the code problem for traces improving the boundaries abstract the code problem for traces given a finite set x decide whether every element in x has a unique factorization over x is decidable if the independence relation equals p 4 the line graph on four nodes additionally it is undecidable for a particular independence relation that does not have c 4 the cycle on four nodes as induced subgraph these results improve on the previously known boundaries of un decidable cases for this problem

boundedness of inhomogeneous journe s type operators on multi parameter local hardy spaces abstract in this paper we establish a boundedness criterion of a class of inhomogeneous journe s type of multi parameter singular integral operators on multi parameter local hardy spaces h p r n 1 r n k recently introduced in ding et al 2019 the consideration of such inhomogeneous journe s type of multi parameter singular integral operators is motivated by the study of the multi parameter pseudo differential operators on multi parameter local hardy spaces the lack of the vanishing moments of both the corresponding functions in the multi parameter local calderon reproducing formula and the multi parameter local littlewood paley stein functions as well as the lack of vanishing moments of atoms in the multi parameter local hardy spaces h p r n 1 r n k with large size all create substantial difficulties in our proofs

a lower bound criterion for iterated commutators abstract we consider iterated commutators of multiplication by a symbol function and smooth calderon zygmund operators described by fourier multipliers of homogeneity 0 we establish a criterion for a collection of symbols so that the corresponding calderon zygmund operators characterize product bmo by means of iterated commutators we therefore extend in part the line of one parameter results following the work of uchiyama and li as well as the result in several parameters concerning commutators with riesz transforms by lacey petermichl pipher wick

moduli spaces and castelnuovo mumford regularity of sheaves on surfaces we generalize donaldson gieseker li and o grady s results on generic smoothness of the moduli spaces of sheaves with fixed determinant and large discriminant to positive characteristic we also show optimal bounds on the castelnuovo mumford regularity of sheaves on surfaces and we use it to give the first general effective results on irreducibility of the moduli spaces

codes and equations on trees the objective of this paper is to study by new formal methods the notion of tree code introduced by m nivat in cite n92 in particular we introduce the notion of stability for sets of trees closed under concatenation this allows us to give a characterization of tree codes which is very close to the algebraic characterization of word codes in terms of free monoids we further define the stable hull of a set of trees and derive a defect theorem for trees which generalizes the analogous result for words as a consequence we obtain some properties of tree codes having two elements moreover we propose a new algorithm to test whether a finite set of trees is a tree code the running time of the algorithm is polynomial in the size of the input we also introduce the notion of tree equation as a complementary point of view to tree codes the main problem emerging in this approach is to decide the satisfiability of tree equations it is a special case of second order unification and it remains still open

the quantization conjecture revisited a strong version of the quantization conjecture of guillemin and sternberg is proved for a reductive group action on a smooth compact polarized variety x l the cohomologies of l over the git quotient x g equal the invariant part of the cohomologies over x this generalizes the theorem of gs on global sections and strengthens its subsequent extensions jk li to riemannroch numbers remarkable by products are the invariance of cohomology of vector bundles over x g under a small change in the defining polarization or under shift desingularization as well as a new proof of boutot s theorem also studied are equivariant holomorphic forms and the equivariant hodgeto de rham spectral sequences for x and its strata whose collapse is shown one application is a new proof of the borel weil bott theorem of ti for the moduli stack of c bundles over a curve and of analogous statements for the moduli stacks and spaces of bundles with parabolic structures collapse of the hodge to de rham sequences for these stacks is also shown

frobenius splitting and cohomology vanishing for schubert varieties let k be an algebraically closed field of characteristic p 0 and x be a projective variety over k we then have the absolute frobenius f x x and an injection ox f 6x given by f fp f e o9x if this makes ox a direct summand in f ox as an x module we call x a frobenius split variety for such a variety the vanishing theorem for ample line bundles follows trivially from serre s vanishing theorem for tensoring ox f ox by an ample line bundle l and noting that l f x f f l f lp projection formula we get that the map in cohomology h x l h x f lp h x lp is an injection iterating this we see that h x l injects into h x lp for every p but for large v the latter is zero thus frobenius split varieties have quite pleasant properties it also turns out that using duality for the frobenius morphism or equivalently the cartier operator one can give a manageable criterion for frobenius splitting the point here is the local nature of duality the relevance of the compatibility of local and global duality was suggested to us by grothendieck s proof of h x y 0 for a noncomplete variety x of dimension n 4 theorem 6 9 and by kempf s paper 10 by this criterion and the bott samelson demazure desingularisation of schubert varieties it follows very easily that schubert varieties in characteristic p are frobenius split the following vanishing theorem is then an immediate consequence let g be a reductive group over the field k of arbitrary characteristic zero or positive let q be a parabolic subgroup and x c g q a schubert variety let l be an ample line bundle on g q then hi x l 0 for i 0 and the restriction map h0 g q l h0 x l is surjective if char k 0 this is a consequence of the compatible frobenius splitting of x in g q and the char k 0 case is handled by semicontinuity the above result for special schubert varieties has been proved by seshadrimusili lakshmibai 12 13 and by kempf 9 by using characteristic free methods for x g q andersen 1 and haboush 5 have given simple proofs using characteristic p methods

periodicities on trees abstract we introduce the notion of periodicity for k ary labeled trees roughly speaking a tree is periodic if it can be obtained by a sequence of concatenations of a smaller tree plus a remainder the period is the shape of such smaller tree i e the corresponding unlabeled tree this definition reduces to the classical one for string when restricted to the case of unary trees then we define the greatest common divisor of two unlabeled trees and relate right congruences to unlabeled trees this allows us to give a characterization of tree periodicity in terms of right congruences and then to prove a periodicity theorem for trees that is a generalization to trees of the fine and wilf s periodicity theorem for words

harmonic analysis on product spaces rn x r and commute with the dilations p33 8 x2 x2 61x1 62x2 for x1 e rn x2 e rm and 61 82 0 then the corresponding operators are related to many problems in the theory of multiple fourier integrals see zygmund 21 the maximal operator which commutes with the p31382 is the jessen marcinkiewicz and zygmund 14 strong maximal operator ms defined by 1 ms f x1 x2 sup ri a y1 y2 1 dy1 dy2 xl x2 gr jr1

boundedness criterion of journeʼs class of singular integrals on multiparameter hardy spaces abstract this article concerns nonconvolutional type operators also known as journe s type operators associated with a multiparameter family of dilations given by x 1 x 2 x m 1 x 1 2 x 2 m x m where x 1 r n 1 x 2 r n 2 x m r n m and m 3 we are especially interested in the boundedness of such operators on the multiparameter hardy spaces this work is motivated by pipher s result on the boundedness of these operators from the multiparameter h p spaces to l p spaces for 0 p 1 and journe s counter example which shows that the number of parameter plays a crucial role in boundedness of singular integral operators on multiparameter hardy spaces journe s work shows that there is a sharp difference between the situations for two and three or more parameters we establish in this paper the necessary and sufficient conditions under which the singular integral operators in journe s class are bounded on multiparameter h p spaces 0 p 1 with arbitrary number of parameters

groupe de picard des varietes de modules de fibres semi stables sur les courbes algebriques soient x une courbe algebrique projective lisse de genre g 2 sur r d des entiers avec r 2 on note u r d la variete de modules des fibres algebriques semi stables sur x de rang r et de degre d us r d l ouvert de u r d correspondant aux fibres stables on sait que u r d est une variete algebrique projective irreductible et normale si r et d ne sont pas premiers entre eux u r d n est pas lisse sauf dans une exception le cas ou g 2 r 2 et d est pair on supposera dans cet article qu on n est pas dans ce cas on a alors codimu r d u r d us r d 2 et u r d us r d est le lieu des points singuliers de u r d si l est un fibre en droites de degre d sur x on note u r l la sous variete fermee de u r d correspondant aux fibres vectoriels de determinant isomorphe a l on etudie dans cet article les groupes de picard pic u r d et pic u r l on montre d abord que les varietes u r d et u r l sont localement factorielles on calcule ensuite pic u r d et pic u r l qui sont decrits au moyen des diviseurs thetas generalises

on extremal cases of hopcroft s algorithm in this paper we consider the problem of minimization of deterministic finite automata dfa with reference to hopcroft s algorithm hopcroft s algorithm has several degrees of freedom so there can exist different executions that can lead to different sequences of refinements of the set of the states up to the final partition we find an infinite family of binary automata for which such a process is unique whatever strategy is chosen some recent papers cf berstel and carton 2004 3 castiglione et al 2008 6 and berstel et al 2009 1 have been devoted to find families of automata for which hopcroft s algorithm has its worst execution time they are unary automata associated with circular words however automata minimization can be achieved also in linear time when the alphabet has only one letter cf paige et al 1985 14 but such a method does not seem to extend to larger alphabet so in this paper we face the tightness of hopcroft s algorithm when the alphabet contains more than one letter in particular we define an infinite family of binary automata representing the worst case of hopcroft s algorithm for each execution they are automata associated with particular trees and we deepen the connection between the refinement process of hopcroft s algorithm and the combinatorial properties of such trees

globally f regular type of moduli spaces we prove that moduli spaces of semistable parabolic bundles and generalized parabolic sheaves with fixed determinant on a smooth projective curve are of globally f regular type

the generalized verschiebung map for curves of genus 2 let c be a smooth curve and m r c the coarse moduli space of vector bundles of rank r and trivial determinant on c we examine the generalized verschiebung map v r m r c p dashrightarrow m r c induced by pulling back under frobenius our main result is a computation of the degree of v 2 for a general c of genus 2 in characteristic p 2 we also give several general background results on the verschiebung in an appendix

factorisation of generalised theta functions i we prove a version of factorisation relating the space of sections of theta bundles on the moduli spaces of parabolic rank 2 vector bundles on curves of genusg andg 1

rings of invariants of 2 2 matrices in positive characteristic abstract we prove that the rings of invariants of 2 2 matrices over an infinite field are cohen macaulay this result generalizes the similar theorem of mehta and ramadas in odd characteristics our approach is more elementary and it uses only some standard facts from the theory of modules with good filtrations and the theory of determinantal rings

multiparameter riesz commutators it is shown that product bmo of s y a chang and r fefferman defined on the space bbb r d 1 otimes cdots otimes bbb r d t can be characterized by the multiparameter commutators of riesz transforms this extends a classical one parameter result of r coifman r rochberg and g weiss and at the same time extends the work of m lacey and s ferguson and m lacey and e terwilleger on multiparameter commutators with hilbert transforms

the undecidability of the second order unification problem abstract it is shown that there is no effective procedure for determining whether or not two terms of the language of second order logic have a common instance

minimal and complete word unification the fundamental satisfiability problem for word equations has been solved recently by makanin however this algorithm is purely a decision algorithm the main result of this paper solves the complementary problem of generating the set of all solutions specifically the algorithm in this paper generates given a word equation a minimal and complete set of unifiers it stops if this set is finite

der kanonische modul eines invariantenrings abstract we calculate the canonical module of a ring of invariants r of a reductive group acting on an affine variety from that we derive a criterion for the gorenstein property to hold for r in case r is the ring of invariants of a graded ring we obtain lower and upper bounds for the degree of the generating function of r

a constructive proof of the fefferman stein decomposition of bmo rn let s vec r left left f r 1 f cdots r n f right f in h 1 left text r n text r right right subset h 1 left text r n text r n 1 right 22 1 begin gathered s vec r left vec g left g j right j 0 cdots n right in text bmo left text r n text r n 1 right sum limits j 0 n vec r j g j 0 text in bmo left text r n text r n 1 right left text r right end gathered 22 2 j 0

moduli of vector bundles frobenius splitting and invariant theory let x be an irreducible projective curve of genus g over an algebraically closed field of positive characteristic 2 3 in part i we prove adapting the degeneration arguments of n tr that moduli spaces of rank two parabolic bundles on x are frobenius split m r for generic smooth x a similar result holds for x nodal a consequence is the verlinde formula in positive characteristic in part ii we give a direct proof of the fact that the local rings of the moduli spaces are f split and further that they are cohen macaulay this involves showing that the ring of invariants of g copies of 2 x 2 matrices under the adjoint action of sl 2 is f split and cohen macaulay

higher order journe commutators and characterizations of multi parameter bmo abstract we characterize l p boundedness of iterated commutators of multiplication by a symbol function and tensor products of riesz and hilbert transforms we obtain a two sided norm estimate that shows that such operators are bounded on l p if and only if the symbol belongs to the appropriate multi parameter bmo class we extend our results to a much more intricate situation commutators of multiplication by a symbol function and paraproduct free journe operators we show that the boundedness of these commutators is also determined by the inclusion of their symbol function in the same multi parameter bmo class in this sense the tensor products of riesz transforms are a representative testing class for journe operators previous results in this direction do not apply to tensor products and only to journe operators which can be reduced to calderon zygmund operators upper norm estimates of journe commutators are new even in the case of no iterations lower norm estimates for iterated commutators only existed when no tensor products were present in the case of one dimension lower estimates were known for products of two hilbert transforms and without iterations new methods using journe operators are developed to obtain these lower norm estimates in the multi parameter real variable setting

stratified bundles and representation spaces abstract for a given stratified bundle e on x we construct an irreducible closed subvariety n e s of the so called representation space r o x s s p s such that n e s f q contains a dense set of v where v is induced by a representation of 1 e t x and is a frame of v at a given point theorem 3 7 as an application we give a simple proof of the main theorem of 1 and its relative version theorem 4 2

frobenius splitting of hilbert schemes of points on surfaces let x be a quasiprojective smooth surface defined over an algebraically closed field of positive characteristic in this note we show that if x is frobenius split then so is the hilbert scheme hilb n x of n points inx in particular we get the higher cohomology vanishing for ample line bundles on hilb n x when x is projective and frobenius split

two weight characterization for commutators of bi parameter fractional integrals abstract let i be the bi parameter fractional integral operator on r n 1 r n 2 i f x r n 1 r n 2 f y 1 y 2 x 1 y 1 n 1 1 x 2 y 2 n 2 2 d y 0 i n i i 1 2 in this paper we give a characterization of two weight norm inequality for the commutator of i we show that for a p q r n b i l p p l q q b b m o where 1 and 1 p 1 q 1 n 1 1 n 2 it extends the recent one parameter theory to the bi parameter setting we use the modern dyadic methods in which the main idea is to represent continuous operators in terms of dyadic operators moreover by introducing some new full and mixed bi parameter dyadic paraproducts we write the commutator as a finite linear combination of them