a positive mass theorem in three dimensional cauchy riemann geometry in this note we summarize the results from 6 on the positive mass problem in 3 dimensional cr cauchy riemann geometry

c_0 c0 positivity and a classification of closed three dimensional cr torsion solitons a closed cr 3 manifold is said to have c 0 positive pseudohermitian curvature if w c 0 tor x x 0 for any 0 ne x in t 1 0 m we discover an obstruction for a closed cr 3 manifold to possess c 0 positive pseudohermitian curvature we classify closed three dimensional cr yamabe solitons according to c 0 positivity for c 0 1 and the potential function lies in the kernel of paneitz operator moreover we show that any closed three dimensional cr torsion soliton must be the standard sasakian space form at last we discuss the persistence of c 0 positivity along the cr torsion flow starting from a pseudo einstein contact form

pseudo einstein structures on cr manifolds etant donne une variete cr compacte non degeneree on cherche une structure pseudohermitienne associee pour laquelle le tenseur de ricci pseudohermitien est un multiple scalaire de la forme de levi

on the cr analogue of obata s theorem in a pseudohermitian 3 manifold in this paper we first prove the cr analogue of obata s theorem on a closed pseudohermitian 3 manifold with zero pseudohermitian torsion secondly instead of zero torsion we have the cr analogue of li yau s eigenvalue estimate on the lower bound estimate of positive first eigenvalue of the sub laplacian in a closed pseudohermitian 3 manifold with nonnegative cr paneitz operator p0 finally we have a criterion for the positivity of first eigenvalue of the sub laplacian on a complete noncompact pseudohermitian 3 manifold with nonnegative cr paneitz operator the key step is a discovery of integral cr analogue of bochner formula which involving the cr paneitz operator

periodic solutions of the intermediate long wave equation a nonlinear superposition principle periodic stationary wave solutions of the intermediate long wave ilw equation are derived using the bilinear transformation method and a new expression for the dispersion relation is obtained the class of physically important real valued solutions is identified these solutions may be represented as an infinite superposition of solitary wave profiles a property shared by the related korteweg de vries kdv and benjamin ono bo equation this nonlinear superposition principle which has been the subject of various interpretations in the literature is discussed the ilw periodic solution approximates to a sinusoidal wave and a solitary wave in the limits of small and large amplitudes respectively for intermediate amplitudes the solution can be well approximated by either a sine wave or solitary wave in the shallow water kdv limit the ilw periodic solution leads to the familiar cnoidal wave whereas the deep water bo limit yields benjamin s periodic wave a previously unknown expression for the cnoidal wave dispersion relation in terms of theta functions is obtained

smooth solutions of degenerate laplacians on strictly pseudoconvex domains on considere la caracterisation des solutions lisses globales de u 0 sur un domaine general strictement pseudoconvexe ou est le laplacien pour la metrique avec forme de kakler i 2 log 1

differential harnack estimates for time dependent heat equations with potentials in a closed spherical cr 3 manifold abstract in this paper we derive differential harnack estimates for time dependent heat equations with potentials under the cr yamabe flow in a closed spherical cr 3 manifold with positive tanaka webster curvature and vanishing torsion it can be served as a generalization of li yau hamilton inequality for the cr yamabe flow the differential harnack estimate will play an important role in the study of the cr yamabe flow

the special modular transformation for polycnoidal waves of the korteweg de vries equation the modular transformation of the riemann theta function is used to show that the implicit dispersion relation for the n polycnoidal waves of the korteweg de vries equation has a countable infinity of branches for n 2 although the transformation also implies that each branch or mode can be written in a countable infinity of ways it is also shown that there is a unique physical representation for each mode such that the parameters of the theta function can be interpreted as wavenumbers and amplitudes in the limit of either very small or very large amplitude unfortunately the small amplitude physical representation is different by a modular transformation from the large amplitude physical representation for a given mode but this difference explains an apparent paradox as described in the text the general modular transformation expresses the theta function in terms of complex wavenumbers phase speeds and coordinates that have no physical relevance to the korteweg de vries equation but it is shown that for n 2 there is a subgroup here dubbed the special modular transformation which gives a real result this subgroup is explicitly constructed for general n and presented as a table for n 2

theta functions gaussian series and spatially periodic solutions of the korteweg de vries equation it has been shown by novikov funct anal appl 8 236 1974 dubrovin et al russian math surveys 31 59 1976 lax commun pure appl math 28 141 1975 mckean and van moerbeke inv math 30 217 1975 and others that the nonlinear evolution equations which admit solitary waves also have spatially periodic exact solutions polycnoidal waves which can be expressed in terms of multidimensional riemann theta functions here it is shown that via poisson summation the fourier series that define the theta functions can be transformed into an infinite series of gaussian functions because the lowest terms of the gaussian series generate the usual solitary waves it is possible to intimately explore the relationship between solitary waves and these spatially periodic polycnoidal waves also by using the gaussian series one can perturbatively calculate phase velocities and wave structure for the polycnoidal wave even in the strongly nonlinear regime for which the soliton or mult

codimension three nonnegatively curved submanifolds with infinite fundamental group we show that a complete codimension three submanifold m of nonnegative sectional curvature that isometrically splits as bar m times mathbb r has nonnegative curvature operator we apply this result to obtain a classification of codimension three nonflat manifolds of nonnegative sectional curvature and infinite fundamental group

on the periodic solution of the burgers equation a unified approach for the most part the various known expressions for the periodic solution to the burgers equation have been derived by a mixture of ad hoc argument and approximation methods these include the well known solutions due to r d fay and d f parker remarkably these solutions were obtained without recourse to the celebrated hopf cole transformation which linearizes the burgers equation to the heat conduction equation we present here an alternative approach to the periodic problem which uses the fact that the riemann theta functions solve the burgers equation via the hopf cole transformation and thereby provide a unifying framework in which all previous representations of the periodic solution find their natural place yet another expression for the periodic solution in the form of an amplitude modulated sine wave is given and would appear to be new the limiting forms of the periodic solution as t rightarrow 0 embryonic sawtooth profile and t rightarrow infty old age sinusoidal profile are examined in detail the representation due to d f parker which expresses the periodic wave as a superposition of spreading taylor shock profiles is discussed within the context of a nonlinear superposition principle the composite asymptotic expansion due to m b lesser and d g crighton is examined in the light of our results the curious behaviour of this expansion in the embryonic region t ll 1 whereby a first order approximation yields an exact periodic solution the fay solution is explained

the sharp lower bound of the first eigenvalue of the sub laplacian on a quaternionic contact manifold in dimension seven abstract a version of lichnerowicz theorem giving a lower bound of the eigenvalues of the sub laplacian on a compact seven dimensional quaternionic contact manifold is proved assuming a lower bound on the s p 1 s p 1 components of the qc ricci curvature and the positivity of the p function of any eigenfunction the introduced p function and nonlinear c operator are motivated by the paneitz operators defined previously in the riemannian and cr settings and the p function used in the theory of elliptic partial differential equations it is shown that in the case of a seven dimensional compact 3 sasakian manifold the lower bound is reached iff the quaternionic contact manifold is the round 3 sasakian sphere

nonnegativity of the curvature operator and isotropy for isometric immersions positivity of curvature plays a crucial role in riemannian geometry however such positivity conditions can be expressed in different ways of varying strength in the more qualitative parts of riemannian geometry the sectional curvature k and its sign is prevailing the reason is that k enters the fundamental formulas for the geodesic variation in a very direct manner for other questions the curvature operator p and its definiteness is more dominant for example the positive semidefiniteness p 0 is the only known universal condition ensuring the nonnegativity of the general gauss bonnet integrand while k 0 is merely sufficient for dimensions less than 6 moreover for noncompact complete manifolds the gauss bonnet integral and the euler characteristic if existing at all do in general not coincide but their difference is in some instances manageable if the curvature operator is positive semidefinite see theorem 2 f below it is well known that p 0 implies k 0 the converse being true only for dimensions less than 4 so a natural question is what conditions have to be added to conclude from k 0 that p 0 some known results in the realm of intrinsic geometry are listed in sect 2 now if a riemannian manifold m of dimension m is isometrically immersed in another riemannian manifold of dimension rh we may conjecture that for small codimensions p n m there are weaker conditions that in the abstract case to ensure the positive semidefiniteness of p the purpose of the present paper is to initiate some answers in this direction for hypersurfaces p 1 of euclidean space p 0 is directly implied by k 0 since p equals the exterior square of the weingarten map for submanifolds of euclidean space with codimension p 2 weinstein 31 deduced p 0 from k 0 in sect 3 we generalize this result to the semidefinite case and to nonflat ambient spaces also an application to the gauss bonnet formula is given however there is no hope to go beyond p 2 in this manner as is shown by a counterexample at the end of sect 3 for higher codimensions an attack is then made in sect 4

the double cnoidal wave of the korteweg de vries equation an overview earlier work of the author on the spatially periodic solutions of the korteweg de vries equation is here extended via an in depth treatment of a special case the double cnoidal wave is the simplest generalization of the ordinary cnoidal wave discovered by korteweg and de vries in 1895 in the limit of small amplitude the double cnoidal wave is the sum of two noninteracting linear sine waves in the oppositie limit of large amplitude it is the sum of solitary waves of two different heights repeated periodically over all space although special the double cnoidal wave is important because it is but the particular case n 2 of a broad family of solutions known variously as n polycnoidal waves finite gap finite zone solutions waves on a circle or n phase wave trains it has been shown by others that the set of n polycnoidal waves gives the general initial value solution to the korteweg de vries equation this present work is the core of a three part treatment of the double cnoidal wave this part the overview presents graphic examples in all the important parameter regimes explains how collision phase shifts alter the average speed of the two wave phases from the free velocities of the two solitary waves describes the different branches or modes of the double cnoidal wave it is possible to have many solitary waves on each spatial period provided they are of only two distinct sizes and contrasts the results of this work with the very limited numerical calculations of previous authors the second part describes how the problem of numerically calculating the double cnoidal wave can be reduced down to solving four algebraic equations by perturbation theory the third part explains how the so called modular transformation of the riemann theta functions is important in interpreting n polycnoidal waves

double cnoidal waves of the korteweg de vries equation a boundary value approach double cnoidal waves of the korteweg de vries equation are studied by direct solution of the nonlinear boundary value problem these double cnoidal waves which are the spatially periodic generalization of the well known double soliton are exact solutions with two independent phase speeds the equation is written in terms of two phase variables and expanded in two dimensional fourier series the small amplitude solution is obtained via the stokes perturbation expansion this solution is numerically extended to larger amplitude by employing a newton kantorovich continuation in amplitude galerkin algorithm the crests of the finite amplitude solution closely match the sech 2 solitary wave form and the three cases of solitary wave interaction described by lax are identified for the double cnoidal waves this simple approach reproduces specific features such as phase shift upon collision distinction between instantaneous and average phase speeds and a paradox of wavenumbers

construction of lighthill s unitary functions the imbricate series of unity the poisson sum theorem implies that every spatially periodic function f x can be written as a so called imbricate series that is as the superposition of an infinite number of evenly space copies of a pattern function a x the pattern is the fourier transform of a function a n that specifies the usual fourier coefficients because there are many a n which coincide with a given set of fourier coefficients aj when n is an integer the pattern function a x is not unique in this work we show how to construct pattern functions of the form a x f x w x where f x is the periodic function itself the imbrication of a x and where w x is what lighthill called a unitary function we derive an explicit formula to construct all possible unitary functions and tabulate eight examples we also prove that all unitary functions are pattern functions of unity that is for the imbricate series that sums to f x 1 x

collapsing of riemannian manifolds and eigenvalues of laplace operator on etudie la perturbation des valeurs propres de l operateur de laplace sous des deformations des varietes qui ne conservent pas les dimensions

the fefferman metric and pseudo hermitian invariants c fefferman has shown that a real strictly pseudoconvex hypersurface in complex n space carries a natural conformai lorentz metric on a circle bundle over the manifold this paper presents two intrinsic constructions of the metric valid on an abstract cr manifold one is in terms of tautologous differential forms on a natural circle bundle the other is in terms of webster s pseudohermitian invariants these results are applied to compute the connection and curvature forms of the fefferman metric explicitly

numerical solutions of the korteweg di vries equation using the periodic scattering transform m representation abstract we develop a numerical procedure based upon the periodic scattering transform for determining the spatial and temporal evolution of solutions to the korteweg de vries kdv equation with periodic boundary conditions in this approach solutions to kdv are found in terms of the representation i e the wave amplitude is given by a trace formula consisting of a linear superposition of hyperelliptic functions j x t 1 j n which are governed by n nonlinearly coupled ordinary differential equations our approach consists of the numerical integration of these equations using a second order taylor series expansion with variable integration step size it is well known that the j are generalizations of elliptic functions to the hyperelliptic class nevertheless we find that the j x t bear little or no resemblance to elliptic functions except in the small amplitude linear limit or for the case of a single soliton we study in particular the most difficult numerical case which occurs for the n soliton wave train solution of the periodic kdv equation

the sharp upper bounds for the first positive eigenvalue of the kohn laplacian on compact strictly pseudoconvex hypersurfaces we give sharp and explicit upper bounds for the first positive eigenvalue lambda 1 box b of the kohn laplacian on compact strictly pseudoconvex hypersurfaces in mathbb c n 1 in terms of their defining functions as an application we show that in the family of real ellipsoids lambda 1 box b has a unique maximum value at the cr sphere

on the cr obata theorem and some extremal problems associated to pseudoscalar curvature on the real ellipsoids in mathbb c n 1 this paper studies the cr version of obata theorem on a pseudo hermitian cr manifold m the main result of the paper is proving that cr obata theorem holds on real ellipsoid e a with contact form 1 2i a a where a z z 2 re n j 1 a j z 2 j 1 with a j 1 1

solitons cnoidal waves and nonlinear interactions in shallow water ocean surface waves abstract we analyze shallow water surface wave data from the adriatic sea using a nonlinear generalization of fourier analysis based upon the periodic inverse scattering transform in the function representation for the korteweg de vries kdv equation while linear fourier analysis superposes sine waves the nonlinear fourier approach superposes cnoidal waves the travelling wave solution to kdv plus their mutual nonlinear interactions a new procedure is presented for the nonlinear low pass and band pass filtering of measured wave trains we apply the approach to a measured time series and discuss the dynamics of solitons and the physics of the nonlinear interactions in terms of global spatio temporal phase shifts amongst the cnoidal waves

semiclassical quantization of the periodic toda chain gutzwiller s semiclassical quantization scheme for the trace of green s function is applied to the periodic toda chain we obtain a set of algebraic equations that determine the energy levels arising from a special periodic orbit namely the single cnoidal wave solution our formulae show a simple dependence on the number of particles n in the chain n merely occurs as a parameter we perform the soliton limit of our equations and get a semiclassical correction to first order in h to the dispersion relation e e p of a soliton on the infinite chain which is in remarkable agreement with the bethe ansatz result the classical data which enter into the semiclassical quantization formula are of interest in their own right we give a complete treatment of the linear stability analysis of a single cnoidal wave and also some new expressions for its dispersion relation which expresses the frequency v as a function of the wavenumber k

perturbation series for the double cnoidal wave of the korteweg de vries equation by means of the theorems proved earlier by the author the problem of the double cnoidal wave of the korteweg de vries equation is reduced to four algebraic equations in four unknowns two of the unknowns are the nonlinear phase speeds c1 and c2 another is a physically irrelevant integration constant the fourth unknown is the off diagonal element of the symmetric 2 2 theta matrix which in turn gives the explicit coefficients of the riemann theta function the double cnoidal wave u x t is then obtained by taking the second x derivative of the logarithm of the theta function two separate forms of these four nonlinear residual equations are given one is obtained from the fourier series of the theta function and is useful for small wave amplitude the other is based on the gaussian series of the theta function and is highly efficient in the large amplitude regime where the double cnoidal wave is the sum of two solitary waves both sets of residual equations can be solved via perturbation theory and

li yau gradient estimate and entropy formulae for the cr heat equation in a closed pseudohermitian 3 manifold in this paper we derive two subgradient estimates of the cr heat equation in a closed pseudohermitian 3 manifold which are served as the cr version of the li yau gradient estimate with its applications we first get a subgradient estimate of the logarithm of the positive solution of the cr heat equation secondly we have the harnack inequality and upper bound estimate for the heat kernel finally we obtain perelman type entropy formulae for the cr heat equation

new series for the cosine lemniscate function and the polynomialization of the lemniscate integral we discuss the numerical computation of the cosine lemniscate function and its inverse the lemniscate integral these were previously studied by bernoulli euler gauss abel jacobi and ramanujan we review general elliptic formulas for this special case and provide some novelties we show that a fourier series by ramanujan converges twice as fast as the standard elliptic cosine fourier series specialized to this case the so called imbricate series however converges geometrically fast over the entire complex plane we derive two new expansions the rational plus fourier series converges much faster than ramanujan s for real z each term is asymptotically 12 400 times smaller than its immediate predecessor coslem z 4b q 1 q cos bz 1 q 2 4qcos 2 bz n 1 q n 1 2 1 1 q 2 n 1 1 cos 2n 1 bz where q exp p is the elliptic nome k 1 85 is the complete elliptic integral of the first kind for a modulus m 1 2 and b p k2 the rational imbricate series is uniformly valid over the complex plane but converges twice as fast as the sech imbricate series coslem z 4bq 1 q j q 1 q cos bz 1 q 2 4qcos 2 b z jpi where p 4 2 k is the period in both the real and imaginary directions we devise a new approximation for the lemniscate integral for real argument as the arccosine of a chebyshev series and show that 17 terms yield about 15 digits of accuracy for complex argument we show that the lemniscate integral can be found to near machine precision assumed as sixteen decimal digits by computing the roots of a polynomial of degree thirteen alternatively newton s iteration converges in three iterations with an initialization accurate to four decimal places that is the chosen root of a cubic equation