estimates for the extremal sections of lpn balls abstract the problem of finding the maximal hyperplane section of b p n where p 2 has been open for a long time it is known that the answer depends on both p and n in this paper using the well known equivalence between hyperplane sections and the isotropic constant of a body we give an upper bound estimate for the volume of hyperplane sections of normalized l p n balls that does not depend on n and p in addition on the basis of results of meyer pajor and schmuckenschlager we show further the corresponding extremal body and hyperplane section when this volume attains its minimum

small solutions to inhomogeneous linear equations over number fields we consider a system of m independent inhomogeneous linear equations in n m variables having coefficients in an algebraic number field k we give a best possible lower bound on the inhomogeneous height of a solution vector in k n and determine when a solution exists in o s n where o s is the ring of s integers in k if such a system has a solution vector in o s n we show that it has a solution in o s n such that the inhomogeneous height of is relatively small we give an explicit upper bound for this height in terms of the heights of the matrices defining the linear system our method uses geometry of numbers over adele spaces and local to global arguments

small zeros of quadratic forms over number fields let f be a nontrivial quadratic form in n variables with coefficients in a number field k and let a be a k x n matrix over k we show that if the simultaneous equations f x 0 and ax 0 hold on a subspace x of dimension l and l is maximal then such a subspace x can be found with the height of x relatively small in particular the height of x can be explicitly bounded by an expression depending on the height of f and the height of a we use methods from geometry of numbers over adele spaces and local to global techniques which generalize recent work of h p schlickewei

small zeros of quadratic forms outside a union of varieties let f be a quadratic form in n 2 variables defined on a vector space vk n over a global field k and zk n be a finite union of varieties defined by families of homogeneous polynomials over k we show that if v z contains a nontrivial zero of f then there exists a linearly independent collection of small height zeros of f in v z where the height bound does not depend on the height of z only on the degrees of its defining polynomials as a corollary of this result we show that there exists a small height maximal totally isotropic subspace w of the quadratic space v f such that w is not contained in z our investigation extends previous results on small zeros of quadratic forms including cassels theorem and its various generalizations the paper also contains an appendix with two variations of siegel s lemma all bounds on height are explicit

a probabilistic approach to the geometry of the lᵨⁿ ball this article investigates by probabilistic methods various geometric questions on b n p the unit ball of ln p we propose realizations in terms of independent random variables of several distributions on b n p including the normalized volume measure these representations allow us to unify and extend the known results of the sub independence of coordinate slabs in b n p as another application we compute moments of linear functionals on b n p which gives sharp constants in khinchine s inequalities on b n p and determines the 2 constant of all directions on b n p we also study the extremal values of several gaussian averages on sections of b n p including mean width and l norm and derive several monotonicity results as p varies applications to balancing vectors in l 2 and to covering numbers of polyhedra complete the exposition

entropy numbers s numbers and eigenvalue problems abstract we establish inequalities between entropy numbers and approximation numbers for operators acting between banach spaces furthermore we derive inequalities between eigenvalues and entropy numbers for operators acting on a banach space the results are compared with the classical inequalities of bernstein and jackson

gelfand numbers of operators with values in a hilbert space in the paper we prove two inequalities involving gelfand numbers of operators with values in a hilbert space the first inequality is a rademacher version of the main result in pa to 1 which relates the gelfand numbers of an operator from a banach spacex intol 2 n with a certain rademacher average for the dual operator the second inequality states that the gelfand numbers of an operatoru froml 1 n into a hilbert space satisfy the inequality k 1 2 c k u leqq c parallel u parallel log 1 n k 1 2 wherec is a universal constant several applications of these inequalities in the geometry of banach spaces are given

the interacting multiple model algorithm for systems with markovian switching coefficients an important problem in filtering for linear systems with markovian switching coefficients dynamic multiple model systems is the management of hypotheses which is necessary to limit the computational requirements a novel approach to hypotheses merging is presented for this problem the novelty lies in the timing of hypotheses merging when applied to the problem of filtering for a linear system with markovian coefficients the method is an elegant way to derive the interacting multiple model imm algorithm evaluation of the imm algorithm shows that it performs well at a relatively low computational load these results imply a significant change in the state of the art of approximate bayesian filtering for systems with markovian coefficients

existence of correlated equilibria an elementary proof based on linear duality is provided for the existence of correlated equilibria in finite games the existence result is then extended to infinite games including some that possess no nash equilibria

lower bounds for norms of products of polynomials via bombieri inequality in this paper we give a different interpretation of bombieri s norm this new point of view allows us to work on a problem posed by beauzamy about the behavior of the sequence sn p supqn pqn 2 where p is a fixed m homogeneous polynomial and qn runs over the unit ball of the hilbert space of n homogeneous polynomials we also study the factor problem for homogeneous polynomials defined on c and we obtain sharp inequalities whenever the number of factors is no greater than n in particular we prove that for the product of homogeneous polynomials on infinite dimensional complex hilbert spaces our inequality is sharp finally we use these ideas to prove that any set zk k 1 of unit vectors in a complex hilbert space for which sup z 1 z z1 z zn is minimum must be an orthonormal system

extremal slabs in the cube and the laplace transform abstract we study the volume of symmetric slabs in the unit cube we show that for t 3 4 the slab parallel to a face has the minimal volume among all symmetric slabs with width t for large width we prove the asymptotic extremality of the slab orthogonal to the main diagonal the proof is based on certain concavity properties of the laplace transform and on several limit theorems from probability the central limit theorem and classical principles of moderate and large deviations finally we extend some of the results to more general classes of bodies

the multiplicity of binary recurrences of the recurrence relation 0 1 it follows from the skolem mahler lech theorem that if u then one at least of and is a root of unity consequently we call um nondegenerate if none of these quantities is a root of unity in the thirties m ward conjectured that a nondegenerate binary recurrence sequence of integers um has multiplicity u 5 5 this conjecture was proved by kubota 4 kubota also proved that any nondegenerate binary recurrence um with elements um in a number eld k of degree d has multiplicity u 5 c d cf 5 beukers and tijdeman 2 established for this case the bound

holomorphic semi groups and the geometry of banach spaces according to a fundamental theorem of dvoretzky cf 24 25 any infinite dimensional banach space x contains for each integer n and each e 0 a subspace xn which is 1 e isomorphic to 12n in the classical banach spaces lp and 1p with 1 p x it is well known that this result can be improved these spaces contain uniformly complemented ln 7 s this means that we can find subspaces xn as above with the additional property that there exist projections p x xn such that supn 11 pn 11 cx moreover it is also well known that this property no longer holds for 11 or lo this motivated the question cf 13 to determine which infinite dimensional banach spaces contain uniformly complemented 1 n s we will prove below that all uniformly convex spaces have this property moreover we will show that a banach space x verifies this property locally see definition 2 10 for more precision if and only if x does not contain l n s uniformly progressively the work of previous authors cf 14 12 and particularly 5 has reduced such so called geometric questions to a purely analytic problem consisting of verifying in x a certain inequality referred to as k convexity in this paper we connect this inequality with the holomorphy of a certain semi group of operators on lp x and we show how this leads to the results already mentioned

a simple adaptive procedure leading to correlated equilibrium we propose a new and simple adaptive procedure for playing a game regret matching in this procedure players may depart from their current play with probabilities that are proportional to measures of regret for not having used other strategies in the past it is shown that our adaptive procedure guarantees that with probability one the empirical distributions of play converge to the set of correlated equilibria of the game

extremal properties of central half spaces for product measures we deal with the isoperimetric and the shift problem for subsets of measure 1 2 in product probability spaces we prove that the canonical central half spaces are extremal in particular cases products of log concave measures on the real line satisfying precise conditions and products of uniform measures on spheres or balls as a corollary we improve the known log sobolev constants for euclidean balls we also give some new results about the related question of estimating the volume of sections of unit balls of lp sums of minkowski spaces

sections of the unit ball of ipn let bnp xi rn i li n xi p 1 1 p and let ek be a k dimensional subspace of rn it is proved that if p 2 resp then vol bpn ek vol bpk resp we give some applications to linear forms

a quantitative version of the absolute subspace theorem the celebrated subspace theorem of w m schmidt 12 says the following subspace theorem let l1 ln be linearly independent linear forms in n variables with real or complex algebraic coefficients suppose 0 consider the inequality 1 1 l1 x ln x 0 put 1 5 q c x r li x qi 1 i n then the parametric subspace theorem can be stated as follows parametric subspace theorem let c be a fixed tuple satisfying 1 3 let 0 then there are finitely many proper linear subspaces t1 tb of q n such that for any q which is sufficiently large there exists a subspace ti t1 tb with 1 6 q q c z ti

sharp comparison of moments and the log concave moment problem abstract this article investigates sharp comparison of moments for various classes of random variables appearing in a geometric context in the first part of our work we find the optimal constants in the khintchine inequality for random vectors uniformly distributed on the unit ball of the space l q n for q 2 complementing past works that treated q 0 2 as a byproduct of this result we prove an extremal property for weighted sums of symmetric uniform distributions among all symmetric unimodal distributions in the second part we provide a one to one correspondence between vectors of moments of symmetric log concave functions and two simple classes of piecewise log affine functions these functions are shown to be the unique extremisers of the p th moment functional under the constraint of a finite number of other moments being fixed which is a refinement of the description of extremisers provided by the generalised localisation theorem of fradelizi and guedon 2006 7

on polynomial exponential equations let m k be the set of absolute values of k normalized such that they extend the standard or a p adic absolute value of q let s c m k be the subset consisting of the archimedean absolute values as well as of the non archimedean absolute values v e m k having i ol 4 1 for some ej thus s is a finite set an s unit is an element e k having i 1 1 for every v s then in particular the o and therefore every a txe with x e z is an s unit therefore 1 2 is related to the s unit equation

on the structure of reduced kernel lattice bases lattice based reformulation techniques have been used successfully both theoretically and computationally one such reformulation is obtained from the kernel lattice associated with an input matrix some of the hard instances in the literature that have been successfully tackled by lattice based techniques have randomly generated input since the considered instances are very hard even in low dimension less experience is available for larger instances recently we have studied larger instances and observed that the lll reduced basis of the kernel lattice has a specific sparse structure in particular this translates into a map in which some of the original variables get a rich translation into a new variable space whereas some variables are only substituted in the new space if an original variable is important in the sense of branching or cutting planes this variable should be translated in a nontrivial way in this paper we partially explain through a probabilistic analysis the obtained struct

gaussian processes and mixed volumes on demontre la conjecture de dudley si l exposant du volume d un sous ensemble symetrique et convexe compact d un espace de hilbert est 1 alors k est un ensemble g c

extremal probabilistic problems and hotelling s t 2 test under a symmetry condition we consider the hotelling t 2 statistic for an arbitrary d dimensional sample if the sampling is not too deterministic or inhomogeneous then under the zero means hypothesis the limiting distribution for t 2 is chi 2 d it is shown that a test for the orthant symmetry condition introduced by efron can be constructed which does not differ essentially from the one based on chi 2 d and at the same time is applicable not only to large random homogeneous samples but to all multidimensional samples the main results are not limit theorems but exact inequalities corresponding to the solutions to certain extremal problems the following auxiliary result itself may be of interest chi d sqrt d 1 has a monotone likelihood ratio

mesures d independance lineaire de logarithmes dans un groupe algebrique commutatif this work falls within the theory of linear forms in logarithms over a connected and commutative algebraic group defined over the field of algebraic numbers overline mathbb q let g be such a group let w be a hyperplane of the tangent space at the origin of g defined over overline mathbb q and u be a complex point of this tangent space such that the image of u by the exponential map of the lie group g is an algebraic point then we obtain a lower bound for the distance between u and w which improves the results known before and which is in particular the best possible for the height of the hyperplane w the proof rests on baker s method and hirata s reduction as well as a new arithmetic argument chudnovsky s process of variable change which enables us to give a precise estimate of the ultrametric norms of some algebraic numbers built during the proof

a generalized localization theorem and geometric inequalities for convex bodies in this article we generalize a localization theorem of lovasz and simonovits random walks in a convex body and an improved volume algorithm random struct algorithms 4 4 1993 359 412 which is an important tool to prove dimension free functional inequalities for logconcave measures in a previous paper fradelizi and guedon the extreme points of subsets of s concave probabilities and a geometric localization theorem discrete comput geom 31 2004 327 335 we proved that the localization may be deduced from a suitable application of krein milman s theorem to a subset of log concave probabilities satisfying one linear constraint and from the determination of the extreme points of its convex hull here we generalize this result to more constraints give some necessary conditions satisfied by such extreme points and explain how it may be understood as a generalized localization theorem finally using this new localization theorem we solve an open question on the comparison of the volume of sections of non symmetric convex bodies in r n by hyperplanes a surprising feature of the result is

complexity of makanin s algorithm the exponent of periodicity is an important factor in estimates of complexity of word unification algorithms we prove that the exponent of periodicity of a minimal solution of a word equation is of order 2 1 07d where d is the length of the equation we also give a lower bound 2 0 29d so our upper bound is almost optimal and exponentially better than the original bound 6d 22d4 2 consequently our result implies an exponential improvement of known upper bounds on complexity of word unification algorithms

sequential monte carlo methods for multiple target tracking and data fusion the classical particle filter deals with the estimation of one state process conditioned on a realization of one observation process we extend it here to the estimation of multiple state processes given realizations of several kinds of observation processes the new algorithm is used to track with success multiple targets in a bearings only context whereas a jpdaf diverges making use of the ability of the particle filter to mix different types of observations we then investigate how to join passive and active measurements for improved tracking

a new method for the nonlinear transformation of means and covariances in filters and estimators this paper describes a new approach for generalizing the kalman filter to nonlinear systems a set of samples are used to parametrize the mean and covariance of a not necessarily gaussian probability distribution the method yields a filter that is more accurate than an extended kalman filter ekf and easier to implement than an ekf or a gauss second order filter its effectiveness is demonstrated using an example

the shattering dimension of sets of linear functionals we evaluate the shattering dimension of various classes of linear functionals on various symmetric convex sets the proofs here relay mostly on methods from the local theory of normed spaces and include volume estimates factorization techniques and tail estimates of norms viewed as random variables on euclidean spheres the estimates of shattering dimensions can be applied to obtain error bounds for certain classes of functions a fact which was the original motivation of this study although this can probably be done in a more traditional manner we also use the approach presented here to determine whether several classes of linear functionals satisfy the uniform law of large numbers and the uniform central limit theorem

concurrent particle filtering and data association using game theory for tracking multiple maneuvering targets we propose a particle filtering technique to track multiple maneuvering targets in the presence of clutter we treat data association and state estimation which are the two important sub problems in tracking as separate problems we develop a game theoretic framework to solve the data association in which we model each tracker as a player and the set of measurements as strategies we develop utility functions for each player and then use a regret based learning algorithm to find the equilibrium of this game the game theoretic approach allows us to associate measurements to all the targets simultaneously further in contrast to the traditional monte carlo data association algorithms that use samples of the association vector obtained from a proposal distribution our method finds the association in a deterministic fashion we then use monte carlo sampling on the reduced dimensional state of each target independently and thereby mitigate the curse of dimensionality problem that is known to occur in particle filtering we provide a number of numerical results to demonstrate the performance of our proposed filtering algorithm

a tutorial on particle filters for online nonlinear non gaussian bayesian tracking increasingly for many application areas it is becoming important to include elements of nonlinearity and non gaussianity in order to model accurately the underlying dynamics of a physical system moreover it is typically crucial to process data on line as it arrives both from the point of view of storage costs as well as for rapid adaptation to changing signal characteristics in this paper we review both optimal and suboptimal bayesian algorithms for nonlinear non gaussian tracking problems with a focus on particle filters particle filters are sequential monte carlo methods based on point mass or particle representations of probability densities which can be applied to any state space model and which generalize the traditional kalman filtering methods several variants of the particle filter such as sir asir and rpf are introduced within a generic framework of the sequential importance sampling sis algorithm these are discussed and compared with the standard ekf through an illustrative example

the calculation of posterior distributions by data augmentation comment a noniterative sampling importance resampling alternative to the data augmentation algorithm for creating a few imputations when fractions of missing information are modest the sir algorithm

duality of metric entropy abstractfor two convex bodies kand t in r n the covering number of kbyt denoted n k t is de ned as the minimal number of translates oft needed to cover k let us denote by k the polar body of k andby d the euclidean unit ball in r n we prove that the two functionsof t n k td and n d tk are equivalent in the appropriate sense uniformly over symmetric convex bodies k r n and over n n inparticular this veri es the duality conjecture for entropy numbers of linearoperators posed by pietsch in 1972 in the central case when either thedomain or the range of the operator is a hilbert space 1 introduction for two convex bodies kand t in r n the covering number of kby t denoted n k t is de ned as the minimal number of translates of tneeded to cover kn k t min n x 1 x n r n k i n x i t we denote by dthe euclidean unit ball in r n in this paper we prove thefollowing duality result for covering numbers theorem 1 main theorem there exist two universal constants

a note on the volume of sections of bnp abstract let b p n denote the unit ball in l p n with p 1 we prove that vol n 1 h b p n vol n b p n n 1 n for any n 1 dimensional subspace h of r n this is a consequence of bounding the isotropy constant of b p n above by 1 12 and we show that one can replace 1 12 by a possibly smaller number for n 2

weyl numbers in sequence spaces and sections of unit balls using techniques of a geometrical nature the following result is proved denoting by iqp m lpm lqm the natural embedding and by xk the kth weyl number if 0 p q 2 there are positive constants c1 and c2 such that for all k m n with k m2 c1k1q 1p xk iqp m c2k1q 1p the upper estimate being even valid for all k 1 m as a consequence of the approach used some results about sections of unit balls are also derived namely volh h bpm volk bpk for 0 p 2 where bpm bpk are the closed unit balls centred at zero of the spaces lpm and lpk respectively h is a k dimensional subspace of lpm and volk volh denote lebesgue measures in rk and h respectively

bounded height in pencils of finitely generated subgroups in this paper we prove a general bounded height result for specializations in finitely generated subgroups varying in families which complements and sharpens the toric mordell lang theorem by replacing finiteness by emptyness for the intersection of varieties and subgroups all moving in a pencil except for bounded height values of the parameters and excluding identical relations more precisely an instance of the results is as follows consider the torus scheme g r m c over a curve c defined over q and let be a subgroup scheme generated by finitely many sections satisfying some necessary conditions further let v be any subscheme then there is a bound for the height of the points p c q such that for some which does not generically lie in v p lies in the fiber vp we further offer some direct diophantine applications to illustrate once again that the results implicitly contain information absent from the previous bounds in this context

small zeros of quadratic forms we give upper and lower bounds for zeros of quadratic forms in the rational real and p adic fields for example given r 0 s 0 there are infinitely many forms a with integer coefficients in r s variables of the type r s i e equivalent over r to x2 xr2 3 x2 j such that every nontrivial integer zero x has lxl fr 28 where f is the maximum modulus of the coefficients of a

volume ratio and other s numbers of operators related to local properties of banach spaces abstract volume ratio numbers are studied and inequalities relating them to other s numbers of operators are established characterizations of spaces of weak cotype 2 in terms of s numbers and of operator ideal norms are obtained

on the volume of non central sections of a cube abstract let q n be the cube of side length one centered at the origin in r n and let f be an affine n d dimensional subspace of r n having distance to the origin less than or equal to 1 2 where 0 d n we show that the n d dimensional volume of the section q n f is bounded below by a value c d depending only on the codimension d but not on the ambient dimension n or a particular subspace f in the case of hyperplanes d 1 we show that c 1 1 17 is a possible choice we also consider a complex analogue of this problem for a hyperplane section of the polydisc

computing correlated equilibria in multi player games we develop polynomial time algorithms for finding correlated equilibria a well studied notion of rationality that generalizes the nash equilibrium in a broad class of succinctly representable multiplayer games encompassing graphical games anonymous games polymatrix games congestion games scheduling games local effect games as well as several generalizations our algorithm is based on a variant of the existence proof due to hart and schmeidler and employs linear programming duality the ellipsoid algorithm markov chain steady state computations as well as application specific methods for computing multivariate expectations over product distributions for anonymous games and graphical games of bounded tree width we provide a different polynomial time algorithm for optimizing an arbitrary linear function over the set of correlated equilibria of the game in contrast to our sweeping positive results for computing an arbitrary correlated equilibrium we prove that optimizing over correlated equilibria is np hard in all of the other classes of games that we consider

a further improvement of the quantitative subspace theorem in 2002 evertse and schlickewei 11 obtained a quantitative version of the so called absolute parametric subspace theorem this result deals with a parametrized class of twisted heights one of the consequences of this result is a quantitative version of the absolute subspace theorem giving an explicit upper bound for the number of subspaces containing the solutions of the diophantine inequality under consideration in the present paper we further improve evertse s and schlickewei s quantitative version of the absolute parametric subspace theorem and deduce an improved quantitative version of the absolute subspace theorem we combine ideas from the proof of evertse and schlickewei which is basically a substantial refinement of schmidt s proof of his subspace theorem from 1972 22 with ideas from faltings and wustholz proof of the subspace theorem 14 a new feature is an interval result which gives more precise information on the distribution of the heights of the solutions of the system of inequalities considered in the subspace theorem

isomorphic steiner symmetrization this paper proves that there exist 3n steiner symmetrizations that transform any convex set k n into an isomorphic euclidean ball i e if vol k vol dn where dn is the standard euclidean unit ball then k can be transformed into a body k such that c1dn k c2dn where c1 c2 are numerical constants moreover for any c 2 cn symmetrizations are also enough

on the distribution of points in projective space of bounded height in this paper we consider the uniform distribution of points in compact metric spaces we assume that there exists a probability measure on the borel subsets of the space which is invariant under a suitable group of isometries in this setting we prove the analogue of weyl s criterion and the erdos tur an inequality by using orthogonal polynomials associated with the space and the measure in particular we discuss the special case of pro jective space over completions of number elds in some detail an invariant measure in these projective spaces is introduced and the explicit formulas for the orthogonal polynomials in this case are given finally using the analo gous erdos tur an inequality we prove that the set of all projective points over the number eld with bounded arakelov height is uniformly distributed with respect to the invariant measure as the bound increases