Nonstandard analysis | Infinity | Real closed field
The history of calculus is fraught with philosophical debates about the meaning and logical validity of fluxions or infinitesimal numbers. The standard way to resolve these debates is to define the operations of calculus using epsilon–delta procedures rather than infinitesimals. Nonstandard analysis instead reformulates the calculus using a logically rigorous notion of infinitesimal numbers. Nonstandard analysis originated in the early 1960s by the mathematician Abraham Robinson. He wrote: ... the idea of infinitely small or infinitesimal quantities seems to appeal naturally to our intuition. At any rate, the use of infinitesimals was widespread during the formative stages of the Differential and Integral Calculus. As for the objection ... that the distance between two distinct real numbers cannot be infinitely small, Gottfried Wilhelm Leibniz argued that the theory of infinitesimals implies the introduction of ideal numbers which might be infinitely small or infinitely large compared with the real numbers but which were to possess the same properties as the latter. Robinson argued that this law of continuity of Leibniz's is a precursor of the transfer principle. Robinson continued: However, neither he nor his disciples and successors were able to give a rational development leading up to a system of this sort. As a result, the theory of infinitesimals gradually fell into disrepute and was replaced eventually by the classical theory of limits. Robinson continues: ... Leibniz's ideas can be fully vindicated and ... they lead to a novel and fruitful approach to classical Analysis and to many other branches of mathematics. The key to our method is provided by the detailed analysis of the relation between mathematical languages and mathematical structures which lies at the bottom of contemporary model theory. In 1973, intuitionist Arend Heyting praised nonstandard analysis as "a standard model of important mathematical research". (Wikipedia).
Overview of non probability sampling; advantages and disadvantages, types. Check out my e-book, Sampling in Statistics, which covers everything you need to know to find samples with more than 20 different techniques: https://prof-essa.creator-spring.com/listing/sampling-in-statistics
From playlist Sampling
A Gentle Introduction to Non-Parametric Statistics (15-1)
We are now going to look at a special class of tests that give us the ability to do statistical analyses in circumstances when parametric tests just won’t do. They are called non-parametric statistics. Parametric statistics like t tests and ANOVA compare groups using scale-level data. Non-
From playlist WK15 Chi-Square & Non-Parametric Alternatives - Online Statistics for the Flipped Classroom
This video is brought to you by the Quantitative Analysis Institute at Wellesley College. The material is best viewed as part of the online resources that organize the content and include questions for checking understanding: https://www.wellesley.edu/qai/onlineresources
From playlist Intro To Non-Parametric Tests (updated)
Parametric and nonparametric tests
Parametric tests are most commonly used in healthcare research. They include tests such as Student's t-test and ANOVA. There is, however a rich set of non-parametric tests that are much more appropriate to use in certain circumstances.
From playlist Learning medical statistics with python and Jupyter notebooks
There is a written typo at 8:03 in this video. I write that the linearity assumption of a linear regression is "usually true," but that's not true. What I meant is what I actually said in the audio, which is that the assumption is usually not true. This video is brought to you by the Quan
From playlist Applied Data Analysis and Statistical Inference
Transformation matrix with respect to a basis | Linear Algebra | Khan Academy
Courses on Khan Academy are always 100% free. Start practicing—and saving your progress—now: https://www.khanacademy.org/math/linear-algebra/alternate-bases/change-of-basis/v/lin-alg-transformation-matrix-with-respect-to-a-basis Finding the transformation matrix with respect to a non-stan
From playlist Alternate coordinate systems (bases) | Linear Algebra | Khan Academy
(PP 6.2) Multivariate Gaussian - examples and independence
Degenerate multivariate Gaussians. Some sketches of examples and non-examples of Gaussians. The components of a Gaussian are independent if and only if they are uncorrelated.
From playlist Probability Theory
Using nonstandard natural numbers in Ramsey Theory - M. Di Nasso - Workshop 1 - CEB T1 2018
Mauro Di Nasso (Pisa) / 01.02.2018 In Ramsey Theory, ultrafilters often play an instrumental role. By means of nonstandard models, one can reduce those third-order objects (ultrafilters are sets of sets of natural numbers) to simple points. In this talk we present a nonstandard technique
From playlist 2018 - T1 - Model Theory, Combinatorics and Valued fields
Regression Analysis by Dr. Soumen Maity,Department of Mathematics,IIT Kharagpur.For more details on NPTEL visit http://nptel.ac.in
From playlist IIT Kharagpur: Regression Analysis | CosmoLearning.org Mathematics
Standard and Nonstandard Comparisons of Relative Trace Formulas - Yiannis Sakellaridis
Yiannis Sakellaridis Rutgers, The State University of New Jersey March 1, 2013 The trace formula has been the most powerful and mainstream tool in automorphic forms for proving instances of Langlands functoriality, including character relations. Its generalization, the relative trace formu
From playlist Mathematics
Sampling Techniques & Cautions (Full Length)
I define and discuss the differences of observational studies and experiments. I then discuss the difference between a sample and a census. I introduce two types of sampling techniques that yield biased results...Voluntary Response and Convenience Sampling. I discuss Stratified Random S
From playlist AP Statistics
Alternate basis transformation matrix example | Linear Algebra | Khan Academy
Courses on Khan Academy are always 100% free. Start practicing—and saving your progress—now: https://www.khanacademy.org/math/linear-algebra/alternate-bases/change-of-basis/v/lin-alg-alternate-basis-tranformation-matrix-example Example of finding the transformation matrix for an alternate
From playlist Alternate coordinate systems (bases) | Linear Algebra | Khan Academy
Two Nonstandard Infinite Geometric Series (visual proof)
My previous wordless Wednesday video showing the sum of the powers of 1/7 generated some interesting questions about using the same technique for nonstandard geometric series. Here we include a short, animated visual proof demonstrating the infinite sum of the powers of 2/7 and 3/11; from
From playlist Geometric Sums
Find a Polynomial Given B-coordinates and a Nonstandard Basis
This video explains how to determine a polynomial given the B-coordinates and a nonstandard basis.
From playlist Transformations of General Vector Spaces
Primordial Black Holes and Gravitational Waves In Non-Standard Cosmologies by Sukannya Bhattacharya
PROGRAM LESS TRAVELLED PATH OF DARK MATTER: AXIONS AND PRIMORDIAL BLACK HOLES (ONLINE) ORGANIZERS: Subinoy Das (IIA, Bangalore), Koushik Dutta (IISER, Kolkata / SINP, Kolkata), Raghavan Rangarajan (Ahmedabad University) and Vikram Rentala (IIT Bombay) DATE: 09 November 2020 to 13 Novemb
From playlist Less Travelled Path of Dark Matter: Axions and Primordial Black Holes (Online)
Find the B-coordinates of a Vector Given Coordinates Relative to Standard Basis (R3)
This video explains how to find the standard basis coordinates of a vector given a nonstandard basis and nonstandard coordinates. https://mathispower4u.com
From playlist Vectors: Change of Basis
Find the B-coordinates of a Vector Given Coordinates Relative to Standard Basis (R3)
This video explains how to find the standard basis coordinates of a vector given a nonstandard basis and nonstandard coordinates.
From playlist Vectors: Change of Basis
Discussions of circular inference (a.k.a. biased selection, a.k.a. double-dipping) and how to avoid it during statistical analyses. For more online courses about programming, data analysis, linear algebra, and statistics, see http://sincxpress.com/
From playlist OLD ANTS #8) Statistics
Find a Vector Given Basis Vectors and B-coordinates (R2)
This video explains how to find the standard basis coordinates of a vector given a nonstandard basis and nonstandard coordinates.
From playlist Vectors: Change of Basis