distribution of values of bounded generalized polynomials a generalized polynomial is a real valued function which is obtained from conventional polynomials by the use of the operations of addition multiplication and taking the integer part a generalized polynomial mapping is a vector valued mapping whose coordinates are generalized polynomials we show that any bounded generalized polynomial mapping u zd rl has a representation u n f n x n zd where f is a piecewise polynomial function on a compact nilmanifold x x x and is an ergodic zd action by translations on x this fact is used to show that the sequence u n n zd is well distributed on a piecewise polynomial surface mathcal s subset mathbf r l with respect to the borel measure on mathcal s that is the image of the lebesgue measure under the piecewise polynomial function defining mathcal s as corollaries we also obtain a von neumann type ergodic theorem along generalized polynomials and a result on diophantine approximations extending the work of van der corput and of furstenberg weiss

on the number of quintic fields we show that the number of quintic number fields whose discriminant does not exceed x in absolute value is bounded by a constant times x1 e for any e 0 this may be compared with the conjecture that the number of such fields is asymptotic to a constant times x as x tends to infinity

the concomitants of a prehomogeneous vector space abstract the classical method of radial separation of variables is applied to gain some information about the concomitants also known as relatively equivariant polynomial maps or covariants of a class of prehomogeneous vector spaces particular attention is payed to concomitants that are themselves prehomogeneous the results are applied to classify all prehomogeneous concomitants of several examples of interest in arithmetic

initiation of damage in the kauffman model an alternate method is presented for initiating damage to the kauffman model a numerical study suggests that this modification increases the efficiency of generating damage clusters without changing their fractal properties

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

uniqueness and stability in determining the heat radiative coefficient the initial temperature and a boundary coefficient in a parabolic equation we consider an inverse parabolic problem we prove that the heat radiative coefficient the initial temperature and a boundary coefficient can be simultaneously determined from the final overdetermination provided that the heat radiative coefficient is a priori known in a small subdomain moreover we establish a stability estimate for this inverse problem

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

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

solvability of strongly nonlinear boundary value problems for second order differential inclusions abstract in this paper we study a problem for a second order differential inclusion with dirichlet neumann and mixed boundary conditions the equation is driven by a nonlinear not necessarily homogeneous differential operator satisfying certain conditions and containing as a particular case the p laplacian operator we prove the existence of solutions both for the case in which the multivalued nonlinearity has convex values and for the case in which it has not convex values the presence of a maximal monotone operator in the equation make the results applicable to gradient systems with non smooth time invariant convex potential and differential variational inequalities

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

bounds for periodic solutions of vector second order nonlinear differential equations upper bounds for the amplitudes and lower bounds for the periods of periodic solutions of coupled second order nonlinear differential equations are found the results are expressed in a solution of a positive operator equation which is associated with the considered system cases are indicated for which the obtained bounds are precise and surprising behavior of the amplitude and periods of such systems under insertion of certain terms is revealed the results are illustrated by examples

a geometrical interpretation of the chaotic state of inhomogeneous deterministic cellular automata abstract we propose a geometrical interpretation of the chaotic state of inhomogeneous cellular automata from the rules of the cellular automaton we construct a network the percolating phase of this network coincides with the chaotic phase of the cellular automaton we also report numerical tests of these ideas on several cellular automata

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

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

revisiting the nilpotent polynomial hales jewett theorem abstract answering a question posed by bergelson and leibman in 6 we establish a nilpotent version of the polynomial hales jewett theorem that contains the main theorem in 6 as a special case important to the formulation and the proof of our main theorem is the notion of a relative syndetic set relative with respect to a closed non empty subsets of g 25 as a corollary of our main theorem we prove an extension of the restricted van der waerden theorem to nilpotent groups which involves nilprogressions

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

homotopy perturbation method a new nonlinear analytical technique in this paper a new perturbation method is proposed in contrast to the traditional perturbation methods this technique does not require a small parameter in an equation in this method according to the homotopy technique a homotopy with an imbedding parameter email protected 0 1 is constructed and the imbedding parameter is considered as a small parameter so the method is called the homotopy perturbation method which can take the full advantages of the traditional perturbation methods and homotopy techniques to illustrate its effectiveness and its convenience a duffing equation with high order of nonlinearity is used the result reveals that its first order of approximation obtained by the proposed method is valid uniformly even for very large parameter and is more accurate than the perturbation solutions

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

polynomial extensions of van der waerden s and szemeredi s theorems an extension of the classical van der waerden and szemeredi the orems is proved for commuting operators whose exponents are polynomials as a consequence for example one obtains the following result let s zl be a set of positive upper banach density let p1 n pk n be polynomials with rational coefficients taking integer values on the integers and satisfying pi 0 0 i 1 k then for any v1 vk zl there exist an integer n and a vector u zl such that u pi n vi s for each i k department of mathematics ohio state university columbus ohio 43210 e mail address vitaly math ohio state edu department of mathematics technion haifa 23000 israel e mail address sashal techunix technion ac il current address department of mathematics stanford university stanford california 94305 e mail address leibman math stanford edu license or copyright restrictions may apply to redistribution see http www ams org journal terms of use

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

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

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

variational iteration method new development and applications variational iteration method has been favourably applied to various kinds of nonlinear problems the main property of the method is in its flexibility and ability to solve nonlinear equations accurately and conveniently in this paper recent trends and developments in the use of the method are reviewed major applications to nonlinear wave equation nonlinear fractional differential equations nonlinear oscillations and nonlinear problems arising in various engineering applications are surveyed the confluence of modern mathematics and symbol computation has posed a challenge to developing technologies capable of handling strongly nonlinear equations which cannot be successfully dealt with by classical methods variational iteration method is uniquely qualified to address this challenge the flexibility and adaptation provided by the method have made the method a strong candidate for approximate analytical solutions this paper outlines the basic conceptual framework of variational iteration technique with application to nonlinear problems both achievements and limitations are discussed with direct reference to approximate solutions for nonlinear equations a new iteration formulation is suggested to overcome the shortcoming a very useful formulation for determining approximately the period of a nonlinear oscillator is suggested examples are given to illustrate the solution procedure

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

non trivial periodic solutions of a non linear hill s equation with positively homogeneous term th ee xistence of non trivial periodic solutions of a general family of second order differential equations whose main model is a hill s equation with a cubic nonlinear term arising in different physical applications is proved c

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 homotopy perturbation method applied for solving complex valued differential equations with strong cubic nonlinearity abstract in this paper the homotopy perturbation method is adopted for solving a complex valued second order strongly nonlinear differential equation homotopy with an imbedding parameter p 0 1 is constructed the perturbation procedure with parameter p transforms the strongly nonlinear differential equation into a system of linear complex valued differential equations whose solutions give the approximate solution of the initial differential equation to illustrate the effectiveness and convenience of the suggested procedure a duffing equation with strong cubic nonlinearity is considered the periodic solution in the first approximation is obtained the solution is compared with the exact one and shows good agreement

quotients of some prehomogeneous vector spaces abstract we give quotients v g spec c v g for some prehomogeneous vector spaces g h v

a connection between the percolation transition and the onset of chaos in the kauffman model it is demonstrated numerically that the percolation transition of the unstable sites and the onset of chaos in the kauffman model 1969 happens for the same value of the bias on the randomly chosen rules in the two dimensional triangular lattice and in the three and four dimensional hypercubic lattices the percolation thresholds for the stable sites are different for the three and presumably also four dimensional lattices on the triangular lattice the differences between the thresholds are too small to be resolved the fractal dimension of the damage is calculated and also the spreading time on the four dimensional hypercubic lattice

the slice method for g torsors abstract the notion of a g n slice of a g variety was introduced by p i katsylo in the early 80 s for an algebraically closed base field of characteristic 0 slices also known under the name of relative sections have ever since provided a fundamental tool in invariant theory allowing reduction of rational or regular invariants of an algebraic group g to invariants of a simpler group we refine this notion for a g scheme over an arbitrary field and use it to get reduction of structure group results for g torsors namely we show that any g n slice of a versal g scheme gives surjective maps h 1 l n h 1 l g in fppf cohomology for infinite fields l containing f we show that every stabilizer in general position h for a geometrically irreducible g variety v gives rise to a g n g h slice in our sense the combination of these two results is applied in particular to obtain a striking new upper bound on the essential dimension of the simply connected split algebraic group of type e 7

set polynomials and polynomial extension of the hales jewett theorem an abstract hales jewett type extension of the polynomial van der waerden theorem bl is established theorem let r d q n there exists n n such that for any r coloring of the set of subsets of v 1 n d 1 q there exist a set a v and a nonempty set 1 n such that a d 1 q and the subsets a a d 1 a d 2 a d q are all of the same color this polynomial hales jewett theorem contains refinements of many combinatorial facts as special cases the proof is achieved by introducing and developing the apparatus of set polynomials polynomials whose coefficients are finite sets and applying the methods of topological dynamics

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

nonmetrizable topological dynamics and ramsey theory applying ideas from topological dynamics in compact metric spaces to the stone cech compactification of a discrete semigroup several new proofs of old results and some new results in ramsey theory are obtained in particular two ultrafilter proofs of van der waerden s theorem are given an ultrafilter approach to central sets sets which are combinatorially rich is developed this enables us to show that for any partition of the positive integers one cell is both additively and multiplicatively central also a fortuitous answer to a question of ellis is obtained

the homotopy perturbation method for nonlinear oscillators with discontinuities in the paper the homotopy perturbation method is applied to the nonlinear oscillators with discontinuities only one iteration leads to high accuracy of the solutions

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

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

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

orbit closure diagram for the space of quadruples of quinary alternating forms abstract we construct the hasse diagram for the closure ordering of the orbits of the irreducible reduced prehomogeneous vector space of quadruples of quinary alternating forms in five variables

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

on the existence of solutions for strongly nonlinear differential equations the objectives of this paper are twofold firstly to prove the existence of an approximate solution in the mean for some nonlinear differential equations we also investigate the behavior of the class of solutions which may be associated with the differential equation secondly we aim to implement the homotopy perturbation method hpm to find analytic solutions for strongly nonlinear differential equations

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

stability estimate for the hyperbolic inverse boundary value problem by local dirichlet to neumann map abstract in this paper we consider the stability of the inverse problem of determining a function q x in a wave equation t 2 u u q x u 0 in a bounded smooth domain in r n from boundary observations this information is enclosed in the hyperbolic dynamic dirichlet to neumann map associated to the solutions to the wave equation we prove in the case of n 2 that q x is uniquely determined by the range restricted to a subboundary of the dirichlet to neumann map whose stability is a type of double logarithm

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

a modular compactification of m_ 1 n from a_ infty structures we show that a certain moduli space of minimal a structures coincides with the modular compactification m1 n n 1 of m1 n constructed by smyth in 22 in addition we describe these moduli spaces and the universal curves over them by explicit equations prove that they are normal and gorenstein show that their picard groups have no torsion and that they have rational singularities if and only if n 11

upper critical dimension of kauffman cellular automata random boolean networks on nearest neighbour d dimensional lattices are argued to belong to the universality class of directed percolation with quenched disorder in d 1 dimensions hansen s computer simulations for d 4 are thus interpreted as being at the upper critical dimension

are damage spreading transitions generically in the universality class of directed percolation we present numerical evidence for the fact that the damage spreading transition in the domany kinzel automaton found by martinset al is in the same universality class as directed percolation we conjecture that also other damage spreading transitions should be in this universality class unless they coincide with other transitions as in the ising model with glauber dynamics and provided the probability for a locally damaged state to become healed is not zero

fractal dimensions in three dimensional kauffman cellular automata the kauffman random networks of automata are studied on a simple cubic lattice by computer simulations each automaton follows random rules depending on its six neighbours and fixed in time a transition between the frozen and the chaotic phase is observed and the fractal dimension of the asymptotic actual damage at the phase transition is measured

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

some groups of type vf a group is of type vf if it contains a finite index subgroup which has a finite classifying space we construct groups of type vf in which the centralizers of some elements of finite order are not of type vf and groups of type vf containing infinitely many conjugacy classes of finite subgroups from these examples it follows that a group g of type vf need not admit a finite type or finite classifying space for proper actions sometimes also called the universal proper g space we construct groups g for which the minimal dimension of a universal proper g space is strictly greater than the virtual cohomological dimension of g each of our groups embeds in some general linear group over the rational integers applications to algebraic k theory of group algebras and topological k theory of group c algebras are also considered the groups are constructed as finite extensions of bestvina brady groups

adiabatic limits of η invariants and the meyer functions we construct a function on the orbifold fundamental group of the moduli space of smooth theta divisors which we call the meyer function for smooth theta divisors in the construction we use the adiabatic limits of the invariants of the mapping torus of theta divisors we shall prove that the meyer function for smooth theta divisors cobounds the signature cocycle and we determine the values of the meyer function for the dehn twists in particular we give an analytic construction of the meyer function of genus two

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