Boundary value problem analyzed.

<p>(A) confocal micrograph of cardiomyocyte, (B) schematic of cell-in-gel experiment (contracted configuration of cell exaggerated).</p

Normalized strain energy for the inhomogeneous inclusion problem versus modulus ratio .

<p>The strain energies are normalized by the inclusion modulus and inclusion volume and are calculated for the baseline case (, ).</p

Inhomogeneous inclusion results (baseline case , ) versus modulus ratio .

<p>(A) strain magnitudes are normalized by the magnitude of transformation strain , (B) stress components , mean stress , and maximum shear stress , each normalized by inclusion modulus .</p

Axial strain knockdown factor for the inhomogeneous inclusion.

<p>Axial constrained strain of the inclusion normalized by transformation strain is plotted against matrix/inclusion modulus ratio for several inclusion aspect ratios with fixed.</p

Benefit function determines optimal distribution.

<p>(<b>A</b>), benefit landscape defined by <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0143475#pone.0143475.e026" target="_blank">Eq (11)</a> when <i>f</i>(<i>L</i>) is cubic (<i>γ</i> = 4). (<b>B</b>), normal distributions with <i>s</i> = 0.242 (red, dashed curve) or 0.354 (blue, dot-dashed curve). The former is the magic number for this benefit function while the latter is the magic number of the benefit function in the tulip farmer case. (<b>C</b>) shows the corresponding outputs of these populations; black curve is <i>f</i>(<i>L</i>). Sloppy Algorithm output matches <i>f</i>(<i>L</i>) exactly. Heavy black curve in panel (<b>A</b>) lies along the ridge of maximal benefit. Red (with white dots added for clarity) and blue curves on the benefit landscape are the benefit corresponding to the outputs in panel (<b>B</b>). (<b>D</b>), benefit landscape defined by <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0143475#pone.0143475.e026" target="_blank">Eq (11)</a> when <i>f</i>(<i>L</i>) is sigmoidal. (<b>E</b>), magic number lognormal (red, dashed curve) and magic number normal (blue, dot-dashed curve) population decision models. The magic numbers are <i>s</i>_<i>m</i> = 0.559 (shape factor) for the lognormal and <i>s</i>_<i>m</i> = 0.271 (standard deviation) for the normal distribution. (<b>F</b>), outputs from corresponding distributions; black curve is <i>f</i>(<i>L</i>). Heavy black curve in panel (<b>D</b>) mark the ridge of maximal benefit. Red (with white dots) curve is the benefit for the lognormal distribution and the blue curve is for the normal distribution. Cyan-colored curves in (<b>A</b>) and (<b>D</b>) show where <i>B</i>(<i>L</i>, <i>ν</i>) = 0; points closer to the demand axis are positive.</p

Nomenclature.

<p>Nomenclature.</p

Convergence of the Sloppy Algorithm.

<p>Convergence of <math><msub><mi>λ</mi><mo>¯</mo><mi>i</mi></msub></math> (<b>A</b>) and <i>s</i>_<i>i</i> (<b>B</b>) when the Sloppy Algorithm was used with different initial values of <i>s</i>; <i>s</i>₀ = 10⁻⁵, light green, square; magic number, red, circle; 0.5, green, up-triangle; 0.7, blue, diamond; 1.0, magenta, down-triangle. Dashed black line marks the correct values of <i>L</i> = 0.723 and <i>s</i>* = 0.377.</p

Optimizing Population Variability to Maximize Benefit

<div><p>Variability is inherent in any population, regardless whether the population comprises humans, plants, biological cells, or manufactured parts. Is the variability beneficial, detrimental, or inconsequential? This question is of fundamental importance in manufacturing, agriculture, and bioengineering. This question has no simple categorical answer because research shows that variability in a population can have both beneficial and detrimental effects. Here we ask whether there is a certain level of variability that can maximize benefit to the population as a whole. We answer this question by using a model composed of a population of individuals who independently make binary decisions; individuals vary in making a yes or no decision, and the aggregated effect of these decisions on the population is quantified by a benefit function (e.g. accuracy of the measurement using binary rulers, aggregate income of a town of farmers). Here we show that an optimal variance exists for maximizing the population benefit function; this optimal variance quantifies what is often called the “right mix” of individuals in a population.</p></div

Sloppy rulers without Sloppy Algorithm.

<p>(<b>A</b>), length estimate (<i>Q</i>(<i>L</i>, <i>s</i>)) when <i>s</i> is fixed to 0 (light green), 0.1, 0.2, magic number (red), 0.4, 0.5, 0.7 (blue), 1 (magenta), and 2. Perfect fit falls on the diagonal line (circles). (<b>B</b>), performance <i>P</i>(<i>s</i>) given by <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0143475#pone.0143475.e021" target="_blank">Eq (8)</a>. Maximum occurs at the magic number <math><mrow><msqrt><mn>2</mn></msqrt><mo>/</mo><mn>4</mn><mo>≈</mo><mn>0</mn><mo>.</mo><mn>35</mn></mrow></math> (red circle).</p

Farmer example.

<p>(<b>A</b>) Probability density function (<i>ϕ</i>) of farmers’ sensitivities. <i>s</i> = 0.01, green (solid, left axis); magic number, red (dashed, right axis); 0.8, blue (dash-dot, right axis). (<b>B</b>) Farmers output relative to demand. Solid black line shows perfect matching between output and demand. Curves’ colors and line patterns match those in A. (<b>C</b>), benefit landscape, <i>B</i>(<i>L</i>, <i>ν</i>). Thick white line lies on the ridge where <i>B</i> is maximized. Cyan curve near the <i>L</i>-axis is where <i>B</i> = 0; <i>B</i> < 0 for points below the curve (closer to <i>L</i>-axis) and <i>B</i> > 0 above the curve. Other colored curves are the benefits derived from farmer tulip output shown in panel (<b>B</b>). Black, Sloppy Algorithm; red, <i>s</i> = magic number; green, 0.01; and blue, 0.8. (<b>D</b>) Performance as a function of population variability <i>s</i>. Circles mark <i>s</i> = 0.01 (green), magic number (red), and 0.8 (blue).</p