generalised eisenhart lift of the toda chain the toda chain of nearest neighbour interacting particles on a line can be described both in terms of geodesic motion on a manifold with one extra dimension the eisenhart lift or in terms of geodesic motion in a symmetric space with several extra dimensions we examine the relationship between these two realisations and discover that the symmetric space is a generalised multi particle eisenhart lift of the original problem that reduces to the standard eisenhart lift such generalised eisenhart lift acts as an inverse kaluza klein reduction promoting coupling constants to momenta in higher dimension in particular isometries of the generalised lift metric correspond to energy preserving transformations that mix coordinates and coupling constants a by product of the analysis is that the lift of the toda lax pair can be used to construct higher rank killing tensors for both the standard and generalised lift metrics

the projective geometry of freudenthal s magic square abstract we connect the algebraic geometry and representation theory associated to freudenthal s magic square we give unified geometric descriptions of several classes of orbit closures describing their hyperplane sections and desingularizations and interpreting them in terms of composition algebras in particular we show how a class of invariant quartic polynomials can be viewed as generalizations of the classical discriminant of a cubic polynomial

on twistor transformations and invariant differential operator of simple lie group g2 2 the twistor transformations associated to the simple lie group g2 are described explicitly we consider the double fibration g2 p2 g2 b g2 p1 where p1 and p2 are two parabolic subgroups of g2 and b is a borel subgroup and its local version h2 f h1 where h1 is the heisenberg group of dimension 5 embedded in the coset space g2 p1 f cp1 h1 and h2 contains the nilpotent lie group h2 of step three the baker campbell hausdorff formula is used to parametrize the coset spaces coordinates charts their transition functions and the fibers of the projection as complex curves we write down the relative de rham sequence on f along the fibers and push it down to h1 to get a family of matrix valued differential operators dk then we establish a kind of penrose correspondence for g2 the kernel of dk is isomorphic to the first cohomology of the sheaf o k over h2 we also give the penrose type integral transformation u pf for f o k which gives solutions to equations dku 0 when restricted to the re

special prehomogeneous vector spaces associated to f4 e6 e7 e8 and simple jordan algebras of rank 3 abstract a simple complex lie algebra has a natural grading g g 2 g 1 g 0 g 1 g 2 associated to the highest root it is related to the quaternionic real form of g the real rank of the associated symmetric space is shown to be at most 4 when the rank is equal to 4 i e g a n c n g 2 the semi simple lie algebra m g 0 g 0 is shown to be the conformal algebra of a rank 3 semi simple jordan algebra v if g is one of the four exceptional lie algebras f4 e6 e7 e8 then v is simple the vector space g 1 is a prehomogeneous vector space under the action of m c where m int m is the adjoint group of m it admits a non zero m invariant polynomial of degree 4 conversely to any simple jordan algebra v of rank 3 is naturally associated a representation of a twofold covering of its conformal group co v on a vector space w such that w is a prehomogeneous vector space under the action of co v c an invariant polynomial of degree 4 is explicitly constructed a geometric description of all the orbits of co v in w is given

levels of distribution for sieve problems in prehomogeneous vector spaces in our companion paper 28 we developed an efficient algebraic method for computing the fourier transforms of certain functions defined on prehomogeneous vector spaces over finite fields and we carried out these computations in a variety of cases here we develop a method based on fourier analysis and algebraic geometry which exploits these fourier transform formulas to yield level of distribution results in the sense of analytic number theory such results are of the shape typically required for a variety of sieve methods as an example of such an application we prove that there are gg frac x log x quartic fields whose discriminant is squarefree bounded above by x and has at most eight prime factors

the dynamic stiffness matrix of two dimensional elements application to kirchhoff s plate continuous elements this paper describes a procedure for building the dynamic stiffness matrix of two dimensional elements with free edge boundary conditions the dynamic stiffness matrix is the basis of the continuous element method then the formulation is used to build a kirchhoff rectangular plate element gorman s method of boundary condition decomposition and levy s series are used to obtain the strong solution of the elementary problem a symbolic computation software partially performs the construction of the dynamic stiffness matrix from this solution the performances of the element are evaluated from comparisons with harmonic responses of plates obtained by the finite element method

explicit orbit classification of reducible jordan algebras and freudenthal triple systems we determine explicit orbit representatives of reducible jordan algebras and their corresponding freudenthal triple systems this work has direct application to the classification of extremal black hole solutions of mathcal n 2 4 locally supersymmetric theories of gravity coupled to an arbitrary number of abelian vector multiplets in d 4 5 space time dimensions

on the davenport heilbronn theorems and second order terms we give simple proofs of the davenport heilbronn theorems which provide the main terms in the asymptotics for the number of cubic fields having bounded discriminant and for the number of 3 torsion elements in the class groups of quadratic fields having bounded discriminant we also establish second main terms for these theorems thus proving a conjecture of roberts our arguments provide natural interpretations for the various constants appearing in these theorems in terms of local masses of cubic rings

the structure of the invariants of perfect lie algebras the structure of the invariants of perfect lie algebras with nontrivial centre is analysed it is shown that if the radical of a semidirect sum of a semisimple and a nilpotent lie algebra has a one dimensional centre then the defining representation r contains a copy of the trivial representation d0 of using this fact a criterion can be deduced to eliminate variables and to characterize the casimir operators of an algebra by means of certain subrepresentations r of r for rank 1 levi subalgebras all representations leading to perfect lie algebras with a radical isomorphic to a heisenberg lie algebra are determined we prove that the number of casimir operators of such an algebra is fixed for any dimension and moreover that they can be explicitly computed by means of determinantal formulae obtained from the brackets of the algebra for rank n 2 levi part a stabilization result of this nature is also given

a jordan algebra approach to the cubic eiconal equation we establish a natural correspondence between the equivalence classes of cubic solutions of a cubic eiconal equation and the isomorphy classes of cubic jordan algebras

structure of internal modules and a formula for the spherical vector of minimal representations abstract we develop a formula for the spherical vector of minimal representations of simply laced chevalley groups

the mean number of 3 torsion elements in the class groups and ideal groups of quadratic orders we determine the mean number of 3 torsion elements in the class groups of quadratic orders where the quadratic orders are ordered by their absolute discriminants moreover for a quadratic order o we distinguish between the two groups cl3 o the group of ideal classes of order 3 and i3 o the group of ideals of order 3 we determine the mean values of both cl3 o and i3 o as o ranges over any family of orders defined by finitely many or in suitable cases even infinitely many local conditions as a consequence we prove the surprising fact that the mean value of the difference cl3 o i3 o is equal to 1 regardless of whether one averages over the maximal orders in complex quadratic fields or over all orders in such fields or indeed over any family of complex quadratic orders defined by local conditions for any family of real quadratic orders defined by local conditions we prove similarly that the mean value of the difference cl3 o 1 3 i3 o is equal to 1 independent of the family

higher composition laws iv the parametrization of quintic rings rst three parts of this series we considered quadratic cubic and quartic rings i e rings free of ranks 2 3 and 4 over z respectively and found that various algebraic structures involving these rings could be completely parametrized by the integer orbits of an appropriate group representation on a vector space these orbit results are summarized in table 1 in particular the theories behind the parametrizations of quadratic cubic and quartic rings noted in items 2 9 and 13 of table 1 were seen to closely parallel the classical developments of the solutions to the quadratic cubic and quartic equations respectively despite the quintic having been shown to be unsolvable nearly two centuries ago by abel it turns out there still remains much to be said regarding the integral theory of the quintic although a solution naturally still is not possible we show in this article that it is nevertheless possible to completely parametrize quintic rings indeed a theory just as complete as in the quadratic cubic and quartic cases exists also in the case of the quintic in fact we present here a unied theory of ring parametrizations which includes the cases n 2

4 dimensional black holes from kaluza klein theories in this paper we consider generalizations in 4 dimensions of the einstein maxwell equations which typically arise from kaluza klein theories we specify conditions such that stationary solutions lead to non linear models for symmetric spaces using both this group theoretic structure and some properties of harmonic maps we are able to generalize many of the known existence and uniqueness theorems for black holes in einstein maxwell theory to this more general setting

some groups of transformations defined by jordan algebras i it is well known that the lie algebra of derivations of the exceptional jordan algebra m 3 8 over an algebraically closed field of characteristic 0 or over the field of reals is the exceptional lie algebra f 4 of killing cartan and that the lie algebra of linear transformations in m 3 8 leaving a certain cubic norm form defined on m 3 8 invariant is the exceptional lie algebra e 6 chevalley schafer 7 freudenthal 11 the program envisaged in the present series of papers is the extension of these results and their group analogues to arbitrary central simple jordan algebras the present paper gives the definition of the groups and lie algebras in the general case and gives their determination for special jordan algebras with some restrictions on the base field in a second paper we plan to consider the groups of automorphisms of arbitrary reduced exceptional simple jordan algebras and in a third paper we hope to study the generalized groups of types e 6 which arise from these jordan algebras

quasi conformal actions quaternionic discrete series and twistors su 2 1 and g 2 2 quasi conformal actions were introduced in the physics literature as a generalization of the familiar fractional linear action on the upper half plane to hermitian symmetric tube domains based on arbitrary jordan algebras and further to arbitrary freudenthal triple systems in the mathematics literature quaternionic discrete series unitary representations of real reductive groups in their quaternionic real form were constructed as degree 1 cohomology on the twistor spaces of symmetric quaternionic kahler spaces these two constructions are essentially identical as we show explicitly for the two rank 2 cases su 2 1 and g2 2 we obtain explicit results for certain principal series quaternionic discrete series and minimal representations of these groups including formulas for the lowest k types in various polarizations we expect our results to have applications to topological strings black hole micro state counting and to the theory of automorphic forms

a realization of the lie algebra associated to a kantor triple system we present a nonlinear realization of the 5 graded lie algebra associated to a kantor triple system any simple lie algebra can be realized in this way starting from an arbitrary 5 grading in particular we get a unified realization of the exceptional lie algebras f4 e6 e7 e8 in which they are respectively related to the division algebras r c h o

singularity of type d_4 arising from four qubit systems an intriguing correspondence between four qubit systems and simple singularity of type d4 is established we first consider the algebraic variety x of separable states within the projective hilbert space then cutting x with a specific hyperplane h we prove that the x hypersurface defined from the section x h x has an isolated singularity of type d4 it is also shown that this is the worst possible isolated singularity that one can obtain by this construction moreover it is demonstrated that this correspondence admits a dual version by proving that the equation of the dual variety of x which is nothing but the cayley hyperdeterminant of type 2 2 2 2 can be expressed in terms of the stochastic local operation and classical communication invariant polynomials as the discriminant of the miniversal deformation of the d4 singularity as a consequence of this correspondence one obtains a finer grained classification of entanglement classes of four qubit systems

freudenthal triple systems by root system methods abstract for certain lie algebras g we can use a grading g g 2 g 1 g 0 g 1 g 2 and define a quartic form and a skew symmetric bilinear form on g 1 thereby constructing a freudenthal triple system the structure of the freudenthal triple system is examined using root system methods available in the lie algebra context in the cases g e 8 where g 1 is the minuscule representation of e 7 and g d 4 we determine the groups stabilizing the quartic form and both the quartic and bilinear forms

on a canonical form of a 3 3 hermitian matrix over the ring of integral split octonions abstract we show that a 3 3 hermitian matrix over the ring of integral split octonions has a unique canonical diagonal form under the action of the group e 6 6 z this confirms the conjecture of j maldacena g moore and a strominger hep th 9903163 on a standard form of a charge vector

simple space time symmetries generalizing conformal field theory we study simple space time symmetry groups g which act on a space time manifold m g h which admits a g invariant global causal structure we classify pairs g m which share the following additional properties of conformal field theory 1 the stability subgroup h of o m is the identity component of a parabolic subgroup of g implying factorization h man where m generalizes lorentz transformations a dilatations and n special conformal transformations 2 special conformal transformations n act trivially on tangent vectors v tom the allowed simple lie groups g are the universal coverings of su m m so 2 d sp l r so 4n and e7 25 and h are particular maximal parabolic subgroups they coincide with the groups of fractional linear transformations of euclidean jordan algebras whose use as generalizations of minkowski space time was advocated by gunaydin mod phys lett a 8 1407 1993 all these groups g admit positive energy representations it will also be shown that the classical confor

non interleaving bisimulation equivalences on basic parallel processes we show polynomial time algorithms for deciding hereditary history preserving bisimilarity in o n 3logn and history preserving bisimilarity in o n 6 on the class basic parallel processes the latter algorithm also decides a number of other non interleaving behavioural equivalences e g distributed bisimilarity which are known to coincide with history preserving bisimilarity on this class the common general scheme of both algorithms is based on a fixpoint characterization of the equivalences for tree like labelled event structures the technique for realizing the greatest fixpoint computation in the case of hereditary history preserving bisimilarity is based on the revealed tight relationship between equivalent tree like labelled event structures in the case of history preserving bisimilarity a technique of deciding classical bisimilarity on acyclic petri nets is used

decidability of performance equivalence for basic parallel processes we study an extension of the class of basic parallel processes bpp in which actions are durational and urgent and parallel components have independent local clocks the main result is decidability of strong bisimilarity known also as performance equivalence in this class this extends the earlier decidability result for plain bpp by christensen et al our decision procedure is based on decidability of the validity problem for presburger arithmetic we prove also polynomial complexity in positive duration fragment thus properly extending a previous result by berard et al both ill timed and well timed semantics are treated

dynamic behaviour of structures in large frequency range by continuous element methods abstract the continuous element method is presented in the context of the harmonic response of beam assemblies a general formulation is described from the displacement solution of the elementary problem a direct computation of elementary dynamic stiffness matrices is presented in the present formulation distributed loadings are taken into account in the case of more complex geometries for which many coupling phenomena occur an explicit formulation is no more conceivable in this case a numerical approach is presented this approach allows an algorithmic computation of exact dynamic stiffness matrices this method called numerical continuous element allows one to consider the coupled vibrations of curved beams and those of helical beams the validation of this numerical method is achieved by comparisons with the harmonic response of various beams obtained by a finite element approach finally a comparison between eigenfrequencies obtained experimentally and numerically for a straight beam and a helical beam has been made to evaluate the performances of the method

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

minimal representations spherical vectors and exceptional theta series theta series for exceptional groups have been suggested as a possible description of the eleven dimensional quantum supermembrane we present explicit formulae for these automorphic forms whenever the underlying lie group g is split or complex and simply laced specifically we review and construct explicitly the minimal representation of g generalizing the schrodinger representation of symplectic groups we compute the spherical vector in this representation i e the wave function invariant under the maximal compact subgroup which plays the role of the summand in the automorphic theta series we also determine the spherical vector over the complex field we outline how the spherical vector over the p adic number fields provides the summation measure in the theta series postponing its determination to a sequel of this work the simplicity of our result is suggestive of a new born infeld like description of the membrane where u duality is realized non linearly our results may also be used in constructing quantum mechanical systems with spectrum generating symmetries

generalized conformal realizations of kac moody algebras we present a construction which associates an infinite sequence of kac moody algebras labeled by a positive integer n to one single jordan algebra for n 1 this reduces to the well known kantor koecher tits construction our generalization utilizes a new relation between different generalized jordan triple systems together with their known connections to jordan and lie algebras applied to the jordan algebra of hermitian 3 3 matrices over the division algebras r c h o the construction gives the exceptional lie algebras f4 e6 e7 e8 for n 2 moreover we obtain their infinite dimensional extensions for n 3 in the case of 2 2 matrices the resulting lie algebras are of the form so p n q n and the concomitant nonlinear realization generalizes the conformal transformations in a spacetime of signature p q

minimal unitary representation of su 2 2 and its deformations as massless conformal fields and their supersymmetric extensions we study the minimal unitary representation minrep of so 4 2 over a hilbert space of functions of three variables obtained by quantizing its quasiconformal action on a five dimensional space the minrep of so 4 2 which coincides with the minrep of su 2 2 similarly constructed corresponds to a massless conformal scalar in four space time dimensions there exists a one parameter family of deformations of the minrep of su 2 2 for positive negative integer values of the deformation parameter one obtains positive energy unitary irreducible representations corresponding to massless conformal fields transforming in 0 2 2 0 representation of the sl 2 c subgroup we construct the supersymmetric extensions of the minrep of su 2 2 and its deformations to those of su 2 2 n the minimal unitary supermultiplet of su 2 2 4 in the undeformed case simply corresponds to the massless n 4 yang mills supermultiplet in four dimensions for each given nonzero integer value of one obtains a uniq

jordan pairs e6 and u duality in five dimensions by exploiting the jordan pair structure of u duality lie algebras in d 3 and the relation to the super ehlers symmetry in d 5 we elucidate the massless multiplet structure of the spectrum of a broad class of d 5 supergravity theories both simple and semi simple euclidean rank 3 jordan algebras are considered theories sharing the same bosonic sector but with different supersymmetrizations are also analyzed

classification of simple linearly compact kantor triple systems over the complex numbers abstract simple finite dimensional kantor triple systems over the complex numbers are classified in terms of satake diagrams we prove that every simple and linearly compact kantor triple system has finite dimension and give an explicit presentation of all the classical and exceptional systems

the exceptional lie algebra e7 25 multiplets and invariant differential operators in the present paper we continue the project of systematic construction of invariant differential operators on the example of the non compact exceptional algebra e7 25 our choice of this particular algebra is motivated by the fact that it belongs to a narrow class of algebras which we call conformal lie algebras which have very similar properties to the conformal algebras of n dimensional minkowski spacetime this class of algebras is identified and summarized in a table another motivation is related to the ads cft correspondence we give the multiplets of indecomposable elementary representations including the necessary data for all relevant invariant differential operators

conformal and quasiconformal realizations of exceptional lie groups we present a nonlinear realization of e8 8 on a space of 57 dimensions which is quasiconformal in the sense that it leaves invariant a suitably defined light cone in 57 this realization which is related to the freudenthal triple system associated with the unique exceptional jordan algebra over the split octonions contains previous conformal realizations of the lower rank exceptional lie groups on generalized space times and in particular a conformal realization of e7 7 on 27 which we exhibit explicitly possible applications of our results to supergravity and m theory are briefly mentioned

special entangled quantum systems and the freudenthal construction we consider special quantum systems containing both distinguishable and identical constituents it is shown that for these systems the freudenthal construction based on cubic jordan algebras naturally defines entanglement measures invariant under the group of stochastic local operations and classical communication slocc for this type of multipartite entanglement the slocc classes can be explicitly given these results enable further explicit constructions of multiqubit entanglement measures for distinguishable constituents by embedding them into identical fermionic ones we also prove that the plucker relations for the embedding system provide a sufficient and necessary condition for the separability of the embedded one we argue that this embedding procedure can be regarded as a convenient representation for quantum systems of particles which are really indistinguishable but for some reason they are not in the same state of some inner degree of freedom

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

d4 modular forms in this paper we study modular forms on two simply connected groups of type d 4 over q one group g s is a globally split group of type d 4 viewed as the group of isotopies of the split rational octonions the other g c is the isotopy group of the rational nonsplit octonions we study automorphic forms on g s in analogy to the work of gross gan and savin on g 2 namely we study automorphic forms whose component at infinity corresponds to a quaternionic discrete series representation we study automorphic forms on g c using gross s formalism of algebraic modular forms finally we follow work of gan savin gross rallis and others to study an exceptional theta correspondence connecting modular forms on g c and g s this can be thought of as an octonionic generalization of the jacquet langlands correspondence

multiplicative richness of additively large sets in zd abstract in their proof of the ip szemeredi theorem a far reaching extension of the classic theorem of szemeredi on arithmetic progressions furstenberg and katznelson 14 introduced an important class of additively large sets called ip r sets which underlies recurrence aspects in dynamics and is instrumental to enhanced formulations of combinatorial results the authors recently showed that additive ip r subsets of z d are multiplicatively rich with respect to every multiplication on z d without zero divisors e g multiplications induced by degree d number fields in this paper we explain the relationships between classes of multiplicative largeness with respect to different multiplications on z d we show for example that in contrast to the case for z there are infinitely many different notions of multiplicative piecewise syndeticity for subsets of z d when d 2 this is accomplished by using the associated algebra representations to prove the existence of sets which are large with respect to some multiplications while small with respect to others in the process we give necessary and sufficient conditions for a linear transformation to preserve a class of multiplicatively large sets one consequence of our results is that additive ip r sets are multiplicatively rich in infinitely many genuinely different ways we conclude by cataloging a number of sources of additive ip r sets from combinatorics and dynamics

proof of the positive mass theorem ii the positive mass theorem states that for a nontrivial isolated physical system the total energy which includes contributions from both matter and gravitation is positive this assertion was demonstrated in our previous paper in the important case when the space time admits a maximal slice here this assumption is removed and the general theorem is demonstrated abstracts of the results of this paper appeared in 11 and 13

an accurate method for free vibration analysis of structures with application to plates in this work the continuous element method which has been used as an alternative to the finite element method of vibration analysis of frames is applied to more general structures like 3 d continuum and rectangular plates the method is based on the concept of the so called impedance matrix giving in the frequency domain the linear relation between the generalized displacements of the boundaries and the generalized forces exerted on these boundaries for a 3 d continuum the concept of impedance matrix is introduced assuming a particular kind of boundary conditions for rectangular plates this new development leads to the solution of vibration problems for boundary conditions other than the simply supported ones

arithmetic of arithmetic coxeter groups in the 1990s j h conway published a combinatorial geometric method for analyzing integer valued binary quadratic forms bqfs using a visualization he named the topograph conway revisited the reduction of bqfs and the solution of quadratic diophantine equations such as pell s equation it appears that the crux of his method is the coincidence between the arithmetic group p g l 2 z and the coxeter group of type 3 there are many arithmetic coxeter groups and each may have unforeseen applications to arithmetic we introduce conway s topograph and generalizations to other arithmetic coxeter groups this includes a study of arithmetic flags and variants of binary quadratic forms

process rewrite systems many formal models for infinite state concurrent systems are equivalent to special classes of rewrite systems we classify these models by their expressiveness and define a hierarchy of classes of rewrite systems we show that this hierarchy is strict with respect to bisimulation equivalence the most general and most expressive class of systems in this hierarchy is called process rewrite systems prs they subsume petri nets pa processes and pushdown processes and are strictly more expressive than any of these intuitively prs can be seen as an extension of petri nets by subroutines that can return a value to their caller we show that the reachability problem is decidable for prs it is even decidable if there is a reachable state that satisfies certain properties that can be encoded in a simple logic thus prs are more expressive than petri nets but not turing powerful

freudenthal triple systems in arbitrary characteristic abstract let q v k be the form of degree 4 and h the alternating bilinear form that together give rise to a freudenthal triple system when the characteristic of k is different from 2 and 3 we give a form of degree 4 on a central cover of v over an arbitrary field in arbitrary characteristic and determine the group of similitudes of this group coincides with the intersection of the group of similitudes of q and the isometry group of h when the characteristic is different from 2 and 3 and has the expected structure of a suitable levi subgroup in general we extend known results about the stabilizers of elements of v and orbits of the group of similitudes and show in particular the existence of an exotic set of orbits when char k 2 and k is not perfect

binary quartic forms having bounded invariants and the boundedness of the average rank of elliptic curves we prove a theorem giving the asymptotic number of binary quartic forms having bounded invariants this extends to the quartic case the classical results of gauss and davenport in the quadratic and cubic cases respectively our techniques are quite general and may be applied to counting integral orbits in other representations of algebraic groups we use these counting results to prove that the average rank of elliptic curves over q when ordered by their heights is bounded in particular we show that when elliptic curves are ordered by height the mean size of the 2 selmer group is 3 this implies that the limsup of the average rank of elliptic curves is at most 1 5

jordan algebras exceptional groups and bhargava composition abstract we consider an integral version of the freudenthal construction relating jordan algebras and exceptional algebraic groups we show how this construction is related to higher composition laws of m bhargava in number theory we propose an algorithmic approach to studying orbit spaces of groups underlying bhargava s higher composition laws using this method we discover two new examples of spaces sharing similar properties and indicate several more examples of spaces where such composition laws may be introduced

how many rational points does a random curve have a large part of modern arithmetic geometry is dedicated to or motivated by the study of rational points on varieties for an elliptic curve over q the set of rational points forms a finitely generated abelian group the ranks of these groups when ranging over all elliptic curves are conjectured to be evenly distributed between rank 0 and rank 1 with higher ranks being negligible we will describe these conjectures and discuss some results on bounds for average rank highlighting the recent work of bhargava and shankar

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

classical nonlinear σ models on grassmann manifolds of compact or noncompact type grassmannian models are reexamined in the light of a new geometrical result namely the cartan immersion of a riemannian symmetric space g h into its isometry group g is not always a diffeomorphism onto the set m introduced by eichenherr and forger nucl phys b 164 528 1980 this is the case in particular for grassmann manifolds of all types their set m is shown to be disconnected and its structure is completely analyzed as a consequence a new constraint must be introduced in the backlund transformation bt method proposed by harnad saint aubin and shnider commun math phys 92 329 1984 it is shown however to always be satisfied for grassmann manifolds of compact type i e g compact but the problem remains open in most other cases on the other hand the bt method is extended to the euclidean regime

mermin pentagrams arising from veldkamp lines for three qubits we study the geometry of the space of mermin pentagrams objects that are used to rule out the existence of noncontextual hidden variable theories as alternatives to quantum theory it is shown that this space of 12 096 possible pentagrams is organized into 1008 families with each family containing a double six of pentagrams the 1008 families are connected to a special set of veldkamp lines in the veldkamp space for three qubits an object well known to finite geometers but has only been introduced to physics recently due to the transitive action of the symplectic group on this set of veldkamp lines it is enough to study only one canonical double six configuration of pentagrams we prove that the geometry of this double six configuration is encapsulated in the weight diagram of the 20 dimensional irreducible representation of the group as an interesting by product of our approach we show that mermin pentagrams and a class of mermin squares labelled by three qubit pauli operators are inherently related we conjecture that by studying the representation theoretic content of other veldkamp lines of the veldkamp space for n qubits makes it possible to find new contextual configurations in a systematic manner

undecidability of domino games and hhp bisimilarity history preserving bisimilarity hp bisimilarity and hereditary history preserving bisimilarity hhp bisimilarity are behavioural equivalences taking into account causal relationships between events of concurrent systems their prominent feature is that they are preserved under action refinement an operation important for the top down design of concurrent systems it is shown that in contrast to hp bisimilarity checking hhp bisimilarity for finite labelled asynchronous transition systems is undecidable by a reduction from the halting problem of 2 counter machines to make the proof more transparent a novel intermediate problem of checking domino bisimilarity for origin constrained tiling systems is introduced and shown undecidable it is also shown that the unlabelled domino bisimilarity problem is undecidable which implies undecidability of hhp bisimilarity for unlabelled finite asynchronous systems moreover it is argued that the undecidability of hhp bisimilarity holds for finite elementary net systems and 1 safe petri nets

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

structurable algebras and groups of type e6 and e7 abstract it is well known that every group of type f 4 is the automorphism group of an exceptional jordan algebra and that up to isogeny all groups of type 1 e 6 with trivial tits algebras arise as the isometry groups of norm forms of such jordan algebras we describe a similar relationship between groups of type e 6 and groups of type e 7 and use it to give explicit descriptions of the homogeneous projective varieties associated to groups of type e 7 with trivial tits algebras

on d_ ell extensions of odd prime degree ell generalizing the work of a morra and the authors we give explicit formulas for the dirichlet series generating function of d l extensions of odd prime degree l with given quadratic resolvent over the course of our proof we explain connections between our formulas and the ankeny artin chowla conjecture the ohno nakagawa relation for binary cubic forms and other topics

exceptional lie algebras su 3 and jordan pairs part 2 zorn type representations a representation of the exceptional lie algebras reflecting a simple unifying view based on realizations in terms of zorn type matrices is presented the role of the underlying jordan pair and jordan algebra content is crucial in the development of the structure each algebra contains three jordan pairs sharing the same lie algebra of automorphisms and the same external su 3 symmetry the applications in physics are outlined

freudenthal ranks ghz versus w the hilbert space of three qubit pure states may be identified with a freudenthal triple system every state has an unique freudenthal rank ranging from 1 to 4 which is determined by a set of automorphism group covariants it is shown here that the optimal success rates for winning a three player non local game varying over all local strategies are strictly ordered by the freudenthal rank of the shared three qubit resource

the conformal factor in the sas einstein maxwell field equations and a central extension of a formal loop group we consider a relation between the conformal factor in the stationary axisymmetric sas einstein maxwell field equations and a central extension of a formal loop group which is described by a group 2 cocycle on the formal loop group the corresponding 2 cocycle on the lie algebra of the formal loop group is the one which describes an affine lie algebra as a result we see that the space of formal solutions with conformal factors is a homogeneous space of a central extension of the hauser group

observables in terms of connection and curvature variables for einstein s equations with two commuting killing vectors einsteins equations with two commuting killing vectors and the associated lax pair are considered the equations for the connection a 1 where the variable spectral parameter are considered a