Articles and PreprintsCopyright (c) 2014 Southern Illinois University Carbondale All rights reserved.
http://opensiuc.lib.siu.edu/math_articles
Recent documents in Articles and Preprintsen-usTue, 29 Jul 2014 18:43:18 PDT3600Efficient Variable Step Size Approximations for Strong Solutions of Stochastic Differential Equations with Additive Noise and Time Singularity
http://opensiuc.lib.siu.edu/math_articles/112
http://opensiuc.lib.siu.edu/math_articles/112Thu, 10 Jul 2014 12:32:16 PDT
We consider stochastic differential equations with additive noise and conditions on the coefficients in those equations that allow a time singularity in the drift coefficient. Given a maximum step size, ℎ^{∗}, we specify variable (adaptive) step sizes relative to ℎ^{∗} which decrease as the time node points approach the singularity. We use an Euler-type numerical scheme to produce an approximate solution and estimate the error in the approximation. When the solution is restricted to a fixed closed time interval excluding the singularity, we obtain a global pointwise error of order 𝑂(ℎ^{∗}). An order of error 𝑂(ℎ^{∗𝑝}) for any 𝑝 < 1 is obtained when the approximation is run up to a time within ℎ^{∗𝑞} of the singularity for an appropriate choice of exponent 𝑞. We apply this scheme to Brownian bridge, which is defined as the nonanticipating solution of a stochastic differential equation of the type under consideration. In this special case, we show that the global pointwise error is of order 𝑂(ℎ^{∗}), independent of how close to the singularity the approximation is considered.
]]>
Harry Randolph Hughes et al.On a Diophantine Equation That Generates All Integral Apollonian Gaskets
http://opensiuc.lib.siu.edu/math_articles/111
http://opensiuc.lib.siu.edu/math_articles/111Mon, 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.
]]>
Jerzy KocikState and Feedback Linearizations of Single-Input Control Systems
http://opensiuc.lib.siu.edu/math_articles/110
http://opensiuc.lib.siu.edu/math_articles/110Thu, 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.
]]>
Issa Amadou TallAnalytic Normal Forms and Symmetries of Strict Feedforward Control Systems
http://opensiuc.lib.siu.edu/math_articles/109
http://opensiuc.lib.siu.edu/math_articles/109Thu, 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.
]]>
Issa Amadou Tall et al.Feedback Linearizable Feedforward Systems: A Special Class
http://opensiuc.lib.siu.edu/math_articles/108
http://opensiuc.lib.siu.edu/math_articles/108Fri, 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.
]]>
Issa Amadou TallNorm Euclidean Quaternionic Orders
http://opensiuc.lib.siu.edu/math_articles/107
http://opensiuc.lib.siu.edu/math_articles/107Thu, 08 Jul 2010 06:04:18 PDT
We determine the norm Euclidean orders in a positive definite quaternion algebra over Q.
]]>
Robert W. FitzgeraldDescent Construction for GSpin Groups: Main Results and Applications
http://opensiuc.lib.siu.edu/math_articles/106
http://opensiuc.lib.siu.edu/math_articles/106Thu, 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).
]]>
Joseph Hundley et al.On <em>k</em>-minimum and <em>m</em>-minimum Edge-Magic Injections of Graphs
http://opensiuc.lib.siu.edu/math_articles/105
http://opensiuc.lib.siu.edu/math_articles/105Tue, 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}).
]]>
John P. McSorley et al.Sun's Conjectures on Fourth Powers in the Class Group of Binary Quadratic Forms
http://opensiuc.lib.siu.edu/math_articles/104
http://opensiuc.lib.siu.edu/math_articles/104Thu, 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.
]]>
Robert W. FitzgeraldOn Spin <em>L</em>-functions for <em>GSO</em><sub>10</sub>
http://opensiuc.lib.siu.edu/math_articles/103
http://opensiuc.lib.siu.edu/math_articles/103Thu, 12 Mar 2009 15:09:19 PDT
In this paper we construct a Rankin-Selberg integral which represents the Spin_{10} X StL-function attached to the group GSO_{10} X PGL_{2}. We use this integral representation to give some equivalent conditions for a generic cuspidal representation on GSO_{10} to be a functorial lift from the group G_{2} X PGL_{2}.
]]>
David Ginzburg et al.Descent Construction for <em>GSpin</em> Groups – Even Case
http://opensiuc.lib.siu.edu/math_articles/102
http://opensiuc.lib.siu.edu/math_articles/102Sat, 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}.
]]>
Joseph Hundley et al.Descent Construction for <em>GSpin</em> Groups – Odd Cuspidal Case
http://opensiuc.lib.siu.edu/math_articles/101
http://opensiuc.lib.siu.edu/math_articles/101Sat, 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}.
]]>
Joseph Hundley et al.The Adjoint <em>L</em>-function for <em>GL</em><sub>5</sub>
http://opensiuc.lib.siu.edu/math_articles/100
http://opensiuc.lib.siu.edu/math_articles/100Sat, 07 Mar 2009 16:01:52 PST
We describe two new Eulerian Rankin-Selberg integrals, using the same Eisenstein series defined on the group E_{8}, and cuspidal representations from GL_{5} and GSpin_{11}, 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.
]]>
David Ginzburg et al.Multi-variable Rankin-Selberg Integrals for Orthogonal Groups
http://opensiuc.lib.siu.edu/math_articles/99
http://opensiuc.lib.siu.edu/math_articles/99Sat, 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 GSO_{8}(A). We also relate these two conditions with the functorial lift from the exceptional group G_{2} to GSO_{8}. The main new ingridient is a new multivariable Rankin-Selberg integral which represents the above two L functions.
]]>
David Ginzburg et al.Descent Construction for <em>GSpin</em> Groups–Odd Case
http://opensiuc.lib.siu.edu/math_articles/98
http://opensiuc.lib.siu.edu/math_articles/98Sat, 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}.
]]>
Joseph Hundley et al.The Adjoint <em>L</em>-function of <em>SU</em><sub>2,1</sub>
http://opensiuc.lib.siu.edu/math_articles/97
http://opensiuc.lib.siu.edu/math_articles/97Sat, 07 Mar 2009 15:45:14 PSTJoseph HundleySpin L-functions for <em>GSO</em><sub>10</sub> and <em>GSO</em><sub>12</sub>
http://opensiuc.lib.siu.edu/math_articles/96
http://opensiuc.lib.siu.edu/math_articles/96Sat, 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.
]]>
Joseph HundleyA New Tower of Rankin-Selberg Integrals
http://opensiuc.lib.siu.edu/math_articles/95
http://opensiuc.lib.siu.edu/math_articles/95Sat, 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 E_{6}, and find out which L-functions they represent. The heuristics also predict the existence of a fourth integral.
]]>
David Ginzburg et al.The Spin <em>L</em>-Function of Quasi-Split <em>D</em><sub>4</sub>
http://opensiuc.lib.siu.edu/math_articles/94
http://opensiuc.lib.siu.edu/math_articles/94Sat, 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 D_{4}. This proves the meromorphic continuation of this L-function. When the quasi-split group of type D_{4} is associated to a cubic field extention, this L-function cannot be analyzed by the Langlands-Shahidi method.
]]>
Wee Teck Gan et al.Counting Structures in the Möbius Ladder
http://opensiuc.lib.siu.edu/math_articles/93
http://opensiuc.lib.siu.edu/math_articles/93Thu, 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.
]]>
John P. McSorley