methods of infinite dimensional morse theory for geodesics on finsler manifolds abstract we prove the shifting theorems of the critical groups of critical points and critical orbits for the energy functionals of finsler metrics on hilbert manifolds of h 1 curves and two splitting lemmas for the functionals on banach manifolds of c 1 curves two results on critical groups of iterated closed geodesics are also proved their corresponding versions on riemannian manifolds are based on the usual splitting lemma by gromoll and meyer 1969 our approach consists in deforming the square of the finsler metric in a lagrangian which is smooth also on the zero section and then in using the splitting lemma for nonsmooth functionals that the author recently developed in lu 2011 0000 2013 the argument does not involve finite dimensional approximations and any palais result in palais 1966 as an application we extend to finsler manifolds a result by v bangert and w klingenberg 1983 about the existence of infinitely many geometrically distinct closed geodesics on a compact riemannian manifold

the total coordinate ring of a wonderful variety abstract we study the cone of effective divisors and the total coordinate ring of wonderful varieties with applications to their automorphism group we show that the total coordinate ring of any spherical variety is obtained from that of the associated wonderful variety by a base change of invariant subrings

corrigendum to the conley conjecture for hamiltonian systems on the cotangent bundle and its analogue for lagrangian systems j funct anal 256 9 2009 2967 3034 abstract in lines 8 11 of lu 2009 18 p 2977 we wrote for integer m 3 if m is c m smooth and c m 1 smooth l r t m r satisfies the assumptions l1 l3 then the functional l is c 2 smooth bounded below satisfies the palais smale condition and all critical points of it have finite morse indexes and nullities see 1 prop 4 1 4 2 and 4 however as proved in abbondandolo and schwarz 2009 2 the claim that l is c 2 smooth is true if and only if for every t q the function v l t q v is a polynomial of degree at most 2 so the arguments in lu 2009 18 are only valid for the physical hamiltonian in 1 2 and corresponding lagrangian therein in this note we shall correct our arguments in lu 2009 18 with a new splitting lemma obtained in lu 2011 20

the logarithmic cotangent complex we define the cotangent complex of a morphism of fine log schemes prove that it is functorial and construct under certain restrictions a transitivity triangle we also discuss its relationship with deformation theory

adaptive methods for linear programming decoding detectability of failures of linear programming lp decoding and the potential for improvement by adding new constraints motivate the use of an adaptive approach in selecting the constraints for the underlying lp problem in this paper we make a first step in studying this method and show that by starting from a simple lp problem and adaptively adding the necessary constraints the complexity of lp decoding can be significantly reduced in particular we observe that with adaptive lp decoding the sizes of the lp problems that need to be solved become practically independent of the density of the parity check matrix we further show that adaptively adding extra constraints such as constraints based on redundant parity checks can provide large gains in the performance

uber bewertungen welche unter einer reduktiven gruppe invariant sind letg be a reductive group defined over an algebraically closed fieldk and letx be ag variety in this paper we studyg invariant valuationsv of the fieldk of rational functions onx these objects are fundamental for the theory of equivariant completions ofx letb be a borel subgroup andu the unipotent radical ofb it is proved thatv is uniquely determined by its restriction toku then we study the set of invariant valuations having some fixed restrictionv0 tokb ifv0 is geometric i e induced by a prime divisor then this set is a polyhedron in some vector space in characteristic zero we prove that this polyhedron is a simplicial cone and in fact the fundamental domain of finite reflection groupwx thus the classification of invariant valuations is almost reduced to the classification of valuations ofkb

morse theory for continuous functionals abstract using the notion of weak slope introduced by m degiovanni and m marzocchi 1994 ann mat pura appl 167 73 100 and also independently by c katriel 1994 ann inst h poincare anal non lineaire11 189 209 we obtain some results in morse theory for continuous functionals having isolated critical points the main tools are deformation properties generalizing classical ones for smooth functionals and refining those of j n corvellec m degiovanni and m marzocchi 1993 topol methods nonlinear anal 1 151 171 we obtain for example generalized morse relations through the notion of critical group which has a natural link with the weak slope these results are developed in view of applications to non smooth variational problems

multiple periodic points of the poincare map of lagrangian systems on tori in this paper suppose l t x p 1 over 2 a x p cdot p v t x a is positive definite and symmetric and both a and v are c 3 and 1 periodic in all of their variables we prove that the poincare map i e the time 1 solution map of the lagrangian system

asymptotic enumeration methods for analyzing ldpc codes we show how asymptotic estimates of powers of polynomials with nonnegative coefficients can be used in the analysis of low density parity check ldpc codes in particular we show how these estimates can be used to derive the asymptotic distance spectrum of both regular and irregular ldpc code ensembles we then consider the binary erasure channel bec using these estimates we derive lower bounds on the error exponent under iterative decoding of ldpc codes used over the bec both regular and irregular code structures are considered these bounds are compared to the corresponding bounds when optimal maximum likelihood ml decoding is applied

mathbb a 1 connected varieties of rank one over nonclosed fields in this paper we prove two results regarding the arithmetics of separably mathbb a 1 connected varieties of rank one first we prove that over a large field there is an mathbb a 1 curve through any rational point of the boundary if the boundary divisor is smooth and separably rationally connected secondly we generalize a theorem of hassett tschinkel for the zariski density of integral points over function fields of curves

the approximate maximum likelihood certificate the confidence in the reliability of a codeword output by some not necessarily optimal decoding algorithm is discussed a new property which relies on the linear programming lp decoder the approximate maximum likelihood certificate amlc is introduced to address this issue as follows first the channel output vector is decoded by some symmetric decoder d e g belief propagation or min sum algorithm decoding second the channel output vector is decoded by lp decoding third if the decoding result of d is a codeword its lp value is compared to the lp value of the lp decoding result the latter need not be a codeword if these two values are close the amlc holds using upper bounding techniques we show that the conditional frame error probability given that the amlc holds is with some degree of confidence below a threshold in channels with low noise this threshold is orders of magnitude lower than the simulated frame error rate and our bound holds with a very high degree of confidence this is in stark contrast with standard monte carlo simulation which would require excessively long runs to demonstrate like performance when the amlc holds our approach thus provides the decoder with extra error detection capability which is especially important in applications requiring high data integrity

a1 curves on log k3 surfaces abstract in this paper we study a 1 curves on log k3 surfaces we classify all genuine log k3 surfaces of type ii which admit countably infinite a 1 curves

on the energy functional on finsler manifolds and applications to stationary spacetimes in this paper we first study some global properties of the energy functional on a non reversible finsler manifold in particular we present a fully detailed proof of the palais smale condition under the completeness of the finsler metric moreover we define a finsler metric of randers type which we call fermat metric associated to a conformally standard stationary spacetime we shall study the influence of the fermat metric on the causal properties of the spacetime mainly the global hyperbolicity moreover we study the relations between the energy functional of the fermat metric and the fermat principle for the light rays in the spacetime this allows one to obtain existence and multiplicity results for light rays using the finsler theory finally the case of timelike geodesics with fixed energy is considered

using linear programming to decode binary linear codes a new method is given for performing approximate maximum likelihood ml decoding of an arbitrary binary linear code based on observations received from any discrete memoryless symmetric channel the decoding algorithm is based on a linear programming lp relaxation that is defined by a factor graph or parity check representation of the code the resulting lp decoder generalizes our previous work on turbo like codes a precise combinatorial characterization of when the lp decoder succeeds is provided based on pseudocodewords associated with the factor graph our definition of a pseudocodeword unifies other such notions known for iterative algorithms including stopping sets irreducible closed walks trellis cycles deviation sets and graph covers the fractional distance d sub frac of a code is introduced which is a lower bound on the classical distance it is shown that the efficient lp decoder will correct up to spl lceil d sub frac 2 spl rceil 1 errors and that there are codes with d sub frac spl omega n sup 1 spl epsi an efficient algorithm to compute the fractional distance is presented experimental evidence shows a similar performance on low density parity check ldpc codes between lp decoding and the min sum and sum product algorithms methods for tightening the lp relaxation to improve performance are also provided

automorphisms root systems and compactifications of homogeneous varieties let g be a complex semisimple group and let h c g be the group of fixed points of an involutive automorphism of g then x g h is called a symmetric variety in cp de concini and procesi have constructed an equivariant compactification x which has a number of remarkable properties some of them being i the boundary is the union of divisors d1 dr ii there are exactly 2r orbits their closures are the intersections di1 n ndi even schematically in particular there is only one closed orbit iii in case g is of adjoint type all orbit closures are smooth it is called the wonderful embedding of x or a complete symmetric variety and is the foundation for most deeper results about x independently luna and vust developed in lv a general theory of equivariant compactifications of homogeneous varieties under a connected reductive group g in particular they realized the reason which makes symmetric varieties behave so nicely a borel subgroup b has an open dense orbit in g h varieties with this property are called spherical luna and vust were able to describe all equivariant compactifications of them in terms of combinatorial data very similar to torus embeddings which are actually a special case they obtained in particular that every spherical embedding has only finitely many orbits nevertheless the reason for the existence of a compactification with properties i iii remained mysterious then brion and pauer established a relation with the automorphism group they proved in bp a spherical variety x g h possesses an equivariant compactification with exactly one closed orbit if and only if autg x ng h h is finite in this case there is a unique one which dominates all others the wonderful compactification x they also showed that the orbits of x correspond to the faces of a strictly convex polyhedral cone z then properties i and ii above are equivalent to z being simplicial this fact is much deeper and was proved by brion in brl in fact he showed much more let f be the set of characters of b which are the characters of a rational b eigenfunction on x this is a finitely generated free abelian group then the cone z is a subset of the real vector space hom f r brion showed that there is a finite reflection group wx acting on f such that z is one of its weyl chambers in case of a symmetric variety wx is its little weyl group

the conley conjecture for hamiltonian systems on the cotangent bundle and its analogue for lagrangian systems abstract in this paper the conley conjecture which was recently proved by franks and handel j franks m handel periodic points of hamiltonian surface diffeomorphism geom topol 7 2003 713 756 for surfaces of positive genus hingston n hingston subharmonic solutions of hamiltonian equations on tori ann math in press for tori and ginzburg v l ginzburg the conley conjecture arxiv math sg 0610956v1 for closed symplectically aspherical manifolds is proved for c 1 hamiltonian systems on the cotangent bundle of a c 3 smooth compact manifold m without boundary of a time 1 periodic c 2 smooth hamiltonian h r t m r which is strongly convex and has quadratic growth on the fibers namely we show that such a hamiltonian system has an infinite sequence of contractible integral periodic solutions such that any one of them cannot be obtained from others by iterations if h also satisfies h t q p h t q p for any t q p r t m it is shown that the time 1 map of the hamiltonian system if exists has infinitely many periodic points siting in the zero section of t m if m is c 5 smooth and dim m 1 h is of c 4 class and independent of time t then for any 0 the corresponding system has an infinite sequence of contractible periodic solutions of periods of integral multiple of such that any one of them cannot be obtained from others by iterations or rotations these results are obtained by proving similar results for the lagrangian system of the fenchel transform of h l r t m r which is proved to be strongly convex and to have quadratic growth in the velocities yet

improved linear programming decoding of ldpc codes and bounds on the minimum and fractional distance we examine ldpc codes decoded using linear programming lp four contributions to the lp framework are presented first a new method of tightening the lp relaxation and thus improving the lp decoder is proposed second we present an algorithm which calculates a lower bound on the minimum distance of a specific code this algorithm exhibits complexity which scales quadratically with the block length third we propose a method to obtain a tight lower bound on the fractional distance also with quadratic complexity and thus less than previously existing methods finally we show how the fundamental lp polytope for generalized ldpc codes and nonbinary ldpc codes can be obtained

performance bounds for nonbinary linear block codes over memoryless symmetric channels the performance of nonbinary linear block codes is studied in this paper via the derivation of new upper bounds on the block error probability under maximum likelihood ml decoding the transmission of these codes is assumed to take place over a memoryless and symmetric channel the new bounds which are based on the gallager bounds and their variations are applied to the gallager ensembles of nonbinary and regular low density parity check ldpc codes these upper bounds are also compared with sphere packing lower bounds this study indicates that the new upper bounds are useful for the performance evaluation of coded communication systems which incorporate nonbinary coding techniques

infinitely many periodic solution orbits of autonomous lagrangian systems on tori we prove that for any 0 the autonomous lagrangian system ddtlx x x lx x x 0 on the standard n dimensional torus possesses infinitely many geometrically distinct contractible periodic solution orbits whose periods are integer multiples of provided that the lagrangian function satisfies l x p 12ap p v x where a is a positive definite symmetric real matrix and v is c3 and 1 periodic in all of its variables

families of rationally connected varieties recall that a proper variety x is said to be rationally connected if two general points p q x are contained in the image of a map g p x this is clearly a birationally invariant property when x is smooth this turns out to be equivalent to the a priori weaker condition that two general points can be joined by a chain of rational curves and also to the a priori stronger condition that for any finite subset x there is a map g p x whose image contains and such that gtx is an ample bundle

variations on the gallager bounds connections and applications there has been renewed interest in deriving tight bounds on the error performance of specific codes and ensembles based on their distance spectrum we discuss many reported upper bounds on the maximum likelihood ml decoding error probability and demonstrate the underlying connections that exist between them in addressing the gallager bounds and their variations we focus on the duman and salehi see ieee trans commun vol 46 p 717 723 1998 variation which originates from the standard gallager bound a large class of efficient bounds or their chernoff versions is demonstrated to be a special case of the generalized second version of the duman and salehi bounds implications and applications of these observations are pointed out including the fully interleaved fading channel resorting to either matched or mismatched decoding the proposed approach can be generalized to geometrically uniform nonbinary codes finite state channels bit interleaved coded modulation systems and it can be also used for the derivation of upper bounds on the conditional decoding error probability

the existence of two closed geodesics on every finsler 2 sphere in this paper we prove that for every finsler metric on s2 there exist at least two distinct prime closed geodesics

on the asymptotic weight and stopping set distribution of regular ldpc ensembles in this correspondence we estimate the variance of weight and stopping set distribution of regular low density parity check ldpc ensembles using this estimate and the second moment method we obtain bounds on the probability that a randomly chosen code from regular ldpc ensemble has its weight distribution and stopping set distribution close to respective ensemble averages we are able to show that a large fraction of total number of codes have their weight and stopping set distribution close to the average

rationally connected varieties over local fields let x be a proper variety defined over a field k following ma we say that two points x x x k are r equivalent if they can be connected by a chain of rational curves defined over k cf 4 1 in essence two points are r equivalent if they are obviously rationally equivalent several authors have proved finiteness results over local and global fields cubic hypersurfaces ma sd linear algebraic groups ct sa vo1 gi vo2 rational surfaces ct co ct1 ct sk1 quadric bundles and intersections of two quadrics ct sa sd pa su r equivalence is only interesting if there are plenty of rational curves on x at least over k such varieties have been studied in the series of papers komi mo1 ko mi mo3 see also ko1 there are many a priori different ways of defining what plenty of rational curves should mean fortunately many of these turn out to be equivalent and this leads to the notion of rationally connected varieties see ko mi mo2 ko1 iv 3 ko2 4 1 2

stable logarithmic maps to deligne faltings pairs i we introduce a new compactication of the space of relative stable maps this approach uses logarithmic geoemetry in the sense of kato fontaineillusie without taking expansions of the target the underlying structures of the stable logarithmic maps are stable in the usual sense

lower bounds on the error rate of ldpc code ensembles the ensemble of regular low definition parity check ldpc codes is considered using concentration results on the weight distribution lower bounds on the error rate of a random code in the ensemble are derived these bounds hold with some confidence level combining these results with known lower bounds on the error exponent confidence intervals on the error exponent under maximum likelihood ml decoding are obtained over a large range of channel parameter and transmission rate values when the graph connectivity is sufficiently large the upper bound of the interval approaches the lower bound and the probability that the error exponent is within the interval can be arbitrarily close to one in fact in this case the true error exponent approaches the maximum between the random coding and the expurgated random coding exponents with probability that approaches one

rationally connected varieties over finite fields let x be a geometrically rational or more generally separably rationally connected variety over a finite field k we prove that if k is large enough then x contains many rational curves defined over k as a consequence we prove that r equivalence is trivial on x if k is large enough these results imply the following conjecture of j l colliot thelene let y be a rationally connected variety over a number field f for a prime p let y p denote the corresponding variety over the local field f p then for almost all primes p the chow group of 0 cycles on y p is trivial and r equivalence is trivial on y p

every rationally connected variety over the function field of a curve has a rational point in a paper from 1992 kollar miyaoka and mori posed the following question given a proper flat morphism f p x with target a nonsingular curve and whose geometric generic fiber is rationally connected does it follow that f admits a regular section in the case that the ground field is an algebraically closed field of characteristic zero this was answered affirmatively by graber harris and starr using a topological argument we prove that f admits a regular section when the ground field is an algebraically closed field of arbitrary characteristic the key ingredients in the proof are the following a proposition 2 1 which shows that after taking the reduced fiber product of f with a generically etale morphism y x we may assume that the smooth locus of f intersects every geometric fiber and b proposition 1 1 which is a purely algebraic analogue of the moving branch points argument in graber harris starr the reader is cautioned that our definition of separably rationally connected differs slightly from the original definition