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1988

Volume 19, Issue 6, pp. 1259-1493


Iterates of Maps with Symmetry

Pascal Chossat and Martin Golubitsky

SIAM J. Math. Anal. 19, pp. 1259-1270 (12 pages) | Cited 15 times

Online Publication Date: July 17, 2006

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In this paper the elementary aspects of bifurcation of fixed points, period doubling, and Hopf bifurcation for iterates of equivariant mappings are discussed. The most interesting of these is an algebraic formulation of the hypotheses of Ruelle’s theorem (D. Ruelle [1973], “Bifurcations in the presence of a symmetry group,” Arch. Rational Mech. Anal., 51, pp. 136–152) on Hopf bifurcation in the presence of symmetry.
In the last sections this result is used to show that Hopf bifurcation from standing waves in a system of ordinary differential equations with $O(2)$ symmetry can lead directly to motion on an invariant $3$-torus; indeed, depending on the exact symmetry of the standing waves, one might expect to see three invariant $3$-tori emanating from such a bifurcation. The unexpected third frequency comes from drift along the torus of standing waves whose existence is forced by the $O(2)$ symmetry.

Heteroclinic Orbits and Chaotic Dynamics in Planar Fluid Flows

Andrea Louise Bertozzi

SIAM J. Math. Anal. 19, pp. 1271-1294 (24 pages) | Cited 11 times

Online Publication Date: July 17, 2006

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An extension of the planar Smale–Birkhoff homoclinic theorem to the case of a heteroclinic saddle connection containing a finite number of fixed points is presented. This extension is used to find chaotic dynamics present in certain time-periodic perturbations of planar fluid models. Specifically, the Kelvin–Stuart cat’s eye flow is studied, a model for a vortex pattern found in shear layers. A flow on the two-torus with Hamiltonian $H_0 = (2\pi )^{ - 1} \sin (2\pi x_1 )\cos (2\pi x_2 )$ is studied, as well as the evolution equations for an elliptical vortex in a three-dimensional strain flow.

Bifurcations of Relative Equilibria in the $N$-Body and Kirchhoff Problems

Kenneth R. Meyer and Dieter S. Schmidt

SIAM J. Math. Anal. 19, pp. 1295-1313 (19 pages) | Cited 7 times

Online Publication Date: July 17, 2006

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The bifurcations of a one-parameter family of relative equilibria in the $N$-body problem are studied using normal form theory, Lie transforms, and an algebraic processor. The one-parameter family consists of $N - 1$ bodies of mass 1 at the vertices of a regular polygon and one body of mass $m$ at the centroid. As $N$ increases there are more and more values of the mass parameter $m$ where the relative equilibrium is degenerate. For $N \leqq 13$ each of these degenerates gives rise to a bifurcation and a new relative equilibrium. This is established using a computer-aided proof. A similar analysis is carried out for the $N$-vortex problem of Kirchhofl.

An Existence Result for the Electropainting Problem

Pierluigi Colli and Luc Oswald

SIAM J. Math. Anal. 19, pp. 1314-1323 (10 pages)

Online Publication Date: July 17, 2006

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A time-dependent family of harmonic problems with boundary condition $h({{\partial \varphi } /{\partial \nu }}) = \varphi $, where $h$ is a function dependent on the history of $\varphi $, models an electropaint process. It is proven that the problem has a weak solution $\{ \varphi (x,t),h(x,t)\} $ and $\lim _{t \to \infty } \varphi (x,t)$ exists and coincides with the solution of an appropriate Signorini problem.

Exact Estimates for Potentials

Martin Schechter

SIAM J. Math. Anal. 19, pp. 1324-1328 (5 pages) | Cited 1 time

Online Publication Date: July 17, 2006

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For $\lambda \geqq 0$ it is determined precisely which functions $V(x) \geqq 0$ on ${\bf R}^n $ satisfy an inequality of the form \[ (Vu,u) \leqq C_\lambda (V)\left( {\| {\nabla u} \|^2 + \lambda ^2 \| u \|^2 } \right),\quad u \in C_0^\infty \] for some constant $C_\lambda (V)$. The value of the smallest such constant is found. Inequalities of this type are important in the study of the Schrödinger equation. An pplication is given.

Nonlocal Variational Problems in Nonlinear Electromagneto-Elastostatics

Robert C. Rogers

SIAM J. Math. Anal. 19, pp. 1329-1347 (19 pages) | Cited 4 times

Online Publication Date: July 17, 2006

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The effects of arbitrary applied electric and magnetic fields on unshielded, nonlinear, deformable, polarizable, magnetizable, nonconducting bodies are studied. Both monotone materials (dielectric, paramagnetic, etc.) and classical ferromagnetic materials are considered. The lack of shielding forces us to consider unknown fields outside of the material. This leads to nonlocal (“shape-dependent”) effects. The work of Ball is extended [“Convexity conditions and existence theorems in nonlinear elasticity,” Arch. Rat. Mech. Anal., 63 (1977), pp. 337–403] to get an existence theory using direct methods of the calculus of variations.

The Integral Representation of the Positive Solutions of the Generalized Weinstein Equation on a Quarter-Space

Ömer Akin

SIAM J. Math. Anal. 19, pp. 1348-1354 (7 pages)

Online Publication Date: July 17, 2006

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The present paper gives a unique integral representation for the positive solutions to the generalized Weinstein equation \[ L[u] = L_{p,q} [u] \equiv \sum\limits_{i = 1}^n {u_{x_i x_i } } + \frac{p}{{x_{n - 1} }}u_{x_{n - 1} } + \frac{q}{{x_n }}u_{x_n } .\] This integral representation is explicitly described in terms of hypergeometric functions. The methods used are those of potential theory; the technique of Martin plays a particularly crucial role.

Existence of Solutions to the Stommel–Charney Model of the Gulf Stream

V. Barcilon, P. Constantin, and E. S. Titi

SIAM J. Math. Anal. 19, pp. 1355-1364 (10 pages) | Cited 13 times

Online Publication Date: July 17, 2006

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The existence of weak solutions to the equations proposed by Stommel [Trans. Amer. Geophys. Union, 29 (1948), pp. 202–206] and Charney [Proc. Nat. Acad. Sci. U.S.A., 41 (1955), pp. 731–740] as a model of the Gulf Stream are established by means of the method of artificial viscosity.

Multiple Traveling Waves in a Combustion Model

S. P. Hastings

SIAM J. Math. Anal. 19, pp. 1365-1371 (7 pages)

Online Publication Date: July 17, 2006

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A model reaction scheme, consisting of two simple competing reactions $A \to P_1 $ and $A \to P_2 $, is studied using Arrhenius kinetics with a cut-off to handle the cold boundary difficulty. It is shown that for appropriate values of the parameters in the problem, the model equations have three distinct traveling wave solutions. The middle solution, presumably unstable, is obtained from a singularly perturbed problem by rigorous matching.

Maxwell Equations in Polarizable Media

Bernardo Cockburn and Patrick Joly

SIAM J. Math. Anal. 19, pp. 1372-1390 (19 pages) | Cited 2 times

Online Publication Date: July 17, 2006

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The resolution of Maxwell equations in polarizable conductive media led to the resolution of a linear integrodifferential system. A method for the numerical approximation of this system was proposed in [B. Cockburn, SIAM J. Sci. Statist. Comput., 6 (1985), pp. 843–852]. Here the mathematical results that justify this method are given.

Formation of Shocks for a Single Conservation Law

Shizuo Nakane

SIAM J. Math. Anal. 19, pp. 1391-1408 (18 pages) | Cited 3 times

Online Publication Date: July 17, 2006

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The initial value problem for an equation of scalar conservation law in several space dimensions is considered. By the method of characteristics, the solution of this problem with $C^\infty $-initial datum is concretely constructed. Generally, this solution becomes multivalued in finite time. By virtue of the theory of singularities of $C^\infty $-mappings, its structure as a multivalued function is completely revealed. The entropy solution is constructed by making it single-valued. In this process, shocks occur. Shock surfaces are constructed by using the stable manifold theory. Thus propagation of shocks is described.

On the Strongly Damped Wave Equation: $u_{tt} - \Delta u - \Delta u_t + f(u) = 0$

Dang Dinh Ang and Alain Pham Ngoc Dinh

SIAM J. Math. Anal. 19, pp. 1409-1418 (10 pages) | Cited 4 times

Online Publication Date: July 17, 2006

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The following initial boundary value problem is considered \[ \begin{gathered} u_{tt} - \Delta u - \lambda \Delta u_t + f(u) = 0,\quad (x,t) \in \Omega \times ] {0,T} [,\quad \lambda > 0, \hfill \\ u = 0\quad {\text{on }}\partial \Omega \times [0,T), \hfill \\ u(x,0) = w_0 (x),\qquad u_t (x,0) = w_1 (x) \hfill \\ \end{gathered} \] where $\Omega $ is a bounded domain in $R^N $ with a sufficiently regular boundary $\partial \Omega $. In Part 1, a theorem on local existence and uniqueness is proved for $w_0 $ in $H_0^1 (\Omega )$ and $w_1 $ in $L^2 (\Omega )$, under a certain Lipschitzian condition on $f$.
In Part 2, the question of global existence and asymptotic behavior for $t \to \infty $ is studied, under more restrictive conditions, namely $1 \leqq N \leqq 3$, $f \in C^1 (\mathbb{R},\mathbb{R})$, $f(0) = 0$, and $f' \geqq - c$ with $c > 0$ “small” and $w_0 \in H_0^1 (\Omega ) \cap H^2 (\Omega )$, $w_1 \in L^2 (\Omega )$. It is proved that under these conditions, a unique solution $u(t)$ exists on $\mathbb{R}_ + $ such that $\| {u_t (t)} \|$ and $\| {\Delta u (t)} \|$ decay exponentially to $0$ as $t \to \infty $. ($\| \cdot \|$ denotes the $L^2 (\Omega )$ norm.) The method followed in this paper is that of successive linearizations (Part 1) and Galerkin (Part 2).

On the Nodal Sets of the Eigenfunctions of the String Equation

Chao-Liang Shen

SIAM J. Math. Anal. 19, pp. 1419-1424 (6 pages) | Cited 11 times

Online Publication Date: July 17, 2006

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In this paper the nodal sets of the eigenfunctions of the string equation are investigated. For $n$ sufficiently large it is found that the shortest nodal domain of the $n$th eigenfunction must be one of the neighboring nodal domains of the maximum points, and the longest nodal domain of the $n$th eigenfunction must be one of the neighboring nodal domains of the minimum points of the density function of the string equation. A limit formula for the ratio of the longest length and the shortest length of the nodal domains of the $n$th eigenfunction is also proved, and some average formulae for the nodal domains are derived.

A Resolution Method for Riccati Differential Systems Coupled in Their Quadratic Terms

L. Jodar and H. Abou-Kandil

SIAM J. Math. Anal. 19, pp. 1425-1430 (6 pages) | Cited 1 time

Online Publication Date: July 17, 2006

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By means of algebraic transformations a Riccati differential matrix system coupled in its quadratic terms is reduced to another one for which the successive approximation method is available. An iterative algorithm for solving the problem and an error upper bound for the approximation are given.

On Smoothest Interpolants

A. Pinkus

SIAM J. Math. Anal. 19, pp. 1431-1441 (11 pages) | Cited 2 times

Online Publication Date: July 17, 2006

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This paper is concerned with the problem of characterizing those functions of minimum $L^p $-norm on their $n$th derivative, $1 \leqq p \leqq \infty $, that sequentially take on the given values $(e_i )_1^N $. For $p = \infty $ the unique minimizing function is characterized. For $p < \infty $ fairly explicit necessary conditions are given.

Extension of Szegö’s Theorem on the Sections of Univalent Functions

Stephan Ruscheweyh

SIAM J. Math. Anal. 19, pp. 1442-1449 (8 pages) | Cited 1 time

Online Publication Date: July 17, 2006

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In this paper a far-reaching extension of Szegö’s theorem on the univalence of partial sums of the power series expansion of univalent functions in the class $\mathcal{S}$ is given. In particular, it is shown that the property “univalent” can be replaced by the stronger one “starlike univalent” and that the conclusion is not only true for $\mathcal{S}$ but also for the closed convex hull of $\mathcal{S}$. The paper concludes with the discussion of a new conjecture on $\mathcal{S}$, stronger than the former “Bieberbach conjecture.”

On the Zeros of Derivatives of Bessel Functions

Lee Lorch and Peter Szego

SIAM J. Math. Anal. 19, pp. 1450-1454 (5 pages)

Online Publication Date: July 17, 2006

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Bessel functions of the first and second kind are denoted as usual by $J_\nu (x)$, $Y_\nu (x)$, the general cylinder function $AJ_\nu (x) + BY_\nu (x)$, where $A$, $B$ are independent of $x$ and $\nu $, by $C_\nu (x)$, their respective positive zeros and those of their derivatives by $j_{\nu k} $, $y_{\nu k} $, $c_{\nu k} $, $j'_{\nu k} $, $y'_{\nu k} $, $c'_{\nu k} $, etc. It is shown here that for $ - 1 < \nu < 0$, (i) $j'_{\nu k} $ increases in $\nu ,k = 1,2, \cdots $, (ii) $j'_{\nu 1} > j'_{11} = 1.84 \cdots $, and (iii) $( - 1)^k J'''_\nu (j'_{\nu k} ) > 0$. It is also established that $c'_{\nu k} - c_{\nu l} $ increases in $\nu > 0$, provided $c'_{\nu k} > c_{\nu l} > \nu > 0$, where the ranks $k$, $l$ may or may not be equal but are kept fixed as $\nu $ varies. Further, (iii′) $( - 1)^k J'''_\nu (j'_{\nu k} ) < 0$ for $0 < \nu \leqq 1$, $k = 1,2, \cdots $; $J'''_0 (j'_{01} ) = 0$, $( - 1)^k J'''_0 (j'_{0k} ) < 0$, $k = 2,3, \cdots $.

A Combinatorial Interpretation of the Integral of the Product of Legendre Polynomials

J. Gillis, J. Jedwab, and D. Zeilberger

SIAM J. Math. Anal. 19, pp. 1455-1461 (7 pages) | Cited 3 times

Online Publication Date: July 17, 2006

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Denote by $P_n (x)$ the Legendre polynomial of degree $n$ and let \[ I_{n_1 , \cdots ,n_k } = \int_{ - 1}^1 {P_{n_1 } (x)} \cdots P_{n_k } (x)\,dx. \]$I_{n_1 , \cdots ,n_k } $ is written as a sum involving binomial coefficients and the sum is interpreted via a combinatorial model. This makes possible a combinatorial proof of a number of general theorems concerning $I_{n_1 , \cdots ,n_k } $, not all of which seem analytically straightforward, including a direct combinatorial derivation of the known formula for $I_{a,b,c} $ and the expression of $I_{a,b,c,d} $ as a simple finite sum. In addition, a number of apparently new combinatorial identities are obtained.

A Beta Integral Associated with the Root System $G_2 $

F. G. Garvan

SIAM J. Math. Anal. 19, pp. 1462-1474 (13 pages) | Cited 3 times

Online Publication Date: July 17, 2006

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Some conjectures of Askey are proven that have to do with adding roots in the Macdonald-Morris conjecture for $G_2 $. This is done by extending Aomoto’s proof of Selberg’s integral. This yields a new proof of the Macdonald–Morris root system conjecture for $G_2 $ which should extend to other root systems.

Une $q$-Intégrale de Selberg et Askey

Laurent Habsieger

SIAM J. Math. Anal. 19, pp. 1475-1489 (15 pages) | Cited 7 times

Online Publication Date: July 17, 2006

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We prove a conjecture by R. Askey (“Some basic hypergeometric extensions of integrals of Selberg and Andrews,” SIAM J. Math. Anal., 11(1980), pp. 938–951) on a basic extension of Selberg’s integral: \[ \int_0^1 \cdots \int_0^1 {\mathop \prod \limits_{1 \leqq i < j \leqq n} } \left| {t_i - t_j } \right|^{2z} \mathop \prod \limits_{i = 1}^n t_i^{x - 1} \left(1 - t_i \right)^{y - 1} dt_1 \cdots dt_n . \] We deduce from this a conjecture due to Morris about the constant term in the expansion of \[ \mathop \prod \limits_{j = 1}^l \left( {{{x_0 } / {x_j }}} \right)_a \left( {{{qx_j } / {x_0 }}} \right)_b \mathop \prod \limits_{1 \leqq i < j \leqq l} \left( {{{x_i } / {x_j }}} \right)_c \left( {{{qx_j } / {x_i }}} \right)_c ,\] where $(x)_k = (1 - x)(1 - qx) \cdots (1 - q^k x)$. In the appendix there can be found a proof of another conjecture by Askey related to the Dyson $q$-conjecture.

A Proof of Ramanujan’s Identity by Use of Loop Integrals

Katsuhisa Mimachi

SIAM J. Math. Anal. 19, pp. 1490-1493 (4 pages) | Cited 2 times

Online Publication Date: July 17, 2006

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Ramanujan’s identity means the following: \[ \sum_{n = - \infty }^{ + \infty } {\frac{{(a;q)_n }}{{(b;q)_n }}x^n = } \frac{{(ax;q)_\infty ({q / {ax;q}})_\infty ({b / {a;q}})_\infty (q;q)_\infty }}{{(x;q)_\infty ({b / {ax;q}})_\infty ({q / {a;q}})_\infty (b;q)_\infty }},\] where $(a;q)_\infty = \prod _{j = 0}^{ + \infty } (1 - aq^j )$, $(a;q)_n = {{(a;q)_\infty } /{(aq^n ;q)_\infty }}$ for $ - \infty < n < + \infty $, and $| {{b / a}} | < | x | < 1$, $| q | < 1$. This identity plays an important role in the theory of “$q$-analysis” (see, for example, [1], [3]). Various proofs of it are known ([2], [4], etc.). The aim of this paper is to derive the identity by another method, that of loop integrals.
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