proof of a conjecture of polya on the zeros of successive derivatives of real entire functions we prove polya s conjecture of 1943 for a real entire function of order greater than 2 with finitely many non real zeros the number of non real zeros of the nth derivative tends to infinity as n to infty we use the saddle point method and potential theory combined with the theory of analytic functions with positive imaginary part in the upper half plane

strong traces for solutions of multidimensional scalar conservation laws in this paper we consider multidimensional scalar conservation laws without bv estimates defined in a subset d we show that with a non degeneracy hypothesis on the flux we can define a strong notion of trace at the boundary of reached by l 1 convergence

structure of entropy solutions for multi dimensional scalar conservation laws an entropy solution u of a multi dimensional scalar conservation law is not necessarily in bv even if the conservation law is genuinely nonlinear we show that u nevertheless has the structure of a bv function in the sense that the shock location is codimension one rectifiable this result highlights the regularizing effect of genuine nonlinearity in a qualitative way it is based on the locally finite rate of entropy dissipation the proof relies on the geometric classification of blow ups in the framework of the kinetic formulation

velocity averaging kinetic formulations and regularizing effects in quasi linear pdes we prove in this paper new velocity averaging results for second order multidimensional equations of the general form l x v f x v g x v where l x v a v x b v x these results quantify the sobolev regularity of the averages v f x v v dv in terms of the nondegeneracy of the set

multi dimensional burgers equation with unbounded initial data well posedness and dispersive estimates the cauchy problem for a scalar conservation law admits a unique entropy solution when the datum u 0 is a bounded measurable function kru hkov the semi group s t t geqq 0 is contracting in the l 1 distance for the multi dimensional burgers equation we show that s t t geqq 0 extends uniquely as a continuous semi group over l p mathbb r n whenever 1 leqq p 0 we prove that s t maps l p mathbb r n into l q mathbb r n these results are based upon new dispersive estimates the ingredients are on the one hand compensated integrability and on the other hand a degiorgi type iteration

non analyticity of the callan symanzik β function of two dimensional o n models we discuss the analytic properties of the callan symanzik function g associated with the zero momentum four point coupling g in the two dimensional 4 model with o n symmetry using renormalization group arguments we derive the asymptotic behaviour of g at the fixed point g we argue that g g o g g 1 7 for n 1 and g g o 1 log g g for n 3 our claim is supported by an explicit calculation in the ising lattice model and by a 1 n calculation for the two dimensional 4 theory we discuss how these non analytic corrections may give rise to a slow convergence of the perturbative expansion in powers of g

drift diffusion equations with fractional diffusion and the quasi geostrophic equation motivated by the critical dissipative quasi geostrophic equation we prove that drift diffusion equations with l 2 initial data and minimal assumptions on the drift are locally holder continuous as an application we show that solutions of the quasi geostrophic equation with initial l 2 data and critical diffusion 1 2 are locally smooth for any space dimension

reality of the zeros of an entire function and its derivatives in 1914 p6lya raised the problem of classifying the entire functions which together with all their derivatives have only real zeros in earlier work hellerstein and williamson settled this problem for entire functions which are real on the real axis we complete the classification in all cases and show that it is sufficient to consider the function and its first two derivatives introduction in 12 and 13 g polya posed the problem of classifying all entire functions which have along with all their derivatives only real zeros he conjectured that such a function must have one of the following forms 0 1 f z aebz 0 2 f z a eicz eid 0 3 f z azme az jii l1 ez an where a and b are eomplex constants a 0 b c d and the an are real i l a2 x and m is a nonnegative integer in 10 b ja levin and i v ostrovskii showed that if f is a real i e real on the real axis entire function with only real zeros satisfying the growth condition 0 4 limsup loglog m r f rlog r x then the zeros cn of f satisfy 0 5 2 i m llcn 0oc in particular f has infinitely many nonreal zeros polya s conjecture was completely affirmed for real entire functions by two of the present authors in 6 and 7 where the cases of functions of finite order and of infinite order growth slower than that of 0 4 were successfully treated these results coupled with those of levin and ostrovskii showed that if f f and f have only real zeros and f is real entire then f is in the laguerre p6lya class i e of the form 0 3 received by the editors august 14 1981 and in revised form december 31 1981 1980 mathematics subject classification primary 30d30 secondary 30d35 partially supported by nsf grant mcs 8003108 2part of research towards ph d degree university of wisconsin 1983 american mathematical society 0002 9947 82 0000 0099 04 00

on the number of nonreal zeros of real entire functions and the fourier polya conjecture this paper is concerned with a general theorem on the number of nonreal zeros of transcendental functions j fourier formulated the theorem in his work analyse des equations determineesin 1831 but he did not give a proof roughly speaking the theorem states that if a real entire function f x can be expressed as a product of linear factors then we can count the nonreal zeros of f x by observing the behavior of the derivatives of f x on the real axis alone as we shall see in the sequel this theorem completely justifies his former argument by which he tried to prove that the function j0 2 x has only real zeros it seems that no complete proof of the theorem is known and no general theorem has been published that justifies the argument later in 1930 g polya published a paper entitled some problems connected with fourier s work on transcendental equations p3 in this paper polya conjectured two hypothetical theorems that are closely related to fourier s unproved theorem in fact he conjectured three but he proved that two of them are equivalent to each other the first hypothetical theorem is a modernized formulation of the theorem and it justifies fourier s argument completely the second conjecture was proved in 1990 but it is impossible to justify the argument using the conjecture alone in the present paper we prove polya s formulation of the theorem his first conjecture as well as its extensions give a very simple and direct proof of the second conjecture mentioned above and exhibit some applications of the results in particular we completely justify fourier s argument by our general theorems acknowledgments professor fefferman has encouraged and helped us to publish this paper the authors truly thank him for this the authors also thank professor csordas for his kind interest in the results and his valuable suggestions on the proof of the polya wiman conjecture 1 historical introduction in this section we briefly explain our results as well as their background a real entire function is an entire function that assumes only real values on the real axis fourier s unproven theorem asserts that we can know the number of nonreal zeros of such real entire functions by counting their critical points which are defined as follows let f x be a real analytic function defined in an open

on the structure of l infty entropy solutions to scalar conservation laws in one space dimension we prove that if u is the entropy solution to a scalar conservation law in one space dimension then the entropy dissipation is a measure concentrated on countably many lipschitz curves this result is a consequence of a detailed analysis of the structure of the characteristics in particular the characteristic curves are segments outside a countably 1 rectifiable set and the left and right traces of the solution exist in a c 0 sense up to the degeneracy due to the segments where f 0 we prove also that the initial data is taken in a suitably strong sense and we give some examples which show that these results are sharp

decay of entropy solutions of nonlinear conservation laws we are concerned with the asymptotic behavior of entropy solutions of nonlinear conservation laws the main objective of this paper is to present an analytical approach and to explore its applications to studying the large time behavior of periodic entropy solutions of hyperbolic conservation laws the asymptotic decay of periodic solutions of nonlinear hyperbolic conservation laws is an important nonlinear phenomenon it is observed that the genuine nonlinearity of equations forces the waves of each characteristic family to interact vigorously and to cancel each other the insightful analysis of glimm lax gl for scalar equations and 2 2 systems has indicated that the resultant mutual cancellation of interacting shock and rarefaction waves of the same family induces the decay of periodic solutions such a result was first shown by lax l1 in 1957 for one dimensional convex scalar conservation laws dafermos d1 applying his uniform processes proved the decay result for the case that the set of inflection points of the flux does not have an accumulation point the glimm lax theory gl indicates that for 2 2 strictly hyperbolic and genuinely nonlinear systems any periodic glimm solution decays like o 1 t this result was proved by using the approximate characteristic method in the glimm difference solutions provided that the oscillation of the corresponding initial data is small recently using the method of generalized characteristics dafermos d3 showed that any periodic solution with local bounded variation and small oscillation for 2 2 systems decays asymptotically with the same detailed structure pas found by lax l1 for the

duality symmetry strong coupling expansion and universal critical amplitudes in two dimensional phi 4 field models we show that the exact beta function g in the continuous two dimensional g4 model possesses the kramers wannier duality symmetry the duality symmetry transformation d g such that d g d g g is constructed and the approximate values of g computed from the duality equation d g g are shown to agree with the available numerical results the calculation of the beta function g for the 2d scalar g4 field theory based on the strong coupling expansion is developed and the expansion of g in powers of g 1 is obtained up to order g 8 the numerical values calculated for the renormalized coupling constant g are in reasonably good agreement with the best modern estimates recently obtained from the high temperature series expansion and with those known from the perturbative four loop renormalization group calculations the application of cardy s theorem for calculating the renormalized isothermal coupling constant gc of the 2d ising model and the related universal critical amplitudes is also discussed

universal surface tension and critical isotherm amplitude ratios in three dimensions the universal surface tension and critical isotherm amplitude ratios are studied numerically for three dimensional ising models modern estimates of the critical temperature and exponents allow reliable evaluation of the critical surface tension amplitude k using recent monte carlo data for the simple cubic lattice likewise the amplitudes cc for the susceptibility and 1c for the second moment correlation length on the critical isotherm have been re estimated using existing series expansions the method of inhomogeneous differential approximants also yields a direct estimate of the correction to scaling exponent c on the critical isotherm which via scaling corresponds to the thermal correction exponent 0 55 5 this supports previous estimates and the stronger conclusion 0 54 3 for the universal ratios we estimate k 1 2 0 0965 2 cc b 1c 1 0 93 25 and c cc 1c 1 2 1 17 2 where b 1 and c are the amplitudes of the spontaneous magnetization and second moment correlation length and of the susceptibility above tc

on the zeros of the derivatives of real entire functions and wiman s conjecture let f be an entire function of finite order real on the real axis and possessing only real zeros a classical problem proposed by g p6lya 9 10 and a wiman 1 is to determine from the hadamard canonical representation of the function the number of nonreal zeros of the higher derivatives of f in this paper we shall prove the following conjecture of wiman which has been open since 1915

regularity of the moments of the solution of a transport equation let u u x v satisfy the transport equation u v xu f x rn v rnwhere f belongs to some space of type lp dx d v where is a positive bounded measure on rn we study the resulting regularity of the moment u x v d v in terms of fractional sobolev spaces for example counter examples are given in order to test the optimality of our results