on a class of nonlinear dispersive dissipative interactions abstract we study a prototype dissipative dispersive equation ut a um x un xxx uk xx a consts which represents a wide variety of interactions at the critical value k m n 2 which separates dispersive and dissipation dominated phenomena these effects are in a detailed balance and the patterns formed do not depend on the amplitude in particular when m n 2 k 1 the equation can be transformed into a form free of convection and dissipation making it accessible to analysis both bounded and unbounded oscillations as well as solitary waves are found a variety of exact solutions are presented with a notable example being a solitary doublet for n 1 and a 2 3 2 the problem may be mapped into a linear equation leading to rational periodic or aperiodic solutions among others

existence of weak backward solutions to a generalized hele shaw flow moving boundary problem in the original hele shaw problem n 2 and u 6 the dirac measure at the origin a couple of essentially equivalent weak formulations of this problem have been given in 4 8 91 and in 3 f or a variant of the above problem in the case n 2 it has been proved in 4 81 that a classical solution this notion been made precise in two different ways in 4 81 also is a weak solution and in 4 8 91 and 3 that a weak solution exists and is unique for the time interval 0 0 for arbitrary given initial domains do it has also been proved 4 81 that at least under some extra hypotheses which probably are fulfilled for almost all f 0 in nonpathological cases ad consists of analytic curves when t 0 if d t 2 0 is a weak solution all the above results except the last one generalize without much trouble to arbitrary n b 2 as noted also in 9 section 31 some variant of the last result probably also holds for n 2 with analytic curve replaced by real analytic hypersurface no results seem to have been published on the existence or uniqueness of weak solutions for time intervals of the kind t 0 t 0 with do d given i e backward solutions the above mentioned results on the analyticity of dd show or indicate that a backward solution

restriction operators balayage and doubly orthogonal systems of analytic functions abstract systems of analytic functions which are simultaneously orthogonal over each of two domains were apparently first studied in particular cases by walsh and szego and in full generality by bergman in principle these are very interesting objects allowing application to analytic continuation that is not restricted as weierstrassian continuation via power series either by circular geometry or considerations of locality however few explicit examples are known and in general one does not know even gross qualitative features of such systems the main contribution of the present paper is to prove qualitative results in a quite general situation it is by now very well known that the phenomenon of double orthogonality is not restricted to bergman spaces of analytic functions nor even indeed has it any intrinsic relation to analyticity its essence is an eigenvalue problem arising whenever one considers the operator of restriction on a hilbert space of functions on some set to a subset thereof provided this restriction is injective and compact however in this paper only hilbert spaces of analytic functions are considered especially bergman spaces in the case of the hardy spaces fisher and micchelli discovered remarkable qualitative features of doubly orthogonal systems and we have shown how based on the classical potential theoretic notion of balayage and its modern generalizations one can deduce analogous results in the bergman space set up but with restrictions imposed on the geometry of the considered domains and measures these were not needed in the fisher micchelli analysis but are necessary here as shown by examples from a more constructive point of view we study the bergman restriction operator between the unit disk and a compactly contained quadrature domain and show that the representing kernel of this operator is rational and it is expressible as an inversion followed by a logarithmic derivative in terms of the polynomial equation of the boundary of the inner domain

a class of quantizers with dc free quantization noise and optimal immunity to nonlinearity induced spurious tones fractional n phase locked loops plls typically use noise shaping coarse quantizers to control their instantaneous output frequency the resulting quantization noise and its running sum inevitably get distorted by non ideal analog components within the pll which induces undesirable spurious tones in the pll s output signal a recently proposed quantizer called a successive requantizer has been shown to mitigate this problem its quantization noise and the running sum of its quantization noise can be subjected to up to fifth order and third order nonlinear distortion respectively without inducing spurious tones this paper extends the previously published successive requantizer results to enable the design of successive requantizers whose quantization noise running sum sequences attain such immunity to nonlinearity induced spurious tones up to arbitrarily high orders of distortion the extended results are used to design example successive requantizers whose quantization noise and quantization noise running sum sequences have optimally reduced susceptibility to nonlinearity induced spurious tones

codes for zero spectral density at zero frequency corresp in pulse amplitude modulation pam digital transmission systems line encoding is used for shaping the spectrum of the encoded symbol sequence to suit the frequency characteristics of the transmission channel in particular it is often required that the encoded symbol sequence have a zero mean and spectral density vanishing at zero frequency we show that the finite running digital sum condition is a necessary and sufficient condition for this to occur the result holds in particular for alphabetic codes which are the most widely used line codes

weierstrass s criterion and compact solitary waves weierstrass s theory is a standard qualitative tool for single degree of freedom equations used in classical mechanics and in many textbooks in this brief report we show how a simple generalization of this tool makes it possible to identify some differential equations for which compact and even semicompact traveling solitary waves exist in the framework of continuum mechanics these differential equations correspond to bulk shear waves for a special class of constitutive laws

compact solitary waves in linearly elastic chains with non smooth on site potential it was recently observed by saccomandi and sgura that one dimensional chains with nonlinear elastic interaction and regular on site potential can support compact solitary waves i e travelling solitary waves with strictly compact support in this paper we show that the same applies to chains with linear elastic interaction and an on site potential which is continuous but non smooth at minima some different features arise in particular the speed of compact solitary waves is not uniquely fixed by the equation we also discuss several generalizations of our findings

a digital requantizer with shaped requantization noise that remains well behaved after nonlinear distortion a major problem in oversampling digital to analog converters and fractional n frequency synthesizers which are ubiquitous in modern communication systems is that the noise they introduce contains spurious tones the spurious tones are the result of digitally generated quantized signals passing through nonlinear analog components this paper presents a new method of digital requantization called successive requantization special cases of which avoids the spurious tone generation problem sufficient conditions are derived that ensure certain statistical properties of the quantization noise including the absence of spurious tones after nonlinear distortion a practical example is presented and shown to satisfy these conditions

tri hamiltonian duality between solitons and solitary wave solutions having compact support a simple scaling argument shows that most integrable evolutionary systems which are known to admit a bi hamiltonian structure are in fact governed by a compatible trio of hamiltonian structures we demonstrate how their recombination leads to integrable hierarchies endowed with nonlinear dispersion that supports compactons solitary wave solutions having compact support or cusped and or peaked solitons a general algorithm for effecting this duality between classical solitons and their nonsmooth counterparts is illustrated by the construction of dual versions of the modified korteweg de vries equation the nonlinear schr odinger equation the integrable boussinesq system used to model the two way propagation of shallow water waves and the ito system of coupled nonlinear wave equations these hierarchies include a remarkable variety of interesting integrable nonlinear differential equations textcopyright 1996 the american physical society

properties of some balayage operators with applications to quadrature domains and moving boundary problems in this paper we define and derive some properties of a fairly wide class of balayage operators we also apply our results to obtain geometric information about quadrature domains and solutions of certain moving boundary problems e g for hele shaw flows actually it was our interest in these applications that was the starting point for the present investigations and the balayage operators were developed rather as a tool our balayage operators are denoted f fp fp r and they depend on an open set r c i and a measure p in r actually we always assume that p has a density also denoted p in l r what f does is replace a given measure p by the nearest one in the energy norm v f p which satisfies v 5 p in r this presupposes that fi has finite energy if this is not the case there is another definition which is the one we will actually use among all distributions u in irn satisfying u i 17 the newtonian potential of p in irn and au i p in r there is a largest one u vp then f p ai by definition the operator of interest in the above mentioned applications is f with r irn p 1 the definition of f y when u has finite energy can be formulated as an elliptic variational inequality or linear complementarity problem and this is one of the standard tools when dealing with free boundary problems of the above kind see l 4 however we think that the balayage point of view is quite natural in our context the reason for considering the balayage operators for more general r and p than r rn p 1 is firstly that we need them in the proof of one of our theorems theorem 4 1 secondly and perhaps more important is that we think that the more general operators have quite a lot of intrinsic interest in particular as they turn out to contain classical balayage of positive measures as a special case namely with r bounded and p 0 recently we also learnt that discrete or numerical versions of these operators have been developed and applied in geophysics during the last three decades by dimiter zidarov who calls the balayage process partial gravi equivalent mass scattering see 5 and also for example 6 the paper is organized as follows in section 2 we define and establish the basic properties of our operators at the end we also explain their relation to quadrature domains and certain moving boundary problems section 3 is devoted to one single theorem theorem 3 1 and its corollaries these results concern the f with r lrn p 1 and give rather good geometric

compactlike discrete breathers in systems with nonlinear and nonlocal dispersive terms discrete breathers with purely anharmonic short range interaction potentials localize superexponentially becoming compactlike we analyze their spatial localization properties and their dynamical stability several branches of solutions are identified one of them connects to the well known page and sievers takeno lattice modes another one connects with the compacton solutions of rosenau the absence of linear dispersion allows for extremely long lived time quasiperiodic localized excitations adding long range anharmonic interactions leads to an extreme case of competition between length scales defining the spatial breather localization we show that short and long range interaction terms competition results in the appearance of several characteristic crossover lengths and essentially breaks the concept of compactness of the corresponding discrete breathers

applications of variational inequalities to the existence theorem on quadrature domains in this paper we shall study quadrature domains for the class of subharmonic functions by using the theory of variational inequalities we shall give a new proof of the existence and uniqueness theorem as an application we deal with hele shaw flows with a free boundary and show that their two weak solutions one of which was defined by the author using quadrature domains and the other was defined by gustafsson 3 using variational inequalities are identical with each other introduction in a previous paper 7 the author has defined the quadrature domains of positive measures for the class of subharmonic functions and studied their applications to complex function theory let v be a finite positive measure on the two dimensional euclidean space r2 let sl1 q be the class of subharmonic functions in an open set q which are integrable with respect to the two dimensional lebesgue measure m a nonempty open set q is called a quadrature domain of v for class sl1 if qi v is concentrated in q namely v qc 0 where qc denotes the complement of q qii jus s dv 1 a e in a connected open set w with finite area f 0 a e in wc and jf dm m w then q fm sl1 0 and there exists a minimum domain w in q fm sl1 namely q e q jm sl1 if and only if w c q and m q w o received by the editors october 5 1981 and in revised form february 23 1982 1980 mathematics subject classification primary 3 1aos 3 1bos secondary 35a15

discrete compactons some exact results in this letter we use the method of constructing exact solutions on lattices proposed by kinnersley and described in schmidt 1979 phys rev b 20 4397 to obtain exact compacton solutions in discrete models we examine the linear stability of such solutions both for the bright compacton and for the dark compacton cases we focus on a quantization condition that the width of the profile should satisfy we also use this quantization condition to examine the possibility of compact coherent structures travelling in discrete settings our results are obtained for sinusoidal profiles and then generalized to elliptic functions of arbitrary modulus the possibility of multi compacton solutions is considered

traveling wave solutions of the bbm like equations in this work we apply the factorization technique to the benjamin bona mahony like equations b m n in order to get traveling wave solutions we will focus on some special cases for which m n and we will obtain these solutions in terms of the special forms of weierstrass functions

a novel characteristic parameter approach for analysis and design of linear components in nonlinear systems nonlinearity always brings difficulties to the analysis and design of practical systems this study shows that for a wide class of nonlinear systems described by nonlinear differential equations the nonlinear output spectrum can be expressed as a polynomial function with respect to linear model parameters based on the recently developed concepts of parametric characteristic analysis and nonlinear characteristic output spectrum ncos function the coefficients of the new ncos function are independent of the linear parameters of interest and can be directly determined with proposed procedures this reveals an explicit and analytical relationship between the output spectrum of a nonlinear system and its linear components and therefore provides a convenient tool for the analysis and design of nonlinear systems using only linear components or linear feedback control while simultaneously fully considering inherent nonlinear influence this result is a useful complement to the ncos function based analysis and design recently developed in the literature the effectiveness and usefulness of the results are demonstrated through an application to the analysis of an amplifier with a multi tone input for suppressing intermodulation distortion it is shown that the new results reveal a straightforward insight into the parametric optimization of the amplifier for suppression of intermodulation distortion

the formation of the dead core in parabolic reaction diffusion problems on donne des conditions pour l existence d une region a concentration de reactant nulle pour des problemes paraboliques non lineaires de reaction diffusion

global solutions to bubble growth in porous media abstract we study a moving boundary problem modeling an injected fluid into another viscous fluid the viscous fluid is withdrawn at infinity and governed by darcy s law we present solutions to the free boundary problem in terms of time derivative of a generalized newtonian potentials of the characteristic function of the bubble this enables us to show that the bubble occupies the entire space as the time tends to infinity if and only if the internal generalized newtonian potential of the initial bubble is a quadratic polynomial howison 1985 7 and dibenedetto and friedman 1986 2 studied such behavior but for bounded bubbles we extend their results to unbounded bubbles

approximating a bandlimited function using very coarsely quantized data a family of stable sigma delta modulators of arbitrary order digital signal processing has revolutionized the storage and transmission of audio and video signals as well as still images in consumer electronics and in more scientific settings such as medical imaging the main advantage of digital signal processing is its robustness although all the operations have to be implemented with of necessity not quite ideal hardware the a priori knowledge that all correct outcomes must lie in a very restricted set of well separated numbers makes it possible to recover them by rounding off appropriately bursty errors can compromise this scenario as is the case in many communication channels as well as in memory storage devices making the perfect data unrecoverable by rounding off in this case knowledge of the type of expected contamination can be used to protect the data prior to transmission or storage by encoding them with error correcting codes this is done entirely in the digital domain these advantages have contributed to the present widespread use of digital signal processing many signals however are not digital but analog in nature audio signals for instance correspond to functions f t modeling rapid pressure oscillations which depend on the continuous time t i e t ranges over or an interval in and not over a discrete set and the range of f typically also fills an interval in f or this reason the first step in any digital processing of such signals must consist in a conversion of the analog signal to the digital world usually abbreviated as a d conversion for different types of signals different a d schemes are used in this paper we restrict our attention to a particular class of a d conversion schemes adapted to audio signals note that at the end of the chain after the signal has been processed stored retrieved transmitted all in digital form it needs to be reconverted to an analog signal that can be understood by a human hearing system we thus need a d a conversion there

a fundamental limitation of dc free quantization noise with respect to nonlinearity induced spurious tones fractional n phase locked loops plls are widely used to synthesize local oscillator signals for modulation and demodulation in communication systems such plls generate and subsequently lowpass filter dc free quantization noise as part of their normal operation unfortunately the quantization noise and its running sum inevitably are subjected to nonlinear distortion from analog circuit imperfections which causes spurious tones in the pll output signal that can degrade communication system performance this paper presents the first general mathematical analysis of this phenomenon it proves that if the running sum of the quantization noise t n satisfies tlow t n thigh for all n where tlow and thigh are integers then subjecting t n to kth order distortion for at least one k 1 2 3 thigh tlow will result in spurious tones for most fractional n pll output frequencies regardless of how the quantization is performed it also shows that quantizers exist which are optimal in the sense that subjecting the running sum of their quantization noise to th order distortion for any k 1 2 3 thigh tlow 1 does not result in any spurious tones in a typical fractional n pll the larger the range of t n the greater the power of the pll s phase noise so these results imply a fundamental tradeoff between phase noise power and spurious tones in plls

a variational inequality approach to hele shaw flow with a moving boundary this paper is concerned with the study of a mathematical model of the injection of fluid into a finite hele shaw cell the mathematical problem is one of solving laplace s equation in an unknown region whose boundary changes with time by a transformation of the dependent variable an elliptic variational inequality formulation of the moving boundary problem is obtained the variational inequality is shown to have a unique solution up to the time at which the cell is filled regularity results for the solution of the inequality are obtained by studying a penalty approximation of the inequality

super exponential decay and holomorphic extensions for semilinear equations with polynomial coefficients abstract we show that all eigenfunctions of linear partial differential operators in r n with polynomial coefficients of shubin type are extended to entire functions in c n of finite exponential type 2 and decay like exp z 2 for z in conic neighbourhoods of the form im z re z we also show that under semilinear polynomial perturbations all nonzero homoclinics keep the super exponential decay of the above type whereas a loss of the holomorphicity occurs namely we show holomorphic extension into a strip z c n im z t for some t 0 the proofs are based on geometrical and perturbative methods in gelfand shilov spaces the results apply in particular to semilinear schrodinger equations of the form u x 2 u u f x u u our estimates on homoclinics are sharp in fact we exhibit examples of solutions of with super exponential decay which are meromorphic functions the key point of our argument being the celebrated great picard theorem in complex analysis

dithering and its effects on sigma delta and multistage sigma delta modulation the spectrum of the quantization error in a dithered sigma delta modulator is derived under the constraint that the dithering signal does not cause overload the results apply to dc sinusoidal and more general quasi stationary signals it is shown in the case of a simple sigma delta modulation that no overload dithering can smooth the error spectrum and can make the quantization error asymptotically uncorrelated with the input it does not however make the quantization error white in the case of multistage sigma delta modulation with the appropriate dithering the quantization error becomes white even for a system with only two stages the signal to quantization noise ratio sqnr is derived for sigma delta and multistage sigma delta oversampled analog to digital conversion with additive dithering simulation results are presented to support the theoretical analysis