radon cosine and sine transforms on real hyperbolic space new pointwise inversion formulae are obtained for the d dimensional totally geodesic radon transform on the n dimensional real hyperbolic space 1 d n 1 in terms of polynomials of the laplace beltrami operator and intertwining fractional integrals similar results are established for hyperbolic cosine and sine transforms

series representations for some mathematical constants abstract the authors apply a classical series identity involving the psi or digamma function with a view to deriving series representations for a number of known mathematical constants several closely related consequences and results are also considered

the h function associated with a certain class of feynman integrals motivated by some further examples of the use of feynman integrals which arise in perturbation calculations of the equilibrium properties of a magnetic model of phase transitions a generalization of the familiar h function of fox 1961 was proposed recently by inayat hussain 1987 a brief discussion of contour selection and convergence conditions of the integrals involved is presented it is also shown how some recent work of buschman 1990 can be applied to derive various recurrence relations for the general h function finally some relevant comments are made on the validity and novelty of two summation formulae for the clausenian hypergeometric function which were derived by inayat hussain

absolutely continuous functions with values in a banach space abstract let be an open subset of r n n 1 and let x be a banach space we prove that absolutely continuous functions f x are continuous and differentiable in some sense almost everywhere in

a certain family of infinite series associated with digamma functions abstract the sums of several interesting infinite series were recently expressed in terms of the psi or digamma functions the object of this paper is to present a systematic account of these and of numerous similar or more general series whose sums can be found in the literature in various equivalent forms some relevant unifications and further generalizations are also indicated

intersections of intrinsic submanifolds in the heisenberg group in the first heisenberg group we show that the intersection of two intrinsic submanifolds with linearly independent horizontal normals locally coincides with the image of an injective continuous curve the key tool is a chain rule that relies on a recent result by dafermos

partial regularity of almost minimizing rectifiable g chains in hilbert space we adapt to an infinite dimensional ambient space e r reifenberg s epiperimetric inequality and a quantitative version of d preiss second moments computations to establish that the set of regular points of an almost mass minimizing rectifiable g chain in l is dense in its support whenever the group g of coefficients is so that g g g is discrete and closed

partial regularity for mass minimizing currents in hilbert spaces recently the theory of currents and the existence theory for plateau s problem have been extended to the case of finite dimensional currents in infinite dimensional manifolds or even metric spaces see acta math 185 2000 1 80 and also proc lond math soc 3 106 2013 1121 1142 adv calc var 7 2014 227 240 for the most recent developments in this paper in the case when the ambient space is hilbert we provide the first partial regularity result in a dense open set of the support for n dimensional integral currents which locally minimize the mass our proof follows with minor variants indiana univ math j 31 1982 415 434 implementing lipschitz approximation and harmonic approximation without indirect arguments and with estimates which depend only on the dimension n and not on codimension or dimension of the target space

uniformization of two dimensional metric surfaces we establish uniformization results for metric spaces that are homeomorphic to the euclidean plane or sphere and have locally finite hausdorff 2 measure applying the geometric definition of quasiconformality we give a necessary and sufficient condition for such spaces to be qc equivalent to the euclidean plane disk or sphere moreover we show that if such a qc parametrization exists then the dilatation can be bounded by 2 as an application we show that the euclidean upper bound for measures of balls is a sufficient condition for the existence of a 2 qc parametrization this result gives a new approach to the bonk kleiner theorem on parametrizations of ahlfors 2 regular spheres by quasisymmetric maps

area formula for centered hausdorff measures in metric spaces abstract motivated by an example in magnani in press we study inside a separable metric space x d the relations between centered and non centered m dimensional densities of a radon measure in x and their relations with spherical and centered spherical m dimensional hausdorff measures eventually we give an application to finite perimeter sets in carnot groups

sub riemannian vs euclidean dimension comparison and fractal geometry on carnot groups we solve gromov s dimension comparison problem for hausdorff and box counting dimension on carnot groups equipped with a carnot caratheodory metric and an adapted euclidean metric the proofs use sharp covering theorems relating optimal mutual coverings of euclidean and carnot caratheodory balls and elements of sub riemannian fractal geometry associated to horizontal self similar iterated function systems on carnot groups inspired by falconer s work on almost sure dimensions of euclidean self affine fractals we show that carnot caratheodory self similar fractals are almost surely horizontal as a consequence we obtain explicit dimension formulae for invariant sets of euclidean iterated function systems of polynomial type jet space carnot groups provide a rich source of examples

rectifiable and flat g chains in a metric space chains in a complete metric space which have coefficients in a normed abelian group g are studied an m dimensional rectifiable chain is the lipschitz push forward of a region in bbb r m equipped with a measurable g valued density flat chains are obtained by completion using a certain flat norm on polyhedral or lipschitz chain approximations numerous basic results of geometric measure theory for these chains are derived including the rectifiability of finite mass flat chains provided that g contains no nonconstant lipschitz curves the work here generalizes and uses many ideas from the 1999 paper of b white on at g chains in bbb r n and the 2000 paper of l ambrosio and b kirchheim on currents in a metric space the use of flat and rectifiable chains in geometric variational problems or in defining geometric homology theories may reveal geometric properties of spaces

asymptotic relation for the transition density of the three dimensional markov random flight on small time intervals we consider the markov random flight varvec x t t 0 in the three dimensional euclidean space mathbb r 3 with constant finite speed c 0 and the uniform choice of the initial and each new direction at random time instants that form a homogeneous poisson flow of rate lambda 0 series representations for the conditional characteristic functions of varvec x t corresponding to two and three changes of direction are obtained based on these results an asymptotic formula as t rightarrow 0 for the unconditional characteristic function of varvec x t is derived by inverting it we obtain an asymptotic relation for the transition density of the process we show that the error in this formula has the order o t 3 and therefore it gives a good approximation on small time intervals whose lengths depend on lambda an asymptotic formula as t rightarrow 0 for the probability of being in a three dimensional ball of radius r ct is also derived estimate of the accuracy of the approximation is analysed

quasiconformal mappings on the heisenberg group on etudie les applications conformes et quasi conformes sur le groupe de heisenberg la metrique invariante a gauche naturelle sur cet espace est non isotrope sous des dilatations et ne peut se decrire par une metrique de riemann ceci engendre de nouveaux phenomenes

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

dimensions of attractors in pinched skew products we study dimensions of strange non chaotic attractors and their associated physical measures in so called pinched skew products introduced by grebogi and his coworkers in 1984 our main results are that the hausdorff dimension the pointwise dimension and the information dimension are all equal to one although the box counting dimension is known to be two the assertion concerning the pointwise dimension is deduced from the stronger result that the physical measure is rectifiable our findings confirm a conjecture by ding grebogi and ott from 1989

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

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

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

structure of measures in lipschitz differentiability spaces we prove the equivalence of two seemingly very di erent ways of generalising rademacher s theorem to metric measure spaces the rst was introduced by cheeger and is based upon di erentiation with respect to another xed chart func tion the second approach is new for this generality and originates in some ideas of alberti it is based upon forming partial derivatives along a very rich structure of lipschitz curves analogous to the di erentiability theory of euclidean spaces by examining this structure further we naturally arrive to several descriptions of lipschitz di erentiability spaces

some transformation and reduction formulas for hypergeometric series in several variables abstract an alternative elementary proof of a certain hypergeometric identity which was posed recently as a problem leads naturally first to a multiple series extension of the problem and then also to its related multivariable hypergeometric transformation and summation formulas a systematic analysis of several general multiple series identities analogous to these hypergeometric transformations is presented many of the general results obtained are shown to unify and extend numerous transformation and reduction formulas for various classes of double and multiple hypergeometric series

a variant of gromov s problem on holder equivalence of carnot groups abstract it is unknown if there exists a locally holder homeomorphism f r 3 h 1 for any 1 2 2 3 although the identity map r 3 h 1 is locally 1 2 holder more generally gromov asked given k and a carnot group g for which does there exist a locally holder homeomorphism f r k g here we equip a carnot group g with the carnot caratheodory metric in 2014 balogh hajlasz and wildrick considered a variant of this problem these authors proved that if k n there does not exist an injective 1 2 holder mapping f r k h n that is also locally lipschitz as a mapping into r 2 n 1 for their proof they use the fact that h n is purely k unrectifiable for k n in this paper we will extend their result from the heisenberg group to model filiform groups and carnot groups of step at most three we will now require that the carnot group is purely k unrectifiable the main key to our proof will be showing that 1 2 holder maps f r k g that are locally lipschitz into euclidean space are weakly contact proving weak contactness in these two settings requires understanding the relationship between the algebraic and metric structures of the carnot group we will use coordinates of the first and second kind for carnot groups

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

generalized language equations with multiple solutions abstract a generalized language equation x f x is an equation in a variable x over an alphabet a where the language function f is expressible in terms of arbitrary set operations including complementation and concatenation from the left by a constant i e a language in previous work these equations were studied provided an easily verifiable condition was met in this case solutions always exist and are always unique in the present paper the condition is dropped as a consequence our equations may have zero one or more up to uncountably infinitely many solutions we give a representation theorem for these equations and use it to classify them for equations in one variable a complete classification is given including parametric representations of all solutions for systems of m equations in m variables the notion of 1 reducible substitution graph is introduced and employed to reduce the problem of existence of solutions of the entire system to the same problem for a single equation the paper concludes with a discussion of the implications if one assumes that all constants are regular languages and poses some open problems

rectifiability and lipschitz extensions into the heisenberg group denote by mathbb h n the 2n 1 dimensional heisenberg group we show that the pairs mathbb r k mathbb h n and mathbb h k mathbb h n do not have the lipschitz extension property for k n

rectifiability of flat chains in banach spaces with coefficients in zp in this paper we study the behaviour of a class of currents with coefficients in zp in metric spaces obtained by taking a suitable quotient of currents with coefficients in z we prove under fairly general assumption on the ambient space the rectifiability of the so called slice mass and under more restrictive assumptions the rectifiability of the mass measure

on a measure theoretic area formula we review some classical differentiation theorems for measures showing how they can be turned into an integral representation of a borel measure with respect to a fixed caratheodory measure we focus our attention on the case when this measure is the spherical hausdorff measure giving a metric measure area formula our aim is to use certain covering derivatives as generalized densities some consequences for the sub riemannian heisenberg group are also pointed out