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1975

Volume 29, Issue 4, pp. 571-787


On Minimum Cost Networks with Nonlinear Costs

J. Soukup

SIAM J. Appl. Math. 29, pp. 571-581 (11 pages) | Cited 3 times

Online Publication Date: July 12, 2006

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This paper studies the topology of the minimum cost network, which connects a given set of points in $R_2 $ The cost of each edge of the network is assumed to be a nonlinear function of its length. The cost function can be of a rather general nature—convex, concave, discontinuous, etc. On the assumption that the network has no loops and no additional vertices which are adjacent to less than 3 edges, the minimum cost network exists. Theorem 1 describes the main result which is an upper estimate of the number of edges which may be adjacent to one vertex. It is shown that most common cost functions lead to a topology similar to that connected with the linear cost function. In addition to the given points, the minimum cost network has some additional vertices characterized by 3 edges adjacent to each of them (so called “Steiner points”). The actual angles between the edges do not have to be exactly the same as those which obtain for the Steiner points of the network with linear costs. However, a lower limit for the angle between two edges is derived, which applies to the edges adjacent to the given points.

Diffraction of Elastic Waves by a Rigid-Smooth Wedge

S. H. Zemell

SIAM J. Appl. Math. 29, pp. 582-596 (15 pages) | Cited 2 times

Online Publication Date: July 12, 2006

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The scattering of cylindrical dilatational and vertically polarized shear waves (P- and SV-waves) by a rigid-smooth wedge is investigated by means of transform methods and the so-called edge condition. The exact solutions are sums of an acoustic and elastic term. The former has been known for a long time while the latter is presented in closed form for both time-harmonic and transient line sources. In addition to being dominant near the apex of the wedge in some instances, the elastic terms due to the influence of the edge can be significant throughout space. As a check on our results, Kostrov’s plane wave solution is retrieved by appropriately moving a line source to infinity.

Spherical Means of Solutions of Partial Differential Equations in a Conical Region

Lu Ting

SIAM J. Appl. Math. 29, pp. 597-623 (27 pages)

Online Publication Date: July 12, 2006

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The spherical means of the solutions of a linear partial differential equation $Lu = f$ in a conical region are studied. The conical region is bounded by a surface generated by curvilinear $\xi $ lines and by two truncating $\xi $ surfaces. The spherical mean is the average of $u$ over a constant $\xi $ surface. Conditions on the linear differential operator, $L$, and on the orthogonal coordinates $\xi $, $\eta $ ,$\xi $ are established so that the problem for the determination of the spherical mean of the solution subjected to the appropriate boundary and initial conditions can be reduced to a problem with only one space variable. Conditions are then established so that the spherical mean of the solution in one conical region will be proportional to that of a known solution in another conical region. Applications to various problems of mathematical physics and their physical interpretations are presented.

Generators for Certain Alternating Groups with Applications to Cryptography

Don Coppersmith and Edna Grossman

SIAM J. Appl. Math. 29, pp. 624-627 (4 pages) | Cited 5 times

Online Publication Date: July 12, 2006

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A set of block ciphers is described which can readily be adapted for computer encipherment of data. The subgroup of permutations generated by any one of these ciphers is shown to be the alternating group, in all cases of interest, suggesting that such systems have a high level of security.

The Method of Averaging and Domains of Stability for Integral Manifolds

David E. Gilsinn

SIAM J. Appl. Math. 29, pp. 628-660 (33 pages) | Cited 4 times

Online Publication Date: July 12, 2006

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Liapunov’s direct method is a standard and effective approach to computing the domain of stability (or region of attraction) of an autonomous ordinary differential equation. In this paper the author investigates domains of stability of integral manifolds of solutions generated by nonlinear mechanical and electrical oscillatory systems with many degrees of freedom. These manifolds are families of solutions that exhibit stronger stability properties than individual solutions. The problem of estimating the domain of stability of an asymptotically stable integral manifold is reduced to computing the domain of stability of an associated autonomous system of differential equations. This is done by applying the method of averaging to the system generating the integral manifold thus removing angular and time dependences. The stability region of this associated system is then computed and a result is established showing that this region is contained in the stablity region of the original system. Several examples, including a coupled van der Pol system of oscillators, are considered.

Orthonormal Expansions of Angular Momentum Functions

P. L. Corio

SIAM J. Appl. Math. 29, pp. 661-664 (4 pages) | Cited 1 time

Online Publication Date: July 12, 2006

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A method is given for expanding an arbitrary function of a component of angular momentum in terms of orthonormal operators. The orthonormal operators are given explicitly in terms of Chebyshev’s polynomials. A general formula for the expansion coefficients is derived and applied to the rotation operator.

The Linearized Boltzmann Collision Operator for Cut-Off Potentials

Hans Birger Drange

SIAM J. Appl. Math. 29, pp. 665-676 (12 pages) | Cited 3 times

Online Publication Date: July 12, 2006

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Boundedness and compactness of integral operators arising from the linearized Boltzmann collision operator are investigated for a wide class of angular and radial cut-off potentials.

On Norms and Singular Values of Rectangular Matrices

Khursheed Alam

SIAM J. Appl. Math. 29, pp. 677-679 (3 pages)

Online Publication Date: July 12, 2006

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Let $A$ and $B$ be $m \times n$ matrices with real elements. Upper and lower bounds are given for the sum of the squares of the lengths of a given number of column (row) vectors of $A$ in terms of the characteristic roots of $A'A$. The value of $\inf \| {A - B} \|$ is derived where $\| \cdot \|$denotes the Euclidean norm and the infimum is taken with respect to all $m \times n$ matrices $B$ of rank $r\leqq {\operatorname{rank}} A$.

On Piecewise Affine Mappings in $R^n $

Werner C. Rheinboldt and James S. Vandergraft

SIAM J. Appl. Math. 29, pp. 680-689 (10 pages) | Cited 5 times

Online Publication Date: July 12, 2006

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Let $H_1 , \cdots ,H_p $, be given hyperplanes which divide $R^n $ into finitely many closed, convex polytopes $\bar C_1 , \cdots \bar C_q $, and consider a continuous, piecewise affine mapping $F:R^n \to R^n ,Fx = A_j x + a^J ,\forall x \in \bar C_J ,j = 1, \cdots ,q$, with nonsingular $A_J \in L( {R^n } )$. Such functions arise, for instance, in the piecewise linear analysis of nonlinear resistive electric networks. We prove here that $F$ is surjective if the signs of the determinants of all matrices $A_j $ are the same, and that then the Katzenelson algorithm for solving $Fx = b$ will always reach a solution. This extends recent results of Fujisawa and Kuh who proved a homeomorphism theorem for piecewise affine mappings and considered the Katzenelson algorithm in that case. We also augment their basic theorem in another direction by showing that when all $A_j $ are $P$- or $M$-matrices then $F$ is a (surjective) $P$- or $M$-function, respectively. In the $M$-function case this implies, for example, the global convergence of the nonlinear Gauss–Seidel process.

Oscillatory Criteria for a General Second Order Functional Differential Equation

J. B. Garner

SIAM J. Appl. Math. 29, pp. 690-698 (9 pages)

Online Publication Date: July 12, 2006

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Conditions are given for $f$ under which all nontrivial continuable solutions of \[ y'' + f\left( {t,y,y'} \right) = 0 \] are oscillatory on $[ {0,\infty } )$ and not of one sign on any ray $[ {a,\infty } )$. Then, by modifying these conditions slightly, it becomes apparent that these results remain valid for the second order functional differential equation\[ y'' + f( {t,y,( t ),y( {u_2 ( t )} } ), \cdots ,y( {u_n ( t )} ),y'( t ),y'( v _2 {( t )} ),\cdots ,y'( {v _m {( t )} )} ) = 0 \] By considering special cases of these equations, it is shown how these results amend and generalize a large number of oscillatory criteria in the literature for second order differential equations.

The Interaction Energy, Field Strength and Force Acting on a Pair of Dielectric Spheres Embedded in a Dielectric Medium

Dieter K. Ross

SIAM J. Appl. Math. 29, pp. 699-707 (9 pages) | Cited 1 time

Online Publication Date: July 12, 2006

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A rigorous mathematical method is found for finding the electrostatic interaction energy, the field strength and the force acting on a pair of nonconducting dielectric spheres embedded in an aqueous solution of dielectric constant 78.30. A numerical method is found which amounts to a method of successive approximations that is stable even when the two spheres are at contact separation.

Constraint Sets of Geometric Programs Characterized by Auxiliary Problems

Willy Gochet and Yves Smeers

SIAM J. Appl. Math. 29, pp. 708-718 (11 pages) | Cited 2 times

Online Publication Date: July 12, 2006

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This paper refines a previous classification scheme by using auxiliary problems of the dual geometric program. This new scheme provides a one-to-one correspondence between the possible states of primal and dual geometric programs. Moreover, a characterization of superconsistency, strong inconsistency and subconsistency without consistency is obtained in terms of properties of the dual problem.

Perturbation of Two-Dimensional Predator-Prey Equations with an Unperturbed Critical Point

H. I. Freedman and P. Waltman

SIAM J. Appl. Math. 29, pp. 719-733 (15 pages) | Cited 6 times

Online Publication Date: July 12, 2006

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The paper investigates the question of the existence of periodic solutions of the predator-prey equations of Lotka–Volterra type under perturbations. The focus here is on the case where (i) the first derivatives of the perturbation terms are zero at the critical point and (ii) the critical point remains fixed. In this case, the problem is one of perturbation in the small parameter and not of “higher order.” This case was left open in a previous investigation of the authors. Conditions are given for the existence of a periodic solution and the stability of the solution is investigated.

Domains of Attraction for Reciprocals of Powers of Random Variables

Jesse M. Shapiro

SIAM J. Appl. Math. 29, pp. 734-739 (6 pages) | Cited 1 time

Online Publication Date: July 12, 2006

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The problem of finding to which domain of attraction the reciprocal of a positive power of a random variable belongs is considered. Under general conditions,, it is shown that this reciprocal always belongs to the domain of attraction of some stable law. The characteristic exponent' of this stable law is given in terms of the power to which the original random variable is raised.
Special attention is given to the case where the positive power is one. In this case; the problem is that of finding the limit distribution of normed sums of reciprocals of a sequence of independent identically distributed random variables. Under general conditions, the limit distribution is shown to be the Cauchy law, and the exact norming constants are found.

Optimal Gambling Systems Under Discounting and Disbursement

Barry Alan Pasternack

SIAM J. Appl. Math. 29, pp. 740-750 (11 pages)

Online Publication Date: July 12, 2006

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This paper is concerned with the study of gambling situations in which the time element involved is accounted for. The approach is to assume that either a discounting of capital or an exponentially growing disbursement is incurred by the gambler. The discount case accounts for inflation while the disbursement case can be thought of as an annuity which pays its members a fixed amount adjusted for inflation. The study concerns simple coin tossing games. Criteria for favorability and optimal strategies for a number of different objectives are developed.

A Bounding Technique for Polynomial Functions

Michael A. Crane

SIAM J. Appl. Math. 29, pp. 751-754 (4 pages) | Cited 1 time

Online Publication Date: July 12, 2006

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A technique is presented for obtaining bounds for a polynomial function of one variable. Consider an $n$th order polynomial function$g( x )$, defined over the interval $a_0 \leqq x\leqq a_{n + 1} $ and having unknown coefficients $z_1 ,z_2 , \cdots ,z_n $. Suppose that upper and lower bounds for $g( x )$ are known at the points $x = a_i ,1\leqq i\leqq n$, where $a_0 \leqq a_1 < a_2 < \cdots < a_n \leqq a_{n + 1} .$ Then upper and lower bounds can be obtained for the entire function $g$ over the interval $a_0 \leqq x\leqq a_{n + 1} $ The upper and lower bounds for $g$ are found to be piecewise-polynomials which pass through appropriate points selected from among the upper and lower bounds at the points $a_1 ,\,a_2 , \cdots ,a_n .$

An Inequality for Discrete Convex Hulls and Applications

H. S. Witsenhausen

SIAM J. Appl. Math. 29, pp. 755-762 (8 pages)

Online Publication Date: July 12, 2006

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For a set $S$ in $R_ + ^n $, let $S_k = \{ {k^{ - 1} \sum _{\alpha = 1}^k x_\alpha | {x_\alpha \in S,\alpha = 1, \cdots ,k} } \}$ be its discrete convex hull of order $k$. Then the infimum of $\| x \|_\infty $ over $S_k $ cannot exceed the infimum, over the convex hull of $k$, of $k^{ - 1} \| x \|_1 + ( {1 - k^{ - 1} } )\| x \|_\infty $, and this bound is sharp.
One application is the proof of a conjecture of Graham and Garey arising from their work on multiprocessor scheduling. Another application is to two person zero sum games. If the minimizes is restricted to a single use of a random device producing one of $k$ equiprobable outputs, the lowest expected payoff he can guarantee is bounded from above in terms of the value of a game, whose payoff matrix is obtained from the given payoff matrix by multiplication by the matrix $k^{ - 1} E + ( {1 - k^{ - 1} } )I$, where $E$ is the all ones matrix.
Another consequence is that for positive integers $n$, $v$, $k$ and real $x_\alpha ^i \geqq 0,i = 1, \cdots ,n,\alpha = 1, \cdots ,\nu $, there exist nonnegative integers $m^1 , \cdots m^{\nu } $, of which at most $n$ are positive, with $\sum _{\alpha = 1}^{\nu } m^\alpha = k$ and there exist real $q_j \geqq 1, j = 1, \cdots ,n$, with $\sum _{j = 1}^n q_j = n + k - 1$ such that for $i = 1, \cdots ,n,\alpha = 1, \cdots ,\nu $,\[ \sum\limits_{\beta = 1}^{\nu } {m^\beta x_\beta ^i } \leqq \sum\limits_{j = 1}^n {q_j x_\alpha ^i } . \]

Modern Developments in Transonic Flow

Julian D. Cole

SIAM J. Appl. Math. 29, pp. 763-787 (25 pages) | Cited 1 time

Online Publication Date: July 12, 2006

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A survey is given of transonic small disturbance theory. Basic equations, shock relations, similarity laves, lift and drag integrals are derived., The airfoil boundary value problem is formulated. Finite difference methods and computational algorithms are described. Results are compared with other calculation methods and experiments.
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