a block equation solver for large unsymmetric matrices arising in the boundary integral equation method an equation solver for the large unsymmetric systems of linear equations arising in the application of the boundary integral equation method to problems of linear elasticity in homogeneous or piecewise homogeneous solids is presented the solver uses gaussian elimination the advantages of the solver over many existing eliminational or iterational methods are that it ignores many of the large groups of zeros that occur in piecewise homogeneous work and also restrains growth of the number of non zero matrix entries memory requirements and computation times therefore are reduced for many piecewise homogeneous problems

design sensitivity analysis of boundary element substructures the ability to reduce or condense a three dimensional model exactly and then iterate on this reduced size model representing the parts of the design that are allowed to change in an optimization loop is discussed the discussion presents the results obtained from an ongoing research effort to exploit the concept of substructuring within the structural shape optimization context using a boundary element analysis bea formulation the first part contains a formulation for the exact condensation of portions of the overall boundary element model designated as substructures the use of reduced boundary element models in shape optimization requires that structural sensitivity analysis can be performed a reduced sensitivity analysis formulation is then presented that allows for the calculation of structural response sensitivities of both the substructured reduced and unsubstructured parts of the model it is shown that this approach produces significant computational economy in the design sensitivity analysis and reanalysis process by facilitating the block triangular factorization and forward reduction and backward substitution of smaller matrices the implementatior of this formulation is discussed and timings and accuracies of representative test cases presented

boundary element shape optimization system for aircraft structural components the design sensitivities are obtained economically by implicit differentiation of the boundary integral equations the mesh generation and regeneration is done using a parametric and auxiliary geometry concept that allows the original mesh to remain adequate for a wide range of subsequent evolving geometries as the optimization proceeds

iterative solution techniques in boundary element analysis iterative techniques for the solution of the algebraic equations associated with the direct boundary element analysis bea method are discussed continuum structural response analysis problems are considered employing single and multi zone boundary element models with and without zone condensation the impact on convergence rate and computer resource requirements associated with the sparse and blocked matrices resulting in multi zone bea is studied both conjugate gradient and generalized minimum residual preconditioned iterative solvers are applied for these problems and the performance of these algorithms is reported included is a quantification of the impact of the preconditioning utilized to render the boundary element matrices solvable by the respective iterative methods in a time competitive with direct methods to characterize the potential of these iterative techniques we discuss accuracy storage and timing statistics in comparison with analogous information from direct sparse blocked matrix factorization procedures matrix populations that experience block fill in during the direct decomposition process are included with different degrees of preconditioning iterative equation solving is shown to be competitive with direct methods for the problems considered

on a4 b4 c4 d4 on utilise les courbes elliptiques pour determiner une infinite de solutions de l equation a 4 b 4 c 4 d 4 en entiers copremiers a partir de 2682440 4 153656639 4 18796760 4 20615673 4 on donne ainsi un contre exemple de la conjecture d euler concernant la generalisation du grand theoreme de fermat

some counting questions for matrices with restricted entries we obtain upper bounds for the number of arbitrary and symmetric matrices with integer entries in a given box in an arbitrary location and a given determinant we then apply these bounds to estimate the number of matrices in such boxes which have an integer eigenvalues finally we outline some open questions

plane curves in boxes and equal sums of two powers let f z x1 x2 x3 be an absolutely irreducible form of degree d producing a plane curve in p the central aim of this paper is to analyze the density of rational points on such curves which are contained in boxes with unequal sides we shall see below how such considerations may be used to obtain new paucity results for equal sums of two powers suppose that p p1 p2 p3 for fixed real numbers 1 p1 p2 p3 say then we define

counting rational points on hypersurfaces for any n geq 2 let f in mathbb z x 1 ldots x n be a form of degree d geq 2 which produces a geometrically irreducible hypersurface in mathbb p n 1 this paper is concerned with the number n f b of rational points on f 0 which have height at most b for any varepsilon 0 we establish the estimate n f b o b n 2 varepsilon whenever either n leq 5 or the hypersurface is not a union of lines here the implied constant depends at most upon d n and varepsilon

the density of rational points near hypersurfaces we establish a sharp asymptotic formula for the number of rational points up to a given height and within a given distance from a hypersurface our main innovation is a bootstrap method that relies on the synthesis of poisson summation projective duality and the method of stationary phase this has surprising applications to counting rational points lying on the manifold indeed we are able to prove an analogue of serre s dimension growth conjecture originally stated for projective varieties in this general setup as another consequence of our main counting result we solve the generalized baker schmidt problem in the simultaneous setting for hypersurfaces

power free values of polynomials on symmetric varieties given a symmetric variety y defined over q and a non zero polynomial with integer coefficients we use techniques from homogeneous dynamics to establish conditions under which the polynomial can be made r free for a zariski dense set of integral points on y we also establish an asymptotic counting formula for this set in the special case that y is a quadric hypersurface we give explicit bounds on the size of r by combining the argument with a uniform upper bound for the density of integral points on general affine quadrics defined over q

numerical characterization of the kahler cone of a compact kahler manifold the goal of this work is to give a precise numerical description of the kahler cone of a compact kahler manifold our main result states that the kahler cone depends only on the intersection form of the cohomology ring the hodge structure and the homology classes of analytic cycles if x is a compact kahler manifold the kahler cone k of x is one of the connected components of the set p of real 1 1 cohomology classes which are numerically positive on analytic cycles i e y p 0 for every irreducible analytic set y in x p dim y this result is new even in the case of projective manifolds where it can be seen as a generalization of the well known nakai moishezon criterion and it also extends previous results by campana peternell and eyssidieux the principal technical step is to show that every nef class which has positive highest self intersection numberx n 0 contains a kahler current this is done by using the calabi yau theorem and a mass concentration technique for monge ampere equations the main result admits a number of variants and corollaries including a description of the cone of numerically effective 1 1 classes and their dual cone another important consequence is the fact that for an arbitrary deformation x s of compact kahler manifolds the kahler cone of a very general fibre xt is independent of t i e invariant by parallel transport under the 1 1 component of the gauss manin connection

on a theorem of castelnuovo and the equations defining space curves introduction 491 w notat ion and conventions 493 w a regularity theorem 494 w a rationality theorem 498 w the existence of secant lines 501 w open problems 505 references 506

on thue s equation and more precisely for the number of primitive solutions to 1 1 that is solutions in coprime integers x y the first important results on the number of solutions of thue s equation were obtained by c l siegel in the case r 3 and in the binomial case f x y a x r b y r in view of his results and comments we should attribute to siegel the problem to decide whether the number of primitive solutions to 1 1 for irreducible f of degree r 3 could be bounded by a function depending only on r and h but otherwise independent of f in 1983 siegel s question was answered in the affirmative by j h evertse 2 as a special case of his results he also treats equations in number fields evertse obtains the bound

on the representation of integers by binary forms let f be a binary form with integer coefficients non zero discriminant and degree d with d at least 3 let r f z denote the number of integers of absolute value at most z which are represented by f we prove that there is a positive number c f such that r f z is asymptotic to c f z frac 2 d

boundary element shape sensitivity analysis for thermal problems with nonlinear boundary conditions implicit differentiation of the discretized boundary integral equations governing the conduction of heat in solid objects subjected to nonlinear boundary conditions is shown to generate an accurate and economical approach for the computation of shape sensitivities for this class of problems this approach involves the employment of analytical derivatives of boundary element kernel functions with respect to shape design variables a formulation is presented that can consistently account for both temperature dependent convection and radiation boundary conditions several iterative strategies are presented for the solution of the resulting sets of nonlinear equations and the computational performances examined in detail multizone analysis and zone condensation strategies are demonstrated to provide substantive computational economies in this process for models with either localized nonlinear boundary conditions or regions of geometric insensitivity to design variables a series of nonlinear example problems are presented that have closed form solutions 39 refs

counting rational points on algebraic varieties for any n 2 let z pn be a geometrically integral algebraic variety of degree d this article is concerned with the number nz b of q rational points on z which have height at most b for any e 0 we establish the estimate nz b od e n bdimz e provided that d e as indicated the implied constant depends at most on d e and n

counting rational points on biquadratic hypersurfaces abstract an asymptotic formula is established for the number of rational points of bounded anticanonical height which lie on a certain zariski open subset of an arbitrary smooth biquadratic hypersurface in sufficiently many variables the proof uses the hardy littlewood circle method

zeta functions and solutions of falconer type problems for self similar subsets of ℤn this paper uses the method of zeta functions to solve falconer type problems about sets of k simplices whose endpoints belong simultaneously to a self similar subset f of z k n and a disc b x of a large radius x assuming that the similarity transformations pairwise commute we study four euclidean metric invariants of the simplices the most basic and frequently studied of which is the distance between endpoints of a 1 simplex for each we introduce a zeta function derive its functional properties and apply such information to derive a lower bound on the upper minkowski dimension of f which guarantees that the number of distinct metric invariants must be unbounded as x mathematics subject classifications 11m41 28a80 52c10 11j69 11m32

explicit uniform estimation of rational points ii hypersurface coverings by using the slope method we obtain an explicit uniform estimation for the density of rational points in an arithmetic projective variety with given degree and dimension embedded in a given arithmetic projective space

on the density of integer points on generalised markoff hurwitz and dwork hypersurfaces we use bounds of mixed character sums modulo a square free integer q of a special structure to estimate the density of integer points on the hypersurface begin aligned f 1 x 1 cdots f n x n a x 1 k 1 ldots x n k n end aligned for some polynomials f i in mathbb z x and nonzero integers a and k i i 1 ldots n in the case of begin aligned f 1 x cdots f n x x 2 quad text and quad k 1 cdots k n 1 end aligned the above hypersurface is known as the markoff hurwitz hypersurface while for begin aligned f 1 x cdots f n x x n quad text and quad k 1 cdots k n 1 end aligned it is known as the dwork hypersurface our results are substantially stronger than those known for general hypersurfaces

on the number of solutions of polynomial congruences and thue equations has only finitely many solutions in integers x and yv in the first part of this paper we shall establish upper bounds for the number of solutions of 1 in coprime integers x and y under the assumption that the discriminant d f of f is nonzero for most integers h these bounds improve upon those obtained by bombieri and schmidt in 5 in the course of proving these bounds we shall establish a result on polynomial congruences that extends earlier work of nagell 30 ore 32 sandor 33 and huxley 19 in fact we shall establish an upper bound for the number of solutions of a polynomial congruence that is in general best possible in the second part we shall address the problem of finding forms f for which 1 has many solutions for arbitrarily large integers h finally we shall obtain upper bounds for the number of solutions of certain thue mahler and ramanujan nagell equations by appealing to estimates of evertse gy6ry stewart and tijdeman 17 for the number of solutions of s unit equations

the number of integral points on arcs and ovals integral lattice points and that the exponent and constant are best possible however swinnerton dyer 10 showed that the preceding result can be substantially improved if we start with a fixed c strictly convex arc and consider the number of lattice points on t the dilation of by a factor t t 1 this of course is the same as asking for rational points mn n n on as n in fact swinnerton dyer proves a bound of type t zz c e t 3 5 e

semianalytical structural sensitivity formulation in boundary elements developpement d une methode semi analytique qui combine la differenciation implicite des equations integrales frontieres discretisees et une perturbation a une variable etude de milieux continus a symetrie axiale et elastiques plans 2d incluant des forces de volume de type gravite et centrifuge l integration numerique d une nouvelle classe de noyaux de sensibilite des solutions fondamentales est evitee et la mise en uvre est facile presentation de resultats numeriques pour l etude de l effet des perturbations dans les variables du modele

the density of rational points on curves and surfaces let c be an irreducible projective curve of degree d in mathbb p 3 defined over overline mathbb q it is shown that c has o varepsilon d b 2 d varepsilon rational points of height at most b for any varepsilon 0 uniformly for all curves c this result extends an estimate of bombieri and pila duke math j 59 1989 337 357 to projective curves for a projective surface s in mathbb p 3 of degree d ge 3 it is shown that there are o varepsilon d b 2 varepsilon rational points of height at most b of which at most o varepsilon d b 52 27 varepsilon do not lie on a rational line in s for non singular surfaces one may reduce the exponent to 4 3 16 9d for d 4 or 5 or max 1 3 sqrt d 2 d 1 for d ge 6 even for the surface x 1 d x 2 d x 3 d x 4 d this last result improves on the previous best known as a further application it is shown that almost all integers represented by an irreducible binary form f x y in mathbb z x y have essentially only one such representation this extends a result of hooley j reine angew math 226 1967 30 87 which concerned cubic forms only the results are not restricted to projective surfaces and as an application of other results in the paper it is shown that x 1 x 2 x 3 in mathbb n 3 x 1 d x 2 d x 3 d n ll varepsilon d n theta d varepsilon with theta frac 2 sqrt d frac 2 d 1 when d ge 8 this provides the first non trivial bound for the number of representations as a sum of three d th powers

on the density of rational and integral points on algebraic varieties let x n be a projective geometrically integral variety over of dimension r and degree d 4 suppose that there are only finitely many r 1 planes over on x the main result of this paper is a proof of the fact that the number n x b of rational points on xwhich have height at most b satisfies for any 0 the implied constant depends at most on d n and

rational points of bounded height on projective surfaces we give upper bounds for the number of rational points of bounded height on the complement of the lines on projective surfaces

explicit uniform estimation of rational points i estimation of heights by using the slope method in arakelov geometry we study the complexity of the singular locus of an arithmetic projective variety and explicit estimations of the arithmetic hilbert samuel function

non linear heat conduction in composite bodies a boundary element formulation the present paper discusses the numerical solution of steady state non linear heat conduction problems in composite bodies by using the boundary element method two kinds of non linearities are considered the temperature dependence of the thermal conductivity and boundary conditions of the radiative type by introducing the integral of conductivity as a new variable the governing equation of the problem becomes linear in the transform space transformed boundary conditions of the dirichlet and neumann types are also linear but convective boundary conditions become non linear also discontinuities arise in the value of the integral of conductivity across the interface between materials with different properties since continuity of temperature is imposed the problem is numerically solved by discretizing the external and interface boundaries of the region under consideration with constant boundary elements and applying an iterative scheme of the newton raphson type

boundary formulations for shape sensitivity of temperature dependent conductivity problems used in concert with the kirchhoff transformation implicit differentiation of the discretized boundary integral equations governing the conduction of heat in solids with temperature dependent thermal conductivity is shown to generate an accurate and economical approach for computation of shape sensitivities for problems with specified temperature and heat flux boundary conditions a linear problem results for both the analysis and sensitivity analysis in problems with either convection or radiation boundary conditions a non linear problem is generated several iterative strategies are presented for the solution of the resulting sets of non linear equations and the computational performances examined in detail multi zone analysis and zone condensation strategies are demonstrated to provide substantive computational economies in this process for models with either localized non iinear boundary conditions or regions of geometric insensitivity to design variables a series of non linear example problems is presented that have closed form solutions exact anaytical expressions tor the shape sensitivities associated with these problems are developed and these are compared with the sensitivities computed using the boundary element formulation

transient thermoelasticity and other body force effects in boundary element shape sensitivity analysis several new formulations for structural design sensitivity analysis dsa of elastic solids are presented that consistently account for steady and transient thermal loading along with centrifugal and gravity loading the approach employs implicit differentiation of the discretized boundary integral equations and facilitates the production of accurate and efficient computational procedures that obviate the need for matrix factorizations in the dsa step for centrifugal gravity and general thermal loading a new dsa concept employing a particular integral technique is developed for centrifugal gravity and steady thermal effects a surface integration approach is also presented and the two methods compared alternative strategies for modelling the behaviour of temperature distributions of evolving shapes undergoing shape optimization and the ramifications of these strategies in the dsa process are also discussed it is shown that for certain classes of thermoelastic problems shape optimization can be performed without the need to incorporate thermal analysis or dsa into the overall process a thermoelastic surface stress sensitivity analysis formulation is presented that does not require additional numerical integration a new thermoelastic sharp corner traction component consistent sensitivity analysis formulation is also presented this technique is shown to work for any initial stress distribution numerical sensitivity results are presented for many problems with analytical solutions

the energy of a kahler class on admissible manifolds on a compact complex manifold of kahler type the energy e of a kahler class is given by the squared l 2 norm of the projection onto the space of holomorphic potentials of the scalar curvature of any kahler metric representing the said class and any one such metric whose scalar curvature has squared l 2 norm equal to e must be an extremal representative of a strongly extremal metric is an extremal metric representing a critical point of e when restricted to the set of kahler classes of fixed positive top cup product we study the existence of strongly extremal metrics and critical points of e on certain admissible manifolds producing a number of nontrivial examples of manifolds that carry this type of metrics and where in many of the cases the class that they represent is one other than the first chern class and some examples of manifolds where these special metrics and classes do not exist we also provide a detailed analysis of the gradient flow of e on admissible ruled surfaces show that this dynamical system can be extended to one beyond the kahler cone and analyze the convergence of solution paths at infinity in terms of conditions on the initial data in particular proving that for any initial data in the kahler cone the corresponding path is defined for all t and converges to a unique critical class of e as time approaches infinity

a gradient based parameter free approach to shape optimization in shape optimization the independent node movement approach wherein finite element node coordinates are used directly as design variables allows the most freedom for shape change and avoids the time consuming parameterization preprocess however this approach lacks a length scale control that is necessary to ensure a well posed shape optimization problem and avoid numerical instability motivated by the success of filtering techniques that impose minimum length scales in topology optimization we propose a scheme with consistent filtering to introduce a length scale and thereby ensure smoothness in shape optimization while preserving the advantages of the independent node movement approach

representations of integers by systems of three quadratic forms it is classically known that the circle method produces an asymptotic for the number of representations of a tuple of integers n1 nr by a system of quadratic forms q1 qr in k variables as long as k is sufficiently large with respect to r reducing the required number of variables remains a significant open problem in this work we consider the case of three forms and improve on the classical result by reducing the number of required variables to k 10 for almost all tuples under a non singularity assumption on the forms q1 q2 q3 to accomplish this we develop a three dimensional analogue of kloosterman s circle method in particular capitalizing on geometric properties of appropriate systems of three quadratic forms

kahler einstein metrics with positive scalar curvature in this paper we prove that the existence of kahler einstein metrics implies the stability of the underlying kahler manifold in a suitable sense in particular this disproves a long standing conjecture that a compact kahler manifold admits kahler einstein metrics if it has positive first chern class and no nontrivial holomorphic vector fields we will also establish an analytic criterion for the existence of kahler einstein metrics our arguments also yield that the analytic criterion is satisfied on stable kahler manifolds provided that the partial c 0 estimate posed in t6 is true

the practical computation of areas associated with binary quartic forms we derive formulas for practically computing the area of the region f z y 1 defined by a binary quartic form f x y r x y these formulas which involve a particular hypergeometric function are useful when estimating the number of lattice points in certain regions of the type f x y h and will likely find application in many contexts we also show that for forms f of arbitrary degree the maximal size of the area of the region f x y 1 normalized with respect to the discriminant of f and taken with respect to the number of conjugate pairs of f x 1 increases as the number of conjugate pairs decreases and we give explicit numerical values for these normalized maxima when f is a quartic form

additive representation in short intervals ii sums of two like powers we establish that for almost all natural numbers n there is a sum of two positive integral cubes lying in the interval n n 7 18 varepsilon n here the exponent 7 18 lies half way between the trivial exponent 4 9 stemming from the greedy algorithm and the exponent 1 3 constrained by the number of integers not exceeding x that can be represented as the sum of two positive integral cubes we also provide analogous conclusions for sums of two positive integral k th powers when k ge 4

the density of rational points on non singular hypersurfaces ii for any integers d n geq 2 let x subset mathbb p n be a non singular hypersurface of degree d that is defined over the rational numbers the main result in this paper is a proof that the number of rational points on x which have height at most b is o b n 1 varepsilon for any varepsilon 0 the implied constant in this estimate depends at most upon d varepsilon and n

boundary element implicit differentiation equations for design sensitivities of axisymmetric structures abstract design sensitivity analysis of axisymmetric elastic media is formulated using boundary elements the kernels for sensitivity matrices are obtained through implicit differentiation of the corresponding boundary element elasticity kernels the singular terms are obtained by applying the boundary displacements and tractions and their respective sensitivities for the rigid body motion mode and the inflation mode the equations for the recovery of sensitivities of axisymmetric boundary stresses are presented as a check on accuracy the approach is applied to a series of examples for which analytical elasticity solutions are available the predictions for both displacement and stress sensitivities are accurate additional examples are provided to demonstrate the versatility of the present approach

the dynamics of the energy of a kahler class the energy of a kahler class on a compact complex manifold m j of kahler type is the infimum of the squared l2 norm of the scalar curvature over all kahler metrics representing the class we study general properties of this functional and define its gradient flow over all kahler classes represented by metrics of fixed volume when besides the trivial holomorphic vector field of m j all others have no zeroes we extend it to a flow over all cohomology classes of fixed top cup product we prove that the dynamical system in this space defined by the said flow does not have periodic orbits that its only fixed points are critical classes of a suitably defined extension of the energy function and that along solution curves in the kahler cone the energy is a monotone function if the kahler cone is forward invariant under the flow solutions to the flow equation converge to a critical point of the class energy function we show that this is always the case when the manifold has a signed first chern class we characterize the forward stability of the kahler cone in terms of the value of a suitable time dependent form over irreducible subvarieties of m j we use this result to draw several geometric conclusions including the determination of optimal dimension dependent bounds for the squared l2 norm of the scalar curvature functional