Transposition generation of alternating permutations

A permutation _{1 2}... _n is alternating if ₁< ₂> ₃< ₄.... Alternating permutations are counted by the Euler numbers. Here we show that alternating permutations can be listed so that successive permutations differ by a transposition, ifn is odd. Extensions and open problems are mentioned.Research supported by the Natural Sciences and Engineering Research Council of Canada under grant A3379.     

Generation of alternating groups by pairs of conjugates

Considering the conjugacy classes of the alternating group of degreen, those classes that contain a pair of generators are in the majority. In fact, the proportion of such classes is 1 –(n), and(n) 0 asn .     

Generating alternating permutations lexicographically

A permutation _{1 2} ... _n is alternating if ₁<₂>₃<₄ .... We present a constant average-time algorithm for generating all alternating permutations in lexicographic order. Ranking and unranking algorithms are also derived.Research supported by the Natural Sciences and Engineering Research Council of Canada under grant A3379.     

The <Emphasis Type="Italic">K</Emphasis>(π,1)-conjecture for the affine braid groups

The complement of the hyperplane arrangement associated to the (complexified) action of a finite, real reflection group on ⁿ is known to be a K(,1) space for the corresponding Artin group $\Cal A$. A long-standing conjecture states that an analogous statement should hold for infinite reflection groups. In this paper we consider the case of a Euclidean reflection group of type à _n and its associated Artin group, the affine braid group $\tilde{\Cal A}$. Using the fact that $\tilde{\Cal A}$ can be embedded as a subgroup of a finite type Artin group, we prove a number of conjectures about this group. In particular, we construct a finite, $n$-dimensional K(,1)-space for $\tilde{\Cal A}$, and use it to prove the K(,1) conjecture for the associated hyperlane complement. In addition, we show that the affine braid groups are biautomatic and give an explicit biautomatic structure.     

A characterization of the generalized twisted field planes of characteristic ≥5

Let be a non-Desarguesian semifield plane of orderp ⁿ, p a prime number 5 andn3, and let denote the group induced by the autotopism groupG of on the line at infinity. We prove that is a generalized twisted field plane if, and only if, has an element of order (p ^k–1)((p ⁿ–1)/(p ^m–1)), for some integersk andm, wherek | m, m | n, andm.This work was supported in part by NSF grants RII-9014056, component IV of the EPSCoR of Puerto Rico grant and ARO grant for Cornell MSI     

Teichmüller curves in moduli space,Eisenstein series and an application to triangular billiards

Summary There exists a Teichmüller disc _n containing the Riemann surface ofy ²+x ⁿ =1, in the genus [n–1/2] Teichmüller space, such that the stabilizer of _n in the mapping class group has a fundamental domain of finite (Poincaré) volume in _n . Application is given to an asymptotic formula for the length spectrum of the billiard in isosceles triangles with angles (/n, /n,n–2/n) and to the uniform distribution of infinite billiard trajectories in the same triangles.

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Research supported by NSF-DMS-8521620     

Classes caractéristiques d''une π-algèbre et suite spectrale en K-théorie bivariante

Résumé Pour tout groupe discret et pour toute -algèbre D, la C ^*-algèbre D(E) (dont la définition exacte est donnée dans la section 4) est la version équivariante de la C ^*-algèbre C(B, D) des fonctions continues sur B, le classifiant du groupe, à valeurs dans D et qui s'annulent à l'infini. Si D désigne une autre -algèbre, nous définissons une suite spectrale en K-théorie bivariante dont les premiers termes sont donnés par les groupes H ^p (B, KK(D, D)) et qui converge (lorsque B est de dimension finie) vers KK(B; D(E), D(E)). Cette suite spectrale généralise celle de Kasparov mais est obtenue de manière différente: en étendant la définition des quasihomomorphismes aux C(X)-algèbres (X est une espace topologique localement compact), on a recours à des méthodes homotopiques telles les décompositions de Postnikov et le calcul des groupes d'homotopie des espaces d'équivalences d'homotopie. Sous certaines hypothèses, ces mÊmes constructions nous permettent de définir, pour toute -algèbre D, une obstruction, appelée classe secondaire de la -algèbre D, qui détermine la différentielle d ₂ de la suite spectrale de Kasparov.

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For all discrete group and all -algebra D, the C ⁺-algebra D(E) (whose exact definition is given in Section 4) is the equivariant version of the C ^*-algebra C(B, D) of continuous functions from B (the classifiant of the group) to D, vanishing at infinity. If D is another -algebra, we define a spectral sequence in bivariant K-theory whose first terms are given by the groups H ^p (B, KK(D, D)) and which converges (if B of finite dimension) to KK(B; D(E), D(E)). This spectral sequence generalises the spectral sequence given by Kasparov but it is obtained in a quite different way: by extending the definition of quasihomomorphisms to the C(X)-algebras (where X is a locally compact topological space), we use homotopical methods, like Postnikov decompositions and the calculus of homotopy groups of spaces of homotopy equivalences. Furthermore, under certain hypotheses, with these constructions, we define an obstruction, called the secondary class of the -algebra D, which determines the differential d ₂ of the Kasparov spectral sequence.

     

A conjecture on semi-field planes. II

Conclusion The results in the previous sections lend strong support to the conjecture made in the Introduction. Furthermore, if the long-standing conjecture concerning the solvability of autotopism groups for semi-field planes is correct then the probability of our conjecture being true is greatly increased. In any case the existence of a semi-field plane for which u() = 2, 3, or 4 would provide a counterexample to the earlier conjecture.There are examples of semi-field planes with u() = %. As mentioned in Example 2 of Section 3, one of the semi-field planes of order 16 has u()-5. For that plane, the five orbits of the autotopism group G in (G) have lengths 27, 36, 54, 54, 54. The union of the orbit having length 36 and one of those having 54 is the union of the points in (G) on 6 lines through a vertex U and the union of the remaining three orbits consists of the 135 points on the remaining 9 lines through U. There are also non-Desarguesian A-planes in which u() = 5; the semi-field plane of order 3⁴ coordinatized by the twisted field of Albert has u() = 5.Supported in part by NSF Grants No. MPS 75-05260 and MPS 76-06661     

On depth first search strees inm-out digraphs

We consider depth first search (DFS for short) trees in a class of random digraphs: am-out model. Let _i be thei ^th vertex encountered by DFS andL(i, m, n) be the height of _i in the corresponding DFS tree. We show that ifi/n asn, then there exists a constanta(,m), to be defined later, such thatL(i, m, n)/n converges in probability toa(,m) asn. We also obtain results concerning the number of vertices and the number of leaves in a DFS tree.     

High-Energy Relations Between Elastic Scattering Amplitudes of Particles and Antiparticles

We find conditions under which the ratio of particle and antiparticle total scattering cross sections as s. If the forward elastic scattering amplitudes become purely real asymptotically, then their ratio tends to –1. We prove that /_–(s)1 for ₊(s)/m ², where m is the -meson mass. We show that the asymptotic relations obtained have finite-energy analogues for some processes.     

ApproximatingL 2-invariants by their finite-dimensional analogues

LetX be a finite connectedCW-complex. Suppose that its fundamental group is residually finite, i.e. there is a nested sequence ... _m + 1 _m ... of in normal subgroups of finite index whose intersection is trivial. Then we show that thep-thL ²-Betti number ofX is the limit of the sequenceb _p(X_m)/[: _m ] whereb _p(X_m) is the (ordinary)p-th Betti number of the finite covering ofX associated with _m .     

Local Recognition Of Non-Incident Point-Hyperplane Graphs

Let be a projective space. By H() we denote the graph whose vertices are the non-incident point-hyperplane pairs of , two vertices (p,H) and (q,I) being adjacent if and only if p I and q H. In this paper we give a characterization of the graph H() (as well as of some related graphs) by its local structure. We apply this result by two characterizations of groups G with PSL _n ( )GPGL _n ( ), by properties of centralizers of some (generalized) reflections. Here is the (skew) field of coordinates of .     

The steinitz lemma inl inf∞ sup2

Let (a, b) be a pair of non-negative numbers such that (1)a, b1 and (2)a+b3. Letu ₁,...,u _n be a sequence of vectors from the set {(x, y)R ²: |x|, |y|1}, withu ₁+...+u _n =0. It is shown that there is a permutation of indices such that all partial sumsu ₍₁₎+...+u _(k) lie in the rectangle |x|a, |y|b. Conditions (1) and (2) are also necessary.     

Finite presentability of Steinberg groups over group rings

LetA be a finitely generated commutative -algebra with Krull dimensiond, and let be an arbitrary finite group. It is proved that the Steinberg groupSt _n (A) is finitely presented whenevern4. If, in addition,nd+3, andK ₁ (A) andK ₂ (A) are finitely generated, thenE _n (A) andGL _n (A) are finitely presented.The Project supported by National Natural Science Foundation of China.     

Collineation groups whose order is divisible by a p-primitive divisor

Summary Let be a translation plane of order q³ with kernel GF(q). Our main result is that the translation complement of cannot contain a group G such that G/Z(G)=A₇. This removes a possible exception to the results in our paper Collineation groups irreducible on the components of a translation plane.We also show that the assumptions of the above paper can be relaxed slightly.Both authors supported in part by NSF Grant No. MCS76-06661 A01     

The co-existence of elations and homologies of prime order

N. Krier and J. C. D. S. Yaqub have proved that if a projective plane admits an involutory homology and an involutory elation, then does not belong to the Lenz-Barlotti class I₁, and belongs to the class I₂. In this paper, we find the classification of projective planes having a homology of orderp and an elation of orderq, wherep andq are primes.This is based on a part of the doctoral dissertation of A. Solai Raju. The work was supported by a Senior Research Fellowship of the CSIR, India.     

Isolated points in duals of certain locally compact groups

Summary LetG be a separable locally compact group with dual space. consists of all equivalence classes of irreducible unitary representations ofG, and is endowed with the Fell-topology. We study the topological properties in of the square-integrable representations ofG. [ is square-integrable provided there is a coordinate functiong((g)v, v),gG, for which is inL ²(G) w.r.t. left Haar measure onG.]SupposeG contains an open normal subgroupN of the formeKN ⁿ e whereK is compact. (All groups with a compact invariant neighborhood of the identity, [IN] groups, satisfy this condition.) In this case we show that if is square-integrable then {} is an open point of.Finally, our techniques are used to prove this result for arbitrary (non connected) nilpotent Lie groups.     

The random-cluster model on the complete graph

Summary The random-cluster model of Fortuin and Kasteleyn contains as special cases the percolation, Ising, and Potts models of statistical physics. When the underlying graph is the complete graph onn vertices, then the associated processes are called mean-field. In this study of the mean-field random-cluster model with parametersp=/n andq, we show that its properties for any value ofq(0, ) may be derived from those of an Erds-Rényi random graph. In this way we calculate the critical point _c (q) of the model, and show that the associated phase transition is continuous if and only ifq2. Exact formulae are given for _C (q), the density of the largest component, the density of edges of the model, and the free energy. This work generalizes earlier results valid for the Potts model, whereq is an integer satisfyingq2. Equivalent results are obtained for a fixed edge-number random-cluster model. As a consequence of the results of this paper, one obtains large-deviation theorems for the number of components in the classical random-graph models (whereq=1).     

Joint similarity problems and the generation of operator algebras with bounded length

There is a pair of commuting operators (T ₁,T ₂) on Hilbert space such that eachT ₁ andT ₂ is similar to a contraction but the pair (T ₁,T ₂) is not similar to a pair of contractions. There is a pair of commuting unitarizable representations (₁,₂) on the free group withN2 generators such that (₁,₂) is not similar to a pair of unitary representations. In connection with these examples, we introduce and study a notion of length for aC ^*-algebra (or an operator algebra) generated by two subalgebras, which is analogous to the minimum length of a word in the generators of a group.Partially supported by the N.S.F.     

An algebraic model of fibration with the fiberK(π,n)-space

For a fibration with the fiberK(,n)-space, the algebraic model as a twisted tensor product of chains of the base with standard chains ofK(,n)-complex is given which preserves multiplicative structure as well. In terms of this model the action of then-cohomology of the base with coefficients in on the homology of fibration is described.