Aesthetical issues with stochastic evaluation.

<p>This research uses a stochastic computational tool (2D-C) for characterizing images in order to examine similarities and differences among artworks. 2D-C is measures the degree of variability (change in variability vs. scale) in images using stochastic analysis.</p><p>Apparently, beauty is not easy to quantify, even with stochastic measures. The meaning of beauty is linked to the evolution of human civilization and the analysis of the connection between the observer and the beauty (art, nature) has always been of high interest in both philosophy and science. Even though this analysis has mostly been considered part of the so-called social studies and humanities, mathematicians have also been involved. Mathematicians are generally not specialized to contribute, through their expertise, in sociopolitical analysis of messages and motivations of art but have been consistently applying mathematical knowledge, which is their expertise, in trying to explain aesthetics. In most of these analyses, the question at hand is if what is pleasing to the eye or not can be explained though mathematics.</p><p>Historically, it is known that from the time of the ancient Egyptian civilization a mathematic rule of the analogies of human body as models of beauty had been developed, and later in ancient Greece, the mathematicians Pythagoras and Euclid were the first known to have searched for a common rule (canon) existing in shapes that are perceived as beautiful. Euclid's Elements (c. 300 BC), for example, contains the first known definition of the “golden ratio”.</p><p>The opinions of later philosophers on this pursuit of mathematicians in the analysis of aesthetics were more varied. Leibniz, for example, believed that there is a norm behind every aesthetic feeling which we simply don’t know how to measure. On the contrary, Descartes supports that instead of regarding the aesthetic quality as an inherent quality of a physical object, the distinction of mind and nature have allowed humans to incorporate their own subjective feelings in determining their aesthetic preferences.</p><p>Thus many artists knew and apply math and geometry in their artwork, many philosophers tried to connect math and arts. Hence, it might be interesting to examine art work through a stochastic view. Stochastic analyses of the examined artworks are presented using climacograms and through stochastic evaluation with 2D-C we try to quantify some aspects of the artists’ expression. </p>

Supports of Extremal Doubly Stochastic Measures

A doubly stochastic measure on the unit square is a Borel probability measure whose horizontal and vertical marginals both coincide with the Lebesgue measure. The set of doubly stochasticmeasures is convex and compact so its extremal points are of particular interest. The problem number 111 of Birkhoò is to provide a necessary and suõcient condition on the support of a doubly stochastic measure to guarantee extremality. It was proved by Beneš and Štepán that an extremal doubly stochastic measure is concentrated on a set which admits an aperiodic decomposition. Hestir and Williams later found a necessary condition which is nearly sufficient by further refining the aperiodic structure of the support of extremal doubly stochastic measures. Our objective in this work is to provide a more practical necessary and nearly sufficient condition for a set to support an extremal doubly stochastic measure