A generalization of addition formulae

Let R denote the set of reals, J a real interval and X a real linear space. We determine the functions g : X J, M : J R and H : J ² R satisfying the equationg(x+M(g(x))y)=H(g(x),g(y)),under the assumptions that g is continuous on rays, M is continuous, and H is symmetric. As a consequence we obtain characterizations of some groups and semigroups.     

The intersection of the powers of the radical in Noetherian P.I. rings

LetR be a ring and J its radical. DefineJ ₁=∩Jⁿ, J₂=∩J ₁ ⁿ ,…,… J_k=∩J _k−1 ⁿ .... It is shown that in a ringR satisfying a polynomial identity and the ascending chain condition on ideals,J _k =0 for some appropriatek. The work of the first author was supported by an NSF grant at the University of Chicago. The work of the second author was supported by an NSF grant at the University of California, San Diego.     

Right radicals for right distributively generated near-rings

For right near-rings the left representation has always been considered the natural one. However, Hanna Neumann [6] constructed her right near-rings by writing the reduced free group on the left of the near-ring. In [2] and [8] Neumann's ideas are placed in a more general setting in the sense that right R-groups are used to define radical-like objects in the near-ring R. The right 0-radical ^r J ₀(R) and the right half radical ^r J _½(R) are introduced in [2] where it is shown that for distributively generated (d.g.) near-rings R with a multiplicative identity and satisfying the descending chain condition for left R-subgroups ^r J ₀(R) = J ₂(R), the 2-radical from left representation. In this article we introduce the right 2-radical, ^r J ₂(R) for d.g. near-rings and discuss some of its properties. In particular, we show that for all finite d.g. near-rings with identity J ₂(R) = ^r J ₂(R).     

More on b-injectors of sporadic groups +

A B-Injector in an arbitrary finite group G is defined as a maximal nilpo-tent subgroup of G, containing a subgroup A of G of maximal order satisfying class(A) ≥ 2. Among other results the B-Injectors of the sporadic groups J₁,J₂,J₄, M₂₄ are determined.     

Singular spherical maximal operators on a class of two step nilpotent lie groups

LetH ⁿ ≅ℝ²ⁿ ⋉ℝ be the Heisenberg group and letμ _t be the normalized surface measure for the sphere of radiust in ℝ²ⁿ . Consider the maximal function defined byM f=sup _t>0|f*μ _t |. We prove forn≥2 thatM defines an operator bounded onL ^p (H ⁿ ) provided thatp>2n/(2n−1). This improves an earlier result by Nevo and Thangavelu, and the range forL ^p boundedness is optimal. We also extend the result to a more general class of surfaces and to groups satisfying a nondegeneracy condition; these include the groups of Heisenberg type. The second author was supported in part by the National Science Foundation.     

<Emphasis Type="Italic">OD</Emphasis>-Characterization of the automorphism groups of <Emphasis Type="Italic">O</Emphasis><Stack><Subscript>10</Subscript><Superscript>±</Superscript></Stack>(2)

The prime graph of a finite group was introduced by Gruenberg and Kegel. The degree pattern of a finite group G associated to its prime graph was introduced in [1] and denoted by D(G). The group G is called k-fold OD-characterizable if there exist exactly k non-isomorphic groups H satisfying conditions (1) |G| = |H| and (2) D(G) = D(H). Moreover, a 1-fold OD-characterizable group is simply called an OD-characterizable group. Till now a lot of finite simple groups were shown to be OD-characterizable, and also some finite groups especially the automorphism groups of some finite simple groups were shown not being OD-characterizable but k-fold OD-characterizable for some k > 1. In the present paper, the authors continue this topic and show that the automorphism groups of orthogonal groups O ₁₀⁺(2) and O ₁₀⁻(2) are OD-characterizable.     

Projective Summands in Tensor Products of Simple Modules of Finite Dimensional Hopf Algebras

Abstract

Let H be a finite dimensional Hopf algebra over a field k. We show that H contains a unique maximal Hopf ideal J _w (H) contained in J(H), the Jacobson radical of H. We give various characterizations of J _w (H), for example J _w (H) = Ann _H ((H/J(H))^?n ) for all large enough n. The smallest positive integer n with this property is denoted by l _w (H). We prove that l _w (H) equals the smallest number n such that (H/J(H))^?n contains every projective indecomposable H/J _w (H)-module as a direct summand. This also equals the minimal n such that the tensor product of n suitable simple H-modules contains the projective cover of the trivial H/J _w (H)-module as a direct summand. We define projective homomorphisms between H-modules, which are used to obtain various reciprocity laws for tensor products of simple H-modules and their projective indecomposable direct summands. We also discuss some consequences of our general results in case H = kG is a group algebra of a finite group G and k is a field of characteristic p.     

Invariant curves and boundedness of solutions of a class of reversible systems

In this paper, the boundedness of all solutions for the following planar reversible system Ju ′ = ?H (u) + G (u) + h (t) (1) is discussed, where the function H (u) ∈ C²(?², ?⁺) is positive for u ≠ 0 and positively (q, p)‐quasihomogeneous of quasi‐degree pq, G ∈ C⁵ is bounded, h ∈ C⁶ is 2π ‐periodic and J is the standard symplectic matrix. (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

NOTE The Johnson Graphs Satisfy a Distance Extension Property

G has property if whenever F and H are connected graphs with and |H|=|F|+1, and and are isometric embeddings, then there is an isometric embedding such that . It is easy to construct an infinite graph with for all k, and holds in almost all finite graphs. Prior to this work, it was not known whether there exist any finite graphs with . We show that the Johnson graphs J(n,3) satisfy whenever , and that J(6,3) is the smallest graph satisfying . We also construct finite graphs satisfying and local versions of the extension axioms studied in connection with the Rado universal graph. Received June 9, 1998     

Upper Bounds for Neumann–Schatten Norms

Let H and H _aux be Hilbert spaces, H a nonnegative self-adjoint operator in H,,s>0 and J a bounded linear transformation from the Hilbert space D(H ^s/2) (equipped with the graph scalar product of H ^s/2) to H _aux. It is shown that the operator J(H+)^–t belongs to the Neumann–Schatten class of order p=2+2(u–t)/(t–s/2) provided s/2<t<u,t–s/2<u–t and J(H+)^–u is Hilbert–Schmidt operator. An upper bound for the pth order Neumann–Schatten norm of J(H+)^–t is derived. If J is a closed operator from D(H ^1/2) to H _aux and D(J)D(H) then there exists a unique self-adjoint operator H ^J in H such that D(H ^J )D(J) and ( . Conditions which are sufficient in order that the operator (H ^J +)^–1–(H+)^–1 is compact and conditions which are sufficient in order that the wave operators W ^±(H ^J ,H) exist and are complete are derived. Instead of (Jf,Jg)_aux also certain other perturbation terms, not by necessity nonnegative, are considered. The special case when H equals the operator (–) ^r in L ²(R ^d ) for any strictly positive real number and H ^J equals (–) ^r + for some suitably chosen measure is discussed in detail. In particular, new results on existence and completeness of the wave operators W ^±(–+,–) are obtained.     

On the homotopy of the stable mapping class group

By considering all surfaces and their mapping class groups at once, it is shown that the classifying space of the stable mapping class group after plus construction, BΓ_∞ ⁺, has the homotopy type of an infinite loop space. The main new tool is a generalized group completion theorem for simplicial categories. The first deloop of BΓ_∞ ⁺ coincides with that of Miller [M] induced by the pairs of pants multiplication. The classical representation of the mapping class group onto Siegel's modular group is shown to induce a map of infinite loop spaces from BΓ_∞ ⁺ to K-theory. It is then a direct consequence of a theorem by Charney and Cohen [CC] that there is a space Y such that BΓ_∞ ⁺≃Im J _(1/2)×Y, where Im J _(1/2) is the image of J localized away from the prime 2. Oblatum 23-X-1995 &19-XI-1996     

The quantum orbifold cohomology of weighted projective spaces

We calculate the small quantum orbifold cohomology of arbitrary weighted projective spaces. We generalize Givental’s heuristic argument, which relates small quantum cohomology to S ¹-equivariant Floer cohomology of loop space, to weighted projective spaces and use this to conjecture an explicit formula for the small J-function, a generating function for certain genus-zero Gromov–Witten invariants. We prove this conjecture using a method due to Bertram. This provides the first non-trivial example of a family of orbifolds of arbitrary dimension for which the small quantum orbifold cohomology is known. In addition we obtain formulas for the small J-functions of weighted projective complete intersections satisfying a combinatorial condition; this condition naturally singles out the class of orbifolds with terminal singularities.     

Uniform and natural existence proofs for Janko‚s sporadic groups J2 and J3

In this article both sporadic Janko-groups J₂ and J₃ are constructed from their common involution centralizer H @ 2¹⁺⁴ H \cong 2^{1+4} : A₅ within one single run of Michler‘s deterministic algorithm described in [11]. At the beginning we choose the -- in some sense -- most natural point to start from, and in the end we realize that Michler‘s algorithm does not necessary lead us to a simple group but sometimes to a covering group of a simple group.     

Association Schemes and Fusion Algebras (An Introduction)

We introduce the concept of fusion algebras at algebraic level, as a purely algebraic concept for the fusion algebras which appear in conformal field theory in mathematical physics. We first discuss the connection between fusion algebras at algebraic level and character algebras, a purely algebraic concept for Bose-Mesner algebras of association schemes. Through this correspondence, we establish the condition when the matrix S of a fusion algebra at algebraic level is unitary or symmetric. We construct integral fusion algebras at algebraic level, from association schemes, in particular from group association schemes, whose matrix S is unitary and symmetric. Finally, we consider whether the modular invariance property is satisfied or not, namely whether there exists a diagonal matrix T satisfying the condition (ST)³ = S ². We prove that this property does not hold for some integral fusion algebras at algebraic level coming from the group association scheme of certain groups of order 64, and we also prove that the (nonintegral) fusion algebra at algebraic level obtained from the Hamming association scheme H(d, q) has the modular invariance property.     

Krein H* - Triple Systems

In this paper, the authors initiate the study of a class of triple systems called Krein H^*-triple systems and show that a semisimple Krein H^*-triple system is an orthogonal direct sum of simple Krein H^*-triple systems. A separable Krein H_*-triple system satisfying the condition of regularity is the closure of an increasing union of regular Krein H^*-triple systems.This research is supported by the University Grants Commission of India.     

On the degree of commutativity of p ‐groups of maximal class

Let G be a p ‐group of maximal class of order p^m , p ≠ 2, and c (G) the degree of commutativity of G. Let c₀ be the nonnegative residue of c modulo p – 1. In this paper, by using only Lie algebra techniques, we prove that 2c ≥ m – 2p + c 0 + 1. Also, we give examples of Lie algebras satisfying the following equalities: In addition, there exist examples of p ‐groups of maximal class satisfying 2c = m – 2p + c₀ + 3 for each c₀ ∈ [2, p – 2] (see [6, Theorem 4.5]). (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Multiplicity Results for Second Order Nonlinear Problems with Maximum and Minimum

A functional differential equation of the type where F: C¹(J) → L₁(J) is a unbounded operator, is considered. Sufficient conditions for the existence of at least two different solutions satisfying boundary conditions min{x(t): t ? J} = α, max{x(t): t ? J} = β are given.