Integral Lie powers of a lattice for the cyclic group of order 2

Let L_n denote the n-th homogeneous component of the free Lie ring L(W) on a given \Bbb ZC₂{{\Bbb Z}}C_{2}-lattice W. This paper gives explicit formulae for the multiplicities of the three indecomposable \Bbb ZC₂{{\Bbb Z}}C_{2}-lattices in a Krull-Schmidt decomposition of L_n. In the case where W is a free \Bbb ZC₂{{\Bbb Z}}C_{2}-lattice, L_n is shown to have no non-zero direct summand on which C₂ acts trivially - this extends a result of R. M. Bryant for the special case where W is the regular \Bbb ZC₂{{\Bbb Z}}C_{2}-lattice. As an application, the structure of the higher dimensional modules associated to a non-cyclic free presentation of C₂ is determined.     

Quadratic isoperimetric inequality for mapping tori of polynomially growing automorphisms of free groups

We prove that the mapping torus F_n \rtimes_f \Bbb Z F_n \rtimes_\phi {\Bbb Z} of a polynomially growing automorphism f: F_n ? F_n \phi : F_n \to F_n of finitely generated free group F_n satisfies the quadratic isoperimetric inequality.     

Representations of nilpotent Lie algebras

Let (L,[p]) a finite dimensional nilpotent restricted Lie algebra of characteristic p 3 3, c ? L^*p \geq 3, \chi \in L^* a linear form. In this paper we study the representation theory of the reduced universal enveloping algebra u(L,c)u(L,\chi ). It is shown that u(L,c)u(L,\chi ) does not admit blocks of tame representation type. As an application, we prove that the nonregular AR-components of u(L,c)u(L,\chi ) are of types \Bbb Z [A_￥ ]\Bbb Z [A_\infty ] or \Bbb Z [A_n]/(t)\Bbb Z [A_n]/(\tau ).     

The dimension monoid of a lattice

We introduce the dimension monoid of a lattice L, denoted by Dim L. The monoid Dim L is commutative and conical, the latter meaning that the sum of any two nonzero elements is nonzero. Furthermore, Dim L is given along with the dimension map, D\Delta from L2L to Dim L, which has the intuitive meaning of a distance function. The maximal semilattice quotient of Dim L is isomorphic to the semilattice Con_c L of compact congruences of L; hence Dim L is a precursor of the congruence lattice of L. Here are some additional features of this construction: ¶¶ (1) Our dimension theory provides a generalization to all lattices of the von Neumann dimension theory of continuous geometries. In particular, if L is an irreducible continuous geometry, then Dim L is either isomorphic to \Bbb Z⁺\Bbb Z^+ or to \Bbb R⁺\Bbb R^+.¶ (2) If L has no infinite bounded chains, then Dim L embeds (as an ordered monoid) into a power of \Bbb Z⁺è{￥}.{\Bbb Z}^{+}\cup \{\infty\}.¶ (3) If L is modular or if L has no infinite bounded chains, then Dim L is a refinement monoid.¶ (4) If L is a simple geometric lattice, then Dim L is isomorphic to \Bbb Z⁺\Bbb Z^+, if L is modular, and to the two-element semilattice, otherwise.¶ (5) If L is an à₀\aleph_0-meet-continuous complemented modular lattice, then both Dim L and the dimension function D\Delta satisfy (countable) completeness properties.¶¶ If R is a von Neumann regular ring and if L is the lattice of principal right ideals of the matrix ring M₂ (R), then Dim L is isomorphic to the monoid V (R) of isomorphism classes of finitely generated projective right R-modules. Hence the dimension theory of lattices provides a wide lattice-theoretical generalization of nonstable K-theory of regular rings.     

Blocks with cyclic defect of Hecke orders of Coxeter groups

Let B\cal B be a p-block of cyclic defect of a Hecke order over the complete ring \Bbb Z[q] _{áq-1,p ?}\Bbb {Z}[q] _{\langle q-1,p \rangle}; i.e. modulo áq-1 ?\langle q-1 \rangle it is a p-block B of cyclic defect of the underlying Coxeter group G. Then B\cal B is a tree order over \Bbb Z[q]_{áq-1, p ?}\Bbb {Z}[q]_{\langle q-1, p \rangle } to the Brauer tree of B. Moreover, in case B\cal B is the principal block of the Hecke order of the symmetric group S(p) on p elements, then B\cal B can be described explicitly. In this case a complete set of non-isomorphic indecomposable Cohen-Macaulay B\cal B-modules is given.     

Invariants of free Lie algebras

Let k be a principal ideal domain with identity and characteristic zero. For a positive integer n, with n \geqq 2n \geqq 2, let H(n) be the group of all n x n matrices having determinant ±1\pm 1. Further, we write SL(n) for the special linear group. Let L be a free Lie algebra (over k) of finite rank n. We prove that the algebra of invariants L^B(n) of B(n), with B(n) ? { H(n), SL(n)}B(n) \in \{ H(n), {\rm SL}(n)\} , is not a finitely generated free Lie algebra. Let us assume that k is a field of characteristic zero and let áSem(n) ?\langle {\rm Sem}(n) \rangle be the Lie subalgebra of L generated by the semi-invariants (or Lie invariants) Sem(n). We prove that áSem(n) ?\langle {\rm Sem}(n) \rangle is not a finitely generated free Lie algebra which gives a positive answer to a question posed by M. Burrow [4].     

Rational surfaces and moduli spaces of vector bundles on rational surfaces

Let X be a smooth algebraic surface, L ? Pic(X) L \in \textrm{Pic}(X) and H an ample divisor on X. Set M_X,H(2; L, c₂) the moduli space of rank 2, H-stable vector bundles F on X with det(F) = L and c₂(F) = c₂. In this paper, we show that the geometry of X and of M_X,H(2; L, c₂) are closely related. More precisely, we prove that for any ample divisor H on X and any L ? Pic(X) L \in \textrm{Pic}(X) , there exists n₀ ? \mathbbZ n_0 \in \mathbb{Z} such that for all n₀ \leqq c₂ ? \mathbbZ n_0 \leqq c_2 \in \mathbb{Z} , M_X,H(2; L, c₂) is rational if and only if X is rational.     

Arithmetic progressions in sumsets

We prove several results concerning arithmetic progressions in sets of integers. Suppose, for example, that a \alpha and b \beta are positive reals, that N is a large prime and that C,D í \Bbb Z/N\Bbb Z C,D \subseteq {\Bbb Z}/N{\Bbb Z} have sizes gN \gamma N and dN \delta N respectively. Then the sumset C + D contains an AP of length at least e^{c ?{log} N} e^{c \sqrt{\rm log} N} , where c > 0 depends only on g \gamma and d \delta . In deriving these results we introduce the concept of hereditary non-uniformity (HNU) for subsets of \Bbb Z/N\Bbb Z {\Bbb Z}/N{\Bbb Z} , and prove a structural result for sets with this property.     

The Problem of Freeness for Euler Monoids and Möbius Groups

The Euler monoid E_n = {(a,b,t) epsilon Z³ : a² + b² = tⁿ, n S 1, is free if and only if n is odd (Theorem 1). We extend the results of Lyndon and Ullman, and Beardon concerning the set of those rational numbers mu epsilon (-2,2) for which the matrix Möbius group G_mu generated by A= and B = is not free (Theorems 2, 3, 4).     

The differential Hilbert series of a local algebra

The differential Hilbert series of a commutative local algebra R/R₀ which is essentially of finite type is the generating function of the numerical function which associates with each n ? \Bbb N n\in \Bbb N the minimal number of generators of the algebra Pⁿ_R/R0P^n_{R/R_0} of principal parts of order n, considered as an R-module. It can be expressed as a rational function over the integers. We wish to compute this rational function in terms of other invariants of the local algebra or at least give estimates of it. We obtain formulas which generalize wellknown facts about the minimal number of generators of the module of Kähler differentials.     

Covering and packing in \Bbb Zn{\Bbb Z}^n and \Bbb Rn{\Bbb R}^n, (I)

Given a finite subset A{\cal A} of an additive group \Bbb G{\Bbb G} such as \Bbb Zⁿ{\Bbb Z}^n or \Bbb Rⁿ{\Bbb R}^n , we are interested in efficient covering of \Bbb G{\Bbb G} by translates of A{\cal A} , and efficient packing of translates of A{\cal A} in \Bbb G{\Bbb G} . A set S ì \Bbb G{\cal S} \subset {\Bbb G} provides a covering if the translates A + s{\cal A} + s with s ? Ss \in {\cal S} cover \Bbb G{\Bbb G} (i.e., their union is \Bbb G{\Bbb G} ), and the covering will be efficient if S{\cal S} has small density in \Bbb G{\Bbb G} . On the other hand, a set S ì \Bbb G{\cal S} \subset {\Bbb G} will provide a packing if the translated sets A + s{\cal A} + s with s ? Ss \in {\cal S} are mutually disjoint, and the packing is efficient if S{\cal S} has large density. In the present part (I) we will derive some facts on these concepts when \Bbb G = \Bbb Zⁿ{\Bbb G} = {\Bbb Z}^n , and give estimates for the minimal covering densities and maximal packing densities of finite sets A ì \Bbb Zⁿ{\cal A} \subset {\Bbb Z}^n . In part (II) we will again deal with \Bbb G = \Bbb Zⁿ{\Bbb G} = {\Bbb Z}^n , and study the behaviour of such densities under linear transformations. In part (III) we will turn to \Bbb G = \Bbb Rⁿ{\Bbb G} = {\Bbb R}^n .     

On the infinitesimal affine rigidity of ellipsoids in the n-space

Due to R. Schneider 1967 an ellipsoid E in the affine space \Bbb Aⁿ\Bbb A^n is affinely rigid, i.e. every other ovaloid F in \Bbb Aⁿ\Bbb A^n with the same affine Blaschke metric as for E equals E up to an equiaffine motion of E. Due to M. Kozlowski 1985 resp. W. Blaschke 1922 for n = 3 ellipsoids are moreover S-rigid resp. infinitesimally S-rigid in the sense of equal resp. infinitesimally equal affine scalar curvature S (unknown until now for n >3). - In this article it is proved that ellipsoids in \Bbb Aⁿ\Bbb A^n are also infinitesimally S-rigid for any n.     

Analytic Fourier integral operators, Monge-Ampère equation and holomorphic factorization

We will show that the factorization condition for the Fourier integral operators I_r ^m (X,Y;L )I_\rho ^\mu (X,Y;\it\Lambda ) leads to a parametrized parabolic Monge-Ampère equation. For an analytic operator, the fibration by the kernels of the Hessian of phase function is shown to be analytic in a number of cases, by considering a more general continuation problem for the level sets of a holomorphic mapping. The results are applied to obtain L^p-continuity for translation invariant operators in \Bbb Rⁿ{\Bbb R}^n with n ￡ 4n\leq 4 and for arbitrary \Bbb Rⁿ{\Bbb R}^n with dp_X×Y|_L ￡ n+2d\pi _{X\times Y}|_\Lambda \leq n+2.     

A note on discrete heat conduction

If the longitudinal line method is applied to the Cauchy problem u_t = u_xx, u(0, x) = u₀(x) with a bounded function u₀, one is led to a linear initial value problem v￠(t)=A v(t), v(0)=wv'(t)=A v(t),\, v(0)=w in l^￥ (\Bbb Z)l^\infty (\Bbb Z). Using Banach limit techniques we study the asymptotic behaviour of the solutions of these problems as t tends to infinity.     

Simple Explicit Formulas for the Frame-Stewart Numbers

Several different approaches to the multi-peg Tower of Hanoi problem are equivalent. One of them is Stewart's recursive formula ¶¶ S (n, p) = min {2S (n₁, p) + S (n-n₁, p-1) | n₁, n-n₁ ? \mathbbZ⁺}. S (n, p) = min \{2S (n_1, p) + S (n-n_1, p-1)\mid n_1, n-n_1 \in \mathbb{Z}^+\}. ¶¶In the present paper we significantly simplify the explicit calculation of the Frame-Stewart's numbers S(n, p) and give a short proof of the domain theorem that describes the set of all pairs (n, n₁), such that the above minima are achieved at n₁.     

L^p -uniqueness for Dirichlet operators with singular potentials

We study the problem of strong uniqueness in L^p for the Dirichlet operator perturbed by a singular complex-valued potential. First we construct the generator -H_p of a C₀-semigroup in L^p, with H_p extending the restriction of the perturbed Dirichlet operator to the set of smooth functions. The corresponding sesquilinear form in L² is not assumed to be sectorial. Then we reveal sufficient conditions on the logarithmic derivative # of the measure rdx \rho dx and the potential q which ensure that -H_p is the only extension of D+b·?-q \upharpoonright_C0^￥ \Delta +\beta \cdot \nabla -q \upharpoonright_{C_0^{\infty}} which generates a C₀-semigroup on L^p. The method of a priori estimates of solutions to corresponding differential equations is employed.     

Integration of the lifting formulas and the cyclic homology of the algebras of differential operators

We integrate the Lifting cocycles Y_2n+1, Y_2n+3, Y_2n+5,? ([Sh1,2]) \Psi_{2n+1}, \Psi_{2n+3}, \Psi_{2n+5},\ldots\,([\rm Sh1,2]) on the Lie algebra Dif_n of holomorphic differential operators on an n-dimensional complex vector space to the cocycles on the Lie algebra of holomorphic differential operators on a holomorphic line bundle l \lambda on an n-dimensional complex manifold M in the sense of Gelfand--Fuks cohomology [GF] (more precisely, we integrate the cocycles on the sheaves of the Lie algebras of finite matrices over the corresponding associative algebras). The main result is the following explicit form of the Feigin--Tsygan theorem [FT1]:¶¶ H^·_Lie(\frak g\frak l^fin_￥(Dif_n);\Bbb C) = ù^·(Y_2n+1, Y_2n+3, Y_2n+5,? ) H^\bullet_{\rm Lie}({\frak g}{\frak l}^{\rm fin}_\infty({\rm Dif}_n);{\Bbb C}) = \wedge^\bullet(\Psi_{2n+1}, \Psi_{2n+3}, \Psi_{2n+5},\ldots\,) .     

A pinching theorem for minimal hypersurfaces in a sphere

We study compact minimal hypersurfaces Mⁿ in Sⁿ⁺¹S^{n+1} with two distinct principal curvatures and prove that if the squared norm S of the second fundamental form of Mⁿ satisfies S \geqq nS \geqq n, then S o nS \equiv n and Mⁿ is a minimal Clifford torus.     

Reducibility of polynomials a0(x) + a1 (x)y + a2 (x) y2 modulo p

Let f=a₀(x)+a₁(x)y+a₂(x)y² ? \Bbb Z[x,y]f=a_0(x)+a_1(x)y+a_2(x)y^2\in {\Bbb Z}[x,y] be an absolutely irreducible polynomial of degree m in x. We show that the reduction f mod p will also be absolutely irreducible if p 3 c_m·H(f)^e_mp\ge c_m\cdot H(f)^{e_m} where H (f) is the height of f and e₁ = 4,e₂ = 6, e₃ = 6 [2/3]{2}\over{3} and e_m = 2 m for m S 4. We also show that the exponents e_m are best possible for m 1 3m\ne 3 if a plausible number theoretic conjecture is true.