optimal las vegas reduction from one way set reconciliation to error correction suppose we have two players a and c where player a has a string s 0 u 1 and player c has a string t 0 u 1 and none of the two players knows the other s string assume that s and t are both over an integer alphabet 0 1 where the first string contains n non zero entries we would wish to answer the following basic question assuming that s and t differ in at most k positions how many bits does player a need to send to player c so that he can recover s with certainty further how much time does player a need to spend to compute the sent bits and how much time does player c need to recover the string s this problem has a certain number of applications for example in databases where each of the two parties possesses a set of n key value pairs where keys are from the universe u and values are from and usually n u in this paper we show a time and message size optimal las vegas reduction from this problem to the problem of systematic error correction of k errors for strings of length n over an alphabet of size 2 log log u n the additional running time incurred by the reduction is linear expected randomized for player a and linear worst case deterministic for player c but the correction works with certainty when using the popular reed solomon codes the reduction gives a protocol that transmits o k log u log bits and runs in time o n polylog n log u log for all values of k the time is expected for player a encoding time and worst case for player c decoding time the message size is optimal whenever k u 1 1

knottedness is in np modulo grh abstract given a tame knot k presented in the form of a knot diagram we show that the problem of determining whether k is knotted is in the complexity class np assuming the generalized riemann hypothesis grh in other words there exists a polynomial length certificate that can be verified in polynomial time to prove that k is non trivial grh is not needed to believe the certificate but only to find a short certificate this result complements the result of hass lagarias and pippenger that unknottedness is in np our proof is a corollary of major results of others in algebraic geometry and geometric topology

sensitivity analysis with plate and shell finite elements this paper addresses the problem of calculating sensitivity data by direct methods for isoparametric plate or shell elements sensitivity parameters of interest include intrinsic properties such as material modulus and plate thickness as well as geometry variables which influence the size and shape of a structure the sensitivity calculation therefore must consider the parametric mapping within an element as well as the influence of geometric variables on the orientation of an element in space the methods presented specialize directly to continuum elements in which the co ordinate transformation is omitted or to simple structural members situated arbitrarily in space numerical examples are presented which illustrate the accuracy of the proposed techniques and the effect of discretization error on computed sensitivities

dynamics of the bilinear mindlin plate element plate and shell finite elements based upon mindlin plate theory and evaluated by one point quadrature are attractive because of their effectiveness for both thick and thin shells at minimal computing cost however these elements are rank deficient and must be stabilized to obtain reliable behaviour methods for achieving full rank of the stiffness and for stabilizing element behaviour in static analysis and in explicit dynamic calculations exist and are quite effective this paper addresses the remaining issue of controlling spurious modes of response in vibration analysis and implicit dynamic solutions several alternatives for the element mass formulation are examined in detail we show that non physical dynamic modes present a potential problem with most mass matrix formulations and that spurious modes other than the familiar hourglassing motion are possible a combination of projection methods and reduced quadrature is suggested which eliminates these deficiencies and produces accurate numerical results the remaining techniques investigated give rise to anomalous behaviour which make them unsuitable for general use

a fractional order darcy s law abstract by using spatial averaging methods in this work we derive a darcy s type law from a fractional newton s law of viscosity which is intended to describe shear stress phenomena in non homogeneous porous media as a prerequisite towards this end we derive an extension of the spatial averaging theorem for fractional order gradients the usage of this tool for averaging continuity and momentum equations yields a darcy s law with three contributions i similar to the classical darcy s law a term depending on macroscopic pressure gradients and gravitational forces ii a fractional convective term induced by spatial porosity gradients and iii a fractional brinkman type correction in the three cases the corresponding permeability tensors should be computed from a fractional boundary value problem within a representative cell consistency of the resulting darcy s type law is demonstrated by showing that it is reduced to the classical one in the case of integer order velocity gradients and homogeneous porous media

hidden structure in the randomness of the prime number sequence abstract we report a rigorous theory to show the origin of the unexpected periodic behavior seen in the consecutive differences between prime numbers we also check numerically our findings to ensure that they hold for finite sequences of primes that would eventually appear in applications finally our theory allows us to link with three different but important topics the hardy littlewood conjecture the statistical mechanics of spin systems and the celebrated sierpinski fractal

on the bounded sum of digits discrete logarithm problem in finite fields in this paper we study the bounded sum of digits discrete logarithm problem in finite fields our results are concerned primarily with fields fqn where n q 1 the fields are called kummer extensions of fq it is known that we can efficiently construct an element g with order exponential in n let s q bullet be the function from integers to the sum of digits in their q ary expansions we first present an algorithm that given ge 0 leq e qn finds e in random polynomial time provided that sq e n we then show that the problem is solvable in random polynomial time for most of the exponent e with sq e 1 32 n by exploring an interesting connection between the discrete logarithm problem and the problem of list decoding of reed solomon codes and applying the guruswami sudan algorithm as far as we are aware our algorithm is the first one which can solve discrete logarithms of 2 log 1 epsilon q n many instances in polynomial time for infinite many constant characteristic fields fqn furthermore since every finite field has an extension of reasonable degree which is a kummer extension our result reveals an unexpected property of the discrete logarithm problem namely the bounded sum of digits discrete logarithm problem in any given finite field becomes polynomial time solvable in certain low degree extensions as a side result we obtain a sharper lower bound on the number of congruent polynomials generated by linear factors than the one based on the stothers mason abc theorem we also prove that in the field fqq 1 the bounded sum of digits discrete logarithm with respect to g can be computed in random time o f w log4 qq 1 where f is a subexponential function and w is the bound on the q ary sum of digits of the exponent hence the problem is fixed parameter tractable these results are shown to be generalized to artin schreier extension fpp where p is a prime

comparison between the variational and implicit differentiation approaches to shape design sensitivities the most commonly used approach to the shape design sensitivity problem results from the implicit differentiation of the discretized equilibrium equations the most general implementation of this technique requires that finite differences be used to differentiate the element stiffness matrices proper choice of the step size is necessary to obtain high levels of accuracy and to avoid round off errors furthermore since it is necessary to operate on the element matrices this method is difficult to implement into a general purpose finite element program a more recent shape design sensitivity formulation based upon variational calculus avoids having to differentiate the discretized equations and results in an analytical expression for the derivative the approach is based upon the total derivative of the variational state equation and uses an adjoint variable technique for design sensitivity analysis only structural response data on the boundary of the structure is necessary thereby making implementation into a general purpose program less difficult this paper attempts to compare the two different techniques and point out the similarities two test problems are shown to demonstrate the accuracy of the variational approach

hardness randomness tradeoffs for bounded depth arithmetic circuits in this paper we show that lower bounds for bounded depth arithmetic circuits imply derandomization of polynomial identity testing for bounded depth arithmetic circuits more formally if there exists an explicit polynomial f that cannot be computed by a depth d arithmetic circuit of small size then there exists an efficient deterministic black box algorithm to test whether a given depth d 5 circuit that computes a polynomial of relatively small individual degrees is identically zero or not in particular if we are guaranteed that the tested circuit computes a multilinear polynomial then we can perform the identity test efficiently to the best of our knowledge this is the first hardness randomness tradeoff for bounded depth arithmetic circuits the above results are obtained using the arithmetic nisan wigderson generator of kabanets and impagliazzo together with a new theorem on bounded depth circuits which is the main technical contribution of our work this theorem deals with polynomial equations of the form p x 1 dots x n y equiv0 and shows that if p has a circuit of depth d and size s and if the polynomial f x 1 dots x n satisfies p x 1 dots x n f equiv0 then f has a circuit of depth d 3 and size mathrm poly s m r where m is the total degree of f and r is the degree of y in p this circuit for f can be found probabilistically in time mathrm poly s m r in the other direction we observe that the methods of kabanets and impagliazzo can be used to show that derandomizing identity testing for bounded depth circuits implies lower bounds for the same class of circuits more formally if we can derandomize polynomial identity testing for bounded depth circuits then nexp does not have bounded depth arithmetic circuits that is either mathrm nexp not subseteq p mathrm poly or the permanent is not computable by polynomial size bounded depth arithmetic circuits

quantum computing postselection and probabilistic polynomial time i study the class of problems efficiently solvable by a quantum computer given the ability to postselect on the outcomes of measurements i prove that this class coincides with a classical complexity class called pp or probabilistic polynomial time using this result i show that several simple changes to the axioms of quantum mechanics would let us solve pp complete problems efficiently the result also implies as an easy corollary a celebrated theorem of beigel reingold and spielman that pp is closed under intersection as well as a generalization of that theorem due to fortnow and reingold this illustrates that quantum computing can yield new and simpler proofs of major results about classical computation

on taking square roots without quadratic nonresidues over finite fields we present a novel idea to compute square roots over finite fields without being given any quadratic nonresidue and without assuming any unproven hypothesis the algorithm is deterministic and the proof is elementary in some cases the square root algorithm runs in o log 2 q bit operations over finite fields with q elements as an application we construct a deterministic primality proving algorithm which runs in o log 3 n for some integers n

primality and identity testing via chinese remaindering we give a simple and new randomized primality testing algorithm by reducing primality testing for number n to testing if a specific univariate identity over zn holds we also give new randomized algorithms for testing if a multivariate polynomial over a finite field or over rationals is identically zero the first of these algorithms also works over zn for any n the running time of the algorithms is polynomial in the size of arithmetic circuit representing the input polynomial and the error parameter these algorithms use fewer random bits and work for a larger class of polynomials than all the previously known methods for example the schwartz zippel test schwartz 1980 zippel 1979 chen kao and lewin vadhan tests chen and kao 1997 lewin and vadhan 1998

random walk on the prime numbers the one dimensional random walk rw where steps up and down are performed according to the occurrence of special primes is defined some quantities characterizing rw are investigated the mean fluctuation function f l displays perfect power law dependence f l l1 2 indicating that the defined rw is not correlated the number of returns of this special rw to the origin is investigated it turns out that this single very special realization of rw is a typical one in the sense that the usual characteristics used to measure rw take values close to the ones averaged over all random walks this fact suggests that random numbers of good quality could be obtained by means of rw on prime numbers the fractal structure on the subset of primes is also found

first occurrence prime gaps an ongoing search for first occurrence prime gaps continues an ongoing search for first occurrence prime gaps is being carried out which extends all previous work done on this subject to date this search has found all such gaps for primes up to 7 263 x 1013 first occurrence prime gaps had previously been known for primes less than 4 444 x 1012 2 several gaps larger than the previously largest gap of 682 not a first occurrence found by weintraub 4 have been found computer programs were written in fortran and cal cray assembly language on a cray 2 supercomputer the computation was conducted as follows a sufficient number of primes were generated to perform a sieve odd numbers beginning with 3 were sieved one block at a time where each block was chosen to contain 40 000 000 numbers based on system resource availability the even numbers were eliminated during initialization of each block one number was stored per 64 bit computer word after sieving each block the differences between consecutive primes were calculated and stored this was accomplished by loading 64 elements of the sieved block at a time into a vector register a 64 bit vector mask was created containing l s for corresponding nonzero values in the vector register if the vector mask was zero the next 64 values of the sieved block were loaded into the vector register if the mask was nonzero an instruction to count the number of leading zeros was executed to get the offset from the beginning of the vector register for the next prime a subtraction of the previous prime was done thus arriving at the gap the leftmost 1 of the vector mask was cleared and the method repeated beginning with checking if the vector mask was zero this was repeated until the entire block was processed the last prime in each block was saved in order to calculate the difference between that prime and the first prime in the next block to make sure no gaps were missed the time to perform the sieve for each block and to calculate all gaps generated was about 10 5 seconds for numbers in the range of 7 2 x 1013 the largest prime in the last block processed to date is 72635119999997 so the table of first occurrence prime gaps is complete to that prime the following table lists all the first occurrence prime gaps found this table agrees with all previously published results 2 3 1 the maximal first occurrence prime gaps are marked with an asterisk received june 8 1987 revised november 16 1987 1980 mathematics subject classification 1985 revision primary 11a41 1989 american mathematical society 0025 5718 89 1 00 s 25 per page

redesign of plates by large admissible perturbations the problem of structural redesign of plates for static deflection and modal dynamics objectives is formulated and solved by the method of large admissible perturbations the perturbation approach to redesign is first used to develop response equations for the objective plate design based on its specifications and the baseline plate design the equations of the objective state are strongly nonlinear implicit expressions of the variable plate thickness a large admissible perturbations algorithm is developed to solve the plate redesign problem and define the optimal objective state the latter is reached incrementally with a prediction correction scheme without repeated finite element analyses systematic numerical applications in redesign of a cantilever plate of 216 degrees of freedom are used to investigate the effects of number of extracted modes and redesign variables it is shown that the large admissible perturbations theory can be used efficiently to redesign plates for multiple specifications that require changes to the baseline design and its response of the order of 100 nomenclature cy admixture coefficient for participation of the j th mode to changes in the th mode e young s modulus k k global and generalized stiffness matrices ke stiffness matrix of element or group of elements related to property e

similarity in the statistics of prime numbers abstract we present numerical evidence for regularities in the distribution of gaps between primes when these are divided into congruence families in dirichlet s classification the histograms for the distribution of gaps of families are scale invariant

lee yang theorems and the complexity of computing averages we study the complexity of computing average quantities related to spin systems such as the mean magnetization and susceptibility in the ferromagnetic ising model and the average dimer count or average size of a matching in the monomer dimer model by establishing connections between the complexity of computing these averages and the location of the complex zeros of the partition function we show that these averages are p hard to compute and hence under standard assumptions computationally intractable in the case of the ising model our approach requires us to prove an extension of the famous lee yang theorem from the 1950s

models of flux in porous media with memory some data on the flow of fluids in rocks exhibit properties which may not be interpreted with the classic theory of propagation of pressure and of fluids in porous media bell and nur 1978 roeloffs 1988 based on darcy s law which states that the flux is proportional to the pressure gradient concerning the fluids some may react chemically with the medium enlarging the pores some carry solid particles which may obstruct some of the pores and finally some may precipitate minerals in the pores diminishing their size or even closing them as in geothermal areas these phenomena create a spatially variable pattern of mineralization and permeability changes that can be localized in order to obtain a better representation of the flux and of the pressure of fluids darcy s law is modified introducing general memory formalisms operating on the flow as well as on the pressure gradient which imply a filtering of the pressure gradient without singularities we also modify the second constitutive equation of diffusion which relates the density variations of the fluid to the pressure by introducing a rheology in the fluid also represented by memory formalisms operating on the pressure as well as on the density variations the memory formalisms are then specified as derivatives of fractional order the equations used here for the diffusion of fluids are different from the classic ones however the equation governing the diffusion of the pressure is the same as that of the flux as in the classic case for technical reasons the majority of the studies on diffusion is devoted to the diffusion of the pressure of the fluid rather than to the flux in this paper we shall devote our attention to studying the flux and its spectral properties in a practical example seeing that the memory used implies a low pass filtering of the flux or a band pass centered in the low frequency range a half space is considered where the boundary values are applied to the plane limiting it and the problems solved are the computation of the green function of the flux in the cases when 1 a pressure constant in time is applied on the boundary plane and 2 a periodic pressure is applied to the boundary plane while the half space is initially at zero pressure in both cases we found closed form formulae for the flux and its spectrum a discussion follows concerning the mode of determination of the parameters of memory formalisms ruling the diffusion using the observed pressure and or the flux at several frequencies in problem 2 concerning the flux it is tentatively seen that when the medium is oil rock and the fluid is water for a pressure of 105 pa at the boundary and derivative of order 0 1 the flux at a distance of 0 1 km from the boundary plane is 1700 kg h 1

1 ƒ noise in the distribution of prime numbers the fourier transform of the signal given by the number of primes contained in the successive intervals of equal length l 216 65 536 up to n 238 2 749 1011 was performed it turns out that the power spectrum displays the 1 behaviour with the exponent 1 64 this slope does not depend on the length of the sampled intervals which suggests some kind of self similarity in the distribution of primes

the gaps between the gaps some patterns in the prime number sequence abstract it has long been known that the gaps between consecutive prime numbers cluster on multiples of 6 recently it was shown that the frequency of the gaps between the gaps is lower for multiples of 6 than for other values p kumar et al information entropy and correlation in prime numbers arxiv cond mat 0303110 this paper investigates higher moments of the prime number series and shows that they exhibit certain peculiarities in order to remove doubts as to whether these peculiarities are related to the known clustering of the gaps on multiples of 6 the results are compared to a benchmark series of simulated gaps

searching for primes in the digits of π many people are fascinated by p vast amounts of human and computer resourceshave been spent producing billions of its digits similarly many people are intrigued by primes how many primes are there within the digits of p how many can we expect to find we present the results of some computer searches and develop a theory to predict how many of these primes extensive computer searches are likely to find

iterated absolute values of differences of consecutive primes let dc n p the nth prime for n 1 and let dk x n dk n dk n 1 for k 0 n 1 a well known conjecture usually ascribed to gilbreath but actually due to proth in the 19th century says that dk 1 for all k 1 this paper reports on a computation that verified this conjecture for k tt 1013 3 x 10 it also discusses the evidence and the heuristics about this conjecture it is very likely that similar conjectures are also valid for many other integer sequences

applications of statistical mechanics in number theory abstract the links between statistical physics and number theory are discussed first the attempts to prove the riemann hypothesis by means of the suitable spin model and the lee yang theorem about zeros of the partition function are shortly reviewed next the analogies between random walks and prime numbers are mentioned in the last section the partition function of the system whose energies are defined by the distances between consecutive primes is calculated the arguments are given that such a prime numbers gas behaves like a set of noninteracting harmonic oscillators

fractional calculus model for temperature and pressure waves in fluid saturated porous rocks we study a fractional time derivative generalization of a previous natale salusti model about nonlinear temperature and pressure waves propagating in fluid saturated porous rocks their analytic solutions i e solitary shock waves characterized by a sharp front are here generalized introducing a formalism that allows memory mechanisms in realistic wave propagation in porous media we must take into account spatial or temporal variability of permeability diffusivity and other coefficients due to the system history such a rock fracturing or fine particulate migration could affect the rock and its pores we therefore take into account these phenomena by introducing a fractional time derivative to simulate a memory conserving formalism we also discuss this generalized model in relation to the theory of dynamic permeability and tortuosity in fluid saturated porous media in such a realistic model we obtain exact solutions of burgers equation with time fractional derivatives in the inviscid case

on the fractal distribution of primes and prime indexed primes by the binary image analysis abstract in this paper the distribution of primes and prime indexed primes pips is studied by mapping primes into a binary image which visualizes the distribution of primes these images show that the distribution of primes and pips is similar to a cantor dust moreover the self similarity with respect to the order of pips already proven in batchko 2014 can be seen as an invariance of the binary images the index of primes plays the same role of the scale for fractals so that with respect to the index the distribution of prime indexed primes is characterized by the self similarity alike any other fractal in particular in order to single out the scale dependence the pips fractal distribution will be evaluated by limiting to two parameters fractal dimension and lacunarity that are usually used to measure the fractal nature because of the invariance of the corresponding binary plots the fractal dimension and lacunarity of primes distribution are invariant with respect to the index of pips

peaks and gaps spectral analysis of the intervals between prime numbers abstract prime numbers seem to be distributed somewhat randomly throughout the sequence of natural numbers it is known however that gaps between consecutive primes exhibit a preponderance towards 6 and its multiples and recently it was shown that the gaps between the gaps also exhibit a peculiar pattern in this paper higher order gaps are analyzed using fourier decomposition the resulting power spectra reveal additional patterns in the prime number sequence

optimization of boundary conditions for maximum fundamental frequency of vibrating structures a sensitivity formula of eigenvalues with respect to the change of boundary conditions is derived using the material derivative concept based on a variational formulation the change of boundary conditions is described with the introduction of the tangential component of the design velocity field used in shape design simply supported and partially welded plates are taken as numerical examples to check the accuracy of the sensitivity formula the sensitivites of the distinct and multiple eigenvalues calculated by the formulas are compared with those calculated by finite differences optimal support locations are then determined by use of a gradient based optimization method it is shown that a crossing of eigenvalues can occur in the solution process

a deterministic version of pollard s p 1 algorithm in this article we present applications of smooth numbers to the unconditional derandomization of some well known integer factoring algorithms we begin with pollard s p 1 algorithm which finds in random polynomial time the prime divisors p of an integer n such that p 1 is smooth we show that these prime factors can be recovered in deterministic polynomial time we further generalize this result to give a partial derandomization of the k th cyclotomic method of factoring k 2 devised by bach and shallit we also investigate reductions of factoring to computing euler s totient function we point out some explicit sets of integers n that are completely factorable in deterministic polynomial time given n these sets consist roughly speaking of products of primes p satisfying with the exception of at most two certain conditions somewhat weaker than the smoothness of p 1 finally we prove that o ln n oracle queries for values of are sufficient to completely factor any integer n in less than exp 1 o 1 ln n 1 3 ln ln n 2 3 deterministic time

generalized compound scaling algorithm and application to minimum weight design of plate structures this paper presents the generalized compound scaling algorithm the optimum designs are reached by simply scaling the design variables to an optimum intersection of multiple constraints a four noded isoparametric plate element is used for modeling the structure

computing prime harmonic sums 1801241484456448000 andthenusedaprecomputedtableofsumsintheschoenfeldinterval therebyavoiding interpolation or binary search and multiple evaluations of s x todiscovers 1801241230056600467 3 99999 99999 99999 99966 ands 1801241230056600523 4 00000 00000 00000 00021 there are no primes between these two xvalues this computation took roughly one week on two workstations franz mertensjosephine mitchell lowell schoenfeld

application of the nonlocal darcy law to the propagation of nonlinear thermoelastic waves in fluid saturated porous media abstract in the propagation of nonlinear waves in fluid saturated porous media a key role is played by the darcy law in many cases one should take into account a realistic spatial or temporal variability of rock parameters as permeability porosity diffusivity to this purpose we here introduce a nonlocal modified darcy law mdl by considering a fractional derivative generalization of the classical case as an application we here study the effect of a generalized darcy law on the propagation of nonlinear thermoelastic waves in porous media in more details we discuss an application of this mdl to an early natale and salusti model about quick and strong pressure and temperature waves in fluid saturated rocks the original model leads to the burgers equation while with the mdl here we obtain a nonlocal formulation of the burgers equation we moreover find the analytic solution in the case that diffusion plays a secondary role with this nonlocal model we obtain a richer analysis of realistic characteristic of such transient phenomena in particular the spatial delays in wave propagation

construction of exact solutions for fractional order differential equations by the invariant subspace method the invariant subspace method for constructing particular solutions is modified for fractional differential equations it allows one to reduce a fractional partial differential equation to a system of nonlinear ordinary fractional differential equations point symmetries of such systems are used to construct their solutions which generate solutions of the original fractional partial differential equation

permanent does not have succinct polynomial size arithmetic circuits of constant depth we show that over fields of characteristic zero there does not exist a polynomial p n and a constant free succinct arithmetic circuit family f n using division by constants where f n has size at most p n and depth o 1 such that f n computes the nxn permanent a circuit family f n is succinct if there exists a nonuniform boolean circuit family c n with o logn many inputs and size n o 1 such that c n can correctly answer direct connection language queries about f n succinctness is a relaxation of uniformity to obtain this result we develop a novel technique that further strengthens the connection between black box derandomization of polynomial identity testing and lower bounds for arithmetic circuits from this we obtain the lower bound by giving an explicit construction computable in the polynomial hierarchy of a hitting set for arithmetic circuits