nonlinear fractional dynamics on a lattice with long range interactions abstract a unified approach has been developed to study nonlinear dynamics of a 1d lattice of particles with long range power law interaction a classical case is treated in the framework of the generalization of the well known frenkel kontorova chain model for the non nearest interactions quantum dynamics is considered following davydov s approach for molecular excitons in the continuum limit the problem is reduced to dynamical equations with fractional derivatives resulting from the fractional power of the long range interaction fractional generalizations of the sine gordon nonlinear schrodinger and hilbert schrodinger equations have been found there exists a critical value of the power s of the long range potential below the critical value s 3 s 1 2 we obtain equations with fractional derivatives while for s 3 we have the well known nonlinear dynamical equations with space derivatives of integer order long range interaction impact on the quantum lattice propagator has been studied we have shown that the quantum exciton propagator exhibits transition from the well known gaussian like behavior to a power law decay due to the long range interaction a link between 1d quantum lattice dynamics in the imaginary time domain and a random walk model has been discussed

convergence of the complex envelope of bandlimited ofdm signals orthogonal frequency division multiplexing ofdm systems have been used extensively in wireless communications in recent years thus there is significant interest in analyzing the properties of the transmitted signal in such systems in particular a large amount of work has focused on analyzing the variation of the complex envelope of the transmitted signal and on designing methods to minimize this variation in this paper it is established that the complex envelope of a bandlimited uncoded ofdm signal converges weakly to a gaussian random process as the number of subcarriers goes to infinity this shows that the properties of the ofdm signal will asymptotically approach those of a gaussian random process over any finite time interval the convergence proof is then extended to two important cases namely coded ofdm systems and systems with an unequal power allocation across subcarriers

simultaneous multifrequency inversion of full waveform seismic data we present a simultaneous multifrequency inversion approach for seismic data interpretation this algorithm inverts all frequency data components simultaneously a data weighting scheme balances the contributions from different frequency data components so the inversion process does not become dominated by high frequency data components which produce a velocity image with many artifacts a gauss newton minimization approach achieves a high convergence rate and an accurate reconstructed velocity image by introducing a modified adjoint formulation we can calculate the jacobian matrix efficiently allowing the material properties in the perfectly matched layers pmls to be updated automatically during the inversion process this feature ensures the correct behavior of the inversion and implies that the algorithm is appropriate for realistic applications where a priori information of the background medium is unavailable two different regularization schemes an l2 norm and a weighted l2 norm function a

on the distribution of the peak to average power ratio in ofdm signals the distribution of the peak to average power ratio papr in strictly band limited orthogonal frequency division multiplexing ofdm signals is studied assuming that the base band ofdm signal is characterized as a band limited complex gaussian process we first attempt to derive the exact distribution of the papr in the band limited ofdm signals since this distribution cannot be expressed in a closed form we further develop a simple closed form approximation based on the level crossing rate analysis comparisons of the proposed distributions with those obtained by computer simulations show good agreement and convergence with an increase in the number of subcarriers

fractional variational calculus and the transversality conditions this paper presents the euler lagrange equations and the transversality conditions for fractional variational problems the fractional derivatives are defined in the sense of riemann liouville and caputo the connection between the transversality conditions and the natural boundary conditions necessary to solve a fractional differential equation is examined it is demonstrated that fractional boundary conditions may be necessary even when the problem is defined in terms of the caputo derivative furthermore both fractional derivatives the riemann liouville and the caputo arise in the formulations even when the fractional variational problem is defined in terms of one fractional derivative only examples are presented to demonstrate the applications of the formulations

reduced rank approximations to the far field transform in the gridded fast multipole method the fast multipole method fmm has been shown to have a reduced computational dependence on the size of finest level groups of elements when the elements are positioned on a regular grid and fft convolution is used to represent neighboring interactions however transformations between plane wave expansions used for fmm interactions and pressure distributions used for neighboring interactions remain significant contributors to the cost of fmm computations when finest level groups are large the transformation operators which are forward and inverse fourier transforms with the wave space confined to the unit sphere are smooth and well approximated using reduced rank decompositions that further reduce the computational dependence of the fmm on finest level group size the adaptive cross approximation aca is selected to represent the forward and adjoint far field transformation operators required by the fmm however the actual error of the aca is found to be greater than that predicted using traditional estimates and the aca generally performs worse than the approximation resulting from a truncated singular value decomposition svd to overcome these issues while avoiding the cost of a full scale svd the aca is employed with more stringent accuracy demands and recompressed using a reduced truncated svd the results show a greatly reduced approximation error that performs comparably to the full scale truncated svd without degrading the asymptotic computational efficiency associated with aca matrix assembly

a balancing method for pmepr reduction in ofdm signals a relation is established between the strength of a binary code over the alphabet 1 1 and its ability to reduce peak to mean envelope power ratio pmepr in n subcarrier ofdm signals based on this relation a method is proposed to deterministically bound pmepr of such signals using coordinate wise multiplication by a balancing vector bv chosen from a code of given strength a practical probabilistic scheme considering a small number of candidate codewords is devised for this scheme estimates on the pmepr reduction achievable with arbitrary high probability are derived in particular the scheme provides for large n pmepr of lnn 2 01lnlnn with ln2 middot log2n 2 1 bits of redundancy the failure probability at most e n and testing n lnlnn candidate bvs finally several practical settings are considered for example for quaternary phase shift keying n 128 the scheme with 36 bits of redundancy 18 redundant subcarriers by testing only 4 bvs provides over 2 db pmepr reduction for any failure rate below 10 2 5

peak value estimation of bandlimited signals from their samples noise enhancement and a local characterization in the neighborhood of an extremum the paper addresses the problem of estimating the peak value of bandlimited signals from their samples with and without oversampling this problem has significant relevance to orthogonal frequency division multiplexing ofdm signal processing and system design in particular an upper bound on the peak value is established given the peak value of the samples and the oversampling rate moreover it is shown that the bounds are sharp for all practical rates by constructing bandlimited signals taking on this bound the proof also provides a local characterization of bandlimited signals in the neighborhood of an extremum a different analysis examines the effect of small errors in the samples it is shown that oversampling can provide robust recovery in the sense that small errors in the samples lead to small errors in the reconstructed signal again an upper bound is derived relating the peak error in the samples and the peak error in the signals furthermore both problems are shown to be coupled and put in a unifying context the bounds are compared and applied to problems concerning ofdm

analysis of four parameter fractional derivative model of real solid materials the introduction of fractional derivatives into the constitutive equation of the differential operator type of linear solid materials has led to the development of the so called fractional derivative models one of these models characterized by four parameters has been found usable for describing the variation of dynamics elastic and damping properties in a wide frequency range provided that there is only one loss peak in this paper this four parameter model is theoretically analyzed the effect of the parameters on the frequency curves is demonstrated and it is shown that there is a strict relation between the dispersion of the dynamic modulus the loss peak and the slope of the frequency curves the half value bandwidth of the loss modulus frequency curve is investigated and conditions are developed to establish the applicability of the model for a class of materials moreover it is shown that the model can be used to predict the frequency dependences of dynamic properties for a wide range even if measurements are made in only a narrow frequency range around the loss peak

comparison of te and tm inversions in the framework of the gauss newton method the gauss newton inversion method in conjunction with a regularized formulation of the inverse scattering problem is used to invert transverse electric te and transverse magnetic tm data the utilized data sets consist of experimental data provided by the institut fresnel as well as synthetic data the te inversion outperformed the tm inversion when utilizing near field scattering data collected using only a few transmitters and receivers however very little difference was found between te and tm inversions when using far field scattering data it is conjectured that the reason for the better performance of the near field te result is that the near field te data contains more information than the near field tm data at each receiver point in all cases considered herein the te inversion required equal or fewer iterations than the tm inversion the per iteration computational complexity of both te and tm inversions is discussed in the framework of the gauss newton inversion method actual costs are consistent with the computational complexity analysis that is given

mechanics with fractional derivatives lagrangian and hamiltonian mechanics can be formulated to include derivatives of fractional order f riewe phys rev 53 1890 1996 lagrangians with fractional derivatives lead directly to equations of motion with nonconservative classical forces such as friction the present work continues the development of fractional derivative mechanics by deriving a modified hamilton s principle introducing two types of canonical transformations and deriving the hamilton jacobi equation using generalized mechanics with fractional and higher order derivatives the method is illustrated with a frictional force proportional to velocity in contrast to conventional mechanics with integer order derivatives quantization of a fractional derivative hamiltonian cannot generally be achieved by the traditional replacement of momenta with coordinate derivatives instead a quantum mechanical wave equation is proposed that follows from the hamilton jacobi equation by application of the correspondence principle

generalized bounds on the crest factor distribution of ofdm signals with applications to code design in this paper generalized bounds on the crest factor cf distribution in orthogonal frequency division multiplexing ofdm transmission for both independent and dependent subcarriers are derived here the latter situation represents the coded case for independent subcarriers a general path for bounding practical constellations is provided moreover a complete characterization of their asymptotic behavior is devised and discussed the results are shown to carry over to the spherical constellations improving on recent results for dependent subcarriers the focus is mainly on binary codes where bounds on the cf distribution are obtained in terms of the distance distributions and their duals the asymptotic behavior of codes is analyzed and it is shown that the upper bound on the effective crest factor of a large class of bose chaudhuri hocquenghem bch codes behaves asymptotically as radiclogn finally two applications of the results to code design are presented first fixed phase shifts on the subcarriers for all codewords are used and an algorithm to calculate the phase shifts is designed that way it is proved that the effective cf of any binary code can be scaled to be of order radiclogn for large n without sacrificing on rate furthermore the same approach is applied to calculation of the phases of redundant subcarriers for each codeword it is shown by simulations that the values can be effectively chosen so that the cf is significantly reduced with nonexponential complexity

focusing and imaging using eigenfunctions of the scattering operator an inverse scattering method that uses eigenfunctions of the scattering operator is presented this approach provides a unified framework that encompasses eigenfunction methods of focusing and quantitative image reconstruction in arbitrary media scattered acoustic fields are described using a compact normal operator the eigenfunctions of this operator are shown to correspond to the far field patterns of source distributions that are directly proportional to the position dependent contrast of a scattering object conversely the eigenfunctions of the scattering operator specify incident wave patterns that focus on these effective source distributions these focusing properties are employed in a new inverse scattering method that represents unknown scattering media using products of numerically calculated fields of eigenfunctions a regularized solution to the nonlinear inverse scattering problem is shown to result from combinations of these products so that the products comprise a natural basis for efficient and accurate reconstructions of unknown inhomogeneities the corresponding linearized problem is solved analytically resulting in a simple formula for the low pass filtered scattering potential the linear formula is analytically equivalent to known filtered backpropagation formulas for born inversion and at least in the case of small scattering objects has advantages of computational simplicity and efficiency a similarly efficient and simple formula is derived for the nonlinear problem in which the total acoustic pressure can be determined based on an estimate of the medium computational results illustrate focusing of eigenfunctions on discrete and distributed scattering media quantitative imaging of inhomogeneous media using products of retransmitted eigenfunctions inverse scattering in an inhomogeneous background medium and reconstructions for data corrupted by noise

transition to chaos in discrete nonlinear schrodinger equation with long range interaction discrete nonlinear schrodinger dnls equation describes a chain of oscillators with nearest neighbor interactions and a specific nonlinear term we consider its modification with long range interaction through a potential proportional to 1 l1 with fractional 2 and l as a distance between oscillators this model is called dnls it exhibits competition between the nonlinearity and a level of correlation between interacting far distanced oscillators that is defined by the value of we consider transition to chaos in this system as a function of and nonlinearity it is shown that decreasing of with respect to nonlinearity stabilize the system connection of the model to the fractional generalization of the nls called fnls in the long wave approximation is also discussed and some of the results obtained for dnls can be correspondingly extended to the fnls

the multi component coupled burgers hierarchy of soliton equations and its multi component integrable couplings system with two arbitrary functions abstract a new loop algebra g m is constructed which is devoted to establishing an isospectral problem by making use of tu scheme the multi component coupled burgers hierarchy is obtained furthermore an expanding loop algebra f m of the loop algebra g m is presented it follows that the multi component integrable coupling system of the multi component coupled burgers hierarchy is worked out in general the method can be applied to other evolution equations hierarchies

full wave three dimensional microwave imaging with a regularized gauss newton method theory and experiment a reconstruction algorithm is detailed for three dimensional full vectorial microwave imaging based on newton type optimization the goal is to reconstruct the three dimensional complex permittivity of a scatterer in a homogeneous background from a number of time harmonic scattered field measurements the algorithm combines a modified gauss newton optimization method with a computationally efficient forward solver based on the fast fourier transform method and the marching on in source position extrapolation procedure a regularized cost function is proposed by applying a multiplicative additive regularization to the least squares datafit this approach mitigates the effect of measurement noise on the reconstruction and effectively deals with the non linearity of the optimization problem it is furthermore shown that the modified gauss newton method converges much faster than the broyden fletcher goldfarb shanno quasi newton method promising quantitative reconstructions from both simulated and experimental data are presented the latter data are bi static polarimetric free space measurements provided by institut fresnel marseille france

the fractional boltzmann transport equation the fractional boltzmann transport equation is derived making use of the fractional hamilton s equations based on the fractional actionlike variational approach by simply defining a distribution function and inspecting its time derivative many important results in statistical physics can be derived

brief paper stability and resonance conditions of elementary fractional transfer functions elementary fractional transfer functions are studied in this paper some basic properties of elementary transfer functions of the first kind are recalled then two main results are presented regarding elementary fractional transfer functions of the second kind written in a canonical form and characterized by a commensurate order a pseudo damping factor and a natural frequency first stability conditions are established in terms of the pseudo damping factor and the commensurate order as a corollary to matignon s stability theorem they extend the previous result into conditions that are simpler to check then resonance conditions are established numerically in terms of the commensurate order and the pseudo damping factor and give interesting information on the frequency behavior of fractional systems it is shown that elementary transfer functions of the second kind might have up to two resonant frequencies moreover three abaci are given allowing to determine the pseudo damping factor and the commensurate order for respectively a desired normalized gain at each resonance a desired phase at each resonance and a desired normalized first or second resonant frequency

on multicarrier signals where the pmepr of a random codeword is asymptotically logn multicarrier signals exhibit a large peak to mean envelope power ratio pmepr in this correspondence without using a gaussian assumption we derive lower and upper probability bounds for the pmepr distribution when the number of subcarriers n is large even though the worst case pmepr is of the order of n the main result is that the pmepr of a random codeword c c sub 1 c sub n is logn with probability approaching one asymptotically for the following three general cases i c sub i s are independent and identically distributed i i d chosen from a complex quadrature amplitude modulation qam constellation in which the real and imaginary part of c sub i each has i i d and even distribution not necessarily uniform ii c sub i s are i i d chosen from a phase shift keying psk constellation where the distribution over the constellation points is invariant under spl pi 2 rotation and iii c is chosen uniformly from a complex sphere of dimension n based on this result it is proved that asymptotically the varshamov gilbert vg bound remains the same for codes with pmepr of less than logn chosen from qam psk constellations

novel low complexity slm schemes for papr reduction in ofdm systems selected mapping slm schemes are commonly employed to reduce the peak to average power ratio papr in orthogonal frequency division multiplexing ofdm systems it has been shown that the computational complexity of the traditional slm scheme can be substantially reduced by adopting conversion vectors obtained by using the inverse fast fourier transform ifft of the phase rotation vectors in place of the conventional ifft operations c l wang and y ouyang low complexity selected mapping schemes for peak to average power ratio reduction in ofdm systems ieee trans signal process vol 53 no 12 pp 4652 4660 dec 2005 to ensure that the elements of these phase rotation vectors have an equal magnitude conversion vectors should have the form of a perfect sequence this paper presents three novel classes of perfect sequence each of which comprises certain base vectors and their cyclically shifted versions three novel low complexity slm schemes are then proposed based upon the unique structures of these perfect sequences it is shown that while the papr reduction performances of the proposed schemes are marginally poorer than that of the traditional slm scheme the three schemes achieve a substantially lower computational complexity

multiobjective fractional variational calculus in terms of a combined caputo derivative abstract the study of fractional variational problems in terms of a combined fractional caputo derivative is introduced necessary optimality conditions of euler lagrange type for the basic isoperimetric and lagrange variational problems are proved as well as transversality and sufficient optimality conditions this allows to obtain necessary and sufficient pareto optimality conditions for multiobjective fractional variational problems

on the peak to mean envelope power ratio of phase shifted binary codes the peak to mean envelope power ratio pmepr of a code employed in orthogonal frequency division multiplexing ofdm systems can be reduced by permuting its coordinates and by rotating each coordinate by a fixed phase shift motivated by some previous designs of phase shifts using suboptimal methods the following question is considered in this paper for a given binary code how much pmepr reduction can be achieved when the phase shifts are taken from a 2h ary phase shift keying 2h psk constellation a lower bound on the achievable pmepr is established which is related to the covering radius of the binary code generally speaking the achievable region of the pmepr shrinks as the covering radius of the binary code decreases the bound is then applied to some well understood codes including nonredundant bpsk signaling bch codes and their duals reed muller codes and convolutional codes it is demonstrated that most presumably not optimal phase shift designs from the literature attain or approach our bound

vector continued fractions using a generalized inverse a real vector space combined with an inverse involution for vectors is sufficient to define a vector continued fraction whose parameters consist of vector shifts and changes of scale the choice of sign for different components of the vector inverse permits construction of vector analogues of the jacobi continued fraction these vector jacobi fractions are related to vector and scalar valued polynomial functions of the vectors which satisfy recurrence relations similar to those of orthogonal polynomials the vector jacobi fraction has strong convergence properties which are demonstrated analytically and illustrated numerically

on a jump type stochastic fractional partial differential equation with fractional noises abstract we study a class of stochastic fractional partial differential equations of order 1 driven by a pure jump levy space time white noise and a fractional noise we prove the existence and uniqueness of the global mild solution by the fixed point principle under some suitable assumptions

fast inverse scattering solutions using the distorted born iterative method and the multilevel fast multipole algorithm the distorted born iterative method dbim computes iterative solutions to nonlinear inverse scattering problems through successive linear approximations by decomposing the scattered field into a superposition of scattering by an inhomogeneous background and by a material perturbation large or high contrast variations in medium properties can be imaged through iterations that are each subject to the distorted born approximation however the need to repeatedly compute forward solutions still imposes a very heavy computational burden to ameliorate this problem the multilevel fast multipole algorithm mlfma has been applied as a forward solver within the dbim the mlfma computes forward solutions in linear time for volumetric scatterers the typically regular distribution and shape of scattering elements in the inverse scattering problem allow the method to take advantage of data redundancy and reduce the computational demands of the normally expensive mlfma setup additional benefits are gained by employing kaczmarz like iterations where partial measurements are used to accelerate convergence numerical results demonstrate both the efficiency of the forward solver and the successful application of the inverse method to imaging problems with dimensions in the neighborhood of ten wavelengths

on a stochastic fractional partial differential equation driven by a levy space time white noise we study the existence and uniqueness of the global mild solution for a stochastic fractional partial differential equation driven by a levy space time white noise moreover the flow property for the solution is also studied

spectrally precoded ofdm by bilinearly transforming the rectangularly pulsed multicarrier basis signals new basis sets containing full response continuous basis signals are developed for constructing spectrally efficient orthogonal frequency division multiplexing ofdm signals with or without cyclic prefix cp or zero padding zp both multiple product and linear combination formats are presented for these basis sets in the multiple product format the developed full response basis signals are shown to be zero valued at interval edges and thus can be used to construct continuous phase ofdm signals it is analytically shown that the constructed continuous phase ofdm signals exhibit relatively small power spectral sidelobes which fall off as f 4 and can thus provide much higher spectral efficiency than the traditional rectangularly pulsed ofdm signal in the linear combination format the proposed basis signals are shown to be the linear combinations of rectangularly pulsed multicarrier basis signals and thereby allow for the efficient realization of fast fourier transform fft specifically a spectrally precoded signaling structure is developed in conjunction with inverse fft and guard insertion to construct continuous phase ofdm signals with or without cp or zp the corresponding spectral decoders are also developed to enable zero forcing zf demodulation for spectrally precoded ofdm with the zero forcing demodulation the spectrally precoded ofdm schemes are shown to outperform in average error rate characteristics the conventional rectangularly pulsed and correlatively coded ofdm schemes on various delay spread channels

quantitative imaging using a time domain eigenfunction method an inverse scattering method that uses eigenfunctions of a scattering operator at a single frequency is extended to include the full range of frequencies present in the incident pulse waveform the resulting so called time domain eigenfunction method is shown to yield a modulated version of the scattering potential the potential is recovered by a demodulation process using cross correlation with a reference including an adaptive delay in the reference is shown to compensate partially for the linearization of the born approximation and to extend its valid range the k space window of the time domain solution is expressed in terms of the incident waveform and shown to be smoother than that of a single frequency solution the time domain method is examined using both calculated and measured data in the calculations an exact solution for scattering from one or multiple nonconcentric cylinders is used to obtain the scattered field in the measurements a novel ring transducer system was employed to obtain the incident and total fields the results of simulations and experiments show that the method is robust and accurate for the size of objects considered and that the point resolution approaches one half the wavelength at the pulse center frequency

fractional ginzburg landau equation for fractal media abstract we derive the fractional generalization of the ginzburg landau equation from the variational euler lagrange equation for fractal media to describe fractal media we use the fractional integrals considered as approximations of integrals on fractals some simple solutions of the ginzburg landau equation for fractal media are considered and different forms of the fractional ginzburg landau equation or nonlinear schrodinger equation with fractional derivatives are presented the agrawal variational principle and its generalization have been applied

the trace identity a powerful tool for constructing the hamiltonian structure of integrable systems a new approach to hamiltonian structures of integrable systems is proposed by making use of a trace identity for a variety of isospectral problems that can be unified to one model x u it is shown that both the related hierarchy of evolution equations and the hamiltonian structure can be obtained from the same solution of the equation vx u v

dynamics with low level fractionality abstract the notion of fractional dynamics is related to equations of motion with one or a few terms with derivatives of a fractional order this type of equation appears in the description of chaotic dynamics wave propagation in fractal media and field theory for the fractional linear oscillator the physical meaning of the derivative of order 2 is dissipation in systems with many spacially coupled elements oscillators the fractional derivative along the space coordinate corresponds to a long range interaction we discuss a method of constructing a solution using an expansion in n with small and positive integer n the method is applied to the fractional linear and nonlinear oscillators and to fractional ginzburg landau or parabolic equations

ultrasound tomography for simultaneous reconstruction of acoustic density attenuation and compressibility profiles a fast and efficient forward scattering solver is developed for use in ultrasound tomography the solver is formulated so as to enable the calculation of scattering from large and relatively high contrast objects with inhomogeneous physical properties that vary simultaneously in acoustic attenuation compressibility and density it is based on the method of moments in conjunction with a novel implementation of the conjugate gradient algorithm which requires the use of the adjoints of the scattering operators the solver takes advantage of the symmetric block toeplitz matrix with symmetric toeplitz blocks property of the green s function matrix to increase efficiency and only stores the first row of this matrix to reduce memory requirements this row is then used for the matrix vector multiplication using the fast fourier transform technique thus resulting in the computational complexity of o n log n the marching on source technique is also used to provide a good initial guess which allows the conjugate gradient technique to converge faster than initializing with an arbitrary guess this feature is important in tomographic inversion algorithms which require that the object to be imaged be interrogated via several incident fields forward scattering and inversion examples based on the conjugate gradient least squares regularized born iterative method are shown in two dimensions for objects varying in all three physical properties

fractional generalization of gradient and hamiltonian systems we consider a fractional generalization of hamiltonian and gradient systems we use differential forms and exterior derivatives of fractional orders we derive fractional generalization of helmholtz conditions for phase space examples of fractional gradient and hamiltonian systems are considered the stationary states for these systems are derived

integrable evolution systems based on gerdjikov ivanov equations bi hamiltonian structure finite dimensional integrable systems and n fold darboux transformation a spectral problem and the associated gerdjikov ivanov gi hierarchy of nonlinear evolution equations is presented as a reduction the well known gi equation of derivative nonlinear schrodinger equations is obtained it is shown that the gi hierarchy is integrable in a liouville sense and possesses bi hamiltonian structure moreover the spectral problem can be nonlinearized as a finite dimensional completely integrable system under the bargmann constraint between the potentials and the eigenfunctions in particular an explicit n fold darboux transformation for the gi equation is constructed with the help of a gauge transformation of spectral problems and a reduction technique some explicit solitonlike solutions of the gi equation are given by applying its darboux transformation

the complex envelope of a bandlimited ofdm signal converges weakly to a gaussian random process orthogonal frequency division multiplexing ofdm systems have been used extensively in wireless communications applications in recent years thus there is significant interest in analyzing the properties of the transmitted signal in such systems in particular a large amount of recent work has focused on analyzing the variation of the complex envelope of the transmitted signal and on designing methods to minimize this variation in this paper it is established that the complex envelope of a bandlimited uncoded ofdm signal converges weakly to a gaussian random process as the number of subcarriers goes to infinity this establishes that the properties of the ofdm signal will asymptotically approach those of a gaussian random process over any finite time interval the symbol length in a bandlimited ofdm system will eventually exceed any finite time interval as the number of subcarriers approaches infinity however practical interest is in how asymptotic approximations apply for a finite number of carriers and hence the convergence proof is reasonable motivation for considering how the extremal value theory of gaussian random processes might provide accurate approximations for the distribution of the peak to mean envelope power ratio pmepr of practical ofdm systems indeed numerical results are presented that indicate that the resulting simple expressions are accurate for a wide range of the distribution for moderate numbers of subcarriers the important extensions of the analytical and numerical results to coded ofdm systems as well as systems with unequal power allocation across subcarriers are also presented

fractional embedding of differential operators and lagrangian systems this paper is a contribution to the general program of embedding theories of dynamical systems following our previous work on the stochastic embedding theory developed with darses c r acad sci ser i math 342 333 2006 preprint ihes 06 27 p 87 2006 we define the fractional embedding of differential operators and ordinary differential equations we construct an operator combining in a symmetric way the left and right riemann liouville fractional derivatives for lagrangian systems our method provides a fractional euler lagrange equation we prove developing the corresponding fractional calculus of variations that such equation can be derived via a fractional least action principle we then obtain naturally a fractional noether theorem and a fractional hamiltonian formulation of fractional lagrangian systems all these constructions are coherents i e the embedding procedure is compatible with the fractional calculus of variations we then extend our results to cover the ostrogradski

fractional schrodinger equation some properties of the fractional schrodinger equation are studied we prove the hermiticity of the fractional hamilton operator and establish the parity conservation law for fractional quantum mechanics as physical applications of the fractional schrodinger equation we find the energy spectra of a hydrogenlike atom fractional bohr atom and of a fractional oscillator in the semiclassical approximation an equation for the fractional probability current density is developed and discussed we also discuss the relationships between the fractional and standard schrodinger equations

a general finite element formulation for fractional variational problems abstract this paper presents a general finite element formulation for a class of fractional variational problems fvps the fractional derivative is defined in the riemann liouville sense for fvps the euler lagrange and the transversality conditions are developed in the fractional finite element formulation ffef presented here the domain of the equations is divided into several elements and the functional is approximated in terms of nodal variables minimization of this functional leads to a set of algebraic equations which are solved using a numerical scheme three examples are considered to show the performance of the algorithm results show that as the number of discretization is increased the numerical solutions approach the analytical solutions and as the order of the derivative approaches an integer value the solution for the integer order system is recovered for unspecified boundary conditions the numerical solutions satisfy the transversality conditions this indicates that for the class of problems considered the numerical solutions can be obtained directly from the functional and there is no need to solve the fractional euler lagrange equations thus the formulation extends the traditional finite element approach to fvps

incomplete normalization of probability on multifractals abstract this work is an extension of the incomplete probability theory from the simple case of monofractals previously studied to the more general case of multifractals that can occur in the phase space without equiprobable partition

amplitude and sign adjustment for peak to average power reduction in this letter we propose a method to reduce the peak to mean envelope power ratio pmepr of multicarrier signals by modifying the constellation for m ary phase shift keying constellations we minimize the maximum of the multicarrier signal over the sign and amplitude of each subcarrier in order to find an efficient solution to the aforementioned nonconvex optimization problem we present a suboptimal solution by first optimizing over the signs and then optimizing over the amplitudes given the signs we prove that the minimization of the maximum of a continuous multicarrier signal over the amplitude of each subcarrier can be written as a convex optimization problem with linear matrix inequality constraints we also generalize the idea to other constellations such as 16 quadrature amplitude modulation simulation results show that by an average power increase of 0 21 db and not sending information over the sign of each subcarrier pmepr can be decreased by 5 1 db for a system with 128 subcarriers

existence of codes with constant pmepr and related design recently several coding methods have been proposed to reduce the high peak to mean envelope ratio pmepr of multicarrier signals it has also been shown that with probability one the pmepr of any random codeword chosen from a symmetric quadrature amplitude modulation phase shift keying qam psk constellation is logn for large n where n is the number of subcarriers therefore the question is how much reduction beyond logn can one asymptotically achieve with coding and what is the price in terms of the rate loss in this paper by optimally choosing the sign of each subcarrier we prove the existence of q ary codes of constant pmepr for sufficiently large n and with a rate loss of at most log sub q 2 we also obtain a varsharmov gilbert type upper bound on the rate of a code given its minimum hamming distance with constant pmepr for large n since ours is an existence result we also study the problem of designing signs for pmepr reduction motivated by a derandomization algorithm suggested by spencer we propose a deterministic and efficient algorithm to design signs such that the pmepr of the resulting codeword is less than clogn for any n where c is a constant independent of n for symmetric q ary constellations this algorithm constructs a code with rate 1 log sub q 2 and with pmepr of clogn with simple encoding and decoding simulation results for our algorithm are presented

six standard deviations suffice given n sets on n elements it is shown that there exists a two coloring such that all sets have discrepancy at most knl 2 k an absolute constant this improves the basic probabilistic method with which k c ln n l 2 the result is extended to n finite sets of arbitrary size probabilistic techniques are melded with the pigeonhole principle an alternate proof of the existence of rudin shapiro functions is given showing that they are exponential in number given n linear forms in n variables with all coefficients in 1 1 it is shown that initial values pl p e 0 1 may be approximated by el e 0 1 so that the forms have small error 1 we state our main result first in the language of linear forms theorem 1 let 1 1 li xl x l ailxl xe ai txn 1 i n be n linear forms in n variables with all laijl 1 then there exist e1 esl e 1 1 such that 1 2 li el e l ks for all i 1 i n here k is an absolute constant when all aij e 0 1 we may consider e1 aij as the incidence matrix for a family of n sets on n elements that is we may set ai j aij 1 given a two coloring say red and blue of 1 n the discrepancy disc x of a set x c 1 n is defined as the number of red points in x minus the number of blue points in x if we interpret si 1 as meaning i is to be colored red and ej 1 as blue then we obtain the following result corollary 2 let al an c 1 n then there exists a two coloring of 1 n so that 1 3 disc aj kw foralli 1 i n received by the editors july 1 1984 1980 muthemcltics subject clclssific ltion primary 05b20 secondary 41a28 42a61 this work was initiated while the author was an irex exchange fellow at the mathematical institute budapest and completed with support of the national science foundation for the many kindnesses and for the creative research environment provided by institute staff and colleagues koszonom szepen t1985 american mathematical society 0002 9947 85 1 00 25 per page

a numerical study based on an implicit fully discrete local discontinuous galerkin method for the time fractional coupled schrodinger system in this paper we develop and analyze an implicit fully discrete local discontinuous galerkin ldg finite element method for solving the time fractional coupled schrodinger system the method is based on a finite difference scheme in time and local discontinuous galerkin methods in space through analysis we show that our scheme is unconditionally stable and the l 2 error estimate has the convergence rate o h k 1 dt 2 dt a 2h k 1 2 for the linear case extensive numerical results are provided to demonstrate the efficiency and accuracy of the scheme

nonlinear microwave imaging for breast cancer screening using gauss newton s method and the cgls inversion algorithm breast cancer screening using microwave imaging is emerging as a new promising technique as a supplement to x ray mammography to create tomographic images from microwave measurements it is necessary to solve a nonlinear inversion problem for which an algorithm based on the iterative gauss newton method has been developed at dartmouth college this algorithm determines the update values at each iteration by solving the set of normal equations of the problem using the tikhonov algorithm in this paper a new algorithm for determining the iteration update values in the gauss newton algorithm is presented which is based on the conjugate gradient least squares cgls algorithm the iterative cgls algorithm is capable of solving the update problem by operating on just the jacobian and the regularizing effects of the algorithm can easily be controlled by adjusting the number of iterations the new algorithm is compared to the gauss newton algorithm with tikhonov regularization and is shown to reconstruct images of similar quality using fewer iterations

fractional variational calculus in terms of riesz fractional derivatives this paper presents extensions of traditional calculus of variations for systems containing riesz fractional derivatives rfds specifically we present generalized euler lagrange equations and the transversality conditions for fractional variational problems fvps defined in terms of rfds we consider two problems a simple fvp and an fvp of lagrange results of the first problem are extended to problems containing multiple fractional derivatives functions and parameters and to unspecified boundary conditions for the second problem we present lagrange type multiplier rules for both problems we develop the euler lagrange type necessary conditions which must be satisfied for the given functional to be extremum problems are considered to demonstrate applications of the formulations explicitly we introduce fractional momenta fractional hamiltonian fractional hamilton equations of motion fractional field theory and fractional optimal control the formulations presented and the resulting equations are similar to the formulations for fvps given in agrawal 2002 j math anal appl 272 368 2006 j phys a math gen 39 10375 and to those that appear in the field of classical calculus of variations these formulations are simple and can be extended to other problems in the field of fractional calculus of variations

inverse scattering from arbitrary two dimensional objects in stratified environments via a green s operator an important problem in geophysics medical imaging and nondestructive imaging today is the construction of a practical accurate and efficient means of imaging geophysical anomalies tumours or material defects in layered media this paper discusses such a method the use of a stratified green s function for the solution of the forward problem is detailed this forward problem is then incorporated into an efficient and accurate inversion algorithm based on optimization the method is nonperturbative unlike diffraction tomography which relies on linearization to make the problem tractable in the inversion a pair of lippmann schwinger like integral equations are solved simultaneously via the galerkin procedure for the unknown total internal fields and speed distribution the computational burden is high but made manageable by utilizing biconjugate gradients fast fourier transforms and sinc basis functions to speed up the solution of the forward problem the size and contrasts for which the me

on the peak to average power of ofdm signals based on oversampling orthogonal frequency division multiplexing ofdm introduces large amplitude variations in time which can result in significant signal distortion in the presence of nonlinear amplifiers we introduce a new bound for the peak of the continuous envelope of an ofdm signal based on the maximum of its corresponding oversampled sequence it is shown to be very tight as the oversampling rate increases the bound is then used to derive a closed form probability upper bound for the complementary cumulative distribution function of the peak to mean envelope power ratio of uncoded ofdm signals for sufficiently large numbers of subcarriers as another application of the bound for oversampled sequences we propose tight relative error bounds for computation of the peak power using two main methods the oversampled inverse fast fourier transform and the method introduced for coded systems based on minimum distance decoding of the code

cross entropy based sign selection algorithms for peak to average power ratio reduction of ofdm systems sign selection uses a set of subcarrier signs to reduce the peak to average power ratio par of orthogonal frequency division multiplexing ofdm however the computational complexity worst case is exponential in n the number of subcarriers suboptimal sign selection algorithms achieving different tradeoffs between the par reduction and complexity have thus been developed for example the derandomization method achieves high par reduction of o log n with relatively high complexity of o n2 on the other hand selective mapping slm and partial transmit sequences pts sacrifice the achievable par reduction for lower complexity in this paper we develop two new cross entropy ce based sign selection algorithms our algorithms simultaneously updates the probabilities of the signs of all subcarriers the first algorithm obtains a par lower than the above methods with a complexity level of o n2 however if the number of iterations is fixed this algorithm obtains the same par reduction as derandomization but with o n log n complexity practical par reduction algorithms require that the extra cost of par reduction must be small therefore we propose the second algorithm which adaptively adjusts the probability of elite samples and stops whenever a par threshold is reached our second algorithm achieves up to 95 complexity savings over the first with only a 0 4 db par reduction loss the simulations confirm the complexity advantages of the proposed algorithms compared to slm and derandomization

enhancement of the krylov subspace regularization for microwave biomedical imaging although krylov subspace methods provide fast regularization techniques for the microwave imaging problem they cannot preserve the edges of the object being imaged and may result in an oscillatory reconstruction to suppress these spurious oscillations and to provide an edge preserving regularization we use a multiplicative regularizer which improves the reconstruction results significantly while adding little computational complexity to the inversion algorithm we show the inversion results for a real human forearm assuming the 2 d transverse magnetic illumination and a cylindrical object assuming the 2 d transverse electric illumination

fractional actionlike variational problems fractional actionlike variational problems have recently gained importance in studying dynamics of nonconservative systems in this note we address multidimensional fractional actionlike problems of the calculus of variations

fractional differential forms a generalization of exterior calculus is considered by allowing the partial derivatives in the exterior derivative to assume fractional orders that is a fractional exterior derivative is defined this is found to generate new vector spaces of finite and infinite dimension fractional differential form spaces the definitions of closed and exact forms are extended to the new fractional form spaces with closure and integrability conditions worked out for a special case coordinate transformation rules are also computed the transformation rules are different from those of the standard exterior calculus due to the properties of the fractional derivative the metric for the fractional form spaces is given based on the coordinate transformation rules all results are found to reduce to those of standard exterior calculus when the order of the coordinate differentials is set to one

high rate codes with bounded pmepr for bpsk and other symmetric constellations in this letter we consider the problem of constructing high rate codes with low peak to mean envelope power ratio pmepr for multicarrier signals assuming coefficients of the multicarrier signal are chosen from a symmetric q ary constellation we construct codes with rate 1 1 r logq2 and pmepr of less than crlogn for any r and n where n is the number of subcarriers and c is a constant independent of n and r the construction is based on dividing n subcarriers into n r groups of r subcarriers and choosing a sign for each group to minimize the pmepr the signs are chosen using a variation of the algorithm proposed by the authors in previous papers for large n we can in fact construct a code with a rate of 1 o 1 logn and pmepr of less than clog2n for binary phase shift keying modulated signals this partially solves the problem posed by litsyn and implies a construction of 2n 2 codewords with pmepr less than 2clogn

a formulation of noether s theorem for fractional problems of the calculus of variations abstract fractional or non integer differentiation is an important concept both from theoretical and applicational points of view the study of problems of the calculus of variations with fractional derivatives is a rather recent subject the main result being the fractional necessary optimality condition of euler lagrange obtained in 2002 here we use the notion of euler lagrange fractional extremal to prove a noether type theorem for that we propose a generalization of the classical concept of conservation law introducing an appropriate fractional operator

the fast multipole method and fourier convolution for the solution of acoustic scattering on regular volumetric grids the fast multipole method fmm is applied to the solution of large scale three dimensional acoustic scattering problems involving inhomogeneous objects defined on a regular grid the grid arrangement is especially well suited to applications in which the scattering geometry is not known a priori and is reconstructed on a regular grid using iterative inverse scattering algorithms or other imaging techniques the regular structure of unknown scattering elements facilitates a dramatic reduction in the amount of storage and computation required for the fmm both of which scale linearly with the number of scattering elements in particular the use of fast fourier transforms to compute green s function convolutions required for neighboring interactions lowers the often significant cost of finest level fmm computations and helps mitigate the dependence of fmm cost on finest level box size numerical results demonstrate the efficiency of the composite method as the number of scattering elements in each finest level box is increased

the restaurant at the end of the random walk recent developments in the description of anomalous transport by fractional dynamics fractional dynamics has experienced a firm upswing during the past few years having been forged into a mature framework in the theory of stochastic processes a large number of research papers developing fractional dynamics further or applying it to various systems have appeared since our first review article on the fractional fokker planck equation metzler r and klafter j 2000a phys rep 339 1 77 it therefore appears timely to put these new works in a cohesive perspective in this review we cover both the theoretical modelling of sub and superdiffusive processes placing emphasis on superdiffusion and the discussion of applications such as the correct formulation of boundary value problems to obtain the first passage time density function we also discuss extensively the occurrence of anomalous dynamics in various fields ranging from nanoscale over biological to geophysical and environmental systems

fractional systems and fractional bogoliubov hierarchy equations we consider the fractional generalizations of the phase volume volume element and poisson brackets these generalizations lead us to the fractional analog of the phase space we consider systems on this fractional phase space and fractional analogs of the hamilton equations the fractional generalization of the average value is suggested the fractional analogs of the bogoliubov hierarchy equations are derived from the fractional liouville equation we define the fractional reduced distribution functions the fractional analogs of the vlasov equation and the debye radius are considered

clipping noise guided sign selection for par reduction in ofdm systems the peak to average power ratio par of orthogonal frequency division multiplexing ofdm systems can be reduced by using an optimal set of subcarrier signs however this sign selection is a hard discrete optimization problem we therefore consider the use of the clipping noise generated when the ofdm signal is clipped at a given threshold level to find a good set of signs the key idea of clipping noise guided sign selection cgs is to iteratively flip the signs of those subcarriers with high levels of clipping noise in each iteration the key task is to determine the number and locations of such subcarriers we develop suitable criteria for this task and derive cgs algorithms that can handle both unitary e g m ary phase shift keying and nonunitary e g m ary quadrature amplitude modulation signal constellations the simulation results show that the par reduction of cgs is about 1 db larger than that of derandomization and tone reservation for a 256 subcarrier system and is about 1 2 db larger than that of partial transmit sequence pts and selective mapping slm cgs also removes the error floor due to nonlinear amplifiers

a generalized fractional kn equation hierarchy and its fractional hamiltonian structure a generalized hamiltonian structure of the fractional soliton equation hierarchy is presented by using differential forms and exterior derivatives of fractional orders we construct the generalized fractional trace identity through the riemann liouville fractional derivative an example of the fractional kn soliton equation hierarchy and hamiltonian structure is presented which is a new integrable hierarchy and possesses hamiltonian structure

acoustic scattering for 3d multi directional periodic structures using the boundary element method an efficient boundary element formulation is proposed to solve three dimensional exterior acoustic scattering problems with multi directional periodicity the multi directional periodic acoustic problem is represented as a multilevel block toeplitz matrix by exploiting the toeplitz structure the computational time and storage requirements to construct and to solve the linear system of equations arising from the boundary element formulation are significantly reduced the generalized minimal residual method is implemented to solve the linear system of equations to efficiently calculate the matrix vector product in the iterative algorithm the original matrix is embedded into a multilevel block circulant matrix a multi dimensional discrete fourier transform is then employed to accelerate the matrix vector product the proposed approach is applicable to a periodic acoustic problem for any arbitrary shape of the structure in both full space and half space two case studies involving sonic crystal barriers