numerical solution to a sideways parabolic equation the inverse heat conduction problem can be considered to be a sideways parabolic equation in the quarter plane this is a model of a problem where one wants to determine the temperature on both sides of a thick wall but one side is inaccessible to measurements a numerical procedure for this severely ill posed problem is suggested which consists of two steps namely a mollification of the data and a marching difference scheme the numerical method is proved to be stable several computational results are presented and discussed copyright 2001 john wiley sons ltd

a family of calabi yau varieties and potential automorphy we prove potential modularity theorems for l adic representations of any dimension from these results we deduce the sato tate conjecture for all elliptic curves with nonintegral j invariant defined over a totally real field

identification of dynamic green s functions in structural networks this paper presents the mathematical and computational basis of a method for experimentally identifying the dynamic behavior of linear structural systems the method consists of exciting the structure with specific pulses and reconstructing the dynamic green s functions by deconvolution of the measured response the reconstruction procedure involves the solution of an ill posed integral equation in the presence of noise inverse gaussian functions which have mathematical properties that render the deconvolution problem tractable are used as probing pulses two examples are presented for which dynamic green s functions are reconstructed from synthetic response data 1 a nondispersive medium consisting of a mechanical slider with free body motion and 2 a network that may be representative of a large orbiting structure with rigidly connected torsional members and timoshenko beams it is shown that residual effects of noise that may be detectable in the reconstructed green s functions are effectively smoothed out when these functions are used to predict the response to forces such as may be induced by control systems finally an example is presented of deconvolution implemented in slow motion

infinitely divisible pulses continuous deconvolution and the characterization of linear time invariant systems this paper addresses the problem of determining the impulse response of a linear time invariant system by probing the system with a causal c infty approximation to the dirac delta function we analyze the ill posed deconvolution problem which results from a wide choice of possibly multimodal infinitely divisible probe pulses the notion of infinite divisibility is shown to play a key role when the system s response is suspected of having nondifferentiable singularities we reformulate the volterra integral equation as a cauchy problem for a linear partial differential equation in two independent variables and introduce the concepts of partial and continuous deconvolution we then show that partial deconvolution of the output waveform results in l infty error bounds for the regularized solution and its derivatives under l 2 a priori bounds on the data noise and the unknown system response using the poisson summation formula and fft algorithms we construct an efficient computational alg

impulse response acquisition as an inverse heat conduction problem the impulse response of a linear time invariant system can be obtained by deconvolution of the response to a physically realizable smooth approximation to the dirac delta function as shown in siam j app math 47 1987 pp 892 927 it is advantageous to synthesize infinitely divisible probe waveforms and recover the system s impulse response by solving the related cauchy problem for a generalized diffusion equation the present paper is concerned with perturbations of the originally designed pulse shape caused by amplifiers transducers and other interfacing devices that may destroy infinite divisibility it is shown that in a large number of cases a simple linear transformation of the noisy output data can be found that converts the deconvolution problem with the perturbed pulse back into one for which the diffusion theory applies the search for this transformation is accomplished in the fourier domain by comparing the perturbed pulse with the designed pulse as a by product it is shown

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

hopf bifurcation at a degenerate stationary pitchfork soit e un espace de hilbert reel on considere l existence de petites solutions periodiques pour une equation d evolution sur e donnee par du dt g u u e r avec g 0 0 r

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

vinogradov s mean value theorem via efficient congruencing we obtain estimates for vinogradov s integral that for the rst time approach those conjectured to be the best possible several applications of these new bounds are provided in particular the conjectured asymptotic formula in waring s problem holds for sums of s kth powers of natural numbers whenever s 2k 2 2k 3

vinogradov s mean value theorem via efficient congruencing ii we apply the efficient congruencing method to estimate vino gradov s integral for moments of order 2s with 1 6 s 6 k 2 1 thereby we show that quasi diagonal behaviour holds when s o k 2 we obtain near optimal estimates for 1 6 s 6 1 k 2 k and optimal estimates for s k 2 1 in this way we come half way to proving the main conjecture in two different directions there are consequences for estimates of weyl type and in several allied applications thus for example the anticipated asymptotic formula in waring s problem is established for sums of s kth powers of natural numbers whenever s 2k 2 2k 8 k 6 estimates stemming from vinogradov s mean value theorem deliver bounds for exponential sums of large degree both in mean and pointwise beyond the competence of alternate approaches the ubiquity of such exponential sums in analytic number theory in the analysis for example of the riemann zeta function in waring s problem and beyond accounts for the high pro file of vinogradov s methods in the associated literature in recent work we established a version of vinogradov s mean value theorem which achieves an essentially optimal upper bound with a number of variables only twice the number conjectured to be best possible see 19 for systems of degree k previous estimates missed such a bound by a factor of order logk our earlier approach provides no upper bounds when the number of variables is smaller precluding the possibility of applications involving the finer features of these mean values our goal in this paper is to remedy this deficiency at the same time strengthening our previous conclusions it transpires that we are able to come within a hair s breadth of proving the main conjecture concerning vino gradov s mean value theorem in half of the basic interval of relevant moments such developments illustrate the flexibility of the new efficient congruencing method introduced in 19 we now introduce some notation when k n and r k define

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

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

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

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

hopf bifurcation from a turning point on considere une equation d evolution du dt g u 0 dans un espace de hilbert reel e dependant d un parametre reel variant dans un intervalle r on etudie la bifurcation des solutions periodiques a partir d un point de rebroussement

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

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

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

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

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

a bifurcation theorem for potential operators abstract a sufficient condition for bifurcation of real solutions of g u 0 where g r d e d e is given in terms of the crossing number of the eigenvalue perturbation of the linearization along the trivial solutions 0 this number describes how many eigenvalues leave the negative real axis wen the parameter varies from negative to positive values whereas in general only an odd crossing number entails bifurcation for potential operators in a hilbert space e a nonzero crossing number is sufficient for bifurcation the proof uses a lyapunov schmidt reduction and conley s index theory

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

variation of the unit root along the dwork family of calabi yau varieties we study the variation of the unit roots of members of the dwork families of calabi yau varieties over a finite field by the method of dwork katz and also from the point of view of formal group laws a p adic analytic formula for the unit roots away from the hasse locus is obtained

impulse response functions for elastic structures with rigid body degrees of freedom abstract a procedure based on the theory of generalized functions is developed which allows the calculation of impulse response functions for discrete and continuous elastic systems with structural damping and rigid body degrees of freedom for any degree of freedom the procedure divides the fourier transform of the impulse response function into two parts the first part contains the singularity due to the rigid body degree of freedom its time domain counterpart is obtained in closed form the second part exhibits no singularities and its time domain counterpart is obtained by numerical integration using a fast fourier transform routine the procedure is applied to two systems for which closed form solutions are also obtained a system consisting of two point masses connected by an elastic spring which slide without friction on a horizontal plane and a non dispersive mechanical slider with elastic deformations due to shear strain the agreement between the results yielded by the proposed procedure and the closed form solutions is shown to be excellent the procedure is then used to calculate the impulse response function for the tip displacement of a dispersive medium consisting of a flexible robot manipulator with the mechanical properties of a timoshenko beam

interaction of periodic and stationary bifurcation from multiple eigenvalues on considere des equations d evolution non lineaires du dt g u o dependant d un parametre reel dans un espace de hilbert reel e

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