the complex lorentzian leech lattice and the bimonster ii let d be the incidence graph of the projective plane over f3 the artin group of the graph d maps onto the bimonster and a complex hyperbolic reflection group acting on 13 dimensional complex hyperbolic space y the generators of the artin group are mapped to elements of order 2 resp 3 in the bimonster resp let y y be the complement of the union of the fixed points of reflections in daniel allcock has conjectured that the orbifold fundamental group of y surjects onto bimonster in this article we study the reflection group we show that th e artin group of d maps to the orbifold fundamental group of y thus answering a question in allcock s article a monstrous proposal and taking one step towards the proof of allcock s conjecture the finite group aut d acts on y we make a detailed study of the complex hyperbolic line fixed by the subgroup l3 3 aut d we define meromorphic automorphic forms on y invariant under with poles along mirrors we show that the restriction of these forms to the complex hyperbolic line fixed by l3 3 gives meromorphic modular forms of level 13

on the y555 complex reflection group abstract we give a computer free proof of a theorem of basak describing the group generated by 16 complex reflections of order 3 satisfying the braid and commutation relations of the y 555 diagram the group is the full isometry group of a certain lattice of signature 13 1 over the eisenstein integers z 1 3 along the way we enumerate the cusps of this lattice and classify the root and niemeier lattices over z 1 3

on automorphisms of enriques surfaces an enriques surface over an algebraically closed field k of characteristic 4 2 is a nonsingular projective surface f with hi f gv h2 f or 0 2kv 0 the unramified double cover of f defined by the torsion class k v is a k3 surface f a nonsingular projective surface with hi f gr 0 kr 0 the study of enriques surfaces is equivalent to the study of k3 surfaces with a fixed point free involution z in particular the automor phism group aut f of f is isomorphic to the group aut ff z z where aut f z is the centralizer of z in the automorphism group aut f of ft in the case k the field of complex numbers the study of aut f is based on the global torelli theorem for k3 surfaces proven by i piatetski shapiro and i shafarevich in 19 it follows from this theorem that up to a finite group the group aut ff is isomorphic to the quotient group o pic f w where o pic f is the orthogonal group of the picard lattice of ff and w its normal subgroup generated by the reflections into the classes of nonsingular rational curves for a generic enriques surface f this theorem allows to compute aut f see 3 and also 17 where this result is not stated explicitly for an arbitrary f the relation between f and ff does not help since it is very difficult to compute the action of z in pie if however by other means we can prove the following analog of piatetski shapiro and shafarevich s result

locally toroidal polytopes of rank 6 and sporadic groups abstract we augment the list of finite universal locally toroidal regular polytopes of type 3 3 4 3 3 due to p mcmullen and e schulte adding as well as removing entries this disproves a related long standing conjecture our new universal polytope is related to a well known y shaped presentation for the sporadic simple group fi 22 and admits s 4 o 8 2 s 3 as the automorphism group we also discuss further extensions of its quotients in the context of y shaped presentations as well we note that two known examples of finite universal polytopes of type 3 3 4 3 3 are related to y shaped presentations of orthogonal groups over f 2 mixing construction is used in a number of places to describe covers and 2 covers

on k3 double planes covering enriques surfaces the moduli space of nodal enriques surfaces is irreducible of dimension 9 a nodal enriques surface is shown to be the quotient of a double cover of the plane by a lift of the cremona involution we also show that this gives a straightforward proof of the known description of the automorphism group for the generic such surface

higher toroidal regular polytopes abstract quad a regular polytope is locally toroidal if its minimal sections which are not of spherical type are toroids the locally toroidal polytopes of rank 4 have been extensively discussed and their classification is now nearly complete in this paper the locally toroidal polytopes of higher rank are investigated again an almost complete classification of these regular polytopes is obtained as well as sporadic examples there are several infinite families

fast algorithms for brownian matrices brownian motion is one of the most common models used to represent nonstationary signals the covariance matrix of a discrete time brownian motion has a very particular structure and is called a brownian matrix this note presents a number of results concerning linear problems appearing in digital signal processing with brownian matrices in particular it is shown that fast algorithms used for toeplitz matrices are simpler and faster for brownian matrices examples are given to illustrate the different results presented in the note

reflection groups in algebraic geometry after a brief exposition of the theory of discrete reflection groups in spherical euclidean and hyperbolic geometry as well as their analogs in complex spaces we present a survey of appearances of these groups in various areas of algebraic geometry

a look ahead block schur algorithm for diagonal plus semiseparable matrices abstract a look ahead recursive algorithm for the block triangular factorization of matrices represented as a sum of diagonal and semiseparable ones is derived this factorization is used for fast solving of the corresponding linear system for the case of sizes of diagonal blocks not depending on dimension of original matrix the algorithm obtained has linear complexity for a wide class of matrices the algorithm exhibits stable behavior

a modification of the dewilde van der veen method for inversion of finite structured matrices abstract we study a class of block structured matrices r r ij i j 1 n with a property that the solution of the corresponding system rx y of linear algebraic equations may be performed for o n arithmetic operations in this paper for finite invertible matrices we analyze in detail factorization and inversion algorithms these algorithms are related to those suggested by p m dewilde and a j van der veen time varying systems and computations kluwer academic publishers new york 1998 for a class of finite and infinite matrices with a small hankel rank the algorithms presented here are more transparent and are a modification of the algorithms from the above reference the approach and the proofs are essentially different from those in the above mentioned reference the paper contains also analysis of complexity and results of numerical experiments

smooth rational curves on enriques surfaces let s be an enriques surface over an algebraically closed field k of arbitrary characteristic p recall that this means that s is a connected smooth projective surface whose canonical class is numerically trivial and second betti number equal to 10 4 it is well known that generically over k e an enriques surface does not contain nonsingular rational curves this can be seen for example by considering the period space for such surfaces i 3 also it is known that if s contains such a curve then again generically it contains infinitely many of them this can be seen for example by viewing an enriques surface as an elliptic surface whose jacobian surface is a rational elliptic surface assuming that the latter is general enough its translation group is infinite and acts on s by automorphisms thus the existence of one such curve implies the existence of infinitely many in this paper we prove the following rather surprising result

inversion formulas and linear complexity algorithm for diagonal plus semiseparable matrices abstract matrices represented as a sum of diagonal and semiseparable ones are considered here these matrices belong to the class of structured matrices which arises in numerous applications fast o n algorithms for their inversion were developed before under additional restrictions which were a source of instability our aim is to eliminate these restrictions and to develop reliable and stable numerical algorithms in this paper the case of semiseparable matrices of order one is considered

quotients of infinite reflection groups let w be the weyl group of a finite dimensional complex simple lie algebra the structure of w is quite well known see 2 3 for instance in particular w is finite and w o2 w is isomorphic to a symmetric group or an orthogonal or symplectic group over the field of two elements it is natural to consider certain infinite analogues w p q r of such weyl groups and inquire about their structure these are reflection groups defined by diagrams t p q r of the form

enriques surfaces analytic discriminants and borcherds s φ function in bor 96 borcherds constructed a non vanishing weight 4 modular form on the moduli space of marked polarized enriques surface of degree 2 by considering the twisted denominator function of the fake monster lie algebra associated to an automorphism of order 2 of the leech lattice fixing an 8 dimensional subspace in jt 94 and jt 96 we defined and studied a meromorphic multi valued modular form of weight 2 which we call the k3 analytic discriminant on the moduli space of marked polarized k3 surfaces of degree 2d in certain cases including when where p k are distinct primes our meromorphic form is actually a holomorphic form our construction involves a determinant of the laplacian on a polarized k3 surface with respect to the calabi yau metric together with the l 2 norm of the image of the period map with respect to a properly scaled holomorphic two form since the universal cover of any enriques surface is a k3 surface we can restrict the k3 analytic discriminant to the moduli space of degree 2 enriques surfaces the main result of this paper is the observation that the square of our degree 2 analytic discriminant viewed as a function on the moduli space of degree 2 enriques surfaces is equal to the borcherd s function up to a universal multiplicative constant this result generalizes known results in the study of generalized kac moody algebras and elliptic curves and suggests further connections with higher dimensional calabi yau varieties specifically those which can be realized as complete intersections in some possibly weighted projective space

fast o n complexity algorithms for diagonal innovation matrices in general the direct solution of an n dimensional system of linear equations requires o n3 arithmetic operations frequently in high data rate signal processing fast algorithms of complexity lower than o n3 are required to solve a large system of linear equations in several interesting applications the specific structure of the coefficient matrix associated with the linear system may be used to reduce the required number of operations fast algorithms have been developed when the coefficient matrix is a toeplitz matrix a hankel matrix or when it can be represented as a sum of toeplitz and hankel matrices the arithmetic complexity associated with these fast algorithms is o n2 in this paper fast algorithms of o n are presented that can be used for a class of matrices called diagonal innovation matrices dim previous results for this class of matrices require an arithmetic complexity of o n2 a number of results are presented regarding linear systems having dim coefficient matrices and several special cases are examined the effect of recursively increasing the order of the coefficient matrix on the number of operations is studied and some observations are made

a note on diagonal innovation matrices by noting that a recently introduced class of so called diagonal innovation matrices has a form previously encountered in state space estimation problems new linear complexity algorithms are presented for their factorization and inversion

an exact recursion for the composite nearest neighbor degeneracy for a 2 n lattice space a set theoretic argument is utilized to develop a recursion relation that yields exactly the composite nearest neighbor degeneracy for simple indistinguishable particles distributed on a 2 n lattice space the associated generating functions as well as the expectation of the resulting statistics are also treated

enriques surfaces and jacobian elliptic k3 surfaces this paper proposes a new geometric construction of enriques surfaces its starting point are k3 surfaces with jacobian elliptic fibration which arise from rational elliptic surfaces by a quadratic base change the enriques surfaces obtained in this way are characterised by elliptic fibrations with a rational curve as bisection which splits into two sections on the covering k3 surface the construction has applications to the study of enriques surfaces with specific automorphisms it also allows us to answer a question of beauville about enriques surfaces whose brauer groups show an exceptional behaviour in a forthcoming paper we will study arithmetic consequences of our construction

automorphisms of enriques surfaces o introduction the aim of this note is to compute the group aut y of biholomorphic auto morphisms for the general enriques surface y the basic tool is the global to relli theorem for projective k3 surfaces as it was given by piatetski shapiro and shafarevich 11 and refined by burns and rapaport 2 the essential result is that in contrast to the case of curves aut y is big for general y and small for special y in this paper we consider the complex case only recall that an enriques surface y is a projective complex surface with universal double cover a k3 surface one knows that h2 y z 7z 2 2 and that the cup product provides

alexander invariant for perturbations of fredholm operators abstract the alexander type invariant for the bifurcation problems involving perturbations of linear fredholm operators of nonnegative index is discussed and compared with another homotopy invariant called the bifurcation index detecting global bifurcations of solutions to multiparameter equations

on irregular threefolds and fourfolds with numerically trivial canonical bundle we prove that for a smooth projective irregular threefold x with k x equiv 0 and a nef and big divisor l on x ml p gives a birational map for all m ge 3 and all p in mathrm pic 0 x we also use the same method to deal with fourfolds and prove that for a smooth projective irregular fourfold x with k x equiv 0 and an ample divisor l on x ml p gives a birational map for all m ge 5 and all p in mathrm pic 0 x these results are optimal

efficient solution of a toeplitz plus hankel coefficient matrix system of equations there are well known fast algorithms such as the levinson recursion for solving linear equations with a toeplitz or hankel coefficient matrix this paper extends the saving obtained by the levinson recursion for inversion of a toeplitz matrix to problems involving the inversion of matrices which are the sum of toeplitz and hankel matrices

normal subgroups in the cremona group let k be an algebraically closed field we show that the cremona group of all birational transformations of the projective plane mathbb p mathbf k 2 is not a simple group the strategy makes use of hyperbolic geometry geometric group theory and algebraic geometry to produce elements in the cremona group that generate non trivial normal subgroups

on two parameter global bifurcation of periodic solutions to a class of differential variational inequalities abstract in this paper by using the method of integral guiding functions a multiparameter global bifurcation theorem for differential inclusions with the periodic condition is proved it is shown how the abstract result can be applied to the study of the two parameter global bifurcation of periodic solutions for a class of differential variational inequalities

on the picard group of enriques surfaces an enriques surface over an algebraically closed field k of characteristic 4 2 is a non singular projective surface s with h s d s h2 s cs 0 and 2ks 0 the unramitied double cover defined by ks is a k3 surface r a non singular projective surface with hi r d r 0 kr 0 illusie has shown i that the group of divisors modulo numerical equivalence is isomorphic to the enriques lattice u e 8 1 where u and e s 1 denote respectively the unique even unimodular lattices of index of inertia 1 1 and 0 8 the purpose of this note is to use this isomorphism to study the picard group of s we prove the existence of certain configurations of irreducible curves of arithmetic genus 0 or 1 and deduce from them the existence of certain projective models for s and r for example we prove the following results

y groups via transitive extension abstract motivated by an earlier observation by j h fischer around 1980 b conway conjectured that the coxeter diagram y555 on fig 1 together with a single additional so called spider relation ab1c1ab2c2ab3c3 10 1 form a presentation for the wreath product of the monster group m and a group of order 2 this conjecture was proved by s p norton and the author in 1990 the original proof was rather involved relying on simple connectedness results for certain diagram geometries on numerous data obtained by coset enumeration on a computer and on some delicate calculations with subgroups coming from the 26 node theorem in the present work we follow an inductive approach to the identification of y groups by considering larger y groups as transitive extensions of smaller ones along these lines we obtain an alternative identification proof for the y groups which is almost computer free we refer to only one result of double coset enumeration our approach provides a uniform understanding of the y groups particularly of features such as centres and redundant generators

a ring of brownian matrices abstract conditions are given for the product of two brownian matrices to be brownian a special class of brownian matrices is then introduced which forms a ring and it is shown that the determinant the eigenvalues and the eigenvectors for matrices in this class can all be determined in o n flops the inverse of a nonsingular matrix in this class is also found in o n flops compared with o n 2 for general brownian matrices generalized inverses when the matrix is singular are shown to belong to the class and are also found in o n flops a connection with magic square matrices is also discussed

string c groups as transitive subgroups of sn abstract if is a string c group which is isomorphic to a transitive subgroup of the symmetric group s n other than s n and the alternating group a n then the rank of is at most n 2 1 with finitely many exceptions which are classified it is conjectured that only the symmetric group has to be excluded

a global bifurcation index for set valued perturbations of fredholm operators abstract in the paper the homotopy invariant detecting global bifurcations of solutions to multi parameter equations involving general set valued perturbations of linear fredholm operators of non negative index is introduced some applications to the existence problems for differential inclusions are provided

rational surfaces with a large group of automorphisms we classify rational surfaces for which the image of the automorphisms group in the group of linear transformations of the picard group is the largest possible this answers a question raised by arthur coble in 1928 and can be rephrased in terms of periodic orbits of a birational action of an infinite coxeter group on ordered point sets in the projective plane modulo projective equivalence we also outline the classification of non rational surfaces with large automorphism groups

noncommutative cyclic covers and maximal orders on surfaces abstract in this paper we construct various examples of maximal orders on surfaces including some del pezzo orders some ruled orders and some numerically calabi yau orders the method of construction is a noncommutative version of the cyclic covering trick these noncommutative cyclic covers are very computable and we give a formula for their ramification data this often allows us to determine if a maximal order described via ramification data can be constructed as a noncommutative cyclic cover the construction also has applications to brauer severi varieties and in the quaternion case we show how to obtain some brauer severi varieties from g hilbert schemes of p 1 bundles

iterative methods for restoring noisy images an iterative image restoration algorithm is presented that is based on the best linear mean square estimate of the wiener filter technique and the iterative approach of the first order stationary linear jacobi method the iterative algorithm incorporates a priori knowledge concerning the image and noise statistics directly into the iterative procedure it also assumes the image covariance to be described by a separable first order markov field and exploits the resulting toeplitz structure the usefulness and validity of this algorithm are demonstrated by implementation and testing on real images the advantages offered by the algorithm are that it is computationally efficient since only a small number of computations per pixel per iteration is required it requires a small amount of memory it has a fast rate of convergence and it exhibits neither high noise sensitivity nor significant loss of resolution

approximate controllability of differential inclusions in hilbert spaces abstract in this paper controllability for the system originating from semilinear functional differential equations in hilbert spaces is studied we consider the problem of approximate controllability of semilinear differential inclusion assuming that semigroup generated by the linear part of the inclusion is compact and under the assumption that the corresponding linear system is approximately controllable by using resolvent of controllability gramian operator and fixed point theorem sufficient conditions have been formulated and proved an example is presented to illustrate the utility and applicability of the proposed method