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

correction a class of renyi information estimators for multidimensional densities and hq 1 f m fq x dx q 1 of an unknown probability measure with density on rm with respect to the lebesgue measure though kth nearest neighbor distances in a sample x xjy i i d with the density the results in 1 about the asymptotic unbiasedness and consistency of the estimator proposed for iq e fq l x frm fq x dx are correct for q 1 but for q 0 fmm x rf x dx rc see 2 a correct version of the convergence results of ln k q to for 0 g 1 and having unbounded support can be obtained by using results on the sub additivity of euclidean functionals see 3 and can be formulated as follows 2

a novel approach to an impulsive feedback control with and without memory involvement abstract the present work studies the effect of an impulsive perturbation to classical and fractional classes of initial value problems scattering properties of the time sequences corresponding to the impulses are studied in the both integer and non integer order cases it is shown that the sign changing property of the perturbation function determines the mentioned scattering property of the kick times

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

one class classifiers based on entropic spanning graphs one class classifiers offer valuable tools to assess the presence of outliers in data in this paper we propose a design methodology for one class classifiers based on entropic spanning graphs our approach also takes into account the possibility to process nonnumeric data by means of an embedding procedure the spanning graph is learned on the embedded input data and the outcoming partition of vertices defines the classifier the final partition is derived by exploiting a criterion based on mutual information minimization here we compute the mutual information by using a convenient formulation provided in terms of the alpha jensen difference once training is completed in order to associate a confidence level with the classifier decision a graph based fuzzy model is constructed the fuzzification process is based only on topological information of the vertices of the entropic spanning graph as such the proposed one class classifier is suitable also for data characterized by complex geometric structures we provide experiments on well known benchmarks containing both feature vectors and labeled graphs in addition we apply the method to the protein solubility recognition problem by considering several representations for the input samples experimental results demonstrate the effectiveness and versatility of the proposed method with respect to other state of the art approaches

entropic integrals of hyperspherical harmonics and spatial entropy of d dimensional central potentials the information entropy of a single particle in a quantum mechanical d dimensional central potential is separated in two parts one depends only on the specific form of the potential radial entropy and the other depends on the angular distribution spatial entropy the latter is given by an entropic like integral of the hyperspherical harmonics which is expressed in terms of the entropy of the gegenbauer polynomials this entropy is expressed in terms of the values of the quadratic logarithmic potential of gegenbauer polynomials cn t at the zeros of these polynomials then this potential for integer is given as a finite expansion of chebyshev polynomials of even order whose coefficients are shown to be wilson polynomials

higher order asymptotic distribution of the eigenvalues of nondefinite sturm liouville problems with one turning point in this paper we derive the higher order asymptotic distribution of the positive eigenvalues associated with a linear real second order equation y x q x y 0 of sturm liouville type on a b with dirichlet boundary condition i e y a y b 0 where 1 is chosen so that the boundary problem is non definite

existence and uniqueness of solutions for impulsive fractional differential equations motivated by some recent developments in the existence theory of impulsive fractional differential equations in this paper we present a general method for converting an impulsive fractional differential equation to an equivalent integral equation the applicability of the method is demonstrated by considering some boundary value problems for impulsive fractional differential equations

kullback leibler divergence measure of intermittency application to turbulence for generic systems exhibiting power law behaviors and hence multiscale dependencies we propose a simple tool to analyze multifractality and intermittency after noticing that these concepts are directly related to the deformation of a probability density function from gaussian at large scales to non gaussian at smaller scales our framework is based on information theory and uses shannon entropy and kullback leibler divergence we provide an extensive application to three dimensional fully developed turbulence seen here as a paradigmatic complex system where intermittency was historically defined and the concepts of scale invariance and multifractality were extensively studied and benchmarked we compute our quantity on experimental eulerian velocity measurements as well as on synthetic processes and phenomenological models of fluid turbulence our approach is very general and does not require any underlying model of the system although it can probe the relevance of such a model

ica using spacings estimates of entropy this paper presents a new algorithm for the independent components analysis ica problem based on an efficient entropy estimator like many previous methods this algorithm directly minimizes the measure of departure from independence according to the estimated kullback leibler divergence between the joint distribution and the product of the marginal distributions we pair this approach with efficient entropy estimators from the statistics literature in particular the entropy estimator we use is consistent and exhibits rapid convergence the algorithm based on this estimator is simple computationally efficient intuitively appealing and outperforms other well known algorithms in addition the estimator s relative insensitivity to outliers translates into superior performance by our ica algorithm on outlier tests we present favorable comparisons to the kernel ica fast ica jade and extended infomax algorithms in extensive simulations we also provide public domain source code for our algorithms

comparing and weighting imperfect models using d probabilities abstractwe propose a new approach for assigning weights to models using a divergence based method d probabilities relying on evaluating parametric models relative to a nonparametric bayesian ref

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

entropic properties of d dimensional rydberg systems abstract the fundamental information theoretic measures the renyi r p and tsallis t p entropies p 0 of the highly excited rydberg quantum states of the d dimensional d 1 hydrogenic systems which include the shannon entropy p 1 and the disequilibrium p 2 are analytically determined by use of the strong asymptotics of the laguerre orthogonal polynomials which control the wavefunctions of these states we first realize that these quantities are derived from the entropic moments of the quantum mechanical probability r densities associated to the rydberg hydrogenic wavefunctions n l r which are closely connected to the l p norms of the associated laguerre polynomials then we determine the n asymptotics of these norms in terms of the basic parameters of our system the dimensionality d the nuclear charge and the hyperquantum numbers n l of the state by use of recent techniques of approximation theory finally these three entropic quantities are analytically and numerically discussed in terms of the basic parameters of the system for various particular states

higher order asymptotic formula for the eigenvalues of sturm liouville problem with n turning points abstract in this paper we investigate the asymptotic behavior of the differential equation y r x q x y 0 0 x 1 where 0 1 contains a finite number of zeros of r x the so called turning points is a real parameter and the function q x is bounded and integrable in 0 1 using a technique used previously in b j harris s t talarico on the eigenvalues of second order linear differential equations with fractional transition points math proc r ir acad ser a 99 1 1999 29 38 we derive the higher order asymptotic distribution of the positive eigenvalues associated with this equation for the dirichlet problem i e y 0 y 1 0 this method is different from eberhard s method w eberhard g freiling the distribution of the eigenvalues for second order eigenvalues problems in the presence of an arbitrary number of turning points results math 21 1992 24 41 they have used asymptotic solution in order to obtain asymptotic distribution of the eigenvalues in most differential equations with variable coefficient it is impossible to obtain an exact solution so we want to obtain asymptotic distribution of the eigenvalues without solving equation note that in similar case the leading term of the asymptotic distribution of positive and negative eigenvalues was derived previously by atkinson and mingarelli f v atkinson a mingarelli asymptotics of the number of zeros and the eigenvalues of general weighted sturm liouville problems j reine angew math 395 1986 380 93

on quantizer distortion and the upper bound for exponential entropy a sharp upper bound is derived for the exponential entropy in the class of absolutely continuous distributions with specific standard deviation and an exact description of the extremal distributions this result is interpreted as determining the least favorable cases for certain methods of quantization of analog sources it is known that for a large class of quantizers both zero memory and vector the rth power distortion as well as some other distortion criteria are bounded below by a constant depending on r multiplied by a certain integral of the source s probability density it is pointed out that this bound can be rewritten in terms of the exponential entropy the exponential entropy measures the quantitative extent or range of the source distribution this fact gives a physical interpretation of the indicated limits of quantizer performance further elucidated by the main result

ensemble estimators for multivariate entropy estimation the problem of estimation of density functionals like entropy and mutual information has received much attention in the statistics and information theory communities a large class of estimators of functionals of the probability density suffer from the curse of dimensionality wherein the mean squared error decays increasingly slowly as a function of the sample size t as the dimension d of the samples increases in particular the rate is often glacially slow of order o t d where 0 is a rate parameter examples of such estimators include kernel density estimators k nearest neighbor k nn density estimators k nn entropy estimators intrinsic dimension estimators and other examples in this paper we propose a weighted affine combination of an ensemble of such estimators where optimal weights can be chosen such that the weighted estimator converges at a much faster dimension invariant rate of o t1 furthermore we show that these optimal weights can be determined by solving a convex optimization problem which can be performed offline and does not require training data we illustrate the superior performance of our weighted estimator for two important applications 1 estimating the panter dite distortion rate factor and 2 estimating the shannon entropy for testing the probability distribution of a random sample

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

estimating mutual information we present two classes of improved estimators for mutual information m x y from samples of random points distributed according to some joint probability density mu x y in contrast to conventional estimators based on binnings they are based on entropy estimates from k nearest neighbor distances this means that they are data efficient with k 1 we resolve structures down to the smallest possible scales adaptive the resolution is higher where data are more numerous and have minimal bias indeed the bias of the underlying entropy estimates is mainly due to nonuniformity of the density at the smallest resolved scale giving typically systematic errors which scale as functions of k n for n points numerically we find that both families become exact for independent distributions i e the estimator m x y vanishes up to statistical fluctuations if mu x y mu x mu y this holds for all tested marginal distributions and for all dimensions of x and y in addition we give estimators for redundancies between more than two random variables we compare our algorithms in detail with existing algorithms finally we demonstrate the usefulness of our estimators for assessing the actual independence of components obtained from independent component analysis ica for improving ica and for estimating the reliability of blind source separation

estimation of nonlinear functionals of densities with confidence this paper introduces a class of k nearest neighbor k nn estimators called bipartite plug in bpi estimators for estimating integrals of nonlinear functions of a probability density such as shannon entropy and renyi entropy the density is assumed to be smooth have bounded support and be uniformly bounded from below on this set unlike previous k nn estimators of nonlinear density functionals the proposed estimator uses data splitting and boundary correction to achieve lower mean square error specifically we assume that t i i d samples xi rd from the density are split into two pieces of cardinality m and n respectively with m samples used for computing a k nn density estimate and the remaining n samples used for empirical estimation of the integral of the density functional by studying the statistical properties of k nn balls explicit rates for the bias and variance of the bpi estimator are derived in terms of the sample size the dimension of the samples and the underlying probability distribution based on these results it is possible to specify optimal choice of tuning parameters m t k for maximizing the rate of decrease of the mean square error the resultant optimized bpi estimator converges faster and achieves lower mean squared error than previous k nn entropy estimators in addition a central limit theorem is established for the bpi estimator that allows us to specify tight asymptotic confidence intervals

directed information graphs for the granger causality of multivariate time series abstract in this paper we investigate the links between strong granger causality and directed information theory for multivariate time series based on the decomposition of conditional directed information we propose a definition of granger causality including instantaneous variables in the conditional set which can avoid the spurious causality the directed information graphs are presented to describe the granger causality and instantaneous coupling the structure learning of the graph models is based on the leonenko s k nn estimator of the statistics and a permutation test of the significant finally we demonstrate the numerical implementation of these techniques on linear and nonlinear time series

binless strategies for estimation of information from neural data we present an approach to estimate information carried by experimentally observed neural spike trains elicited by known stimuli this approach makes use of an embedding of the observed spike trains into a set of vector spaces and entropy estimates based on the nearest neighbor euclidean distances within these vector spaces l f kozachenko and n n leonenko probl peredachi inf 23 9 1987 using numerical examples we show that this approach can be dramatically more efficient than standard bin based approaches such as the direct method s p strong r koberle r r de ruyter van steveninck and w bialek phys rev lett 80 197 1998 for amounts of data typically available from laboratory experiments

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

dimensional regularization of renyi s statistical mechanics abstract we show that typical renyi s statistical mechanics quantifiers exhibit poles we are referring to the partition function z and the mean energy u renyi s entropy is characterized by a real parameter the poles emerge in a numerable set of rational numbers belonging to the line physical effects of these poles are studied by appeal to dimensional regularization as usual interesting effects are found as for instance gravitational ones in particular negative specific heats

global stability of impulsive fractional differential equations abstract in the present paper the global stability of systems of impulsive differential equations of fractional order with impulse effect at fixed moments of time is considered using piecewise continuous functions of the type of lyapunov s functions and a new fractional comparison principle sufficient conditions for various types of global stability are proved

quantum expectation values of d dimensional rydberg hydrogenic states by use of laguerre and gegenbauer asymptotics the radial position and momentum p 1 3 expectation values of the d dimensional rydberg hydrogenic states i e states where the electron has a large hyperquantum number n are rigorously determined by means of powerful tools of the modern approximation theory relative to the asymptotics of the varying orthogonal laguerre and gegenbauer polynomials which control the corresponding wavefunctions in position and momentum spaces

renyi entropies characterizing the shape and the extension of the phase space representation of quantum wave functions in disordered systems we discuss some properties of the generalized entropies called renyi entropies and their application to the case of continuous distributions in particular it is shown that these measures of complexity can be divergent however their differences are free from these divergences thus enabling them to be good candidates for the description of the extension and the shape of continuous distributions we apply this formalism to the projection of wave functions onto the coherent state basis i e to the husimi representation we also show how the localization properties of the husimi distribution on average can be reconstructed from its marginal distributions that are calculated in position and momentum space in the case when the phase space has no structure i e no classical limit can be defined numerical simulations on a one dimensional disordered system corroborate our expectations

practical bounds on image denoising from estimation to information recently in a previous work we proposed a way to bound how well any given image can be denoised the bound was computed directly from the noise free image that was assumed to be available in this work we extend the formulation to the more practical case where no ground truth is available we show that the parameters of the bounds namely the cluster covariances and level of redundancy for patches in the image can be estimated directly from the noise corrupted image further we analyze the bounds formulation to show that these two parameters are interdependent and they along with the bounds formulation as a whole have a nice information theoretic interpretation as well the results are verified through a variety of well motivated experiments

the hausdorff entropic moment problem our aim in this paper is twofold first to find the necessary and sufficient conditions to be satisfied by a given sequence of real numbers n n 0 to represent the entropic moments 0 a x ndx of an unknown non negative decreasing and differentiable a e density function x with a finite interval support these moments are called entropic moments because they are closely connected with various information entropies renyi tsallis second we outline an efficient method for the reconstruction of the density function from the knowledge of its first n entropic moments

lipschitz stability criteria for functional differential systems of fractional order in this paper a class of fractional functional differential equations is investigated using differential inequalities and lyapunov like functions lipschitz stability uniform lipschitz stability and global uniform lipschitz stability criteria are proved since the problem of lipschitz stability of dynamic systems is relevant in various contexts including many inverse and control problem our results can be applied in the qualitative investigations of many practical problems of diverse interest

on almost periodic processes in uncertain impulsive delay models of price fluctuations in commodity markets we present an impulsive price model for a single commodity market with delays and uncertain terms impulsive perturbations are realized at fixed moments of time and are proposed to model price shocks in the case of continuous time representation to do so the paper resorts to the theory of impulsive differential equations uncertain terms are due to modeling errors measurement inaccuracy mutations in the fluctuation processes and so on we investigate conditions under which the extended model is capable of generating a stable almost periodic process

a class of renyi information estimators for multidimensional densities a class of estimators of the renyi and tsallis entropies of an unknown distribution f in r m is presented these estimators are based on the k th nearest neighbor distances computed from a sample of n i i d vectors with distribution f we show that entropies of any order q including shannon s entropy can be estimated consistently with minimal assumptions on f moreover we show that it is straightforward to extend the nearest neighbor method to estimate the statistical distance between two distributions using one i i d sample from each

asymptotic theory of greedy approximations to minimal k point random graphs let spl chi sub n x sub 1 x sub n be an independent and identically distributed i i d sample having multivariate distribution p we derive almost sure a s limits for the power weighted edge weight function of greedy approximations to a class of minimal graphs spanning k of the n samples the class includes minimal k point graphs constructed by the partitioning method of ravi sundaram marathe rosenkrantz and ravi see proc 5th annu acm siam symp discrete algorithms arlington va p 546 55 1994 where the edge weight function satisfies the quasi additive property of redmond and yukich see ann appl probab vol 4 no 4 p 1057 73 1994 in particular this includes greedy approximations to the k point minimal spanning tree k mst steiner tree k st and the traveling salesman problem k tsp an expression for the influence function of the minimal weight function is given which characterizes the asymptotic sensitivity of the graph weight to perturbations in the underlying distribution the influence function takes a form which indicates that the k point minimal graph in d 1 dimensions has robustness properties in r sup d which are analogous to those of rank order statistics in one dimension a direct result of our theory is that the log weight of the k point minimal graph is a consistent nonparametric estimate of the renyi entropy of the distribution p possible applications of this work include analysis of random communication network topologies estimation of the mixing coefficient in spl epsiv contaminated mixture models outlier discrimination and rejection clustering and pattern recognition robust nonparametric regression two sample matching and image registration

sums of functions of nearest neighbor distances moment bounds limit theorems and a goodness of fit test abstract the limiting behavior of sums of functions of nearest neighbor distances is studied for an m dimensional sample a central limit theorem and moment bounds for such sums and an invariance principle for the empirical process of nearest neighbor distances are both established as a consequence the asymptotic behavior of a practicable goodness of fit test is obtained based on nearest neighbor distances

minimization of the renyi entropy production in the stationary states of the brownian process with matched death and birth rates we analyze the fleming viot process the system is confined in a box whose boundaries act as a sink of brownian particles the death rate at the boundaries is matched by the branching birth rate in the system and thus the number of particles is kept constant we show that such a process is described by the renyi entropy whose production is minimized in the stationary state the entropy production in this process is a monotonically decreasing function of time irrespective of the initial conditions the first laplacian eigenvalue is shown to be equal to the renyi entropy production in the stationary state as an example we simulate the process in a two dimensional box

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

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

nonlocal initial value problems for differential equations with hilfer fractional derivative in this paper we discuss the existence of solutions to nonlocal initial value problem for differential equations with hilfer fractional derivative to begin with we establish an equivalent mixed type integral equation for our problem then we utilize basic properties of hilfer fractional calculus and fixed point methods to derive three fundamental existence results in the weighted space of continuous functions some examples are given to illustrate our results

exact time dependent solutions of the renyi fokker planck equation and the fokker planck equations related to the entropies proposed by sharma and mittal abstract exact time dependent solutions representing generalized one dimensional ornstein uhlenbeck processes are derived from nonlinear fokker planck equations and the corresponding diffusion processes are studied the corresponding systems are related to the renyi entropy and entropies proposed by sharma and mittal in the former case normal diffusion is found with a diffusion rate depending on the order of the renyi entropy in the latter case we find anomalous diffusion processes

asymptotic moments of near neighbour distance distributions let c be a compact convex body in r m and consider a set of points selected at random from c according to some well behaved sampling distribution we obtain an asymptotic expression for the positive moments of the k th near neighbour distance distribution as the number of points increases to infinity

bias reduction and metric learning for nearest neighbor estimation of kullback leibler divergence asymptotically unbiased nearest neighbor estimators for kl divergence have recently been proposed and demonstrated in a number of applications with small sample sizes however these nonparametric methods typically suer from high estimation bias due to the non local statistics of empirical nearest neighbor information in this paper we show that this non local bias can be mitigated by changing the distance metric and we propose a method for learning an optimal mahalanobis type metric based on global information provided by approximate parametric models of the underlying densities in both simulations and experiments we demonstrate that this interplay between parametric models and nonparametric estimation methods signicantly improves the accuracy of the nearest neighbor kl divergence estimator

asymptotics of the number of zeros and of the eigenvalues of general weighted sturm liouville problems soit n le nombre de zeros d une solution de l equation de sturm liouville ponderee generale p x y r x q x y 0 sur 0 b ou 0 b p q r 0 b r et 1 p q r l 0 b on donne une demonstration directe de l estimation asymptotique de n

is there a stable hydrogen atom in higher dimensions the schrodinger equation in higher dimensions is considered it consists of the kinetic energy part given by the corresponding laplace operator and a term describing the interaction with the electrostatic field of a point charge from rutherford type scattering experiments one can conclude that the potential of a point charge is 1 r irrespective of the dimension of the space where the experiment is carried through also the structure of the kinetic energy is unchanged in higher dimensions so that one is lead to the result that there exist stable atoms in higher spatial dimensions d 4 the solutions and energy eigenvalues to this schrodinger equation in higher dimensions are presented as a consequence the dimensionality of space can be read off from the spectral scheme of atoms the three dimensionality of space is a consequence of the existence of the lyman series another consequence is that the maxwell equations in higher dimensions must be modified in order to have the 1 r potential as solution for