On the number of conjugacy classes of maximal subgroups in a finite soluble group

We show that for many formations \frak F\frak F, there exists an integer n = [`(m)](\frak F)n = \overline m(\frak F) such that every finite soluble group G not belonging to the class \frak F\frak F has at most n conjugacy classes of maximal subgroups belonging to the class \frak F\frak F. If \frak F\frak F is a local formation with formation function f, we bound [`(m)](\frak F)\overline m(\frak F) in terms of the [`(m)](f(p))(p ? \Bbb P )\overline m(f(p))(p \in \Bbb P ). In particular, we show that [`(m)](\frak N^k) = k+1\overline m(\frak N^k) = k+1 for every nonnegative integer k, where \frak N^k\frak N^k is the class of all finite groups of Fitting length ￡ k\le k.     

On $\frak {F}$-normal Fitting classes of finite soluble groups

Let \frak X, \frak F,\frak X\subseteqq \frak F\frak {X}, \frak {F},\frak {X}\subseteqq \frak {F}, be non-trivial Fitting classes of finite soluble groups such that G_{\frak X}G_{\frak {X}} is an \frak X\frak {X}-injector of G for all G ? \frak FG\in \frak {F}. Then \frak X\frak {X} is called \frak F\frak {F}-normal. If \frak F=\frak S_p\frak {F}=\frak {S}_{\pi }, it is known that (1) \frak X\frak {X} is \frak F\frak {F}-normal precisely when \frak X^*=\frak F^*\frak {X}^{\ast }=\frak {F}^{\ast }, and consequently (2) \frak F í \frak X\frak N\frak {F}\subseteq \frak {X}\frak {N} implies \frak X^*=\frak F^*\frak {X}^{\ast }=\frak {F}^{\ast }, and (3) there is a unique smallest \frak F\frak {F}-normal Fitting class. These assertions are not true in general. We show that there are Fitting classes \frak F\not = \frak S_p\frak {F}\not =\frak {S}_{\pi } filling property (1), whence the classes \frak S_p\frak {S}_{\pi } are not characterized by satisfying (1). Furthermore we prove that (2) holds true for all Fitting classes \frak F\frak {F} satisfying a certain extension property with respect to wreath products although there could be an \frak F\frak {F}-normal Fitting class outside the Lockett section of \frak F\frak {F}. Lastly, we show that for the important cases \frak F=\frak Nⁿ, n\geqq 2\frak {F}=\frak {N}^{n},\ n\geqq 2, and \frak F=\frak S_p1?\frak S_pr, p_i \frak {F}=\frak {S}_{p_{1}}\cdots \frak {S}_{p_{r}},\ p_{i} primes, there is a unique smallest \frak F\frak {F}-normal Fitting class, which we describe explicitly.     

On permutable subgroups of finite groups

Let \frak Z \frak Z be a complete set of Sylow subgroups of a finite group G, that is, for each prime p dividing the order of G, \frak Z \frak Z contains exactly one and only one Sylow p-subgroup of G. A subgroup H of a finite group G is said to be \frak Z \frak Z -permutable if H permutes with every member of \frak Z \frak Z . The purpose here is to study the influence of \frak Z \frak Z -permutability of some subgroups on the structure of finite groups. Some recent results are generalized.     

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\,) .     

On matrix near-rings

Let R be a right near-ring with identity and M_n(R) be the near-ring of n 2 n matrices over R in the sense of Meldrum and Van der Walt. In this paper, M_n(R) is said to be s\sigma-generated if every n 2 n matrix A over R can be expressed as a sum of elements of X_n(R), where X_n(R)={f_ij^r | 1\leqq i, j\leqq n, r ? R}X_n(R)=\{f_{ij}^r\,|\,1\leqq i, j\leqq n, r\in R\}, is the generating set of M_n(R). We say that R is s\sigma-generated if M_n(R) is s\sigma-generated for every natural number n. The class of s\sigma-generated near-rings contains distributively generated and abstract affine near-rings. It is shown that this class admits homomorphic images. For abelian near-rings R, we prove that the zerosymmetric part of R is a ring, so the class of zerosymmetric abelian s\sigma-generated near-rings coincides with the class of rings. Further, for every n, there is a bijection between the two-sided subgroups of R and those of M_n(R).     

On the singularities of residual intertwining operators

Let G be a reductive algebraic group defined over \Bbb Q {\Bbb Q} . Let P, P' be parabolic subgroups of G, defined over \Bbb Q {\Bbb Q} , and let _boxclose_boxclose, a_P') t \in W({\frak a}_{P}, {\frak a}_{P'}) . In this paper we study the intertwining operator M_P￠|P(t,l), l ? \frak a^*_{P,\Bbb C} M_{P' \vert P}(t,\lambda),\,\lambda \in {\frak a}^*_{P,{\Bbb C}} , acting in corresponding spaces of automorphic forms. One of the main results states that each matrix coefficient of M_P￠|P(t,l) M_{P' \vert P}(t,\lambda) is a meromorphic function of order ￡ n + 1 \le n + 1 , where n = dim G. Using this result, we further investigate the rank one intertwining operators, in particular, we study the distribution of their poles.     

A rigidity theorem for the pair ${\cal q}{\Bbb C} P^n$ (complex hyperquadric, complex projective space)

Given a compact Kähler manifold M of real dimension 2n, let P be either a compact complex hypersurface of M or a compact totally real submanifold of dimension n. Let q\cal q (resp. \Bbb R Pⁿ{\Bbb R} P^n) be the complex hyperquadric (resp. the totally geodesic real projective space) in the complex projective space \Bbb C Pⁿ{\Bbb C} P^n of constant holomorphic sectional curvature 4l \lambda . We prove that if the Ricci and some (n-1)-Ricci curvatures of M (and, when P is complex, the mean absolute curvature of P) are bounded from below by some special constants and volume (P) / volume (M) ￡\leq volume (q\cal q)/ volume (\Bbb C Pⁿ)({\Bbb C} P^n) (resp. ￡\leq volume (\Bbb R Pⁿ)({\Bbb R} P^n) / volume (\Bbb C Pⁿ)({\Bbb C} P^n)), then there is a holomorphic isometry between M and \Bbb C Pⁿ{\Bbb C} P^n taking P isometrically onto q\cal q (resp. \Bbb R Pⁿ{\Bbb R} P^n). We also classify the Kähler manifolds with boundary which are tubes of radius r around totally real and totally geodesic submanifolds of half dimension, have the holomorphic sectional and some (n-1)-Ricci curvatures bounded from below by those of the tube \Bbb R Pⁿ_r{\Bbb R} P^n_r of radius r around \Bbb R Pⁿ{\Bbb R} P^n in \Bbb C Pⁿ{\Bbb C} P^n and have the first Dirichlet eigenvalue not lower than that of \Bbb R Pⁿ_r{\Bbb R} P^n_r.     

Complex finitary simple Lie algebras

An algebra is called finitary if it consists of finite-rank transformations of a vector space. We classify finitary simple Lie algebras over an algebraically closed field of zero characteristic. It is shown that any such algebra is isomorphic to one of the following¶ (1) a special transvection algebra \frak t(V,P)\frak t(V,\mit\Pi );¶ (2) a finitary orthogonal algebra \frak fso (V,q)\frak {fso} (V,q); ¶ (3) a finitary symplectic algebra \frak fsp (V,s)\frak {fsp} (V,s).¶Here V is an infinite dimensional K-space; q (respectively, s) is a symmetric (respectively, skew-symmetric) nondegenerate bilinear form on V; and P\Pi is a subspace of the dual V^* whose annihilator in V is trivial: 0={v ? V | Pv=0}0=\{{v}\in V\mid \Pi {v}=0\}.     

Zyklische kubische monogene Erweiterungen der ganzalgebraischen Zahlen eines quadratischen Körpers

D'après [6] et [7] l'anneau des entiers du corps quadratique Q(?d), d \not = -3{\bf Q}(\sqrt {d}), d \not = -3, possède une extension cyclique cubique monogène (de discriminant 1) si, et seulement si, l'équation diophantienne¶¶ 4m³ = y²d + 274m^3 = y^2d + 27 a une solution avec d \not o 21d \not \equiv 21 (mod 36) et m \not o 3m \not \equiv 3 (mod 9).¶¶ On démontre ici que pour qu'une telle extension existe il faut que 3 divise h (d) et, lorsque d o 1d \equiv 1 (mod 8), d'où (2) = \frak p₁\frak p₂(2) = \frak p_1\frak p_2 où \frak p₁\frak p_1 et \frak p₂\frak p_2 sont deux idéaux premiers distincts de A_d, que la classe [\frak p₁][\frak p_1] de \frak p₁\frak p_1 dans le groupe de classes de Q(?d){\bf Q}(\sqrt {d}) ne soit pas un cube. Pour |d||d| < 100'000 cela élimine 68,37 % des valeurs restantes, les valeurs éliminées passent ainsi de 90 à 97 %.¶ De plus d ne doit pas être de la forme pq ou -3 pq pour lesquels le symbole d'Aigner T(p *q)T(p \star q) vale -1. L'article comporte aussi deux corrections, des résultats complétant [6] et [7], parus dans une thèse, et d'autres (en particulier l'indépendance des critères et des résultats numériques) parus ailleurs.     

Banach space properties forcing a reflexive, amenable Banach algebra to be trivial

It is an open problem whether an infinite-dimensional amenable Banach algebra exists whose underlying Banach space is reflexive. We give sufficient conditions for a reflexive, amenable Banach algebra to be finite-dimensional (and thus a finite direct sum of full matrix algebras). If \frak A {\frak A} is a reflexive, amenable Banach algebra such that for each maximal left ideal L of \frak A {\frak A} (i) the quotient \frak A/L {\frak A}/L has the approximation property and (ii) the canonical map from \frak A \check? L^{^} {\frak A} \check{\otimes} L^\perp to (\frak A / L) \check? L^{^} ({\frak A} / L) \check{\otimes} L^\perp is open, then \frak A {\frak A} is finite-dimensional. As an application, we show that, if \frak A {\frak A} is an amenable Banach algebra whose underlying Banach space is an \scr L^p {\scr L}^p -space with p ? (1,￥) p\in (1,\infty) such that for each maximal left ideal L the quotient \frak A/L {\frak A}/L has the approximation property, then \frak A {\frak A} is finite-dimensional.     

Quantization of Lie bialgebras and shuffle algebras of Lie algebras

To any field \Bbb K \Bbb K of characteristic zero, we associate a set (\mathbbK) (\mathbb{K}) and a group G₀(\Bbb K) {\cal G}_0(\Bbb K) . Elements of (\mathbbK) (\mathbb{K}) are equivalence classes of families of Lie polynomials subject to associativity relations. Elements of G₀(\Bbb K) {\cal G}_0(\Bbb K) are universal automorphisms of the adjoint representations of Lie bialgebras over \Bbb K \Bbb K . We construct a bijection between (\mathbbK)×G₀(\Bbb K) (\mathbb{K})\times{\cal G}_0(\Bbb K) and the set of quantization functors of Lie bialgebras over \Bbb K \Bbb K . This construction involves the following steps.? 1) To each element v \varpi of (\mathbbK) (\mathbb{K}) , we associate a functor \frak a?\operatornameSh^v(\frak a) \frak a\mapsto\operatorname{Sh}^\varpi(\frak a) from the category of Lie algebras to that of Hopf algebras; \operatornameSh^v(\frak a) \operatorname{Sh}^\varpi(\frak a) contains U\frak a U\frak a .? 2) When \frak a \frak a and \frak b \frak b are Lie algebras, and r_{\frak a\frak b} ? \frak a?\frak b r_{\frak a\frak b} \in\frak a\otimes\frak b , we construct an element ?^v (r_{\frak a\frak b}) {\cal R}^{\varpi} (r_{\frak a\frak b}) of \operatornameSh^v(\frak a)?\operatornameSh^v(\frak b) \operatorname{Sh}^\varpi(\frak a)\otimes\operatorname{Sh}^\varpi(\frak b) satisfying quasitriangularity identities; in particular, ?^v(r_{\frak a\frak b}) {\cal R}^\varpi(r_{\frak a\frak b}) defines a Hopf algebra morphism from \operatornameSh^v(\frak a)^* \operatorname{Sh}^\varpi(\frak a)^* to \operatornameSh^v(\frak b) \operatorname{Sh}^\varpi(\frak b) .? 3) When \frak a = \frak b \frak a = \frak b and r_\frak a ? \frak a?\frak a r_\frak a\in\frak a\otimes\frak a is a solution of CYBE, we construct a series r^v(r_\frak a) \rho^\varpi(r_\frak a) such that ?^v(r^v(r_\frak a)) {\cal R}^\varpi(\rho^\varpi(r_\frak a)) is a solution of QYBE. The expression of r^v(r_\frak a) \rho^\varpi(r_\frak a) in terms of r_\frak a r_\frak a involves Lie polynomials, and we show that this expression is unique at a universal level. This step relies on vanishing statements for cohomologies arising from universal algebras for the solutions of CYBE.? 4) We define the quantization of a Lie bialgebra \frak g \frak g as the image of the morphism defined by ?^v(r^v(r)) {\cal R}^\varpi(\rho^\varpi(r)) , where r ? \mathfrakg ?\mathfrakg^* r \in \mathfrak{g} \otimes \mathfrak{g}^* .<\P>     

A rigidity result for highest order local cohomology modules

Let M be a finitely generated faithful module over a noetherian ring R of dimension d < ￥ \infty and let \mathfrak a \subseteqq R {\mathfrak a} \subseteqq R be an ideal. We describe the (finite) set Supp_R(H_{\mathfrak a}^d (M)) = Ass_R(H_{\mathfrak a}^d (M)) \textrm{Supp}_R(H_{\mathfrak a}^d (M)) = \textrm{Ass}_R(H_{\mathfrak a}^d (M)) of primes associated to the highest local cohomology module H_{\mathfrak a}^d (M) H_{\mathfrak a}^d (M) in terms of the local formal behaviour of \mathfrak a {\mathfrak a} . If R is integral and of finite type over a field, Supp_R(H_{\mathfrak a}^d (M)) \textrm{Supp}_R(H_{\mathfrak a}^d (M)) is the set of those closed points of X = Spec(R) whose fibre under the normalization morphism n: X￠? X \nu : X' \rightarrow X contains points which are isolated in n^-1(Spec(R/\mathfrak a)) \nu^{-1}(\textrm{Spec}(R/{\mathfrak a})) .     

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.     

White noise eliminates instability

Let x₁,..., x_n be points in the d-dimensional Euclidean space E^d with || x_i-x_j|| ￡ 1\| x_{i}-x_{j}\| \le 1 for all 1 \leqq i,j \leqq n1 \leqq i,j \leqq n, where || .||\| .\| denotes the Euclidean norm. We ask for the maximum M(d,n) of \mathop?_i, j=1ⁿ|| x_i-x_j|| ²\textstyle\mathop\sum\limits _{i,\,j=1}^{n}\| x_{i}-x_{j}\| ^{2} (see [4]). This paper deals with the case d = 2. We calculate M(2, n) and show that the value M(2, n) is attained if and only if the points are distributed as evenly as possible among the vertices of a regular triangle of edge-length 1. Moreover we give an upper bound for the value \mathop?_i, j=1ⁿ|| x_i-x_j|| \textstyle\mathop\sum\limits _{i,\,j=1}^{n}\| x_{i}-x_{j}\| , where the points x₁,...,x_n are chosen under the same constraints as above.     

Efficient Testing of Large Graphs

Let P be a property of graphs. An e\epsilon -test for P is a randomized algorithm which, given the ability to make queries whether a desired pair of vertices of an input graph G with n vertices are adjacent or not, distinguishes, with high probability, between the case of G satisfying P and the case that it has to be modified by adding and removing more than en²\epsilon n^2 edges to make it satisfy P. The property P is called testable, if for every e\epsilon there exists an e\epsilon -test for P whose total number of queries is independent of the size of the input graph. Goldreich, Goldwasser and Ron [8] showed that certain individual graph properties, like k-colorability, admit an e\epsilon -test. In this paper we make a first step towards a complete logical characterization of all testable graph properties, and show that properties describable by a very general type of coloring problem are testable. We use this theorem to prove that first order graph properties not containing a quantifier alternation of type ``"$\forall \exists ' are always testable, while we show that some properties containing this alternation are not.     

On normed products of operator ideals which contain $ \frak L_2 $ as a factor

We investigate quasi-Banach operator ideal products (\frak A°\frak B,A°B) (\frak A\circ\frak B,\textbf{A}\circ \textbf{B}) which contain (\frak L₂, L₂) (\frak L_2, \textbf{L}_2) as a factor. In particular, we ask for conditions which guarantee that A°B \textbf{A}\circ \textbf{B} is even a norm if each factor of the product is a 1-Banach ideal. In doing so, we reveal the strong influence of the existence of such a norm in relation to the accessibility of the product ideal and the structure of its factors.     

Sharp bounds for the Bernoulli numbers

We determine the best possible real constants a\alpha and b\beta such that the inequalities [(2(2n)!)/((2p)²ⁿ)] [1/(1-2^a-2n)] \leqq |B_2n| \leqq [(2(2n)!)/((2p)²ⁿ)] [1/(1-2^b-2n)]{2(2n)! \over(2\pi)^{2n}} {1 \over 1-2^{\alpha -2n}} \leqq |B_{2n}| \leqq {2(2n)! \over (2\pi )^{2n}}\, {1 \over 1-2^{\beta -2n}}hold for all integers n\geqq 1n\geqq 1. Here, B₂, B₄, B₆,... are Bernoulli numbers.     

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.     

On the existence of certain measures appearing in distance geometry

Abstract. We prove the following result: Let X be a compact connected Hausdorff space and f be a continuous function on X x X. There exists some regular Borel probability measure m\mu on X such that the value of¶¶ ò\limit _X f(x,y)dm(y)\int\limit _X f(x,y)d\mu (y) is independent of the choice of x in X if and only if the following assertion holds: For each positive integer n and for all (not necessarily distinct) x₁,x₂,...,x_n,y₁,y₂,...,y_n in X, there exists an x in X such that¶¶ ?_i=1ⁿ f(x_i,x)=?_i=1ⁿ f(y_i,x).\sum\limits _{i=1}^n f(x_i,x)=\sum\limits _{i=1}^n f(y_i,x).     

On Ricci curvature of isotropic and Lagrangian submanifolds in complex space forms

Let M ⁿ be a Riemannian n-manifold. Denote by S(p) and [`(Ric)](p)\overline {Ric}(p) the Ricci tensor and the maximum Ricci curvature on M <sup>n</sup> at a point p ? M<sup>n</sup>p\in M^n, respectively. First we show that every isotropic submanifold of a complex space form [(M)\tilde]<sup>m</sup>(4 c)\widetilde M^m(4\,c) satisfies S ￡ ((n-1)c+ [(n<sup>2</sup>)/4] H<sup>2</sup>)gS\leq ((n-1)c+ {n^2 \over 4} H^2)g, where H<sup>2</sup> and g are the squared mean curvature function and the metric tensor on M <sup>n</sup>, respectively. The equality case of the above inequality holds identically if and only if either M <sup>n</sup> is totally geodesic submanifold or n = 2 and M <sup>n</sup> is a totally umbilical submanifold. Then we prove that if a Lagrangian submanifold of a complex space form [(M)\tilde]<sup>m</sup>(4 c)\widetilde M^m(4\,c) satisfies [`(Ric)] = (n-1)c+ [(n²)/4] H²\overline {Ric}= (n-1)c+ {n^2 \over 4} H^2 identically, then it is a minimal submanifold. Finally, we describe the geometry of Lagrangian submanifolds which satisfy the equality under the condition that the dimension of the kernel of second fundamental form is constant.