equation hence permitting to extend the concept of information entropy as introduced by wehrl for the coherent states to the squeezed coherent states it is shown that the entropy functionals can be used as a measure of quantum correlations between the phase space variables for a given wavefunction we also present several properties and discuss the use of the entropic inequality as a concept that complements the heisenberg uncertainty relationship

microscopic analysis of clausius duhem processes given a thermodynamic process which carries a system from one equilibrium state to another we construct a quantity whose average over an ensemble of microscopic realizations of the process depends only on these end states even if at intermediate times the system is out of equilibrium this result 1 can be used to express the entropy difference between two equilibrium states in terms of an irreversible process connecting them 2 leads to two statistical statements of the clausius duhem inequality and 3 can be generalized to situations in which the system begins and or ends in nonequilibrium states

generalized entropy as a measure of quantum uncertainty abstract a generalized entropy is used in order to advance a different form of expressing the uncertainty principle of quantum mechanics we consider the generalized entropic formulation for different pairs of incompatible observables in particular we study the number phase entropic uncertainty measure for the case of coherent states within the pegg barnett theory we also tackle the situation of operators with continuous spectra where a correlation functional is calculated in terms of generalized joint and marginal entropies for harmonic oscillator wavefunctions

the infinite number of generalized dimensions of fractals and strange attractors abstract we show that fractals in general and strange attractors in particular are characterized by an infinite number of generalized dimensions dq q 0 to this aim we develop a rescaling transformation group which yields analytic expressions for all the quantities dq we prove that lim q 0 dq fractal dimension d limq 1dq information dimension and dq 2 correlation exponent v dq with other integer q s correspond to exponents associated with ternary quaternary and higher correlation functions we prove that generally dq dq for any q q for homogeneous fractals dq dq a particularly interesting dimension is dq for two examples feigenbaum attractor generalized baker s transformation we calculate the generalized dimensions and find that d is a non trivial number all the other generalized dimensions are bounded between the fractal dimension and d

on a generalized entropic uncertainty relation in the case of the qubit we revisit generalized entropic formulations of the uncertainty principle for an arbitrary pair of quantum observables in two dimensional hilbert space r nyi entropy is used as an uncertainty measure associated with the distribution probabilities corresponding to the outcomes of the observables we derive a general expression for the tight lower bound of the sum of r nyi entropies for any couple of positive entropic indices thus we have overcome the h lder conjugacy constraint imposed on the entropic indices by riesz thorin theorem in addition we present an analytical expression for the tight bound inside the square 0 1 2 2 in the plane and a semi analytical expression on the line it is seen that previous results are included as particular cases moreover we present a semi analytical suboptimal bound for any couple of indices in all cases we provide the minimizing states

entropic measures of joint uncertainty effects of lack of majorization we compute renyi entropies for the statistics of a noisy simultaneous observation of two complementary observables in two dimensional quantum systems the relative amount of uncertainty between two states depends on the uncertainty measure used these results are not reproduced by a more standard duality relation we show that these behaviors are consistent with the lack of majorization relation between the corresponding statistics

about closedness by convolution of the tsallis maximizers in this paper we study the stability under convolution of the maximizing distributions of the tsallis entropy under energy constraint called hereafter tsallis distributions these distributions are shown to obey three important properties a stochastic representation property an orthogonal invariance property and a duality property as a consequence of these properties the behaviour of tsallis distributions under convolution is characterized at last a special random convolution called kingman convolution is shown to ensure the stability of tsallis distributions

fisher information renyi entropy power and quantum phase transition in the dicke model abstract fisher information renyi entropy power and fisher renyi information product are presented for the dicke model there is a quantum phase transition in this quantum optical model it is pointed out that there is an abrupt change in the fisher information renyi entropy power the fisher shannon and renyi lengths at the transition point it is found that these quantities diverge as the characteristic length c 1 4 around the critical value of the coupling strength c for any value of the parameter

entropy and complexity analysis of dirac delta like quantum potentials abstract the dirac delta like quantum mechanical potentials are frequently used to describe and interpret numerous phenomena in many scientific fields including atomic and molecular physics condensed matter and quantum computation the entropy and complexity properties of potentials with one and two dirac delta functions are here analytically calculated and numerically discussed in both position and momentum spaces we have studied the information theoretic lengths of fisher renyi and shannon types as well as the cramer rao fisher shannon and lmc shape complexities of the lowest lying stationary states of one delta and twin delta they allow us to grasp and quantify different facets of the spreading of the charge and momentum of the system far beyond the celebrated standard deviation

uncertainty in quantum measurements the object of this letter is to show that except in the case of canonically conjugate observables the generalized heisenberg inequality does not properly express the quantum uncertainty principle it is in general too weak an inequality is obtained which does express the principle

the fluctuation dissipation theorem in the framework of the tsallis statistics in the framework of a generalized statistical mechanics introduced recently by tsallis 1993 we derive a generalized form of the fluctuation dissipation theorem which expresses a relation between extended susceptibilities and equilibrium fluctuations to achieve this we consistently propose a generalized functional form for the instantaneous distribution function the present theorem recovers as particular cases the corresponding generalized relations already obtained for the specific heat in terms of the generalized energy fluctuations and for the susceptibility of a magnetic system under the action of a uniform magnetic field

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

bezier curves in the modification of boundary integral equations bie for potential boundary values problems the paper presents a modification of the classical boundary integral equation method biem for two dimensional potential boundary values problems the proposed modification consists in describing the boundary geometry by means of bezier curves as a result of this analytical modification of the biem a new parametric integral equation system pies was obtained the kernels of these equations include the geometry of the boundary this new pies is no longer defined on the boundary as in the case of the biem but on the straight line for any given domain the solution of the new pies does not require a boundary discretization since it can be reduced merely to an approximation of boundary functions to solve this pies a pseudospectral method has been proposed and the results obtained were compared with exact solutions

superspace geometry the exact uncertainty relationship between complementary aspects after reviewing the standard uncertainty relations due to heisenberg 1927 robertson 1929 and schrodinger 1930 as well as the relations of deutsch and of maassen and uffink including the so called entropic relations the author presents a complete account of the uncertainty relationship between complementary aspects in terms of superspace geometry an approach not hitherto employed two incompatible properties a sigma alpha aalpha mod alpha alpha mod and n sigma nnn mod n n mod belong to a pair of complementary aspects defined by two orthonormal bases mod alpha and mod n in the hilbert space h if the state is mod psi then p alpha mod alpha mod psi mod 2 is the probability of obtaining the value aalpha in a measurement of a and p n mod n mod psi mod 2 is the probability of obtaining the value nn in a measurement of n the two aspects are characterised relative to mod psi by the numbers so called purities pi sigma alpha p alpha 2 and pi sigma n p n 2 both or 1 he gives a complete characterisation of the uncertainty relationship between a and n more precisely between their aspects in terms of the range of joint values of pi pi for arbitrary initial states pure as well as mixed a theorem of lenard is given an alternative proof employing only elementary superspace geometry the results depend on two angles phi m minimal angle and phi m maximal angle between the two aspects exact expressions for phi m and phi m are obtained in terms of the overlap matrix lambda lambda alpha n mod alpha mod n mod 2 as a corollary he finds the uncertainty relation for a pure state mod psi pi pi or 1 1 g 1 1 g cos phi m where g dim h and a sharper one for mixed states pi pi 2 is obtainable if and only if the intersection of the aspects holds a pure state if phi m pi 2 maximal incompatibility then pi pi or 1 1 g is a special case of a stronger relation sigma mu g o pi mu 2 which one obtains for g 1 maximally incompatible aspects by means of a thereom of ivanovic

possible generalization of boltzmann gibbs statistics with the use of a quantity normally scaled in multifractals a generalized form is postulated for entropy namelys q k 1 i 1 w p i q q 1 whereq characterizes the generalization andp i are the probabilities associated withw microscopic configurations w the main properties associated with this entropy are established particularly those corresponding to the microcanonical and canonical ensembles the boltzmann gibbs statistics is recovered as theq 1 limit

stable non gaussian random processes stochastic models with infinite variance stable random variables on the real line multivariate stable distributions stable stochastic integrals dependence structures of multivariate stable distributions non linear regression complex stable stochastic integrals and harmonizable processes self similar processes chentsov random fields introduction to sample path properties boundedness continuity and oscillations measurability integrability and absolute continuity boundedness and continuity via metric entropy integral representation historical notes and extensions

multivariable fluctuation theorems in the steady state cycle kinetics of single enzyme with competing substrates we prove the detailed and integral multivariable fluctuation theorems in the steady state cycle kinetics of single enzyme with competing substrates for any arbitrary time interval 0 t it is shown that the moment generating function for the stochastic number of each enzymatic cycle follows the multivariable fluctuation theorems in the form of kurchan lebowitz spohn type symmetry these symmetry relations associated with different variables are independent of each other which may help experimentally determine the thermodynamic affinities from the sample trajectories separately furthermore we also obtain the kawasaki equalities for the fluctuating chemical work done within each enzymatic cycle the derivation here is directly based on the generalized haldane equalities which say that the forward and backward conditional dwell times for each enzymatic cycle have identical distributions these symmetries are independent of each other which may help experimentally determine the thermodynamic affinities from the sample trajectories separately

the generalized differential quadrature rule for initial value differential equations abstract the generalized differential quadrature rule gdqr proposed recently by the authors is applied here to solve initial value differential equations of the 2nd to 4th order differential quadrature expressions are derived based on the gdqr for these equations the hermite interpolation functions are used as trial functions to obtain the explicit weighting coefficients for an easy and efficient implementation of the gdqr the numerical solutions for example problems demonstrate that the gdqr has high efficiency and accuracy a detailed discussion on the present method is presented by comparing with other existing methods the present method can be extended to other types of differential equation systems

physical origins of entropy production free energy dissipation and their mathematical representations a unifying mathematical theory of nonequilibrium thermodynamics of stochastic systems in terms of master equations is presented as generalizations of isothermal entropy and free energy two functions of state play central roles the gibbs entropy s and the relative entropy f which are related via the stationary distribution of the stochastic dynamics s satisfies the fundamental entropy balance equation ds dt e p ensuremath h d t with entropy production rate e p ensuremath ge 0 and heat dissipation rate h d while df dt ensuremath f d ensuremath le 0 for closed systems that satisfy detailed balance t e p t f d t for open systems one has t e p t f d t q hk t where the housekeeping heat q hk ensuremath ge 0 was first introduced in the phenomenological nonequilibrium steady state thermodynamics put forward by oono and paniconi q hk represents the irreversible work done by the surrounding to the system that is kept away from reaching equilibrium hence entropy production e p consists of free energy dissipation associated with spontaneous relaxation i e self organization f d and active energy pumping that sustains the open system q hk the amount of excess heat involved in the relaxation q ex h d ensuremath q hk f d ensuremath t ds dt two kinds of irreversibility and the meaning of the arrow of time emerge quasistationary processes adiabaticity and maximum principle for entropy are also generalized to nonequilibrium settings

uncertainty relation for fisher information of d dimensional single particle systems with central potentials an uncertainty fisher information relation in quantum mechanics is derived for multidimensional single particle systems with central potentials it is based on the concept of fisher information in the two complementary position and momentum spaces which is a gradient functional of the corresponding probability distributions the lower bound of the product of position and momentum fisher informations is shown to depend on the orbital and magnetic quantum numbers of the physical state and the space dimensionality applications to various elementary systems is discussed

imposition of boundary conditions by modifying the weighting coefficient matrices in the differential quadrature method one of the important issues in the implementation of the differential quadrature method is the imposition of the given boundary conditions there may be multiple boundary conditions involving higher order derivatives at the boundary points the boundary conditions can be imposed by modifying the weighting coefficient matrices directly however the existing method is not robust and is known to have many limitations in this paper a systematic procedure is proposed to construct the modified weighting coefficient matrices to overcome these limitations the given boundary conditions are imposed exactly furthermore it is found that the numerical results depend only on those sampling grid points where the differential quadrature analogous equations of the governing differential equations are established the other sampling grid points with no associated boundary conditions are not essential copyright 2002 john wiley sons ltd

free vibration analysis of isosceles triangular mindlin plates by the triangular differential quadrature method the triangular differential quadrature method is applied to the free flexural vibration of isosceles triangular mindlin plates the first six frequencies are sought and the convergence of triangular differential quadrature method in vibrational analysis of triangular mindlin plates is examined in comparison with available results good to excellent agreement is achieved for triangular plates with combinations of clamped and simply supported boundary conditions the implementation of various other boundary conditions in triangular differential quadrature analysis is also discussed and the sensitivity of solution to the implementation of corner conditions in vibrational analysis is highlighted

some extensions of the uncertainty principle abstract we study the formulation of the uncertainty principle in quantum mechanics in terms of entropic inequalities extending results recently derived by bialynicki birula i bialynicki birula formulation of the uncertainty relations in terms of the renyi entropies physical review a 74 5 2006 052101 and zozor et al s zozor c vignat on classes of non gaussian asymptotic minimizers in entropic uncertainty principles physica a 375 2 2007 499 517 those inequalities can be considered as generalizations of the heisenberg uncertainty principle since they measure the mutual uncertainty of a wave function and its fourier transform through their associated renyi entropies with conjugated indices we consider here the general case where the entropic indices are not conjugated in both cases where the state space is discrete and continuous we discuss the existence of an uncertainty inequality depending on the location of the entropic indices and in the plane our results explain and extend a recent study by luis a luis quantum properties of exponential states physical review a 75 2007 052115 where states with quantum fluctuations below the gaussian case are discussed at the single point 2 2

application of differential quadrature to static analysis of structural components the numerical technique of differential quadrature for the solution of linear and non linear partial differential equations first introduced by bellman and his associates is applied to the equations governing the deflection and buckling behaviour of one and two dimensional structural components separate transformations are used for higher order derivatives as suggested by mingle thus extending the method to treat fourth order equations and to include multiple boundary conditions in the respective co ordinate directions results are obtained for various boundary and loading conditions and are compared with existing exact and numerical solutions by other methods the application of differential quadrature to this class of problems is seen to lead to accurate results with relatively small computational effort

squeezed states generalized hermite polynomials and pseudo diffusion equation abstract we show that the wavefunctions pq n of the harmonic oscillator in the squeezed state representation have the generalized hermite polynomials as their natural orthogonal polynomials these wavefunctions lead to generalized poisson distribution pn pq which satisfy an interesting pseudo diffusion equation p n p q 1 4 2 p 2 1 2 2 q 2 p 2 p q in which the squeeze parameter plays the role of time th entropies sn have minima at the unsqueezed states 1 which means that squeezing or stretching decreases the correlation between momentum p and position q

entropic uncertainty measures for large dimensional hydrogenic systems the entropic moments of the probability density of a quantum system in position and momentum spaces describe not only some fundamental and or experimentally accessible quantities of the system but also the entropic uncertainty measures of renyi type which allow one to find the most relevant mathematical formalizations of the position momentum heisenberg s uncertainty principle the entropic uncertainty relations it is known that the solution of difficult three dimensional problems can be very well approximated by a series development in 1 d in similar systems with a non standard dimensionality d moreover several physical quantities of numerous atomic and molecular systems have been numerically shown to have values in the large d limit comparable to the corresponding ones provided by the three dimensional numerical self consistent field methods the d dimensional hydrogenic atom is the main prototype of the physics of multidimensional many electron systems in this work we rigorously determine the lea

chaotic hypothesis onsager reciprocity and fluctuation dissipation theorem it is shown that the chaoticity hypothesis recently introduced in statistical mechanics which is analogous to ruelle s principle for turbulence implies the onsager reciprocity and the fluctuation dissipation theorem in various reversible models for coexisting transport phenomena

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