random polynomials having few or no real zeros consider a polynomial of large degree n whose coefficients are independent identically distributed nondegenerate random variables having zero mean and finite moments of all orders we show that such a polynomial has exactly k real zeros with probability n b o 1 as n through integers of the same parity as the fixed integer k 0 in particular the probability that a random polynomial of large even degree n has no real zeros is n b o 1 the finite positive constant b is characterized via the centered stationary gaussian process of correlation function sech t 2 the value of b depends neither on k nor upon the specific law of the coefficients under an extra smoothness assumption about the law of the coefficients with probability n b o 1 one may specify also the approximate locations of the k zeros on the real line the constant b is replaced by b 2 in case the i i d coefficients have a nonzero mean

cubic correspondences arising from g 2 we use the automorphic realization of the minimal representation the theta representation of the three fold cover of g 2 to construct a correspondence of automorphic forms between members of dual pairs inside g 2 the dual pairs under consideration are sl2 sl2 and sl3 z 3 in the first case we obtain the shimura correspondence between the three fold cover of sl2 and sl2 through explicit integral formulae in the second case we decompose theta over the three fold cover of sl3 both locally and globally the constituents are parametrized by their central characters and are cuspidal except essentially when the central character is trivial we thus obtain cuspidal theta representations and the unramified theta representation of the three fold cover of sl3

secondary terms in counting functions for cubic fields we prove the existence of secondary terms of order x5 6 in the davenport heilbronn theorems on cubic fields and 3 torsion in class groups of quadratic fields for cubic fields this confirms a conjecture of datskovsky wright and roberts we also prove a variety of generalizations including to arithmetic progressions where we discover a curious bias in the secondary term roberts s conjecture has also been proved independently by bhargava shankar and tsimerman in contrast to their work our proof uses the analytic theory of zeta functions associated to the space of binary cubic forms developed by shintani and datskovsky wright

paper zeros of sampled systems the zeros of the discrete time system obtained when sampling a continuous time system are explored theorems for the limiting zeros for large and small sampling periods are given conditions which guarantee that the sampled system has stable zeros are also presented

error estimates for the davenport heilbronn theorems we obtain the first known power saving remainder terms for the theorems of davenport and heilbronn on the density of discriminants of cubic fields and the mean number of 3 torsion elements in the class groups of quadratic fields in addition we prove analogous error terms for the density of discriminants of quartic fields and the mean number of 2 torsion elements in the class groups of cubic fields these results prove analytic continuation of the related dirichlet series to the left of the line r s 1

singular invariant tempered distributions on regular prehomogeneous vector spaces revue des espaces vectoriels prehomogenes distributions temperees invariantes distributions temperees avec un parametre meromorphe puissances complexes des invariants relatifs soit p x un polynome homogene irreductible a coefficients reels sur un espace vectoriel v soit s x v p x 0 se decompose en un nombre fini de g 1 orbites avec g 1 g gl v p g x p x on pose v 1 uv 2 u uv e v s la decomposition en composantes connexes on donne une condition necessaire et suffisante pour qu une distribution temperee g 1 invariante a support dans s puisse s obtenir comme une combinaison lineaire finie de coefficients de laurent negatifs de la distribution p x s vd i

a conjecture on coincidence among the zeta functions associated with the space of binary cubic forms about twenty five years ago t shintani defined and studied four dirichlet series whose coefficients are class numbers of integral binary cubic forms by using the theory of prehomogeneous vector spaces in this paper we give a conjecture that two of these dirichlet series are essentially the same as the remaining two series the conjecture is based on our calculation of the first two hundred nonzero coefficients of the four dirichlet series and is also consistent with the known functional equation and residues at poles if the conjecture is true we get simpler symmetric functional equations than previously known namely if we take a certain linear combination of shintani s dirichlet series that a single function is invariant under the variable change of s into 1 s

on the convergence of the adelic zeta functions associated to irreducible regular prehomogeneous vector spaces introduction let g x be an irreducible regular prehomogeneous vector space defined over an algebraic number field k d denotes the derived group of g x is an affine n space affn f x e k xl xn is the relative invariant f gx v g f x for every g e g where vi e hom g gm y x f 1 0 the subscript a denotes the adelization relative to k the subscript k denotes the taking of k rational points let aug be a haar measure on ga consider the following integral

the cassels tate pairing on polarized abelian varieties let a e be a principally polarized abelian variety defined over a global field k and let a be its shafarevich tate group let a nd denote the quotient of a by its maximal divisible subgroup cassels and tate constructed a nondegenerate pairing a nd a nd q z if a is an elliptic curve then by a result of cassels the pairing is alternating but in general it is only antisymmetric using some new but equivalent definitions of the pairing we derive general criteria deciding whether it is alternating and whether there exists some alternating nondegenerate pairing on a nd these criteria are expressed in terms of an element c a nd that is canonically associated to the polarization e in the case that a is the jacobian of some curve a down to earth version of the result allows us to determine effectively whether a if finite is a square or twice a square we then apply this to prove that a positive proportion in some precise sense of all hyperelliptic curves of even genus g 2 over q have a jacobian with nonsquare if finite for example it appears that this density is about 13 for curves of genus 2 the proof makes use of a general result relating global and local densities this result can be applied in other situations

regularized orbital integrals for representations of 2 given a finite dimensional representation of sl 2 f on a vector space v defined over a local field f of characteristic zero we produce a regularization of orbital integrals and determine when the resulting distribution is non trivial

reconstructing sliding surfaces for siso variable structure output feedback control systems using inverse method there exists a dynamic mapping relationship between the sliding variable and the output variable in a linear time invariant lti single input single output siso system this relationship is invariant of the matched disturbances which lies in the control space in this note a continuous time dynamic inverse compensator is proposed to reconstruct the sliding variable from the system output this inverse compensator is stable and causal provided that the original siso system is minimum phase and that the input output relative degree is less than or equal to one in case the system relative degree is greater than one a digital approximation for the inverse compensator is introduced to track the sliding variable in discrete time to the accuracy of o t where t is the sampling period thus the discrete time variable structure control vsc law can be implemented without full state accessibility

an o t sup 2 boundary layer in sliding mode for sampled data systems the use of a discontinuous control law typically sign functions in a sampled data system will bring about chattering phenomenon in the vicinity of the sliding manifold leading to a boundary layer with thickness o t where t is the sampling period however by proper consideration of the sampling phenomenon in the discrete time sliding mode control design the thickness of the boundary layer can be reduced to o t sup 2 in contrast to discontinuous control for continuous time vss the discrete time sliding mode control need not be of switching type

racines orthogonales formes quadratiques et orbites d algebres de lie simples graduees abstract we give a unified orbital decomposition for irreducible regular prehomogeneous vector spaces of parabolic type graded by strongly orthogonal roots of same length with one dimensional corresponding root space on a local or global field of characteristic 0 in terms of a class of quadratic forms related to a root system weyl action and the invariants of the prehomogeneous vector spaces

higher composition laws ii on cubic analogues of gauss composition in our first article 2 we developed a new view of gauss composition of binary quadratic forms which led to several new laws of composition on various other spaces of forms moreover we showed that the groups arising from these composition laws were closely related to the class groups of orders in quadratic number fields while the spaces underlying those composition laws were closely related to certain exceptional lie groups in this paper our aim is to develop analogous laws of composition on certain spaces of forms so that the resulting groups yield information on the class groups of orders in cubic fields that is we wish to obtain genuine cubic analogues of gauss composition the fundamental object in our treatment of quadratic composition 2 was the space of 2 x 2 x 2 cubes of integers in particular gauss composition arose from the three different ways of slicing a cube a into two 2 x 2 matrices mi ni i 1 2 3 each such pair mi ni gives rise to a binary quadratic form qa x y qi x y defined by qi x y det mix niy the cube law of 2 declares that as a ranges over all cubes the sum of qi q2 q3 is zero it was shown in 2 that the cube law gives a law of addition on binary quadratic forms that is equivalent to gauss composition various other invariant theoretic constructions using the space of 2 x 2 x 2 cubes led to several new composition laws on other spaces of forms furthermore we showed that each of these composition laws gave rise to groups that are closely related to the class groups of orders in quadratic fields based on the quadratic case described above our first inclination for the cubic case might be to examine 3 x 3 x 3 cubes of integers a 3 x 3 x 3 cube c can be sliced in three different ways into three 3 x 3 matrices li mi ni i 1 2 3 we may therefore obtain from c three ternary cubic forms fl x y z f2 x y z f3 x y z defined by

the adelic zeta function associated with the space of binary cubic forms with coefficients in a function field in this paper we study the adelic zeta function associated with the prehomogeneous vector space of binary cubic forms defined over a function field we establish its rationality find its poles and residues and a simple functional equation that this zeta function satisfies introduction the subject of integral equivalence of binary cubic forms although not as popular or well developed as the theory of integral equivalence of binary quadratic forms has received a fair amount of attention the first paper on this topic 9 written by eisenstein appeared in 1844 later on several other first rate mathematicians tried their hand at the subject among others the list includes arndt cayley hermite and davenport the classical method of attacking this problem is to draw upon a one to one discriminant preserving relation between binary cubic and binary quadratic forms which assigns to a cubic form its hessian and then use the reduction theory for binary quadratic forms to obtain results on class numbers of binary cubic forms the best of the classical results are due to davenport 5 who showed that e h ir n 2 36x o x15 16 nax e h ir n v2 12x o x 5116 nax where h ir n is the class number of irreducible binary cubic forms of discriminant n the credit for a new surge of interest in the study of integral equivalence of binary cubic forms must indisputably go to shintani for it was his 1972 paper 25 that put this study on modern footing in that paper shintani introduced a certain zeta function which as it turns out decomposes into a linear combination of products of dirichlet series whose coefficients are weighted class numbers of binary cubic forms and certain mellin transforms by studying analytic properties of the zeta function shintani was able to improve the error term in davenport s result to o x576 it is of interest to note that the zeta function and the dirichlet series it received by the editors september 3 1985 1980 mathematics subject classification 1985 revision primary 12a70 secondary iocio 12a90 1987 american mathematical society 0002 9947 87 1 00 25 per page

gauss composition for p1 and the universal jacobian of the hurwitz space of double covers abstract in this paper we give an explicit description of the moduli space of line bundles on hyperelliptic curves including singular curves we study the universal jacobian j 2 g n of degree n line bundles over the hurwitz stack of double covers of p 1 by a curve of genus g our main results are the construction of a smooth irreducible universally closed but not separated moduli compactification j bd 2 g n of j 2 g n whose points we describe simply and explicitly as sections of certain vector bundles on p 1 a description of the global geometry and moduli properties of these stacks and a computation of the picard groups of j bd 2 g n and j 2 g n in the cases when n g is even an important ingredient of our work is the parametrization of line bundles on double covers by binary quadratic forms this parametrization generalizes the classical number theoretic correspondence between ideal classes of quadratic rings and integral binary quadratic forms which in particular gives the group law on integral binary quadratic forms first discovered by gauss

prehomogeneous vector spaces and field extensions letk be an infinite field of characteristic not equal to 2 3 5 in this paper we construct a natural map from the set of orbits of certain prehomogeneous vector spaces to the set of isomorphism classes of galois extensions ofk which are splitting fields of equations of certain degrees and prove that the inverse image of this map corresponds bijectively with conjugacy classes of galois homomorphisms

twists of the iwasawa tate zeta function the intent of this paper is to develop the theory of a simple variation of the adelic zeta function first described by tate 14 and iwasawa 9 in independent efforts to simplify hecke s analysis of his l functions attached to gr6bencharakters our twists of the iwasawa tate zeta function arise in the framework of a more general theory of zeta functions associated to representations of algebraic groups many of the basic problems of this general theory have already been solved by yukie and the author see 20 and 21 1 but still await appearance in final written form thus it seems worthwhile to give here a general statement of these problems before proceeding to the very special case considered in this paper let k be a global field i e algebraic number field or function field of a curve defined over a finite field let n and a denote the ring of adeles and the group of ideles of k respectively let co be a quasicharacter of the idele group trivial on the embedded group k of nonzero elements of k the group f2 f2 k of all such quasicharacters co has a natural riemann surface structure let l i denote the usual idele norm on a and for s tl2 let co be the principal quasicharacter co t ltl let re co be the real part of co f2 defined as the unique real number a such that l o i co for any reductive linear algebraic group g defined over k let g denote the adelizat ion of g over k as defined in 16 and let gk be the group of k rational points of g embedded as a discrete group in ga for simplicity we shall fix g as g1 now although the theory to be described below can also be effected for any reductive algebraic group with at least one nontrivial k rational character let p g g1 v be an irreducible k rational representation of g in the finite dimensional vector space v of dimension m let i be the identity matrix in g by schur s lemma there is an integer d such that p ti tdp i for all scalars t we shall always assume that d 0 let dg be any convenient left invariant measure on the quotient ga gk let v be the schwartz bruhat space of functions on v for the time being let vk be any gk invariant subset of vk the zeta function associated to the representation p may now be defined

universal single input single output siso sliding mode controllers with finite time convergence an universal controller is constructed formulated in input output terms only which causes the output of any uncertain smooth single input single output siso minimum phase dynamic system with known relative degree to vanish in finite time this allows exact tracking of arbitrary real time smooth signals only one parameter is to be adjusted since the approach is based on higher order finite time convergence sliding modes the control can be made arbitrarily smooth providing for the arbitrarily high tracking accuracy order with respect to the sampling step

dynamic output feedback variable structure control for system stabilization simplex type dynamic output feedback variable structure control for stabilizing multivariable linear time invariant plants is analysed two types of controllers are considered a compensator type and an observer type it is shown that while the compensator type controller stabilizes the plant only if the plant is minimum phase the observer type controller does not face the same restriction the design of low order less than the order of the plant observer type controllers is also considered

covers of modules over a fixed ring abstract in this note we find conditions under which it is possible to prove the existence of relative injective covers of any module over the fixed ring r g by means of relative injective covers of modules over the base ring r the same problem is treated for flat covers

on variable structure output feedback controllers for uncertain dynamic systems a class of variable structure output feedback controllers for uncertain dynamic systems with bounded uncertainties is proposed no statistical information about the uncertain elements is assumed a variable structure systems approach and a geometric approach to the analysis and synthesis of system zeros are employed in the synthesis of the proposed controllers

output feedback variable structure control design this paper describes a procedure for variable structure output feedback control vsofc for a class of multivariable linear time invariant systems the equivalent control method of utkin in the output feedback mode yields a reduced order system exhibiting output feedback equivalent dynamics using the kimura davison sufficient conditions for pole placement using output feedback a sufficient condition is derived to assign the eigenvalues of the reduced order system the observability and controllability of the reduced system is discussed two switching surface design algorithms are introduced based on both a static output feedback and on an eigenstructure assignment techniques a control design is discussed as well

two cover descent on hyperelliptic curves we describe an algorithm that determines a set of unramified covers of a given hyperelliptic curve with the property that any rational point will lift to one of the covers in particular if the algorithm returns an empty set then the hyperelliptic curve has no rational points this provides a relatively efficiently tested criterion for solvability of hyperelliptic curves we also discuss applications of this algorithm to curves of genus 1 and to curves with rational points

higher composition laws i a new view on gauss composition and quadratic generalizations two centuries ago in his celebrated work disquisitiones arithmeticae of 1801 gauss laid down the beautiful law of composition of integral binary quadratic forms which would play such a critical role in number theory in the decades to follow even today two centuries later this law of composition still remains one of the primary tools for understanding and computing with the class groups of quadratic orders it is hence only natural to ask whether higher analogues of this composition law exist that could shed light on the structure of other algebraic number rings and fields this article forms the first of a series of four articles in which our aim is precisely to develop such higher composition laws in fact we show that gauss s law of composition is only one of at least fourteen composition laws of its kind which yield information on number rings and their class groups in this paper we begin by deriving a general law of composition on 2 2 2 cubes of integers from which we are able to obtain gauss s composition law on binary quadratic forms as a simple special case in a manner reminiscent of the group law on plane elliptic curves we also obtain from this composition law on 2 2 2 cubes four further new laws of composition these laws of composition are defined on 1 binary cubic forms 2 pairs of binary quadratic forms 3 pairs of quaternary alternating 2 forms and 4 senary six variable alternating 3 forms more precisely gauss s theorem states that the set of sl2 z equivalence classes of primitive binary quadratic forms of a given discriminant d has an inherent group structure the five other spaces of forms mentioned above including the space of 2 2 2 cubes also possess natural actions by special linear groups over z and certain products thereof we prove that just like gauss s space of binary quadratic forms each of these group actions has the following remarkable properties first each of these six spaces possesses only a single polynomial invariant for the corresponding group action which we call the discriminant this discriminant invariant is found to take only values that

relations among dirichlet series whose coefficients are class numbers of binary cubic forms we study the zeta functions for the space of binary cubic forms introduced by shintani the zeta function is defined for each invariant lattice we classify the invariant lattices and investigate explicit relationships between the zeta functions associated with those lattices we also study the analytic properties of the zeta functions and rewrite shintani s functional equation in a self dual form using an explicit relation

full order observers for linear systems with unknown inputs this note presents a simple method to design a full order observer for linear systems with unknown inputs the necessary and sufficient conditions for the existence of the observer are given

from picard groups of hyperelliptic curves to class groups of quadratic fields let c be a hyperelliptic curve defined over mathbb q whose weierstrass points are defined over extensions of mathbb q of degree at most three and at least one of them is rational generalizing a result of r soleng in the case of elliptic curves we prove that any line bundle of degree 0 on c which is not torsion can be specialised into ideal classes of imaginary quadratic fields whose order can be made arbitrarily large this gives a positive answer for such curves to a question by agboola and pappas