nombre de points de hauteur bornee sur les surfaces de del pezzo de degre 5 we state manin s conjecture in the particular case of the split del pezzo s surfaces of degree 5 over q we show that for an open set u v nu q b card p u q h p b c b log b 4 b the constant c is the one conjectured by e peyre sommaire

asymptotic trace formula for the hecke operators given integers m n and k we give an explicit formula with an optimal error term with square root cancelation for the petersson trace formula involving the mth and nth fourier coefficients of an orthonormal basis of s k left n right the weight k newforms with fixed square free level n provided that 4 pi sqrt mn k o left k frac 1 3 right moreover we establish an explicit formula with a power saving error term for the trace of the hecke operator mathcal t n on s k left n right averaged over k in a short interval by bounding the second moment of the trace of mathcal t n over a larger interval we show that the trace of mathcal t n is unusually large in the range 4 pi sqrt n k o left n frac 1 6 right as an application for any fixed prime p coprime to n we show that there exists a sequence k n of weights such that the error term of weyl s law for mathcal t p is unusually large and violates the prediction of arithmetic quantum chaos in particular this generalizes the result of gamburd et al j eur math soc 1 1 51 85 1999 theorem 1 4 with an improved exponent

scattering from a cylindrically conformal slotted waveguide array antenna a numerical method is developed to investigate electromagnetic scattering by a cylindrically conformal waveguide fed slot array the problem is first formulated in terms of integral equations using the equivalence principle the integral equations are then solved using the method of moments mom in conjunction with global sinusoidal basis functions and galerkin s testing procedure the mom solution requires the evaluation of the generalized admittance matrices involving various dyadic green s functions the slow convergence of the series associated with the summation of waveguide modes is accelerated using the kummer transformation and the slow convergence of the series associated with the summation of the exterior modes is avoided by using the asymptotic solutions with proper treatment for singular integrals the evaluation of the excitation vector and scattered field is also accelerated using watson s transformation and asymptotic solutions numerical results are presented to illustrate the scattering characteristics of the cylindrically conformal waveguide fed slot arrays such as the effects of curvature slot thickness and waveguide termination on the radar cross section rcs of the arrays

quadratic forms and systems of forms in many variables let f 1 ldots f r be quadratic forms with integer coefficients in n variables when n ge 9r and the variety v f 1 ldots f r is a smooth complete intersection we prove an asymptotic formula for the number of integer points in an expanding box at which these forms simultaneously vanish which in particular implies the hasse principle for v f 1 ldots f r previous work in this direction required n to grow at least quadratically with r we give a similar result for r forms of degree d conditional on an upper bound for the number of solutions to an auxiliary inequality in principle this result may apply as soon as n d2 dr in the case that d ge 3 several strategies are available to prove the necessary upper bound for the auxiliary inequality in a forthcoming paper we use these ideas to apply the circle method to nonsingular systems of forms with real coefficients

universal torsors of del pezzo surfaces and homogeneous spaces let cox sr be the homogeneous coordinate ring of the blow up sr of in r general points i e a smooth del pezzo surface of degree 9 r we prove that for r 6 7 proj cox sr can be embedded into gr pr where gr is an algebraic group with root system given by the primitive picard lattice of sr and pr gr is a certain maximal parabolic subgroup

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

manin s conjecture for certain biprojective hypersurfaces using the circle method we count integer points on complete intersections in biprojective space in boxes of different side length provided the number of variables is large enough depending on the degree of the defining equations and certain loci related to the singular locus having established these asymptotics we deduce asymptotic formulas for rational points on such varieties with respect to the anticanonical height function in particular we establish a conjecture of manin for certain smooth hypersurfaces in biprojective space of sufficiently large dimension

density of rational points on a quadric bundle in mathbb p 3 times mathbb p 3 we establish an asymptotic formula for the number of rational points of bounded anticanonical height which lie on a certain zariski dense subset of the biprojective hypersurface x1y12 x4y42 0 in p3 p3 this confirms the modified manin conjecture for this variety in which the removal of a thin set of rational points is allowed

a new form of the circle method and its application to quadratic forms if the coefficients r n satisfy suitable arithmetic conditions the behaviour of f will be determined by an appropriate rational approximation a q to with small values of q usually producing large values of f when lies in an interval a q a q with q small a major arc one hopes to estimate f asymptotically while if the corresponding q is large for the minor arcs one hopes that f will be small at least on average one can distinguish two different types of application of the circle method the first type following the above pattern uses one of a variety of mean value techniques hua s inequality for example to give an upper bound for the minor arc contribution results on waring s problem are generally proved this way for example the second type of application on which we shall be focusing our attention avoids the minor arcs completely and treats f asymptotically for all here one cannot take advantage of the mean value ideas used in the first type of problem the savings come instead from the availability of good estimates for complete exponential sums kloosterman s work 12 on quaternary quadratic forms and the papers of the author 5 and hooley 8 and 9 on cubic forms are of this latter type we should point out that such applications do not include any problems in which f is the generating function for a polynomial of degree 4 or more there are

on the hasse principle for quartic hypersurfaces we establish the hasse principle for smooth projective quartic hypersurfaces of dimension greater than or equal to 28 defined over q

interval observer design for continuous time linear parameter varying systems abstract this paper studies interval observer design for linear parameter varying systems subject to uncertainties in parameters input and output based on a general assumption that these uncertainties are unknown but bounded by known intervals a novel interval observer is presented to estimate the bounds of state to attenuate the effect of uncertainties and obtain tight state intervals a novel l norm based design method is proposed numerical simulations are conducted to demonstrate the performance of the proposed method

subconvexity for additive equations pairs of undenary cubic forms we investigate pairs of diagonal cubic equations with integral coefficients for a class of such diophantine systems with 11 or more variables we are able to establish that the number of integral solutions in a large box is at least as large as the expected order of magnitude

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

manin s conjecture for quartic del pezzo surfaces with a conic fibration an asymptotic formula is established for the number of q rational points of bounded height on a non singular quartic del pezzo surface with a conic bundle structure

on the representation of integers by quadratic forms let n 4 and let q 2 z x1 xn be a non singular quadratic form when q is indefinite we provide new upper bounds for the least non trivial integral solution to the equation q 0 and when q is positive definite we provide improved upper bounds for the greatest positive integer k for which the equation q k is insoluble in integers despite being soluble modulo every prime power

optimal strong approximation for quadratic forms for a nondegenerate integral quadratic form f x1 xd in d 5 variables we prove an optimal strong approximation theorem let be a fixed compact subset of the affine quadric f x1 xd 1 over the real numbers take a small ball b of radius 0 0 finally assume that an integral vector 1 d mod m is given then we show that there exists an integral solution x x1 xd of f x n such that xi imodm and xn b provided that all the local conditions are satisfied we also show that 4 is the best possible exponent moreover for a nondegenerate integral quadratic form in four variables we prove the same result if n is odd and n r 1m 6 based on our numerical experiments on the diameter of lps ramanujan graphs and the expected square root cancellation in a particular sum that appears in remark 6 8 we conjecture that the theorem holds for any quadratic form in four variables with the optimal exponent 4

sieving rational points on varieties an upper bound sieve for rational points on suitable varieties isdeveloped together with applications tocounting rational points in thin sets to local solubility in families and to the notion of friable rational pointswith respect to divisors in the special case of quadrics sharper estimates areobtained by developing a version of the selberg sieve for rational points

subconvex bounds on mathrm gl _3 via degeneration to frequency zero for a fixed cusp form pi on mathrm gl 3 mathbb z and a varying dirichlet character chi of prime conductor q we prove that the subconvex bound begin aligned l left pi otimes chi tfrac 1 2 right ll q 3 4 delta end aligned holds for any delta 1 36 this improves upon the earlier bounds delta 1 1612 and delta 1 308 obtained by munshi using his mathrm gl 2 variant of the delta method the method developed here is more direct we first express chi as the degenerate zero frequency contribution of a carefully chosen summation formula a la poisson after an elementary amplification step exploiting the multiplicativity of chi we then apply a sequence of standard manipulations reciprocity voronoi cauchy schwarz and the weil bound to bound the contributions of the nonzero frequencies and of the dual side of that formula

the equivalent parameters for the radiating slot on a sectoral waveguide formulas for the field components of the te sub pq mode and the dominant mode te sub 11 in a sectoral waveguide are given using the equivalence principle the electric field distribution on the aperture surface of a narrow longitudinal slot cut in the curved broad wall of a sectoral waveguide is solved by the moment method galerkin s method important results such as the scattering parameters the equivalent shunt admittance and the resonant conductance and length are studied the method used in this paper has good precision and has been adopted in the design of a linear slot array

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

simultaneous integer values of pairs of quadratic forms we prove that a pair of integral quadratic forms in 5 or more variables will simultaneously represent almost all pairs of integers that satisfy the necessary local conditions provided that the forms satisfy a suitable nonsingularity condition in particular such forms simultaneously attain prime values if the obvious local conditions hold the proof uses the circle method and in particular pioneers a two dimensional version of a kloosterman refinement

consensus seeking in multiagent systems under dynamically changing interaction topologies this note considers the problem of information consensus among multiple agents in the presence of limited and unreliable information exchange with dynamically changing interaction topologies both discrete and continuous update schemes are proposed for information consensus this note shows that information consensus under dynamically changing interaction topologies can be achieved asymptotically if the union of the directed interaction graphs have a spanning tree frequently enough as the system evolves

cubic hypersurfaces and a version of the circle method for number fields a version of the hardy littlewood circle method is developed for number fields k q and is used to show that non singular projective cubic hypersurfaces over k always have a k rational point when they have dimension at least 8

a fast domain decomposition method for solving three dimensional large scale electromagnetic problems an efficient algorithm based on domain decomposition method ddm and partial basic solution vectors pbsv technique is proposed for solving three dimensional 3 d large scale finite periodic electromagnetic problems such as photonic or electromagnetic bandgap structures frequency selective surfaces the entire computational domain is divided into many smaller nonoverlapping subdomains a robin type condition is introduced at the interfaces between subdomains to enforce the field continuity with the help of a set of dual unknowns each subdomain can be tackled independently because of geometric repetitions all the sudomains can be classified into a few building blocks which can be dealt with by an improved pbsv algorithm thus the original problem becomes a much smaller one which involves the unknowns only at the interfaces the resulting linear system of equations is solved by a block symmetric successive over relaxation ssor preconditioned krylov subspace method once the unknowns at the interfaces have been obtained the final solution on each subdomain can easily be calculated independently some numerical examples are provided and show the method is scalable with the number of subdomains

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

towards a minimal order distributed observer for linear systems abstract in this paper we consider the distributed estimation problem for continuous time linear time invariant lti systems a single linear plant is observed by a network of local observers each local observer in the network has access to only part of the output of the observed system but also receives information on the state estimates of its neighbors each local observer should in this way generate an estimate of the plant state in this paper we study the problem of existence of a reduced order distributed observer we show that if the observed system is observable and the network graph is a strongly connected directed graph then a distributed observer exists with state space dimension equal to n n i 1 n p i where n is the number of network nodes n is the state space dimension of the observed plant and p i is the rank of the output matrix of the observed output received by the i th local observer in the case of a single observer this result specializes to the well known minimal order observer in classical observer design

rational points on quartic hypersurfaces let x be a projective non singular quartic hypersurface of dimension 39 or more which is defined over we show that x is non empty provided that x is non empty and x has p adic points for every prime p 2009 walter de gruyter

rational points of bounded height on fano varieties a prime pe7z let v be an algebraic variety defined over f and li a metrized line bundle on v i e a system l consisting of a line bundle l and a family of banach v adic metrics on l f for all v s satisfying well known conditions cf 1 chap vi then we can define a height function hl x f i r by the formula h x is x 1 where s is a local section of l non vanishing

rational solutions of pairs of diagonal equations one cubic and one quadratic we obtain an essentially optimal estimate for the moment of order 32 3 of the exponential sum having argumentx 3 x 2 subject to modest local solubility hypotheses we thereby establish that pairs of diagonal diophantine equations one cubic and one quadratic possess non trivial integral solutions whenever the number of variables exceeds 10

accurate asymptotic solution for the surface field due to apertures in a conducting cylinder asymptotic solutions for the surface field due to a magnetic point source on a cylinder can simplify the computational task in the analysis of mutual coupling e g in conformal arrays although several asymptotic solutions are at present available their accuracy is limited the aim here is to obtain a new approximate solution with improved accuracy an asymptotic solution of the exact modal solution for the field on a cylinder which contains terms up to k 0 t 2 where t is the distance from the source is derived approximate terms up to k 0 t 3 are included from the solution for a source on a plane conductor and an earlier asymptotic solution the new approximate solution consists of the exact solution for a plane conductor modulated by a fock function plus a contribution associated with the cylinder which vanishes in the limit of infinite radius typically for cylinders of radius greater than one and half wavelengths it agrees to within 0 5 db in magnitude and 1 deg in phase of the modal solution results for dominant mode mutual and self admittances of rectangular apertures on a cylinder obtained from this solution are presented a formula is also given for calculating self admittances from an asymptotic solution there is excellent agreement between admittances computed from the asymptotic solution and the modal solution one exception is the self susceptance of an axial slot where the error is larger than expected although the error is low enough to be acceptable in most practical cases

design of distributed lti observers for state omniscience consider that an autonomous linear time invariant lti plant is given and that each member of a network of lti observers accesses a portion of the output of the plant the dissemination of information within the network is dictated by a pre specified directed graph in which each vertex represents an observer this paper proposes a distributed estimation scheme that is a natural generalization of consensus in which each observer computes its own state estimate of the plant using only the portion of the output vector accessible to it and the state estimates of other observers that according to the graph are available to it unlike straightforward high order solutions in which each observer broadcasts its measurements throughout the network the average dimension of the state of each observer in the proposed scheme does not exceed the order of the plant plus one we determine necessary and sufficient conditions for the existence of a parameter choice for which the proposed scheme attains asymptotic omniscience of the state of the plant at all observers the conditions reduce to certain detectability requirements that imply that if omniscience is not possible under the proposed scheme then it is not viable under any other scheme including higher order lti nonlinear and time varying ones subject to the same graph

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

height zeta functions of equivariant compactifications of unipotent groups we prove manin s conjecture for bi equivariant compactifications of unipotent groups 2015 wiley periodicals inc