Applied Mathematics Commons^™

Positive Solution Curves Of Semipositone Problems With Concave Nonlinearities, Alfonso Castro, Sudhasree Gadam, Ratnasingham Shivaji

All HMC Faculty Publications and Research

We consider the positive solutions to the semilinear equation:

-Δu(x) = λf(u(x)) for x ∈ Ω

u(x) = 0 for x ∈ ∂Ω

where Ω denotes a smooth bounded region in R^N (N > 1) and λ > 0. Here f :[0, ∞)→R is assumed to be monotonically increasing, concave and such that f(0) < 0 (semipositone). Assuming that f'(∞) ≡ lim t→∞ f'(t) > 0, we establish the stability and uniqueness of large positive solutions in terms of (f(t)/t)'. When Ω is a ball, we determine the exact number of positive solutions for each λ > 0. We also obtain the geometry of the branches of positive solutions completely and establish how …

Positive Solutions For A Semilinear Elliptic Problem With Critical Exponent, Ismail Ali, Alfonso Castro

All HMC Faculty Publications and Research

No abstract provided in article.

The Steady Boundary Layer Due To A Fast Vortex, Andrew J. Bernoff, Harald J. H. M. Van Dongen, Seth Lichter

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A point vortex located above and convected parallel to a wall is an important model of the process by which a boundary layer becomes unstable due to external disturbances. Often it has been assumed that the boundary layer due to the passage of the vortex is inherently unsteady. Here we show that for a vortex convected by a uniform shear flow, there is a steady solution when the speed of the vortex c_v is sufficiently fast. The existence of the steady solution is demonstrated analytically in the limit of large vortex velocity (c_v→∞) and numerically …

Existence Results For Semipositone Systems, V. Anuradha, Alfonso Castro, Ratnasingham Shivaji

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We study existence of positive solutions to the coupled-system of boundary value problems of the form

-Δu(x) = λf(x,u,v); x ∈ Ω

-Δv(x) = λg(x,u,v); x ∈ Ω

u(x) = 0 = v(x); x ∈ ∂Ω

where λ > 0 is a parameter, Ω is a bounded domain in R^N; N ≥ 1 with a smooth boundary ∂Ω and f,g are C^1 function with at least one of f(x_0,0,0) or g(x_0,0,0) being negative for some x_0 ∈ Ω (semipositone). We establish our existence results using the method of sub-super solutions. We also discuss non-existence results for λ small.

Branches Of Radial Solutions For Semipositone Problems, Alfonso Castro, Sudhasree Gadam, Ratnasingham Shivaji

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We consider the radially symmetric solutions to the equation −Δu(x) = λƒ(u(x)) for x ∈ Ω, u(x) = 0 for x ∈ ∂Ω, where Ω denotes the unit ball in R^N (N > 1), centered at the origin and λ > 0. Here ƒ: R→R is assumed to be semipositone (ƒ(0) < 0), monotonically increasing, superlinear with subcritical growth on [0, ∞). We establish the structure of radial solution branches for the above problem. We also prove that if ƒ is convex and ƒ(t)/(tƒ'(t)−ƒ(t)) is a nondecreasing function then for each λ > 0 there exists at most one positive solution u such that (λ, u) belongs to the unbounded branch of positive solutions. Further when ƒ(t) = tp − k, k > 0 and 1 < p < (N + 2)/(N − 2), we prove that the set of positive solutions is connected. Our results are motivated by and extend the developments in [4].

Distortion And Evolution Of A Localized Vortex In An Irrotational Flow, Joseph F. Lingevitch, Andrew J. Bernoff

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This paper examines the interaction of an axisymmetric vortex monopole, such as a Lamb vortex, with a background irrotational flow. At leading order, the monopole is advected with the background flow velocity at the center of vorticity. However, inhomogeneities of the flow will cause the monopole to distort. It is shown that a shear‐diffusion mechanism, familiar from the study of mixing of passive scalars, plays an important role in the evolution of the vorticity distribution. Through this mechanism, nonaxisymmetric vorticity perturbations which do not shift the center of vorticity are homogenized along streamlines on a Re1/3 time scale, much faster …

Descartes And Problem-Solving, Judith V. Grabiner

Pitzer Faculty Publications and Research

What can Descartes' Geometry teach us about problem solving?

Rapid Relaxation Of An Axisymmetric Vortex, Andrew J. Bernoff, Joseph F. Lingevitch

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In this paper it is argued that a two‐dimensional axisymmetric large Reynolds number (Re) monopole when perturbed will return to an axisymmetric state on a time scale (Re^1/3) that is much faster than the viscous evolution time scale (Re). It is shown that an arbitrary perturbation can be broken into three pieces; first, an axisymmetric piece corresponding to a slight radial redistribution of vorticity; second, a translational piece which corresponds to a small displacement of the center of the original vortex; and finally, a nonaxisymmetric perturbation which decays on the Re^1/3 time scale due to a shear/diffusion …

Applied Mathematics Should Be Taught Mixed, Gary I. Brown

Humanistic Mathematics Network Journal

No abstract provided.

Uniqueness Of Stable And Unstable Positive Solutions For Semipositone Problems, Alfonso Castro, Sudhasree Gadam

All HMC Faculty Publications and Research

Abstract not included in this article.

Optical Properties Of Human Uterus At 630 Nm, Steen J. Madsen, Bruce J. Tromberg, Yona Tadir, Pius Wyss, Lars O. Svaasand, Richard C. Haskell

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The optical properties of normal and fibriotic human uteri were determined using frequency-domain and steady-state techniques .

Frequency-Domain Photon Migration In Turbid Media, Bruce J. Tromberg, Steen J. Madsen, Curtis Chapman, Lars O. Svaasand, Richard C. Haskell

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An analytical model is presented for the propagation of diffuse photon density waves in turbid media. The frequency- and wavelength-dependence of photon density waves are measured using Frequency-domain Photon Migration (FDPM). Media optical properties, including absorption, transport, and fluorescence relaxation times are calculated from experimental results.

Optimal Blackjack Strategy With "Lucky Bucks", Arthur T. Benjamin, Eric Huggins '91

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In the casino game blackjack or "21," mathematically determined best plays have been calculated by various mathematicians and gambling experts. These optimal playing strategies all assume that the casino pays even money on bets (excluding when the player has a "blackjack"). However, many casinos offer the player "lucky bucks" that pay the player either 3-to-2 or 2-to-1. In the usual game, the player's expected loss is under 1¢ per dollar bet. in this paper, we derive optimal strategies under luck-buck conditions, giving the player an expected gain of 26¢ or 55¢ per dollar bet.

A Bifurcation Theorem And Applications, Alfonso Castro, Jorge Cossio

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In this paper we give a sufficient condition on the nonlinear operator N for a point (λ, u) to be a local bifurcation point of equations of the form u + λL^-1(N(u)) = 0, where L is a linear operator in a real Hilbert space, L has compact inverse, and λ ∈ R is a parameter. Our result does not depend on the variational structure of the equation or the multiplicity of the eigenvalue of the linear operator L. Applications are made to systems of differential equations and to the existence of periodic solutions of nonlinear second order …

The Hermeneutics Of Mathematical Modeling, David Tudor

Humanistic Mathematics Network Journal

No abstract provided.

Mathematics For Life And Society, Miriam Lipschutz-Yevick

Humanistic Mathematics Network Journal

No abstract provided.

On A Lower Bound For The Redundancy Of Reliable Networks With Noisy Gates, Nicholas Pippenger

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A proof is provided that a logarithmic redundancy factor is necessary for the reliable computation of the parity function by means of a network with noisy gates. This result was first stated by R.L. Dobrushin and S.I. Ortyukov (1977). However, the authors believe that the analysis given by Dobrushin and Ortyukov is not entirely correct. The authors establish the result by following the same steps and by replacing the questionable part of their analysis with entirely new arguments.

Graphs, Maneuvers, And Turnpikes, Arthur T. Benjamin

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We address the problem of moving a collection of objects from one subset of Z^m to another at minimum cost. We show that underfairly natural rules for movement assumptions, if the origin and destination are far enough apart, then a near optimal solution with special structure exists: Our trajectory from the originto the destination accrues almost all of its cost repeatingat most m different patterns of movement. Directions for related research are identified.

Nonnegative Solutions For A Class Of Radially Symmetric Nonpositone Problems, Alfonso Castro, Ratnasingham Shivaji

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We consider the existence of radially symmetric non-negative solutions for the boundary value problem

\begin{displaymath}\begin{array}{*{20}{c}} { - \Delta u(x) = \lambda f(u(x))\qua... ...\\ {u(x) = 0\quad \left\Vert x \right\Vert = 1} \\ \end{array} \end{displaymath}

where $ \lambda > 0,f(0) < 0$ (non-positone), $ f' \geq 0$ and $ f$ is superlinear. We establish existence of non-negative solutions for $ \lambda $ small which extends some work of our previous paper on non-positone problems, where we considered the case $ N = 1$. Our work also proves a recent conjecture by Joel Smoller and Arthur Wasserman.

Nonnegative Solutions To A Semilinear Dirichlet Problem In A Ball Are Positive And Radially Symmetric, Alfonso Castro, Ratnasingham Shivaji

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We prove that nonnegative solutions to a semilinear Dirichlet problem in a ball are positive, and hence radially symmetric. In particular, this answers a question in [3] where positive solutions were proven to be radially symmetric. In section 4 we provide a sufficient condition on the geometry of the domain which ensures that nonnegative solutions are positive in the interior.

Turnpike Structures For Optimal Maneuvers, Arthur T. Benjamin

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This dissertation is concerned with problems of optimally maneuvering a collection of objects ("pieces") from one location to another, subject to various restrictions on the allowable movements. We illustrate and prove that when the "distance" from the origin to the destination is large, and the movement rules and environment satisfy certain "homogeneity" properties, there exist near-optimal trajectories with very special (turnpike) structure.

These results are obtained by representing the problem through a configuration graph. Here, we have a node for each configuration and an arc for every "different" legal move. Each arc is endowed with a scalar weight …

Viscous Cross-Waves: An Analytical Treatment, Andrew J. Bernoff, L. P. Kwok, Seth Lichter

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Viscous effects on the excitation of cross‐waves in a semi‐infinite box of finite depth and width are considered. A formalism using matched asymptotic expansions and an improved method of computing the solvability condition is used to derive the relative contributions of the free‐surface, sidewall, bottom, and wavemaker viscous boundary layers. This analysis yields an expression for the damping coefficient previously incorporated on heuristic grounds. In addition, three new contributions are found: a viscous detuning of the resonant frequency, a slow spatial variation in the coupling to the progressive wave, and a viscous correction to the wavemaker boundary condition. The wavemaker …

The Centrality Of Mathematics In The History Of Western Thought, Judith V. Grabiner

Pitzer Faculty Publications and Research

This article explores the interplay of mathematics and philosophy in Western thought as well as applications to other fields.

Multiple Solutions For A Dirichlet Problem With Jumping Nonlinearities Ii, Alfonso Castro, Ratnasingham Shivaji

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No abstract provided for this article.

Stability Of Steady Cross-Waves: Theory And Experiment, Seth Lichter, Andrew J. Bernoff

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A bifurcation analysis is performed in the neighborhood of neutral stability for cross waves as a function of forcing, detuning, and viscous damping. A transition is seen from a subcritical to a supercritical bifurcation at a critical value of the detuning. The predicted hysteretic behavior is observed experimentally. A similarity scaling in the inviscid limit is also predicted. The experimentally observed bifurcation curves agree with this scaling.

Nonnegative Solutions For A Class Of Nonpositone Problems, Alfonso Castro, Ratnasingham Shivaji

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In the recent past many results have been established on non-negative solutions to boundary value problems of the form

-u''(x) = λf(u(x)); 0 < x < 1,

u(0) = 0 = u(1)

where λ>0, f(0)>0 (positone problems). In this paper we consider the impact on the non-negative solutions when f(0)<0. We find that we need f(u) to be convex to guarantee uniqueness of positive solutions, and f(u) to be appropriately concave for multiple positive solutions. This is in contrast to the case of positone problems, where the roles of convexity and concavity were interchanged to obtain similar results. We further establish the existence of non-negative solutions with interior zeros, which did not exist in positone problems.