noncoherent capacity of underspread fading channels we derive bounds on the noncoherent capacity of wide sense stationary uncorrelated scattering wssus channels that are selective both in time and frequency and are underspread i e the product of the channel s delay spread and doppler spread is small the underspread assumption is satisfied by virtually all wireless communication channels for input signals that are peak constrained in time and frequency we obtain upper and lower bounds on capacity that are explicit in the channel s scattering function are accurate for a large range of bandwidth and allow to coarsely identify the capacity optimal bandwidth as a function of the peak power and the channel s scattering function we also obtain a closed form expression for the first order taylor series expansion of capacity in the infinite bandwidth limit and show that our bounds are tight in the wideband regime for input signals that are peak constrained in time only and hence allowed to be peaky in frequency we provide upper and lower bounds on the infinite bandwidth capacity our lower bound is closely related to a result by viterbi 1967 we find cases where the bounds coincide and hence the infinite bandwidth capacity is characterized exactly the analysis in this paper is based on a discrete time discrete frequency approximation of wssus time and frequency selective channels this discretization takes the underspread property of the channel explicitly into account

gravity capillary waves in finite depth on flows of constant vorticity this paper considers two dimensional periodic gravity capillary waves propagating steadily in finite depth on a linear shear current constant vorticity a perturbation series solution for steady periodic waves accurate up to the third order is derived using a classical stokes expansion procedure which allows us to include surface tension effects in the analysis of wave current interactions in the presence of constant vorticity the analytical results are then compared with numerical computations with the full equations the main results are i the phase velocity is strongly dependent on the value of the vorticity ii the singularities wilton singularities in the stokes expansion in powers of wave amplitude that correspond to a bond number of 1 2 and 1 3 which are the consequences of the non uniformity in the ordering of the fourier coefficients are found to be influenced by vorticity iii different surface profiles of capillary gravity waves are computed and the effect of vorticity on those profiles is shown to be important in particular that the solutions exhibit type 2 like wave features characterized by a secondary maximum on the surface profile with a trough between the two maxima

on the spectral efficiency of noncoherent doubly selective block fading channels in this paper we consider noncoherent single antenna communication over doubly selective block fading channels with discrete block fading interval n in our noncoherent setup neither the transmitter nor the receiver know the channel fading coefficients though both know the channel statistics in particular we consider discrete time channels whose impulse response trajectories obey a complex exponential basis expansion model with uncorrelated coefficients and we show that such a model holds in the limit n for pulse shaped transmission reception over certain wide sense stationary uncorrelated scattering channels first we show that when the inputs are chosen from continuous distributions the channel s multiplexing gain i e capacity pre log factor equals max 0 1 n delay n dopp n for discrete delay spread n delay and discrete doppler spread n dopp next for the case of strictly doubly selective fading i e n dopp 1 and n delay 1 we establish that for cyclic prefixed affine pilot aided transmission pat schemes designed to minimize the mean squared error mse attained by pilot aided minimum mse channel estimation the pre log factor of the achievable rate is less than the channel s multiplexing gain we then provide guidelines for the design of pat schemes whose achievable rate pre log factor equals the channel s multiplexing gain and construct an example

communication on the grassmann manifold a geometric approach to the noncoherent multiple antenna channel we study the capacity of multiple antenna fading channels we focus on the scenario where the fading coefficients vary quickly thus an accurate estimation of the coefficients is generally not available to either the transmitter or the receiver we use a noncoherent block fading model proposed by marzetta and hochwald see ibid vol 45 p 139 57 1999 the model does not assume any channel side information at the receiver or at the transmitter but assumes that the coefficients remain constant for a coherence interval of length t symbol periods we compute the asymptotic capacity of this channel at high signal to noise ratio snr in terms of the coherence time t the number of transmit antennas m and the number of receive antennas n while the capacity gain of the coherent multiple antenna channel is min m n bits per second per hertz for every 3 db increase in snr the corresponding gain for the noncoherent channel turns out to be m 1 m t bits per second per hertz where m min m n t 2 the capacity expression has a geometric interpretation as sphere packing in the grassmann manifold

analyticity of periodic traveling free surface water waves with vorticity we prove that the prole of a periodic traveling wave propagating at the surface of water above a at bed in a ow with a real analytic vorticity must be real analytic provided the wave speed exceeds the horizontal uid velocity throughout the ow the real analyticity of each streamline be

nonlinear oblique boundary value problems for nonlinear elliptic equations on considere le probleme aux valeurs limites a derivee oblique non lineaire pour des equations uniformement elliptiques quasi lineaires et totalement non lineaires d ordre 2 les operateurs elliptiques satisfont des conditions de structure naturelle on donne des theoremes d existence pour les solutions

steady periodic water waves with constant vorticity regularity and local bifurcation this paper studies periodic traveling gravity waves at the free surface of water in a flow of constant vorticity over a flat bed using conformal mappings the free boundary problem is transformed into a quasilinear pseudodifferential equation for a periodic function of one variable the new formulation leads to a regularity result and by use of bifurcation theory to the existence of waves of small amplitude even in the presence of stagnation points in the flow

regularity of traveling periodic stratified water waves with vorticity abstract we prove real analyticity of all the streamlines including the free surface of a steady stratified flow of water over a flat bed in the absence of stagnation points with a holder continuous bernoulli function and a holder continuously differentiable density function furthermore we show that if the bernoulli function and the density function possess some gevrey regularity of index s then the stream function admits the same gevrey regularity throughout the fluid domain in particular if the gevrey index s equals to 1 then we obtain analyticity of the stream function the regularity results hold for three distinct physical regimes capillary capillary gravity and gravity water waves

bounds on the capacity of ofdm underspread frequency selective fading channels the analysis of the channel capacity in the absence of prior channel knowledge noncoherent channel has gained increasing interest in recent years yet the noncoherent channel capacity is still unknown for the general case in this paper we derive bounds on the capacity of the noncoherent underspread complex gaussian orthogonal frequency division multiplexing wide sense stationary channel with uncorrelated scattering under a peak power constraint or a constraint on the second and fourth moments of the transmitted signal these bounds are characterized only by the system signal to noise ratio snr and by a newly defined quantity termed effective coherence time intuitively the effective coherence time can be interpreted as the length of a block in the block fading model in which a system with the same snr will achieve the same capacity as in the analyzed channel unlike commonly used coherence time definitions it is shown that the effective coherence time depends on the snr and is a nonincreasing function of it we show that for low snr the capacity is proportional to the effective coherence time while for higher snr the coherent channel capacity can be achieved provided that the effective coherence time is large enough

computation of the inverse laplace transform based on a collocation method which uses only real values we develop a numerical algorithm for inverting a laplace transform lt based on laguerre polynomial series expansion of the inverse function under the assumption that the lt is known on the real axis only the method belongs to the class of collocation methods c methods and is applicable when the lt function is regular at infinity difficulties associated with these problems are due to their intrinsic ill posedness the main contribution of this paper is to provide computable estimates of truncation discretization conditioning and roundoff errors introduced by numerical computations moreover we introduce the pseudoaccuracy which will be used by the numerical algorithm in order to provide uniform scaled accuracy of the computed approximation for any x with respect to e x these estimates are then employed to dynamically truncate the series expansion in other words the number of the terms of the series acts like the regularization parameter which provides the trade off between errors with the aim to validate the reliability and usability of the algorithm experiments were carried out on several test functions

steady periodic waves and formal stability for fixed depth rotational equatorial flows abstract in this paper we apply the bifurcation theory to study the steady two dimensional periodic equatorial water waves for the f plane approximation we consider the waves with vorticity which propagate with a specified fixed depth between the thermocline and the upper boundary of the centre layer moreover we present a functional j and its first variation corresponds to the exact equations in the transformed variables using the second variation of j we prove a formal stability result for the bifurcation inducing the laminar solution

on the pressure transfer function for solitary water waves with vorticity in this paper we analyse the role which the pressure function on the sea bed plays in determining solitary waves with vorticity we prove that the pressure function on the flat bed determines a unique real analytic solitary wave solution to the governing equations given a real analytic vorticity distribution in particular the pressure function on the flat bed prescribes a unique surface profile for the resulting solitary water wave

gaussian fading is the worst fading the capacity of peak power limited single antenna noncoherent flat fading channels with memory is considered the emphasis is on the capacity pre log i e on the limiting ratio of channel capacity to the logarithm of the signal to noise ratio snr as the snr tends to infinity it is shown that among all stationary and ergodic fading processes of a given spectral distribution function and whose law has no mass point at zero the gaussian process gives rise to the smallest pre log the assumption that the law of the fading process has no mass point at zero is essential in the sense that there exist stationary and ergodic fading processes whose law has a mass point at zero and that give rise to a smaller pre log than the gaussian process of equal spectral distribution function an extension of these results to multiple input single output miso fading channels with memory is also presented

existence and stability of multidimensional transonic flows through an infinite nozzle of arbitrary cross sections we establish the existence and stability of multidimensional steady transonic flows with transonic shocks through an infinite nozzle of arbitrary cross sections including a slowly varying de laval nozzle the transonic flow is governed by the inviscid potential flow equation with supersonic upstream flow at the entrance uniform subsonic downstream flow at the exit at infinity and the slip boundary condition on the nozzle boundary our results indicate that if the supersonic upstream flow at the entrance is sufficiently close to a uniform flow there exists a solution that consists of a c1 subsonic flow in the unbounded downstream region converging to a uniform velocity state at infinity and a c1 multidimensional transonic shock separating the subsonic flow from the supersonic upstream flow the uniform velocity state at the exit at infinity in the downstream direction is uniquely determined by the supersonic upstream flow and the shock is orthogonal to the nozzle boundary at every point of their intersection in order to construct such a transonic flow we reformulate the multidimensional transonic nozzle problem into a free boundary problem for the subsonic phase in which the equation is elliptic and the free boundary is a transonic shock the free boundary conditions are determined by the rankine hugoniot conditions along the shock we further develop a nonlinear iteration approach and employ its advantages to deal with such a free boundary problem in the unbounded domain we also prove that the transonic flow with a transonic shock is unique and stable with respect to the nozzle boundary and the smooth supersonic upstream flow at the entrance

capacity bounds for peak constrained multiantenna wideband channels bounds are derived on the noncoherent capacity of a very general class of multiple input multiple output fading channels that are selective in time and frequency as well as correlated in space the bounds apply to peak constrained inputs they are explicit in the channel s scattering function are useful for a large range of bandwidth and allow one to coarsely identify the capacity optimal combination of bandwidth and number of transmit antennas furthermore a closed form expression is obtained for the first order taylor series expansion of capacity in the limit of infinite bandwidth from this expression it is concluded that in the wideband regime i it is optimal to use only one transmit antenna when the channel is spatially uncorrelated ii rank one statistical beamforming is optimal if the channel is spatially correlated and iii spatial correlation be it at the transmitter the receiver or both is beneficial

on the global existence and wave breaking criteria for the two component camassa holm system considered herein is a two component camassa holm system modeling shallow water waves moving over a linear shear flow a wave breaking criterion for strong solutions is determined in the lowest sobolev space hs s 32 by using the localization analysis in the transport equation theory moreover an improved result of global solutions with only a nonzero initial profile of the free surface component of the system is established in this sobolev space hs

stability properties of steady water waves with vorticity we present two stability analyses for exact periodic traveling water waves with vorticity the first approach leads in particular to linear stability properties of water waves for which the vorticity decreases with depth the second approach leads to a formal stability property for long water waves that have small vorticity and amplitude although we do not use a small amplitude or long wave approximation 2006 wiley periodicals inc

on fixed input distributions for noncoherent communication over high snr rayleigh fading channels it is well known that independent and identically distributed gaussian inputs scaled appropriately based on the signal to noise ratio snr achieve capacity on the additive white gaussian noise awgn channel at all values of snr in this correspondence we consider the question of whether such good input distributions exist for frequency nonselective rayleigh fading channels assuming that neither the transmitter nor the receiver has a priori knowledge of the fading coefficients in this noncoherent regime for a gauss markov model of the fading channel we obtain explicit mutual information bounds for the gaussian input distribution the fact that gaussian input generates bounded mutual information motivates the search for better choices of fixed input distributions for high rate transmission over rapidly varying channels necessary and sufficient conditions are derived for characterizing such distributions for the worst case scenario of memoryless fading using the criterion that the mutual information is unbounded as the snr gets large examples of both discrete and continuous distributions that satisfy these conditions are given a family of fixed input distributions with mutual information growth rate of o loglogsnr sup 1 u u 0 are constructed it is also proved that there does not exist a single fixed input distribution that achieves the optimal mutual information growth rate of loglogsnr

trimodal steady water waves we construct three dimensional families of small amplitude gravity driven rotational steady water waves of finite depth the solutions contain counter currents and multiple crests in each minimal period each such wave is generically a combination of three different fourier modes giving rise to a rich and complex variety of wave patterns the bifurcation argument is based on a blow up technique taking advantage of three parameters associated with the vorticity distribution the strength of the background stream and the period of the wave

multiple eigenvalue bifurcation for fredholm operators on fait le point sur les resultats de bifurcation locale et globale et on considere la stabilite des branches de bifurcation

on the information dimension of stochastic processes in 1959 renyi proposed the information dimension and the d dimensional entropy to measure the information content of general random variables this paper proposes a generalization of information dimension to stochastic processes by defining the information dimension rate as the entropy rate of the uniformly quantized stochastic process divided by minus the logarithm of the quantizer step size 1 m in the limit as m to infty it is demonstrated that the information dimension rate coincides with the rate distortion dimension defined as twice the rate distortion function r d of the stochastic process divided by log d in the limit as d downarrow 0 it is further shown that among all multivariate stationary processes with a given matrix valued spectral distribution function sdf the gaussian process has the largest information dimension rate and the information dimension rate of multivariate stationary gaussian processes is given by the average rank of the derivative of the sdf the presented results reveal that the fundamental limits of almost zero distortion recovery via compressible signal pursuit and almost lossless analog compression are different in general

capacity results for block stationary gaussian fading channels with a peak power constraint a peak power limited single antenna block stationary gaussian fading channel is studied where neither the transmitter nor the receiver knows the channel state information but both know the channel statistics this model subsumes most previously studied gaussian fading models the asymptotic channel capacity in the high signal to noise ratio snr regime is first computed and it is shown that the behavior of the channel capacity depends critically on the channel model for the special case where the fading process is symbol by symbol stationary it is shown that the codeword length must scale at least logarithmically with snr in order to guarantee that the communication rate can grow logarithmically with snr with decoding error probability bounded away from one an expression for the capacity per unit energy is also derived furthermore it is shown that the capacity per unit energy is achievable using temporal on off signaling with optimally allocated on symbols where the optimal on symbol allocation scheme may depend on the peak power constraint

on the asymptotic capacity of stationary gaussian fading channels we consider a peak power limited single antenna flat complex gaussian fading channel where the receiver and transmitter while fully cognizant of the distribution of the fading process have no knowledge of its realization upper and lower bounds on channel capacity are derived with special emphasis on tightness in the high signal to noise ratio snr regime necessary and sufficient conditions in terms of the autocorrelation of the fading process are derived for capacity to grow double logarithmically in the snr for cases in which capacity increases logarithmically in the snr we provide an expression for the pre log i e for the asymptotic ratio between channel capacity and the logarithm of the snr this ratio is given by the lebesgue measure of the set of harmonics where the spectral density of the fading process is zero we finally demonstrate that the asymptotic dependence of channel capacity on the snr need not be limited to logarithmic or double logarithmic behaviors we exhibit power spectra for which capacity grows as a fractional power of the logarithm of the snr

symmetry of steady periodic gravity water waves with vorticity we prove that steady periodic two dimensional rotational gravity water waves with a monotone surface profile between troughs and crests have to be symmetric about the crest irrespective of the vorticity distribution within the fluid

a rate splitting approach to fading channels with imperfect channel state information as shown by medard the capacity of fading channels with imperfect channel state information can be lower bounded by assuming a gaussian channel input x with power p and by upper bounding the conditional entropy h x y hat h by the entropy of a gaussian random variable with variance equal to the linear minimum mean square error in estimating x from y hat h we demonstrate that using a rate splitting approach this lower bound can be sharpened by expressing the gaussian input x as the sum of two independent gaussian variables x 1 and x 2 and by applying medard s lower bound first to bound the mutual information between x 1 and y while treating x 2 as noise and by applying it a second time to the mutual information between x 2 and y while assuming x 1 to be known we obtain a capacity lower bound that is strictly larger than medard s lower bound we then generalize this approach to an arbitrary number l of layers where x is expressed as the sum of l independent gaussian random variables of respective variances p ell ell 1 dotsc l summing up to p among all such rate splitting bounds we determine the supremum over power allocations p ell and total number of layers l this supremum is achieved for l rightarrow infty and gives rise to an analytically expressible capacity lower bound for gaussian fading this novel bound is shown to converge to the gaussian input mutual information as the signal to noise ratio snr grows provided that the variance of the channel estimation error h hat h tends to zero as the snr tends to infinity

on the existence of extreme waves and the stokes conjecture with vorticity this is a study of singular solutions of the problem of traveling gravity water waves on flows with vorticity we show that for a certain class of vorticity functions a sequence of regular waves converges to an extreme wave with stagnation points at its crests we also show that for any vorticity function the profile of an extreme wave must have either a corner of 120 or a horizontal tangent at any stagnation point about which it is supposed symmetric moreover the profile necessarily has a corner of 120 if the vorticity is nonnegative near the free surface

an induced additive noise model for memoryless rayleigh fading channels this correspondence investigates a noncoherent discrete time memoryless rayleigh fading channel a logarithmic transform converts it into an induced channel with additive noise that is independent of the channel input from this perspective it is natural and convenient for us to revisit several known results and gain new insights in particular we specify a class of simple log scale uniform channel input distributions that performs well for moderate to high signal to noise ratio snr furthermore a continuous amplitude log scale uniform distribution is asymptotically capacity achieving in the sense that the difference between the resulting mutual information and the channel capacity approaches zero as snr becomes large we also extend the induced additive noise channel approach to memoryless multiple antenna channels possibly with transmit but without receive spatial correlation

particle trajectories beneath small amplitude shallow water waves in constant vorticity flows abstract we investigate the particle trajectories in a constant vorticity shallow water flow over a flat bed as periodic waves propagate on the water s free surface within the framework of small amplitude waves we find the solutions of the nonlinear differential equations system which describes the particle motion in the considered case and we describe the possible particle trajectories depending on the relation between the initial data and the constant vorticity some particle trajectories are undulating curves to the right or to the left others are loops with forward drift or with backward drift others can follow some peculiar shapes

bernoulli free boundary problems in strip like domains and a property of permanent waves on water of finite depth we study weak solutions for a class of free boundary problems which includes as a special case the classical problem of travelling gravity waves on water of finite depth we show that such problems are equivalent to problems in fixed domains and study the regularity of their solutions we also prove that in very general situations the free boundary is necessarily the graph of a function

asymptotic capacity analysis for adaptive transmission schemes under general fading distributions asymptotic comparisons of ergodic channel capacity at high and low snrs are provided for several adaptive transmission schemes over fading channels with general distributions including optimal power and rate adaptation rate adaptation only channel inversion and its variants analysis of the high snr pre log constants of the ergodic capacity reveals the existence of constant capacity difference gaps among the schemes with a pre log constant of 1 closed form expressions for these high snr capacity difference gaps are derived which are proportional to the snr loss between these schemes in db scale the largest one of these gaps is found to be between the optimal power and rate adaptation scheme and the channel inversion scheme based on these expressions it is shown that the presence of space diversity or multi user diversity makes channel inversion arbitrarily close to achieving optimal capacity at high snr with sufficiently large number of antennas or users a low snr analysis also reveals that the presence of fading provably always improves capacity at sufficiently low snr compared to the additive white gaussian noise awgn case numerical results are shown to corroborate our analytical results

a flow force reformulation for steady irrotational water waves abstract the classical irrotational water wave problem described by euler equations with a nonlinear free surface boundary condition and influenced by gravity over a flat bed is considered exploiting the monotonicity property of the flow force lines with depth the unknown boundary problem is transformed into a problem with fixed domain a flow force function formulation of the irrotational water wave problem is developed variational approach is used to prove the existence of small amplitude irrotational traveling wave solutions local bifurcation results have been proved relying on crandall rabinowitz bifurcation theorem

existence of capillary gravity water waves with piecewise constant vorticity abstract in this paper we construct small amplitude periodic capillary gravity water waves with a piecewise constant vorticity distribution they describe water waves traveling on superposed linearly sheared currents that have different vorticities this is achieved by associating to the height function formulation of the water wave problem a diffraction problem where we impose suitable transmission conditions on each line where the vorticity function has a jump the solutions of the diffraction problem found by using local bifurcation theory are the desired solutions of the hydrodynamical problem

sparse bayesian learning for the laplace transform inversion in dynamic light scattering a new method is described using the sparse bayesian learning sbl algorithm of tipping to obtain an optimal and reliable solution to the laplace transform inversion in dynamic light scattering dls the linear inverse problem in dls has numerical solutions that depend on their domains and dimensions for a given domain and dimension a sparse solution in an sbl framework is the most probable solution and can be used for classifying a system of objects by a few relevant values recently we have shown that the sbl algorithm of tipping is suitable for studying cataract in ocular lenses by describing the opacity of a lens with a few dominant sizes of crystallin proteins in the lens however since the sparseness of sbl solutions cannot reflect a true system we need to develop a method by using the sbl algorithm to give a true description and at the same time a useful classification of the opacity of lenses we generate a set of sparse solutions of different domains but of the same dimension and then superimpose them to give a general solution with its dimension treated as a regularization parameter an optimal solution which provides a reliable description of a particle system is determined by the l curve criterion for selecting the suitable value of the regularization parameter the performance of our method is evaluated by analyzing simulated data generated from unimodal and bimodal distributions from the reconstructed distributions we see that our method gives high resolution comparable to the sophisticated bryan s maximum entropy algorithm which gives better resolution than contin our method is then applied to experimental dls data of the ocular lenses of a fetal calf and a rhesus monkey to obtain optimal particle size distributions of crystallins and the crystallin aggregates in the lenses we conclude by discussing possible improvements on our method for analyzing dls data and for solving any linear inverse problem by an sbl algorithm

constant vorticity water flows in the equatorial β plane approximation with centripetal forces abstract in this paper we consider three dimensional equatorial flows with constant vorticity beneath a wave train and above a flat bed in the plane approximation with centripetal forces as a striking upshot we show that the equatorial flow is necessarily irrotational and the free surface is necessarily flat if it exhibits a constant vorticity owing to the presence of centripetal forces for the case exhibiting the surface tension we prove that there are no equatorial flows exhibiting a constant vorticity

computational error bounds for laplace transform inversion based on smoothing splines abstract in the numerical methods for the laplace transform inversion the errors quantification in computational processes is a crucial issue in this paper we propose two inversion methods based on smoothing splines combined with a procedure for the derivation of error bounds in particular we numerically study the impact of the fitting error amplification through the analysis of several sources of error and their propagation

exact solitary water waves with vorticity the solitary water wave problem is to find steady free surface waves which approach a constant level of depth in the far field the main result is the existence of a family of exact solitary waves of small amplitude for an arbitrary vorticity each solution has a supercritical parameter value and decays exponentially at infinity the proof is based on a generalized implicit function theorem of the nash moser type the first approximation to the surface profile is given by the kdv equation with a supercritical value of the surface tension coefficient a family of small amplitude solitary waves of depression with subcritical parameter values is constructed for an arbitrary vorticity

local well posedness and stability of peakons for a generalized dullin gottwald holm equation abstract we establish the local well posedness for a generalized dullin gottwald holm equation by using kato s theory furthermore the orbital stability of the peaked solitary waves is also proved

the fading number of single input multiple output fading channels with memory we derive the fading number of stationary and ergodic not necessarily gaussian single input multiple output simo fading channels with memory this is the second term after the double logarithmic term of the high signal to noise ratio snr expansion of channel capacity the transmitter and receiver are assumed to be cognizant of the probability law governing the fading but not of its realization it is demonstrated that the fading number is achieved by independent and identically distributed i i d circularly symmetric inputs of squared magnitude whose logarithm is uniformly distributed over an snr dependent interval the upper limit of the interval is the logarithm of the allowed transmit power and the lower limit tends to infinity sublogarithmically in the snr the converse relies inter alia on a new observation regarding input distributions that escape to infinity lower and upper bounds on the fading number for gaussian fading are also presented these are related to the mean squared errors of the one step predictor and the one gap interpolator of the fading process respectively the bounds are computed explicitly for stationary mth order autoregressive ar m gaussian fading processes

short packet communications over multiple antenna rayleigh fading channels motivated by the current interest in ultra reliable low latency machine type communication systems we investigate the tradeoff between reliability throughput and latency in the transmission of information over multiple antenna rayleigh block fading channels specifically we obtain finite blocklength finite snr upper and lower bounds on the maximum coding rate achievable over such channels for a given constraint on the packet error probability numerical evidence suggests that our bounds delimit tightly the maximum coding rate already for short blocklengths packets of about 100 symbols furthermore our bounds reveal the existence of a tradeoff between the rate gain obtainable by spreading each codeword over all available time frequency spatial degrees of freedom and the rate loss caused by the need of estimating the fading coefficients over these degrees of freedom in particular our bounds allow us to determine the optimal number of transmit antennas and the optimal number of time frequency diversity branches that maximize the rate finally we show that infinite blocklength performance metrics such as the ergodic capacity and the outage capacity yield inaccurate throughput estimates

on some properties of velocity field for two dimensional rotational steady water waves abstract we investigate the properties of velocity field for steady periodic water waves over a flat bed for a generalized class of c 1 vorticity functions results are proved by exploiting the maximum principles and are based on dubreil jacotin transformation of the fluid domain some properties of velocities interior to the fluid domain and the locations of maximum minimum horizontal fluid velocities are investigated and proved for rotational water waves without any restriction to amplitude for u c and a monotonically varying positive vorticity u y v x 0 the location of maximal horizontal velocity has been proven to be the crest generalizing some of the recently proved results

on the achievable rate of stationary rayleigh flat fading channels with gaussian inputs in this work a discrete time stationary rayleigh flat fading channel with unknown channel state information at transmitter and receiver side is studied the law of the channel is presumed to be known to the receiver for independent identically distributed i i d zero mean proper gaussian input distributions the achievable rate is investigated the main contribution of this paper is the derivation of two new upper bounds on the achievable rate with gaussian input symbols one of these bounds is based on the one step channel prediction error variance but is not restricted to peak power constrained input symbols like known bounds moreover it is shown that gaussian inputs yield the same pre log as the peak power constrained capacity the derived bounds are compared with a known lower bound on the capacity given by deng and haimovich and with bounds on the peak power constrained capacity given by sethuraman et al finally the achievable rate with i i d gaussian input symbols is compared to the achievable rate using a coherent detection in combination with a solely pilot based channel estimation

surface waves on steady perfect fluid flows with vorticity this is a theory of two dimensional steady periodic surface waves on flows under gravity in which the given data are three quantities that are independent of time in the corresponding evolution problem the volume of fluid per period the circulation per period on the free stream line and the rearrangement class equivalently the distribution function of the vorticity field a minimizer of the total energy per period among flows satisfying these three constraints is shown to be a weak solution of the surface wave problem for which the vorticity is a decreasing function of the stream function this decreasing function can be thought of as an infinite dimensional lagrange multiplier corresponding to the vorticity rearrangement class being specified in the minimization problem note that functional dependence of vorticity on the stream function was not specified a priori but is part of the solution to the problem and ensures the flow is steady to illustrate the idea with a minimum of technical difficulties the existence of nontrivial waves on the surface of a fluid flowing with a prescribed distribution of vorticity and confined beneath an elastic sheet is proved the theory applies equally to irrotational flows and to flows with locally square integrable vorticity 2011 wiley periodicals inc

a uniqueness result for periodic traveling waves in water of finite depth it is shown that in water of finite depth the surface profile eta of a periodic traveling wave uniquely determines the corresponding flow in the body of the fluid this holds for rotational flow as long as the vorticity function gamma psi satisfies the condition gamma psi max xis an element ofr eta 2 x pi 2 this condition is also shown to be sharp

capacity of noncoherent time selective rayleigh fading channels the capacity of noncoherent time selective rayleigh fading channels is studied under various models for the variations in time the study includes both single input and single output siso and multiple input and multiple output mimo systems a block fading model is first considered where the channel changes correlatively over each block period of length t and independently across blocks the predictability of the channel is characterized through the rank q of the correlation matrix of the vector of channel gains in each block this model includes as special cases the standard block fading model where the channel remains constant over block periods q 1 and models where the fading process has finite differential entropy rate q t the capacity is initially studied for long block lengths and some straightforward but interesting asymptotes are established for the case where q is kept fixed as t spl rarr spl infin it is shown that the noncoherent capacity converges to the coherent capacity for the case where both t q spl rarr spl infin with q t being held constant a bound on the capacity loss due to channel unpredictability is established the more interesting scenario of large signal to noise ratio snr is then explored in detail for siso systems useful upper and lower bounds on the large snr asymptotic capacity are derived and it is shown that the capacity grows logarithmically with snr with a slope of t q spl rarr t for q t next in order to facilitate the analysis of mimo systems the rank q block fading model is specialized to the case where each t symbol block consists of q subblocks of length l with the channel remaining constant over each subblock and changing correlatively across subblocks for this model it is shown that the log snr growth behavior of the capacity is the same as that of the standard block fading model with block length l finally the siso and mimo channel models are generalized to allow the fading process to be correlated across blocks in a stationary and ergodic manner it is shown that the log snr growth behavior of the capacity is not affected by the correlation across blocks

the second variation of the stream function energy of water waves with vorticity abstract we compute the second variation of the stream function energy of two dimensional steady free surface gravity water waves with vorticity in the stream function formulation we prove that for nonpositive vorticity the second variation of the stream function energy at extreme waves with stokes corner asymptotics cannot be nonnegative in any small neighbourhood of a given isolated stagnation point the particular form of our second variation suggests however the possibility that certain singularities in the case of nonzero vorticity might be constructible as minimizers of the stream function energy

the fading number of multiple input multiple output fading channels with memory the fading number of a general not necessarily gaussian regular multiple input multiple output mimo fading channel with arbitrary temporal and spatial memory is derived the channel is assumed to be noncoherent i e neither receiver nor transmitter have knowledge about the channel state but they only know the probability law of the fading process the fading number is the second term in the asymptotic expansion of channel capacity when the signal to noise ratio snr tends to infinity it is related to the border of the high snr region with double logarithmic capacity growth

capacity pre log of noncoherent simo channels via hironaka s theorem we find the capacity pre log of a temporally correlated rayleigh block fading single input multiple output simo channel in the noncoherent setting it is well known that for block length l and rank of the channel covariance matrix equal to q the capacity pre log in the single input single output siso case is given by 1 q l here q l can be interpreted as the pre log penalty incurred by channel uncertainty our main result reveals that by adding only one receive antenna this penalty can be reduced to 1 l and can hence be made to vanish for the block length l even if q l remains constant as l intuitively even though the siso channels between the transmit antenna and the two receive antennas are statistically independent the transmit signal induces enough statistical dependence between the corresponding receive signals for the second receive antenna to be able to resolve the uncertainty associated with the first receive antenna s channel and thereby make the overall system appear coherent the proof of our main theorem is based on a deep result from algebraic geometry known as hironaka s theorem on the resolution of singularities

a new formulation of the water wave problem for stokes waves of constant vorticity we consider steady symmetric gravity water waves on finite depth with constant vorticity and a monotone surface profile between crests and troughs the problem is transformed into one concerning the vertical velocity a representation formula for the stream function in terms of the surface and the vorticity is presented and we show that the surface can be determined from the vertical velocity

the fading number of memoryless multiple input multiple output fading channels in this correspondence we derive the fading number of multiple input multiple output mimo flat fading channels of general not necessarily gaussian regular law without temporal memory the channel is assumed to be noncoherent i e neither receiver nor transmitter have knowledge about the channel state but they only know the probability law of the fading process the fading number is the second term after the double logarithmic term of the high signal to noise ratio snr expansion of channel capacity hence the asymptotic channel capacity of memoryless mimo fading channels is derived exactly the result is then specialized to the known cases of single input multiple output simo multiple input single output miso and single input single output siso fading channels as well as to the situation of gaussian fading

modulational instability in the whitham equation with surface tension and vorticity abstract we study modulational stability and instability in the whitham equation combining the dispersion relation of water waves and a nonlinearity of the shallow water equations and modified to permit the effects of surface tension and constant vorticity when the surface tension coefficient is large we show that a periodic traveling wave of sufficiently small amplitude is unstable to long wavelength perturbations if the wave number is greater than a critical value and stable otherwise similarly to the benjamin feir instability of gravity waves in the case of weak surface tension we find intervals of stable and unstable wave numbers whose boundaries are associated with the extremum of the group velocity the resonance between the first and second harmonics the resonance between long and short waves and a resonance between dispersion and the nonlinearity for each constant vorticity we show that a periodic traveling wave of sufficiently small amplitude is unstable if the wave number is greater than a critical value and stable otherwise moreover it can be made stable for a sufficiently large vorticity the results agree with those based upon numerical computations or formal multiple scale expansions to the physical problem

dispersion relations for fixed mean depth flows with two discontinuities in vorticity abstract we derive here the dispersion relation for small amplitude steady two dimensional periodic water waves that propagate over a flat bed with a specified and fixed mean depth the water flow has a discontinuous vorticity distribution reflecting an isolated rotational layer of fluid we then use the dispersion relation to obtain necessary and sufficient conditions for local bifurcation to occur moreover a stability analysis of the laminar flows is presented

singularities of steady axisymmetric free surface flows with gravity we consider a steady axisymmetric solution of the euler equations describing the irrotational flow without swirl of an incompressible inviscid fluid acted on by gravity and with a free surface we analyze stagnation points as well as points on the axis of symmetry at points on the axis of symmetry that are not stagnation points constant velocity motion is the only blowup profile consistent with the invariant scaling of the equation this suggests the presence of downward pointing cusps at those points at stagnation points on the axis of symmetry the unique blowup profile consistent with the invariant scaling of the equation is the garabedian pointed bubble solution with water above air thus at stagnation points on the axis of symmetry with no water above the stagnation point the invariant scaling of the equation cannot be the right scaling a finer blowup analysis of the velocity field yields that in the case when the surface is described by an injective curve the velocity field scales almost like and is asymptotically given by with a nonzero constant c0 the last result relies on a frequency formula in combination with a concentration compactness result for the axially symmetric euler equations by delort while the concentration compactness result alone does not lead to strong convergence in general we prove the convergence to be strong in our application 2014 wiley periodicals inc

capacity bounds via duality with applications to multiple antenna systems on flat fading channels a technique is proposed for the derivation of upper bounds on channel capacity it is based on a dual expression for channel capacity where the maximization of mutual information over distributions on the channel input alphabet is replaced with a minimization of average relative entropy over distributions on the channel output alphabet we also propose a technique for the analysis of the asymptotic capacity of cost constrained channels the technique is based on the observation that under fairly mild conditions capacity achieving input distributions escape to infinity the above techniques are applied to multiple antenna flat fading channels with memory where the realization of the fading process is unknown at the transmitter and unknown or only partially known at the receiver it is demonstrated that for high signal to noise ratio snr the capacity of such channels typically grows only double logarithmically in the snr to better understand this phenomenon and the rates at which it occurs we introduce the fading number as the second order term in the high snr asymptotic expansion of capacity and derive estimates on its value for various systems it is suggested that at rates that are significantly higher than the fading number communication becomes extremely power inefficient thus posing a practical limit on practically achievable rates upper and lower bounds on the fading number are also presented for single input single output siso systems the bounds coincide thus yielding a complete characterization of the fading number for general stationary and ergodic fading processes we also demonstrate that for memoryless multiple input single output miso channels the fading number is achievable using beam forming and we derive an expression for the optimal beam direction this direction depends on the fading law and is in general not the direction that maximizes the snr on the induced siso channel using a new closed form expression for the expectation of the logarithm of a noncentral chi square distributed random variable we provide some closed form expressions for the fading number of some systems with gaussian fading including siso systems with circularly symmetric stationary and ergodic gaussian fading the fading number of the latter is determined by the fading mean fading variance and the mean squared error in predicting the present fading from its past it is not directly related to the doppler spread for the rayleigh ricean and multiple antenna rayleigh fading channels we also present firm nonasymptotic upper and lower bounds on channel capacity these bounds are asymptotically tight in the sense that their difference from capacity approaches zero at high snr and their ratio to capacity approaches one at low snr

the trajectories of particles in stokes waves analyzing a free boundary problem for harmonic functions we show that there are no closed particle paths in an irrotational inviscid traveling wave propagating at the surface of water over a flat bed within a period each particle experiences a backward forward motion with a slight forward drift

capillary gravity water waves modified flow force formulation abstract the classical irrotational capillary gravity water wave problem described by the euler equations with a nonlinear free surface boundary condition over a flat bed is considered a modified flow force has been defined and a new formulation of capillary gravity waves in the framework of the modified flow force function has been developed using bifurcation theory the local existence of waves of small amplitude is proved

oversampling increases the pre log of noncoherent rayleigh fading channels we analyze the capacity of a continuous time time selective rayleigh block fading channel in the high signal to noise ratio regime the fading process is assumed stationary within each block and to change independently from block to block further more its realizations are not known a priori to the transmitter and the receiver noncoherent setting a common approach to analyzing the capacity of this channel is to assume that the receiver performs matched filtering followed by sampling at symbol rate symbol matched filtering this yields a discrete time channel in which each transmitted symbol corresponds to one output sample liang veeravalli 2004 showed that the capacity of this discrete time channel grows logarithmically with the signal to noise ratio snr with a capacity pre log equal to 1 q n here n is the number of symbols transmitted within one fading block and q is the rank of the covariance matrix of the discrete time channel gains within each fading block in this paper we show that sym bol matched filtering is not a capacity achieving strategy for the underlying continuous time channel specifically we analyze the capacity pre log of the discrete time channel obtained by oversam pling the continuous time channel output i e by sampling it faster than at symbol rate we prove that by oversampling by a factor two one gets a capacity pre log that is at least as large as 1 1 n since the capacity pre log corresponding to symbol rate sampling is 1 q n our result implies indeed that symbol matched filtering is not capacity achieving at high snr

small amplitude stokes and solitary gravity water waves with an arbitrary distribution of vorticity this paper presents an existence theory for small amplitude stokes and solitary wave solutions to the classical water wave problem in the absence of surface tension and with an arbitrary distribution of vorticity the hydrodynamic problem is formulated as an infinite dimensional hamiltonian system in which the horizontal spatial coordinate is the time like variable a centre manifold technique is used to reduce the system to a locally equivalent hamiltonian system with one degree of freedom for values of a dimensionless parameter a near its critical value alpha the phase portrait of the reduced system contains a homoclinic orbit for alpha alpha the corresponding solutions of the water wave problem are respectively a solitary wave of elevation and a family of stokes waves c 2008 elsevier b v all rights reserved less

a free boundary approach to two dimensional steady capillary gravity water waves we prove that suitable weak solutions of the two dimensional steady capillary gravity water wave problem even in the presence of vorticity are smooth on a technical level solutions are closely related to critical points of the mumford shah functional so that our main task is to exclude cusps pointing into the water phase we prove that cusps do not exist and that the free surface is smooth

impact of feedback and side information on the asymptotic capacity of single input multiple output fading channels with memory an analytic expression for the asymptotic capacity of noncoherent single input multiple output regular fading channels with memory and with partial receiver side information is derived and is shown to remain unchanged by causal or acausal side information at the transmitter and by a noiseless feedback link in particular the corresponding fading numbers are identical furthermore the asymptotic capacity of a single input single output nonregular gaussian fading channel with memory is investigated and it is shown that the prelog is unaffected by noiseless feedback

upper bound on the slope of steady water waves with small adverse vorticity abstract we consider the angle of inclination with respect to the horizontal of the profile of a steady 2d inviscid symmetric periodic or solitary water wave subject to gravity there is an upper bound of 31 15 in the irrotational case 1 and an upper bound of 45 in the case of favorable vorticity 13 on the other hand if the vorticity is adverse the profile can become vertical we prove here that if the adverse vorticity is sufficiently small then the angle still has an upper bound which is slightly larger than 45

new solutions for solving problem of particle trajectories in linear deep water waves via lie group method the nonlinear equations of the two dimensional inviscid incompressible fluid in a constant gravitational field describing the wave propagation on the water surface are considered the lie group method has been applied for determining symmetry reductions of the system of partial differential equations lie group method starts out with a general infinitesimal group of transformations under which the given partial differential equations are invariant the determining equations are a set of linear differential equations the solution of which gives the transformation function or the infinitesimals of the dependent and independent variables after the group has been determined a solution to the given partial differential equations may be found from the invariant surface condition such that its solution leads to similarity variables that reduce the number of independent variables of the system effects of the wavelength l and time t on the particle path have been studied and the results are plotted

exact steady periodic water waves with vorticity we consider the classical water wave problem described by the euler equations with a free surface under the influence of gravity over a flat bottom we construct two dimensional inviscid periodic traveling waves with vorticity they are symmetric waves whose profiles are monotone between each crest and trough we use bifurcation and degree theory to construct a global connected set of such solutions c 2004 wiley periodicals inc

existence of wilton ripples for water waves with constant vorticity and capillary effects in this paper we study the water wave problem with capillary effects and constant vorticity when stagnation points are not excluded when the constant vorticity is close to certain critical values we show that there exist wilton ripples solutions of the water wave problem with two crests and two troughs per minimal period they form smooth secondary bifurcation curves that emerge from primary bifurcation branches that contain a laminar flow solution and consist of symmetric waves of half of the period of the wilton ripples at some nonlaminar solution we also prove that any wilton ripple contains an internal critical layer provided its minimal period is sufficiently small

new integrable hierarchy its parametric solutions cuspons one peak solitons and m w shape peak solitons in this paper we propose a new completely integrable hierarchy particularly in the hierarchy we draw two new soliton equations 1 mt 12 1 m2 xxx 12 1 m2 x 2 mt mx u2 ux2 2m2ux 0 m u uxx the first one is the second positive member in the hierarchy while the second one is the second negative member in the hierarchy both equations can be derived from the two dimensional euler equation by using the approximation procedure all equations in the hierarchy are proven to have bi hamiltonian operators and lax pairs through solving a crucial matrix equation moreover we develop parametric solutions of the entire hierarchy through constructing two kinds of constraints one is for all negative members of the hierarchy on a symplectic submanifold and the other is for all positive members in the standard symplectic space the most interesting things are both equations possess new type of peaked solitons continuous and piecewise smooth w m shape peak soliton solutions in addition we find new cusp solit

a penalization method for calculating the flow beneath traveling water waves of large amplitude we suggest an algorithm for the calculation of large amplitude traveling water waves in irrotational flows and in flows with constant and nonconstant vorticity respectively the algorithm is based on a quite general partial differential equation constrained optimization formulation maximizing the wave amplitude subject to the pde constraint that the euler equations hold numerical examples illustrate the performance of the algorithm for irrotational flows and in flows with constant vorticity where the results are compared with the literature as a proof of principle we also consider a test case with a nonconstant vorticity

periodic traveling gravity water waves with discontinuous vorticity we use elliptic theory to prove the existence of steady two dimensional periodic waterwaves of large amplitude in a flowwith an arbitrary bounded but discontinuous vorticity this is achieved by developing a local and global bifurcation construction of weak solutions of the elliptic partial differential equations that are relevant to this hydrodynamical context

on the sensitivity of continuous time noncoherent fading channel capacity the noncoherent capacity of stationary discrete time fading channels is known to be very sensitive to the fine details of the channel model more specifically the measure of the support of the fading process power spectral density psd determines if noncoherent capacity grows logarithmically with the signal to noise ratio snr or slower than logarithmically such a result is unsatisfactory from an engineering point of view as the support of the psd cannot be determined through measurements the aim of this paper is to assess whether for general continuous time rayleigh fading channels this sensitivity has a noticeable impact on capacity at snr values of practical interest to this end we consider the general class of band limited continuous time rayleigh fading channels that satisfy the wide sense stationary uncorrelated scattering wssus assumption and are in addition under spread we show that for all snr values of practical interest the noncoherent capacity of every channel in this class is close to the capacity of an additive white gaussian noise channel with the same snr and bandwidth independently of the measure of the support of the scattering function the 2 d channel psd our result is based on a lower bound on noncoherent capacity which is built on a discretization of the channel input output relation induced by projecting onto weyl heisenberg sets this approach is interesting in its own right as it yields a mathematically tractable way of dealing with the mutual information between certain continuous time random signals

error analysis of a collocation method for numerically inverting a laplace transform in case of real samples in s cuomo l d amore a murli m r rizzardi computation of the inverse laplace transform based on a collocation method which uses only real values j comput appl math 198 1 2007 98 115 the authors proposed a collocation method c method for real inversion of laplace transforms lt based on the truncated laguerre expansion of the inverse function f n x e s x k 0n 1c ke b xl k 2bx where s b are parameters and c k k n are the maclaurin coefficients of a function depending on the lt the computational kernel of a c method is the solution of a vandermonde linear system where the right hand side is obtained evaluating the lt on the real axis the bjorck pereira algorithm has been used for solving the vandermonde linear system providing a computable componentwise error bound on the solution for an inversion problem on discrete data f is known on a pre assigned set of points we refer to these points as samples of f only and the major challenge is to deal with a significative loss of information a natural approach to overcome this intrinsic difficulty is to construct a suitable fitting model that approximates the given data in this case we show that such approach leads to a c method with perturbed right hand side and then we use again the bjorck pereira algorithm starting from the error introduced by the fitting model we study its propagation in order to determine the maximum attainable accuracy on f n moreover we derive a computable error bound that allows to get the suitable value of the parameter n that gives the maximum attainable accuracy