integer division in linear time with bounded fan in a binary algorithm for division of an m n bit integer by an n bit integer is presented the algorithm produces the m 1 bit quotient and the n bit remainder in time o m n two hardware implementations one using combinational logic in cellular arrays and one employing systolic arrays are given these implementations are designed for modularity and regularity and thus are suitable for vlsi systems an important property of these implementations is that decisions are based on only one bit of the operands thus fan in and length of connecting wires are bounded independently of operand size in addition the systolic implementation has area o n which is the best possible

sub gaussian estimates of heat kernels on infinite graphs we prove that a two sided sub gaussian estimate of the heat kernel on an infinite weighted graph takes place if and only if the volume growth of the graph is uniformly polynomial and the green kernel admits a uniform polynomial decay

high speed low complexity systolic designs of novel iterative division algorithms in gf 2 sup m we extend the binary algorithm invented by stein and propose novel iterative division algorithms over gf 2 sup m for systolic vlsi realization while algorithm ebg is a basic prototype with guaranteed convergence in at most 2m 1 iterations its variants algorithms ebd and ebdf are designed for reduced complexity and fixed critical path delay respectively we show that algorithms ebd and ebdf can be mapped to parallel in parallel out systolic circuits with low area time complexities of o m sup 2 loglogm and o m sup 2 respectively compared to the systolic designs based on the extended euclid s algorithm our circuits exhibit significant speed and area advantages

convexity at infinity in cartan hadamard manifolds and applications to the asymptotic dirichlet and plateau problems we study the asymptotic dirichlet and plateau problems on cartan hadamard manifolds satisfying the so called strict convexity abbr sc condition the main part of the paper consists in studying the sc condition on a manifold whose sectional curvatures are bounded from above and below by certain functions depending on the distance to a fixed point in particular we are able to verify the sc condition on manifolds whose curvature lower bound can go to infty and upper bound to 0 simultaneously at certain rates or on some manifolds whose sectional curvatures go to infty faster than any prescribed rate these improve previous results of anderson borbely and ripoll and telichevsky we then solve the asymptotic plateau problem for locally rectifiable currents with mathbb z 2 multiplicity in acartan hadamard manifold satisfying the sc condition given any compact topologically embedded k 1 dimensional submanifold of partial infty m 2 le k le n 1 as the boundary data we also solve the asymptotic plateau problem for locally rectifiable currents with mathbb z multiplicity on any rotationally symmetric manifold satisfying the sc condition given a smoothly embedded submanifold as the boundary data these generalize previous results of anderson bangert and lang moreover we obtain new results on the asymptotic dirichlet problem for a large class of pdes in particular we are able to prove the solvability of this problem on manifolds with super exponential decay to infty of the curvature

area bounds for free boundary minimal surfaces in a geodesic ball in the sphere abstract we extend to higher dimensions earlier sharp bounds for the area of two dimensional free boundary minimal surfaces contained in a geodesic ball of the round sphere this follows work of brendle and fraser schoen in the euclidean case

fundamental solutions for second order subelliptic operators estimation de la solution fondamentale d un operateur aux derivees partielles lineaire du second ordre sur une variete compacte avec une mesure reguliere

expansions in generalized eigenfunctions of selfadjoint operators let a quantum mechanical system be in the normalized state f measuring the observable quantity which is represented by h we have the probability i l 2 to find the value a n and the probability density to find a 2 in the continuous spectrum is ic 2 l 2 b let for example h a v be a one body hamiltonian i e v x 0 for xj oo we expect the t to be bounded or at most slowly increasing plane waves for v 0 on the other hand for values e a h the solutions t of a v t e 7 should grow fast exponentially at infinity these conjectures can be summarized to

essential self adjointness generalized eigenforms and spectra for the neumann problem on g manifolds let m be a complex manifold with boundary satisfying a subelliptic estimate which is also the total space of a principal g bundle with g a lie group and compact orbit space m g here we investigate the neumann laplacian on m we show that it is essentially self adjoint on its restriction to compactly supported smooth forms moreover we relate its spectrum to the existence of generalized eigenforms an energy belongs to if there is a subexponentially bounded generalized eigenform for this energy vice versa there is an expansion in terms of these well behaved eigenforms so that spectrally almost every energy comes with such a generalized eigenform

mean convex hulls and least area disks spanning extreme curves we show that for any extreme curve in a 3 manifold m there exist a canonical mean convex hull containing all least area disks spanning the curve similar result is true for asymptotic case in open image in new window such that for any asymptotic curve open image in new window there is a canonical mean convex hull containing all minimal planes spanning applying this to quasi fuchsian manifolds we show that for any quasi fuchsian manifold there exist a canonical mean convex core capturing all essential minimal surfaces on the other hand we also show that for a generic c3 smooth curve in the boundary of c3 smooth mean convex domain in 3 there exist a unique least area disk spanning the curve

isoperimetric inequalities for complete proper minimal submanifolds in hyperbolic space the classical isoperimetric inequality for a domain sigma subset mathbb r k with smooth boundary partial sigma is displaystyle k k omega k mathrm vol sigma k 1 leq mathrm vol partial sigma k where equality holds if and only if sigma is a ball in mathbb r k

double basis multiplicative inversion over gf 2 sup m inversion over galois fields is much more difficult than the corresponding multiplication efficient computation of inverses in gf 2 sup m is considered by solving a set of linear equations over the ground field gf 2 the proposed algorithm uses two separate bases for the representation of its input and output elements and has low computational complexity the algorithm is also suitable for hardware implementation using vlsi technologies

interior curvature estimates and the asymptotic plateau problem in hyperbolic space we show that for a very general class of curvature functions defined in the positive cone the problem of finding a complete strictly locally convex hypersurface in h satisfying f 0 1 with a prescribed asymptotic boundary at infinity has at least one smooth solution with uniformly bounded hyperbolic principal curvatures moreover if is euclidean starshaped the solution is unique and also euclidean starshaped while if is mean convex the solution is unique we also show via a strong duality theorem that analogous results hold in de sitter space a novel feature of our approach is a global interior curvature estimate

generalized eigenfunctions for waves in inhomogeneous media abstract many wave propagation phenomena in classical physics are governed by equations that can be recast in schrodinger form in this approach the classical wave equation e g maxwell s equations acoustic equation elastic equation is rewritten in schrodinger form leading to the study of the spectral theory of its classical wave operator a self adjoint partial differential operator on a hilbert space of vector valued square integrable functions physically interesting inhomogeneous media give rise to nonsmooth coefficients we construct a generalized eigenfunction expansion for classical wave operators with nonsmooth coefficients our construction yields polynomially bounded generalized eigenfunctions the set of generalized eigenvalues forming a subset of the operator s spectrum with full spectral measure

a hardware algorithm for modular multiplication division a mixed radix 4 2 algorithm for modular multiplication division suitable for vlsi implementation is proposed the algorithm is based on montgomery method for modular multiplication and on the extended binary gcd algorithm for modular division both algorithms are modified and combined into the proposed algorithm so that almost all the hardware components are shared the new algorithm carries out both calculations using simple operations such as shifts additions and subtractions the radix 2 signed digit representation is used to avoid carry propagation in all additions and subtractions a modular multiplier divider based on the algorithm performs an n bit modular multiplication division in o n clock cycles where the length of the clock cycle is constant and independent of n the modular multiplier divider has a linear array structure with a bit slice feature and can be implemented with much smaller hardware than that necessary to implement both multiplier and divider separately

scattering by obstacles of finite capacity abstract for very general generators of diffusion semigroups we show that the essential and absolutely continuous spectra do not change when one adds an extra dirichlet boundary condition on a small set this is done by proving that the corresponding semigroup differences are hilbert schmidt or trace class respectively our method consists in a factorization argument which is based on calculating the semigroup difference via the feynman kac formula we also derive trace class estimates for differences of resolvent powers provided the underlying semigroup has finite local dimension

new systolic architectures for inversion and division in gf 2 sup m we present two systolic architectures for inversion and division in gf 2 sup m based on a modified extended euclidean algorithm our architectures are similar to those proposed by others in that they consist of two dimensional arrays of computing cells and control cells with only local intercell connections and have o m sup 2 area time product however in comparison to similar architectures both our architectures have critical path delays that are smaller gate counts that range from being considerably smaller to only slightly larger and latencies that are identical for inversion but somewhat larger for division one architecture uses an adder or an m l bit ring counter inside each control cell while the other architecture distributes the ring counters into the computing cells thereby reducing each control cell to just two gates

intrinsic metrics for non local symmetric dirichlet forms and applications to spectral theory we present a study of what may be called an intrinsic metric for a general regular dirichlet form for such forms we then prove a rademacher type theorem for strongly local forms we show existence of a maximal intrinsic metric under a weak continuity condition and for dirichlet forms with an absolutely continuous jump kernel we characterize intrinsic metrics by bounds on certain integrals we then turn to applications on spectral theory and provide for measure perturbation of general regular dirichlet forms an allegretto piepenbrinck type theorem which is based on a ground state transform and a shnol type theorem our setting includes laplacian on manifolds on graphs and stable processes

eigenvalue estimates for minimal surfaces in hyperbolic space this paper gives an upper bound for the first eigenvalue of the universal cover of a complete stable minimal surface in hyperbolic space and a sharper one for least area disks

moving centre monotonicity formulae for minimal submanifolds and related equations abstract monotonicity formulae play a crucial role for many geometric pdes especially for their regularity theories for minimal submanifolds in a euclidean ball the classical monotonicity formula implies that if such a submanifold passes through the centre of the ball then its area is at least that of the equatorial disk recently brendle and hung proved a sharp area bound for minimal submanifolds when the prescribed point is not the centre of the ball which resolved a conjecture of alexander hoffman and osserman their proof involves asymptotic analysis of an ingeniously chosen vector field and the divergence theorem in this article we prove a sharp moving centre monotonicity formula for minimal submanifolds which implies the aforementioned area bound we also describe similar moving centre monotonicity formulae for stationary p harmonic maps mean curvature flow and the harmonic map heat flow

on the geometric and topological rigidity of hyperbolic 3 manifolds theorem 0 1 let n be a closed hyperbolic 3 manifold containing an embedded hyperbolic tube of radius log 3 2 549306 about a closed geodesic then i iff m n is a homotopy equivalence where m is an irreducible 3 manifold then f is homotopic to a homeomorphism ii if f g m s n are homotopic homeomorphisms then f is isotopic to g iii the space of hyperbolic metrics on n is path connected

a carry free algorithm for finding the greatest common divisor of two integers abstract we investigate a variant of the so called binary algorithm for finding the gcd greatest common divisor of two numbers which requires no comparisons we show that when implemented with carry save hardware it can be used to find the modulo b inverse of an n bit binary integer in a time proportional to n using only registers of length proportional to n such a hardware implementation of this algorithm set up for finding inverses with respect to a 336 bit modulus b would have applications in the currently expanding field of secure data transmission and storage in such an implementation multiplication in linear time both modulo b and ordinary would come along as a by product because multiplication can be achieved by a sequence of nine inversions some additions and negations

hypersurfaces of constant mean curvature in hyperbolic space with prescribed asymptotic boundary at infinity in this paper we study the problem of finding smooth complete hypersurfaces of constant mean curvature in hyperbolic space with a prescribed asymptotic boundary at infinity our main results are proved by deriving a priori global gradient estimates based on the classical maximum principle for elliptic equations

on the dirichlet problem for minimal graphs in hyperbolic space soit r n n 2 un domaine borne on etudie le probleme de dirichlet suivant f f i f j f ij 1 df 2 n f 0 dans f 0 dans f 0 sur on considere le comportement a la frontiere de la solution

analysis on graphs and noncommutative geometry abstract we study the form of the continuous time heat kernel for a second order discrete laplacian on a weighted graph the analysis is shown to be closely related to the theory of symmetric markov semigroups on noncommutative l p spaces and to the noncommutative geometry of connes the paper obtains better pointwise upper bounds on the heat kernels than those previously known by the use of a novel metric on the graph in certain cases it is shown that the new estimates are optimal of their type the metric is investigated at some length and compared with other known metrics on graphs

binary ternary plus minus modular inversion in rns a fast rns modular inversion for finite fields arithmetic has been published at ches 2013 conference it is based on the binary version of the plus minus euclidean algorithm in the context of elliptic curve cryptography i e 160 550 bits finite fields it significantly speeds up modular inversions in this paper we propose an improved version based on both radix 2 and radix 3 this new algorithm leads to 30 percent speed up for a maximal area overhead about 4 percent on virtex 5 fpgas

systolic vlsi arrays for polynomial gcd computation the problem of finding a greatest common divisor gcd of any two nonzero polynomials is fundamental to algebraic and symbolic computations as well as to the decoder implementation for a variety of error correcting codes this paper describes new systolic arrays that can lead to efricient vlsi solutions to both the gcd problem and the extended gcd problem

higher rank hyperbolicity the large scale geometry of hyperbolic metric spaces exhibits many distinctive features such as the stability of quasi geodesics the morse lemma the visibility property and the homeomorphism between visual boundaries induced by a quasi isometry we prove a number of closely analogous results for spaces of rank n ge 2 in an asymptotic sense under some weak assumptions reminiscent of nonpositive curvature for this purpose we replace quasi geodesic lines with quasi minimizing locally finite n cycles of r n volume growth prime examples include n cycles associated with n quasiflats solving an asymptotic plateau problem and producing unique tangent cones at infinity for such cycles we show in particular that every quasi isometry between two proper text cat 0 spaces of asymptotic rank n extends to a class of n 1 cycles in the tits boundaries

on computing multiplicative inverses in gf 2 sup m the design of a modular standard basis inversion for galois fields gf 2 sup m based on euclid s algorithm for computing the greatest common divisor of two polynomials is presented the asymptotic complexity is linear with m both in computation time and area requirement thus resulting in an at complexity of o m sup 2 this is a significant improvement over the best previous proposal which achieves at complexity of only o m sup 3

explicit constants for gaussian upper bounds on heat kernels on utilise des inegalites logarithmiques de sobolev pour obtenir des bornes superieures gaussiennes sur les noyaux de la chaleur associes a des operateurs d ordre 2 elliptiques et hypoelliptiques

asymptotic behavior of area minimizing currents in hyperbolic space on etudie le comportement asymptotique des courants localement rectifiables minimisant l aire dans un espace hyperbolique on considere l aspect theorie de la mesure geometrique et l aspect equation aux derivees partielles

density and spectrum of minimal submanifolds in space forms let varphi m m rightarrow n n be a minimal proper immersion in an ambient space suitably close to a space form mathbb n n k of curvature k le 0 in this paper we are interested in the relation between the density function theta r of m and the spectrum of its laplace beltrami operator in particular we prove that if theta r has subexponential growth when k 0 or sub polynomial growth k 0 along a sequence then the spectrum of m m is the same as that of the space form mathbb n m k notably the result applies to anderson s smooth solutions of plateau s problem at infinity on the hyperbolic space independently of their boundary regularity we also give a simple condition on the second fundamental form that ensures m to have finite density in particular we show that minimal submanifolds with finite total curvature in the hyperbolic space also have finite density

renormalized area and properly embedded minimal surfaces in hyperbolic 3 manifolds we study the renormalized area functional mathcal a in the ads cft correspondence specifically for properly embedded minimal surfaces in convex cocompact hyperbolic 3 manifolds and somewhat more broadly poincare einstein spaces our main results include an explicit formula for the renormalized area of such a minimal surface y as an integral of local geometric quantities as well as formulae for the first and second variations of mathcal a which are given by integrals of global quantities over the asymptotic boundary loop of y all of these formulae are also obtained for a broader class of nonminimal surfaces the proper setting for the study of this functional when the ambient space is hyperbolic requires an understanding of the moduli space of all properly embedded minimal surfaces with smoothly embedded asymptotic boundary we show that this moduli space is a smooth banach manifold and develop a mathbb z valued degree theory for the natural map taking a minimal surface to its boundary curve we characterize the nondegenerate critical points of mathcal a for minimal surfaces in mathbb h 3 and finally discuss the relationship of mathcal a to the willmore functional

quantum error correcting codes and 4 dimensional arithmetic hyperbolic manifolds using 4 dimensional arithmetic hyperbolic manifolds we construct some new homological quantum error correcting codes they are low density parity check codes with linear rate and distance ne their rate is evaluated via euler characteristic arguments and their distance using z2 systolic geometry this construction answers a question of zemor on cayley graphs surface codes and the limits of homological coding for quantum error correction in proceedings of second international workshop on coding and cryptology iwcc lecture notes in computer science vol 5557 2009 pp 259 273 who asked whether homological codes with such parameters could exist at all

geometry and analysis of dirichlet forms ii abstract given a regular strongly local dirichlet form e under assumption that the lower bound of the ricci curvature of bakry emery the local doubling and local poincare inequalities are satisfied we obtain that i the intrinsic differential and distance structures of e coincide ii the cheeger energy functional ch d e is a quadratic norm this shows that ii is necessary for the riemannian ricci curvature defined by ambrosio gigli savare to be bounded from below this together with some recent results of ambrosio gigli savare yields that the heat flow gives a gradient flow of boltzman shannon entropy under the above assumptions we also obtain an improvement on kuwada s duality theorem for dirichlet forms under the assumptions of doubling and poincare inequalities finally dirichlet forms are constructed to show that doubling and poincare inequalities are not enough to obtain either i or ii above that is the lower bound of the bakry emery curvature condition is essential

quasi minimizing surfaces in hyperbolic space let h 3 g denote 3 dimensional hyperbolic space with constant curvature 1 we are interested in surfaces f in h 3 with the following quasi minimality property if it f h 3 0 1 which does not depend on the homotopy if this condition is satisfied with q 1 then f is called homotopically minimizing a surface f c h 3 with this property is not necessarily area minimizing in contrast to the situation in higher dimensions cf 13 our main result is

an asymptotic theorem for minimal surfaces and existence results for minimal graphs in mathbb h 2 times mathbb r in this paper we prove a general and sharp asymptotic theorem for minimal surfaces in mathbb h 2 times mathbb r as a consequence we prove that there is no properly immersed minimal surface whose asymptotic boundary is a jordan curve homologous to zero in partial infty mathbb h 2 times mathbb r such that is contained in a slab between two horizontal circles of partial infty mathbb h 2 times mathbb r with width equal to we construct vertical minimal graphs in mathbb h 2 times mathbb r over certain unbounded admissible domains taking certain prescribed finite boundary data and certain prescribed asymptotic boundary data our admissible unbounded domains in mathbb h 2 times 0 are non necessarily convex and non necessarily bounded by convex arcs each component of its boundary is properly embedded with zero one or two points on its asymptotic boundary satisfying a further geometric condition

localization of classical waves i acoustic waves we consider classical acoustic waves in a medium described by a position dependent mass density x we assume that x is a reandom perturbation of a periodic function 0 x and that the periodic acoustic operator a 0 nabla cdot tfrac 1 varrho 0 x nabla has a gap in the spectrum we prove the existence of localized waves i e finite energy solutions of the acoustic equations with the property that almost all of the wave s energy remains in a fixed bounded region of space at all times with probability one localization of acoustic waves is a consequence of anderson localization for the self adjoint operators a nabla cdot tfrac 1 varrho 0 x nabla onl2 d we prove that in the random medium described by x the random operatora exhibits anderson localization inside the gap in the spectrum ofa0 this is shown even in situations when the gap is totally filled by the spectrum of the random opertor we can prescribe random environments that ensure localization in almost the whole gap

lipschitz algebras and derivations ii exterior differentiation abstract basic aspects of differential geometry can be extended to various non classical settings lipschitz manifolds rectifiable sets sub riemannian manifolds banach manifolds wiener space etc although the constructions differ in each of these cases one can define a module of measurable 1 forms and a first order exterior derivative we give a general construction which applies to any metric space equipped with a finite measure and produces the desired result in all of the above cases it agrees with cheeger s construction when the latter is defined it also applies to an important class of dirichlet spaces where however the known first order differential calculus in general differs from ours although the two are related

bootstrap multiscale analysis and localization in random media we introduce an enhanced multiscale analysis that yields subexponentially decaying probabilities for bad events for quantum and classical waves in random media we obtain exponential decay for the resolvent of the corresponding random operators in boxes of side l with probability higher than 1 e l for any 0 1 the starting hypothesis for the enhanced multiscale analysis only requires the verification of polynomial decay of the finite volume resolvent at some sufficiently large scale with probability bigger than 1 d is the dimension note that from the same starting hypothesis we get conclusions that are valid for any 0 1 this is achieved by the repeated use of a bootstrap argument as an application we use a generalized eigenfunction expansion to obtain strong dynamical localization of any order in the hilbert schmidt norm and better estimates on the behavior of the eigenfunctions

erratum on the dirichlet problem for minimal graphs in hyperbolic space r n 0 that can be identified with the asymptotic infinity of hn 1 let nn 1 hn 1 be a closed embedded n 1 dimensional submanifold then there exists a complete absolutely area minimizing integral n current asymptotic to nn 1 at infinity see for example 1 2 it was also noted in 1 2 that in the case n 6 currents obtained are smoothly embedded complete hypersurfaces in the case n 7 as in the euclidean case there can be a singular set of hausdorff dimension at most n 7

modular multiplication without trial division let n 1 we present a method for multiplying two integers called n residues modulo n while avoiding division by n n residues are represented in a nonstandard way so this method is useful only if several computations are done modulo one n the addition and subtraction algorithms are unchanged 1 description some algorithms 1 2 4 5 require extensive modular arith metic we propose a representation of residue classes so as to speed modular multiplication without affecting the modular addition and subtraction algorithms other recent algorithms for modular arithmetic appear in 3 6 fix n 1 define an a residue to be a residue class modulo n select a radix r coprime to n possibly the machine word size or a power thereof such that r n and such that computations modulo r are inexpensive to process let r l and n be integers satisfying 0 n then return t n else return t to validate redc observe mn tn n tmod r so t is an integer also tr tmod n so t tr x mod n thirdly 0 t mn rn rn so 0 t 2n if r and n are large then t mn may exceed the largest double precision value one can circumvent this by adjusting m so r m 0 given two numbers x and y between 0 and n 1 inclusive let z redc xy then z xy r x mod n so xr l yr x zrx mod n also 0 z n so z is the product of x and y in this representation other algorithms for operating on n residues in this representation can be derived from the algorithms normally used the addition algorithm is unchanged since xr x yr x zr x mod n if and only if x y z mod n also unchanged are

a computer algorithm for calculating the product ab modulo m it is possible to find the smallest nonnegative integer r congruent modulo m to the product ab of two nonnegative integers without dividing by m in multiple precision arithmetic doing away with the division cuts the calculation time by varying amounts depending on machine architecture it also cuts storage space

regularity at infinity for area minimizing hypersurfaces in hyperbolic space equipped with the hyperbolic metric y z dxz dy2 a standard compactification of ih involves adding the boundary r 0 w so that i i is simply the one point compactification of the euclidean closed half space r 0 suppose 0 7 any interior singularities of m must remain in a bounded region of hyperbolic space near points of f in the euclidean topology m w f may thus be described as the graph of a function this function is the solution of an interesting partial differential equation that becomes degenerate along the part of the boundary corresponding to f the second author has recently established l a boundary higher regularity result for this equation which implies in particular that m w f is r if f is cgk for k 2 3 finally in case f bounds a star

expansion in generalized eigenfunctions for laplacians on graphs and metric measure spaces we consider an arbitrary selfadjoint operator on a separable hilbert space to this operator we construct an expansion in generalized eigenfunctions in which the original hilbert space is decomposed as a direct integral of hilbert spaces consisting of general eigenfunctions this automatically gives a plancherel type formula for suitable operators on metric measure spaces we discuss some growth restrictions on the generalized eigenfunctions for laplacians on locally finite graphs the generalized eigenfunctions are exactly the solutions of the cor responding difference equation

complete minimal varieties in hyperbolic space it is a basic problem in the subject of minimal varieties to determine the class of complete minimal and area minimizing varieties m k in a given riemannian manifold n besides being of their own intrinsic interest the cases r of euclidean space and mn sn of the sphere are fundamental in understanding the local structure of minimal varieties due to the existence of tangent cones to general minimal varieties there has been much study of the complete areaminimizing varieties in these cases in particular relating to the bernstein conjecture and minimizing cones e g l2 f1 nevertheless many deep problems remain unsolved this paper is centered around the question of existence of complete areaminimizing varieties in hyperbolic space h here by way of contrast to the situation in ir n or s we find a comparatively complete solution to the problem of existence in fact the main result of this paper exhibits a wealth of complete area minimizing varieties in h in all dimensions and codimensions more precisely we have