exactly solvable potentials and the concept of shape invariance the author analyses the relationship between shape invariance and exactly solvable potentials using earlier results he shows that all actually known shape invariant potentials with a single parameter can be recovered quite simply extension to the case of shape invariant potentials with two parameters as well as to non shape invariant but exactly solvable potentials are also possible

independence and symbolic independence of nonstationary heartbeat series during atrial fibrillation abstract heartbeat intervals during atrial fibrillation are commonly believed to form a series of almost independent variables the series extracted from 24 h holter recordings show a nonstationary behavior because of nonstationarity it is difficult to give a quantitative measure of independence in this paper we use and compare two methods for this the first is a classical method which models a nonstationary series using a linear gaussian state space model in this framework the independence is tested on the stationary sequence of the residuals the second method codes data into permutations and tests the uniformity of their distribution this test assumes as null hypothesis a weaker form of independence which we call symbolic independence we discuss some advantages of symbolic independence in the context of heartbeat series we analyze the time series of heartbeat intervals from 24 h holter recordings of nine subjects with chronic atrial fibrillation and find that the detrended series is a zero or one memory process for 83 of regular segments and is symbolically independent for 93 of segments

sharp estimates for solutions of multi bubbles in compact riemann surfaces in this paper we consider a sequence of multibubble solutions u k of the equation 0 1 0 u k k he uk f m he u k d 1 0 in m where h is a c 2 positive function in a compact riemann surface m and k is a constant satisfying lim k k 8m for some positive integer m 1 we prove among other things that k 8m 2 m m j 1h 1 p k j 0 log h p k j 8m 2k p k j k j e k j o e k j where p k j are centers of the bubbles of u k and k j are the local maxima of u k after adding a constant this yields a uniform bound of solutions as k converges to 8m from below provided that 0 log h p k j 8m 2k p k j 0 it generalizes a previous result due to ding jost li and wang 18 and nolasco and tarantello 31 which says that any sequence of minimizers u k is uniformly bounded if k 0 for any maximum point p of the sum of 2 log h and the regular part of the green function where k is the gaussian curvature of m the analytic work of this paper is the first step toward computing the topological degree of 0 1 which was initiated by li 24

estimation of 2 d noisy fractional brownian motion and its applications using wavelets the two dimensional 2 d fractional brownian motion fbm model is useful in describing natural scenes and textures most fractal estimation algorithms for 2 d isotropic fbm images are simple extensions of the one dimensional 1 d fbm estimation method this method does not perform well when the image size is small say 32 spl times 32 we propose a new algorithm that estimates the fractal parameter from the decay of the variance of the wavelet coefficients across scales our method places no restriction on the wavelets also it provides a robust parameter estimation for small noisy fractal images for image denoising a wiener filter is constructed by our algorithm using the estimated parameters and is then applied to the noisy wavelet coefficients at each scale we show that the averaged power spectrum of the denoised image is isotropic and is a nearly 1 f process the performance of our algorithm is shown by numerical simulation for both the fractal parameter and the image estimation applications to coastline detection and texture segmentation in a noisy environment are also demonstrated

the separation of a coupled system of differential equations in quantum mechanics the authors discuss various aspects of the methodology of the separation operation of a coupled system of differential equations in quantum mechanics namely the darboux transformation the t and n matrices approach for the case of symmetric and non symmetric coupling of the equations the analysis is also extended to systems with more than two equations with in each case specific examples illustrating their implementation in practical situations

wild binary segmentation for multiple change point detection we propose a new technique called wild binary segmentation wbs for consistent estimation of the number and locations of multiple change points in data we assume that the number of change points can increase to infinity with the sample size due to a certain random localisation mechanism wbs works even for very short spacings between the change points and or very small jump magnitudes unlike standard binary segmentation on the other hand despite its use of localisation wbs does not require the choice of a window or span parameter and does not lead to a significant increase in computational complexity wbs is also easy to code we propose two stopping criteria for wbs one based on thresholding and the other based on what we term the strengthened schwarz information criterion we provide default recommended values of the parameters of the procedure and show that it offers very good practical performance in comparison with the state of the art the wbs methodology is implemented in the r package wbs available on cran in addition we provide a new proof of consistency of binary segmentation with improved rates of convergence as well as a corresponding result for wbs

two dimensional solutions of a mean field equation on flat tori abstract we study the mean field equation on the flat torus t c z z u e u t e u 1 t 0 where is a real parameter for a general flat torus we obtain the existence of two dimensional solutions bifurcating from the trivial solution at each eigenvalue up to a multiplicative constant t of laplace operator on the torus in the space of even symmetric functions we further characterize the subset of all eigenvalues through which only one bifurcating curve passes finally local convexity near bifurcating points of the solution curves are obtained

offdiagonal complexity a computationally quick complexity measure for graphs and networks abstract a vast variety of biological social and economical networks shows topologies drastically differing from random graphs yet the quantitative characterization remains unsatisfactory from a conceptual point of view motivated from the discussion of small scale free networks a biased link distribution entropy is defined which takes an extremum for a power law distribution this approach is extended to the node node link cross distribution whose nondiagonal elements characterize the graph structure beyond link distribution cluster coefficient and average path length from here a simple and computationally cheap complexity measure can be defined this offdiagonal complexity odc is proposed as a novel measure to characterize the complexity of an undirected graph or network while both for regular lattices and fully connected networks odc is zero it takes a moderately low value for a random graph and shows high values for apparently complex structures as scale free networks and hierarchical trees the odc approach is applied to the helicobacter pylori protein interaction network and randomly rewired surrogates

constant q curvature metrics in arbitrary dimension working in a given conformal class we prove existence of constant q curvature metrics on compact manifolds of arbitrary dimension under generic assumptions the problem is equivalent to solving a nth order non linear elliptic differential or integral equation with variational structure where n is the dimension of the manifold since the corresponding euler functional is in general unbounded from above and below we use critical point theory jointly with a compactness result for the above equation

noise reduction in chaotic time series data a survey of common methods this paper surveys some of the methods that have been suggested for reducing noise in time series data whose underlying dynamical behavior can be characterized as low dimensional chaos although the procedures differ in details all of them must solve three basic problems how to reconstruct an attractor from the data how to approximate the dynamics in various regions on the attractor and how to adjust the observations to satisfy better the approximations to the dynamics all current noise reduction methods have similar limitations but the basic problems are reasonably well understood the methods are an important tool in the experimentalist s repertoire for data analysis in our view they should be used more widely particularly in studies of attractor dimension lyapunov exponents prediction and control

supersymmetry operator transformations and exactly solvable potentials a large class of potentials can be solved algebraically by using supersymmetry and shape invariance the authors apply operator transformations f transformations to these algebraically solvable problems to obtain a larger class of solvable potentials the natanzon class of potentials which are not shape invariant the important condition which is independent of supersymmetry for finding new potentials from old ones using operator transformations is that the resulting schrodinger equation has a potential which does not depend on the state as a special case of the f transformation they rederive the previously known connection between the 3d harmonic oscillator the hydrogen atom and the morse potential they also discuss the lack of commutivity of susy and the f transformations

darboux transformations for the schrodinger equation in three dimensions the author applies darboux transformations to the schrodinger equation in three dimensions and derives the general form of the potentials and potential differences which can be treated by these transformations in particular the author establishes the existence of a new class of potentials whose solutions are related to each other by these transformations

families of strictly isospectral potentials for any potential v x which holds n bound states the authors use repeated supersymmetry transformations to construct an n parameter family of strictly isospectral potentials identical eigenenergies reflection and transmission amplitudes for all family members they investigate how the shape and behaviour of various isospectral potentials changes as the n parameters are varied contained in this family are potentials with n widely separated wells each of which holds a single bound state

qualitative properties of solutions to some nonlinear elliptic equations in r 2 o introduction elliptic equations in this paper we investigate properties of the solutions to the au r x e u x x r 2 for functions r x which are positive near infinity equations of this kind arise from a variety of situations such as from prescribing gaussian curvature in geometry i 2 and from combustion theory in physics 3 recently a series ofworks have been sought to understand the existence and the qualitative properties of the solutions of ni i 5 and ni cheng 4 1 considered the case where r x is nonpositive mcowen 6 1 and aviles 7 investigated the situation where r x 0 in some order as ixl in our previous paper 1 we consider a special case where r is a constant we proved that the solutions are radially symmetric and hence classified all the solutions in this paper we consider more general functions r x first we obtain the asymptotic behavior of the solution near infinity consequently we prove that all the solutions satisfy an identity which is somewhat of a generalization of the well known kazdan warner condition finally using the asymptotic behavior together with the further development of the method employed in our previous paper i 1 we show that all the solutions are radially symmetric provided r is radially symmetric and nonincreasing this part can be viewed as the completion of 1 throughout this paper we assume that the function r x is positive near infinity in 1 we study the asymptotic behavior of the solution u x of let fl 1 2n r2 r x e dx under some appropriate conditions we show that the solutions approach

testing and locating variance changepoints with application to stock prices abstract this article explores testing and locating multiple variance changepoints in a sequence of independent gaussian random variables assuming known and common mean this type of problem is very common in applied economics and finance a binary procedure combined with the schwarz information criterion sic is used to search all of the possible variance changepoints existing in the sequence the simulated power of the proposed procedure is compared to that of the cusum procedure used by inclan and tiao to cope with variance changepoints the sic and unbiased sic for this problem are derived to obtain the percentage points of the sic criterion the asymptotic null distribution of a function of the sic is obtained and then the approximate percentage points of the sic are tabulated finally the results are applied to the weekly stock prices the unknown but common mean case is also outlined at the end

coupled channel scattering and separation of coupled differential equations by generalized darboux transformations general methods for separation of coupled differential equations are applied in coupled channel scattering theory with appropriate asymptotic conditions it is shown that the problems with thresholds cannot be separated finally it is pointed out that separation of the coupled channel s wave problem without thresholds can be realized by the use of generalized darboux transformations

a generalized cole hopf transformation for nonlinear odes we introduce a hybrid cole hopf darboux transformation to relate the solutions of nonlinear and linear second order differential equations and derive classification and sufficient condition for this correspondence we explore physical applications of this correspondence to nonlinear oscillations the duffing equation and a nonlinear form of the schr dinger equation for the nonrelativistic hydrogen atom in addition we show that solutions of some nonlinear second order equations are related to the special functions of mathematical physics through this transformation these nonlinear equations can be viewed as the class of special nonlinear equations which correspond to the linear differential equations which define the special functions of mathematical physics

optimal detection of changepoints with a linear computational cost in this article we consider the problem of detecting multiple changepoints in large datasets our focus is on applications where the number of changepoints will increase as we collect more data for example in genetics as we analyze larger regions of the genome or in finance as we observe time series over longer periods we consider the common approach of detecting changepoints through minimizing a cost function over possible numbers and locations of changepoints this includes several established procedures for detecting changing points such as penalized likelihood and minimum description length we introduce a new method for finding the minimum of such cost functions and hence the optimal number and location of changepoints that has a computational cost which under mild conditions is linear in the number of observations this compares favorably with existing methods for the same problem whose computational cost can be quadratic or even cubic in simulation studies we show that our new method can

computational applications of nonextensive statistical mechanics computational applications of the nonextensive entropy s q and nonextensive statistical mechanics a current generalization of the boltzmann gibbs bg theory are briefly reviewed the corresponding bibliography is provided as well

factorization of systems of differential equations it is shown that the classical infeld hull factorization method can be extended to coupled systems of second order equations a complete solution of the factorization equations in two dimensions is given and a partial enumeration of factorizable systems is made

supersymmetry and potentials with bound states at arbitrary energies ii for pt i see ibid vol 19 p 2297 1986 it has been shown previously that a potential v0 chi in one dimension which supports no bound states may be used as a reference potential from which by successive applications of the concept of a supersymmetric partner to a given hamiltonian it is possible to find a potential vn chi which supports any specified number n of bound states at any chosen energies ej j 1 n the reflection coefficient of vn is related to the reflection coefficient of v0 various alternative representations of the potentials constructed by this procedure are presented an illustrative example in which vn is constructed by using a sech2 chi barrier as the reference potential is discussed

generalized entropy arising from a distribution of q indices it is by now well known that the boltzmann gibbs bg entropy can be usefully generalized using the nonextensive entropies which have been applied to a wide range of phenomena however it seems that even more general entropies could be useful in order to describe other complex physical systems a task which has already been undertaken in the literature following this approach we introduce here a quite general entropy based on a distribution of q indices thus generalizing s q we establish some general mathematical properties for the new entropic functional and explore some examples we also exhibit a procedure for finding given any entropic functional the q indices distribution that produces it finally on the road to establishing a quite general statistical mechanics we briefly address possible generalized constraints under which the present entropy could be extremized in order to produce canonical ensemble like stationary state distributions for hamiltonian systems

preseismic electromagnetic signals in terms of complexity there is a recent thesis in the literature that an important organization of a physical system precedes a catastrophic event in this context one can search for signatures that imply the transition from a normal state to a main catastrophic event e g earthquake experimental techniques are thus useful in corroborating theories from observed data for example recent results indicate that preseismic electromagnetic time series contain information characteristic of an ensuing earthquake event hereby we attempt to demonstrate that an easily computable complexity measure such as t complexity or approximate entropy gives evidence of state changes leading to the point of global instability the appearance of a precatastrophic state is characterized by significant lower complexity in terms of t complexity and approximate entropy the present study confirms the conclusions of previous works based on an independent linear fractal spectral analysis this convergence between nonlinear and linear analysis provides a more reliable detection concerning the emergence of the last phase of the earthquake preparation process more precisely we claim that our results suggest an important principle significant complexity decrease and accession of persistency in electromagnetic em time series can be confirmed at the tail of the preseismic em emission which could be used as diagnostic tools for the earth s impending crust failure direct laboratory and field experimental data as well as theoretical arguments support the conclusions of the present analysis 2006 the american physical society

darboux transformations for schrodinger equations in two variables darboux transformations in one variable form the basis for the factorization methods and have numerous applications to geometry nonlinear equations and susy quantum mechanics in spite of this wide range of applications the theory of darboux transformations in two variables and its elegant relationship to analytic complex functions has not been recognized in the literature to close this gap we develop in this paper the theory of darboux transformation in the context of schrodinger equations in two variables this yields a constructive algorithm to determine the relationship between potential functions which are related by darboux transformations

separation of coupled systems of differential equations by darboux transformations the author shows that certain classes of coupled schrodinger equations in one dimension which appear in various physical applications can be uncoupled by the application of an appropriate darboux transformation

the darboux transformation and solvable double well potential models for schrodinger equations the darboux transformation a method used to transform a schrodinger type equation to a schrodinger equation with a new potential is discussed an exactly solvable double well potential model for the one dimensional schrodinger equation is obtained

long term memory in stock market prices a test for long term memory that is robust to short range dependence is developed it is a modification of the r s statistic and the relevant asymptotic sampling theory is derived via functional central limit theory contrary to previous findings when applied to daily and monthly stock returns indexes over several time periods this test yields no evidence of long range dependence once short range dependence is accounted for monte carlo experiments show that the modified r s test has power against at least two specific models of long term memory suggesting that models with short range dependence may adequately capture the behavior of historical stock returns copyright 1991 by the econometric society

the theory of coupled differential equations in supersymmetric quantum mechanics there is a close relationship between the theory of coupled differential equations and supersymmetric quantum mechanics the authors set up the bridge connecting these fields with two theorems concerning both the coupling and noncoupling cases

variance change point detection for fractional brownian motion based on the likelihood ratio test abstract fractional brownian motion is one of the main stochastic processes used for describing the long range dependence phenomenon for self similar processes it appears that for many real time series characteristics of the data change significantly over time such behaviour one can observe in many applications including physical and biological experiments in this paper we present a new technique for the critical change point detection for cases where the data under consideration are driven by fractional brownian motion with a time changed diffusion coefficient the proposed methodology is based on the likelihood ratio approach and represents an extension of a similar methodology used for brownian motion the process with independent increments here we also propose a statistical test for testing the significance of the estimated critical point in addition to that an extensive simulation study is provided to test the performance of the proposed method

the atkinson prufer transformation and the eigenvalue problem for coupled systems of schrodinger equations the matrix generalisation of the prufer transformation introduced by atkinson is applied to a coupled system of radial schrodinger equations it is shown that the phase functions corresponding to the matrix case exhibit properties analogous to those of the prufer phase function encountered in the scalar case rigorous theorems are established which allow the authors to determine the eigenvalues of the original schrodinger system with an arbitrary accuracy provided that the asymptotic behaviour of the phase functions is known the possibility of obtaining the phase functions by means of the integration of an appropriate system of nonlinear first order differential equations is briefly discussed

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

approach to general methods for fitting and their sensitivity abstract this paper examines the performance of various fitting methods between the curves of two explicit functions the approximating test function is considered to be one parametrical first and multi parametrical directly after the widely used least square deviations constitutional method based on the euclidean norm is not unique methods based on q norm for q 1 can also be defined emphasis on these methods especially for q 1 is placed furthermore any functional fulfilling the norm s preconditions induces a metric for deviations supporting a respective method for fitting through the minimization of total deviations this dissertation addresses also the sensitivity of each method that is a measure of how abrupt the variation of the total deviations near its minimum is we show that the least square method does not indispensably perform the largest sensitivity in regard to the alternative methods based on other q norms in addition we represent the explicit general expression of normal equations from which the fitting can be achieved and the sensitivity from which one can positively extract the suitable norm for a given model

dynamical entropies applied to stochastic resonance we will show that sr is accompanied by an extremal behavior of these quantities when sweeping noise intensity over the resonance region we will prove that an increase of applied noise at first leads to an increase of temporal order in the output sequence simultaneously the predictability of the output sequence will improve in the resonance region from the kullback entropy analysis we will learn that for a certain value of noise intensity the input and output distributions of binary subsequences maximally match moreover input and output subsequences registered at equal times exhibit maximum correlation finally we relate these results to the syn

heuristic segmentation of a nonstationary time series is the standard deviation of the mean values of the segments and s e is the standard deviation of the fluctuations within a segment furthermore we determine that the performance of the algorithm is generally not affected by uncorrelated noise spikes or by weak long range temporal correlations of the fluctuations within segments

a sup inf inequality for some nonlinear elliptic equations involving exponential nonlinearities let u be a solution of the equation u v x eu in a domain r2 where 0 a v b and v is lipschitz continuous we prove that sup u can be controlled in terms of inf u more precisely supku inf u c a b k v l for any compact subset k this extends an earlier result of shafrir who obtained a similar conclusion when v 1

moving planes moving spheres and a priori estimates abstract in this paper we use the method of moving planes and the method of moving spheres to obtain a priori estimates for the solutions of semi linear elliptic equations using the method of moving planes we establish a sharper estimate on the solutions for prescribing scalar curvature equations with indefinite curvature functions and thus generalized a resent result of lin applying the method of moving spheres we give a different proof for a well known sup inf inequality established by brezis li and shafrir

theory of coupled differential equations the problem of decoupling a system of linear coupled differential equations without changing the order of the equations will be considered starting from a theorem on the separation of the coupled equations the author continues the investigations in the general case where the conditions set up by this theorem are not satisfied in deriving a method of separation of the equations in the strong coupling case an example on a system of two coupled schrodinger equations will be taken up as an illustration of this method

a statistical measure of complexity with nonextensive entropy abstract a statistical measure of complexity utilising the concept of entropy or information is proposed our way in this study is to use a nonextensive entropy instead of an extensive additive shannon entropy in the definition but can be characterised as a difference between the qth order renyi entropy and the second one furthermore we devise a conditional joint and mutual complexity measure as a coherent possibility the behavior of the measure for the logistic map shows that it is more sensitive to nonextensivity at the transition point ac 3 8284 than any other values when 0

use of cumulative sums of squares for retrospective detection of changes of variance abstract this article studies the problem of multiple change points in the variance of a sequence of independent observations we propose a procedure to detect variance changes based on an iterated cumulative sums of squares icss algorithm we study the properties of the centered cumulative sum of squares function and give an intuitive basis for the icss algorithm for series of moderate size i e 200 observations and beyond the icss algorithm offers results comparable to those obtained by a bayesian approach or by likelihood ratio tests without the heavy computational burden required by these approaches simulation results comparing the icss algorithm to other approaches are presented

self similar sequences and universal scaling of dynamical entropies symbol sequences play a prominent role in the context of symbolic dynamics important features of a dynamical system are reflected by related statistics of subsequences a dynamical behavior giving rise to a self similar attractor and universal scaling relations expressed by critical exponents will lead to self similar statistics of subsequences in the present paper we show how self similar distributions of subsequences i e temporal self similarity can be connected with a scaling relation for dynamical entropies moreover the effect of slightly perturbing perfectly self similar sequences by contaminating them with noise is investigated the achieved results are of importance for physical processes marking the borderline between order and chaos textcopyright 1996 the american physical society

on a sharp sobolev type inequality on two dimensional compact manifolds motivated by the asymptotic analysis of double vortex condensates in the chern simons higgs theory we construct a suitable minimizing sequence for a sharp sobolev inequality a la moser for two dimensional compact manifolds as a consequence we first obtain a direct proof of the sharp character of such an inequality secondly and more interestingly we use such minimizing sequence to show that for the flat torus the corresponding extremal problem attains its infimum

generalisation of the theory of coupled differential equations the author generalises a method derived from the theorem on the separation of coupled equations which has been stated previously to solve the case of two equations it will be shown that this generalisation is always possible with a sequence of special transformations in terms of scaling parameters the case of a system of three coupled differential equations is considered in some detail including a discussion on the search and use of these parameters

isospectral susy potentials the authors introduce a new linear transformation in order to construct isospectral potentials within the framework of supersymmetric quantum mechanics under this transformation the adjoint character of the ladder operator is destroyed but with preservation of the energy spectrum leading to some interesting consequences such as the concept of pseudo partnership of the two components connection with the darboux theorem as well as with some other recent methods of construction is analysed

a quantization property for blow up solutions of singular liouville type equations abstract motivated by the study of self dual vortices in gauge field theory cf vortices and monopoles birkhauser boston 1980 solitons in field theory and nonlinear analysis monographs in mathematics springer new york 2001 we consider a concentrating solution sequence u k satisfying u k z 2 k v k z e u k in b 1 z 2 k v k z e u k z 0 weakly in the sense of measures in b 1 where i k r k ii v k c 0 1 b 1 0 a v k b and v k a in b 1 we prove that necessarily 8 n 8 1 8 z the result is sharp as shown by explicit examples and should be compared with that obtained by li shafrir ind univ math j 43 4 1994 1255 concerning the situation where k 0 k n

extension of the theorem on the separation of a system of coupled differential equations we develop the idea presented in a previous paper by including the case of non identical coupling terms and show that under some conditions separation of the equations is also possible

testing for homogeneity of variance in time series long memory wavelets and the nile river 1 we consider the problem of testing for homogeneity of variance in a time series with long memorystructure wedemonstratethat atest whose nullhypothesis isdesigned tobewhite noisecan in fact be applied on a scale by scale basis to the discrete wavelet transform of long memory processes in particular we show that evaluating a normalized cumulative sum of squares test statistic using critical levels for the null hypothesis of white noise yields approximately the same null hypothesis rejection rates when applied to the discrete wavelet transform of samples from a fractionally differenced process the point at which the test statistic using a nondecimated version of the discrete wavelet transform achieves its maximum value can be used to estimate the time of the unknown variance change we apply our proposed test statistic on five time series derived from the historical record of nile river yearly minimum water levels covering 622 1922 a d each series exhibiting various degrees of serial correlation including long memory in the longest subseries spanning 622 1284 a d the test confirms an inhomogeneity of variance at short time scales and identifies the change point around 720 a d which coincides closely with the construction of a new device around 715 a d for measuring the nile river the test also detects a change in variance for a record of only 36 years index terms 1829 hydrology groundwater hydrology 1869 hydrology stochastic processes 3299 mathematical geophysics general or miscellaneous

the integrated periodogram for stable processes we study the asymptotic behavior of the integrated periodogram for alpha stable linear processes for alpha is an element of 1 2 we prove a functional limit theorem for the integrated periodogram the limit is an alpha stable analogue to the brownian bridge we apply our results to investigate some specific goodness of fit tests for heavy tailed linear processes

a sup inf inequality near r 0 let be a domain in r2 not necessarily bounded consider the semi linear elliptic equation u r x eu x we prove that for any compact subset k of there is a constant c such that the inequality 1 supku inf u c holds for all solutions u this type of inequality was first established by brezis li and shafrir h brezis y y li i shafrir a sup inf inequality for some nonlinear elliptic equations involving exponential nonlinearities j funct anal 115 1993 344 358 under the assumption that r x is positive and bounded away from zero it has become a useful tool in estimating the solutions of semi linear elliptic equations either in euclidean spaces or on riemannian manifolds see h brezis y y li i shafrir a sup inf inequality for some nonlinear elliptic equations involving exponential nonlinearities j funct anal 115 1993 344 358 c c chen c s lin a sharp sup inf inequality for a nonlinear elliptic equation in r2 comm anal geom 6 1998 1 19 w chen c li gaussian curvature in the negative case proc amer math soc 131 2003 741 744 w chen c li indefinite elliptic problems with critical exponent in advances in non linear pde and related areas world scientific 1998 pp 67 79 y y li i shafrir blow up analysis for solutions of u veu in dimension two indiana univ math j 43 1994 1255 1270 in brezis li and shafrir s result the constant c depends on the lower bound of the function r x in this paper we remove this restriction and extend the inequality to the case where r x is allowed to have zeros so that it can be applied to obtain a priori estimates for a broader class of equations as we will illustrate in the last section the key to prove this inequality is the analysis of asymptotic decay near infinity of solutions for a corresponding limiting equation in r2 which is interesting and useful in its own right

a theorem on the separation of a system of coupled differential equations considers the separation of a system of finite linear coupled differential equations the conditions which govern this separation in the case of a system of two coupled equations are discussed it is then shown how these results may be extended to the case of a system of a finite number of coupled equations for which a general theorem on the separability is formulated

a search for shape invariant solvable potentials the author investigates a simple method of constructing potentials which is related to the work of bhattacharjie and sudarshan 1962 and for which the schrodinger equation can be solved in terms of known special functions it turns out that this method can be related to supersymmetric quantum mechanics and this relationship can help to decide which special functions satisfying linear homogeneous second order differential equations can be solutions of the schrodinger equation with potentials of the form v x w2 x w x the author illustrates this procedure with the example of orthogonal polynomials and obtains explicit expressions of wavefunctions of a wide class of shape invariant potentials

the maximum likelihood method for testing changes in the parameters of normal observations 1 4 ha 1 oh2 2 o n and ali 2 a nt n for some r e 0 1 has received considerable attention in the literature under ha we can have change only in the mean the variances remain the same page 1954 1955 considered a very simple procedure for testing ho against h sen and srivastava 1975a b srivastava and worsley 1986 yao and davis 1986 james james and siegmund 1987 and worsley 1986 derived tests for ho against ha using the likelihood ratio yao and davis 1986 obtained the asymptotic distribution of the likelihood ratio for further results the asymptotic distributions of these and related statistics we refer to csorgo and horvath 1988 and gombay and horvath 1990 the maximum likelihood method can be used when we want to test ho against the more general alternative ha it is easy to show that the likelihood

approach to block entropy modeling and optimization abstract this paper introduces an approach to the block entropy modeling of stationary signals the block entropies constructed according to the tsallis generalized formalism are optimized with respect to the block length and the partition of the signal value domain to appropriately measure the complexity of the signal the optimal partition is known to be addressed by the euclidean mean expressing the signal optimized level based on the least squares fitting method however this is valid only for random signal values following a symmetric distribution alternatively and within the framework of fitting methods based on non euclidean metrics we implement the q th order means to consistently describe the optimal signal level and to clearly present the difference between the optimal signal level and the optimal partition the signal optimization is utilized for detecting the distribution modes developing a technique being resistant to the noise corruption that can be useful for detecting the optimal partition moreover the mechanisms that affect the optimal partition are identified and thoroughly investigated we first consider random signals following an arbitrary distribution where the block entropy modeling reveals that the optimal partition is located at the median thereafter we consider persistent signals specifying a degree of determinism where the optimal partition is found to be driven far from the median and close to the persistent mode the existence of persistent modes of small hitting time is the key point of this dissertation highlighting their implications on the block entropy modeling finally efforts towards block entropy modeling of non stationary signals are discussed

parameter identification for mixed fractional brownian motions with the drift parameter abstract this paper deals with the problem of estimating the unknown parameters in the drift mixed fractional brownian motion based on discrete sampled observations we construct two different estimators for the drift parameter based on the approximation by random walks of the driving noise and prove the consistency of these two estimators some numerical simulations are also presented to illustrate the performance of these maximum likelihood estimators empirical examples are given to illustrate the potential applications of these proposed estimators

nonparametric segmentation of nonstationary time series the nonstationary evolution of observable quantities in complex systems can frequently be described as a juxtaposition of quasistationary spells given that standard theoretical and data analysis approaches usually rely on the assumption of stationarity it is important to detect in real time series intervals holding that property with that aim we introduce a segmentation algorithm based on a fully nonparametric approach we illustrate its applicability through the analysis of real time series presenting diverse degrees of nonstationarity thus showing that this segmentation procedure generalizes and allows one to uncover features unresolved by previous proposals based on the discrepancy of low order statistical moments only

fractional langevin equation we investigate fractional brownian motion with a microscopic random matrix model and introduce a fractional langevin equation we use the latter to study both subdiffusion and superdiffusion of a free particle coupled to a fractal heat bath we further compare fractional brownian motion with the fractal time process the respective mean square displacements of these two forms of anomalous diffusion exhibit the same power law behavior here we show that their lowest moments are actually all identical except the second moment of the velocity this provides a simple criterion that enable us to distinguish these two non markovian processes

non euclidean normed statistical mechanics abstract this analysis introduces a possible generalization of statistical mechanics within the framework of non euclidean metrics induced by the l p norms the internal energy is interpreted by the non euclidean l p normed expectation value of a given energy spectrum the presented non euclidean adaptation of statistical mechanics involves finding the stationary probability distribution in the canonical ensemble by maximizing the boltzmann gibbs and tsallis entropy under the constraint of internal energy the derived non euclidean canonical probability distributions are respectively given by an exponential and by a q deformed exponential of a power law dependence on energy states the case of the continuous energy spectrum is thoroughly examined the canonical probability distribution is analytically calculated for a power law density of energy the relevant non euclidean normed kappa distribution is also derived this analysis exposes the possible values of the q or indices which are strictly limited to certain ranges depending on the given l p norm the equipartition of energy in each degree of freedom and the extensivity of the internal energy are also shown surprisingly the physical temperature coincides with the kinetically defined temperature similar to the euclidean case finally the connection with thermodynamics arises through the well known standard classical formalisms

the role of constraints within generalized nonextensive statistics the gibbs jaynes path for introducing statistical mechanics is based on the adoption of a specific entropic form sand of physically appropriate constraints for instance for the usual canonical ensemble one adopts i ii ipi 1 and iii ei eigenvalues of the hamiltonian u1 internal energy equilibrium consists in optimizing s1 with regard to pi in the presence of constraints ii and iii within the recently introduced nonextensive statistics i is generalized into sq k 1 ipiq q 1 q 1 reproduces s1 ii is maintained and iii is generalized in a manner which might involve piq in the present effort we analyze the consequences of some special choices for iii and their formal and practical implications for the various physical systems that have been studied in the literature to illustrate some mathematically relevant points we calculate the specific heat respectively associated with a nondegenerate two level system as well as with the classical and quantum harmonic oscillators