representation zeta functions of compact p adic analytic groups and arithmetic groups we introduce new methods from p adic integration into the study of representation zeta functions associated to compact p adic analytic groups and arithmetic groups they allow us to establish that the representation zeta functions of generic members of families of p adic analytic pro p groups obtained from a global perfect lie lattice satisfy functional equations in the case of semisimple compact p adic analytic groups we exhibit a link between the relevant p adic integrals and a natural filtration of the locus of irregular elements in the associated semisimple lie algebra defined by centraliser dimension based on this algebro geometric description we compute explicit formulae for the representation zeta functions of principal congruence subgroups of the groups sl 3 o where o is a compact discrete valuation ring of characteristic 0 and of the corresponding unitary groups these formulae combined with approximative clifford theory allow us to determine the abscissae of convergence of representation zeta functions associated to arithmetic subgroups of algebraic groups of type a 2 assuming a conjecture of serre on the congruence subgroup problem we thereby prove a conjecture of larsen and lubotzky on lattices in higher rank semisimple groups for algebraic groups of type a 2 defined over number fields

newton polyhedra unstable faces and the poles of igusa s local zeta function in this paper we examine when the order of a pole of igusa s local zeta function associated to a polynomial f is smaller than expected we carry out this study in the case that f is sufficiently non degenerate with respect to its newton polyhedron f and the main result of this paper is a proof of one of the conjectures of denef and sargos our technique consists in reducing our question about the polynomial f to the same question about polynomials f where are faces of f depending on the examined pole and f is obtained from f by throwing away all monomials of f whose exponents do not belong to secondly we obtain a formula for igusa s local zeta function associated to a polynomial f with unstable which shows that in this case the upperbound for the order of the examined pole is obviously smaller than expected

ideal spatial adaptation by wavelet shrinkage summary with ideal spatial adaptation an oracle furnishes information about how best to adapt a spatially variable estimator whether piecewise constant piecewise polynomial variable knot spline or variable bandwidth kernel to the unknown function estimation with the aid of an oracle offers dramatic advantages over traditional linear estimation by nonadaptive kernels however it is a priori unclear whether such performance can be obtained by a procedure relying on the data alone we describe a new principle for spatially adaptive estimation selective wavelet reconstruction we show that variable knot spline fits and piecewise polynomial fits when equipped with an oracle to select the knots are not dramatically more powerful than selective wavelet reconstruction with an oracle we develop a practical spatially adaptive method riskshrink which works by shrinkage of empirical wavelet coefficients riskshrink mimics the performance of an oracle for selective wavelet reconstruction as well as it is possible to do so a new inequality in multivariate normal decision theory which we call the oracle inequality shows that attained performance differs from ideal performance by at most a factor of approximately 2 log n where n is the sample size moreover no estimator can give a better guarantee than this within the class of spatially adaptive procedures riskshrink is essentially optimal relying only on the data it comes within a factor log 2 n of the performance of piecewise polynomial and variableknot spline methods equipped with an oracle in contrast it is unknown how or if piecewise polynomial methods could be made to function this well when denied access to an oracle and forced to rely on data alone

analytic properties of zeta functions and subgroup growth it has become somewhat of a cottage industry over the last fifteen years to understand the rate of growth of the number of subgroups of finite index in a group g although the story began much before the recent activity grew out of a paper by dan segal in 36 the story so far has been well documented in lubotzky s subsequent survey paper in 30 in 24 the second author of this article segal and smith introduced the zeta function of a group as a tool for understanding this growth of subgroups let an g be the number of subgroups of index n in the finitely generated group g and sn g a1 g an g be the number of subgroups of index n or less the zeta function is defined as the dirichlet series with coefficients an g and has a natural interpretation as a noncommutative generalization of the dedekind zeta function of a number field

motivic local density we develop a theory of local densities and tangent cones in a motivic framework extending work by cluckers comte loeser about p adic local density we prove some results about geometry of definable sets in henselian valued fields of characteristic zero both in semi algebraic and subanalytic languages and study lipschitz continuous maps between such sets we prove existence of regular stratifications satisfying analogous of verdier condition w f using cluckers loeser theory of motivic integration we define a notion of motivic local density with values in the grothendieck ring of the theory of the residue sorts we then prove the existence of a distinguished tangent cone and that one can compute the local density on this cone endowed with appropriate motivic multiplicities as an application we prove a uniformity theorem for p adic local density

geometric proofs of theorems of ax kochen and ersov we give an algebraic geometric proof of the theorem of ax and kochen on p adic diophantine equations in many variables unlike ax kochen s proof ours does not use any notions from mathematical logic and is based on weak toroidalization of morphisms we also show how this geometric approach yields new proofs of the ax kochen er v s ov transfer principle for local fields and of quantifier elimination theorems of basarab and pas

vanishing of principal value integrals on surfaces principal value integrals are associated to multi valued rational differential forms with normal crossings support on a non singular algebraic variety we prove their vanishing on rational surfaces in the context of a conjecture of denef jacobs as an appli cation we obtain a strong vanishing result for candidate poles of p adic and motivic igusa zeta functions

iterated vanishing cycles convolution and a motivic analogue of a conjecture of steenbrink we prove a motivic analogue of steenbrink s conjecture 25 conjecture 2 2 on the hodge spectrum proved by m saito in 21 to achieve this we construct and compute motivic iterated vanishing cycles associated with two functions we are also led to introduce a more general version of the convolution operator appearing in the motivic thom sebastiani formula throughout the article we use the framework of relative equivariant grothendieck rings of varieties endowed with an algebraic torus action

functional equations and uniformity for local zeta functions of nilpotent groups we investigate in this paper the zeta function inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i associated to a nilpotent group introduced in gss this zeta function counts the subgroups h whose profinite completion is isomorphic to the profinite completion inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 02i by representing inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 03i as an integral with respect to the haar measure on the algebraic automorphism group g of the lie algebra associated to and by generalizing some recent work of igusa i we give under some assumptions on an explicit finite form for inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 04i in terms of the combinatorial data of the root system of g and information about the weights of various representations of g as a corollary of this finite form we are able to prove 1 a certain uniformity in p confirming a question raised in gss and 2 a functional equation that the local factors satisfy inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 05i this functional equation is perhaps the most important result of the paper as it is a new feature of the theory of zeta functions of groups

on the smallest poles of igusa s p adic zeta functions let k be a p adic field we explore igusa s p adic zeta function which is associated to a k analytic function on an open and compact subset of k n first we deduce a formula for an important coefficient in the laurent series of this meromorphic function at a candidate pole afterwards we use this formula to determine all values less than 1 2 for n 2 and less than 1 for n 3 which occur as the real part of a pole

on density estimation in the view of kolmogorov s ideas in approximation theory the paper deals with upper and lower bounds for the quality of probability density estimation connections are established between these problems and the theory of approximation of functions particularly it is demonstrated how some of kolmogorov s concepts work

on the distribution of points of bounded height on equivariant compactifications of vector groups we prove asymptotic formulas for the number of rational points of bounded height on smooth equivariant compactifications of the affine space

on euler characteristics associated to exceptional divisors let k be an algebraically closed field and f e k x1 x fix an embedded resolution h x an i of f i o and denote by ei i e s the irreducible components of h l f l o with multiplicity ni in the 0 0 visor of f o h put also es ei uj1iej and denote by x ei its euler characteristic several conjectures concerning igusa s local zeta function and the topological zeta function of f motivate the study of euler characteristics associated to subsets uie tei of uiese1 which are maximal connected with respect to the property that dini for all i e t here d e n d 1 we prove that if h maps uiet ei to a point then

on affine plane curve evolution abstract an affine invariant curve evolution process is presented in this work the evolution studied is the affine analogue of the euclidean curve shortening flow evolution equations for both affine and euclidean invariants are developed an affine version of the classical euclidean isoperimetric inequality is proved this inequality is used to show that in the case of affine evolution of convex plane curves the affine isoperimetric ratio is a non decreasing function of time convergence of this affine isoperimetric ratio to the ellipse s value 8 2 as well as convergence in the hausdorff metric of the evolving curve to an ellipse is also proved

a morphological scheme for mean curvature motion and applications to anisotropic diffusion and motion of level sets this paper introduces a discrete scheme for mean curvature motion using a morphological image processing approach an axiomatic approach of image processing and the mean curvature partial differential equation pde are briefly presented then the properties of the proposed scheme are studied in particular consistency and convergence are proved the applications of mean curvature motion in image denoising and form evolution are developed and experiences are presented

igusa type functions associated to finite formed spaces and their functional equations we study symmetries enjoyed by the polynomials enumerating non degenerate flags in finite vector spaces equipped with a non degenerate alternating bilinear hermitian or quadratic form to this end we introduce igusa type rational functions encoding these polynomials and prove that they satisfy certain functional equations some of our results are achieved by expressing the polynomials in question in terms of what we call parabolic length functions on coxeter groups of type a while our treatment of the orthogonal case exploits combinatorial properties of integer compositions and their refinements we formulate a precise conjecture how in this situation too the polynomials may be described in terms of parabolic length functions

user s guide to viscosity solutions of second order partial differential equations the notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems existence theorems and theorems about continuous dependence may now be proved by very efficient and striking arguments the range of important applications of these results is enormous this article is a self contained exposition of the basic theory of viscosity solutions

the classification of spinors under over finite fields the spinors of a 14 dimensional vector space v are studied with respect to the group gspin 14 of the 14 dimensional vector space v over finite fields f q results are given as follows 1 the decomposition of the space of spinors into gspin 14 equivalence classes or orbits over f q 2 the structure of the fixer of gspin 14 for each orbit as an f q group

on the arithmetic of a singular invariant introduction we know by schouten 13 and gurevich 1 that the ring of invariants under sl7 acting on the space x of trivectors in aff7 the affine 7 space is generated by an irreducible homogeneous polynomial f x of degree 7 with rational coefficients we normalize f x up to sign by the condition that its coefficients are relatively prime integers this is the singular invariant in the title if k is a number field the image g of gl7 in gl x is a connected irreducible k subgroup of gl x acting transitively on y x f f1 0 and f gx v g f x for everyg in g with a rational character v of g we take a hecke quasicharacter x of k and introduce a zeta distribution z w in the adelization xa of x as

asymptotically exact nonparametric hypothesis testing in sup norm and at a fixed point for the signal in gaussian white noise model we consider the problem of testing the hypothesis h 0 f 0 the signal f is zero against the nonparametric alternative h 1 f where is a set of functions on r 1 of the form f f f c here is a holder or sobolev class of functions f is either the sup norm of f or the value of f at a fixed point c 0 is a constant is the minimax rate of testing and 0 is the asymptotic parameter of the model we find exact separation constants c 0 such that a test with the given summarized asymptotic errors of first and second type is possible for c c and is not possible for c c we propose asymptotically minimax test statistics

functional equations for zeta functions of groups and rings we introduce a new method to compute explicit formulae for various zeta functions associated to groups and rings the specific form of these formulae enables us to deduce local functional equations more precisely we prove local functional equations for the subring zeta functions associated to rings the subgroup conjugacy and representation zeta functions of finitely generated torsion free nilpotent or t groups and the normal zeta functions of t groups of class 2 in particular we solve the two problems posed in 9 section 5 we deduce our theorems from a blueprint result on certain p adic integrals which generalises work of denef and others on igusa s local zeta function the malcev correspondence and a kirillov type theory developed by howe are used to linearise the problems of counting subgroups and representations in t groups respectively

on zeta functions of modular representations of a discrete group abstract it is proved that the generating function defined by the numbers of isomorphism classes of absolutely irreducible representations of a finitely generated group g into gl d f q n for n 1 is a rational function it is also proved that the generating function defined by weighted sums over isomorphism classes of representations of g into gl d f q n for n 1 is meromorphic over both the complex numbers and the p adic complex numbers

the heat equation shrinks embedded plane curves to round points soit c 0 s 1 r 2 une courbe lisse plongee dans le plan alors c s 1 0 t r 2 existe en satisfaisant c t k n ou k est la courbure de c et n est son vecteur unite normal entrant c t est lisse pour tout t il converge vers un point quand t t et sa forme limite quand t t est un cercle rond avec convergence dans norme c

computing topological zeta functions of groups algebras and modules ii abstract building on our previous work we develop the first practical algorithm for computing topological zeta functions of nilpotent groups possibly non associative algebras and modules while we previously depended upon non degeneracy assumptions the theory developed here allows us to overcome these restrictions in various interesting cases

definable sets motives and p adic integrals 0 1 let x be a scheme reduced and separated of finite type over z for p a prime number one may consider the set x zp of its zp rational points for every n in n there is a natural map n x zp x z p assigning to a zp rational point its class modulo p the image yn p of x zp by n is exactly the set of z p rational points which can be lifted to zp rational points denote by nn p the cardinality of the finite set yn p by a result of the first author 7 the poincare series

a simple proof of convergence for an approximation scheme for computing motions by mean curvature we prove the convergence of an approximation scheme recently proposed by bence merriman and osher for computing motions of hypersurfaces by mean curvature our proof is based on viscosity solutions methods

on signal detection in the presence of weakly correlated noise over fading channels in cognitive radio networks in order to avoid interference form secondary users to the primary license holders of the spectrum reliable spectrum sensing is necessary in scenarios where the noise samples are correlated the spectrum sensing methods optimized considering impairment by independent noise samples will not provide optimum performances to address this issue a locally optimum detection method for random signals under a weakly correlated noise model over fading channels is proposed the probabilities of false alarm and detection of the proposed detector in the low signal to noise ratio regime are analyzed the average probabilities are calculated over different channel gains numerical and simulation results demonstrate the superiority of the proposed method over the known energy detection method with comparable complexities furthermore we consider the scenario where the estimated correlation is different from the real correlation and investigate the effect of this correlation mismatch on the probabilities of false alarm and detection of the proposed method

the local geometry of testing in ellipses tight control via localized kolmogorov widths mean testing over ellipses is a decision problem that arises in several applications including non parametric goodness of fit testing signal detection in cognitive radio and regression function testing in reproducing kernel hilbert spaces we study the local geometry of such testing problems with compound alternatives given samples of a gaussian random vector the goal is to distinguish whether the mean is equal to a known vector within an ellipse or equal to some other unknown vector in the ellipse while past work on such problems has focused on the difficulty in a global sense we study difficulty in a way that is localized to each vector within the ellipse our main result is to give sharp upper and lower bounds on the localized minimax testing radius in terms of an explicit formula involving the kolmogorov width of the ellipse intersected with a euclidean ball when applied to particular examples our general theorems yield interesting rates that were not known before as a particular case for testing in sobolev ellipses of smoothness alpha we demonstrate rates that vary from sigma 2 frac 4 alpha 4 alpha 1 corresponding to the classical global rate to the faster rate sigma 2 frac 8 alpha 8 alpha 1 achievable for vectors at favorable locations within the ellipse we also show that the optimal test for this problem is achieved by a linear projection test that is based on an explicit lower dimensional projection of the observation vector

the heat equation shrinking convex plane curves soient m et m des varietes de riemann et f m m une application reguliere si m est une courbe convexe plongee dans le plan r 2 l equation de la chaleur contracte m a un point

local zeta functions of certain prehomogeneous vector spaces in which k is a p adic field k is an absolute value on k ok is the ring of integers of k x is an affine space with an ok structure and f x is a polynomial on x we take as f x the discriminant f x1 x2 of the binary cubic form n ux1 vx2 in u v where x1 x2 are in a simple k split jordan algebra a of degree 3 with n as its generic norm and we shall determine z s in general if f x is a polynomial on an affine space v equipped with a nondegenerate quadratic form q x and if f x y is the coefficient of a variable t in f x ty then a morphism v v is defined as q x y f x y for every x y in v in this notation the above n x can be characterized up to a normalization as a cubic form satisfying x n x x cf springer 18 in general if gi is a subgroup of gl vi for i 1 2 then we shall denote by g1 0 g2 0 the image of g1 x g2 x in gl v1 0 v20 in this and other standard notations the above f x1 x2 becomes a generator of the ring of relative invariants of

zeta functions of groups zeros and friendly ghosts we consider zeta functions defined as euler products of w p p s over all primes p where w x y is a polynomial such zeta functions arise naturally in the context of zeta functions of algebraic groups and nilpotent groups the analytic behavior of such euler products is controlled by something we call the ghost polynomial of w x y the ghost is defined from the zeros of w p y as p ranges over all primes when the euler product of the ghost is meromorphic on inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i an important property in relation to the analytic behavior of the original zeta function we say that w x y has a friendly ghost we give an explicit expression for the ghosts of classical reductive groups and prove that they are friendly we also consider the ghosts of exceptional reductive groups and zeta functions of nilpotent groups the concept of the ghost implies a new dichotomy for the representations of an algebraic group according to whether their associated zeta function has a friendly ghost or not

zeta functions for analytic mappings log principalization of ideals and newton polyhedra in this paper we provide a geometric description of the possi ble poles of the igusa local zeta function z s f associated to an analytic mapping f f1 fl u k n k l and a locally constant function with support in u in terms of a log principalizaton of the k x ideal if f1 fl typically our new method provides a much shorter list of possible poles compared with the previous methods we determine the largest real part of the poles of the igusa zeta function and then as a corollary we ob tain an asymptotic estimation for the number of solutions of an arbitrary sys tem of polynomial congruences in terms of the log canonical threshold of the subscheme given by if we associate to an analytic mapping f f1 fl a newton polyhedron f and a new notion of non degeneracy with respect to f the novelty of this notion resides in the fact that it depends on one newton polyhedron and khovanskii s non degeneracy notion depends on the newton polyhedra of f1 fl by constructing a log principalization we give an explicit list for the possible poles of z s f l 1 in the case in which f is non degenerate with respect to f

rational points on compactifications of semi simple groups we prove manin s conjecture concerning the distribution of rational points of bounded height and its refinement by peyre for wonderful compactifications of semi simple algebraic groups over number fields the proof proceeds via the study of the associated height zeta function and its spectral expansion

a family of class 2 nilpotent groups their automorphisms and pro isomorphic zeta functions the pro isomorphic zeta function zeta wedge gamma s of a finitely generated nilpotent group gamma is a dirichlet generating function that enumerates finite index subgroups whose profinite completion is isomorphic to that of gamma such zeta functions can be expressed as euler products of p adic integrals over the mathbb q p points of an algebraic automorphism group associated to gamma in this way they are closely related to classical zeta functions of algebraic groups over local fields

zeta functions and oscillatory integrals for meromorphic functions abstract in the 70 s igusa developed a uniform theory for local zeta functions and oscillatory integrals attached to polynomials with coefficients in a local field of characteristic zero in the present article this theory is extended to the case of rational functions or more generally meromorphic functions f g with coefficients in a local field of characteristic zero this generalization is far from being straightforward due to the fact that several new geometric phenomena appear also the oscillatory integrals have two different asymptotic expansions the usual one when the norm of the parameter tends to infinity and another one when the norm of the parameter tends to zero the first asymptotic expansion is controlled by the poles with negative real parts of all the twisted local zeta functions associated to the meromorphic functions f g c for certain special values c the second expansion is controlled by the poles with positive real parts of all the twisted local zeta functions associated to f g

local rademacher complexities we propose new bounds on the error of learning algorithms in terms of a data dependent notion of complexity the estimates we establish give optimal rates and are based on a local and empirical version of rademacher averages in the sense that the rademacher averages are computed from the data on a subset of functions with small empirical error we present some applications to classification and prediction with convex function classes and with kernel classes in particular

poles of the topological zeta function associated to an ideal in dimension two to an ideal in mathbb c x y one can associate a topological zeta function this is an extension of the topological zeta function associated to one polynomial but in this case we use a principalization of the ideal instead of an embedded resolution of the curve in this paper we will study two questions about the poles of this zeta function first we will give a criterion to determine whether or not a candidate pole is a pole it turns out that we can know this immediately by looking at the intersection diagram of the principalization together with the numerical data of the exceptional curves afterwards we will completely describe the set of rational numbers that can occur as poles of a topological zeta function associated to an ideal in dimension two the same results are valid for related zeta functions as for instance the motivic zeta function