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4434Cogent Mathematics
https://www.merlot.org/merlot/viewMaterial.htm?id=1113081
Cogent Mathematics is a fully peer-reviewed, open access, international scholarly journal with a mission to help researchers communicate with a global audience and interact with experts across the mathematics community and beyond. Cogent Mathematics considers original research articles in any of the following branches of mathematical and statistical research: Apllied and Interdisciplinary Mathematics; Computational Science; Pure Mathematics; and Statistics. Cogent Mathematics open access articles are distributed under a Creative Commons Attribution (CC-BY) 4.0 International license agreement.Sat, 26 Dec 2015 19:54:02 -0800Infinite Reflections
https://www.merlot.org/merlot/viewMaterial.htm?id=280379
The author offers reflections on specific questions mathematicians and philosophers have asked about the infinite over the centuries. He examines why explorers of the infinite, even in its strictly mathematical forms, often find it to be sublime.Tue, 6 Nov 2007 09:29:39 -0800MathPages
https://www.merlot.org/merlot/viewMaterial.htm?id=83263
This site contains several hundred articles concerned with mathematics and physics. General topics include Number Theory, Combinatorics, Geometry, Algebra, Calculus & Differential Equations, Probability & Statistics, Set Theory & Foundations, Reflections on Relativity, History, and Physics. The articles under each general heading are highly varied, many are quite advanced, and there is no apparent organizational scheme. For example, under Calculus & Differential Equations there is a proof that pi is irrational, a examination of the Limit Paradox, a discussion of Ptolemy's Orbit, and an historical review of the cycloid among many other articles. Visitors can browse by topics or search by keyword. (Anyone with information on the identity of the site author please contact the MERLOT submitter.)Tue, 8 Mar 2005 00:00:00 -0800The Teacher's Guide
https://www.merlot.org/merlot/viewMaterial.htm?id=674464
Provides free worksheets, printouts, lesson plans, SMARTBoard templates, and more. Also provides free Math Interactive Sites.Tue, 17 Jul 2012 13:02:42 -0700White Hole, Black Whole, and The Book
https://www.merlot.org/merlot/viewMaterial.htm?id=89727
Intellectual space is defined as the set of all proofs of mathematical logic, contained in The Book of Erdos. Physical and intellectual spaces are visualized making use of concepts from Intuitive Set Theory.Tue, 26 Jun 2001 00:00:00 -0700A Crash Course in the Mathematics of Infinite Sets
https://www.merlot.org/merlot/viewMaterial.htm?id=280696
This is pretty much what it says it is. This course is written in a lively and engaging style. It starts with elementary set theory and quickly builds up to a discussion of the Continuum Hypothesis with quite a few proofs along the way.Wed, 7 Nov 2007 20:00:13 -0800A Robust Ranking Methodology based on Diverse Calibration of AdaBoost
https://www.merlot.org/merlot/viewMaterial.htm?id=941890
This video was recorded at European Conference on Machine Learning and Principles and Practice of Knowledge Discovery in Databases (ECML PKDD), Athens 2011. In subset ranking, the goal is to learn a ranking function that approximates a gold standard partial ordering of a set of objects (in our case, relevance labels of a set of documents retrieved for the same query). In this paper we introduce a learning to rank approach to subset ranking based on multi-class classification. Our technique can be summarized in three major steps. First, a multi-class classification model (AdaBoost.MH) is trained to predict the relevance label of each object. Second, the trained model is calibrated using various calibration techniques to obtain diverse class probability estimates. Finally, the Bayes-scoring function (which optimizes the popular Information Retrieval performance measure NDCG), is approximated through mixing these estimates into an ultimate scoring function. An important novelty of our approach is that many different methods are applied to estimate the same probability distribution, and all these hypotheses are combined into an improved model. It is well known that mixing different conditional distributions according to a prior is usually more efficient than selecting one "optimal" distribution. Accordingly, using all the calibration techniques, our approach does not require the estimation of the best suited calibration method and is therefore less prone to overfitting. In an experimental study, our method outperformed many standard ranking algorithms on the LETOR benchmark datasets, most of which are based on significantly more complex learning to rank algorithms than ours.Sun, 8 Feb 2015 21:19:19 -0800A Set Theory for Scientists and Engineers (youtube video)
https://www.merlot.org/merlot/viewMaterial.htm?id=383787
Engineers know that they can land a man on the moon without using the Lebesgue integral and they will never encounter Skolem paradox in their nuclear reactor design. Intuitive Set Theory (IST) defined here, de-emphasizes concepts that are not required by scientists in their practical work.AXIOM OF COMBINATORIAL SETS: A set as important as the powerset of Cantor is what I call the combinatorial set of \aleph_0, which is defined as the set of all subsets of \aleph_0 with cardinality \aleph_0. Axiom of Combinatorial Sets (ACS) says that \aleph_1 is equal to the combinatorial set of \aleph_0. Even though, the combinatorial set is a subset of the powerset, it can be shown that powerset and combinatorial set have the same cardinality.AXIOM OF iNFINITESIMALS: First of all, let us note that corresponding to every real recursive number it is possible to visualize an infinitesimal attached to it. We will illustrate this with an example. Consider the number 2/3 written as an infinite binary sequence 0.101010... and its finite terminations 0.1, 0.101, 0.10101, ... which can be used to represent the intervals (1/2,2/3), (5/8,2/3), (21/32, 2/3), ... respectively. Note that the length of the interval decreases monotonically when the length of the termination increases and the cardinality of the set of points inside these intervals remain constant at 2^\aleph_0. From this, we can say that an infinitesimal is what we get when we visualize the interval corresponding to the entire nonterminating sequence, and this infinitely small interval contains 2^\aleph_0 points in it. The Axiom of Infinitesimals (AI) says that the unit interval is a set, with cardinality \aleph_0, of infinitesimals. We call an infinitesimal an relement and the elements in it figments, claiming that not even the axiom of choice can pick a figment from an relement.INTUITIVE SET THEORY: We define IST as the theory we get when AI and ACS are added to ZF theory. The discerning reader will easily recognize that the notion of a figment will not allow nonLebesgue measurable sets in IST. Also, the fact that \aleph_0 is the cardinality of the set of infinitesimals in a unit interval, provides us with a way to circumvent the Skolem paradox.IN A NUTSHELL: If only relements are allowed in set theory, it is enough for scientists for all practical purposes. If all elements of ZF theory are allowed, then set theorists can live happily in "Cantor's heaven״.Tue, 19 May 2009 04:30:12 -0700Ackermann Functions and Transfinite Ordinals
https://www.merlot.org/merlot/viewMaterial.htm?id=316548
An important part of Cantor's set theory, which forms the foundations of mathematics, is the concept of transfinite ordinals. A systematic way of writing the sequence of ordinals is given.Sun, 15 Jun 2008 22:51:20 -0700b-Bit Minwise Hashing for Estimating Three-Way Similarities
https://www.merlot.org/merlot/viewMaterial.htm?id=975481
This video was recorded at Video Journal of Machine Learning Abstracts - Volume 1. Computing two-way and multi-way set similarities is a fundamental problem. This study focuses on estimating 3-way resemblance (Jaccard similarity) using b-bit minwise hashing. While traditional minwise hashing methods store each hashed value using 64 bits, b-bit minwise hashing only stores the lowest b bits (where b>= 2 for 3-way). The extension to 3-way similarity from the prior work on 2-way similarity is technically non-trivial. We develop the precise estimator which is accurate and very complicated; and we recommend a much simplified estimator suitable for sparse data. Our analysis shows that $b$-bit minwise hashing can normally achieve a 10 to 25-fold improvement in the storage space required for a given estimator accuracy of the 3-way resemblance.Tue, 10 Feb 2015 13:41:03 -0800