An addition-subtraction chain, a generalization of addition chains to include subtraction, is a sequence a0, a1, a2, a3, ... that satisfies An addition-subtraction chain for n, of length L, is an addition-subtraction chain such that . That is, one can thereby compute n by L additions and/or subtractions. (Note that n need not be positive. In this case, one may also include a−1 = 0 in the sequence, so that n = −1 can be obtained by a chain of length 1.) By definition, every addition chain is also an addition-subtraction chain, but not vice versa. Therefore, the length of the shortest addition-subtraction chain for n is bounded above by the length of the shortest addition chain for n. In general, however, the determination of a minimal addition-subtraction chain (like the problem of determining a minimum addition chain) is a difficult problem for which no efficient algorithms are currently known. The related problem of finding an optimal addition sequence is NP-complete (Downey et al., 1981), but it is not known for certain whether finding optimal addition or addition-subtraction chains is NP-hard. For example, one addition-subtraction chain is: , , , . This is not a minimal addition-subtraction chain for n=3, however, because we could instead have chosen . The smallest n for which an addition-subtraction chain is shorter than the minimal addition chain is n=31, which can be computed in only 6 additions (rather than 7 for the minimal addition chain): Like an addition chain, an addition-subtraction chain can be used for addition-chain exponentiation: given the addition-subtraction chain of length L for n, the power can be computed by multiplying or dividing by x L times, where the subtractions correspond to divisions. This is potentially efficient in problems where division is an inexpensive operation, most notably for exponentiation on elliptic curves where division corresponds to a mere sign change (as proposed by Morain and Olivos, 1990). Some hardware multipliers multiply by n using an addition chain described by n in binary: n = 31 = 0 0 0 1 1 1 1 1 (binary). Other hardware multipliers multiply by n using an addition-subtraction chain described by n in Booth encoding: n = 31 = 0 0 1 0 0 0 0 −1 (Booth encoding). (Wikipedia).

Determine a Subtraction Problem Modeled on a Number Line

This video explains how to write an subtraction equation from a number line model. http://mathispower4u.com

From playlist Addition and Subtraction of Whole Numbers

3B Addition and Subtraction Examples-YouTube sharing.mov

Some examples of matrix addition and subtraction.

From playlist Linear Algebra

An introduction to subtraction, the terms and concepts involved, and subtraction as the opposite of addition. Some example problems are carefully worked and explained. From the Prealgebra course by Derek Owens. This course is available online at http://www.LucidEducation.com.

From playlist Prealgebra Chapter 1 (Complete chapter)

The Addition and Subtraction Formulas for Trig are AWESOME PART 2

The Addition and Subtraction Formulas for Trig are AWESOME! In this first video, we look at some of the patterns used to remember them. #Trigisawesome For other videos, check this playlist: https://youtube.com/playlist?list=PLntYGYK-wJE3TbusouawjDYeSKHAO9Oms

From playlist Addition and Subtraction Trig Formulas

Basic Whole number Subtraction Using the Number Line

The video explains how to model basic whole number subtraction using the number line.

From playlist Addition and Subtraction of Whole Numbers

Addition and Subtraction of Fractions

http://mathispower4u.wordpress.com/

From playlist Adding and Subtracting Fractions

Solving a one step equation with subtraction

👉 Learn how to solve a one step equation. An equation is a statement stating that two values are equal. A one step equation is an equation whose solution can be obtained by performing only one step of operation on the equation. To solve a one step addition/subtraction equation, we isolate

From playlist How to Solve One Step Equations with Addition

Subtract two vectors algebraically and numerically

Learn how to add/subtract vectors. Vectors can be added, subtracted and multiplied. To add or subtract two or more vectors, we simply add each of the corresponding components of the vectors. #trigonometry#vectors #vectors

From playlist Vectors

Ex: Matrix Addition and Subtraction

This video explains how to perform matrix addition and subtraction. Site: http://mathispower4u.com

From playlist Introduction to Matrices and Matrix Operations

Chain of Responsibility Design Pattern

Get the Code: http://goo.gl/hpssM Welcome to my Chain of Responsibility Design Pattern Tutorial! Wow, that was a mouthful! This pattern has a group of objects that are expected to between them be able to solve a problem. If the first Object can't solve it, it passes the data to the next

From playlist Java Video Tutorial

Level 1 Chartered Financial Analyst (CFA ®): Holding period, money-, and time-weighted returns

Session 2, Reading 7 (Part 2): The holding period return (HPR) is given by [P(t) + D - P(0)]/P(0). The HPR does not account for the time interval, so importantly it is annualized; for example, a 15.50% HPR over 5 years is much less impressive than over one month. The time-weighted return (

From playlist Level 1 Chartered Financial Analyst (CFA ®) Volume 1

MATH1011 Calculus Section 2.3 Questions 2 to 5 (a selection)

In this video we apply the rules of differentiation to calculate a range of derviatives. Presented by Thanom Shaw of the School of Mathematics and Statistics, UNSW.

From playlist MATH1011 Calculus Problems

How 3D Printing Could Decentralize (Almost) Everything

This is how 3D printing could solve shortages and decentralize production. https://www.patreon.com/thehatedone Monero: 84DYxU8rPzQ88SxQqBF6VBNfPU9c5sjDXfTC1wXkgzWJfVMQ9zjAULL6rd11ASRGpxD1w6jQrMtqAGkkqiid5ef7QDroTPp Bitcoin: 1FuKzwa5LWR2xn49HqEPzS4PhTMxiRq689 Ethereum: 0x6aD936198f87582

From playlist Analyses

⛓→∂🤖 [Chain Rule to Auto Diff] - How to Rewrite Chain Rule to Auto Differentiation(Auto Grad) #SoME1

We'll code a basic version of automatic differentiation, starting from a version of the chain rule. This video bridges a definition of the chain rule from mathematics to an implementation of auto diff in computer science. 'Auto Diff' is what makes training many machine learning models pos

From playlist Summer of Math Exposition Youtube Videos

Two-dimensional power law behavior in one-dimensional NiTe2O5 by Yoon Seok Oh

Program The 2nd Asia Pacific Workshop on Quantum Magnetism ORGANIZERS: Subhro Bhattacharjee, Gang Chen, Zenji Hiroi, Ying-Jer Kao, SungBin Lee, Arnab Sen and Nic Shannon DATE: 29 November 2018 to 07 December 2018 VENUE: Ramanujan Lecture Hall, ICTS Bangalore Frustrated quantum magne

From playlist The 2nd Asia Pacific Workshop on Quantum Magnetism

Johan Segers: Modelling multivariate extreme value distributions via Markov trees

CONFERENCE Recording during the thematic meeting : "Adaptive and High-Dimensional Spatio-Temporal Methods for Forecasting " the September 26, 2022 at the Centre International de Rencontres Mathématiques (Marseille, France) Filmmaker: Guillaume Hennenfent Find this video and other talks

From playlist Probability and Statistics

❖ Lots of Different Derivative Examples! ❖

Thanks to all of you who support me on Patreon. You da real mvps! $1 per month helps!! :) https://www.patreon.com/patrickjmt !! Please consider supporting me on Patreon! Be a Patron of Mathematics! https://www.patreon.com/patrickjmt?ty=h Buy my book!: '1001 Calculus Problems for Dummies

From playlist Calculus / First Semester - Limits, Continuity, Derivatives

Lec 11 | MIT 7.012 Introduction to Biology, Fall 2004

Molecular Biology 2 (Prof. Eric Lander) View the complete course: http://ocw.mit.edu/7-012F04 License: Creative Commons BY-NC-SA More information at http://ocw.mit.edu/terms More courses at http://ocw.mit.edu

From playlist MIT 7.012 Introduction to Biology, Fall 2004

Solving a one step equation using subtraction

👉 Learn how to solve a one step equation. An equation is a statement stating that two values are equal. A one step equation is an equation whose solution can be obtained by performing only one step of operation on the equation. To solve a one step addition/subtraction equation, we isolate

From playlist How to Solve One Step Equations with Addition

What is a Manifold? Lesson 9: The Tangent Space-Definition

What is a Manifold? Lesson 9: The Tangent Space-Definition This lesson is longer than the others because it is rather technical. I made a slip up at about minute 56 which is annotated in the video and mentioned in the first comment.

From playlist What is a Manifold?