Sets of real numbers | Mathematical structures | Fractals | Iterated function system fractals
An ordinary fractal string is a bounded, open subset of the real number line. Such a subset can be written as an at-most-countable union of connected open intervals with associated lengths written in non-increasing order; we also refer to as a fractal string. For example, is a fractal string corresponding to the Cantor set. A fractal string is the analogue of a one-dimensional "fractal drum," and typically the set has a boundary which corresponds to a fractal such as the Cantor set. The heuristic idea of a fractal string is to study a (one-dimensional) fractal using the "space around the fractal." It turns out that the sequence of lengths of the set itself is "intrinsic," in the sense that the fractal string itself (independent of a specific geometric realization of these lengths as corresponding to a choice of set ) contains information about the fractal to which it corresponds. For each fractal string , we can associate to a geometric zeta function : the Dirichlet series . Informally, the geometric zeta function carries geometric information about the underlying fractal, particularly in the location of its poles and the residues of the zeta function at these poles. These poles of (the analytic continuation of) the geometric zeta function are then called complex dimensions of the fractal string , and these complex dimensions appear in formulae which describe the geometry of the fractal. For fractal strings associated with sets like Cantor sets, formed from deleted intervals that are rational powers of a fundamental length, the complex dimensions appear in an arithmetic progression parallel to the imaginary axis, and are called lattice fractal strings (For example, the complex dimensions of the Cantor set are , which are an arithmetic progression in the direction of the imaginary axis). Otherwise, they are called non-lattice. In fact, an ordinary fractal string is Minkowski measurable if and only if it is non-lattice. A generalized fractal string is defined to be a local positive or complex measure on such that for some , where the positive measure is the total variation measure associated to . These generalized fractal strings allow for lengths to be given non-integer multiplicities (among other possibilities), and each ordinary fractal string can be associated with a measure that makes it into a generalized fractal string. (Wikipedia).
In physics, string theory is a theoretical framework in which the point-like particles of particle physics are replaced by one-dimensional objects called strings. String theory describes how these strings propagate through space and interact with each other. On distance scales larger than
From playlist Physics
Experimenting and seeing what we can do with strings
From playlist Computer Science
What is the goal of string theory?
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From playlist Science Unplugged: String Theory
Make your own number for #MegaFavNumbers
Every string is a number. Get the number for your string at https://angyongen.github.io/MegaFavoriteNumberGenerator/ or check out the code at https://github.com/angyongen/MegaFavoriteNumberGenerator
From playlist MegaFavNumbers
stringr: Basic String Manipulation
The stringr library is part of the R tidyverse and provides a range of convenience functions for working with character strings. In this first lesson of the stringr series, we look at several basic string manipulation functions. stringr Series Code Notebook: https://www.kaggle.com/hamelg
From playlist stringr
What is the definition of scientific notation
👉 Learn about scientific notations. Scientific notation is a convenient way of writing very large or very small numbers. A number written in scientific notation is of the form a * 10^n where a is the first non-zero number between 1 and 10, (1 included) and n is the number of digits up to t
From playlist Scientific Notation | Learn About
The stringr library is part of the R tidyverse and provides a range of convenience functions for working with character strings. In this lesson, we learn how to use stringr to string interpolation: filling values into a string based on stored variables, calculations, function calls and dat
From playlist stringr
This video introduces big strings and provides the formulas need to determine the total number of n-bit strings and how to determine the number of n-bit strings with a given weight.
From playlist Counting (Discrete Math)
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From playlist Science Unplugged: String Theory
!!Con West 2019 - Michael Malis: Generating fractals … with SQL queries!!!
Presented at !!Con West 2019: http://bangbangcon.com/west SQL databases can do a lot. They are fantastic at making it easy to work with large amounts of data. One of the lesser-known capabilities of SQL databases is that they can be used to generate fractals! In this talk, we’ll take a l
From playlist !!Con West 2019
!!Con West 2019 - Michael Malis: Generating fractals … with SQL queries!!!
Presented at !!Con West 2019: http://bangbangcon.com/west SQL databases can do a lot. They are fantastic at making it easy to work with large amounts of data. One of the lesser-known capabilities of SQL databases is that they can be used to generate fractals! In this talk, we’ll take a l
From playlist !!Con West 2019
Xie Chen - Foliated Fracton and Beyond - IPAM at UCLA
Recorded 30 August 2021. Xie Chen of the California Institute of Technology presents "Foliated Fracton and Beyond" at IPAM's Graduate Summer School: Mathematics of Topological Phases of Matter. Abstract: This talk will introduce the foliation idea to characterize type I fracton models. We
From playlist Graduate Summer School 2021: Mathematics of Topological Phases of Matter
Broadcasted live on Twitch -- Watch live at https://www.twitch.tv/simuleios
From playlist research
Recursively Defined Sets - An Intro
Recursively defined sets are an important concept in mathematics, computer science, and other fields because they provide a framework for defining complex objects or structures in a simple, iterative way. By starting with a few basic objects and applying a set of rules repeatedly, we can g
From playlist All Things Recursive - with Math and CS Perspective
Order Within Chaos: How this game creates a fractal | Summer of Math Exposition #1
How does a seemingly random universe create something seemingly concrete and certain? In this video, we will unpack the reasoning behind this phenomena, while exploring the field of fractals. Chapters: 0:00 Introduction 0:25 The Chaos Game 1:44 Sierpinski's Triangle 3:14 Self-simil
From playlist Summer of Math Exposition Youtube Videos
8.5: L-Systems - The Nature of Code
This video covers the basics of L-System algorithms and how they can be applied to "turtle graphics" drawing in Processing. http://natureofcode.com Contact: http://twitter.com/shiffman/ (If I reference a link or project and it's not included in this description, please let me know!) Re
From playlist The Nature of Code: Simulating Natural Systems
Geometer Explains One Concept in 5 Levels of Difficulty | WIRED
Computer scientist Keenan Crane, PhD, is asked to explain fractals to 5 different people; a child, a teen, a college student, a grad student, and an expert. Still haven’t subscribed to WIRED on YouTube? ►► http://wrd.cm/15fP7B7 Listen to the Get WIRED podcast ►► https://link.chtbl.com
From playlist Tutorials and Lectures
!!Con 2016 - Plants are Recursive!!: Using L-Systems to Generate Realistic Weeds By Sher Minn Chong
Plants are Recursive!!: Using L-Systems to Generate Realistic Weeds By Sher Minn Chong Plants and programming are more related than you might realize! Trees, shrubs and weeds may look complex, but with the magic of recursion, we can describe them with just a few simple rules. We'll explo
From playlist !!Con 2016
David Aasen - Topological Defect Networks for Fracton Models - IPAM at UCLA
Recorded 30 August 2021. David Aasen of Microsoft Station Q presents "Topological Defect Networks for Fracton Models" at IPAM's Graduate Summer School: Mathematics of Topological Phases of Matter. Learn more online at: http://www.ipam.ucla.edu/programs/summer-schools/graduate-summer-school
From playlist Graduate Summer School 2021: Mathematics of Topological Phases of Matter