pair correlation densities of inhomogeneous quadratic forms under explicit diophantine conditions on 2 w eprove that the local two point correlations of the sequence given by the values m 2 n 2 with m n 2 are those of a poisson process this partly confirms a conjecture of berry and tabor 2 on spectral statistics of quantized integrable systems and also establishes a particular case of the quantitative version of the oppenheim conjecture for inhomogeneous quadratic forms of signature 2 2 the proof uses theta sums and ratner s classification of measures invariant under unipotent flows

free path lengths in quasicrystals previous studies of kinetic transport in the lorentz gas have been limited to cases where the scatterers are distributed at random e g at the points of a spatial poisson process or at the vertices of a euclidean lattice in the present paper we investigate quasicrystalline scatterer configurations which are non periodic yet strongly correlated a famous example is the vertex set of a penrose tiling our main result proves the existence of a limit distribution for the free path length which answers a question of wennberg the limit distribution is characterised by a certain random variable on the space of higher dimensional lattices and is distinctly different from the exponential distribution observed for random scatterer configurations the key ingredients in the proofs are equidistribution theorems on homogeneous spaces which follow from ratner s measure classification

the riemann problem for the leray burgers equation abstract for riemann data consisting of a single decreasing jump we find that the leray regularization captures the correct shock solution of the inviscid burgers equation however for riemann data consisting of a single increasing jump the leray regularization captures an unphysical shock this behavior can be remedied by considering the behavior of the leray regularization with initial data consisting of an arbitrary mollification of the riemann data as we show for this case the leray regularization captures the correct rarefaction solution of the inviscid burgers equation additionally we prove the existence and uniqueness of solutions of the leray regularized equation for a large class of discontinuous initial data all of our results make extensive use of a reformulation of the leray regularized equation in the lagrangian reference frame the results indicate that the regularization works by bending the characteristics of the inviscid burgers equation and thereby preventing their finite time crossing

a distribution free procedure for comparing receiver operating characteristic curves from a paired experiment summary a distribution free permutation test procedure is proposed for comparing receiver operating characteristic curves based on continuous data from a paired design the method tests the hypothesis that the two curves are identical for all operating points unlike previously proposed methods which test the equivalence of the areas under the curves the new test is shown by simulation to have very similar operating characteristics to the standard method based on comparisons of the areas when the curves are parallel but markedly superior power when the curves cross that is when the curves are different but have similar areas the prospects of generalising the approach to unpaired experiments and to comparisons of ordinal rating data are discussed

the quantitative behaviour of polynomial orbits on nilmanifolds a theorem of leibman asserts that a polynomial orbit g n g n z on a nilmanifold g g is always equidistributed in a union of closed sub nilmanifolds of g g in this paper we give a quantitative version of leibman s result describing the uniform distribution properties of a finite polynomial orbit g n g n n in a nilmanifold more specifically we show that there is a factorisation g eg where e n is smooth n g n z is periodic and rational and g n g n p is uniformly distributed up to a specified error d inside some subnilmanifold g g of g g for all sufficiently dense arithmetic progressions p n our bounds are uniform in n and are polynomial in the error tolerance d in a companion paper we shall use this theorem to establish the mobius and nilsequences conjecture from an earlier paper of ours

linear equations in primes consider a system of nonconstant affine linear forms 1 t d no two of which are linearly dependent let n be a large integer and let k n n d be convex a generalisation of a famous and difficult open conjecture of hardy and littlewood predicts an asymptotic as n for the number of integer points n d k for which the integers 1 n t n are simultaneously prime this implies many other well known conjectures such as the twin prime conjecture and the weak goldbach conjecture it also allows one to count the number of solutions in a convex range to any simultaneous linear system of equations in which all unknowns are required to be prime in this paper we conditionally verify this asymptotic under the assumption that no two of the affine linear forms 1 t are affinely related this excludes the important binary cases such as the twin prime or goldbach conjectures but does allow one to count nondegenerate configurations such as arithmetic progressions our result assumes two families of conjectures which we term the inverse gowers norm conjecture gi s and the mobius and nilsequences conjecture mn s where s 1 2 is the complexity of the system and measures the extent to which the forms i depend on each other the case s 0 is somewhat degenerate and follows from the prime number theorem in aps roughly speaking the inverse gowers norm conjecture gi s asserts the gowers u s 1 norm of a function f n 1 1 is large if and only if f correlates with an s step nilsequence while the mobius and nilsequences conjecture mn s asserts that the mobius function is strongly asymptotically orthogonal to s step nilsequences of a fixed complexity these conjectures have long been known to be true for s 1 essentially by work of hardy littlewood and vinogradov and were established for s 2 in two papers of the authors thus our results in the case of complexity s 2 are unconditional in particular we can obtain the expected asymptotics for the number of 4 term progressions p 1 p 2 p 3 p 4 n of primes and more generally for any nondegenerate problem involving two linear equations in four prime unknowns

effective equidistribution for closed orbits of semisimple groups on homogeneous spaces we prove effective equidistribution with polynomial rate for large closed orbits of semisimple groups on homogeneous spaces under certain technical restrictions notably the acting group should have finite centralizer in the ambient group the proofs make extensive use of spectral gaps and also of a closing lemma for such actions

confidence intervals for a binomial proportion thirteen methods for computing binomial confidence intervals are compared based on their coverage properties widths and errors relative to exact limits the use of the standard textbook method x n 1 96 square root of x n 1 x n n or its continuity corrected version is strongly discouraged a commonly cited rule of thumb stating that alternatives to exact methods may be used when the estimated proportion p is such that np and n 1 p both exceed 5 does not ensure adequate accuracy score limits are easily calculated from closed form solutions to quadratic equations and can be used at all times based on coverage functions the continuity corrected score method is recommended over exact methods its conservative nature should be kept in mind as should the wider fluctuation of actual coverage that accompanies omission of the continuity correction

comparing diagnostic tests a simple graphic using likelihood ratios the diagnostic abilities of two or more diagnostic tests are traditionally compared by their respective sensitivities and specificities either separately or using a summary of them such as youden s index several authors have argued that the likelihood ratios provide a more appropriate if in practice a less intuitive comparison we present a simple graphic which incorporates all these measures and admits easily interpreted comparison of two or more diagnostic tests we show using likelihood ratios and this graphic that a test can be superior to a competitor in terms of predictive values while having either sensitivity or specificity smaller a decision theoretic basis for the interpretation of the graph is given by relating it to the tent graph of hilden and glasziou statistics in medicine 1996 finally a brief example comparing two serodiagnostic tests for lyme disease is presented published in 2000 by john wiley sons ltd

on a regularization of the compressible euler equations for an isothermal gas abstract we consider a leray type regularization of the compressible euler equations for an isothermal gas the regularized system depends on a small parameter 0 using riemann invariants we prove the existence of smooth solutions for the regularized system for every 0 the regularization mechanism is a non linear bending of characteristics that prevents their finite time crossing we prove that in the 0 limit the regularized solutions converge strongly however based on our analysis and numerical simulations the limit is not the unique entropy solution of the euler equations the numerical method used to support this claim is derived from the riemann invariants for the regularized system this method is guaranteed to preserve the monotonicity of characteristics

weakly nonlinear high frequency waves abstract in this paper we derive a method for finding small amplitude high frequency solutions to hyperbolic systems of quasilinear partial differential equations our solution is the sum of two parts i a superposition of small amplitude high frequency waves ii a slowly varying mean solution each high frequency wave displays nonlinear distortion of the wave profile and shocks may form shock conditions are derived for conservative systems different high frequency waves do not interact provided the frequencies and wave numbers of two waves are not linearly related to those of a third the mean solution is found by solving a linear partial differential equation this method generalizes whitham s nonlinearization technique 9 for single waves to problems where many waves are present we obtain these results by generalizing a scheme first proposed by choquet bruhat 1 which employs the method of multiple scales author

regret graphs diagnostic uncertainty and youden s index regret is the difference in outcome between the action we took and the best action we could in retrospect have taken tent graphs representing decision problems under uncertainty in terms of expected regret offer an instructive geometric supplement to standard formulae allow instant calculations and suggest as we shall illustrate novel algebraic properties for instance the problem of finding the pre test probability range in which it is worthwhile doing a diagnostic test becomes trivial even when test costs depend on the unknown disease status we take the opportunity to explore the non trivial algebraic properties of this problem the graphs also provide a simple way to illustrate and read off the expected value of information and expected value of perfect information this property is used to derive a clinical interpretation of youden s index sensitivity specificity 1 namely it is the maximum proportional reduction in expected regret achieved by the test finally we examine the relationship between youden s index and the area under an roc curve

on raghunathan s measure conjecture this paper represents the last in our three part series on raghunathan s measure conjecture see r1 r2 for parts i and ii more specifically let g be a real lie group all groups in this paper are assumed to be second countable with the lie algebra r a discrete subgroup of g and ir g r g the projection lwg rg the group g acts by right translations on r g x x g x e r g g e g let gt be a borel probability measure on r g define

a non iterative accurate asymptotic confidence interval for the difference between two proportions i propose a new confidence interval for the difference between two binomial probabilities that requires only the solution of a quadratic equation the procedure is based on estimating the variance of the observed difference at the boundaries of the confidence interval and uses least squares estimation rather than maximum likelihood as previously suggested the proposed procedure is non iterative agrees with the conventional test of equality of two binomial probabilities and even for fairly small sample sizes appears to yields actual 95 per cent confidence intervals with mean or median probabilities of coverage very close to 0 95 the yates continuity correction appears to generate confidence intervals with the conditional probability of coverage at least equal to nominal levels 1997 by john wiley sons ltd

the mobius function is strongly orthogonal to nilsequences we show that the mobius function n is strongly asymptotically or thogonal to any polynomial nilsequence f g n n2n here g is a sim ply connected nilpotent lie group with a discrete and cocompact subgroup so g is a nilmanifold g z g is a polynomial sequence and f g r is a lipschitz function more precisely we show that j 1 n p n n 1 n f g n j f g a log a n for all a 0 in particu lar this implies the mobius and nilsequence conjecture mn s from our earlier paper for every positive integer s this is one of two major in gredients in our programme to establish a large number of cases of the generalised hardy littlewood conjecture which predicts how often a collec tion 1 t z d z of linear forms all take prime values the proof is a relatively quick application of the results in our recent companion paper we give some applications of our main theorem we show for exam ple that the mobius function is uncorrelated with any bracket polynomial such as n p 3bn p 2c we also obtain a result about the distribution of nilse quences a n x n2n as n ranges only over the primes

an effective ratner equidistribution result for operatorname sl 2 mathbb r ltimes mathbb r 2 let g sl 2 r proportional to r 2 be the affine special linear group of the plane and set gamma sl 2 z proportional to z 2 we prove a polynomially effective asymptotic equidistribution result for the orbits of a 1 dimensional nonhorospherical unipotent flow on gamma g

the distribution of free path lengths in the periodic lorentz gas and related lattice point problems the periodic lorentz gas describes the dynamics of a point particle in a periodic array of spherical scatterers and is one of the fundamental models for chaotic diffusion in the present paper we investigate the boltzmann grad limit where the radius of each scatterer tends to zero and prove the existence of a limiting distribution for the free path length we also discuss related problems such as the statistical distribution of directions of lattice points that are visible from a fixed position

comparative analysis of two rates in this paper we examine comparative analysis of rates with a view to each of the usual comparative parameters rate difference rd rate ratio rr and odds ratio or and with particular reference to first principles for rd and rr we show the prevailing statistical practices to be rather poor we stress the need for restricted estimation of variance in the chi square function underlying interval estimation and also point estimation and hypothesis testing for rr analysis we propose a chi square formulation analogous to that for rd and thus one which obviates the present practice of log transformation and its associated use of taylor series approximation of the variance as for or analysis we emphasize that the chi square function introduced by cornfield for unstratified data and extended by gart to the case of stratified analysis is based on the efficient score and thus embodies its optimality properties we provide simulation results to evince the better performance of the proposed parameter constrained procedures over the traditional ones

invariant distributions and time averages for horocycle flows there are infinitely many obstructions to existence of smooth solutions of the cohomological equation uu f where u is the vector field generating the horocycle flow on the unit tangent bundle sm of a riemann m surface of finite area and f is a given function on sm we study the sobolev regularity of these obstructions construct smooth solutions of the cohomological equation and derive asymptotics for the ergodic averages of horocycle flows

orbits of discrete subgroups on a symmetric space and the furstenberg boundary let x be a symmetric space of noncompact type and let be a lattice in the isometry group of x we study the distribution of orbits of acting on the symmetric space x and its geometric boundary x generalizing the main equidistribution result of margulis s thesis m theorem 6 to higher rank symmetric spaces more precisely for any y x and b x we investigate the distribution of the set y b 1 in x x it is proved in particular that the orbits of in the furstenberg boundary are equidistributed and that the orbits of in x are equidistributed in sectors defined with respect to a cartan decomposition our main tools are the strong wavefront lemma and the equidistribution of solvable flows on homogeneous spaces which we obtain using shah s result s corollary 1 2 based on ratner s measure classification theorem r1 theorem 1

on the uniform equidistribution of long closed horocycles it is well known that on any given hyperbolic surface of finite area a closed horocycle of length becomes asymptotically equidistributed as in this paper we prove that any subsegment of length greater than 1 2 e of such a closed horocycle also becomes equidistributed as the exponent 1 2 e is the best possible and improves upon a recent result by hejhal he3 we give two proofs of the above result our second proof leads to explicit information on the rate of convergence we also prove a result on the asymptotic joint equidistribution of a finite number of distinct subsegments having equal length proportional to

quantitative version of the oppenheim conjecture for inhomogeneous quadratic forms a quantitative version of the oppenheim conjecture for inhomogeneous quadratic forms is proved we also give an application to eigenvalue spacing on flat 2 tori with aharonov bohm flux

gaps in n mod 1 and ergodic theory cut the unit circle s r z at the points 1 2 n where x x mod 1 and let j1 jn denote the complementary intervals or gaps that remain we show that in contrast to the case of random points whose gaps are exponentially distributed the lengths ji n are governed by an explicit piecewise real analytic distribution f t dt with phase transitions at t 1 2 and t 2 the gap distribution is related to the probability p t that a random unimodular lattice translate r meets a fixed triangle st of area t in fact p t f t the proof uses ergodic theory on the universal elliptic curve e sl2 r r 2 sl2 z z and ratner s theorem on unipotent invariant measures

sparse equidistribution problems period bounds and subconvexity we introduce a geometric method to bound periods of automorphic forms the key features of this method are the use of equidistribution results in place of mean value theorems and the systematic use of mixing and the spectral gap applications are given to equidistribution of sparse subsets of horocycles and to equidistribution of cm points to subconvexity of the triple product period in the level aspect over number fields which implies subconvexity for certain standard and rankin selberg l functions and to bounding fourier coefficients of automorphic forms

expanding translates of curves and dirichlet minkowski theorem on linear forms we show that a multiplicative form of dirichlet s theorem on simulta neous diophantine approximation as formulated by minkowski cannot be improved for almost all points on any analytic curve on r k which is not contained in a proper affine subspace such an investigation was initiated by davenport and schmidt in the late sixties the diophantine problem is then settled via showing that certain sequence of expanding translates of curves on the homogeneous space of unimodular lattices in r k 1 gets equidistributed in the limit we use ratner s theorem on unipotent flows linearization techniques and a new observation about intertwined linear dynamics of various sl m r s contained in sl k 1 r

asymptotic evolution of smooth curves under geodesic flow on hyperbolic manifolds extending the earlier results for analytic curve segments in this article we describe the asymptotic behaviour of evolution of a finite segment of a c n smooth curve under the geodesic flow on the unit tangent bundle of a finite volume hyperbolic n manifold in particular we show that if the curve satisfies certain natural geometric conditions the pushforward of the parameter measure on the curve under the ge odesic flow converges to the normalized canonical riemannian measure on the tangent bundle in the limit we also study the limits of geodesic evolution of shrinking segments we use ratner s classification of ergodic invariant measures for unipo tent flows on homogeneous spaces of so n 1 and an observation re lating local growth properties of smooth curves and dynamics of linear sl 2 r actions