# mathematica Published articles for mathematica. This is one page of public article previews, not the complete archive. Follow Next page to continue. Summaries are not the original full articles. ## Elliptic Curves as Elementary Equations DevFeed: [Elliptic Curves as Elementary Equations]() Original publisher: [Read original article]() Published: 2014-02-10T10:00:40Z Content type: tutorial Language: en Sources: [Jeremy Kun]() Topics: [Mathematics](), [Cryptography](), [Security]() Tags: [algebra](), [algebraic-geometry](), [cryptography](), [elliptic-curves](), [encryption](), [function](), [mathematica](), [mathematics](), [security](), [systems]() ### AI overview This introductory tutorial explains elliptic curves as the sets of solutions to cubic polynomial equations in two variables. It motivates their study through algebraic geometry and their role in encryption, then begins developing the elliptic curve group structure using an elementary example involving pyramids of spheres. ### Source excerpt Finding solutions to systems of polynomial equations is one of the oldest and deepest problems in all of mathematics. This is broadly the domain of algebraic geometry, and mathematicians wield some of the most sophisticated and abstract tools available to attack these problems. The elliptic curve straddles the elementary and advanced mathematical worlds in an interesting way. On one hand, it's easy to describe in elementary terms: it's the set of solutions to a cubic function of two variables. ## k-Means Clustering and Birth Rates DevFeed: [k-Means Clustering and Birth Rates]() Original publisher: [Read original article]() Published: 2013-02-04T17:54:20Z Content type: tutorial Language: en Sources: [Jeremy Kun]() Topics: [clustering](), [Machine learning](), [Algorithms](), [Mathematics]() Tags: [algorithm](), [clustering](), [heuristic-algorithm](), [machine-learning](), [mathematica](), [mathematics](), [metrics](), [partition](), [programming](), [python](), [unsupervised-learning]() ### AI overview This tutorial introduces the clustering problem, formalizes k-means clustering as a partitioning problem over points in a metric space, explains why finding an exact solution is difficult, and describes using a heuristic algorithm instead. ### Source excerpt A common problem in machine learning is to take some kind of data and break it up into "clumps" that best reflect how the data is structured. A set of points which are all collectively close to each other should be in the same clump. A simple picture will clarify any vagueness in this: cluster-example Here the data consists of points in the plane. There is an obvious clumping of the data into three pieces, and we want a way to automatically determine which points are in which clumps. ## The Cellular Automaton Method for Cave Generation DevFeed: [The Cellular Automaton Method for Cave Generation]() Original publisher: [Read original article]() Published: 2012-07-29T20:01:18Z Content type: article Language: en Sources: [Jeremy Kun]() Topics: [Cellular automaton](), [procedural flowers](), [Game Development](), [Simulation](), [Computing]() Tags: [article](), [cellular-automata](), [computing](), [game-development](), [graphics](), [javascript](), [mathematica](), [procedural-generation](), [programming](), [simulation](), [video-games]() ### AI overview This article examines a practical application of cellular automata: procedural generation of cave-like levels in video games. It explains how randomized level generation can reduce hand-crafted design effort and increase replay value, placing the technique in the broader history of procedural content generation. ### Source excerpt Dear reader, this post has an interactive simulation! We encourage you to play with it as you read the article below. In our series of posts on cellular automata, we explored Conway's classic Game of Life and discovered some interesting patterns therein. And then in our primers on computing theory, we built up a theoretical foundation for similar kinds of machines, including a discussion of Turing machines and the various computational complexity classes surrounding them. ## Numerical Integration DevFeed: [Numerical Integration]() Original publisher: [Read original article]() Published: 2012-01-08T18:08:42Z Content type: tutorial Language: en Sources: [Jeremy Kun]() Topics: [function](), [Code]() Tags: [article](), [calculus](), [function](), [mathematica](), [mathematics](), [numerical-analysis](), [programming](), [simpson-s-rule](), [trapezoidal-rule]() ### AI overview This article explains numerical integration by revisiting definite integrals, Riemann sums, and partition-based approximations. It introduces the problem of approximating a function's definite integral and begins with the left Riemann sum. ### Source excerpt Rectangles, Trapezoids, and Simpson's I just wrapped up a semester of calculus TA duties, and I thought it would be fun to revisit the problem of integration from a numerical standpoint. In other words, the goal of this article is to figure out how fast we can approximate the definite integral of a function $ f:\mathbb{R} \to \mathbb{R}$. Intuitively, a definite integral is a segment of the area between a curve $ f$ and the $ x$-axis, where we allow area to be negative when $ f(x) < 0$. ## Introduction to Cellular Automata DevFeed: [Introduction to Cellular Automata]() Original publisher: [Read original article]() Published: 2011-06-29T21:08:45Z Content type: tutorial Language: en Sources: [Jeremy Kun]() Topics: [Cellular automaton](), [Automaton](), [Finite-state machine]() Tags: [cellular-automata](), [computability-theory](), [conus](), [conway](), [life](), [mathematica](), [mathematics](), [patterns](), [programming](), [rules](), [simulation](), [turing-machine]() ### AI overview An introductory tutorial on cellular automata, explaining cells, states, transition rules, simultaneous updates, and a one-dimensional binary example. ### Source excerpt Cellular Automata There is a long history of mathematical models for computation. One very important one is the Turing Machine, which is the foundation of our implementations of actual computers today. On the other end of the spectrum, one of the simpler models of computation (often simply called a system) is a cellular automaton. Surprisingly enough, there are deep connections between the two. But before we get ahead of ourselves, let's see what these automata can do. ## Well Orderings and Search DevFeed: [Well Orderings and Search]() Original publisher: [Read original article]() Published: 2011-06-14T11:18:04Z Content type: tutorial Language: en Sources: [Jeremy Kun]() Topics: [Algorithm](), [Sorting](), [ordering](), [implementation]() Tags: [algorithm](), [axiom-of-choice](), [binary-search](), [mathematica](), [mathematics](), [optimization](), [ordering](), [pseudocode](), [recursion](), [sorting](), [well-ordering]() ### AI overview This tutorial explains binary search on sorted lists, including its recursive structure and O(log n) runtime. It then introduces strict total and well orders to explain why sorting and comparison work. ### Source excerpt Binary Search Binary search is perhaps the first and most basic nontrivial algorithm a student learns. For the mathematicians out there, binary search is a fast procedure to determine whether a sorted list contains a particular element. Here is a pseudocode implementation: # Binary Search: # Given a list L, sorted via the total order <, and a sought # element x, return true iff L contains x. function binarySearch(L, x, <): # base case if(length(L) == 1): return L[0] == x middleIndex = floor(length(L) / 2) if (L[middleIndex] == x): return true # inductive step, with ellipsis notation meaning slices of L # from the beginning and to the end, respectively if (x < L[middleIndex]): return binarySort(L[.