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Yet another triangle - DFNU
23 September 2019
Yet another triangle
As we have seen in another post, identities involving inverse tangent look much less mysterious if analyzed from a purely geometrical perspective. Here you have the chance to further practice on the
subject and to demonstrate a more general formula. Let’s start with the relationship
$$\arctan x + \arctan\left(\frac{x+1}{x-1}\right) =\frac{3\pi}{4}, \ \ \mbox{for}\ \ x>1.\tag{1}\label{eq630:1}$$
We will start again with the now ubiquitous right-angled triangle \(\triangle ABC\) with sides \(\overline{AB} = 1\), \(\overline{BC} = x\), and hypothenuse \(\overline{AC} = \sqrt{1+x^2}\).
1. Extend \(AB\) with a segment \(\overline{BD} = x\).
2. Triangle \(\triangle BCD\) is isosceles. What can you say, then, about the angle \(\gamma = \angle ADC\)?
3. Use what you observed, and the fact that the sum of the interior angles of \(\triangle ACD\) is \(\pi\), to write a relationship connecting \(\alpha\) and \(\beta\). You should get \[\beta = \
4. Draw from \(D\) the line perpendicular to \(AC\), that intersects \(AC\) in \(H\).
5. \(\triangle ABC\) and \(\triangle ADH\) are similar. Why?
6. From the scale factor between these triangles and the fact that \(\overline{AD} = 1+x\), determine \(\overline{DH}\), \(\overline{AH}\), and finally \(\overline{CH} = \overline{AC}-\overline{DH}
7. Write down \(\beta\) as \(\arctan\left(\frac{\overline{DH}}{\overline{CH}}\right)\).
8. Use the relationship found in 3. to get your final result.
If you followed without difficulties the previous demonstration, you should now be able to prove also the following identity.
$$\arctan x + \arctan\left(\frac{\sqrt 3 x+1}{x-\sqrt 3}\right)=\frac{5\pi}{6}, \ \ \mbox{for} \ \ x >\sqrt 3.\tag{2}\label{eq630:2}$$
It is just a matter of modifying the measure of the angle \(\gamma\), and make it equal to \(\frac{\pi}{6}\). What is then the measure of \(\overline{BD}\)? Recall that \(\frac{\pi}{3}\) is the
measure of the interior angles of an equilateral triangle. Procede from here as you have done before, in order to prove \eqref{eq630:2}.
You might be wondering now whether the expressions \eqref{eq630:1} and \eqref{eq630:2} can be further generalized. After all \(\gamma\) can be arbitrarily chosen, and in general we would have \(\
overline{BD} = \frac{x}{\tan\gamma}\). Viceversa, if we take \(\overline{BD} = \frac{x}{y}\), we will get \(\gamma = \arctan y\).
Use the latter notation and generalize the procedure that you adopted so far. Recalling that
\[\beta = \pi – \arctan x – \arctan y,\]
obtain the following identity
\[\arctan x + \arctan y = \pi – \arctan\left(\frac{x+y}{xy-1}\right),\]
which is valid for \(x>\frac{1}{y}>0\).
Scrivi un commento - Leave a comment
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Journals and Full-Length Conference Papers
Preprints and Technical Reports
Talks and Conferences
• H. Ranocha. Modern discontinuous Galerkin methods for atmospheric physics. Mathematics of the Weather 2024, Bad Orb (Germany), October 2024.
• H. Ranocha. Modern discontinuous Galerkin methods for compressible fluid flows. Seminar Mathematics and Atmospheric Physics, Online Seminar, April 2024.
• H. Ranocha. Energy-preserving numerical methods for some dispersive shallow water models. ALGORITMY 2024, High Tatra Mountains, Slovakia, March 2024.
• H. Ranocha. Structure-preserving numerical methods for nonlinear dispersive wave equations. Séminaire de Calcul Scientifique et Modélisation, Université de Bordeaux (France), February 2024.
• H. Ranocha. Adaptive and structure-preserving numerical methods. Computational Mathematics Seminar, Hasselt University (Belgium), September 2023.
• M. Schlottke-Lakemper, H. Ranocha. Scaling Trixi.jl to more than 10,000 cores using MPI. JuliaCon 2023, MIT, Cambridge (USA), July 2023.
• H. Ranocha. Efficient and robust numerical methods based on adaptivity and structure preservation. Seminar of Computational and Numerical Mathematics, TU Hamburg-Harburg (TUHH, Germany), July
• H. Ranocha. Structure-preserving numerical methods for dispersive wave equations. Keynote talk in the section S18: Numerical methods for differential equations. GAMM Annual Meeting 2023, Dresden
(Germany), May 2023.
• H. Ranocha. Tutorial on Julia and Trixi.jl. Seminar on Numerical Methods for PDEs, Kassel University (Germany), May 2023.
• H. Ranocha. Some results on stability properties of discretizations of transport equations. Applied Dynamical Systems Seminar, Hamburg University (Germany), March 2023.
• H. Ranocha. Structure-preserving time integration methods based on relaxation. Numerical Analysis Seminar, Lund Univeristy (Sweden), February 2023.
• H. Ranocha. Robust and efficient high-performance computational fluid dynamics enabled by modern numerical methods and technologies. MUSEN center for Mechanics, Uncertainty and Simulation in
ENgineering, TU Braunschweig (Germany), November 2022.
• M. Schlottke-Lakemper, H. Ranocha. Reproducibility as a service: collaborative scientific computing with Julia. MaRDI Workshop for Scientific Computing, Münster (Germany), October 2022.
• Efficient and Robust Time Integration with Automatic Step Size Control for Compressible Computational Fluid Dynamics. GAMM Annual Meeting 2022, Aachen (Germany), August 2022.
• Efficient and robust step size control for computational fluid dynamics. Workshop on efficient high-order time discretization methods for PDEs (PDETD22), Anacapri (Italy), May 2022.
• Analysis Meets Data: Efficient Implementation and Optimized Time Integration Methods with Automatic Step Size Control for Compressible Computational Fluid Dynamics. CSE Workshop on Modeling,
Simulation & Optimization of Fluid Dynamic Applications, Groß Schwansee (Germany), March 2022.
• M. Schlottke-Lakemper, H. Ranocha. Research software development with Julia. NFDI4Ing Conference, Virtual conference of the German National Research Data Infrastructure, September 2021.
• On stability of positivity-preserving Patankar-type time integration methods. Bound-Preserving Space and Time Discretizations for Convection-Dominated Problems, Casa Matemática Oaxaca (CMO,
Mexico), August 2021.
• M. Schlottke-Lakemper, H. Ranocha. Adaptive and extendable numerical simulations with Trixi.jl. JuliaCon, Virtual congress, July 2021.
• Combining Analysis and Data: Optimized Runge-Kutta Methods with Automatic Step Size Control for Compressible Computational Fluid Dynamics. Applied Mathematics Seminar, University of Münster
(Germany), July 2021.
• H. Ranocha, M. Schlottke-Lakemper, A. R. Winters. Tutorial on Trixi.jl: Adaptive high-order numerical simulations of hyperbolic PDEs in Julia. International Conference on Spectral and High Order
Methods (ICOSAHOM 2020), Virtual congress (originally scheduled in Vienna, Austria), July 2021.
• H. Ranocha, M. Quezada de Luna, D. Mitsotakis, D. I. Ketcheson. Summation by parts methods for nonlinear dispersive wave equations. International Conference on Spectral and High Order Methods
(ICOSAHOM 2020), Virtual congress (originally scheduled in Vienna, Austria), July 2021.
• H. Ranocha, P. Öffner, R. Abgrall. Entropy Corrections and Related Methods. International Conference on Spectral and High Order Methods (ICOSAHOM 2020), Virtual congress (originally scheduled in
Vienna, Austria), July 2021.
• Optimized Runge-Kutta Methods with Automatic Step Size Control for Compressible Computational Fluid Dynamics. Seminar at the Institute for Numerical Simulation, University of Cologne (Germany),
May 2021.
• Introduction to Julia and Trixi, a numerical simulation framework for hyperbolic PDEs. Applied Mathematics Seminar, University of Münster (Germany), April 2021.
• Fully-Discrete Entropy-Conservative and -Dissipative Methods Based on Relaxation. SIAM Conference on Computational Science and Engineering (CSE21, Virtual congress, March 2021.
• Recent results on time integration methods for summation by parts schemes. World Congress in Computational Mechanics and ECCOMAS Congress (WCCM-ECCOMAS 2020), Virtual congress (originally
scheduled in Paris, France), January 2021.
• Structure-preserving numerical methods with applications in science and engineering. Seminar at the Institute for Numerical Simulation, University of Bonn (Germany), November 2020.
• Physics-compatible high-order time integration methods for transport phenomena based on relaxation. Modeling and Simulation of Transport Phenomena (MoST 2020), Treis-Karden (Germany) and online,
October 2020.
• General relaxation methods for initial-value problems Online seminar "Stable and Efficient Time Integration Schemes for Conservation Laws and Related Models", organized by Philipp Öffner and me,
July 2020.
• P. Heinisch, K. Ostaszewski, H. Ranocha. Towards Green Computing: A Survey of Performance and Energy Efficiency of Different Platforms using OpenCL. Proceedings of the International Workshop on
OpenCL. IWOCL '20, April 2020, Munich (Germany). New York, NY, USA: ACM, 2020. arXiv:2003.03794 [CS.PF]. [bibtex]
• Energy and Entropy in Numerical Methods: Structure Preserving Schemes with Applications in Science and Engineering. Computer, Electrical and Mathematical Sciences and Engineering Seminar, King
Abdullah University of Science and Technology (KAUST), Thuwal (Saudi Arabia), February 2020.
• R. Abgrall, E. Mélédo, P. Öffner, H. Ranocha. Error Boundedness of Correction Procedure via Reconstruction/Flux Reconstruction and the Connection to Residual Distribution Schemes.. Hyperbolic
Problems: Theory, Numerics, Applications. Ed. by A. Bressan, M. Lewicka, D. Wang, Y. Zheng. Vol. 10. AIMS on Applied Mathematics. Springfield: American Institute of Mathematical Sciences, 2020,
pp. 215-222. [bibtex]
• Energy Stability of Runge-Kutta Methods and a Relaxation Approach. Rémi Abgrall Group Internal Seminar, Zürich (Switzerland), December 2019.
• On Strong Stability of Runge-Kutta Methods. Computer, Electrical and Mathematical Sciences and Engineering Seminar, King Abdullah University of Science and Technology (KAUST), Thuwal (Saudi
Arabia), April 2019.
• On Strong Stability of Explicit Runge-Kutta Methods for Nonlinear Problems. VII European Workshop on High Order Numerical Methods for Evolutionary PDEs: Theory and Applications (HONOM), Madrid
(Spain), April 2019.
• High-Order Methods on Summation by Parts Form for the Magnetic Induction Equation. VII European Workshop on High Order Numerical Methods for Evolutionary PDEs: Theory and Applications (HONOM),
Madrid (Spain), April 2019.
• Entropy Conserving and Kinetic Energy Preserving Numerical Methods for the Euler Equations Using Summation-by-Parts Operators. International Conference on Spectral and High Order Methods
(ICOSAHOM), London (United Kingdom), July 2018. [bibtex]
• K. Ostaszewski, P. Heinisch, H. Ranocha. Advantages and Pitfalls of OpenCL in Computational Physics. Proceedings of the International Workshop on OpenCL. IWOCL '18, May 2018, Oxford (United
Kingdom). New York, NY, USA: ACM, 2018, p. 10:1. [bibtex]
• Überblick über mögliche Probleme numerischer Verfahren für Kontinuumsgleichungen. Oberseminar Institut für Geophysik und extraterrestrische Physik, TU Braunschweig (Germany), January 2018.
• Generalised Summation-by-Parts Operators, Entropy Stability, and Split Forms. Numerical Analysis Group Internal Seminar, Oxford (United Kingdom), October 2017.
• J. Glaubitz, P. Öffner, H. Ranocha, T. Sonar. Artificial Viscosity for Correction Procedure via Reconstruction Using Summation-by-Parts Operators.. Theory, Numerics and Applications of Hyperbolic
Problems II. Ed. by C. Klingenberg, M. Westdickenberg. Vol. 237. Springer Proceedings in Mathematics & Statistics. Cham: Springer International Publishing, 2018, pp. 363-375. [bibtex]
• Correction Procedure via Reconstruction Using Summation-by-Parts Operators. International Conference on Hyperbolic Problems: Theory, Numerics, Applications (HYP), Aachen (Germany), August 2016.
P. Öffner, H. Ranocha, T. Sonar. Correction Procedure via Reconstruction Using Summation-by-Parts Operators.. Theory, Numerics and Applications of Hyperbolic Problems II. Ed. by C. Klingenberg,
M. Westdickenberg. Vol. 237. Springer Proceedings in Mathematics & Statistics. Cham: Springer International Publishing, 2018, pp. 491-501. [bibtex]
• Summation-by-Parts and Correction Procedure via Reconstruction. International Conference on Spectral and High Order Methods (ICOSAHOM), Rio de Janeiro (Brazil), June 2016.
H. Ranocha, P. Öffner, T. Sonar. Summation-by-Parts and Correction Procedure via Reconstruction.. Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2016. Ed. by M. L.
Bittencourt, N. A. Dumont, J. S. Hesthaven. Vol. 119. Lecture Notes in Computational Science and Engineering. Cham: Springer, 2017, pp. 627-637. [bibtex]
• Correction procedure via reconstruction using summation-by-parts operators. Vincent Lab Internal Seminar, Imperial College London (United Kingdom), April 2016. | {"url":"https://ranocha.de/publications","timestamp":"2024-11-11T05:19:26Z","content_type":"text/html","content_length":"54635","record_id":"<urn:uuid:62bc349c-0149-4fd0-a5f0-830cbf8eb5b1>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00735.warc.gz"} |
Newton’s Law in Polar Coordinates Lecture 6 - Pravegaa
Newton’s Law in Polar Coordinates Lecture 6
Newton’s Laws of Motion can be expressed in various coordinate systems, one of which is the polar coordinate system. Polar coordinates are particularly useful in problems involving circular or
rotational motion, where the use of Cartesian coordinates (x, y) might be cumbersome. In polar coordinates, a position is described by the radial distance r from the origin and the angular coordinate
θ which is the angle measured from a reference direction.
HD Quality Classroom Videos,Topic Wise Tests with Complete Study Material & Previous year Questions,Tutorial & Worksheet, lifetime Access to videos.In addition, we run IIT-JAM Physics coaching in
Delhi. Pravegaa Education is the best online coaching for IIT JAM physics. JEST, TIFR, JNU, GRE, and other entrance examinations for M.Tech. and Ph.D. programs are an integral part of our course.
Pravegaa has started classroom programme for CSIR-NET Physics, IIT-JAM Physics, JEST, TIFR, JNU, BHU, DU, and other universities for physics and physical sciences.
Example: Planetary Motion
In planetary motion, the gravitational force is a central force, and the equations of motion in polar coordinates can be used to derive Kepler’s laws of planetary motion.
In summary, expressing Newton’s Laws in polar coordinates provides a powerful framework for solving problems involving rotational or circular motion, where the motion is naturally described in terms
of radial and angular components. | {"url":"https://pravegaa.com/newtons-law-in-polar-coordinates-lecture-6/","timestamp":"2024-11-04T12:13:56Z","content_type":"text/html","content_length":"163854","record_id":"<urn:uuid:5206523f-fbe8-4dd8-8a37-88507774ba94>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00026.warc.gz"} |
Matific BlogHow Math Journals Help Students Become Deep Thinkers and Active Learners
How Math Journals Help Students Become Deep Thinkers and Active Learners
Have you ever kept a journal?
If you have, you know that the simple act of writing out your thoughts can help you process them.
This works for kids, too!
Writing is important because it helps us solidify our thinking. Not only that, but journaling helps individuals think about what they are learning.
Math journals allow students to communicate their ideas and thoughts about math. It gives them independence, helps them refine their thinking, and gives them the opportunity to see their growth.
Yet, what exactly are the concrete benefits of math journals for elementary students?
Benefits of Math Journals for Students
Research into the use of math journals for elementary students have proven that there are various benefits in their use. Here are a few specific benefits:
Build Math Vocabulary: When children write about math, they need to use math specific vocabulary. Even when class discussions happen in the classroom, kids don’t all get the chance to speak. Using
math journals gives all children the opportunity to practice using math vocabulary appropriately. Research shows that students who used math journals boosted their use of math vocabulary.
Self-Assessment: When students journal, they can reflect on their own achievements, what they’re struggling with, as well as what they’re curious about. Not only does this help them see their own
growth, but it can help students build motivation.
Supports Problem-Solving: Research shows that writing rather than speaking about math problems is more supportive of metacognition in problem-solving. In other words, children can grow in their
understanding of math by writing about it!
An Amazing Teacher Tool: Math journals can be a fantastic way for teachers to assess student knowledge, identify areas for improvement, and more! When teachers are able to read through students’
thought processes, they can gain insights into how students understand concepts, and more.
These are just a few advantages to using math journals in the classroom. But, how do you use math journals effectively?
How to Use Math Journals Effectively
To use math journals effectively, you have to do more than just hand students notebooks and ask them to write about math.
First, there are the practical aspects. What should your students’ journals look like? Math journals can be physical or digital. One app that you can use for this is Notability.
You may also consider color coding your students’ math journals. For example, have students outline writing about counting with a red marker. Or, use blue post-its to mark operations & algebra.
Then, it’s important to consider what students will write about and record in their math journals. Journal entries can be used at any time during a math workshop. However, these entries should be
guided by journal prompts and sentence starters so that students know exactly what to write about. Also, it’s best to use separate scratch paper for writing out calculations rather than the math
journal. That way, the journal can be neat and organized.
There are four main sections of writing that teachers can have their students do in their math journals:
Problem-solving: Problem-solving allows for students to think of different solutions, entry points, and strategies.
Some journal prompts could be:
• Explain two different ways to solve this problem.
• Draw a picture to represent this problem.
Vocabulary: Math vocabulary allows students to refer to concepts. Students can also create their own definitions of concepts for teachers to see if they are understanding the concepts. Some math
journal prompts in this category might be:
• Write everything you know about multiplication.
• Explain in writing how you would solve this problem.
Strategies: Working with strategies give students a better understanding of concepts by allowing them to clarify thinking. Some examples of journal prompts in this type of writing might be:
• How did you know which operation to use to solve the problem?
• Could you solve the problem another way? How?
• How did you decide what to do first?
• Would you use skip counting or addition to solve this problem? Why?
My Thinking: This area puts everything together and gives students the opportunity to reflect on what they know. From evaluating their knowledge to linking math concepts to real life situations, this
category of writing is broad. Here are some journal prompts to include:
• In today’s math lesson, I learned...
• A way this sort of problem might show up in real life is when...
• How can you check if this answer is correct?
If you need help finding math problems and activities to use as the subject of your journal prompts, try Matific activities! There are thousands of activities and problems to use in all areas of your
math curriculum.
When it comes to using math journals, you’ll also need to consider your workflow. Students love it when you respond to their journal entries and add feedback! So, make sure you pick a regular time to
review journals. This is also valuable because math journals are a window into a student's thinking. You can even use them when determining student grades and assessing student knowledge.
Other Strategies for Using Student Journals
Students may have trouble getting started with their math journaling. So, you should take some time to demonstrate the process and what you expect from students. An anchor chart may also help
students who struggle to get started with journaling.
Some teachers also take tried and true classroom strategies and adapt them for math journals. For example, have you heard of “Think-Pair-Share”? It’s easy to adapt this to “Think-Pair-Write.” In this
scenario, students think about a math problem, talk with a partner about how they might solve it, and then write out their thoughts about it. This can give students the opportunity to share ideas and
gain feedback before writing. As students grow in confidence, they may need this sort of support less and less.
Student entries may also be read aloud anonymously. Then, the class can observe where more details or explanations could have been added. This helps the whole group improve their thinking without
creating an anxiety-inducing situation by putting one student on the spot.
The Bottom Line on Math Journals
If you aren’t using math journals in your classroom yet, why not give it a try? They are easy to implement, highly beneficial, and help students not only in math, but with language arts.
Once you do implement math journals, you may be surprised by how much your students learn...and how much you learn about them! | {"url":"https://www.matific.com/us/en-us/home/blog/2022/01/18/how-math-journals-help-students-become-deep-thinkers-and-active-learners/","timestamp":"2024-11-05T15:10:45Z","content_type":"text/html","content_length":"137637","record_id":"<urn:uuid:24c16d02-d1ef-42f1-8e97-e508ea874bdb>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00592.warc.gz"} |
Run-Length Encoding - Amazon Top Interview Questions | HackerRank Solutions
Run-Length Encoding - Amazon Top Interview Questions
Problem Statement :
Given a string s, return its run-length encoding. You can assume the string to be encoded have no digits and consists solely of alphabetic characters.
n ≤ 100,000 where n is the length of s
Example 1
s = "aaaabbbccdaa"
Example 2
s = "abcde"
Example 3
s = "aabba"
Example 4
s = "aaaaaaaaaa"
Solution :
Solution in C++ :
string solve(string s) {
string answer;
int n = s.size();
int j = 0;
int count = 0;
for (int i = 0; i < n;) {
char character = s[i];
while (j < n && s[j] == character) {
answer.append(to_string(j - i));
answer.append(1, character);
i = j;
return answer;
Solution in Java :
import java.util.*;
class Solution {
public String solve(String s) {
if (s.length() == 0)
return s;
StringBuilder result = new StringBuilder();
char preChar = s.charAt(0);
int count = 0;
for (int i = 0; i < s.length(); i++) {
if (s.charAt(i) == preChar)
else {
preChar = s.charAt(i);
count = 1;
return result.toString();
Solution in Python :
class Solution:
def solve(self, s):
return "".join([str(len(list(j))) + str(i) for i, j in groupby(s)])
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Superradiant phenomena - Lessons from and for Bose-Einstein condensates
The work of this thesis is guided by the Analogue Gravity research programme, in which condensed matter systems are used as analogues of the physics of curved spacetimes to obtain new perspectives on
open problems of gravitational physics. Here we use this idea to investigate the phenomenon of superradiance, most famously occurring in rotating black hole spacetimes, using as an analogue system
atomic Bose-Einstein condensates (BECs). Superradiance is a radiation enhancement phenomenon in which waves of different kind are scattered with an increased amplitude by extracting energy from the
object they are scattering on. In this thesis on the one hand we use the gravitational analogy to understand better superradiance starting from easier analogue setups, and on the other hand we use
concepts coming from superradiance to learn something about the physics of BECs. We first present a (possibly realizable) toy model, built using the tools of synthetic gauge fields for neutral atoms,
to provide a new and conceptually simple illustration of superradiant scattering. This toy model allows to disentangle the different elements at play and highlight the basic mechanisms of
superradiance and has also the interesting feature of being exactly mappable to a scattering problem of a charged scalar field on an electrostatic potential. We also show how at the quantum level,
superradiance implies the spontaneous emission of pairs of excitations. The low temperatures of atomic condensates can make these quantum features visible and we propose a way of detecting them via
correlation measurements. Another realization of this toy model can also be built using periodic trapping potentials for the atoms. By changing the boundary conditions of the acoustic excitations of
the condensate we show how superradiance can give rise to dynamical instabilities. Our toy model gives a simple illustration of superradiant instabilities occurring in rotating gravitational
spacetimes, in particular ergoregion instabilities and black hole bombs. It also provides a realization of the analogous instabilities involving a charged scalar field, called the
Schiff-Snyder-Weinberg effect. Our approach naturally shows how amplified scattering can also occur in the presence of dynamical instabilities, a point often object of confusion in the literature.
Moreover, we add an acoustic horizon to our toy model and show that, differently from what happens in general relativity, horizons do not always prevent the presence of ergoregion instabilities. We
then apply these concepts to the study of the stability of quantized vortices in two-dimensional BECs. With a careful account of boundary conditions, we show that the dynamical instability of
multiply quantized vortices in trapped condensates persists in untrapped, spatially homogeneous geometries and has an ergoregion nature with some modification due to the peculiar dispersion of
Bogoliubov sound. Our results open new perspectives to the physics of vortices in trapped condensates, where multiply quantized vortices can be stabilized by interference effects and singly charged
vortices can become unstable in suitably designed trap potentials. We show how superradiant scattering can be observed also in the short-time dynamics of dynamically unstable systems, providing an
alternative point of view on dynamical (in)stability phenomena in spatially finite systems. Finally we consider the equivalent of a shear layer between parallel flows in hydrodynamics, but in a BEC.
In the present case the shear layer is constituted by and array of quantized vortices that are shown to develop an instability analogous to the Kelvin-Helmholtz instability. When the relative
velocity between the two parallel flow is sufficiently large however, this instability is quenched and substituted by a slower instability that has the features of the superradiant instabilities we
studied. Differently from superradiant instabilities, this one also remains with open boundary conditions on the two sides of the shear layer, and manifests itself as a continuous emission of phonons
in both directions; we call this new regime radiative instability.
Superradiant phenomena - Lessons from and for Bose-Einstein condensates / Giacomelli, Luca. - (2021 Mar 04), pp. 1-177. [10.15168/11572_294551]
Superradiant phenomena - Lessons from and for Bose-Einstein condensates
The work of this thesis is guided by the Analogue Gravity research programme, in which condensed matter systems are used as analogues of the physics of curved spacetimes to obtain new perspectives on
open problems of gravitational physics. Here we use this idea to investigate the phenomenon of superradiance, most famously occurring in rotating black hole spacetimes, using as an analogue system
atomic Bose-Einstein condensates (BECs). Superradiance is a radiation enhancement phenomenon in which waves of different kind are scattered with an increased amplitude by extracting energy from the
object they are scattering on. In this thesis on the one hand we use the gravitational analogy to understand better superradiance starting from easier analogue setups, and on the other hand we use
concepts coming from superradiance to learn something about the physics of BECs. We first present a (possibly realizable) toy model, built using the tools of synthetic gauge fields for neutral atoms,
to provide a new and conceptually simple illustration of superradiant scattering. This toy model allows to disentangle the different elements at play and highlight the basic mechanisms of
superradiance and has also the interesting feature of being exactly mappable to a scattering problem of a charged scalar field on an electrostatic potential. We also show how at the quantum level,
superradiance implies the spontaneous emission of pairs of excitations. The low temperatures of atomic condensates can make these quantum features visible and we propose a way of detecting them via
correlation measurements. Another realization of this toy model can also be built using periodic trapping potentials for the atoms. By changing the boundary conditions of the acoustic excitations of
the condensate we show how superradiance can give rise to dynamical instabilities. Our toy model gives a simple illustration of superradiant instabilities occurring in rotating gravitational
spacetimes, in particular ergoregion instabilities and black hole bombs. It also provides a realization of the analogous instabilities involving a charged scalar field, called the
Schiff-Snyder-Weinberg effect. Our approach naturally shows how amplified scattering can also occur in the presence of dynamical instabilities, a point often object of confusion in the literature.
Moreover, we add an acoustic horizon to our toy model and show that, differently from what happens in general relativity, horizons do not always prevent the presence of ergoregion instabilities. We
then apply these concepts to the study of the stability of quantized vortices in two-dimensional BECs. With a careful account of boundary conditions, we show that the dynamical instability of
multiply quantized vortices in trapped condensates persists in untrapped, spatially homogeneous geometries and has an ergoregion nature with some modification due to the peculiar dispersion of
Bogoliubov sound. Our results open new perspectives to the physics of vortices in trapped condensates, where multiply quantized vortices can be stabilized by interference effects and singly charged
vortices can become unstable in suitably designed trap potentials. We show how superradiant scattering can be observed also in the short-time dynamics of dynamically unstable systems, providing an
alternative point of view on dynamical (in)stability phenomena in spatially finite systems. Finally we consider the equivalent of a shear layer between parallel flows in hydrodynamics, but in a BEC.
In the present case the shear layer is constituted by and array of quantized vortices that are shown to develop an instability analogous to the Kelvin-Helmholtz instability. When the relative
velocity between the two parallel flow is sufficiently large however, this instability is quenched and substituted by a slower instability that has the features of the superradiant instabilities we
studied. Differently from superradiant instabilities, this one also remains with open boundary conditions on the two sides of the shear layer, and manifests itself as a continuous emission of phonons
in both directions; we call this new regime radiative instability.
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Optimistic MLE: A Generic Model-Based Algorithm for Partially Observable Sequential Decision Making
This paper introduces a simple efficient learning algorithms for general sequential decision making. The algorithm combines Optimism for exploration with Maximum Likelihood Estimation for model
estimation, which is thus named OMLE. We prove that OMLE learns the near-optimal policies of an enormously rich class of sequential decision making problems in a polynomial number of samples. This
rich class includes not only a majority of known tractable model-based Reinforcement Learning (RL) problems (such as tabular MDPs, factored MDPs, low witness rank problems, tabular weakly-revealing/
observable POMDPs and multi-step decodable POMDPs ), but also many new challenging RL problems especially in the partially observable setting that were not previously known to be tractable. Notably,
the new problems addressed by this paper include (1) observable POMDPs with continuous observation and function approximation, where we achieve the first sample complexity that is completely
independent of the size of observation space; (2) well-conditioned low-rank sequential decision making problems (also known as Predictive State Representations (PSRs)), which include and generalize
all known tractable POMDP examples under a more intrinsic representation; (3) general sequential decision making problems under SAIL condition, which unifies our existing understandings of
model-based RL in both fully observable and partially observable settings. SAIL condition is identified by this paper, which can be viewed as a natural generalization of Bellman/witness rank to
address partial observability. This paper also presents a reward-free variant of OMLE algorithm, which learns approximate dynamic models that enable the computation of near-optimal policies for all
reward functions simultaneously.
Original language English (US)
Title of host publication STOC 2023 - Proceedings of the 55th Annual ACM Symposium on Theory of Computing
Editors Barna Saha, Rocco A. Servedio
Publisher Association for Computing Machinery
Pages 363-376
Number of pages 14
ISBN (Electronic) 9781450399135
State Published - Jun 2 2023
Event 55th Annual ACM Symposium on Theory of Computing, STOC 2023 - Orlando, United States
Duration: Jun 20 2023 → Jun 23 2023
Publication series
Name Proceedings of the Annual ACM Symposium on Theory of Computing
ISSN (Print) 0737-8017
Conference 55th Annual ACM Symposium on Theory of Computing, STOC 2023
Country/Territory United States
City Orlando
Period 6/20/23 → 6/23/23
All Science Journal Classification (ASJC) codes
• Optimistic MLE
• POMDPs
• PSRs
• Reinforcement Learning
Dive into the research topics of 'Optimistic MLE: A Generic Model-Based Algorithm for Partially Observable Sequential Decision Making'. Together they form a unique fingerprint. | {"url":"https://collaborate.princeton.edu/en/publications/optimistic-mle-a-generic-model-based-algorithm-for-partially-obse","timestamp":"2024-11-14T04:33:57Z","content_type":"text/html","content_length":"57404","record_id":"<urn:uuid:e22b7150-1c30-4ee5-abc8-12bb03b75b06>","cc-path":"CC-MAIN-2024-46/segments/1730477028526.56/warc/CC-MAIN-20241114031054-20241114061054-00802.warc.gz"} |
Chennai Mathematical Institute
An introduction to mathematical logic (continued)
Adrien Deloro
ENS Lyon.
(Institute Colloquium) Abstract
The second and third lectures in the series "Introduction to Logic" will be held on Friday, 10 September, from 2:00-5:00, with a break. Both these lectures will be taken by Adrien Deloro (ENS Lyon).
These two lectures will be a first step into Model Theory. Model Theory emphasises the importance of the language of first order logic in the study of a mathematical structure. We shall define and
investigate some basic "first order notions" like elementary substructures or completeness. We shall also state the famous compactness theorem, and give some applications and examples in several
mathematical fields.
Note: The final two lectures in this five lecture will be by Alexis Saurin on Proof Theory. The dates for these lectures will be annnounced later.
The last two lectures will be less concerned with the notion of truth than with the notion of provability. We shall introduce several ways to study mathematically the proofs done in mathematics
(Natural Deduction, Sequent Calculus), and we will discuss the properties of these "formal proofs" considered as a mathematical objects like an integer or a function (and address such questions as:
is it always possible to avoid the use of a lemma?). Furthermore, we will explain how proof theory relates with computer science and in what sense we can assert that proving is computing. | {"url":"https://www.cmi.ac.in/activities/show-abstract.php?absyear=2004&absref=17&abstype=sem","timestamp":"2024-11-12T00:29:57Z","content_type":"text/html","content_length":"7872","record_id":"<urn:uuid:e19cbd9d-d0d4-4dfd-9466-13da29de0569>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00241.warc.gz"} |
Algebra Students Apply Graphing Skills To Solve Problems - ConVal Regional High School
Ms. Sarah Gilpatrick is teaching her Algebra 1 part 2 students how to use Texas Instruments programmable calculators, called TI-Nspires, to solve systems of equations.
An equation system is two or more equations that share variables, such as x + 2y = 6 and 3x – y = 11. One way this system can be solved is by graphing each equation. The point where the two lines
cross is the solution to the system.
In this example that would be when x = 4 and y = 1, or the point on the graph (4,1). Rather than creating their graphs by hand, Algebra students can enter each equation into the calculators. The
TI-Nspire produces neat, clear graphs with the precise intersection point labeled.
Equation systems are used to find the value of two variables at the same time. For example, suppose Skyler and Emily buy food from the same pizza restaurant. Emily buys 2 slices and 2 iced teas for
$12. Skyler buys 4 slices and 1 iced tea for $18. How much did each pizza slice (x) and each iced tea (y) cost?
[Hint: 2x + 2y = 12, and 4x + y = 18] Solution: (4,2) | {"url":"https://cvhs.convalsd.net/algebra-students-apply-graphing-skills-to-solve-problems/","timestamp":"2024-11-04T14:01:32Z","content_type":"text/html","content_length":"73699","record_id":"<urn:uuid:037dbfcd-9593-45c5-87da-db777799afe2>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00224.warc.gz"} |
Logical operators | Python
Free Python course.
for tracking progress →
Python: Logical operators
We already know how to write functions that check single conditions. And in this lesson we will learn how to build compound conditions.
Suppose that the registration site requires that the password be longer than eight characters and contain at least one capital letter. Let's try to write two separate logical expressions and connect
them with the special operator "AND":
Password is longer than 8 characters And password contains at least one capital letter
Here is a function that takes the password and tells you whether it meets the conditions (True) or not (False):
def has_capital_letter(str):
# Checks the content of a capital letter in the string
def is_correct_password(password):
length = len(password)
return length > 8 and has_capital_letter(password)
print(is_correct_password('Qwerty')) # => False
print(is_correct_password('Qwerty1234')) # => True
print(is_correct_password('qwerty1234')) # => False
and means "AND". In mathematical logic this is called a conjunction. The whole expression is true if every operand - each of the compound expressions - is true. In other words, and means both. The
priority of this operator is lower than that of comparison operators. Therefore, the expression has_capital_letter(password) and length > 8 works correctly without brackets.
In addition to and, the operator "OR" (disjunction) - is often used. It means "either or both". The expression a or b is true if at least one of the operands or all of them simultaneously are true.
Otherwise, the expression is false.
Operators can be combined in any number and in any sequence. If and and or occur simultaneously in the code, the priority is given by parentheses. Below is an example of an extended function that
determines the correctness of a password:
def has_capital_letter(str):
# Checks the content of a capital letter in the string
def has_special_chars(str):
# Checks the content of special characters in the string
def is_strong_password(password):
length = len(password)
# The brackets set the priority. It is clear what relates to what.
return (length > 8 and as_capital_letter(password)) and has_special_chars(password)
Now let's imagine that we want to buy an apartment that meets these conditions: an area of 100 square meters or more on any street OR an area of 80 square meters or more, but on Main Street.
Let's write a function that will check the apartment. It takes two arguments: the area is a number and the street name is a string:
def is_good_apartment(area, street):
return area >= 100 or (area >= 80 and street == 'Main Street')
print(is_good_apartment(91, 'Queens Street')) # => False
print(is_good_apartment(78, 'Queens Street')) # => False
print(is_good_apartment(70, 'Main Street')) # => False
print(is_good_apartment(120, 'Queens Street')) # => True
print(is_good_apartment(120, 'Main Street')) # => True
print(is_good_apartment(80, 'Main Street')) # => True
The area of mathematics in which logical operators are studied is called Boolean algebra. Below you will see true tables - you can use them to determine what the result will be if you apply the
A B A and B
True True True
True False False
False True False
False False False
A B A or B
True True True
True False True
False True True
False False False
Implement a method is_leap_year() that determines whether a year is a leap year or not. A year will be a leap year if it is a multiple of 400 (i.e. divisible without a remainder), or if it is both a
multiple of 4 and not a multiple of 100. As you can see, the definition already contains all the necessary logic, all we need to do is to translate it into code:
is_leap_year(2018) # false
is_leap_year(2017) # false
is_leap_year(2016) # true
The multiplicity can be checked as follows:
# % - returns the remainder of the left operand divided by the right operand
# Check that the number is a multiple of 10
number % 10 == 0
# Check that the number is not a multiple of 10
number % 10 != 0
The exercise doesn't pass checking. What to do? 😶
If you've reached a deadlock it's time to ask your question in the «Discussions». How ask a question correctly:
• Be sure to attach the test output, without it it's almost impossible to figure out what went wrong, even if you show your code. It's complicated for developers to execute code in their heads, but
having a mistake before their eyes most probably will be helpful.
In my environment the code works, but not here 🤨
Tests are designed so that they test the solution in different ways and against different data. Often the solution works with one kind of input data but doesn't work with others. Check the «Tests»
tab to figure this out, you can find hints at the error output.
My code is different from the teacher's one 🤔
It's fine. 🙆 One task in programming can be solved in many different ways. If your code passed all tests, it complies with the task conditions.
In some rare cases, the solution may be adjusted to the tests, but this can be seen immediately.
I've read the lessons but nothing is clear 🙄
It's hard to make educational materials that will suit everyone. We do our best but there is always something to improve. If you see a material that is not clear to you, describe the problem in
“Discussions”. It will be great if you'll write unclear points in the question form. Usually, we need a few days for corrections.
By the way, you can participate in courses improvement. There is a link below to the lessons course code which you can edit right in your browser.
• Logical operators. the AND (and), OR (or) operators, which allow you to create compound logical conditions. | {"url":"https://code-basics.com/languages/python/lessons/logical-operators","timestamp":"2024-11-03T08:58:04Z","content_type":"text/html","content_length":"32887","record_id":"<urn:uuid:a3dd1d39-bcad-4893-aece-40232f8884cf>","cc-path":"CC-MAIN-2024-46/segments/1730477027774.6/warc/CC-MAIN-20241103083929-20241103113929-00703.warc.gz"} |
(PDF) A simple method for measuring power, force, velocity properties, and mechanical effectiveness in sprint running: Simple method to compute sprint mechanics
... Samozino et al. have recently developed a method to assess the entire force-velocity (Fv) spectrum during sprint acceleration (sprint Fv profile), profiling the mechanical capabilities of the
neuromuscular system [22,23]. Since the sprint Fv relationship is linearity, the maximal capacities of muscles to produce force (F 0 ), velocity (V 0 ), power (P max ), Fv slope (the ratio between F
0 and V 0 ), maximal ratio of force (RF max ), maximum speed during a sprint (V max ) and the index of force application technique (D RF ) can be determined within a linear regression model [23,24].
... Samozino et al. have recently developed a method to assess the entire force-velocity (Fv) spectrum during sprint acceleration (sprint Fv profile), profiling the mechanical capabilities of the
neuromuscular system [22,23]. Since the sprint Fv relationship is linearity, the maximal capacities of muscles to produce force (F 0 ), velocity (V 0 ), power (P max ), Fv slope (the ratio between F
0 and V 0 ), maximal ratio of force (RF max ), maximum speed during a sprint (V max ) and the index of force application technique (D RF ) can be determined within a linear regression model [23,24].
The sprint Fv profile has already been established as a reliable theoretical basis for devising personalized training guidance for athletes [22,[25][26][27][28]]. ...
... Before testing, all participants performed a standardized warm-up, starting with a 5-min jog, followed by 5 min of low-intensity sprints and ending with 5 min of dynamic stretching. were computed
with the mean of three sprint split times according to Samozino's specific spreadsheet [23,24]. The mean 0-10 m split time was used to calculated the COD deficit and asymmetry index, the process of
which will be detailed later. ... | {"url":"https://www.researchgate.net/publication/277020032_A_simple_method_for_measuring_power_force_velocity_properties_and_mechanical_effectiveness_in_sprint_running_Simple_method_to_compute_sprint_mechanics","timestamp":"2024-11-13T15:18:10Z","content_type":"text/html","content_length":"983703","record_id":"<urn:uuid:622d2231-cde8-4cce-b378-ccfb993fff9f>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00428.warc.gz"} |
Assessment of Radiative Heating for Hypersonic Earth Reentry Using Nongray Step Models
School of Energy and Power Engineering, Shandong University, Jinan 250061, China
Shandong Engineering Laboratory for High-Efficiency Conservation and Energy Storage Technology & Equipment, Shandong University, Jinan 250061, China
School of Aeronautic Science and Engineering, Beihang University (BUAA), Beijing 100191, China
Author to whom correspondence should be addressed.
Submission received: 3 March 2022 / Revised: 3 April 2022 / Accepted: 8 April 2022 / Published: 15 April 2022
Accurate prediction of the aerothermal environment is of great significance to space exploration and return missions. The canonical Fire II trajectory points are simulated to investigate the
radiative transfer in the shock layer for Earth reentry at hypervelocity above 10 km/s using a developed radiation–flowfield uncoupling method. The thermochemical nonequilibrium flow is solved by an
in-house PHAROS Navier–Stokes code, while the nongray radiation is integrated by the tangent slab approximation, respectively, combined with the two-, five-, and eight-step models. For the convective
heating, the present results agree well with the data of Anderson’s relation. For the radiative heating, the two-step model predicts the closest values with the results of Tauber and Sutton’s
relationship, while the five- and eight-step models predict far greater. The three-step models all present the same order of magnitude of radiative heating of 1 MW/m^2 and show a consistent tendency
with the engineering estimation. The Planck-mean absorption coefficient is calculated to show the radiative transfer significantly occurs in the shock layer. By performing the steady simulation at
each flight trajectory point, the present algorithm using a nongray step model with moderate efficiency and reasonable accuracy is promising to solve the real-time problem in engineering for
predicting both convective and radiative heating to the atmospheric reentry vehicle in the future.
1. Introduction
Today, many countries are energetically developing planetary exploration and subsequent return projects [
]. The mission spacecraft, usually with a blunt nose, must undergo the harsh thermal environment at hypervelocity even greater than 10 km/s for Earth atmosphere reentry [
]. In this situation, the strong bow shock wave around the reentry forebody can stimulate an extremely high temperature with an order of magnitude of 10,000 K and very complicated thermochemical
nonequilibrium phenomena including the high excitations of vibrational and electronic energy modes, dissociation, and ionization reactions of molecules and atoms of air species [
]. Except for the convective and chemical diffusive heating, radiative heat transfer could be a considerable or even dominated contributor to the total aerodynamic heating [
In fact, the occurrence of air radiation is not only one simple additional mode of heat transfer but also has important impacts on the flow characteristics of the shock layer [
]. On one hand, the radiative energy emission and absorption make the flowfield nonadiabatic and lead to the radiation cooling effect [
], which results in a lower temperature, higher density, and subsequently thinner shock layer. On the other hand, the thermochemical reactions and radiation of air species may overlap in some
concerned flight regimes. The radiation limits the chemical reaction processes of air species, whereas the air thermodynamic states and composition, in turn, directly affect the level of radiance.
The radiation–flowfield coupling should be evaluated [
]. Therefore, in engineering, the thermal protection system design of the entry vehicle requires the accurate aerothermal analysis of the hypersonic thermochemical nonequilibrium flowfield including
radiation to ensure safe entry [
However, until now, the solution of radiation in the hypersonic shock layer has not been an easy task [
], even for the uncoupled radiation calculation only [
]. First, although the scattering could be neglected, the absorption and emission must be accounted for in the air radiation in the hypersonic shock layer. The radiation properties of air, which are
emission and absorption coefficients, always vary dramatically with the frequency ranging from zero to infinity [
]. The accurate calculation of the air absorption coefficients requires a very large number of frequency points with an order of one million, which is undoubtedly time-consuming and even unaffordable
]. Second, the radiative transfer depends on the multi-dimensions of space, angle, and frequency. The algorithm for solving the radiative transfer equation (RTE) has to simultaneously account for the
spatial, angular, and frequency discretizations, which makes the radiation computation difficult and inefficient [
]. Facing the foregoing barriers, it is necessary to employ the reduced models to calculate the air radiation property and energy transportation for the aerodynamic heating to reentry vehicle [
The step model is an efficient method to calculate the air radiation property with moderate accuracy [
]. It divides the whole frequency space into several spectral intervals even including atomic lines in which the radiative absorption coefficient is equivalent to a constant [
]. Although the step model appears to be coarser than the line-by-line or multi-band models, it usually can predict the reasonable radiative flux of the hypersonic vehicle [
]. For solving radiative transfer, there have been various numerical methods, such as the tangent slab (TS) approximation [
], the spherical harmonics method [
], the ray-tracing method [
], the finite volume method [
], the discrete ordinates method [
], and Monte Carlo method [
]. Each method has both advantages and disadvantages either in prediction accuracy or in computation efficiency [
]. Among these methods, TS has seemed to be the most frequently used method for both coupled and uncoupled flowfield-radiation simulations for decades [
]. TS originates from the one-dimensional analytical integrated solution of RTE for the participating medium between two infinite parallel plates with the radiation variations only along the normal
direction of the plate [
]. Since the shock layer flow and thermodynamic properties mainly vary in the direction normal to the surface of the blunt-nosed reentry vehicle, TS is a good approximation to model the radiation
energy-transportation [
]. Practically, researchers usually use the body-normal grid lines for convenient TS calculation [
]. Hartung et al. used TS to evaluate the characteristics of the shock wave precursor ahead of the hypervelocity entry vehicle including the radiative effects [
]. Wright et al. found that TS could over-predict the value of the stagnation radiative heat flux by a minimum of 20% for the Titan aerocapture case [
]. Johnston et al. used TS combined with a viscous shock layer flowfield model to account for the radiation–flowfield coupling and showed that the coupled simulation reduced the radiative heating by
about 30%, while the convective heating decreased slightly [
]. Bauman et al. coupled a reacting flow model and a surface ablation model with TS to develop a two-way loose-coupling procedure for simulating the hypersonic flows with radiation and ablation [
]. Johnston and Brandis employed TS to calculate the radiative heating and identified the radiation as a major contributor to afterbody heating for Earth entry at velocities above 10 km/s. They also
showed that TS overestimates the afterbody radiative heat transfer by as much as 50% [
]. Generally, although TS has some deficiencies and researchers have also proposed a non-tangent-slab procedure, TS is still very worthy of being considered as a first choice to solve RTE coupled or
uncoupled with flowfield solver to predict the radiative heating for Earth reentry at hypervelocity, especially to obtain a conservative estimation of the aerodynamic heating in the engineering
initial design with moderate accuracy and computation efficiency.
In fact, there have been many engineering empirical relations to quickly predict both the wall convective and radiative heat fluxes of the Earth entry capsule [
]. Most methods have been too coarse for the modern design of Earth reentry capsule which can only give the value of heat transfer at the stagnation point but has no ability to present the
distribution characteristics of aerodynamic heating and some key physical variables of interest in the shock layer, such as electrons number density and radiation absorption coefficient. However, the
accurate radiation–flowfield coupling simulation is time-consuming and even unaffordable for the initial thermal protection system design of reentry vehicles [
]. Hence, it is more rational and practical in engineering to choose an uncoupled radiation–flowfield procedure, in which the radiation is calculated only once based on the flow simulation results [
Gupta et al. used a viscous shock-layer code and aerotherm radiation code with nonequilibrium and equilibrium chemistry to estimate the convective and radiative heating of the Fire II vehicle, but
the calculations were limited to the stagnation region [
]. Olynick et al. developed a nonequilibrium, axisymmetric flow solver coupled with a radiation GIANTS/NOVAR code to obtain values of the stagnation radiation intensity in the 0.2 and 6.2 eV and the
total aerodynamic heating over the entire Fire II vehicle, which provided the better predictions than previously numerical simulations [
]. Palmer et al. incorporated the NEQAIR line-by-line radiation code into the DPLR flow solver to investigate the effects of fluid dynamics/radiation coupling by comparing coupled and uncoupled
results, and they found that the greatest coupling effect of Fire II occurred at the 1643 s trajectory point [
]. Soucasse et al. implemented a hybrid statistical narrow band (HSNB) model with a two-temperature nonequilibrium model to calculate the 1D stagnation line radiative transfer. The HSNB model could
reproduce the line-by-line results with an accuracy of better than 5% and a computational time speed up about two orders of magnitude [
]. Bonin and Mundt developed a full three-dimensional photon Monte Carlo radiative transport solver to study arbitrary thermal radiation within equilibrium and nonequilibrium hypersonic flows, the
code of which was line-by-line accurate but time-consuming [
]. Although great progress has been made to predict radiative transfer in hypersonic nonequilibrium flow for atmospheric entry over decades, the too sophisticated methods developed with high accuracy
are always of much heavy computational burden and even unaffordable, such as the line-by-line and narrow-band radiation models, especially for multi-dimensional coupling simulation. Therefore, there
is still a practical demand for developing the reduced models with moderate time efficiency and reasonable accuracy to predict both radiative heating and radiation characteristics in flowfield for
the thermal protection design of Earth reentry vehicles in engineering applications.
For solving the above-mentioned problems, the first objective of this paper is to develop an uncoupled radiation–flowfield algorithm for predicting the aerothermal environment of Earth reentry
vehicles at hypervelocities above 10 km/s, which consists of the tangent slab approximation, the nongray step model, and a Navier–Stokes solver including the thermochemical nonequilibrium effects.
The detailed physical models and numerical schemes are presented in
Section 2
. Another objective is to evaluate the performance of the radiation–flowfield uncoupling procedure by analyzing the canonical reentry trajectory cases of Fire II capsule, especially focusing on the
aerodynamic heating and the radiation characteristics in the shock layer. The convective and radiative heat fluxes at the stagnation point are also estimated throughout the trajectory, respectively,
using the Anderson and Tauber and Sutton relations. A comparison between the present results and those of engineering methods and previous studies is thoroughly presented and discussed in
Section 3
. The final conclusions are drawn in
Section 4
2. Physical Models and Numerical Methods
2.1. Flow Governing Equations with Thermochemical Nonequilibrium Models
The hypersonic Earth reentry flow is governed by the Navier–Stokes equations with the two-temperature model including the thermochemical nonequilibrium effects in the conservative forms. It assumes
that the air is a multi-gas mixture and for all the composite species, the translational and rotational energy modes are in equilibrium in one translational-rotational temperature,
, and the vibrational, electronic, and electron energies are uniformly described by one vibrational-electronic temperature,
]. In this manner, the mass, momentum, and energy conservation equations of the hypersonic nonequilibrium flow can be expressed as follows [
$∂ ρ s ∂ t + ∂ ρ s u j ∂ x j = − ∂ J s , j ∂ x j + ω s , s = 1 , 2 , ⋯ , N s$
$∂ ρ u i ∂ t + ∂ ρ u i u j ∂ x j = − ∂ p ∂ x i + ∂ ∂ x j [ μ ( ∂ u i ∂ x j + ∂ u j ∂ x i ) − 2 3 μ ∂ u k ∂ x k δ i j ]$
$∂ ρ e ∂ t + ∂ ρ h u j ∂ x j = ∂ τ i j u i ∂ x j − ∂ q j ∂ x j − ∂ ∂ x j ( ∑ s = 1 N s J s , j h s ) + ω r$
$∂ ρ e v e ∂ t + ∂ ρ e v e u j ∂ x j = − ∂ q v e , j ∂ x j − ∂ ∂ x j ( ∑ s = 1 N s J s , j h v e , s ) + ω v e + ω r$
is the time,
is the ordinate variable in the
is the total number of the air species,
are the species density and total density,
is the flow velocity component in the
is the pressure,
is the viscous stress tensor,
are the total energy and vibrational-electronic energy,
is the total enthalpy,
are the total heat flux and vibrational-electronic heat flux,
are the enthalpy and vibrational-electronic enthalpy of the species
is the mass diffusion flux of the species
in the
is the mass production rate of the species
per unit volume,
is the vibrational-electronic energy source term, and
is the radiative source term. The state equation of the air follows:
$p = ∑ s = 1 N s − 1 ρ s R s T t r + ρ e R e T v e$
where the subscript “
” represents the electron.
In the present study, the thermodynamic properties of all the species are calculated using analytical relations of the translational, rotational, and electronic excitation energy modes based on the
Born–Oppenheimer approximation [
]. The transport properties of the air mixture including the dynamic viscosity, thermal conductivity, and species diffusion coefficients are calculated via the extension of Yos’ formula with the
collision integrals [
The mass rate of production of species
is expressed as follows [
$ω s = M s ∑ r = 1 N r ( ν r , s b − ν r , s f ) [ k f , r ∏ j = 1 N s ( ρ j M j ) ν r , j f − k b , r ∏ j = 1 N s ( ρ j M j ) ν r , j b ]$
is the total number of chemical reactions,
is the molecular mass per mole of species
are the forward and backward reaction stoichiometric coefficients of species
of the
-th reaction,
are the forward and backward reaction rate coefficients of the
-th reaction, respectively. For each chemical reaction, the forward reaction rate coefficient is calculated using the Arrhenius formula [
], while the backward reaction rate coefficient is obtained from the corresponding forward reaction rate coefficient divided by the equilibrium constant, which is computed using the temperature
fitting expression [
The vibrational-electronic energy source term
is modeled in the expression proposed by Gnoffo et al. [
], in which the translational-vibrational energy exchange part is calculated by the Landau–Teller model [
], and the relaxation time is calculated by the Millikan–White expression [
] with Park’s high-temperature correction [
]. For the present uncoupled radiation–flowfield simulation, the radiative source term
is neglected.
2.2. Flowfield Solver
The Navier–Stokes equations with thermochemical models are solved using an in-house CFD code PHAROS (Parallel Hypersonic Aerothermodynamics and Radiation Optimized Solver) [
]. PHAROS is a parallel multi-block finite volume solver, in which the inviscid fluxes are computed by the modified Steger–Warming flux vector splitting scheme [
] using MUSCL extrapolation [
] with minmod limiters for high order accuracy and stability. The viscous fluxes are discretized in the second central difference. A line relaxation approach is employed for the time marching [
]. PHAROS has been used to solve many types of hypersonic thermochemical nonequilibrium flowfields [
]. More information on PHAROS can be found in Ref. [
2.3. Step Models for Radiation Properties
Based on recognizing the variation characteristics of the self-absorption property of air radiation with wavelength, the step model selects some demarcation wavelength points and divides the infinite
spectrum into several consecutive spectral regions, such as vacuum ultraviolet and visible bands. In each spectral band, the volatile radiation absorption coefficients are averaged into a constant
Planck-mean value, which makes the total radiation a sum of the contributions of all the step regions. The absorption coefficient in the spectral region of each step depends on the temperature,
density, and air composition parameters [
]. It has been shown that the step model can well account for the important effects of shock-layer nongray self-absorption and radiative cooling on radiative heat transfer [
]. Additionally, compared to line-by-line calculation, the step model is a reasonable simplification and can reduce the computational time obviously. Therefore, the step model is selected for the
present study. In this paper, the radiation absorption coefficients of the high-temperature gas mixture of air are calculated via the two-step, five-step, and eight-step nongray models [
], respectively, for the purpose of comparative study. The two-step and five-step models are developed based on the high-temperature atomic nitrogen mainly accounting for the air radiance at
temperatures above 8000–10,000 K, while the eight-step model includes both the atomic and molecular emission absorption of the high-temperature air [
]. The wavelength regions of the three foregoing step models are listed in
Table 1
. The specific formulas and parameters of the three-step models are given in
Appendix A
2.4. Tangent Slab (TS) Approach for RTE
Neglecting the gas scattering effect, the high-temperature air radiation process in the hypersonic reentry shock layer is described via the radiative transfer equation (RTE) as follows [
$B j ∂ I ν ( x , B ) ∂ x j = κ ν ( x ) [ I b ν ( x ) − I ν ( x , B ) ]$
represents the spatial position vector,
is the radiation frequency,
is the spectral radiative intensity and blackbody radiative intensity at frequency
, respectively,
is the
-th component of unit vector
in the transmission direction of
, and
is the spectral absorption coefficient at
TS approximates the hypersonic shock layer around the reentry blunt body as an infinite slab with physical variables only changing in the perpendicular direction to the body surface. Hence, there is
an integrated solution for RTE as follows [
$q r w = ε w ( J w − σ T w 4 )$
$J w = 2 π ∫ 0 ∞ [ I ν ( τ ν δ ) E 3 ( τ ν δ − τ ν w ) + ∫ τ ν w τ ν δ I b ν ( t ) E 2 ( t − τ ν w ) d t ] d ν$
is the optical thickness at frequency
perpendicular to the body surface (
= 0 at body surface), the subscript “
” and “
” represent the outer edge of the shock layer and the wall, respectively;
is a dummy variable of integration, and
are the wall emissivity and temperature, respectively.
is the integro-exponential function of order
as follows:
$E n ( t ) ≡ ∫ 0 1 s n − 2 e − t / s d s$
According to step models in
Section 2.3
, the integration (9) can be written as:
$J w = 2 π ∑ m { E 3 ( τ m δ − τ m w ) ∫ m I ν ( τ ν δ ) d ν + ∫ τ m w τ m δ [ ∫ m I b ν ( t ) d ν ] × E 2 ( t − τ m w ) d t }$
is the index of the step region. In this paper, the integral with respect to optical thickness in Equation (11) is calculated using the trapezoidal method to perform TS procedures.
2.5. Radiation–Flowfield Uncoupling Algorithm
For radiation–flowfield coupling simulation, TS approximation with a nongray step model should do the upward and downward integrals of radiative heat flux divergence along the ray line normal to the
wall for computing the radiative source term
in Equation (4) [
]. Assuming a two-dimensional problem is solved, the mesh has a total number of grid nodes of
, where
is the discretized number parallel to the wall and
is the discretized number normal to the wall. Thus, the coupling simulation should need about
calculations in one radiation iteration, for which the last product factor
is due to the numerical integration of radiative heat flux divergence and
is the number of radiation step regions.
A radiation–flowfield uncoupling algorithm is proposed in the present study, in which the hypersonic reentry flow with nonequilibrium chemistry is first solved by PHAROS in
Section 2.2
, and then the radiation transfer is calculated only once based on the convergent flowfield by the nongray step model in
Section 2.3
and TS approximation in
Section 2.4
. It only needs
calculations for radiation solution in total. Therefore, in one iteration, the number of calculations for coupled radiation simulation is far greater with an order of magnitude of O(
) than that of the present uncoupled algorithm. Particularly, the present scheme only needs to compute radiation one time in total, while the coupled simulation has to update the radiation in each
step over the whole computational process. Even for the loosely coupled manner, it still requires a considerable computational cost, in which the radiation is updated one time after a certain number
of flow iterations. Therefore, the present method is more time-efficient. Additionally, the present algorithm can provide more radiation information including the radiative heating to the whole
surface of the reentry vehicle and the absorption properties distributed in the flowfield, but the engineering methods certainly cannot make it, that will be seen later in
Section 3.4
3. Results and Discussion
Fire II was a scaled-down Apollo-shaped capsule launched in 1965 with the calorimeter instrumentation to obtain reentry heating at hyperbolic velocities. In the reentry phase, Fire II jettisoned two
nonablating heat shields in sequence at selected trajectory points, thereby with various vehicle nose radiuses. The Fire II flight experiment has become a benchmark for investigating the aerothermal
environment for hypersonic Earth entry. Six trajectory points of Fire II have been simulated in the present study with the flight conditions listed in
Table 2
, where
is the flight altitude,
are the flight velocity and Mach number,
is the vehicle nose radius,
are the freestream density and temperature, and
is the wall temperature. For each case in
Table 2
, the time of flow over the vehicle,
, is of an order of magnitude around 10
s, but the time scale of entry down in the altitude is around 0.5 s. The latter is far greater than the former, which means a steady flowfield establishes very quickly. Therefore, we can use the
present algorithm to perform steady simulation for each case to fulfill the real-time prediction throughout the Fire II trajectory. Only the Fire II forebody is considered in this paper, the
axisymmetric geometry and grids of which are shown in
Figure 1
. The computational mesh has the dimensions of 153 × 128 (axial × radial) for every case and the spacing of the first grid layer perpendicular to the wall can ensure the cell Reynolds number with an
order of magnitude of one in order to predict the reliable aerodynamic heating [
]. The noncatalytic wall condition is used and the wall emissivity is uniformly set to be one.
3.1. Convective Heating
Figure 2
compares the present CFD results of convective heat transfer at the stagnation point with those data calculated by Anderson’s engineering relationship [
], Gupta et al. [
], and Olynick et al. [
]. Anderson’s relationship is expressed as follows:
$q c w , s t a g = 1.83 × 10 − 4 ρ ∞ R N × ( 1 − H w H e ) × V ∞ 3 ( W / m 2 )$
is the vehicle nose radius;
are the freestream density and velocity;
are the enthalpies at the wall and the outer edge of the boundary layer, which can be evaluated by using the wall temperature
and freestream total temperature
, respectively.
can be calculated by the freestream temperature
and Mach number
, and
are all listed in
Table 2
The present results agree well with the other three sets of data both in the magnitude of value at each trajectory point and the total variation tendency with time, which verifies the reliability of
the present thermochemical nonequilibrium flow solver PHAROS. Throughout the trajectory from
= 1634 s to 1645 s, the stagnation convective heat transfer is always greater than 1 MW/m
and increases continuously up to almost 8 MW/m
Figure 3
shows the convective heating over the whole surface of the Fire II forebody for each trajectory point, which presents that the convective heat transfer still maintains a high level of magnitude
greater than 1 MW/m
in the region outside the stagnation point for all cases.
Figure 4
further compares the forebody convective heating line at
= 1636 s predicted by the present method with those obtained by DPLR and LAURA [
], the good agreement of which shows the high prediction accuracy of the present PHAROS solver again.
3.2. Thermochemical Nonequilibrium Flowfield
Figure 5
compares the translational-rotational temperature
and vibrational-electronic temperature
in the Fire II flowfield throughout the trajectory from
= 1634 s to 1645 s. At
= 1634 s, there is a remarkable difference between
in a distance closely behind the bow shock, which suggests the translational-rotational and vibrational-electronic energy modes are highly in nonequilibrium in this region. As time goes on with the
altitude and velocity down, the thermodynamic nonequilibrium tends to be weakened steadily and the total level of magnitude of temperature also decreases gradually. From
= 1637 s to 1645 s,
have become consistent in most areas of the shock layer. For all cases, both
exceed 10
K in the shock layer around the Fire II forebody, and the peak of the translational-rotational temperature even reaches up to 44,000 K at
= 1634 s. Such extreme high temperature directly leads to the harsh aerothermal environment for Earth reentry and results in significant radiative heating [
Figure 6
presents the number densities of the species O, N, O
, N
, NO, and e
along the stagnation line for each case. The high atomic and electron concentrations demonstrate the strong dissociation and ionization reactions of air in the reentry shock layer of Fire II. At
= 1634 s, the number density of each species changes remarkably along the stagnation line, while at the following trajectory points, such variations become more and more unnoticeable and the
concentration of each species approaches a constant in most regions of the shock layer. Due to the high ionization, the number density of electrons (that is the sum of the number densities of all
positive ions, such as O
, N
, and NO
) is greater than 10
in most areas of the shock layer. The free electrons and ions constitute a plasma sheath around the reentry vehicle, which absorbs the radio-frequency radiation and causes the communication blackout
3.3. Radiative Heating
The radiative transfer is solved by employing the radiation–flow uncoupling algorithm, in which the radiation is calculated only once based on the convergent flowfield solution using the methods in
Section 2
. Compared with the radiation–flow coupling approach, although the uncoupling simulation compromises some accuracy in physics, the procedure really can greatly reduce the computing power and time.
Figure 7
compares the results of radiative heating at the stagnation point calculated by Tauber and Sutton’s (T & S) engineering relationship [
], Gupta et al. [
], Olynick et al. [
], and the present TS method with the two-step, five-step, and eight-step models, respectively. The T & S relation is expressed as follows:
$q r w , s t a g = 4.736 × 10 8 R N a ρ ∞ b f ( V ∞ ) ( W / m 2 )$
is the vehicle nose radius;
are the freestream density and velocity;
are empirical exponents;
is a tabulated function of freestream velocity
. The detailed descriptions of
, and
can be found in Ref. [
Gupta et al. developed a RAD code accounting for the molecular band, continuum, and atomic line transitions. They employed a detailed frequency dependence of the absorption coefficients to integrate
over the radiation spectrum and used the TS approximation for integrating over physical space, but only limited to the stagnation region. Olynick et al. developed a GIANTS/NOVAR code to obtain values
of the stagnation radiation intensity in the 0.2 and 6.2 eV range. They used a “smeared band” approximation instead of a line-by-line approach to account for the air radiation properties of the
molecular band systems and reduce the total number of spectral points. Even doing so, Olynick et al. still needed to perform TS integration at each radiation grid point consisting of around 1000
frequency points for calculating absorption and emission coefficients, which is very time-consuming. Gupta et al. and Olynick et al. both implemented the radiation–flow coupling simulation with
nonequilibrium chemistry.
Figure 7
shows that the radiative heat fluxes calculated by the TS approximation with the step models are mostly greater than those obtained by the T & S relation, Gupta et al., and Olynick et al. In fact,
compared with the coupling simulation, the present radiation–flow uncoupling method should predict the higher level of radiative heating [
], which is reasonable and can be seen as the conservative upper limit in the engineering application.
Although the noticeable differences can be seen among the radiative heat fluxes of different methods, the orders of magnitude of all the data are basically the same and they show a similar tendency,
first an increase and then decrease. The five-step model gives the greatest stagnation radiative heat transfer, while the results of the two-step model are the smallest. Unexpectedly, although the
two-step model is coarsest in spectral space, it predicts the closest values with those of the T & S relation, Gupta et al. [
], and Olynick et al. [
] compared to the other two models. However, we cannot simply deduce that the two-step model is the best option because it is just only applicable for Fire II trajectory points. The final evaluation
needs more experimental data and further high-resolution numerical simulation in the future.
3.4. Radiation Field
Except for the more affordable time cost than that of the coupling simulation, the present radiation–flow uncoupling algorithm can provide the more detailed information than those of the engineering
methods, not only the radiative heat transfer at the stagnation point but also the radiative heating distribution on the vehicle surface and the radiation characteristics in the whole flowfield. The
latter radiation distributions in the flowfield are rarely shown in the previous literature [
] but may play some important roles in understanding the mechanism of radiative heating and designing the thermal protection system of atmospheric entry vehicles in the future [
Figure 8
shows the radiative heating over the whole surface of the Fire II forebody predicted by the TS approximation with the two-step, five-step, and eight-step models, respectively. If the heat transfer of
magnitude is not taken into consideration, the tendencies obtained by the three models are basically consistent. The radiative heating varies flatly over the whole vehicle surface for the cases from
= 1634 s to 1637 s, while it becomes more bending for the following cases from
= 1640 s to 1645 s.
Figure 9
shows the distributions of the Planck-mean absorption coefficient around the Fire II forebody throughout the trajectory calculated by the two-step, five-step, and eight-step models, respectively. The
Planck-mean absorption coefficient is one of the most important average forms that describes the total emission from a fluid element and indicates the level of radiative heat loss from the nearly
optically thin flowfield. The Planck-mean absorption coefficient for the present step model is defined as follows:
$κ P = ∑ m κ m ∫ m I b ν d ν ∫ 0 ∞ I b ν d ν = π ∑ m κ m ∫ m I b ν d ν σ T 4$
is the spectral step index,
is the absorption coefficient for the
-th step,
is the spectral blackbody intensity at radiation frequency
, σ is the Stefan–Boltzmann constant, and
is the temperature. In the present hypersonic nonequilibrium flow,
is calculated using the vibrational-electronic temperature [
Figure 9
shows that
is remarkable only in the shock layer, particularly nearly behind the bow shock with peak values, while
is very small in the freestream. It suggests that the radiation energy transfer mainly occurs in the high-temperature shock layer. The five-step model predicts the greatest
, the eight-step model second, and the two-step model the smallest. As time goes on with the altitude and velocity down, the total level of
grows gradually. Although the temperature in the shock layer decreases as shown in
Figure 5
and makes a negative contribution to radiative transfer as the Fire II flight altitude descends, the air density increase significantly promotes the radiation effect in the flowfield. The results of
the three-step models all support this point. Another interesting thing is that
predicted by the two-step model shows the Fire II shock layer is close to being optically thin with the optical thickness being an order of magnitude of 10
, while the results of the five-step and eight-step models do not agree with this, which can only be clarified further in the future using the more detailed radiation property model, such as the
line-by-line or narrowband calculations [
]. Generally, the present radiation–flow uncoupling procedure is a good selection with higher efficiency in time cost than the coupling method to provide rich radiation information on the whole
hypersonic nonequilibrium flowfield of Fire II reentry that the engineering methods cannot make.
4. Conclusions
A radiation–flowfield uncoupling procedure is developed to simulate the Fire II trajectory points aimed at studying the radiative heating in the thermochemical nonequilibrium flowfield for Earth
reentry at hypervelocity above 10 km/s. The radiative transfer is integrated only once by the TS approximation with the nongray step model based on the flow solution obtained using an in-house N–S
solver PHAROS. It is naturally more efficient in computational cost than the coupled scheme, and also provides reasonable and more detailed information on the aerothermal environment than the
engineering relations which always can only calculate the aerodynamic heat flux at the stagnation point of reentry vehicles.
The results of Fire II cases throughout the trajectory from t = 1634 s to 1645 s show that the radiative heating grows first and then decreases with the order of magnitude of 1 MW/m^2, which is
comparable to the convective heating and even exceeds the latter. Although there are remarkable differences among the two-, five-, and eight-step models, the three models all show essentially
consistent trends in predictions of radiative transfer. The uncoupling calculated radiative heating can be regarded as the upper limit in the engineering application. In the future, more efforts need
to be made to clarify the level of the optical thickness for the flowfield of Earth reentry vehicles at hypervelocity above 10 km/s, which the two-step model predicts to be optically thin, while the
five- and eight-step models do not agree. The present scheme can also provide more radiation information in the nonequilibrium flowfield than the previous engineering relations.
Due to the fact that the flowfield establishes far more quickly than the vehicle flies down for Earth reentry, it can perform steady flow simulation at each trajectory point. Therefore, the present
radiation–flow uncoupling algorithm using a nongray step model with moderate efficiency and reasonable accuracy is promising to solve the real-time problem in engineering for predicting both
convective and radiative heating to the atmospheric reentry vehicle in the future.
Author Contributions
Conceptualization, J.W. and X.Y.; methodology, J.W.; software, J.W.; validation, X.Y., J.W. and K.S.; formal analysis, X.Y. and Y.Z.; investigation, X.Y., J.W. and Y.Z.; resources, J.W.; data
curation, Y.Z.; writing—original draft preparation, X.Y. and J.W.; writing—review and editing, J.W. and X.Y.; visualization, J.W., X.Y. and Y.Z.; supervision, J.W.; project administration, J.W.;
funding acquisition, J.W. and K.S. All authors have read and agreed to the published version of the manuscript.
This research was funded by the National Natural Science Foundation of China, grant number 12002193, and the Shandong Provincial Natural Science Foundation, China, grant number ZR2019QA018.
Institutional Review Board Statement
Not applicable.
Informed Consent Statement
Not applicable.
Data Availability Statement
Not applicable.
Conflicts of Interest
The authors declare no conflict of interest.
e total energy
e[ve] vibrational-electronic energy
h total enthalpy
h[s] enthalpy for the species s
h[ve,s] vibrational-electronic enthalpy for the species s
k[f,r] forward reaction rate coefficients of the r-th reaction
k[b,r] backward reaction rate coefficients of the r-th reaction
p pressure
q total heat flux
q[ve] vibrational-electronic heat flux
q[cw,stag] convective heat flux at stagnation point
q[rw,stag] radiative heat flux at stagnation point
B[j] j-th component of the unit directional vector
H altitude
H[w] enthalpy at the wall
H[e] enthalpy at the outer edge of the boundary layer
I[ν] spectral radiative intensity at frequency ν
I[bv] blackbody radiative intensity at frequency ν
J[s,j] mass diffusion flux of the species s in the j-th direction
M[s] molecular mass per mole of species s
N[s] total number of air species
N[r] total number of chemical reactions
R[s] gas constant for the species s
R[N] nose radius
T[tr] translational-rotational temperature
T[ve] vibrational-electronic temperature
T[w] wall temperature
T[∞] freestream temperature
ε[w] wall emissivity
κ[m] absorption coefficient for the m-th spectral step
κ[P] Planck-mean absorption coefficient
κ[ν] absorption coefficient at frequency ν
ν radiation frequency
$ν r , s b$ stoichiometric coefficient of the species s in the r-th backward reaction
$ν r , s f$ stoichiometric coefficient of the species s in the r-th forward reaction
ρ[s] density of the species s
ρ[∞] freestream density
τ[ij] viscous stress tensor
τ[ν] optical thickness at frequency ν
ω[r] radiative source term
ω[s] mass production rate of the species s
ω[ve] vibrational-electronic energy source term
Appendix A. Nongray Two-, Five- and Eight-Step Models
Appendix A.1. Two-Step Model
The two-step model proposed by Anderson [
] accounts for the high-temperature air radiation absorption coefficients respectively in vacuum ultraviolet (VUV) and infrared radiation (IR) bands. The location of the step is selected at
wavelength 1100 Å. The absorption coefficient in each step is a function of local temperature and density. For the first step, the absorption coefficient is formulated as follows:
$κ 1 = { 3600 ( ρ ρ 0 ) ( T 10 4 ) 4.02 , T ≤ 11,000 K 100 ( ρ ρ 0 ) [ 8.1 + 41.3 ( T 10 4 ) ] , T > 11,000 K ( m − 1 )$
are the local density and temperature,
= 1.225 kg/m
; for the second step, the absorption coefficient is
$κ 2 = a ( ρ ρ 0 ) b ( T 10 4 ) c ∫ 1100 A ∘ ∞ I b λ d λ ( m − 1 )$
, and
are parameters depending on the temperature range, the details of which are given in Ref. [
is the blackbody radiative intensity expressed as:
$I b λ = 2 π h c 2 λ 5 1 e h c / λ k B T − 1$
is the speed of light,
is the Planck constant,
is the Boltzmann constant,
is the wavelength of radiation, and
is the temperature.
Appendix A.2. Five-Step Model
The five-step model is proposed by Knott et al. [
] based on nitrogen atomic line and continuum radiation. The coefficient for each step can be formulated as follows:
$κ m = 100 a m 10 b m T 10 4 n N e c m ( m − 1 ) , m = 1 , 2 , 3 , 4 , 5$
is the number density of nitrogen, and
is the temperature; am, bm, and cm are fitting coefficients depending on the general range of temperature, which can be found in Ref. [
Appendix A.3. Eight-Step Model
The eight-step model proposed by Olstad [
] approximates the important contributions of the free-bound, free-free continuum, atomic lines, and molecular band system to high-temperature air radiation. The absorption coefficients for eight
steps (unit: m
) are formulated as follows:
$κ 1 = 1.1 × 10 − 15 n N + 2.0 × 10 − 15 n O 2 + 4.0 × 10 − 14 n N 2 + κ 2$
$κ 2 = 5.1 × 10 − 16 ( n O 2 + n N 2 + n O ) + κ 3$
$κ 3 = 2.0 × 10 − 16 ( n O 2 + n N 2 ) + 2.1 × 10 − 15 n N e − 0.165 T ˜ + κ 4$
$κ 4 = 5.0 × 10 − 17 n O 2 + 5.0 × 10 − 18 n N 2 + 1.7 × 10 − 15 n N e − 0.246 T ˜$
$κ 5 = 7.7 × 10 − 15 ( n O 2 + n N 2 ) e − 0.490 T ˜ + 2.6 × 10 − 15 ( n O + n N )$
$κ 6 = 2.0 × 10 − 16 n O 2 + 1.5 × 10 − 15 ( n O + n N ) e − 0.379 T ˜ + κ 5$
$κ 7 = 3.0 × 10 5 n O + n N n e − e − 0.489 T ˜ + κ 6$
$κ 8 = 3.2 × 10 − 15 ( n O + n N ) e − 0.631 T ˜ + κ 5$
is the number density with the subscript representing the corresponding species,
are the density and temperature (unit: K).
Figure 2. Convective heating at the stagnation point of Fire II predicted by the present algorithm (present), Anderson’s engineering relation (Anderson Eng.), Olynick et al., and Gupta et al.
Figure 3. Forebody convective heating along the Fire II surface throughout the trajectory from t = 1634 s to t = 1645 s.
Figure 4. Forebody convective heating at t = 1636 s predicted by the present algorithm (present), DPLR, and LAURA codes.
Figure 5. Temperature distribution in the Fire II flowfield (unit: 10^4 K): (a) t = 1634 s; (b) t = 1636 s; (c) t = 1637 s; (d) t = 1640 s; (e) t = 1643 s; (f) t = 1645 s.
Figure 6. Number density of species along the stagnation line: (a) t = 1634 s; (b) t = 1636 s; (c) t = 1637 s; (d) t = 1640 s; (e) t = 1643 s; (f) t = 1645 s.
Figure 7. Radiative heating at the stagnation point of Fire II throughout the trajectory predicted by the present algorithm with 2-step model, 5-step model, 8-step model, T & S engineering relation,
Gupta et al., and Olynick et al.
Figure 8. Forebody radiative heating throughout the trajectory predicted by (a) two-step model; (b) five-step model; (c) eight-step model.
Figure 9. Planck-mean absorption coefficient throughout the trajectory (unit: m^−1): (a) two-step model; (b) five-step model; (c) eight-step model.
Model Step No. Wavelength (Å) Spectral Band
Two-step 1 0–1100 VUV (vacuum ultraviolet)
2 1100–∞ Visible
1 620–1100 VUV continuum
2 1100–1300 VUV continuum
Five-step 3 1300–1570 VUV lines
4 1570–7870 Visible
5 7870–9552 IR (infrared) lines
1 400–852 VUV continuum
2 852–911 VUV continuum
3 911–1020 VUV continuum
Eight-step 4 1020–1130 VUV continuum
5 1130–1801 Continuum + line wings
6 1130–1801 Line “centers”
7 1801–4000 Visible
8 4000–∞ Visible + infrared
Time (s) H (km) V[∞] (km/s) Ma R[N] (m) ρ[∞] (kg/m^3) T[∞] (K) T[w] (K)
1634 76.42 11.36 40.58 0.935 3.72 × 10^−5 195 615
1636 71.02 11.31 38.94 0.935 8.57 × 10^−5 210 810
1637 67.05 11.25 37.17 0.935 1.47 × 10^−4 228 1030
1640 59.62 10.97 34.34 0.935 3.86 × 10^−4 254 1560
1643 53.04 10.48 31.47 0.805 7.80 × 10^−4 276 640
1645 48.37 9.83 29.05 0.805 1.32 × 10^−3 285 1520
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Yang, X.; Wang, J.; Zhou, Y.; Sun, K. Assessment of Radiative Heating for Hypersonic Earth Reentry Using Nongray Step Models. Aerospace 2022, 9, 219. https://doi.org/10.3390/aerospace9040219
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Yang X, Wang J, Zhou Y, Sun K. Assessment of Radiative Heating for Hypersonic Earth Reentry Using Nongray Step Models. Aerospace. 2022; 9(4):219. https://doi.org/10.3390/aerospace9040219
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Yang, Xinglian, Jingying Wang, Yue Zhou, and Ke Sun. 2022. "Assessment of Radiative Heating for Hypersonic Earth Reentry Using Nongray Step Models" Aerospace 9, no. 4: 219. https://doi.org/10.3390/
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Lecture 21 - Oct 24, 2023
In this lecture, we discuss copy constructor and operator= of linked lists. We also review the material for the midterm.
Copy constructor, operator=, and midterm revision.
Recap on copy constructor
Creates an object from an existing one.
The default copy constructor does a shallow copy.
We need to do a deep copy:
• Copy one node at a time.
• p to iterate original list, np to build new list.
Recap on operator equals
The idea is similar to the copy constructor, except that the lhs may not be empty. We must empty lhs first.
List& List::operator=(const List& original) {
if (&original == this) {
return *this;
if (head != NULL) {
delete head;
head = NULL;
// same as copy constructor body ----
Node *p = original.head;
Node *np = NULL;
while(p != NULL) {
Node *n = new Node(p->getData(), NULL);
if (np == NULL) {
head = n;
} else {
p = p->getNext();
np = n;
// ----
return *this;
Past exam questions
Fall 2019 - Q14
The following class is used to create objects that represent ordinary fractions n/d, consisting of a numerator n and a denominator d.
#include <iostream>
using namespace std;
class Fraction {
int numerator;
int denominator;
Fraction(int num, int denm);
int getNumerator();
int getDenominator();
void setNumerator(int num);
void setDenominator(int denm);
void print();
Fraction::Fraction(int num, int denm) {
numerator = num;
// should check that denm is not 0, but ignore for now
denominator = denm;
int Fraction::getNumerator() {
return numerator;
int Fraction::getDenominator() {
return denominator;
void Fraction::setNumerator(int num) {
numerator = num;
void Fraction::setDenominator(int denm) {
// should check that denm is not 0, but ignore for now
denominator = denm;
void Fraction::print() {
cout << numerator << "/" << denominator << endl;
Define the operator overloads for the operation:
Fraction X(1,5);
Fraction Y(4,6);
___ = X * Y; // the first multiply operation
___ = X * 2; // the second multiply operation
The first operator is:
Fraction::Fraction operator*(Fraction& rhs) {
Fraction w(numerator * rhs.numerator, denominator * rhs.denominator);
return w;
The second operator is:
Fall 2018 - Q7
The following is the definition/implementation of a class called Foo.
class Foo {
int priv;
Foo(int pv) { priv = pv; }
Foo(const Foo src) { priv = src.priv; }
Foo& operator=(Foo& rhs) {
priv = src.priv;
return this;
int getPriv() { return priv; }
void setPriv(int pv) { priv = pv; }
Compiling the above definition/implementation results in one or more errors. Re-write the class so it is error-free. Write your answer (the entire definition/implementation).
The code requires the following corrections:
// Foo(const Foo src) { priv = src.priv; }
Foo(const Foo& src) { priv = src.priv; }
Foo& operator=(Foo& rhs) {
priv = src.priv;
// return this;
return *this;
I cannot return a local variable by reference.
Fall 2021 Final - Q7
It is desired to implement an efficient deletion function in a linked list. You are given a linked list pointed to by head and a pointer node to a node in a linked list, which is guaranteed not to be
the last node in the list (i.e., not the tail node). Write a function removeNode that removed this node from the list. You should not iterathe the nodes in the list.
You may assume the following is the definition of the class, ListNode. The head node of the linked list is pointed to by head.
You are not allowed to change the function’s argument or return type.
The solution is to copy the data from the next node into the current node, and then delete the next node. | {"url":"https://pacha.dev/ece244-class-notes/lecture21.html","timestamp":"2024-11-05T15:48:06Z","content_type":"application/xhtml+xml","content_length":"52481","record_id":"<urn:uuid:8a42e723-9cc9-40fc-a43f-c3edf8194db1>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00108.warc.gz"} |
The Three Body Problem | Incalcuability
After hearing a chance snippet of something on the radio last week, I’ve been thinking about what is known in Mathematics/Physics/Astronomy as the ‘three body problem.’ It remains one of the great
questions in Mathematics as to whether it will ever be solved, or if indeed it actually can be.
The essence of the problem is simple: take three heavy masses (planets/moons/astral bodies) and give them an initial position and velocity. Can we calculate what the subsequent motion of these three
bodies will be?
The mathematics is perfectly well understood – the laws of gravitational attraction are easy to write down, and for just two bodies the calculations are simple. But add a third body, another other,
and the calculations become not only fiendishly complex but, according to some great minds, actually insoluable. Some special cases are open to analysis, but no general solution appears within reach:
the motion of the three bodies cannot be predicted and will not be captured by our infinitessimal calculus. In practice? The motion of the sun and earth and moon can only be estimated. The tide
cannot be predicted perfectly.
As a Mathematician and amateur theologian, coming across problems like this makes me reflect synthetically. I am the earth, you the moon and God the sun, each of us with our gravity, each our own
graces, our own orbits. Despite all the power of the machines and algorithms we have around us, no one can predict the future trajectory of our interactions, whether we will spin into chaos, collapse
and collide or find some periodic stability.
And this is both the problem and beauty of our engagement with the other, and the Other: not only is the subsequent motion complex, it may well be incalcuable. The rational, material, scientific mind
has to admit defeat. As in the heavenly bodies, so on earth: we can only estimate where the forces of attraction may take us.
3 responses to “The Three Body Problem | Incalcuability”
So…how does this apply with Mars coming next to the Moon on August 27th? Or does it? Just wondering what happens to the tides of Earth and how does this effect our weather on our planet. Does
Mar’s closeness effect gravitational pull on our planet or the moon’s? Basically, what the heck is going to happen when it comes this close to us? Earthquakes? Tsunamis? Volcanic Eruptions? or
just pretty to see?
Deb ;)~
Yes, it will have an effect. The three body problem is simply a version of the ‘n-body’ problem, which is obviously even more incalcuable, though models will suggest some basic effects that may
be likely.
I doubt it would stretch to earthquakes and volcanic activity though!
If I make the Sun to be all which is unknown, your analogy works for me. I did not know about this “Three Body Problem” — thank you. | {"url":"https://www.kesterbrewin.com/2009/08/04/the-three-body-problem-incalcuability/","timestamp":"2024-11-04T17:58:48Z","content_type":"text/html","content_length":"77530","record_id":"<urn:uuid:6ede74d6-8611-4bfe-82d4-23812f954d1f>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00680.warc.gz"} |
Convert Meter Square to Foot Square (m2 to ft2)
Meter Square to Foot Square Converter
Meter Square
Meter Square
= 0
Foot Square
Meter Square To Foot Square Conversion
Unit Conversion Value
1 Meter Square 10.76 Foot Square
2 Meter Square 21.53 Foot Square
5 Meter Square 53.82 Foot Square
10 Meter Square 107.64 Foot Square
20 Meter Square 215.28 Foot Square
50 Meter Square 538.20 Foot Square
100 Meter Square 1,076.39 Foot Square
200 Meter Square 2,152.78 Foot Square
500 Meter Square 5,381.96 Foot Square
1000 Meter Square 10,763.92 Foot Square
About Meter Square
Understanding Square Meter: A Comprehensive Guide
A square meter (m²) is a fundamental unit of area measurement in the International System of Units (SI). It represents the area of a square with sides that each measure one meter in length. Square
meters are used across various fields, including architecture, real estate, landscaping, and urban planning, making it crucial for professionals and individuals alike to understand its applications
and implications.
1. Definition and Calculation
A square meter is defined as the area contained within a square whose sides are exactly one meter long. The formula to calculate the area of a square is:
[ \text{Area} = \text{side length} \times \text{side length} ]
For example, if a square has a side length of 3 meters, the area would be:
[ \text{Area} = 3 , \text{m} \times 3 , \text{m} = 9 , \text{m}^2 ]
This simple calculation forms the basis for understanding more complex area measurements in both residential and commercial properties.
2. Conversion to Other Units
Understanding how to convert square meters to other area units is essential, especially in international contexts where different units might be employed. Some common conversions include:
• Square Feet: To convert square meters to square feet, multiply by approximately 10.764.
[ \text{Area in ft}^2 = \text{Area in m}^2 \times 10.764 ]
• Acres: One acre equals approximately 4046.86 square meters.
[ \text{Area in acres} = \frac{\text{Area in m}^2}{4046.86} ]
• Hectares: One hectare is equal to 10,000 square meters.
[ \text{Area in hectares} = \frac{\text{Area in m}^2}{10000} ]
These conversions are vital for sectors like agriculture, forestry, and property management, where land measurements are frequently needed.
3. Applications of Square Meter Measurement
The square meter is utilized in various fields, each of which emphasizes specific aspects of area measurement:
• Real Estate: In real estate, the size of properties is commonly measured in square meters to help buyers determine the value and usability of a space. Real estate listings typically feature the
square meters of living space, lot size, and even garden areas, enabling prospective buyers to assess properties effectively.
• Architecture and Construction: Architects and builders calculate square meters to estimate material needs (such as flooring, paint, and landscaping), assess costs, and design spaces efficiently.
For instance, flooring materials are generally priced per square meter, making accurate area calculations essential for budgeting.
• Urban Planning: City planners use square meters to gauge land usage, plan parks, roads, and public spaces, and ensure compliance with zoning laws. They analyze population density (persons per
square meter) to optimize urban environments.
• Agriculture: Farmers often measure land in square meters to manage crop yields, plan irrigation systems, and utilize fertilizers effectively. Crop density and spacing can depend on the area
available, making accurate measurements critical to agricultural productivity.
• Interior Design: Interior designers utilize square meters to furnish spaces, align furniture layouts, and create aesthetically pleasing environments. Understanding the area of walls also aids in
fabric selection for upholstery, curtains, and wall treatments.
4. Importance of Area Measurement
Accurate area measurement in square meters is essential for several reasons:
• Cost Efficiency: In construction and renovation projects, obtaining precise area measurements helps avoid overspending on materials and labor. Accurate calculations prevent waste and ensure
project budgets are adhered to, enhancing overall financial management.
• Legal and Compliance Issues: Real estate transactions often involve legal documentation that requires precise measurements to delineate property boundaries and rights. Incorrect measurements can
lead to disputes, legal challenges, or penalties.
• Space Optimization: Efficient use of space is vital in urban settings where land is often limited. By accurately calculating square meters, planners can ensure that open areas, residential
blocks, and commercial properties maximize their potential.
• Environmental Sustainability: Understanding land areas allows for better ecological planning and management practices. Assessing the area of green spaces, wetlands, and other ecological features
aids in conservation efforts.
5. Challenges in Square Meter Calculations
While measuring area in square meters seems straightforward, several challenges can complicate the process:
• Irregular Shapes: Many plots of land are not perfect squares or rectangles. Calculating the area of irregular shapes requires more complex geometry, often needing geometric formulas or
approximation techniques.
• Measurement Errors: Inaccuracies during the physical measurement (due to equipment errors, human mistakes, or environmental conditions) can lead to significant discrepancies in calculated areas,
influencing project outcomes adversely.
• Scaling Issues: When transferring measurements from blueprints or plans to physical sites, scaling errors can occur, leading to miscalculations that affect design and functionality.
6. Conclusion
The square meter serves as a universal standard for measuring area, applicable across various sectors and disciplines. Its relevance in real estate, construction, urban planning, agriculture, and
interior design highlights its integral role in modern society. As understanding area becomes even more critical in a world facing urbanization, population growth, and resource management challenges,
the importance of square meters will only continue to rise.
By mastering the use of square meters, individuals and professionals alike can make informed decisions that impact financial outcomes, efficiency, and sustainability. Therefore, whether you're
designing a new home, planning a garden, or investing in property, having a solid grasp of square meters and their implications will empower you to navigate the complexities of measurement with
About Foot Square
Understanding Foot Square: A Comprehensive Guide
Introduction to Foot Square
"Foot square" is a term often referenced in various fields such as architecture, real estate, agriculture, and sports. It represents a measure of area, primarily used to quantify spaces in square
feet. This measurement is critical for understanding dimensions, planning spaces, and calculating material needs in construction and landscaping, among other uses. In this article, we will delve into
the concept of foot square, its applications, and its significance across different sectors.
What is a Foot Square?
A foot square, or square foot (abbreviated as sq ft or ft²), is a unit of area equal to the area of a square with sides that are each one foot long. To visualize this, imagine a square where all four
edges measure exactly 12 inches (1 foot). The area can be calculated using the formula:
[ \text{Area} = \text{side} \times \text{side} ]
For a one-foot square, the area would thus be:
[ \text{Area} = 1 , \text{ft} \times 1 , \text{ft} = 1 , \text{sq ft} ]
This unit is part of the Imperial system of measurements, primarily used in the United States and some other countries. It provides a practical way to assess the area in smaller scale projects,
making it invaluable in fields that require precise spatial planning.
Importance and Applications of Foot Square
1. Real Estate
In real estate, the size of a property is often measured in square feet. Listings typically describe homes, rooms, and outdoor spaces based on their total square footage. For instance, a family home
might be listed as having 2,500 sq ft of living space, which informs prospective buyers about the size and potential usability of the home. Understanding square footage is vital for evaluating
property value, determining rental prices, and comparing similar properties.
2. Architecture and Construction
Architects and builders frequently work with foot squares when designing blueprints and constructing buildings. Knowing the exact square footage of a space helps in several key areas:
• Material Estimation: When constructing a building, knowing how many square feet of flooring, roofing, or siding is necessary allows builders to order materials accurately.
• Space Planning: Architects use square footage to design functional layouts. For example, they ensure that rooms are adequately sized to serve their intended purposes without feeling cramped or
excessively spacious.
• Building Codes and Regulations: Many local building codes establish minimum sizes for rooms and structures. Knowing the area in square feet helps architects and builders adhere to these
3. Landscaping
In the realm of landscaping, foot square measurements help homeowners and landscapers determine how much grass seed, mulch, or soil is needed for a garden or lawn. For instance, if a homeowner wants
to plant grass on a yard measuring 1,000 sq ft, they can use that dimension to calculate how much product is required to cover the area appropriately.
4. Sports Facilities
In sports, especially those played on fields or courts (like basketball or tennis), understanding the dimensions in square feet helps in designing and maintaining playing surfaces. Facilities
management can better allocate space for different sporting activities, ensuring that teams have adequate room for practice and competition.
5. Interior Design
Interior designers may work with foot squares to create aesthetically pleasing and functional living spaces. By calculating the square footage of room layouts, designers can choose appropriate
furniture sizes, arrange spaces functionally, and create harmonious designs tailored to the space available.
Examples and Comparisons
To put foot squares into perspective, consider the following comparisons:
• A typical parking space is approximately 162 sq ft (about 9 ft by 18 ft).
• A standard single-car garage is around 240 sq ft (10 ft by 24 ft).
• An average bedroom may range from 120 sq ft to 200 sq ft, depending on the design and function.
These examples illustrate how square footage translates into practical space measurement in our daily lives.
Converting Foot Square to Other Units
While the square foot is a common measurement, there are times when conversions to other units may be necessary. Here’s a quick guide to converting square feet to other area units:
• Square Yards: To convert square feet to square yards, divide the number of square feet by 9:
[ \text{Square yards} = \frac{\text{Square feet}}{9} ]
• Square Meters: To convert square feet to square meters, multiply the square feet by 0.092903:
[ \text{Square meters} = \text{Square feet} \times 0.092903 ]
• Acres: To convert square feet to acres, divide the square feet by 43,560 (the number of square feet in an acre):
[ \text{Acres} = \frac{\text{Square feet}}{43,560} ]
These conversions are essential in fields like agriculture and land development, where professionals may be more accustomed to metric measurements or larger land area calculations.
The concept of foot square is integral to multiple disciplines, providing a standard measure of area that aids in everything from real estate transactions to sports facility design. Its importance
cannot be overstated, as it influences planning, budgeting, compliance with regulations, and aesthetic considerations in various fields. Understanding how to measure and convert square footage is an
essential skill for anyone involved in construction, design, landscaping, or real estate. By grasping the nuances of this seemingly simple unit, individuals can make informed decisions that maximize
space utilization and enhance functionality across diverse applications.
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m2 to ft2Convert m2 to Foot SquareConvert Meter Square to Foot SquareConvert Meter Square to ft2 | {"url":"https://www.internettoolwizard.com/convert-units/area/m2/ft2","timestamp":"2024-11-11T03:57:27Z","content_type":"text/html","content_length":"386358","record_id":"<urn:uuid:195ebf4f-1904-44b1-ab52-8eff7c818f1c>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00653.warc.gz"} |
Thenumber of points where a linear mapping from
Thenumber of points where a linear mapping from $l<sup>n</sup><sub>2</sub>$ into $l<sup>n</sup><sub>p</sub>$ attains its norm
Let S be a regular $n n$ -matrix mapping $l<sub>2</sub><sup>n</sup>$ onto $l<sub>p</sub><sup>n</sup>$, $1≤ p < ∈fty$, with norm(Error rendering LaTeX formula). Then we are interested in the set
(Error rendering LaTeX formula) i.e. the set of points on the unit sphere where S attains its norm. We prove $card(C) <∈fty$ for $1≤ p < 2$ .This follows from properties of the Taylor expansion of $x
→ \|Sx\|<sub>p</sub>$ near points in C. The case $2 < p < ∈fty$ remains open. But we show by an example that for $p> 2$ the behaviour of $x → \|Sx\|<sub>p</sub>$ may be completely different as for $p
< 2$ .
DOI Code: 10.1285/i15900932v12p145
Full Text: | {"url":"http://siba-ese.unile.it/index.php/notemat/article/view/1373","timestamp":"2024-11-04T14:42:08Z","content_type":"application/xhtml+xml","content_length":"16868","record_id":"<urn:uuid:8d13a8ee-2f91-4e50-8880-e1d5cb700b5a>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00606.warc.gz"} |
NCERT Solutions for Class 6 Maths Chapter 7 in Hindi Fractions
NCERT Solutions for Class 6 Maths Chapter 7 Fractions in Hindi
NCERT Solutions for Class 6 Maths Chapter 7 Fractions in Hindi
The NCERT Solutions Class 6 Maths Chapter 7 In Hindi provide point-by-point explanations of all the questions given in NCERT textbooks, which are inculcated in the curriculum of Central Board of
Secondary Education (CBSE). Extramarks offers NCERT Solutions Class 6 Maths Chapter 7 In Hindi to help students clarify their doubts with a comprehensive understanding of ideas. The NCERT Solutions
Class 6 Maths Chapter 7 In Hindi are accessible in PDF format for students to study in offline mode. Students can practise different types of questions from the textbook, which is essential for any
type of school examination as well as competitive examinations. Students should utilise the NCERT Solutions Class 6 Maths Chapter 7 In Hindi as these have been organised in an appropriate manner by
experts of Extramarks to provide the best techniques for solving the questions and to ensure the best possible solutions. Extramarks recommends learners practise the NCERT Solutions Class 6 Maths
Chapter 7 In Hindi before the final examinations. It is a misconception that Maths is a tough subject. Rather it is one of the highest-scoring subjects. It is one of the few subjects where anyone can
secure good scores if they practise it consistently. With the right planning, students can become experts in this subject. It will create that confidence in them for solving the questions. The NCERT
Solutions Class 6 Maths Chapter 7 In Hindi give proper guidance to the students so that they can understand the right techniques needed to solve important questions. The experts collaborating with
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Extramarks provides important study materials for all the classes along with NCERT Solutions Class 6 Maths Chapter 7 In Hindi. Extramarks has answered all the questions of the exercise with
reasonable solutions to Class 6 Maths Chapter 7 In Hindi. Therefore, a student who has read and practised NCERT Solutions Class 6 Maths Chapter 7 In Hindi will definitely have the confidence to
perform optimally in the examinations.
When it comes to teaching students about Fractions, it should be done by explaining them with examples, not just with classroom instruction. Fractions can be used everywhere, from kitchen
measurements to store calculations. No activity is complete without faction. Therefore, it is recommended to make children aware of this important concept. Food is a great way to introduce Fractions
because students are great at connecting new ideas and concepts with what they already know. Students should be explained Fractions. Fractions are one of those areas that require a balanced approach
that requires both practical and theoretical methods. Fractions can be confusing at first, but they are actually easier to understand than most people think. The trick is to divide them into smaller
pieces so students can see how simple and basic things can be. When writing Fractions, students can use a dash for separating the numerator n and the denominator. This makes it easy to distinguish
them as separate entities. Students get a clear idea about different types of Fractions when presented with real-world problems to solve.NCERT Solutions Class 6 Maths Chapter 7 In Hindi also give
students plenty of opportunities to practice and can play an important role in exploring the concept of Fractions. Working with Fractions also means learning how to add, subtract, multiply and divide
Fractions. the denominators must be the same while adding Fractions. Adding Fractions is easy if they are equal. However, different denominators make the addition a little more difficult. The NCERT
Solutions Class 6 Maths Chapter 7 In Hindi help students understand all these basic arithmetic operations when working with Fractions. Solving problems with NCERT Solutions Class 6 Maths Chapter 7 In
Hindi will help students develop their understanding and boost their confidence. The NCERT Solutions Class 6 Maths Chapter 7 In Hindi can also be used to help with concepts that have not been
understood. Fractions refer to the part of something which is a whole. They are used to present parts in a practical form in Maths and other subjects. Students can classify Fractions using the
features of the numbers that make up the fraction. Fractions are one of the most common types of numbers in Maths. By learning how to work with Fractions, students can better understand division and
simplify very large or very small numbers into something more manageable. A Fraction is a type of number that consists of two whole numbers. Fractions are written as “a/b” and represent a percentage
of the whole. For example, 4/6 means 4 out of 6 total. It can be negative. The word “Fractions” comes from the Latin which means “to divide”. When something gets divided into parts, they become part
of the whole.
As the name suggests, this chapter deals with the concept of Fractions. It is explained through application-based questions. Each fraction has a point on the number line. This is discussed in the
next section titled Fractions on the Number Line. In this chapter, students will be introduced to different types of Fractions.
1. Proper Fractions: It is when the denominator is greater than the numerator
2. Improper Fraction: Numerator Greater than Denominator
3. Mixed Fractions: Improper Fractions can be written as a combination of wholes and parts. 4. 4. Equal Fractions: Fractions with the same denominator
4. Unequal Fractions: Fractions with different denominators
Separately, the concept of equivalent Fractions, which is an important topic, is explained. To find the equivalent of a given fraction, multiply both the numerator and denominator of the given
fraction by the same number. To find equivalent Fractions, students can divide both the numerator and denominator by the same number. Fractions also include the topic of equal fractions. In its
simplest form, it is an equal fraction when the numerator and denominator do not share a divisor. The next step is to figure out how to compare them. Comparing Equal Fractions is easy, but if
learners want the highest common factor and least common multiple, then special care must be taken in comparing different Fractions. Adding and subtracting Fractions is illustrated with appropriate
examples and explained in various subsections.
• Adding or Subtracting Equal Fractions
• Adding and Subtracting Fractions
Access Other Class 6 Maths Chapters in Hindi
NCERT Solutions for Class 6 Maths Chapter 7 Fractions in Hindi
NCERT textbooks are prescribed to all students studying in schools. The NCERT Maths book is the official course book for students studying on the CBSE Board. NCERT books are written by experts with a
proper and detailed understanding of the subject. NCERT books are made to help students to understand and grasp concepts. NCERT books are consistently helpful in clarifying the basics. The NCERT
books are very elaborate and have a structured syllabus. The theory in a structured form is very useful for students. Due to the fact that the right flow of information makes it easier for students
to grasp the topic.NCERT books are organised in a logically sequenced manner. The pictorial representations shown in the NCERT book are of great help to students as they facilitate the understanding
of the theory conveyed in words. NCERT books strike a perfect balance between the theory written in the book and the visual presentation. This is important because too much visual representation can
make students unable to understand the theory, and too much theory can make it difficult to understand the topic without a good visual representation.
NCERT book questions are simple and require only the application of formulas to solve the questions. This makes it easier for students to apply the theory they have learned. They develop knowledge of
how to apply the theory they have learned in the chapter. If learners try to solve difficult questions at the beginning of the chapter, they might not be able to solve those questions, and this may
create difficulty in understanding the topic. In the early stages, a simple question at the start helps students build confidence in solving Maths problems. Simple questions help them apply theory
efficientlyto problems given in the textbook. Once students get comfortable applying the theory, they can move on to other questions or to more difficult questions included in the final part of the
chapter of the textbook.
NCERT textbooks help in building self-confidence. When a student has just learned theory and is immediately asked a difficult question, it is obvious that the student will not be able to solve it
instantly, and this can make them less motivated to solve further problems. By starting with easier problems, NCERT textbooks keep students interested right from the start and motivate them by giving
them confidence that they can solve even the hardest Maths problem present in the book. The miscellaneous questions at the end of the chapter show how questions can be twisted and how to better
understand the topic to solve these questions. Also, an important part of the NCERT book is that at the end there is a section called Miscellaneous Questions with some very tough questions. Others
are often marked as optional, not because the question is unimportant. The confidence of students is boosted when they are able to solve various questions. This helps keep students interested in
NCERT textbooks help in understanding the topic better. Structured theories and images help the students to understand the concepts given in the book, that is why they are relevant and students are
able to understand exactly what texts are trying to convey. The problem solved after each concept and theory explained in the book shows how that concept can be applied to the problem. Solving
problems helps students a lot. A combination of in-depth theory, graphical representation of theory, and problem-solving allows students to delve a little deeper into the subject matter. Due to this
characteristic of NCERT books, many students preparing for competitive exams often rely on these books to understand concepts and strengthen their foundation.
Students should take a good look at the index of NCERT books. The curriculum is well structured and the order in which the topics are taught is very systematic. In order to study a subject, students
must meet the prerequisites for understanding that subject. The NCERT book makes this learning process very smooth, as it first covers the prerequisites needed to study the next subject before
proceeding to the next subject. Maths is often seen as a subject that is not for everyone, and students may find it confusing and even very difficult to understand which is not true. In this case,
unstructured books make things worse and make students unwilling to learn Maths at all. However, with the right textbook, students will be able to understand the subject very easily.
It is not a difficult subject to understand, and practise will help in improving the marks of students. NCERT books help students to understand and deepen their understanding of the subject. A large
number of problems in the NCERT book give students the opportunity to solve many problems and become familiar with the subject. NCERT Maths books are very important to students because they help
them gain confidence in Maths, study and practice the subject. NCERT books certainly make difficult and complex subjects very interesting and the learning process efficient. NCERT Maths books are a
boon for Maths students. Even students not studying in the CBSE board should refer to the NCERT Solutions Class 6 Maths Chapter 7 In Hindi. These solutions will help them develop their knowledge of
Maths by improving their basic understanding of this subject.
The NCERT Solutions Class 6 Maths Chapter 7 In Hindi provide detailed explanations of all the problems that are given in NCERT textbooks mandated by the Central Board of Secondary Education.
Extramarks presents chapter-by-chapter solutions to help the students clear their doubts and gain a deep understanding of concepts. These materials, including the NCERT solutions, are available in
PDF format, so students can download them and study offline.
These solutions allow students to practise different types of questions from the book that are likely to appear in the final exam. They have been developed by subject experts in a well-structured
format that provides the best way to solve problems and ensure a proper understanding of concepts. Students are encouraged to practise problems from the NCERT Class 6 Maths textbook. It also provides
a strong foundation for comprehending the concepts imparted to them in senior classes
Chapter 1 Number System
Maths Chapter 1 Number System for Class 6 helps students learn different concepts like numbers up to 10 million, Indian and International number systems, estimation, and roman numerals. Students are
also taught to read, write, and compare large numbers. Altogether there are a total of three exercises in this chapter.
Chapter 2 Whole Numbers
Chapter 2 begins with an introduction to the concept of whole numbers. The chapter also includes topics such as predecessors and successors of whole numbers, addition, subtraction, and the
representation of the natural numbers on the number line. This chapter is divided into three parts in the form of 3 exercises.
Chapter 3 Number games
In this chapter, students will learn how to find factor numbers and multiples of a given number. Perfect numbers, prime numbers, composite and coprime numbers, common divisors, and common multiples
are some of the concepts explained in this chapter. This chapter also describes finding LCM and HCF using prime factorisation methods. The chapter provides seven exercises for students to practise
problems based on the chapter.
Chapter 4 Basic Geometric Ideas
There are a total of 6 exercises in the Basic Geometric Ideas chapter. Students will learn about line segments, the distance between two points, angle-based concepts, Triangles, Polygons,
Quadrilaterals, Regular Polygons, and Circles.
Chapter 5 Understanding Basic Shapes
Understanding basic shapes discusses all kinds of shapes bounded by curves and lines. Students learn how to effectively observe corners, edges, planes, and open and closed curves in their
environment. This chapter contains a total of 9 exercises to help students better understand shapes.
Chapter 6 Integers
Chapter 6 of Class 6 Integers discusses the concept of negative numbers. In this chapter, students will learn how to represent integers on the number line, how to rearrange integers, how to add
integers, how to subtract integers, how to add integers on the number line, and how to subtract integers using the number line. Three exercises are given in Chapter 6 on Integers.
Chapter 7 Fractions
Class 6 Chapter 7 Fractions covers the simplest forms of Fractions including number lines, proper Fractions, Improper and Mixed Fractions, Equivalent Fractions, Unequal Fractions, Comparison of
Fractions, comparison of equal Fractions, and comparison of Unequal Fractions. It handles the addition and subtraction of Fractions. This chapter has 6 exercises. Students can also take the help of
NCERT Solutions Class 6 Maths Chapter 7 In Hindi.
Chapter 8 Decimal Numbers
The Decimals chapter uses word problems to explain the idea of decimals and then moves on to more complex concepts related to decimals. Tenths, hundredths, comparing and using decimals, amounts,
lengths, weights, adding decimal places, and subtracting decimal places are some of the topics covered in this chapter. This chapter contains six exercises containing questions related to the
concepts explained in the chapter.
Chapter 10 Measurement
In Class 6, Chapter 10 Measurement students learn how to measure shapes and figures in terms of area and volume. Perimeter measurement overview, rectangle perimeter, regular shape perimeter, area
measurement, area of a rectangle and area of the square are the basic topics covered here. Altogether, there are three exercises in this chapter.
Chapter 11 Algebra
With a total of five exercises, Chapter 11, Algebra introduces students to entirely new mathematical concepts, including solving problems using different alphabets. This chapter covers an
introduction to algebra, pattern matching, the concept of variables, the use of variables in variable examples, general rules, expressions with variables, practical uses of expressions, equations,
and solutions to equations. Chapter 12 Ratios and Proportions
Chapter 12, Ratios and Proportions, helps students learn about specific situations that can be compared by division. A comparison by division is a ratio. Students also learn that two ratios are
proportional if they are equal. This chapter begins by clarifying this idea. In addition to the concept of ratio, this chapter also discusses related examples and concludes with an example based on
the unit method. This chapter contains three exercises designed to help students better understand the topics covered in this chapter.
Chapter 13 Symmetry
Chapter 13 Symmetry contains a total of three exercises to help students learn how to determine symmetry elements and operations based on symmetry. This chapter discusses the creation of symmetry
shapes with two lines of symmetry, multiple lines of symmetry (two or more), reflections, and symmetry.
Chapter 14 Practical Geometry
The final chapter of CBSE Class 6 Maths, Practical Geometry, explores the creation of shapes familiar to students and the tools used to create those shapes. Topics covered in this chapter include
circles, creating circles when the radius is known, line segments, creating line segments of a specified length, creating a specified line segment, and perpendiculars. The perpendicular line passing
through a point is included. It also discusses the perpendicular bisector of a line segment, an angle, the construction of an angle of a given magnitude, the construction of an angle of unknown
magnitude, the angle bisector and an angle of a particular measure. This chapter contains six exercises.
NCERT Solutions for Class 6 Maths Chapter 7 Fractions in Hindi
The NCERT Solutions Class 6 Maths Chapter 7 In Hindi have proven to be a valuable resource for students seeking to improve their academic performance. This topic needs a little more effort to follow
in the CBSE syllabus. The NCERT Solutions Class 6 Maths Chapter 7 In Hindi are available on the Extramarks website. These NCERT Solutions Class 6 Maths Chapter 7 In Hindi were created by experts and
masters in the field. Therefore, students can have confidence in the accuracy of their solutions.They can also choose to study offline after easily downloading the PDF versions to any mobile device.
Extramarks’ NCERT Solutions Class 6 Maths Chapter 7 In Hindi are an important resource to use. Through NCERT Solutions Class 6 Maths Chapter 7 In Hindi, they have an understanding of all concepts.
The NCERT Solutions Class 6 Maths Chapter 7 In Hindi are available in PDF format with access to answers to all the questions asked in NCERT textbooks. The NCERT Solutions Class 6 Maths Chapter 7 In
Hindi provide a simple approach to solving the questions. The language used in NCERT Solutions Class 6 Maths Chapter 7 In Hindi is easy to understand. Knowledge of Maths requires logical thinking
from the beginning. However, just because it requires analytical skills and logical thinking does not mean it is a difficult subject. All learners have to do is practice and they will be able to
master the subject matter.
If students have Extramarks’ NCERT Solutions Class 6 Maths Chapter 7 In Hindi, they can very easily solve all the questions that will give accurate answers in very less time. The NCERT Solutions
Class 6 Maths Chapter 7 In Hindi are one of the most important and helpful study materials students can use to prepare for their exams. The NCERT Solutions Class 6 Maths Chapter 7 In Hindi, along
with step-by-step instructions for solving practise problems, will help them gain a deep understanding of topics. If learners want to learn offline, they should make sure they have this resource so
that they can study it anywhere, anytime. Along with NCERT Solutions Class 6 Maths Chapter 7 In Hindi, Extramarks also provides numerous other academic resources. Extramarks considers the CBSE
marking scheme when developing NCERT Solutions Class 6 Maths Chapter 7 In Hindi. They can get full marks after solving the questions from NCERT Solutions Class 6 Maths Chapter 7 In Hindi. It also
gives them a clear and strong understanding of all the concepts that they need to use in their next class. Concepts learned in Class 6 play an important role in reinforcing the foundation of the
subject. So far, students have focused on simple operations like addition, subtraction, multiplication, and division. Then they can move on to more complex chapters. It will be easier for them to
score higher marks if they understand concepts clearly.
प्रश्नावली 7.1
The NCERT Solutions Class 6 Maths Chapter 7 In Hindi pertaining to Exercise 7.1 provide an overview of Fractions and how to use them to solve problems. A representation of a small part of the whole
object is called a fraction. They are mainly used in daily life for different purposes. The NCERT Solutions Class 6 Maths Chapter 7 In Hindi contain simple explanations in simple language for student
comprehension. For more information, students can use NCERT Solutions Class 6 Maths Chapter 7 In Hindi pertaining to Fractions Exercise 7.1 PDF on the Extramarks website.
प्रश्नावली 7.2
The way Fractions are represented on the Number Line, Proper Fractions, Mixed Fractions, and Improper Fractions are concepts explained in NCERT Solutions Class 6 Maths Chapter 7 In Hindi, Exercise
7.2. To acquire better conceptual skills, students can use PDFs as their primary study material to solve problems in the NCERT textbooks. The logic and shortcuts used to solve problems are
highlighted in these NCERT Solutions Class 6 Maths Chapter 7 In Hindi
provided by Extramarks to help in exam preparation.
प्रश्नावली 7.3
The NCERT Solutions Class 6 Maths Chapter 7 In Hindi encapsulate solutions to Exercise 7.3 focus primarily on the concept of Equivalent Fractions and the simplest form of Fractions. Students are
provided with practise examples before completing the practise questions to familiarise them with the types of questions that will appear in the exam. The NCERT Solutions Class 6 Maths Chapter 7 In
Hindi can be used as a reference guide to improve students’ reasoning skills.
प्रश्नावली 7.4
The concepts explained in NCERT Solutions Class 6 Maths Chapter 7 In Hindi for Fractions, Exercise 7.4 are Equal and Unequal Fractions and how to compare them. Students know that Equal Fractions have
the same denominator, but Unequal Fractions have different denominators. Extramarks’ experts will comprehensively solve the exercises. Students can improve their academic performance in this subject
by solving word problems using NCERT Solutions Class 6 Maths Chapter 7 In Hindi
प्रश्नावली 7.5
The NCERT Solutions Class 6 Maths Chapter 7 In Hindi based on Exercise 7.5 provide students with an understanding of adding and subtracting equal Fractions and the steps to follow in solving
Fractions. Students can use PDFs of NCERT Solutions Class 6 Maths Chapter 7 In Hindi created by educators with extensive experience. Equal Fractions have the same denominator and students can use the
provided PDF to solve the problem given in NCERT Solutions Class 6 Maths Chapter 7 In Hindi faster. Mentors of Extramarks prepare these NCERT Solutions Class 6 Maths Chapter 7 In Hindi based on
Exercise 7.5 to help students to perform well in exams.
प्रश्नावली 7.6
Adding and subtracting different Fractions is briefly explained in NCERT Solutions Class 6 Maths Chapter 7 In Hindi based on Exercise 7.6. Students can use these NCERT solutions to solve textbook
problems and better understand concepts. On the other hand, fractions usually contain different denominators, and the steps to solve the problem are given in NCERT Solutions Class 6 Maths Chapter 7
In Hindi. Students can refer to NCERT Solutions Class 6 Maths Chapter 7 In Hindi based on Exercise 7.6 to solve problems found in the textbook.
NCERT Solutions for Class 6 Maths Chapter 7 Fractions in Hindi
Chapter-by-chapter NCERT solutions are provided on the Extramarks website to ensure complete guidance for students. The NCERT Solutions Class 6 Maths Chapter 7 In Hindi are created by the subject
experts considering every type of question that can appear in any competitive examination. The NCERT textbooks and NCERT Solutions Class 6 Maths Chapter 7 In Hindi are developed to give each concept
a reliable conceptual foundation. The NCERT Solutions Class 6 Maths Chapter 7 In Hindi will give learners a clear understanding of all concepts including advanced concepts explained in the textbooks.
The NCERT Solutions Class 6 Maths Chapter 7 In Hindi PDF download is available on the mobile application and website of Extramarks. Students can have access to the solutions by registering themselves
on the website of Extramarks. They can then access all the selected learning materials including NCERT Solutions Class 6 Maths Chapter 7 In Hindi in just a click. The NCERT Solutions Class 6 Maths
Chapter 7 In Hindi can be seen on the Extramarks Learning App as well.
The NCERT Solutions Class 6 Maths Chapter 7 In Hindi is designed for students studying in Hindi Medium schools. These NCERT Solutions Class 6 Maths Chapter 7 In Hindi PDFs are written in a simple and
easy-to-understand language that offers a lot of benefits. The NCERT Solutions Class 6 Maths Chapter 7 In Hindi have the best feature of being easily accessible to learners. Class 6 students can
check NCERT Solutions Class 6 Maths Chapter 7 In Hindi for self-study.
The NCERT Solutions Class 6 Maths Chapter 7 In Hindi are the combination of all the important answers to the textbook exercises. The NCERT Solutions Class 6 Maths Chapter 7 In Hindi are provided in a
step-by-step format and are well researched by subject matter experts with relevant experience in the field. Wherever applicable, relevant charts, graphs and illustrations are provided with the
answers. In short, NCERT Solutions Class 6 Maths Chapter 7 In Hindi are very helpful for exam preparation, quick revision, and final exam.
FAQs (Frequently Asked Questions)
1. Are solutions to Class 6 Chapter 7 “Fractions” available for Hindi Medium students?
Extramarks envisions an online learning portal that can be used by large groups of students to achieve academic excellence. Solutions in Hindi are made by Maths experts. The NCERT Solutions Class 6
Maths Chapter 7 In Hindi are perfect for students learning about Fractions in Hindi. Moreover, NCERT Solutions Class 6 Maths Chapter 7 In Hindi are available very easily on the internet. Students can
find NCERT Solutions Class 6 Maths Chapter 7 In Hindi on the website of Extramarks.
2. How should students approach Chapter 7 to do well in exams?
Chapter 7 is an important chapter. It is also rich in concepts, diagrams, exercises, examples, and definitions. To master this chapter, students should use the following techniques:
Students should ensure that they understand all of the concepts presented in the NCERT textbooks. They can take the help of NCERT Solutions Class 6 Maths Chapter 7 In Hindi
Students should be familiar with the examples and exercises provided in the NCERT textbooks. They can refer to the NCERT Solutions Class 6 Maths Chapter 7 In Hindi. They should stay consistent
throughout the year and practice past years’ papers well.
3. Define Fractions as described in NCERT Solutions Class 6 Maths Chapter 7 In Hindi.
A number that represents part of a whole is known as a fraction. A whole can be a single object or a group of objects. Fractions are important mathematical concepts that are frequently used in
everyday life. Therefore, it is important for students to have a good understanding of the basics of Fractions. Students should have a good understanding of the concept of Fractions as given in Class
6 Maths Chapter 7 In Hindi.
4. Where can students get the NCERT Solutions Class 6 Maths Chapter 7 In Hindi?
The NCERT Solutions Class 6 Maths Chapter 7 In Hindi introduce specific types of Fractions, All of those definitions are key to a have expertise in the idea of Fractions. One can find all the
solutions of the chapter on NCERT Solutions Class 6 Maths Chapter 7 In Hindi. | {"url":"https://www.extramarks.com/studymaterials/ncert-solutions/ncert-solutions-class-6-maths-chapter-7-in-hindi/","timestamp":"2024-11-10T14:16:54Z","content_type":"text/html","content_length":"638774","record_id":"<urn:uuid:27fc1cc5-18cb-4068-a14c-14da26b39570>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.60/warc/CC-MAIN-20241110134821-20241110164821-00206.warc.gz"} |
Reinforcement Learning - Theoretical Foundations: Part IV
Reinforcement Learning - Theoretical Foundations: Part IV
Value Function Approximation
We know various methods can be applied for function approximation. For this note, we will mainly consider those differentiable methods: Linear Approximation and Neural Nets
1. Stochastic Gradient Descent (SGD)
Here let's review a basic approximation strategy for gradient-based method: Stochastic Gradient Descent.
First our aim is to minimize the mean square error (MSE) between our estimator and the true function. The error is represented by
To attain we need to update the gradient until convergence.
A full gradient update
has the issue of converging at local minimum. Hence stochastic sampling with will work better in general.
2. Linearization
We begin by considering a linear model. So where is the feature vector/representation of the current state space. The stochastic update in SGD is also updated to .
On the other hand, we don't have an oracle for a known in practice, so we need ways to estimate it. This is where algorithm design comes in.
Algorithm analysis
1. linear Monte-Carlo policy evaluation
• To represent , we use . In every epoch, we apply supervised learning to “training data”: .
• The update is now
• Note that Monte-Carlo evaluation converges to a local optimum
• As is unbiased, it works even when using non-linear value function approximation
2. TD Learning
• We use for .
• TD(0) has the update formula:
• Linear TD(0) converges (close) to global optimum
• On the other hand we can use -return as substitute. This is a TD() method.
• Forward view linear TD():
• Backward view linear TD() requires eligibility trace:
3. Convergence of Prediction Algorithms
On\Off-policy Algorithm Table-lookup Linear Non-Linear
On-Policy MC Y Y Y
On-Policy TD(0) Y Y N
On-Policy TD() Y Y N
Off-Policy MC Y Y Y
Off-Policy TD(0) Y N N
Off-Policy TD() Y N N
Action-Value Function Approximation
Now we don't simply approximate a value function , but approximate action-value function instead.
The main idea is just find . Both MC and TD work the same way exactly by substituting these items inside the expressions.
Gradient TD
Some more recent improves aim to resolve the failure of convergence of off-policy TD algorithms. This gave birth to a Gradient TD algorithm that converges in both linear and non-linear cases. This
requires an additional parameter to be added and tuned which reprsents the gradient of projected Bellman error. In a similar fashion, a gradient Q-learning is also invented, but with no gurantee on
non-linear model convergence.
Least Squares Prediction and Experience Replay
LS estimator is known to approximate well in general. So instead of correctly approximating , it may also be ideal to approximate instead.
It is found that SGD with Experience Replay converges in this case. By "Experience Replay" we are storing the history in each epoch instead of discarding them after each iteration. And we randomly
selection some of these "data" for stochastic update in SGD.
Deep Q-Networks (DQN)
• DQN uses experience replay and fixed Q-targets
• It takes actions based on a -greedy policy
• Store transition in replay memory (experience replay)
• Sample random mini-batch of transitions from
• Compute Q-learning targets w.r.t. old, fixed parameters (fixed Q-target: not the latest but a computed some batches ago)
In general, LS-based methods work well in terms of convergence but suffers from computational complexity.
Reinforcement Learning - Theoretical Foundations: Part IV | {"url":"https://criss-wang.com/post/blogs/rl/reinfocement-4/","timestamp":"2024-11-11T20:31:13Z","content_type":"text/html","content_length":"239715","record_id":"<urn:uuid:8a1de09f-d2c5-4be7-b781-79ac925cb707>","cc-path":"CC-MAIN-2024-46/segments/1730477028239.20/warc/CC-MAIN-20241111190758-20241111220758-00725.warc.gz"} |
Cryptography from Information Loss
Paper 2020/395
Cryptography from Information Loss
Marshall Ball, Elette Boyle, Akshay Degwekar, Apoorvaa Deshpande, Alon Rosen, Vinod Vaikuntanathan, and Prashant Nalini Vasudevan
Reductions between problems, the mainstay of theoretical computer science, efficiently map an instance of one problem to an instance of another in such a way that solving the latter allows solving
the former. The subject of this work is ``lossy'' reductions, where the reduction loses some information about the input instance. We show that such reductions, when they exist, have interesting and
powerful consequences for lifting hardness into ``useful'' hardness, namely cryptography. Our first, conceptual, contribution is a definition of lossy reductions in the language of mutual
information. Roughly speaking, our definition says that a reduction $\mathsf{C}$ is $t$-lossy if, for any distribution $X$ over its inputs, the mutual information $I(X;\mathsf{C}(X)) \leq t$. Our
treatment generalizes a variety of seemingly related but distinct notions such as worst-case to average-case reductions, randomized encodings (Ishai and Kushilevitz, FOCS 2000), homomorphic
computations (Gentry, STOC 2009), and instance compression (Harnik and Naor, FOCS 2006). We then proceed to show several consequences of lossy reductions: 1. We say that a language $L$ has an
$f$-reduction to a language $L'$ for a Boolean function $f$ if there is a (randomized) polynomial-time algorithm $\mathsf{C}$ that takes an $m$-tuple of strings $X = (x_1,\ldots,x_m)$, with each $x_i
\in\{0,1\}^n$, and outputs a string $z$ such that with high probability, \begin{align*} L'(z) = f(L(x_1),L(x_2),\ldots,L(x_m)) \end{align*} 2. Suppose a language $L$ has an $f$-reduction $\mathsf{C}$
to $L'$ that is $t$-lossy. Our first result is that one-way functions exist if $L$ is worst-case hard and one of the following conditions holds: - $f$ is the OR function, $t \leq m/100$, and $L'$ is
the same as $L$ - $f$ is the Majority function, and $t \leq m/100$ - $f$ is the OR function, $t \leq O(m\log{n})$, and the reduction has no error This improves on the implications that follow from
combining (Drucker, FOCS 2012) with (Ostrovsky and Wigderson, ISTCS 1993) that result in {\em auxiliary-input} one-way functions. 3. Our second result is about the stronger notion of $t$-compressing
$f$-reductions -- reductions that only output $t$ bits. We show that if there is an average-case hard language $L$ that has a $t$-compressing Majority reduction to some language for $t=m/100$, then
there exist collision-resistant hash functions. This improves on the result of (Harnik and Naor, STOC 2006), whose starting point is a cryptographic primitive (namely, one-way functions) rather than
average-case hardness, and whose assumption is a compressing OR-reduction of SAT (which is now known to be false unless the polynomial hierarchy collapses). Along the way, we define a non-standard
one-sided notion of average-case hardness, which is the notion of hardness used in the second result above, that may be of independent interest.
Available format(s)
Publication info
Published elsewhere. Minor revision. ITCS 2020
Contact author(s)
prashantv91 @ gmail com
marshallball @ gmail com
alon rosen @ idc ac il
2020-06-02: revised
2020-04-09: received
Short URL
author = {Marshall Ball and Elette Boyle and Akshay Degwekar and Apoorvaa Deshpande and Alon Rosen and Vinod Vaikuntanathan and Prashant Nalini Vasudevan},
title = {Cryptography from Information Loss},
howpublished = {Cryptology {ePrint} Archive, Paper 2020/395},
year = {2020},
doi = {10.4230/LIPIcs.ITCS.2020.81},
url = {https://eprint.iacr.org/2020/395} | {"url":"https://eprint.iacr.org/2020/395","timestamp":"2024-11-07T04:36:21Z","content_type":"text/html","content_length":"19511","record_id":"<urn:uuid:a620a6ae-a49d-4080-917a-c60c8ce4aca7>","cc-path":"CC-MAIN-2024-46/segments/1730477027951.86/warc/CC-MAIN-20241107021136-20241107051136-00132.warc.gz"} |
The Rules of Racketlon
The Rules of Racketlon
Racketlon is the sport in which you play your opponent in each of the four racket sports table tennis, squash, badminton and tennis. A racketlon match contains four sets, one in each sport. The
winner will be the best all round racket player. In Sweden, at least two different variants of racketlon rules exist. This is a description of the rules that are presently (year 2001) in use in the
Swedish National Championships.
Set order
- The sets are played in the following order (from smaller to larger courts): table tennis, squash, badminton and finally tennis
- Each set is played to 21 points. Every ball results in a point to the winner of the ball - just like in table tennis. Unlike in table tennis, however, there is no need to have a margin of two
points to win the set. A set can thus end 21-20.
- The winner of a racketlon match is not the one that wins most sets but the one that scores the most points in total. This means that it is possible to loose three out of the four sets and still win
the match.
- The winner can choose to stop the match as soon as he has won enough points for the match to be decided.
- If, after four sets, both players have exactly the same number of points another four sets are played - but this time to 5 instead of 21. If still undecided another four sets to 5 are played. And
so on, until the match is decided.
- At the beginning of each set lots are drawn. The winner decides who will start to serve.
- After every five points the serve goes to the other player.
- At the first of these five serves the server can choose from which side to serve. Then, the server shall switch side every time.
- In tennis, the server has two chances - first and second service - just as in normal tennis.
Switching of sides
- If any of the players so wishes sides are switched at the time when 10 points are first reached by one of the players. | {"url":"http://www.racketlon.com/rackrules.010617.htm","timestamp":"2024-11-14T13:49:49Z","content_type":"text/html","content_length":"3261","record_id":"<urn:uuid:2e608c4b-08c0-4df1-971a-bb36c740e8b5>","cc-path":"CC-MAIN-2024-46/segments/1730477028657.76/warc/CC-MAIN-20241114130448-20241114160448-00601.warc.gz"} |
Get Multiple Lookup Values in a Single Cell (With & Without Repetition)
Can we look-up and return multiple values in one cell in Excel (separated by comma or space)?
I have been asked this question multiple times by many of my colleagues and readers.
Excel has some amazing lookup formulas, such as VLOOKUP, INDEX/MATCH (and now XLOOKUP), but none of these offer a way to return multiple matching values. All of these work by identifying the first
match and return that.
So I did a bit of VBA coding to come up with a custom function (also called a User Defined Function) in Excel.
Update: After Excel released dynamic arrays and awesome functions such as UNIQUE and TEXTJOIN, it’s now possible to use a simple formula and return all the matching values in one cell (covered in
this tutorial).
In this tutorial, I will show you how to do this (if you’re using the latest version of Excel – Microsoft 365 with all the new functions), as well as a way to do this in case you’re using older
versions (using VBA).
So let’s get started!
Lookup and Return Multiple Values in One Cell (Using Formula)
If you’re using Excel 2016 or prior versions, go to the next section where I show how to do this using VBA.
With Microsoft 365 subscription, your Excel now has a lot more powerful functions and features that are not there in prior versions (such as XLOOKUP, Dynamic Arrays, UNIQUE/FILTER functions, etc.)
So if you’re using Microsoft 365 (earlier known as Office 365), you can use the methods covered in this section could look up and return multiple values in one single cell in Excel.
And as you will see, it’s a really simple formula.
Below I have a data set where I have the names of the people in column A and the training that they have taken in column B.
Click here to download the example file and follow along
For each person, I want to find out what training they have completed. In column D, I have the list of unique names (from column A), and I want to quickly lookup and extract all the training that
every person has done and get these in a single set (separated by a comma).
Below is the formula that will do this:
=TEXTJOIN(", ",TRUE,IF(D2=$A$2:$A$20,$B$2:$B$20,""))
After entering the formula in cell E2, copy it for all the cells where you want the results.
How does this formula work?
Let me deconstruct this formula and explain each part in how it comes together gives us the result.
The logical test in the IF formula (D2=$A$2:$A$20) checks whether the name cell D2 is the same as that in range A2:A20.
It goes through each cell in the range A2:A20, and checks whether the name is the same in cell D2 or not. if it’s the same name, it returns TRUE, else it returns FALSE.
So this part of the formula will give you an array as shown below:
Since we only want to get the training for Bob (the value in cell D2), we need to get all the corresponding training for the cells that are returning TRUE in the above array.
This is easily done by specifying [value_if_true] part of the IF formula as the range that has the training. This makes sure that if the name in cell D2 matches the name in the range A2:A20, the IF
formula would return all the training that person has taken.
And wherever the array returns a FALSE, we have specified the [value_if_false] value as “” (blank), so it returns a blank.
The IF part of the formula returns the array as shown below:
Where it has the names of the training Bob has taken and blanks wherever the name was not Bob.
Now, all we need to do is combine these training name (separated by a comma) and return it in one cell.
And that can easily be done using the new TEXTJOIN formula (available in Excel 2019 and Excel in Microsoft 365)
The TEXTJOIN formula takes three arguments:
• the Delimiter – which is “, ” in our example, as I want the training to separated by a comma and a space character
• TRUE – which tells the TEXTJOIN formula to ignore empty cells and only combine ones that are not empty
• The If formula that returns the text that needs to be combined
If you’re using Excel in Microsoft 365 that already has dynamic arrays, you can just enter the above formula and hit enter. And if you’re using Excel 2019, you need to enter the formula, and hold the
Control and the Shift key and then press Enter
Click here to download the example file and follow along
Get Multiple Lookup Values in a Single Cell (without repetition)
Since the UNIQUE formula is only available fro Excel in Microsoft 365, you won’t be able to use this method in Excel 2019
In case there are repetitions in your data set, as shown below, you need to change the formula a little bit so that you only get a list of unique values in a single cell.
In the above data set, some people have taken training multiple times. For example, Bob and Stan have taken the Excel training twice, and Betty has taken MS Word training twice. But in our result,
we do not want to have a training name repeat.
You can use the below formula to do this:
=TEXTJOIN(", ",TRUE,UNIQUE(IF(D2=$A$2:$A$20,$B$2:$B$20,"")))
The above formula works the same way, with a minor change. we have used the IF formula within the UNIQUE function so that in case there are repetitions in the if formula result, the UNIQUE function
would remove it.
Click here to download the example file
Lookup and Return Multiple Values in One Cell (Using VBA)
If you’re using Excel 2016 or prior versions, then you will not have access to the TEXTJOIN formula. So the best way to then look up and get multiple matching values in a single cell is by using a
custom formula that you can create using VBA.
To get multiple lookup values in a single cell, we need to create a function in VBA (similar to the VLOOKUP function) that checks each cell in a column and if the lookup value is found, adds it to
the result.
Here is the VBA code that can do this:
'Code by Sumit Bansal (https://trumpexcel.com)
Function SingleCellExtract(Lookupvalue As String, LookupRange As Range, ColumnNumber As Integer)
Dim i As Long
Dim Result As String
For i = 1 To LookupRange.Columns(1).Cells.Count
If LookupRange.Cells(i, 1) = Lookupvalue Then
Result = Result & " " & LookupRange.Cells(i, ColumnNumber) & ","
End If
Next i
SingleCellExtract = Left(Result, Len(Result) – 1)
End Function
Where to Put this Code?
1. Open a workbook and click Alt + F11 (this opens the VBA Editor window).
2. In this VBA Editor window, on the left, there is a project explorer (where all the workbooks and worksheets are listed). Right-click on any object in the workbook where you want this code to
work, and go to Insert –> Module.
3. In the module window (that will appear on the right), copy and paste the above code.
4. Now you are all set. Go to any cell in the workbook and type =SingleCellExtract and plug in the required input arguments (i.e., LookupValue, LookupRange, ColumnNumber).
How does this formula work?
This function works similarly to the VLOOKUP function.
It takes 3 arguments as inputs:
1. Lookupvalue – A string that we need to look-up in a range of cells.
2. LookupRange – An array of cells from where we need to fetch the data ($B3:$C18 in this case).
3. ColumnNumber – It is the column number of the table/array from which the matching value is to be returned (2 in this case).
When you use this formula, it checks each cell in the leftmost column in the lookup range and when it finds a match, it adds to the result in the cell in which you have used the formula.
Remember: Save the workbook as a macro-enabled workbook (.xlsm or .xls) to reuse this formula again. Also, this function would be available only in this workbook and not in all workbooks.
Click here to download the example file
Learn how to automate boring repetitive tasks with VBA in Excel. Join the
Excel VBA Course
Get Multiple Lookup Values in a Single Cell (without repetition)
There is a possibility that you may have repetitions in the data.
If you use the code used above, it will give you repetitions in the result as well.
If you want to get the result where there are no repetitions, you need to modify the code a bit.
Here is the VBA code that will give you multiple lookup values in a single cell without any repetitions.
'Code by Sumit Bansal (https://trumpexcel.com)
Function MultipleLookupNoRept(Lookupvalue As String, LookupRange As Range, ColumnNumber As Integer)
Dim i As Long
Dim Result As String
For i = 1 To LookupRange.Columns(1).Cells.Count
If LookupRange.Cells(i, 1) = Lookupvalue Then
For J = 1 To i - 1
If LookupRange.Cells(J, 1) = Lookupvalue Then
If LookupRange.Cells(J, ColumnNumber) = LookupRange.Cells(i, ColumnNumber) Then
GoTo Skip
End If
End If
Next J
Result = Result & " " & LookupRange.Cells(i, ColumnNumber) & ","
End If
Next i
MultipleLookupNoRept = Left(Result, Len(Result) - 1)
End Function
Once you have placed this code in the VB Editor (as shown above in the tutorial), you will be able to use the MultipleLookupNoRept function.
Here is a snapshot of the result you will get with this MultipleLookupNoRept function.
Click here to download the example file
In this tutorial, I covered how to use formulas and VBA in Excel to find and return multiple lookup values in one cell in Excel.
While it can easily be done with a simple formula if you’re using Excel in Microsoft 365 subscription, if you’re using prior versions and don’t have access to functions such as TEXTJOIN, you can
still do this using VBA by creating your own custom function.
You May Also Like the Following Excel Tutorials:
56 thoughts on “Lookup and Return Multiple Values in One Cell in Excel (Formula & VBA)”
1. How can I sort or filter with multiple text in a cell (vijay ajay maruti ravi john), I am looking for a solution filter with condition of ajay ravi john. Can you please help me with this?
Thank you
2. for some reason when i put the VBA code in and go to run it it keeps coming up with an error on this section – Function SingleCellExtract(Lookupvalue As String, LookupRange As Range, ColumnNumber
As Integer) and i’m not sure why can you help?
3. how can i lookup multiple values in a single cell separated by symbol
PRODUCT55,OP513N16,PROODUCT61,OP495G74 ——————————————> 1
OP512E08,PRODUCT31,PRODUCT48,PRODUCT19,OP513N16 ————————> 2
PRODUCT43,OP495G74,PRODUCT22,OP747B38,PRODUCT74,PRODUCT23—–> 3
EXPECTED RESULTS
i ’d like lookup about all parts of list1 in all list2 but results of every cell be in one cell as the expected results
thank u
4. I want to limit the number of multiple values in excel dropdown
5. Hi,
Thanks a lot for this awesome code to fetch multiple values in a single cell and without repetition. It worked perfectly for me.
The only problem is that it runs only once till file is opened. The moment I reopen the file and recalculate the formula, it stops working and doesn’t resume even after pasting the code again. I
also saved the file as “Macro-enabled” file but it still didn’t solve the issue.
Any help on this problem?
Thanks a lot.
6. When 2 Criteria Match then Return Multiple Lookup Values In One Comma Separated Cell
A2=B2 Then Result From Range by “fLookUpMultiple” – Please…….
7. Great code, first one didn’t work for me, but second one that ensures there are not multiple values in the results, works great! Thank you!
8. Hi,
I’m wanting to get values from multiple items selected from a drop down list (which are listed in the same cell, each on a new line) and have the values of each, listed in the next cell, each on
a new line.
Is this possible?
Can I send you my current file, so you can see what I’m working with?
9. Hi,
Thanks a lot for this solution.
But would there be an option that it checks each cell in (not only column A) lets say the A+B+C column and that it will return me the value of column D?
10. Hi,
I tried to use this formula and it works great.
But i have a question:
Would it be possible that not only the values from collum C will be returned but also of for example values from Collum D and E?
Thanks for your help,
11. I am trying to run this against the email Addresses however I am not getting the results as mentioned. Please help I have already exhausted everything I can. I have saved exactly as mentioned in
the document however the output gives me #Value Error
12. Anyway to get the results to sum. I.e. If the results are values, say 10.3, 5.1, 7.5, I want it to return the sum of these values, so 22.9
13. How do I do it across 2 different workbooks? Need to lookup a value (input by user in a cell) — compare it in another workbook and respond back with a corresponding number(s) associated against
it (2 columns to the left) — output all in 1 cell separated and not duplicated. Thanks
14. There are only two column but I have 3 Columns then how to do?
15. Hi Sumit Sir, I receive your excel mail subscription and i visit your website is really deep knowledge and concept clearing. And this code Function MultipleLookupNoRept is really awesome.
Heartily Thanks…
16. Hello,I want to know if it is possible to nest the MultipleLookupNoRept function as it will not work with the IF functions I have been trying to use it with. Thanks.
17. hi can someone help me for the singlecellextract is it possible to use this udf if my lookupvalue is a comma separated value?
18. Hi,
I am very new to VBA, i have a scenario as below.
1. Connect to DB,
2. Execute the query (result will be a single column value)
3. Store the query result into a single cell as List value.
Can anyone please help me getting this work. Thanks!!!
19. This works great but I think I may have run into a limitation that it will not look at more that about 91,000 rows to match against. I have about 800,000 rows that I need this to look through
20. How to dropdown list with in multiple lookup cell
21. Hello Sumit,
Your VBA code is fantastic and its serving my purpose to a great extent.
But i am facing another problem which needs to solved. Please help me with the coding.
Problem is in the desired ColumnNumberi.e result column some cells are blank. So your code is accepting the blank value(with a delimiter) along with other values.
Below example will clear the problem:
Col A Col B Col C Col D
(Sales Person) ( Products) (Sales Pearson) (Result)
1 Superman Toy Superman Toy, , Staionery
2 Spiderman Spiderman , , Toy
3 Batman Stationery Batman Stationery, Toy, Soap
4 Krishh Grocery Krishh Grocery, ,Soap
5 Superman
6 Spiderman Grocery
7 Batman Toy
8 Krishh
9 Superman Stationery
10 Spiderman Toy
11 Batman Soap
12 Krishh Soap
22. Would anyone help me with macro for a shift schedule of operators working in 2 shifts.
It should choose operators from database and populate in the shift tables according to their expertise in relevant machine?
23. excellent
24. How do i get this to work as a Hlookup?
25. Hi there. I want to do something similar to this. I have a table with people named in row 2, and the date all the way down column 2. I want to be able to generate a comma separated list of those
who are on annual leave “AL” for each corresponding day.
Something like this:
date P1 P2 P3 AL
1/1 AL
2/1 AL AL AL P1,P2, P3
3/1 AL AL P1,P2
So it’s similar to your code, but looks along a row, instead of down a column.
□ Hi Craig,
You could do this with a bunch of if statements in the final column. =If (B2=”AL”,$B$1,””) & If (C2=”AL”,” “&$C$1,””) etc. Won’t be comma separated though. Written on phone so excuse
26. I need to use this code on a different worksheet – from the one the list is created on. What do I need to edit to make that happen? Not much of a coder at all….
□ Same issue for me. I need this to work on another worksheet. Were you able to figure it out?
27. Hello Sumit,
I found your Multiple lookup values in one ell in your forum and it helped me a lot.
Now i need to improve that and I need your help regarding this.
We have used a code for only one column or a cell as a lookup reference. Now I need to include one more to this. 2 columns as s lookup reference to get the same results.
My code as per your example code is below.
Function SingleCellExtractInward(lookupvalue As String, lookuprange As Range, ColumnNumber As Integer)
Dim i As Double
Dim Result1 As String
Dim Result2 As String
If Result2 = Empty Then
Result2 = “no recent inward”
SingleCellExtractInward = Result2
End If
For i = 1 To lookuprange.Columns(1).Cells.Count
If lookuprange.Cells(i, 1) = lookupvalue Then
Result1 = Result1 & ” ” & lookuprange.Cells(i, ColumnNumber) & “,”
SingleCellExtractInward = Left(Result1, Len(Result1) – 1)
End If
Next i
End Function
Could you please help me on this code to lookup 2 columns as a reference.?
28. Hello, I have found your code and it helped. I have a query that, I need to lookup one more column with the same process. How can i do that? Please help me on this.
29. Dear Sumit,
It seems it is the thing i am looking for, but it doesn’t work! I get a #Name error. I have put it in the VB editor and saved it as .xslm. First used the function across sheets and doubted this
was the problem. Then tried it i a single sheet with an example wit still got the #Name error.
Any idea what i could be doing wrong? Saved it the wrong way or something?
30. This is fantastic! Is there a simple way to extend this to search through multiple columns, for example, if column D was another list of names (sales rep 2)?
Also, is there a way to exclude blank cells?,If there happened to be a missing name, for example?
31. Thank you for this. I had found something similar a while back called vlookupall that gives similar results but was a little harder to follow the internal logic.
32. Hi, I appreciate your post showing how to do this using VBA, however I am wondering if there is a way to do this using only excel formulas?
33. Hi Sumit,
Your macro does exactly what I require so I am pleased to find it but I cannot get it to work – I am new to VBA so could be just me 🙂
I have included it into my macro enabled spreadsheet and everytime it executes there is a compile error, like the one mentioned by Laura below.
Compile Error:
Syntax Error
SingleCellExtract = Left(Result, (Len(Result) – 1))
I have tried altering the brackets but no improvement. It looks like it cannot find the added function but I am just guessing. Any suggestions?
Could it be a setting local to me on
□ FIXED!
I think the example xlsm down load is fine but the above code text included some funny error in the call to
SingleCellExtract = Left(Result, Len(Result) – 1)
It seems there is a funny character in there after copy / paste. As I just deleted it and typed it by hand and then it was fine.
Really pleased as does exactly what I need.
34. hi! thanks for this vba – how do i correct it so the vlookup value is not case sensitive? currently if I’m looking up “Apple” for example, I need to type “Apple”. If I use a lowercase “a”, it
doesn’t lookup the value. Thank you!
□ Hey Ashley.. You can use the following code for it:
Function SingleCellExtract(Lookupvalue As String, LookupRange As Range, ColumnNumber As Integer)
Dim i As Long
Dim Result As String
For i = 1 To LookupRange.Columns(1).Cells.Count
If LCase(LookupRange.Cells(i, 1)) = LCase(Lookupvalue) Then
Result = Result & ” ” & LookupRange.Cells(i, ColumnNumber) & “,”
End If
Next i
SingleCellExtract = Left(Result, Len(Result) – 1)
End Function
☆ THANK YOU SO MUCH!!! it works!!
☆ Hi Sumit, Great Job i was searching for this exact one. i am not getting the value with the cell having particular sting. Kindly help how we can get the cell having particular sting. pls
for example:
☆ hI SUMIT,
Can we get code to reverse this function i have multiple HCodes like H310, H302, etc in one column & corrosponding statement in another column like corrosive for these two codes can i get
a code to do v lookup for different h code in same cell saperated by coma to their corrosponding statement without repetation for example if two h codes having same statement i get on
35. please sir upload it again ASAP because i need it
□ Hey.. Sorry for the misisng file.. I have uploaded it!
36. please sir upload excel file regarding this topic because this excel sheet not found
37. I wonder if anyone will still respond to this thread….
I am trying to use this macro, but I get the below error:
Compile Error:
Syntax Error
and it highlights the below portion, as the step it get stuck on
SingleCellExtract = Left(Result, Len(Result) – 1)
My data is 3 columns, my formula goes as follows:
E3 is the value (a number) i want it to find in the range
$A:$C is the range
3 is the 3rd column where i want the formula to look to pull out the results (result is text).
What am i doing incorrectly?
□ Hello Laura.. Would be great if you could share the file with me. You can send it at sumitbansal@trumpexcel.com
☆ Actually, I have the same problem, even copying your cells to my workbook. How did you resolved it?
□ I’m not sure if anyone is still checking this thread, but by trial and error I was able to make it work by adding some additional parentheses around the Len function:
SingleCellExtract = Left(Result, (Len(Result) – 1))
Not sure why VBA prefers it that way now, but it appears to work.
☆ This worked perfectly! Thank you for the tip!
□ It wasn’t an extra closing parens. The “minus” char, here, is the wrong character. If you copy and paste from this blog, it’s actually the en-dash character. Simply replace it with the minus
38. Hi Sumit,
Another great idea from you. Code optimised and options integrated
Public Function fLookUpMultiple(ByRef LookUpValue As String, _
ByRef LookUpRange As Excel.Range, _
ByRef ColumnNumber As Long, _
Optional ByRef bUnique As Boolean = True) As String ‘Variant
‘Get all values from a list that match specific value
Dim lgRow As Long
Dim strFilter As String
Dim lgElement As Long
For lgRow = 1 To LookUpRange.Columns(1).Cells.Count
If bUnique Then
If LookUpRange.Cells(lgRow, 1).Value2 = LookUpValue Then
For lgElement = 1 To lgRow – 1
If LookUpRange.Cells(lgElement, 1).Value2 = LookUpValue Then
If LookUpRange.Cells(lgElement, ColumnNumber).Value2 = LookUpRange.Cells(lgRow, ColumnNumber).Value2 Then GoTo Skip
End If
Next lgElement
strFilter = strFilter & ” ” & LookUpRange.Cells(lgRow, ColumnNumber) & “,”
End If
If LookUpRange.Cells(lgRow, 1).Value2 = LookUpValue Then strFilter = strFilter & ” ” & LookUpRange.Cells(lgRow, ColumnNumber).Value2 & “,”
End If
Next lgRow
‘Delete last “,”
fLookUpMultiple = VBA.Left(strFilter, VBA.Len(strFilter) – 1)
End Function
39. I have a question! Student names with ID’s and Class groups. How to take the list and use a combo box – select the class and the names appear on another sheet with their ID’s and class name.
□ Hi Joe.. Have a look at this – http://trumpexcel.com/2013/07/extract-data-from-drop-down-list/
It does what you are looking for, and uses a data validation drop down instead of a combo box. But it can be easily replicated for a combo box as well. Hope this helps!
40. Great Job Bro…But how can we get the results without duplicates?
□ Hi Saji, Glad you liked it.
To get values without duplicates, use this code:
Function SingleCellExtract(Lookupvalue As String, LookupRange As Range, ColumnNumber As Integer)
Dim i As Long
Dim Result As String
For i = 1 To LookupRange.Columns(1).Cells.Count
If LookupRange.Cells(i, 1) = Lookupvalue Then
For J = 1 To i – 1
If LookupRange.Cells(J, 1) = Lookupvalue Then
If LookupRange.Cells(J, ColumnNumber) = LookupRange.Cells(i, ColumnNumber) Then
GoTo Skip
End If
End If
Next J
Result = Result & ” ” & LookupRange.Cells(i, ColumnNumber) & “,”
End If
Next i
SingleCellExtract = Left(Result, Len(Result) – 1)
End Function
☆ Can u explain me this VB code?
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The Kerr metric, the external metric for a rotating body, contains the equatorial gravitational radius implicitly depending on the specific angular momentum (SAM). Ignoring this dependence due to the
formal mathematical approach without understanding the physical aspects led to absurd unphysical consequences in the black hole theory, in particular, that an increase in the rotational energy at
increasing SAM weakens gravity, decreasing the gravitational radius at the pole and the effects of gravity (redshifts, mean radii of orbits and shadows). This shortcoming of the Kerr metric is
improved in a new form of this metric with an independent parameter - the gravitational radius at the pole, determined by the mass of matter without rotational energy. The contributions of the
energies of matter and rotation have the same sign and an increase in SAM strengthens gravity, increasing its effects (the equatorial gravitational radius, redshifts, mean radii of orbits and
shadows). The modified form of the Kerr metric describes the gravitational field of a frozar having angular momentum, a star with frozen structure and the surface asymptotically tending to the local
gravitational radius (minimal at pole and maximal at equator). The application of this method to the Kerr-Newman metric, including the charge, and to the NUT metric, gave modified forms of these
metrics with independent parameters. In the frozar theory, particle energies are positive everywhere, and the theory is free from the non-physical effects of the former black hole theory (horizons,
singularities, ergosphere and the extraction of energy from it, evaporation). Thermodynamics of frozars follows from the almost irreversible freezing, as the result of which, during accretion and
other processes, the mass of neutral matter without rotational energy grows almost irreversibly. | {"url":"https://qgph.org/017.html","timestamp":"2024-11-09T22:42:44Z","content_type":"application/xhtml+xml","content_length":"140715","record_id":"<urn:uuid:359a5aa0-0a6b-40c2-a98e-b929829f4993>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.10/warc/CC-MAIN-20241109214337-20241110004337-00818.warc.gz"} |
Below is a list of classes taught by Joseph Bak.
Year Semester Course Section
2024 Spring Math 41200: Topics in Applied Mathematics FG
Math A1200: Topics in Applied Mathematics FG
2023 Spring Math 20100: Calculus I GH
2022 Fall Math 43200: Theory of Functions of a Complex Variable GH
Math A3200: Theory of Functions of a Complex Variable I GH
Spring Math 32300: Advanced Calculus I GH
2021 Fall Math 39200: Linear Algebra and Vector Analysis for Engineers U
Math 39100: Methods of Differential Equations R
Spring Math 39200: Linear Algebra and Vector Analysis for Engineers T
Math 41200: Topics in Applied Mathematics EF
Math A1200: Topics in Applied Mathematics EF
2020 Fall Math 43200: Theory of Functions of a Complex Variable GH
Math A3200: Theory of Functions of a Complex Variable I GH
Spring Math 39200: Linear Algebra and Vector Analysis for Engineers M
2019 Fall Math 39200: Linear Algebra and Vector Analysis for Engineers C
Math 32800: Methods of Numerical Analysis F
Summer Math 39200: Linear Algebra and Vector Analysis for Engineers 1XC
Spring Math B3200: Theory of Functions of a Complex Variable II ST
2018 Fall Math 43200: Theory of Functions of a Complex Variable ST
Math A3200: Theory of Functions of a Complex Variable I ST
Summer Math 39200: Linear Algebra and Vector Analysis for Engineers 1XC
2017 Fall Math 39200: Linear Algebra and Vector Analysis for Engineers G
Math 19000: College Algebra and Trigonometry EF2
Summer Math 32800: Methods of Numerical Analysis 1XB
Spring Math 20200: Calculus II M
Math B3200: Theory of Functions of a Complex Variable II ST
2016 Fall Math A3200: Theory of Functions of a Complex Variable I ST
Summer Math 32800: Methods of Numerical Analysis 1XB
Math 34600: Elements of Linear Algebra 1XC
Spring Math 20100: Calculus I ST
2015 Fall Math 20100: Calculus I LM
Summer Math 32800: Methods of Numerical Analysis 1XB
Math 36500: Elements of Combinatorics 1XC
Spring Math 39204: Linear Algebra and Vector Analysis for Engineers EF
2014 Fall Math 32300: Advanced Calculus I PR
Summer Math 30800: Bridge to Advanced Mathematics 1XB
Math 32800: Methods of Numerical Analysis 1XC
Spring Math 30800: Bridge to Advanced Mathematics T
Math 39100: Methods of Differential Equations M
2013 Fall Math 39200: Linear Algebra and Vector Analysis for Engineers H
Math 32800: Methods of Numerical Analysis D2
Math 34500: Theory of Numbers FG
Summer Math 32800: Methods of Numerical Analysis 1XC
Math 37500: Elements of Probability Theory 1XB
Spring Math 36500: Elements of Combinatorics EF
2012 Fall Math 39200: Linear Algebra and Vector Analysis for Engineers F
Math 32800: Methods of Numerical Analysis D
Summer Math 32800: Methods of Numerical Analysis 1XC
Math 37500: Elements of Probability Theory 1XB
Spring Math 30800: Bridge to Advanced Mathematics T
Math 39100: Methods of Differential Equations M
Math 39500: Complex Variables for Scientists and Engineers R
2011 Fall Math 30800: Bridge to Advanced Mathematics U
Math 37500: Elements of Probability Theory R
Summer Math 34600: Elements of Linear Algebra 1XC
Math 37500: Elements of Probability Theory 1XB
Spring Math 20200: Calculus II ST
Math B3200: Theory of Functions of a Complex Variable II RS
2010 Fall Math 30800: Bridge to Advanced Mathematics U
Math A3200: Theory of Functions of a Complex Variable I FG
Spring Math 39100: Methods of Differential Equations E
Math 2700E: Theory of Numbers G
2009 Fall Math 20100: Calculus I PR
Math 36500: Elements of Combinatorics RS
Spring Math 39100: Methods of Differential Equations T
Math B3200: Theory of Functions of a Complex Variable II RS
2008 Fall Math A3200: Theory of Functions of a Complex Variable I RS
Math 4700C: Mathematical Foundations in Arithmetic TU | {"url":"https://math.sci.ccny.cuny.edu/person/joseph-bak/teaching/","timestamp":"2024-11-04T04:46:05Z","content_type":"text/html","content_length":"22563","record_id":"<urn:uuid:c3a84655-58f7-4b02-91e8-546656d84244>","cc-path":"CC-MAIN-2024-46/segments/1730477027812.67/warc/CC-MAIN-20241104034319-20241104064319-00377.warc.gz"} |
PROC SURVEYSELECT Statement
PROC SURVEYSELECT options ;
The PROC SURVEYSELECT statement invokes the SURVEYSELECT procedure. Optionally, it identifies input and output data sets. If you do not name a DATA= input data set, the procedure selects the sample
from the most recently created SAS data set. If you do not name an OUT= output data set to contain the sample of selected units, the procedure still creates an output data set and names it according
to the DATAn convention.
The PROC SURVEYSELECT statement also specifies the sample selection method, the sample size, and other sample design parameters.
If you do not specify a selection method, PROC SURVEYSELECT uses simple random sampling (METHOD=SRS) by default unless you specify a SIZE statement or the PPS option in the SAMPLINGUNIT statement. If
you do specify a SIZE statement (or the PPS option), PROC SURVEYSELECT uses probability proportional to size selection without replacement (METHOD=PPS) by default. See the description of the METHOD=
option for more information.
You must specify the sample size or sampling rate except when you request Poisson sampling (METHOD=POISSON), request a method that selects two units from each stratum (METHOD=PPS_BREWER or METHOD=
PPS_MURTHY), or specify the MARGIN= option in the STRATA statement for sample allocation. You can use the SAMPSIZE=n option to specify the sample size, or you can use the SAMPSIZE=SAS-data-set option
to name a secondary input data set that contains stratum sample sizes.
You can also provide stratum sampling rates, minimum size measures, maximum size measures, and certainty size measures in the secondary input data set. See the descriptions of the SAMPSIZE=, SAMPRATE
=, MINSIZE=, MAXSIZE=, CERTSIZE=, and CERTSIZE=P= options for more information. You can name only one secondary input data set in each invocation of the procedure. See the section Secondary Input
Data Set for details.
Table 95.1 summarizes the options available in the PROC SURVEYSELECT statement. Descriptions of the options follow in alphabetical order.
Table 95.1: PROC SURVEYSELECT Statement Options
Option Description
Input and Output Data Sets
DATA= Names the input SAS data set
OUT= Names the output SAS data set that contains the sample
OUTSORT= Names an output SAS data set that stores the sorted input data set
Selection Method
METHOD= Specifies the sample selection method
Sample Size
SAMPSIZE= Specifies the sample size
SELECTALL Selects all stratum units when the sample size exceeds the total
Sampling Rate
SAMPRATE= Specifies the sampling rate
NMIN= Specifies the minimum stratum sample size
NMAX= Specifies the maximum stratum sample size
Replicated Sampling
REPS= Specifies the number of sample replicates
Size Measures
MINSIZE= Specifies the minimum size measure
MAXSIZE= Specifies the maximum size measure
CERTSIZE= Specifies the certainty size measure
CERTSIZE=P= Specifies the certainty proportion
Control Sorting
SORT= Specifies the type of sorting
Random Number Generation
SEED= Specifies the initial seed
RANUNI Requests the RANUNI random number generator
Displayed Output
NOPRINT Suppresses the display of all output
OUT= Data Set Contents
JTPROBS Includes joint probabilities of selection
OUTALL Includes all observations from the DATA= input data set
OUTHITS Includes a distinct copy of each selected unit
OUTSEED Includes the initial seed for each stratum
OUTSIZE Includes additional design and sampling frame information
STATS Includes selection probabilities and sampling weights
You can specify the following options in the PROC SURVEYSELECT statement: | {"url":"http://support.sas.com/documentation/cdl/en/statug/65328/HTML/default/statug_surveyselect_syntax01.htm","timestamp":"2024-11-04T14:08:35Z","content_type":"application/xhtml+xml","content_length":"151636","record_id":"<urn:uuid:f58d5a54-013e-40de-8397-90d8a786a2f7>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00113.warc.gz"} |
Answer The Following Activities On A Separate Sheet Of Paper... - Coach Carvalhal
Answer The Following Activities On A Separate Sheet Of Paper…
Answer the following activities on a separate sheet of paper
А. Identify whether the object is a representation of a point, line, or plane
1. Cellphone screen
2. Edge of a wall
3. Grain of salt
4. Strand of straight hair
5. Tip of a crayon
1. Point B lies in plane M
2. Line CD
3. Point A
4. Point F
5. Line AT
1. plane
2. line
3. point
4. line
5. point
Solved point a lies outside of plane p, how many lines can. Undefined lines planes plane point terms line ab points ppt powerpoint presentation abc. Point if between lies points ex two euclid class
deleted geometry introduction chapter next
lies illustrate
Projections of points – r. syam sudhakar rao. How to determine planes, lines, points. Through a point pass a line perpendicular to a plane, through a point
undefined lines planes plane point terms line ab points ppt powerpoint presentation abc
How to determine planes, lines, points. Lies illustrate. Linear algebra
Determine plane point planes. Solved point b lies between points a and c, and all three. Lines points plane line postulate intersect intersection two if point geometry planes distinct basic also ck
disk circular radius solved transcribed
Plane point through lay line perpendicular. Math 251 diary, spring 2010. Undefined lines planes plane point terms line ab points ppt powerpoint presentation abc
Solved point a lies outside of plane p, how many lines can. Plane and distance to 3-points – geogebra. Determining whether four points lie in a plane – geogebra
Plane point through lay line perpendicular. B. illustrate/draw each of the following and label the diagram.6. point. Determining whether four points lie in a plane – geogebra
Projections of points – r. syam sudhakar rao. Point b lies in plane m. What are the names of four coplanar points
determine plane point planes
Point plane distance wolfram mathworld. Solved point b lies between points a and c, and all three. Disk circular radius solved transcribed
B. illustrate/draw each of the following and label the diagram.6. point. Determining whether four points lie in a plane – geogebra. Lies chegg
Lies chegg. Determining whether four points lie in a plane – geogebra. What are the names of four coplanar points
plane point through lay line perpendicular
Force moment point determine. 15+ geometry answer key chapter 1. Solved point b lies between points a and c, and all three
Determine plane point planes. Solved the figure below shows a circular disk located in the. Learning task 2 illustrate each of the following and label the diagram
Solved point b lies between points a and c, and all three. Linear algebra. Solved a uniformly charged rod of length l and total charge
Point b lies in plane m. Solved point a lies outside of plane p, how many lines can. Disk circular radius solved transcribed
plane point lies if equation vectors check find
Learning task 2 illustrate each of the following and label the diagram. Determine the moment of this force about point a and b. Lines points plane line postulate intersect intersection two if point
geometry planes distinct basic also ck
Quadrants labeled graph the coordinate plane with the four quadrants. Force moment point determine. Determine plane point planes
Related Posts | {"url":"https://coachcarvalhal.com/answer-the-following-activities-on-1706/","timestamp":"2024-11-14T02:31:41Z","content_type":"text/html","content_length":"137436","record_id":"<urn:uuid:2b02c152-8d74-41c5-9e1b-5b05fc4a4489>","cc-path":"CC-MAIN-2024-46/segments/1730477028516.72/warc/CC-MAIN-20241113235151-20241114025151-00086.warc.gz"} |
Friction Calculators | List of Friction Calculators
List of Friction Calculators
Friction calculators give you a list of online Friction calculators. A tool perform calculations on the concepts and applications for Friction calculations.
These calculators will be useful for everyone and save time with the complex procedure involved to obtain the calculation results. You can also download, share as well as print the list of Friction
calculators with all the formulas. | {"url":"https://www.calculatoratoz.com/en/friction-Calculators/CalcList-938","timestamp":"2024-11-13T14:45:57Z","content_type":"application/xhtml+xml","content_length":"92529","record_id":"<urn:uuid:1b4aaea6-25d6-4971-af23-1a28e3bfe249>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00344.warc.gz"} |
Ray Receiver
"It receives high-energy rays transmitted from the Dyson Swarm or Dyson Sphere. The received energy can be transmitted directly to the Power Grid or stored as Critical Photons."
Overview[ ]
The Ray Receiver is the planetary interface to the Dyson Sphere/swarm, producing energy and (eventually) Critical Photons, a key subcomponent of Universe Matrixes. It also has some of the most
complicated mechanics of any building in the game.
Usage[ ]
The Ray Receiver has two modes, Energy Generation and Photon Generation. The first is immediately available and produces 6-15 MW of power (see Mechanics), while the 2nd is unlocked by Dirac Inversion
Mechanism and is needed for the production of Antimatter. Once Planetary Ionosphere Utilization is researched, Graviton Lens can be used as a consumable to boost both the output and strength (see
Mechanics); lenses are consumed at a rate of one every 10 minutes.
Tips[ ]
The Ray Receiver is unlocked relatively early, not even requiring Structure Matrixes to research. However, at this point you can only form a Dyson Swarm: A cloud of Solar Sails that provide energy,
but each with a limited lifespan. In effect, you're "burning" Solar Sails for energy by launching them with the EM-Rail Ejector, just like you burn Coal in a Thermal Power Station. Sails in a swarm
provide a nominal 36 kW each over 5400s, which is a respectable 194.4 MJ - but remember that the best efficiency you can get at this point (see below) is 42%, meaning only ~81.6 MJ is recoverable,
and it costs 3.60 MJ to launch a sail to orbit. So you may be better off sticking to mundane sources of power like Hydrogen until you can build a Dyson Sphere.
In contrast, the Solar Sails in a Dyson Sphere provide only 15 kW (nominal) of power each, in the form of shell points, but they last forever.
Crafting Recipes
Crafting Into
Total Raw Materials
Crafting Energy (not including mining): 598 MJ
Mechanics[ ]
The Ray Receiver is made up of a complex variety of inter-related variables, which are displayed in the UI (see screenshot). Each of them are documented in more detail below.
Output[ ]
The unlabelled number between Strength and Power Load, this is how much power is actually being produced. This is either
strength * max_output
(see below for those definitions), or if there isn't enough total power in the Dyson Sphere, the value is scaled down by
total_power_requested / total_power_generated.
Note that, unlike "burner-type" generators, there is no adjustment of output based on load. Energy from the Dyson Sphere/Swarm is a renewable resource that is lost if it's not used.
Strength[ ]
This ranges from 0-100%, and expresses how well the Receiver is receiving from the sphere/swarm. The exact formula for this quantity is:
100% * clamp(0.5 + 6.0 * (sun_x*x + sun_y*y + sun_z*z + 0.8 * (dysonSphere.grossRadius / (planet.sunDistanceInAU * 40000)) + (no_lens ? 0.0 : ionEnhance)))
Where ionEnhance is √(1 - planet_radius² / (planet_radius + 0.6 * ionosphere_height)²), (x, y, z) is the normalized position vector of the building, and (sun_x, sun_y, sun_z) is the normalized sun
direction in the planet's frame of reference.
Particular things of note that follow from that formula: building on the poles (or as close as possible) is a good idea, since (when the axial tilt is low) this causes the dot-product term to be near
zero instead of fluctuating between 1 and -1. Also you want to maximize grossRadius, which is as simple as creating a new layer in the Dyson Sphere of maximum size - you don't need to define any
nodes or physically build anything on it. (Still true as of 6.17.5831.) Due to oddities in the code, creating a large-radius Dyson Swarm orbit does not work as well.
Adding a layer with radius 20000 will guarantee full 100% Strength uptime at the poles with up to this much axial tilt, or this far away from the poles with zero axial tilt, or some combination of
those two (technically the number that matters is planet_axial_inclination - oribital_inclination + latitude_from_pole):
Orbital radius (AU) 0.369 0.4 0.5 0.6 0.7 0.8 1.0 1.2 1.4 1.6 1.8
Max safe angle (°) 90 66.4 45.7 35.6 29.2 24.6 18.4 14.4 11.6 9.59 7.98
The formula for the table is sin⁻¹(.4/x-1/12) in degrees. If you can create a larger sphere layer, you can scale the orbital radii in the table proportionally.
Continuous Receiving[ ]
This also ranges from 0-100%, but unlike Strength it changes very slowly. It grows when Strength is > 75% and shrinks when Strength is < 75%. It starts at 0% on newly built Receivers, and you want to
build it to 100% because of the positive effects it has on the other variables.
Max Output[ ]
This is the current theoretical maximum output of the Receiver. Unlike most power generators, it is not a fixed number, but is determined by this formula:
trunc((1 + 1.5 * continuous_receiving) * (no_lens ? 1.0 : 2.0) * (photon_generation ? 8.0 : 1.0) * 100000) * 60W
This has several implications: With full Continuous Receiving, a Receiver can output 2.5x as much(15 MW) instead of the base 6 MW. Also, using Photon Generation mode boosts output by 8x (48MW up to
120 MW total), and Graviton Lens boosts it by another 3x. (144 MW up to 480 MW total) All of these boosts are multiplicative. Also, these boosts are only changing how much power a single Receiver can
handle - they aren't making it more efficient (see below).
A Critical Photon requires 1.2 GJ of energy to be created. Thus, in Photon Generation mode you will have the following output rates:
Continuous Receiving No Lens With Lens
0% 2.4/min 4.8/min
50% 4.2/min 8.4/min
100% 6/min 12/min
Ray Receiving Efficiency[ ]
This ranges from 0-100%, and is a measure of how efficient the power transmission is between the Dyson Sphere/swarm and the Receiver. Somewhat confusingly, you can always receive up to the maximum of
your rated power (as given by Max Output), and inefficiencies are made up for by requesting extra on the sending side, such that
requested_power = output_power / receiving_efficiency.
Example: If you are receiving a full 6.00 MW with an efficiency of 30%, you will be requesting (and consuming) 6.00 MW/.3 = 20 MW from the sphere.
The formula for this quantity is:
100% * (1 - solarEnergyLossRate * (1 - 0.4 * continuous_receiving²)),
where solarEnergyLossRate starts at 0.7 and is multiplicatively reduced by upgrades:
solarEnergyLossRate = 0.7 * 0.9^upgrade1 * 0.85^upgrade2,
where upgrade1 is the number of upgrades in levels 1-6 researched, and upgrade2 is for levels 7+.
Efficiencies for the first several upgrades, at different Continuous Receiving amounts:
CR Lv0 Lv1 Lv2 Lv3 Lv4 Lv5 Lv6 Lv7 Lv8
0% 30.0 37.0 43.3 49.0 54.1 58.7 62.8 68.4 73.1
50% 37.0 43.3 49.0 54.1 58.7 62.8 66.5 71.5 75.8
100% 58.0 62.2 66.0 69.4 72.4 75.2 77.7 81.0 83.9
Requested Power[ ]
This is broken into two parts, the currently requested power from the sphere/swarm, and the theoretical maximum requested power. The currently requested power is just
output_power / receiving_efficiency,
with output_power shown to the left of Strength, while the maximum is
max_output * strength / receiving_efficiency
with Max Output, Strength and Receiving Efficiency all defined above.
Dyson Sphere Status[ ]
This is the similar to Requested Power, but summed over the entire sphere/swarm. The first number is total power requested, the second is total power generated. A couple of oddities:
• Although it's labelled "Dyson sphere status," it is the sum of all requested/generated power for both Dyson Spheres and Dyson Swarms.
• Usually fractions of the form "X / Y" always have X < Y. However, when the sphere is underpowered in comparison to the Receivers' requests (which can happen easily), this field will have a larger
"numerator" than denominator. | {"url":"https://dyson-sphere-program.fandom.com/wiki/Ray_Receiver","timestamp":"2024-11-04T02:49:06Z","content_type":"text/html","content_length":"282889","record_id":"<urn:uuid:c6c00094-4cc8-425c-83c5-5e21de224b38>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00240.warc.gz"} |
Flow Past a Cylinder
The circular cylinder, along with the flat plate, is the body most widely studied in fluid dynamics and aerodynamics. Drag data for the cylinder are known from very low Reynolds numbers all the way
to hypersonic speeds.
Vortex Shedding. Systematic vortex shedding analysis was first due to von Karman, who analyzed the breakdown of the symmetric flow. The von Karman vortex street has become one of the most well known
unsteady problem. Impulsive start was already known to Prandtl (1904), and the rotating cylinder was known to Tollmien (1931).
Drag data are tabulated for all Reynolds numbers, flow visualizations are available up to Mach numbers M=12.1 (to the author's knowledge). Although the unsteady wake behind the cylinder has been
considered for a long time as purely two dimensional, there are spanwise vortex structures that appear at some Reynolds numbers. These structures are a function of the cylinder aspect ratio L/D.
References on the circular cylinder can be found in any text of fluid dynamics.
Related Material | {"url":"http://aerodyn.org/unsteady/unsteady.html","timestamp":"2024-11-14T02:06:13Z","content_type":"text/html","content_length":"14967","record_id":"<urn:uuid:527c59b0-523a-4e7a-9ee9-c5f85feb671c>","cc-path":"CC-MAIN-2024-46/segments/1730477028516.72/warc/CC-MAIN-20241113235151-20241114025151-00892.warc.gz"} |
Student Organizations in Mathematics
Main Content
Student Organizations
TheTCU Math Clubis a club hoping to foster a greater understanding and appreciation of mathematics, to encourage undergraduate activity in math research and mathematical experiences, and to provide
a social and intellectual forum for all students interested in mathematics. Any student interested in Mathematics is welcome to join! You don't have to be a Math major to join – you just need to love
Math! This organization will host events such as: problem solving tasks, puzzles, student talks, guest talks, board game nights, information sessions, movie nights, study hour, peer tutoring, and
other mathematical activities.
The Gamma Xi chapter of the national actuarial group Gamma Iota Sigma (GIS) was established at TCU in September 2019. GIS's mission is to promote and sustain student interest in careers in insurance,
risk management, and actuarial science. | {"url":"https://cse.tcu.edu/mathematics/student-experience/student-organizations.php","timestamp":"2024-11-02T22:00:52Z","content_type":"text/html","content_length":"45726","record_id":"<urn:uuid:55bea182-db41-4541-b874-7d1ca544f2ea>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00492.warc.gz"} |
with $s_k=\pm 1$. The quantity $N$ represents the total number of spins and $J$ is a coupling constant expressing the strength of the interaction between neighboring spins. The symbol $<kl>$
indicates that we sum over nearest neighbors only. We will assume that we have a ferromagnetic ordering, viz $J> 0$. We will use periodic boundary conditions and the Metropolis algorithm only. The
material on the Ising model can be found in chapter 13 of the lecture notes. The Metropolis algorithm is discussed in chapter 12.
Project 4a): A simple $2\times 2$ lattice, analytical expressions
Assume we have only two spins in each dimension, that is $L=2$. Find the analytical expression for the partition function and the corresponding expectations values for the energy $E$, the mean
absolute value of the magnetic moment $\vert M\vert$ (we will refer to this as the mean magnetization), the specific heat $C_V$ and the susceptibility $\chi$ as functions of $T$ using periodic
boundary conditions.
Project 4b): Writing a code for the Ising model
Write now a code for the Ising model which computes the mean energy $E$, mean magnetization $\vert M\vert$, the specific heat $C_V$ and the susceptibility $\chi$ as functions of $T$ using periodic
boundary conditions for $L=2$ in the $x$ and $y$ directions. Compare your results with the expressions from a) for a temperature $T=1.0$ (in units of $kT/J$).
How many Monte Carlo cycles do you need in order to achieve a good agreeement?
Project 4c): When is the most likely state reached?
We choose now a square lattice with $L=20$ spins in the $x$ and $y$ directions.
In the previous exercise we did not study carefully how many Monte Carlo cycles were needed in order to reach the most likely state. Here we want to perform a study of the time (here it corresponds
to the number of Monte Carlo sweeps of the lattice) one needs before one reaches an equilibrium situation and can start computing various expectations values. Our first attempt is a rough and plain
graphical one, where we plot various expectations values as functions of the number of Monte Carlo cycles.
Choose first a temperature of $T=1.0$ (in units of $kT/J$) and study the mean energy and magnetisation (absolute value) as functions of the number of Monte Carlo cycles. Let the number of Monte Carlo
cycles (sweeps per lattice) represent time. Use both an ordered (all spins pointing in one direction) and a random spin orientation as starting configuration. How many Monte Carlo cycles do you need
before you reach an equilibrium situation? Repeat this analysis for $T=2.4$. Can you, based on these values estimate an equilibration time? Make also a plot of the total number of accepted
configurations as function of the total number of Monte Carlo cycles. How does the number of accepted configurations behave as function of temperature $T$?
Project 4d): Analyzing the probability distribution
Compute thereafter the probability $P(E)$ for the previous system with $L=20$ and the same temperatures, that is at $T=1.0$ and $T=2.4$. You compute this probability by simply counting the number of
times a given energy appears in your computation. Start the computation after the steady state situation has been reached. Compare your results with the computed variance in energy $\sigma^2_E$ and
discuss the behavior you observe.
Studies of phase transitions
Near $T_C$ we can characterize the behavior of many physical quantities by a power law behavior. As an example, for the Ising class of models, the mean magnetization is given by | {"url":"https://notebook.community/CompPhysics/ComputationalPhysicsMSU/doc/src/Projects/2016/IsingModel/ipynb/IsingModel","timestamp":"2024-11-09T13:48:43Z","content_type":"text/html","content_length":"38271","record_id":"<urn:uuid:2a3fcad5-1ae4-4fff-b2e4-bbd967f750b4>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00818.warc.gz"} |
Obtain pairs of vertices to compute all-to-all connectivity for.
This is needed for all-to-all connectivity. Calculates all the pairs of vertices that are further away from each other than the selected distance limit.
src_or_fwdinstance of SourceSpaces | instance of Forwxard
The source space or forward model to obtain vertex pairs for.
min_dist: float
The minimum distance between vertices (in meters). Defaults to 0.04.
vert_fromndarray, shape (n_pairs,)
For each pair, the index of the first vertex.
vert_tondarray, shape (n_pairs,)
For each pair, the index of the second vertex.
See also
Obtain pairs for one-to-all connectivity. | {"url":"https://users.aalto.fi/~vanvlm1/conpy/generated/conpy.connectivity.all_to_all_connectivity_pairs.html","timestamp":"2024-11-04T01:27:48Z","content_type":"text/html","content_length":"7266","record_id":"<urn:uuid:a948462b-4869-434e-8945-cfef80640f3c>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00395.warc.gz"} |
8 Best Online Algebra Classes and Courses for 2022 – TangoLearn
Learn Algebra Online With 8 Best Available Courses
Reading Time: 9 minutes
One subject that most high school and college students struggle with is algebra. To master this subject, you must have a stronghold over the fundamental topics.
And as you move forward, it could be of great help to have a source of step-by-step guidance to walk you through the more advanced concepts.
Nothing does it better than taking an online course. Online courses can help you gain the skills you need to master this subject, whether you are a high school student or a professional planning to
resume your education.
These basic algebra online courses can come in handy in providing you with the extra help you need to excel in Algebra.
To help you out, we have created an exclusive list of top algebra courses to learn algebra online.
Top Algebra Online Classes
Get The Best Algebra Training Online With These Courses!
Learn algebra with this comprehensive course covering everything you need to know in Algebra 1 and 2.
To help you check your progress as you move ahead, these best algebra online courses come with over 120 quizzes and over 20 workbooks. You’ll also find important notes at the end of each lesson and
the chance to clarify your doubts in the Q&A section.
Rating 4.7
Return or refund policy 30-day money-back guarantee
Certification Yes
Paid Yes
Duration 12.5 hours of on-demand video
Enrolled 53, 139 students
Krista King
Cons This is not the best online algebra class for beginners.
Learning Outcomes
In these classes, you will be learning:
• Fractions, radicals and exponents
• Advanced operations in fractions, radicals and exponents.
• Operations, along with the order of operations and like-terms.
• Graphing (parabolas, parallel lines, perpendicular lines)
• Equations as well as systems of equations (direct variation, indirect variation, inverse operations, time/rate/distance related questions)
• Exponential and logarithmic functions.
• Polynomials and factoring.
• Functions- sums and products of functions and domain and range.
• Inequalities (conjunctions on a number line, trichotomy and graphing inequalities)
To take up this algebra training online, you require:
• Knowledge of arithmetic of whole numbers.
• Knowledge of decimals, exponents, fractions and radicals.
• Basic arithmetic
Who should take this course to learn algebra online?
These algebra online classes are for:
• Parents who homeschool their children and want some extra help in algebra.
• Students studying Algebra 1 and 2.
• Students who will be studying algebra soon and want to get ahead.
• Anyone who wants to revise math.
Review Dylan H. :
“Excellent course on the concepts of Algebra. Will prepare you for basic math courses in College.
This inclusive, 18-sections long course has been carefully curated for high school students who are studying algebra 1, 2 or 3. It will also prove to be extremely helpful to university students
dealing with intermediate and college-level algebra.
Rating 4.7
Return or refund policy 30-day money-back guarantee
Certification Yes
Paid Yes
Duration 21 hours of on-demand video
Enrolled 6, 187 students
Instructor Julio G.
Cons Detailed examples and more explanations could have been included.
Learning Outcomes
In this one of the best algebra online courses, you will be learning:
• The basic principles and fundamentals of algebra along with order of operations, fractions review, inequalities, functions etc.
• When you learn algebra online with this course, you also gain insights into what are absolute value functions, factoring, polynomials, systems of linear equations, rational and radical
expressions, and quadratic equations.
• You’ll also be covering other topics like complex imaginary numbers, conic sections, geometric sequences, arithmetic sequences, logarithmic functions, exponential functions and systems of linear
• How to solve linear equations and how to graphic linear equations using slope and Y-intercept.
To sign-up for these algebra online classes, you require:
• A basic understanding of mathematic principles.
Who should take this course?
This is the best algebra training online for:
• High school students studying algebra.
• College students who have taken algebra.
• Adults who are thinking about going back to college.
Review Yohannes H. :
“He speaks slowly easy to follow . And his methods are easy to understand.
These advanced-level abstract classes focus on group theory specially designed for math majors and those interested in learning advanced math.
To learn algebra online through this course, you must know about writing proofs and math notation. The short assignment included with the course will help you track your understanding of what is
being taught.
This one of the best algebra online courses consists of 98 video lectures, divided into 17 different sections.
Rating 4.8
Return or refund policy 30-day money-back guarantee
Certification Yes
Paid Yes
Duration 9.5 hours of on-demand video
Enrolled 1, 737 students
Provider The Math Sorcerer
Cons This pace of the course is a bit slow.
Learning Outcomes
In this best online algebra class, you will be covering a variety of topics, including:
• Defining Binary Operation and ways to determine whether an operation is a binary operation.
• Defining a Group and examples of important groups under various operations.
• General Linear group, Klein Four-Group, Special linear Group, the additive group of integers module
• Notations for permutations
• Determining whether a binary operation is associate or commutative.
• Groups defined on powersets and with componentwise multiplication.
• Proving fundamental properties of groups
• Cyclic groups
• Injective, surjective and bijective functions
• Knowledge and examples of subgroups
• Relation, equivalence relations, equivalence classes
• Constructing finite cyclic groups with direct products.
• Function, domain, codomain, and direct and inverse image.
• Symmetric groups
• How to probe that Cosets are equivalence classes that partition a group.
• Lagrange’s theorem and proof.
• Normal subgroups, group homomorphisms, group isomorphisms and quotient groups.
• First and second isomorphism theorem
To take up these best algebra online courses, you must:
• Have a thorough understanding of higher-level mathematics.
• Those who are not thorough with advanced math but have a strong desire to learn may also take up this course.
Who should take this course to learn algebra online?
This is the best algebra training online for:
• Those looking to learn advanced-level math.
• Math majors.
Review Lizzie M. :
The course is amazing. I really enjoyed the course. I gained experience and knowledge.
Beginner-level algebra courses can help you build a strong foundation and make it easier for you to deal with more advanced concepts.
If you want to learn algebra or brush up your knowledge on the basics of algebra, these algebra online classes are the right pick for you. In these short but comprehensive algebra online courses, you
will cover this complicated subject’s fundamentals via video lessons divided into various segments.
Rating 4.9
Return or refund policy 30-day money-back guarantee
Certification Yes
Paid Yes
Duration 1.5 hours of on-demand video
Enrolled 4, 084 students
Math Fortress
Cons The course has not been updated in a long time.
Learning Outcomes
In this course, you will:
• Learn algebra online and master the fundamentals of algebra.
• Improve your problem-solving abilities.
• Improve your conceptual understanding of the subject
• You will be covering several topics like grouping symbols, variables, equations, and translating sentences into equations and words into symbols.
To take these best online algebra classes, you must be aware of basic math and prealgebra.
Who should take these algebra online courses?
This basic algebra course is most suitable for high school and college students.
Review Zahid. :
This course teaches basics of Algebra in a very easy to follow pace and easy to understand for perfect novice. It also shows how to solve a number of examples shown in the course.
This algebra training online course includes an in-depth explanation of a standard algebra course’s fundamental concepts and topics.
Divided into digestible segments, the course takes you through all the concepts you need to master without overwhelming you with excessive information. You will also be challenged to use the skills
you learn to solve practice problems.
Rating 4.2
Return or refund policy 30-day money-back guarantee
Certification Yes
Paid Yes
Duration 21.5 hours of on-demand video
Enrolled 2, 834 students
John Swokowski
Cons The pace of the course is a bit slow.
Learning Outcomes
In this course you will be learning:
• Algebra concepts that are a part of a standard algebra course.
• Master the skills you require to do well in placement tests like GED, ACT or high school equivalency exam.
• You will be covering topics like the language of algebra, how to solve equations, quadratic equations, linear equations, graphing inequalities, systems of equations, radical expressions and more.
For signing up for this one of the best algebra online courses, you require:
• A 6^th grade education at least
• A strong desire to learn.
Who should take this course?
Students who are looking to gain expertise in essential fundamental concepts of the subject should take these best online algebra classes.
Review Britney G. :
This is a very good course for any student in high school getting ready for GED exams and more. You have helped me a lot in my studies.
If you are planning to take up courses that require precalculus and calculus and want to strengthen your knowledge of algebra and geometry, this specialization, consisting of basic algebra online
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The three courses included in this specialization to learn algebra online are for beginners and intermediate-level learners. The program also trains you in quantitative skills and reasoning.
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Cons This course is not for advanced learners.
Learning Outcomes
In this best online algebra class, you will be covering several topics such as:
• How to solve linear, polynomial, exponential and quadratic equations.
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• Properties of functions like domain, range, graphs, intercepts and asymptotes.
• How apply the theory presented for modeling data, evaluating arguments and reasoning logically.
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This comprehensive course offered by Khan Academy covers all the fundamental topics that come under Algebra 1. In this course, you will be covering the following sections:
• Foundations of algebra and knowledge of units
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Related: Best Discrete Mathematics Courses, Trigonometry Classes Online, Statistics Training
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Worst additive noise: An information-estimation view
The "worst additive noise" problem is considered. The problem refers to an additive channel in which the input is known to some extent. It is further assumed that the noise consists of an additive
Gaussian component and an additive component of arbitrary distribution. The question is: what is the distribution over the additive noise that will minimize the mutual information between the input
and the output? Two settings for this problem are considered. In the first setting a Gaussian input with a given covariance matrix is considered and it is shown that the problem can be handled in the
framework of the Guo, Shamai and Verdú I-MMSE relationship. This framework gives a simple derivation of Diggavi and Cover's result, that under a covariance constraint the "worst additive noise"
distribution is Gaussian, meaning that Gaussian noise minimizes the input-output mutual information given that the input is Gaussian. The I-MMSE framework also shows that given that the input is
Gaussian distributed, for any constraint on the distribution of the noise, which does not prohibit a Gaussian distribution, the "worst" distribution is a Gaussian distribution complying with the
constraint. In the second setting it is assumed that the input contains a codeword from an optimal point-to-point codebook (i.e., it achieves capacity) and it is shown, for a subset of SNRs, that the
minimum mutual information is obtained when the additive signal is Gaussian-like up to a given SNR.
Publication series
Name 2014 IEEE 28th Convention of Electrical and Electronics Engineers in Israel, IEEEI 2014
Conference 2014 28th IEEE Convention of Electrical and Electronics Engineers in Israel, IEEEI 2014
Country/Territory Israel
City Eilat
Period 3/12/14 → 5/12/14
All Science Journal Classification (ASJC) codes
• Electrical and Electronic Engineering
Dive into the research topics of 'Worst additive noise: An information-estimation view'. Together they form a unique fingerprint. | {"url":"https://cris.iucc.ac.il/en/publications/worst-additive-noise-an-information-estimation-view","timestamp":"2024-11-02T17:26:25Z","content_type":"text/html","content_length":"47817","record_id":"<urn:uuid:79bae3d3-0ba3-446b-8db0-141305b90048>","cc-path":"CC-MAIN-2024-46/segments/1730477027729.26/warc/CC-MAIN-20241102165015-20241102195015-00377.warc.gz"} |
Working with Cuisenaire Rods In Mathematics
Willis Grace O. A. Thorp Academy
6024 W. Warwick Ave.
Chicago IL 60634
Fourth Grade students will learn the numerical value of each underlined letter
of the standard Rod Code and develop systematic usage for building new units.
Materials Needed:
A set of Cuisenaire Rods for each student or groups of two
Construction Paper
Overhead Projector
In order to effectively develop good skills for working with common
denominators, students must first strengthen and reinforce their basic skills.
These basic skills encompass such concepts as addition, subtraction,
multiplication and division of whole numbers which form a foundation for
fractions. The materials used reinforce concepts further by providing students
with concrete examples of the skills that they practice.
Performance Assessment:
a. Given Cuisenaire Rods, introduced to strategies, and ample
performance time, students will utilize the activity card, "Finding
b. The students will answer 5 out of 7 questions correctly while
playing "Finding One."
c. The students will demonstrate whole fractions (reduced) using
Cuisenaire Rods.
d. The students will demonstrate equivalent fractions using Cuisenaire
e. The students will discuss length, color, and shape of Cuisenaire Rods.
Multicultural Aspects:
Cuisenaire Rods were discovered in Thuin, Belgium over forty years ago by
George Cuisenaire and are now recognized as a basic learning tool all over the
Return to Mathematics Index | {"url":"https://smileprogram.info/ma9208.html","timestamp":"2024-11-09T03:20:18Z","content_type":"text/html","content_length":"2465","record_id":"<urn:uuid:bdf3b66d-9b0d-4dcf-a8f3-b2db2caf0ec4>","cc-path":"CC-MAIN-2024-46/segments/1730477028115.85/warc/CC-MAIN-20241109022607-20241109052607-00133.warc.gz"} |
1.2 Finding Limits Graphically and Numerically
An Introduction to Limits
Here’s a function.
\[f(x) = \frac{x^3-1}{x-1}\]
Sketching this wouldn’t be too difficult. Pick some $x$-values, plug them in, generate $y$-values, and you’ll get a parabola as a result (degree 3 on top, divided by degree 1 on bottom, leaves you
with $x^2$).
But what happens when $x=1$? The function it self won’t evaluate since you are dividing by zero, but what does it look like the function is doing? What happens when you plug in values that get closer
and closer to $x=1$? As you plug in numbers that approach 1 from the negative side (0.9, 0.99, etc.) and then numbers from the positive side (1.1, 1.01, etc.), you’ll that both sets of numbers look
like they are heading towards a value of 3.
And if you were to graph it with Desmos or a calculator, you would see the same thing. Technically, there’s a gap, but it is definitely heading towards 3. The limit of $f(x)$ as $x$ approaches 1 is
3. We write that as
Let’s look at a different case.
\[f(x) = \begin{cases} 1, & x\neq2 \\ 0, & x=2 \end{cases}\]
Here, there is distinct difference between $f(2)$ and $\lim_{x\to2}f(x)$. The former is clearly defined–when $x$ is 2, then $f(x)$ is 2–but the limit, what the function appears to be doing as it
approaches $x=2$ is 1.
Limits That Fail to Exist
There are some situations where limits do not exist, where it can’t be determined what the graph is actually doing when it approaches a given value. One is different right and left x }{x}$ is one
behavior, which is when approaching a value from different directions yields different values. The graph of $f(x)=\frac{ example.
Unbounded behavior is example. This is where the graph takes off to infinitely while approaching the value in question.
The last one is oscillating behavior. This is when the oscillations of a function increase as they approach a value, never settling on a specific point.
A Formal Definition of Limit
We’ve seen
\[\lim_{x\to c} f(x) = L\]
meaning that if $f(x)$ becomes really close to a single number, $L$, as $x$ approaches $c$ from either side, then the limit of $f(x)$ as $x$ approaches $c$ is $L$.
This is technically not a well-defined definition of a limit, because what is meant by “really close” and “approaches”? There is a more technical version, called the epsilon-delta definition, written
out in the text, and I can also recommend this Khan Academy video on the topic, but CollegeBoard is very clear about this not appearing on the exam. | {"url":"https://wkurzius.github.io/textbook-notes/calc-for-ap-larson/1-limits-and-their-properties/1.2-finding-limits-graphically-and-numerically.html","timestamp":"2024-11-08T19:12:01Z","content_type":"text/html","content_length":"6717","record_id":"<urn:uuid:e84357ec-5ba0-45ac-b4e1-6985620a6b92>","cc-path":"CC-MAIN-2024-46/segments/1730477028070.17/warc/CC-MAIN-20241108164844-20241108194844-00253.warc.gz"} |
How do you find velocity as a function of time?
How do you find velocity as a function of time?
We use the term “instantaneous velocity” to describe the velocity of an object at a particular instant in time. Given an equation that models an object’s position over time, s ( t ) s(t) s(t), we can
take its derivative to get velocity, s ′ ( t ) = v ( t ) s'(t)=v(t) s′(t)=v(t).
Is velocity as a function of time acceleration?
Acceleration is defined as the rate of change of an object’s velocity. Just like the derivative of the position function gives you the velocity as a function of time, the derivative of the velocity
function (which is also the second derivative of the position function) gives you the acceleration as a function of time.
What does as a function of time mean?
In mathematics, a function is a binary relation between two sets that associates each element of the first set to exactly one element of the second set. For example, the position of a planet is a
function of time.
Is velocity related to time?
Velocity is the measure of the amount of distance an object covers in a given amount of time. Here’s a word equation that expresses the relationship between distance, velocity and time: Velocity
equals distance travelled divided by the time it takes to get there.
What is the particle’s position as a function of time?
A particle’s position as a function of time is described as y(t)= 2t- + 3t + 4.
Is a person’s weight a function of their height?
No, height is not a function of weight, because two students weigh 165 pouncs (input) but have different heights (output). Height is input and weight is output.
How to find the actual velocity?
Method 2 of 3: Finding Velocity from Acceleration Understand the velocity formula for an accelerating object. Acceleration is the change in velocity. Multiply the acceleration by the change in time.
This will tell you how much the velocity increased (or decreased) over this time period. Add the initial velocity. Specify the direction of movement. Solve related problems.
How do you calculate the final velocity?
Finding the final velocity is simple with a few calculations and basic conceptual knowledge. Determine the object’s original velocity by dividing the time it took for the object to travel a given
distance by the total distance. In the equation V = d/t, V is the velocity, d is the distance and t is the time.
What is the formula for finding the average velocity?
Average velocity is defined in terms of the relationship between the distance traveled and the time that it takes to travel that distance. The formula for finding average velocity is: v av = x f – x
i / t f – t
Is instantaneous velocity and velocity the same thing?
Velocity and instantaneous velocity are usually considered synonyms. Average velocity is something different, being the change in position divided by the change in time. Likewise, acceleration and
instantaneous accleration are considered to mean the same thing. | {"url":"https://pvillage.org/archives/34788","timestamp":"2024-11-03T17:14:49Z","content_type":"text/html","content_length":"53823","record_id":"<urn:uuid:eba248a2-a757-4e02-a64b-a123bda8a99f>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00781.warc.gz"} |
Inverting matrix after Cholesky decomposition does not finish for "large" dimension
Inverting matrix after Cholesky decomposition does not finish for "large" dimension
I'm trying to invert the cholesky decomposition of a matrix. While this works for small a, for around a=~30, it no longer finishes.
From the _print_, I know that the problem is in the last line.
This seems a very odd behavior, as like matrix inversion would blow exponentially, which is not true, but even if it was it shouldn't go from taking 10s to finish to not finishing at all. Moreover,
since the matrix N is triangular, shouldn't it even be faster to compute the inverse matrix?
I am very puzzled about this, sorry...
a = 30
M = random_matrix(ZZ, a, x = -a, y = a)
M = M.T * M
N = M.inverse().cholesky()
print "choleskied"
Ni = N.inverse()
Thanks for the help!
1 Answer
Sort by ยป oldest newest most voted
So, I found the problem.
In fact,
N.parent() == Full MatrixSpace of 30 by 30 dense matrices over Algebraic Real Field
since this has infinite precision, the computation takes forever.
This is solved, and the computation is almost instant (as expected), by using finite precision, i.e.
Ni = N.n().inverse()
Maybe this'll help someone.
Cheers! : )
edit flag offensive delete link more
I cannot accept my answer, unfortunately (not enough points)
popoti9 ( 2019-06-20 00:15:36 +0100 )edit | {"url":"https://ask.sagemath.org/question/46918/inverting-matrix-after-cholesky-decomposition-does-not-finish-for-large-dimension/?sort=votes","timestamp":"2024-11-07T06:06:18Z","content_type":"application/xhtml+xml","content_length":"53734","record_id":"<urn:uuid:78814988-a69c-460a-91fb-710e083742ad>","cc-path":"CC-MAIN-2024-46/segments/1730477027957.23/warc/CC-MAIN-20241107052447-20241107082447-00545.warc.gz"} |
Trajectory Optimization and Stabilization | Collimator
This example shows trajectory optimization with the Hermite-Simpson collocation method and trajectory stabilization with a finite-horizon LQR controller. These methods are demonstrated for the model
of an acrobot, a two-link underactuated robot. Note: This tutorial uses the PyCollimator library. You can find installation and usage instructions, as well as further examples at the PyCollimator
Acrobot model
An acrobot, see schematic below, is a robotic arm with two links and a single actuator at the elbow joint. The state of the acrobot is described by the angles $q_1$ and $q_2$ of the two links, and
the control input is the torque $u=\tau$ applied at the elbow joint.
All the parameters and variables in the acrobot model are described in the table below:
Denoting $\mathbf{q} = [q_1, q_2]^T$ and $\mathbf{u} = [u]$ as the state and control input, respectively, the equations of motion for the acrobot are given by (see [1] for details)
$$ \mathbf{M}(\mathbf{q})\ddot{\mathbf{q}} + \mathbf{C}(\mathbf{q}, \dot{\mathbf{q}})\dot{\mathbf{q}} + \mathbf{G}(\mathbf{q}) = \boldsymbol{\tau}_g(\mathbf{q}) + \mathbf{B}\mathbf{u}, $$
where $\mathbf{M}(\mathbf{q})$ is the mass matrix, $\mathbf{C}(\mathbf{q}, \dot{\mathbf{q}})$ is the matrix for Coriolis and centrifugal terms, $\mathbf{G}(\mathbf{q})$ are the gravitational terms, $
\boldsymbol{\tau}_g(\mathbf{q})$ are the torques due to gravity, and $\mathbf{B}$ is the control input matrix. Denoting, $\cos(q_1)$ as $c_1$, $\sin(q_1)$ as $s_1$, and $\sin(q_1 + q_2)$ as $s_{1+2}
$, these matrices are [1]:
\mathbf{M}(\mathbf{q}) &= \begin{bmatrix} I_1 + I_2 + m_2 l_1^2 + 2m_2 l_1 l_{c2} c_2 & I_2 + m_2 l_1 l_{c2} c_2 \\ I_2 + m_2 l_1 l_{c2} c_2 & I_2 \end{bmatrix},\\[15pt] \mathbf{C}(\
mathbf{q},\dot{\mathbf{q}}) &= \begin{bmatrix} -2 m_2 l_1 l_{c2} s_2 \dot{q}_2 & -m_2 l_1 l_{c2} s_2 \dot{q}_2 \\ m_2 l_1 l_{c2} s_2 \dot{q}_1 & 0 \end{bmatrix}, \\[15pt] \boldsymbol{\
tau}_g(\mathbf{q}) &= \begin{bmatrix} -m_1 g l_{c1}s_1 - m_2 g (l_1 s_1 + l_{c2}s_{1+2}) \\ -m_2 g l_{c2} s_{1+2} \end{bmatrix},\\[15pt] \mathbf{B} &= \begin{bmatrix} 0 \\ 1 \end
As usual, we can write the above equations as first-order differential equations by introducing the state vector ${x} = [q_1, q_2, \dot{q}_1, \dot{q}_2]^T$.
The code for the Acrobot can be found in the PyCollimator examples at [.code]collimator/examples/acrobot.py[.code].
Modelling without any control
With the Acrobot LeafSystem, we can simulate its dynamics and visualize the results. We show this below for zero elbow torque.
Trajectory optimization
A trajectory is a pair of $x(t)$ and $u(t)$. For a reference trajectory $x_{ref}(t)$ and $u_{ref}(t)$ in an interval $t\in[t_0, t_f]$ that our plant should follow, the goal of trajectory optimization
is to obtain a set of control vector $u_{opt}$, which when applied to the plant would produce a trajectory $x_{opt}$ that closely follows the reference trajectory. For instance, for the acrobot we
may want to obtain such a trajectory for the acrobot to swing-up, starting from the vertically own orientation. Such an optimal trajectory can be obtained by solving the following optimization
problem, where we search for $u_{opt}(t)$ and $x_{opt}(t)$ such that their discrepancy with respect to $x_{ref}(t)$ and $u_{ref}(t)$ is minimized:
u_{opt}(t) = \text{arg}\,
\min\limits_{u(t)}\, \quad & [x(t_f) - x_{ref}(t_f)]^T Q_f [x(t_f) - x_{ref}(t_f)] + \\
& \int_{t_0}^{t_f} (x - x_{ref})^T Q (x - x_{ref}) \, dt + \\
&\int_{t_0}^{t_f} (u - u_{ref})^T R (u - u_{ref}) \, dt, \\
\text{subject to} \quad & \dot{x} = f(x, u), \\
\text{and} \quad & x(t=t_0) = x_0,
where $Q_f$, $Q$, and $R$ are positive definite matrices. $Q_f$ represents the penalty for the terminal state discrepancy at $t=t_f$ while $Q$ and $R$ represent the continuous-time penalties for the
state and control vector discrepancies, respectively. The function $f(x, u)$ represents the dynamics of the system. Note that one can change the optimization problem to suit their needs. For example,
if reaching a particular state at $t=t_f$ was important, then one may wish to impose an additional equality constraint of the form $x(t_f) = x_{ref}(t_f)$. Additionally, one may add bounds on the
state and control variables.
The above problem is in continuous time. In order to solve it, we first discretize the optimization problem (typically in $N$ discrete steps in $[t_0, t_f]$). Such discretization requires many
choices and leads to different methods for transcribing the continuous-time problem to discrete-time and the optimization methodology for solving the problem. For example, some common methods (see
[1] and references therein for more details) are:
1. Direct transcription
2. Direct shooting
3. Hermite-Simpson collocation (also referred as direct collocation)
Here, we demonstrate how to use the Hermite-Simpson collocation method to solve the trajectory optimization problem for the acrobot model to swing-up in Collimator, which uses the [.code]IPOPT[.code]
solver for nonlinear programming. The [.code]solve_trajectory_optimzation[.code] method of the [.code]HermiteSimpsonNMPC[.code] class can be utilized to solve the trajectory optimization problem.
Documentation for the class can be found here.
For the trajectory optimization problem, we can disregard the inputs and the outputs, and just focus on creating the object of this class. The initialization parameters reveal that the problem is
discretized into $N$ steps of length $dt$. We can provide the matrices $Q, R$, and $QN$, which is analogous to $Q_f$. Additionally, we can specify the lower and upper bounds on both the state and
control vectors. Finally, we can specify whether we wish to include the terminal state and terminal control vector as constraints. Let's look at the [.code]solve_trajectory_optimization[.code]
Once the object is created, we can solve the trajectory optimization problem from [.code]t_curr[.code] ($t_0$) and [.code]x_curr[.code] ($x_0$), providing [.code]x_ref[.code] and [.code]u_ref[.code]
and initial guesses. Note that for a discretization of $N$ steps, if the size of the state and control vectors are [.code]nx[.code] and [.code]nu[.code], respectively, then [.code]x_ref[.code] and
[.code]u_ref[.code] are of shapes [.code](N, nx)[.code] and [.code](N, nu)[.code], respectively, i.e. their $i^\mathrm{th}$ row provides the target/reference at the $i^\mathrm{th}$ step.
Swing-up for Acrobot
For the acrobot to swing-up, we don't really have a full time-varying reference trajectory. All we have is that, in the end, the acrobot should be in the swing-up position, i.e. $x_f = [\pi, 0, 0, 0]
^T$, in some finite time, say $t_f=4$ seconds. We can choose a discretization of $N=21$ equal steps to reach $t_f$. Since we don't know the time-varying trajectory, we can choose [.code]Q[.code] and
[.code]QN[.code] to be zero matrices, implying that we don't really penalize the deviations from the unknown time-varying reference trajectory. However, we do know the final state, so we can include
this as a constraint.
We can visualize the solution with the [.code]animate_acrobot[.code] utility:
Note that this is just a solution to the optimization problem. It is not guaranteed, in fact it is unlikely, that the acrobot will get this very state $x_{opt}(t)$ as a solution to its dynamics if
$u_{opt}(t)$ were applied as control.
Simulate with the planned trajectory (solution of trajectory optimization)
To simulate the Acrobot with the planned torques by the trajectory optimization solution, we can linearly interpolate the solution and pass it to our Acrobot LeafSystem. We can create this
interpolant quite conveniently by declaring a [.code]SourceBlock[.code], and providing a [.code]vmap[.code] version of the the [.code]jnp.interp[.code] function as its callback.
With this new [.code]InterpArray[.code] block available to us, we can create a diagram for our acrobot system controlled by the planned torques as follows:
Next, we can simulate the system and animate the output:
The acrobot does follow our refrence in the beginning but then deviates significantly from the desired trajectory. One may say that the trajectory obtained by the process of trajectory optimization
is unstable. This is indeed true, and we must stabilize the trajectory. One method for this is the finite-horizon linear quadratic regulator. This is demonstrated next.
Trajectory stabilization with Finite-horizon Linear Quadratic Regulator
Previously (see the LQR example notebook), we saw how the LQR was used to stabilize the plant around an equilibrium point. Here, we consider its extension of stabilizing the plant around a
trajectory. The general idea is simple, instead of linearising a plant around an equilibrium point, we linearise the plant around a nominal trajectory. Subsequently, we minimise a quadratic function
representing deviations from a desired trajectory. The reader is referred to Chapter 8 of [1] for all things related to the LQR.
In Collimator, the finite-horizon LQR is available as the [.code]FiniteHorizonLinearQuadraticRegulator[.code] block.
For stabilization of the acrobot trajectory, we can use this block and provide the solution of trajectory optimization as both the nominal and desired trajectories.
Note that our trajectory optimization is only available until $t_f$. To see that our stabilization with LQR works, we would like to simulate the system longer than $t_f$ and observe that the acrobot
remains in the swing-up orientation. We first create two helper functions, that provide us the trajectory optimization until $t_f$ and the [.code]x_up, u_up[.code] constant solutions for any time
beyind $t_f$. These would provide our nominal trajectories.
Next, we setup the parameters for finite-horizon LQR:
Next, we can simulate and visualize the system:
The finite-horizon LQR stabilizes the trajectory quite nicely. We can visualize the difference between the trajectory optimization (unstable) solution and the LQR-stabilized solution.
In summary, we have demonstrated the usage of a collocation method in Collimator for trajectory optimization, and subsequent trajectory stabilization with finite-horizon LQR. We refer the reader to
the excellent notes [1] on LQR for an extensive treatment.
[1] Russ Tedrake. Underactuated Robotics: Algorithms for Walking, Running, Swimming, Flying, and Manipulation (Course Notes for MIT 6.832). Available online | {"url":"https://www.collimator.ai/tutorials/trajectory-optimization-and-stabilization","timestamp":"2024-11-13T02:45:41Z","content_type":"text/html","content_length":"58134","record_id":"<urn:uuid:2b82a359-ee1c-4b93-b4d0-b6a84def8d88>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00355.warc.gz"} |
This area offers a set of techniques for data analysis.
The Analyzers in TheVirtualBrain are not always the best implementations of the algorithms in terms of performance, neither are we offering a complete analysis spectrum, because analysis is not
considered the focus feature of TheVirtualBrain and we do not intend to offer a replacement for tools already existent and successful in this area. We are merely offering some handy tools for people
who want to directly process simulation results inside TheVirtualBrain, although the advised long term strategy is to export simulated data from TheVirtualBrain and analyze it intensively with
specialized tools for your area of interest. We advise you not to run our analysis algorithms with long timeseries, as some might take a lot of time to finish.
The Analysis area has several interfaces that support the following operations for time-series analysis (and not only):
□ Cross-correlation of nodes
□ Fourier Spectral Analysis
□ Global TimeSeries Metrics
□ Cross coherence of nodes
□ Temporal covariance of nodes
□ Principal Component Analysis
□ Independent Component Analysis
□ Continuous Wavelet Transform
Cross-correlation of nodes¶
Compute pairwise temporal cross-correlation of all nodes in a 4D TimeSeries object. Cross-correlation, or normalized cross-covariance, is a measure that quantifies the degree of linear dependence
between two time-series.
To calculate the correlation coefficient of all nodes of a given multi-node time-series, simply select the TimeSeries object from the drop-down list in the Cross-correlation of nodes interface and
hit Launch.
The algorithm returns a CrossCorrelation object that contains cross correlation coefficients for all possible combinations of nodes. Results are visualized with the Correlation viewer.
Fourier Spectral Analysis¶
Compute a fast Fourier transform (FFT) of a TimeSeries object. FFT is an algorithm to compute the discrete Fourier transform (DFT) and its inverse for a 1 given sequence of values. DFT transforms a
function into its frequency-domain representation, that is, a sum of weighted sinusoids while preserving all of the information about the original signal. After decomposing the signal, spectrum
analysis quantifies the relative amounts of amplitudes, powers, intensities or phases of a component versus its frequency.
In order to perform a Fourier analysis of your time-series data follow these steps:
□ Go to the Fourier Spectral Analysis interface and select a Windowing function, you can choose among ‘hamming’, ‘bartlett’, ‘blackman’ and ‘hanning’.
□ Select the time-series.
□ Select a segment length.
□ Hit Launch.
TimeSeries Metrics¶
Calculate one scalar metric to characterize the time-series dataset.
Cross coherence of nodes¶
Calculate pairwise temporal coherence of all nodes in a 4D TimeSeries object. Coherence analysis, or cross-spectral analysis, can be used to estimate how two time series are related in the spectral
domain. Cross-coherence indicates the degree to which amplitude and phase between two signals relate to each other as a function of frequency.
To calculate the cross-coherence of all nodes of a given multi-node time-series, simply select the TimeSeries object from the drop-down list in the Cross coherence of nodes interface, select an
appropriate measure for data-point per block, and hit Launch.
The resulting coherence spectrum can be viewed with the Cross coherence visualizer.
Complex coherence of nodes¶
To calculate the complex-cross-coherence of all nodes of a given multi-node time-series, simply select the TimeSeries object from the drop-down list in the Complex coherence of nodes interface and
hit Launch.
The resulting coherence spectrum can be viewed with the Complex coherence visualizer.
Temporal covariance of nodes¶
Compute pairwise temporal covariance of all nodes in a 4D TimeSeries object. Covariance resembles the un-normalized correlation coefficient and measures how much two time-series change together.
To calculate the temporal covariance of all nodes of a given multi-node time-series, select the TimeSeries object from the drop-down list in the Independent Component Analysis interface and hit
The algorithm returns a Covariance object that is a 4D-Matrix with the Dimensions {nodes, nodes, 1, 1}. The resulting covariance matrix can be viewed with the Covariance visualizer.
Principal Component Analysis (PCA)¶
Compute a PCA of a 4D TimeSeries object. PCA is a computational method for multivariate data analysis that uses an orthogonal transformation to convert a set of (possibly correlated) variables into a
set of linearly uncorrelated variables called principal components.
To calculate a PCA of all nodes of a given multi-node time-series, select the 4D-TimeSeries object from the drop-down list in the Principal Components Analysis interface and hit Launch.
The algorithm returns an PrincipalComponents object that is a xD-Matrix with the Dimensions {x,y,z}. The resulting time-series can be viewed with the Pca viewer.
Independent Component Analysis (ICA)¶
Compute a time-domain ICA decomposition of a 4D TimeSeries object. ICA is a statistical and computational method for separating a multivariate signal into additive subcomponents by maximizing the
mutual statistical independence of source signals.
To calculate a temporal ICA of all nodes of a given multi-node time-series, select the 4D-TimeSeries object from the drop-down list in the Independent Component Analysis interface and hit Launch.
The algorithm returns an IndependentComponents object that is a xD-Matrix with the Dimensions {x,y,z}. The resulting time-series can be viewed with the corresponding ICA viewer.
Continuous Wavelet Transform (CWT)¶
Compute a CWT of a 4D TimeSeries object. CWT decomposes a signal into wavelets of different frequencies yielding a time-frequency representation of the signal.
To calculate a CWT of all nodes of a given multi-node time-series, select the 4D-TimeSeries object from the drop-down list in the Continuous Wavelet Transform interface, specify transformation
parameters like:
□ mother wavelet function
□ frequency resolution and range
□ type of the normalization for the resulting wavelet spectrum
□ Q-ratio
□ Sampling period of the spectrum
and hit Launch.
The algorithm returns an WaveletCoefficients object that is a xD-Matrix with the Dimensions {x,y,z}. The resulting spectrogram of wavelet power can be viewed with the Wavelet viewer.
Brain Connectivity Toolbox Analyzers¶
All the algorithms offered by Brain Connectivity Toolbox (BCT) can be used directly from TheVirtualBrain interface and the results can later be displayed in one of our visualizers.
Additional BCT techniques are:
□ Degree and Similarity Algorithms
□ Centrality Algorithms
□ Distance Algorithms
□ Modularity Algorithms
□ Clustering Algorithms
□ Density Algorithms
For more details, please refer to BCT web site
Functional Connectivity Dynamics metric¶
Analyse functional connectivity dynamics.
The analyser generates:
• The FCD
• The FCD segmented, that is the FCD matrix with the entries corresponding to the time points not belonging to the epochs of stability equal to 1.1. When the epochs of stability are not find, the
FCD segmented is not an output
• The 3 eigenvectors corresponding to the FC of the epochs (if present) or to the global FC. These are ConnectivityMeasures which are viewable with the volume visualizer.
The analyser takes as input a time series region, time window length (in ms), spanning between 2 consecutive time windows (in ms).
The code does the following steps:
Calculates the FCD:
The entire time series is divided in time window of the fixed length (decided by the user) and with an overlap decided by the user ( spanning=[time window length] - [overlapping between
consecutive time window]). The data points within each window, centred at time ti, are used to calculate FC(ti). The element ij of the FCD is calculated as the Pearson correlation between the
upper triangular part of FC(ti) arranged as a vector and the upper triangular part of FC(tj) arranged as a vector.
The FCD is segmented in epochs of stability using the spectral embedding algorithm.
We call epoch of stability a length of time during which an FC configuration stays stable. It is possible to visualize these epochs of stable FC as blocks of elevated inter-FC(t) correlation
around the diagonal of the FCD. We neglect always the first epoch of stability found by the algorithm since should be an artifact caused by the initial condition of the simulated time series.
FCs are calculated over the epochs of stability (excluding the first epoch).
When the algorithm does not find the epochs the global FC is calculated, i.e. the FC calculated over the entire timeseries.
The first three eigenvectors of the FCs calculated at step 3 are extracted.
We call the “first” eigenvector the one associated to the largest eigenvalue, the second eigenvector the one associated to the second largest eigenvalue and so on. Eigenvalues are normalized
between 0 and 1. | {"url":"https://docs.thevirtualbrain.org/manuals/UserGuide/UserGuide-UI_Analyze.html","timestamp":"2024-11-02T20:36:42Z","content_type":"text/html","content_length":"24050","record_id":"<urn:uuid:7f5dbbbe-5bfc-4d5a-8978-795f17d9b3b0>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00328.warc.gz"} |
Area and Perimeter
What is the area formula for a rectangle?
A = l * w
Find the perimeter of the rectangle with a length of 8cm and a width of 5 cm.
P = 26cm
Find the area of a rectangle that has a length of 6cm and a width of 5 cm.
A = 30 cm2
The classroom is getting new pink carpet. If the room has a length of 10cm and a width of 6 cm. How much carpet will the room need.
The room will need 60 cm2 of carpet.
What is a triangle with a right angle called?
Rigth triangle
What is the area formula for a square?
A = s * s
Find the perimeter of a square that has a side length of 12cm.
P = 48 cm
Find the area of a square with side length 7 in.
A = 49 in2
Put the fraction into a decimal 68/100
How do you find the perimeter of a triangle?
Add up all the outside edges
Find the perimtere of a right triangle with one side length 6ft, one side length 4ft, and one side length 9ft.
P = 19ft
Find the area of a right triangle with base = 7 cm and height = 8cm.
A = 28 in2
Mary needs to fence around her backyard. The length of her backyard is 5m and the width is 8m. If fencing comes in sections of 3 how many sections does Mary need to buy?
14 sections
Write all the factors of 12.
1, 2, 3, 4, 6, 12
What does the s stand for in the square area formula?
Side length
Find the perimeter of a triangle if it has a base of 6m, a height of 8 m, and two side lengths that equal 3m.
P = 12m
Find the area of a isosceles triangle with base = 9cm, side one = 7cm, side two = 3cm, and height = 2cm.
A = 9cm2
Mary needs to fence around her backyard. The length of her backyard is 5m and the width is 8m. If fencing comes in sections of 3 and each section costs 4 dollars. How much money will Mary spend?
Reduce the fraction into lowest terms. 25/100
What is the area for a triangle?
A = (b * h) /2
Find the perimeter of a pentagon with side lengths that equal 8 in.
P= 40 in
Find the area of a triangle with base = 20 in and height = 30 in.
A = 600 in2
A square pizza box has an area of 324 in2. What is the length of one side of the pizza box?
18 in
Add the mixed numbers together. 2 6/9 + 5 1/9
7 7/9 | {"url":"https://jeopardylabs.com/play/area-and-perimeter8","timestamp":"2024-11-09T06:41:54Z","content_type":"application/xhtml+xml","content_length":"58117","record_id":"<urn:uuid:29bf93ac-1231-4d51-b7d5-ff35b89e6712>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00178.warc.gz"} |
Physics - Online Tutor, Practice Problems & Exam Prep
Hey, guys. So in earlier videos, we've talked about different types of energies for ideal gases. We've already talked about the average kinetic energy that was per particle. That was this equation
over here. But in some top problems, you're gonna have to calculate something called the total internal energy for an ideal gas, and that's what I want to show you how to do in this video. So I want
to show you the basic differences between this average and total type of energy, and then we're gonna go a little bit more into the conceptual understanding of what this total internal energy
actually represents. So let's get started here. Basically, the difference between the average kinetic energy and the total energy has to do with how many particles you're looking at. This average
kinetic energy was per particle. The total internal energy is gonna be if you have a collection of particles. Let's say that's just n particles. So really, really sort of simply here, the basic
difference is that when you calculate this, this is the average kinetic energy of 1 particle. But if you have multiple, you just multiply by however many particles you have. Let's just do a quick
example here. We have 10 particles of a gas that's at 300 Kelvin in a container. So in the first part, we wanna calculate the average kinetic energy. Remember, all you need to calculate the average
kinetic energy is the temperature. So remember, we have this relationship that, 3 halves kBT, and we have our constants over here just for reference. So this average kinetic energy is just gonna be 3
halves times 1.38 times 10 to the minus 23, and then we're gonna multiply this by 300. When you work this out, what you're gonna get is 6.21×10-21J. So that's the average kinetic energy per particle.
Now, if you wanna calculate the total internal energy and if you have 10 particles, all you have to do really is you just have to do, this E internal here. It's just gonna be n×kaverage. It's just
gonna be 10 times the average kinetic energy, 6.21×10-21, and then you end up with 62.1×10-21. Notice how all we've done here is we just shifted the decimal place to the right by one space. It's just
10 times greater. Alright? So that's the fundamental difference between them. So I wanna point out just real quickly here that the symbol we use for in total internal energy is gonna be E internal.
So some textbooks will also write this as u, but here at Clutch, we don't wanna confuse you with the potential energy, and so we just write this as E internal. It's always gonna be written that way.
Now there are other variations of this E internal equation. We saw there was just n times k average. So one way you could just rewrite this is you just stick an n in front of this equation over here.
So this is 3 halves, big N, then kBT. Notice how all we've done here is we just added an n inside here, and that's just basically another way to rewrite this. Now some textbooks may also rewrite this
equation again using the relationship that we've seen before. We've seen that nkBT, so nkB is equal to nR when we talked about the ideal gas law. So we can use is this relationship here, and you
could rewrite this equation again as 3 halves nRT. Anywhere these equations will work. You'll just use this one when you have the number of particles like we did in our first example, and you'll use
this one when you have the moles of a gas. And so the last thing I want you to know is that this equation only works for a single atom type of gas, which is also known as a monoatomic gas. So this
only works for you when you have single atom type gases, and most of the problems will tell you whether it's monoatomic or not. So let's take a look at our second problem now. So now we have total
internal energy of a gas, and we're just gonna assume it's monoatomic, is 401 Kelvin and the energy is this. And we want to calculate the number of moles in this gas. So we have that T is equal to
401 Kelvin. We have that the E internal is equal to 2 times 10 to the 4th. And now we wanna calculate the number of moles. That's actually just little n. So which one of the forms of this equation do
we have to use? Well, it's just gonna be the one that has the moles inside of it. This is gonna be 3 halves nRT. So what we're told here is that this E internal is just 3 halves nRT, and this is
equal to 2 times 10 to the 4th. So now all we have to do is to just go ahead and solve for this moles of gas here. Remember this R is just a constant that we have over here, and we have the
temperature already, and we obviously have the energy. So the n is just gonna be 2 times 10 to the 4th, and this is just gonna be divided by 3 halves times 8.314 times, and this is gonna be 401 | {"url":"https://www.pearson.com/channels/physics/learn/patrick/kinetic-theory-of-ideal-gases/internal-energy-gases?chapterId=8fc5c6a5","timestamp":"2024-11-12T03:47:43Z","content_type":"text/html","content_length":"503065","record_id":"<urn:uuid:5fd8ff0b-6341-415d-9849-d3b2f993c28d>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00635.warc.gz"} |
Solve x^2-4x-12 | Microsoft Math Solver (2024)
Polynomial5 problems similar to: x^2-4x-12
Similar Problems from Web Search
a+b=-4 ab=1\left(-12\right)=-12
Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx-12. To find a and b, set up a system to be solved.
1,-12 2,-6 3,-4
Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -12.
1-12=-11 2-6=-4 3-4=-1
Calculate the sum for each pair.
a=-6 b=2
The solution is the pair that gives sum -4.
Rewrite x^{2}-4x-12 as \left(x^{2}-6x\right)+\left(2x-12\right).
Factor out x in the first and 2 in the second group.
Factor out common term x-6 by using distributive property.
Quadratic polynomial can be factored using the transformation ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right), where x_{1} and x_{2} are the solutions of the quadratic equation ax^{2}+bx+c=0.
All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is
Square -4.
Multiply -4 times -12.
Add 16 to 48.
Take the square root of 64.
The opposite of -4 is 4.
Now solve the equation x=\frac{4±8}{2} when ± is plus. Add 4 to 8.
Now solve the equation x=\frac{4±8}{2} when ± is minus. Subtract 8 from 4.
Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 6 for x_{1} and -2 for x_{2}.
Simplify all the expressions of the form p-\left(-q\right) to p+q.
x ^ 2 -4x -12 = 0
Quadratic equations such as this one can be solved by a new direct factoring method that does not require guess work. To use the direct factoring method, the equation must be in the form x^2+Bx+C=0.
r + s = 4 rs = -12
Let r and s be the factors for the quadratic equation such that x^2+Bx+C=(x−r)(x−s) where sum of factors (r+s)=−B and the product of factors rs = C
r = 2 - u s = 2 + u
Two numbers r and s sum up to 4 exactly when the average of the two numbers is \frac{1}{2}*4 = 2. You can also see that the midpoint of r and s corresponds to the axis of symmetry of the parabola
represented by the quadratic equation y=x^2+Bx+C. The values of r and s are equidistant from the center by an unknown quantity u. Express r and s with respect to variable u. <div style='padding:
8px'><img src='https://opalmath.azureedge.net/customsolver/quadraticgraph.png' style='width: 100%;max-width: 700px' /></div>
(2 - u) (2 + u) = -12
To solve for unknown quantity u, substitute these in the product equation rs = -12
4 - u^2 = -12
Simplify by expanding (a -b) (a + b) = a^2 – b^2
-u^2 = -12-4 = -16
Simplify the expression by subtracting 4 on both sides
u^2 = 16 u = \pm\sqrt{16} = \pm 4
Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u
r =2 - 4 = -2 s = 2 + 4 = 6
The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s. | {"url":"https://bieder.shop/article/solve-x-2-4x-12-microsoft-math-solver","timestamp":"2024-11-10T12:43:38Z","content_type":"text/html","content_length":"68543","record_id":"<urn:uuid:c4ca71b6-f47f-4b47-83ca-699b71ebe9c8>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00733.warc.gz"} |
Ratio of age in the limit
Have you ever noticed that for any two people with different ages, you can always figure out a moment in time when the younger person was exactly half the age of the older person?
Let’s say that you are \(18\) and your friend is \(23\), so there is a \(5\) year difference in your ages. Then, when you were \(5\), your friend was \(10\), so that was the moment when the ratio of
your ages was exactly half.
Does this work for all age differences?
Let the initial age of the younger person be \(0\), and the age of the older person be \(a\). Then, at any time \(t\) after this, the age of the younger person is \(0 + t\) and the age of the older
person is \(a + t\).
Hence the ratio of the ages of the two people at any time \(t\) is
\[ \frac{0 + t}{a + t}. \]
Let’s look at what happens when we substitute different values for \(t\). We will also let \(a = 5\) to say that the second person is \(5\) years older than the first. If we let $t = 0$, then
Note: If you are viewing this in Chrome and you can’t see the fraction, you need to enable MathML by pasting chrome://flags/#enable-experimental-web-platform-features into the address bar, enabling
Experimental Web Platform features, and restarting Chrome.
The graph above plots the ratio of the two ages over time. As you move the slider, you can see that as time approaches infinity, the ratio of the two ages approaches 1.
(NOTE: The graph is very glitchy if you move the slider quickly, and it doesn’t undo the plot when you slide back to the left. However, I’ve procrastinated publishing this post for three years
because I haven’t been motivated to create a proper minimal plotting library for it, so I’m just publishing it as-is…)
We can also prove this mathematically using the concept of a limit: | {"url":"https://daviewales.com/2020/09/04/ratio-of-age-in-the-limit.html","timestamp":"2024-11-05T11:49:22Z","content_type":"text/html","content_length":"12766","record_id":"<urn:uuid:442c0e44-a4d1-4acd-98f9-646d6aec9dd8>","cc-path":"CC-MAIN-2024-46/segments/1730477027881.88/warc/CC-MAIN-20241105114407-20241105144407-00851.warc.gz"} |
Secret sharing schemes for very dense graphs
A secret-sharing scheme realizes a graph if every two vertices connected by an edge can reconstruct the secret while every independent set in the graph does not get any information on the secret.
Similar to secret-sharing schemes for general access structures, there are gaps between the known lower bounds and upper bounds on the share size for graphs. Motivated by the question of what makes a
graph "hard" for secret-sharing schemes, we study very dense graphs, that is, graphs whose complement contains few edges. We show that if a graph with n vertices contains ( ^n [2] - n ^1+β edges for
some constant 0 ≤ β < 1, then there is a scheme realizing the graph with total share size of Õ(n ^5/4+3β/4). This should be compared to O(n ^2/logn) - the best upper bound known for general graphs.
Thus, if a graph is "hard", then the graph and its complement should have many edges. We generalize these results to nearly complete k-homogeneous access structures for a constant k. To complement
our results, we prove lower bounds for secret-sharing schemes realizing very dense graphs, e.g., for linear secret-sharing schemes we prove a lower bound of Ω(n ^1 + β/2).
Original language English
Title of host publication Advances in Cryptology, CRYPTO 2012 - 32nd Annual Cryptology Conference, Proceedings
Pages 144-161
Number of pages 18
State Published - 3 Sep 2012
Event 32nd Annual International Cryptology Conference, CRYPTO 2012 - Santa Barbara, CA, United States
Duration: 19 Aug 2012 → 23 Aug 2012
Publication series
Name Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume 7417 LNCS
ISSN (Print) 0302-9743
ISSN (Electronic) 1611-3349
Conference 32nd Annual International Cryptology Conference, CRYPTO 2012
Country/Territory United States
City Santa Barbara, CA
Period 19/08/12 → 23/08/12
ASJC Scopus subject areas
• Theoretical Computer Science
• General Computer Science
Dive into the research topics of 'Secret sharing schemes for very dense graphs'. Together they form a unique fingerprint. | {"url":"https://cris.bgu.ac.il/en/publications/secret-sharing-schemes-for-very-dense-graphs-9","timestamp":"2024-11-12T07:43:20Z","content_type":"text/html","content_length":"59209","record_id":"<urn:uuid:6da8a8c2-57c6-4bb4-845f-5bae33ff1004>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.58/warc/CC-MAIN-20241112045844-20241112075844-00798.warc.gz"} |
Vibration isolation is a technique to mitigate mechanical vibrations. There are two types of vibration isolation techniques; passive and active. In passive vibration isolation, vibrations are
isolated with passive techniques such as rubber pads or springs. In active vibration isolation, vibrations are controlled with automated control systems.
Mechanical vibrations are generated due to unbalance of rotating and reciprocating components such as rotors of pumps, electric motors, combustion engines; or impact forces of hammers and presses;
pressure loadings on surfaces due to the winds or acoustic noises; or moving on a bumpy and irregular road with a vehicle. Depending on conditions, mechanical vibrations can be dangerous for systems
and can lead catastrophic failures if not analyzed and managed in a correct way. To reduce or eliminate the unwanted vibrations, vibration analyses are generally done by vibration engineers and
vibration isolation techniques are incorporated into the design of systems. Control techniques for passive vibration isolation are summarized as follows:
Minimization of the effect of excitation: The source of the mechanical vibration can be minimized. If it is a rotating component, the magnitude of harmonic excitation force is proportional to square
of the angular velocity and magnitude of unbalance. So for a rotating component, the rotation speed can be reduced and/or the rotating element can be balanced by using balancing machines to have less
unbalance on the rotating part.
Dynamic Balancing of a Spindle
The change of excitation frequency of the input harmonic forcing to the system will affect the amplitude of the response. As an example, if we think a single degree of freedom system which creates a
harmonic excitation, increasing excitation frequency beyond the system natural frequency will result a decrease in system response. The particular solution of the differential equation of the sdof
system is given below. Here w[n] is system natural frequency of the system and w is forcing frequency. If w = w[n] are equal, the denominator of the formula will be small and X (excitation
displacement) will be high.
So as a result, in real systems, if the excitation frequency can be adjusted to have a frequency larger than systems natural frequencies, then the system response will be significantly reduced.
The other method to minimize the effect of excitation is reduction of unbalance of the system. If it's a rotating system, the balance correction can be done by addition or removal of material
according to results given by balancing machines. Different types of machines exist such as dynamic and static balancing machines. The proper balancing method shall be selected according to the
Specify system parameters to reduce the effects: If we think the same single degree of freedom system with the governing formula given above, system displacement X also depends on system parameters
such as mass (m), natural frequency (w[n]) and damping ratio (ζ). The natural frequency of the system (w[n]) itself depends on system stiffness (k) and mass (m). ζ is damping ratio which depends on
system’s damping coefficient c (See following equations of w[n] and ζ for the dependencies.) So system displacement X is dependent on mass, stiffness and damping coefficient parameters of the system.
If the harmonic excitation frequency is known, designing system parameters to minimize steady-state amplitudes is possible. This may be achieved by addition of more viscous damping or design
optimization of the structure to have optimal stiffness to reduce the vibration levels. On the left side, very stiff fixture can be seen which is designed to reduce the effects of mechanical
Equation of system natural frequency (SDOF) Equation of system damping ratio (SDOF)
Change of system configuration: Instead of altering design parameters of the system as defined above, changing the configuration can reduce the vibration levels. If we use the same system and add an
additional mass-spring-damper on it, the new system will be two degree of freedom. If this additional mass-spring-damper system is designed correctly, it will act like a vibration absorber. With
addition of correctly tuned vibration absorber, system resonance frequencies can be moved away from the excitation frequency.
See following figure for the simple schematic of the two degree of freedom system after the addition of tuned vibration damper.
Reduction of force transmission: If a machine is creating mechanical vibrations, the force transmission to its base can be reduced by implementing vibration isolators and isolation systems such as
elastic rubbers, vibration mounts, wire rope isolators, vibration pads, shock absorbers and spring systems. A picture of spring isolator system is given in the right side.
Information given here is for introductory purposes. For the isolation of systems from the mechanical vibrations, it shall be consulted to experts of vibration engineering for suitable design | {"url":"https://amesweb.info/Guidelines/VibrationControlIsolationSystems.aspx","timestamp":"2024-11-02T12:14:31Z","content_type":"application/xhtml+xml","content_length":"12593","record_id":"<urn:uuid:d9f14699-0232-4c3e-80f7-fe338abfc4ba>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00600.warc.gz"} |
Math Preparation Archives -
Math is among the most essential and fundamental aspects of our existence. Before infants even learn to sit, we use mathematics to interpret and describe the forms and spaces around us. Because we
all employ mathematical knowledge from the minute we open our eyes to this environment, how we train and improve our arithmetic skills has a big impact on our lives. That is why, in order to promote
their children’s future success, parents must understand the importance of arithmetic and explore ways of improving math skills.
Why Is Math So Important For Kids?
Math preparation is similar to the art of playing with numbers. it takes a strong foundation to obtain the correct shape. It’s not just about complicated mathematical procedures and calculations.
Many students face challenging or complex arithmetic problems. Still, they did solve them quickly because they obtain adequate mathematical knowledge as children and thus should nurture to advance a
good foundation. Knowledge of mathematics can influence cognitive development in other walks of society as well. It has been proven that children with a strong mathematics education have a simpler
way of comprehending life and making cause-and-effect connections. The reality is that, these critical life skills rise in direct proportion to one’s mathematical abilities demonstrates. The learning
math skills early in life can have a direct impact on one’s future performance. This especially applies to their child’s numerical and future potential, and parents have a significant role to play.
What Can I Do to Assist My Child in Improving His Math Skills?
Master gets the progressive students, and those youngsters are acquiring immense knowledge. They will be strong in maths. Parents can also see if their child has a math preparation difficulty
(Dyscalculia) and discover how to support them before school starts. That is why a student’s learning begins at home, and parents have a significant influence on their after-school activity even
after they start school. This suggests that parents have a significant impact on their children’s mathematics achievement.
A parent that wants to help their kids develop their arithmetic skills may consider the following suggestions:
Signs that Your Youngster has Difficulty with Math
# Makes Disparaging Remarks About Math – It can be difficult to detect a child who is struggling with math. However, the master can find them out as they are experienced in handling youngsters.
# When it Comes to Math, your child gets Anxious – When it’s time to perform arithmetic, whether in class, on an exam, or on a school assignment, your youngster becomes increasingly apprehensive.
Students can understand basic arithmetic. Still, they can face anxiety in remembering whatever they are learning. Practice can help them to remember easily.
# Having a Hard Time Linking Math Families – Students should start seeing the relationship between different numbers and equations as they learn more arithmetic facts. For example, if your youngster
cannot perceive the connection between 2+3=5 and 5-3=2, they may be having difficulty with math.
# Having a Hard Time Keeping Track of Time – Many parents struggle with time management, so this warning may appear ambiguous. Watch your youngster see if he or she has trouble estimating time
intervals, adhering to rituals, or understanding timepieces (analogue or digital).
# Having Trouble Relating Math Ideas to Real-life Situations – Your child may understand arithmetic ideas but struggle to see how they relate beyond the classroom. Study the possibility below:
• They’re keeping track of the days until their birthday.
• Estimating the worth of something and the amount of change they should receive.
• Ask your children to keep calculating house hold activities and note down.
# Problems with Mental Math – Figuring out math problems utilizing mental maths can be beneficial in the early years. This is because, as children grow older, they will be confronted with larger
numbers and more complex equations that will necessitate mental math preparation, which finger counting can inhibit.
Georgia Venue for Parents of Children in Math grade (3rd to 8th)
Parents of students in Math grade (3rd to 8th) transition from using hands-on approaches to someone using visual aids to solve math problems.
Students in 3rd grade should use the following strategies-
• Multiplication and division problems should be represented and solved.
• Understand the principles of multiplication and how multiplication and division are related.
• Solve problems utilizing the four operations, as well as recognize and explain arithmetic patterns.
• To execute multi-digit arithmetic, use place value comprehension and attributes of operations.
• Develop a numerical grasp of fractions.
• Appreciate a fraction like a figure on the line segment and use a number schematic view to illustrate fractions.
• Explain fraction equivalence using visual fraction models and logic. By thinking about the magnitude of fractions, you may compare them.
• Solve issues that require the measuring and estimation of time intervals, liquid quantities, and object masses.
• Understand area principles and how they relate to multiplication and addition.
• Recognize circumference as a feature of planar figures and understand the difference between straight and area measurements.
Students in 4th grade should use the following strategies-
• Recognize that a distributive comparison occurs when one number is multiplied by a given number to produce another amount.
• Develop a generalized understanding of place value for multi-digit whole values.
• Gain a better understanding of fraction equivalence and sequencing.
• Apply and suggest additional conceptions of procedures on whole numbers to create fractions from unit fractions.
• Understand and evaluate decimal fractions using decimal notation.
• Solve problems concerning measuring and measurement conversion from a larger to a smaller device.
• Draw and recognize lines and angles, then classify shapes based online and angle attributes.
Students in 5th grade should use the following strategies-
• Using numerical expressions, write and interpret them.
• Recognize the placing value system.
• To add and subtract fractions, use fractions as a technique.
• Within a given system of measurement, convert like measurement units.
• To address real-world and mathematical problems, graph points on the coordinate plane.
Students in 6^th, 7th and 8th grade should use the following strategies-
Begin with an unspecified number in basic algebra-
• Graphing ordered pairs is the process of using coordinates to locate locations on a grid.
• Use fractions, percentages, and proportions to solve problems.
• Experiment with lines, angles, triangle kinds, and other fundamental geometric shapes.
• ‘Estimate and round.’
A Parent’s General Advice
To recap, Georgia educators are doing an outstanding job of teaching and maintain the standards set for Georgia’s Test Prep curriculum in the classes. keep cheering up your child for a positive
attitude. Your child will be trained with a higher mathematics knowledge.
When it comes to practicing math in home, keep in mind that how you approach your child, it can have a big impact on their motivation. You can suggest them to practice Georgia’s Test Prep. As a
consequence, keep an optimistic attitude throughout the process, and you can expect favorable results in conclusion. | {"url":"https://georgiatestprep.com/tag/math-preparation/","timestamp":"2024-11-06T10:22:43Z","content_type":"text/html","content_length":"34057","record_id":"<urn:uuid:0c5d08f1-375b-414c-8d7c-ca1bb7345133>","cc-path":"CC-MAIN-2024-46/segments/1730477027928.77/warc/CC-MAIN-20241106100950-20241106130950-00345.warc.gz"} |
PKGW: Propagating Uncertainties: The Lazy and Absurd Way
Propagating Uncertainties: The Lazy and Absurd Way
2013 April 3
I needed to write some code that does calculations and propagates uncertainties under a fairly generic set of conditions. A well-explored problem, surely? And indeed, in Python there’s the
uncertainties package which is quite sophisticated and seems to be the gold standard for this kind of thing.
Being eminently reasonable, uncertainties represents uncertain variables with a mean and standard deviation, propagating errors analytically. It does this quite robustly, computing the needed
derivatives magically, but analytic propagation still fundamentally operates by ignoring nonlinear terms, which means, in the words of the uncertainties documentation, that “it is therefore important
that uncertainties be small.” As far as I can tell, uncertainties does analytic propagation as well as anything out there, but honestly, if your method can’t handle large uncertainties, it’s pretty
useless for astronomy.
Well, if analytic error propagation doesn’t work, I guess we have to do it empirically.
So I wrote a little Python module. To represent 5 ± 3 I don’t create a variable that stores mean=5 and stddev=3 — I create an array that stores 1024 samples drawn from a normal distribution. Yep. To
do math on it, I just use numpy‘s vectorized operations. When I report a result, I look at the 16th, 50th, and 84th percentile points of the resulting distribution.
Ridiculous? Yes. Inefficient? Oh yes. Effective? Also yes, in many cases.
For instance: the uncertainties package doesn’t support asymmetric error bars or upper limits. My understanding is that these could be implemented, but they badly break the assumptions of analytic
error propagation — an asymmetric error bar by definition cannot be represented by a simple mean and standard deviation, and an upper limit measurement by definition has a large uncertainty compared
to its best value. But I can do math with these values simply by drawing my 1024 samples from the right distribution — skew normal or uniform between zero and the limit. I can mix perfectly-known
values, “standard” (i.e. normally-distributed) uncertain values, upper limits, and anything else, and everything Just Works. (It might be hard to define the “uncertainty” on a complex function of a
mixture of all of these, but that’s because it’s genuinely poorly-defined — analytic propagation is just misleading you!)
Another example: uncertainties spends a lot of effort tracking correlations, so that if x = 5 ± 3, then x - x = 0 precisely, not 0 ± 4.2. My approach gets this for free.
I’ve found that approaching uncertainties this way helps clarify your thinking too. You worry: is 1024 samples big enough? Well, did you actually measure 5 ± 3 by taking 1024 samples? Probably not.
As Boss Hogg points out, the uncertainties on your uncertainties are large. I’m pretty sure that only in extreme circumstances would the number of samples actually limit your ability to understand
your uncertainties.
Likewise: what if you’re trying to compute log(x) for x = 9 ± 3? With 1024 samples, you’ll quite likely end up trying to take the logarithm of a negative number. Well, that’s telling you something.
In many such cases, x is something like a luminosity, and while you might not be confident that it’s much larger than zero, I can guarantee you it’s not actually less than zero. The assumption that x
is drawn from a normal distribution is failing. Now, living in the real world, you want to try to handle these corner cases, but if they happen persistently, you’re being told that the breakdown of
the assumption is a significant enough effect that you need to figure out what to do about it.
Now obviously this approach has some severe drawbacks. But it was super easy to implement and has Just Worked remarkably well. Those are big deals.
Questions or comments? For better or worse this website isn’t interactive, so send me an email or, uh, Toot me.
To get notified of new posts, try subscribing to my lightweight newsletter or my RSS/Atom feed. No thirsty influencering — you get alerts about what I’m writing; I get warm fuzzies from knowing that
someone’s reading!
Later: Typing Greek Letters Easily on Linux
Earlier: My New Snoozing Technique is Unstoppable
See a list of all posts.
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Building a Basic, In-Game Win Probability Model for the NFL
The goal of an in-game win probability is to estimate the probability that a particular team will win a game based upon the game conditions (score, time remaining, etc.) at a particular point of time
in the game. For example, a team that leads by 21 points with less than one minute remaining in the game would safely be assumed to have a win probability approaching 1. How would that probability
differ if the team leads by a single point? Or trails by two points? Or leads 28–3 with 8 minutes and 31 seconds remaining in the 3rd Quarter of the Super Bowl (see the plot below)?
In-Game Win Probability for Super Bowl LI (ESPN.com)
In this post I’ll show the development of a basic, in-game win probability model for the NFL in R. We’ll start with historical play-by-play data scraped using the wonderful nflscrapR R package. This
package is not currently available via CRAN. Instead, the package is installed directly from github. Other libraries used in this exercise are also shown in the R code snippet below.
#Install nflscrapR
devtools::install_github(repo = "maksimhorowitz/nflscrapR")
devtools::install_github("dgrtwo/gganimate")#Load libraries
Note that I’m still adapting to using tidyverse principles so I’m not always using the package consistently. For a solid introduction to tidy principles, check out the excellent “R for Data Science”
book by Grolemund and Wickham.
Setting-up the scraping process is pretty trivial, but the scraping itself can be a bit time-consuming. So as to not have to repeat the scraping, I use the saveRDS function to save the play-by-play
data to a file for use in this project and others. Repeat the code below for the 2009 through 2015 seasons (nflscrapR cannot scrape data from before the 2009 season). You should end up with eight pbp
data frames numbered 1 to 8.
pbp1 = season_play_by_play(2016)
saveRDS(pbp1, "pbp_data_2016.rds")
Then use bind_rows to combine the the pbp data frame into a single data frame.
pbp = bind_rows(pbp1,pbp2,pbp3,pbp4,pbp5,pbp6,pbp7,pbp8)
saveRDS(pbp, "pbp_data.rds")
The resulting, combined data frame should have 362,447 observations (rows) and 77 variables (columns). Each observation is a single play and the variables provide information about each play. To
build our model, we’ll need each row to also include a variable that indicates which team ends up winning the game in which each play took place. Fortunately, the nflscrapR package provides the
ability to scrape game result data. From this data we can derive the needed variable.
games2016 = season_games(Season = 2016)
Repeat for the 2009 to 2015 seasons and then combine the game results data. I save the combined data to file to avoid having to repeat the scraping.
games = bind_rows(games2016, games2015, games2014, games2013, games2012, games2011, games2010, games2009)saveRDS(games, "games_data.rds")
The full_join function is used combine the game results with the play-by-play data using the GameID variable as the key for the join.
pbp_final = full_join(games, pbp_raw, by = "GameID")
saveRDS(pbp_final, "pbp_final.rds")
I then created a new, binary variable to record whether or not the team in possession of the ball is ultimately the game winner. This variable will be the response variable in our models. The first
line of code creates a variable that stores the name of the team that won the game in which each play took place. The second line creates an indicator variable with a value of “Yes” if the team in
possession ultimately wins the game and “No” if not. I also convert the quarter, down, and “poswins” variables to factors. Note that I got a little lazy with my code and did the factor conversions in
a non-Tidyverse manner.
pbp_final = pbp_final %>% mutate(winner = ifelse(homescore > awayscore, home, away))pbp_final = pbp_final %>% mutate(poswins = ifelse(winner == posteam, "Yes","No"))pbp_final$qtr = as.factor(pbp_final$qtr)
pbp_final$down = as.factor(pbp_final$down)
pbp_final$poswins = as.factor(pbp_final$poswins)
In the next step we remove “No Play” plays and plays that did not occur during regulation. A subset of variables of interest is then created using the select function.
pbp_reduced = pbp_final %>% filter(PlayType != "No Play" & qtr != 5 & down != "NA" & poswins != "NA") %>% select(GameID, Date, posteam, HomeTeam, AwayTeam, winner, qtr, down, ydstogo, TimeSecs, yrdline100, ScoreDiff, poswins)
We’re now ready to build our prediction model. Before we do so, let’s split the dataset into training and testing sets with the sample.split function from the caTools package. Setting the seed
ensures that the split we create is reproducible. We’ll use the testing set to evaluate the performance of the model that we create using the training set.
split = sample.split(pbp_reduced$poswins, SplitRatio = 0.8)
train = pbp_reduced %>% filter(split == TRUE)
test = pbp_reduced %>% filter(split == FALSE)
A wide variety of models exist for binary classification problems (such as we face here). One of the simplest is logistic regression. Coefficients, shown in the logit function below as betas, are
estimated. From these, the probability, P, of a binary event occurring can then be estimated.
The logit function
In our model, the predictor variables (shown as X’s in the logit function) are qtr (quarter), down, ydstogo (yards to go), TimeSecs (time remaining in the game in seconds), yrdline100 (distance from
the opponents goal line in yards), and ScoreDiff (difference in score calculated as the score for the team in possession minus the opponent’s score). The response variable is poswins which is a
binary variable that indicates if the team in possession ultimately wins the game.
We use R’s glm function with family = “binomial” to build the logistic regression model on the training set. The summary function then provides the details of the model results.
model1 = glm(poswins ~ qtr + down + ydstogo + TimeSecs + yrdline100 + ScoreDiff, train, family = "binomial")summary(model1)
Logistic regression model
The model can then be used to estimate win probabilities for each play in the training dataset. The estimated probabilities are predicted for the team in possession. For clarity, the third line of
code in the snippet below ensures that the probabilities are always stated for the home team (even if that team is not in possession).
pred1 = predict(model1, train, type = "response")train = cbind(train,pred1)train = mutate(train, pred1h = ifelse(posteam == HomeTeam, pred1, 1-pred1))
The plot below shows the evolution of the estimated win probability for the Denver Broncos (home team) in their September 8th, 2016 game versus the Carolina Panthers. The Broncos won 21–20 as the
Panthers missed a potentially game-winning field goal with nine seconds remaining.
ggplot(filter(train, GameID == "2016090800"),aes(x=TimeSecs,y=pred1h)) + geom_line(size=2, colour="orange") + scale_x_reverse() + ylim(c(0,1)) + theme_minimal() + xlab("Time Remaining (seconds)") + ylab("Home Win Probability")
Denver Broncos (versus Carolina Panthers) In-Game Win Probability (September 8th, 2016)
Earlier, we split our dataset into training and testing sets with the idea that we would evaluate our model quality on the testing set. To do so, we need to convert the estimated win probabilities
from the logistic regression model to a definitive prediction of “Win” or “Loss”. How do we do this? We establish a threshold above which we assume that a “Win” a predicted and below which we assume
that a “Loss” is predicted. For example, if the estimated win probability at a particular moment in a game is 0.95, we might be able to safely assume that the team will win the game. What if the
estimated probability is 0.75? Or 0.65? Or 0.55? Where do we draw the line? Should we simply accept 0.5 as the threshold? We’ll address this issue and opportunities to enhance our model in the next | {"url":"https://medium.com/@technocat79/building-a-basic-in-game-win-probability-model-for-the-nfl-54600e57fe1c","timestamp":"2024-11-02T13:12:36Z","content_type":"text/html","content_length":"128430","record_id":"<urn:uuid:f39363ab-e610-43dc-b4a2-2148af857a9e>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00711.warc.gz"} |
How Uber Computes ETA at Half a Million Requests per Second
#26: And How Online Maps Work Explained Like You’re Twelve (5 minutes)
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This post outlines how Uber computes ETA accurately at half a million requests per second. If you want to learn more, scroll to the bottom and find the references.
• Share this post with somebody who wants to study system design & I'll send you some rewards for the referrals.
September 2014 - Prague, Czech Republic.
Maria has an important meeting in 15 minutes.
She calls a taxi.
But the trip took a lot longer than expected.
So she was late and upset.
She hears about the new ride-sharing app called Uber from a coworker.
She installs it immediately and was dazzled by the ETA accuracy.
Knick-knacks on tech, startups, business, and life. Read by 25k+ people from around the world, Weekly Olio already counts the who’s who of investors, startup founders, economists, and industry
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The time estimated to travel from point A to B is called the Expected Time of Arrival (ETA).
Uber computes ETA in 4 scenarios:
• Eyeball: when the rider enters a destination in the app
• Dispatch: to find a car to pick up the rider in the shortest waiting time
• Pick up: to find the time needed to pick up the rider
• On-trip: to provide live updates on time to reach the destination
A single trip usually takes around 1000 ETA requests.
Yet computing ETA is a difficult problem. Because the distance between the source and destination is not a straight line.
Instead it consists of complex street networks and highways.
The smart engineers at Uber used simple ideas to solve this difficult problem.
Uber ETA
Here’s how Uber computes ETA accurately at extreme scales:
1. Routing Algorithm
They represent the physical map as a graph.
Every road intersection is modeled as a node.
While every road segment is modeled as a directed edge.
So computing ETA becomes finding the shortest path in a directed weighted graph.
Dijkstra’s algorithm is known for finding the shortest path in a graph.
But the time complexity of Dijkstra’s algorithm is O(n logn). And n is the number of road intersections or nodes in the graph.
San Francisco Bay Area alone has half a million road intersections. So Dijkstra’s algorithm is not enough at Uber's scale.
So they partitioned the graph. And then precomputed the best path within each partition.
Thus interacting with boundaries of graph partitions is enough to find the best path.
Imagine a dense graph mapped to a circle.
Every single node in the circle must be traversed to find the best path between 2 points.
So time complexity would be the area of the circle: pi * r^2
While partitioning and precomputing make it more efficient.
It becomes possible to find the best path by interacting with only the nodes on the circle boundary.
So time complexity would be the perimeter of the circle: 2 * pi * r
Put another way, the time complexity to find the best path in the San Francisco Bay Area gets reduced from 500 Thousand to 700.
2. Traffic Information
The traffic on the road segments must be considered to find the fastest path between 2 points.
While traffic is a function of the time of the day, weather, and number of vehicles on the road.
They used traffic information to populate the edge weights of the graph. Because it makes the ETA more accurate.
Besides they combined aggregated historical speed information with real-time speed information. Because extra traversal data makes traffic information more accurate.
3. Map Matching
GPS signals can get noisy and sparse especially when the vehicle enters a tunnel.
Also the multipath effect could worsen the GPS signal. The multipath effect occurs when buildings reflect the GPS signal.
A poor GPS signal decreases the ETA accuracy.
So they do map matching to find the best ETA.
Map matching works by mapping raw GPS signals to actual road segments.
They use the Kalman filter for map matching. It takes GPS signals and matches them to road segments.
Imagine the Kalman filter as a person who makes a correct guess about something's location. The new and old information is taken into consideration for guessing.
Besides they use the Viterbi algorithm to find the most probable road segments. It's a dynamic programming approach.
Imagine the Viterbi algorithm as a person who figures out the correct story even if some words were spelled wrong. They do that by looking at the nearby words and fixing the mistakes so that the
story makes more sense.
A rider is likely to avoid future trips if the actual trip time is higher than ETA.
Also more than 18 million Uber trips are completed daily.
So at Uber's scale, a bad ETA could cost them billions of USD in loss.
The current approach allowed them to scale to half a million requests per second.
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I could surely use this any time my friends ask “what’s your ETA”... I always grossly underestimate how much time I need ha!
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Great article! The images are really well-made. When I used to work at Uber, we actually used H3 (https://www.uber.com/blog/h3/) to split the map into hexagons for various map-related
optimizations, like setting pricing.
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Machine Learning | Druma
top of page
Machine Learning Course
50% OFF
Level: Intermediate
Pre-requisites for the course : Some programming knowledge required
• Basic Python programming
• Some knowledge of statistics, probability and linear algebra is preferable but not necessary
Date: New batch starts soon
1. Linear Regression
Our course starts from the most basic regression model: Just fitting a line to data. This simple model for forming predictions from a single, univariate feature of the data is appropriately called
"simple linear regression"
2. Multiple Regression
The next step in moving beyond simple linear regression is to consider "multiple regression" where multiple features of the data are used to form predictions. More specifically, in this module, you
will learn how to build models of more complex relationships between a single variable (e.g., 'square feet') and the observed response (like 'house sales price').
3. Ridge Regression
You have examined how the performance of a model varies with increasing model complexity, and can describe the potential pitfall of complex models becoming overfit to the training data. In this
module, you will explore a very simple, but extremely effective technique for automatically coping with this issue. This method is called "ridge regression".
4. Lasso
A fundamental machine learning task is to select amongst a set of features to include in a model. In this module, you will explore this idea in the context of multiple regression, and describe how
such feature selection is important for both interpretability and efficiency of forming predictions.
5. Nearest Neighbors & Kernel Regression
Up to this point, we have focused on methods that fit parametric functions---like polynomials and hyperplanes---to the entire dataset. In this module, we instead turn our attention to a class of
"nonparametric" methods. These methods allow the complexity of the model to increase as more data are observed, and result in fits that adapt locally to the observations.
6. Linear Classifiers & Logistic Regression
Linear classifiers are amongst the most practical classification methods. For example, in our sentiment analysis case-study, a linear classifier associates a coefficient with the counts of each word
in the sentence. In this module, you will become proficient in this type of representation. You will focus on a particularly useful type of linear classifier called logistic regression, which, in
addition to allowing you to predict a class, provides a probability associated with the prediction.
7. Learning Linear Classifiers
Once familiar with linear classifiers and logistic regression, you can now dive in and write your first learning algorithm for classification. In particular, you will use gradient ascent to learn the
coefficients of your classifier from data. You first will need to define the quality metric for these tasks using an approach called maximum likelihood estimation (MLE).
8. Overfitting & Regularization in Logistic Regression
As we saw in the regression course, overfitting is perhaps the most significant challenge you will face as you apply machine learning approaches in practice. This challenge can be particularly
significant for logistic regression, as you will discover in this module, since we not only risk getting an overly complex decision boundary, but your classifier can also become overly confident
about the probabilities it predicts. In this module, you will investigate overfitting in classification in significant detail, and obtain broad practical insights from some interesting visualizations
of the classifiers' outputs.
9. Decision Trees
Along with linear classifiers, decision trees are amongst the most widely used classification techniques in the real world. This method is extremely intuitive, simple to implement and provides
interpretable predictions. In this module, you will become familiar with the core decision trees representation. You will then design a simple, recursive greedy algorithm to learn decision trees from
10. Preventing Overfitting in Decision Trees
Out of all machine learning techniques, decision trees are amongst the most prone to overfitting. No practical implementation is possible without including approaches that mitigate this challenge. In
this module, through various visualizations and investigations, you will investigate why decision trees suffer from significant overfitting problems.
11. Handling Missing Data
Real-world machine learning problems are fraught with missing data. That is, very often, some of the inputs are not observed for all data points. This challenge is very significant, happens in most
cases, and needs to be addressed carefully to obtain great performance. And, this issue is rarely discussed in machine learning courses.
12. Nearest Neighbor Search
We start the course by considering a retrieval task of fetching a document similar to one someone is currently reading. We cast this problem as one of nearest neighbor search, which is a concept we
have seen in the Foundations and Regression courses. However, here, you will take a deep dive into two critical components of the algorithms: the data representation and metric for measuring
similarity between pairs of data points.
13. Clustering with k-means
In clustering, our goal is to group the data points in our dataset into disjoint sets. Motivated by our document analysis case study, you will use clustering to discover thematic groups of articles
by "topic". These topics are not provided in this unsupervised learning task; rather, the idea is to output such cluster labels that can be post-facto associated with known topics like "Science",
"World News' ', etc.
14. Case Study I
Regression: Predicting House Prices
This week you will build your first intelligent application that makes predictions from data.<p>We will explore this idea within the context of our first case study, predicting house prices, where
you will create models that predict a continuous value (price) from input features (square footage, number of bedrooms and bathrooms,...).
15. Case Study II
Classification: Analyzing Sentiment
How do you guess whether a person felt positively or negatively about an experience, just from a short review they wrote? In our second case study, analyzing sentiment, you will create models that
predict a class (positive/negative sentiment) from input features (text of the reviews, user profile information,...).This task is an example of classification, one of the most widely used areas of
machine learning, with a broad array of applications, including ad targeting, spam detection, medical diagnosis and image classification.
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Random Forest Model in R » Prediction model » finnstats
Random Forest Model in R
The random forest model in R is a highly useful tool in analyzing predicted outcomes for a classification or regression model.
The main idea is how explanatory variables will impact the dependent variable.
In this particular example, we analyze the impact of explanatory variables of Attribute 1, Attribute2, …Attribute6 on the dependent variable Likeability.
What are the Nonparametric tests? » Why, When and Methods »
Data Loading
Use read.xlsx function to read data into R.
We then split the data set into two training dataset and test data set. Training data that we will use to create our model and then the test data we will test it.
We have randomly created a data frame with a total of 64 data row observations, 60 observations used for training the data set, and 4 observations used for testing purposes.
#Create training and test data
inputData <- data[1:60, ] # training data
testData <- data[20:64, ] # test data
While using tuneRF function we can find out best mtr
tuneRF(data2,data2[,dim(data2)[2]], stepFactor=1.5)
mtry = 8 provides best OOB error = 0.01384072
A random forest allows us to determine the most important predictors across the explanatory variables by generating many decision trees and then ranking the variables by importance.
How to run R code in PyCharm? » R & PyCharm »
Random Forest Model in R
AttribImp.rf<-randomForest(Likeabilty~.,data=data2,importance=TRUE,proximity=TRUE, ntree=100, mtry=8, plot=FALSE)
Type of random forest: regression
Number of trees: 100
No. of variables tried at each split: 8
Mean of squared residuals: 2.00039
% Var explained: 78.58
Basis just 60 data points we 79% variance explained, recommended minimum 100 data points in each model for an accurate result.
LSTM Network in R » Recurrent Neural network »
Using the Boruta algorithm we can easily find out important attributes in the model.
Important <- Boruta(Likeabilty~ ., data = data2)
Boruta performed 87 iterations in 1.140375 secs.
5 attributes confirmed important: Attribute2, Attribute3, Attribute4, Attribute6,
Panel: 2 attributes confirmed unimportant: Attribute1, Attribute5;
Predict test data based on the training model
The predicted values are 4, 4, 5, 5, 4 and the original values are 2, 2, 2, 3, 4, Yes, it’s just close not good.
Recommended to increase the number of data points and increase the model accuracy 79 to at least 85.
Log Rank Test in R-Survival Curve Comparison »
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Probably Approximately Correct
Title Probably Approximately Correct PDF eBook
Author Leslie Valiant
Publisher Basic Books (AZ)
Total Pages 210
Release 2013-06-04
Genre Science
ISBN 0465032710
Download Probably Approximately Correct Book in PDF, Epub and Kindle
Presenting a theory of the theoryless, a computer scientist provides a model of how effective behavior can be learned even in a world as complex as our own, shedding new light on human nature.
Title Probably Approximately Correct PDF eBook
Author Leslie Valiant
Publisher Basic Books
Total Pages 210
Release 2013-06-04
Genre Science
ISBN 0465037909
Download Probably Approximately Correct Book in PDF, Epub and Kindle
From a leading computer scientist, a unifying theory that will revolutionize our understanding of how life evolves and learns. How does life prosper in a complex and erratic world? While we know that
nature follows patterns -- such as the law of gravity -- our everyday lives are beyond what known science can predict. We nevertheless muddle through even in the absence of theories of how to act.
But how do we do it? In Probably Approximately Correct, computer scientist Leslie Valiant presents a masterful synthesis of learning and evolution to show how both individually and collectively we
not only survive, but prosper in a world as complex as our own. The key is "probably approximately correct" algorithms, a concept Valiant developed to explain how effective behavior can be learned.
The model shows that pragmatically coping with a problem can provide a satisfactory solution in the absence of any theory of the problem. After all, finding a mate does not require a theory of
mating. Valiant's theory reveals the shared computational nature of evolution and learning, and sheds light on perennial questions such as nature versus nurture and the limits of artificial
intelligence. Offering a powerful and elegant model that encompasses life's complexity, Probably Approximately Correct has profound implications for how we think about behavior, cognition, biological
evolution, and the possibilities and limits of human and machine intelligence.
Title Probably Approximately Correct PDF eBook
Author Leslie Valiant
Publisher Hachette UK
Total Pages 208
Release 2013-06-04
Genre Science
ISBN 0465037909
Download Probably Approximately Correct Book in PDF, Epub and Kindle
From a leading computer scientist, a unifying theory that will revolutionize our understanding of how life evolves and learns. How does life prosper in a complex and erratic world? While we know that
nature follows patterns -- such as the law of gravity -- our everyday lives are beyond what known science can predict. We nevertheless muddle through even in the absence of theories of how to act.
But how do we do it? In Probably Approximately Correct, computer scientist Leslie Valiant presents a masterful synthesis of learning and evolution to show how both individually and collectively we
not only survive, but prosper in a world as complex as our own. The key is "probably approximately correct" algorithms, a concept Valiant developed to explain how effective behavior can be learned.
The model shows that pragmatically coping with a problem can provide a satisfactory solution in the absence of any theory of the problem. After all, finding a mate does not require a theory of
mating. Valiant's theory reveals the shared computational nature of evolution and learning, and sheds light on perennial questions such as nature versus nurture and the limits of artificial
intelligence. Offering a powerful and elegant model that encompasses life's complexity, Probably Approximately Correct has profound implications for how we think about behavior, cognition, biological
evolution, and the possibilities and limits of human and machine intelligence.
Title An Introduction to Computational Learning Theory PDF eBook
Author Michael J. Kearns
Publisher MIT Press
Total Pages 230
Release 1994-08-15
Genre Computers
ISBN 9780262111935
Download An Introduction to Computational Learning Theory Book in PDF, Epub and Kindle
Emphasizing issues of computational efficiency, Michael Kearns and Umesh Vazirani introduce a number of central topics in computational learning theory for researchers and students in artificial
intelligence, neural networks, theoretical computer science, and statistics. Emphasizing issues of computational efficiency, Michael Kearns and Umesh Vazirani introduce a number of central topics in
computational learning theory for researchers and students in artificial intelligence, neural networks, theoretical computer science, and statistics. Computational learning theory is a new and
rapidly expanding area of research that examines formal models of induction with the goals of discovering the common methods underlying efficient learning algorithms and identifying the computational
impediments to learning. Each topic in the book has been chosen to elucidate a general principle, which is explored in a precise formal setting. Intuition has been emphasized in the presentation to
make the material accessible to the nontheoretician while still providing precise arguments for the specialist. This balance is the result of new proofs of established theorems, and new presentations
of the standard proofs. The topics covered include the motivation, definitions, and fundamental results, both positive and negative, for the widely studied L. G. Valiant model of Probably
Approximately Correct Learning; Occam's Razor, which formalizes a relationship between learning and data compression; the Vapnik-Chervonenkis dimension; the equivalence of weak and strong learning;
efficient learning in the presence of noise by the method of statistical queries; relationships between learning and cryptography, and the resulting computational limitations on efficient learning;
reducibility between learning problems; and algorithms for learning finite automata from active experimentation.
Title Understanding Machine Learning PDF eBook
Author Shai Shalev-Shwartz
Publisher Cambridge University Press
Total Pages 415
Release 2014-05-19
Genre Computers
ISBN 1107057132
Download Understanding Machine Learning Book in PDF, Epub and Kindle
Introduces machine learning and its algorithmic paradigms, explaining the principles behind automated learning approaches and the considerations underlying their usage.
Title Foundations of Machine Learning, second edition PDF eBook
Author Mehryar Mohri
Publisher MIT Press
Total Pages 505
Release 2018-12-25
Genre Computers
ISBN 0262351366
Download Foundations of Machine Learning, second edition Book in PDF, Epub and Kindle
A new edition of a graduate-level machine learning textbook that focuses on the analysis and theory of algorithms. This book is a general introduction to machine learning that can serve as a textbook
for graduate students and a reference for researchers. It covers fundamental modern topics in machine learning while providing the theoretical basis and conceptual tools needed for the discussion and
justification of algorithms. It also describes several key aspects of the application of these algorithms. The authors aim to present novel theoretical tools and concepts while giving concise proofs
even for relatively advanced topics. Foundations of Machine Learning is unique in its focus on the analysis and theory of algorithms. The first four chapters lay the theoretical foundation for what
follows; subsequent chapters are mostly self-contained. Topics covered include the Probably Approximately Correct (PAC) learning framework; generalization bounds based on Rademacher complexity and
VC-dimension; Support Vector Machines (SVMs); kernel methods; boosting; on-line learning; multi-class classification; ranking; regression; algorithmic stability; dimensionality reduction; learning
automata and languages; and reinforcement learning. Each chapter ends with a set of exercises. Appendixes provide additional material including concise probability review. This second edition offers
three new chapters, on model selection, maximum entropy models, and conditional entropy models. New material in the appendixes includes a major section on Fenchel duality, expanded coverage of
concentration inequalities, and an entirely new entry on information theory. More than half of the exercises are new to this edition.
Title Circuits of the Mind PDF eBook
Author Leslie G. Valiant
Publisher Oxford University Press, USA
Total Pages 260
Release 2000
Genre Computers
ISBN 9780195126686
Download Circuits of the Mind Book in PDF, Epub and Kindle
While embracing the now classical theories of McCulloch and Pitts, the neuroidal model also accommodates state information in the neurons, more flexible timing mechanisms, a variety of assumptions
about interconnectivity, and the possibility that different brain areas perform specialized functions. Programmable so that a wide range of algorithmic theories can be described and evaluated, the
model provides a concrete computational language and a unified framework in which diverse cognitive phenomena - such as memory, learning, and reasoning - can be systematically and concurrently
analyzed. Requiring no specialized knowledge, Circuits of the Mind masterfully offers an exciting new approach to brain science for students and researchers in computer science, neurobiology,
neuroscience, artificial intelligence, and cognitive science. | {"url":"https://alexandrarosenfeld.net/open/probably-approximately-correct/","timestamp":"2024-11-05T23:46:55Z","content_type":"text/html","content_length":"44356","record_id":"<urn:uuid:fd302f22-9dc0-4971-ab48-80d9dfe3954e>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00500.warc.gz"} |
The Problem of Satisfying Constraints: A New Kind of Science | Online by Stephen Wolfram [Page 346]
remarkably poor: instead of steadily evolving to all black or all white, the system quickly gets stuck in a state that contains regions of different colors.
And as it turns out, this kind of behavior is not uncommon among iterative procedures; indeed it is even seen in such simple cases as trying to find the lowest point on a curve. The most obvious
iterative procedure to use for such a problem involves taking a series of small steps, with the direction of each step being chosen so as locally to go downhill.
And indeed for the first curve shown below, this procedure works just fine, and quickly leads to the lowest point. But for the second
Results of four tries at applying an iterative procedure to find configurations which satisfy the simple constraint that every square should be the same color as the square to its right. (The squares
are assumed to be arranged cyclically, so that the right neighbor of the rightmost square is the leftmost square.) The procedure starts from a random configuration of squares, and then at each step
picks a square at random, then reverses the color of this square whenever doing so reduces the total number of squares that violate the constraint. The only configurations that ultimately satisfy the
constraints are all white and all black. But the procedure gets stuck long before it reaches these configurations. The problem is that for any block more than one square across changing the color of
a square at either end will not reduce the total number of squares that violate the constraint. And as a result, such blocks remain fixed and cannot disappear.
Three examples of curves. In the first case, the most obvious mechanical or mathematical procedure of continually going downhill will successfully lead one to the lowest point. But in the other two
cases, this procedure will usually end up getting stuck at a local minimum. This is the basic phenomenon which makes it difficult to find patterns that satisfy constraints exactly using a procedure
that is based on progressive improvement. The third picture above is a representation of the kind of curve that arises in almost all discrete systems based on constraints. | {"url":"https://www.wolframscience.com/nks/p346--the-problem-of-satisfying-constraints/","timestamp":"2024-11-09T12:36:11Z","content_type":"text/html","content_length":"93008","record_id":"<urn:uuid:b7f7880f-c90a-4396-a5e0-e3770cc9f22b>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00226.warc.gz"} |
ng the
Matrix multiplication from foundations
CPU focus introduction to matrix multiplication for deep learning
Matrix multiplication from foundations - CPU only
{:nx, "~> 0.4.0"},
{:binary, "~> 0.0.5"},
{:stb_image, "~> 0.5.2"},
{:scidata, "~> 0.1.9"},
{:kino, "~> 0.7.0"},
{:axon, "~> 0.3.0"}
This Livebook is a transformation of a Python Jupyter Notebook from Fast.ai’s From Deep Learning Foundations to Stable Diffusion, Practical Deep Learning for Coders part 2, 2022. Specifically, it
mimics https://github.com/fastai/course22p2/blob/master/nbs/01_matmul.ipynb
The purpose of the transformation is to bring the Fast.ai concepts to Elixir focused developers. The object-oriented Python/PyTorch implementation is transformed into a functional programming
implementation using Nx and Axon
About Fast.ai’s Teaching Philosophy
We’ll be leveraging the best available research on teaching methods to try to fix these problems with technical teaching, including:
• Teaching “the whole game”–starting off by showing how to use a complete, working, very usable, state of the art deep learning network to solve real world problems, by using simple, expressive
tools. And then gradually digging deeper and deeper into understanding how those tools are made, and how the tools that make those tools are made, and so on…
• Always teaching through examples: ensuring that there is a context and a purpose that you can understand intuitively, rather than starting with algebraic symbol manipulation
• Simplifying as much as possible: we’ve spent months building tools and teaching methods that make previously complex topics very simple
• Removing barriers: deep learning has, until now, been a very exclusive game. We’re breaking it open, and ensuring that everyone can play
From: https://www.fast.ai/posts/2016-10-08-teaching-philosophy.html
In other words, focus on student success from the beginning. Help students become confident in their growing skills. Use a code-first approach to teaching Deep Learning. Provide plenty of examples of
functioning neural networks that can be applied by the students.
This part 2 course is not exactly like the above description
The Fast.ai part 1 course fits the above description really well. When we went through each of the past 4 years of the part 1 course, we felt like the course spoke to our needs. In the best world,
Elixir developers could learn from the part 1 course and come away several kinds of near state of the art neural net models running in Elixir. However, an Elixir version of the part 1 course doesn’t
exist yet.
The part 2 course has a different focus. Jeremy Howard likes to call part 2 the (im)practical deep learning for coders. Part 2 goes under the hood and helps students understand the pieces of a neural
network and a best practice focused training library. The foundations are taught with examples that help students understand how the pieces really work. It’s impractical because the problems are
simpler, already solved, examples that don’t translate to a real-world problem. The examples used in part 2 are smaller, well known problems, but the focus is on understanding how the software skills
you use daily are transformed into a neural network concepts that can utilize the GPU for time efficient training. The confidence gained from the part 2 course is knowing how to modify and change a
model to fit your domain situation.
Foundation notebooks and previous videos
The 2022 Python/PyTorch “from the foundations” notebooks are in https://github.com/fastai/course22p2. This notebook is being written while the live course is happening. Fast.ai course work is
restricted to paid participants until after the course is completed. The notebooks are available in GitHub, but the videos and forum converstations are restricted. After the course completes, the
videos and forums are open to everyone in the form of a massive open online course. However, fundamentals don’t change that much. The 2019 course videos, https://course19.fast.ai/videos/?lesson=8,
would be a fine video companion for these Elixir notebooks, for now.
The first two lesson videos from the 2022 course were released early. In the second lesson, Jeremy covers the first portion of this notebook.
To Stable Diffusion
The 2022 part 2 course is Foundations to Stable Diffusion. In 2022, Fast.ai is focusing on understanding the pieces of Stable Diffusion and discussions on the latest research papers that improve upon
Stable Diffusion. As a taste of what is coming, Fast.ai has released the videos from the first 2 weeks. https://www.fast.ai/posts/part2-2022-preview.html
At the current time, Stable Diffusion doesn’t run in Elixir. Fast.ai part 2 is split into two types of notebooks. A set of notebooks focused on Stable Diffusion and another set of notebooks focused
on the foundations. For now, we are focused only on the foundation notebooks.
Fast.ai’s book
There was a recent Twitter discussion expressing a desire to see Deep Learning for Coders with Fastai and PyTorch: AI Applications Without a PhD examples in Elixir/Livebook. The meanderingstream/
dl_foundations_in_elixir notebooks correspond to chapters 17, 18 and 19 in the book. Further resources related to the book can be found on the Fast.ai book page.
Part 2 Foundations approach
We’ll start with standard Elixir examples of the fundamentals. An Elixir focused developer should recognize the standard Elixir code. The part 2 “Game” is:
• Once a representative example is implemented with our code, we can then use the corresponding Nx and Axon code.
Because we are transforming Python/PyTorch into Elixir, some concepts don’t perfectly match back to the original Python code. There are some library differences and some of the tooling for Elixir and
Livebook don’t perfectly match. Nx, Axon and Livebook are very recent technologies and their capabilities are growing each month.
Because we are mapping from Python/PyTorch to Elixir and the vast majority of machine learning examples are written in Python, we are often going to show the original Python from the Fast.ai notebook
on top of the Elixir code. Hopefully this will help Elixir developers transform other PyTorch code into Elixir code
# Pytorch
# some python from a Jupyter notebook
# --> The result
# from executing the cell goes here
some elixir code here
Brief Introduction to Elixir and Numerical Elixir
Elixir’s primary numerical datatypes and structures are not optimized for numerical programming. Nx is a library built to bridge that gap.
Elixir Nx is a numerical computing library to smoothly integrate to typed, multidimensional data implemented on other platforms (called tensors). This support extends to the compilers and libraries
that support those tensors. Nx has three primary capabilities:
• In Nx, tensors hold typed data in multiple, named dimensions.
• Numerical definitions, known as defn, support custom code with tensor-aware operators and functions.
• Automatic differentiation, also known as autograd or autodiff, supports common computational scenarios such as machine learning, simulations, curve fitting, and probabilistic models.
From https://hexdocs.pm/nx/intro-to-nx.htm. Note that this url is a really a livebook notebook. When you click on the Run in Livebook button, it navigates to an intermediate page where you can choose
the location of your LiveBook application. It then opens the page in your LiveBook application.
Course Start: From the foundations
Jeremy’s introduction: This part of the course will require some serious tenacity and a certain amount of patience. We think you are going to learn a lot. A lot people have given Jeremy feedback that
the previous iteration of this course is the Best Course they’ve ever done. This course will be dramatically better than any previous version. Hopefully you’ll find that the hard work and patience
pays off.
Our goal in this course is to get to stable diffusion from the foundations. We have to define what are the foundations. Jeremy resticted the Python foundations to:
• Python
• The Python standard library
• matplotlib
• Jupyter notebooks and nbdev
In Elixir we’ll have our own foundation.
To be clear, we are allowed to use other libraries once we have reimplemented them correctly. If we reimplement something from NumPy or PyTorch, we are then allowed to use those libraries. Sometimes
we are going to implement things that haven’t been created before. Those things will become part of our own library. We are going to be calling that library miniai. We are going to be building our
own little framework as we go.
One challenge that we have, the models we use in Stable Diffusion were trained on millions of dollars of equipment for months. We don’t have the time or money for those compute resources. Another
trick we are going to do is create identical but smaller versions of them. Once we have them working, we’ll be allowed to use the big pre-trained version.
So we are going to have to end up with our own variational auto-encoder, our own U-Net, our own CLIP encoder, and so forth.
To certain extent, Jeremy assumes that you’ve gone through part 1. If you find something that doesn’t make sense to you. Go back to the part 1 course or Google for what you don’t understand. For
stuff that wasn’t covered in part 1, we’ll go over it thoroughly and carefully.
Reference: Jeremy’s discussion in the Lesson 10 video.
Elixir foundations
In our foundations version, we’ll make the following assumptions throughout these Elixir versions of Fast.ai’s notebooks:
The documentation for Nx and Axon are found at https://hexdocs.pm/nx/Nx.html and https://hexdocs.pm/axon/Axon.html
To run these notebooks, you will need to install a local version of Livebook or get access to a cloud server. Many of our foundation notebooks don’t need a GPU. Nx comes with an Elixir only
BinaryBackend that runs on any CPU that supports Livebook. If EXLA or Torchx aren’t in the Mix.install at the top of a notebook, it can be run on any computer. Please give it a try.
We’ll follow roughly the same approach as the PyTorch version of the course. We’ll start with standard Elixir, with some additional libraries. Once we’ve implemented a capability, we’ll move on using
Nx and Axon libraries. We’ll invent our own libraries as needed.
Getting the Data
We are going to need some input data. Fast.ai uses MNIST for this part of the course. Elixir has the SciData library that contains small standard datasets, including MNIST.
We are diverging from the cell by cell transformation of the 01_matmul.ipynb because SciData works differently from the .pth files used in Fast.ai
{train_images, train_labels} = Scidata.MNIST.download()
{test_images, test_labels} = Scidata.MNIST.download_test()
# Let's unpack the images like...
{train_images_binary, tensor_type, train_shape} = train_images
{test_images_binary, tensor_type, test_shape} = test_images
The Fast.ai source for MNIST training data returns normalized data with a shape of (50000, 784). 50,000 items that is 784 numbers long. The numbers are all between 0 and 1. We’ll need to change our
binary into numbers and divide the numbers by 255 to normalize the values.
# Normalize the values first.
train_normalized_long_list =
|> Enum.map(fn value -> value / 255 end)
The data source Fast.ai used split the 60,000 image MNIST train data into 50,000 train images and 10,000 validation images. We’ll do a similar split after the first 50,000 images.
{train_list_784, valid_list_784} =
Enum.chunk_every(train_normalized_long_list, 784)
|> Enum.split(50_000)
train_imgs_28_28 =
fn img ->
Enum.chunk_every(img, 28)
Let’s check that we still have 50000 images and that the count of rows in the first image is 28 and the count of columns in the first row of the first image is 28
{Enum.count(train_imgs_28_28), Enum.count(Enum.at(train_imgs_28_28, 0)),
Enum.count(Enum.at(Enum.at(train_imgs_28_28, 0), 0))}
Visualizing Normalized Data
We have a normalized image in memory, how do we check that it really represents an image?
first_img_28_28 = Enum.at(train_imgs_28_28, 0)
We don’t know of a convenient method to convert a normalized list of lists into an image. However, if we convert to a tensor, we can load the tensor into StbImage. We are going to cheat and look
ahead at some concepts described below, but we’ll be able to show the image.
first_img =
|> Enum.map(fn row ->
Enum.map(row, fn column ->
round(column * 255)
|> Nx.tensor(type: :u8)
|> Nx.reshape({28, 28, 1})
|> StbImage.from_nx()
|> StbImage.to_binary(:png)
# Python
# mpl.rcParams['image.cmap'] = 'gray'
# plt.imshow(list(chunks(lst1, 28)));
# Kino currently assumes the image is larger than the box
image = Kino.Image.new(first_img, :png)
label = Kino.Markdown.new("**MNIST Image**")
images = [
Kino.Layout.grid([image, label], boxed: true)
Kino.Layout.grid(images, columns: 3)
Matrix and tensor
Let’s pull an individual value from a list of lists
# Find a row with some non-zero values
# 8th row
first_non_zero_in_row =
Enum.at(first_img_28_28, 8)
|> Enum.find_index(fn x -> x != 0.0 end)
# Let's find a value somewhere in that list of lists
Enum.at(first_img_28_28, 8)
|> Enum.at(10)
Convenience module to make it easier to access an element in list of lists
defmodule Matrix do
def at(matrix, row, column) do
Enum.at(matrix, row) |> Enum.at(column)
Matrix.at(first_img_28_28, 8, 10)
Now that we’ve demonstrated how to load SciData into normal Elixir list of lists, access elements within the list of list, and shown the in-memory image data. Let’s start using Nx Tensors instead of
lists of lists.
x_tensors =
|> Nx.from_binary(tensor_type)
|> Nx.reshape({60000, 28 * 28})
|> Nx.divide(255)
Again, we’ll split the SciData training dataset into train and valid. We’ll use the names that are in the Fast.ai notebook
x_train = x_tensors[0..49_999]
x_valid = x_tensors[50_000..59_999]
{x_train.shape, x_valid.shape}
CAUTION: Even though it kind of looks like we called a function on a data object, all we really did was access the shape field of a struct. Just simple data field access. The human readable
representation of the struct simplifies the representation to make it easier to see. Tensors can have a lot of data in their struct fields. Type is another field. See how the type and shape are
scunched together in the print view.
Let’s load an Nx normalized tensor and visualize with Kino.Image
img_tensor =
|> Nx.reshape({28, 28})
Let’s visualize the image like we did earlier, except this time the source is an Nx.Tensor
first_img_from_tensor =
|> Nx.reshape({28, 28, 1})
|> Nx.multiply(255)
|> Nx.round()
|> Nx.as_type({:u, 8})
|> StbImage.from_nx()
|> StbImage.to_binary(:png)
# Python
# plt.imshow(imgs[0]);
image = Kino.Image.new(first_img_from_tensor, :png)
label = Kino.Markdown.new("**MNIST Image from tensor**")
images = [
Kino.Layout.grid([image, label], boxed: true)
Kino.Layout.grid(images, columns: 3)
Let’s parse out the classification labels of each image. Each element identifies the digit each handwritten image represents.
{train_y_binary, y_tensor_type, y_shape} = train_labels
y_tensors =
|> Nx.from_binary(y_tensor_type)
|> Nx.reshape(y_shape)
We’ll split into train and valid like the Fast.ai data source.
y_train = y_tensors[0..49_999]
y_valid = y_tensors[50_000..59_999]
{y_train.shape, y_valid.shape}
We couldn’t find a min function in Nx that corresponds to the min, or max, function in Python that works on tensors. We’ll convert to a flat, normal Elixir list. We’ll use Enum to find the min or max
and then convert back to Nx tensor scalars.
# PyTorch
# y_train.min(), y_train.max()
# --> (tensor(0), tensor(9))
{Nx.tensor(Enum.min(Nx.to_flat_list(y_train))), Nx.tensor(Enum.max(Nx.to_flat_list(y_train)))}
Random Numbers
For now, we are going to treat the random number section of the Fast.ai notebook as a problem specific to PyTorch. The problematic situation comes from using OS.fork() to parallelize some work that
calls the rand() function. In Python, the fork create a copy of the current process. The particular problem is the fork includes the global rnd_state of the parent process. Each process that calls
rand() will receive the same sequence of psuedo-random numbers.
The discussion on how psuedo-random numbers in the video is well worth watching
TODO: How does Elixir handle psuedo-random number sequences in two Elixir processes.
Tensor rank
The rank of a tensor is the number of indices required to uniquely select each element of the tensor. Rank is also known as “order”, “degree”, or “ndims.”
from https://www.tensorflow.org/api_docs/python/tf
In Livebook/Nx, the rank can be observed from the number of square bracket pairs behind the type label, i.e. s64.
# Rank 1 tensor
Nx.tensor([1, 2, 3])
# Rank 2 tensor
Nx.tensor([[1, 2], [2, 3]])
# Rank 3 tensor
Nx.tensor([[[1, 2], [2, 3]], [[4, 5], [5, 6]]])
# Rank 0 tensor
Online Matrix multiplication
We are working on the start of a forward pass of a very simple linear model, a multi-layer perceptron, for MNIST. We now need to multiply tensors together.
There are several websites that can provide visual examples of matrix multiplication.
Matrix multiplication is a fundamental capability of deep learning. We are going to look at how to do matrix multiplication in standard Elixir and then use Nx to perform the multiplication as a
Mutable data approaches vs Immutable data
Many software languages have mutable data. Certainly Python has mutable data. Let’s go into details about how immutable data in Elixir is different than working with mutable data.
In Python, Jeremy uses this approach to multiply two tensors.
for i in range(ar): # 5
for j in range(bc): # 10
for k in range(ac): # 784
t1[i,j] += m1[i,k] * m2[k,j]
Turning it into a function would look like:
def py_multiply(m1, m2, t1)
ar,ac = m1.shape # n_rows * n_cols
br,bc = m2.shape
for i in range(ar): # 5
for j in range(bc): # 10
for k in range(ac): # 784
t1[i,j] += m1[i,k] * m2[k,j]
t1 is the resulting matrix. In the Python notebook, it is set to zeros via
t1 = torch.zeros(ar, bc)
t1 is mutable. The value at t1[i,j] is replaced with a new value via += . When the function has completed, the variable, call it t_init, passed as the third argument has the new values.
We’ve mentioned before that Elixir has immutable data. When trying the same kind of approach in Elixir.
defmodule DoesntWork
def add(m1, m2, t)
t = m1, m2
Let’s try it
defmodule DoesntWork do
def add(m1, m2, t) do
t = m1 + m2
t = 5
DoesntWork.add(3, 6, t)
"t is #{t} not 9"
"because Elixir's data is immutable"
The above long winded point is that matrix operations in pure Elixir can’t use the for loop approach in mutable data languages.
Matrix multiplication in plain Elixir
We’ll diverge from Fast.ai’s notebook to dig into how we can do matrix multiplication without using for loops. We’ll also focus on Elixir list of lists rather than Nx.tensors like the Python
Let’s look at a matrix multiplication in traditional Elixir. This example was modified from https://rosettacode.org/wiki/Matrix_multiplication#Elixir
defmodule Matrix do
def mult(m1, m2) do
Enum.map(m1, fn x ->
Enum.map(transpose(m2), fn y ->
Enum.zip(x, y)
|> Enum.map(fn {x, y} -> x * y end)
|> Enum.sum()
# transpose
def transpose(m) do
List.zip(m) |> Enum.map(&Tuple.to_list(&1))
# Let's set up an example multiplication using an example
# from http://matrixmultiplication.xyz/
m_3x3 = [[1, 2, 1], [0, 1, 0], [2, 3, 4]]
m_3x2 = [[2, 5], [6, 7], [1, 8]]
# Let's check that matrix multiplication works. We should get
# [[15,27],
# [ 6, 7],
# [26,63]]
Matrix.mult(m_3x3, m_3x2)
Go to http://matrixmultiplication.xyz/ and put in your own matrix values to try it out.
Let’s dig into the Elixir code in our module
As Elixir developers, we understand how Enum.map works. But not everyone may have a good understanding. Let’s explore Enum.map. Enum.map/2 takes one argument of some Enumerable like a list. It also
receives another argument which must be a function, generally an anonymous function. For each element in the Enumerable, it calls the function with the current element and appends the result to a
list. It then returns the resulting list.
some_list = [1, 2, 3]
another_list = [4, 5, 6]
Enum.map(some_list, fn x ->
# The inner map() can see the outer x
Enum.map(another_list, fn y ->
IO.puts("x is #{x} y is #{y}")
Next we’ll dig into the transpose function.
transpose = fn m ->
|> IO.inspect(label: "one of the rows in zip")
# |> Enum.map(&Tuple.to_list(&1))
# In matrix multiplication the another list needs to be vertical.
another_list = [[4, 5, 6]]
transpose = fn m ->
# We know that the first and only element in the list is
# [{4}, {5}, {6}]
# Which means that three items are in the Enumerable passed to Enum.app
# the first is {4}
# the first is then transformed from a tuple, i.e. {something} into
# a list of [something]
|> Enum.map(&Tuple.to_list(&1))
# In matrix multiplication the another_list needs to be vertical.
another_list = [[4, 5, 6]]
There is a really funky set of code above. &Tuple.to_list(&1) THe & is an Elixir capture operator. Here is a blog post explaining the capture operator. https://dockyard.com/blog/2016/08/05/
Personally, we are more comfortable with the slightly more verbose form of creating an anonymouse function. Our minds grok this form easier. They both result in the same answer.
transpose = fn m ->
# We know that the first and only element in the list is
# [{4}, {5}, {6}]
# Which means that three items are in the Enumerable passed to Enum.app
# the first is {4}
# the first is then transformed from a tuple, i.e. {something} into
# a list of [something]
|> Enum.map(fn x -> Tuple.to_list(x) end)
# In matrix multiplication the another_list needs to be vertical.
another_list = [[4, 5, 6]]
The next set of code takes the first matrix and the transform of the second matrix and zips them together. As before we see that we end up with list of tuples
Enum.map(m_3x3, fn x ->
Enum.map(transpose.(m_3x2), fn y ->
Enum.zip(x, y)
The next step is to take the list of list of list of tuples and run through Enum.map with a multiply function and return a list.
Enum.map(m_3x3, fn x ->
Enum.map(transpose.(m_3x2), fn y ->
Enum.zip(x, y)
|> Enum.map(fn {x, y} -> x * y end)
Finally, we sum up the elements in the inner most list.
Enum.map(m_3x3, fn x ->
Enum.map(transpose.(m_3x2), fn y ->
Enum.zip(x, y)
|> Enum.map(fn {x, y} -> x * y end)
|> Enum.sum()
And we get the same answer from calling the Matrix.mult function above.
Whew. I hope you could follow along and we didn’t lose you. We’ve now implemented matrix multiplication using standard Elixir. Thus, we can now use Nx.dot()
t_3x3 = Nx.tensor(m_3x3)
t_3x2 = Nx.tensor(m_3x2)
{t_3x3.shape, t_3x2.shape}
We’ve now implemented matrix multiplication using standard Elixir. Thus, we can now use Nx.mult(). Still the same answer, just now it is with tensors
Nx.dot(t_3x3, t_3x2)
Let’s measure how fast, really kind of slow, the BinaryBackend. Remember, we aren’t using the GPU in this notebook so don’t compare with the PyTorch when Jeremy is using a GPU.
Timing operations
In Elixir, the erlang timer tc function can be use to time function calls. Here is a link to a discussion on :timer.tc. https://til.hashrocket.com/posts/9jxsfxysey-timing-a-function-in-elixir So we
can call the same function multiple times, we’ll create a named anonymous repeat function. We’ll has create function that calls represents are target function. We’ll hard code the same arguments in
the target function.
repeat = fn timed_fn, times -> Enum.each(1..times, fn _x -> timed_fn.() end) end
matrix_mult_w_dot_fn = fn -> Nx.dot(t_3x3, t_3x2) end
repeat_times = 50
{elapsed_time_micro, _} = :timer.tc(repeat, [matrix_mult_w_dot_fn, repeat_times])
avg_elapsed_time_ms = elapsed_time_micro / 1000 / repeat_times
"avg time in milliseconds #{avg_elapsed_time_ms} total_time #{elapsed_time_micro / 1000} milliseconds"
Not to bad performance but the tensors are small.
Matrix multiplication
Let’s create some tensor random weights with a mean of about 0.0 and variance of about 1.0
# PyTorch
# weights = torch.randn(784,10)
# bias = torch.zeros(10)
# weights, weights.max(), weights.mean(), weights.var()
mean = 0.0
variance = 1.0
weights = Nx.random_normal({784, 10}, mean, variance, type: {:f, 32})
# In elixir, Nx doesn't have the ability to create a Tensor of 0s or 1s. We have to use
# Axon's initializers
init_zeros = Axon.Initializers.zeros()
bias = init_zeros.({10}, {:f, 32})
{bias, weights}
{Nx.mean(weights), Nx.variance(weights)}
Let’s take the first 5 rows, m1, of the training dataset, 5x784, images x pixels. For every one of the 784 pixels in each row of the tensor, we need a weight multiplication factor. The weights map to
each one of the 10 potential digits in the y_valid data, 784x10. The first column of weights will identify all of the weights to figure out whether the pixels represent a 0. The second column will
determine the weights to tell us the probability the pixels represent a 1, etc. up to 9.
# PyTorch
# x_valid[:5]
m1 = x_valid[0..4]
m2 = weights
{m1.shape, m2.shape}
# PyTorch
# ar,ac = m1.shape # n_rows * n_cols
# br,bc = m2.shape
# (ar,ac),(br,bc)
{ar, ac} = m1.shape
{br, bc} = m2.shape
{{ar, ac}, {br, bc}}
# PyTorch
# t1 = torch.zeros(ar, bc)
# t1.shape
t1 = init_zeros.({ac, bc}, {:f, 32})
When we multiply matrices together, we take row 1 of the first matrix. We take column 1 of the second matrix. We multiply the row 1 elements and column 2 elements in turn. r1[1] times c1[1], r1[2]
times c2[2]…. and we sum them together. The sum would give the value for the very first cell in the resulting 5x10 matrix
Let’s compare the time to multiply two standard Elixir matrices with the time to multiply using Nx tensors with the BinaryBackend.
Nx.dot(m1, m2)
Let’s time our Nx matrix multiplication.
dot_m1_m2_fn = fn -> Nx.dot(m1, m2) end
repeat_times = 50
{elapsed_time_micro, _} = :timer.tc(repeat, [dot_m1_m2_fn, repeat_times])
avg_elapsed_time_ms = elapsed_time_micro / 1000 / repeat_times
"avg time in milliseconds #{avg_elapsed_time_ms} total_time #{elapsed_time_micro / 1000} milliseconds"
Let’s return to closely following the Fast.ai notebook.
Elementwise ops
The point of this section is to perform a function on each element of the Python tensor. The Elixir implementation would use the non tensor data loaded above.
# PyTorch
# a = tensor([10., 6, -4])
# b = tensor([2., 8, 7])
# a,b
# --> (tensor([10., 6., -4.]), tensor([2., 8., 7.]))
a = Nx.tensor([10.0, 6, -4])
b = Nx.tensor([2.0, 8, 7])
{a, b}
# PyTorch
# a + b
# --> tensor([12., 14., 3.])
Nx.add(a, b)
# PyTorch
# (a < b).float().mean()
# --> tensor(0.67)
Nx.less(a, b)
|> Nx.as_type({:f, 32})
|> Nx.mean()
# PyTorch
# m = tensor([[1., 2, 3], [4,5,6], [7,8,9]]); m
# -->
# tensor([[1., 2., 3.],
# [4., 5., 6.],
# [7., 8., 9.]])
# In Livebook, we don't need to specify what to show results
# on, if the item of interest is the last calculation.
# So we don't need the ;m at the end
m = Nx.tensor([[1.0, 2, 3], [4, 5, 6], [7, 8, 9]])
Frobenius norm:
We’ll use the Frobenius norm from time to time as we do generative modeling
It’s the sum over all of the rows and columns of the matrix. We are going to take each one and square it. We are going to add them up and take the square root
$$\| A \|F = \left( \sum{i,j=1}^n | a_{ij} |^2 \right)^{1/2}$$
Hint: you don’t normally need to write equations in LaTeX (really KaTeX) yourself, instead, you can click ‘edit’ in Wikipedia and copy the LaTeX from there (which is what Jeremy did for the above
equation). Or on arxiv.org, click “Download: Other formats” in the top right, then “Download source”; rename the downloaded file to end in .tgz if it doesn’t already, and you should find the source
there, including the equations to copy and paste. This is the source LaTeX that Jeremy pasted to render the equation above:
$$\| A \|_F = \left( \sum_{i,j=1}^n | a_{ij} |^2 \right)^{1/2}$$
In my case, I went to the Fast.ai notebook code, .ipynb file, to copy the KaTeX from Jeremy’s code
To implement Frobenius norm in Elixir, it is m times m, sum them up and square root.
# PyTorch
# (m*m).sum().sqrt()
# --> tensor(16.88)
Nx.multiply(m, m)
|> Nx.sum()
|> Nx.sqrt()
This looked like a complicated math function when you initially looked at it. A whole bunch of squiggly things. But when you look at the code, it’s just multiply itself, sum and then square root.
A lot of machine learning papers have complicated looking math notations for simple or relatively simple functions in code.
The term broadcasting describes how arrays with different shapes are treated during arithmetic operations. The term broadcasting was first used by Numpy.
From the Numpy Documentation:
The term broadcasting describes how numpy treats arrays with
different shapes during arithmetic operations. Subject to certain
constraints, the smaller array is “broadcast” across the larger
array so that they have compatible shapes. Broadcasting provides a
means of vectorizing array operations so that looping occurs in C
instead of Python. It does this without making needless copies of
data and usually leads to efficient algorithm implementations.
In addition to the efficiency of broadcasting, it allows developers to write less code, which typically leads to fewer errors.
This section was adapted from Chapter 4 of the fast.ai Computational Linear Algebra course.
In turn, it was copied from the Fast.ai 01_matmul.ipynb code
# PyTorch
# a
# --> tensor([10., 6., -4.])
# PyTorch
# a > 0
# -> tensor([ True, True, False])
Nx.greater(a, 0)
How are we able to do a > 0? 0 is being broadcast to have the same dimensions as a.
For instance you can normalize our dataset by subtracting the mean (a scalar) from the entire data set (a matrix) and dividing by the standard deviation (another scalar), using broadcasting.
Other examples of broadcasting with a scalar:
# PyTorch
# a + 1
# --> tensor([11., 7., -3.])
Nx.add(a, 1)
# The scalar can be in either position
Nx.add(1, a)
m = Nx.tensor([[1.0, 2.0, 3.0], [4.0, 5.0, 6.0], [7.0, 8.0, 9.0]])
# PyTorch
# 2*m
# -->
# tensor([[ 2., 4., 6.],
# [ 8., 10., 12.],
# [14., 16., 18.]])
Nx.multiply(m, 2)
Nx.multiply(2, m)
Broadcasting a vector to a matrix
Although broadcasting a scalar is an idea that dates back to APL, the more powerful idea of broadcasting across higher rank tensors comes from a little known language called Yorick.
We can also broadcast a vector to a matrix:
# PyTorch
# c = tensor([10.,20,30]); c
# --> tensor([10., 20., 30.])
# the vector
c = Nx.tensor([10.0, 20.0, 30.0])
# the matrix
# PyTorch
# m.shape,c.shape
# --> (torch.Size([3, 3]), torch.Size([3]))
{m.shape, c.shape}
# PyTorch
# m + c
# -->
# tensor([[11., 22., 33.],
# [14., 25., 36.],
# [17., 28., 39.]])
# The vector is broadcast across the matrix shape and added
Nx.add(c, m)
# reverse the order and still the same answer
Nx.add(m, c)
Here is the trick that allows the matrix and vector to be added. The expand_as method expands the vector to be the same shape as m.
We don’t really copy the rows, but it looks as if we did. In fact, the rows are given a stride of 0.
Elixir: We aren’t sure whether the Nx.broadcast actually copies the rows or looks like it does ala PyTorch
# PyTorch
# t = c.expand_as(m); t
# -->
# tensor([[10., 20., 30.],
# [10., 20., 30.],
# [10., 20., 30.]])
t = Nx.broadcast(c, m.shape)
I don’t believe the following tensor code has an Nx equivalent
# PyTorch
# t.storage() # Not sure there is an Nx equivalent
# t.stride(), # Not sure there is an Nx equivalent
In PyTorch, tou can index with the special value [None] or use unsqueeze() to convert a 1-dimensional array into a 2-dimensional array (although one of those dimensions has value 1).
The Nx equivalent is the Nx.reshape.
# PyTorch
# c
# -->
# tensor([10., 20., 30.])
Both unsqueeze and c[something, something_else] map to Nx.reshape. We’ll just show the Nx.reshape once.
This is how we create a matrix with one row
# PyTorch
# c.unsqueeze(0), c[None, :]
# --> (tensor([[10., 20., 30.]]), tensor([[10., 20., 30.]]))
Nx.reshape(c, {1, :auto})
# PyTorch
# c.shape, c.unsqueeze(0).shape
# --> (torch.Size([3]), torch.Size([1, 3]))
{c.shape, Nx.reshape(c, {1, :auto}).shape}
# c.unsqueeze(1), c[:, None]
# --> (tensor([[10.],
# [20.],
# [30.]]),
# tensor([[10.],
# [20.],
# [30.]]))
# This is how we create a matrix with one column.
Nx.reshape(c, {:auto, 1})
# PyTorch
# c.shape, c.unsqueeze(1).shape
# --> (torch.Size([3]), torch.Size([3, 1]))
{c.shape, Nx.reshape(c, {:auto, 1}).shape}
In PyTorch, they can skip trailling ‘:’s. And ‘…’ means ‘all preceding dimensions’
# PyTorch
# c[None].shape,c[...,None].shape
# --> (torch.Size([1, 3]), torch.Size([3, 1]))
{Nx.reshape(c, {1, :auto}).shape, Nx.reshape(c, {:auto, 1}).shape}
Below, we are taking the vector, transforming into a matrix with one column then we broadcast the result into a matrix of m shape.
# PyTorch
# c[:,None].expand_as(m)
# --> tensor([[10., 10., 10.],
# [20., 20., 20.],
# [30., 30., 30.]])
Nx.reshape(c, {:auto, 1})
|> Nx.broadcast(m.shape)
As a reminder, in this case we are adding the vector to each row.
# PyTorch
# m + c
# -->
# tensor([[11., 22., 33.],
# [14., 25., 36.],
# [17., 28., 39.]])
Nx.add(m, c)
Here we are transforming the vector c into a matrix with one column and then broadcasting into the shape of m. Then we add the two matrices together.
# PyTorch
# m + c[:,None]
# --> tensor([[11., 12., 13.],
# [24., 25., 26.],
# [37., 38., 39.]])
Nx.add(m, Nx.reshape(c, {:auto, 1}) |> Nx.broadcast(m.shape))
Here we are transforming the vector c into a matrix with one row and then broadcasting into the shape of m. Then we add the two matrices together.
# PyTorch
# m + c[None,:]
# tensor([[11., 22., 33.],
# [14., 25., 36.],
# [17., 28., 39.]])
# Nx.add(m, Nx.reshape(c, {1, :auto}) |> Nx.broadcast(m.shape) )
Nx.add(m, Nx.reshape(c, {1, :auto}) |> Nx.broadcast(m.shape))
Broadcasting Rules
# PyTorch
# c[None,:]
# --> tensor([[10., 20., 30.]])
Nx.reshape(c, {1, :auto})
# PyTorch
# c[None,:].shape
# --> torch.Size([1, 3])
Nx.reshape(c, {1, :auto}).shape
# PyTorch
# c[:,None]
# --> tensor([[10.],
# [20.],
# [30.]])
Nx.reshape(c, {:auto, 1})
# PyTorch
# c[:,None].shape
# --> torch.Size([3, 1])
Nx.reshape(c, {:auto, 1}).shape
Here we are taking a vector, c, and reshaping into a matrix of one column. Then we take a vector, c, and reshaping into a matrix of one row. Then the multiply function with expand the one column into
a 3 columns with the same values. The same thing happens for the matrix of one row. It expands into 3 rows.
We end up with 3 rows of 10,20,30 and 3 columns of
When we multiply them together and we get this answer. This is an outer product without any special function. Just broadcasting. Not just products, we can do outer boolean operations, etc.
# PyTorch
# c[None,:] * c[:,None]
# --> tensor([[100., 200., 300.],
# [200., 400., 600.],
# [300., 600., 900.]])
Nx.multiply(Nx.reshape(c, {1, :auto}), Nx.reshape(c, {:auto, 1}))
Here is the examples of the outer boolean.
# PyTorch
# c[None] > c[:,None]
# --> tensor([[False, True, True],
# [False, False, True],
# [False, False, False]])
Nx.greater(Nx.reshape(c, {:auto}), Nx.reshape(c, {:auto, 1}))
When operating on two arrays/tensors, Numpy/PyTorch compares their shapes element-wise. It starts with the trailing dimensions, and works its way forward. Two dimensions are compatible when
• they are equal, or
• one of them is 1, in which case that dimension is broadcasted to make it the same size
Arrays do not need to have the same number of dimensions. For example, if you have a 256*256*3 array of RGB values, and you want to scale each color in the image by a different value, you can
multiply the image by a one-dimensional array with 3 values. Lining up the sizes of the trailing axes of these arrays according to the broadcast rules, shows that they are compatible:
Image (3d array): 256 x 256 x 3
Scale (1d array): 3
Result (3d array): 256 x 256 x 3
The numpy documentation includes several examples of what dimensions can and can not be broadcast together.
Matmul using Nx
As a reminder, we defined these tensors further back in the notebook.
tr = Nx.dot(x_valid, weights)
Using the default BinaryBackend, the above dot() function returns in about 28 seconds on our Linux computer. Not nearly as quick as the broadcast example in the Fast.ai course.
Let’s explore how this same matrix multiplication works for different backends next. To keep things simple and focused, we’ll stop this notebook here and create different notebooks to focus on Nx on
XLA using the CPU and XLA using the GPU.
• I haven’t explored the difference between Nx.dot and Nx.multiply. In light of the what we learned so far, when would multiply be more appropriate?
• Need to explore swapping out backends.
• Demonstrate EXLA CPU backend, speed improvements vs BinaryBackend. Demonstrate EXLA GPU backend. Would like to demonstrate how to switch from CPU to GPU and back to CPU in the same notebook.
• Would like to demonstrate TorchScript backend. TorchX hasn’t had as much focus as the XLA backend. The dynamic UNet from Fast.ai probably won’t work well in XLA. If so, then TorchX might prove
fastai, livebook, axon, foundations, matrix_multiplication, deep_learning | {"url":"http://alongtheaxon.com/blog/01a_matmul_using_CPU","timestamp":"2024-11-11T16:02:15Z","content_type":"text/html","content_length":"143622","record_id":"<urn:uuid:a2bf5e5e-0bc6-48c9-a4e5-fd297a10dd54>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00369.warc.gz"} |
Computer Scientists Expand the Frontier of Verifiable Knowledge | Quanta Magazine
Imagine someone came along and told you that they had an oracle, and that this oracle could reveal the deep secrets of the universe. While you might be intrigued, you’d have a hard time trusting it.
You’d want some way to verify that what the oracle told you was true.
This is the crux of one of the central problems in computer science. Some problems are too hard to solve in any reasonable amount of time. But their solutions are easy to check. Given that, computer
scientists want to know: How complicated can a problem be while still having a solution that can be verified?
Turns out, the answer is: Almost unimaginably complicated.
In a paper released in April, two computer scientists dramatically increased the number of problems that fall into the hard-to-solve-but-easy-to-verify category. They describe a method that makes it
possible to check answers to problems of almost incomprehensible complexity. “It seems insane,” said Thomas Vidick, a computer scientist at the California Institute of Technology who wasn’t involved
in the new work.
The research applies to quantum computers — computers that perform calculations according to the nonintuitive rules of quantum mechanics. Quantum computers barely exist now but have the potential to
revolutionize computing in the future.
The new work essentially gives us leverage over that powerful oracle. Even if the oracle promises to tell you answers to problems that are far beyond your own ability to solve, there’s still a way to
ensure the oracle is telling the truth.
Until the End of the Universe
When a problem is hard to solve but easy to verify, finding a solution takes a long time, but verifying that a given solution is correct does not.
For example, imagine someone hands you a graph — a collection of dots (vertices) connected by lines (edges). The person asks you if it’s possible to color the vertices of the graph using only three
colors, such that no connected vertices have the same color.
5W Infographics for Quanta Magazine
This “three-color” problem is hard to solve. In general, the time it takes to find a three-coloring of a graph (or determine that none exists) increases exponentially as the size of the graph
increases. If, say, finding a solution for a graph with 20 vertices takes 3^20 nanoseconds — a few seconds total — a graph with 60 vertices would take on the order of 3^60 nanoseconds, or about 100
times the age of the universe.
But let’s say someone claims to have three-colored a graph. It wouldn’t take long to check whether their claim is true. You’d just go through the vertices one by one, examining their connections. As
the graph gets bigger, the time it takes to do this increases slowly, in what’s called polynomial time. As a result, a computer doesn’t take much longer to check a three-coloring of a graph with 60
vertices than it does to check a graph with 20 vertices.
“It’s easy, given a proper three-coloring, to check that it works,” said John Wright, a physicist at the Massachusetts Institute of Technology who wrote the new paper along with Anand Natarajan of
In the 1970s computer scientists defined a class of problems that are easy to verify, even if some are hard to solve. They called the class “NP,” for nondeterministic polynomial time. Since then, NP
has been the most intensively studied class of problems in computer science. In particular, computer scientists would like to know how this class changes as you give the verifier new ways to check
the truth of a solution.
The Right Questions
Prior to Natarajan and Wright’s work, verification power had increased in two big leaps.
To understand the first leap, imagine that you’re colorblind. Someone places two blocks on the table in front of you and asks whether the blocks are the same or different colors. This is an
impossible task for you. Moreover, you can’t verify someone else’s solution.
But you’re allowed to interrogate this person, whom we’ll call the prover. Let’s say the prover tells you that the two blocks are different colors. You designate one block as “Block A” and the other
as “Block B.” Then you place the blocks behind your back and randomly switch which hand holds which block. Then you reveal the blocks and ask the prover to identify Block A.
If the blocks are different colors, this couldn’t be a simpler quiz. The prover will know that Block A is, say, the red block and will correctly identify it every single time.
But if the blocks are actually the same color — meaning the prover erred in saying that they were different colors — the prover can only guess which block is which. Because of this, it will only be
possible for the prover to identify Block A 50 percent of the time. By repeatedly probing the prover about the solution, you will be able to verify whether it’s correct.
David Sella (Natarajan); Soya Park (Wright)
“The verifier can send the prover questions,” Wright said, “and maybe at the end of the conversation the verifier can become more convinced.”
In 1985 a trio of computer scientists proved that such interactive proofs can be used to verify solutions to problems that are more complicated than the problems in NP. Their work created a new class
of problems called IP, for “interactive polynomial” time. The same method used to verify the coloring of two blocks can be used to verify solutions to much more complicated questions.
The second major advance took place in the same decade. It follows the logic of a police investigation. If you have two suspects you believe committed a crime, you’re not going to question them
together. Instead, you’ll interrogate them in separate rooms and check each person’s answers against the other’s. By questioning them separately, you’ll be able to reveal more of the truth than if
you had only one suspect to interrogate.
“It’s impossible for [two suspects] to form some sort of distributed, consistent story because they simply don’t know what answers the other is giving,” Wright said.
In 1988 four computer scientists proved that if you ask two computers to separately solve the same problem — and you interrogate them separately about their answers — you can verify a class of
problems that’s even larger than IP: a class called MIP, for multi-prover interactive proofs.
With a multi-prover interactive approach, for example, it’s possible to verify three-colorings for a sequence of graphs that increase in size much faster than the graphs in NP. In NP, graph sizes
increase at a linear rate — the number of vertices might grow from 1 to 2 to 3 to 4 and so on — so that the size of a graph is never hugely disproportionate to the amount of time needed to verify its
three-coloring. But in MIP, the number of vertices in a graph grows exponentially — from 2^1 to 2^2 to 2^3 to 2^4 and so on.
As a result, the graphs are too big even to fit in the verifying computer’s memory, so it can’t check three-colorings by running through the list of vertices. But it’s still possible to verify a
three-coloring by asking the two provers separate but related questions.
In MIP, the verifier has enough memory to run a program that allows it to determine whether two vertices in the graph are connected by an edge. The verifier can then ask each prover to state the
color of one of the two connected vertices — and it can cross-reference the provers’ answers to make sure the three-coloring works.
The expansion of hard-to-solve-but-easy-to-verify problems from NP to IP to MIP involved classical computers. Quantum computers work very differently. For decades it’s been unclear how they change
the picture — do they make it harder or easier to verify solutions?
The new work by Natarajan and Wright provides the answer.
Quantum Cheats
Quantum computers perform calculations by manipulating quantum bits, or “qubits.” These have the strange property that they can be entangled with one another. When two qubits — or even large systems
of qubits — are entangled, it means that their physical properties play off each other in a certain way.
In their new work, Natarajan and Wright consider a scenario involving two separate quantum computers that share entangled qubits.
This kind of setup would seem to work against verification. The power of a multi-prover interactive proof comes precisely from the fact that you can question two provers separately and cross-check
their answers. If the provers’ answers are consistent, then it’s likely they’re correct. But two provers sharing an entangled state would seem to have more power to consistently assert incorrect
And indeed, when the scenario of two entangled quantum computers was first put forward in 2003, computer scientists assumed entanglement would reduce verification power. “The obvious reaction of
everyone, including me, is that now you’re giving more power to the provers,” Vidick said. “They can use entanglement to correlate their answers.”
Despite that initial pessimism, Vidick spent several years trying to prove the opposite. In 2012, he and Tsuyoshi Ito proved that it’s still possible to verify all the problems in MIP with entangled
quantum computers.
Natarajan and Wright have now proved that the situation is even better than that: A wider class of problems can be verified with entanglement than without it. It’s possible to turn the connections
between entangled quantum computers to the verifier’s advantage.
To see how, remember the procedure in MIP for verifying three-colorings of graphs whose sizes grow exponentially. The verifier doesn’t have enough memory to store the whole graph, but it does have
enough memory to identify two connected vertices, and to ask the provers the colors of those vertices.
With the class of problems Natarajan and Wright consider — called NEEXP for nondeterministic doubly exponential time — the graph sizes grow even faster than they do in MIP. Graphs in NEEXP grow at a
“doubly exponential” rate. Instead of increasing at a rate of powers of 2 — 2^1, 2^2, 2^3, 2^4 and so on — the number of vertices in the graph increases at a rate of powers of powers of 2 — $latex 2^
{2^1}, 2^{2^2}, 2^{2^3}, 2^{2^4}$ and so on. As a result, the graphs quickly become so big that the verifier can’t even identify a single pair of connected vertices.
5W Infographics for Quanta Magazine
“To label a vertex would take 2^n bits, which is exponentially more bits than the verifier has in its working memory,” Natarajan said.
But Natarajan and Wright prove that it’s possible to verify a three-coloring of a doubly-exponential-size graph even without being able to identify which vertices to ask the provers about. This is
because you can make the provers come up with the questions themselves.
The idea of asking computers to interrogate their own solutions sounds, to computer scientists, as advisable as asking suspects in a crime to interrogate themselves — surely a foolish proposition.
Except Natarajan and Wright prove that it’s not. The reason is entanglement.
“Entangled states are a shared resource,” Wright said. “Our entire protocol is figuring out how to use this shared resource to generate connected questions.”
If the quantum computers are entangled, then their choices of vertices will be correlated, producing just the right set of questions to verify a three-coloring.
At the same time, the verifier doesn’t want the two quantum computers to be so intertwined that their answers to those questions are correlated (which would be the equivalent of two suspects in a
crime coordinating their false alibis). Another strange quantum feature handles this concern. In quantum mechanics, the uncertainty principle prevents us from knowing a particle’s position and
momentum simultaneously — if you measure one property, you destroy information about the other. The uncertainty principle strictly limits what you can know about any two “complementary” properties of
a quantum system.
Natarajan and Wright take advantage of this in their work. To compute the color of a vertex, they have the two quantum computers make complementary measurements. Each computer computes the color of
its own vertex, and in doing so, it destroys any information about the other’s vertex. In other words, entanglement allows the computers to generate correlated questions, but the uncertainty
principle prevents them from colluding when answering them.
“You have to force the provers to forget, and that’s the main thing [Natarajan and Wright] do in their paper,” Vidick said. “They force the prover to erase information by making a measurement.”
Their work has almost existential implications. Before this new paper, there was a much lower limit on the amount of knowledge we could possess with complete confidence. If we were presented with an
answer to a problem in NEEXP, we’d have no choice but to take it on faith. But Natarajan and Wright have burst past that limit, making it possible to verify answers to a far more expansive universe
of computational problems.
And now that they have, it’s unclear where the limit of verification power lies.
“It could go much further,” said Lance Fortnow, a computer scientist at the Georgia Institute of Technology. “They leave open the possibility that you could take another step.” | {"url":"https://www.quantamagazine.org/computer-scientists-expand-the-frontier-of-verifiable-knowledge-20190523/?fbclid=IwAR3ipSoKRwKSGXG91Ia9g5V7Aj977LvmLS-EVPbqFK4yeLEL8eoXBTDMe_Q","timestamp":"2024-11-07T22:14:39Z","content_type":"text/html","content_length":"225357","record_id":"<urn:uuid:5ed361e8-748f-4fde-b0a1-8164c2f0bdad>","cc-path":"CC-MAIN-2024-46/segments/1730477028017.48/warc/CC-MAIN-20241107212632-20241108002632-00011.warc.gz"} |
O Archimedes' Theorem, also called “Archimedes' Principle” (Law of Buoyancy) refers to the experience of the great Greek physicist-mathematician: Archimedes of Syracuse.
So, from the “gravity specific”, Archimedes' theorem allows us to calculate the value of the vertical and upward force (buoyancy force) that makes a body lighter inside a fluid.
Thus, the postulate of Archimedes says:
“every body immersed in a fluid receives an upward thrust equal to the weight of the volume of the fluid displaced, for this reason, the bodies denser than water sink, while the less dense float
This explains why when we are immersed in water, whether at the beach or in the pool, the perception we have is that we are lighter in the water than outside of it, which explains the strength thrust
(E) acting in the opposite direction to the weight force (P).
THE buoyant force (impulsion) is a force hydrostatic is Vector greatness (has module, direction and direction) represented by the letter F with an arrow above the letter.
The buoyant force designates the net force exerted by the fluid on a given body.
In the International System (SI) of Units, thrust is measured by the Newton unit (N). Thus, to calculate the buoyant force, the following formula is used:
E= df. Vfd.g
from where,
df: fluid density
Vfd: fluid volume
g: Acceleration of gravity
Thus, it is important to emphasize that if the density if the body is greater than the density of the fluid, the body will sink; if the density of the body is equivalent to the density of the fluid,
the body will be in equilibrium with the fluid; and finally, if the density of the body is less than the density of the fluid, the body will float on the surface of the fluid.
In other words, if the buoyant force (E) is less than the weight force (P), the body will sink; if the buoyant force (E) has the same intensity as the weight force (P) the body will neither rise nor
fall, remaining in balance; finally, if the buoyancy force is greater than the weight force (P), the body will rise to the surface.
Note that in the International System (SI) the density of fluid is measured in kilograms per cubic meter (kg/m³), O volume in cubic meters (m³) and the acceleration of gravity in meters per second
squared (m/s²).
Read more:
• thrust formula
• Hydrostatics
• Hydrostatic Pressure
• Stevin's Theorem
• Pascal's Principle
• Physics Formulas
• Density exercises
• The Dead Sea is a lake with a lot of salt, located in the Middle East and, therefore, has a very high density. Note that the greater the density of the fluid, the greater the acting buoyancy
force; this explains why people have great facility to float in this place, since the acting buoyancy force has greater intensity than the weight force.
Test your knowledge with questions about the topic in Hydrostatic Exercises. | {"url":"https://forma-slova.com/en/articles/26456-archimedes-theorem-law-of-buoyancy","timestamp":"2024-11-10T18:13:13Z","content_type":"text/html","content_length":"14510","record_id":"<urn:uuid:32f6dbc4-a1b0-48e9-abda-40bd1f0120a1>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.61/warc/CC-MAIN-20241110170046-20241110200046-00787.warc.gz"} |
Quantum and Post-Quantum Cryptography | Computer Science Blog @ HdM Stuttgart
In a world where political activists and dissidents get persecuted by authoritarian governments, strong cryptography is more necessary than ever. But the general public benefits from it as well.
Identity theft, banking fraud and cyber bullying can happen to anybody. The most effective protection is to not make sensitive material available to anybody. Unfortunately some people have an “I have
nothing to hide” mentality. But would you post your opened mail to your garden fence? Just because most people are not doing illegal activities, some information is better kept private to stay safe
from the aforementioned crimes.
In times where government agencies but also black hat hackers have access to a wide variety of personal information, the best protections are transparent and mathematically proven encryption
algorithms. Currently we have two main forms of encryption:
Symmetric Cryptography
Two parties (in cryptography usually called Alice and Bob) want to secretly share information between them. Alice encrypts the data with a secret key and Bob can decrypt the data with the same key.
The data that is sent between them is called ciphertext. A good encryption algorithm can produce ciphertext that looks pretty much random. Changing a single bit in the data should completely change
the ciphertext as well. This makes sure that an attacker that eavesdrops on the communication channel cannot determine the secret key, even if they might know parts of the unencrypted message. So far
so good, or is it? The main problem is how to share the key between the communicating parties. They might sit on the opposite sides of the world. To overcome this issue, we have to make use of
another form of encryption.
Asymmetric Cryptography
It uses not only one, but a public and a private key. The names already imply the secrecy of these keys. To encrypt a message, Alice uses Bob’s public key which is known to everybody. Once the
message is encrypted, it can only be decrypted with the private key, which is only known to Bob. This is enforced through trapdoor functions. These are mathematical functions that can be computed
easily in one direction, but not in the reverse direction. However, with enough computational power, it is still possible! Since asymmetric encryption is generally slower than symmetric encryption,
it is only used to share a secret key, which is then used for symmetric encryption algorithms.
This has worked really well for the past decades. But although many researchers are working on keeping encryption safe and accessible for everybody, just as many work on breaking currently used
algorithms. That is why ideally we want a way of encrypting data, which is not only safe from our current most powerful computers, but from any possible computer in the future. Here Quantum
Cryptography comes into play. Correctly implemented it is unbreakable because it is protected by the laws of physics.
Quantum Cryptography
Before we learn more about this highly anticipated encryption wonderland, we have to dive a little bit into the world of quantum physics. Now a famous quote, widely attributed to Richard Feynman,
says, “If you think you understand quantum mechanics, you don’t understand quantum mechanics.” Luckily we only really need to understand one simple law of quantum mechanics. Heisenberg’s uncertainty
principle states that “the more precisely the position is determined, the less precisely the momentum is known, and conversely”. This is in regards to quantum particles, where any form of measuring
them will inevitably result in a change of their quantum state. Going a bit further, we arrive at the No-Cloning Theorem. Because our measurement of quantum particles will alter them, we cannot make
an identical copy of their quantum state. We simply don’t know what the particle was like, before we interacted with it. This is the fundamental principle behind quantum cryptography.
Quantum Key Distribution
Quantum cryptography has many different applications. The most important one and sometimes even used synonymously to quantum cryptography, is called Quantum Key Distribution. It still uses regular
symmetric encryption but the secret key is shared with quantum mechanics.
BB84 Protocol
Named after the inventors Charles Bennett and Gilles Brassard, it makes use of the polarization of light. Light consists of photons which can be sent out individually from an emitter. They can also
be given a polarization, which is the direction in which they oscillate. When they travel through a polarized filter, they interact with it depending on their orientation.
Figure 1 shows four ways of polarized photons and two different filter orientations. Photons which are aligned with the filter can pass through and be seen as light by a detector. This is registered
as a 1. When the orientations are perpendicular to each other, no light will pass through. The result is a 0. But if the orientations are at a 45° angle to each other, the output cannot be previously
determined. The photon will randomly pass through with a 50% chance. This property of photons can be used to generate a secret key between two parties.
Alice sends photons to Bob which are polarized in a random direction (out of the possible variants). Bob chooses a random filter orientation and measures the outcome. He tells Alice which filters he
used and she answers with the positions where he used the correct ones (marked in green). These have an orientation to the photon where the output is deterministic. All other photons produce a random
output and are therefore worthless. The key consists of the bits that were measured by the correct filter. Alice can also calculate the key since she knows which photons she sent and what filters Bob
used. An attacker cannot calculate the key because he doesn’t know the polarization of the original photons sent by Alice.
Figure 3 shows an eavesdropper (Eve) on the quantum channel. Eve will use a random filter and send a photon to Bob, which is polarized according to the output she measured and the filter she used.
Now Bob has a 75% chance to receive the outcome that was originally intended by Alice. But in 25% of the cases, the measured bit will differ between Alice and Bob. To find an eavesdropper on the
channel, they have to compare parts of the key, which can then obviously not be used anymore. The quantum channel also has to be authenticated. Otherwise a Man-In-The-Middle attack is possible.
Ekert Protocol
The Ekert protocol uses quantum entanglement to distribute a key. Two quantum particles can be created in such a way that they have an entangled property, like their spin. If one particle is measured
to be spin up, we know that the other one is spin down. By measuring probabilities, it can be proven that the state of an entangled property will only collapse into a fixed value once one particle is
measured. Then the other particle will immediately show the opposite value, even if the particles are light years apart. This does not violate Einstein’s postulate that information cannot be
transmitted faster than the speed of light, since it is not possible to transmit data via quantum entanglement. But it can be used to share a random key between two parties. This is the principle of
the Ekert protocol.
Quantum key exchange has been achieved through optical fibre over a distance of 400 km and through the air over 144 km. It was also tested with a satellite connection, where the satellite sent out
entangled photons. More practical applications include a bank transfer in Vienna in 2004 and the transmission of election results in Switzerland in 2007. But the main problem with using this
technology over long distances is the fact that the signal cannot be repeated or routed. A direct connection between the communicating partners is necessary. This makes it difficult to use on a
global scale.
Post-Quantum Cryptography
Properly implemented, our current ways of using cryptography on the internet are sufficient for today’s technology. But in the future, asymmetric cryptography is threatened by quantum computers. This
form of cryptography mainly relies on three mathematical problems that are currently unsolvable in polynomial time: Integer factorization, the discrete logarithm and the elliptic-curve discrete
logarithm. But Peter Shor already proposed a quantum algorithm in 1994 that can factorize integers in polynomial time. Once powerful enough quantum computers exist, current asymmetric cryptography
algorithms like RSA or the Diffie-Hellman key exchange can be broken. This is a major threat to the current way the internet works. Therefore, a lot of research has already been invested into quantum
safe encryption algorithms or Post-Quantum Cryptography.
Hash-Based Signature Systems
Digital signatures rely on asymmetric cryptography and are therefore also not future-proof. Hash functions however are safe from quantum computers, based on the current understanding. Using hashing
for digital signatures was already invented by Ralph Merkle in 1979, but there are major disadvantages to currently used signature schemes, which is why it never became popular.
To use this signature scheme securely, the private key can only be used once, since parts of the key are revealed to the receivers. The public key is a hashed version of the private key. To sign a
message, parts of the private key are sent with the message data. The receiver of the message also hashes the private key and compares it to the public key and the original message data.
Code-Based Encryption Systems
Robert McEliece invented a code-based encryption system in 1978. It uses error correcting codes. Encryption data with the public key adds errors to the ciphertext, which can be reverted with the
private key. The main disadvantages are the very large key sizes of 512 kilobit in the standard configuration. There are endeavours to reduce the key size, to make this the main public-key encryption
system for post-quantum cryptography.
Quantum cryptography makes use of the No-Cloning theorem, which postulates that quantum states cannot be identically copied. This makes it physically safe from eavesdroppers. Quantum Key Exchange
allows the distribution of keys for classic symmetric encryption algorithms like the One-Time Pad. Unfortunately, a direct connection between the communicating partners is always necessary. This
makes it difficult to implement on a global scale.
But quantum cryptography is not necessary to achieve encryption, which is safe from quantum computers. Although classic asymmetric encryption algorithms can be broken by quantum computers, new
algorithms for encryption and digital signatures are already worked upon.
Research Questions
Quantum and post-quantum cryptography still require a lot of research to overcome their problems. Quantum computers are just now starting to become a thing, and many things are still unclear about
them. These questions came to mind when researching this topic:
• How can a global quantum cryptography network be implemented?
• Can photons be propelled without interacting with them?
• Which algorithms can quantum computers implement better than conventional computers?
• Can hashing be broken by quantum computers?
Further Reading
1. Bernstein, D. et al (2009). Post-Quantum Cryptography. Springer-Verlag Berlin Heidelberg
2. https://www.technologyreview.com/s/601787/quantum-cryptographers-set-400k-distance-record/ [Accessed 2018-08-01]
3. https://www.sciencenews.org/article/global-quantum-communication-top-science-stories-2017-yir [Accessed 2018-08-01]
4. https://www.youtube.com/watch?v=ZuvK-od647c [Accessed 2018-08-01]
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Nonconservative Forces
Learning Objectives
By the end of this section, you will be able to:
• Define nonconservative forces and explain how they affect mechanical energy.
• Show how the principle of conservation of energy can be applied by treating the conservative forces in terms of their potential energies and any nonconservative forces in terms of the work they
Nonconservative Forces and Friction
Forces are either conservative or nonconservative. Conservative forces were discussed in Conservative Forces and Potential Energy. A nonconservative force is one for which work depends on the path
taken. Friction is a good example of a nonconservative force. As illustrated in Figure 1, work done against friction depends on the length of the path between the starting and ending points. Because
of this dependence on path, there is no potential energy associated with nonconservative forces. An important characteristic is that the work done by a nonconservative force adds or removes
mechanical energy from a system. Friction, for example, creates thermal energy that dissipates, removing energy from the system. Furthermore, even if the thermal energy is retained or captured, it
cannot be fully converted back to work, so it is lost or not recoverable in that sense as well.
How Nonconservative Forces Affect Mechanical Energy
Mechanical energy may not be conserved when nonconservative forces act. For example, when a car is brought to a stop by friction on level ground, it loses kinetic energy, which is dissipated as
thermal energy, reducing its mechanical energy. Figure 2 compares the effects of conservative and nonconservative forces. We often choose to understand simpler systems such as that described in
Figure 2a first before studying more complicated systems as in Figure 2b.
How the Work-Energy Theorem Applies
Now let us consider what form the work-energy theorem takes when both conservative and nonconservative forces act. We will see that the work done by nonconservative forces equals the change in the
mechanical energy of a system. As noted in Kinetic Energy and the Work-Energy Theorem, the work-energy theorem states that the net work on a system equals the change in its kinetic energy, or W[net]
= ΔKE. The net work is the sum of the work by nonconservative forces plus the work by conservative forces. That is,
W[net] = W[nc] + W[c],
so that
W[nc] + W[c] = ΔKE,
where W[nc] is the total work done by all nonconservative forces and W[c] is the total work done by all conservative forces.
Consider Figure 3, in which a person pushes a crate up a ramp and is opposed by friction. As in the previous section, we note that work done by a conservative force comes from a loss of gravitational
potential energy, so that W[c] = −ΔPE. Substituting this equation into the previous one and solving for W[nc] gives
W[nc] = ΔKE + ΔPE.
This equation means that the total mechanical energy (KE + PE) changes by exactly the amount of work done by nonconservative forces. In Figure 3, this is the work done by the person minus the work
done by friction. So even if energy is not conserved for the system of interest (such as the crate), we know that an equal amount of work was done to cause the change in total mechanical energy.
We rearrange W[nc] = ΔKE + ΔPE to obtain
KE[i] + PE[i] + W[nc] = KE[f] + PE[f].
This means that the amount of work done by nonconservative forces adds to the mechanical energy of a system. If W[nc] is positive, then mechanical energy is increased, such as when the person pushes
the crate up the ramp in Figure 3. If W[nc] is negative, then mechanical energy is decreased, such as when the rock hits the ground in Figure 2b. If W[nc] is zero, then mechanical energy is
conserved, and nonconservative forces are balanced. For example, when you push a lawn mower at constant speed on level ground, your work done is removed by the work of friction, and the mower has a
constant energy.
Applying Energy Conservation with Nonconservative Forces
When no change in potential energy occurs, applying KE[i] + PE[i] + W[nc] = KE[f] + PE[f] amounts to applying the work-energy theorem by setting the change in kinetic energy to be equal to the net
work done on the system, which in the most general case includes both conservative and nonconservative forces. But when seeking instead to find a change in total mechanical energy in situations that
involve changes in both potential and kinetic energy, the previous equation KE [i] + PE[i] + W[nc] = KE[f] + PE[f] says that you can start by finding the change in mechanical energy that would have
resulted from just the conservative forces, including the potential energy changes, and add to it the work done, with the proper sign, by any nonconservative forces involved.
Example 1. Calculating Distance Traveled: How Far a Baseball Player Slides
Consider the situation shown in Figure 4, where a baseball player slides to a stop on level ground. Using energy considerations, calculate the distance the 65.0-kg baseball player slides, given that
his initial speed is 6.00 m/s and the force of friction against him is a constant 450 N.
Friction stops the player by converting his kinetic energy into other forms, including thermal energy. In terms of the work-energy theorem, the work done by friction, which is negative, is added to
the initial kinetic energy to reduce it to zero. The work done by friction is negative, because f is in the opposite direction of the motion (that is, θ = 180º, and so cos θ = −1). Thus W[nc] = −fd.
The equation simplifies to
This equation can now be solved for the distance d.
Solving the previous equation for d and substituting known values yields
[latex]\begin{array}{lll}d&=&\frac{mv_{\text{i}}^2}{2f}\\\text{ }&=&\frac{(65.0\text{ kg})(6.00\text{ m/s})^2}{(2)(450\text{ N})}\\\text{ }&=&2.60\text{ m}\end{array}\\[/latex]
The most important point of this example is that the amount of nonconservative work equals the change in mechanical energy. For example, you must work harder to stop a truck, with its large
mechanical energy, than to stop a mosquito.
Example 2. Calculating Distance Traveled: Sliding Up an Incline
Suppose that the player from Example 1 is running up a hill having a 5.00º incline upward with a surface similar to that in the baseball stadium. The player slides with the same initial speed.
Determine how far he slides.
In this case, the work done by the nonconservative friction force on the player reduces the mechanical energy he has from his kinetic energy at zero height, to the final mechanical energy he has by
moving through distance d to reach height h along the hill, with h = d sin 5.00º. This is expressed by the equation KE + PE[i] + W[nc] = KE [f] + PE[f].
The work done by friction is again W[nc] = −fd; initially the potential energy is PE[i] = mg · 0 = 0 and the kinetic energy is [latex]\text{KE}_{\text{i}}=\frac{1}{2}mv_{\text{i}}^2\\[/latex]; the
final energy contributions are KE[f] = 0 for the kinetic energy and PE[f] = mgh = mgd sin θ for the potential energy.
Substituting these values gives
Solve this for d to obtain
[latex]\begin{array}{lll}d&=&\frac{\left(\frac{1}{2}\right)mv_{\text{i}}^2}{f+mg\sin\theta}\\&=&\frac{(0.5)(65.0\text{ kg})(6.00\text{ m/s})^2}{450\text{ N}+(65.0\text{ kg})\left(9.80\text{ m/s}^2\
right)\sin(5.00^{\circ})}\\&=&2.31\text{ m}\end{array}\\[/latex]
As might have been expected, the player slides a shorter distance by sliding uphill. Note that the problem could also have been solved in terms of the forces directly and the work energy theorem,
instead of using the potential energy. This method would have required combining the normal force and force of gravity vectors, which no longer cancel each other because they point in different
directions, and friction, to find the net force. You could then use the net force and the net work to find the distance d that reduces the kinetic energy to zero. By applying conservation of energy
and using the potential energy instead, we need only consider the gravitational potential energy mgh, without combining and resolving force vectors. This simplifies the solution considerably.
Making Connections: Take-Home Investigation—Determining Friction from the Stopping Distance
This experiment involves the conversion of gravitational potential energy into thermal energy. Use the ruler, book, and marble from the “Making Connections” section of Gravitational Potential Energy.
In addition, you will need a foam cup with a small hole in the side, as shown in Figure 6. From the 10-cm position on the ruler, let the marble roll into the cup positioned at the bottom of the
ruler. Measure the distance d the cup moves before stopping. What forces caused it to stop? What happened to the kinetic energy of the marble at the bottom of the ruler? Next, place the marble at the
20-cm and the 30-cm positions and again measure the distance the cup moves after the marble enters it. Plot the distance the cup moves versus the initial marble position on the ruler. Is this
relationship linear?
With some simple assumptions, you can use these data to find the coefficient of kinetic friction μ[k] of the cup on the table. The force of friction f on the cup is μ[k]N, where the normal force N is
just the weight of the cup plus the marble. The normal force and force of gravity do no work because they are perpendicular to the displacement of the cup, which moves horizontally. The work done by
friction is fd. You will need the mass of the marble as well to calculate its initial kinetic energy.
It is interesting to do the above experiment also with a steel marble (or ball bearing). Releasing it from the same positions on the ruler as you did with the glass marble, is the velocity of this
steel marble the same as the velocity of the marble at the bottom of the ruler? Is the distance the cup moves proportional to the mass of the steel and glass marbles?
PhET Explorations: The Ramp
Explore forces, energy and work as you push household objects up and down a ramp. Lower and raise the ramp to see how the angle of inclination affects the parallel forces acting on the file cabinet.
Graphs show forces, energy and work.
Section Summary
• A nonconservative force is one for which work depends on the path.
• Friction is an example of a nonconservative force that changes mechanical energy into thermal energy.
• Work W[nc] done by a nonconservative force changes the mechanical energy of a system. In equation form, W[nc] = ΔKE + ΔPE or, equivalently, KE[i] + PE[i] + W[nc] = KE[f] + PE[f].
• When both conservative and nonconservative forces act, energy conservation can be applied and used to calculate motion in terms of the known potential energies of the conservative forces and the
work done by nonconservative forces, instead of finding the net work from the net force, or having to directly apply Newton’s laws.
Problems & Exercises
1. A 60.0-kg skier with an initial speed of 12.0 m/s coasts up a 2.50-m-high rise as shown in Figure 7. Find her final speed at the top, given that the coefficient of friction between her skis and
the snow is 0.0800. (Hint: Find the distance traveled up the incline assuming a straight-line path as shown in the figure.)
2. (a) How high a hill can a car coast up (engine disengaged) if work done by friction is negligible and its initial speed is 110 km/h? (b) If, in actuality, a 750-kg car with an initial speed of
110 km/h is observed to coast up a hill to a height 22.0 m above its starting point, how much thermal energy was generated by friction? (c) What is the average force of friction if the hill has a
slope 2.5º above the horizontal?
nonconservative force: a force whose work depends on the path followed between the given initial and final configurations
friction: the force between surfaces that opposes one sliding on the other; friction changes mechanical energy into thermal energy
Selected Solutions to Problems & Exercises
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Equality (mathematics) explained
In mathematics, equality is a relationship between two quantities or, more generally, two mathematical expressions, asserting that the quantities have the same value, or that the expressions
represent the same mathematical object. Equality between and is written, and pronounced " equals ". In this equality, and are the members of the equality and are distinguished by calling them
left-hand side or left member, and right-hand side or right member. Two objects that are not equal are said to be distinct.
A formula such as
where and are any expressions, means that and denote or represent the same object.
For example,
are two notations for the same number. Similarly, using
set builder notation
since the two
s have the same elements. (This equality results from the
axiom of extensionality
that is often expressed as "two sets that have the same elements are equal".
The truth of an equality depends on an interpretation of its members. In the above examples, the equalities are true if the members are interpreted as numbers or sets, but are false if the members
are interpreted as expressions or sequences of symbols.
An identity, such as
means that if is replaced with any number, then the two expressions take the same value. This may also be interpreted as saying that the two sides of the equals sign represent the same
(equality of functions), or that the two expressions denote the same
(equality of polynomials).
^[3] ^[4]
The word is derived from the Latin aequālis ("equal", "like", "comparable", "similar"), which itself stems from aequus ("equal", "level", "fair", "just").^[5]
Basic properties
for every, one has .
for every and, if, then .
for every,, and, if and, then .^[6] ^[7]
Informally, this just means that if, then can replace in any mathematical expression or formula.
, if, then
For example:
□ Given real numbers, and, if, then
. (Here,
. A
unary operation
□ Given real numbers, and, if
, then
. (Here,
. A
binary operation
□ Given real-valued functions and over some variable, if
, then
$\fracg(a) = \frach(a)$
. (Here,
$f(x) = \frac$
. An operation over functions (i.e. an
), called the
, those first three properties make equality an
equivalence relation
. In fact, equality is the unique equivalence relation on
equivalence class
es are all
Equality as predicate
In logic, a predicate is a proposition which may have some free variables. When A and B are not fully specified or depend on some variables, equality is a predicate, which may be true for some values
and false for other values. Equality is a binary relation (i.e., a two-argument predicate) which may produce a truth value (true or false) from its arguments. In computer programming, equality is
called a Boolean-valued expression, and its computation from the two expressions is known as comparison.
See main article: Identity (mathematics). When A and B may be viewed as functions of some variables, then A = B means that A and B define the same function. Such an equality of functions is sometimes
called an identity. An example is
Sometimes, but not always, an identity is written with a
triple bar
An equation is the problem of finding values of some variable, called, for which the specified equality is true. Each value of the unknown for which the equation holds is called a of the given
equation; also stated as the equation. For example, the equation
has the values
as its only solutions. The terminology is used similarly for equations with several unknowns.
An equation can be used to define a set. For example, the set of all solution pairs
of the equation
forms the
unit circle
analytic geometry
; therefore, this equation is called .
An identity is an equality that is true for all values of its variables in a given domain.^[11] An "equation" may sometimes mean an identity, but more often than not, it a subset of the variable
space to be the subset where the equation is true. There is no standard notation that distinguishes an equation from an identity, or other use of the equality relation: one has to guess an
appropriate interpretation from the semantics of expressions and the context.^[12]
See also: Equation solving
In logic
In mathematical logic and mathematical philosophy, equality is often described through the following properties:^[13] ^[14] ^[15]
. It is the first of the historical three laws of thought.
(a=b)\impliesl[\phi(a) ⇒ \phi(b)r]
(with a free variable), if
, then
For example: For all real numbers and, if, then implies (here,
These properties offer a formal reinterpretation of equality from how it is defined in standard Zermelo–Fraenkel set theory (ZFC) or other formal foundations. In ZFC, equality only means that two
sets have the same elements. However, mathematicians don't tend to view their objects of interest as sets. For instance, many mathematicians would say that the expression "
" (see
) is an
abuse of notation
or meaningless. This is a more abstracted framework which is grounded in ZFC (that is, both
s can be proved within ZFC), but is closer to how most mathematicians use equality.
Note that this says "Equality implies these two properties" not that "These properties define equality"; this is intentional. This makes it an incomplete axiomatization of equality. That is, it does
not say what equality is, only what "equality" must satify. However, the two axioms as stated are still generally useful, even as an incomplete axiomatization of equality, as they are usually
sufficient for deducing most properties of equality that mathematicians care about.^[16] (See the following subsection)
If these properties were to define a complete axiomatization of equality, meaning, if they were to define equality, then the converse of the second statement must be true. The converse of the
Substitution property is the identity of indiscernibles, which states that two distinct things cannot have all their properties in common. In mathematics, the identity of indiscernibles is usually
rejected since indiscernibles in mathematical logic are not necessarily forbidden. Set equality in ZFC is capable of declairing these indiscernibles as not equal, but an equality defined by these
properties is not. Thus these properties form a strictly weaker notion of equality than set equality in ZFC. Outside of pure math, the identity of indiscernibles has attracted much controversy and
criticism, especially from corpuscular philosophy and quantum mechanics.^[17] This is why the properties are said to not form a complete axiomatization.
However, apart from cases dealing with indiscernibles, these properties taken as axioms of equality are equivalent to equality as defined in ZFC.
These are sometimes taken as the definition of equality, such as in some areas of first-order logic.^[18]
Derivations of basic properties
• Reflexivity of Equality: Given some set with a relation induced by equality (
), assume
. Then
by the Law of identity, thus
The Law of identity is distinct from reflexivity in two main ways: first, the Law of Identity applies only to cases of equality, and second, it is not restricted to elements of a set. However, many
mathematicians refer to both as "Reflexivity", which is generally harmless.^[19]
• Symmetry of Equality: Given some set with a relation induced by equality (
), assume there are elements
such that
. Then, take the formula
. So we have
. Since
by assumption, and
by Reflexivity, we have that
• Transitivity of Equality: Given some set with a relation induced by equality (
), assume there are elements
such that
. Then take the formula
. So we have
. Since
by symmetry, and
by assumption, we have that
, assume there are elements and from its
such that, then take the formula
. So we have
(a=b)\implies[(f(a)=f(a)) ⇒ (f(a)=f(b))]
by assumption, and
by reflexivity, we have that
This is also sometimes included in the axioms of equality, but isn't necessary as it can be deduced from the other two axioms as shown above.
Approximate equality
There are some logic systems that do not have any notion of equality. This reflects the undecidability of the equality of two real numbers, defined by formulas involving the integers, the basic
arithmetic operations, the logarithm and the exponential function. In other words, there cannot exist any algorithm for deciding such an equality (see Richardson's theorem).
The binary relation "is approximately equal" (denoted by the symbol
) between
real number
s or other things, even if more precisely defined, is not transitive (since many small differences can add up to something big). However, equality
almost everywhere is
A questionable equality under test may be denoted using the
Relation with equivalence, congruence, and isomorphism
See main article: Equivalence relation, Isomorphism, Congruence relation and Congruence (geometry).
Viewed as a relation, equality is the archetype of the more general concept of an equivalence relation on a set: those binary relations that are reflexive, symmetric and transitive. The identity
relation is an equivalence relation. Conversely, let R be an equivalence relation, and let us denote by x^R the equivalence class of x, consisting of all elements z such that x R z. Then the relation
x R y is equivalent with the equality x^R = y^R. It follows that equality is the finest equivalence relation on any set S in the sense that it is the relation that has the smallest equivalence
classes (every class is reduced to a single element).
In some contexts, equality is sharply distinguished from equivalence or isomorphism. For example, one may distinguish fractions from rational numbers, the latter being equivalence classes of
fractions: the fractions
are distinct as fractions (as different strings of symbols) but they "represent" the same rational number (the same point on a number line). This distinction gives rise to the notion of a quotient
Similarly, the sets
are not equal sets – the first consists of letters, while the second consists of numbers – but they are both sets of three elements and thus isomorphic, meaning that there is a bijection between
them. For example
However, there are other choices of isomorphism, such as
and these sets cannot be identified without making such a choice – any statement that identifies them "depends on choice of identification". This distinction, between equality and isomorphism, is of
fundamental importance in category theory and is one motivation for the development of category theory.
In some cases, one may consider as equal two mathematical objects that are only equivalent for the properties and structure being considered. The word congruence (and the associated symbol
) is frequently used for this kind of equality, and is defined as the quotient set of the
isomorphism class
es between the objects. In
for instance, two geometric shapes are said to be
equal or congruent
when one may be moved to coincide with the other, and the equality/congruence relation is the isomorphism classes of
between shapes. Similarly to isomorphisms of sets, the difference between isomorphisms and equality/congruence between such mathematical objects with properties and structure was one motivation for
the development of
category theory
, as well as for
homotopy type theory
univalent foundations
^[21] ^[22] ^[23]
Equality in set theory
See main article: Axiom of extensionality.
Equality of sets is axiomatized in set theory in two different ways, depending on whether the axioms are based on a first-order language with or without equality.
Set equality based on first-order logic with equality
In first-order logic with equality, the axiom of extensionality states that two sets which contain the same elements are the same set.^[24]
Incorporating half of the work into the first-order logic may be regarded as a mere matter of convenience, as noted by Lévy.
"The reason why we take up first-order predicate calculus with equality is a matter of convenience; by this we save the labor of defining equality and proving all its properties; this burden is now
assumed by the logic."^[25]
Set equality based on first-order logic without equality
In first-order logic without equality, two sets are defined to be equal if they contain the same elements. Then the axiom of extensionality states that two equal sets are contained in the same sets.^
See also
• Book: Kleene, Stephen Cole. Stephen Cole Kleene. 1967. Mathematical Logic. 2002. Dover Publications. Mineola, New York. 978-0-486-42533-7.
• Book: Lévy, Azriel . Azriel Lévy . 978-0-486-42079-0 . 2002 . 1979 . Basic set theory. Dover Publications. Mineola, New York.
• Book: Saunders Mac Lane . Saunders . Mac Lane . Garrett . Birkhoff . Garrett Birkhoff . Algebra . Third . American Mathematical Society . Providence, Rhode Island . 1999. 1967.
□ Book: Mendelson, Elliott. Elliott Mendelson. Introduction to Mathematical Logic. 1964. Van Nostrand Reinhold. New York.
• Book: Rosser, John Barkley . J. Barkley Rosser. Logic for mathematicians. Dover Publication. Mineola, New York. 2008. 1953. 978-0-486-46898-3.
• Book: Shoenfield . Joseph Robert . Joseph R. Shoenfield . Mathematical Logic . 1967 . . 2nd . 978-1-56881-135-2 . 2001.
Notes and References
1. Equation. Springer Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Equation&oldid=32613
2. Pratt, Vaughan, "Algebra", The Stanford Encyclopedia of Philosophy (Winter 2022 Edition), Edward N. Zalta & Uri Nodelman (eds.), URL: https://plato.stanford.edu/entries/algebra/#Laws
3. Web site: Definition of EQUAL . 2020-08-09 . . en . 2020-09-15 . https://web.archive.org/web/20200915001915/https://www.merriam-webster.com/dictionary/equal . live .
4. Book: Stoll, Robert R.. 1963. Set Theory and Logic. Dover Publications. San Francisco, CA. 978-0-486-63829-4.
5. Book: 3-87144-118-X . Lilly Görke . Mengen - Relationen - Funktionen . Zürich . Harri Deutsch . 4th . 1974 . Here: sect.3.5, p.103.
6. Equality axioms. Springer Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Equality_axioms&oldid=46837
7. Web site: Identity – math word definition – Math Open Reference. www.mathopenref.com. 2019-12-01.
8. Equation. Springer Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Equation&oldid=32613
9. Equation. Springer Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Equation&oldid=32613
10. Web site: Marcus . Solomon . Watt . Stephen M. . What is an Equation? . 2019-02-27 . Solomon Marcus .
11. Equality axioms. Springer Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Equality_axioms&oldid=46837
12. Deutsch, Harry and Pawel Garbacz, "Relative Identity", The Stanford Encyclopedia of Philosophy (Fall 2024 Edition), Edward N. Zalta & Uri Nodelman (eds.), forthcoming URL: https://
13. Forrest, Peter, "The Identity of Indiscernibles", The Stanford Encyclopedia of Philosophy (Winter 2020 Edition), Edward N. Zalta (ed.), URL: https://plato.stanford.edu/entries/
14. Equality axioms. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Equality_axioms&oldid=46837
15. Encyclopedia: French . Steven . 2019 . Identity and Individuality in Quantum Theory . Stanford Encyclopedia of Philosophy . 1095-5054 .
16. Equality axioms. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Equality_axioms&oldid=46837
17. Eilenberg . S. . Mac Lane . S. . 1942 . Group Extensions and Homology . Annals of Mathematics . 43 . 4 . 757–831 . registration . 10.2307/1968966 . 1968966 . 0003-486X . JSTOR.
18. Web site: Marquis . Jean-Pierre . 2019 . Category Theory . . . 26 September 2022.
19. Book: Hofmann . Martin . Twenty Five Years of Constructive Type Theory . Streicher . Thomas . 1998 . Clarendon Press . 978-0-19-158903-4 . Sambin . Giovanni . Oxford Logic Guides . 36 . 83–111 .
The groupoid interpretation of type theory . 1686862 . Thomas Streicher . Smith . Jan M. . https://books.google.com/books?id=pLnKggT_In4C&pg=PA83. | {"url":"https://everything.explained.today/Equality_(mathematics)/","timestamp":"2024-11-10T06:05:23Z","content_type":"text/html","content_length":"49212","record_id":"<urn:uuid:70694052-ad6c-4a8a-b156-ee4ac1aefb5b>","cc-path":"CC-MAIN-2024-46/segments/1730477028166.65/warc/CC-MAIN-20241110040813-20241110070813-00320.warc.gz"} |
Symmetry entities define planes of symmetry within a model so that morphs can be applied in a symmetric fashion.
Symmetries do not have an active or export state.
There are two basic symmetry groups: reflective and non-reflective. Symmetries can be combined, but you must be careful not to create confusing symmetrical arrangements. Symmetries can also be
applied to unconnected domains. In this case, the symmetric handle linking works the same as that for connected domains, but the influences between handles and nodes for non-reflective symmetries do
not extend across to all domains.
Reflective Symmetries
Reflective symmetries link handles in a symmetric fashion so that the movements of one handle will be reflected and applied to the symmetric handles. You can also use reflective symmetries to reflect
morphs performed on domains when using the alter dimensions.
Reflective symmetries are one plane, two plane, three plane, and cyclical.
One Plane
A mirror is placed at the origin perpendicular to the selected axis (default = x-axis).
Two Plane
Two mirrors are placed at the origin perpendicular to the selected axis and the subsequent axis (that is x and y, y and z, z and x) (default = x and y-axis).
Three Plane
Three mirrors are placed at the origin perpendicular to all three axes.
Two mirrors are placed along the selected axis (default = z-axis) and running through the origin with a given angle in between that is a factor of 360. The result is a wedge that is reflected a
certain number of times about the selected axis.
Reflective symmetries can be defined as either unilateral or multilateral and either approximate or enforced.
Unilateral Symmetries
One side governs the other, but not vice versa.
For example, handles created and morphs applied to handles on the positive side of the symmetry are reflected onto the other side or sides of the symmetry, but handles created or morphs applied
to handles on the other side or sides of the symmetry are not reflected.
Multilateral Symmetries
All sides govern all other sides.
For example, a handle created or a morph applied to a handle on any side is reflected to all the other sides.
Approximate Symmetries
Contain handles that are not symmetric to other handles. This option is best for asymmetrical, but similar, domains or for a cyclical symmetry applied to a mesh that sweeps through an arc but not
a full circle.
For example, handles created on any side of the symmetry are not reflected to the other sides.
Enforced Symmetries
Cannot contain handles that are not symmetric on all other sides.
For example, handles created or deleted on any side of the symmetry are created or deleted on the other sides so that the symmetry is maintained. When a reflective symmetry is created with the
enforced option, additional handles may also be created to meet the enforcement requirements.
Note: Handles created due to the enforced symmetry may not be located on any mesh, however, they will always be assigned to the nearest domain and will affect nodes in that domain.
Non-Reflective Symmetries
Non-reflective symmetries change the way that handles influence nodes as well as link the symmetric handles so that the movement of one affects the others.
The handles for a domain with non-reflective symmetry will act as if they are the shape of the symmetry type. For instance, a domain with linear symmetry causes handle movements to act on the domain
as if the handle was a line in the direction of the x-axis. A domain with circular symmetry causes handle movements to act on the domain as if the handle was a circle centered around the z-axis. The
edges of a domain affect how influences between handles and nodes are calculated. Non-reflective symmetries work best for domains that are shaped like the symmetry type and have a regular mesh. For
example, a circular symmetry works best for a round domain with a concentric mesh.
Non-reflective symmetries are linear, circular, planar, radial 2D, cylindrical, radial + linear, radial 3D, and spherical.
Handle acts as a line drawn through the handle location parallel to the selected axis (default = x-axis).
Handle acts as a circle drawn through the handle position about the selected axis (default = z-axis).
Handle acts as a plane drawn through the handle location perpendicular to the selected axis (default = x-axis).
Radial 2D
Handle acts as a ray drawn through the handle position originating from and extending perpendicular to the selected axis (default = z-axis).
Handle acts as a cylinder drawn through the handle position about the selected axis (default = z-axis).
Radial + Linear
Handle acts as a plane drawn through the handle position extending from the selected axis (default = z-axis).
Radial 3D
Handle acts as a ray drawn through the handle position originating from origin.
Handle acts as a sphere drawn through the handle position centered on the origin. | {"url":"https://2021.help.altair.com/2021.1/hwdesktop/hm/topics/pre_processing/entities/symmetries_r.htm","timestamp":"2024-11-13T16:10:39Z","content_type":"application/xhtml+xml","content_length":"61154","record_id":"<urn:uuid:ba952ccb-6ad9-4bc6-9470-4049b5a09559>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00861.warc.gz"} |
Source for wiki CyclesMedernach version 18
This site is a static rendering of the Trac instance that was used by R7RS-WG1 for its work on R7RS-small (PDF), which was ratified in 2013. For more information, see Home.
Source for wiki CyclesMedernach version 18
== Cycle type ==
Cycles are an immutable ordered, but unindexed, container type similar to circular lists. Unlike lists, however, cycles are fully bidirectional, so many of the procedures are provided in forward and reversed pairs.
=== Constructors and type conversion ===
`(cycle `''element'' ...`)`
Returns a cycle containing ''elements''. Order is preserved.
`(list->cycle `''list''`)`
`(list->cycle/reverse `''list''`)`
Returns a cycle whose elements are the elements of ''list''. Order is preserved (reversed).
`(cycle->list `''cycle''`)`
`(reversed-cycle->list `''cycle''`)`
Returns a list whose elements are those of ''cycle''. Order is preserved (reversed).
`(cycle-unfold `''continue? successor mapper seed''`)`
`(cycle-unfold/reverse `''stop? successor mapper seed''`)`
Start with an empty list. If the result of applying the predicate ''stop?'' to ''seed'' is true, apply `make-cycle` to the list and return the result. (The list need not actually be created.)
Otherwise, apply the procedure ''mapper'' to ''seed''. The value of ''mapper'' is prepended onto the list. Then get a new seed by applying the procedure ''successor'' to ''seed'', and repeat this algorithm. Convert the list to a cycle in forward (reverse) order and return the cycle.
=== Predicates ===
`(cycle? `''obj''`)`
Returns `#t` if ''obj'' is a cycle, and otherwise returns `#f`.
`(cycle-empty? `''obj''`)`
Returns `#t` if ''obj'' is an empty cycle, and otherwise returns `#f`.
`(cycle=? `''equivalence cycle,,1,, cycle,,2,,''`)`
Return `#t` if ''cycle,,1,,'' and ''cycle,,2,,'' contain the same values (in the sense of the ''equivalence'' predicate) in the same order, independent of their rotations; otherwise return `#f`.
Example: `(cycle=? eqv? (cycle 1 2 3) (3 1 2)) => t`
=== Accessors ===
`(cycle-front `''cycle''`)`
Returns the front element of ''cycle''.
Returns the back element of ''cycle''.
`(cycle-take `''cycle k''`)`
`(cycle-take/reverse `''cycle k''`)`
Returns a cycle containing the first ''k'' elements of ''cycle'' in forward (reverse) order.
`(cycle-drop `''cycle k''`)`
`(cycle-drop/reverse `''cycle k''`)`
Returns a cycle containing all but the last ''k'' elements of ''cycle'' in forward (reverse) order.
`(cycle-split-at`''cycle k''`)`
`(cycle-split-at/reverse`''cycle k''`)`
Returns two values, both cycles, containing the first ''k'' elements of ''cycle'' in forward (reverse) order and containing all but the last ''k'' elements of ''cycle'' in forward (reverse) order.
=== Rotation ===
`(cycle-step `''cycle''`)`
`(cycle-step/reverse `''cycle''`)`
Returns a cycle obtained from ''cycle'' by a rotation of a single step forward (backward).
`(cycle-rotate `''cycle k''`)`
`(cycle-rotate/reverse `''cycle k''`)`
Returns a cycle obtained from ''cycle'' by a rotation of ''k'' steps forward (backward), where ''k'' is an exact non-negative integer.
`(cycle-rotate-while `''cycle predicate''`)`
`(cycle-rotate-while/reverse `''cycle predicate''`)`
Returns two values: a cycle obtained from ''cycle'' by a forward (backward) rotation of as many steps as possible while the value of `cycle-front` satisfies `predicate`, and the number of steps.
`(cycle-rotate-until `''cycle predicate''`)`
`(cycle-rotate-until/reverse `''cycle predicate''`)`
Returns two values: a cycle obtained from ''cycle'' by a forward (backward) rotation of as few steps as possible until the value of `cycle-back` satisfies `predicate`, and the number of steps.
=== The whole cycle ===
`(cycle-length `''cycle''`)`
Returns the number of elements in ''cycle''.
`(cycle-reverse `''cycle''`)`
Return a cycle containing the same elements as this cycle but in reverse order. Navigating a reversed cycle forward is the same as navigating the original cycle backward.
`(cycle-count `''cycle predicate''`)`
Returns the number of elements of ''cycle'' which satisfy ''predicate''.
`(cycle-append `''cycle ...''`)`
`(cycle-append/reverse `''cycle ...''`)`
Returns a cycle containing all the elements of ''cycles'' in the order given, each in forward (reverse) order. Note that `cycle-append/reverse` is not the same as appending the ''cycles'' and reversing the result.
`(cycle-zip `''stop? cycle'' ...`)`
Returns a cycle of lists (not cycles) which contain the respective elements of each ''cycle''. The predicate ''stop?'' is invoked on each such list before it is added to the result, and when it returns true, the procedure terminates.
=== Mapping and folding on elements ===
`(cycle-map `''proc n cycle'' ...`)`
`(cycle-map/reverse `''proc n cycle'' ...`)`
It is an error unless ''proc'' is a procedure taking as many arguments as there are ''cycles'' and returning a single value. `cycle-map` applies ''proc'' to the elements of the cycle(s) in forward (reverse) order ''n'' times and returns a cycle of the corresponding results.
`(cycle-for-each `''proc n cycle'' ...`)`
`(cycle-for-each/reverse `''proc n cycle'' ...`)`
It is an error unless ''proc'' is a procedure taking as many arguments as there are ''cycles''. `cycle-for-each` applies ''proc'' to the elements of the cycle(s) in forward (reverse) order ''n'' times and discards any results. Returns an unspecified value.
`(cycle-fold `''proc nil n cycle'' ...`)`
`(cycle-fold/reverse `''proc nil n cycle'' ...`)`
It is an error unless ''proc'' is a procedure taking as many arguments as there are ''cycles'', plus one additional argument, and returning a single value. `cycle-fold` applies ''proc'' ''n'' times to the elements of the cycle(s) in forward (reverse) order and the value previously returned by ''proc''. On the first call to ''proc'', the additional argument is ''nil''. Returns the result of the final call to ''proc''.
== Filtering and partitioning ==
`(cycle-filter `''cycle predicate''`)`
Returns a cycle containing those elements which satisfy ''predicate''. Order is preserved.
`(cycle-remove `''cycle predicate''`)`
Returns a cycle containing those elements which do not satisfy ''predicate''. Order is preserved.
`(cycle-partition `''cycle predicate''`)`
Returns two values, a cycle containing those elements which satisfy ''predicate'', and another cycle containing those elements which do not. Order is preserved.
== Searching ==
`(cycle-any `''cycle predicate''`)`
If any element of ''cycle'' satisfies ''predicate'', the result of ''predicate'' is returned, and `#f` otherwise.
`(cycle-every `''cycle predicate''`)`
If any element of ''cycle'' does not satisfy ''predicate'', `#f` is returned, and `#t` otherwise.
`(cycle-find `''cycle predicate''`)`
`(cycle-find/reverse `''cycle predicate''`)`
Returns the first element of ''cycle'' that satisfies ''predicate'', searching in forward (reverse) order, and `#f` if there is none. Note that it is not possible to use these procedures to determine if a cycle contains `#f`.
`(cycle-take-while `''cycle pred''`)`
`(cycle-take-while/reverse `''cycle pred''`)`
Returns a list containing the first (last) elements of ''cycle'' that satisfy ''pred''.
`(cycle-drop-while `''cycle pred''`)`
`(cycle-drop-while/reverse `''cycle pred''`)`
Returns a list containing all but the first (last) elements of ''cycle'' that satisfy ''pred''.
2015-10-17 03:36:27 | {"url":"https://small.r7rs.org/wiki/CyclesMedernach/18/source.html","timestamp":"2024-11-06T04:29:35Z","content_type":"text/html","content_length":"9410","record_id":"<urn:uuid:1eec166a-0281-4282-90bb-5421934ef7cf>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00648.warc.gz"} |
MUHAMMAD IBN MUHAMMAD AL-FULLANI AL-KISHWANI, GREAT AFRICAN MATHEMATICIAN IN THE EARLY 1700's
Celebrating our African historical personalities,discoveries, achievements and eras as proud people with rich culture, traditions and enlightenment spanning many years.
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MUHAMMAD IBN MUHAMMAD AL-FULLANI AL-KISHWANI, GREAT AFRICAN MATHEMATICIAN IN THE EARLY 1700's
The history of mathematics in the world cannot be full without metioning the great contribution of the early black mathematician especially those from the medieval black Africans. Among these African
mathematicians was Muhammad ibn Muhammad al-Fullani al-Kishwani. Unlike the 18th century Ghanaian Wilhem Anton Amoo, the great philosopher who worked and live in Europe, al-Fullani al-kishwani spent
all his life and career in the Middle East.He was a ma of many talents. He was mathematician, astronomer, mystic and astrologer. He was a Fulani and the Fulanis were the first people to convert to
Islam. He traveled to Egypt and in 1732 he wrote a mathematical scholarly manuscript (in Arabic) of procedures for constructing magic squares up to the order 11.
Muhammad is noted for saying “work in secret” and for saying “Do not give up, for that is ignorance and not according to the rules of this art. Those who know the arts of war and killing cannot
imagine the agony and pain of a practitioner of this honorable science. Like the lover, you cannot hope to achieve success without infinite perseverance.Muhammad died in Cairo in 1741
Ancient fulani farmers
Some historians believe the Fulani emerged from a prehistoric pastoral group that originated in the upper Nile region around 3500 B.C. As the climate of the Sahara grew increasingly harsh, population
pressures drove them to migrate slowly west and south in search of better grazing lands. By the eleventh century the Fulani emerged as a distinct people group in the Sénégambia Valley. Over the
next 400 years they journeyed back east, but south of the Sahara, which had become an inhospitable desert.
Traditionally most Fulani are shepherds or cattle herders, but over time some settled down and, by the nineteenth century, had established a series of kingdoms between Sénégal and Cameroon. The
Fulani have myths about how the nomads and settled rulers emerged..
Muhammad: a Life in Math, Magic, and Religion.
Have you ever wondered how mathematics, magic, and religion are all connected? Look no further than the work of Muhammad ibn Muhammad al-Fullani al-Kishnawi, of Katsina (now Nigeria). Although not
much is known about Muhammad’s life, what we do have are his quotes and written words that reveal to us what type of person and mathematician he became. We also know what type of math Muhammad worked
on through the reading of Africa Counts.
There is still debate as to what year Muhammad was born, however, we do know that his time was spent creating a new way to develop magic squares and completing the five pillars of Islam. His multi
talents as an astronomer, mathematician, mystic, and astrologer helped him during his prolific career. As a member of the Fulani people, he was one of the first groups to be converted to Islam. The
Fulani people have a history as nomadic herders and traders; they also have made an impact on politics and economics throughout West Africa. Additionally, the Fulani people are very independent and
competitive. They have used Islam as well as their competitive spirit to acquisition new lands around present day Nigeria.
Because of Muhammad’s faith, he spent a large portion of his life in the Middle East completing his duties as a devoted Muslim. It is because of this devotion to Islam, that he is recorded as saying,
“work in secret and privacy. The letters are in God’s safekeeping. God’s power is in his names and his secrets, and if you enter his treasury you are in God’s privacy, and you should not spread God’s
secrets indiscriminately.” This quote from Muhammad clearly symbolizes the first pillar of Islam by stating that any inspiration that is given to you by God, stays between you and God until another
is found worthy of this inspiration. This leads us to the conclusion that Muhammad worked independently and led his students to do the same. After completing the fifth pillar of Islam, which is the
pilgrimage to Mecca, Muhammad traveled to Egypt. While there, in 1732, he wrote a manuscript in Arabic about how to complete magic squares of up to an order of eleven. Unfortunately, Muhammad ibn
Muhammad died in Cairo in 1741 before returning to Katsina.
Does it bother you when you believe you have mastered a concept only to discover you have not even come close? Do not worry because some things are not always the way they appear. In the words of
Muhammad, “Do not give up, for that is ignorance and not according to the rules of this art. Those who know the arts of war and killing cannot imagine the agony and pain of a practitioner of this
honorable science. Like the lover, you cannot hope to achieve success without infinite perseverance.” This quote describes the pain and suffering of someone who does not live up to his full potential
by giving up. Muhammad’s statement reveals to us the quality of his work as a mathematician. He was not only devoted to the art of mathematics, but Muhammad wanted his students to understand and join
him in God’s privacy. This could not be achieved without time and energy, devotion, and practice. Without a doubt, giving up is not an option.
Curious as to what this has to do with math, magic, and religion? The answer goes back centuries to a divine turtle Lo Shu in ancient China. On the back of this divine turtle, appeared this
configuration of numbers:
Notice anything magical about this square? Look closely and you will find that all rows, all columns, and the two main diagonals sum to fifteen. This arrangement of numbers in which the columns,
rows, and main diagonals sum to the same number is known as magical squares. For instance, the row consisting of four plus nine plus two is equal to the column of four plus three plus eight, which is
equal to the diagonal of two plus five plus eight. All of these sums are equal to fifteen. The mysterious number fifteen is known as the magical constant. Muhammad’s work in the mathematical arts
consisted of developing a system to come up with higher order magical squares. The order of a magic square is found by counting the number of rows and columns. For example, the magic square that
appeared on the divine turtle Lo Shu, above is of order three. All magic squares have an odd order. The odd order is necessary because an even order square does not comply with every property of a
magic square. For example, one can have an even order in which the columns and rows add to the same. However, the diagonals of the square will not sum to the same magical constant. The numbers will
repeat themselves, and in a true magical square the numbers are used only once. The numbers used in a magic square can be found by multiplying the number of rows by the number of columns. This is
also the same as squaring the order, which is found by counting the number of rows or columns. For instance, if there is a three by three magic square, you will use numbers one through nine.
Muhammad came up with a formula to find the magical constant, the number that is the sum of the rows, columns, and diagonals and a formula to find the middle square. The formula for finding the
magical constant is n(n^2 + 1)/2, where n is equal to the order of the magic square. The second formula that Muhammad developed was
(n^2 + 1)/2. Once again, n is the order of the square and in this formula we can derive the middle number.
Muhammad’s work on magic squares was a beginning to group theory. By group we mean that a set of elements is closed, associative, contains an identity, and contains inverses for each element.
Muhammad noticed that you could perform certain operations such as reflection about an axis or rotations of up to any degree and not change the properties of the square. This meant that out of one
simple square one could now generate a finite number of magic squares and the properties would still hold true. For example, the following magic squares are the same square as above reflected about
the x-axis and rotated ninety degrees.
This square is rotated about the
This square is rotated about an
Ninety-degree angle.
Muhammad proved that combinations of these two reflections are the dihedral group. In other words these two reflections generate the rest of the group. In this case generate means that all
combinations of these two reflections produce a finite number of elements. There are eight distant elements in this group. They include the identity and its inverse and the inverse of every other
element. This group is also associative and is closed under the compositions. Only the square position is reflected, not the numbers. This is so you do not end-up with an E for a three.
Although Muhammad ibn Muhammad al-Fullani al-Kishnawi was not a minority in either race or religion in the western part of Africa, he was considered a minority because of his career as a
mathematician. Also in the mathematical world he was one of the few who were not Anglo-Saxon or Christian. Ideas that included people of African decent could not do mathematical problems, and were
intellectually inferior kept on in the minds of Anglo-Saxons until recently. Despite this Muhammad never once gave up. He persevered through it all, never giving in to the pressures of being a
minority in both race and religion. Muhammad showed the people of the time as well as today that no matter what race, ethnicity, or religion, you should not let this stand in the way of what you want
to do with your life. If Muhammad had let the issues of multiculturalism get in the way he would have never developed the mathematical formulas and concepts of group theory that are still used
hundreds of years later.
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8 comments | {"url":"https://kwekudee-tripdownmemorylane.blogspot.com/2012/07/muhammad-ibn-muhammad-al-fullani-al_28.html","timestamp":"2024-11-07T15:39:04Z","content_type":"text/html","content_length":"147607","record_id":"<urn:uuid:fc5babc8-01b2-454f-880b-bf1402b3bb22>","cc-path":"CC-MAIN-2024-46/segments/1730477028000.52/warc/CC-MAIN-20241107150153-20241107180153-00811.warc.gz"} |
Complexity of DSP magazine articles - Opinions requested
8 years ago
●9 replies●
latest reply 8 years ago
285 views
The IEEE Signal Processing Magazine has a monthly column titled: "Lecture Notes". Those Lecture Notes articles are intended to be "easily accessible, being proper for our DSP students and young
The magazine's January 2017 issue contains a 'Lecture Notes' article titled: "Compressive Privacy: From Information/Estimation Theory to Machine Learning". Counting algebra expressions such as 'i <
L' and
as equations, that Lecture Note contains
* 57 standalone equations,
* approximately 72 equations embedded in the text, and
* approximately 10 equations embedded in the figure captions
for a Total of roughly 140 mathematical equations in one Lecture Note article!!
It seems to me that 140 equations in one magazine article are too many equations for a reader to "keep track of." What's your opinion?
[ - ]
Reply by ●January 26, 2017
No doubt, it is overwhelming much.
One of the reasons I dropped my subscription (I was paying it myself), because not really much was of practical use.
OK, the title is "lecture notes". Probably targeting people more interested in the academic aspect of DSP, rather than in the "mundane" practical implementation for solving real-life problems.
Just an opinion.
[ - ]
Reply by ●January 26, 2017
Good morning,
I think this is a somewhat common ailment of most technical articles today. Non-technical articles too. I get the sense that the authors have the need to treat every aspect of their topic to ensure
that the article has enough "hooks" in order to attract readers and, perhaps, highlight their intellectual "chops" or prowess.
Sometimes these articles are useful starting points for people new to a field, a guide to the terminology and ideas. In these cases the authors have been careful to adhere to this goal and have
written the article appropriately, expressing the general ideas of the area and including only relevant references. They also keep jargon and the number of equations to a minimum.
At times I still find myself trying to read such articles, but I'm getting better at understanding the fact that if the article is about the "100 Important Details of Feature Selection for Optimal
Machine Learning", for example, that there are better, more succinct resources available.
[ - ]
Reply by ●January 26, 2017
I love math, and the higher the equations to text ratio the better. But that is definitely because I intend to become an expert in whatever it is I'm learning about. From the description "proper
for our DSP students and young professionals" I would agree it fails, it is aimed at experts, or people with broad experience who want to become experts.
There are times when math is the most succinct language. It should only take a few equations to describe a problem. Solutions take a lot more, but an introductory article isn't going to have space
to really deal with that (unless it's 20 pages long).
I'm betting a lot of those equations are redundant. This is a style issue in a way - do you want to refer to an equation 3 paragraphs back or just repeat it so the reading flow is not interrupted?
Lots of times the same equation is written with different variables on the left - again redundant. If it's 140 independent equations, nobody can keep track of that, let alone students!
Does the math flow with the text in a way that makes sense? Is it written so you don't have to juggle 5 equations in your head at once to see how a new equation is derived? If it flows, then maybe
it isn't too much. But if it is just a lot equations which are covering a lot of ground so the article fits in as few pages as possible, it's not for students.
Dr. mike
[ - ]
Reply by ●January 26, 2017
the concepts must first be explained without any numbers then fortified/quantified with equations. Not the other way around.
[ - ]
Reply by ●January 26, 2017
Hi Mike. The article I referred to was 10.5 pages in length.
[ - ]
Reply by ●January 26, 2017
14 equations per page is too much. I suspect I would have slept through that lecture! All the technical papers I have on my desk have 5 to 7 per page, and these are not for students. As Fred
points out, "We publish to impress, no?" - sometimes impressions are negative.
[ - ]
Reply by ●January 26, 2017
A few years ago, I tried out the IEEE signal processing magazine subscription for one year, but didn't like the writing style. The articles read like journal publications which were (in my opinion)
almost impossible to follow without a solid background in the area. Many authors jumped directly into the advanced concept, and didn't really provide any practical application info.
T The UK variant (IEE computing and control) was significantly better, by providing well structured articles which were also technical, but I always felt that there was take home message.
Unfortunately, the IEE turned into the IET (in order to attract a wider audience), and as such the quality of the articles went downhill.
Rick, perhaps you can keep up the good work?
[ - ]
Reply by ●January 26, 2017
Like Kaz, I believe that the mathematics should follow physics / understanding. It's a bit like sunglasses: it improves the image only if the image is there to begin with.
I would admit that there could be situations where pure mathematical expressions can lend insights to some. I believe those occasions are rare compared with the opposite.
I have the notion that the simpler the equation, the more likely it might be generalized in a useful way. E=mc^2 comes to mind.
I can well imagine making mathematical mistakes or creating overly simplified mathematical models due to a lack of understanding. But then, much of my universe is made up of things that have to be
So, pages full of complex equations that are made to be as terse as possible are usually of no interest to me. Page charges may be one reason for things to be terse. Elegance of notation is another
reason for things to be terse. Maybe we should adopt the phrase: "Everything should be less terse as possible and no less terse" But some stuff can be read if you understand the notation. I find
myself speaking in my head: "C is contained in E such that ..... etc." Then there are the cases where there are pages and pages of matrices. Not terse and not readable really either. For
mathematicians some things might be important when constructing a proof. For practitioners of the mathematical sciences it's often nonsensical fluff. Oh! But it is impressive!! We publish to
impress, no?
Oh? No? :-)
[ - ]
Reply by ●January 26, 2017
Yeah, I agree with you Rick. To illustrate their point to students and young professionals, a magazine article meant should have a high word to equation ratio.
Of course this is simply my opinion, but I believe much of the writing should be focused on describing the main concepts through the use of analogies and metaphors rather than equations. The easy way
to write is simply plaster equations across your paper, I find that writing technical engaging papers calls for more skillful writing. The writing should be creative as well as technically sound if
you are writing to engage new young audience.
Lastly, the paper should be the catalyst that causes them to pick up a textbook. It is in the text book where the student will become more familiar with the material.
The equations may just deter the semi-interested student or young professional. Thus, by using analogies and metaphors, it offers the opportunity that, should the student not going any further than
the magazine article, they would walk away with a working super-level understanding of the topic at hand. | {"url":"https://dsprelated.com/thread/1651/complexity-of-dsp-magazine-articles-opinions-requested","timestamp":"2024-11-12T15:51:40Z","content_type":"text/html","content_length":"49355","record_id":"<urn:uuid:2cac834b-799f-4590-854f-db80a3aa85b4>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.63/warc/CC-MAIN-20241112145015-20241112175015-00475.warc.gz"} |
Elementary arithmetic functions in WPS Spreadsheet | WPS Office Academy
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Elementary arithmetic functions in WPS Spreadsheet
Uploaded time: October 8, 2021 Difficulty Beginner
Elementary arithmetic functions in WPS Spreadsheet
Elementary arithmetic functions in WPS Spreadsheet
Today I will introduce you to introductory calculation of functions, namely addition, subtraction, multiplication and division. In the table, there are three ways to enter the function, respectively,
input in the formula bar input in the cell and click Insert Function.
So how can we do elementary arithmetic in the table? We can enter the corresponding symbol for the operation in the formula bar or cells. Take this table as an example, we enter = A2 + B2 in cell C2,
press the Enter key to get the result.
Similarly, we select cell D2, enter =A2-B2, and press the Enter key to get the calculation results. Then, we select cell E2, enter“=A2*B2”, and press the Enter key to get the calculation results. The
same goes for division. Select cell F2, enter“=A2/B2”, and press the Enter key to get the result.
Of course, we can also use the method of inserting function to do calculation. If we want to do thesummation calculation, click the Formulas tab, and click theInsert Function button. In the popup
dialog box, find the SUM function, enter the range of cells that needs to be summed, and click OK to quickly complete the summation.
If we want to do the subtraction calculation, select cell D2, open the Insert Function dialog box, find the IMSUB function, enter the range of cells that needs to be operated, and click OK to get the
If we want to do the multiplication calculation, select cell E2, open the Insert Function dialog box, find the PRODUCT function, enter the range of cells that needs to be operated, and click OK to
get the result.
There is no function for performing division calculation, but the division and multiplication are inverse operations. We can convert division into multiplication. Open the Insert Function dialog box,
find the PRODUCT function, enter the range of cells that needs to be operated. Here we enter 1/B2 in the Number2 box, and click OK.
Use the mouse to select cell B2:F2, double-click the fill handle to copy the formula, then we can quickly calculate the required data in batches.
Did you get it?To be office excel advanced, you could learn how to use WPS Office Spreadsheet online in WPS Academy. | {"url":"https://academy.wps.com/academy/elementary-arithmetic-functions-in-wps-spreadsheet/1861724/","timestamp":"2024-11-13T09:28:17Z","content_type":"text/html","content_length":"167420","record_id":"<urn:uuid:bd999c47-a4ea-4d66-b411-c024509df97a>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00615.warc.gz"} |
How Many People Can a Plane Hold? (With a Breakdown by Plane Type)
How Many People Can a Plane Hold?
When you’re planning a trip, one of the first things you need to consider is how you’re going to get there. If you’re flying, you’ll need to know how many people can fit on a plane.
The number of people that a plane can hold varies depending on the type of plane. A small, single-engine plane can hold just a few people, while a large, wide-body jet can hold hundreds.
In this article, we’ll take a look at the different factors that affect how many people can fit on a plane, and we’ll provide some specific examples of how many people different types of planes can
We’ll also discuss some of the reasons why airlines choose to seat different numbers of people on their planes, and we’ll explore the impact that this has on passengers.
So, whether you’re planning a trip for yourself or a group of friends or family, read on to learn more about how many people can fit on a plane!
Plane Passenger Capacity Range
Boeing 747 416-524 13,450 mi
Airbus A380 525-853 15,200 mi
Antonov An-225 250-880 11,000 mi
Factors Affecting the Number of People a Plane Can Hold
The number of people a plane can hold is determined by a number of factors, including the size of the plane, the configuration of the seats, the maximum takeoff weight, and the range of the plane.
Size of the Plane
The size of the plane is the most important factor affecting its capacity. Larger planes can hold more people than smaller planes, simply because they have more space. The average commercial airliner
can hold up to 200 passengers, while a small private plane can only hold a handful of people.
Configuration of the Seats
The configuration of the seats can also affect the number of people a plane can hold. Planes with a more efficient seat configuration can hold more people than planes with a less efficient seat
configuration. For example, a plane with a 3-3 seating configuration can hold more people than a plane with a 2-2 seating configuration.
Maximum Takeoff Weight
The maximum takeoff weight of a plane is the maximum weight that the plane can safely take off with. This weight includes the weight of the plane itself, the weight of the passengers and cargo, and
the weight of the fuel. The maximum takeoff weight of a plane limits the number of people that the plane can hold.
Range of the Plane
The range of a plane is the maximum distance that the plane can fly without refueling. The range of a plane limits the number of people that the plane can hold because the plane needs to carry enough
fuel to reach its destination.
Different Types of Planes and Their Capacity
The capacity of a plane varies depending on the type of plane. The following is a list of different types of planes and their typical capacities:
• Single-engine planes: Single-engine planes are the smallest type of plane and can typically hold up to 4 people.
• Multi-engine planes: Multi-engine planes are larger than single-engine planes and can typically hold up to 19 people.
• Regional jets: Regional jets are mid-sized planes that can typically hold up to 100 people.
• Airliners: Airliners are the largest type of plane and can typically hold up to 200 people.
The number of people a plane can hold is determined by a number of factors, including the size of the plane, the configuration of the seats, the maximum takeoff weight, and the range of the plane.
The type of plane also plays a role in the number of people that the plane can hold. Single-engine planes are the smallest type of plane and can typically hold up to 4 people. Multi-engine planes are
larger than single-engine planes and can typically hold up to 19 people. Regional jets are mid-sized planes that can typically hold up to 100 people. Airliners are the largest type of plane and can
typically hold up to 200 people.
How Many People Can A Plane Hold?
The number of people that a plane can hold depends on a number of factors, including the type of plane, the configuration of the seats, and the weight of the cargo.
Types of Planes
There are two main types of planes: commercial airliners and private jets. Commercial airliners are designed to carry large numbers of passengers, while private jets are designed to carry a smaller
number of passengers in greater comfort.
Configuration of Seats
The configuration of the seats on a plane can also affect the number of people that it can hold. Planes with a more dense configuration, such as those with a 3-3 seating arrangement, can hold more
people than planes with a less dense configuration, such as those with a 2-2 seating arrangement.
Weight of Cargo
The weight of the cargo on a plane can also affect the number of people that it can hold. Planes that are carrying a heavy load of cargo will have less weight available for passengers.
Examples of Plane Capacities
The following table provides some examples of the capacities of different types of planes:
| Plane Type | Capacity |
| Boeing 747 | 416 passengers |
| Airbus A380 | 525 passengers |
| Boeing 737 | 149 passengers |
| Embraer E170 | 76 passengers |
| Cessna Citation CJ4 | 8 passengers |
The History of Airplane Capacity
The Wright brothers’ first plane, the Wright Flyer, could only hold one person. The first commercial airliners, which were introduced in the early 1900s, could hold a few dozen people. Today’s
airliners can hold hundreds of people.
The increase in airplane capacity over the years has been driven by a number of factors, including the development of new technologies, the increasing demand for air travel, and the deregulation of
the airline industry.
The Future of Airplane Capacity
New technologies are making it possible to build planes that can hold more people. For example, the development of composite materials has made it possible to build planes that are lighter and more
fuel-efficient, which allows them to carry more passengers.
The increasing demand for air travel is also driving the need for more capacity. As the world’s population grows, more and more people are flying. This is putting a strain on the existing air
transportation infrastructure, and it is leading to increased congestion and delays.
The deregulation of the airline industry has also contributed to the increase in airplane capacity. In the past, airlines were tightly regulated by governments. This made it difficult for airlines to
compete on price, and it limited the number of flights that were available. However, in recent years, many governments have deregulated their airline industries, which has led to increased
competition and more flights.
The number of people that a plane can hold has increased significantly over the years. This has been driven by a number of factors, including the development of new technologies, the increasing
demand for air travel, and the deregulation of the airline industry. As the world’s population continues to grow, the demand for air travel is likely to continue to increase. This will put a strain
on the existing air transportation infrastructure, and it will lead to increased congestion and delays. However, new technologies and increased competition are likely to help to increase airplane
capacity and to make air travel more affordable and accessible.
How many people can a plane hold?
The number of people that a plane can hold depends on the size of the plane. A small, single-engine plane can hold a few people, while a large, commercial airliner can hold hundreds of people.
What is the largest plane in the world?
The largest plane in the world is the Antonov An-225 Mriya. It has a wingspan of 290 feet and can hold up to 250 passengers.
What is the smallest plane in the world?
The smallest plane in the world is the Mosquito Microlight. It has a wingspan of just 13 feet and can hold one person.
How many people can a private jet hold?
The number of people that a private jet can hold depends on the size of the jet. A small, light jet can hold up to four people, while a large, heavy jet can hold up to 19 people.
How many people can a commercial airliner hold?
The number of people that a commercial airliner can hold depends on the size of the airliner. A small, regional airliner can hold up to 100 people, while a large, long-haul airliner can hold up to
500 people.
What is the average number of people that a plane holds?
The average number of people that a plane holds is around 150 people. This number can vary depending on the size of the plane and the type of flight.
the maximum number of people a plane can hold depends on a variety of factors, including the size of the plane, the number of seats, and the weight restrictions. The largest passenger plane in the
world, the Airbus A380, can hold up to 850 people. However, most commercial planes are much smaller, with a capacity of around 200 passengers. When choosing a plane, it is important to consider the
number of people you need to transport and the weight of your cargo. By understanding the factors that affect a plane’s capacity, you can make an informed decision about the best plane for your
Author Profile
Dale, in his mid-thirties, embodies the spirit of adventure and the love for the great outdoors. With a background in environmental science and a heart that beats for exploring the unexplored,
Dale has hiked through the lush trails of the Appalachian Mountains, camped under the starlit skies of the Mojave Desert, and kayaked through the serene waters of the Great Lakes.
His adventures are not just about conquering new terrains but also about embracing the ethos of sustainable and responsible travel. Dale’s experiences, from navigating through dense forests to
scaling remote peaks, bring a rich tapestry of stories, insights, and practical tips to our blog. | {"url":"https://flyfreshflight.com/how-many-people-can-a-plane-hold/","timestamp":"2024-11-02T03:06:31Z","content_type":"text/html","content_length":"72005","record_id":"<urn:uuid:8932f745-9992-4167-b6f9-89fa65573990>","cc-path":"CC-MAIN-2024-46/segments/1730477027632.4/warc/CC-MAIN-20241102010035-20241102040035-00424.warc.gz"} |
Vertex Weight Mix Modifier
Vertex Weight Mix Modifier
This modifier mixes a second vertex group (or a simple value) into the affected vertex group, using different operations.
This modifier does implicit clamping of weight values in the standard (0.0 to 1.0) range. All values below 0.0 will be set to 0.0, and all values above 1.0 will be set to 1.0.
You can view the modified weights in Weight Paint Mode. This also implies that you will have to disable the Vertex Weight Mix modifier if you want to see the original weights of the vertex group you
are editing.
Vertex Group A, B
□ A: The vertex group to affect.
□ B: The second vertex group to mix into the affected one. Leave it empty if you only want to mix in a simple value.
Invert Weights A/B
Invert the influence of the vertex group.
Default Weight A, B
□ A: The default weight to assign to all vertices not in the given vertex group.
□ B: The default weight to assign to all vertices not in the given second vertex group.
Vertex Set
Choose which vertices will be affected.
Affects all vertices, disregarding the vertex groups content.
Vertex Group A:
Affects only vertices belonging to the affected vertex group.
Vertex Group B:
Affects only vertices belonging to the second vertex group.
Vertex Group A or B:
Affects only vertices belonging to at least one of the vertex groups.
Vertex Group A and B:
Affects only vertices belonging to both vertex groups.
When using All vertices, Vertices from group B or Vertices from one group, vertices might be added to the affected vertex group.
Mix Mode
How the vertex group weights are affected by the other vertex group’s weights.
Replaces affected weights with the second group’s weights.
Adds the values of Group B to Group A.
Subtracts the values of Group B from Group A.
Multiplies the values of Group B with Group A.
Divides the values of Group A by Group B.
Subtracts the smaller of the two values from the larger.
Adds the values together, then divides by 2.
Uses the smallest weight value of VGroup A’s or VGroup B’s weights.
Uses the largest weight value of VGroup A’s or VGroup B’s weights.
Normalize Weights
Scale the weights in the vertex group to keep the relative weight but the lowest and highest values follow the full 0 - 1 range.
Here is and example of using a texture and the mapping curve to generate weights used by the Wave modifier.
The blend-file, TEST_4 scene. | {"url":"https://docs.blender.org/manual/en/3.6/modeling/modifiers/modify/weight_mix.html","timestamp":"2024-11-13T22:47:08Z","content_type":"text/html","content_length":"27629","record_id":"<urn:uuid:203eb0c3-be44-4545-8771-9d162f016740>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00229.warc.gz"} |
How to get all digraphs with loops
How to get all digraphs with loops
I'm trying to count all of the directed graphs on n vertices which have fixed in/out degree, up to isomorphism. I would like to allow loops, though not multiple edges. I can't figure out how to tell
the digraphs iterrator to include the ones with loops, though i see this is an option in the graphs iterator. I would appreciate suggestions to get around this/explanations why it is not an option.
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Let me first answer your last question: this is not an option because nobody implemented it. Sage is a free-software and it seems that the developers who worked on graphs prefered to work on graphs
than digraphs. There is no mathematical reason. If you have time to work on digraphs, please to not hesitate to contribute your code to Sage.
Regarding your first question, let me first notice that being of fixed degree is not an hereditary property (not stable if you remove an edge or a vertex), while the digraphs generator . See the
following two posts for more explanations and hints:
So, if d is the degree you want to select, let me suggest to generate all digraph (up to isomorphism) with in/out degree at most d and then filter the ones that have uniform in/out degree d.
Now, regarding loops, it is clear that if you attach loops to non-isomorphic digraphs, the resulting looped digraphs will remain non-isomorphic. Hence, for each digraph (up to isomorphism) worked
separately, you have to see how to attach loops by being careful that some different choices of vertices to attach loop on may lead to isomorphic looped digraph. Such cases appear when the digraph
has non-trivial automorphisms, and knowing the automorphism group of the graph (and its action on the graph) is enough to determine the classes of loop-attachment. For this, when G is a digraph, you
can do:
sage: G.automorphism_group()
However, since your use case has to deal with the degree constraint and you already had to filter digraphs with degree at most d (not exactly d as explained above), you can rely on this generator,
and filter both conditions simultaneously. The trick is to filter, among the digraphs of degree at most d, the digraphs whose in/out degrees are either (d,d) or (d-1,d-1), and then to attach a loop
to each vertex with degrees (d-1,d-1). The disymmetry d vs d-1 is such that there is no need to deal with the automorphism groups of the digraphs.
I hope there is enough information to find your way, do not hesitate to ask questions and to post your code once it is written.
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Lesson 15
Lots of Fruit (optional)
Lesson Purpose
The purpose of this lesson is for students to write and solve their own Put Together/Take Apart, Both Addends Unknown story problems.
Lesson Narrative
This lesson is optional because it does not address any new mathematical content standards. This lesson does provide students with an opportunity to apply precursor skills of mathematical
modeling. In previous lessons, students represented and solved Put Together/Take Apart, Both Addends Unknown story problems. This lesson builds on students’ experience in the Math Stories center.
In this lesson, students use familiar contexts to generate and solve Put Together/Take Apart, Both Addends Unknown story problems. In the second activity, students are encouraged to find all possible
solutions and use reasoning based on patterns explored in previous lessons (MP8).
When students attend to the mathematical features of a situation, adhere to mathematical constraints, make choices, and translate a mathematical answer back into the context they model with
mathematics (MP4).
Learning Goals
Teacher Facing
• Solve addition and subtraction story problems.
Student Facing
• Let’s make up story problems and solve them.
Required Materials
Materials to Gather
Materials to Copy
Required Preparation
Activity 1:
• Each group of 2 needs 1 connecting cube.
Activity 2:
• Each group of 2 needs at least 10 two-color counters.
CCSS Standards
Building Towards
Lesson Timeline
Warm-up 10 min
Activity 1 20 min
Activity 2 20 min
Lesson Synthesis 10 min
Teacher Reflection Questions
What language did students use as they made up their problems? How has the language that students use progressed throughout the unit?
Print Formatted Materials
Teachers with a valid work email address can click here to register or sign in for free access to Cool Down, Teacher Guide, and PowerPoint materials.
Student Task Statements pdf docx
Lesson Cover Page pdf docx
Teacher Guide Log In
Teacher Presentation Materials pdf docx
Blackline Masters zip
Additional Resources
Google Slides Log In
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Package org.onosproject.net.topology
• Interface Summary
Interface Description
GraphDescription Describes attribute(s) of a network graph.
LinkWeigher Entity capable of determining cost or weight of a specified topology graph edge.
PathAdminService Provides administrative abilities to tailor the path service behaviours.
PathService Service for obtaining pre-computed paths or for requesting computation of paths using the current topology snapshot.
Topology Represents a network topology computation snapshot.
TopologyCluster Representation of an SCC (strongly-connected component) in a network topology.
TopologyEdge Represents an edge in the topology graph.
TopologyGraph Represents an immutable topology graph.
TopologyListener Entity capable of receiving network topology related events.
TopologyProvider Means for injecting topology information into the core.
TopologyProviderRegistry Abstraction of a network topology provider registry.
TopologyProviderService Means for injecting topology information into the core.
TopologyService Service for providing network topology information.
TopologyStore Manages inventory of topology snapshots; not intended for direct use.
TopologyStoreDelegate Topology store delegate abstraction.
TopologyVertex Represents a vertex in the topology graph.
• Class Summary
Class Description
AbstractPathService Helper class for path service.
ClusterId Representation of the topology cluster identity.
DefaultGraphDescription Default implementation of an immutable topology graph data carrier.
DefaultTopologyCluster Default implementation of a network topology cluster.
DefaultTopologyEdge Implementation of the topology edge backed by a link.
DefaultTopologyVertex Implementation of the topology vertex backed by a device id.
GeoDistanceLinkWeight Link weight for measuring link cost using the geo distance between link vertices as determined by the element longitude/latitude annotation.
HopCountLinkWeigher Link weight for measuring link cost as hop count with indirect links being as expensive as traversing the entire graph to assume the worst.
MetricLinkWeight Link weight for measuring link cost using the link metric annotation.
TopologyEvent Describes network topology event. | {"url":"https://api.onosproject.org/2.5.1/apidocs/org/onosproject/net/topology/package-summary.html","timestamp":"2024-11-04T02:42:31Z","content_type":"text/html","content_length":"13612","record_id":"<urn:uuid:e0a4fc39-7f07-4c8b-8360-e621c6a10513>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00272.warc.gz"} |
Cogito Ergo Sum – The Philosophy of René Descartes
On March 31, 1596, French philosopher, mathematician, and writer René Descartes was born. The Cartesian coordinate system is named after him, allowing reference to a point in space as a set of
numbers, and allowing algebraic equations to be expressed as geometric shapes in a two-dimensional coordinate system. He is credited as the father of analytical geometry, the bridge between algebra
and geometry, crucial to the discovery of infinitesimal calculus and analysis. Descartes was also one of the key figures in the Scientific Revolution and has been described as an example of genius.
He has been dubbed the ‘Father of Modern Philosophy’. His Meditations on First Philosophy continues to be a standard text at most university philosophy departments.
“Of all things, good sense is the most fairly distributed: everyone thinks he is so well supplied with it that even those who are the hardest to satisfy in every other respect never desire more
of it than they already have.”,
– Rene Descartes, Discours de la Methode (1637)
Youth and Education
René Descartes was born in the Touraine, France and attended the Jesuit College of La Fleche in 1606. At the school he learned Latin, Greek and studied the philosophies of Aristotele, Plato, the
Stoics, and Cicero. Descartes also studied curiously mathematics, physics, and especially the works of Galileo Galilei.[4] Just like many of Descartes’ ancestors, he was supposed to become a lawyer,
but never actually practiced law or anything like it after graduating in 1616. Instead, Descartes became a soldier as support to Protestant Prince Maurice for some years.
A New Philosophy
One of his first influences depicted Isaac Beeckman, a mathematician and natural philosopher, who met with Descartes while stationed at Breda. According to French scholar Adrien Baillet, on the night
of 10–11 November 1619 (St. Martin’s Day), while stationed in Neuburg an der Donau, Descartes shut himself in a room with an “oven” to escape the cold. While within, he had three dreams and believed
that a divine spirit revealed to him a new philosophy. Upon exiting, he had formulated analytical geometry and the idea of applying the mathematical method to philosophy. He concluded from these
visions that the pursuit of science would prove to be, for him, the pursuit of true wisdom and a central part of his life’s work. Descartes also saw very clearly that all truths were linked with one
another so that finding a fundamental truth and proceeding with logic would open the way to all science. Descartes discovered this basic truth quite soon: his famous “I think, therefore I am“.
The Cartesian System
In these years, Descartes discovered the technique of describing lines through mathematical equations, which led to the combination of both, algebra and geometry. Algebra and analysis evolved step by
step after Descartes’ findings and the coordinate system of algebraic geometry came to be called “Cartesian coordinates” in honor to the scientist. Later on, Descartes enrolled at Leiden University,
studying mathematics and astronomy and then became teacher at Utrecht University.
“Nothing comes out of nothing.”
– Rene Descartes, Principia philosophiae, Part I, Article 49
Principles of Philosophy
In the 1620’s, René Descartes worked on a metaphysical piece on the existence of God, nature, and soul as well as tried to explain the set of parhelia in Rome. He combined both in the work Treatise
on the World, which consisted of three parts. Only two of these, The Treatise of Light and the Treastise of Man survived. The two parts gave a good illustration of the universe as a system including
all of its structures, operations, planet formations, light transmission, and the role of the human on Earth. However, Descartes abandoned his plans to publish the Treatise on the World after Galileo
was condemned. He continued publishing works on philosophy, geometry, meterology and his most famous Discours de la Méthode, demonstrating four rules of thought. Further influential works followed
after 1641, when Descartes published his Mediations on First Philosophy and his Principles of Philosophy.
Discours de la Methode
The key points of Discours de la Méthode are:
• a theory of cognition that only accepts as correct what is verified as plausible by its own step-by-step analysis and logical reflection,
• an ethics according to which the individual must behave conscientiously and morally in the sense of proven social conventions,
• a metaphysics which (by logical proof) accepts the existence of a perfect Creator-God, but leaves little room for church-like institutions,
• a physics which regards nature as regulated by God-given but generally valid laws and makes its rational explanation and thus ultimately its control the task of man.
The philosophical method formulated in detail in the Discours de la méthode of Descartes is summarized in four rules (II. 7-10):
• Scepticism: Do not believe anything that is not so clearly recognized that it cannot be called into doubt.
• Analysis: Solving difficult problems in substeps.
• Construction: progressing from simple to difficult (inductive procedure: from concrete to abstract)
• Recursion: Always check whether the examination is complete.
Cogito ergo sum
During his lifetime, Descartes is now regarded as one of the first to write about the importance of reason in natural sciences rejecting any doubtable ideas. This was illustrated in his famous phrase
‘cogito ergo sum’ (I think, therefore i am) through which he concluded that doubting the existence of a person was already the prove of one’s presence. Descartes was also known for his dualism. He
once wrote that a human body functioned like a machine with material properties and the mind, both interacting at the pineal gland. In other words, this means that the body is controlled by the mind
and vise versa.
“In order to seek truth, it is necessary once in the course of our life, to doubt, as far as possible, of all things.”
– Descartes, René, Principles of Philosophy (1644)
Laying the Foundations for Leibniz and Newton
Through his works, René Descartes was able to set the foundations of the society’s emancipation from the Church, and shifting it from the medieval to the modern period. In mathematics, Descartes was
able to lay the foundations for Leibniz and Newton to develop calculus and he discovered the law of reflection, achieving a critical contribution to the field of optics. One of Descartes’ most
enduring legacies was his development of Cartesian or analytic geometry, which uses algebra to describe geometry. He “invented the convention of representing unknowns in equations by x, y, and z, and
knowns by a, b, and c“. He also “pioneered the standard notation” that uses superscripts to show the powers or exponents. He was first to assign a fundamental place for algebra in our system of
knowledge, using it as a method to automate or mechanize reasoning, particularly about abstract, unknown quantities.
René Descartes passed away on February 11, 1650 in Stockholm. In 1663, Pope Alexander VII set his works on the ‘Index of Prohibited Books’.
Dr. Richard Brown on Descartes’ Method of Doubt.
Richard Brown, Descartes 1: The Method of Doubt, [8]
References and Further Reading:
• [4] The Galileo Affair, SciHi Blog, February 13, 2014.
• [5] Galileo Galilei and his Telescope, SciHi Blog, August 25, 2012.
• [7] Timeline for Rene Descartes, via Wikidata
• [8] Richard Brown, Descartes 1: The Method of Doubt, Richard Brown @ youtube
• [9] Gillespie, A. (2006). Descartes’ demon: A dialogical analysis of ‘Meditations on First Philosophy.’ Theory & Psychology, 16, 761–781.
• [10] Sorrell, Tom (1987). Descartes. Oxford, England: Oxford University Press
• [11] Works by or about René Descartes at Internet Archive
• [12] Herbermann, Charles, ed. (1913). . Catholic Encyclopedia. New York: Robert Appleton Company.
• [13] René Descartes at the Mathematics Genealogy Project
• [14] Biography of Rene Descartes at at MacTutor’s History of mathematics | {"url":"http://scihi.org/cogito-ergo-sum-rene-descartes/","timestamp":"2024-11-12T02:36:51Z","content_type":"text/html","content_length":"660516","record_id":"<urn:uuid:96a9555b-a29c-41a8-8299-8105a2d17cb5>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00852.warc.gz"} |
Ch 05: Partial Fractions: Mathematics FSc Part 1
Notes (Solutions) of Chapter 05: Partial Fractions, Text Book of Algebra and Trigonometry Class XI (Mathematics FSc Part 1 or HSSC-I), Punjab Textbook Board (PTB), Lahore. There are four exercises in
this chapter. Please see the main page of this chapter for MCQs and important question at here.
Please select the Exercise from the list given below.
Here is the list of all exercises of Chapter 05
Here are previous and next chapters | {"url":"https://www.mathcity.org/fsc/fsc_part_1_solutions/ch05/view","timestamp":"2024-11-02T11:22:50Z","content_type":"application/xhtml+xml","content_length":"25785","record_id":"<urn:uuid:4da6c430-a350-486c-8529-683f45a1d535>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00500.warc.gz"} |
Molecular electronic, vibrational and rotational motion - DiVA
angular velocities - Swedish translation – Linguee
See section I.B.1 for a periodic table view. Six or fewer heavy atoms and twenty or fewer total atoms. Exception: Versions 8 and higher have a few substituted benzenes with more than six heavy atoms.
b Recalculated as r e = [505 379.006(51) / µ r B e] 1/2 with units as given in table. Rotational constants are inversely related to moments of inertia: B = h / (8 π 2 c I) Where B is the rotational
constant (cm-1) h is Plancks constant (gm cm 2 /sec) In this video will will use the rotational constant of HI to calculate its bond length. We will follow three steps in doing this calculation.
First we will c (December 23, 1968) This paper li sts, in order of increasing value, the " B" rot ational constants of most of the linear and symmetric top molecules which have been observed by
microwave spectroscopy.
17 Jan 2018 FTIR spectroscopy was used to analyze rotational-vibrational transitions in gas- state HCl and DCl where B is called the rotational constant. Solution for B (cm-1) is the rotational
constant: 1.025*10^32 Using the formulas below to find the moment of inertia (I), and bond length in angstrom (R) Show… David explains the rotational kinematic formulas and does a couple sample These
formulas are for constant angular acceleration, not constant angular velocity. end with now the second part Part B says what was the angular velocity The angular velocity ω is not tied to the
distance from the axis of rotation r unless you want to keep Kr constant. Comment. Introduction to rotational motion review · Next lesson For instance, suppose the initial angular velocity is zero
and the angular acceleration is constant. That is, an object that is rotating at constant angular velocity will remain rotating unless it is acted upon by an external torque.
av R MERIGGIOLA · Citerat av 3 — with their advices, wisdom and constant presence, in spite of the distance. Finally, last but first in to the rotational parameters is reported on the Annex A and B.
In the case of rotational motion, the analogous calculation for power is the The reciprocal of this slope is the torque constant kM (in units of torque per unit Platform G,Fixed
displacement,Size,22,25,28,32,36,40,45,50,56,63,70,80100,Continuous pressure,250 bar,3600 psi,220 bar,3200 psi,195 bar,2800 psi,170 bar ASUB-NEPTUNE SI ZED PLANET T RANSI TING THE M2.5-DWARF G 9-40
3 tle Telescope (KELT), which is consistent with the rotation. Pumps tested at several different, constant rotational frequencies ..
Particle Astrophysics Second Edition - SINP
The curve cycloid.) (b) Determine the velocity components and the accelera-.
2.4 Rotational Spectroscopy. Theoretical Background vibration wavenumber ω in the harmonic approximation. The rotational constant B also has a dependence av S Gavrilyuk · 2009 —
Photoelectron-recoil-induced rotational excitation of the B 2Σ+ u state in N+ his skillful guidance, patience, constant support, and scientific training. I want to The rovibrational analysis of the
band yield a band origin upsilon(0) of 2779.0968(12) cm(-1) together with a value for the upper-state rotational constant B' of A + B. Time.
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Ease of translation and rotation in vertical and horizontal planes. A laser tube which is actively controlled to achieve a constant laser. av I Nakhimovski · Citerat av 26 — 3.2 Rotation of a
Material Particle due to Deformation . a = b · C bijcjkêiêk dot product of two matrices. C1,2 a constant that gives the relative stiffness between nämligen byrådirektör B 3tiöclén, fil lic D Timra
och byråassistent. K Lundqvist Ground-state rotational bands for nuclear rotational band'Structure less constant budget, a sizeable effort has gone into the technical development during the.
Determining the rotational constant B. enables ROTATIONAL ENERGY LEVELS: Diatomic and Linear molecules. B = rotational constant. )1(. )1(. 2.
Naturvårdsverkets författningssamling
Accurate Rotational Constants of CO, HCI, and HF: Spectral Standards for the 0.3- to 6-THz (10- to 200-cm-’) Region I. G. NOLT’ AND J. V. RADOSTITZ Department of Physics, University of Oregon,
Eugene, Oregon 97403 G. DILONARDO Dipartimento di Chimica Fisica ed Inorganica, Universitd di Bologna, 40136 Bologna, Italy K. M. EVENSON, D. A. Rotational Energy. When there is no vibrational motion
we expect the molecule to have the internuclear separation (bond length) R = R. e, and the rotational energy in cm-1. or wavenumbers becomes F(J) = B. e. J(J + 1) with where B. e. is the . Rotational
Constant. and c is the speed of light and h is the Planck’s constant.
Consequently, the rotation frequencies in each vibration state are different from each other.
Sommarjobb helsingborgs kommun 2021
Det svenska kärnvapenprogrammets tekniska - OSTI.GOV
Belova, E., P. B. Chilson, S. Kirkwood, and M. T. Rietveld, A time constant of approximately zero and and are approximately constant. the center of mass and of rotation about the center of mass.
(b). The cylinder is initially slipping av S Larsson · Citerat av 2 — rotational speed, the outlet hole diameter and the binder air tank pressure were evaluated using a statistical (a) Excavation of
test pit; and (b) the column machine used in Strängnäs. (a) Other factors were kept constant as far as possible. Differential Rotation and Magnetism across the HR Diagram. | {"url":"https://hurmanblirrikuuhjj.netlify.app/77699/98137.html","timestamp":"2024-11-03T21:49:49Z","content_type":"text/html","content_length":"16144","record_id":"<urn:uuid:eb3058f0-92b4-4ad5-bded-422905ba9f5c>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00117.warc.gz"} |
Solutions to Systems of 2 Variables
Solutions to Systems of 2 Variables and 3 Variables
Recall that a solution to a system of a linear equation in $n$ variables is an ordered $n$-tuple denoted $(x_1, x_2, ..., x_n) = (s_1, s_2, ..., s_n)$, which is a point of intersections between all
of the equations in the system.
We stated earlier that a system of linear equations can have either one solution, infinitely many solutions, or no solutions. We will prove this later on, but until then, we will look at the various
cases we can run into when dealing with solutions to systems of 2 variables and of 3 variables.
Solutions to Systems of 2 Variables
Consider a system of 2 linear equations in two variables $x, y$.
No Solutions
1. Two parallel lines with no intersections between lines.
One Solution
2. Two lines with a single point of intersection.
Infinite Solutions
3. Two lines that are coincident (the same) with every point on the lines being an intersection.
Solutions to Systems of 3 Variables
Consider a system of 3 linear equations in three variables $x, y, z$. Similarly to systems of 2 variables, systems of 3 variables can have one of three different outcomes when it comes to the number
of solutions the system has, that is: no solutions, one solution, or infinitely many solutions as illustrated:
No Solutions
1. Three parallel planes with no intersections between planes.
2. Two parallel planes and one plane that intersects them.
3. No common intersection between all three planes.
**4. ** Two equations represent the same plane while the third plane is parallel to them.
One Solution
1. All three planes intersect at a common point.
Infinite Solutions
1. All three planes intersect at a common line. All points on that line are solutions to the system.
2. Two planes are coincident while the third plane intersects them at a line. All points on that line are solutions to the system.
3. All three planes are coincident and every point on the plane is a solution to the system. | {"url":"http://mathonline.wikidot.com/solutions-to-systems-of-2-variables-and-3-variables","timestamp":"2024-11-04T21:54:36Z","content_type":"application/xhtml+xml","content_length":"19185","record_id":"<urn:uuid:2cbe86c4-3d24-4829-b471-090af0c3f8b0>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.16/warc/CC-MAIN-20241104194528-20241104224528-00587.warc.gz"} |
Convert Earth's Equatorial Radius to Exameter
Please provide values below to convert Earth's equatorial radius to exameter [Em], or vice versa.
Earth's Equatorial Radius to Exameter Conversion Table
Earth's Equatorial Radius Exameter [Em]
0.01 Earth's equatorial radius 6.37816E-14 Em
0.1 Earth's equatorial radius 6.37816E-13 Em
1 Earth's equatorial radius 6.37816E-12 Em
2 Earth's equatorial radius 1.275632E-11 Em
3 Earth's equatorial radius 1.913448E-11 Em
5 Earth's equatorial radius 3.18908E-11 Em
10 Earth's equatorial radius 6.37816E-11 Em
20 Earth's equatorial radius 1.275632E-10 Em
50 Earth's equatorial radius 3.18908E-10 Em
100 Earth's equatorial radius 6.37816E-10 Em
1000 Earth's equatorial radius 6.37816E-9 Em
How to Convert Earth's Equatorial Radius to Exameter
1 Earth's equatorial radius = 6.37816E-12 Em
1 Em = 156785028911.16 Earth's equatorial radius
Example: convert 15 Earth's equatorial radius to Em:
15 Earth's equatorial radius = 15 × 6.37816E-12 Em = 9.56724E-11 Em
Popular Length Unit Conversions
Convert Earth's Equatorial Radius to Other Length Units | {"url":"https://www.unitconverters.net/length/earth-s-equatorial-radius-to-exameter.htm","timestamp":"2024-11-10T08:06:00Z","content_type":"text/html","content_length":"18971","record_id":"<urn:uuid:284e3702-f25d-4fa8-906b-b5eebd126f20>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00285.warc.gz"} |
We consider the problem to minimize the sum of piecewise-linear convex functions under both linear and nonnegative constraints. We convert the piecewise-linear convex problem into a standard form
linear programming problem (LP) and apply a primal-dual interior-point method for the LP. From the solution of the converted problem, we can obtain the solution of the … Read more
Exactly solving a Two-level Hierarchical Location Problem with modular node capacities
In many telecommunication networks a given set of client nodes must be served by different sets of facilities, providing different services and having different capabilities, which must be located
and dimensioned in the design phase. Network topology must be designed as well, by assigning clients to facilities and facilities to higher level entities, when necessary. … Read more
A mathematical programming model for assessing the design of an optimal airport topology is presented herein. It takes into account the efficient and safe taxiing of aircraft on the ground. We
balance a set of conflicting factors that depend directly on aircraft trajectories on the ground, such as the number of arriving and departing flights … Read more
An inexact interior point method for L1-regularized sparse covariance selection
Sparse covariance selection problems can be formulated as log-determinant (log-det) semidefinite programming (SDP) problems with large numbers of linear constraints. Standard primal-dual
interior-point methods that are based on solving the Schur complement equation would encounter severe computational bottlenecks if they are applied to solve these SDPs. In this paper, we consider a
customized inexact primal-dual … Read more
Minimizing irregular convex functions: Ulam stability for approximate minima
The main concern of this article is to study Ulam stability of the set of $\varepsilon$-approximate minima of a proper lower semicontinuous convex function bounded below on a real normed space $X$,
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On Computation of Performance Bounds of Optimal Index Assignment
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Optimal location of family homes for dual career couples
The number of dual-career couples with children is growing fast. These couples face various challenging problems of organizing their lifes, in particular connected with childcare and time-management.
As a typical example we study one of the difficult decision problems of a dual career couple from the point of view of operations research with a particular … Read more
Strengthening weak sandwich theorems in the presence of inconnectivity
In the paper we consider degree, spectral, and semidefinite bounds on the stability and chromatic numbers of a graph: so-called weak sandwich theorems. We examine the additivity properties of the
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Automatic tuning of GRASP with path-relinking heuristics with a biased random-key genetic algorithm
GRASP with path-relinking (GRASP+PR) is a metaheuristic for finding optimal or near-optimal solutions of combinatorial optimization problems. This paper proposes a new automatic parameter tuning
procedure for GRASP+PR heuristics based on a biased random-key genetic algorithm (BRKGA). Given a GRASP+PR heuristic with N input parameters, the tuning procedure makes use of a BRKGA in a … Read
Generalized differentiation with positively homogeneous maps: Applications in set-valued analysis and metric regularity
We propose a new concept of generalized differentiation of set-valued maps that captures the first order information. This concept encompasses the standard notions of Frechet differentiability,
strict differentiability, calmness and Lipschitz continuity in single-valued maps, and the Aubin property and Lipschitz continuity in set-valued maps. We present calculus rules, sharpen the
relationship between the Aubin … Read more | {"url":"https://optimization-online.org/2010/02/","timestamp":"2024-11-13T21:33:27Z","content_type":"text/html","content_length":"106982","record_id":"<urn:uuid:31372d48-728f-4ea8-87c1-62b494b64186>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00145.warc.gz"} |
TUTORIAL 3: Dispersion mapping as an alternative to hemodynamic lag
The test data for this tutorial consists of a pre-processed BOLD dataset acquired using a 7 tesla Philips MRI system with corresponding end-tidal CO2 traces measured using a 3rd generation RespirAct
computer controlled gas delivery system (https://thornhillmedical.com/research/respiract-ra-mr/). The respiratory protocol consisted of a baseline period followed by a hypercapnic block and then a
final baseline period. Unzip the data file and update the data path (/Data) in the included Matlab live script (.mlx) file.
TUTORIAL 3: Dispersion mapping as an alternative to hemodynamic lag
This tutorial comprises a template data analysis script that utilizes native matlab functions as well as functions from the seeVR toolbox. If you use any part of this process or functions from this
toolbox in your work please cite the following article:
... and toolbox
loadTimeseries: wrapper function to load nifti timeseries data using native matlab functions
loadMask: wrapper function to load nifti mask/image data using native matlab functions
meanTimeseries: calculates the average time-series signal intensity in a specific ROI
normTimeseries: normalizes time-series data to a specified baseline period
remLV: generates a mask that can be used to isolate and remove large vessel signal contributions
denoiseData: temporally de-noises data using a wavelet or moving-window based method (toolbox dependent)
filterData: perfomes gaussian smoothing operation on image/time-series data
lagCVR: calculates CVR and hemodynamic lag using a cross-correlation or lagged-GLM approach. Hemodynamic maps and optimized BOLD regressor are output.
fitTau: fits the dispersion time constant and associated model parameters
MRI Data Properties
This example uses BOLD data acquired using a Philips 7 tesla MRI scanner with the
following parameters:
Scan resolution (x, y): 136 133
Scan mode: MS
Repetition time [ms]: 3000
Echo time [ms]: 25
FOV (ap,fh,rl) [mm]: 217.600 68.800 192.000
Scan Duration [sec]: 369
Slices: 43
Volumes: 120
EPI factor: 47
Slice thickness: 1.6mm
Slice gap: none
In-place resolution (x, y): 1.5mm
MRI Data Processing
Simple pre-processing of MRI data was done using FSL (https://fsl.fmrib.ox.ac.uk/fsl/fslwiki/) as follows
1) motion correction (MCFLIRT)
2) calculate mean image (FSLMATHS -Tmean)
3) brain extraction (BET f = 0.2 -m)
4) tissue segmentation on BOLD image (FAST -t 2 -n 3 -H 0.2 -I 6 -l 20.0 -g --nopve -o)
Analysis Pipeline
1: Setup options structure and load data
The seeVR functions share parameters using the 'opts' structure. In the current version this is implemented as a global variable (subect to change in the future). To get started, first initialize the
global opts struct. Nifti data can be loaded using the loadTimeseries/LoadMask wrapper functions based on the native matlab nifti functions (or 'loadImageData' using provided nifti functions - see
which one works for you). This function will also initialize certain variables in the opts structure including opts.TR (repetition time). opts.dyn (number of volumes), opts.voxelsize (resolution),
and opts.info ( if using loadTimeseries then opts.headers is initialized and used by saveImageData to save timeseries, parameter maps and masks in opts.headers.ts/map/mask).
*If you use your own functions to load your imaging data, ensure that you also fill the above fields in the opts structure to ensure maintained functionality (especially opts.TR). If functions throw
errors, it is usually because a necessary option is not specified before the function call.
Startup - use 'ctrl + enter' to run individual code blocks
% initialize the opts structure (!)
addpath(genpath('ADDPATH TO seeVR toolbox'));
% Set the location for the MRI data
datadir = 'ADDPATH to DATA';
% Load motion corrected data
filename = 'BOLD_masked_mcf.nii.gz';
[BOLD,INFO] = loadTimeseries(datadir, filename);
file = 'BOLD_mean_brain_seg_0.nii.gz';
[GMmask,INFOmask] = loadMask(datadir, file);
file = 'BOLD_mean_brain_seg_1.nii.gz';
[WMmask,~] = loadMask(datadir, file);
file = 'BOLD_mean_brain_seg_2.nii.gz';
[CSFmask,~] = loadMask(datadir, file);
file = 'BOLD_mean_brain_mask.nii.gz';
[WBmask,~] = loadMask(datadir, file);
2: Setup directories for saving output
Pay special attention to the savedir, resultsdir and figdir as you can use these to organize the various script outputs.
Setup necessary directories
% specify the root directory to save results
% specify a results directory to save parameter maps - this
% directory can be changed for multiple runs etc.
opts.resultsdir = fullfile(datadir,'RESULTS'); mkdir(opts.resultsdir);
% specify a sub directory to save certain figures - this
% directory can be changed for multiple runs etc.
opts.figdir = fullfile(datadir,'FIGURES'); mkdir(opts.figdir);
3: Take a quick look at the data
Use the 'meanTimeseries' function with any mask to look at the ROI timeseries. To see the differences between GM, WM and whole-brain timeseries, use the 'normTimeseries' function before plotting
using the 'meanTimeseries' function. As you can see in the figure, there are three signal peaks corresponding to three hypercapnic periods delivered using a RespirAct system.
Visualize ROI timeseries
% Normalized BOLD data to 20 volumes in baseline period
% if no index is provided, baseline can be selected manually
nBOLD = normTimeseries(BOLD, WBmask, [5 25]);
TS1 = meanTimeseries(nBOLD, GMmask);
%establish new xdata vector for plotting
xdata = opts.TR:opts.TR:opts.TR*size(nBOLD,4);
plot(xdata, TS1, 'k'); title('time-series data');
ylabel('absolute signal'); xlabel('time (s)'); xlim([0 xdata(end)])
TS2 = meanTimeseries(nBOLD, WMmask);
set(gcf, 'Units', 'pixels', 'Position', [200, 500, 600, 160]);
4: Load end-tidal gas traces
For simplicity I have already aligned the gas traces. Simply load them from the example dataset. If loading your own data, you can use the 'resampletoTR' function to interpolate breathing data to the
same temporal resolution as your MRI data. For alignment, use the trAlign function (see tutorial 1). If you need help getting your physiological data loaded in, email me and I can write you a
5: Removing contributions using a large vessel mask
CVR data can often be weighted by contributions from large vessels that may overshadow signals of interest. Particulary, for very high resolution acquisitions at high field strength. We can modify
the whole brain mask to exclude CSF and large vessel contributions using the 'remLV' function ('remove large vessels'). This can be done by specifying the percentile threshold from which to remove
voxels, or if the appropriate toolbox is not available, a manual threshold value.
% Supply necessary options
% Define the cutoff percentile (higher values are removed; default = 98);
% If the stats toolbox is not present, then a manual threshold must
% be applied. This can vary depending on the data (i.e. trial and error).
[mWBmask] = remLV(BOLD,WBmask,opts);
6: Temporally de-noise data
To remove spikes from motion or high frequency noise these try the 'denoiseData' function. When the wavelet toolbox is available, this function uses a wavelet-based approach. Otherwise a more simple
moving window approach is applied, however this can result in a temporal shift. Low-pass filtering is another option - this can be done using the bandpassfilt.m function.
Temporal de-noising
%If no wavelet toolbox, then moving average is applied. See
opts.wdlevel = 2; %level of de-noising (higher number = greater effect)
opts.family = 'db4'; %family - can opitimize based on data
denBOLD = denoiseData(BOLD, WBmask, opts);
7: Smooth and normalize data
There are several tunable smoothing options ranging from standard gaussian to edge-preserving bilateral smoothing. For MAC users (sorry) this is restricted to gaussian but you can easily replace with
your own spatial smoothing algorithms.
Spatial smoothing
% 'guided' gaussian smoothing (for MAC see smthData function)
opts.filter = 'gaussian';
opts.spatialdim = 3; %default = 2
% Normalize data to first 15 baseline images. *NB if nornIdx is not
% supplied, you will be asked to manually select baseline indices.
nBOLD = normTimeseries(denBOLD,mWBmask,normIdx);
%use modified mask from step 5 as input to avoid smoothing large vessel
guideImg = squeeze(BOLD(:,:,:,1));
sBOLD = filterData(nBOLD, guideImg, mWBmask, opts);
8) Dispersion Mapping (including lag mapping for comparison)
Hemodynamic lag calculations (see tutorial #1) provides information on possible blood flow delays. Generally, the lag is expressed in seconds or TR but is not always reflective of true temporal
effects. Often, correlations can be weighed by the time-to-peak of the BOLD signal response. This can be seen when comparing white matter and grey matter time courses. The white matter signal rises
more slowly and this leads to longer delays. However, the onset of the white matter response is often close to that of the GM. This effect is can be termed 'signal dispersion', and may be related to
draining vein effects. See: https://doi.org/10.1016/j.neuroimage.2021.118771 and https://pubmed.ncbi.nlm.nih.gov/26126862/.
The 'fitTau.m' function can be used to calculate the voxel-wise dispersion. This function also outputs a scaling map that can be considered as a 'disperson-corrected' CVR map.
%First we will generate an initial probe using the lagCVR function.
%This probe will represent the 'fastest' intrinsic responses to the
%vasoactive stimulus and will already show inherent changes that
% occur as the CO2 bolus moves from the lungs to the brain and then
opts.glm_model = 1; %this can take a while - to explore and increase processing speed just use the corr_model. NB that turning this off will cause problems with some plots below.
% Factor by which to temporally interpolate data. Better for picking up
% lags between TR. Higher value means longer processing time and more RAM
opts.interp_factor = 1; %default is 4
% The correlation threshold is an important parameter for generating the
% optimized regressor. For noisy data, a too low value will throw an error.
% Ideally this should be set as high as possible, but may need some trial
opts.corrthresh = 0.7; %default is 0.7
% Thresholds for refinening optimized regressors. If you make this range too large
% it smears out your regressor and lag resolution is lost. When using a CO2
% probe, the initial 'bulk' alignment becomes important here as well. A bad
% alignment will mean this range is also maybe not appropriate or should be
% widened for best results. (ASSUMING TR ~1s!, careful).
opts.lowerlagthresh = -2; %default is -3
opts.upperlagthresh = 2; %default is 3
% Lag thresholds (in units of TR) for lag map creation. Since we are looking at a healthy
% brain, this can be limited to between 20-60TRs. For impariment you can consider
% to raise the opper threshold to between 60-90TRs (ASSUMING TR ~1s!, careful).
opts.lowlag = -3; %setup lower lag limit; negative for misalignment and noisy correlation
opts.highlag = 25; %setups upper lag limit; allow for long lags associated with pathology
% Perform hemodyamic analysis
% The lagCVR function saves all maps and also returns them in a struct for
% further analysis. It also returns the optimized probe when applicable.
Lets load our motion parameters to compare the effect of adding them for lag mapping
cd(datadir) % Go to our data directory
mpfilename = 'BOLD_masked_mcf.par'; % Find MCFLIRT motion parameter file
nuisance = load(mpfilename); % Load nuisance regressors (translation, rotation etc.)
drift_term = 1:1:size(sBOLD,4);
%concatenate motion params with drift term
np = [nuisance drift_term'];
[newprobe, maps] = lagCVR(GMmask, mWBmask, sBOLD, CO2trace, np, opts);
rmse = 1.2101e+03
rmse = 0.0092
rmse = 0.0057
rmse = 0.0041
passes = 1
perc = 4.9737
passes = 2
perc = 2.6942
passes = 3
perc = 2.0019
passes = 4
perc = 1.6810
passes = 1
perc = 8.8149
passes = 2
perc = 4.7155
passes = 3
perc = 3.7246
passes = 4
perc = 3.3259
passes = 5
perc = 3.1388
passes = 6
perc = 3.0378
passes = 7
perc = 2.9726
passes = 8
perc = 2.9285
passes = 9
perc = 2.9049
passes = 10
perc = 2.8881
Fitting Dispersion
Dispersion fitting has been simplified compared the the older tutorial. The option to generate a look-up table including onset is still possible using the convHRF2 and firHRF2 functions.
% The advantage of using the global opts struct is that the variables used
% for a particular processing run (including all defaults set within
% functions themselves) can be saved to compare between runs.
[tau_maps] = fitTau(CO2trace, sBOLD, mWBmask, opts)
passes = 1
passes = 2
passes = 3
passes = 4
save([opts.resultsdir,'processing_options.mat'], 'opts');
9: Plot results
Compare CVR with Lag-Corrected CVR
opts.scale = [-0.4 0.4]; % this is the expected data range. Default is [-5 5]
opts.row = 5; % this is the nr of rows. More rows means more images. Too many images will throw an error
opts.col = 6; % this is the nr of columns. This should be an even number. For 6 cols, 3 will be source images and 3 will be the param map.
opts.step = 1; % This value is multiplied by 2 in the function. So step = 2 means you jump 4 slices between images.
opts.start = 8; % this is the starting image
%rotate input images to display correctly
sourceImg = imrotate(guideImg,90); %use the guide image we used for smoothing. Alternatively a mean BOLD image or single time-point
paramMap1 = imrotate(maps.XCORR.CVR.bCVR, 90); % use the basic CVR map calculated using LagCVR
mask = imrotate(mWBmask, 90);
map = flip(brewermap(128, 'Spectral')); %use the spectral colormap (or any other you like)
paramMap2 = imrotate(maps.XCORR.CVR.cCVR, 90);
%PLOT DIFFERENCE - high values indicate where CVR estimate may have been
%improved after considering lag
opts.scale = [-0.02 0.02];
plotMap(sourceImg,mask,(paramMap2 - paramMap1),map,opts);
Compare LAG with TAU map
paramMap1 = imrotate(maps.GLM.optiReg_lags, 90); % use the basic CVR map calculated using LagCVR
paramMap2 = imrotate(tau_maps.expHRF.tau, 90); % use the basic CVR map calculated using LagCVR | {"url":"https://www.seevr.nl/tutorials/dispersion-mapping/","timestamp":"2024-11-12T07:08:06Z","content_type":"text/html","content_length":"593775","record_id":"<urn:uuid:3f24bb78-aadc-4c8d-afac-2e052b93669c>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.58/warc/CC-MAIN-20241112045844-20241112075844-00485.warc.gz"} |
A group sequential design provides interim analyses before the formal completion of a trial. The monitoring process provides possible early stopping for either positive or negative results and thus
reduces the time to complete the trial. With a specified number of stages, the design creates critical values such that at each interim analysis, a hypothesis can be rejected, accepted, or continued
to the next time point. At the final stage, a hypothesis is either rejected or accepted. Usually, the critical values are derived such that the specified overall Type I and Type II error probability
levels are maintained in the design.
For example, to test a null hypothesis
Armitage, McPherson, and Rowe (1969) showed that repeated significance tests at a fixed level on accumulating data increase the probability of obtaining a significant result under the null
hypothesis. For example, with a significance level
Pocock (1977) applied these repeated significance tests to group sequential trials with equally spaced information levels and derives a constant critical value on the standardized normal
O’Brien and Fleming (1979) proposed a sequential procedure that has boundary values decrease over the stages on the standardized normal
Wang and Tsiatis (1987), Emerson and Fleming (1989) and Pampallona and Tsiatis (1994) generalized the Pocock and O’Brien-Fleming methods to the power family, where a power parameter is used to allow
a continuous set of designs between the Pocock and O’Brien-Fleming methods.
Kittelson and Emerson (1999) extended the methods in the power family even further to the unified family, which also includes the exact triangular method. The shape and location of each of the four
boundaries can be independently specified in the unified family methods.
Whitehead and Stratton (1983) and Whitehead (1997, 2001) developed triangular methods by adapting tests for continuous monitoring to discrete monitoring. With early stopping to reject or accept the
null hypothesis in a one-sided test, the derived continuation region has a triangular shape for the score-scaled boundaries. Only elementary calculations are needed to derive the boundary values for
Whitehead’s triangular methods.
For a sequential design, you can derive the
Refer to Jennison and Turnbull (2000, pp. 5–11) for a more detailed history of group sequential methods.
The following three types of methods are available in the SEQDESIGN procedure to derive boundaries in a sequential design:
• fixed boundary shape methods, which derive boundaries with specified boundary shapes. These include the unified family method and Haybittle-Peto method.
• Whitehead methods, which adjust the boundaries from continuous monitoring for discrete monitoring
• error spending methods
You can use the SEQDESIGN procedure to specify methods from the same group for each design. A different method can be specified for each boundary separately, but all methods in a design must be from
the same group.
Fixed Boundary Shape Methods
The fixed boundary shape methods include the unified family method (Kittelson and Emerson 1999) and the Haybittle-Peto method (Haybittle 1971; Peto et al. 1976). The unified family methods derive the
boundary values with the specified boundary shape. The unified family methods include the Pocock method (Pocock 1977), the O’Brien-Fleming method (O’Brien and Fleming 1979), the power family method
(Wang and Tsiatis 1987; Emerson and Fleming 1989; Pampallona and Tsiatis 1994), and the triangular method (Kittelson and Emerson 1999). See the section Unified Family Methods for a detailed
description of the methods that use the unified family approach.
The Haybittle-Peto method uses a value of Haybittle-Peto Method for a detailed description of the Haybittle-Peto method.
Whitehead Methods
The Whitehead methods (Whitehead and Stratton 1983; Whitehead 1997, 2001) derive the boundary values by adapting the continuous monitoring tests to the discrete monitoring of group sequential tests.
The Type I error probability and power corresponding to the resulting boundaries are extremely close but differ slightly from the specified values because of the approximations used in deriving the
tests (Jennison and Turnbull 2000, p. 106). The SEQDESIGN procedure provides the BOUNDARYKEY= option to adjust the boundary value at the final stage for the exact Type I or Type II error probability
level. See the section Whitehead Methods for a detailed description of Whitehead’s methods.
Error Spending Methods
An error spending method (Lan and DeMets 1983) uses the error spending function to specify the error spending at each stage and then uses these error probabilities to derive the boundary values. You
can specify these errors explicitly or with an error spending function for these cumulative errors. See the section Error Spending Methods for a detailed description of the error spending methods.
Error spending methods derive boundary values at each stage sequentially and require much more computation than other types of methods for group sequential trials with a large number of stages,
especially for a two-sided asymmetric design with early stopping to accept
The sample size requirement for some applicable tests can also be computed in the procedure. After the actual data from a clinical trial are collected, you can then use the boundary information
created in the SEQDESIGN procedure to perform a group sequential test in the SEQTEST procedure. | {"url":"http://support.sas.com/documentation/cdl/en/statug/63347/HTML/default/statug_seqdesign_sect013.htm","timestamp":"2024-11-05T13:53:20Z","content_type":"application/xhtml+xml","content_length":"27910","record_id":"<urn:uuid:500ef31d-19b0-4357-9731-785636a44127>","cc-path":"CC-MAIN-2024-46/segments/1730477027881.88/warc/CC-MAIN-20241105114407-20241105144407-00448.warc.gz"} |
[Solved] The areas of three adjacent faces of a cuboid are x, y, z. I
The areas of three adjacent faces of a cuboid are x, y, z. If the volume is V, then V^2 will be equal to:
Answer (Detailed Solution Below)
Option 4 : (xyz)
CG TET Paper 1 Full Test 1
1.4 K Users
150 Questions 150 Marks 150 Mins
Length = l
Breadth = b
Height = h
Formula Used:
The volume of the cuboid = Length × Breadth × Height
Let the areas of three faces of cuboid as x, y, z
So, Let length of cuboid be = l
Breadth of cuboid be = b
Height of cuboid be = h
Let, x = l × b
y = b × h
z = h × l
Multiplying above three equations
⇒ xyz = l^2b^2h^2 --- (i)
Given that ‘V’ is volume of cuboid
V = l.b.h
⇒ V^2 = l2b2h2
∴ V^2 = xyz (using (i) )
Latest CG TET Updates
Last updated on Sep 19, 2024
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-> Prepare for the exam with CG TET Previous Year Papers. | {"url":"https://testbook.com/question-answer/the-areas-of-three-adjacent-faces-of-a-cuboid-are--61c41ce532eee36ad7d895d9","timestamp":"2024-11-10T06:21:58Z","content_type":"text/html","content_length":"197500","record_id":"<urn:uuid:2e1ce5e2-ba73-4019-abd7-f26b893e6100>","cc-path":"CC-MAIN-2024-46/segments/1730477028166.65/warc/CC-MAIN-20241110040813-20241110070813-00746.warc.gz"} |
Math Help
Intuitively, characteristic is saying about how big is the number on an exponential scale. If you remember your scientific notation, the -2 in log(0.0624) = -2 + 0.7952 is exactly the same thing as
0.0624 = 6.24 * 10^-2. So the idea is, in the grand scheme of every number in the world, 0.0624 is somewhere between 0.1 (characteristic = -1) and 0.01 (characteristic = -2).
Mantissa is saying how close/far away is the number from either end of the bracket - let's ignore for now.
The point about looking at characteristics, and why do we need to know logs in the first place, is that we are freakin' good at analyzing linear relations (you know, slope, intercepts, those kind of
stuff), but most things in nature behaves exponentially. Logarithms turn exponential relations into linear relations.
Anyhow, that's a generally approachable first introduction to logs. For maths folks though, we just know log by definition.
You might want to watch this video by 3blue1brown on logarithms.
(edited 2 months ago)
Intuitively, characteristic is saying about
how big is the number on an exponential scale
. If you remember your scientific notation, the
in log(0.0624) = -2 + 0.7952 is exactly the same thing as 0.0624 = 6.24 * 10^
. So the idea is,
in the grand scheme of every number in the world
, 0.0624 is somewhere between 0.1 (characteristic = -1) and 0.01 (characteristic = -2).
Mantissa is saying how close/far away is the number from either end of the bracket - let's ignore for now.
The point about looking at characteristics, and why do we need to know logs in the first place, is that
we are freakin' good at analyzing linear relations
(you know, slope, intercepts, those kind of stuff), but most things in nature behaves exponentially. Logarithms turn exponential relations into linear relations.
Anyhow, that's a generally approachable first introduction to logs. For maths folks though, we just know log by definition.
You might want to watch
this video by 3blue1brown on logarithms.
Quick Reply | {"url":"https://www.thestudentroom.co.uk/showthread.php?t=7517942&p=99847221&page=1#post99847221","timestamp":"2024-11-07T19:19:13Z","content_type":"text/html","content_length":"394835","record_id":"<urn:uuid:d09cd311-0af6-4714-b61a-097086977d8a>","cc-path":"CC-MAIN-2024-46/segments/1730477028009.81/warc/CC-MAIN-20241107181317-20241107211317-00332.warc.gz"} |
Discriminant Validity Assessment and Heterotrait-monotrait Ratio of Correlations (HTMT)
The purpose of the discriminant validity assessment is to verify that a reflective construct exhibits stronger relationships with its own indicators than with those of any other construct in the PLS
path model (Hair et al., 2022).
Brief Description
Discriminant validity assessment has become a generally accepted prerequisite for analyzing relationships between reflectively measured constructs. In the context of variance-based structural
equation modeling, such as partial least squares structural equation modeling (PLS-SEM),
• the Fornell-Larcker criterion and
• the analysis of cross-loadings are considered outdated methods for assessing discriminant validity.
Henseler, Ringle and Sarstedt (2015) demonstrated through a simulation study that these approaches do not reliably detect the lack of discriminant validity in common research situations. These
authors therefore propose an alternative approach, based on the multitrait-multimethod matrix, to assess discriminant validity: the heterotrait-monotrait ratio of correlations (HTMT). Henseler,
Ringle and Sarstedt (2015) substantiate this approach’s superior performance by means of a Monte Carlo simulation study, in which they compare the new approach to the Fornell-Larcker criterion and
the assessment of (partial) cross-loadings. Finally, they provide guidelines on how to handle discriminant validity issues in variance-based structural equation modeling.
Henseler, Ringle and Sarstedt (2015)
provide detailed explanations of the HTMT criterion for discriminant validity assessment in variance-based structural equations modeling. Also see the Appendix of the OPEN ACCESS article by
Ringle et al. (2023)
for some HTMT improvements such as HTMT+.
Discriminant Validity Assessment in SmartPLS
When running the PLS and PLSc algorithm in SmartPLS, the results report includes discriminant validity assessment outcomes, in the section “Quality Criteria”. The following results are provided:
• the Fornell-Larcker criterion,
• cross-loadings, and
• the HTMT criterion results.
We recommend using the HTMT criterion to assess discriminant validity. If the HTMT value is below 0.90, discriminant validity has been established between two reflectively measured constructs.
HTMT bootstrapping: If you like to obtain the HTMT_Inference results, you need to run the bootstrapping procedure. After choosing the -> Calculate -> Bootstrapping in SmartPLS, the start dialog
opens. It is important that you select “Complete (slower)" under the "Amount of results" option in the bootstrapping start dialog. Under "Test type", you should use the option one-tailed. Thereby,
you can test in accordance with Franke & Sarstedt (2019) if the HTMT value is significantly below the critical value of 0.9 to establish discriminant validity. In the bootstrapping results report,
locate the bootstrapped HTMT criterion results in the "Quality Criteria" section. Verify if the upper bound of "Confidence intervals bias corrected" is below the critical HTMT value.
Please note:
In SmartPLS 3.2.1 and later version, the HTMT criterion computation differs from the equation given by Henseler, Ringle and Sarstedt (2015). Instead of using the correlations between indicators,
SmartPLS uses the absolute value of the correlation between indicators. For example, when instead of using 0.1, 0.2 and -0.3, which results in an average correlation of 0 and causes problems in the
original HTMT equation, SmartPLS uses 0.1, 0.2 and 0.3, which results in an average correlation of 0.2. In consequence, the HTMT criterion is normed between 0 and 1 in SmartPLS and no issues result
from negative correlations. For further details on this version (i.e., HTMT+) see the Appendix of the OPEN ACCESS article by
Ringle et al. (2023)
Cite correctly
Please always cite the use of SmartPLS!
Ringle, Christian M., Wende, Sven, & Becker, Jan-Michael. (2024). SmartPLS 4. Bönningstedt: SmartPLS. Retrieved from https://www.smartpls.com | {"url":"https://smartpls.com/documentation/algorithms-and-techniques/discriminant-validity-assessment/","timestamp":"2024-11-12T15:10:20Z","content_type":"text/html","content_length":"670936","record_id":"<urn:uuid:efe10892-9767-4332-9219-56bca4d61cf5>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.63/warc/CC-MAIN-20241112145015-20241112175015-00574.warc.gz"} |
Fibonacci numbers and the golden ratio
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This lecture covers the definition of the Fibonacci sequence, different interpretations of Fibonacci numbers, identities related to Fibonacci numbers, the golden ratio, explicit formulas for
Fibonacci numbers, and the properties of Fibonacci numbers. The instructor explains the relationship between Fibonacci numbers and the golden ratio, provides examples of Fibonacci sequences, and
demonstrates how to compute Fibonacci numbers using generating functions.
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Does the Turnaround Tuesday trading strategy work?(Insights)
Last Updated on 23 July, 2024 by Trading System
The Turnaround Tuesday is one of the most well-known effects in the stock market. But is it a myth or fact? Well, that is what this post is set to find out. It seems buying on weakness on a Monday is
a good trading strategy. But before we go into that, let’s find out what Turnaround Tuesday means.
What does Turnaround Tuesday mean?
There are potential trading strategies based on the day of the week effect. For example, we have noticed that the stock market tends to change direction on Tuesdays and move in the opposite direction
of the move on Mondays. This is what is referred to as the Turnaround Tuesday.
If the market is down on a Monday, it is quite likely that it will rise on Tuesday and the days after it. On the other hand, if Monday is a strong up day, the days after it may have little movement
or a decline.
Turnaround Tuesday Strategy 1
The strategy is as follows:
1. Today is Monday.
2. The close must be at least 1% lower than Friday’s close.
3. Enter at the close if one and two are true.
4. Exit at the close on Tuesday.
Below is the equity curve of this simple trading strategy:
The result is as follows:
• 163 trades
• 7% average gain per trade
• 9% CAGR
• 63% win ratio
• The average gain per winner is 1.75%
• The average loss per losing trade is -1.05%
• Exposure/time in the market is 2.25%
Turnaround Tuesday Strategy 2
Here are the rules of the strategy:
1. Today is Monday.
2. The close must be lower than the open.
3. The IBS must be below 0.2.
4. Enter at the close if 1-3 are true.
5. Sell at Tuesday’s close.
See the compounded equity curve in SPY from 1993 until September 2021:
These are the parameters of the result:
• 247 trades made
• 41% average gain per trade
• CAGR is 3.5%
• The win ratio is 60%
• The average gain per winner is 1.23%
• The average loss per losing trade is -0.82%
• Exposure/time in the market is 3.4%
See the table below for the profits produced by the same strategy on the different weekdays. Note that 1 represents buying at the Monday close, 2 represents Tuesday’s close, and so on:
Obviously, the return is substantially higher buying at Monday’s close than any other day. It is twice as good as buying on the Tuesday close, not to mention the other days which are even less
As you can see, Turnaround Tuesday is by no means a myth. For the last 30 years, buying weakness on Mondays has turned out to be a profitable strategy.
Read more similar articles here on The Robust Trader or on Quantified Strategies
Video about turnaround tuesday.
What is Turnaround Tuesday in the stock market, and is it a reliable trading strategy?
Turnaround Tuesday refers to a stock market phenomenon where the market tends to change direction on Tuesdays, moving opposite to the direction on Mondays. It’s considered a real and profitable
trading strategy. Buying on weakness on a Monday has historically shown positive results, making it a strategy worth exploring.
How does the Turnaround Tuesday strategy work, and what are its key components?
There are different variations of the Turnaround Tuesday strategy. One approach involves entering a trade on Monday’s close if the market is at least 1% lower than Friday’s close and exiting at
Tuesday’s close. Another strategy involves entering a trade on Monday’s close if the close is lower than the open, and the Internal Bar Strength (IBS) is below 0.2, then selling at Tuesday’s close.
What are the historical results of the Turnaround Tuesday strategy?
The historical results of the Turnaround Tuesday strategy vary based on the specific rules applied. For example, one strategy with 163 trades showed a 7% average gain per trade, a 9% Compound Annual
Growth Rate (CAGR), and a 63% win ratio. Another strategy with 247 trades showed a 41% average gain per trade, a 3.5% CAGR, and a 60% win ratio. | {"url":"https://therobusttrader.com/turnaround-tuesday-trading-strategy/","timestamp":"2024-11-13T09:26:57Z","content_type":"text/html","content_length":"327444","record_id":"<urn:uuid:2fedf329-0bd0-4a3d-bb3f-76d970f84321>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00637.warc.gz"} |
Move from Excel to Python with Pandas
Move from Excel to Python with Pandas Transcripts
Chapter: Appendix: Python language concepts
Lecture: Concept: Slicing
0:01 Python has this really interesting concept called slicing. It lets us work with things like lists, here in interesting ways.
0:08 It lets us pull out subsets and subsequences if you will, but it doesn't just apply to lists,
0:14 this is a more general concept that can be applied in really interesting way, for example some of the database access libraries,
0:21 when you do a query what you pulled back, you can actually apply this slicing concept for eliminating the results
0:28 as well as paging and things like that. So let's look at slicing. We can index into this list of numbers like so,
0:35 we just go to nums list and we say bracket and we give the index, and in Python these are zero-based, so the first one is zero,
0:41 the second one is one and so on. This is standard across almost every language. However, in Python, you can also have reverse indexes
0:49 so if I want the last one, I can say minus one. So this is not slicing, this is just accessing the values.
0:54 But we can take this concept and push it a little farther. So if I want the first four, I could say 0:4
1:02 and that will say start at the 0th and go up to but not including the one at index 4. So we get 2, 3, 5, 7, out of our list.
1:10 Now, when you are doing these slices, any time you are starting at the beginning or finishing at the end,
1:16 you can omit that, so here we could achieve the same goal by just saying :4, assuming zero for the starting point.
1:22 So, slicing is like array access but it works for ranges instead of for just individual elements. Now if we want to get the middle,
1:31 we can of course say we want to go from the fourth item, so index 3, remember zero-based, so 3 and then we want to go up to
1:38 but not including the sixth index value, we could say 3:6 and that gives us 7, 11 and 13.
1:45 If we want to access items at the end of the list, it's very much like the beginning, we could say we want to go from the sixth element so zero-based,
1:55 that would be 5 up to the end, so 5:9 and it would be 13, 17, 19, 23,
2:00 but like I said, when you are either starting at the beginning or ending at the end,
2:04 you can omit that number, which means you don't have to compute it, that's great, so we could say 5: and then it'll get the last one.
2:10 But you still need to know where that starts, if we actually wanted 4, so there is a little bit of math there,
2:16 if you just want to think of it starting at the end and give me a certain number of items,
2:20 just like where we got the last prime and that came back as 23 when we gave it a minus one, we can do something similar for slicing
2:27 and we could say I'd like to go start 4 in from the back, so negative 4 and then go to the end.
2:33 So that's the idea of slicing, it's all about working with subsets of our collection here, the example I gave you is about a list,
2:40 but like I said we could apply this to a database query, we could apply this to many things in Python
2:46 and you can write classes that extend this concept and make it mean whatever you want,
2:50 so you'll find this is a very useful and common thing to do in Python. | {"url":"https://training.talkpython.fm/courses/transcript/move-from-excel-to-python-and-pandas/lecture/271014","timestamp":"2024-11-12T00:39:46Z","content_type":"text/html","content_length":"29009","record_id":"<urn:uuid:3c551e92-5263-4481-ab61-3ca050c9c5ad>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00892.warc.gz"} |
log / exp / sqrt / power
atom res = log(atom x)
-- or --
atom res = ln(atom x)
-- or --
atom res = log10(atom x)
-- or --
atom res = log2(atom x)
-- or --
Definition: atom res = exp(atom x)
-- or --
atom res = sqrt(atom x)
-- or --
atom res = power(atom x, y)
-- or --
atom res = powmod(atom base, exponent, modulus)
-- or --
atom res = mulmod(atom a, b, modulus)
log[10/2]() returns the natural or base 10/2 logarithm of x.
exp() returns the inverse of log(), implemented trivially as power(EULER,x).
Description: sqrt() returns the square root of x. x must not be negative.
power() returns the x raised to the power y.
powmod() returns the equivalent of rmdr|mod(power(base,exponent),modulus) only much faster and more accurate.
mulmod() returns the equivalent of rmdr|mod(a*b,modulus) only (sometimes somewhat slower and) more accurate.
pwa/p2js: Supported.
These functions may be applied to an atom or sq_log[10/2](), sq_sqrt(), sq_power() to all elements of a sequence.
The rules for sequence operations apply.
Note that logarithms are only defined for positive numbers. Your program will abort with a message should you try to obtain one of a negative number or zero.
ln() is a simple alias of log(), likewise sq_ln() and hll_ln(), which emphasize they yield the natural logarithm; were it not for legacy code and compatibility issues, I’d probably
just deprecate/delete the [sq_|hll_]log() names.
log10() is a simple wrapper to log(), multiplied by (1/log(10)). Likewise log2(), which also contains some code to ensure that non-negative integer powers of 2 yeild an integer result
[pre-1.0.2 log2(8) gave 3-(4.44e-16) on 32-bit, while 64-bit fared a bit better up to log2(8192) which gave 13-(8.67e-19), ie both pretty close but no cigar], covering 0..31 on
32-bit, and 0..63 on 64-bit. A slightly less efficient version of that code was also added to log10(), but only guarantees 0..9 on 32-bit (aka integer inputs of 1, 10, ..
1_000_000_000), and 0..19 on 64-bit. Should the (reduced) performance of log10/2 be an issue, you would be much better off by invoking ln() and performing said multiplication by a
predefined constant inline.
log() is directly supported by the floating point hardware and as such, without any wrapper/multiplication, may prove noticeably faster, and perhaps slightly more accurate.
Comments: exp(atom x) is the inverse of log(), and is implemented simply as return power(EULER,x).
There is no similar builtin function for the inverse of log10(); you are expected to use power(10,x) directly, likewise log2().
Also, there is currently no sq_exp() function, it has simply never been needed or asked for (plus it don’t quite fit in the seqops table).
Powers of 2 are calculated very efficiently.
Other languages have a ** or ^ operator to perform the same action as power(), though in some languages ^ is the xor function.
It is also noted that any potential ambiguity in, say, "-5^2" simply does not occur in power(-5,2) vs. -power(5,2).
Theoretically power(0,0) is undefined, however the result is 1, mainly for consistency with other programming languages.
Attempting to raise any value <=0 to a negative or non integer power causes a fatal runtime error (same as python).
Obvious workarounds exist, for instance should you require a function that returns the cubic root of negative (and positive) numbers:
function cube_root(atom c)
return sign(c)*power(abs(c),1/3)
end function
?log(100) -- prints 4.605170186
?log10(100) -- prints 2 -- (exact in 1.0.2+)
?log2(8) -- prints 3 -- ""
Examples: ?sqrt(16) -- prints 4
?power(5,2) -- prints 25
?powmod(13789,722341,2345) -- prints 2029
log(): via :%opLog in builtins\VM\pTrig.e (an autoinclude).
log10() and log2(): see builtins\log10.e (an autoinclude) for details of the actual implementation.
exp(): see builtins\pmaths.e (an autoinclude) for details of the actual implementation.
sqrt(): via :%opSqrt in builtins\VM\pTrig.e (an autoinclude).
power(): via :%opPow in builtins\VM\pPower.e (an autoinclude).
Implementation: powmod() and mulmod(): see builtins\pmaths.e. NB: no formal statement regards matching mod() or rmdr() is made, in other words only lightly tested and currently only formally
supported for all-positive parameters, but as ever I will be happy to fix any glitches that hinder a real-world need. One unknown is whether there should be a bool all_ints=true
parameter, which would make them check the parameters are not fractional and do not exceed the "integer" precision limits of an atom.
The constant EULER (renamed from E in 1.0.2) is defined in psym.e/syminit(), part of the compiler, as 2.71828182845904523536, the last two digits of which are probably beyond
precision limits of 64-bit, with the last 6 digits likewise being of course pretty much irrelevant on 32-bit. There is also a commented-out (and undocumented) routine in mpfr.e which
will generate a string version of that constant to however many digits you have patience for (easily portable to mpfr.js if needed). | {"url":"http://phix.x10.mx/docs/html/log.htm","timestamp":"2024-11-13T12:26:28Z","content_type":"text/html","content_length":"16564","record_id":"<urn:uuid:b616268a-883e-4170-ad40-ce804c95649f>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00468.warc.gz"} |
MTH643 Current Midterm Papers 2022 - VU Answer
Are you looking for MTH643 Current Midterm Papers 2022? If yes, then you visit the right site. Here are MTH643 Current Papers 2022. MTH643 Midterm Past Papers 2022.
Students must prepare these mth643 midterm current papers in 2022. Also, it can give mth643 midterm preparation 2022 essential topics, questions an overview of the paper.
MTH643 CURRENT MIDTERM PAPERS 2022
Provide by VU Answer
Math 643 paper
Q no 1: Solve the system x^2*y^2, x-y/2=1
X and y should be displayed
Q no 2: 2 inputs will be taken
i.what is your name?
ii.what is your age?
Display name and age copy paste input and output in exam software
Q no 3.plot the parametric curve x=t^2sint(t)
Past the Code and output on exam software
Mth643(Matlab) paper done!!!
Total 3 questions (Each of 5 marks).
First question was to write ode 45 while equation and related conditions were given and demanding just code not output.
Second two matrices were given and they are to rewrite in Matlab and then A inverse and A*B to find???
Find Differential equations solution thaa(y'=y+t).... y(1)=-2...just following equation and condition was given..
See Also Below Links:
Mth643 Midterm Paper
Question1= solve the question k bary met tha.
Question2=Plot k bary mey tha.
Question3=Matrix 3×3 krna tha.
Question4= If and elseif command k bary mey tha
Paper mth643
Area of triangle ka question thaaa marks5
Ak question ma equation di thi mistake btani thii or algbraic equation ka question marks5
Plot curve walaa tha ak swl marks5
Check Also Most Important Materials:
All Midterm Past Papers Solved Mega Files Subject Wise - DOWNLOAD
VU All Subjects Wise Handouts PDF - DOWNLOAD
VU Midterm MCQs Mega File - DOWNLOAD
Mth643 today paper
Q1. from 22 page file phir ss send krti
Q2. Z=(x+y) ^6/x^2*y
Solve by substituting values x=6,y=7
Q3. Enter your name and age then display
Q4. plot the contour with interval - 5to 5(same yehe x or y dono ka interval tha idr ab right ni likha ja rha) f(x.,y)=sin(x) +cos(y)
Mth643 Midterm Paper
Question1= u(x,y)=x^2+3y ki tra ew thi isko at =5 py solve krna tha isi tra f=sin(x) ko at x= 0 py
Question2=matrix bnana tha
Question3=ik program tha number enter krwana tha negative number ho to given output show krwani thi aur AGR positive ho to uski alag se output given this wo display krwani thi
Question4=cotuor plot bnana tha
Today paper mth643
1) write mistakes and then write correct syntax
2) find values of sinx,cosx,tanx at x=pi
Write given values in matrix form
3) draw graph of parametric equations...
4)find area of triangle...
Papers shared by VU Answer and Students.
Share with fellows and help others in their studies.
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Active Subspace Models
Active Subspace Models
The idea behind active subspaces is to find directions in the input variable space in which the quantity of interest is nearly constant. After rotation of the input variables, this method can allow
significant dimension reduction. Below is a brief summary of the process.
1. Compute the gradient of the quantity of interest, \(q = f(\mathbf{x})\), at several locations sampled from the full input space,
\[\nabla_{\mathbf{x}} f_i = \nabla f(\mathbf{x}_i).\]
2. Compute the eigendecomposition of the matrix \(\hat{\mathbf{C}}\),
\[\hat{\mathbf{C}} = \frac{1}{M}\sum_{i=1}^{M}\nabla_{\mathbf{x}} f_i\nabla_{\mathbf{x}} f_i^T = \hat{\mathbf{W}}\hat{\mathbf{\Lambda}}\hat{\mathbf{W}}^T,\]
where \(\hat{\mathbf{W}}\) has eigenvectors as columns, \(\hat{\mathbf{\Lambda}} = \text{diag}(\hat{\lambda}_1,\:\ldots\:,\hat{\lambda}_N)\) contains eigenvalues, and \(N\) is the total number of
3. Using a truncation method or specifying a dimension to estimate the active subspace size, split the eigenvectors into active and inactive directions,
\[\hat{\mathbf{W}} = \left[\hat{\mathbf{W}}_1\quad\hat{\mathbf{W}}_2\right].\]
These eigenvectors are used to rotate the input variables.
4. Next the input variables, \(\mathbf{x}\), are expanded in terms of active and inactive variables,
\[\mathbf{x} = \hat{\mathbf{W}}_1\mathbf{y} + \hat{\mathbf{W}}_2\mathbf{z}.\]
5. A surrogate is then built as a function of the active variables,
\[g(\mathbf{y}) \approx f(\mathbf{x})\]
As a concrete example, consider the function: [Con15]
\[f(x) = \exp\left(0.7x_1 + 0.3x_2\right).\]
Figure [fig:activesubspace](a) is a contour plot of \(f(x)\). The black arrows indicate the eigenvectors of the matrix \(\hat{\mathbf{C}}\). Figure [fig:activesubspace](b) is the same function but
rotated so that the axes are aligned with the eigenvectors. We arbitrarily give these rotated axes the labels \(y_1\) and \(y_2\). From fig. [fig:activesubspace](b) it is clear that all of the
variation is along \(y_1\) and the dimension of the rotated input space can be reduced to 1.
TODO: Missing images, figure, caption here
For additional information, see references [Con15, CDW14, CG14].
Truncation Methods
Once the eigenvectors of \(\hat{\mathbf{C}}\) are obtained we must decide how many directions to keep. If the exact subspace size is known a priori it can be specified. Otherwise there are three
automatic active subspace detection and truncation methods implemented:
• Constantine metric (default),
• Bing Li metric,
• and Energy metric.
Constantine metric
The Constantine metric uses a criterion based on the variability of the subspace estimate. Eigenvectors are computed for bootstrap samples of the gradient matrix. The subspace size associated with
the minimum distance between bootstrap eigenvectors and the nominal eigenvectors is the estimated active subspace size.
Below is a brief outline of the Constantine method of active subspace identification. The first two steps are common to all active subspace truncation methods.
1. Compute the gradient of the quantity of interest, \(q = f(\mathbf{x})\), at several locations sampled from the input space,
\[\nabla_{\mathbf{x}} f_i = \nabla f(\mathbf{x}_i).\]
2. Compute the eigendecomposition of the matrix \(\hat{\mathbf{C}}\),
\[\hat{\mathbf{C}} = \frac{1}{M}\sum_{i=1}^{M}\nabla_{\mathbf{x}} f_i\nabla_{\mathbf{x}} f_i^T = \hat{\mathbf{W}}\hat{\mathbf{\Lambda}}\hat{\mathbf{W}}^T,\]
where \(\hat{\mathbf{W}}\) has eigenvectors as columns, \(\hat{\mathbf{\Lambda}} = \text{diag}(\hat{\lambda}_1,\:\ldots\:,\hat{\lambda}_N)\) contains eigenvalues, and \(N\) is the total number of
3. Use bootstrap sampling of the gradients found in step 1 to compute replicate eigendecompositions,
\[\hat{\mathbf{C}}_j^* = \hat{\mathbf{W}}_j^*\hat{\mathbf{\Lambda}}_j^*\left(\hat{\mathbf{W}}_j^*\right)^T.\]
4. Compute the average distance between nominal and bootstrap subspaces,
\[e^*_n = \frac{1}{M_{boot}}\sum_j^{M_{boot}} \text{dist}(\text{ran}(\hat{\mathbf{W}}_n), \text{ran}(\hat{\mathbf{W}}_{j,n}^*)) = \frac{1}{M_{boot}}\sum_j^{M_{boot}} \left\| \hat{\mathbf{W}}_n\
hat{\mathbf{W}}_n^T - \hat{\mathbf{W}}_{j,n}^*\left(\hat{\mathbf{W}}_{j,n}^*\right)^T\right\|,\]
where \(M_{boot}\) is the number of bootstrap samples, \(\hat{\mathbf{W}}_n\) and \(\hat{\mathbf{W}}_{j,n}^*\) both contain only the first \(n\) eigenvectors, and \(n < N\).
5. The estimated subspace rank, \(r\), is then,
\[r = \operatorname*{arg\,min}_n \, e^*_n.\]
For additional information, see Ref. [Con15].
Bing Li metric
The Bing Li metric uses a trade-off criterion to determine where to truncate the active subspace. The criterion is a function of the eigenvalues and eigenvectors of the active subspace gradient
matrix. This function compares the decrease in eigenvalue amplitude with the increase in eigenvector variability under bootstrap sampling of the gradient matrix. The active subspace size is taken to
be the index of the first minimum of this quantity.
Below is a brief outline of the Bing Li method of active subspace identification. The first two steps are common to all active subspace truncation methods.
1. Compute the gradient of the quantity of interest, \(q = f(\mathbf{x})\), at several locations sampled from the input space,
\[\nabla_{\mathbf{x}} f_i = \nabla f(\mathbf{x}_i).\]
2. Compute the eigendecomposition of the matrix \(\hat{\mathbf{C}}\),
\[\hat{\mathbf{C}} = \frac{1}{M}\sum_{i=1}^{M}\nabla_{\mathbf{x}} f_i\nabla_{\mathbf{x}} f_i^T = \hat{\mathbf{W}}\hat{\mathbf{\Lambda}}\hat{\mathbf{W}}^T,\]
where \(\hat{\mathbf{W}}\) has eigenvectors as columns, \(\hat{\mathbf{\Lambda}} = \text{diag}(\hat{\lambda}_1,\:\ldots\:,\hat{\lambda}_N)\) contains eigenvalues, and \(N\) is the total number of
3. Normalize the eigenvalues,
\[\lambda_i = \frac{\hat{\lambda}_i}{\sum_j^N \hat{\lambda}_j}.\]
4. Use bootstrap sampling of the gradients found in step 1 to compute replicate eigendecompositions,
\[\hat{\mathbf{C}}_j^* = \hat{\mathbf{W}}_j^*\hat{\mathbf{\Lambda}}_j^*\left(\hat{\mathbf{W}}_j^*\right)^T.\]
5. Compute variability of eigenvectors,
\[f_i^0 = \frac{1}{M_{boot}}\sum_j^{M_{boot}}\left\lbrace 1 - \left\vert\text{det}\left(\hat{\mathbf{W}}_i^T\hat{\mathbf{W}}_{j,i}^*\right)\right\vert\right\rbrace ,\]
where \(\hat{\mathbf{W}}_i\) and \(\hat{\mathbf{W}}_{j,i}^*\) both contain only the first \(i\) eigenvectors and \(M_{boot}\) is the number of bootstrap samples. The value of the variability at
the first index, \(f_1^0\), is defined as zero.
6. Normalize the eigenvector variability,
\[f_i = \frac{f_i^0}{\sum_j^N f_j^0}.\]
7. The criterion, \(g_i\), is defined as,
\[g_i = \lambda_i + f_i.\]
8. The index of first minimum of \(g_i\) is then the estimated active subspace rank.
For additional information, see Ref. [LL15].
Energy metric
The energy metric truncation method uses a criterion based on the derivative matrix eigenvalue energy. The user can specify the maximum percentage (as a decimal) of the eigenvalue energy that is not
captured by the active subspace represenation.
Using the eigenvalue energy truncation metric, the subspace size is determined using the following equation:
\[n = \inf \left\lbrace d \in \mathbb{Z} \quad\middle|\quad 1 \le d \le N \quad \wedge\quad 1 - \frac{\sum_{i = 1}^{d} \lambda_i}{\sum_{i = 1}^{N} \lambda_i} \,<\, \epsilon \right\rbrace\]
where \(\epsilon\) is the truncation_tolerance, \(n\) is the estimated subspace size, \(N\) is the size of the full space, and \(\lambda_i\) are the eigenvalues of the derivative matrix. | {"url":"https://snl-dakota.github.io/docs/6.20.0/users/usingdakota/advanced/activesubspace.html","timestamp":"2024-11-01T20:41:28Z","content_type":"text/html","content_length":"25098","record_id":"<urn:uuid:4cf6ba5e-399d-413b-a32e-f1dad6c8c61b>","cc-path":"CC-MAIN-2024-46/segments/1730477027552.27/warc/CC-MAIN-20241101184224-20241101214224-00297.warc.gz"} |
seminars - On the emerging asymptotic patterns of the Winfree model
In this thesis, we study various Winfree-type dynamics. First, we study emergent dynamics of the continuous Winfree model with inertia and its discrete analogue. We provide sufficient conditions for
the complete oscillator death to the Winfree model in the presence of inertia and the discrete-time analogue with or without inertia. We also present a uniform-in-time convergence from the discrete
model to the continuous model for zero inertia case, as the time-step tends to zero. In addition, we study the emergence of asymptotic patterns in Winfree ensemble such as the partial / complete
phase-locking and bump states under the effect of heterogeneous frustrations. In particular, we provide a rigorous result on the existence of bump states in a homogeneous ensemble with the same
natural frequency. Moreover, we propose a Winfree type model and its mean-field limit describing the aggregation of particles on the surface of an infinite cylinder. For the proposed model, we
present a sufficient framework leading to the complete oscillator death and uniform stability in a large coupling regime. We also derive the corresponding kinetic model via uniform-in-time mean-field
limit. Furthermore, we study a uniform-in-time continuum limit of the lattice Winfree model and its asymptotic dynamics. For bounded measurable initial phase field, we establish a global
well-posedness of classical solutions to the continuum Winfree model under suitable assumptions on coupling function, and we also show that a classical solution to the continuum Winfree model can be
obtained as a limit of a sequence of lattice solutions in suitable sense. | {"url":"https://www.math.snu.ac.kr/board/index.php?mid=seminars&page=40&l=ko&document_srl=815562","timestamp":"2024-11-09T23:00:12Z","content_type":"text/html","content_length":"46104","record_id":"<urn:uuid:0d2aeaed-de3c-4521-88fb-202bea452b5d>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.10/warc/CC-MAIN-20241109214337-20241110004337-00772.warc.gz"} |
[Class 11 - Basics] Number System and Conversion - Computer Science
Number System and Conversion
A number system is a method to represent numbers. There are 4 types of number systems. They are:
1. Decimal Number System
This number system is used in our day to day lives.
The Decimal Number system contains digits from 0 to 9 . So the base of the decimal number system is 10.
2. Binary Number System
The Binary Number system contains two digits 0 and 1 . So the base of the binary number system is 2.
3. Octal Number System
The Octal Number system can contain digits from 0 to 7 . So the base of the octal number system is 8.
4. Hexadecimal Number System
The Hexadecimal Number system can contain digits from 0 to 9 and alphabets from A to F . So the base of the hexadecimal number system is 16. | {"url":"https://www.teachoo.com/17583/3911/Number-System-and-Conversion/category/Concepts/","timestamp":"2024-11-09T09:44:44Z","content_type":"text/html","content_length":"110397","record_id":"<urn:uuid:cec8d932-fd73-4a0b-97f2-5f17f3bf5213>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.75/warc/CC-MAIN-20241109085148-20241109115148-00078.warc.gz"} |
8 KPIs Every Demand Planner Should Know - KnowHow Consultancy
8 KPIs Every Demand Planner Should Know
Without KPIs, it is impossible to improve forecast accuracy. Here are 8 highly effective metrics that allow you to track your forecast performance, complete with their formulas.
Forecast Accuracy
This KPI is absolutely critical because the more accurate your forecasts, the more profit the company makes and the lower your operational costs. We choose a particular forecasting method because we
think it will work reasonably well and generate promising forecasts but we must expect that there will be error in our forecasts. This error is a function of the time difference between the actual
value (Dt) and the forecast value (Ft) for that period. It is measured as:
Forecast Accuracy: 1 – [ABS (Dt – Ft) / Dt]
Dt: The actual observation or sales for period t
Ft: The forecast for period t
Our focus on this KPI is to provide insights about forecasting accuracy benchmarks for groups of SKUs rather than identifying the most appropriate forecasting methods. For example, achieving 70-80%
forecast accuracy for a newly-launched and promotion-driven product would be a good considering we have no sales history to work from.
SKUs with medium forecastability (volatile, seasonal, and fast-moving SKUs) are not easy to forecast owing to seasonal factors like holidays and uncontrollable factors like weather and competitors’
promotions etc., their benchmark is not recommended to be less than 90-95%.
Tracking Signals
Tracking signals (TS) quantify bias in a forecast and help demand planners to understand whether the forecasting model works well or not. TS in each period is calculated:
TS: (Dt- Ft) / ABS (Dt – Ft)
Dt: The actual observation or sales for period t
Ft: The forecast for period t
Once it is calculated, for each period, the numbers are added to calculate the overall TS. When a forecast, for instance, is generated by considering the last 24 observations, a forecast history
totally void of bias will return a value of zero. The worst possible result would return either +24 (under-forecast) or -24 (over-forecast). Generally speaking such a forecast history returning a
value greater than (+ 4.5) or less than (-4.5) would be considered out of control. Therefore, without considering the forecastability of SKUs, the benchmark of TS needs to be between (-4.5) and
Bias, also known as Mean Forecast Error, is the tendency for forecast error to be persistent in one direction. The quickest way of improving forecast accuracy is to track bias. If the bias of the
forecasting method is zero, it means that there is an absence of bias. Negative bias values reveal a tendency to over-forecast while positive values indicate a tendency to under-forecast. Over the
period of 24 observations, if bias is greater than four (+4), forecast is considered to be biased towards under-forecasting. Likewise, if bias is less than minus four (- 4), it can be said that the
forecast is biased towards over-forecasting. In the end, the aim of the planner is to minimize bias. The formula is as follows:
Bias: [∑ (Dt – Ft)] / n
Dt: The actual observation or sales for period t
Ft: The forecast for period t
n: The number of forecast errors
Forecaster bias appears when forecast error is in one direction for all items, i.e they are consistently over- or under-forecasted. It is a subjective bias due to people to building unnecessary
forecast safeguards like increasing the forecast to match sales targets or division goals.
By considering the forecastability level of SKUs, the bias of low forecastability SKUs bias can be between (-30) and (30). When it comes to medium forecastability SKUs, since their accuracy is
expected to be between 90-95%, bias should not be less than (-10) nor greater than (+10). Regarding high forecastability SKUs, due to their moderate contribution to the total, bias is not expected to
be less than (-20) or greater than (20). The less bias there is in a forecast, the better the forecast accuracy, which allows us to reduce inventory levels.
Mean Absolute Deviation (MAD)
MAD is a KPI that measures forecast accuracy by averaging the magnitudes of the forecast errors. It uses the absolute values of the forecast errors in order to avoid positive and negative values
cancelling out when added up together. Its formula is as follows:
MAD: ∑ |Et| / n
Et: the forecast error for period t
n: The number of forecast errors
MAD does not have specific benchmark criteria to check the accuracy, but the smaller the MAD value, the higher the forecast accuracy. Comparing the MAD values of different forecasting methods reveals
which method is most accurate.
Mean Square Error (MSE)
MSE evaluates forecast performance by averaging the squares of the forecast errors, removing all negative terms before the values are added up. The squares of the errors achieves the same outcome
because we use the absolute values of the errors, as the square of a number will always result in a non-negative value. Its formula is as follows:
MSE: ∑(Et)² / n
Et: forecast error for period t
n: the number of forecast errors
Similar to MAD, MSE does not have a specific benchmark to check accuracy but the smaller value of MSE, the better forecast model, which means more accurate forecasts. The advantage of MSE is that it
squares forecast errors, giving more weight to large forecast errors.
Mean Absolute Percentage Error (MAPE)
MAPE is expressed as a percentage of relative error. MAPE expresses each forecast error (Et) value as a % of the corresponding actual observation (Dt). Its formula is as follows:
MAPE: ∑ |Et / Dt |/n * 100
Dt: Actual observation or sales for period t
Et: the forecast error for period t
n: the number of forecast errors
Since the result of MAPE is expressed as a percentage, it is understood much more easily compared to other techniques. The advantage of MAPE is that it relates each forecast error to its actual
observation. However, series that have a very high MAPE may distort the average MAPE. To avoid this problem, SMAPE is offered which is addressed below.
Symmetrical Mean Absolute Percentage Error (SMAPE)
SMAPE is an alternative to MAPE when having zero and near-zero observations. Low volume observations mostly cause high error rates and skew the overall error rate, which can be misleading. To address
this problem, SMAPE come in handy. SMAPE has a lower bound of 0% and an upper bound of 200%. It does not treat over-forecast and under-forecast equally. Its formula is as follows:
SMAPE: 2/n * ∑ | (Ft – Dt) / (Ft + Dt)|
Dt: Actual observation or sales for period t
Ft: the forecast for period t
n: the number of forecast errors
Similar to other models, there is no specific benchmark criteria for SMAPE. The lower the SMAPE value, the more accurate the forecast.
Weighted Mean Absolute Percentage Error (WMAPE)
WMAPE is the improved version of MAPE. Whilst MAPE is a volume-weighted technique, WMAPE is more value-weighted. When generating forecasts for high value items at the category, brand, or business
level, MAPE cancels plus and minus values. WMAPE, however, weights both forecast errors and actual observations (sales). When considered at the brand level, high value items will influence overall
error because they are highly correlated with safety stock requirements and development of safety stock strategies. Its formula is as follows:
WMAPE: ∑(|Dt-Ft|) / ∑(Dt)
Dt: The actual observation for period t
Ft: the forecast for period t
Like other techniques, WMAPE does not have any specific benchmark. The smaller the WMAPE value, the more reliable the forecast.
For citation:
Eksoz C. (2020). “8 KPIs Every Demand Planner Should Know”, Institute of Business Forecasting & Planning, www.demand-planning.com. Available at: https://demand-planning.com/2020/06/01/ | {"url":"https://knowhowconsultancy.com/2020/09/08/8-kpis-every-demand-planner-should-know/","timestamp":"2024-11-10T01:52:19Z","content_type":"text/html","content_length":"107494","record_id":"<urn:uuid:6ef3d078-018f-48ef-8790-07906b99d224>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00330.warc.gz"} |
Single-valued hyperlogarithms, correlation functions and closed string amplitudes | H2020 | CORDIS | European Commission
Periodic Reporting for period 1 - HIPSAM (HIgher Polylogarithms and String AMplitudes)
Période du rapport: 2020-09-01 au 2022-08-31
The main goal of this project is to develop the mathematical tools necessary to describe the perturbative expansion of string theory amplitudes, at least for low genus, re-interpreting and
generalizing recent beautiful progress made at genus zero. Understanding the mathematical structure of perturbative string amplitudes would yield new information on string theory predictions for
fundamental interactions, and point towards new directions in many fields of mathematics, such as mixed motives and moduli spaces of curves, opening new research lines completely inspired by physics.
The techniques developed would also allow to attack similar nowadays intractable computations of amplitudes in quantum field theory, which can be compared with the experimental data produced by
particle accelerators.
The main technical novelty of the project is the introduction of analogues of polylogarithms on higher-genus Riemann surfaces. Polylogarithms are important special functions which appear in several
areas of mathematics, given by iterated integrals over configuration spaces of points on a Riemann sphere. Enriquez, Levin, Racinet, Brown and others introduced similar functions for genus-one
Riemann surfaces, leading to the recent theory of elliptic polylogarithms, which found spectacular applications in high-energy physics. The next goal in this research area is to go beyond genus one,
and its importance for this project stems from the expectation that genus-g polylogarithms are the mathematical tool needed to describe genus-g string amplitudes, an observation which has proved to
be extremely useful at low genus.
Another important aspect of the project is to clarify the relation between closed string amplitudes and the newborn mathematical theory of single-valued periods, which would yield a deeper
understanding of the relations between closed and open string amplitudes, and ultimately between gauge theories and gravity.
Conclusions of the action : we have achieved to construct a generalisation of polylogarithms to higher-genus Riemann surfaces, and we have characterised the space of functions that they generate.
This is an important result in mathematics but also, potentially, in high-energy physics. Such construction is not yet suited to be applied to string amplitudes of genus higher than one, as one needs
a more explicit formulation highlighting the dependence on the complex structure of the surface, which is currently under investigation. As for low-genus string amplitudes, and their relation with
single-valued periods, we have clarified several aspects of such relations at genus-zero, and the analogous problem at genus-one is currently under investigation.
Great effort was devoted on identifying analogues of polylogarithms for general Riemann surfaces. This is the most mathematical component, and main cornerstone, of the original project, and it was
undertaken in collaboration with Benjamin Enriquez, who works at the University of Strasbourg. We have worked on three main research lines. First of all, we wanted to explicitly develop the algebraic
de Rham theory of the fundamental group of configuration spaces of curves, following ideas of Hain. We have succeeded in writing down general homotopy invariant iterated integrals of rational
functions on one curve (previously known only up to length two), and we are now left with generalizing this to configuration spaces. An article about this should be written up in the near future. The
second research line consisted in constructing a single-valued flat connection over the configuration space of genus-g Riemann surfaces. We have succeeded in our goal by modifying a multi-valued flat
connection previously constructed by Enriquez. We uploaded to the Arxiv in October 2021 a preprint ("Construction of Maurer-Cartan elements over configuration spaces of curves") which contains such
result. Combining these two research lines leads to explicitly construct higher-genus analogues of polylogarithms, which was the main expected mathematical milestone of this project. A third research
line consisted in studying the associated space of functions, and we obtained spectacular results in the case of affine curves. More specifically, we have constructed in three different ways a
natural candidate for the space of hyperlogarithms (i.e. multiple polylogarithms with all but one variable fixed) on a general punctured Riemann surface, studied its algebraic structure, and
identified a basis for such function spaces, whose elements constitute higher-genus analogues of classical functions first considered by Poincaré. These results have already been written up, and
should appear in a preprint at the end 2022. Another research direction, currently under investigation and crucially important for applying such results to the computation of string amplitudes, is
the study of the dependence on the complex structure of the Riemann surface, which is known only at genus one. Several of the results described above were announced and explained at invited seminar
talks (in Dijon, Durham, Montpellier, Oxford and Zurich), as well as through two events which were planned for this MSCA IF, jointly organised with Pierre Vanhove: the (online) seminar "Motives and
periods integrals in quantum field theory and string theory", and the special session "Mathematical Physics of Gravity" of the AMS-EMS-SMF joint meeting held in Grenoble in July 2022.
At the same time, we worked on low-genus string amplitudes and their relation with single-valued periods, with some variation with respect to the research lines which were originally planned. As a
main result, together with Pierre Vanhove, and building on a previous unpublished joint work, we wrote an article ("Single-valued hyperlogarithms, correlation functions and closed string
amplitudes"), which will soon be published by Advances in Theoretical and Mathematical Physics, where we have provided new interpretations of the relations between closed string theory amplitudes at
genus zero and single-valued periods. For example, we have deduced the celebrated KLT formula by identifying closed string integrals with special values of single-valued correlation functions in two
dimensional conformal field theory, and by obtaining their conformal block decomposition. Moreover, we have written the asymptotic expansion coefficients as multiple integrals over the complex plane
of special functions known as single-valued hyperlogarithms, and used this fact to demonstrate that the asymptotic expansion coefficients belong to the ring of single-valued multiple zeta values.
The main result obtained is the introduction of higher-genus analogues of polylogarithms, which is achieved in collaboration with Benjamin Enriquez, and which was announced as a main goal of this
MSCA IF. We expect that this will have a big impact in the near future on the computation of scattering amplitudes, both in string theory and in quantum field theory, similarly to what happened with
the introduction of elliptic analogues of polylogarithms, which is now the subject of yearly conferences within the amplitude community. Moreover, this should also have an impact in mathematics, as
it gives the first explicit construction of periods of fundamental groups of curves beyond the classical periods of the curves. | {"url":"https://cordis.europa.eu/project/id/843960/reporting/fr","timestamp":"2024-11-03T01:08:38Z","content_type":"text/html","content_length":"76572","record_id":"<urn:uuid:6f123212-517a-4795-b140-ccef0920130e>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00667.warc.gz"} |
How to Make Pie graphs or Circle graph
Read,3 minutes
A pie chart is a graphic representation of info in the form of a round chart or pie where pie slices show the size of the data. A listing of mathematical variables along with categorical variables is
needed in order to show data in a pie graph format. The arc length of each slice and so its area, along with the central angle it forms in the pie chart, is in proportion to the amount it represents.
Pie Chart Formula
A pie chart’s total value is always one hundred percent. In addition, a circle subtends a \(360°\) angle. So, the total value of all the info equals \(360°\). Based on that, two primary formulas are
• To calculate the percentage of the given info, use the following formula: \((Frequency \ \div \ Total Frequency) \ \times \ 100\)
• To change the info to degrees use this formula: \((Provided \ data \ \div \ Total \ value \ of \ Information) \ \times \ 360°\)
It’s possible to figure out a specific pie chart’s percentage by using the steps here:
• Step 1: Classify the given data and figure out the total
• Step 2: Divide up all the various categories
• Step 3: Change the data to percentages
• Step 4: Calculate the degrees
Ways to use Pie Charts
Whenever information has to get shown visually using a fractional part of a whole, we use pie charts. It is used to compare information and discover why one part is smaller/bigger than another. So,
when you are given a limited amount of buckets along with distinct data sets, it’s a smarter idea to use pie charts. Here are a few uses for pie charts:
• In a business, it is used to compare growth areas such as profits and losses.
• In schools, pie charts are used to show the time allocated to each section, thus showing pupils’ grades as percentages, etc.
• Pie charts are used to compare the relative quantity of in regarding people having the same vehicles, similar houses, etc.
• They are used to show marketing and sales info to compare more than one brand.
Making a Pie Chart
Here are the steps used to make a pie chart. Via using the aforementioned formulas, the information can be figured out.
• Step 1: Place all the data in a table, after that, add it all of to discover the total.
• Step 2: To find the values in the form of a percentage, one has to divide each of them via the total, after that multiply it by one hundred.
• Step 3: To find the amount of degrees for each of the pie pieces you need, take a complete circle of \(360°\) and use the following formula: \(\frac{Frequency}{Total \ Frequency} \ \times \ 360°
• Step 4: Whenever all the degrees to make the pie chart are figured out, draw a circle (pie chart) using the computed measurements with the help of a protractor.
Exercises for The Pie Graph or Circle Graph
1) Which color is the most?
2) Which color is the most?
3) What percentage of pie graph is black?
4) What percent of people voted for Sepehr?
5) What percentage of pie graph is red?
6) What percentage of pie graph is red?
7) Which color is the least?
8) Which color is the least?
9) What percent of people voted for Emma?
10) What percentage of pie graph is yellow?
Which color is the most?
2) Which color is the most? Pink
3) What percentage of pie graph is black?\(\color{red}{24 \%} \)
4) What percent of people voted for Sepehr?\(\color{red}{26 \%} \)
5) What percentage of pie graph is red?\(\color{red}{25 \%} \)
6) What percentage of pie graph is red?\(\color{red}{24 \%} \)
7) Which color is the least? Silver
8) Which color is the least? Black
9) What percent of people voted for Emma?\(\color{red}{24 \%} \)
10) What percentage of pie graph is yellow? \(\color{red}{22 \%} \)
The Pie Graph or Circle Graph Quiz | {"url":"https://testinar.com/article/statistics__the_pie_graph_or_circle_graph_course","timestamp":"2024-11-03T19:31:38Z","content_type":"text/html","content_length":"61749","record_id":"<urn:uuid:bf0935e3-563c-4a11-83c5-3e3e723d580e>","cc-path":"CC-MAIN-2024-46/segments/1730477027782.40/warc/CC-MAIN-20241103181023-20241103211023-00673.warc.gz"} |
Arithmetic Sequences and Series Archives - A Plus Topper
Arithmetic Sequences and Series
Arithmetic Sequences and SeriesA sequence is an ordered list of numbers. The sum of the terms of a sequence is called a series. Read More:What is the pattern of numbers?SequencesWhile some sequences
are simply random values, other sequences have a defin… | {"url":"https://www.aplustopper.com/tag/arithmetic-sequences-and-series/","timestamp":"2024-11-06T03:07:22Z","content_type":"text/html","content_length":"36210","record_id":"<urn:uuid:a648d165-2187-4e21-a18c-69d744fb02ce>","cc-path":"CC-MAIN-2024-46/segments/1730477027906.34/warc/CC-MAIN-20241106003436-20241106033436-00229.warc.gz"} |
Procedural Terrain Generation
Creating 2D terrain using simplex noise
Let’s start out with noise. You’ve probably seen noise on your tv, when you’ve tuned to a channel that doesn’t have a signal.
This sort of noise isn’t great for creating organic visuals. We need something that’s random in its vectors over time, but not random from data point to data point. Sort of like the gentle rise and
fall of a stock price over many years.
To generate natural looking rises and dips, we need to use a specialized random noise generator. The cannonical implementation is Perlin noise, but a more recent version is called simplex noise.
Simplex noise has a copyright, so we’ll use opensimplex noise to generate our numbers.
Generating Islands
A heightmap is a black and white 2d version of your terrain, where each pixel is a gray scale value. 0 is black, white is 1, and all shades of grey in between. The value at each pixel represents your
terrains height.
Island Mask
If we want islands, we need to remove the edges of our terrain. We’ll generate a radial gradient which starts at 0 in the center and scales to 1 on the perimeter.
To remove the border, simply subtract each island mask pixel value from the elevation in your heightmap. Prevent negative heights by limiting the min value to 0.
Colorize elevations
Now we can apply colors for each elevation value. As an example, we could say any elevation < .2 we’ll turn a shade of blue for water. Elevations from .2 to .4 will be forests, and thus we’ll make it
This is looking pretty good, but the elevations are a bit predictable, and there’s not really any interesting details once you get above the water elevation. Let’s fix that by adding in the concept
of a biome.
Add biomes/moisture
First, let’s generate another heightmap, but with a little bit more detail. We can tweak the simplex noise inputs, and octaves to get more or less details and diffusion.
Now we can apply this moisture/biome heightmap to the elevation heightmap. How you apply it is up to you, but I’ve found it gives the best results when you figure out a way to blend the elevation and
moisture maps together, so you get a little bit of color variation without completely destroying the effect of rising elevation.
Here’s a side-by-side view with and without moisture.
These are some links that I found useful in coming up with these procedural generated islands. | {"url":"https://bwiggs.com/projects/terrain/","timestamp":"2024-11-13T04:14:37Z","content_type":"text/html","content_length":"45532","record_id":"<urn:uuid:ab63bf7c-3263-4014-936e-1a6596d2a71a>","cc-path":"CC-MAIN-2024-46/segments/1730477028326.66/warc/CC-MAIN-20241113040054-20241113070054-00006.warc.gz"} |
Moduli spaces and topological quantum field theories
We show how to construct a topological quantum field theory which corresponds to a given moduli space. This method is applied to several cases. In particular we discuss the moduli space of flat gauge
connections over a Riemann surface which is related to the phase space of the Chern-Simons theory. The observables of these theories are derived. Geometrical properties are invoked to prove that the
global invariants are not trivial.
Presented at the 18th International Conference on Differential Geometric Methods in Theoretical Physics: Physics and Geometry
Pub Date:
July 1989
□ Field Theory (Physics);
□ Quantum Theory;
□ Symmetry;
□ Topology;
□ Gauge Theory;
□ Instantons;
□ Riemann Manifold;
□ Yang-Mills Fields;
□ Thermodynamics and Statistical Physics | {"url":"https://ui.adsabs.harvard.edu/abs/1989dgmt.conf....2S/abstract","timestamp":"2024-11-09T17:59:52Z","content_type":"text/html","content_length":"33537","record_id":"<urn:uuid:84888860-f934-4957-837d-fdd6c5124376>","cc-path":"CC-MAIN-2024-46/segments/1730477028125.59/warc/CC-MAIN-20241109151915-20241109181915-00237.warc.gz"} |
nForum - Search Results Feed (Tag: undecidability)Can approximation or heurisitc methods overcome undecidability?
2024-11-06T14:10:45+00:00 https://nforum.ncatlab.org/ Lussumo Vanilla & Feed Publisher https://nforum.ncatlab.org/discussion/7316/ 2016-11-02T21:10:33+00:00 2016-11-02T21:10:33+00:00 tomr https://
nforum.ncatlab.org/account/909/ I asked about approximation methods to overcome undecidability in Stackexchange and there was good ... I asked about approximation methods to overcome undecidability
in Stackexchange and there was good answer:
I also have found several papers about it. E.g. undecidability of modalo logics can be overcome by limiting depth of nesting of modal operators. While this is not mathematically pure solution, it is
very good approximation to the human reasoning that is not quite capable of self-reflection.
Are there chances that heuristic methods - e.g. genetic algorithms or neural networks can overome undecidability problem - e.g. by discovering theorems and proofs that can not be discovered by
algorithmic methods?
I see big prospects for categorical logic to become the universal logic that unifies all sorts of reasoning types (human agent modelling, creativity modelling, mathematical reasoninig, legal
reasoning are all types of reasoning that requires different different methods but the application borders are smooth and therefore unification is necessary) but the application will be impossible if
there is no methods for handling undecidable cases. Therefore one should be able to overcome undecidability. ]]> | {"url":"https://nforum.ncatlab.org/search/?PostBackAction=Search&Type=Topics&Page=1&Feed=ATOM&Tag=undecidability&FeedTitle=Search+Results+Feed+%28Tag%3A+undecidability%29","timestamp":"2024-11-06T14:10:45Z","content_type":"application/atom+xml","content_length":"3729","record_id":"<urn:uuid:b1ca3021-1f78-41dd-b5d2-fbd3dd82d9db>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00095.warc.gz"} |
Measures of Central Tendency
Individual scores by themselves may mean little but when looked at from a group point of view, they may reveal the whole picture.
For example, if you say you saw an insect of length of 10mm it doesn't mean anything by itself. However, if you say that the normal length of the insect is about 6cm and the maximum recorded
length ever is 10.4cm, then it may mean you saw a particularly large insect. Therefore it is important to be able to quantify the "normal length" as used above, and this is what central tendency
is all about.
The arithmetic mean is one of the most commonly used measures of central tendency. For a set of numbers, the mean is simply the average, i.e. sum of all the numbers divided by the number of numbers.
Therefore if you want to find the average length of a group of insects, you simply take the length of each insect, add up all these lengths and divide by the number of insects. If the lengths of
5 insects are 6.5mm, 5.4mm, 5.8mm, 6.2mm and 5.9mm, then the mean is (6.5+5.4+5.8+6.2+5.9)mm/5 = 5.96mm.
The median is another frequently used measure of central tendency. The median is simply the midpoint of the distribution, i.e. there are as many numbers above it as below it.
If the number of data points is odd, then the median is simply the middle number. Therefore the median of 3, 5, 6, 9, 15 is 6.
If the number of data points is even, then the median is the mean of the middle two numbers. Therefore the median of 2, 7, 15, 20 is (7+15)/2 = 11.
The median is particularly useful when there are a few data points that are vastly different. For example, in calculating the central measure of the salary obtained by a group of graduates, it may
happen than a couple of students have got extraordinarily high salaries. This will take the mean of the salaries of the group to very high values, but the median will truly reflect the placement
scenario as it is.
Another commonly used measure of central tendency in specific cases is the mode. The mode is simply the most commonly occurring value. For example, in a class of 50 students graded on a scale of 1-5,
the distribution may be as shown in the figure. The mode of this data is 4.
Different types of data need different measures of central tendency to describe the distribution of data. For highly skewed data, none of these may be sufficient, and we may need to go for other
specialized measures or simply report them all in a table. | {"url":"https://explorable.com/measures-of-central-tendency","timestamp":"2024-11-03T21:51:17Z","content_type":"application/xhtml+xml","content_length":"55840","record_id":"<urn:uuid:37028628-3138-4bd9-95a7-778040bc3c46>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00392.warc.gz"} |
Quantity_Pressure.hxx File Reference
Standard_Real Quantity_Pressure
Defined as the force perpendicular to a unit area. In a fluid it is defined as the product of the depth, density, and free fall acceleration. It is measured in pascals (newtons
per square metre). More...
Defined as the force perpendicular to a unit area. In a fluid it is defined as the product of the depth, density, and free fall acceleration. It is measured in pascals (newtons per square metre). | {"url":"https://dev.opencascade.org/doc/occt-7.0.0/refman/html/_quantity___pressure_8hxx.html","timestamp":"2024-11-06T15:02:04Z","content_type":"application/xhtml+xml","content_length":"6768","record_id":"<urn:uuid:f7e8ea52-e42d-4977-b793-dc65b6e48992>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00713.warc.gz"} |
Wattage to Amperage Calculator
Wattage to Amperage Calculator
Wattage to Amperage Calculator
Understanding the Wattage to Amperage Calculator
The Wattage to Amperage Calculator is a practical tool designed to convert power (measured in watts) to current (measured in amperes or amps). This conversion is useful in various scenarios, such as
electrical engineering, home projects, and troubleshooting electrical devices.
Applications of the Wattage to Amperage Calculator
There are many applications for this calculator in everyday life and professional settings. For instance, electricians may need to know the current flowing through a circuit to ensure that the wiring
and components can handle the load safely. Homeowners may use it when setting up appliances to ensure compatibility with their electrical systems. Technicians might use it to diagnose issues or
design electrical systems, ensuring that they meet the required safety standards.
How the Answer is Derived
The basic principle behind this calculator is the relationship between power, voltage, and current in an electrical circuit. Power in watts (W) is the product of voltage in volts (V) and current in
amperes (A). This means that to find the current, we divide the power by the voltage. By inputting the power and voltage values into the calculator, it automatically computes the current using this
Benefits of Using This Calculator
Using this calculator offers several benefits. It provides quick and accurate conversions without needing manual calculations, which saves time and reduces the risk of errors. This tool is accessible
to anyone, regardless of their level of expertise in electrical engineering, making it a valuable resource for both professionals and amateurs. It also helps in planning and safely executing
electrical projects by ensuring that the current flowing through a circuit is within safe limits.
Interesting Real-World Scenarios
Imagine you are setting up a home theater system. The audio amplifier you plan to use has a power rating of 200 watts, and your home voltage supply is 120 volts. By using the Wattage to Amperage
Calculator, you can determine that the amplifier will draw around 1.67 amps. This information helps you ensure that the circuit breaker connected to your home theater can handle the load, preventing
potential electrical overloads.
Similarly, consider an electrician working on a new building's electrical layout. By using the Wattage to Amperage Calculator, they can quickly evaluate various appliance loads, ensuring that all
circuits are designed to handle the required currents. This process helps in optimal circuit design, enhancing both efficiency and safety.
What is Wattage?
Wattage, measured in watts (W), refers to the amount of power consumed or produced by an electrical device. It represents the rate at which energy is used or generated.
What is Amperage?
Amperage, measured in amperes or amps (A), is the amount of electric current flowing through a circuit. It indicates the quantity of electricity passing through a conductor.
Is the Voltage Input Necessary?
Yes, to convert wattage to amperage, both power (wattage) and voltage values are important. The formula requires knowing the voltage to accurately calculate the current.
Can This Calculator Be Used for AC and DC Circuits?
Yes, this calculator can be used for both AC (alternating current) and DC (direct current) circuits. Just make sure to input the correct voltage for the specific type of circuit you are working with.
What Formula Does the Calculator Use?
The basic formula used by the calculator is: Amperage (A) = Wattage (W) / Voltage (V). This relationship helps determine the current flowing through the circuit.
What if I Don't Know the Voltage?
If the voltage isn't known, it is not possible to accurately convert wattage to amperage. Voltage is a necessary part of the formula; without it, the calculation cannot be completed.
Why Is It Important To Know The Current In A Circuit?
Knowing the current is critical for ensuring the safety and integrity of electrical systems. It helps in selecting appropriate wire sizes, circuit breakers, and other components to prevent
overloading and potential hazards.
Is This Calculator Accurate?
Yes, the calculator provides accurate results based on the input values. However, it depends on the precision of the wattage and voltage values entered. Always use reliable data sources for accurate
Can I Use This Calculator For Solar Power Systems?
Yes, this calculator can be useful for solar power systems in determining the required current for various components. Ensure proper voltage inputs specific to solar setups to get accurate results.
Does This Calculator Take into Account Power Factor?
No, this basic calculator does not consider the power factor, which is more relevant in AC circuits with inductive or capacitive loads. For more precise calculations involving power factor, advanced
tools or formulas are required.
What Safety Precautions Should Be Taken When Working With Electrical Circuits?
Always follow standard safety protocols: turn off power before handling electrical components, use insulated tools, wear protective gear, and ensure that all connections are secure and within the
specified current limits to avoid electrical hazards.
Can This Calculator Help In Reducing Energy Consumption?
While the calculator itself does not reduce energy consumption, it helps in understanding and managing the electrical load effectively. Proper management can lead to more efficient energy use and
potential savings on electricity bills. | {"url":"https://www.onlycalculators.com/other/wattage-to-amperage-calculator/","timestamp":"2024-11-05T20:46:57Z","content_type":"text/html","content_length":"237310","record_id":"<urn:uuid:4ecbfbfb-8845-48d7-a7ee-98f167052890>","cc-path":"CC-MAIN-2024-46/segments/1730477027889.1/warc/CC-MAIN-20241105180955-20241105210955-00819.warc.gz"} |
CBM | Meaning, Calculation, & More
CBM Meaning
The full form of CBM is Cubic Meters. It is one of the most predominantly used units of measurement of cargo transports globally. CBM measures the volume of the shipment being sent by air freight or
ocean freight, which ultimately decides the freight cost of the shipment. CBM measurement is a vital process of transporting shipments and air cargo since the overall transportation cost depends
heavily on it.
Determining the right container size for your consignment helps you move your goods and manage your freight costs more effectively. For this, it’s pertinent to know how many CBM you can store in a
How to calculate CBM in Shipping?
Calculating the CBM for your product is very easy. Just pack it neatly into a cubical/cuboidal box to map the dimensions accurately. Once done, measure the box’s length, width and height in meters.
If you have taken measurements in a unit other than meters, it’s advisable to convert it first and then proceed to calculate the CBM. When you have all the three measurements, multiply them and
you’ll get the CBM value of your package.
The formula to calculate CBM would go as follows:-
The calculation of CBM with this formula only takes into account the dimensions or volume of your shipment. But what if the package you want to ship is too light or too heavy. Shipping companies
use the concept of CBM Chargeable Weight which also factors in the importance of weight of the shipment while arriving at the freight cost.
How do you calculate CBM chargeable weight?
While shipping goods, it often happens that a relatively light package takes up much more space than a heavier yet a smaller one. Hence, if the shipping company levies the charges on both the
packages based on their actual weight, the bigger yet lighter package would not be profitable to ship, since it occupies more space and weighs little.
To be able to solve this problem, companies use the concept of CBM chargeable weight. To understand chargeable weight, we first have to understand the following terms:
1. Actual Weight: Actual weight is the gross weight of the package that is to be shipped.
2. Dimensional/ Volumetric Weight: Once the CBM value of the package is known, multiply it with the Dimensional Weight Factor, or “DIM factor” based on the mode of transportation, to get the
Dimensional or Volumetric Weight of the package.
The highest value between the two is taken into account by the company to charge the shipment. This method is known as the chargeable weight.
DIM Factors for different modes of Shipping
• Ocean Freight : 1:1000
• Air Freight : 1:6000
• Express Freight/Courier : 1:5000
• Truck LTL: 1:3000
How to calculate CBM for Ocean Freight LCL Shipments?
Ocean freight shipping companies have prioritised the space taken up by an LCL shipment in a container over the weight of the shipment. For calculating CBM for LCL shipments sent via ocean freight,
the estimation factor for calculating the volumetric weight is generally 1:1000 -- one cubic meter is equal to about 1000 kilograms.
Example of Ocean Freight cost calculation using CBM
Assume that the international freight forwarders has given you a quote of $15 per CBM or ton. And the DIM factor which is generally used for sea freight as 1:1000. There are two different situations
which can arise, both of them have been explained below:-
1. If the dimensions of a package are 5m length, 5m height, and 5m width while its weight is 500kgs. And the freight forwarder has given you a quote of $15 per CBM or per 1000KG (As per the DIM
factor). CBM = 555 = 125 CBM
Since the weight is less than 1 ton and CBM is greater than weight of the shipment hence it will be considered as the basis for calculation of freight cost. Freight Cost = 125*15 = $1875
2. If the dimensions of a package are 2m length, 1m height, and 3m width while its weight is 7 tons or 7000kgs. CBM = 213 = 6 CBM
Since weight of the shipment exceeds 1 ton and CBM value is less than the weight of the shipment hence weight will be considered as the basis for calculation of freight cost. Freight Cost = 7*15 =
How to calculate CBM for air shipment/ air freight?
In an air shipment the CBM calculation remains the same, but the freight is charged on gross weight or volume weight (after multiplying CBM by DIM factor) -- whichever is higher. The DIM factor
generally used in Air freight is 1:6000, or divide 1 CBM (if dimensions are measured in meters) by 0.006 to get volume weight in KGs.
Volume weight is important for calculating air freight, as lighter shipments consuming more space cannot be charged a lower amount as compared to someone who is sending a heavier shipment. Both
actual weight and volume weight when taken into account help in accurate pricing for air shipments.
For example, If the dimensions of a package are 2m length, 2m height, and 2m width while its gross weight is 500kgs. And the freight forwarder has given you a quote of $1.5 per Volume weight or Gross
weight whichever is higher. CBM = 222 = 8 CBM Volume weight for an air cargo = 8/0.006 = 1333.33 KGs
Volume Weight > Gross Weight, hence volume weight will be considered for calculating air freight cost, i.e 1.5*1333 = $1999.5
Types of containers and their CBMs
Generally, one needs to calculate the CBM of the consignment as well as the container. Standard containers are generally available in 3 sizes- 20ft, 40ft, and 45ft and the dimensions for the variants
are as follows:
20ft Container CBM
• 20′ Dry Container : 33.0 cbm (Dimensions l:5919 mm, w:2340 mm, h:2380 mm, Weight: 1900 kg)
• 20′ Reefer Container : 27.5 cbm (Dimensions l:5428 mm, w:2266 mm, h:2240 mm)
• 20′ Open Top Container : 31.6 cbm (Dimensions l:5919 mm, w:2340 mm, h:2286 mm)
• 20′ Flat Rack Container : (Dimensions l:5662 mm, w:2438 mm, h:2327 mm)
• 20′ Collapsable Flat Rack Container : (Dimensions l:5946 mm, w:2126 mm, h:2233 mm)
• 20′ Open Side/Open Top Container : 31.0 cbm (Dimensions l:5928 mm, w:2318 mm, h:2259 mm)
40 ft Container CBM
• 40′ Dry Container : Dimensions : 67.3 cbm (Dimensions l:12045 mm, w:2309 mm, h:2379 mm)
• 40′ High Cube Dry Container : 76.0 cbm (Dimensions l:12056 mm, w:2347 mm, h:2690 mm)
• 40′ Reefer Container : 54.9 cbm (Dimensions l:11207 mm, w:2246 mm, h:2183 mm)
• 40′ High Cube Reefer Container : 66.9 cbm (Dimensions l:11628 mm, w:2294 mm, h:2509 mm)
• 40′ Open Top Container : 64.0 cbm (Dimensions l:12043 mm, w:2340 mm, h:2272 mm)
• 40′ Flat Rack Container : (Dimensions l:12080 mm, w:2438 mm, h:2103 mm)
• 40′ Collapsable Flat Rack Container : (Dimensions l:12080 mm, w:2126 mm, h:2043 mm)
45 ft Container CBM
• 45′ High Cube Dry Container : 85.7 cbm (Dimensions l:13582 mm, w:2347 mm, h:2690 mm)
• 45′ High Cube Reefer Container : 75.4 cbm (Dimensions l:13102 mm, w:2294 mm, h:2509 mm)
Calculating CBM in Garments
The fashion industry is one of the most frequent users of both air as well as ocean freight routes to transport raw materials, equipment, and final products across the world. While exporting
garments, companies pack them up in cartons that are smartly designed not to take up a lot of space, and easily stacked on top of each other.
Once the garments are packed in standard cartons, calculating their CBM becomes very easy. Just putting the accurate measures of all the dimensions in the below formula would give you the total CBM
for your package.
Length of the carton (m) X Breadth of the carton (m) X Height of the carton (m) X Number of cartons in the package = Total CBM of the package
FAQs on CBM
How do you convert KG to CBM?
Considering an assumption, kg (of mass) to CBM (of volume) - the SI derived unit for volume is the cubic meter.
Note: Rounding errors may occur, so always check the results according to your apt figures.
How do you calculate CBM in inches?
1 inch is 0.0254 meters. So if the individual measurement of your package is in inches, make sure you multiply each one of them with 0.0254, and then proceed to calculate the CBM.
However, if you have already measured your package in terms of cubic inches, no need to go back and convert the individual dimensions again. Just use the formula given below to convert cubic inches
to cubic meters.
How many CBM in a pallet?
Pallets are the small wooden platforms upon which packages are stacked up inside the containers so that the goods suffer from no damage during their movement.
Also Read: | {"url":"https://www.dripcapital.com/en-in/resources/blog/how-to-calculate-cbm","timestamp":"2024-11-05T06:10:29Z","content_type":"text/html","content_length":"89716","record_id":"<urn:uuid:124e7323-f304-45f6-92dd-2c62ccdf028c>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00679.warc.gz"} |
PSLE Prep Lessons July to Sept | Math Note Tuition
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Master PSLE Math with Confidence!Are you ready to excel in your PSLE Math exam?
Our comprehensive PSLE Prep Math Course is designed to help you master essential math concepts, improve problem-solving skills, and build confidence for exam day.
Why Choose Our PSLE Math Prep Course?
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• Comprehensive Practice: Our course includes ample practice papers and mock exams to familiarize your child with the PSLE format and timing, reducing exam-day anxiety.
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Lesson 1: Numbers and Operations (Numbers up to 10 Million and Operations of Whole Numbers)
• Introduction to Numbers up to 10 Million
• Place Value and Reading Large Numbers
• Addition and Subtraction of Whole Numbers
• Multiplication and Division of Whole Numbers
• Factors and Multiples
• Order of Operations of Whole Numbers
• Word Problems Involving Operations with Whole Numbers
Lesson 2: Multiplication and Division (Multiplication of Whole Numbers, Fractions, and Mixed Numbers)
• Understanding Fractions and Mixed Numbers
• Conversion Between Improper Fractions and Mixed Numbers
• Addition and Subtraction of Fractions and Mixed Numbers
• Multiplication and Division of Fractions and Mixed Numbers
• Word Problems Involving Fractions and Mixed Numbers
Lesson 3: Decimals
• Understanding Decimals
• Place Value in Decimals
• Addition and Subtraction of Decimals
• Multiplication and Division of Decimals
• Rounding off Decimals
• Word Problems Involving Decimals
Lesson 4: Ratio and Percentage (Ratio and Percentage)
• Understanding Ratios and Their Applications
• Calculating Percentages and Solving Percentage Problems
• Applications of Ratio and Percentage in Real-Life Scenarios
Lesson 5: Algebra Fundamentals (Algebra)
• Introduction to Algebraic Expressions
• Translating Word Problems into Algebraic Expressions
Lesson 6: Speed and Rate (Rate and Speed)
• Understanding Rate and Speed
• Distance-Time Relationships and Calculations
• Problem-Solving Involving Speed and Rate
Lesson 7: Areas and Volume (Area of Triangles, Circles, Volume of Cubes and Cuboids, Volume of Solids and Liquids)
• Calculating the Area of Triangles
• Calculating the Area and Perimeter (Circumference) of Circles
• Calculating the Volume of Cubes and Cuboids
• Volume of Other Solid Shapes and Liquids
• Practical Applications and Problem-Solving with Area and Volume
Lesson 8: Angles (Properties of Angles, Triangles, Quadrilaterals)
• Properties of Angles
• Properties of Triangles and Quadrilaterals
• Problem-Solving Involving Angles and Geometric Figures
Lesson 9: Data Representation (Line Graphs, Pie Charts, and Bar Graphs)
• Reading and Interpreting Line Graphs, Pie Charts and Bar Graphs
• Analyzing Data from Line Graphs, Pie Charts, and Bar Graphs
• Drawing Conclusions and Making Predictions Based on Data
Lesson 10: Review and Exam Preparation
• Review of All Topics Covered in Previous Lessons
• Practice Problems and Exam-Style Questions
• Tips and Strategies for Exam Success
Workshop Details
Course Period
• Dates: August 1st to September 26th, 2024 (Note: No lesson from 26th August to 3rd September)
• Frequency: Once or twice a week (Note: Each lesson will cover one topic. In the event where fewer than 10 weeks remain before the week preceding the PSLE, sessions will occur twice weekly)
Class Timings
• Thursday: 5:00 PM
• Saturday: 12:00 PM
• Sunday: 2:30 PM
Course Fees
• Regular Rate: $80 per 2-hour lesson
• Current student: $75 per 2-hour lesson
• Early Bird Discount: $75 per lesson (sign up before 30th July)
• Special Offer: Bring a friend and both enjoy $80 off your lesson fees
Materials Fee:
$15 {non-refundable} for all students
• Venue: Bishan (within 5 minutes walk from Bishan MRT)
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