improved lower bounds for the discrepancy of inversive congruential pseudorandom numbers the inversive congruential method with prime modulus for generating uniform pseudorandom numbers is studied lower bounds for the discrepancy of k tuples of successive pseudorandom numbers are established which improve earlier results of niederreiter moreover the present proof is substantially simpler than the earlier one

inversive congruential pseudorandom numbers distribution of triples this paper deals with the inversive congruential method with power of two modulus m for generating uniform pseudorandom numbers statistical independence properties of the generated sequences are studied based on the distribution of triples of successive pseudorandom numbers it is shown that on the average over the parameters in the inversive congruential method the discrepancy of the corresponding point sets in the unit cube is of an order of magnitude between m 1 2 and m 1 2 logm 3 the method of proof relies on a detailed discussion of the properties of certain exponential sums

on the distribution of inversive congruential pseudorandom numbers in parts of the period the inversive congruential method is an attractive alternative to the classical linear congruential method for pseudorandom number generation in this paper we present the first nontrivial bounds on the discrepancy of individual sequences of inversive congruential pseudorandom numbers in parts of the period the proof is based on a new bound for certain incomplete exponential sums

existence of maximal surfaces in asymptotically flat spacetimes we prove the existence of maximal surfaces in asymptotically flat spacetime satisfying an interior condition this uses a priori estimates which can also be applied to prescribed mean curvature surfaces in cosmological spacetimes and the dirichlet problem

lower bounds for the discrepancy of inversive congruential pseudorandom numbers with power of two modulus the inversive congruential method with modulus m 2w for the generation of uniform pseudorandom numbers has recently been introduced the discrepancy z l of a tuples of consecutive pseudorandom numbers generated by such a generator with maximal period length m 2 is the cru cial quantity for the analysis of the statistical independence properties of these pseudorandom numbers by means of the serial test it is proved that for a pos itive proportion of the inversive congruential generators with maximal period length the discrepancy ij2 is at least of the order of magnitude m xl2 for all k 2 this shows that the bound d 2 2 0 m 1 2 logw 2 established by the second author is essentially best possible

on the period length of congruential pseudorandom number sequences generated by inversions abstract congruential pseudorandom number sequences generated by inversions have been studied recently these sequences do not show the undesirable lattice structure of the linear congruential method the necessary and sufficient condition for the generated sequence to have the maximal period length was given by eichenauer 1988 for the case of 2e modulus generalization of this result to the case of an arbitrary prime power modulus is obtained

multiplicative congruential random number generators with modulus 2 an exhaustive analysis for 32 and a partial analysis for 48 this paper presents the results of a search to find optimal maximal period multipliers for multiplicative congruential random number generators with moduli 2 32 and 2 48 here a multiplier is said to be optimal if the distance between adjacent parallel hyperplanes on which k tuples lie does not exceed the minimal achievable distance by more than 25 percent for k 2 6

the fitness evaluation strategy in particle swarm optimization abstract the particle swarm optimization pso computational method has recently become popular however it has limitations it may trap into local optima and cause the premature convergence phenomenon especially for multimodal and high dimensional problems in this paper we focus on investigating the fitness evaluation in terms of a particle s position particularly we find that the fitness evaluation strategy in the standard pso has two drawbacks i e two steps forward and one step back and two steps back and one step forward in addition we propose a general fitness evaluation strategy gfes by which a particle is evaluated in multiple subspaces and different contexts in order to take diverse paces towards the destination position as demonstrations of gfes a series of psos with gfes are presented experiments are conducted on several benchmark optimization problems the results show that gfes is effective at handling multimodal and high dimensional problems

the serial test for congruential pseudorandom numbers generated by inversions two types of congruential pseudorandom number generators based on inversions were introduced recently we analyze the statistical independence properties of these pseudorandom numbers by means of the serial test the results show that these pseudorandom numbers perform satisfactorily under the serial test the methods of proof rely heavily on bounds for character sums such as the weil stepanov bound for character sums over finite fields

the instability of naked singularities in the gravitational collapse of a scalar field one of the fundamental unanswered questions in the general theory of relativity is whether naked singularities that is singular events which are visible from inflnity may form with positive probability in the process of gravitational collapse the conjecture that the answer to this question is in the negative has been called cosmic censorship the present paper which is a continuation of the work in 1 2 3 and 4 addresses this question in the context of the spherical gravitational collapse of a scalar fleld the problem of a spherically symmetric self gravitating scalar fleld is formulated in terms of a 2 dimensional quotient space time manifold q with boundary see 1 the boundary ofq corresponds to the set of flxed points of the group action the center of symmetry which is a timelike geodesic i the manifoldq is endowed with a lorentzian metric gab an area radius function i i r

explicit inversive congruential pseudorandom numbers with power of two modulus an explicit version of the inversive congruential method with power of two modulus for generating uniform pseudorandom numbers is introduced statistical independence properties of the generated sequences are studied by means of the serial test the method of proof relies on a detailed analysis of certain exponential sums

construction of inversive congruential pseudorandom number generators with maximal period length abstract the inversive congruential method for generating uniform pseudorandom numbers is a particularly attractive alternative to linear congruential generators with their well known inherent deficiencies like the unfavourable coarse lattice structure in higher dimensions in the present paper the modulus in the inversive congruential method is chosen as a power of an arbitrary odd prime the existence of inversive congruential generators with maximal period length is proved by a new constructive characterization of these generators

the serial test for pseudo random numbers generated by the linear congruential method we consider the linear congruential method for pseudo random number generation and establish effective criteria for the choice of parameters in this method which guarantee statistical almost independence of successive pseudo random numbers applications to numerical integration are also discussed

statistical independence of a new class of inversive congruential pseudorandom numbers linear congruential pseudorandom numbers show several undesirable regularities which can render them useless for certain stochastic simulations this was the motiviation for important recent developments in nonlinear congruential methods for generating uniform pseudorandom numbers it is particularly promising to achieve nonlinearity by employing the operation of multiplicative inversion with respect to a prime modulus in the present paper a new class of such inversive congruential generators is introduced and analyzed it is shown that they have excellent statistical independence properties and model true random numbers very closely the methods of proof rely heavily on weil stepanov bounds for rational exponential sums 39 refs

on a nonlinear congruential pseudorandom number generator a nonlinear congruential pseudorandom number generator with modulus m 2 w is proposed which may be viewed to comprise both linear as well as inversive congruential generators the condition for it to generate sequences of maximal period length is obtained it is akin to the inversive one and bears a remarkable resemblance to the latter

on generalized inversive congruential pseudorandom numbers the inversive congruential method with prime modulus for generating uniform pseudorandom numbers has several very promising properties very recently a generalization for composite moduli has been introduced in the present paper it is shown that the generated sequences have very attractive statistical independence properties

pseudorandom vector generation by the multiple recursive matrix method pseudorandom vectors are of importance for parallelized simulation methods in this paper we carry out an in depth analysis of the multiple recursive matrix method for the generation of uniform pseudorandom vectors which was introduced in an earlier paper of the author we study in particular the periodicity properties the lattice structure and the behavior under the serial test for sequences of pseudorandom vectors generated by this method

quadratic congruential pseudorandom numbers distribution of lagged pairs abstract the present paper deals with the quadratic congruential method with power of two modulus for generating uniform pseudorandom numbers statistical independence properties of the generated sequences x n n 0 are studied based on the distribution of lagged pairs x n x n 2 upper and lower bounds for the discrepancy of the corresponding point sets in the unit square are established

boundary harmonic coordinates on manifolds with boundary in low regularity in this paper we prove the existence of h 2 regular coordinates on riemannian 3 manifolds with boundary assuming only l 2 bounds on the ricci curvature l 4 bounds on the second fundamental form of the boundary and a positive lower bound on the volume radius the proof follows by extending the theory of cheeger gromov convergence to include manifolds with boundary in the above low regularity setting the main tools are boundary harmonic coordinates together with elliptic estimates and a geometric trace estimate and a rigidity argument using manifold doubling assuming higher regularity of the ricci curvature we also prove corresponding higher regularity estimates for the coordinates

on the lattice structure of a nonlinear generator with modulus 2 a abstract nonlinear congruential pseudorandom number generators based on inversions have been introduced and analysed recently these generators do not show the simple lattice structure of the widely used linear congruential generators which are too regular for certain simulation purposes in the present paper a nonlinear congruential generator based on inversions with respect to a power of two modulus is considered it is shown that the set of points formed by consecutive pseudorandom numbers has a more complicated lattice structure it forms a superposition of shifted lattices the corresponding lattice bases are explicitly determined and analysed

on the discrepancy of quadratic congruential pseudorandom numbers abstract one of the alternatives to linear congruential pseudorandom number generators with their undesirable lattice structure is the quadratic congruential method which is due to knuth in the present paper the statistical independence properties of pairs of consecutive pseudorandom numbers generated according to this method are analysed by means of the serial test upper bounds for the discrepancy of these pairs are established which are essentially best possible the results show that the quadratic congruential method performs uniformly satisfactorily if a reasonable choice of one of the parameters is made the method of proof relies heavily on the evaluation of certain exponential sums

a nonlinear congruential pseudorandom number generator with power of two modulus a nonlinear congruential pseudorandom number generator is studied where the modulus is a power of two investigation of this generator was suggested by knuth 7 a simple necessary and sufficient condition is given for this generator to have the maximal period length

a study of particle swarm optimization particle trajectories particle swarm optimization pso has shown to be an efficient robust and simple optimization algorithm most of the pso studies are empirical with only a few theoretical analyses that concentrate on understanding particle trajectories these theoretical studies concentrate mainly on simplified pso systems this paper overviews current theoretical studies and extend these studies to investigate particle trajectories for general swarms to include the influence of the inertia term the paper also provides a formal proof that each particle converges to a stable point an empirical analysis of multi dimensional stochastic particles is also presented experimental results are provided to support the conclusions drawn from the theoretical findings

statistical independence properties of pseudorandom vectors produced by matrix generators abstract the uniformity test and the serial test are standard statistical tests for equidistribution and statistical independence respectively in sequences of uniform pseudorandom numbers we introduce analogues of these tests for sequences of uniform pseudorandom vectors and we apply these tests to matrix generators for uniform pseudorandom vector generation the results show that these generators behave well under these tests provided that the matrix in the generation algorithm is chosen carefully the essential condition is that a certain figure of merit attached to the matrix be large we also prove a general theorem which guarantees the existence of matrices with large figure of merit

lower bounds for the discrepancy of inversive congruential pseudorandom numbers the inversive congruential method is a uniform pseudorandom number generator which was introduced recently for a prime modulus p the discrepancy d k of k tuples of successive pseudorandom numbers generated by this method determines the statistical independence properties of these pseudorandom numbers it was shown earlier by the author that d k o p 1 2 logp k for 2 2

the localised bounded l 2 l2 curvature theorem in this paper we prove a localised version of the bounded l 2 curvature theorem of klainerman et al invent math 202 1 91 216 2015 more precisely we consider initial data for the einstein vacuum equations posed on a compact spacelike hypersurface sigma with boundary and show that the time of existence of a classical solution depends only on an l 2 bound on the ricci curvature an l 4 bound on the second fundamental form of partial sigma subset sigma an h 1 bound on the second fundamental form and a lower bound on the volume radius at scale 1 of sigma our localisation is achieved by first proving a localised bounded l 2 curvature theorem for small data posed on b 0 1 and then using the scaling of the einstein equations and a low regularity covering argument on sigma to reduce from large data on sigma to small data on b 0 1 the proof uses the author s previous works and the bounded l 2 curvature theorem as black boxes

quadratic congruential pseudorandom numbers distribution of triples the present paper deals with the quadratic congruential method with power of two modulus for generating uniform pseudorandom numbers statistical independence properties of pairs and triples of successive terms in the generated sequences are studied based on the discrepancy of the corresponding point sets in the main part of the paper a known upper bound for the discrepancy of pairs is improved and new upper bounds for the discrepancy of triples are established known lower bounds for the corresponding discrepancies are recalled the method of proof relies on a detailed discussion of the properties of certain exponential sums

inversive congruential pseudorandom numbers avoid the planes nonlinear congruential pseudorandom number generators based on inversions have recently been introduced and analyzed these generators do not show the lattice structure of the widely used linear congruential method in the present paper it is proved that the points formed by d consecutive pseudorandom numbers of an inversive congruential generator with prime modulus possess an even stronger property any hyperplane in space contains at most d of these points that is to say the hyperplane spanned by d arbitrary points of an inversive congruential generator contains no further points this feature makes the inversive congruential method particularly attractive for simulation problems where linear structures within the generated points should be avoided

equidistribution properties of quadratic congruential pseudorandom numbers this paper deals with the quadratic congruential method for generating uniform pseudorandom numbers equidistribution properties of the generated pseudorandom number sequences over parts of the period are considered based on the discrepancy of corresponding point sets an upper bound for the average value of these discrepancies is established

compound inversive congruential pseudorandom numbers an average case analysis the present paper deals with the compound or generalized inversive congruential method for generating uniform pseudorandom numbers which has been introduced recently equidistribution and statistical independence properties of the generated sequences over parts of the period are studied based on the discrepancy of certain point sets the main result is an upper bound for the average value of these discrepancies the method of proof is based on estimates for exponential sums

a remark on the discrepancy of quadratic congruential pseudorandom numbers abstract the linear congruential method for generating uniform pseudorandom numbers has several known deficiencies which can render the generated sequences useless for certain simulation purposes one of the alternatives is knuth s quadratic congruential method recently it was proved that pairs of quadratic congruential pseudorandom numbers have uniformly excellent statistical independence properties in the present paper it is shown that unfortunately a similar result cannot be obtained for k tuples with k 3 it is proved that for a positive percentage of the possible generators the discrepancy of these k tuples is too large the method of proof relies on the evaluation of certain exponential sums in view of the present result it becomes doubtful whether the quadratic congruential method is absolutely suitable for generating uniform pseudorandom numbers

the serial test for a nonlinear pseudorandom number generator let m 2 and g m 1 3 m 1 a sequence y n y n g m is obtained by the formula y n 1 ay n b cy n mod m the sequence x n x n y n m is a sequence of pseudorandom numbers of the maximal period length m 2 if and only if a c 1 mod 4 b 2 mod 4 in this note the uniformity is investigated by the 2 dimensional serial test for the sequence we follow closely the method of papers by eichenauer herrmann and niederreiter

lattice rules for multiple integration and discrepancy upper and lower bounds for the discrepancy of nodes in lattice rules for multidimensional numerical integration are established in this way the applicability of lattice rules is extended to nonperiodic integrands

are there hyperbolas in the scatter plots of inversive congruential pseudorandom numbers abstract given a scatter plot of overlapping pairs xn xn 1 of pseudorandom numbers and the information that it stems from either a linear congruential or an inversive congruential generator with small modulus it is usually no problem to guess the type of the generator the points of a linear generator form a grid or lattice whereas inversive and explicit inversive congruential pseudorandom numbers exhibit hyperbola like structures we give an analytical description of this most eye catching structural element of inversive congruential generators

on the discrepancy of quadratic congruential pseudorandom numbers with power of two modulus the present paper deals with the quadratic congruential method with power of two modulus m 2 for generating uniform pseudorandom numbers it is shown that for a positive fraction of the quadratic congruential sequences with maximal period length m and all dimensions k 3 the discrepancy of k tuples of successive pseudorandom numbers is of an order of magnitude at least m 13 this order of magnitude is too large in view of the law of the iterated logarithm for discrepancies since the discrepancy of truly random numbers should be roughly m 12 log log m 12 the proof of the main result relies on a thorough evaluation of certain exponential sums

on a new class of pseudorandom numbers for simulation methods abstract the deficiencies of classical methods for uniform pseudorandom number generation have recently led to the development of alternative methods such as nonlinear congruential methods a particularly attractive nonlinear congruential method is the explicit inversive congruential method recently introduced by eichenauer herrmann we establish several new results on the structural and statistical properties of pseudorandom numbers generated by this method these results demonstrate that this method is eminently suitable for the generation of parallel streams of pseudorandom numbers with desirable properties

the bounded l 2 curvature conjecture this is the main paper in a sequence in which we give a complete proof of the bounded l 2 curvature conjecture more precisely we show that the time of existence of a classical solution to the einstein vacuum equations depends only on the l 2 norm of the curvature and a lower bound on the volume radius of the corresponding initial data set we note that though the result is not optimal with respect to the scaling of the einstein equations it is nevertheless critical with respect to its causal geometry indeed l 2 bounds on the curvature is the minimum requirement necessary to obtain lower bounds on the radius of injectivity of causal boundaries we note also that while the first nontrivial improvements for well posedness for quasilinear hyperbolic systems in spacetime dimensions greater than 1 1 based on strichartz estimates were obtained in bahouri and chemin am j math 121 1337 1777 1999 imrn 21 1141 1178 1999 tataru am j math 122 349 376 2000 jams 15 2 419 442 2002 klainerman and rodnianski duke math j 117 1 1 124 2003 and optimized in klainerman and rodnianski ann math 161 1143 1193 2005 smith and tataru ann math 162 291 366 2005 the result we present here is the first in which the full structure of the quasilinear hyperbolic system not just its principal part plays a crucial role to achieve our goals we recast the einstein vacuum equations as a quasilinear so 3 1 valued yang mills theory and introduce a coulomb type gauge condition in which the equations exhibit a specific new type of null structure compatible with the quasilinear covariant nature of the equations to prove the conjecture we formulate and establish bilinear and trilinear estimates on rough backgrounds which allow us to make use of that crucial structure these require a careful construction and control of parametrices including l 2 error bounds which is carried out in szeftel parametrix for wave equations on a rough background i regularity of the phase at initial time arxiv 1204 1768 2012 parametrix for wave equations on a rough background ii construction of the parametrix and control at initial time arxiv 1204 1769 2012 parametrix for wave equations on a rough background iii space time regularity of the phase arxiv 1204 1770 2012 parametrix for wave equations on a rough background iv control of the error term arxiv 1204 1771 2012 as well as a proof of sharp strichartz estimates for the wave equation on a rough background which is carried out in szeftel sharp strichartz estimates for the wave equation on a rough background arxiv 1301 0112 2013 it is at this level that our problem is critical indeed any known notion of a parametrix relies in an essential way on the eikonal equation and our space time possesses barely the minimal regularity needed to make sense of its solutions