On circle diffeomorphisms with discontinuous derivatives and quasi-invariance subgroups of Malliavin-Shavgulidze measures

This note is devoted to construction of the domain of the regular representation of Diff₊(S¹). We consider a subgroup of diffeomorphism group such that the Malliavin-Shavgulidze measure is quasi-invariant with respect to the left action of the subgroup. The measure appears to be quasi-invariant with respect to the left action of diffeomorphisms with discontinuous second derivative. We derive an expression for quasi-invariance density and obtain an additional PSL(2,R) symmetry breaking term.     

Infrared Variables for the SU(3) Yang–Mills Field

We generalize the self-dual parameterization of the SU(2) Yang–Mills field proposed by Niemi and Faddeev for describing the infrared limit of the theory to the case of the gauge group SU(3). We demonstrate that the duality property intrinsic to the SU(2) gauge field cannot be transferred automatically to the higher-rank group case. We interpret the algebraic structures appearing in the Lagrangian for the new compact variables in terms of the group products SU(2)³.     

The Exceptional Compact Symmetric Spaces <Emphasis Type="Italic">G</Emphasis><Subscript>2</Subscript> and <Emphasis Type="Italic">G</Emphasis><Subscript>2</Subscript>/<Emphasis Type="Italic">SO</Emphasis>(4) as Tubes

In the present paper we discuss in detail the cohomogeneity one isometric actions of the Lie groups SU(3) × SU(3) and SU(3) on the exceptional compact symmetric spaces G₂ and G₂/SO(4), respectively. We show that the principal orbits coincide with the tubular hypersurfaces around the totally geodesic singular orbits, and the symmetric spaces G₂ and G₂/SO(4) can be thought of as compact tubes around SU(3) and P², respectively. Moreover, we determine the radii of these tubes and describe the shape operators of the principal orbits. Finally, we apply these results to compute the volumes of the two symmetric spaces.The author was partially supported by the Hungarian National Science and Research Foundation OTKA T032478.     

The Exceptional Compact Symmetric Spaces G 2 and G 2/ SO(4) as Tubes

In the present paper we discuss in detail the cohomogeneity one isometric actions of the Lie groups SU(3) × SU(3) and SU(3) on the exceptional compact symmetric spaces G₂ and G₂/SO(4), respectively. We show that the principal orbits coincide with the tubular hypersurfaces around the totally geodesic singular orbits, and the symmetric spaces G₂ and G₂/SO(4) can be thought of as compact tubes around SU(3) and P², respectively. Moreover, we determine the radii of these tubes and describe the shape operators of the principal orbits. Finally, we apply these results to compute the volumes of the two symmetric spaces.     

A new continuation method for complementarity problems with uniformP-functions

The complementarity problem with a nonlinear continuous mappingf from the nonnegative orthantR ₊ ⁿ ofR ⁿ intoR ⁿ can be written as the system of equationsF(x, y) = 0 and(x, y) R ₊ ²ⁿ , whereF denotes the mapping from the nonnegative orthantR ₊ ²ⁿ ofR ²ⁿ intoR ₊ ⁿ × Rⁿ defined byF(x, y) = (x ₁y₁,,x_ny_n, f₁(x) – y₁,, f_n(x) – y_n) for every(x, y) R ₊ ²ⁿ . Under the assumption thatf is a uniformP-function, this paper establishes that the mappingF is a homeomorphism ofR ₊ ²ⁿ ontoR ₊ ⁿ × Rⁿ. This result provides a theoretical basis for a new continuation method of tracing the solution curve of the one parameter family of systems of equationsF(x, y) = tF(x ⁰, y⁰) and(x, y) R ₊ ²ⁿ from an arbitrary initial point(x ⁰, y⁰) R ₊ ²ⁿ witht = 1 until the parametert attains 0. This approach is an extension of the one used in the polynomially bounded algorithm recently given by Kojima, Mizuno and Yoshise for solving linear complementarity problems with positive semi-definite matrices.     

Quasinormalität in topologischen Gruppen

LetQ be a subgroup of the locally compact groupG. Q is called a topologically quasinormal subgroup ofG, ifQ is closed and for each closed subgroupA ofG. We prove: If the compact elements ofG form a proper subgroup, compact topologically quasinormal subgroups ofG are subnormal of defect 2. IfG is connected, compact topologically quasinormal subgroups ofG are normal. IfG/G ₀ is compact, connected topologically quasinormal subgroups ofG are normal.     

Some remarks on subgroups determined by certain ideals in integral group rings

LetG be a group,ZG the integral group ring ofG andI(G) its augmentation ideal. Subgroups determined by certain ideals ofZG contained inI(G) are identified. For example, whenG=HK, whereH, K are normal subgroups ofG andH∩K⊆ζ(H), then the subgroups ofG determined byI(G)I(H)I(G), andI ³(G)I(H) are obtained. The subgroups of any groupG with normal subgroupH determined by (i)I ²(G)I(H)+I(G)I(H)I(G)+I(H)I²(G), whenH′⊆[H,G,G] and (ii)I(G)I(H)I(G) when degH ²(G/H′, T)≤1, are computed. the subgroup ofG determined byI ⁿ(G)+I(G)I(H) whenH is a normal subgroup ofG withG/H free Abelian is also obtained     

Covering times of random walks on bounded degree trees and other graphs

The motivating problem for this paper is to find the expected covering time of a random walk on a balanced binary tree withn vertices. Previous upper bounds for general graphs ofO(|V| |E|)⁽¹⁾ andO(|V| |E|/d _min)⁽²⁾ imply an upper bound ofO(n ²). We show an upper bound on general graphs ofO( |E| log |V|), which implies an upper bound ofO(n log² n). The previous lower bound was (|V| log |V|) for trees.⁽²⁾ In our main result, we show a lower bound of (|V| (log _{d max} |V|)²) for trees, which yields a lower bound of (n log² n). We also extend our techniques to show an upper bound for general graphs ofO(max{E T_i} log |V|).     

On Infinite Groups with Given Properties of the Norm of Infinite Subgroups

We investigate the relationship between the norm N _G() of infinite subgroups of an infinite group G and the structure of this group. We prove that N _G() is Abelian in the nonperiodic case, and a locally finite group is a finite extension of a quasicyclic subgroup if N _G() is a non-Dedekind group. In both cases, we describe the structure of the group G under the condition that the subgroup N _G() has finite index in G.     

The asymptotic σ-algebra of a recurrent random walk on a locally compact group

Let μ be a probability measure on a locally compact second countable groupG defining a recurrent (but not necessarily Harris) random walk. Denote byG ^∞ the space of paths and byB ^(a)the asymptotic σ-algebra. Let the starting measure be equivalent to the Haar measure and writeQ for the corresponding Markov measure onG ^∞. We prove thatL ^∞(G^∞, B^(a), Q) is in a canonical way isomorphic toL ^∞(G/N) whereN is the smallest closed normal subgroup ofG such that μ(zN)=1 for somez∈G. The groupG/N is either a finite cyclic group with generatorzN or a compact abelian group having the cyclic group as a dense subgroup. As a corollary we obtain that the set of all φ∈L ¹(G) such that coincides with the kernel of the canonical mapping ofL ¹(G)ontoL ¹(G/N). In particular, when μ is aperiodic, i.e.,G=N, then the random walk is mixing: for every φ∈L ¹(G) with ∝ φ=0.     

Centrally large subgroups of finite p-groups

Let S be a finite p-group. We say that an abelian subgroup A of S is a large abelian subgroup of S if |A||A^*| for every abelian subgroup A^* of S. We say that a subgroup Q of S is a centrally large subgroup, or CL-subgroup, of S if |Q||Z(Q)||Q^*||Z(Q^*)| for every subgroup Q^* of S. The study of large abelian subgroups and variations on them began in 1964 with Thompson's second normal p-complement theorem [J.G. Thompson, Normal p-complements for finite groups, J. Algebra 1 (1964) 43–46]. Centrally large subgroups possess some similar properties. In 1989, A. Chermak and A. Delgado [A. Chermak, A. Delgado, A measuring argument for finite groups, Proc. Amer. Math. Soc. 107 (1989) 907–914] studied several families of subgroups that include centrally large subgroups as a special case. In this paper, we extend their work to prove some further properties of centrally large subgroups. The proof uses an analogue for finite p-groups of an application of Borel's Fixed Point Theorem for algebraic groups.     

Classes caracteristiques de Γ(G,H)-structures et finitude de leur evaluation

LetG be a Lie group andH a closed subgroup ofG. We denote by (G,H) the groupoïd of germs of left translations ofG over the homogeneous spaceG/H.LetV be a compact manifold andx the universal characteristic class of dimensionk which belongs to the vector spaceH _cont ^k ((G, H)).The evaluation ofx over all the (G, H)-structures overV determines a subsetA _{(G, H)} (x, V) of the vector spaceH ^k (V;).We show that in some cases this set is finite.     

Isomorphisms ofp-adic group rings

SupposeP is the ring ofp-adic integers,G is a finite group of orderp ⁿ , andPG is the group ring ofG overP. IfV _p (G) denotes the elements ofPG with coefficient sum one which are of order a power ofp, it is shown that the elements of any subgroupH ofV _p (G) are linearly independent overP, and if in additionH is of orderp ⁿ , thenPGPH. As a consequence, the lattice of normal subgroups ofG and the abelianization of the normal subgroups ofG are determined byPG.     

The Atiyah-Segal completion theorem for C*-algebras

We generalize the Atiyah-Segal completion theorem to C ^*-algebras as follows. Let A be a C ^*-algebra with a continuous action of the compact Lie group G. If K _* ^G (A) is finitely generated as an R(G)-module, or under other suitable restrictions, then the I(G)-adic completion K _* ^G (A) is isomorphic to RK _*([A C(EG)]^G), where RK _* is representable K-theory for - C ^*-algebras and EG is a classifying space for G. As a corollary, we show that if and are homotopic actions of G, and if K _*(C ^* (G,A,)) and K _*(C ^* (G,A,)) are finitely generated, then K _*(C ^*(G,A,))K_*(C ^* (G,A,)). We give examples to show that this isomorphism fails without the completions. However, we prove that this isomorphism does hold without the completions if the homotopy is required to be norm continuous.This work was partially supported by an NSF Graduate Fellowship and by an NSF Postdoctoral Fellowship.     

A new type of C -numerical range arising in quantum computing

We consider a new type of numerical range motivated by recent applications in quantum computing. We term the object of interest local C -numerical rangeW_loc(C, A) of A. It is obtained by replacing the special unitary group in the definition of the C -numerical range by the so-called local subgroup of SU (2^N ), i.e. by the N -fold tensor product SU (2) ⊗ · · · ⊗ SU(2) of unitary (2 × 2)-matrices. First, it is shown that the local C -numerical range has rather unusual geometric properties compared to the ordinary one, e.g. it is in general neither star-shaped nor simply connected. Then two numerical algorithms, a Newton and a conjugate gradient method on the Lie group SU (2) ⊗ · · · ⊗ SU (2), are demonstrated to maximize the real part of W_loc(C, A) which also gives a Euclidean measure of the so-called pure-state entanglement in quantum computing. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the classification of elliptic surfaces withq=1

We classify, up to isomorphism, elliptic surfaces with irregularity one having exactly one singular fiber (necessarily of typeI ₆ ^* ). All of them turn out to be elliptic modular surfaces (Shioda [11]), so that the problem is indirectly equivalent to classifying certain subgroups ofSL ₂(Z). These surfaces are then used to produce examples of (elliptic) surfaces withq=1, anyp _g 1, which have maximal Picard number (see Persson [7] for the caseq=0). Finally, the classification yields some interesting relationships between hypergeometric functions, theta functions, and certain automorphic forms.Supported in part by NSF DMS-8501724     

The Higher K-Theory of Real Curves

If X is a smooth curve defined over the real numbers , we show that K _n (X) is the sum of a divisible group and a finite elementary Abelian 2-group when n 2. We determine the torsion subgroup of K _n (X), which is a finite sum of copies of and 2, only depending on the topological invariants of X() and X(), and show that (for n 2) these torsion subgroups are periodic of order 8.     

On the pronormality of subgroups of odd index in finite simple symplectic groups

A subgroup H of a group G is pronormal if the subgroups H and H ^g are conjugate in 〈H,H ^g 〉 for every g ∈ G. It was conjectured in [1] that a subgroup of a finite simple group having odd index is always pronormal. Recently the authors [2] verified this conjecture for all finite simple groups other than PSL _n (q), PSU _n (q), E ₆(q), ² E ₆(q), where in all cases q is odd and n is not a power of 2, and P Sp_2n (q), where q ≡ ±3 (mod 8). However in [3] the authors proved that when q ≡ ±3 (mod 8) and n ≡ 0 (mod 3), the simple symplectic group P Sp²ⁿ (q) has a nonpronormal subgroup of odd index, thereby refuted the conjecture on pronormality of subgroups of odd index in finite simple groups.The natural extension of this conjecture is the problem of classifying finite nonabelian simple groups in which every subgroup of odd index is pronormal. In this paper we continue to study this problem for the simple symplectic groups P Sp_2n (q) with q ≡ ±3 (mod 8) (if the last condition is not satisfied, then subgroups of odd index are pronormal). We prove that whenever n is not of the form 2 ^m or 2 ^m (2^2k +1), this group has a nonpronormal subgroup of odd index. If n = 2 ^m , then we show that all subgroups of P Sp_2n (q) of odd index are pronormal. The question of pronormality of subgroups of odd index in P Sp_2n (q) is still open when n = 2 ^m (2^2k + 1) and q ≡ ±3 (mod 8).