https://www.stat.uchicago.edu/~lekheng/meetings/mathofranking/ref/saaty1.pdf

Decision-making with the AHP: Why is the principal eigenvector necessary

In this paper it is shown that the principal eigenvector is a necessary representation of the priorities derived from a positive reciprocal pairwise comparison judgment matrix A ¼ð aijÞ when A is a small perturbation of a consistent matrix. When providing numerical judgments, an individual attempts to estimate sequentially an underlying ratio scale and its equivalent consistent matrix of ratios. Near consistent matrices are essential because when dealing with intangibles, human judgment is of necessity inconsistent, and if with new information one is able to improve inconsistency to near consistency, then that could improve the validity of the priorities of a decision. In addition, judgment is much more sensitive and responsive to large rather than to small perturbations, and hence once near consistency is attained, it becomes uncertain which coefficients should be perturbed by small amounts to transform a near consistent matrix to a consistent one. If such perturbations were forced, they could be arbitrary and thus distort the validity of the derived priority vector in representing the underlying decision. 2002 Elsevier Science B.V. All rights reserved.

/pdf/decision-aiding-decision-making-with-the-ahp-why-is-the-405jg232t4.pdf

Decision Aiding Decision-making with the AHP: Why is the principal eigenvector necessary

An overview of the analytic hierarchy process and its applications

Self-actuating materials capable of transforming between three-dimensional shapes have applications in areas as diverse as biomedicine, robotics, and tunable micro-optics. We introduce a method of photopatterning polymer films that yields temperature-responsive gel sheets that can transform between a flat state and a prescribed three-dimensional shape. Our approach is based on poly(N-isopropylacrylamide) copolymers containing pendent benzophenone units that allow cross-linking to be tuned by irradiation dose. We describe a simple method of halftone gel lithography using only two photomasks, wherein highly cross-linked dots embedded in a lightly cross-linked matrix provide access to nearly continuous, and fully two-dimensional, patterns of swelling. This method is used to fabricate surfaces with constant Gaussian curvature (spherical caps, saddles, and cones) or zero mean curvature (Enneper’s surfaces), as well as more complex and nearly closed shapes.

https://science.sciencemag.org/content/sci/335/6073/1201.full.pdf

Designing responsive buckled surfaces by halftone gel lithography.

The Ricci flow of special geometries Special and limit solutions Short time existence Maximum principles The Ricci flow on surfaces Three-manifolds of positive Ricci curvature Derivative estimates Singularities and the limits of their dilations Type I singularities The Ricci calculus Some results in comparison geometry Bibliography Index.

/pdf/the-ricci-flow-an-introduction-34asb07sd2.pdf

The Ricci Flow: An Introduction

In [4], R. Hamilton has proved that if a compact manifold M of dimension three admits a C Riemannian metric g0 with positive Ricci curvature, then it also admits a metric g with constant positive sectional curvature, and is thus a quotient of the sphere S. In fact, he shows that the original metric can be deformed into the constant-curvature metric by requiring that, for t ≥ 0, x ∈ M and g = g(t, x),

/pdf/deforming-metrics-in-the-direction-of-their-ricci-tensors-1nfhemd4fg.pdf

Deforming metrics in the direction of their Ricci tensors

© Gauthier-Villars (Éditions scientifiques et médicales Elsevier), 1981, tous droits réservés. L’accès aux archives de la revue « Annales scientifiques de l’É.N.S. » (http://www. elsevier.com/locate/ansens) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

/pdf/some-regularity-theorems-in-riemannian-geometry-1bc7og7ry6.pdf

Some regularity theorems in riemannian geometry

Existence of metrics with prescribed Ricci curvature: Local theory

(2002). Vector Calculus and the Topology of Domains in 3-Space. The American Mathematical Monthly: Vol. 109, No. 5, pp. 409-442.

Dennis DeTurck

Papers

Deforming metrics in the direction of their Ricci tensors

Some regularity theorems in riemannian geometry

Existence of metrics with prescribed Ricci curvature: Local theory

Vector calculus and the topology of domains in 3-space

Isospectral deformations II: Trace formulas, metrics, and potentials