Hopfield Networks are models of content addressable memory. In other words, they are dynamical systems designed to encode $N$ arbitrary strings and then converge to a point of perfect recall of any one of these strings if stimulated with any noised version of one of these strings. The usual textbook presentation on Hopfield networks begins by presenting the dynamical system without any motivation and only then going on to study its properties.
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Arguably, one of the most powerful developments in early modern applied mathematics is that of gradient descent, which is a technique for solving programs of the form
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Consider some set of points ${(x_i, y_i)}$ . To make statistically
significant predictions of $y$ given $x$, we would like to learn the
distribution $P(y|x, \mu_ {y(x)},\sigma)$. In this article, we will
derive an algorithm to learn the $y$ for the case when
$\mu_y \propto x$. Specifically, we make the following assumptions
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This post will continue a running theme on my blog, namely showing how mathematical structures provide powerful analogies for conceptually representing and manipulating complex flows of logic in a sugya of Gemara. Here, I want to show that concepts from linear algebra, specifically the notion of linear independence, vector spaces, and bases, offer a useful structural metaphor for studying sequences of kal vechomer limudim in Gemara.
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The following very nasty question came up in my physics homework this
week:
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My brother and I just began the first perek of Gemara Bava Kamma. Bava Kamma initializes by listing four instances, which it calls avot, for which the halachot of nezikin apply. Rashi^{1} notes the term avot , literally translated as father, denotes category. The four categories of nezikin damages listed by the mishna are learned directly from Parshat Mishpatim. In subsequent dapim, the Gemara will derive the corollories of these overarching avot nezikin categories.
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The formal system of mathematics can be used as an innovative tool in the study of Gemara. It offers great value as a technique for the student to use in their endeavors to dissect and develop a greater appreciation for the Gemara’s highly complex and nuanced logical structure.
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