large gaps between consecutive prime numbers let g x denote the size of the largest gap between consecutive primes below x answering a question of erd os we show that g x f x logx log logx log log log logx log log logx 2 where f x is a function tending to innity with x our proof combines existing arguments with a random construction covering a set of primes by arithmetic progressions as such we rely on recent work on the existence and distribution of long arithmetic progressions consisting entirely of primes

on polynomial factorial diophantine equations we study equations of the form p x n and show that for some classes of polynomials p the equation has only finitely many solutions this is the case say if p is irreducible of degree greater than 1 or has an irreducible factor of relatively large degree this is also the case if the factorization of p contains some large power s of irreducible s for example we can show that the equation x r x 1 n has only finitely many solutions for r 4 but not that this is the case for 1 r 3 although it undoubtedly should be we also study the equation p x h n where h n is one of several other highly divisible sequences proving again that for various classes of polynomials these equations have only finitely many solutions

stress resultant geometrically nonlinear shell theory with drilling rotations part i a consistent formulation abstract in this work we present a consistent theoretical framework for a novel stress resultant geometrically nonlinear shell theory the main feature of the present shell theory development which stands in contrast with the classical developments in the shell theory is the presence of a rotation component around the shell normal so called drilling rotation in the description of shell finite rotations the relationship of the proposed theory with a finite deformation theory of a three dimensional continuum with independent rotation field is clearly identified the latter is shown to be an important link which facilitates the proper choice of the shell constitutive model and a proper construction of the regularized form of the theory capable of supporting the drilling rotations the corresponding linearized form of the present shell theory is discussed in the closure

on the turan number for the hexagon abstract a long standing conjecture of erd s and simonovits is that ex n c 2 k the maximum number of edges in an n vertex graph without a 2 k gon is asymptotically 1 2 n 1 1 k as n tends to infinity this was known almost 40 years ago in the case of quadrilaterals in this paper we construct a counterexample to the conjecture in the case of hexagons for infinitely many n we prove that ex n c 6 3 5 2 5 1 4 3 n 4 3 o n 0 5338 n 4 3 we also show that ex n c 6 n 4 3 o n 0 6272 n 4 3 if n is sufficiently large where is the real root of 16 3 4 2 3 0 this yields the best known upper bound for the number of edges in a hexagon free graph the same methods are applied to find a tight bound for the maximum size of a hexagon free 2 n n bipartite graph

linear size optimal q ary constant weight codes and constant composition codes an optimal constant composition or constant weight code of weight w has linear size if and only if its distance d is at least 2w 1 when d 2w the determination of the exact size of such a constant composition or constant weight code is trivial but the case of d 2w 1 has been solved previously only for binary and ternary constant composition and constant weight codes and for some sporadic instances this paper provides a construction for quasicyclic optimal constant composition and constant weight codes of weight w and distance 2w 1 based on a new generalization of difference triangle sets as a result the sizes of optimal constant composition codes and optimal constant weight codes of weight w and distance 2w 1 are determined for all such codes of sufficiently large lengths this solves an open problem of etzion the sizes of optimal constant composition codes of weight w and distance 2w 1 are also determined for all w 6 except in two cases

staged and constrained two dimensional guillotine cutting problems an and or graph approach abstract an and or graph approach has been proposed by morabito et al to solve non staged and unconstrained two dimensional guillotine cutting problems basically this approach consists of representing cutting patterns as complete paths in an and or graph where nodes and arcs correspond to rectangles and cuts respectively and choosing a search strategy to traverse or enumerate the nodes of the graph in this present paper the and or graph approach is extended to solve staged and constrained problems computational experiments with examples from the literature are performed and indicate that this approach generates good and fast solutions using a microcomputer in addition to the fact that the and or graph approach can handle important constraints it can be easily extended to solve cutting and packing problems with multiple dimensions

the distribution of prime ideals of imaginary quadratic fields let q x y be a primitive positive definite quadratic form with integer coefficients then for all s t e r 2 there exist m n z 2 such that q m n is prime and q m s n t q s t 0 53 1 this is deduced from another result giving an estimate for the number of prime ideals in an ideal class of an imaginary quadratic number field that fall in a given sector and whose norm lies in a short interval

strip generation algorithms for constrained two dimensional two staged cutting problems abstract the constrained two dimensional cutting c tdc problem consists of determining a cutting pattern of a set of n small rectangular piece types on a rectangular stock plate of length l and width w as to maximize the sum of the profits of the pieces to be cut each piece type i i 1 n is characterized by a length l i a width w i a profit or weight c i and an upper demand value b i the upper demand value is the maximum number of pieces of type i which can be cut on rectangle l w in this paper we study the two staged fixed orientation c tdc noted fc 2tdc it is a classical variant of the c tdc where each piece is produced in the final cutting pattern by at most two guillotine cuts and each piece has a fixed orientation we solve the fc 2tdc problem using several approximate algorithms that are mainly based upon a strip generation procedure we evaluate the performance of these algorithms on instances extracted from the literature

integer linear programming models for 2 staged two dimensional knapsack problems we are given a unique rectangular piece of stock material s with height h and width w and a list of m rectangular shapes to be cut from s each shape s type i i 1 m is characterized by a height a width a profit and an upper bound ub i indicating the maximum number of items of type i which can be cut we refer to the two dimensional knapsack tdk as the problem of determining a cutting pattern of s maximizing the sum of the profits of the cut items in particular we consider the classical variant of tdk in which the maximum number of cuts allowed to obtain each item is fixed to 2 and we refer to this problem as 2 staged tdk 2tdk for the 2tdk problem we present two new integer linear programming models we discuss their properties and we compare them with other formulations in terms of the lp bound they provide finally both models are computationally tested within a standard branch and bound framework on a large set of instances from the literature by reinforcing them with the addition of linear inequalities to eliminate symmetries

numerical integration of non linear elastic multi body systems this paper is concerned with the modelling of nonlinear elastic multi body systems discretized using the finite element method the formulation uses cartesian co ordinates to represent the position of each elastic body with respect to a single inertial frame the kinematic constraints among the various bodies of the system are enforced via the lagrange multiplier technique the resulting equations of motion are stiff non linear differential algebraic equations the integration of these equations presents a real challenge as most available techniques are either numerically unstable or present undesirable high frequency oscillations of a purely numerical origin an approach is proposed in which the equations of motion are discretized so that they imply conservation of the total energy for the elastic components of the system whereas the forces of constraint are discretized so that the work they perform vanishes exactly the combination of these two features of the discretization guarantees the stability of the numerical integration process for non linear elastic multi body systems examples of the procedure are presented

a branch and cut and price algorithm for one dimensional stock cutting and two dimensional two stage cutting abstract the one dimensional cutting stock problem 1d csp and the two dimensional two stage guillotine constrained cutting problem 2d 2cp are considered in this paper the gilmore gomory models of these problems have very strong continuous relaxations providing a good bound in an lp based solution approach in recent years there have been several efforts to attack the one dimensional problem by lp based branch and bound with column generation called branch and price and by general purpose c hv a tal g omory cutting planes in this paper we investigate a combination of both approaches i e the lp relaxation at each branch and price node is strengthened by c hv a tal g omory and g omory mixed integer cuts the branching rule is that of branching on variables of the gilmore gomory formulation tests show that for 1d csp general purpose cuts are useful only in exceptional cases however for 2d 2cp their combination with branching is more effective than either approach alone and mostly better than other methods from the literature

on approximately symmetric informationally complete positive operator valued measures and related systems of quantum states we address the problem of constructing positive operator valued measures povms in finite dimension n consisting of n2 operators of rank one which have an inner product close to uniform this is motivated by the related question of constructing symmetric informationally complete povms sic povms for which the inner products are perfectly uniform however sic povms are notoriously hard to construct and despite some success of constructing them numerically there is no analytic construction known we present two constructions of approximate versions of sic povms where a small deviation from uniformity of the inner products is allowed the first construction is based on selecting vectors from a maximal collection of mutually unbiased bases and works whenever the dimension of the system is a prime power the second construction is based on perturbing the matrix elements of a subset of mutually unbiased bases moreover we construct vector systems in cn which are almost orthogonal and which might turn out

the difference between consecutive primes ii with enough effort the value of x0 could be determined effectively the paper has much in common with 1 in particular we use the sieve method of harman 4 5 we no longer use zero density estimates however but rather mean value results on dirichlet polynomials similar to those that give rise to such estimates compare for example work of iwaniec and pintz 9 and baker harman and pintz 2 much of the improvement over 1 arises from the use of watt s theorem 11 on a particular kind of mean value more accurate estimates for six dimensional integrals are also used to good effect there is in addition a device which uses a two dimensional sieve to get an asymptotic formula for a one dimensionally sieved set see lemmas 16 17 unfortunately these lemmas which would be of great signi cance for v 0 53 are not very numerically signi cant when v drops to 0 525 the same applies to the roale reversals discussed below let us introduce enough notation to permit an outline of the proof when e is a nite sequence of positive integers counted with multiplicity we write jej for the number of terms of e and ed fm dm 2 eg let p z y

long gaps between primes let pn denote the n th prime we prove that max pn 1 x pn 1 pn log x log log x log log log log x log log log x for sufficiently large x improving upon recent bounds of the first three and fifth authors and of the fourth author our main new ingredient is a generalization of a hypergraph covering theorem of pippenger and spencer proven using the rodl nibble method

on optimal locally repairable codes with super linear length in this paper locally repairable codes which have optimal minimum hamming distance with respect to the bound presented by prakash et al are considered new upper bounds on the length of such optimal codes are derived the new bounds apply to more general cases and have weaker requirements compared with the known ones in this sense they both improve and generalize previously known bounds further optimal codes are constructed whose length is order optimal with respect to the new upper bounds notably the length of the codes is super linear in the alphabet size

the distribution of integers with a divisor in a given interval we determine the order of magnitude of h x y z the number of integers n x having a divisor in y z for all x y and z we also study hr x y z the number of integers n x having exactly r divisors in y z whenr 1 we establish the order of magnitude ofh1 x y z for allx y z satisfying z x 1 2 for every r 2 c 1 and 0 we determine the order of magnitude of hr x y z uniformly for y large and y y logy log 4 1 z min y c x 1 2 as a consequence of these bounds we settle a 1960 conjecture of erd os and some conjectures of tenenbaum one key element of the proofs is a new result on the distribution of uniform order statistics

note on the group edge irregularity strength of graphs abstract we investigate the group edge irregularity strength esg g of graphs i e the smallest value of s such that taking any abelian group g of order s there exists a function f v g g such that the sums of vertex labels at every edge are distinct in this note we provide some the upper bounds on esg g as well as for edge irregularity strength es g and harmonious order har g

non linear dynamics of three dimensional rods exact energy and momentum conserving algorithms the long term dynamic response of non linear geometrically exact rods under going finite extension shear and bending accompanied by large overall motions is addressed in detail the central objective is the design of unconditionally stable time stepping algorithms which exactly preserve fundamental constants of the motion such as the total linear momentum the total angular momentum and for the hamiltonian case the total energy this objective is accomplished in two steps first a class of algorithms is introduced which conserves linear and angular momentum this result holds independently of the definition of the algorithmic stress resultants second an algorithmic counterpart of the elastic constitutive equations is developed such that the law of conservation of total energy is exactly preserved conventional schemes exhibiting no numerical dissipation symplectic algorithms in particular are shown to lead to unstable solutions when the high frequencies are not resolved compared to conventional schemes there is little if any additional computational cost involved in the proposed class of energy momentum methods the excellent performance of the new algorithm in comparison to other standard schemes is demonstrated in several numerical simulations

computational aspects of vector like parametrization of three dimensional finite rotations theoretical and computational aspects of vector like parametrization of three dimensional finite rotations which uses only three rotation parameters are examined in detail in this work the relationship of the proposed parametrization with the intrinsic representation of finite rotations via an orthogonal matrix is clearly identified careful considerations of the consistent linearization procedure pertinent to the proposed parametrization of finite rotations are presented for the chosen model problem of reissner s non linear beam theory pertaining details of numerical implementation are discussed for the simplest choice of the finite element interpolations for a 2 node three dimensional beam element a number of numerical simulations in three dimensional finite rotation analysis are presented in order to illustrate the proposed approach

geometrically exact 3d beam theory implementation of a strain invariant finite element for statics and dynamics geometrically exact 3d beam theory has been used as a basis for development of a variety of finite element formulations it has recently become apparent that the important requirement of objectivity of adopted strain measures although provided by the theory itself does not automatically extend to a finite element formulation in this paper we present a new finite element formulation of the geometrically exact 3d beam theory specifically designed to preserve the objectivity of the adopted strain measures in order to do so the current local rotations are interpolated in a manner similar to that adopted in co rotational approaches however no approximations typical for co rotational approaches are introduced into the procedure so in contrast to co rotational formulations the present formulation fully preserves the geometric exactness of the theory a range of numerical examples serves to illustrate the problem and to assess the formulation

a kinematically exact space finite strain beam model finite element formulation by generalized virtual work principle the present paper presents a novel finite element formulation for static analysis of linear elastic spatial frame structures extending the formulation given by simo and vu quoc a geometrically exact rod model incorporating shear and torsion warping deformation int j solids structures 27 3 1991 371 3931 along the lines of the work on the planar beam theory presented by saje a variational principle for finite planar deformation of straight slender elastic beams internat j solids and structures 26 1990 887 9001 we apply exact non linear kinematic relationships of the space finite strain beam theory assuming the bernoulli hypothesis and neglecting the warping deformations of the cross section finite displacements and rotations as well as finite extensional shear torsional and bending strains are accounted for in the formulation a deformed configuration of the beam is described by the displacement vector of the deformed centroid axis and an orthonormal moving frame rigidly attached to the cross section of the beam the position of the moving frame relative to a fixed reference frame is specified by an orthogonal matrix parametrized by the rotational vector which rotates the moving frame from an arbitrary position into the deformed configuration in one step also the incremental rotational vector is introduced which rotates the moving frame from the configuration obtained at the previous iteration step into the current configuration of the beam its components relative to the fixed global coordinate system are taken to be the rotational degrees of freedom at nodal points because in 3 d space both the axial and the follower moments are non conservative not the variational principle but the principle of virtual work has been introduced as a basis for the finite element discretization here we have proposed the generalized form of the principle of virtual work by including exact kinematic equations by means of a procedure similar to that of lagrangian multipliers this makes possible the elimination of the displacement vector field from the principle so that the three components of the incremental rotational vector field remain the only functions to be approximated in the finite element implementation of the principle other researchers on the other hand employ the three components of the incremental rotational vector field and the three components of the incremental displacement vector field as a result more accurate and efficient family of beam finite elements for the non linear analysis of space frames has been obtained a one field formulation results in the fact that in the present finite elements the locking never occurs any combination of deformation states is described equally precisely this is in contrast with the elements developed in literature where in order to avoid the locking a reduced numerical integration has to be applied which unfortunately diminishes the accuracy of the solution polynomials have been chosen for the approximation of the components of the rotational vector in this case the order of the numerical integration can rationally be estimated and the computer program can be coded in such a way that the degree of polynomials need not be limited to a particular value the newton method is used for the iterative solution of the non linear equilibrium equations in an non equilibrium configuration the tangent stiffness matrix obtained by the linearization of governing equations using the directional derivative is non symmetric even for conservative loadings only upon achieving an equilibrium state the tangent stiffness matrix becomes symmetric thus obtained tangent stiffness matrix can be symmetrized without affecting the rate of convergence of the newton method for non conservative loadings however the tangent stiffness matrix is always non symmetric the numerical examples demonstrate capability of the present formulation to determine accurately the non linear behaviour of space frames in numerical examples the out of plane buckling loads are determined and the whole pre and post critical load displacement paths of a cantilever and a right angle frame are traced these in the analysis of space beams standard verification example problems show excellent accuracy of the solution even when employing only one element to describe the displacements of the size of the structure itself the rotations of 2n and extensional strains much beyond the realistic values of linear elastic material

heuristic for the rectangular two dimensional single stock size cutting stock problem with two staged patterns abstract two staged patterns are often used in manufacturing industries to divide stock plates into rectangular items a heuristic algorithm is presented to solve the rectangular two dimensional single stock size cutting stock problem with two staged patterns it uses the column generation method to solve the residual problems repeatedly until the demands of all items are satisfied each pattern is generated using a procedure for the constrained single large object placement problem to guarantee the convergence of the algorithm the computational results of benchmark and practical instances indicate the following 1 the algorithm can solve most instances to optimality with the gap to optimality being at most one plate for those solutions whose optimality is not proven and 2 for the instances tested the algorithm is more efficient on average in reducing the number of plates used than a published algorithm and a commercial stock cutting software package

best first search methods for constrained two dimensional cutting stock problems best first search is a widely used problem solving technique in the field of artificial intelligence the method has useful applications in operations research as well here we describe an application to constrained two dimensional cutting stock problems of the following type a stock rectangle s of dimensions l w is supplied there are n types of demanded rectangles r1 r2 rn with the ith type having length li width wi value vi and demand constraint bi it is required to produce from the stock rectangle s ai copies of ri 1 i n to maximize a1v1 a2v2 anvn subject to the constraints ai bi only orthogonal guillotine cuts are permitted all parameters are integers a best first tree search algorithm based on wang s bottom up approach is described that guarantees optimal solutions and is more efficient than existing methods

interpolation of rotational variables in nonlinear dynamics of 3d beams the formulation of dynamic procedures for three dimensional 3 d beams requires extensive use of the algebra pertaining to the non linear character of the rotation group in space the corresponding extraction procedure to obtain the rotations that span a time step has certain limitations which can have a detrimental eect on the overall stability of a time integration scheme the paper describes two algorithms for the dynamics of 3 d beams which dier in their manifestation of the above limitation the rst has already been described in the literature and involves the interpolation of iterative rotations while an alternative formulation which eliminates the above eect by design requires interpolation of incremental rotations theoretical arguments are backed by numerical results similarities between the conventional and new formulation are pointed out and are shown to be big enough to enable easy transformation of the conventional formulation into the new one 1998 john wiley sons ltd

efficient cm constructions of elliptic curves over finite fields we present an algorithm that on input of an integer n 1 together with its prime factorization constructs a finite field f and an elliptic curve e over f for which e f has order n although it is unproved that this can be done for all n a heuristic analysis shows that the algorithm has an expected run time that is polynomial in 2 n log n where n is the number of distinct prime factors of n in the cryptographically relevant case where n is prime an expected run time o logn 4 e can be achieved we illustrate the efficiency of the algorithm by constructing elliptic curves with point groups of order n 10 2004 and n nextprime 102004 10 2004 4863

vinogradov s theorem with almost equal summands let 11 20 we prove that every sufficiently large odd integer n can be written as a sum of three primes n p1 p2 p3 with pi n 3 n for i 1 2 3

diophantine approximation with arithmetic functions i we prove a strong simultaneous diophantine approximation theorem for values of additive and multiplicative functions provided that the functions have certain regularity on the primes

an excursion into large rotations the present discourse develops an enlarged exploration of the matrix formulation of finite rotations in space initiated in 1 it is shown how a consistent but subtle matrix calculus inevitably leads to a number of elegant expressions for the transformation or rotation matrix t appertaining to a rotation about an arbitrary axis also analysed is the case of multiple rotations about fixed or follower axes particular attention is paid to an explicit derivation of a single compound rotation vector equivalent to two consecutive arbitrary rotations this theme is discussed in some detail for a number of cases semitangential rotations for which commutativity holds first proposed in 2 3 are also considered furthermore an elementary geometrical analysis of large rotations is also given finally we deduce in an appendix using a judicious reformulation of quarternions the compound pseudovector representing the combined effect of n rotations in the author s opinion the present approach appears preferable to a pure vectorial scheme and even more so to an indicial formulation and is computationally more convenient

the dh kd algorithm a hybrid approach for unconstrained two dimensional cutting problems abstract we study both weighted and unweighted unconstrained two dimensional guillotine cutting problems we develop a hybrid approach which combines two heuristics from the literature the first one dh uses a tree search procedure introducing two strategies depth first search and hill climbing the second one kd is based on a series of one dimensional knapsack problems using dynamic programming techniques the dh kd algorithm starts with a good initial lower bound obtained by using the kd algorithm at each level of the tree search the proposed algorithm uses also the kd algorithm for constructing new lower bounds and uses another one dimensional knapsack for constructing refinement upper bounds the resulting algorithm can be seen as a generalization of the two heuristics and solves large problem instances very well within small computational time our algorithm is compared to morabito et al s algorithm the unweighted case and to beasley s 2 approach the weighted case on some examples taken from the literature as well as randomly generated instances

on rigid components and joint constraints in nonlinear dynamics of flexible multibody systems employing 3d geometrically exact beam model abstract in this work we discuss the finite element implementation of the internal constraints in a three dimensional 3d geometrically exact beam model which can be formulated as holonomic constraint relationships model problems chosen for a more detailed consideration include a general joint constraint between beams and the beam connected to a rigid component appropriate modifications of the standard form of the geometrically exact beam element arrays are carried out in order to impose explicitly these kinds of constraints into time integration schemes for nonlinear dynamics consequently only the minimum number of unknowns is retained for the global set of nonlinear equations to be solved thus avoiding the use of the extra variables the lagrange multipliers and the pertinent difficulties in integrating the system of differential algebraic equations a number of numerical simulations considering dynamic analysis of multibody systems with rigid flexible components and joint constraints are presented in order to illustrate versatility of the proposed procedure

irreducible algebraic integers in short intervals we study distribution of irreducible algebraic integers in short intervals and prove that if the class number of an algebraic number field k exceeds 2 every interval of the form x x x with a fixed 1 2 contains absolute value of the norm of an irreducible algebraic integer from k provided x x 0 the constant x 0 effectively depends on k and

a nested decomposition approach to a three stage two dimensional cutting stock problem we consider the cutting of rectangular order pieces into stock pieces of specified width and length the cutting process involves three stages of orthogonal guillotine cutting stock pieces are cut into sections that are cut into slits that are cut into order pieces restrictions imposed on the cutting process make the combinatorial structure of the problem more complex but limit the scope of solution space the objective of the problem is mainly to minimize waste but our model also accounts for other issues such as aging stock pieces urgent or optional orders and fixed setup costs our solution approach involves a nested decomposition of the problem and the recursive use of the column generation technique we use a column generation formulation of the problem gilmore and gomory 1965 and the cutting pattern generation subproblem is itself solved using a column generation algorithm lp based lower bounds on the minimum cost are computed and by rounding the lp solution a feasible solution and associated upper bound is obtained this approach could in principle be used in a branch and bound search to solve the problem to optimality we report computational results for industrial instances the algorithm is being used in industry as a production planning tool

unconditionally stable algorithms for rigid body dynamics that exactly preserve energy and momentum we show that for rigid body dynamics the mid point rule formulated in body co ordinates exactly conserves energy and the norm of the angular momentum for incremental force free motions but fails to conserve the direction of the angular momentum vector further we show that the mid point rule formulated in the spatial representation is in general physically and geometrically meaningless an alternative algorithm is developed which exactly preserves energy and the total spatial angular momentum in incremental force free motions the implicit version of this algorithm is unconditionally stable and second order accurate the explicit version conserves exactly angular momentum in incremental force free motions numerical simulations are presented which illustrate the excellent performance of the proposed procedure even for incremental rotations over 65 degrees the procedure is directly applicable to transient dynamic calculations of geometrically exact rods and shells

extrema of graph eigenvalues abstract in 1993 hong asked what are the best bounds on the k th largest eigenvalue k g of a graph g of order n this challenging question has never been tackled for any 2 k n in the present paper tight bounds are obtained for all k 2 and even tighter bounds are obtained for the k th largest singular value k g some of these bounds are based on taylor s strongly regular graphs and others on a method of kharaghani for constructing hadamard matrices the same kind of constructions are applied to other open problems like nordhaus gaddum problems of the kind how large can k g k g be these constructions are successful also in another open question how large can the ky fan norm 1 g k g be ky fan norms of graphs generalize the concept of graph energy so this question generalizes the problem for maximum energy graphs in the final section several results and problems are restated for 1 1 matrices which seem to provide a more natural ground for such research than graphs many of the results in the paper are paired with open questions and problems for further study