subelliptic p harmonic maps into spheres and the ghost of hardy spaces the story begins with the paper of muller 59 who for the sake of an application to nonlinear elasticity proved that if the jacobian determinant ju of a sobolev map u w 1 n loc rn rn is nonnegative then it belongs locally to l log l the result is quite intriguing since a priori holder inequality implies only that ju l1 and one does not suspect any higher integrability if one does not assume that ju is nonnegative then as coifman lions meyer and semmes 14 have proved ju belongs to the local hardy space h 1 loc since a nonnegative function belongs to the local hardy space if and only if it belongs locally to l log l the result of coifman lions meyer and semmes generalizes that of muller in fact coifman lions meyer and semmes proved more namely for 1 p the scalar product of a divergence free vector field e lp rn rn and a curl free vector field b lp p 1 rn rn belongs to the hardy space h 1 a priori holder inequality implies integrability of this expression only note that the jacobian is of this form the discovery that many algebraic expressions involving partial derivatives belong to the hardy space turned out to be important and widely applicable in nonlinear partial differential equations this was first shown by helein 40 41 42 let us describe briefly his result consider maps u bn s m from the n dimensional ball to the mdimensional sphere such that the p energy of u given by the functional

the classification of complete stable area stationary surfaces in the heisenberg group h1 abstract we prove that any c 2 complete orientable connected stable area stationary surface in the sub riemannian heisenberg group h 1 is either a euclidean plane or congruent to the hyperbolic paraboloid t x y

an upper bound on the zero error list coding capacity presents an upper bound on the zero error list coding capacity of discrete memoryless channels using this bound we show that the list 3 capacity of the 4 3 channel is at most 0 3512 bits improving the best previous bound the relation of the bound to earlier similar bounds in particular to korner s 1986 graph entropy bound is discussed

the gauss green theorem for fractal boundaries v is the 1 vectorfield dual to if 1 fi dx1 dxi dxn then v f1 fn there has been considerable effort in the literature e g jk m p to extend this formula to permit integrands of less regularity by generalizing the lebesgue integral on the other hand invariably the situations in which 1 holds require fairly strong hypotheses on the boundary e g that it should have sigmafinite n 1 measure or that the gradient of the characteristic function of be a vector valued measure with finite total variation f p however there is a natural way to expand the validity of 1 to much more general boundaries while still using the ordinary lebesgue integral this is the topic of the present paper for the case of lipschitz forms the results of this paper follow readily from whitney s theory of flat chains w2 however his approach to the gauss green theorem is not widely appreciated because he focused on chains and cochains where effectively 1 is used to define the exterior derivative in hn we extend whitney s method to treat the more general holder case only the case n 2 is discussed

a finite element approximation for a class of degenerate elliptic equations in this paper we exhibit a finite element method fitting a suitable geometry naturally associated with a class of degenerate elliptic equations usually called grushin type equations in a plane region and we discuss the related error estimates

inegalites de sobolev produit sur les groupes de lie nilpotents on generalise dans le cadre des groupes de lie nilpotents reels et connexes l inegalite classique due a gagliardo et nirenberg

the morse sard theorem for clarke critical values abstract the morse sard theorem states that the set of critical values of a c k smooth function defined on a euclidean space r d has lebesgue measure zero provided k d this result is hereby extended for generalized critical values of continuous selections over a compactly indexed countable family of c k functions it is shown that these functions are lipschitz continuous and the set of their clarke critical values is null

isoperimetric inequality under kahler ricci flow let bf m g t be a k ahler ricci flow with positive first chern class first we prove a uniform isoperimetric inequality for all time in the process we also prove a cheng yau type log gradient bound for positive harmonic functions on bf m g t without assuming the ricci curvature is bounded from below

weighted sobolev poincare inequalities and pointwise estimates for a class of degenerate elliptic equations in this paper we prove a sobolev poincare inequality for a class of function spaces associated with some degenerate elliptic equations these estimates provide us with the basic tool to prove an invariant harnack inequality for weak positive solutions in addition holder regularity of the weak solutions follows in a standard way let y z 7 ai aij9 a be a second order degenerate elliptic operator in divergence form with measurable coefficients in this paper we shall obtain pointwise estimates for the weak solutions of su 0 h61der continuity of the weak solutions and harnack inequality for nonnegative solutions let us recall that the original results for elliptic operators were obtained by de giorgi nash and moser an extensive bibliography about the degenerate case can be found in fli fl2 fs to introduce the results of the present paper let us recall some recent results in fl1 fl2 a suitable metric d is associated with the differential operator y in such a way that we obtain a new geometry which is natural for the degenerate operator as the euclidean geometry is natural for the laplace operator or more precisely as a suitable riemannian geometry is natural for a secondorder elliptic operator in the smooth case this idea is contained in many papers we refer to fp nsw the basic results in fl 1 fl2 are obtained via a precise description of this geometry under suitable technical hypotheses on the coefficients whose aim is to give a nonsmooth formulation of the hormander hypoellipticity condition for sum of squares operators we note that the same idea is used in nsw s j v to obtain pointwise estimates for sum of squares operators on the other hand a different class of degenerate elliptic operators is considered in fks instead of a geometrical degeneracy a measure degeneracy is allowed a typical example of this class is given by yu div wo x vu where cl is a weight function belonging to the a2 class of muckenhoupt unified results for a class containing both the operators in fl 1 and in fks have received by the editors august 1 1989 1980 mathematics subject classification 1985 revision primary 46e35 35j70 partially supported by g n a f a of c n r and m u r s t italy 1991 american mathematical society 0002 9947 91 1 00 25 per page

fine properties of functions with bounded variation in carnot caratheodory spaces abstract we study properties of functions with bounded variation in carnot caratheodory spaces we prove their almost everywhere approximate differentiability and we examine their approximate discontinuity set and the decomposition of their distributional derivatives under an additional assumption on the space called property r we show that almost all approximate discontinuities are of jump type and we study a representation formula for the jump part of the derivative

asymptotic error free partitioning over noisy boolean multiaccess channels in this paper we consider the problem of partitioning active users in a manner that facilitates multi access without collision the setting is of a noisy synchronous boolean and multi access channel where k active users out of a total of n users seek channel access a solution to the partition problem places each of the n users in one of k groups or blocks such that no two active nodes are in the same block we consider a simple but non trivial and illustrative case of k 2 active users and study the number of steps t used to solve the partition problem by random coding and a suboptimal decoding scheme we show that for any t geq c 1 xi 1 log n where c 1 and xi 1 are positive constants independent of n and where xi 1 can be arbitrary small the partition problem can be solved with error probability p e vphantom r a n to 0 for large n under the same scheme we also bound t from the other direction establishing that for any t leq c 2 xi 2 log n the error probability p e n to 1 for large n again c 2 and xi 2 are constants and xi 2 can be arbitrarily small these bounds on the number of steps are lower than the tight achievable lower bound in terms of t geq c g xi log n for group testing in which all active users are identified rather than just partitioned thus partitioning may prove to be a more efficient approach for multi access than group testing

monotone sobolev functions in planar domains level sets and smooth approximation we prove that almost every level set of a sobolev function in a planar domain consists of points jordan curves or homeomorphic copies of an interval for monotone sobolev functions in the plane we have the stronger conclusion that almost every level set is an embedded 1 dimensional topological submanifold of the plane here monotonicity is in the sense of lebesgue the maximum and minimum of the function in an open set are attained at the boundary our result is an analog of sard s theorem which asserts that for a c 2 smooth function in a planar domain almost every value is a regular value as an application using the theory of p harmonic functions we show that monotone sobolev functions in planar domains can be approximated uniformly and in the sobolev norm by smooth monotone functions

on the weak continuity of elliptic operators and applications to potential theory in this paper we establish weak continuity results for quasilinear elliptic and subellip tic operators of divergence form acting on corresponding classes of subharmonic functions these results are analogous to our earlier results for fully nonlinear fc hessian operators from the weak continuity we derive various potential theoretic results including capacity estimates potential esti mates and the wiener criterion for regular boundary points our methods make substantial use of harnack inequalities for solutions 1 introduction there are numerous papers on the local behavior of solu tions subsolutions and supersolutions of second order quasilinear elliptic diver gence structure partial differential equations as well as extensions to degenerate elliptic and subelliptic equations in this paper we show that many properties including the quasi continuity of subharmonic functions asymptotic behavior of fundamental solutions removability of singularities wiener criteria for regular boundary points can be derived via harnack inequalities for solutions the two main ingredients in our approach which follow from our treatment of the fully nonlinear hessian operators in 45 46 47 together with that of labutin 30 are a weak convergence result and a pointwise potential estimate for subharmonic functions the weak convergence result is analogous to that for hessian operators 45 46 while the potential estimate extends that found for quasilinear elliptic operators by kilpelainen and maly 27 weak convergence results in the theory of nonlinear partial differential equa tions have been addressed in the survey 13 in this paper we treat nonlinear elliptic divergence structure operators for which harnack type inequalities are available to state our result we need an appropriate notion of subharmonic func tion in the fully nonlinear case 46 we employed the viscosity notion based on differential inequalities for smooth test functions for quasilinear divergence structure equations the traditional definition using the homogeneous equation is more convenient accordingly we shall say that a continuous function w in a domain q in euclidean space ir is harmonic with respect to an operator or c harmonic in short if it is a solution of the homogeneous equation c u 0

boolean compressed sensing and noisy group testing the fundamental task of group testing is to recover a small distinguished subset of items from a large population while efficiently reducing the total number of tests measurements the key contribution of this paper is in adopting a new information theoretic perspective on group testing problems we formulate the group testing problem as a channel coding decoding problem and derive a single letter characterization for the total number of tests used to identify the defective set although the focus of this paper is primarily on group testing our main result is generally applicable to other compressive sensing models

holder domains and poincare domains a domain d c rd of finite volume is said to be a p poincare domain if there is a constant mp d so that fu udii dx d then d is a p poincare domain this answers a question of axler and shields regarding the image of the unit disk under a holder continuous conformal mapping we also consider geometric conditions which imply that the imbedding of the sobolev space w p d lp d is compact and prove that this is the case for a holder domain d

regularity of c 1 smooth surfaces with prescribed p mean curvature in the heisenberg group we consider a c1 smooth surface with prescribed p or h mean curvature in the 3 dimensional heisenberg group assuming only the prescribed p mean curvature h in c 0 we show that any characteristic curve is c2 smooth and its line curvature equals h in the nonsingular domain by introducing characteristic coordinates and invoking the jump formulas along characteristic curves we can prove that the legendrian or horizontal normal gains one more derivative therefore the seed curves are c2 smooth we also obtain the uniqueness of characteristic and seed curves passing through a common point under some mild conditions respectively these results can be applied to more general situations

area stationary surfaces in the heisenberg group h1 we consider area stationary surfaces perhaps with a volume constraint in the heisenberg group h1 endowed with its carnot caratheodory distance by analyzing the first variation of area we characterize c2 area stationary surfaces as those with mean curvature zero or constant if a volume preserving condition is assumed and such that the characteristic curves meet orthogonally the singular curves moreover a minkowski type formula relating the area the mean curvature and the volume is obtained for volume preserving area stationary surfaces enclosing a given region as a consequence of the characterization of area stationary surfaces we refine the bernstein type theorem given in jih hsin cheng jenn fang hwang andrea malchiodi paul yang minimal surfaces in pseudohermitian geometry ann sc norm super pisa cl sci 5 4 1 2005 129 177 mr 2165405 and nicola garofalo scott pauls the bernstein problem in the heisenberg group arxiv math dg 0209065 to describe all c2 entire area stationary graphs over the xy plane in a calibration argument shows that these graphs are globally area minimizing finally by using the description of the singular set in jih hsin cheng jenn fang hwang andrea malchiodi paul yang minimal surfaces in pseudohermitian geometry ann sc norm super pisa cl sci 5 4 1 2005 129 177 mr 2165405 the characterization of area stationary surfaces and the ruling property of constant mean curvature surfaces we prove our main results where we classify area stationary surfaces in with or without a volume constraint and non empty singular set in particular we deduce the following counterpart to alexandrov uniqueness theorem in euclidean space any compact connected c2 surface in area stationary under a volume constraint must be congruent to a rotationally symmetric sphere obtained as the union of all the geodesics of the same curvature joining two points as a consequence we solve the isoperimetric problem in assuming c2 smoothness of the solutions 13 completely reducible hypersurfaces in a pencil pages 672 688 j v pereira s yuzvinsky preview pdf 324 k related articles 14 modules over motivic cohomology pages 689 727 oliver rondigs paul arne ostvaer preview pdf 303 k related articles 15 the fundamental group of period domains over finite fields pages 728 741 sascha orlik preview pdf 154 k related articles articles 1 15

regular submanifolds graphs and area formula in heisenberg groups we describe intrinsically regular submanifolds in heisenberg groups h low dimensional and low codimensional submanifolds turn out to be of a very different nature the first ones are legendrian surfaces while low codimensional ones are more general objects possibly non euclidean rectifiable nevertheless we prove that they are graphs in a natural group way as well as that an area formula holds for the intrinsic hausdorff measure finally they can be seen as federer fleming currents given a natural complex of differential forms on h

instability of graphical strips and a positive answer to the bernstein problem in the heisenberg group h1 in the first heisenberg group h 1 with its sub riemannian structure generated by the horizontal subbundle we single out a class of c 2 non characteristic entire intrinsic graphs which we call strict graphical strips we prove that such strict graphical strips have vanishing horizontal mean curvature i e they are h minimal and are unstable i e there exist compactly supported deformations for which the second variation of the horizontal perimeter is strictly negative we then show that modulo left translations and rotations about the center of the group every c 2 entire hminimal graph with empty characteristic locus and which is not a vertical plane contains a strict graphical strip combining these results we prove the conjecture that in h 1 the only stable c 2 hminimal entire graphs with empty characteristic locus are the vertical planes

isoperimetric and sobolev inequalities for carnot caratheodory spaces and the existence of minimal surfaces after hormander s fundamental paper on hypoellipticity 54 the study of partial differential equations arising from families of noncommuting vector fields has developed significantly in this paper we study some basic functional and geometric properties of general families of vector fields that include the hormander type as a special case similar to their classical counterparts such properties play an important role in the analysis of the relevant differential operators both linear and nonlinear to motivate our results we recall some classical inequalities let e c r be a caccioppoli set a measurable set having a locally finite perimeter then one has the isoperimetric inequality

sharp moser trudinger inequality on the heisenberg group at the critical case and applications let h cn r be the n dimensional heisenberg group q 2n 2 be the homogeneous dimension of h q qq 1 and z 4 t2 14 be the homogeneous norm of z t h then we prove the following sharp moser trudinger inequality on h theorem 1 6 there exists a positive constant q q 2 n 12 q 12 q2 1 n 1 q 1 such that for any pair satisfying 0 q 0 q 1 q there holds sup u 1 1 h1 exp u q q 1 k 0q 2 kk u kq q 1 c q the constant q 1 q is best possible in the sense that the supremum is infinite if q 1 q here is any positive number and u 1 h hu q h u q 1 q our result extends the sharp moser trudinger inequality by cohn and lu 2001 19 on domains of finite measure on h and sharpens the recent result of cohn et al 2012 18 where such an inequality was studied for the subcritical case q 1 q we carry out a completely different and much simpler argument than that in cohn et al 2012 18 to conclude the critical case our method avoids using the rearrangement argument which is not available in an optimal way on the heisenberg group and can be used in more general settings such as riemanian manifolds appropriate metric spaces etc as applications we establish the existence and multiplicity of nontrivial nonnegative solutions to certain nonuniformly subelliptic equations of q laplacian type on the heisenberg group theorems 1 8 1 9 1 10 and 1 11 divh hu q 2 hu v u q 2u f u eh with nonlinear terms f of maximal exponential growth exp u qq 1 as u in particular when e 0 the existence of a nontrivial solution is also given

constant mean curvature surfaces in sub riemannian geometry we investigate the minimal and isoperimetric surface problems in a large class of sub riemannian manifolds the so called vertically rigid spaces we construct an adapted connection for such spaces and using the variational tools of bryant griffiths and grossman derive succinct forms of the euler lagrange equations for critical points for the associated variational problems using the euler lagrange equations we show that minimal and isoperimetric surfaces satisfy a constant horizontal mean curvature conditions away from characteristic points moreover we use the formalism to construct a horizontal second fundamental form ii0 for vertically rigid spaces and as a first application use ii0 to show that minimal surfaces cannot have points of horizontal positive curvature and that minimal surfaces in carnot groups cannot be locally strictly horizontally geometrically convex we note that the convexity condition is distinct from others currently in the literature

on the hausdorff dimension of the mather quotient under appropriate assumptions on the dimension of the ambient manifold and the regularity of the hamiltonian we show that the mather quotient is small in term of hausdorff dimension then we present applications in dynamics

area stationary surfaces inside the sub riemannian three sphere we consider the sub riemannian metric gh on mathbb s 3 given by the restriction of the riemannian metric of curvature 1 to the plane distribution orthogonal to the hopf vector field we compute the geodesics associated to the carnot caratheodory distance and we show that depending on their curvature they are closed or dense subsets of a clifford torus we study area stationary surfaces with or without a volume constraint in mathbb s 3 g h by following the ideas and techniques by ritore and rosales area stationary surfaces in the heisenberg group mathbb h 1 arxiv math dg 0512547 we introduce a variational notion of mean curvature characterize stationary surfaces and prove classification results for complete volume preserving area stationary surfaces with non empty singular set we also use the behaviour of the carnot caratheodory geodesics and the ruling property of constant mean curvature surfaces to show that the only c2 compact connected embedded surfaces in mathbb s 3 g h with empty singular set and constant mean curvature h such that h sqrt 1 h 2 is an irrational number are clifford tori finally we describe which are the complete rotationally invariant surfaces with constant mean curvature in mathbb s 3 g h

on zero error communication via quantum channels in the presence of noiseless feedback we initiate the study of zero error communication via quantum channels when the receiver and the sender have at their disposal a noiseless feedback channel of unlimited quantum capacity generalizing shannon s zero error communication theory with instantaneous feedback we first show that this capacity is only a function of the linear span of choi kraus operators of the channel which generalizes the bipartite equivocation graph of a classical channel and which we dub non commutative bipartite graph then we go on to show that the feedback assisted capacity is non zero allowing for a constant amount of activating noiseless communication if and only if the non commutative bipartite graph is non trivial and give a number of equivalent characterizations this result involves a far reaching extension of the conclusive exclusion of quantum states we then present an upper bound on the feedback assisted zero error capacity motivated by a conjecture originally made by shannon and proved later by ahlswede we demonstrate that this bound to have many good properties including being additive and given by a minimax formula we also prove a coding theorem showing that this quantity is the entanglement assisted capacity against an adversarially chosen channel from the set of all channels with the same choi kraus span which can also be interpreted as the feedback assisted unambiguous capacity the proof relies on a generalization of the postselection lemma de finetti reduction that allows to reflect additional constraints and which we believe to be of independent interest this capacity is a relaxation of the feedback assisted zero error capacity however we have to leave open the question of whether they coincide in general we illustrate our ideas with a number of examples including classical quantum channels and weyl diagonal channels and close with an extensive discussion of open questions

strongly stable surfaces in sub riemannian 3 space forms abstract a surface of constant mean curvature cmc equal to h in a sub riemannian 3 manifold is strongly stable if it minimizes the functional area 2 h volume up to second order in this paper we obtain some criteria ensuring strong stability of surfaces in sasakian 3 manifolds we also produce new examples of c 1 complete cmc surfaces with empty singular set in the sub riemannian 3 space forms by studying those ones containing a vertical line as a consequence we are able to find complete strongly stable non vertical surfaces with empty singular set in the sub riemannian hyperbolic 3 space m 1 in relation to the bernstein problem in m 1 we discover strongly stable c entire minimal graphs in m 1 different from vertical planes these examples are in clear contrast with the situation in the first heisenberg group where complete strongly stable surfaces with empty singular set are vertical planes finally we analyze the strong stability of cmc surfaces of class c 2 and non empty singular set in the sub riemannian 3 space forms when these surfaces have isolated singular points we deduce their strong stability even for variations moving the singular set

weighted poincare and sobolev inequalities and estimates for weighted peano maximal functions 1 introduction in this paper we derive local poincare and sobolev inequalities involving two weight functions the estimates we obtain include those proved by fabes kenig and serapioni in 8 for the one weight function case we also consider the notion of weighted peano derivatives and show that the peano derivative formed with a given weight function can be estimated in terms of a maximal function involving a second weight function in the special case of ordinary lebesgue measure results of this kind lead to the theorems of a p calderon and a zygmund about local differentiability of functions in sobolev spaces and of functions which are of bounded variation in the sense of tonelli see 3 and 4 all our results are based on variants of the basic estimate in 5 but we do not presuppose any facts from 5 let v x be a weight function on r i e let v x be nonnegative and locally integrable with respect to lebesgue measure we shall use the notation v e se v x dx ordinary lebesgue measure will be denoted ie l if bh x is the open ball with center x and radius h we say v satisfies the doubling condition with respect to lebesgue measure if

a lusin type theorem for gradients abstract we prove that for every borel vector field f there exists a function u of class c 1 whose gradient du agrees with f outside a set of arbitrary small measure

some polynomials over q t and their galois groups examples of polynomials with galois group over q t corresponding to every transitive group through degree eight are calculated constructively demonstrating the existence of an infinity of extensions with each galois group over q through degree eight the methods used which for the most part have not appeared in print are briefly discussed

lq l holder continuity for quasilinear parabolic equations associated to sobolev derivations abstract let x be a locally compact metrizable space endowed with a couple of equivalent finite radon measures m and and let e be a hilbert c monomodule over c0 x we consider a class of abstract nonlinear parabolic equations defined as follows let be a closed derivation from l2 x m to l2 e and tt be the strongly continuous nonlinear semigroup naturally associated in the sense of brezis 1973 to the convex l s c functional e u x u x p d x where is the natural modulus function associated to e the generator of the semigroup considered is a natural generalization of the usual p laplacian operator we suppose that a suitable sobolev like inequality of the form u l2d d 2 x m c u l2 e holds true for some d 2 with p 2 d then tt is a nonlinear markov semigroup in the sense that it is order preserving and nonexpansive on each lq x m for any q 2 and moreover it satisfies ttu ttv l x m cm x t u v lq x m for all q 2 and suitable constants depending only on p q d examples include the semigroup generated by the p laplacian on finite measure manifolds with boundary and homogeneous dirichlet boundary conditions as well as by p laplacian like operators associated both to regular sub riemannian structures and to systems of possibly singular or degenerate vector fields satisfying the appropriate sobolev inequalities

l1 poincare inequalities for differential forms on euclidean spaces and heisenberg groups abstract in this paper we prove interior poincare and sobolev inequalities in euclidean spaces and in heisenberg groups in the limiting case where the exterior resp rumin differential of a differential form is measured in l 1 norm unlike for l p p 1 the estimates are doomed to fail in top degree the singular integral estimates are replaced with inequalities which go back to bourgain brezis in euclidean spaces and to chanillo van schaftingen in heisenberg groups

gagliardo nirenberg inequalities for differential forms in heisenberg groups the l 1 sobolev inequality states that for compactly supported functions u on the euclidean n space the l n n 1 norm of a compactly supported function is controlled by the l 1 norm of its gradient the generalization to differential forms due to lanzani and stein and bourgain and brezis is recent and states that a the l n n 1 norm of a compactly supported differential h form is controlled by the l 1 norm of its exterior differential du and its exterior codifferential delta u in special cases the l 1 norm must be replaced by the mathcal h 1 hardy norm we shall extend this result to heisenberg groups in the framework of an appropriate complex of differential forms

tamely ramified subfields of division algebras for any number field k it is unknown which finite groups appear as galois groups of extensions l k such that l is a maximal subfield of a division algebra with center k a k division algebra for k q the answer is described by the long standing q admissibility conjecture we extend a theorem of neukirch on embedding problems with local constraints in order to determine for every number field k what finite solvable groups g appear as galois groups of tamely ramified maximal subfields of k division algebras generalizing liedahl s theorem for metacyclic g and sonn s solution of the q admissibility conjecture for solvable groups

zero error capacity under list decoding shannon s zero error channel capacities c sub of c sub o with and without noiseless feedback are generalized to list decoding the receiver lists l messages erring if the correct one is not listed the corresponding capacities c sub of l c sub o l are nondecreasing in l for an i letter input alphabet c sub of l attains its maximum at l i 1 a lower bound to c sub o l approaches that maximum as l increases

self improving properties of john nirenberg and poincare inequalities on spaces of homogeneous type abstract we give a condition which ensures that if one inequality of sobolev poincare type is valid then other stronger inequalities of a similar type also hold including weighted versions our main result includes many previously known results as special cases we carry out the analysis in the context of spaces of homogeneous type but the main result is new even in the usual euclidean setting

constructions of generalized superimposed codes with applications to group testing and conflict resolution in multiple access channels in this paper we introduce a parameterized generalization of the well known superimposed codes we give algorithms for their construction and provide non existential results we apply our new combinatorial structures to the efficient solution of new group testing problems and access coordination issues in multiple access channels

smooth approximations without critical points of continuous mappings between banach spaces and diffeomorphic extractions of sets abstract let e f be separable hilbert spaces and assume that e is infinite dimensional we show that for every continuous mapping f e f and every continuous function e e 0 there exists a c mapping g e f such that f x g x e x and d g x e f is a surjective linear operator for every x e we also provide a version of this result where e can be replaced with a banach space from a large class including all the classical spaces with smooth norms such as c 0 l p or l p 1 p and f can be taken to be any banach space such that there exists a bounded linear operator from e onto f in particular for such e f every continuous mapping f e f can be uniformly approximated by smooth open mappings part of the proof provides results of independent interest that improve some known theorems about diffeomorphic extractions of closed sets from banach spaces or hilbert manifolds

rectifiability and perimeter in the heisenberg group in this paper we fully extend to the heisenberg group endowed with its intrinsic carnot caratheodory metric and perimeter the classical de giorgi s rectifiability divergence theorems

analysis on lie groups theorie locale diffusion de la chaleur sur des groupes de lie connexes la theorie globale les inegalites de sobolev bases graduees sur des corps lisses construction des bases graduees estimation locale du noyau diffusion de la chaleur demonstration de l inegalite de sobolev locale formes de dirichlet sur des groupes le semi groupe de convolution les conditions de sobolev modifiees

a critical set with nonnull image has large hausdorff dimension the question of how complicated a critical set must be to have a nonnull image is answered by relating its hausdorff dimension to the holder differentiability of the map this leads to a new extension of the morse sard theorem the main tool is an extended version of morse s lemma

random access communication and graph entropy a probabilistic problem that arises for conflict resolution in random access communication is treated an earlier conjecture is disproved and a technique for finding lower bounds on the number of graphs of given structure needed to cover all edges of a given graph is developed

a bridge between dubovitskiĭ federer theorems and the coarea formula abstract the morse sard theorem requires that a mapping v r n r m is of class c k k max n m 0 in 1957 dubovitski generalized this result by proving that almost all level sets for a c k mapping have h s negligible intersection with its critical set where s max n m k 1 0 here the critical set or m critical set is defined as z v m x r n rank v x m another generalization was obtained independently by dubovitski and federer in 1966 namely for c k mappings v r n r d and integers m d they proved that the set of m critical values v z v m is h q negligible for q m 1 n m 1 k they also established the sharpness of these results within the c k category here we prove that dubovitski s theorem can be generalized to the case of continuous mappings of the sobolev lorentz class w p 1 k r n r d p n k this is the minimal integrability assumption that guarantees the continuity of mappings in this situation the mappings need not to be everywhere differentiable and in order to handle the set of nondifferentiability points we establish for such mappings an analog of the luzin n property with respect to lower dimensional hausdorff content finally we formulate and prove a bridge theorem that includes all the above results as particular cases as a limiting case in this bridge theorem we also establish a new coarea type formula if e x r n rank v x m then e j m v x d x r d h n m e v 1 y d h m y the mapping v is r d valued with arbitrary d and the formula is obtained without any restrictions on the image v r n such as m rectifiability or finiteness with respect to the m hausdorff measure these last results are new also for smooth mappings but are presented here in the general sobolev context the proofs of the results are based on our previous joint papers with j bourgain 2013 2015

area stationary and stable surfaces of class c1 in the sub riemannian heisenberg group h1 abstract we consider surfaces of class c 1 in the 3 dimensional sub riemannian heisenberg group h 1 assuming the surface is area stationary i e a critical point of the sub riemannian perimeter under compactly supported variations we show that its regular part is foliated by horizontal straight lines in case the surface is complete and oriented without singular points and stable i e a second order minimum of perimeter we prove that the surface must be a vertical plane this implies the following bernstein type result a complete locally area minimizing intrinsic graph of a c 1 function in h 1 is a vertical plane

pauls rectifiable and purely pauls unrectifiable smooth hypersurfaces abstract this paper is related to the problem of finding a good notion of rectifiability in sub riemannian geometry in particular we study which kind of results can be expected for smooth hypersurfaces in carnot groups our main contribution will be a consequence of the following result there exists a c hypersurface s without characteristic points that has uncountably many pairwise non isomorphic tangent groups on every positive measure subset the example is found in a carnot group of topological dimension 8 it has hausdorff dimension 12 and so we use on it the hausdorff measure h 12 as a consequence we show that any lipschitz map defined on a subset of a carnot group of hausdorff dimension 12 with values in s has negligible image with respect to the hausdorff measure h 12 in particular we deduce that s cannot be lipschitz parametrizable by countably many maps each defined on some subset of some carnot group of hausdorff dimension 12 as main consequence we have that a notion of rectifiability proposed by s pauls is not equivalent to one proposed by b franchi r serapioni and f serra cassano at least for arbitrary carnot groups in addition we show that given a subset u of a homogeneous subgroup of hausdorff dimension 12 of a carnot group every bi lipschitz map f u s satisfies h 12 f u 0 finally we prove that such an example does not exist in heisenberg groups we prove that all c hypersurfaces in h n with n 2 are countably h n 1 r rectifiable according to pauls definition even with bi lipschitz maps

randomness in secret sharing and visual cryptography schemes secret sharing schemes allow a secret to be shared among a group of participants so that only qualified subsets of participants can recover the secret a visual cryptography scheme vcs is a special kind of secret sharing scheme in which the secret to share consists of an image and the shares consist of xeroxed transparencies which are stacked to recover the shared image in this paper we analyze the relationship between secret sharing schemes and vcss focusing our attention on the amount of randomness required to generate the shares we prove that secret sharing schemes for a set of secrets of size two bsss and vcss are equivalent with respect to the randomness indeed we show how to transform a bss for a given access structure into a vcs for the same access structure while preserving the randomness of the original scheme we provide both upper and lower bounds on the randomness of bsss all vcss presented in this paper allow a perfect reconstruction of black pixels

an improved bound on the zero error list decoding capacity of the 4 3 channel we prove a new upper bound on the size of codes c subseteq 1 2 3 4 n with the property that every four distinct codewords in c have a coordinate where they all differ specifically we provide a self contained proof that such codes have size at most 2 6n 19 o n that is rate bounded asymptotically by 6 19 0 3158 measured in bits this improves the previous best upper bound of 0 3512 due to arikan 1994 which in turn improved the 0 375 bound that followed from general bounds for perfect hashing due to fredman and komlos 1984 and korner and marton 1988 finally using a combination of our approach with a simple idea which exploits powerful bounds on the minimum distance of codes in the hamming space we further improve the upper bound to 0 31477

nonsmooth morse sard theorems abstract we prove that every function f r n r satisfies that the image of the set of critical points at which the function f has taylor expansions of order n 1 and non empty subdifferentials of order n is a lebesgue null set as a by product of our proof for the proximal subdifferential p we see that for every lower semicontinuous function f r 2 r the set f x r 2 0 p f x is l 1 null

sub riemannian calculus on hypersurfaces in carnot groups we develop a sub riemannian calculus for hypersurfaces in graded nilpotent lie groups we introduce an appropriate geometric framework such as horizontal levi civita connection second fundamental form and horizontal laplace beltrami operator we analyze the relevant minimal surfaces and prove some basic integration by parts formulas using the latter we establish general first and second variation formulas for the horizontal perimeter in the heisenberg group such formulas play a fundamental role in the sub riemannian bernstein problem

division algebras with psl 2 q galois maximal subfields abstract if g is a finite group and k is a field then g is k admissible if there exists a g galois extension l k such that l is a maximal subfield of a k division algebra we prove that psl 2 7 is k admissible for any number field which either fails to contain 1 or which has two primes lying over the dyadic prime in addition psl 2 11 is shown to be admissible over q or any number field k with at least two extensions of the dyadic prime indeed there exist infinitely many linearly disjoint admissible extensions for these groups

a geometric inequality in the heisenberg group and its applications to stable solutions of semilinear problems in the heisenberg group framework we obtain a geometric inequality for stable solutions of delta mathbb h u f u in a domain omega subseteq mathbb h more precisely if we denote the horizontal intrinsic hessian by hu the mean curvature of a level set by h its imaginary curvature by p the intrinsic normal by and the unit tangent by we have that int limits omega nabla mathbb h phi 2 nabla mathbb h u 2 quad geq int limits omega cap nabla mathbb h u ne0 left hu 2 langle hu 2 nu nu rangle mathbb h 2 left tyuxu txuyu right right phi 2 quad int limits omega cap nabla mathbb h u ne0 nabla mathbb h u 2 big h 2 big p frac langle huv nu rangle mathbb h nabla mathbb h u big 2 2 langle t nu v rangle mathbb h big phi 2 for any phi in c infty 0 omega stable solutions in the entire mathbb h satisfying a suitably weighted energy growth and such that langle t nu v rangle mathbb h ge0 are then shown to have level sets with vanishing mean curvature

a notable family of entire intrinsic minimal graphs in the heisenberg group which are not perimeter minimizing one of the main objectives of this paper is to unravel a new interesting phenomenon of the sub riemannian bernstein problem with respect to its euclidean ancestor with the purpose of also indicating a possible line of attack toward its solution we show that the global intrinsic graphs 1 2 are unstable critical points of the horizontal perimeter as a consequence of this fact the study of the stability acquires a central position in the problem itself

the weighted isoperimetric type and sobolev type inequalities for hypersurfaces in carnot groups abstract this paper mainly studies hypersurfaces immersed in a sub riemannian step k carnot group g with a density e f where f is a smooth function on g we give the first variation formula about the weighted horizontal area of hypersurfaces furthermore on the base of some work we derive the weighted isoperimetric type and sobolev type inequalities for hypersurfaces

sharp a priori estimates for div curl systems in heisenberg groups abstract in this paper we prove a family of inequalities for differential forms in heisenberg groups h 1 and h 2 that are the natural counterpart of a class of div curl inequalities in de rham s complex proved by lanzani stein and bourgain brezis

sub riemannian calculus and monotonicity of the perimeter for graphical strips we consider the class of minimal surfaces given by the graphical strips mathcal s in the heisenberg group mathbb h 1 and we prove that for points p along the center of mathbb h 1 the quantity frac sigma h mathcal s cap b p r r q 1 is monotone increasing here q is the homogeneous dimension of mathbb h 1 we also prove that these minimal surfaces have maximum volume growth at infinity

no signalling assisted zero error capacity of quantum channels and an information theoretic interpretation of the lovasz number we study the one shot zero error classical capacity of a quantum channel assisted by quantum no signalling correlations and the reverse problem of exact simulation of a prescribed channel by a noiseless classical one quantum no signalling correlations are viewed as two input and two output completely positive and trace preserving maps with linear constraints enforcing that the device cannot signal both problems lead to simple semidefinite programmes sdps that depend only on the choi kraus operator space of the channel in particular we show that the zero error classical simulation cost is precisely the conditional min entropy of the choi jamiolkowski matrix of the given channel the zero error classical capacity is given by a similar looking but different sdp the asymptotic zero error classical capacity is the regularization of this sdp and in general we do not know of any simple form interestingly however for the class of classical quantum channels we show that the asymptotic capacity is given by a much simpler sdp which coincides with a semidefinite generalization of the fractional packing number suggested earlier by aram harrow this finally results in an operational interpretation of the celebrated lovasz vartheta function of a graph as the zero error classical capacity of the graph assisted by quantum no signalling correlations the first information theoretic interpretation of the lovasz number

existence and uniqueness for p area minimizers in the heisenberg group in 3 we studied p mean curvature and the associated p minimal surfaces in the heisenberg group from the viewpoint of pde and differential geometry in this paper we look into the problem through the variational formulation we study a generalized p area and associated p minimizers in general dimensions we prove the existence and investigate the uniqueness of minimizers since this is reduced to solving a degenerate elliptic equation we need to consider the effect of the singular set and this requires a careful study we define the notion of weak solution and prove that in a certain sobolev space a weak solution is a minimizer and vice versa we also give many interesting examples in dimension 2 an intriguing point is that in dimension 2 a c 2 smooth solution from the pde viewpoint may not be a minimizer however this statement is true for higher dimensions due to the relative smallness of the size of the singular set

sensitivity of rough differential equations an approach through the omega lemma the ito map assigns the solution of a rough differential equation a generalization of an ordinary differential equation driven by an irregular path when existence and uniqueness hold by studying how a path is transformed through the vector field which is integrated we prove that the ito map is holder or lipschitz continuous with respect to all its parameters this result unifies and weaken the hypotheses of the regularity results already established in the literature

intrinsic curvature of curves and surfaces and a gauss bonnet theorem in the heisenberg group we use a riemannnian approximation scheme to define a notion of intrinsic gaussian curvature for a euclidean c 2 smooth surface in the heisenberg group mathbb h away from characteristic points and a notion of intrinsic signed geodesic curvature for euclidean c 2 smooth curves on surfaces these results are then used to prove a heisenberg version of the gauss bonnet theorem an application to steiner s formula for the carnot caratheodory distance in mathbb h is provided

convexity and horizontal second fundamental forms for hypersurfaces in carnot groups we use a riemannian approximation scheme to give a characterization for smooth convex functions on a carnot group in the sense of danielli garofalo nhieu or lu manfredi stroffolini in terms of the positive semidefiniteness of the horizontal second fundamental form of their graph