Fibonacci-Cayley Numbers and Repetition Patterns in Genomic DNA

Let F(k) denote the k-th Fibonacci number in the Fibonacci sequence F(0) := 0, F(1) := 1,..., F(k+1) := F(k-1)+F(k). Motivated by proposals regarding putative mechanisms that may be responsible for producing those often observed long repetitive patterns in genomic DNA, we study in this note the Fibonacci-Cayley index fc_x of positive integers x, i.e., the largest integer for which positive integers a, b with x = aF(k-1)+bF(k) exist and show that holds for the arithmetic mean of the indices of the smallest and the largest Fibonacci numbers occurring in the Zeckendorf decomposition AMS Subject Classification: 11B39, 11A99, 92D20.     

Generalized Covering Groups

Let v and be two varieties of groups defined by the sets of laws V and W, respectively. In this paper we construct a -v-covering group of a finite v-perfect group G, and show that every automorphism of G may be lifted to its -v-covering group. These generalize the previous work of the first two authors (1999). We also characterize the -v-irreducible extensions in some sense.AMS Mathematics Subject Classification (2000) Primary 20E34 20E36 Secondary 20E10     

On (2,n)-groups related to fibonacci groups

In this paper, we study certain groupsG generated by two elementsa andb of orders 2 andn respectively subject to one further defining relation, and determine their structure. We also point out certain connections between these groups and the Fibonacci groupsF(r, n).     

Bands of groups with universal properties

For a nonempty setX, a bandB, and a mapping :XB, we construct a band of groups, here called a cryptogroup,F(X,,B) which exhibits some remarkable properties. The first of these is a universal property relative to the classCG of all cryptogroups. In fact,CG is a variety with the operations of multiplication and inversion. For a varietyV of bands, we find a varietyV ⁰ of cryptogroups such that wheneverB is a band free inV ⁰ on the setX with embedding :XB, F(X,,B) is free inV ⁰. IfB is a normal band given as a strong semilattice of rectangular bands, we construct an isomorphic copy ofF(X,, B) which is a strong semilattice of completely simple semigroups. The objectsX, , B) admit the structure of a category, which is then related to the category of cryptogroups and their homomorphisms.This research was supported, in part by, NSERC Grant A4044.     

Parametrization of Triangle Groups as Subgroups of PSL(2, q)

In this paper we are interested in triangle groups (j, k, l) where j = 2 and k = 3. The groups (j, k, l) can be considered as factor groups of the modular group PSL(2, Z) which has the presentation x, y : x² = y³ = 1. Since PSL(2,q) is a factor group of G^k,l,m if -1 is a quadratic residue in the finite field F_q, it is therefore worthwhile to look at (j, k, l) groups as subgroups of PSL(2, q) or PGL(2, q). Specifically, we shall find a condition in form of a polynomial for the existence of groups (2, 3, k) as subgroups of PSL(2, q) or PGL(2, q).Mathematics Subject Classification: Primary 20F05 Secondary 20G40.     

Generic properties of the complementarity problem

Givenf: R ₊ ⁿ R ⁿ , the complementarity problem is to find a solution tox 0,f(x) 0, and x, f(x) = 0. Under the condition thatf is continuously differentiable, we prove that for a generic set of such anf, the problem has a discrete solution set. Also, under a set of generic nondegeneracy conditions and a condition that implies existence, we prove that the problem has an odd number of solutions.This work was partially supported by N.S.F. Grants GP-8007 and 010185.     

On a generalization of the Cauchy functional equation

Summary While looking for solutions of some functional equations and systems of functional equations introduced by S. Midura and their generalizations, we came across the problem of solving the equationg(ax + by) = Ag(x) + Bg(y) + L(x, y) (1) in the class of functions mapping a non-empty subsetP of a linear spaceX over a commutative fieldK, satisfying the conditionaP + bP P, into a linear spaceY over a commutative fieldF, whereL: X × X Y is biadditive,a, b K\{0}, andA, B F\{0}. Theorem.Suppose that K is either R or C, F is of characteristic zero, there exist A ₁,A ₂,B ₁,B ₂, F\ {0}with L(ax, y) = A ₁ L(x, y), L(x, ay) = A ₂ L(x, y), L(bx, y) = B ₁ L(x, y), and L(x, by) = B ₂ L(x, y) for x, y X, and P has a non-empty convex and algebraically open subset. Then the functional equation (1)has a solution in the class of functions g: P Y iff the following two conditions hold: L(x, y) = L(y, x) for x, y X, (2)if L 0, then A ₁ =A ₂,B ₁ =B ₂,A = A ₁ ² ,and B = B ₁ ² . (3) Furthermore, if conditions (2)and (3)are valid, then a function g: P Y satisfies the equation (1)iff there exist a y ₀ Y and an additive function h: X Y such that if A + B 1, then y ₀ = 0;h(ax) = Ah(x), h(bx) =Bh(x) for x X; g(x) = h(x) + y ₀ + 1/2A ₁ ^-1 B ₁ ^-1 L(x, x)for x P.     

On the isomorphisms and automorphism groups of circulants

Denote byC _n(S) the circulant graph (or digraph). LetM be a minimal generating element subset ofZ _n, the cyclic group of integers modulon, and In this paper, we discuss the problems about the automorphism group and isomorphisms ofC _n(S). When M S , we determine the automorphism group ofC _n(S) and prove that for any T if and only ifT = S, where is an integer relatively prime ton. The automorphism groups and isomorphisms of some other types of circulant graphs (or digraphs) are also considered. In the last section of this paper, we give a relation between the isomorphisms and the automorphism groups of circulants.     

Further results on the covering radii of the Reed-Muller codes

LetR(r, m) by therth order Reed-Muller code of length2 ^m , and let (r, m) be its covering radius. We obtain the following new results on the covering radius ofR(r, m): 1. (r+1,m+2) 2(r, m)+2 if 0rm–2. This improves the successive use of the known inequalities (r+1,m+2)2(r+1,m+1) and (r+1,m+1) (r, m).2.(2, 7)44. Previously best known upper bound for (2, 7) was 46. 3. The covering radius ofR(1,m) inR(m–1,m) is the same as the covering radius ofR(1,m) inR(m–2,m) form4.     

Bayesian priors based on a parameter transformation using the distribution function

One of the tasks of the Bayesian consulting statistician is to elicit prior information from his client who may be unfamiliar with parametric statistical models. In some cases it may be more illuminating to base a prior distribution for parameter on the transformed version F(/), where F is the data distribution function and v is a designated reference value, rather than on directly. This approach is outlined and explored in various directions to assess its implications. Some applications are given, including general linear regression and transformed linear models.     

Several identities in the Catalan triangle

In this paper, we first establish several identities for the alternating sums in the Catalan triangle whose (n, p) entry is defined by B _{n, p} = $ \tfrac{p} {n}\left( {_{n - p}^{2n} } \right) $ \tfrac{p} {n}\left( {_{n - p}^{2n} } \right) . Second, we show that the Catalan triangle matrix C can be factorized by C = FY = ZF, where F is the Fibonacci matrix. From these formulas, some interesting identities involving B _{n, p} and the Fibonacci numbers F _n are given. As special cases, some new relationships between the well-known Catalan numbers C _n and the Fibonacci numbers are obtained, for example:

On Asymptotic Dimension of Groups Acting on Trees

We prove the following.THEOREM. Let be the fundamental group of a finite graph of groups with finitely generated vertex groups G _v having asdim G _v n for all vertices v. Then asdim n+1.This gives the best possible estimate for the asymptotic dimension of an HNN extension and the amalgamated product.     

An example of pointwise non-convergence of iterated conditional expectation operators

Given ∈, we construct a sequence , … of Borel sub-sigma-algebras on the unit interval with the following property. Suppose the identity functionf(x)=x is transformed by successive conditioning on , then , then , Then the lim sup, with respect ton, will exceed (pointwise almost-everywhere) 1−∈ and its lim inf will be less than ∈. The sequence of functions also will fail to converge in the . This contrasts with the long-open conjecture that if all the come from a finite set of sigma-algebras, then the resulting sequence of functions must converge in . J. L. King was partially supported by NSF grant DMS-9112595.     

Williamson Matrices and a Conjecture of Ito's

We point out an interesting connection between Williamson matrices and relative difference sets in nonabelian groups. As a consequence, we are able to show that there are relative (4t, 2, 4t, 2t)-difference sets in the dicyclic groups Q _8t = a, b|a ^4t = b ⁴ = 1, a ^2t = b ², b ^-1ab = a^-1 for all t of the form t = 2^a · 10 ^b · 26 ^c · m with a, b, c 0, m 1\ (mod 2), whenever 2m-1 or 4m-1 is a prime power or there is a Williamson matrix over _m. This gives further support to an important conjecture of Ito IT5 which asserts that there are relative (4t, 2, 4t, 2t)-difference sets in Q _8t for every positive integer t. We also give simpler alternative constructions for relative (4t, 2, 4t, 2t)-difference sets in Q _8t for all t such that 2t - 1 or 4t - 1 is a prime power. Relative difference sets in Q _8t with these parameters had previously been obtained by Ito IT1. Finally, we verify Ito's conjecture for all t 46.     

The Geometry of Two Generator Groups: Hyperelliptic Handlebodies

A Kleinian group naturally stabilizes certain subdomains and closed subsets of the closure of hyperbolic three space and yields a number of different quotient surfaces and manifolds. Some of these quotients have conformal structures and others hyperbolic structures. For two generator free Fuchsian groups, the quotient three manifold is a genus two solid handlebody and its boundary is a hyperelliptic Riemann surface. The convex core is also a hyperelliptic Riemann surface. We find the Weierstrass points of both of these surfaces. We then generalize the notion of a hyperelliptic Riemann surface to a hyperelliptic three manifold. We show that the handlebody has a unique order two isometry fixing six unique geodesic line segments, which we call the Weierstrass lines of the handlebody. The Weierstrass lines are, of course, the analogue of the Weierstrass points on the boundary surface. Further, we show that the manifold is foliated by surfaces equidistant from the convex core, each fixed by the isometry of order two. The restriction of this involution to the equidistant surface fixes six generalized Weierstrass points on the surface. In addition, on each of these equidistant surfaces we find an orientation reversing involution that fixes curves through the generalized Weierstrass points.Mathematics Subject Classifications (2000). primary 30F10, 30F35, 30F40; secondary 14H30, 22E40.     

A Generalization of <Emphasis Type="Italic">T</Emphasis>-Groups

This paper identifies a certain class of locally supersoluble groups (called soluble hall-T groups) which contains the soluble T-groups as well as the nilpotent groups. The main result states that the product of a normal soluble hall-T subgroup and a subnormal locally supersoluble subgroup is always locally supersoluble.AMS Subject Classification (1991): 20E25, 20F16, 20F19     

On computing the number of subgroups of a finite Abelian group

We consider the numberN _A (r) of subgroups of orderp ^r ofA, whereA is a finite Abelianp-group of type =₁,₂,..., _l ()), i.e. the direct sum of cyclic groups of order ⁱⁱ. Formulas for computingN _A (r) are well known. Here we derive a recurrence relation forN _A (r), which enables us to prove a conjecture of P. E. Dyubyuk about congruences betweenN _A (r) and the Gaussian binomial coefficient .     

Product integration with the Clenshaw-Curtis points: Implementation and error estimates

Summary This paper is concerned with the practical implementation of a product-integration rule for approximating , wherek is integrable andf is continuous. The approximation is , where the weightsw _ni are such as to make the rule exact iff is any polynomial of degree n. A variety of numerical examples, fork(x) identically equal to 1 or of the form |–x| with >–1 and ||1, or of the form cosx or sinx, show that satisfactory rates of convergence are obtained for smooth functionsf, even ifk is very singular or highly oscillatory. Two error estimates are developed, and found to be generally safe yet quite accurate. In the special casek(x)1, for which the rule reduces to the Clenshaw-Curtis rule, the error estimates are found to compare very favourably with previous error estimates for the Clenshaw-Curtis rule.     

Finite Generalized Tetrahedron Groups with a High-Power Relator

An ordinary tetrahedron group is a group with a presentati on of the form

A Remark on the Rank Conjecture

We prove a result about the action of -operations on the homology of linear groups. We use this to give a sharper formulation of the rank conjecture as well as some shorter proofs of various known results. We formulate a conjecture about how the sharper formulation of the rank conjecture together with another conjecture could give rise to a different point of view on the isomorphism between and K_n^{(p)} (F)$ for an infinite field F, and we prove part of this new conjecture.