Articles and Preprints

On a Diophantine Equation That Generates All Integral Apollonian Gaskets

Jerzy Kocik — Mon, 07 May 2012 13:08:03 PDT

A remarkably simple Diophantine quadratic equation is known to generate all, Apollonian integral gaskets disk packings. A new derivation of this formula is presented here based on inversive geometry. Also, occurrence of Pythagorean triples in such gaskets is discussed.

State and Feedback Linearizations of Single-Input Control Systems

Issa Amadou Tall — Thu, 09 Sep 2010 11:17:52 PDT

In this paper we address the problem of state (resp. feedback) linearization of nonlinear single-input control systems using state (resp. feedback) coordinate transformations. Although necessary and sufficient geometric conditions have been provided in the early eighties, the problems of finding the state (resp. feedback) linearizing coordinates are subject to solving systems of partial differential equations. We will provide here a solution to the two problems by defining algorithms allowing to compute explicitly the linearizing state (resp. feedback) coordinates for any nonlinear control system that is indeed linearizable (resp. feedback linearizable). Each algorithm is performed using a maximum of $n-1$ steps ($n$ being the dimension of the system) and they are made possible by explicitly solving the Flow-box or straightening theorem. We illustrate with several examples borrowed from the literature.

Analytic Normal Forms and Symmetries of Strict Feedforward Control Systems

Issa Amadou Tall et al. — Thu, 09 Sep 2010 08:49:49 PDT

This paper deals with the problem of convergence of normal forms of control systems. We identify a $n$-dimensional subclass of control systems, called \emph{special strict feedforward form}, shortly (SSFF), possessing a normal form which is a smooth (resp. analytic) counterpart of the formal normal form of Kang. We provide a constructive algorithm and illustrate by several examples including the Kapitsa pendulum and the Cart-Pole system. The second part of the paper is concerned about symmetries of single-input control systems. We show that any symmetry of a smooth system in special strict feedforward form is conjugated to a \emph{scaling translation} and any 1-parameter family of symmetries is conjugated to a family of scaling translations along the first variable. We compute explicitly those symmetries by finding the conjugating diffeomorphism. We illustrate our results by computing the symmetries of the Cart-Pole system.

Feedback Linearizable Feedforward Systems: A Special Class

Issa Amadou Tall — Fri, 03 Sep 2010 10:48:43 PDT

The problem of feedback linearizability of systems in feedforward form is addressed and an algorithm providing explicit coordinates change and feedback given. At each step, the algorithm replaces the involutive conditions of feedback linearization by some, easily checkable. We also reconsider type II subclass of linearizable strict feedforward systems introduced by Krstic and we show that it constitutes the only linearizable among the class of quasilinear strict feedforward systems. Our results allow an easy computation of the linearizing coordinates and thus provide a stabilizing feedback controller for the original system among others. We illustrate by few examples including the VTOL.

Norm Euclidean Quaternionic Orders

Robert W. Fitzgerald — Thu, 08 Jul 2010 06:04:18 PDT

We determine the norm Euclidean orders in a positive definite quaternion algebra over Q.

Descent Construction for GSpin Groups: Main Results and Applications

Joseph Hundley et al. — Thu, 13 May 2010 06:05:47 PDT

The purpose of this note is to announce an extension of the descent method of Ginzburg, Rallis, and Soudry to the setting of essentially self dual representations. This extension of the descent construction provides a complement to recent work of Asgari and Shahidi [AS06] on the generic transfer for general Spin groups as well as to the work of Asgari and Raghuram [A-R] on cuspidality of the exterior square lift for representations of GL4. Complete proofs of the results announced in the present note will appear in our forthcoming article(s).

On k-minimum and m-minimum Edge-Magic Injections of Graphs

John P. McSorley et al. — Tue, 10 Nov 2009 16:23:19 PST

An edge-magic total labelling (EMTL) of a graph G with n vertices and e edges is an injection λ:V(G) ∪ E(G)→[n+e], where, for every edge uv ∈ E(G), we have wt_λ(uv)=k_λ, the magic sum of λ. An edge-magic injection (EMI) μ of G is an injection μ : V(G) ∪ E(G) → N with magic sum k_μ and largest label m_μ. For a graph G we define and study the two parameters κ(G): the smallest k_μ amongst all EMI’s μ of G, and m(G): the smallest m_μ amongst all EMI’s μ of G. We find κ(G) for G ∈ G for many classes of graphs G. We present algorithms which compute the parameters κ(G) and m(G). These algorithms use a G-sequence: a sequence of integers on the vertices of G whose sum on edges is distinct. We find these parameters for all G with up to 7 vertices. We introduce the concept of a double-witness: an EMI μ of G for which both k_μ=κ(G) and m_μ=m(G) ; and present an algorithm to find all double-witnesses for G. The deficiency of G, def(G), is m(G)−n−e. Two new graphs on 6 vertices with def(G)=1 are presented. A previously studied parameter of G is κ_EMTL(G), the magic strength of G: the smallest k_λ amongst all EMTL’s λ of G. We relate κ(G) to κ_EMTL(G) for various G, and find a class of graphs B for which κ_EMTL(G)−κ(G) is a constant multiple of n−4 for G ∈B. We specialise to G=K_n, and find both κ(K_n) and m(K_n) for all n≤11. We relate κ(K_n) and m(K_n) to known functions of n, and give lower bounds for κ(K_n) and m(K_n).

Sun's Conjectures on Fourth Powers in the Class Group of Binary Quadratic Forms

Robert W. Fitzgerald — Thu, 07 May 2009 12:59:43 PDT

We prove five of Sun's conjectures on the index of the subgroup of fourth powers in the class group of binary quadratic forms.

On Spin L-functions for GSO₁₀

David Ginzburg et al. — Thu, 12 Mar 2009 15:09:19 PDT

In this paper we construct a Rankin-Selberg integral which represents the Spin₁₀ X St L-function attached to the group GSO₁₀ X PGL₂. We use this integral representation to give some equivalent conditions for a generic cuspidal representation on GSO₁₀ to be a functorial lift from the group G₂ X PGL₂.

Descent Construction for GSpin Groups – Even Case

Joseph Hundley et al. — Sat, 07 Mar 2009 16:07:13 PST

In this paper we provide an extension of the theory of descent of Ginzburg-Rallis-Soudry to the context of “almost orthogonal” representations, that is representations τ with the property that the symmetric square L-function, twisted by some Hecke character ω has a pole. Our theory supplements the recent work of Asgari-Shahidi on the functorial lift from (split and quasisplit forms of) GSpin_2n to GL_2n.

Descent Construction for GSpin Groups – Odd Cuspidal Case

Joseph Hundley et al. — Sat, 07 Mar 2009 16:05:02 PST

In this paper we provide an extension of the theory of descent of Ginzburg-Rallis-Soudry to the context of “almost symplectic” representations, that is representations τ with the property that the exterior square L-function twisted by some Hecke character ω has a pole. Our theory supplements the recent work of Asgari-Shahidi on the functorial lift from GSpin_2n+1 groups to GL_2n.

The Adjoint L-function for GL₅

David Ginzburg et al. — Sat, 07 Mar 2009 16:01:52 PST

We describe two new Eulerian Rankin-Selberg integrals, using the same Eisenstein series defined on the group E₈, and cuspidal representations from GL₅ and GSpin₁₁, respectively. Connections with past work of Ginzburg, Bump-Ginzburg, Jiang-Rallis and others are described. We give some details of how to relate our two integrals via formal manipulations.

Multi-variable Rankin-Selberg Integrals for Orthogonal Groups

David Ginzburg et al. — Sat, 07 Mar 2009 15:55:36 PST

We show a relation between a certain period and the existence of a simple pole for the spin and standard partial L functions corresponding to a generic cuspidal representation defined on the group GSO₈(A). We also relate these two conditions with the functorial lift from the exceptional group G₂ to GSO₈. The main new ingridient is a new multivariable Rankin-Selberg integral which represents the above two L functions.

Descent Construction for GSpin Groups–Odd Case

Joseph Hundley et al. — Sat, 07 Mar 2009 15:53:26 PST

In this paper we provide an extension of the theory of descent of Ginzburg-Rallis-Soudry to the context of essentially symplectic representations, that is representations τ with the property that the exterior square L-function twisted by some Hecke character ω has a pole at s = 1. Our theory supplements the recent work of Asgari-Shahidi on the functorial lift from the general Spin groups GSpin_2n+1 to GL_2n.

The Adjoint L-function of SU_2,1

Joseph Hundley — Sat, 07 Mar 2009 15:45:14 PST

Spin L-functions for GSO₁₀ and GSO₁₂

Joseph Hundley — Sat, 07 Mar 2009 14:46:03 PST

Two multi-variable Rankin-Selberg integrals are studied. They may be regarded as extending the theory begun in [G-H1]. Each is shown to be Eulerian with the unramified contribution given explicitly in terms of partial Langlands L-functions.

A New Tower of Rankin-Selberg Integrals

David Ginzburg et al. — Sat, 07 Mar 2009 14:36:33 PST

We recall the notion of a tower of Rankin-Selberg integrals, and two known towers, making observations of how the integrals within a tower may be related to one another via formal manipulations, and offering a heuristic for how the L-functions should be related to one another when the integrals are related in this way. We then describe three new integrals in a tower on the group E₆, and find out which L-functions they represent. The heuristics also predict the existence of a fourth integral.

The Spin L-Function of Quasi-Split D₄

Wee Teck Gan et al. — Sat, 07 Mar 2009 14:29:25 PST

We construct a multivariable Rankin-Selberg integral for the Spin L-function of a globally generic cuspidal representation of an arbitrary quasi-split group of type D₄. This proves the meromorphic continuation of this L-function. When the quasi-split group of type D₄ is associated to a cubic field extention, this L-function cannot be analyzed by the Langlands-Shahidi method.

Counting Structures in the Möbius Ladder

John P. McSorley — Thu, 05 Feb 2009 07:25:51 PST

The Möbius ladder, M_n, is a simple cubic graph on 2n vertices. We present a technique which enables us to count exactly many different structures of M_n, and somewhat unifies counting in M_n. We also provide new combinatorial interpretations of some sequences, and ask some questions concerning extremal properties of cubic graphs.

Multivariate Matching Polynomials of Cyclically Labelled Graphs

John P. McSorley et al. — Thu, 05 Feb 2009 06:39:29 PST

We consider the matching polynomials of graphs whose edges have been cyclically labelled with the ordered set of t labels {x₁, . . ., x_t}.

We first work with the cyclically labelled path, with first edge label x_i, followed by N full cycles of labels {x₁, . . ., x_t}, and last edge label x_j. Let Φ_i,Nt+j denote the matching polynomial of this path. It satisfies the (τ, Δ)-recurrence: Φ_i,Nt+j = τΦ_i,(N−1)t+j−ΔΦ_i,(N−2)t+j, where τ is the sum of all non-consecutive cyclic monomials in the variables {x₁, . . ., x_t} and Δ = (−1)^t x₁ · · ·x_t. A combinatorial/algebraic proof and a matrix proof of this fact are given. Let G_N denote the first fundamental solution to the (τ, Δ)-recurrence. We express G_N (i) as a cyclic binomial using the Symmetric Representation of a matrix, (ii) in terms of Chebyshev polynomials of the second kind in the variables τ and Δ, and (iii) as a quotient of two matching polynomials. We extend our results from paths to cycles and rooted trees.