one dimensional behavior of singular n dimensional solutions of semilinear heat equations we consider u x t a solution of u t u u p 1 u that blows up at time t where u n 0 t p 1 n 2 p 0 in particular if in addition p 3 then the solution is very close to a superposition of one dimensional solutions as functions of the distance to the blow up set finally we prove that the mere hypothesis that the blow up set is continuous implies that it is c 1 1 2 for any 0

computational analysis and improvement of sirt iterative x ray computed tomography ct algorithms have the potential for producing high quality images but are computationally very demanding especially when applied to high resolution problems focusing on simultaneous iterative reconstruction technique sirt we provide an eigenvalue based scheme for automatically determining a near optimal value of the relaxation parameter this accelerates the convergence rate of sirt to the point where only half the number of iterations normally required is needed we also modify the way sirt uses preconditioning to solve a weighted least squares problem the resulting algorithm which we call psirt is associated with a smaller memory footprint and calls for less data to be communicated in a distributed memory implementation experimental residual norm and timing results are provided based on cone beam micro ct mouse data including for an ordered subsets study

adaptive thresholding of tomograms by projection distance minimization segmentation is an important step to obtain quantitative information from tomographic data sets however it is usually not possible to obtain an accurate segmentation based on a single global threshold instead local thresholding schemes can be applied that use a varying threshold selecting the best local thresholds is not a straightforward task as local image features often do not provide sufficient information for choosing a proper threshold recently the concept of projection distance was proposed by the authors as a new criterion for evaluating the quality of a tomogram segmentation k j batenburg j sijbers automatic threshold selection for tomogram segmentation by reprojection of the reconstructed image in computer analysis of images and patterns in lecture notes in computer science vol 4673 springer berlin heidelberg 2007 pp 563 570 in this paper we describe how projection distance minimization pdm can be used to select local thresholds based on the available projection data from which the tomogram was initially computed the results of several experiments are presented in which our local thresholding approach is compared with alternative thresholding methods these results demonstrate that the local thresholding approach yields segmentations that are significantly more accurate compared to previously published methods in particular when the initial reconstruction contains artifacts

prescribing the center of mass of a multi soliton solution for a perturbed semilinear wave equation abstract we construct a finite time blow up solution for a class of strongly perturbed semilinear wave equation with an isolated characteristic point in one space dimension given any integer k 2 and 0 r we construct a blow up solution with a characteristic point a such that the asymptotic behavior of the solution near a t a shows a decoupled sum of k solitons with alternate signs whose centers in the hyperbolic geometry have 0 as a center of mass for all times although the result is similar to the unperturbed case in its statement our method is new indeed our perturbed equation is not invariant under the lorentz transform and this requires new ideas in fact the main difficulty in this paper is to prescribe the center of mass 0 r we would like to mention that our method is valid also in the unperturbed case and simplifies the original proof by cote and zaag 9 as far as the center of mass prescription is concerned

on self similar singular solutions of the complex ginzburg landau equation we address the open problem of existence of singularities for the complex ginzburg landau equation using a combination of rigorous results and numerical computations we describe a countable family of self similar singularities our analysis includes the supercritical nonlinear schrodinger equation as a special case we also consider the problem of stability of these singularities 2001 john wiley sons inc

profile for the imaginary part of a blowup solution for a complex valued semilinear heat equation abstract in this paper we consider the following complex valued semilinear heat equation t u u u p u c in the whole space r n where p n p 2 we aim at constructing for this equation a complex solution u u 1 i u 2 which blows up in finite time t and only at one blowup point a with the following estimates for the final profile u x t p 1 2 x a 2 8 p ln x a 1 p 1 u 2 x t 2 p p 1 2 p 1 2 x a 2 8 p ln x a 1 p 1 1 ln x a as x a note that the imaginary part is non zero and that it blows up also at point a our method relies on two main arguments the reduction of the problem to a finite dimensional one and a topological argument based on the index theory to get the conclusion up to our knowledge this is the first time where the blowup behavior of the imaginary part is derived in multi dimension

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

construction of a multisoliton blowup solution to the semilinear wave equation in one space dimension we consider the semilinear wave equation with power nonlinearity in one space dimension given a blow up solution with a characteristic point we refine the blow up behavior first derived by merle and zaag we also refine the geometry of the blow up set near a charac teristic point and show that except may be for one exceptional situation it is never symmetric with the respect to the characteristic point then we show that all blow up modalities predicted by those authors do occur more precisely given any integer k 2 and 0 we construct a blow up solution with a characteristic point a such that the asymptotic behavior of the solution near a t a shows a decoupled sum of k solitons with alternate signs whose centers in the hyperbolic geometry have 0 as a center of mass for all times

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

the cubic complex ginzburg landau equation for a backward bifurcation abstract the one dimensional complex ginzburg landau equation cgle with a destabilizing cubic nonlinearity and no saturating higher order terms has stable bounded solutions we consider a simple pedagogical model exhibiting qualitatively the mechanism which may suppress the divergence of the solutions then we investigate the functional form of the blow up collapse solutions immediately before the divergence from this analysis we find analytic boundaries for the existence of collapse solutions in the parameter space of the cgle a comparison with numerical simulations demonstrates that for parameters without collapse solutions the solutions of the cgle remain bounded for all times finally we discuss the implications of our results for the solutions of the cgle when saturating higher order terms are included

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

optimal threshold selection for tomogram segmentation by projection distance minimization grey value thresholding is a segmentation technique commonly applied to tomographic image reconstructions many procedures have been proposed to optimally select the grey value thresholds based on the tomogram data only e g using the image histogram in this paper a projection distance minimization pdm method is presented that uses the tomographic projection data to determine optimal thresholds these thresholds are computed by minimizing the distance between the forward projection of the segmented image and the measured projection data an important contribution of the current paper is the efficient implementation of the forward projection method which makes the use of the original projection data as a segmentation criterion feasible simulation experiments applied to various phantom images show that our proposed method obtains superior results compared to established histogram based projection data methods

blow up profile for the complex ginzburg landau equation abstract we construct a solution to the complex ginzburg landau equation which blows up in finite time t only at one blow up point we also give a sharp description of its blow up profile the proof relies on the reduction of the problem to a finite dimensional one and the use of index theory to conclude two major difficulties arise in the proof the linearized operator around the profile is not self adjoint and it has a second neutral mode in the last section the interpretation of the parameters of the finite dimensional problem in terms of the blow up time and the blow up point gives the stability of the constructed solution with respect to perturbations in the initial data

subset sum cubes and the complexity of primality testing a 2 3 boolean circuit has 3 levels of gates the input level is comprised of or gates each taking as inputs 2 not necessarily distinct literals each of these ors feeds one or more and gates at the second level their outputs form the inputs to a single or gate at the output level using the projection technique of paturi saks and zane it is shown that the smallest 2 3 boolean circuit testing primality for any number given by n binary digits has size 2 n g n where g n o n disjunctive normal form dnf formulas can be considered to be a special case of 2 3 circuits and a bound of this sort applies to them too the argument uses the following number theoretic fact which is established via a modified version of gallagher s larger sieve let at a 2 a z be distinct integers in 1 n if a 0 e 1 a 1 e 2 a 2 e z a z is prime for all choices of e 1 e 2 e z e 0 1 then z 9 2 o 1 log n log log n

convergence properties of the nelder mead simplex method in low dimensions the nelder mead simplex algorithm first published in 1965 is an enormously popular direct search method for multidimensional unconstrained minimization despite its widespread use essentially no theoretical results have been proved explicitly for the nelder mead algorithm this paper presents convergence properties of the nelder mead algorithm applied to strictly convex functions in dimensions 1 and 2 we prove convergence to a minimizer for dimension 1 and various limited convergence results for dimension 2 a counterexample of mckinnon gives a family of strictly convex functions in two dimensions and a set of initial conditions for which the nelder mead algorithm converges to a nonminimizer it is not yet known whether the nelder mead method can be proved to converge to a minimizer for a more specialized class of convex functions in two dimensions

a blowup solution of a complex semi linear heat equation with an irrational power abstract in this paper we consider the following semi linear complex heat equation t u u u p u c in r n with an arbitrary power p p 1 we construct for this equation a complex solution u u 1 i u 2 which blows up in finite time t and only at one blowup point a moreover we also describe the asymptotics of the solution by the following final profiles u x t p 1 2 x a 2 8 p ln x a 1 p 1 u 2 x t 2 p p 1 2 p 1 2 x a 2 8 p ln x a 1 p 1 1 ln x a 0 as x a in addition to that since we also have u 1 0 t and u 2 0 t as t t the blowup in the imaginary part shows a new phenomenon unknown for the standard heat equation in the real case a non constant sign near the singularity with the existence of a vanishing surface for the imaginary part shrinking to the origin in our work we have succeeded to extend for any power p where the non linear term u p is not continuous if p is not integer in particular the solution which we have constructed has a positive real part we study our equation as a system of the real part and the imaginary part u 1 and u 2 our work relies on two main arguments the reduction of the problem to a finite dimensional one and a topological argument based on the index theory to get the conclusion

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

a spectral lower bound technique for the size of decision trees and two level and or circuits a universal lower bound technique for the size and other implementation characteristics of an arbitrary boolean function as a decision tree and as a two level and or circuit is derived the technique is based on the power spectrum coefficients of the n dimensional fourier transform of the function the bounds vary from constant to exponential and are tight in many cases several examples are presented

stable blowup dynamics for the 1 corotational energy critical harmonic heat flow we exhibit a stable finite time blowup regime for the 1 corotational energy critical harmonic heat flow from 2 into a smooth compact revolution surface of 3 that reduces to the semilinear parabolic problem for a suitable class of functions f the corresponding initial data can be chosen smooth well localized and arbitrarily close to the ground state harmonic map in the energy critical topology we give sharp asymptotics on the corresponding singularity formation that occurs through the concentration of a universal bubble of energy at the speed predicted by van den berg hulshof and king our approach lies in the continuation of the study of the 1 equivariant energy critical wave map and schrodinger map with 2 target by merle raphael and rodnianski 2012 wiley periodicals inc

constant depth circuits fourier transform and learnability in this paper boolean functions in 4c0 are studied using harmonic analysis on the cube the main result is that an aco boolean function has almost all of its power spectrum on the low order coefficients an important ingredient of the proof is hastad s switching lemma 8 this result implies several new properties of functions in 4c functions in ac have low average sensitivity they may be approximated well by a real polynomial of low degree and they cannot be pseudorandom function generators perhaps the most interesting application is an o n poiyiog n time algorithm for learning func tions in aco the algorithm observes the behavior of an ac function on o npo y og n randomly chosen inputs and derives a good approximation for the fourier transform of the function this approximation allows the algorithm to predict with high probability the value of the function on other randomly chosen inputs

adart an adaptive algebraic reconstruction algorithm for discrete tomography in this paper we suggest an algorithm based on the discrete algebraic reconstruction technique dart which is capable of computing high quality reconstructions from substantially fewer projections than required for conventional continuous tomography adaptive dart adart goes a step further than dart on the reduction of the number of unknowns of the associated linear system achieving a significant reduction in the pixel error rate of reconstructed objects the proposed methodology automatically adapts the border definition criterion at each iteration resulting in a reduction of the number of pixels belonging to the border and consequently of the number of unknowns in the general algebraic reconstruction linear system to be solved being this reduction specially important at the final stage of the iterative process experimental results show that reconstruction errors are considerably reduced using adart when compared to original dart both in clean and noisy environments

construction of a blow up solution for the complex ginzburg landau equation in a critical case we construct a solution for the complex ginzburg landau equation in some critical case which blows up in finite time t only at one blow up point we also give a sharp description of its profile the proof relies on the reduction of the problem to a finite dimensional one and the use of index theory to conclude the interpretation of the parameters of the finite dimension problem in terms of the blow up point and time allows to prove the stability of the constructed solution

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

classification of singularities for blowing up solutions in higher dimensions consider the cauchy problem p u t u u p when x r n t 0 n 1 u x 0 u 0 x when x r n where p 1 and u 0 x is a continuous nonnegative and bounded function it is known that under fairly general assumptions on u 0 x the unique solution of p u x t blows up in a finite time by which we mean that lim sup sup u x t t t x r n in this paper we shall assume that u x t blows up at x 0 t t and derive the possible asymptotic behaviours of u x t as x t 0 t under general assumptions on the blow up rate

characterization of dispersive chaos and related states of binary fluid convection abstract we describe an extensive series of experiments on traveling wave convection in an ethanol water mixture with separation ratio 0 020 in a long narrow annular container as the stress parameter is quasi statically increased through the range 0 2 above onset this system evolves through a series of complex spatiotemporal states before settling into an extended pattern of steady spatially uniform convective rolls much of the dynamical behavior is influenced by strong nonlinear dispersion which causes the repeated and often irregular appearance and sudden collapse of spatially localized bursts of traveling waves small amplitude states seen very close to onset are analyzed quantitatively using a complex ginzburg landau equation complex dynamical states encountered further above onset are analyzed using a number of classical statistical techniques focusing mainly on correlation functions and fourier spectra in addition we use the technique of biorthogonal decomposition in an effort to extract the basic structures of the patterns and to characterize the complexty of their dynamics

on the stability of critical chemotactic aggregation we consider the two dimensional parabolic elliptic keller segel model of chemotactic aggregation for radially symmetric initial data we show the existence of a stable mechanism of singularity formation and obtain a complete description of the associated aggregation 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

multi bump self similar blow up solutions of the ginzburg landau equation abstract for the ginzburg landau equation gl we establish the existence and local uniqueness of two classes of multi bump self similar blow up solutions for all dimensions 2 d 4 under certain conditions on the coefficients in the equation in numerical simulation and via asymptotic analysis one class of solutions was already found the second class of multi bump solutions is new in the analysis we treat the gl as a small perturbation of the cubic nonlinear schrodinger equation nls the existence result given here is a major extension of results established previously for the nls since for the nls the construction only holds for d close to the critical dimension d 2 the behaviour of the self similar solutions is described by a nonlinear non autonomous ordinary differential equation ode after linearisation this ode exhibits hyperbolic behaviour near the origin and elliptic behaviour asymptotically we call the region where the type of behaviour changes the mid range all of the bumps of the solutions that we construct lie in the mid range for the construction we track a manifold of solutions of the ode that satisfy the condition at the origin forward and a manifold of solutions that satisfy the asymptotic conditions backward to a common point in the mid range then we show that these manifolds intersect transversely we study the dynamics in the mid range by using geometric singular perturbation theory adiabatic melnikov theory and the exchange lemma

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

solution of a nonlinear heat equation with arbitrarily given blow up points we consider the equation where i u is scalar valued and p 1 it has been proven that if u t blows up at time t the blow up points are finite in number and located in i our aim is to prove that this result is optimal that is for any given points x1 xk in i there is a solution u such that its blow up points are exactly x1 xk

the cauchy problem in local spaces for the complex ginzburg landau equation ii contraction methods we continue the study of the initial value problem for the complex ginzburg landau equation with a 0 b 0 g 0 in initiated in a previous paper i we treat the case where the initial data and the solutions belong to local uniform spaces more precisely to spaces of functions satisfying local regularity conditions and uniform bounds in local norms but no decay conditions or arbitrarily weak decay conditions at infinity in in i we used compactness methods and an extended version of recent local estimates 3 and proved in particular the existence of solutions globally defined in time with local regularity of the initial data corresponding to the spaces l r for r 2 or h 1 here we treat the same problem by contraction methods this allows us in particular to prove that the solutions obtained in i are unique under suitable subcriticality conditions and to obtain for them additional regularity properties and uniform bounds the method extends some of those previously applied to the nonlinear heat equation in global spaces to the framework of local uniform spaces

automatic parameter estimation for the discrete algebraic reconstruction technique dart computed tomography ct is a technique for noninvasive imaging of physical objects in the discrete algebraic reconstruction technique dart prior knowledge about the material s densities is exploited to obtain high quality reconstructed images from a limited number of its projections in practice this prior knowledge is typically not readily available here a fully automatic method called projection distance minimization dart pdm dart is proposed in which the optimal grey level parameters are adaptively estimated during the reconstruction process to apply pdm dart only the number of different grey levels should be known in advance simulation as well as real ct experiments show that pdm dart is capable of computing reconstructed images of which the quality is similar to reconstructions computed by conventional dart based on exact prior knowledge thereby eliminating the need for tedious and error prone user interaction

circuit and decision tree complexity of some number theoretic problems we extend the area of applications of the abstract harmonic analysis to lower bounds on the circuit and decision tree complexity of boolean functions related to some number theoretic problems in particular we prove that deciding if a given integer is square free and testing co primality of two integers by unbounded fan in circuits of bounded depth requires superpolynomial size