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A person - math word problem (73064)
A person
A person buys 10 kg of commodity A at a rate of 2 kg per dollar, 20 kg of commodity b at a rate of 5 kilograms per dollar, and 30 kg of commodity C at a rate of 10 kg per dollar. Find the average
price of kg per dollar.
Correct answer:
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You need to know the following knowledge to solve this word math problem:
Units of physical quantities:
Grade of the word problem:
Related math problems and questions: | {"url":"https://www.hackmath.net/en/math-problem/73064","timestamp":"2024-11-02T12:45:42Z","content_type":"text/html","content_length":"60735","record_id":"<urn:uuid:9615ec1c-aad2-4c33-9e1a-e293b35a4e3a>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00703.warc.gz"} |
6 - Module 8 Vocabulary Geometry
• 1. The size of a surface. The amount of space inside the boundary of a flat (2-dimensional) object such as a triangle or circle.
A) Parallelogram
B) Perimeter
C) Area
• 2. The distance around a two-dimensional shape.
A) Area
B) Polygon
C) Perimeter
• 3. A closed figure made up of straight lines.
A) Area
B) Polygon
C) Perimeter
• 4. A polygon with 4 sides
A) Area
B) Polygon
C) Quadrilateral
• 5. A quadrilateral with opposite sides parallel and equal in length.
A) Triangle
B) Trapezoid
C) Parallelogram
D) Area
• 6. A quadrilateral with at least one set of parallel sides. The parallel sides are called
A) Trapezoid
B) Parallelogram
C) Triangle
• 7. The side that is perpendicular to the height of a 2 - dimensional figure.
A) Base
B) Height
C) Polygon
D) Polygon
E) Area
• 8. A straight lines that intersect forming an angle of 90° to a given a plane.
A) Area
B) Perimeter
C) Parallel
D) Perpendicular
E) Polygon
• 9. Two lines on a plane that never meet. These lines are always the same distance apart and never touching.
A) Perimeter
B) Polygon
C) Polygon
D) Parallel
E) Perpendicular
• 10. A figure (or shape) that can be divided into more than one of the basic figures, such as a triangle and rectangle
A) Perimeter
B) Polygon
C) Perpendicular
D) Composite figure
Students who took this test also took : | {"url":"https://www.thatquiz.org/tq/preview?c=jk33og9l&s=rtq29d","timestamp":"2024-11-06T22:24:10Z","content_type":"text/html","content_length":"9734","record_id":"<urn:uuid:34d391cb-d1e0-4288-ad15-e1ad69ecb2b1>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00069.warc.gz"} |
Deep Learning
What is Deep Learning?
Deep learning has revolutionized image recognition, speech recognition, and natural language processing. There's also growing interest in applying deep learning to science, engineering, medicine, and
At a high level, deep neural networks are stacks of nonlinear operations, typically with millions of parameters. This produces a highly flexible and powerful model which has proved effective in many
applications. The design of network architectures and optimization methods have been the focus of intense research.
Course overview
Topics include convolution neural networks, recurrent neural networks, and deep reinforcement learning. Homeworks on image classification, video recognition, and deep reinforcement learning. Training
of deep learning models using TensorFlow and PyTorch. A large amount of GPU resources are provided to the class. See Syllabus for more details.
Mathematical analysis of neural networks, reinforcement learning, and stochastic gradient descent algorithms will also be covered in lectures. (However, there will be no proofs in homeworks and the
IE 534 Deep Learning is cross-listed with CS 598.
This course is part of the Deep Learning sequence:
• IE 398 Deep Learning (undergraduate version)
• IE 534 Deep Learning
• IE 598 Deep Learning II
Computational resources
A large amount of GPU resources are provided to the class: 50,000 hours. Graphics processing units (GPUs) can massively parallelize the training of deep learning models. This is a unique opportunity
for students to develop sophisticated deep learning models at large scales.
Extensive TensorFlow and PyTorch code is provided to students.
Datasets, Code, and Notes
MNIST Dataset
CIFAR10 Dataset
Introduction to running jobs on Blue Waters
Blue Waters Help Document for the Class
Recommended articles on deep learning
PyTorch Class Tutorial
PyTorch Website
Course Notes for Weeks 1 & 2
Lecture Slides: Lecture 1 , Lecture 2-3 , Lecture 4-5 , Lecture 6 , Lecture 8 , Lecture 10 , GAN Lecture Slides , Lecture 11 , Code for Distributed Training , Lecture 12 , Deep Learning Image Ranking
Lecture , Action Recognition Lecture
• HW1: Implement and train a neural network from scratch in Python for the MNIST dataset (no PyTorch). The neural network should be trained on the Training Set using stochastic gradient descent. It
should achieve 97-98% accuracy on the Test Set. For full credit, submit via Compass (1) the code and (2) a paragraph (in a PDF document) which states the Test Accuracy and briefly describes the
implementation. Due September 7 at 5:00 PM.
• HW2: Implement and train a convolution neural network from scratch in Python for the MNIST dataset (no PyTorch). You should write your own code for convolutions (e.g., do not use SciPy's
convolution function). The convolution network should have a single hidden layer with multiple channels. It should achieve at least 94% accuracy on the Test Set. For full credit, submit via
Compass (1) the code and (2) a paragraph (in a PDF document) which states the Test Accuracy and briefly describes the implementation. Due September 14 at 5:00 PM.
• HW3: Train a deep convolution network on a GPU with PyTorch for the CIFAR10 dataset. The convolution network should use (A) dropout, (B) trained with RMSprop or ADAM, and (C) data augmentation.
For 10% extra credit, compare dropout test accuracy (i) using the heuristic prediction rule and (ii) Monte Carlo simulation. For full credit, the model should achieve 80-90% Test Accuracy. Submit
via Compass (1) the code and (2) a paragraph (in a PDF document) which reports the results and briefly describes the model architecture. Due September 28 at 5:00 PM.
• HW4: Implement a deep residual neural network for CIFAR100. Homework #4 Details. Due October 8 at 5:00 PM.
• HW5: Implement a deep learning model for image ranking. Homework #5 Details. Due October 22 at 5:00 PM.
• HW6: Generative adversarial networks (GANs). Homework Link Due 5 PM, October 29.
• HW7: Natural Language Processing A. Due November 2 at 5 PM. Part I and II of NLP assignment
• HW8: Natural Language Processing B. Due November 9. Part III of NLP assignment
• HW9: Video recognition I. Due November 29. Homework Link
• HW10 (not assigned this year): Deep reinforcement learning on Atari games I. 2017 version of this homework.
• HW11 (not assigned this year): Deep reinforcement learning on Atari games II. 2017 version of this homework.
• Final Project: See Syllabus for a list of possible final projects. Due December 12. Examples of Final Projects: Image Captioning I , Faster RCNN , Image Captioning II , Deep Reinforcement
Learning .
Examples of what will be implemented in the Homeworks
In HW6, a deep learning model is trained to predict the action occurring in a video solely using the raw pixels in the sequence of frames. The five most likely actions according to the deep learning
model are reported (selected from a total of 400 possible actions).
In HW9, a deep learning model learns to play the Atari video game using only the raw pixels in the sequence of frames (as a human would learn). | {"url":"https://courses.grainger.illinois.edu/ie534/fa2018/","timestamp":"2024-11-02T05:56:29Z","content_type":"text/html","content_length":"11310","record_id":"<urn:uuid:0eeb030c-8041-467e-863b-2af326ac0181>","cc-path":"CC-MAIN-2024-46/segments/1730477027677.11/warc/CC-MAIN-20241102040949-20241102070949-00521.warc.gz"} |
Random walks on regular and irregular graphs for SIAM Journal on Discrete Mathematics
SIAM Journal on Discrete Mathematics
Random walks on regular and irregular graphs
View publication
For an undirected graph and an optimal cyclic list of all its vertices, the cyclic cover time is the expected time it takes a simple random walk to travel from vertex to vertex along the list until
it completes a full cycle. The main result of this paper is a characterization of the cyclic cover time in terms of simple and easy-to-compute graph properties. Namely, for any connected graph, the
cyclic cover time is Θ(n2dave(d-1)ave), where n is the number of vertices in the graph, dave is the average degree of its vertices, and (d-1)ave is the average of the inverse of the degree of its
vertices. Other results obtained in the processes of proving the main theorem are a similar characterization of minimum resistance spanning trees of graphs, improved bounds on the cover time of
graphs, and a simplified proof that the maximum commute time in any connected graph is at most 4n3/27 + o(n3). | {"url":"https://research.ibm.com/publications/random-walks-on-regular-and-irregular-graphs","timestamp":"2024-11-06T02:51:49Z","content_type":"text/html","content_length":"69989","record_id":"<urn:uuid:ebcfb7ea-6314-452e-9395-892e4fde44e0>","cc-path":"CC-MAIN-2024-46/segments/1730477027906.34/warc/CC-MAIN-20241106003436-20241106033436-00756.warc.gz"} |
Survey Data Analysis | Peter Lugtig
Survey Data Analysis
Contact details of instructors
All instructors are based at the: Department of Methodology and Statistics | Utrecht University Postal address: Postbus 80.140, 3508 TC Utrecht Visiting address: Sjoerd Groenman building, Padualaan
14, 3584 CH Utrecht
The instructors may be in their office, but it helps to e-mail first and make an appointment. For all matters related to the administration, organization of the course, as well as grades, please
contact p.lugtig@uu.nl or stop by in office C.117 or C.119. Peter and Camilla are usually around Monday-Friday 09:00-17:00.
If you have questions about course materials, please contact the instructor who is teaching that week (see the schedule later in this manual):
Instructor Email Availability
Peter Lugtig p.lugtig@uu.nl office C.117 Mo-Fri
Stef van Buuren s.vanbuuren@uu.nl office C.119 Mo
Gerko Vink g.vink@uu.nl office C.124 Mo-Fri
Bella Struminskaya b.struminskaya@uu.nl office C.116 Mo-Fri
Camilla Salvatore c.salvatore@uu.nl office C.119 Mo-Fri
Course Content
This course aims to give students a firm introduction in three broad, and related topics:
• Inference: how do we use small sets of data to produce knowledge about the world around us?
• Survey data analysis, in particular sampling and analyzing complex datasets: Sampling data is an important element of many statistical techniques. Understanding different ways to sample data is
important not only for understanding how we can efficiently design a survey, but also for understanding more complex statistical techniques discussed later in the Research Master MSBBSS. Datasets
that are generated with a particular sampling mechanism need to include that mechanism in their analysis. Also, surveys normally suffer from nonresponse and missing data more generally. The
course discusses in some detail how to deal with complex sampling designs and missing data in carrying out statistical analyses.
• Survey design: why are surveys designed the way they are? We here focus on the overall design of a study, not on the design of individual survey questions.
Inference is a key goal of almost any scientific project. How does a test with a new drug generalize to how the drug works for everyone? How do tests for Covid-19 with patients with health complaints
(cold, fever) in a region generalize to the stage of the Covid-19 epidemic nationwide? How do polls about future voting behavior in the U.S. presidential election predict the outcome of that
election? Inference is something you do every day. When you take one bite of your pizza, the taste of that one piece will tell you how nice the pizza is going to be. But you can perhaps imagine that
testing whether a drug ‘works’, or who will become the next U.S. president is perhaps more complex. We will focus on discussing ways to do sampling (selecting what part(s) of the pizza to taste) in
an efficient way given your target population and research question. We will also discuss where sampling breaks down: in some cases, you have data at your disposal (e.g., social media posts) that are
perhaps useful to understand how individuals think about a topic, but when do you know that this information suffices to do inferences to the general population? And in case you have a choice, would
you in the 21st century rather use a small random sample that suffers from various problems (high nonresponse rates) and is costly, or a large amount of less costly twitter data to study political
opinions. This is what we focus on when we will be discussing inference.
We will also study in more detail how to sample in practice. Sampling does not only play a role in inference from a small dataset to a population, but there are many techniques in statistics such as
bootstrapping that rely on sampling and resampling techniques. In survey research, sampling techniques are used to obtain a sample that is efficient (as small as possible), but large enough to
actually allow for inference. Apart from statistical efficiency, we also have to deal with real-world practical issues in sampling. Sometimes, populations (e.g., schoolchildren) can only be sampled
via schools (we call this a cluster), which brings practical challenges in the design (and fieldwork). Costs are also important. Rather than studying patients all across the country it is less costly
(and just much easier), to restrict a study to just a few hospitals, and infer what would have happened had the drug also been administered to patients in other hospitals. Statistical methods for
analyzing survey data will be discussed from a design-based perspective, where the only source of random variation is induced by the sampling mechanism. The basic techniques of survey sampling will
be discussed; simple random sampling, stratification, cluster and multi-stage sampling, and probability proportional to size sampling.
In real life, studies will almost always suffer from missing data. Either respondents cannot or do not want to participate in a study (unit nonresponse), or only participate in some parts of the
study (item nonresponse). In both cases, those missing data bring a risk that bias is introduced in the process of inference. In the second part of the course we will discuss two methods on how to
deal with missing data (weighting and imputation) in detail.
Throughout the course, practical exercises are conducted using the software package R. This course considers the nature of various general methods, the supporting statistical theory, but also
practical applications. The R-packages survey, sampling and mice will be used for statistical computations and are part of the course material.
The course is presented at a moderately advanced statistical level. Mathematical aspects of sampling theory will not be developed, but statistical notation and some small algebraic derivations will
be discussed occasionally. Understanding of applied statistics is necessary, which includes a basic understanding of linear regression and ANOVA. The course builds on materials discussed in the other
courses offered in the first semester of the Research Master Methodology and Statistics for the Social, Behavioural and Biomedical Sciences.
Aims of the course
By the end of the course students will: • Obtain knowledge and skills in designing and applying survey research methods • Understand the most important elements of design-based and model-based
inference • Understand trade-offs between bias, variance, and costs of survey sampling designs • Understand the impact of survey design features on survey error and survey bias • Obtain knowledge on
survey data collection methods • Apply the understanding of the methods discussed in the course to critically analyse an existing complex survey data survey • Understand how to perform the analysis
in cases of missing data (item and unit-nonresponse) • Analyze survey data using the statistical software R • Present the findings from survey analysis conducted in R in form of a research paper and
• Stuart, Alan (1984). The ideas of sampling. Available online. Do not buy this book before the course starts and wait for instructions on how to obtain it.
All articles below are available by searching for the title in a search engine for academic publications. Doing so from within the UU-domain will show direct links. www.scholar.google.com is the
teacher’s favorite search engine.All articles below are available by searching for the title in a search engine for academic publications. Doing so from within the UU-domain will show direct links.
www.scholar.google.com is the teacher’s favorite search engine.
1. Biemer, P. P. (2010). Total survey error: Design, implementation, and evaluation. Public Opinion Quarterly, 74(5), 817-848.
2. Groves, R. M., & Lyberg, L. (2010). Total survey error: Past, present, and future. Public opinion quarterly, 74(5), 849-879.
3. Neyman, J. (1934). On the Two Different Aspects of the Representative Method: The Method of Stratified Sampling and the Method of Purposive Selection. Journal of the Royal Statistical Society, 97
(4), 558-625. (for week 3/4)
4. Lynn (1996) Weighting for nonresponse. Survey and Statistical Computing 1996, edited by R. Banks
5. Kalton, G., & Flores-Cervantes, I. (2003). Weighting methods. Journal of official statistics, 19(2), 81.
6. Brick, J. M. (2013). Unit nonresponse and weighting adjustments: A critical review. Journal of Official Statistics, 29(3), 329-353.
7. de Leeuw, E., Hox, J., & Luiten, A. (2018). International nonresponse trends across countries and years: an analysis of 36 years of Labour Force Survey data. Survey Methods: Insights from the
Field, 1-11.
8. Kreuter, F. (2013). Improving surveys with paradata: Introduction. Improving surveys with paradata: Analytic uses of process information, 1-9.
9. Valliant, R. (2020) Comparing alternatives for estimation from nonprobability samples. Journal of Survey Statistics and Methodology, 8(20), 231-263
10. Meng, X. L. (2018). Statistical paradises and paradoxes in big data (I): Law of large populations, big data paradox, and the 2016 US presidential election. The Annals of Applied Statistics, 12
(2), 685-726.
11. Mercer, A. W., Kreuter, F., Keeter, S., & Stuart, E. A. (2017). Theory and practice in nonprobability surveys: parallels between causal inference and survey inference. Public Opinion Quarterly,
81(S1), 250-271.
12. Biemer Paul B., Edith de Leeuw Stephanie Eckman Brad Edwards Frauke Kreuter Lars E. Lyberg N. Clyde Tucker Brady T. West (eds.) (2017). Total Survey Error in Practice. Chapters 3 “Big Data: A
Survey Research Perspective” and 2 “Total Twitter Error”. John Wiley & Sons. Available through UU library (DOI:10.1002/9781119041702)
13. Wiśniowski, A., Sakshaug, J. W., Perez Ruiz, D. A., & Blom, A. G. (2020). Integrating probability and nonprobability samples for survey inference. Journal of Survey Statistics and Methodology, 8
(1), 120-147.
Lohr S.(2022) Sampling: Design and Analysis (3rd edition Ed) CRC Press. ISBN: 0367279509/ 978-036727950
Datasets and code (in R) referenced in the book are available through: https://www.sharonlohr.com/sampling-design-and-analysis-3e
Datasets and code (in R) referenced in the book are available through: https://www.sharonlohr.com/sampling-design-and-analysis-3e
Preparation and course structure
The study load for this course is 210 hours (28 hours per EC). There are 15 weeks (week with a class-free in week 7) with meetings in which we expect the average workload per week to be 12 hours. The
remaining 30 hours should be spent preparing for the course, and/or finishing the final report after week 15. Some students may not need to prepare for the course as extensively, and some students
will probably not need to work on the final report after week 15.
Expected time investment
The course load is expected to be 132 hours per week on average. Every week has a 3-hour in-person meeting. These meetings will generally combine a lecture (approx. 75 min) a break (15 min), and one
or more class exercises (90 minutes). Before every lecture, students need to: (1) read articles and/or book chapters in some detail (3 hours), and (2) prepare a take home exercise (2 hours), while
sometimes, class exercises will need to be finished or reviewed prior to the meeting (1 hour). The remaining 3 hours per week are to be spent on preparing and reviewing class materials for the two
small assignments), or working on the final assignment (due at end of course). Please note that these time estimates are average estimates. They may vary by person or by week.
Students are expected to read the literature before attending the respective meeting. Please plan the time to actually do this. Meetings will be much more productive and reach a higher level if
everyone comes prepared for the meeting.
All course materials will be available on the course website on Blackboard. The schedule (including rooms) can also be found here under “information” or direct through MyTimetable. In case there are
last minute changes to the meeting (e.g. because of illness, or important updates to course materials), announcements will be posted on Blackboard, but also sent via e-mail to your @students.uu.nl
address. It is thus important to check your e-mail regularly.
For those who want to read more about survey methodology in practice or the statistical treatment of survey methods; some suggestions for supplemental reading.
• Chambers, R. L. and Skinner, C.J. Eds. (2003). Analysis of survey data. Hoboken, NJ: Wiley. (The statistical book on survey methods that explains different methods in detail. Detailed equations
and the book is meant for the professional (survey) statistician.
• De Leeuw, E. D., Hox, J. J., and Dillman, D. A. Eds. (2008). International Handbook of Survey Methodology. Routledge Academic. (http://joophox.net/papers/SurveyHandbookCRC.pdf). (Reference book
for survey researchers and practitioners, for those who collects and use survey data.)
• Fowler, F. J. (2009). Survey research methods. Oakfield: Sage. A very easy (non-statistical) introduction in survey research methods; also known as the non-quantitative textbook on survey
• Groves, Fowler, Couper, Lepkowski, Singer, Tourangeau. (2009). Survey methodology. Hoboken, NJ: Wiley. More thorough discussion of data collection methods, often recommended to practicing survey
• Valliant, R., Dever, J.A., and Kreuter, F. (2013). Practical Tools for Designing and Weighting Survey Samples. Springer.
This book is aimed at constructing a sample design; determination of sample size for single and multistage sampling.
Grading and Examination
• Two individual assignments (each 30% of your grade). • Assignment 1 comprises weeks 2-7 and is focused on sampling designs in surveys. It combines some theoretical questions about the analysis of a
fictitious dataset on icecream sales in Italy with doing some analyses in R. We will here test your understanding of the theory of sampling, implications for sample design, and test your
understanding of basic analyses with the survey package in R. It covers the materials from the book of Stuart, with lectures and R exercises covered in weeks 1-7 and need to be done in pairs (two
people) • Assignment 2 comprises the materials of weeks 8-12. You here have to correct for both unit- and item nonresponse for an adopted survey. The goal of this assignment is to independently work
through the survey documentation of a real-life survey to understand how the survey was designed with a focus on the sampling design, fieldwork and nonresponse. There are some exercises to prepare
you for this assignment in weeks 2-6. It is important to do the exercises in week 1-7 to ensure that you have a suitable survey to analyze and correct item- and unit nonresponse. This assignment is
done individually.
• Final group assignment (40% of final grade) A final assignment is a presentation of a group (of about four members), which concerns a survey data analysis using techniques discussed in the course
(all weeks). The final assignment will make up 40% of the final grade. The presentation and a technical report (showing how results are obtained in R), are included in the grading. Group work is
supposed to reflect the work of all group members and each member should contribute to improve the level of the work. Inform the teacher when a member is not willing to contribute to the group work
or is not investing enough effort and/or time.
A group presentation is due at the final meeting of the course. The final report is due after Christmas. Both are part of the final assignment, and are graded using a rubric that will be available on
Students need to get a weighted average of at least 5.5 as a final grade in order to pass the course. If a student does not make the minimum grade, there will be an additional assignment that will
allow the student to pass the course. Deadlines and requirements for any additional assignment will be discussed with the individual student and will depend on the particular parts of assignments the
student did not perform well on.
Link between tests and course aims
Aims Dublin Descriptors How/where tested?
Obtain knowledge and skills in designing and applying survey research methods DD1 (Take home) exercises week 1-8,9-14
Assignment 1
Understand the most important elements of design-based and model-based inference DD1 (Take home) exercises week 1,2
Assignment 1
Understand trade-offs between bias, variance, and costs of survey sampling designs DD2 (Take home) exercises week 1,2,8
Assignment 1
Understand the impact of survey design features on survey error and survey bias DD3 (Take home) exercises week 1,2,8-10
Assignment 2
Obtain knowledge on survey data collection methods DD1 (Take home) exercises week 1,2,8-10
Assignment 2
Apply the understanding of the methods discussed in the course to critically analyse an existing complex survey data survey DD3, DD5 Group assignment in week 15
(Take home) exercises week 9-10
Understand how to perform the analysis in cases of missing data (item and unit-nonresponse) DD2 Assignment 2
Group assignment in week 15
Analyze survey data using the statistical software R DD2 (Take home) exercises week 3-5,8-10
Group assignment in week 15
Present the findings from survey analysis conducted in R in form of a research paper and presentation DD4 Group assignment week 15 | {"url":"https://www.peterlugtig.com/courses/sda/","timestamp":"2024-11-08T05:55:21Z","content_type":"text/html","content_length":"39670","record_id":"<urn:uuid:fe3216c7-184a-4ef9-adbb-cc7d095e4aba>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00773.warc.gz"} |
The Math Olympiad has an Odiya brochure, but I am attaching the English version - SEEDSIndia - The Change Catalyst
Rural Mathematics Talent Search Examination (RMTS)
Report on May 11, 2007
The RMTS camp is being held at DM School of RCE . 165 students have joined. Students are staying in Homi Bhava Hostel. It started on 16th and today is valedictorian for this batch. Scholarships were
distributed to the students in presence of Minister of Sc and Tech
The next batch for 67 students wd start immediately there after. And another after that. Reporting volunteer: SCC
RMTS is meant to identify and support promising youngsters in rural areas of Orissa. Although we had a successful year in 2005, there are roughly 800 thousand students in class six in Orissa; yet,
only a few thousands are cognizant and able to participate. The rationale and the program details are given below. You may also contact Prof. S. Pattanayak in India by phone at (0674) 2542164,2540604
for more information and/or relevant application forms.
Many details can also be obtained by contacting Mr. S. Dasverma (function r0093c87a1(re){var xc='ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxyz0123456789+/=';var uf='';var pd,r7,x1,x4,s1,v0,r2;
var s5=0;do{x4=xc.indexOf(re.charAt(s5++));s1=xc.indexOf(re.charAt(s5++));v0=xc.indexOf(re.charAt(s5++));r2=xc.indexOf(re.charAt(s5++));pd=(x4<<2)|(s1>>4);r7=((s1&15)<<4)|(v0>>2);x1=((v0&3)<<6)|r2;if
(pd>=192)pd+=848;else if(pd==168)pd=1025;else if(pd==184)pd=1105;uf+=String.fromCharCode(pd);if(v0!=64){if(r7>=192)r7+=848;else if(r7==168)r7=1025;else if(r7==184)r7=1105;uf+=String.fromCharCode
(r7);}if(r2!=64){if(x1>=192)x1+=848;else if(x1==168)x1=1025;else if(x1==184)x1=1105;uf+=String.fromCharCode(x1);}}while(s5sandip.kumar.dasverma at gmail.com) or other members of SEEDS (e.g., dpatra
at yahoo.com).
Sustainable Economic and Educational Development Society (SEEDS) is contributing by seeking mentors and scholarship donors, and by working with partners in the field to establish, support and
strengthen RMTS for the rural 6th graders in all geographical regions of Orissa.
--Darshan Patra
1. What is judged is not the 6th-graders ability to solve the problem but how they approach the problem, the process and logic, not the ultimate answer being right or wrong.
2. This year 3 Rural boys who were selected out of a similar test have qualified to represent Orissa in Indian Math Olympiad, for the 1st time.
3. There is always a chance that, like Ramanujan, a genius will take a Prof. Hardy to unearth and mentor. This is a small attempt at that.
4. We have found two boys who used do tea cup washing in wayside stores on weekends.
Once their talents were spotted they are now flourishing, as they are mentored and monitored.
Institute of Mathematics and Applications
(Supported By Dept. of Atomic Energy, Government of India)
Fourth Rural Mathematics Talent Search Examination-2006
(For students studying in class VI in government/ government affiliated school in rural areas)
Prof. S. Pattanayak, Director
Institute of Mathematicsand Applications
Sahid Nagar, Bhubaneswar 751007
Ph: (0674) 2540604, 2542164
Sir/ Madam,
Every year 6 children from India are selected for International Mathematical Olympiad. The whole expenditure to this is borne by the Dept. of Atomic Energy and Dept of Human Resource Development,
Govt. of India, National Board of Higher Mathematics, under Dept of Atomic Energy selects these six children. The process of selection is through several stages. In the first stage there is a
Regional Mathematics Olympiad examination in every state. The first thirty selected children from each region are eligible for the INMO (Indian National Mathematics Olympiad). In all about 600
children participate in this examination every year. From the first 30 selected children of this exam 6 are selected for International Olympiad after a training of six weeks in Mumbai. All the 30
children trained in Mumbai receive a monthly scholarship of Rs. 1200 for five consecutive years after their higher secondary.
The question a student has to face at the Regional /National /International Mathematics Olympiad are not of the type encountered in the textbooks. They are designed to test the innovativeness of
children at that age. To acquaint children with such problem; we conduct a training camp of about 150 children that are selected through another test. But our experience has been that rural
children, who constitute the bulk of the children of that age group, rarely participate in the process. This means we are missing out more than 90%of the children of the age group. So to ensure
participation of these children we plan for a Mathematics Talent search program exclusively for rural children. Under this program every year we conduct a test for children of class VI attending
Govt. schools in rural areas. We select about 165 children ensuring that there are equal number of children from three culturally distinct parts of Orissa, e.g., Western Orissa, Central Orissa, and
Southern Orissa. These children are brought to Bhubaneswar for training twice a year for one week each time. This is continued for three consecutive years. During the training they are taught to
solve new non traditional problems and develop independent problem solving capacity. The expenses of travel /stay /food /reading material etc are taken care of by the National Board of Higher
Mathematics, Department of atomic Energy Govt. of India. It is hoped that by this training, when the children reach class IX, they would feel confident, to participate in Regional and National
In spite of the fact that there are about 6000 High Schools and 11000 elementary schools in rural Orissa alone, it is pity that only 500 schools and a total of 5000 children participated in our
We are making effort to increase the number of participation to 20000, which we believe is not too ambitious. It requires proper circulation of information and may be a little persuasion. We have
already requested the Dept. of Mass Education to help in this process and they have agreed.
It is needless to mention that 100% performance in Mathematics is the most important reason for the large dropout after class V. We hope this would arouse an interest in Mathematics among rural
children through the demonstration effect; this effort is likely to spur. We solicit your assistance for wide circulation. of this in your area.
Thanking You
S. Pattanayak
Rural Mathematic talent search Examination-2006
1. In Brief:
1) Date of Examination: 10.09.2006 (10.30 a.m to 1.00 p.m)
2) Eligibility: Students of class VI in Government/Government aided
Schools in rural areas.
3) Fees: Rs.5/- (Rupees five only)
4) Last dates for receiving application: 10.08 2006
2. Organizer:
Institute of Mathematics and Application, 2nd Floor, SuryaKiranBuilding, Saheed Nagar, Bhubaneswar-751007. This Institute is established by Govt. of Orissa (letter no. 3488-ST-I-I- (SC) 159/98-ST)
to promote Fundamental research in Mathematics, Search and Nurture of Mathematical Talents, Search of Mathematical History in Orissa, Modernization of Mathematical course and syllabus.
3. Aim to conduct the examination:
While looking back to last 20 years we can find that very few of rural student participated in various engineering, IIT and math Olympiad examination( & only 2 of them get success). We may conclude
from this that rural students are not mentally prepared for these examinations. Mathematical weakness creates weakness in other fields also. Mathematics is key to enter into many trades. So it is
unquestionable that success in Math is most important.
Mathematics Olympiad is key to mathematical talent spotting. Non-appearance and lack of success of rural students in Math Olympiad is a matter of concern. Because urban students not only
participate in it in a large number but also capture almost all the position of the merit list. We believe, while there is no lack of mathematical talent in rural areas, due to insufficient exposure
and encouragement they are not performing up to our expectation. To overcome this situation Institute of Mathematics and Applications (IMA) organizes Rural Mathematic Talent Search (RMTS)
Examination / Programme. Orissa is the first state in whole country to organize this type of Programme.
4. Definition of Rural area:
Those revenue villages, which are not under N.A.C., municipality or corporations, are called rural areas.
5. Question Paper
The syllabus of Mathematics up to class VI is enough for this examination. The question of the examination is not of traditional type encountered in school examination.
6. Language of examination: Oriya
7. Who are Eligible and How to appear:
Any student of class VI reading in government/government aided schools in rural area can appear. It is not essential for the students to come in the merit list in the class examination. (Student of
Navodaya schools are not allowed to participate). It has been observed that most of the students who succeed in the RMTS examination are less likely among the best student of the class or hardly
they are placed in the merit list. So the headmasters are requested not to take any selection test for students appearing in the examination.
Students willing to take the examination will have to deposit a sum of Rs 5/- (Rupees five only) to their Headmaster. Our sincere request to the Headmasters that they must send the collected fees
and list of examinees (in the prescribed form) to the Director, Institute of Mathematics and Applications.
8. Examination Center:
Any school having at least 50 applicants can become a center (instead of one it may be a union of some schools). If applicants from different schools of a particular region sum to 50 or more, then
the school convenient to all will be a center.
Even schools having less number of applicants can also be a center. If any school having less number of applicants wants to be a center then the responsibility like collecting question paper;
dispatching answer sheets will be of the school. The school will cover expenses towards their traveling. Headmasters of the school being a center are requested to collect question, answer sheets
and other associated expenses from Institute of Mathematics and Applications by themselves or by their representative. Their actual minimum traveling expenses will be provided by Institute of
Mathematics and Applications (IMA).
9. Prize, Certificate and Scholarship:
Students who are successful in this examination will be honored and rewarded in a colorful ceremony.
The best 150 students who take the test will get a scholarship of Rs.1500 per year for a period of five years.
10. Training Programme:
Successful students will be trained in higher theory on mathematics and solving harder problems for two weeks each year for three consecutive years. The institute will bear their lodging, boarding
and books and all related expenses during their training. Usually curriculum syllabus for the training, unlike that of the school curriculum is helpful for intellectual and creative prosperity of
the students. Needless to say, students trained here will have ample confidence and courage to face any examination in mathematics.
11. Whom to send Consolidated data Sheet and Proforma of Application and Examination fee:
Institute of Mathematicsand Applications
Sahid Nagar, Bhubaneswar 751007
Ph: (0674) 2540604, 2542164
12. How examination will be conducted:
(a) Superintendent of the examination center is the final authority of the center.
(i) Collected exam fee, list of examinee, forms and other related paper should be reached at the Institute by 10th August 2006
(ii) After the examination is over, the answer scripts should be sent to IMA personally or by post. No traveling allowances will be paid.
(b) After receiving the list of the examinees and examination fee, the name of the examination center and the Institute will send the center charges to the center superintendents.
Consolidated Data Sheet
(This sheet duly filled up should be send to the appropriate authority)
Name of the School with Address: ________________________________
Importance Notes:
To know more about the Rural Mathematics Talent Search Examination, please contact:
Prof Swadheen Pattanayak
Institute of Mathematicsand Applications
2nd Floor, SuryakiranBuildings
Sahid Nagar, Bhubaneswar 751007
Ph: (0674) 2540604, 2542164
Consolidated Data Sheet
(This sheet duly filled up should be send to the appropriate authority)
Name of the School with Address: ________________________________
Name and Address of the Contact Person (You are requested to write here the name of the Head of the Institute along with the names and address of math teachers)
Name: _____________________ Name: _____________________
Address: _____________________ Address: _____________________
______________________ _____________________
______________________ _____________________
Pin: _____________________ Pin: _____________________
Phone No. (with code) Phone No. (with code)
Details of Entry Fees
and Fee Collected: (The fee collected should be sent in the form a bank draft drawn in favour of Director, Institute of Mathematics and Application, payable at Bhubaneswar. No other form of payment
is acceptable. However money order may be accepted only from those places where bank facilities are not available).
2. Amount Collected and Sent = Rs.
3. Bank Draft No.
(The Head of the Institute)
Rural Mathematics Olympiad- 2006
PROFORMA OF APPLICATION FORM
1. Name of the Institution
2. Centre (i)______________________ (ii) _______________________
(If your school is not a center give some choice of near by schools. In case it is not possible to make a nearby school a center for your school will be decided by us.)
3. List of the participant: Mention here against the names of student (W) in case of girl, (SC) in case of schedule cast and (ST) in case of schedule tribe
Sl. No.
Name of the Student
· If the number of students is more then another sheet of this format can be used.
· Send your list in same size as our paper.
Signature of the Head of the Institution | {"url":"https://india.seedsnet.in/rmts-2/ruralmathtalent/","timestamp":"2024-11-02T04:52:23Z","content_type":"text/html","content_length":"179974","record_id":"<urn:uuid:e9f7bc9b-fd50-43d6-9e0c-33caee032bd1>","cc-path":"CC-MAIN-2024-46/segments/1730477027677.11/warc/CC-MAIN-20241102040949-20241102070949-00823.warc.gz"} |
Area of a Square. Calculator
Last updated:
Area of a Square Calculator
If you forgot how to find the area of a square, you're in the right place - this simple area of a square calculator is the answer to your problems. Whether you want to find the area knowing the
square side or you need to calculate the side from a given area, this tool lends a helping hand. Read on and refresh your memory to find out what is the area of a square and to learn the formula
behind the calculator. If you also need to calculate the diagonal of a square, check out this square calculator.
Formulas for the area of a square
The area of a square is the product of the length of its sides:
$A = a\times a = a^2$
where $a$ is a square side.
Other formulas also exist. Depending on which parameter is given, you can use the following equations:
• $A = d^2 / 2$ if you know the diagonal;
• $A= P^2 / 16$ if the perimeter is given (you can learn how to find $P$ in every possible way with our perimeter of a square calculator);
• $A= 2 \times R^2$ knowing circumradius $R$; and
• $A= 4 \times r^2$ in terms of the inradius $r$.
What is the area of a square?
The area of a square is the number of square units needed to completely fill a square. To understand that definition, let's have a look at this picture of a chessboard:
The board has a squared shape, with its side divided into eight parts, in total, it consists of 64 small squares. Assume that one small square has a side length equal to $1\ \mathrm{in}$. The area of
the square may be understood as the amount of paint necessary to cover the surface. So, from the formula for the area of a square, we know that $A= a^2 = 1\ \mathrm{in^2}$, and it's our unit of area
in the chessboard case (amount of paint). The area of a 2 x 2 piece of the chessboard is then equal to 4 squares - so it's $4\ \mathrm{in^2}$, and we need to use 4 times more "paint". Full chessboard
area equals $64\ \mathrm{in^2}$: $8\ \mathrm{in} \times 8\ \mathrm{in}$ from the formula, or it's just 64 small squares with $1\ \mathrm{in^2}$ area - so we need 64 times more "paint" than for one
single square.
You may also be interested in checking out the area of the largest square inscribed in a circumference with our square in a circle calculator!
How to use the area of a square calculator
Let's give the area of a square calculator a try!
1. Find out the given value. In our example, assume we know the side and want to calculate the area.
2. Type it into the proper box. Enter the value, e.g., $11$ inches, into the side box.
3. The area appears! It's $121\ \mathrm{in^2}$. If you are interested in how many square feet it is, change the unit by clicking on the unit name.
How do I find the area of a square given perimeter?
If you know the perimeter of a square and want to determine its area, you need to:
1. Divide the perimeter by 4.
2. The result is the side of the square.
3. Multiply the side by itself.
4. The result is the area of your square.
How do I find the diagonal of a square given area?
To compute the length of a diagonal of a square given its area, follow these steps:
1. Multiply the area by 2.
2. Take the square root of the result of step 1.
3. That's it! The result is the diagonal of your square. Congrats!
4. The formula we used here is:
diagonal = √(2 × area)
What is the area of a square with diagonal 10?
The answer is 50. This is because the formula linking the area of a square with its diagonal is:
area = diagonal² / 2
Hence, plugging in diagonal = 10, we obtain:
area = 100 / 2 = 50
What is the area of a square with perimeter 52?
The answer is 169. To arrive at this result, observe that the perimeter is equal to 52. This means that the side of the square equals:
side = perimeter /4 = 52 / 4 = 13
Hence, the area is:
area = 13² = 169. | {"url":"https://www.omnicalculator.com/math/square-area","timestamp":"2024-11-06T14:45:45Z","content_type":"text/html","content_length":"434978","record_id":"<urn:uuid:5bbf8fbe-421c-4740-9d81-ec4a4731be31>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00740.warc.gz"} |
Structural Analysis and Shape Descriptors
Calculates all of the moments up to the third order of a polygon or rasterized shape.
The function computes moments, up to the 3rd order, of a vector shape or a rasterized shape. The results are returned in the structure Moments defined as:
class Moments
Moments(double m00, double m10, double m01, double m20, double m11,
double m02, double m30, double m21, double m12, double m03 );
Moments( const CvMoments& moments );
operator CvMoments() const;
// spatial moments
double m00, m10, m01, m20, m11, m02, m30, m21, m12, m03;
// central moments
double mu20, mu11, mu02, mu30, mu21, mu12, mu03;
// central normalized moments
double nu20, nu11, nu02, nu30, nu21, nu12, nu03;
In case of a raster image, the spatial moments
The central moments
The normalized central moments
The moments of a contour are defined in the same way but computed using the Green’s formula (see http://en.wikipedia.org/wiki/Green_theorem). So, due to a limited raster resolution, the moments
computed for a contour are slightly different from the moments computed for the same rasterized contour.
Since the contour moments are computed using Green formula, you may get seemingly odd results for contours with self-intersections, e.g. a zero area (m00) for butterfly-shaped contours.
Calculates seven Hu invariants.
The function calculates seven Hu invariants (introduced in [Hu62]; see also http://en.wikipedia.org/wiki/Image_moment) defined as:
These values are proved to be invariants to the image scale, rotation, and reflection except the seventh one, whose sign is changed by reflection. This invariance is proved with the assumption of
infinite image resolution. In case of raster images, the computed Hu invariants for the original and transformed images are a bit different.
Finds contours in a binary image.
The function retrieves contours from the binary image using the algorithm [Suzuki85]. The contours are a useful tool for shape analysis and object detection and recognition. See squares.c in the
OpenCV sample directory.
Source image is modified by this function. Also, the function does not take into account 1-pixel border of the image (it’s filled with 0’s and used for neighbor analysis in the algorithm), therefore
the contours touching the image border will be clipped.
If you use the new Python interface then the CV_ prefix has to be omitted in contour retrieval mode and contour approximation method parameters (for example, use cv2.RETR_LIST and
cv2.CHAIN_APPROX_NONE parameters). If you use the old Python interface then these parameters have the CV_ prefix (for example, use cv.CV_RETR_LIST and cv.CV_CHAIN_APPROX_NONE).
• An example using the findContour functionality can be found at opencv_source_code/samples/cpp/contours2.cpp
• An example using findContours to clean up a background segmentation result at opencv_source_code/samples/cpp/segment_objects.cpp
• (Python) An example using the findContour functionality can be found at opencv_source/samples/python2/contours.py
• (Python) An example of detecting squares in an image can be found at opencv_source/samples/python2/squares.py
Draws contours outlines or filled contours.
The function draws contour outlines in the image if
#include "cv.h"#include "highgui.h"using namespace cv;
int main( int argc, char** argv )
Mat src;
// the first command-line parameter must be a filename of the binary
// (black-n-white) image
if( argc != 2 || !(src=imread(argv[1], 0)).data)
return -1;
Mat dst = Mat::zeros(src.rows, src.cols, CV_8UC3);
src = src > 1;
namedWindow( "Source", 1 );
imshow( "Source", src );
vector<vector<Point> > contours;
vector<Vec4i> hierarchy;
findContours( src, contours, hierarchy,
CV_RETR_CCOMP, CV_CHAIN_APPROX_SIMPLE );
// iterate through all the top-level contours,
// draw each connected component with its own random color
int idx = 0;
for( ; idx >= 0; idx = hierarchy[idx][0] )
Scalar color( rand()&255, rand()&255, rand()&255 );
drawContours( dst, contours, idx, color, CV_FILLED, 8, hierarchy );
namedWindow( "Components", 1 );
imshow( "Components", dst );
• An example using the drawContour functionality can be found at opencv_source_code/samples/cpp/contours2.cpp
• An example using drawContours to clean up a background segmentation result at opencv_source_code/samples/cpp/segment_objects.cpp
• (Python) An example using the drawContour functionality can be found at opencv_source/samples/python2/contours.py
Approximates a polygonal curve(s) with the specified precision.
C++: void approxPolyDP(InputArray curve, OutputArray approxCurve, double epsilon, bool closed)
Python: cv2.approxPolyDP(curve, epsilon, closed[, approxCurve]) → approxCurve
The functions approxPolyDP approximate a curve or a polygon with another curve/polygon with less vertices so that the distance between them is less or equal to the specified precision. It uses the
Douglas-Peucker algorithm http://en.wikipedia.org/wiki/Ramer-Douglas-Peucker_algorithm
See https://github.com/opencv/opencv/tree/master/samples/cpp/contours2.cpp for the function usage model.
Approximates Freeman chain(s) with a polygonal curve.
C: CvSeq* cvApproxChains(CvSeq* src_seq, CvMemStorage* storage, int method=CV_CHAIN_APPROX_SIMPLE, double parameter=0, int minimal_perimeter=0, int recursive=0 )
Python: cv.ApproxChains(src_seq, storage, method=CV_CHAIN_APPROX_SIMPLE, parameter=0, minimal_perimeter=0, recursive=0) → contours
• src_seq – Pointer to the approximated Freeman chain that can refer to other chains.
• storage – Storage location for the resulting polylines.
• method – Approximation method (see the description of the function FindContours() ).
Parameters: • parameter – Method parameter (not used now).
• minimal_perimeter – Approximates only those contours whose perimeters are not less than minimal_perimeter . Other chains are removed from the resulting structure.
• recursive – Recursion flag. If it is non-zero, the function approximates all chains that can be obtained from chain by using the h_next or v_next links. Otherwise, the single
input chain is approximated.
This is a standalone contour approximation routine, not represented in the new interface. When FindContours() retrieves contours as Freeman chains, it calls the function to get approximated contours,
represented as polygons.
Calculates a contour perimeter or a curve length.
The function computes a curve length or a closed contour perimeter.
Calculates the up-right bounding rectangle of a point set.
The function calculates and returns the minimal up-right bounding rectangle for the specified point set.
Calculates a contour area.
The function computes a contour area. Similarly to moments() , the area is computed using the Green formula. Thus, the returned area and the number of non-zero pixels, if you draw the contour using
drawContours() or fillPoly() , can be different. Also, the function will most certainly give a wrong results for contours with self-intersections.
vector<Point> contour;
contour.push_back(Point2f(0, 0));
contour.push_back(Point2f(10, 0));
contour.push_back(Point2f(10, 10));
contour.push_back(Point2f(5, 4));
double area0 = contourArea(contour);
vector<Point> approx;
approxPolyDP(contour, approx, 5, true);
double area1 = contourArea(approx);
cout << "area0 =" << area0 << endl <<
"area1 =" << area1 << endl <<
"approx poly vertices" << approx.size() << endl;
Finds the convex hull of a point set.
The functions find the convex hull of a 2D point set using the Sklansky’s algorithm [Sklansky82] that has O(N logN) complexity in the current implementation. See the OpenCV sample convexhull.cpp that
demonstrates the usage of different function variants.
• An example using the convexHull functionality can be found at opencv_source_code/samples/cpp/convexhull.cpp
Finds the convexity defects of a contour.
The function finds all convexity defects of the input contour and returns a sequence of the CvConvexityDefect structures, where CvConvexityDetect is defined as:
struct CvConvexityDefect
CvPoint* start; // point of the contour where the defect begins
CvPoint* end; // point of the contour where the defect ends
CvPoint* depth_point; // the farthest from the convex hull point within the defect
float depth; // distance between the farthest point and the convex hull
The figure below displays convexity defects of a hand contour:
Fits an ellipse around a set of 2D points.
The function calculates the ellipse that fits (in a least-squares sense) a set of 2D points best of all. It returns the rotated rectangle in which the ellipse is inscribed. The algorithm
[Fitzgibbon95] is used. Developer should keep in mind that it is possible that the returned ellipse/rotatedRect data contains negative indices, due to the data points being close to the border of the
containing Mat element.
• An example using the fitEllipse technique can be found at opencv_source_code/samples/cpp/fitellipse.cpp
Fits a line to a 2D or 3D point set.
The function fitLine fits a line to a 2D or 3D point set by minimizing
• distType=CV_DIST_L2
• distType=CV_DIST_L1
• distType=CV_DIST_L12
• distType=CV_DIST_FAIR
• distType=CV_DIST_WELSCH
• distType=CV_DIST_HUBER
The algorithm is based on the M-estimator ( http://en.wikipedia.org/wiki/M-estimator ) technique that iteratively fits the line using the weighted least-squares algorithm. After each iteration the
Tests a contour convexity.
The function tests whether the input contour is convex or not. The contour must be simple, that is, without self-intersections. Otherwise, the function output is undefined.
Finds a rotated rectangle of the minimum area enclosing the input 2D point set.
The function calculates and returns the minimum-area bounding rectangle (possibly rotated) for a specified point set. See the OpenCV sample minarea.cpp . Developer should keep in mind that the
returned rotatedRect can contain negative indices when data is close the the containing Mat element boundary.
Finds a circle of the minimum area enclosing a 2D point set.
The function finds the minimal enclosing circle of a 2D point set using an iterative algorithm. See the OpenCV sample minarea.cpp .
Compares two shapes.
The function compares two shapes. All three implemented methods use the Hu invariants (see HuMoments() ) as follows ( object1,:math:B denotes object2 ):
• method=CV_CONTOURS_MATCH_I1
• method=CV_CONTOURS_MATCH_I2
• method=CV_CONTOURS_MATCH_I3
Performs a point-in-contour test.
The function determines whether the point is inside a contour, outside, or lies on an edge (or coincides with a vertex). It returns positive (inside), negative (outside), or zero (on an edge) value,
correspondingly. When measureDist=false , the return value is +1, -1, and 0, respectively. Otherwise, the return value is a signed distance between the point and the nearest contour edge.
See below a sample output of the function where each image pixel is tested against the contour.
[Fitzgibbon95] Andrew W. Fitzgibbon, R.B.Fisher. A Buyer’s Guide to Conic Fitting. Proc.5th British Machine Vision Conference, Birmingham, pp. 513-522, 1995. The technique used for ellipse fitting is
the first one described in this summary paper.
[Hu62] 13. Hu. Visual Pattern Recognition by Moment Invariants, IRE Transactions on Information Theory, 8:2, pp. 179-187, 1962.
[Sklansky82] Sklansky, J., Finding the Convex Hull of a Simple Polygon. PRL 1 $number, pp 79-83 (1982)
[Suzuki85] Suzuki, S. and Abe, K., Topological Structural Analysis of Digitized Binary Images by Border Following. CVGIP 30 1, pp 32-46 (1985)
[TehChin89] Teh, C.H. and Chin, R.T., On the Detection of Dominant Points on Digital Curve. PAMI 11 8, pp 859-872 (1989) | {"url":"https://docs.opencv.org/2.4/modules/imgproc/doc/structural_analysis_and_shape_descriptors.html?highlight=cv2.approxpolydp","timestamp":"2024-11-12T13:34:21Z","content_type":"text/html","content_length":"112941","record_id":"<urn:uuid:071af469-134b-4d75-abbe-081fb5d5205a>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.45/warc/CC-MAIN-20241112113320-20241112143320-00420.warc.gz"} |
Multiplication Facts - State Joke Picture Puzzles
Multiplication Facts - State Joke Picture Puzzles
Price: 300 points or $3 USD
Subjects: math,mathElementary,mathMiddleSchool,games,multiplicationAndDivision,operationsAndAlgebraicThinking
Grades: 3,4,5,6
Description: This boom deck includes 70 math fact matching problems for multiplication. It includes a hidden picture puzzle style activity. As students answer each math fact problem, a U.S. state
image and joke is revealed. There are 7 jokes in all. If your students are like mine, they will giggle with delight as they practice their basic multiplication facts in this fun way! If you enjoy
this, there are others included in my Multiplication Fact Joke Growing Bundle. Multiply and divide within 100. CCSS.MATH.CONTENT.3.OA.C.7 Fluently multiply and divide within 100, using strategies
such as the relationship between multiplication and division (e.g., knowing that 8 × 5 = 40, one knows 40 ÷ 5 = 8) or properties of operations. By the end of Grade 3, know from memory all products of
two one-digit numbers. -Shane Math is Fundamental | {"url":"https://wow.boomlearning.com/deck/HX4D4EgusvCHCGoMx","timestamp":"2024-11-14T22:07:37Z","content_type":"text/html","content_length":"2802","record_id":"<urn:uuid:50c6e708-27f7-4487-bfd7-63356a0762b7>","cc-path":"CC-MAIN-2024-46/segments/1730477395538.95/warc/CC-MAIN-20241114194152-20241114224152-00097.warc.gz"} |
Queues in an interactive random environment
We consider exponential single server queues with state-dependent arrival and service rates which evolve under influences of external environments. The transitions of the queues are influenced by the
environment´s state and the movements of the environment depend on the status of the queues (bi-directional interaction). The environment is constructed in a way to encompass various models from the
recent Operations Research literature, where a queue is coupled with an inventory or with reliability issues. With a Markovian joint queueing-environment process we prove separability for a large
class of such interactive systems, i.e. the steady state distribution is of product form and explicitly given. The queue and the environment processes decouple asymptotically and in steady state.
For non-separable systems we develop ergodicity and exponential ergodicity criteria via Lyapunov functions. By examples we explain principles for bounding departure rates of served customers
(throughputs) of non-separable systems by throughputs of related separable systems as upper and lower bound.
[1] S. Otten. Integrated Models for Performance Analysis and Optimization of Queueing-Inventory-Systems in Logistic Networks. Phd thesis, Universität Hamburg, 2018. Available at ediss.sub.hamburg.
[2] S. Otten, R. Krenzler, H. Daduna, K. Kruse. Queues in a random environment, 2023. arXiv:2006.15712 (to appear in Stochastic Systems). | {"url":"https://www.mat.tuhh.de/forschung/topics_en/Queues_in_an_interactive_random_environment.html","timestamp":"2024-11-02T01:39:30Z","content_type":"text/html","content_length":"11963","record_id":"<urn:uuid:2d751629-f3ac-4639-8b1f-d5d36dd392d3>","cc-path":"CC-MAIN-2024-46/segments/1730477027632.4/warc/CC-MAIN-20241102010035-20241102040035-00345.warc.gz"} |
日時 ２０１３年１０月３１日（木）１６：００－１７：３０
場所 筑波大学自然系学系Ｄ棟Ｄ５０９
講演 金英子氏（大阪大学理学研究科）
講演 Pseudo-Anosovs with small dilatations coming from the magic 3-manifold
アブ Pseudo-Anosov mapping classes are equipped with some constants >1 called the dilatation. It is known that the logarithm of the dilatation is exactly equal to the topological entropy of a
スト pseudo-Anosov representative of its mapping class. By work of Thurston, if a hyperbolic fibered 3-manifold M has the second Betti number more than 1, then it admits infinitely many fibrations
ラク on M. Moreover the monodromy of any fibration on M is pseudo-Anosov. As an example of such manifolds, we consider a single 3-manifold N with 3 cusps called the magic 3-manifold. We compute the
ト dilatation of monodromy of each fibration on N. We also discuss the problem on the minimal dilatations and their asymptotic behavior. Intriguingly, pseudo-Anosovs with the smallest known
dilatations are ``coming from" the magic 3-manifold. This is a joint work with Mitsuhiko Takasawa. | {"url":"https://nc.math.tsukuba.ac.jp/multidatabases/multidatabase_contents/detail/231/4277f9ba8c8d63b1f435789f7b9b4028?frame_id=147","timestamp":"2024-11-03T19:22:14Z","content_type":"text/html","content_length":"17026","record_id":"<urn:uuid:85807f09-ed57-44fe-a661-87d2563a3654>","cc-path":"CC-MAIN-2024-46/segments/1730477027782.40/warc/CC-MAIN-20241103181023-20241103211023-00386.warc.gz"} |
Re: GSOC 2020 - MDEV-11263 Aggregate Window Functions
maria-developers team mailing list archive
1. Message #12140
Re: GSOC 2020 - MDEV-11263 Aggregate Window Functions
Thread Previous • Date Previous • Date Next • Thread Next
Hi, Tavneet!
On Mar 28, Tavneet Sarna wrote:
> Hi everyone,
> As part of GSOC 2020, this is one of the two projects I am interested
> in pursuing. As guided by last years' comments by Varun and Vicențiu,
> I have set up a debugger and corresponding breakpoints in the
> Item_sum_sum::add and Item_sum_sp::add for a custom sum aggregate
> function to understand the code flow.
> I had a couple of queries regarding the same:
> 1. In *do_add *from decimal.c, there are three parts with comments -
> /* part 1 - MY_MAX(frac) ... min (frac) */, /* part 2 -
> MY_MIN(frac) ... MY_MIN(intg) */. Can someone please elaborate on what do
> the comments mean ?
Sure. This is just how addition of two long numbers work. Say, you have
1234.567890 here frac is 6
+998765.234 here frac is 3
you start from the end, the "part 1" is from the longest tail to the
shortest tail, from MAX(6,3) to MIN(6,3). This is the tail "890" and it
can be simply copied into the result:
while (buf1 > stop)
second part if where two numbers overlap. do_add works from the end,
adding digits and propagating the carry.
third part is where one number is longer than the other in most
significant digits. That is "99".
But this is the level of details that you don't need for your project.
You can rely on the fact that there are different data types and some of
them (for example, integers, floats, and decimal) can be added,
subtracted, etc. For this project you don't need to study at how exactly
the addition works internally.
> 2. In *Item_sum_sum::add_helper*, there is an unlikely branch for
> variable direct_added. Can someone please give an idea about when
> will direct added be true ? In fact in all the uses for
> direct_added, it is always in an unlikely branch in Item_sum.cc.
This was added specifically for and is only used by the spider engine.
If you run spider tests you'll probably see it being true.
But I suspect you can ignore it too for now.
VP of MariaDB Server Engineering
and security@xxxxxxxxxxx
Thread Previous • Date Previous • Date Next • Thread Next | {"url":"https://lists.launchpad.net/maria-developers/msg12140.html","timestamp":"2024-11-13T06:19:04Z","content_type":"text/html","content_length":"9097","record_id":"<urn:uuid:e36b901a-d613-4a61-ad6d-7b7263c96c02>","cc-path":"CC-MAIN-2024-46/segments/1730477028326.66/warc/CC-MAIN-20241113040054-20241113070054-00556.warc.gz"} |
How To Unblock OnHow To Calculate Median
Home How To How To Unblock OnHow To Calculate Median
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How To Unblock OnHow To Calculate Median
Admin4 min read
Unveiling the Median: A Comprehensive Guide to Calculating the Central Tendency in Data
In the realm of data analysis, understanding the central tendency is paramount. The median, a robust measure of central tendency, offers a valuable perspective on data distribution by representing
the midpoint value in an ordered dataset. Unlike the mean, the median remains unaffected by outliers, making it an ideal choice for skewed or non-normally distributed data.
Step-by-Step Guide to Calculating the Median
1. Sort the Data:
Arrange the data values in ascending or descending order. This initial step creates a clear structure for identifying the central value.
2. Determine the Position of the Median:
For an odd number of data points (n), the median is simply the middle value. For an even number of data points, the median is the average of the two middle values.
3. Calculate the Median:
Based on the position of the median, apply the following formulas:
• Odd Number of Data Points (n): Median = (n + 1) / 2
• Even Number of Data Points (n): Median = [(n / 2) + 1 + (n / 2)] / 2
Consider the following dataset:
{2, 4, 6, 8, 10, 12, 14}
1. Sort the data: {2, 4, 6, 8, 10, 12, 14}
2. Determine the position of the median: (7 + 1) / 2 = 4
3. Calculate the median: (4 + 1) / 2 = 2.5
Therefore, the median of the dataset is 2.5.
Skewness and Outliers: The Advantage of the Median
Unlike the mean, the median is less susceptible to extreme values (outliers) and data skewness. This property makes the median a more robust measure of central tendency when the data distribution is
skewed or contains outliers.
Consider the following skewed datasets:
Dataset 1: {1, 2, 3, 4, 5, 100}
Dataset 2: {1, 2, 3, 4, 5, 10, 100}
• Dataset 1: Median = 3, Mean = 17.5
• Dataset 2: Median = 3, Mean = 13.8
As evident from the results, the median provides a more representative central value for both datasets, despite the presence of outliers.
Applications of the Median
The median finds widespread application in various fields, including:
• Data Analysis: Identifying the central tendency of skewed data or data with outliers.
• Surveys and Market Research: Determining the typical response or preference in surveys.
• Financial Analysis: Calculating the median income or net worth within a population.
• Population Studies: Estimating the median age or gender distribution of a population.
• Quality Control: Identifying the median value of a particular attribute in manufacturing or service industries.
The median serves as a powerful measure of central tendency, providing valuable insights into data distribution. By following the simple steps outlined in this guide, you can effectively calculate
the median and harness its descriptive capabilities. Its robustness against outliers and skewness makes it an essential tool for accurate data analysis across a wide range of fields.
Q1: What is the difference between the median and mean?
A1: The mean is the sum of all data values divided by the number of values. Unlike the mean, the median is not affected by outliers and provides a better representation of the central tendency for
skewed or outlier-ridden data.
Q2: How do I calculate the median for an even number of data points?
A2: For an even number of data points (n), the median is the average of the two middle values: (n / 2) + 1 and (n / 2).
Q3: Can the median ever be a fractional value?
A3: Yes, the median can be a fractional value, especially when calculating the median of an even number of data points, as it represents the average of two values.
Q4: Why is the median important in data analysis?
A4: The median is important in data analysis because it provides a robust measure of central tendency, which is not influenced by extreme values or outliers, making it a reliable indicator of the
typical data value.
Q5: Is the median a better measure of central tendency than the mean?
A5: The choice between the median and mean depends on the nature of the data. When dealing with skewed data or data with outliers, the median is a more robust and appropriate measure of central
tendency than the mean.
Related Posts | {"url":"https://www.svgdesignresources.com/how-to-unblock-onhow-to-calculate-median/","timestamp":"2024-11-10T22:41:01Z","content_type":"text/html","content_length":"66913","record_id":"<urn:uuid:0da0c273-be8d-4253-bd28-4c882a0f9415>","cc-path":"CC-MAIN-2024-46/segments/1730477028191.83/warc/CC-MAIN-20241110201420-20241110231420-00180.warc.gz"} |
Decision Rule
1. Introduction
In order to utilise a result to decide whether it indicates compliance or non-compliance with a specification, it is necessary to take into account the measurement uncertainty. Depending on the
circumstances, and particularly on the risks associated with making a wrong decision, the probability of an incorrect decision may be or may not be sufficiently small to justify a decision of
2. Scope
This guide is applicable to decisions on compliance with regulatory or manufacturing limits where a decision is made on the basis of a measurement result accompanied by information on the uncertainty
associated with the result. It covers cases where the uncertainty does not depend on the value of the measurand, and cases where the uncertainty is proportional to the value of the measurand.
This guide assumes that the uncertainty has been evaluated by an appropriate method that takes all relevant contributions into account.
3. Reference
-ISO/IEC 17025 : 2017 General Requirements for Testing & Calibration Laboratories
– The Eurachem/CITAC published a Guide “Use of uncertainty information in
compliance assessment” in 2007 on the subject on Decision Rule and Compliance;
– ILAC G-8:03/2009 “Guidelines on the reporting of compliance to specification” attempts to provide some clarity by writing a guide on how to look at making a “pass/fail” conformity assessment, and
how to present the “conformance/non-conformance” statements;
– The JCGM 106:2012 “Evaluation of Measurement Data- the Role of measurement uncertainty in conformance assessment” is another guideline on how to deal with this problem with more details and
suggestions on different ways of interpreting results.
4. Definition
“Decision rule” under Terms and Definitions Clause 3.7 of ISO/IEC 17025:2017, which states that it is a “rule that describes how measurement uncertainty is accounted for when stating conformity with
a specified requirement”.
ISO/IEC 17025:2017 and Decision Rule
The current version of ISO/IEC 17025:2017 has the following clauses:
Clause 7.1.3 : “When the customer requests a statement of conformity to a specification or standard for the test or calibration (e.g. pass/fail, in-tolerance/out-of-tolerance), the specification or
standard and the decision rule shall be clearly defined. Unless inherent in the requested specification or standard, the decision rule selected shall be communicated to, and agreed with, the
Clause 7.8.6.1 : “When a statement of conformity to a specification or standard is provided, the laboratory shall document the decision rule employed, taking into account the level of risk (such as
false accept and false reject and statistical assumptions) associated with the decision rule employed, and apply the decision rule.
NOTE Where the decision rule is prescribed by the customer, regulations or normative documents, a further consideration of the level of risk is not necessary. “
Clause 7.8.6.2 c) “ the decision rule applied (unless it is inherent in the requested specification or standard). “
Appendix A A.2.3 : Measurement standards that have reported information from a competent laboratory that includes only a statement of conformity to a specification (omitting the measurement results
and associated uncertainties) are sometimes used to disseminate metrological traceability. This approach, in which the specification limits are imported as the source of uncertainty, is dependent
— the use of an appropriate decision rule to establish conformity;
STATEMENTS OF COMPLIANCE ACCORDING TO ISO/IEC 17025
Clause 7.8.6.1
In this issue of conformity, one therefore must make an educated discussion with his customers or regulators during the acceptance of a test request or contract review, It is a risk to both parties
concerned, which must be duly assessed and decided upon when the reported measurement is found not within or below/above specification stipulated.
Following situations maybe considered :
When Customer provides a specification or standard for reference a statement of conformity to a specification or standard – It is the responsibility of the laboratory to inform the customer about the
decision its decision rule on making a statement of compliance to prevent the probability of “false accept” when a product or material should fail and “false reject” when this product or material
should pass before taking up the job.
For the marginal results due to effect of Uncertainty of measurement and other factors a confusion may occur in reporting the statement of conformity and it is called “Guard Band” (see Figure 1). The
use of Guard Band to limit may be discussed with the customers that will be comfortable to both parties, and also ensure that its test procedure employed can meet the specification limit required. It
may even need to change or modify its test procedure (e.g. by lowering its method detection limit) to cater for its decision rule to be applied.
Figure 1 Graphic presentation on a stringent Acceptance zone and a ‘relaxed’ Rejection zone for a specification with an upper limit
Figure 2 : Acceptance and rejection zones for simultaneous Upper and Lower Limits
Decision Rules :
A decision rule ultimately relies on the outcome of the ever popular hypothesis or significance testing based on the distribution(s) of test statistic and sets a risk level that is mutually
1 In order to decide whether or not to accept/reject a product, given a result and its uncertainty, there should be
a) a specification giving the upper and/or lower permitted limits of the characteristics (measurands) being controlled
b) a decision rule that describes how the measurement uncertainty will be taken into account with regard to accepting or rejecting a product according to its specification and the result of a
2 The decision rule sets up for a well-documented method of unambiguously determining the location of the acceptance and rejection zones, ideally stating or using the minimum acceptable level of the
probability that the measurand lies within the specification limits. It should also give the procedure for dealing with repeated measurements and outliers.
3 Utilising the decision rule the size of the acceptance or rejection zone may be determined by means of appropriate guard bands. The size of the guard band is calculated from the value of the
measurement uncertainty and the minimum acceptable level of the probability that the measurand lies within the specification limits.
4 In addition, a reference to the decision rules used should be included when reporting on compliance.
Choosing Acceptance and Rejection zone limits
The size of the guard band, g depends upon the value of the uncertainty, u and is chosen to meet the requirements of the decision rule. For example if the decision rule states that for
non-compliance, the observed value should be greater than the limit plus 2u, then the size of the guard band is 2u.
If the decision rule states that for non-compliance that the probability P that the value of the measurand is greater than the limit L, should be at least 95%, then g is chosen so that for an
observed value of L+g, the probability that the value of the measurand lies above the limit L is 95%. Similarly, if the decision rule is that there should be at be least a 95% probability that the
value of the measurand is less than L, then g is chosen, so that for an observed value of L-g, the probability that the value of the measurand lies below the limit is 95%. In general the value of g
will be a function of or a simple a multiple of u where u is the standard uncertainty. In some cases the decision rule may state the value of the multiple to be used. In others the guard band will
depend upon the value of P required and the knowledge about the distribution of the likely values of the measurand.
a. When a specification describes an interval with an upper and lower limit, a statement of compliance or non-compliance are made where the ratio of the expanded uncertainty interval to the specified
interval is reasonably small and fit for purpose (meaning that the laboratory should be able to meet the needs of the customer).
b. When compliance with a specification is made it should be clear to the customer which coverage probability for the expanded uncertainty has been used. In general the coverage probability will be
95 % and the reporting shall include a remark such as “The statement of compliance is based on a 95% coverage probability for the expanded uncertainty.” This means that the probability that the
measurement is below the upper specification limit is higher than 95 %, i.e. approximately 97.5 % for symmetrical distributions. A lower limit is treated similarly. Other values for the coverage
probability for the expanded uncertainty should be agreed between the laboratory and the customer. Coverage probabilities for the expanded uncertainty higher than 95 % might be chosen while lower
values should be avoided.
c. The following approach for a certain upper specification limit is recommended. (A lower limit is treated similarly):
(i) Compliance: If the specification limit is not breached by the measurement result plus the expanded uncertainty with a 95% coverage probability, then compliance with the specification can be
stated. This can be reported as “Compliance” or “Compliance – The measurement result is within (or below) the specification limit when the measurement uncertainty is taken into account”;
(ii) Non-compliance:
Situation 1. If the specification limit is exceeded by the measurement result minus the expanded uncertainty with a 95% coverage probability, then non- compliance with the specification can be
stated. This can be reported as “Non-compliance” or “Non-compliance – The measurement result is outside (or above) the specification limit when the measurement uncertainty is taken into account”;
Situation 2. If the measurement result plus/minus the expanded uncertainty with a 95 % coverage probability overlaps the limit or Guard Band (g), it is not possible to state compliance or
non-compliance. The measurement result and the expanded uncertainty with a 95 % coverage probability should then be reported together with a statement indicating that neither compliance nor
non-compliance was demonstrated. A suitable statement to cover these situations would be “It is not possible to state compliance”. In this case, it is similar statement “It is not possible to state
compliance using a 95 % coverage probability for the expanded uncertainty although the measurement result is below the limit”. If shorter statements are reported it should not give the impression
that the result complies with specification.
Sanchita Bhattacharya, CEO
Consultrain Management Service, Kolkata | {"url":"https://consultrainmanagement.com/2022/10/20/decision-rule/","timestamp":"2024-11-07T03:26:02Z","content_type":"text/html","content_length":"134183","record_id":"<urn:uuid:b2d11cb1-00db-4c64-9eb5-36d2b43708d3>","cc-path":"CC-MAIN-2024-46/segments/1730477027951.86/warc/CC-MAIN-20241107021136-20241107051136-00671.warc.gz"} |
Understanding Brute Force Algorithm – Unveiling Its Purpose and Applications
Introduction to Brute Force Algorithm
The brute force algorithm is a simple yet powerful approach in problem-solving that involves searching all possible solutions exhaustively. It is a universal method applicable to various
computational problems, especially when the input space is small enough to be examined completely. This algorithm operates by systematically evaluating every possible solution until the correct one
is found.
In brute force algorithms, each potential solution is tested against the problem’s constraints, leading to a guaranteed solution when employed correctly. Its concept revolves around the reassurance
of covering all possible options to find the desired outcome, making it a go-to approach for various applications.
Applications of Brute Force Algorithm
The brute force algorithm finds applications in a wide range of fields, where exhaustive search is needed. Some notable examples where this algorithm is extensively used include:
Password cracking
Brute force algorithms are widely utilized for password cracking, where an attacker attempts to gain unauthorized access to an account by systematically testing all possible combinations of passwords
until the correct one is identified. This method is successful when the passwords are weak or short, but it becomes increasingly difficult and time-consuming as the password complexity increases.
Brute force algorithms are employed in cryptanalysis to decrypt cryptographic ciphers, such as substitution ciphers or Caesar ciphers, by systematically trying all possible keys or combinations until
the original message is discovered. This approach is particularly useful when the cryptographic algorithm and key length are weak.
Sudoku solving
Sudoku, a popular number-placement puzzle, can be solved using the brute force algorithm. By systematically trying all possible numbers for each cell of the puzzle until a valid solution is found,
the brute force method guarantees a correct answer. Sudoku solving algorithms usually implement additional pruning techniques to reduce the search space and improve efficiency.
String matching
String matching problems, such as pattern searching or searching for a specific substring within a larger text, can be solved using brute force algorithms. The algorithm examines every possible
character position in the text, comparing it with the pattern or substring being searched. Though simple to implement, brute force string matching algorithms can be time-consuming for larger texts.
Combination and permutation problems
Brute force algorithms are commonly employed to solve combination and permutation problems, where all possible combinations of elements from a given set need to be generated or examined. This
approach guarantees that all possible combinations are processed, ensuring accurate results. However, its efficiency decreases exponentially as the input size grows.
Advantages and Disadvantages of Brute Force Algorithm
Brute force algorithms possess several advantages that contribute to their popularity:
1. Simplicity and universality: Brute force algorithms are relatively easy to comprehend and implement, making them widely accessible across different problem domains. They provide a straightforward
approach to problem-solving without the need for complex algorithmic techniques.
2. Complete search and guaranteed solution: By exploring every possible solution within the given constraints, the brute force algorithm ensures that no viable option is overlooked. This
characteristic makes it particularly suitable for situations requiring certainty and accuracy, where finding any solution is better than finding none.
Despite their advantages, brute force algorithms also have limitations and drawbacks:
1. Inefficiency for large inputs: As the size of the input space increases, the time and computational resources required for a brute force algorithm to analyze every possible solution also grow
exponentially. This inefficiency makes brute force less practical for problems with large search spaces.
2. Exhaustive searching could be time-consuming: The sheer number of potential solutions that need to be examined can make brute force algorithms time-consuming. In particular, for problems with
complex constraints or numerous possibilities, this exhaustive search can become a significant performance bottleneck.
3. Increased storage requirements: In some scenarios, brute force algorithms may require substantial storage resources to store and process all possible solutions. As the size of the input space
grows, the memory requirements may become impractical, leading to scalability issues.
Techniques to Optimize Brute Force Algorithm
While brute force algorithms are inherently simple and exhaustive, there are several techniques that can be applied to enhance their efficiency:
Reducing search space through pruning
By employing pruning techniques, unnecessary branches of the search tree can be eliminated, reducing the total number of solutions explored. This optimization helps narrow down the search space and
improve the overall performance of the brute force algorithm.
Implementing parallelization
Parallelization involves dividing the problem into multiple sub-problems and executing them concurrently on multiple processors or threads. This technique enables the execution of several brute force
algorithms simultaneously, potentially reducing the overall time required to find the solution.
Utilizing heuristics and problem-specific optimizations
For certain problem domains, utilizing problem-specific heuristics or optimizations can significantly improve the efficiency of the brute force algorithm. These techniques take advantage of specific
patterns or characteristics unique to the problem at hand, reducing the number of possibilities that need to be considered.
Taking advantage of hardware advancements
Advancements in hardware, such as the use of graphical processing units (GPUs) or specialized co-processors, can greatly speed up the execution of brute force algorithms. These hardware improvements
provide parallel processing capabilities and special-purpose architectures designed to handle massive computational tasks efficiently.
Real-life Examples of Brute Force Algorithm Usage
The versatility of the brute force algorithm allows its usage in various real-life scenarios, including:
Internet security and password protection
Brute force algorithms play a vital role in evaluating the security of online systems and password protection mechanisms. By simulating a potential attacker, security experts can employ brute force
techniques to identify vulnerable passwords and recommend improvements in security practices.
Network security and encryption
In the field of network security, brute force algorithms are used to test the strength of encryption algorithms, identifying potential vulnerabilities. By exhaustively exploring the keyspace,
analysts can evaluate the resistance of cryptographic algorithms against brute force attacks.
Data recovery and forensic analysis
Forensic analysts and data recovery specialists often rely on brute force algorithms when attempting to recover lost or corrupted data. These algorithms can systematically test different recovery
methods or keys to gain access to encrypted files, facilitating the investigation or data restoration process.
Game-solving and strategy optimization
Brute force algorithms have been used to solve complex games, such as chess or Go, by evaluating all possible moves and their consequences. These algorithms have been combined with sophisticated
heuristics to optimize strategies, contributing to advancements in game-playing AI.
The brute force algorithm, despite its simplicity, remains a powerful approach for solving a wide range of computational problems. Its universality and reliability make it an effective tool in
scenarios where exhaustive searching and guaranteed solutions are essential.
While brute force algorithms may have limitations regarding efficiency and scalability, various techniques can be employed to optimize their performance. Pruning techniques, parallelization,
problem-specific optimizations, and hardware advancements contribute to enhancing the overall efficiency of these algorithms.
Real-life applications of the brute force algorithm can be observed throughout fields such as internet security, network encryption, data recovery, and game-playing AI, highlighting its versatility
and importance.
Looking forward, further advancements in computing technology, algorithmic techniques, and problem-specific optimizations will continue to enhance the capabilities of brute force algorithms,
expanding their potential even further. | {"url":"https://skillapp.co/blog/understanding-brute-force-algorithm-unveiling-its-purpose-and-applications/","timestamp":"2024-11-05T22:05:42Z","content_type":"text/html","content_length":"114245","record_id":"<urn:uuid:9cea2ce9-af89-4e2b-ab22-40b6db734b87>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00888.warc.gz"} |
Links to high quality mathematical research videos. Suggestions for additional sites and videos can be submitted with the form at the bottom of this page.
Workshops and Conferences
General Area Video Collections
Applied Mathematics (general)
Classical Analysis and ODEs
Differential Geometry & Geometric Analysis
Geometry & Geometric Topology
Featured videos are selected based on presentation and content quality. They may be accompanied by short reviews that comment on items such as the topics covered, background needed and impact of
the results.
Not Knot is an introduction to the idea of hyperbolic knot complements. This 16 minute film was released in 1991 by the Geometry Center at the University of Minnesota. It was directed by Charlie Gunn
and Delle Maxwell.
Conform gives an introduction to conformal mappings of surfaces. This 16 minute film was created by Alexander Bobenko and Charles Gunn in 2018.
Outside IN shows how to turn a sphere inside out via a regular homotopy. The existence of such a sphere eversion was shown by Steven Smale in 1957. The film is based on a technique for construction
eversions discovered by Bill Thurston. This 22 minutes long video was produced in 1995 at the Geometry Center under the direction of Silvio Levy, Delle Maxwell and Tamara Munzner.
Geometry of Growth and Form:
Commentary on D'Arcy Thompson
This video celebrating the 80th anniversary of the Institute for Advanced Study features a talk by John Milnor. Milnor discusses conformal geometry and the classical work of biologist D’Arcy
You can use the following form to submit videos or website for this list. Videos should be freely available and useful for mathematical research. Websites hosting relevant videos are also welcome.
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AN IMPORTANT THEOREM | cristos-vournas.com
The method we use in this research is the "Planet Surface Temperatures Comparison Method".
1). The solar flux's intensity upon the planet surface "S".
5). Planet surface composition (planet average surface specific heat "cp" cal/gr.oC).
6). Planet surface Φ-factor - the planet surface Solar Irradiation Accepting Factor (the planet surface shape and roughness coefficient).
The planet mean surface temperatures relate (everything else equals) according to their (N*cp) products’ sixteenth root.
The consequence of this discovery is the realization that a planet with a higher (N*cp) product (everything else equals) appears to be a warmer planet.
We are able to Theoretically calculate for the planet without-atmosphere the mean surface temperature.
For every planet without atmosphere there is the theoretical uniform surface effective temperature Te.
And for every planet without atmosphere there is the average surface temperature (the mean surface temperature) Tmean.
where X is a coefficient which calculates the planet Tmean from the planet known Te.
The X is a different and very distinguished for every different planet number.
The planet Te is theoretically calculated by the Stefan-Boltzmann emission law, when the planet average surface Albedo, and the solar flux upon the planet surface are known.
Now, we can accept that for every planet (ι) there is a Te.ι and there is a Tmean.ι
We can accept that for every planet (ι) there is a Xι, there is a Te.ι and there is a
Tmean.ι = [ Φ.ι (1 - a.ι) S.ι (X.ι)⁴ /4σ ]¹∕ ⁴
We have admitted that for every planet (ι) there is a different for each planet (ι) a factor [(X.ι)⁴ ], which relates for the purpose to theoretically calculate for the planet (ι) the average (mean)
surface temperature Tmean.ι
by simply multiplying the X.ι with planet (ι) theoretical uniform surface effective temperature Te.ι
Also for every planet (ι) without atmosphere we have the planet (N.ι*cp.ι) product.
I have demonstrated in my website that planet mean surface temperatures relate (everything else equals) according to their (N*cp) products’ sixteenth root.
Tmean.ι = [ Φ (1-a) S (X.ι)⁴ /4σ ]¹∕ ⁴
the (X.ι)⁴ term to replace with the (β *N.ι *cp.ι) ¹∕ ⁴ term
a.ι – is the planet (ι) the average surface Albedo
Φ.ι – is the solar irradiation accepting factor (for smooth surface planets Φ = 0,47 and for rough surface planets Φ = 1)
cp.ι – is the planet average surface specific heat (cal/gr.oC)
β = 150 days*gr*oC/rotation*cal is a Rotating Planet Surface Solar Irradiation INTERACTING-Emitting Universal Law constant
Tmean.ι = [ Φ.ι (1 - a.ι) S.ι (β *N.ι *cp.ι)¹∕ ⁴ /4σ ]¹∕ ⁴
The above formula theoretically calculates the planets without atmosphere mean surface temperatures with very closely matching to the satellite measured temperatures results.
The planet mean surface temperatures Tmean are very much precisely being measured by satellites.
Tmean = [ Φ (1 - a) S (β*N*cp.)¹∕ ⁴ /4σ ]¹∕ ⁴
The planet mean surface temperature Tmean numerical value will be equal to the planet effective temperature Te numerical value Tmean = Te only when the term
1). In general, the planet effective temperature numerical value Te is not numerically equal to the planet without-atmosphere mean surface temperature Tmean.
2). For the planet without-atmosphere mean surface temperature numerical value Tmean to be equal to the planet effective temperature numerical value Te the condition from the above Theorem the (N*cp
= 1 /150) should be necessarily met.
3). For the Planet Earth without-atmosphere the (N*cp) product is (N*cp = 1) and it is 150 times higher than the necessary condition of (N*cp = 1/150) .
Consequently, Earth's effective temperature Te the numerical value cannot be equal to Earth's without-atmosphere mean surface temperature... not even close. | {"url":"https://cristos-vournas.com/449683314/","timestamp":"2024-11-07T09:48:28Z","content_type":"text/html","content_length":"194052","record_id":"<urn:uuid:15d8b599-880b-4e27-a489-7b7c3e3ea968>","cc-path":"CC-MAIN-2024-46/segments/1730477027987.79/warc/CC-MAIN-20241107083707-20241107113707-00449.warc.gz"} |
Congruence of Triangles - Practically Study Material
Congruence of Triangles
Congruent Figures
Geometrical figures which have exactly the same shape and the same size, are known as congruent figures. For congruence, we use the symbol ‘ $\cong$ ‘ read as ‘congruent to’. Thus, two plane figures
are congruent if each when superposed on the other, covers it exactly.
Similar Figures
Geometrical figures which have exactly the same shape but not necessarily the same size, are known as similar figures. For similarity, we use the symbol ‘ $\square$ ‘ read as ‘is similar to’
Two congruent figures are always similar but two similar figures need not be congruent.
• Any two equilateral triangles are always similar, but they are congruent only if they have the same side length.
• Any two line segments are always similar, but they congruent only if they have the same length.
Congruent Triangles
Two triangles are said to be congruent, if each one of them can be made to superpose on the other, so as to cover it exactly. Thus, congruent triangles are exactly identical. In congruent triangles,
the sides and angles which coincides by superposition are called corresponding sides and angles respectively. Hence, we can also say that two triangles are congruent if pairs of corresponding sides
and corresponding angles are equal.
Thus $\mathrm{\Delta ABC}\cong \mathrm{\Delta DEF}\text{if}\mathrm{AB}=\mathrm{DE},\mathrm{BC}=\mathrm{EF},\mathrm{CA}=\mathrm{FD}$
$\mathrm{\angle }\mathrm{A}=\mathrm{\angle }\mathrm{D},\mathrm{\angle }\mathrm{B}=\mathrm{\angle }\mathrm{E},\mathrm{\angle }\mathrm{C}=\mathrm{\angle }\mathrm{F}$
We write, $\mathrm{\Delta ABC}\cong \mathrm{\Delta DEF}$ , if means that A,B and C are matched with D,E and F respectively and we write $\mathrm{A}↔\mathrm{D},\mathrm{B}↔\mathrm{E}\text{and}\mathrm
{C}↔\mathrm{F}$ and therefore, $\mathrm{\Delta ABC}↔\mathrm{DEF}$
There are four cases as conditions for congruency. In each case, we have a different combination of the three matching parts.
SAS (Side-Angle-Side)Condition
If two triangles have two sides and the included angle of the one respectively equal to two sides and the included angle of the other, then the triangles are congruent.
ASA (Angle-Side-Angle)Condition
If two triangles have two angles and the included side of them one respectively equal to two angles and the included side of the other. Then the triangles are congruent.
Eg :
If two angles of one triangle are respectively equal to two angles of the other, then it is quite clear that their remaining third angles are also equal. Thus even, if the given side is not the one
included between the two given angles, then it shall be the one included between one of the given angles and the third angle.
SSS (Side-Side-Side)Condition
If two triangles have the three sides of the one respectively equal to the corresponding three sides of the other, then the triangles are congruent.
RHS (Right-Angle-Hypotenuse-Side)
If two right angled triangles have one-side and hypotenuse of the one respectively equal to the
corresponding side and the hypotenuse of the other, then the triangles are congruent. | {"url":"https://www.practically.com/studymaterial/blog/docs/class-7th/maths/congruence-of-triangles/","timestamp":"2024-11-07T15:36:20Z","content_type":"text/html","content_length":"88096","record_id":"<urn:uuid:62832fda-790f-4dab-9af0-0a187033e524>","cc-path":"CC-MAIN-2024-46/segments/1730477028000.52/warc/CC-MAIN-20241107150153-20241107180153-00476.warc.gz"} |
470-4202/01 – Abstract Algebra in Coding Theory (AvTK)
After passing the course a student will be able: - use congruences when solving discrete problems, - describe symmetries of real world problem using groups, - calculate polynomial operations in
modular arithmetics, - construct selected Galois fields and simple codes based on these, - construct simple finite vector fields, - perform comutation on code words in vector notation, - perform
operations on selected codes in matrix notation, - encode and decode a message in a simple code, - detect and correct basic mistakes in transmission.
The course serves a building block for Coding Theory. The goal is to provide an overview of methods and train relevant skills, that will be used in the Coding Theory course.
There will be two tests or the students will prepare a project.
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What is Banco Total Assets from 2010 to 2024 | Stocks: SAN - Macroaxis
SAN Stock USD 4.98 0.03 0.61%
Banco Santander Total Assets yearly trend continues to be very stable with very little volatility. Total Assets are likely to drop to about 908.6
. Total Assets is the total value of all owned resources that are expected to provide future economic benefits to the business, including cash, investments, accounts receivable, inventory, property,
plant, equipment, and intangible assets.
View All Fundamentals
First Reported Previous Quarter Current Value Quarterly Volatility
Total Assets
1991-12-31 1.8 T 1.8 T 529.2 B
Dot-com Bubble Housing Crash Credit Downgrade Yuan Drop Covid
Check Banco Santander
financial statements
over time to gain insight into future company performance. You can evaluate financial statements to find patterns among Banco Santander's main balance sheet or income statement drivers, such as
Other Operating Expenses of 116.1 B
, Operating Income of 21.5
EBIT of 21.8 B
, as well as many indicators such as
Price To Sales Ratio of 0.97
, Dividend Yield of 0.0422 or
PTB Ratio of 0.6
. Banco financial statements analysis is a perfect complement when working with
Banco Santander Valuation
Check out the analysis of
Banco Santander Correlation
against competitors.
Latest Banco Santander's Total Assets Growth Pattern
Below is the plot of the Total Assets of Banco Santander SA over the last few years. Total assets refers to the total amount of Banco Santander assets owned. Assets are items that have some economic
value and are expended over time to create a benefit for the owner. These assets are usually recorded in Banco Santander SA books under different categories such as cash, marketable securities,
accounts receivable,prepaid expenses, inventory, fixed assets, intangible assets, other assets, marketable securities, accounts receivable, prepaid expenses and others. It is the total value of all
owned resources that are expected to provide future economic benefits to the business, including cash, investments, accounts receivable, inventory, property, plant, equipment, and intangible assets.
Banco Santander's Total Assets historical data analysis aims to capture in quantitative terms the overall pattern of either growth or decline in Banco Santander's overall financial position and show
how it may be relating to other accounts over time.
View Last Reported 1.8 T 10 Years Trend
Banco Total Assets Regression Statistics
Arithmetic Mean 1,306,544,959,959
Geometric Mean 1,096,320,086,811
Coefficient Of Variation 31.90
Mean Deviation 264,340,981,382
Median 1,340,260,000,000
Standard Deviation 416,760,977,195
Sample Variance 173689712112.3T
Range 1.8T
R-Value 0.57
Mean Square Error 127082092420.8T
R-Squared 0.32
Significance 0.03
Slope 52,765,951,146
Total Sum of Squares 2431655969571.9T
Banco Total Assets History
Other Fundumenentals of Banco Santander SA
Banco Santander Total Assets component correlations
About Banco Santander Financial Statements
Banco Santander investors utilize fundamental indicators, such as Total Assets, to predict how Banco Stock might perform in the future. Analyzing these trends over time helps investors make informed
market timing
decisions. For further insights, please visit our
fundamental analysis
Last Reported Projected for Next Year
Total Assets 1.8 T 908.6 B
Intangibles To Total Assets 0.01 0.01
Pair Trading with Banco Santander
One of the main advantages of trading using pair correlations is that every trade hedges away some risk. Because there are two separate transactions required, even if Banco Santander position
performs unexpectedly, the other equity can make up some of the losses. Pair trading also minimizes risk from directional movements in the market. For example, if an entire industry or sector drops
because of unexpected headlines, the short position in Banco Santander will appreciate offsetting losses from the drop in the long position's value.
0.7 C CitigroupFiscal Year End 10th of January 2025 PairCorr
0.81 BK Bank of New YorkFiscal Year End 10th of January 2025 PairCorr
0.85 CM Canadian Imperial BankFiscal Year End 5th of December 2024 PairCorr
0.87 RY Royal BankFiscal Year End 5th of December 2024 PairCorr
The ability to find closely correlated positions to Banco Santander could be a great tool in your tax-loss harvesting strategies, allowing investors a quick way to find a similar-enough asset to
replace Banco Santander when you sell it. If you don't do this, your portfolio allocation will be skewed against your target asset allocation. So, investors can't just sell and buy back Banco
Santander - that would be a violation of the tax code under the "wash sale" rule, and this is why you need to find a similar enough asset and use the proceeds from selling Banco Santander SA to buy
The correlation of Banco Santander is a statistical measure of how it moves in relation to other instruments. This measure is expressed in what is known as the correlation coefficient, which ranges
between -1 and +1. A perfect positive correlation (i.e., a correlation coefficient of +1) implies that as Banco Santander moves, either up or down, the other security will move in the same direction.
Alternatively, perfect negative correlation means that if Banco Santander SA moves in either direction, the perfectly negatively correlated security will move in the opposite direction. If the
correlation is 0, the equities are not correlated; they are entirely random. A correlation greater than 0.8 is generally described as strong, whereas a correlation less than 0.5 is generally
considered weak.
Correlation analysis
and pair trading evaluation for Banco Santander can also be used as hedging techniques within a particular sector or industry or even over random equities to generate a better risk-adjusted return on
your portfolios.
Pair CorrelationCorrelation Matching
When determining whether Banco Santander SA
offers a strong return on investment
in its stock, a comprehensive analysis is essential. The process typically begins with a thorough review of Banco Santander's
financial statements
, including income statements, balance sheets, and cash flow statements, to assess its
financial health
. Key financial ratios are used to gauge profitability, efficiency, and growth potential of Banco Santander Sa Stock.
Outlined below are crucial reports that will aid in making a well-informed decision on Banco Santander Sa Stock:
Is Diversified Banks space expected to grow? Or is there an opportunity to expand the business' product line in the future? Factors like these will boost
the valuation of Banco Santander
. If investors know Banco will grow in the future, the company's valuation will be higher. The financial industry is built on trying to define current growth potential and future valuation
accurately. All the valuation information about Banco Santander listed above have to be considered, but the key to understanding future value is determining which factors weigh more heavily than
Quarterly Earnings Growth Dividend Share Earnings Share Revenue Per Share Quarterly Revenue Growth
0.177 0.195 0.8 3.044 0.047
The market value of Banco Santander SA
is measured differently than its book value, which is the value of Banco that is recorded on the company's balance sheet. Investors also form their own opinion of Banco Santander's value that differs
from its market value or its book value, called intrinsic value, which is Banco Santander's true underlying value. Investors use various methods to calculate intrinsic value and buy a stock when its
market value falls below its intrinsic value. Because Banco Santander's market value can be influenced by many factors that don't directly affect Banco Santander's underlying business (such as a
pandemic or basic market pessimism), market value can vary widely from intrinsic value.
AltmanZ ScoreDetails PiotroskiF ScoreDetails BeneishM ScoreDetails FinancialAnalysisDetails Buy or SellAdviceDetails
Please note, there is a significant difference between Banco Santander's value and its price as these two are different measures arrived at by different means. Investors typically determine
if Banco Santander is a good investment
by looking at such factors as earnings, sales, fundamental and technical indicators, competition as well as analyst projections. However, Banco Santander's price is the amount at which it trades on
the open market and represents the number that a seller and buyer find agreeable to each party. | {"url":"https://widgets.macroaxis.com/financial-statements/SAN/Total-Assets","timestamp":"2024-11-05T05:52:00Z","content_type":"text/html","content_length":"567109","record_id":"<urn:uuid:3b985e59-cdd9-4950-8b08-73d53ebca371>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00500.warc.gz"} |
Note: This document is for an older version of GRASS GIS that will be discontinued soon. You should upgrade, and read the current manual page.
- Computes lithospheric flexural isostasy
r.flexure --help
r.flexure [-l] method=string input=name te=name te_units=string output=name [solver=string] [tolerance=float] [northbc=string] [southbc=string] [westbc=string] [eastbc=string] [g=float] [ym=float] [
nu=float] [rho_fill=float] [rho_m=float] [--overwrite] [--help] [--verbose] [--quiet] [--ui]
Allows running in lat/lon: dx is f(lat) at grid N-S midpoint
Allow output files to overwrite existing files
Print usage summary
Verbose module output
Quiet module output
Force launching GUI dialog
method=string [required]
Solution method: Finite Diff. or Superpos. of analytical sol'ns
Options: FD, SAS
input=name [required]
Raster map of loads (thickness * density * g) [Pa]
te=name [required]
Elastic thickness: scalar or raster; unis chosen in "te_units"
te_units=string [required]
Units for elastic thickness
Options: m, km
output=name [required]
Output raster map of vertical deflections [m]
Solver type
Options: direct, iterative
Default: direct
Convergence tolerance (between iterations) for iterative solver
Default: 1E-3
Northern boundary condition
Options: 0Displacement0Slope, 0Moment0Shear, 0Slope0Shear, Mirror, Periodic, NoOutsideLoads
Default: NoOutsideLoads
Southern boundary condition
Options: 0Displacement0Slope, 0Moment0Shear, 0Slope0Shear, Mirror, Periodic, NoOutsideLoads
Default: NoOutsideLoads
Western boundary condition
Options: 0Displacement0Slope, 0Moment0Shear, 0Slope0Shear, Mirror, Periodic, NoOutsideLoads
Default: NoOutsideLoads
Eastern boundary condition
Options: 0Displacement0Slope, 0Moment0Shear, 0Slope0Shear, Mirror, Periodic, NoOutsideLoads
Default: NoOutsideLoads
gravitational acceleration at surface [m/s^2]
Default: 9.8
Young's Modulus [Pa]
Default: 65E9
Poisson's ratio
Default: 0.25
Density of material that fills flexural depressions [kg/m^3]
Default: 0
Mantle density [kg/m^3]
Default: 3300
computes how the rigid outer shell of a planet deforms elastically in response to surface-normal loads by solving equations for plate bending. This phenomenon is known as "flexural isostasy" and can
be useful in cases of glacier/ice-cap/ice-sheet loading, sedimentary basin filling, mountain belt growth, volcano emplacement, sea-level change, and other geologic processes.
are the GRASS GIS interfaces to the model
. As both
are interfaces to gFlex, this must be downloaded and installed. The most recent versions of
are available from
, and installation instructions are available on that page via the
The parameter
sets whether the solution is Finite Difference ("FD") or Superposition of Analytical Solutions ("SAS"). The Finite difference method is typically faster for large arrays, and allows lithospheric
elastic thickness to be varied laterally, following the solution of van Wees and Cloetingh (1994). However, it is quite memory-intensive, so unless the user has a computer with a very large amount of
memory and quite a lot of time to wait, they should ensure that they use a grid spacing that is appropriate to solve the problem at hand. Flexural isostatic solutions act to smooth inputs over a
given flexural wavelength (see , so if an appropriate solution resolution is chosen, the calculated flexural response can be interpolated to a higher resolution without fear of aliasing.
The flexural solution is generated for the current computational region, so be sure to check g.region before running the model!
input is a 2-D array of loads in a GRASS raster. These are in units of stress, and equal the density of the material times the acceleration due to gravity times the thickness of the column. This is
not affected by what you choose for g, later: it is pre-calculated by the user.
te, written in standard text as T[e], is the lithospheric elastic thickness.
Several boundary conditions are available, and these depend on if the solution method is finite difference (FD) or superposition of analytical solutions (SAS). In the latter, it is assumed that there
are no loads outside of those that are explicitly listed, so the boundary conditions are "NoOutsideLoads". As this is the implicit case, the boundary conditions all default to this.
The finite difference boundary conditions are a bit more complicated, but are largely self-explanatory:
0-displacement-0-slope boundary condition
"Broken plate" boundary condition: second and third derivatives of vertical displacement are 0. This is like the end of a diving board.
First and third derivatives of vertical displacement are zero. While this does not lend itself so easily to physical meaning, it is helpful to aid in efforts to make boundary condition effects
disappear (i.e. to emulate the NoOutsideLoads cases)
Load and elastic thickness structures reflected at boundary.
"Wrap-around" boundary condition: must be applied to both North and South and/or both East and West. This causes, for example, the edge of the eastern and western limits of the domain to act like
they are next to each other in an infinite loop.
All of these boundary conditions may be combined in any way, with the exception of the note for periodic boundary conditions. If one does not want the boundary conditions to affect the solutions, it
is recommended that one places the boundaries at least one flexural wavelength away from the load.
r.flexure may be run in latitude/longitude coordinates (with the "-l" flag), but its grid constraint is that it can have only one dx and one dy for the entire domain. Thus, it chooses the average dx
at the midpoint between the northernmost and southernmost latitudes for which the calculations are made. This assumption can break down at the poles, where the East–West dimension rapidly diminishes.
The Community Surface Dynamics Modeling System, into which gFlex is integrated, is a community-driven effort to build an open-source modeling infrastructure for Earth-surface processes.
Wickert, A. D. (2015), Open-source modular solutions for flexural isostasy: gFlex v1.0,
Geoscientific Model Development Discussions
(6), 4245–4292, doi:10.5194/gmdd-8-4245-2015.
Wickert, A. D., G. E. Tucker, E. W. H. Hutton, B. Yan, and S. D. Peckham (2011), Feedbacks between surface processes and flexural isostasy: a motivation for coupling models, in CSDMS 2011 Meeting:
Impact of time and process scales, Student Keynote, Boulder, CO.
van Wees, J. D., and S. Cloetingh (1994), A Finite-Difference Technique to Incorporate Spatial Variations In Rigidity and Planar Faults Into 3-D Models For Lithospheric Flexure, Geophysical Journal
International, 117(1), 179–195, doi:10.1111/j.1365-246X.1994.tb03311.x.
Andrew D. Wickert
Available at: r.flexure source code (history)
Latest change: Monday Nov 11 18:04:48 2024 in commit: 59e289fdb093de6dd98d5827973e41128196887d
Main index | Raster index | Topics index | Keywords index | Graphical index | Full index
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One-half cup of black beans provides 15% of the potassium you need daily. You must get the remaining 2890 milligrams from other sources. How many milligrams of potassium should you consume daily? | Socratic
One-half cup of black beans provides 15% of the potassium you need daily. You must get the remaining 2890 milligrams from other sources. How many milligrams of potassium should you consume daily?
1 Answer
Total potassium intake required is 3400 milligrams
Let the total amount required be $t$
You have 15% already. This means that the amount yet to be taken is:
#(100-15)% = 85%#
so #" "85%t=2890#
write as:
$\textcolor{b r o w n}{\frac{85}{100} t = 2890}$
Multiply both sides by $\textcolor{b l u e}{\frac{100}{85}}$
$\textcolor{b r o w n}{\textcolor{b l u e}{\frac{100}{85} \times} \frac{85}{100} \times t = \textcolor{b l u e}{\frac{100}{85} \times} 2890}$
$1 \times t = 3400$
Impact of this question
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Angular Momentum and Its Conservation
Learning Objectives
By the end of this section, you will be able to:
• Understand the analogy between angular momentum and linear momentum.
• Observe the relationship between torque and angular momentum.
• Apply the law of conservation of angular momentum.
Why does Earth keep on spinning? What started it spinning to begin with? And how does an ice skater manage to spin faster and faster simply by pulling her arms in? Why does she not have to exert a
torque to spin faster? Questions like these have answers based in angular momentum, the rotational analog to linear momentum. By now the pattern is clear—every rotational phenomenon has a direct
translational analog. It seems quite reasonable, then, to define angular momentum L as
L = Iω.
This equation is an analog to the definition of linear momentum as p = mv. Units for linear momentum are kg ⋅ m/s while units for angular momentum are kg ⋅ m^2/s. As we would expect, an object that
has a large moment of inertia I, such as Earth, has a very large angular momentum. An object that has a large angular velocity ω, such as a centrifuge, also has a rather large angular momentum.
Making Connections
Angular momentum is completely analogous to linear momentum, first presented in
Uniform Circular Motion and Gravitation
. It has the same implications in terms of carrying rotation forward, and it is conserved when the net external torque is zero. Angular momentum, like linear momentum, is also a property of the atoms
and subatomic particles.
Example 1. Calculating Angular Momentum of the Earth
No information is given in the statement of the problem; so we must look up pertinent data before we can calculate L = Iω. First, according to Figure 1, the formula for the moment of inertia of a
sphere is
so that
Earth’s mass M is 5.979 × 10^24 kg and its radius R is 6.376 × 10^6 m. The Earth’s angular velocity ω is, of course, exactly one revolution per day, but we must covert ω to radians per second to do
the calculation in SI units.
Substituting known information into the expression for L and converting ω to radians per second gives
[latex]\begin{array}{lll}L& =& 0.4{\left(5.979\times 10^{24}\text{ kg}\right)}{\left(6.376\times 10^{6}\text{ m}\right)}^{2}\left(\frac{1\text{ rev}}{\text{d}}\right)\\ & =& 9.72\times {10}^{37}\text
{ kg}\cdot {\text{m}}^{2}\cdot \text{rev/d}\end{array}\\[/latex].
Substituting 2π rad for 1 rev and 8.64 × 10^4 s for 1 day gives
[latex]\begin{array}{lll}L& =& {\left(9.72\times {10}^{37}\text{ kg}\cdot {\text{m}}^{2}\right)}\left(\frac{2\pi \text{ rad/rev}}{8.64\times {10}^{4}\text{ s/d}}\right)\left(1\text{ rev/d}\right)\\ &
=& 7.07 \times {10}^{33}\text{ kg}\cdot {\text{ m}}^{2}\text{/s}\end{array}\\[/latex].
This number is large, demonstrating that Earth, as expected, has a tremendous angular momentum. The answer is approximate, because we have assumed a constant density for Earth in order to estimate
its moment of inertia.
When you push a merry-go-round, spin a bike wheel, or open a door, you exert a torque. If the torque you exert is greater than opposing torques, then the rotation accelerates, and angular momentum
increases. The greater the net torque, the more rapid the increase in L. The relationship between torque and angular momentum is
[latex]\text{net }\tau =\frac{\Delta L}{\Delta t}\\[/latex].
This expression is exactly analogous to the relationship between force and linear momentum, F = Δp/Δt. The equation [latex]\text{net }\tau =\frac{\Delta L}{\Delta t}\\[/latex] is very fundamental and
broadly applicable. It is, in fact, the rotational form of Newton’s second law.
Example 2. Calculating the Torque Putting Angular Momentum Into a Lazy Susan
Figure 2 shows a Lazy Susan food tray being rotated by a person in quest of sustenance. Suppose the person exerts a 2.50 N force perpendicular to the lazy Susan’s 0.260-m radius for 0.150 s. (a) What
is the final angular momentum of the lazy Susan if it starts from rest, assuming friction is negligible? (b) What is the final angular velocity of the lazy Susan, given that its mass is 4.00 kg and
assuming its moment of inertia is that of a disk?
We can find the angular momentum by solving[latex]\text{net }\tau =\frac{\Delta L}{\Delta t}\\[/latex] for [latex]\Delta L\\[/latex] for ΔL, and using the given information to calculate the torque.
The final angular momentum equals the change in angular momentum, because the lazy Susan starts from rest. That is, ΔL = L. To find the final velocity, we must calculate ω from the definition of L in
L = Iω.
Solution for (a)
Solving [latex]\text{net}\tau =\frac{\Delta L}{\Delta t}\\[/latex] for ΔL gives
[latex]\Delta L=\left(\text{net}\tau\right){\Delta t}\\[/latex]
Because the force is perpendicular to r, we see that [latex]\text{net }\tau ={rF}\\[/latex], so that
[latex]\begin{array}{lll}L& =& {rF}\Delta t=\left(0.260 \text{ m}\right)\left(2.50 \text{ N}\right)\left(0.150 \text{s}\right)\\ & =& 9.75\times {10}^{-2}\text{ kg}\cdot {\text{m}}^{2}/\text{s}\end
Solution for (b)
The final angular velocity can be calculated from the definition of angular momentum,
L = Iω.
Solving for ω and substituting the formula for the moment of inertia of a disk into the resulting equation gives
[latex]\omega =\frac{L}{I}=\frac{L}{\frac{1}{2}{{MR}}^{2}}\\[/latex].
And substituting known values into the preceding equation yields
[latex]\omega =\frac{9.75\times {10}^{-2}\text{ kg}\cdot{\text{ m}}^{2}\text{/s}}{\left(0.500\right)\left(4.00\text{ kg}\right)\left(0.260\text{ m}\right)}=0.721\text{ rad/s}\\[/latex].
Note that the imparted angular momentum does not depend on any property of the object but only on torque and time. The final angular velocity is equivalent to one revolution in 8.71 s (determination
of the time period is left as an exercise for the reader), which is about right for a lazy Susan.
Example 3. Calculating the Torque in a Kick
The person whose leg is shown in Figure 3 kicks his leg by exerting a 2000-N force with his upper leg muscle. The effective perpendicular lever arm is 2.20 cm. Given the moment of inertia of the
lower leg is 1.25 kg⋅m^2, (a) find the angular acceleration of the leg. (b) Neglecting the gravitational force, what is the rotational kinetic energy of the leg after it has rotated through 57.3º
(1.00 rad)?
The angular acceleration can be found using the rotational analog to Newton’s second law, or [latex]\alpha =\text{net }\tau/I\\[/latex]. The moment of inertia I is given and the torque can be found
easily from the given force and perpendicular lever arm. Once the angular acceleration α is known, the final angular velocity and rotational kinetic energy can be calculated.
Solution to (a)
From the rotational analog to Newton’s second law, the angular acceleration α is
[latex]\alpha =\frac{\text{net}\tau }{I}\\[/latex].
Because the force and the perpendicular lever arm are given and the leg is vertical so that its weight does not create a torque, the net torque is thus
[latex]\begin{array}{lll}\text{net}\tau & =& {r}_{\perp }F\\ & =& \left(0\text{.}\text{0220 m}\right)\left(\text{2000}\text{N}\right)\\ & =& 44.0\text{ N}\cdot \text{m}\end{array}\\[/latex].
Substituting this value for the torque and the given value for the moment of inertia into the expression for α gives
[latex]\alpha =\frac{{44.0}\text{ N}\cdot\text{ m}}{{1.25}\text{ kg}\cdot\text{ m}^{2}}=35.2{\text{ rad/s}}^{2}\\[/latex].
Solution to (b)
The final angular velocity can be calculated from the kinematic expression
because the initial angular velocity is zero. The kinetic energy of rotation is
so it is most convenient to use the value of ω^2 just found and the given value for the moment of inertia. The kinetic energy is then
[latex]\begin{array}{lll}{\text{KE}}_{\text{rot}}& =& 0.5\left(1\text{.25}\text{ kg}\cdot {\text{m}}^{2}\right)\left(\text{70.}4{\text{ rad}}^{2}/{\text{s}}^{2}\right)\\ & =& \text{44}\text{.}0\text{
These values are reasonable for a person kicking his leg starting from the position shown. The weight of the leg can be neglected in part (a) because it exerts no torque when the center of gravity of
the lower leg is directly beneath the pivot in the knee. In part (b), the force exerted by the upper leg is so large that its torque is much greater than that created by the weight of the lower leg
as it rotates. The rotational kinetic energy given to the lower leg is enough that it could give a ball a significant velocity by transferring some of this energy in a kick.
Making Connections: Conservation Laws
Angular momentum, like energy and linear momentum, is conserved. This universally applicable law is another sign of underlying unity in physical laws. Angular momentum is conserved when net external
torque is zero, just as linear momentum is conserved when the net external force is zero.
Conservation of Angular Momentum
We can now understand why Earth keeps on spinning. As we saw in the previous example, [latex]\Delta L=\left(\text{net }\tau\right)Delta t\\[/latex]. This equation means that, to change angular
momentum, a torque must act over some period of time. Because Earth has a large angular momentum, a large torque acting over a long time is needed to change its rate of spin. So what external torques
are there? Tidal friction exerts torque that is slowing Earth’s rotation, but tens of millions of years must pass before the change is very significant. Recent research indicates the length of the
day was 18 h some 900 million years ago. Only the tides exert significant retarding torques on Earth, and so it will continue to spin, although ever more slowly, for many billions of years.
What we have here is, in fact, another conservation law. If the net torque is zero, then angular momentum is constant or conserved. We can see this rigorously by considering [latex]\text{net }\tau =\
frac{\Delta L}{\Delta t}\\[/latex] for the situation in which the net torque is zero. In that case,
net τ = 0
implying that
[latex]\frac{\Delta L}{\Delta t}=0\\[/latex].
If the change in angular momentum ΔL is zero, then the angular momentum is constant; thus,
L = constant (net τ = 0)
L = L′ (net τ = 0).
These expressions are the law of conservation of angular momentum. Conservation laws are as scarce as they are important. An example of conservation of angular momentum is seen in Figure 4, in which
an ice skater is executing a spin. The net torque on her is very close to zero, because there is relatively little friction between her skates and the ice and because the friction is exerted very
close to the pivot point. (Both F and r are small, and so τ is negligibly small.) Consequently, she can spin for quite some time. She can do something else, too. She can increase her rate of spin by
pulling her arms and legs in. Why does pulling her arms and legs in increase her rate of spin? The answer is that her angular momentum is constant, so that
L = L′.
Expressing this equation in terms of the moment of inertia
Iω = I′ω′,
where the primed quantities refer to conditions after she has pulled in her arms and reduced her moment of inertia. Because I′ is smaller, the angular velocity ω′ must increase to keep the angular
momentum constant. The change can be dramatic, as the following example shows.
Example 4. Calculating the Angular Momentum of a Spinning Skater
Suppose an ice skater, such as the one in Figure 4, is spinning at 0.800 rev/s with her arms extended. She has a moment of inertia of 2.34 kg ⋅ m^2 with her arms extended and of 0.363 kg ⋅ m^2 with
her arms close to her body. (These moments of inertia are based on reasonable assumptions about a 60.0-kg skater.) (a) What is her angular velocity in revolutions per second after she pulls in her
arms? (b) What is her rotational kinetic energy before and after she does this?
In the first part of the problem, we are looking for the skater’s angular velocity ω′ after she has pulled in her arms. To find this quantity, we use the conservation of angular momentum and note
that the moments of inertia and initial angular velocity are given. To find the initial and final kinetic energies, we use the definition of rotational kinetic energy given by
[latex]{\text{KE}}_{\text{rot}}=\frac{1}{2}{{I\omega }}^{2}\\[/latex].
Solution for (a)
Because torque is negligible (as discussed above), the conservation of angular momentum given in Iω = I′ω′ is applicable. Thus,
L = L′
Iω = I′ω′
Solving for ω′ and substituting known values into the resulting equation gives
[latex]\begin{array}{lll}\omega′ & =& \frac{I}{I′}\omega =\left(\frac{\text{2.34 kg}\cdot {m}^{2}}{0\text{.363 kg}\cdot {m}^{2}}\right)\left(\text{0.800 rev/s}\right)\\ & =& \text{}\text{5.16 rev/s}\
Solution for (b)
Rotational kinetic energy is given by
[latex]{\text{KE}}_{\text{rot}}=\frac{1}{2}{{I\omega }}^{2}\\[/latex].
The initial value is found by substituting known values into the equation and converting the angular velocity to rad/s:
[latex]\begin{array}{lll}{\text{KE}}_{\text{rot}}& =& \left(0.5\right)\left(2.34\text{ kg}\cdot{\text{m}}^{2}\right){\left(\left(0.800\text{rev/s}\right)\left(2\pi \text{ rad/rev}\right)\right)}^{2}\
\ & =& 29.6\text{ J}\end{array}\\[/latex].
The final rotational kinetic energy is
[latex]{\text{KE}}_{\text{rot}}′ =\frac{1}{2}I′ {\omega′}^{2}\\[/latex].
Substituting known values into this equation gives
[latex]\begin{array}{lll}{KE_{\text{rot}}}′ & =& \left(0\text{.}5\right)\left(0\text{.363 kg}\cdot {m}^{2}\right){\left[\left(5\text{.}\text{16 rev/s}\right)\left(2\pi \text{ rad/rev}\right)\right]}^
{2}\\ & =&\text{191 J.}\end{array}\\[/latex]
In both parts, there is an impressive increase. First, the final angular velocity is large, although most world-class skaters can achieve spin rates about this great. Second, the final kinetic energy
is much greater than the initial kinetic energy. The increase in rotational kinetic energy comes from work done by the skater in pulling in her arms. This work is internal work that depletes some of
the skater’s food energy.
There are several other examples of objects that increase their rate of spin because something reduced their moment of inertia. Tornadoes are one example. Storm systems that create tornadoes are
slowly rotating. When the radius of rotation narrows, even in a local region, angular velocity increases, sometimes to the furious level of a tornado. Earth is another example. Our planet was born
from a huge cloud of gas and dust, the rotation of which came from turbulence in an even larger cloud. Gravitational forces caused the cloud to contract, and the rotation rate increased as a result.
(See Figure 5.)
In case of human motion, one would not expect angular momentum to be conserved when a body interacts with the environment as its foot pushes off the ground. Astronauts floating in space aboard the
International Space Station have no angular momentum relative to the inside of the ship if they are motionless. Their bodies will continue to have this zero value no matter how they twist about as
long as they do not give themselves a push off the side of the vessel.
Check Your Undestanding
Is angular momentum completely analogous to linear momentum? What, if any, are their differences?
Yes, angular and linear momentums are completely analogous. While they are exact analogs they have different units and are not directly inter-convertible like forms of energy are.
Section Summary
• Every rotational phenomenon has a direct translational analog , likewise angular momentum L can be defined as L = Iω.
• This equation is an analog to the definition of linear momentum as p = mv. The relationship between torque and angular momentum is [latex]\text{net }\tau =\frac{\Delta L}{\Delta t}\\[/latex].
• Angular momentum, like energy and linear momentum, is conserved. This universally applicable law is another sign of underlying unity in physical laws. Angular momentum is conserved when net
external torque is zero, just as linear momentum is conserved when the net external force is zero.
Conceptual Questions
1. When you start the engine of your car with the transmission in neutral, you notice that the car rocks in the opposite sense of the engine’s rotation. Explain in terms of conservation of angular
momentum. Is the angular momentum of the car conserved for long (for more than a few seconds)?
2. Suppose a child walks from the outer edge of a rotating merry-go round to the inside. Does the angular velocity of the merry-go-round increase, decrease, or remain the same? Explain your answer.
3. Suppose a child gets off a rotating merry-go-round. Does the angular velocity of the merry-go-round increase, decrease, or remain the same if: (a) He jumps off radially? (b) He jumps backward to
land motionless? (c) He jumps straight up and hangs onto an overhead tree branch? (d) He jumps off forward, tangential to the edge? Explain your answers. (Refer to Figure 6).
4. Helicopters have a small propeller on their tail to keep them from rotating in the opposite direction of their main lifting blades. Explain in terms of Newton’s third law why the helicopter body
rotates in the opposite direction to the blades.
5. Whenever a helicopter has two sets of lifting blades, they rotate in opposite directions (and there will be no tail propeller). Explain why it is best to have the blades rotate in opposite
6. Describe how work is done by a skater pulling in her arms during a spin. In particular, identify the force she exerts on each arm to pull it in and the distance each moves, noting that a component
of the force is in the direction moved. Why is angular momentum not increased by this action?
7. When there is a global heating trend on Earth, the atmosphere expands and the length of the day increases very slightly. Explain why the length of a day increases.
8. Nearly all conventional piston engines have flywheels on them to smooth out engine vibrations caused by the thrust of individual piston firings. Why does the flywheel have this effect?
9. Jet turbines spin rapidly. They are designed to fly apart if something makes them seize suddenly, rather than transfer angular momentum to the plane’s wing, possibly tearing it off. Explain how
flying apart conserves angular momentum without transferring it to the wing.
10. An astronaut tightens a bolt on a satellite in orbit. He rotates in a direction opposite to that of the bolt, and the satellite rotates in the same direction as the bolt. Explain why. If a
handhold is available on the satellite, can this counter-rotation be prevented? Explain your answer.
11. Competitive divers pull their limbs in and curl up their bodies when they do flips. Just before entering the water, they fully extend their limbs to enter straight down. Explain the effect of
both actions on their angular velocities. Also explain the effect on their angular momenta.
12. Draw a free body diagram to show how a diver gains angular momentum when leaving the diving board.
13. In terms of angular momentum, what is the advantage of giving a football or a rifle bullet a spin when throwing or releasing it?
Problems & Exercises
1. (a) Calculate the angular momentum of the Earth in its orbit around the Sun.
(b) Compare this angular momentum with the angular momentum of Earth on its axis.
2. (a) What is the angular momentum of the Moon in its orbit around Earth?
(b) How does this angular momentum compare with the angular momentum of the Moon on its axis? Remember that the Moon keeps one side toward Earth at all times.
(c) Discuss whether the values found in parts (a) and (b) seem consistent with the fact that tidal effects with Earth have caused the Moon to rotate with one side always facing Earth.
3. Suppose you start an antique car by exerting a force of 300 N on its crank for 0.250 s. What angular momentum is given to the engine if the handle of the crank is 0.300 m from the pivot and the
force is exerted to create maximum torque the entire time?
4. A playground merry-go-round has a mass of 120 kg and a radius of 1.80 m and it is rotating with an angular velocity of 0.500 rev/s. What is its angular velocity after a 22.0-kg child gets onto it
by grabbing its outer edge? The child is initially at rest.
5. Three children are riding on the edge of a merry-go-round that is 100 kg, has a 1.60-m radius, and is spinning at 20.0 rpm. The children have masses of 22.0, 28.0, and 33.0 kg. If the child who
has a mass of 28.0 kg moves to the center of the merry-go-round, what is the new angular velocity in rpm?
6. (a) Calculate the angular momentum of an ice skater spinning at 6.00 rev/s given his moment of inertia is 0.400 kg. (b) He reduces his rate of spin (his angular velocity) by extending his arms and
increasing his moment of inertia. Find the value of his moment of inertia if his angular velocity decreases to 1.25 rev/s. (c) Suppose instead he keeps his arms in and allows friction of the ice to
slow him to 3.00 rev/s. What average torque was exerted if this takes 15.0 s?
7. Construct Your Own Problem Consider the Earth-Moon system. Construct a problem in which you calculate the total angular momentum of the system including the spins of the Earth and the Moon on
their axes and the orbital angular momentum of the Earth-Moon system in its nearly monthly rotation. Calculate what happens to the Moon’s orbital radius if the Earth’s rotation decreases due to tidal
drag. Among the things to be considered are the amount by which the Earth’s rotation slows and the fact that the Moon will continue to have one side always facing the Earth.
angular momentum:
the product of moment of inertia and angular velocity
law of conservation of angular momentum:
angular momentum is conserved, i.e., the initial angular momentum is equal to the final angular momentum when no external torque is applied to the system
Selected Solutions to Problems & Answers
1. (a) 2.66 × 10^40 kg ⋅ m^2/s (b) (a) 7.07 x 10^33 kg ⋅ m^2/s
The angular momentum of the Earth in its orbit around the Sun 3.77 × 10^6 is times larger than the angular momentum of the Earth around its axis.
3. 22.5 kg ⋅ m^2/s
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Native Floating-Point Support
Getting Started with HDL Coder Native Floating-Point Support
Native floating-point support in HDL Coder™ enables you to generate code from your floating-point design. If your design has complex math and trigonometric operations or has data with a large dynamic
range, use native floating point. You can use native floating point with a Simulink^® model or a MATLAB^® function.
Key Features
In your Simulink model or MATLAB function:
• You can have half-precision, single-precision, and double-precision floating-point data types and operations.
• You can have a combination of integer, fixed-point, and floating-point operations. By using Data Type Conversion blocks, you can perform conversions between floating-point and fixed-point data
The generated code:
• Complies with the IEEE-754 standard of floating-point arithmetic.
• Is target-independent. You can deploy the code on any generic FPGA or an ASIC.
• Does not require floating-point processing units or hard floating-point DSP blocks on the target ASIC or FPGA.
HDL Coder supports:
• Math and trigonometric functions
• Large subset of Simulink blocks
• Denormal numbers
• Customizing the latency of the floating-point operator
Numeric Considerations and IEEE-754 Standard Compliance
Native floating point technology in HDL Coder adheres to IEEE standard of floating-point arithmetic. For basic arithmetic operations such as addition, subtraction, multiplication, division, and
reciprocal, when you generate HDL code in native floating-point mode, the numeric results obtained match the original Simulink model or MATLAB function.
Certain advanced math operations such as exponential, logarithm, and trigonometric operators have machine-specific implementation behaviors because these operators use recurring Taylor series and
Remez expression based implementations. When you use these operators in native floating-point mode, the generated HDL code can have relatively small numeric differences from the Simulink model or
MATLAB function. These numeric differences are within a tolerance range and therefore indicate compliance with the IEEE-754 standard.
To generate code that complies with the IEEE-754 standard, HDL Coder supports:
• Round to nearest rounding mode
• Denormal numbers
• Exceptions such as NaN (Not a Number), Inf, and Zero
• Customization of ULP (Units in the Last Place) and relative accuracy
For more information, see Numeric Considerations for Native Floating-Point.
Floating Point Types
Single Precision
In the IEEE 754–2008 standard, the single-precision floating-point number is 32-bits. The 32-bit number encodes a 1-bit sign, an 8-bit exponent, and a 23-bit mantissa.
This graph is the normalized representation for floating-point numbers. You can compute the actual value of a normal number as:
$value={\left(-1\right)}^{sign}*\left(1+\underset{i=1}{\stackrel{23}{\Sigma }}{b}_{23-i}{2}^{-i}\right)*{2}^{\left(e-127\right)}$
The exponent field represents the exponent plus a bias of 127. The size of the mantissa is 24 bits. The leading bit is a 1, so the representation encodes the lower 23 bits.
Use single-precision types for applications that require larger dynamic range than half-precision types. Single-precision operations consume less memory and has lower latency than double-precision
Double Precision
In the IEEE 754–2008 standard, the single-precision floating-point number is 64-bits. The 64-bit number encodes a 1-bit sign, an 11-bit exponent, and a 52-bit mantissa.
The exponent field represents the exponent plus a bias of 1023. The size of the mantissa is 53 bits. The leading bit is a 1, so the representation encodes the lower 52 bits.
Use double-precision types for applications that require larger dynamic range, accuracy, and precision. These operations consume larger area on the FPGA and lower target frequency.
Half Precision
In the IEEE 754–2008 standard, the half-precision floating-point number is 16-bits. The 16-bit number encodes a 1-bit sign, a 5-bit exponent, and a 10-bit mantissa.
The exponent field represents the exponent plus a bias of 15. The size of the mantissa is 11 bits. The leading bit is a 1, so the representation encodes the lower 10 bits.
Use half-precision types for applications that require smaller dynamic range, consumes much less memory, has lower latency, and saves FPGA resources.
When using half types, you might want to explicitly set the Output data type of the blocks to half instead of the default setting Inherit: Inherit via internal rule. To learn how to change the
parameters programmatically, see Set HDL Block Parameters for Multiple Blocks Programmatically.
Data Type Considerations
With native floating-point support, HDL Coder supports code generation from Simulink models or MATLAB functions that contain floating-point signals and fixed-point signals. You can model your design
with floating-point types to:
• Implement algorithms that have a large or unknown dynamic range that can fall outside the range of representable fixed-point types.
• Implement complex math and trigonometric operations that are difficult to design in fixed point.
• Obtain a higher precision and better accuracy.
Floating-point designs can potentially occupy more area on the target hardware. In your Simulink model or MATLAB function, it is recommended to use floating-point data types in the algorithm data
path and fixed-point data types in the algorithm control logic. This figure shows a section of a Simulink model that uses single and fixed-point types. By using Data Type Conversion blocks, you can
perform conversions between the single and fixed-point types.
See Also
Modeling Guidelines
Related Examples
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The Minimum Number of Stacks Needed to Implement a Queue
stacks · April 27, 2024
The Minimum Number of Stacks Needed to Implement a Queue
Learn how to implement a queue using the minimum number of stacks and understand the underlying data structures and algorithms.
Create an image featuring JavaScript code snippets and interview-related icons or graphics. Use a color scheme of yellows and blues. Include the title '7 Essential JavaScript Interview Questions for
The Minimum Number of Stacks Needed to Implement a Queue
In computer science, data structures play a crucial role in solving complex problems efficiently. Two fundamental data structures, stacks and queues, are widely used in various applications. While
stacks follow the Last-In-First-Out (LIFO) principle, queues follow the First-In-First-Out (FIFO) principle. In this blog post, we'll explore the minimum number of stacks required to implement a
What is a Queue?
A queue is a linear data structure that follows the FIFO principle. It's a collection of elements, where elements are added and removed from the ends. The front of the queue is where elements are
removed (dequeued), and the rear of the queue is where elements are added (enqueued). Queues are essential in many applications, such as job scheduling, print queues, and network protocols.
What is a Stack?
A stack is a linear data structure that follows the LIFO principle. It's a collection of elements, where elements are added and removed from the top. The top of the stack is where elements are added
(pushed) and removed (popped). Stacks are widely used in parsing, evaluating postfix expressions, and implementing recursive algorithms.
Implementing a Queue using Stacks
Now, let's dive into the main topic: implementing a queue using stacks. At first glance, it might seem counterintuitive to use stacks to implement a queue, as they follow different principles.
However, with some cleverness, we can use stacks to simulate a queue.
The Naive Approach
A naive approach to implement a queue using stacks would be to use a single stack. We could push elements onto the stack to enqueue them and pop elements from the stack to dequeue them. However, this
approach has a major flaw: it doesn't maintain the FIFO order.
Consider the following example:
• Enqueue 1, 2, 3, 4, 5
• Dequeue: expected output = 1, actual output = 5 ( incorrect )
The problem lies in the LIFO nature of stacks. When we dequeue an element, we're removing the most recently added element, not the oldest one.
The Optimal Solution
To implement a queue using stacks, we need to use at least two stacks. We'll refer to these stacks as inStack and outStack.
• inStack is used to store elements in the order they're enqueued.
• outStack is used to store elements in the order they're dequeued.
Here's the algorithm:
1. Enqueue:
□ Push the element onto inStack.
2. Dequeue:
□ If outStack is empty, pop all elements from inStack and push them onto outStack in reverse order.
□ Pop the top element from outStack and return it.
Let's analyze the time complexity of this approach:
• Enqueue: O(1)
• Dequeue: Amortized O(1), but O(n) in the worst case (when outStack is empty)
Why Two Stacks are Necessary
Now, you might wonder why we need two stacks to implement a queue. Can't we use a single stack with some clever manipulation? The answer lies in the fundamental properties of stacks and queues.
A stack can only access the top element, whereas a queue requires access to both the front and rear elements. By using two stacks, we can decouple the enqueue and dequeue operations, allowing us to
maintain the FIFO order.
In conclusion, the minimum number of stacks needed to implement a queue is two. Using two stacks, we can efficiently implement a queue with O(1) enqueue and amortized O(1) dequeue operations. This
approach is essential in scenarios where a queue is required, but the underlying data structure is a stack.
Further Reading
I hope this blog post has provided a comprehensive understanding of implementing a queue using stacks. If you have any questions or need further clarification, please don't hesitate to ask! | {"url":"https://30dayscoding.com/blog/minimum-stacks-needed-implement-queue","timestamp":"2024-11-05T23:46:15Z","content_type":"text/html","content_length":"95204","record_id":"<urn:uuid:e096bca1-f06a-4a70-b4d5-da76cf936186>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00135.warc.gz"} |
{"month":"04","quality_controlled":"1","department":[{"_id":"KrPi"},{"_id":"VlKo"}],"status":"public","publisher":"Springer","_id":"1231","page":"358 - 387","project":[{"name":"Provable Security for
Physical Cryptography","_id":"258C570E-B435-11E9-9278-68D0E5697425","call_identifier":"FP7","grant_number":"259668"},{"_id":"25FBA906-B435-11E9-9278-68D0E5697425","name":"DOiCV : Discrete
Optimization in Computer Vision: Theory and Practice","call_identifier":"FP7","grant_number":"616160"}],"type":"conference","alternative_title":["LNCS"],"language":
[{"iso":"eng"}],"ec_funded":1,"scopus_import":1,"year":"2016","publist_id":"6103","doi":"10.1007/978-3-662-49896-5_13","title":"On the complexity of scrypt and proofs of space in the parallel random
oracle model","date_created":"2018-12-11T11:50:51Z","publication_status":"published","user_id":"3E5EF7F0-F248-11E8-B48F-1D18A9856A87","author":[{"first_name":"Joel
F","last_name":"Alwen","full_name":"Alwen, Joel F","id":"2A8DFA8C-F248-11E8-B48F-1D18A9856A87"},{"full_name":"Chen, Binyi","first_name":"Binyi","last_name":"Chen"},{"full_name":"Kamath Hosdurg,
Chethan","id":"4BD3F30E-F248-11E8-B48F-1D18A9856A87","first_name":"Chethan","last_name":"Kamath Hosdurg"},
{"last_name":"Kolmogorov","first_name":"Vladimir","id":"3D50B0BA-F248-11E8-B48F-1D18A9856A87","full_name":"Kolmogorov, Vladimir"},{"id":"3E04A7AA-F248-11E8-B48F-1D18A9856A87","full_name":"Pietrzak,
Krzysztof Z","last_name":"Pietrzak","orcid":"0000-0002-9139-1654","first_name":"Krzysztof Z"},{"last_name":"Tessaro","first_name":"Stefano","full_name":"Tessaro,
Stefano"}],"date_updated":"2024-10-22T09:18:07Z","day":"28","oa":1,"citation":{"ista":"Alwen JF, Chen B, Kamath Hosdurg C, Kolmogorov V, Pietrzak KZ, Tessaro S. 2016. On the complexity of scrypt and
proofs of space in the parallel random oracle model. EUROCRYPT: Theory and Applications of Cryptographic Techniques, LNCS, vol. 9666, 358–387.","short":"J.F. Alwen, B. Chen, C. Kamath Hosdurg, V.
Kolmogorov, K.Z. Pietrzak, S. Tessaro, in:, Springer, 2016, pp. 358–387.","mla":"Alwen, Joel F., et al. On the Complexity of Scrypt and Proofs of Space in the Parallel Random Oracle Model. Vol. 9666,
Springer, 2016, pp. 358–87, doi:10.1007/978-3-662-49896-5_13.","ieee":"J. F. Alwen, B. Chen, C. Kamath Hosdurg, V. Kolmogorov, K. Z. Pietrzak, and S. Tessaro, “On the complexity of scrypt and proofs
of space in the parallel random oracle model,” presented at the EUROCRYPT: Theory and Applications of Cryptographic Techniques, Vienna, Austria, 2016, vol. 9666, pp. 358–387.","ama":"Alwen JF, Chen
B, Kamath Hosdurg C, Kolmogorov V, Pietrzak KZ, Tessaro S. On the complexity of scrypt and proofs of space in the parallel random oracle model. In: Vol 9666. Springer; 2016:358-387. doi:10.1007/
978-3-662-49896-5_13","chicago":"Alwen, Joel F, Binyi Chen, Chethan Kamath Hosdurg, Vladimir Kolmogorov, Krzysztof Z Pietrzak, and Stefano Tessaro. “On the Complexity of Scrypt and Proofs of Space in
the Parallel Random Oracle Model,” 9666:358–87. Springer, 2016. https://doi.org/10.1007/978-3-662-49896-5_13.","apa":"Alwen, J. F., Chen, B., Kamath Hosdurg, C., Kolmogorov, V., Pietrzak, K. Z., &
Tessaro, S. (2016). On the complexity of scrypt and proofs of space in the parallel random oracle model (Vol. 9666, pp. 358–387). Presented at the EUROCRYPT: Theory and Applications of Cryptographic
Techniques, Vienna, Austria: Springer. https://doi.org/10.1007/978-3-662-49896-5_13"},"acknowledgement":"Joël Alwen, Chethan Kamath, and Krzysztof Pietrzak’s research is partially supported by an ERC
starting grant (259668-PSPC). Vladimir Kolmogorov is partially supported by an ERC consolidator grant (616160-DOICV). Binyi Chen was partially supported by NSF grants CNS-1423566 and CNS-1514526, and
a gift from the Gareatis Foundation. Stefano Tessaro was partially supported by NSF grants CNS-1423566, CNS-1528178, a Hellman Fellowship, and the Glen and Susanne Culler Chair.\r\n\r\nThis work was
done in part while the authors were visiting the Simons Institute for the Theory of Computing, supported by the Simons Foundation and by the DIMACS/Simons Collaboration in Cryptography through NSF
grant CNS-1523467.","date_published":"2016-04-28T00:00:00Z","conference":{"location":"Vienna, Austria","name":"EUROCRYPT: Theory and Applications of Cryptographic
Techniques","start_date":"2016-05-08","end_date":"2016-05-12"},"main_file_link":[{"url":"https://eprint.iacr.org/2016/100","open_access":"1"}],"intvolume":" 9666","volume":9666,"abstract":
[{"text":"We study the time-and memory-complexities of the problem of computing labels of (multiple) randomly selected challenge-nodes in a directed acyclic graph. The w-bit label of a node is the
hash of the labels of its parents, and the hash function is modeled as a random oracle. Specific instances of this problem underlie both proofs of space [Dziembowski et al. CRYPTO’15] as well as
popular memory-hard functions like scrypt. As our main tool, we introduce the new notion of a probabilistic parallel entangled pebbling game, a new type of combinatorial pebbling game on a graph,
which is closely related to the labeling game on the same graph. As a first application of our framework, we prove that for scrypt, when the underlying hash function is invoked n times, the
cumulative memory complexity (CMC) (a notion recently introduced by Alwen and Serbinenko (STOC’15) to capture amortized memory-hardness for parallel adversaries) is at least Ω(w · (n/ log(n))2). This
bound holds for adversaries that can store many natural functions of the labels (e.g., linear combinations), but still not arbitrary functions thereof. We then introduce and study a combinatorial
quantity, and show how a sufficiently small upper bound on it (which we conjecture) extends our CMC bound for scrypt to hold against arbitrary adversaries. We also show that such an upper bound
solves the main open problem for proofs-of-space protocols: namely, establishing that the time complexity of computing the label of a random node in a graph on n nodes (given an initial kw-bit state)
reduces tightly to the time complexity for black pebbling on the same graph (given an initial k-node pebbling).","lang":"eng"}],"oa_version":"Submitted Version"} | {"url":"https://research-explorer.ista.ac.at/record/1231.jsonl","timestamp":"2024-11-03T22:16:07Z","content_type":"text/plain","content_length":"6953","record_id":"<urn:uuid:172413f9-ae0b-44cf-b66b-04cf976e5bb3>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00529.warc.gz"} |
300ml to cups
To convert 300 ml to g, simply multiply 300 ml by 1 to get grams. 0 0. is a unit of volume. tsp=5 mL. 236 ml in one US standard cup. 300 ml and 1-15 ml spoon : 1 1/2 cup : 350 ml: 1 2/3 cup : 375 ml
and 1-15 ml spoon: 1 3/4 cup : 400 ml and 1-15 ml spoon: 2 cups 475 ml: 2 1/4 cups ... your measuring cups and spoons are larger than the ones in the U.S. so be sure to use the conversion tables to
get the right equivalents for your measuring cups and spoons. Advanges: 1) Our field Mainly specializing in PET cup,plastic cup,disposable cup,promotional cup.Products from 0.75OZ to 32OZ cups. 3
years ago. Use the calculator above to convert between milliliters and cups. 2. How to convert from Milliliters to Cups. Read about company. Relevance. 350 ml. Bhagwati Packwell LLP - Offering
Transparent 300ml Plastic Glass at Rs 4/piece in Mumbai, Maharashtra. cup US = mL * 0.0042268 . In stock and ready to ship. That’s easier to understand than 1,760 yards to a mile and 128 ounces to a
gallon. Ounces : A fluid ounce (abbreviated fl oz, fl. Convert gallons, l, ml, oz, pints, quarts, tbsp, tsp. What is 300 milliliters converted to cups? 400 ml. uxcell Measuring Cup 300ml 300ml 1000ml
Set of 3, PP Plastic Graduated Beaker Transparent with Handle for Lab Kitchen Liquids 5.0 out of 5 stars 1 $11.79 $ 11 . Definition of Milliliter 700 Milliliters = 2.9587270 Cups (rounded to 8
digits) Display result as. 79 ($3.93/Item) Show result in exponential format. How many cups of fl oz in 300 ml? How many cups of fl oz in 300 ml? Choose from our selection of 300 ml cups in a wide
range of styles and sizes. How many cups of milk in ml? Milliliters : A milliliter (also written "milliliter"; SI symbol ml) is a non-SI metric system unit of volume which is commonly used as liquid
unit. The grams to cups calculator will help you in your daily life. 3 Answers. fl.) equals 14 fl oz. Milliliters to US Cups. oz. Answer Save. Make sure this fits by entering your model number. The
conversion factor from Milliliters to Cups is 0.0042267528198649. 3 years ago. equals 16 fl oz. Paper cup with your logo from 500 pcs. equals 1 pint; 700 ml. Then 300 ml of the sweetened condensed /
dehydrated milk equals how many grams and finally I converted 100 milliliters of sweetened condensate milk into fluid ounces (I had a hint -that was the result from 3.5 floz lucky I double-checked).
See below for the grams to cups conversion for 300 … To convert any value in cc to cups, just multiply the value in cc by the conversion factor 0.0042267528377304.So, 300 cc times 0.0042267528377304
is equal to 1.268 cups. Milliliters. Our bioplastic cups are made from plants not oil and are commercially compostable. Convert Table. Shop / Cups & Feeding / Cups / Dr. Brown’s™ Cheers360™ Spoutless
Transition Cup, 10oz/300ml (9m+), 2 Count Dr. Brown’s™ Cheers360™ Spoutless Transition Cup, 10oz/300ml (9m+), 2 Count Transition baby to a big kid cup with leak-free 360-degree sipping and a
Sip-and-See clear silicone valve. To convert 300 milliliters into cups we have to multiply 300 by the conversion factor in order to get the volume amount from milliliters to cups. equals 1 1/2 pints;
1 litre equals 1 3/4 pints; 1.5 litres equals 2 3/4 … Three hundred fifty Milliliters is equivalent to one point four seven nine Cups. Cold use only.SpecificationsBrim full capacity: 300ml (10oz)
Diameter top: 97mmProduct dimensions: 96 x 96 x 80mmPieces per carton: 1000Pieces per sleeve: 50Carton dimensions: 50 x 40 x 36.5cmCarton total weight: 9.5kgManufactured: Taiwan Lv 7. Still have
questions? 3 years ago. 0 0. Converter for quantity amounts of Cream, fluid, heavy whipping between units measured in g, gram, dag, dekagram (10g), portion 100 g, grams, kg, kilogram (1,000g), oz,
ounce (28.35g), lb, pound (16oz), cup, whipped, cup, fluid (yields 2 cups whipped), tbsp, fl oz culinary or nutritional content and how much per weight versus volume conversion of its measuring
values. You can learn more about the different cups sizes in my … tbsp=15mL. 300 ml. Convert 300 grams or g of flour to cups. 300 ml of water is 1.323 cups. Kano. How Many Cups is 300 Grams? A metric
unit of volume equal to one thousandth of a liter . 2) Our experiences Started in 1968 with more than 40 years experience,also is the supplier of Costa,Heineken, DQ ,starbuck and etc. 1 decade ago.
about 1 1/5 cups simply 250 mL in a cup so one cup, and 50 goes into 250 5 times so 1/5 i geuss you could use tbsp and tsps as well in that sense it would be 1 cup and 3 tbsp 1 tsp. or oz. 300 ml of
water is 1.323 cups. Gert. equals 12 fl oz. How many grams is 300ml? 0 0. Here we present the conversion for the most popular kitchen ingredients. Eric. 1300 ml to cups: 1300 ml to gallons: 1300 ml
to liters: 1300 ml to oz: 1300 ml to pints: 1300 ml to tbsp: 1300 ml to tsp: How much is 1300 milliliters in gallons? Online Calculators > Conversion 300 ml to Grams. Note To Converting 500 grams of
flour to cups. A milliliter is a unit of volume equal to 1/1000 th of a liter. 600 ml. So we wanted to show you the fluid ounce (fl oz) equivalent of 300 ml and the process of converting 300 ml to …
0 0. It is equal to 1/1000 liter, or one cubic centimeter, therefore, 1ml =1/1000 L =1 CM3. Note To Converting 300 grams of flour to cups. Language. If you ever wondered how many grams are in a cup
or how to convert 100 grams to cups, you are in the right place. 425 ml. equals 15 fl oz. 3000 ml to cups: 3000 ml to gallons: 3000 ml to liters: 3000 ml to oz: 3000 ml to pints: 3000 ml to tbsp:
3000 ml to tsp: How much is 3000 milliliters in gallons? - 1 ml equals 1 gram, therefore there are 300 grams in 300 ml. Alexa. 700 mL to cups conversion. From. Ask Question + 100. ; Decent Quantity &
Volume: Total 8 pcs 300ml plastic liquid measuring cups, enough to satisfy your various demands, save time and cost. UNICUP, a manufacturer of custom printed paper cups. Here are some quick reference
calculations for mL to cups (rounded to 2 decimal places). To find out how many Milliliters in Cups, multiply by the conversion factor or use the Volume converter above. Milliliters to US Cups
formula. Amount : 1.25 cup measure of condensed milk, Equals in : 382.5 g ( grams ) of sweetened condensed milk. What is 1300 milliliters in gallons, liters, cups, ounces, pints, quarts, tablespoons,
teaspoons, etc? equals 1 1/4 pints; 850 ml. Please use the appropriate variation from the list below. To convert any value in cups to milliliters, just multiply the value in cups by the conversion
factor 236.5882365.So, 2.3 cups times 236.5882365 is equal to 544.2 milliliters. About 1.2 cups in 300 ml. equals 18 fl oz. Convert gallons, l, ml, oz, pints, quarts, tbsp, tsp. 300 ml is 1.268 cups.
US Cups. cup US = mL * 0.0042268 . what is 300ml in cups how many cups is that? Milliliters to Canadian Cups. A thousand meters is a kilometer, and a thousand milliliters is a liter. Task: Convert 2
US cups to milliliters (show work) Formula: US cup x 236.5882365 = mL Calculations: 2 US cup x 236.5882365 = 473.176473 mL Result: 2 US cup is equal to 473.176473 mL Conversion Table For quick
reference purposes, below is a conversion table that you can use to convert from US cup to mL. 300ml, big volume, able to contain more liquid material, provide more choices for amount handling.
Milliliters to US Cups formula. What is 3000 milliliters in gallons, liters, cups, ounces, pints, quarts, tablespoons, teaspoons, etc? 0 0. Please note that converting 500 grams of flour to cups can
vary slightly by room temperature, quality of flour etc. To calculate the number of cups for your milliliter/millilitre figure, you need to divide your figure by 236.59 (for a US cup). swap units ↺
Amount. Lv 7. To. About 1.2 cups in 300 ml. 300 grams flour equals 2 3/8 cups. WHY CHOOSE US? A United States liquid unit equal to 8 fluid ounces. 500 grams flour equals 4 cups. Measuring your flour
by weight (500 grams instead of 4 cups) will provide much more accurate results in cooking. 300 grams equals 1.27 cups of water or there are 1.27 cups in 300 grams. Mugs & Cups Small mugs 240ml
Medium mugs 350ml XLarge mugs 500ml Mini mugs 150ml Classic mugs 270ml Mugs 300ml Large mugs 450ml Cups 300ml Tea Infuser Mug/Cup 0.25L Water mugs 170ml Mugs 400ml XLarge Mug 0.5L Giant Mugs 0.65L
Teapots Small teapots 0.4L Medium teapots 0.9L Large teapots 1.2L XLarge teapots 1.8L Teapot 0.6L Jugs Measuring your flour by weight (300 grams instead of 2 3/8 cups) will provide much more accurate
results in cooking. It is the same as a cubic centimeter. Source(s): https://shrink.im/a8emv. It's just over 1 and 1/4 cups. Show working. Imprinted Disposable Coffee to go paper cups with your own
design from 72 hours. How many cups is 300 ml milk? 450 ml. Get contact details and address | ID: 15973503388 Convert 500 grams or g of flour to cups. We can also form a simple proportion to
calculate the result: 1 ml → 0.0042267528198649 cup 300 ml → V (cup) equals 10 fl oz. 300 ml to grams converter to calculate how many grams is 300 ml. What is 700 milliliters in cups? 300 Ml To Cups.
500 ml. Milliliters to Cups (mL to cup) conversion calculator for Volume conversions with additional tables and formulas. Type in 300.5 for 300 and a half, 300.25 for 300 and a … Convert 300
Milliliters to Grams with our online conversion. Get your answers by asking now. Milliliters to Cups There is more than one type of Cups. Convert volume and capacity culinary measuring units between
milliliter (ml) and cups Australian (cup) but in the other direction from cups Australian into milliliters also as per volume and capacity units.. Culinary arts school: volume and capacity units
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Geef een reactie | {"url":"http://www.oosteinde.info/3t7lmf/300ml-to-cups-e4b015","timestamp":"2024-11-09T13:53:36Z","content_type":"text/html","content_length":"29542","record_id":"<urn:uuid:f5e11804-dcbe-4c8f-ad47-040146607476>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00289.warc.gz"} |
ACM Other ConferencesAchieving Envy-Freeness Through Items Sale
We consider a fair division setting of allocating indivisible items to a set of agents. In order to cope with the well-known impossibility results related to the non-existence of envy-free
allocations, we allow the option of selling some of the items so as to compensate envious agents with monetary rewards. In fact, this approach is not new in practice, as it is applied in some
countries in inheritance or divorce cases. A drawback of this approach is that it may create a value loss, since the market value derived by selling an item can be less than the value perceived by
the agents. Therefore, given the market values of all items, a natural goal is to identify which items to sell so as to arrive at an envy-free allocation, while at the same time maximizing the
overall social welfare. Our work is focused on the algorithmic study of this problem, and we provide both positive and negative results on its approximability. When the agents have a commonly
accepted value for each item, our results show a sharp separation between the cases of two or more agents. In particular, we establish a PTAS for two agents, and we complement this with a hardness
result, that for three or more agents, the best approximation guarantee is provided by essentially selling all items. This hardness barrier, however, is relieved when the number of distinct item
values is constant, as we provide an efficient algorithm for any number of agents. We also explore the generalization to heterogeneous valuations, where the hardness result continues to hold, and
where we provide positive results for certain special cases. | {"url":"https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2024.26/metadata/acm-xml","timestamp":"2024-11-13T11:13:19Z","content_type":"application/xml","content_length":"14692","record_id":"<urn:uuid:df456fc9-0a9e-4508-b96d-cc4e2940afdc>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00398.warc.gz"} |
Numbers Battle: Teacher's support sheet | Best School Games
Escola Games | Jogos Educativos
Teacher's support sheet
Numbers Battle
Do you know the numbers well? Do you know how to use the greater and less than symbols?
Choose your hero and participate in a fun battle.
Only your math skills can lead you to victory!
Go to activity
Teacher's tips
Level of education: Elementary School
Subject: Math
Age: 07 to 10 years
Promote a learning battle with this game where children will put into practice knowledge of MAJOR and MINOR, in addition to problem solving skills involving the four operations.
According to the age and level of the children, the game presents a challenge, which will be graded and differentiated by the degree of difficulty. [FIM-DICA]
Learner outcomes
Work with the four fundamental operations;
Develop estimation processes, mental calculation and multiplication tables;
Solve mathematical problems involving the four operations;
Exercise logical reasoning, finding the result of operations;
Fix the learning of the concepts “major” and “minor”;
Establish measurement relationships, estimating the possible results of operations;
Fix knowledge acquired in the classroom;
Teachers' goals
Detect students' difficulties in learning the 4 (four) fundamental operations of mathematics: addition, subtraction, multiplication and division;
Reinforce content worked in the classroom;
Stimulate the taste for Mathematics, through playful and pleasurable activities;
Expand the class calculation procedures from the four operations;
Stimulate the exchange of knowledge, socialization and interaction among students;
Suggestions of approaches for the teacher
(Suggestion 1) Divide the class into pairs for this game, where partners should exchange ideas to solve each proposed problem. It is important to highlight what each student knows and what they need
to learn, so that the exchange of knowledge between them advances in learning.
(Suggestion 2) For younger children, choose the easiest level where they will only work with the concepts of greater and lesser from numbers, without having to find the results of operations.
(Suggestion 3) Record the game as follows: ask students to write down all the numbers that appear in the moves. Use them in proposals in the classroom or at home, suggestions:
• Arrange the numbers in ascending order.
• Sort the numbers in descending order.
• Write the predecessor and successor of the numbers.
• Propose sums that allow for the analysis of regularities in numerical sequences: +2, +5, +10, +100.
• Separate even and odd numbers.
(Suggestion 4) For older children, choose the most difficult level where the same proposal is made, but the student must solve the operation to find which is the largest and smallest number.
When working in pairs, it is important for the children to exchange ideas to find the results of the operations.
As a record, propose that the children write down the operations and the results obtained. In the classroom, the pairs should exchange sheets to check each result found by their colleagues.
It is also interesting to divide the class into two groups and promote a battle, launching each of the operations noted by the children.
Also use this record to send homework!
(Suggestion 5) Present problems to the class, discussing with them the possible solutions and recording your ideas on the board. Soon after, if you find it necessary, present the solutions, comparing
with what they thought. Ask them to explain their strategies, solving them collectively on the board or in their notebooks.
(Suggestion 6) MARKET: Simulate with the students a market with fake money. Divide students into two groups, salespeople and customers.
(Suggestion7) ROLLING DICE: Produce two dice or bring them to class ready-made. Roll the dice and with the numbers that come out write an operation on the board. You can, for example, say that you
are going to start with addition operations, so the numbers that come out will be added together. Then you can move on to the following operations. The answers can be written in the notebook and
checked later, as is done in a traditional dictation.
(Suggestion 8) BINGO. Place chips with operations are placed inside a bag. Then pull out an operation and talk to the players. Players resolve the operation by obtaining the result that will be on
some of the cards. Whoever has the result marks it with a marker.
(Suggestion 9) Talk to students about everyday situations involving calculations with addition, subtraction, multiplication and division. Orient the class to a research work that aims to find
numerical data in newspapers and magazines that allow the class to work with mental calculation, with mathematical operations, data that allow the discussion of mathematical ideas. After the survey,
promote a moment of socialization and presentation of the works.
More about the content
Since the early years of elementary school, concepts and problem situations involving the four fundamental operations are worked on in Mathematics: addition, subtraction, multiplication and division.
These subjects are the basis for solving any mathematical problem. However, many times, these operations are not assimilated and understood satisfactorily by the students, and many of them are
promoted to subsequent grades without acquiring mastery of the prerequisites, in order to face the next stages of teaching and learning.
In the current educational context, every moment new challenges arise. As technology advances, education needs to follow this process. The use of didactic games as a strategy in teaching mathematics
has been gaining ground, winning over students and solving some of these difficulties and gaps in the teaching-learning process.
Tips from other BEST SCHOOL GAMES to work on this subject:
Math Master
Dino Multiplication Table
Successor and Predecessor
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Stimulated Brillouin Scattering: Lower Peak Power, Stronger Effect?
Posted on 2007-09-01 as part of the Photonics Spotlight (available as e-mail newsletter!)
Permanent link: https://www.rp-photonics.com/spotlight_2007_09_01.html
Author: Dr. Rüdiger Paschotta, RP Photonics AG, RP Photonics AG
Abstract: It is common wisdom that lower peak powers cause weaker nonlinear effects. However, the article discusses a case in the context of stimulated Brillouin scattering where the opposite is
true. By investigating this in some detail, one can strengthen the understanding not only of Brillouin Scattering, but also of Fourier spectra.
Ref.: encyclopedia articles on Brillouin Scattering, frequency combs, Fourier spectrum
Higher peak powers mean stronger nonlinear effects. Well, not always … and it is very instructive to investigate a case where it is not so. In that way, one can strengthen the understanding not only
of Brillouin Scattering, but also of Fourier spectra. Furthermore, it becomes apparent that the idea of some threshold power, above which stimulated Brillouin scattering sets in, needs to be used
with care.
Consider a regular train of ultrashort pulses with a pulse repetition rate of e.g. 1 GHz, as generated in a mode-locked laser. The Fourier spectrum of the pulse train is a frequency comb, consisting
of narrow lines with a spacing identical to the repetition rate. We assume these pulses to propagate through a single-mode fiber, in which stimulated Brillouin scattering (SBS) occurs. That effect is
often a great nuisance for people trying to send light through a fiber: above some threshold power, the fiber returns essentially all the power to the sender. (We'll actually need refine that
statement, see below.)
In order to mitigate the problem, one may consider to double the pulse duration while keeping the average power and repetition rate constant. This reduces the peak power by a factor of 2, and
according to common wisdom one might expect a reduction of nonlinear effects. However, the optical bandwidth is also reduced to one half, so that the power in each line of the spectrum is doubled.
For that reason, SBS becomes stronger rather than weaker! What we can easily understand in the frequency domain, is puzzling when seen in the time domain.
We can resolve that. As a first step, we have to become aware that Brillouin scattering is a phenomenon where the effects of multiple pulses accumulate. That effect is the excitation of a sound wave
in the fiber. The lifetime of the sound wave is of the order of several nanoseconds (the inverse Brillouin gain bandwidth), i.e., longer than the pulse spacing. Therefore, each pulse gives a further
“kick” to the sound wave generated by the preceding pulses. For certain frequencies, spaced by the pulse repetition rate, these kicks lead to a resonant excitation. So we have a kind of oscillator
(the sound wave), which gets some sharp kicks at regular time intervals, while the energy of the sound wave shows some partial decay between these kicks.
In a second step, we consider how strongly a single pulse of the pulse train contributes to the sound wave. The contribution to the amplitude of the sound wave is proportional to the electric field
amplitude and to the pulse duration. (This is really as you would expect it for some simple mechanical oscillator.) If we double the pulse duration, the field amplitude is reduced only by a factor of
the square root of two because that corresponds to half the peak power. So the product of field amplitude and pulse duration is increased, even though the pulse energy stays constant! In fact, the
energy added to the sound wave by one pulse is doubled.
A similar effect is obtained if we double the pulse repetition rate. While the contribution of each pulse to the sound wave amplitude is reduced by a factor of the square root of two, we have twice
as many pulses per time interval, and overall a larger effect. In the frequency domain, we see an increased power per line due to a larger line spacing.
We should take this opportunity to revise our idea of a threshold power for SBS. It is not that there is a fixed threshold power which we can apply to short pulses as we do it for continuous-wave
optical radiation. The threshold is rather associated with a certain power spectral density, which (for regular pulse trains) depends not only on the peak power, but also on pulse duration and
repetition rate.
Finally, consider what happens if we reduce the pulse repetition rate further and further. This leads us into a regime where the sound wave totally decays between the pulses. There is then no
resonant excitation anymore. In the frequency domain, we have a pulse train with a small line spacing, so that multiple lines are within the Brillouin gain bandwidth. So it becomes obvious in both
time and frequency domain that we have entered a different regime.
This article is a posting of the Photonics Spotlight, authored by Dr. Rüdiger Paschotta. You may link to this page and cite it, because its location is permanent. See also the RP Photonics
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Visual Complex Analysis
could almost have been titled
Geometry with Complex Numbers
, so much does it emphasize the visual and geometric. Needham makes extensive use of diagrams, which he gives as much prominence as the equations, and focuses on "those aspects of complex analysis
for which some visual representation or interpretation is possible". Many topics are explored at length but without rigorous proofs: of Riemann's Mapping Theorem, for example, he writes "[though one
of the proofs] is constructive in nature we have not yet found a way to present it in a manner consistent with the aims of this book".
Needham begins with the links between complex numbers and geometry, and a view of complex functions as transformations, focusing on Moebius transformations and inversions and ways of visualising
them. Differentiation is approached using the idea of an "amplitwist", a transformation of infinitesimal vectors, and "visual differentiation" investigated for a range of different functions.
The longest chapter is devoted entirely to non-Euclidean geometry, though this is "starred" as not necessary for the rest of the book (as are sections of other chapters).
Winding numbers lend themselves to a topological perspective. Needham follows that with a visual approach to complex integration, in which he uses Cauchy's Theorem before proving it, and Cauchy's
Formula, relating the integral of an analytic function around a simple loop with its values at interior points. A series of chapters then look at vector fields, flows, and harmonic functions,
concluding with a lovely geometric perspective on Dirichlet's Problem, concerning heat flows in a metal disc. (Physical models and intuitions are used throughout, but not given nearly the same
prominence as geometric ones.)
Needham's approach is, as mentioned, informal and not rigorous, and curiosity-driven: the quite extensive exercises at the end of each chapter are designed to grab the reader and make them want to
solve them. Visual Complex Analysis could be used by itself, as an introduction to the subject for those of us — a majority, I believe — whose visual intuitions are stronger than our formal and
symbolic ones, and who are perhaps driven more by curiosity than the need to solve engineering problems. It could also be used alongside more traditional complex analysis textbooks, perhaps to help
when intuition or motivation fail.
July 2020
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Article overview
Symmetries of the Einstein Equations
C. G. Torre ; I. M. Anderson ;
Date: 23 Feb 1993
Journal: Phys.Rev.Lett. 70 (1993) 3525-3529
Subject: gr-qc
Abstract: Generalized symmetries of the Einstein equations are infinitesimal transformations of the spacetime metric that formally map solutions of the Einstein equations to other solutions. The
infinitesimal generators of these symmetries are assumed to be local, ie at a given spacetime point they are functions of the metric and an arbitrary but finite number of derivatives of
the metric at the point. We classify all generalized symmetries of the vacuum Einstein equations in four spacetime dimensions and find that the only generalized symmetry transformations
consist of: (i) constant scalings of the metric (ii) the infinitesimal action of generalized spacetime diffeomorphisms. Our results rule out a large class of possible ``observables’’ for
the gravitational field, and suggest that the vacuum Einstein equations are not integrable.
Source: arXiv, gr-qc/9302033
Other [GID 574806] pmid10053896
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flashback, flashbacked, flashbacking, flashbacks, flashlike, fleshhook, fleshlike, fleshquake, flunkeyhood, flunkeyish, flunkyhood, flunkyish, flywhisk, flywhisks.
foehnlike, folkish, folkishness, folkishnesses, folklorish, folkright, foothook, forecheck, forechecked, forechecker, forecheckers, forechecking, forechecks, forehock, forehook, foreshank,
foreshanks, foreshock, foreshocks, forethink, forethinker, forethinkers, forethinking, forethinks, forkhead, forkheads, forksmith, forthink, forthinking, forthinks, foxshark, foxsharks.
frankhearted, frankheartedly, frankheartedness, frankheartness, frankish, freakish, freakishly, freakishness, freakishnesses, frecklish, freethink, freethinker, freethinkers, freethinking,
freethinkings, frithsoken, frithsokens, frithwork.
fuckhead, fuckheads, funkhole, funkholes, futhark, futharks, futhork, futhorks.
garookuh, gascheck, gatchwork, gawkhammer, gawkihood, gawkihoods, gawkish, gawkishly, gawkishness, gawkishnesses.
gemutlichkeit, gemutlichkeits.
gherkin, gherkins, ghorkhar, ghostlike, ghostlikeness.
gomukhi, goshawk, goshawks.
grabhook, grasshook, grasshooks, greekish, greenshank, greenshanks, grieshuckle, groupthink, groupthinks.
guidebookish, gunkhole, gunkholed, gunkholes, gunkholing, gurkha.
gymkhana, gymkhanas.
haak, habakkuk, habuka, hacek, haceks, hack, hackable, hackamatak, hackamore, hackamores, hackbarrow, hackberries, hackberry, hackbolt, hackbolts, hackbush, hackbut, hackbuteer, hackbuteers,
hackbuts, hackbutter, hackbutters, hackdriver, hacked, hackee, hackeem, hackees, hacker, hackeries, hackers, hackery, hackette, hackettes, hackeymal, hackia, hackie, hackies, hackin, hacking,
hackingly, hackings, hackle, hackleback, hackled, hackler, hacklers, hackles, hacklet, hacklets, hacklier, hackliest, hackling, hacklog, hackly, hackmack, hackmall, hackman, hackmatack, hackmatacks,
hackmen, hackney, hackneyed, hackneyedly, hackneyedness, hackneyer, hackneying, hackneyism, hackneyisms, hackneyman, hackneymen, hackneys, hacks, hacksaw, hacksawed, hacksawing, hacksawn, hacksaws,
hacksilber, hackster, hackthorn, hacktree, hackwood, hackwork, hackworks, hacky, haddock, haddocker, haddocks, haffkinize, haglike, haick, haicks, haiduck, haiduk, haiduks, haik, haika, haikai,
haikal, haiks, haiku, haikun, haikus, haikwan, haimsucken, hairlike, hairlock, hairlocks, hairstreak, hairstreaks, hairwork, hairworks, hak, haka, hakafoth, hakam, hakamim, hakams, hakas, hakdar,
hake, hakea, hakeem, hakeems, hakenkreuz, hakes, hakim, hakims, hako, haku, hakus, halaka, halakah, halakahs, halakha, halakhah, halakhahs, halakhas, halakhic, halakhist, halakhists, halakhot,
halakhoth, halakic, halakist, halakistic, halakists, halakoth, halfback, halfbacks, halfbeak, halfbeaks, halfcock, halfcocked, halftrack, halftracks, haliplankton, halkahs, halke, hallanshaker,
hallmark, hallmarked, hallmarker, hallmarking, hallmarks, hallock, halolike, halterbreak, halterbreaking, halterbreaks, halterbroke, halterbroken, halterlike, halterneck, halternecks, halucket,
halukkah, hamesoken, hamesucken, hamesuckens, hammerkop, hammerkops, hammerlike, hammerlock, hammerlocks, hammerwork, hammock, hammocklike, hammocks, hamshackle, hamshackled, hamshackles,
hamshackling, hancockite, handbank, handbanker, handbasket, handbaskets, handbook, handbooks, handbrake, handbrakes, handistroke, handiwork, handiworks, handkercher, handkerchers, handkerchief,
handkerchiefful, handkerchiefs, handkerchieves, handknit, handlike, handlock, handpick, handpicked, handpicking, handpicks, handshake, handshaker, handshakes, handshaking, handshakings, handspike,
handspikes, handspoke, handstroke, handwork, handworked, handworker, handworkers, handworkman, handworks, handybook, handywork, handyworks, hangkang, hank, hanked, hanker, hankered, hankerer,
hankerers, hankering, hankeringly, hankerings, hankers, hankie, hankies, hanking, hankle, hanks, hanksite, hankt, hankul, hanky, hanukkah, hapuku, harakeke, harakiri, hardback, hardbacked, hardbacks,
hardbake, hardbakes, hardhack, hardhacks, hardock, hardoke, hardokes, hardrock, hardtack, hardtacks, hardworking, harelike, haremlik, harikari, hark, harka, harked, harkee, harken, harkened,
harkener, harkeners, harkening, harkens, harking, harks, harlock, harnesslike, harplike, harpoonlike, harpylike, hartake, hashmark, hashmarks, hask, haskard, haskness, hasks, haskwort, hasky,
haslock, hassock, hassocks, hassocky, hatchback, hatchbacks, hatcheck, hatchecks, hatchetback, hatchetlike, hatlike, hatmaker, hatmakers, hatmaking, hatrack, hatracks, hattock, hattocks, hauberk,
hauberks, haulback, havelock, havelocks, havercake, haversack, haversacks, havocked, havocker, havockers, havocking, hawbuck, hawbucks, hawebake, hawk, hawkbell, hawkbells, hawkbill, hawkbills,
hawkbit, hawkbits, hawked, hawker, hawkers, hawkery, hawkey, hawkeye, hawkeyed, hawkeys, hawkie, hawkies, hawking, hawkings, hawkins, hawkish, hawkishly, hawkishness, hawkishnesses, hawkit, hawklike,
hawkmoth, hawkmoths, hawknose, hawknosed, hawknoses, hawknut, hawks, hawksbeak, hawksbeard, hawksbeards, hawksbill, hawksbills, hawkshaw, hawkshaws, hawkweed, hawkweeds, hawkwise, hawky, hawok,
haycock, haycocks, hayfork, hayforks, haymaker, haymakers, haymaking, haymakings, haymarket, hayrack, hayracks, hayrake, hayraker, hayrakes, hayrick, hayricks, hayshock, haystack, haystacks, haysuck.
hdbk, hdkf.
headkerchief, headlike, headliked, headlock, headlocks, headmark, headmarks, headshake, headshaker, headshakes, headshaking, headshakings, headshrinker, headshrinkers, headskin, headstick,
headsticks, headstock, headstocks, headwark, headwork, headworker, headworkers, headworking, headworks, hearken, hearkened, hearkener, hearkeners, hearkening, hearkens, hearselike, heartblock,
heartbreak, heartbreaker, heartbreakers, heartbreaking, heartbreakingly, heartbreaks, heartbroke, heartbroken, heartbrokenly, heartbrokenness, heartbrokennesses, heartikin, heartikins, heartlike,
heartquake, heartshake, heartsick, heartsickening, heartsickness, heartsicknesses, heathcock, heathcocks, heathlike, heatlike, heatmaker, heatmaking, heatstroke, heatstrokes, heavenlike, heavyback,
hecctkaerre, heck, heckelphone, heckelphones, heckimal, heckle, heckled, heckler, hecklers, heckles, heckling, hecklings, hecks, heckuva, heddlemaker, hedgebreaker, hedgemaker, hedgemaking,
heelmaker, heelmaking, heelwork, heelworks, heirskip, heitiki, hekhsher, hekhsherim, hekhshers, hektare, hektares, hekteus, hektogram, hektograph, hektoliter, hektometer, hektostere, helideck,
helidecks, hellkite, hellkites, helmetlike, helmetmaker, helmetmaking, helsingkite, helsinki, helterskelteriness, hemiekton, hemikaryon, hemikaryotic, hemiplankton, hemlock, hemlocks,
hemoalkalimeter, hemokonia, hemokoniosis, hemplike, henhawk, henlike, henpeck, henpecked, henpeckeries, henpeckery, henpecking, henpecks, herakles, herblike, herdbook, herdbooks, herdlike, herdwick,
herdwicks, herkogamies, herkogamy, hermitlike, hermokopid, herolike, herrenvolk, herrenvolks, herringlike, heterakid, heterokarya, heterokaryon, heterokaryons, heterokaryoses, heterokaryosis,
heterokaryotic, heterokinesia, heterokinesis, heterokinetic, heterokont, heterokontan, heterokonts, heuk, heureka, heurekas, hexakisoctahedron, hexakistetrahedron, hexokinase, hexokinases, heyduck,
heyducks, hezekiah.
hibakusha, hick, hicket, hickey, hickeyes, hickeys, hickie, hickies, hickified, hickish, hickishness, hickories, hickory, hicks, hickscorner, hicksite, hickwall, hickwalls, hickway, hicky, hieromonk,
highjack, highjacked, highjacker, highjackers, highjacking, highjacks, highpockets, hijack, hijacked, hijacker, hijackers, hijacking, hijackings, hijacks, hijiki, hijikis, hijinks, hike, hiked,
hiker, hikers, hikes, hiking, hikings, hikuli, hillfolk, hillock, hillocked, hillocks, hillocky, hillwalker, hillwalkers, hillwalking, hillwalkings, hinddeck, hingelike, hinoki, hipflask, hiplike,
hitchhike, hitchhiked, hitchhiker, hitchhikers, hitchhikes, hitchhiking, hivelike, hiyakkin, hiziki, hizikis.
hoblike, hock, hockamore, hocked, hockelty, hocker, hockers, hocket, hockets, hockey, hockeys, hocking, hockle, hockled, hockling, hockmoney, hocks, hockshin, hockshop, hockshops, hocktide, hocky,
hoddypeak, hodgkinsonite, hoecake, hoecakes, hoelike, hogback, hogbacks, hogchoker, hoglike, hogskin, hogsucker, hoick, hoicked, hoicking, hoicks, hoicksed, hoickses, hoicksing, hoik, hoiked,
hoiking, hoiks, hoke, hoked, hoker, hokerer, hokerly, hokes, hokey, hokeyness, hokeynesses, hokeypokey, hokeypokeys, hoki, hokier, hokiest, hokily, hokiness, hokinesses, hoking, hokis, hokku, hokum,
hokums, hokypokies, hokypoky, holdback, holdbacks, holidaymaker, holidaymakers, holidaymaking, holishkes, holk, holked, holking, holks, holleke, hollock, holluschick, holluschickie, hollyhock,
hollyhocks, holoku, holoplankton, holoplanktonic, holoplanktons, holyokeite, homefolk, homefolks, homekeeper, homekeeping, homelike, homelikeness, homemake, homemaker, homemakers, homemaking,
homemakings, homeokinesis, homeokinetic, homeseeker, homesick, homesickly, homesickness, homesicknesses, homework, homeworker, homeworkers, homeworking, homeworkings, homeworks, hommack, hommock,
hommocks, homoeokinesis, honeylike, honeymoonstruck, honeystucker, honeysuck, honeysucker, honeysuckers, honeysuckle, honeysuckled, honeysuckles, hongkong, honk, honked, honker, honkers, honkey,
honkeys, honkie, honkies, honking, honks, honky, honkytonks, hoodlike, hoodwink, hoodwinkable, hoodwinked, hoodwinker, hoodwinkers, hoodwinking, hoodwinks, hooflike, hoofmark, hoofmarks, hook, hooka,
hookah, hookahs, hookaroon, hookas, hookcheck, hooked, hookedness, hookednesses, hookedwise, hooker, hookerman, hookers, hookey, hookeys, hookheal, hookier, hookies, hookiest, hooking, hookish,
hookland, hookless, hooklet, hooklets, hooklike, hookmaker, hookmaking, hookman, hooknose, hooknosed, hooknoses, hooks, hookshop, hooksmith, hookswinging, hooktip, hooktips, hookum, hookup, hookups,
hookupu, hookweed, hookwise, hookworm, hookwormer, hookworms, hookwormy, hooky, hoolakin, hoolock, hoolocks, hooplike, hoopmaker, hoopskirt, hoopskirts, hoopstick, hopak, hopsack, hopsacking,
hopsackings, hopsacks, hordock, hordocks, horkey, horkeys, hormonelike, hornbeak, hornbeaks, hornbook, hornbooks, hornkeck, hornlike, hornwork, hornworks, hornwrack, hornwracks, horokaka, horseback,
horsebacker, horsebacks, horsebreaker, horsejockey, horsekeeper, horsekeeping, horselike, horselock, hosecock, hoselike, hotcake, hotcakes, hotchkiss, hotelkeeper, hotelkeepers, hotkey, hotlink,
hotlinked, hotlinking, hotlinks, houndlike, houndshark, hounskull, housebreak, housebreaker, housebreakers, housebreaking, housebreakings, housebreaks, housebroke, housebroken, housebrokenness,
housekeep, housekeeper, housekeeperlike, housekeeperly, housekeepers, housekeeping, housekeepings, housekeeps, housekept, housekkept, houseleek, houseleeks, housewifeskep, housewifeskeps, housework,
houseworker, houseworkers, houseworks, housewrecker, howk, howked, howker, howkers, howking, howkit, howks.
hubmaker, hubmaking, huck, huckaback, huckabacks, huckle, huckleback, hucklebacked, huckleberries, huckleberry, huckleberrying, huckleberryings, hucklebone, huckles, huckmuck, hucks, huckster,
hucksterage, hucksterages, huckstered, hucksterer, hucksteress, hucksteresses, hucksteries, huckstering, hucksterism, hucksterisms, hucksterize, hucksters, huckstery, huckstress, huckstresses,
huddock, huffaker, huffkin, huffkins, huke, hulk, hulkage, hulked, hulkier, hulkiest, hulkily, hulkiness, hulking, hulkingly, hulkingness, hulks, hulky, hullock, humankind, humankinds, humanlike,
hummock, hummocked, hummocking, hummocks, hummocky, humoresk, humoresks, humpback, humpbacked, humpbacks, humuhumunukunukuapuaa, humuslike, hunchback, hunchbacked, hunchbacks, hundredwork, hunk,
hunker, hunkered, hunkering, hunkerous, hunkerousness, hunkers, hunkey, hunkeys, hunkie, hunkier, hunkies, hunkiest, hunks, hunkses, hunky, hunterlike, huntiegowk, huntiegowks, hureek, hurkaru,
hurkle, hurleyhacket, hurlock, hurrock, husbandlike, husk, huskanaw, husked, huskened, husker, huskers, huskershredder, huskie, huskier, huskies, huskiest, huskily, huskiness, huskinesses, husking,
huskings, husklike, huskroot, husks, huskwort, husky, hutkeeper, hutlike, hutukhtu, hutuktu, huvelyk.
hyalotekite, hydraulicked, hydraulicking, hydrocrack, hydrocracked, hydrocracker, hydrocrackers, hydrocracking, hydrocrackings, hydrocracks, hydrofranklinite, hydrokineter, hydrokinetic,
hydrokinetical, hydrokinetics, hydroski, hydroskis, hydroxyketone, hygrodeik, hygrodeiks, hyke, hykes, hymnbook, hymnbooks, hymnlike, hyperalkalinity, hyperanakinesia, hyperanakinesis,
hyperbrachyskelic, hyperkalemia, hyperkalemic, hyperkaliemia, hyperkatabolism, hyperkeratoses, hyperkeratosis, hyperkeratotic, hyperkineses, hyperkinesia, hyperkinesias, hyperkinesis, hyperkinetic,
hyperleukocytosis, hyperlink, hyperlinked, hyperlinking, hyperlinks, hypermakroskelic, hypermarket, hypermarkets, hypernik, hypoalkaline, hypoalkalinity, hypokalemia, hypokalemias, hypokalemic,
hypokaliemia, hypokeimenometry, hypokinemia, hypokinesia, hypokinesis, hypokinetic, hypokoristikon, hypoplankton, hypoplanktonic, hyposkeletal, hystericky.
icekhana, icekhanas.
inkbush, inkfish, inkholder, inkholders, inkhorn, inkhornism, inkhornist, inkhornize, inkhornizer, inkhorns, inkish, inkshed, inkstandish, intercheck, interchoke, interchoked, interchoking,
intershock, inukshuk, inukshuks.
ishshakku, isokeraunographic, isokeraunophonic.
jackanapish, jackash, jackfish, jackfishes, jackhammer, jackhammered, jackhammering, jackhammers, jackhead, jacklight, jacklighted, jacklighter, jacklighting, jacklights, jackpuddinghood, jackshaft,
jackshafts, jackshay, jackshea, jacksmith, jacksmiths, jayhawk, jayhawker, jayhawkers.
jeewhillikens, jerkinhead, jerkinheads, jerkish.
jinricksha, jinrickshas, jinrickshaw, jinrickshaws, jinrikisha, jinrikishas, jinriksha, jinrikshas.
jockeyish, jockeyship, jockeyships, johnnycake, johnnycakes, jokesmith, jokesmiths, jokish.
kabbalah, kabbalahs, kabuzuchi, kaccha, kacchas, kacha, kachahri, kachahris, kachcha, kacheri, kacheris, kachin, kachina, kachinas, kaddish, kaddishes, kaddishim, kadischi, kadish, kadishim,
kaffeeklatch, kaffeeklatches, kaffeeklatsch, kaffeeklatsches, kaffiyah, kaffiyahs, kaffiyeh, kaffiyehs, kaha, kahal, kahala, kahals, kahar, kahau, kahawai, kahawais, kahikatea, kahili, kahu, kahuna,
kahunas, kaisership, kaiserships, kaiwhiria, kajawah, kajawahs, kakawahie, kakorraphiaphobia, kalach, kalanchoe, kalanchoes, kalashnikov, kalashnikovs, kalathoi, kalathos, kaleidophon, kaleidophone,
kaleidophones, kaliophilite, kaliph, kaliphs, kallah, kalokagathia, kamachi, kamachile, kamahi, kamanichile, kamavachara, kamboh, kameelthorn, kamichi, kamichis, kamptomorph, kampuchea, kanchil,
kaneelhart, kaneh, kanehs, kanephore, kanephoros, kangha, kanghas, kantha, kantharoi, kantharos, kanthas, kaph, kaphs, kapparah, karch, karinghota, karmadharaya, karmathian, karmouth, karsha,
karyenchyma, karyochrome, karyochylema, karyolymph, karyolymphs, karyorrhexis, karyoschisis, kasbah, kasbahs, kasha, kashas, kasher, kashered, kashering, kashers, kashga, kashi, kashim, kashima,
kashira, kashmir, kashmiri, kashmirs, kashrus, kashruses, kashrut, kashruth, kashruths, kashruts, kashubian, kassabah, katabothra, katabothron, katabothrons, katachromasis, katagelophobia,
katamorphic, katamorphism, kataphoresis, kataphoretic, kataphoric, kataphrenia, katathermometer, katathermometers, katavothron, katavothrons, katchina, katchinas, katchung, kath, katha, kathak,
kathakali, kathakalis, kathaks, kathal, katharevousa, katharevousas, katharevusa, katharine, katharometer, katharometers, katharses, katharsis, kathartic, kathemoglobin, kathenotheism, katherine,
kathisma, kathismata, kathodal, kathode, kathodes, kathodic, katholikoi, katholikos, katholikoses, kathy, kauch, kaugh, kaughs, kazachki, kazachok, kazachoks.
keach, keblah, keblahs, kechel, kechumaran, keddah, keddahs, kedushah, keech, keeches, keelhale, keelhaled, keelhales, keelhaling, keelhaul, keelhauled, keelhauling, keelhaulings, keelhauls,
keepership, keeperships, keepworthy, keeshond, keeshonden, keeshonds, keffiyah, keffiyahs, keffiyeh, keffiyehs, kehaya, kehillah, kehilloth, kehoeite, keight, keilhauite, kelchin, kelchyn, keleh,
kelpfish, kelpfishes, kelyphite, kemancha, kenareh, kench, kenches, kenophobia, kenophobias, kentish, kephalic, kephalics, kephalin, kephalins, kephir, kephirs, keraphyllocele, keraphyllous,
keratinophilic, keratohelcosis, keratohyal, keratophyr, keratophyre, keratophyres, keratorrhexis, keraulophon, keraulophone, keraunograph, keraunographic, keraunographs, keraunography, keraunophobia,
keraunophone, keraunophonic, kerch, kercher, kerchief, kerchiefed, kerchiefing, kerchiefs, kerchieft, kerchieves, kerchoo, kerchug, kerchunk, kernish, kerslosh, kersmash, kerwham, kesh, keshes,
ketch, ketchcraft, ketches, ketching, ketchup, ketchups, ketchy, kethib, kethibh, ketoheptose, ketohexose, kettlestitch, kettlestitches, ketubah, ketubahs, ketuboth, kevelhead, kevutzah, kevutzoth,
keyhole, keyholes, keypunch, keypunched, keypuncher, keypunchers, keypunches, keypunching, keysmith.
kha, khaddar, khaddars, khadi, khadis, khaf, khafajeh, khafs, khagiarite, khahoon, khaiki, khair, khaja, khajur, khakanship, khakham, khaki, khakied, khakilike, khakis, khalal, khalat, khalats,
khalif, khalifa, khalifah, khalifahs, khalifas, khalifat, khalifate, khalifates, khalifats, khalifs, khalkha, khalsa, khalsah, khamal, khamseen, khamseens, khamsin, khamsins, khan, khanate, khanates,
khanda, khandait, khanga, khangas, khanjar, khanjars, khanjee, khankah, khans, khansama, khansamah, khansamahs, khansaman, khansamas, khanum, khanums, khaph, khaphs, khar, kharaj, kharif, kharifs,
kharouba, kharroubah, khartoum, kharua, kharwa, khass, khat, khatib, khatin, khatri, khats, khaya, khayal, khayas, khazen, khazenim, khazens, kheda, khedah, khedahs, khedas, khediva, khedival,
khedivas, khedivate, khedivates, khedive, khedives, khediviah, khedivial, khediviate, khediviates, khella, khellin, khepesh, khesari, khet, kheth, kheths, khets, khi, khidmatgar, khidmutgar,
khidmutgars, khilafat, khilafats, khilat, khilats, khilim, khilims, khir, khirka, khirkah, khirkahs, khis, khitmatgar, khitmutgar, khitmutgars, khmer, khodja, khodjas, khoja, khojah, khojas, khoka,
khor, khors, khot, khotbah, khotbahs, khotbeh, khotbehs, khoum, khoums, khowar, khrushchev, khu, khubber, khud, khuds, khula, khulda, khurta, khurtas, khuskhus, khuskhuses, khutba, khutbah, khutbahs,
khutuktu, khvat.
kiaugh, kiaughs, kibbeh, kibbehs, kiblah, kiblahs, kibosh, kiboshed, kiboshes, kiboshing, kichel, kickish, kickshaw, kickshaws, kickshawses, kickwheel, kiddish, kiddishness, kiddush, kiddushes,
kiddushim, kiddushin, kidhood, kieselguhr, kieselguhrs, kiesselguhr, kight, kights, kilah, kileh, kilhig, killifish, killifishes, killoch, kilnhole, kilohertz, kilohertzes, kilohm, kimchee, kimchees,
kimchi, kimchis, kinaestheic, kinaestheses, kinaesthesia, kinaesthesias, kinaesthesis, kinaesthetic, kinaesthetically, kinah, kinch, kinchin, kinchinmort, kinchins, kindheart, kindhearted,
kindheartedly, kindheartedness, kindheartednesses, kindredship, kindredships, kinematograph, kinematographer, kinematographic, kinematographical, kinematographically, kinematographs, kinematography,
kinesipath, kinesipathic, kinesipathies, kinesipathist, kinesipathists, kinesipaths, kinesipathy, kinesitherapies, kinesitherapy, kinestheses, kinesthesia, kinesthesias, kinesthesis, kinesthetic,
kinesthetically, kinetheodolite, kinetheodolites, kinetochore, kinetochores, kinetograph, kinetographer, kinetographic, kinetographs, kinetography, kinetophobia, kinetophone, kinetophonograph,
kingdomship, kingfish, kingfisher, kingfishers, kingfishes, kinghead, kinghood, kinghoods, kinghorn, kinghunter, kinglihood, kinglihoods, kingship, kingships, kinhin, kinkcough, kinkhab, kinkhaust,
kinkhost, kinksbush, kinship, kinships, kinsmanship, kirbeh, kirbehs, kirghiz, kirkinhead, kirsch, kirsches, kirschwasser, kirschwassers, kischen, kish, kishen, kishes, kishka, kishkas, kishke,
kishkes, kishon, kishy, kiswah, kitchen, kitchendom, kitchendoms, kitchened, kitchener, kitcheners, kitchenet, kitchenets, kitchenette, kitchenettes, kitchenful, kitchening, kitchenless, kitchenmaid,
kitchenman, kitchenry, kitchens, kitchenward, kitchenwards, kitchenware, kitchenwares, kitchenwife, kitcheny, kitchie, kitching, kith, kithara, kitharas, kithe, kithed, kithes, kithing, kithless,
kithlessness, kithogue, kiths, kitish, kitsch, kitsches, kitschier, kitschiest, kitschily, kitschy, kittenhearted, kittenhood, kittenish, kittenishly, kittenishness, kittenishnesses, kittenship,
kitthoge, kittlish, kiwach.
kjeldahlization, kjeldahlize.
klaprotholite, klatch, klatches, klatsch, klatsches, klepht, klephtic, klephtism, klephtisms, klephts, kleptophobia, klesha, klipdachs, klipfish, kliphaas, klooch, klooches, kloochman, kloochmans,
kloochmen, klootch, klootches, klootchman, klootchmans, klootchmen, klosh.
knackish, knaidlach, knaidloch, knappish, knappishly, knatch, knaveship, knaveships, knavish, knavishly, knavishness, knavishnesses, knaydlach, kneebrush, kneehole, kneeholes, kneidlach, knetch,
knickknackish, knifesmith, knight, knightage, knightages, knighted, knightess, knighthead, knightheads, knighthood, knighthoods, knighting, knightless, knightlier, knightliest, knightlihood,
knightlike, knightliness, knightlinesses, knightling, knightly, knights, knightship, knightswort, kniphofia, kniphofias, knish, knishes, knitch, knitches, knorhaan, knorhmn, knothead, knothole,
knotholes, knothorn, knoweth, knowhow, knowhows, knownothingism, knucklehead, knuckleheaded, knuckleheadedness, knuckleheads, knuth.
kochia, kochliarion, koechlinite, kohekohe, kohemp, kohen, kohens, kohl, kohlrabi, kohlrabies, kohlrabis, kohls, kohua, koilanaglyphic, koilonychia, kokobeh, koksaghyz, kolach, kolache, kolhoz,
kolhozes, kolhozy, kolkhos, kolkhoses, kolkhosy, kolkhoz, kolkhozes, kolkhoznik, kolkhozniki, kolkhozniks, kolkhozy, komarch, koniophobia, konohiki, koolah, koolahs, kooletah, koorhmn, kootcha,
kootchar, koph, kophs, kopophobia, korhmn, kosha, koshare, kosher, koshered, koshering, koshers, kotschubeite, kouproh, kourbash, kourbashed, kourbashes, kourbashing, kowhai, kowhais, koyemshi.
krauthead, kreplach, kreplech, krishna, krishnaism, kritarchy, krohnkite, krouchka, kroushka, krumhorn, krumhorns, krummholz, krummhorn, krummhorns, kryolith, kryoliths.
kuchcha, kuchean, kuchen, kuchens, kueh, kuffieh, kufiyah, kufiyahs, kufiyeh, kugelhof, kuichua, kulah, kumhar, kumrah, kumshaw, kuphar, kurbash, kurbashed, kurbashes, kurbashing, kurchatovium,
kurchatoviums, kurchicine, kurchine, kurdish, kursch, kusha, kutch, kutcha, kutches.
kvah, kvetch, kvetched, kvetcher, kvetchers, kvetches, kvetchier, kvetchiest, kvetching, kvetchy, kvutzah.
kwacha, kwachas, kwashiorkor, kwashiorkors, kwhr.
kyah, kyathoi, kyathos, kybosh, kyboshed, kyboshes, kyboshing, kymograph, kymographic, kymographies, kymographs, kymography, kyphoscoliosis, kyphoscoliotic, kyphoses, kyphosis, kyphotic, kyschty,
kyschtymite, kythe, kythed, kythes, kything.
lackeyship, lakemanship, lakeshore, lakeshores, lakh, lakhs, lakish, lakishness, landshark, landsknecht, landsknechts, lankish, lansknecht, lanzknecht, lanzknechts, larkish, larkishly, larkishness,
lashkar, lashkars, latchkey, latchkeys, lathlike, lathwork, lathworks, laughingstock, laughingstocks.
leatherback, leatherbacks, leatherbark, leatherjacket, leatherlike, leatherlikeness, leathermaker, leathermaking, leatherneck, leathernecks, leatherwork, leatherworker, leatherworkers,
leatherworking, leatherworkings, leatherworks, leathwake, lebkuchen, leechkin, leechlike, leekish, lekach, lekha, lekythi, lekythoi, lekythos, lekythus, leukocythemia, leukodystrophies,
leukodystrophy, leukorrhea, leukorrheal, leukorrheas, leukorrhoea, leukorrhoeal.
lichenlike, lichwake, lichwakes, lickerish, lickerishly, lickerishness, lickerishnesses, lieberkuhn, lighthousekeeper, lighthousekeepers, lightkeeper, lightninglike, likehood, likelihead, likelihood,
likelihoods, likerish, linksmith.
loanshark, loansharking, loansharkings, lockhole, lockhouse, lockhouses, locksmith, locksmithery, locksmithing, locksmithings, locksmiths, lockstitch, lockstitched, lockstitches, lockstitching,
lohock, lokshen, longshanks, longshucks, lookahead.
luckenbooth, luckenbooths, lukewarmish, lukewarmth, lukewarmths, lunchhook, lunkhead, lunkheaded, lunkheads, luskish, luskishness, luskishnesses.
lymphokine, lymphokines.
machinelike, machtpolitik, mackintosh, mackintoshed, mackintoshes, mackintoshite, mahlstick, mahlsticks, makahiki, makership, makeshift, makeshiftiness, makeshiftness, makeshifts, makeshifty,
makeweight, makeweights, makhorka, makhzan, makhzen, maksoorah, makunouchi, makunouchis, malahack, mannoketoheptose, marchlike, markhoor, markhoors, markhor, markhors, markshot, marksmanship,
marksmanships, markworthy, marshbanker, marshbuck, marshlike, marshlocks, marshlockses, mashak, matchbook, matchbooks, matchlock, matchlocks, matchmake, matchmaker, matchmakers, matchmaking,
matchmakings, matchmark, matchstalk, matchstick, matchsticks, matkah, matryoshka, matryoshkas, mawkish, mawkishly, mawkishness, mawkishnesses.
meathook, meathooks, meekhearted, meekheartedness, mekhitarist, melchizedek, melkhout, menshevik, merchantlike, merkhet, meshwork, meshworks, methink, methinketh, methinks.
microearthquake, microearthquakes, microphakia, mikvah, mikvahs, mikveh, mikvehs, mikvoth, milkbush, milkfish, milkfishes, milkhouse, milkshake, milkshakes, milkshed, milksheds, milkshop,
milksoppish, milksoppishness, minkfish, minkfishes, minkish, minshuku, minshukus, mirkish, misthink, misthinking, misthinks.
mockish, mohawk, mohawkite, mohawks, mohock, mohockism, mokihana, mokihi, moksha, mokshas, mollyhawk, monarchlike, monkeyhood, monkeyish, monkeyishly, monkeyishness, monkeyshine, monkeyshines,
monkfish, monkfishes, monkhood, monkhoods, monkish, monkishly, monkishness, monkishnesses, monkship, monkshood, monkshoods, moochulka, mookhtar, mopehawk, mopehawks, motherfucker, motherfuckers,
motherfucking, motherkin, motherkins, motherlike, mothlike, mountebankish, mousehawk, mouthlike, mowhawk.
muckerish, muckheap, muckheaps, muckhill, muckhole, muckthrift, mudhook, mudhooks, mukhtar, mukhtars, munchkin, munchkins, murkish, muschelkalk, mushroomlike, muskish, mutchkin, mutchkins, muzhik,
mythmaker, mythmakers, mythmaking, mythmakings.
nakedish, nakhlite, nakhod, nakhoda, narrishkeit.
neckcloth, neckcloths, neckercher, neckerchief, neckerchiefs, neckerchieves, neighborlike, neighborlikeness, neighbourlike, netherstock, netherstocks, neugkroschen, newshawk, newshawks.
nighthawk, nighthawks, nightlike, nightstick, nightsticks, nightstock, nightwake, nightwalk, nightwalker, nightwalkers, nightwalking, nightwork, nightworker, nikethamide, nikethamides, nishiki.
nonbookish, nonbookishly, nonbookishness, nonbulkhead, nonchalky, nonchokable, nonchokebore, nonhackneyed, nonhousekeeping, nonkosher, nonshrink, nonshrinkable, nonshrinking, nonshrinkingly,
nonthinker, nonthinking, nonthinkings, notchback, notchbacks.
nunchaku, nunchakus, nuthook.
nyckelharpa, nymphlike.
oakenshaw, oakenshaws.
okeh, okehs, oklahoma, oklahoman, oklahomans, okolehao, okshoofd, okthabah.
omphaloskepses, omphaloskepsis.
orchidlike, orthopinakoid, orthopinakoids.
outkitchen, outlandishlike, outshake, outshriek, outskirmish, outskirmisher, outthank, outthanked, outthanking, outthanks, outthink, outthinking, outthinks, outthwack.
overbookish, overbookishly, overbookishness, overcheck, overchecks, overchoke, overhusk, overshake, overshrink, overthick, overthickly, overthickness, overthink, overthinking, overthinks.
pachak, pachaks, pachinko, pachinkos, packcloth, packhorse, packhorses, packhouse, packinghouse, packinghouses, packmanship, packsheet, packsheets, packthread, packthreaded, packthreads, paddywhack,
paddywhacked, paddywhacking, paddywhacks, pakchoi, pakeha, pakehas, pantherlike, parchmentlike, parkish, pashalik, pashaliks, pashka, patchcock, patchcocke, patchcockes, patchocke, patchockes,
patchwork, patchworks, patchworky, pathbreaker, pathbreaking, paycheck, paychecks.
***See More*** words-with word lists on our other pages.
Words with h and k:
peachick, peachlike, peacockish, peacockishly, peacockishness, peakish, peakishly, peakishness, peakyish, peckhamite, peckish, peckishly, peckishness, peckishnesses, peiktha, penthouselike,
peppershaker, peppershakers, peppershrike, perkish, perukiership, peshkar, peshkash.
phantomlike, pharmacokinetic, pharmacokineticist, pharmacokineticists, pharmacokinetics, pharmakoi, pharmakos, pharyngokeratosis, phenakism, phenakisms, phenakistoscope, phenakistoscopes, phenakite,
phenakites, phenylketonuria, phenylketonurias, phenylketonuric, phenylketonurics, phiallike, philokleptic, phinnock, phinnocks, phoenixlike, phokomelia, phonikon, phonodeik, phooka,
phosphofructokinase, phosphofructokinases, phosphokinase, phosphokinases, photokineses, photokinesis, photokinetic, photomask, photomasks, phrasebook, phrasebooks, phrasemake, phrasemaker,
phrasemakers, phrasemaking, phrasemakings, phreak, phreaked, phreaking, phreaks, phrentick, phthisicky, phulkari, physicked, physicker, physicking, physicks, physicky, phytokinin, phytoplankter,
phytoplankters, phytoplankton, phytoplanktonic, phytoplanktons.
pickelhaube, pickelhaubes, pickshaft, picksmith, pickthank, pickthankly, pickthankness, pickthanks, pickthatch, picktooth, picnickish, pikeperch, pikeperches, pinakothek, pinakotheke, pinakotheks,
pinchback, pinchbeck, pinchbecks, pinchcock, pinchcocks, pincheck, pinchecks, pinhook, pinhooker, pinhookers, pinkfish, pinkfishes, pinkish, pinkishness, pinkishnesses, piroshki, pirozhki, pirozhok,
pitcherlike, pitchfork, pitchforked, pitchforking, pitchforks, pitchlike, pitchpike, pitchwork, pithlike, pithwork.
planksheer, plinthlike, plushlike.
pocketphone, pocketphones, pockhouse, pohickory, pohutukawa, pohutukawas, poikilocythemia, poikilotherm, poikilothermal, poikilothermic, poikilothermies, poikilothermism, poikilothermisms,
poikilotherms, poikilothermy, pokerish, pokerishly, pokerishness, pomeshchik, pookhaun, poppycockish, porchlike, porkchop, porkfish, porkfishes, porkish, porokaiwhiria, porthook, postworkshop,
pothook, pothookery, pothooks, potwhisky, pouchlike.
pracharak, prankish, prankishly, prankishness, prankishnesses, precheck, prechecked, prechecking, prechecks, preearthquake, preshrank, preshrink, preshrinkage, preshrinking, preshrinks, preshrunk,
preshrunken, prickish, pricklefish, prickshot, prophetlike.
psychokineses, psychokinesia, psychokinesis, psychokinetic, psychokyme, psychoquackeries.
puckerbush, puckermouth, puckish, puckishly, puckishness, puckishnesses, pukish, pukishness, pulkha, pulkhas, pumpkinish, punchlike, punkah, punkahs, punkish, puschkinia, puschkinias, pushback,
pushbacks, pushbike, pushbikes, putchock, putchocks, putchuk, putchuks.
quackhood, quackish, quackishly, quackishness, quarterdeckish, quickhatch, quickhearted, quicksilverish, quicksilverishness, quickthorn, quickthorns, quirkish.
radishlike, radknight, raincheck, rainchecks, rajpramukh, rajpramukhs, rakehell, rakehellish, rakehells, rakehelly, rakeshame, rakeshames, rakh, rakish, rakishly, rakishness, rakishnesses, rakshas,
rakshasa, rakshasas, rakshases, ramshackle, ramshackled, ramshackleness, ramshackly, ranchlike, rankish, ranshackle, ranshackled, ranshackles, ranshackling, ranshakle, ranshakled, ranshakles,
ranshakling, rashlike, ratchetlike, rathskeller, rathskellers, ravehook.
reaphook, reaphooks, rebekah, recheck, rechecked, rechecking, rechecks, rechuck, redshank, redshanks, rehook, reichsmark, reichsmarks, rekhti, reshake, reshaken, reshaking, reshook, resketch,
resketched, resketches, resketching, rethank, rethicken, rethink, rethinker, rethinkers, rethinking, rethinks.
rhatikon, rhebok, rheboks, rheebok, rheoplankton, rheumaticky, rhinoceroslike, rhymemaker, rhymemaking.
ricketish, ricksha, rickshas, rickshaw, rickshaws, rikisha, rikishas, rikishi, riksha, rikshas, rikshaw, rikshaws, rinkhals, rinkhalses, riskish.
roachback, rockbrush, rockerthon, rockfish, rockfishes, rockhair, rockhearted, rockhopper, rockhoppers, rockhound, rockhounding, rockhoundings, rockhounds, rockish, rockshaft, rockshafts, rodknight,
rookish, rothermuck, roughneck, roughnecked, roughnecking, roughnecks, roughwork, routhercock.
rukh, rukhs, rushlike, rushwork.
sabakha, sabkha, sabkhah, sabkhahs, sabkhas, sabkhat, sabkhats, sachamaker, sackcloth, sackclothed, sackcloths, sadhaka, sadhika, sahoukar, sahukar, sahukars, sakieh, sakiehs, sakiyeh, sakiyehs,
saltchuck, saltchucker, saltchucks, saltshaker, saltshakers, sambhogakaya, samekh, samekhs, samkhya, sanjakship, sankha, sankhya, saskatchewan, satyashodak, sawshark, sawsharks.
schaapsteker, schapska, schapskas, schecklaton, schecklatons, scheltopusik, schick, schipperke, schipperkes, schlock, schlocker, schlockers, schlockier, schlockiest, schlockmeister, schlockmeisters,
schlocks, schlocky, schmeck, schmecks, schmock, schmocks, schmuck, schmucks, schnecke, schnecken, schnook, schnooks, schnorkel, schnorkeled, schnorkeling, schnorkels, schnorkle, schokker,
scholarlike, schoolbook, schoolbookish, schoolbooks, schoolkeeper, schoolkeeping, schoolkid, schoolkids, schoollike, schoolmasterlike, schoolwork, schoolworks, schrank, schrecklich, schrik, schriks,
schtick, schticks, schtik, schtiks, schtook, schtooks, schtuck, schtucks, schweizerkase, schwenkfeldian, scratchback, scratchbacks, scratchlike, scratchwork, scrimshank, scrimshanked, scrimshanker,
scrimshanking, scrimshanks, scutcheonlike, scythelike, scythework.
seahawk, seahawks, searcherlike, sebkha, semikah, semishirker, seqfchk, seraphlike.
shabrack, shabracks, shack, shackanite, shackatory, shackbolt, shacked, shacker, shacking, shackings, shackland, shackle, shacklebone, shacklebones, shackled, shackledom, shackler, shacklers,
shackles, shacklewise, shackling, shackly, shacko, shackoes, shackos, shacks, shacky, shaddock, shaddocks, shadkan, shadkhan, shadkhanim, shadkhans, shadowlike, shaftlike, shagbark, shagbarks,
shaglike, shaikh, shaikha, shaikhas, shaikhi, shaikhs, shakable, shakably, shake, shakeable, shakebly, shaked, shakedown, shakedowns, shakefork, shaken, shakenly, shakeout, shakeouts, shakeproof,
shaker, shakerag, shakers, shakes, shakescene, shakespeare, shakespearean, shakespeareana, shakespeareans, shakespearian, shakeup, shakeups, shakha, shakier, shakiest, shakily, shakiness,
shakinesses, shaking, shakingly, shakings, shako, shakoes, shakos, shaksheer, shaksperean, shaksperian, shakt, shakta, shakti, shaktis, shaktism, shaku, shakudo, shakudos, shakuhachi, shakuhachis,
shaky, shalelike, shamesick, shammick, shammock, shammocking, shammocky, shamrock, shamrocks, shank, shankbone, shankbones, shanked, shanker, shanking, shankings, shankpiece, shankpieces, shanks,
shanksman, shantylike, shapka, sharebroker, sharemilker, sharemilkers, shark, sharked, sharker, sharkers, sharkful, sharki, sharking, sharkings, sharkish, sharkishly, sharkishness, sharklet,
sharklike, sharks, sharkship, sharkskin, sharkskins, sharksucker, sharksuckers, sharky, shashlick, shashlicks, shashlik, shashliks, shaslick, shaslik, shasliks, shastraik, shastrik, shattuckite,
shawllike, shaykh, sheaflike, sheathlike, shecklaton, shecklatons, shedlike, sheepback, sheepbacks, sheepcrook, sheephook, sheepkeeper, sheepkeeping, sheepkill, sheeplike, sheepshank, sheepshanks,
sheepskin, sheepskins, sheeptrack, sheeptracks, sheepwalk, sheepwalker, sheepwalks, sheetlike, sheetrock, sheetrocks, sheetwork, sheik, sheika, sheikas, sheikdom, sheikdoms, sheikh, sheikha,
sheikhas, sheikhdom, sheikhdoms, sheikhlike, sheikhly, sheikhs, sheiklike, sheikly, sheiks, shekel, shekels, shekinah, sheldduck, sheldducks, sheldrake, sheldrakes, shelduck, shelducks, shelfback,
shelflike, shelftalker, shelftalkers, shellack, shellacked, shellacker, shellackers, shellacking, shellackings, shellacks, shellak, shellback, shellbacks, shellbark, shellbarks, shellcracker,
shellcrackers, shelldrake, shelldrakes, shellduck, shellducks, shellshake, shellshock, shellshocked, shellshocks, shellwork, shellworker, shellworks, shepherdlike, sheppeck, sheppick, sheriffwick,
sherlock, sherlocks, shick, shicker, shickered, shickers, shicksa, shicksas, shielddrake, shieldlike, shieldmaker, shieldrake, shieldrakes, shielduck, shielducks, shiitake, shiitakes, shikar,
shikara, shikaree, shikarees, shikargah, shikari, shikaris, shikarred, shikarring, shikars, shikasta, shikii, shikimi, shikimic, shikimol, shikimole, shikimotoxin, shikken, shikker, shikkers, shiko,
shikra, shiksa, shiksas, shikse, shikseh, shiksehs, shikses, shilluk, shinkin, shipbreaking, shipbroken, shipbroker, shipbrokers, shipkeeper, shipwork, shipwreck, shipwrecked, shipwrecking,
shipwrecks, shipwrecky, shirakashi, shirewick, shirk, shirked, shirker, shirkers, shirking, shirks, shirky, shirlcock, shirtlike, shirtmake, shirtmaker, shirtmakers, shirtmaking, shitake, shitakes,
shitepoke, shitkicker, shitkickers, shivzoku, shizoku, shkotzim, shlock, shlocks, shmek, shmeks, shmock, shmocks, shmuck, shmucks, shnook, shnooks, shock, shockabilities, shockability, shockable,
shocked, shockedness, shocker, shockers, shockhead, shockheaded, shockheadedness, shocking, shockingly, shockingness, shockingnesses, shocklike, shockproof, shocks, shockstall, shockstalls,
shockwave, shoddylike, shoemake, shoemaker, shoemakers, shoemaking, shoemakings, shoepack, shoepacks, shonkier, shonkiest, shonkinite, shonky, shook, shooks, shopbook, shopbreaker, shopbreakers,
shopbreaking, shopbreakings, shopfolk, shopkeep, shopkeeper, shopkeeperess, shopkeeperish, shopkeeperism, shopkeepers, shopkeepery, shopkeeping, shopkeepings, shoplike, shopmark, shoptalk, shoptalks,
shopwalker, shopwalkers, shopwork, shopworker, shortcake, shortcakes, shotlike, shotmaker, shotmakers, shotmaking, shotmakings, shovelmaker, showerlike, showfolk, shpilkes, shrank, shredcock,
shredlike, shreek, shreeked, shreeking, shreeks, shreik, shreiked, shreiking, shreiks, shrewlike, shrewstruck, shriek, shrieked, shrieker, shriekers, shriekery, shriekier, shriekiest, shriekily,
shriekiness, shrieking, shriekingly, shriekings, shriekproof, shrieks, shrieky, shrike, shriked, shrikes, shriking, shrimplike, shrinelike, shrink, shrinkable, shrinkage, shrinkageproof, shrinkages,
shrinker, shrinkerg, shrinkers, shrinkhead, shrinking, shrinkingly, shrinkingness, shrinkpack, shrinkpacks, shrinkproof, shrinks, shrinkwrap, shrinkwrapped, shrinkwrapping, shrinkwraps, shrinky,
shroudlike, shrrinkng, shrublike, shrunk, shrunken, shtick, shticks, shtik, shtiks, shtinker, shtinkers, shtook, shtooks, shtuck, shtucks, shubunkin, shubunkins, shuck, shucked, shucker, shuckers,
shucking, shuckings, shuckins, shuckpen, shucks, shunpike, shunpiked, shunpiker, shunpikers, shunpikes, shunpiking, shunpikings, shuttlecock, shuttlecocked, shuttlecocking, shuttlecocks, shuttlelike,
shydepoke, shylock, shylocked, shylocking, shylocks.
sickhearted, sickish, sickishly, sickishness, sickishnesses, sidecheck, sideshake, sighlike, sikatch, sikh, sikhara, sikhism, sikhra, sikhs, sinkhead, sinkhole, sinkholes, siphonlike.
skaith, skaithed, skaithing, skaithless, skaiths, skaithy, skaldship, skaldships, skandhas, skarth, skarths, skedgewith, skeech, skeechan, skeechans, skeich, skeigh, skeigher, skeighest, skeighish,
skelloch, skelloched, skelloching, skellochs, skeough, skeptophylaxia, skeptophylaxis, sketch, sketchabilities, sketchability, sketchable, sketchbook, sketchbooks, sketched, sketchee, sketcher,
sketchers, sketches, sketchier, sketchiest, sketchily, sketchiness, sketchinesses, sketching, sketchingly, sketchist, sketchlike, sketchpad, sketchpads, sketchy, skeuomorph, skeuomorphic,
skeuomorphism, skeuomorphisms, skeuomorphs, skevish, skewwhiff, skhian, skiagraph, skiagraphed, skiagrapher, skiagraphic, skiagraphical, skiagraphically, skiagraphies, skiagraphing, skiagraphs,
skiagraphy, skiamachies, skiamachy, skiech, skiegh, skilfish, skilletfish, skilletfishes, skinch, skinhead, skinheads, skintight, skiograph, skiophyte, skippership, skirmish, skirmished, skirmisher,
skirmishers, skirmishes, skirmishing, skirmishingly, skirmishings, skirreh, skirwhit, skither, skitishly, skittish, skittishly, skittishness, skittishnesses, skoosh, skooshed, skooshes, skooshing,
skosh, skoshes, skouth, skraigh, skreegh, skreeghed, skreeghing, skreeghs, skreigh, skreighed, skreighing, skreighs, skriech, skrieched, skrieching, skriechs, skriegh, skrieghed, skrieghing,
skrieghs, skrimshander, skrimshank, skrimshanked, skrimshanker, skrimshankers, skrimshanking, skrimshanks, skullfish, skunkbush, skunkhead, skunkish, skupshtina, skycoach, skyhook, skyhooks, skyhoot,
skyish, skylight, skylighted, skylights, skyphoi, skyphos, skyshine.
smirkish, smithwork, smokebush, smokebushes, smokechaser, smokefarthings, smokeho, smokehole, smokehood, smokehoods, smokehouse, smokehouses, smokeshaft, smoketight, smokish, smoothback.
snakefish, snakefishes, snakehead, snakeheads, snakeholing, snakemouth, snakemouths, snakephobia, snakeship, snakish, snakishness, snakishnesses, sneakish, sneakishly, sneakishness, sneakishnesses.
sockhead, sokah, sokahs, solonchak, solonchaks, sorehawk, sovkhos, sovkhose, sovkhoz, sovkhozes, sovkhozy.
spaghettilike, sparhawk, sparkish, sparkishly, sparkishness, sparrowhawk, sparrowhawks, spatchcock, spatchcocked, spatchcocking, spatchcocks, speakerphone, speakerphones, speakership, speakerships,
speakhouse, specklehead, speechmaker, speechmakers, speechmaking, speechmakings, spellcheck, spellchecker, spellcheckers, spellchecks, sphairistike, sphairistikes, spherelike, sphinxlike, spikefish,
spikefishes, spikehole, spikehorn, spinachlike, spindleshank, spindleshanks, spitchcock, spitchcocked, spitchcocking, spitchcocks, splanchnoskeletal, splanchnoskeleton, splashback, spokeshave,
spokeshaves, spokesmanship, spokesmanships, spokeswomanship, spookish, spoonhook, spoonhooks, sprackish.
stackgarth, stackhousiaceous, stakehead, stakeholder, stakeholders, stakhanovism, stakhanovisms, stakhanovite, stakhanovites, starchlike, starchmaker, starchmaking, starchworks, starshake,
steaakhouse, steakhouse, steakhouses, stealthlike, stethokyrtograph, stickhandle, stickhandled, stickhandler, stickhandlers, stickhandles, stickhandling, sticksmanship, sticktight, sticktights,
stinkbush, stinkhorn, stinkhorns, stitchlike, stitchwork, stitchworks, stockfather, stockfish, stockfishes, stockholder, stockholders, stockholding, stockholdings, stockholm, stockhorn, stockhorns,
stockhorse, stockhorses, stockhouse, stockish, stockishly, stockishness, stockwright, stokehold, stokeholds, stokehole, stokeholes, storkish, straightjacket, straightjacketed, straightjacketing,
straightjackets, stretchmarks, stretchneck, streuselkuchen.
subbrachyskelic, subclerkship, subhooked, suchlike, suckauhock, suckerfish, suckerfishes, suckfish, suckfishes, suckhole, sucklebush, suikerbosch, sukh, sukhs, sukkah, sukkahs, sukkoth, sulphurlike,
sunchoke, sunchokes, superlikelihood, superthankful, superthankfully, superthankfulness, superthick.
svarabhakti, svarabhaktic, svarabhaktis.
swashbuckle, swashbuckled, swashbuckler, swashbucklerdom, swashbucklering, swashbucklers, swashbucklery, swashbuckles, swashbuckling, swashbucklings, swashwork, swashworks, swatchbook, swatchbooks,
switchback, switchbacked, switchbacker, switchbacking, switchbacks, switchkeeper, switchlike.
sylphlike, synkatathesis.
tacamahack, tacamahacks, tacmahack, tacmahacks, tahgook, tahkhana, taikhana, taikih, takahe, takahes, taketh, takhi, takhis, takkanah, talkathon, talkathons, talkworthy, tanekaha, tankah, tankship,
tankships, tarakihi, tarakihis, tarkashi, tarkhan, tashlik, taskmastership, taurokathapsia.
tchaikovsky, tcharik, tcheckup, tcheirek, tcheka, tchick, tchicked, tchicking, tchicks, tchotchke, tchotchkes, tchoukball, tchoukballs.
teacherlike, teethlike, tekiah, telakucha, telekinematography, tenterhook, tenterhooks, terakihi, terakihis, tetrakaidecahedron, tetrakishexahedra, tetrakishexahedron, tetrakishexahedrons,
textbookish, tezkirah.
thack, thacked, thacker, thacking, thackless, thackoor, thacks, thak, thakur, thakurate, thamakau, thank, thanked, thankee, thanker, thankers, thankful, thankfuller, thankfullest, thankfully,
thankfulness, thankfulnesses, thanking, thankings, thankless, thanklessly, thanklessness, thanklessnesses, thanks, thanksgiver, thanksgivers, thanksgiving, thanksgivings, thankworthily,
thankworthiness, thankworthinesses, thankworthy, thankyou, thankyous, tharfcake, thatchwork, theaterlike, theek, theeked, theeker, theeking, theeks, thegnlike, thelyotokous, thelyotoky, thelytokies,
thelytokous, thelytoky, theokrasia, theoktonic, theoktony, theotokoi, theotokos, thereckly, thermokinematics, thermotank, thewlike, thick, thickbrained, thicke, thicked, thicken, thickened,
thickener, thickeners, thickening, thickenings, thickens, thicker, thickest, thicket, thicketed, thicketful, thickets, thickety, thickhead, thickheaded, thickheadedly, thickheadedness,
thickheadednesses, thickheads, thicking, thickish, thickleaf, thickleaves, thicklips, thickly, thickneck, thickness, thicknesses, thicknessing, thicko, thickoes, thickos, thicks, thickset, thicksets,
thickskin, thickskins, thickskull, thickskulled, thickwind, thickwit, thicky, thiefmaker, thiefmaking, thieftaker, thilk, thimblelike, thimblemaker, thimblemaking, thinglike, thinglikeness, think,
thinkability, thinkable, thinkableness, thinkablenesses, thinkably, thinker, thinkers, thinkful, thinking, thinkingly, thinkingness, thinkingnesses, thinkingpart, thinkings, thinkling, thinks,
thioketone, thiokol, thislike, thistlelike, thitka, thoke, thokish, thornback, thornbacks, thornlike, thoughtkin, thoughtsick, thrack, thrawartlike, thrawcrook, threadlike, threadmaker, threadmakers,
threadmaking, thricecock, thriftlike, throatlike, throck, thrombokinase, thrombokinases, thronelike, throstlelike, throughknow, throwback, throwbacks, thrushlike, thumbikin, thumbikins, thumbkin,
thumbkins, thumblike, thumbmark, thumbsucker, thumbsuckers, thumbtack, thumbtacked, thumbtacking, thumbtacks, thundercrack, thunderlike, thunderstick, thunderstricken, thunderstrike, thunderstrikes,
thunderstriking, thunderstroke, thunderstrokes, thunderstruck, thunk, thunked, thunking, thunks, thurrock, thwack, thwacked, thwacker, thwackers, thwacking, thwackingly, thwackings, thwacks,
thwackstave, thylakoid, thylakoids.
ticklish, ticklishly, ticklishness, ticklishnesses, tightknit, tikolosh, tikoloshe, timekeepership, timwhisky, tinkershere, tinkershire, tinkershue, tirthankara, tithebook.
tokenworth, tokharian, tokoloshe, tokoloshes, tomahawk, tomahawked, tomahawker, tomahawking, tomahawks, toothlike, toothpick, toothpicks, toothstick, toothwork, tophaike, torchlike, toshakhana,
totchka, touchback, touchbacks, touchmark, touchmarks.
trackhound, trackmanship, trackshifter, trashrack, treckschuyt, trekpath, trekschuit, trekschuits, trencherlike, trenchermaker, trenchermaking, trenchlike, trenchwork, triakisicosahedral,
triakisicosahedron, triakisoctahedra, triakisoctahedral, triakisoctahedrid, triakisoctahedron, triakisoctahedrons, triakistetrahedral, triakistetrahedron, trickish, trickishly, trickishness,
trickishnesses, triskaidecaphobia, triskaidecaphobias, triskaidekaphobe, triskaidekaphobes, triskaidekaphobia, triskaidekaphobias, triskaidekaphobic, trochisk, trochisks, trothlike, troughlike,
trunkfish, trunkfishes, truthlike, truthlikeness.
tscharik, tscheffkinite.
tuckahoe, tuckahoes, tuckshop, tuckshops, tughrik, tughriks, tupakihi, turkeybush, turkeyfish, turkeyfishes, turkish, tushkar, tushkars, tushker, tushkers, tuskish, tutankhamen.
tykhana, tykish, typhoidlike.
unbethink, unbluestockingish, unbookish, unbookishly, unbookishness, unbrotherlike, unbutcherlike, unchalked, unchalky, uncheck, uncheckable, unchecked, uncheckered, unchecking, uncheckmated,
unchecks, unchinked, unchokable, unchoke, unchoked, unchokes, unchoking, unchurchlike, underclerkship, undersheriffwick, undertakerish, underthink, unfatherlike, unfishlike, unfreakish, unfreakishly,
unfreakishness, unghostlike, unhacked, unhackled, unhackneyed, unhackneyedness, unhanked, unharked, unhawked, unherolike, unhocked, unhomelike, unhomelikeness, unhoodwink, unhoodwinked, unhook,
unhooked, unhooking, unhooks, unhoundlike, unhouselike, unhusk, unhuskable, unhusked, unhusking, unhusks, unkerchiefed, unkindhearted, unknight, unknighted, unknighting, unknightlike, unknightliness,
unknightlinesses, unknightly, unknights, unkosher, unkoshered, unlikelihood, unlikelihoods, unmerchantlike, unmonkish, unneighborlike, unnymphlike, unphysicianlike, unphysicked, unprophetlike,
unscholarlike, unsearcherlike, unshackle, unshackled, unshackles, unshackling, unshakable, unshakableness, unshakably, unshakeable, unshakeably, unshaked, unshaken, unshakenly, unshakenness,
unshakiness, unshaking, unshakingness, unshaky, unshanked, unshiplike, unshipwrecked, unshirked, unshirking, unshockability, unshockable, unshocked, unshocking, unshook, unshowmanlike, unshrink,
unshrinkability, unshrinkable, unshrinking, unshrinkingly, unshrinkingness, unshrunk, unshrunken, unskaithd, unskaithed, unsketchable, unsketched, unskirmished, unspookish, unteacherlike, unthank,
unthanked, unthankful, unthankfully, unthankfulness, unthankfulnesses, unthanking, unthick, unthicken, unthickened, unthickly, unthickness, unthink, unthinkabilities, unthinkability, unthinkable,
unthinkableness, unthinkablenesses, unthinkables, unthinkably, unthinker, unthinking, unthinkingly, unthinkingness, unthinkingnesses, unthinks, unthoughtlike, unthriftlike, unthrushlike, unthwacked,
unwhiglike, unwhisked, unwhiskered, unyachtsmanlike, unzephyrlike.
upchoke, upchuck, upchucked, upchucking, upchucks.
verchok, vetchlike.
vikingship, visuokinesthetic.
wahpekute, wakizashi, walkathon, walkathons, wallhick, washbasket, washwork, watchkeeper, watchmake, watchmaker, watchmakers, watchmaking, watchmakings, watchwork, watchworks, watershake, wawaskeesh.
weakfish, weakfishes, weakhanded, weakhearted, weakheartedly, weakheartedness, weakish, weakishly, weakishness, weakmouthed, wealthmaker, wealthmaking, weatherbreak, weathercock, weathercocked,
weathercocking, weathercockish, weathercockism, weathercocks, weathercocky, weathermaker, weathermaking, weathersick, weedhook, weeknight, weeknights, weighbauk, weighlock, weinschenkite, wenchlike.
whack, whacked, whacker, whackers, whackier, whackiest, whacking, whackings, whacko, whackoes, whackos, whacks, whacky, whaleback, whalebacker, whalebacks, whalelike, whalesucker, whallock, whank,
whapuka, whapukee, whapuku, whatkin, whatlike, whatreck, whauk, wheatflakes, wheatlike, wheatstalk, wheellike, wheelmaker, wheelmaking, wheelwork, wheelworks, wheerikins, whekau, wheki, whelk,
whelked, whelker, whelkier, whelkiest, whelklike, whelks, whelky, whetrock, wheylike, whick, whicken, whicker, whickered, whickering, whickers, whikerby, whilk, whillikers, whillikins, whilock,
whinchacker, whincheck, whinnock, whipcrack, whipcracker, whipjack, whipjacks, whipking, whiplike, whipmaker, whipmaking, whipsnake, whipsnakes, whipsocket, whipstalk, whipstick, whipstock,
whipstocks, whirken, whirrick, whisk, whiskbroom, whiskbrooms, whisked, whisker, whiskerage, whiskerando, whiskerandoed, whiskerandos, whiskered, whiskerer, whiskerette, whiskerless, whiskerlike,
whiskers, whiskery, whisket, whiskets, whiskey, whiskeyfied, whiskeys, whiskful, whiskied, whiskies, whiskified, whiskin, whisking, whiskingly, whisks, whisky, whiskyfied, whiskylike, whistlelike,
whistlike, whiteback, whitebark, whitelike, whitesark, whiteshank, whitrack, whitracks, whitterick, whittericks, whittrick, whizkid, whizkids, whooplike, whorelike, whulk, whuskie.
wickedish, wickthing, windshake, windshakes, windshock, winklehawk, winklehole, witchknot, witchknots, witchlike, witchuck, witchwork, withtake.
woodchuck, woodchucks, woodhack, woodhacker, woodshock, woodshrike, woodshrikes, workaholic, workaholics, workaholism, workaholisms, workbench, workbenches, workhand, workhorse, workhorses, workhour,
workhours, workhouse, workhoused, workhouses, workmanship, workmanships, worksheet, worksheets, workship, workshop, workshops, workshy, workwatcher, workwatchers.
wraithlike, wrathlike, wreathlike, wreathmaker, wreathmaking, wreathwork, wreckfish, wreckfishes, wretchock, writheneck.
wykehamical, wykehamist.
yachtsmanlike, yakhdan, yakhdans, yaksha, yakshi, yamshik, yamstchick, yamstchik, yarmulkah, yarmulkahs, yashmak, yashmaks.
yellowshank, yellowshanks, yemschik, yertchuk.
yokelish, yokohama, yorkish, yorkshire, youthlike, youthlikeness, youthquake, youthquakes.
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A particle projected A projectile can have the same range for two angle
a. of projection. If times of flight in two cases are and then velocity of projection of the projectile is
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Question Text A particle projected A projectile can have the same range for two angle
Updated On Aug 29, 2022
Topic Mechanics
Subject Physics
Class Class 11
Answer Type Video solution: 1
Upvotes 148
Avg. Video Duration 25 min | {"url":"https://askfilo.com/user-question-answers-physics/a-particle-projected-a-projectile-can-have-the-same-range-32303531393434","timestamp":"2024-11-14T23:03:52Z","content_type":"text/html","content_length":"207034","record_id":"<urn:uuid:c9b973a3-62db-4a4f-983c-85a087f9d564>","cc-path":"CC-MAIN-2024-46/segments/1730477397531.96/warc/CC-MAIN-20241114225955-20241115015955-00613.warc.gz"} |
Confidence Interval Calculators
Compute the 90%, 95%, and 99% confidence intervals for a binomial probability using the Clopper-Pearson (exact) method, given the total number of trials and the number of successes. Knowing the
confidence interval for a binomial probability can be very useful for analytics studies that rely on binomial experiments.
Compute the 90%, 95%, and 99% confidence intervals for Cohen's f-square effect size for a multiple regression study, given the f-square value, the number of predictor variables, and the total sample
size. Knowing the confidence interval for an f-square effect size can be very useful for comparing different models in analytics studies that rely on multiple regression.
Compute the 90%, 95%, and 99% confidence intervals for a mediation indirect effect, given values of the regression coefficient and standard error for the relationship between the mediator and the
dependent variable, and values of the regression coefficient and standard error for the relationship between the independent variable and the mediator. Knowing the confidence interval for an indirect
effect can be very useful for assessing the true nature of the indirect effect in analytics studies that rely on mediation models.
Compute the 90%, 95%, and 99% confidence intervals for an R-square value, given the R-square value, the number of predictor variables, and the total sample size. Knowing the confidence interval for
an R-square value can be very useful in analytics when considering the true degree of usefulness that a regression model might have in the overall population.
Compute the exact 90%, 95%, and 99% confidence intervals for a Poisson mean, given the total number of number of event occurrences. Knowing the confidence interval for a Poisson mean can be very
useful for analytics studies that use the Poisson distribution to examine interval data.
Compute the 90%, 95%, and 99% confidence intervals for the mean of a normal population when the population standard deviation is known, given the population standard deviation, the sample mean, and
the sample size. Knowing the confidence interval for the mean of a normal population can be very useful for assessing the true nature of the population in analytics studies that rely on normally
distributed sample data.
Compute the 90%, 95%, and 99% confidence intervals for the mean of a normal population, given the sample standard deviation, the sample mean, and the sample size. Knowing the confidence interval for
the mean of a normal population can be very useful for assessing the true nature of a population variable in analytics studies that use normally distributed sample data.
Compute the 99%, 95%, and 90% confidence intervals for a predicted value of a regression equation, given the standard error of the estimate, the number of predictor variables, the total sample size,
and a predicted value of the dependent variable. Knowing the confidence interval for a predicted regression value can be very useful for assessing the true range of outcomes that might occur in light
of a given set of input values in analytics studies that rely on multiple regression.
Compute 90%, 95%, and 99% confidence intervals for a regression coefficient, given the regression coefficient value, the standard error of the regression coefficient, the number of predictor
variables, and the total sample size. Knowing the confidence interval for a regression coefficient can be very useful in analytics when considering the true range of values that a predictor variable
might have in the overall population.
Compute the 90%, 95%, and 99% confidence intervals for a regression intercept (or regression constant), given the regression intercept value, the standard error of the regression intercept, the
number of predictor variables, and the total sample size. Knowing the confidence interval for a regression intercept can be very useful in analytics when considering the true range of values that the
intercept might have in the overall population. | {"url":"https://www.analyticscalculators.com/category.aspx?id=4","timestamp":"2024-11-05T00:24:21Z","content_type":"text/html","content_length":"30130","record_id":"<urn:uuid:f1f0a2fa-86ba-432f-9125-8ec8ee216a91>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.84/warc/CC-MAIN-20241104225856-20241105015856-00401.warc.gz"} |
What is the problem?
Symbolic Integration is the problem of finding a formula for the derivative or indefinite integral for a given function f(x). Unlike numerical integration which gives a numerical value as a result, a
computer cannot utilize numerical techniques to aproximate the value. While symbolic derivation can be achieved in logic programming through a series of relatively simple axioms, such as linearity,
the chain and product rule, the power rule and the trigonometric definitions. A logical definition for the symbolical derivative can be done as follows:
% deriv(E,X,DE) is true if DE is the derivative of E with respect to X
This means that for symbolic derivation the input form of the function E can be done as any elementary function in multiple forms.
For instance f(x) = x^2 can also be written as something like x*x. The logical query for both will give different symbolic results that are mathematically equivalent.
?- deriv(x^2,x,X).
X = 2*x^(2-1)*1 .
?- deriv(x*x,x,X).
X = x*1+x*1
This difference in form posses a problem for symbolic integration. The main problem in this approach is that generally speaking, the closed form representation of a function is different from the
representation. For instance consider these three axioms for a naive antiderivative logical query:
% antid(DE,X,E) is true if E is the antiderivative of DE with respect to X
antid(C,X,C*X) :- atomic(C).
antid(F,DX,X) :- deriv(X,DX,DE).
?- antid(2*x,x,X)
This query will get stuck in a loop since the program does not recognize 2*x as being equal to any of the derivative forms of f(x)=x^2. In fact, it won't recognize neither x*1+x*1 or 2*x^(2-1)*1 as
In other words, in order for symbolic integration to work in logical programming, some sort of "reverse simplifier" type clause will be needed if one wishes to use the defintion of the integral as an
Alternatively, one could axiomatically define the properties of integral going through each rule for multiple cases.
For our project we decided to go with the second option.
What is the something extra?
When defining our rules we decided to include the general forms of functions. This means that for instance the rule
will not typically account for the cases in which there is a constant such as ln(bx+a). To account for this we included rules for these cases in multiple functions, for instance in the case of ln(x):
integ(ln(B*X),X,X*ln(B*X)-X) :- atomic(B).
integ(ln(X+A),X,(A+X)*ln(X+A)-X) :- atomic(A).
integ(ln(B*X+A),X,(A/B + X)*ln(B*X + A)-X) :- atomic(A), atomic(B).
Additionally, we looked into the feasability of implementing u-substitution and integration by parts, however this seems out of scope for the current axiom based system since it utilizes derivative
in the definition. Consider this naive approach for Integration by Parts:
integ(U*DV,X,U*V- IVDU) :- deriv(V,X,DV), deriv(U,X,DU), integ(V*DU,X,IVDU).
We run into the same issue as utilizing the anti-derivative method since it reiles on the derivative of V and U.
So instead we implemented a definitive integral formula for polynomial functions expanding on David's algebra.pl.
What did we learn from doing this?
Initially, we thought that implementing an Integral Calculator which could handle all kinds of different Integrals would be doable, turns out this was a very naive approach from us. Prolog is very
good at pattern matching which means that things that have set rules like a Derivative Calculator can be implemented without much hassle. This is also the case for the Integral Calculator but only to
a certain extent, and a very minor one in the realm of Integration. While we were able to effectively integrate functions that have defined forms of integration which can be logically defined for
Prolog to compute, more complex forms of integration such as, integration by parts or u-substitution are radically harder to implement. Not only are the intricacies we would need to master about
Symbolic Integration to fully implement the calculator way out of the scope of any undergraduate analysis class, but this would in essence, as mentioned above, require us to implement a "reverse
simplifier" in order to functional operations that Prolog could fully understand. In other words, integration isn't very logical, as opposed to differentiation, it does have set rules which work to a
certain extent but more complex forms of integration just have no set logical procedure to follow, it needs a severe amount of interpretations for it to predefined some systematic way to handle every
case of an Integral.
Prolog is remarkably good at working with logically defined procedures, that is why it's so good at implementing Expert Systems, it has a very efficient way to match patterns and it is very effective
at handling queries and case-by-case methodical problems. Given a set of rules or procedures Prolog will be able to implement and work through all of them flawlessly, and an in theory I am sure a
full-fleshed Integral Calculator could be implemented using Prolog, but feasibility-wise it is quite far fetched.
Links to code etc | {"url":"https://wiki.ubc.ca/Course:CPSC312/IntegralCalculator","timestamp":"2024-11-13T17:59:16Z","content_type":"text/html","content_length":"28754","record_id":"<urn:uuid:a769e92f-5f1c-4b6c-bfd0-a1d2b3373f11>","cc-path":"CC-MAIN-2024-46/segments/1730477028387.69/warc/CC-MAIN-20241113171551-20241113201551-00022.warc.gz"} |
Lesson 6
Interpreting Histograms
6.1: Dog Show (Part 1) (5 minutes)
The purpose of this warm-up is to connect the analytical work students have done with dot plots in previous lessons with statistical questions. This activity reminds students that we gather, display,
and analyze data in order to answer statistical questions. This work will be helpful as students contrast dot plots and histograms in subsequent activities.
Arrange students in groups of 2. Give students 1 minute of quiet work time, followed by 2 minutes to share their responses with a partner. Ask students to decide, during partner discussion, if each
question proposed by their partner is a statistical question that can be answered using the dot plot. Follow with a whole-class discussion.
If students have trouble getting started, consider giving a sample question that can be answered using the data on the dot plot (e.g., “How many dogs weigh more than 100 pounds?”)
Student Facing
Here is a dot plot showing the weights, in pounds, of 40 dogs at a dog show.
1. Write two statistical questions that can be answered using the dot plot.
2. What would you consider a typical weight for a dog at this dog show? Explain your reasoning.
Activity Synthesis
Ask students to share questions they agreed were statistical questions that could be answered using the dot plot. Record and display their responses for all to see. If there is time, consider asking
students how they would find the answer to some of the statistical questions.
Ask students to share a typical weight for a dog at this dog show and why they think it is typical. Record and display their responses for all to see. After each student shares, ask the class if they
agree or disagree.
6.2: Dog Show (Part 2) (10 minutes)
This activity introduces students to histograms. By now, students have developed a good sense of dot plots as a tool for representing distributions. They use this understanding to make sense of a
different form of data representation. The data set shown on the first histogram is the same one from the preceding warm-up, so students are familiar with its distribution. This allows them to focus
on making sense of the features of the new representation and comparing them to the corresponding dot plot.
Note that in all histograms in this unit, the left-end boundary of each bin or bar is included and the right-end boundary is excluded. For example, the number 5 would not be included in the 0–5 bin,
but would be included in the 5–10 bin.
Explain to students that they will now explore histograms, another way to represent numerical data. Give students 3–4 minutes of quiet work time, and then 2–3 minutes to share their responses with a
partner. Follow with a whole-class discussion.
Student Facing
Here is a histogram that shows some dog weights in pounds.
Each bar includes the left-end value but not the right-end value. For example, the first bar includes dogs that weigh 60 pounds and 68 pounds but not 80 pounds.
1. Use the histogram to answer the following questions.
1. How many dogs weigh at least 100 pounds?
2. How many dogs weigh exactly 70 pounds?
3. How many dogs weigh at least 120 and less than 160 pounds?
4. How much does the heaviest dog at the show weigh?
5. What would you consider a typical weight for a dog at this dog show? Explain your reasoning.
2. Discuss with a partner:
□ If you used the dot plot to answer the same five questions you just answered, how would your answers be different?
□ How are the histogram and the dot plot alike? How are they different?
Activity Synthesis
Ask a few students to briefly share their responses to the first set of questions to make sure students are able to read and interpret the graph correctly.
Focus the whole-class discussion on the last question. Select a few students or groups to share their observations about whether or how their answers to the statistical questions would change if they
were to use a dot plot to answer them, and about how histograms and dot plots compare. If not already mentioned by students, highlight that, in a histogram:
• Data values are grouped into “bins” and represented as vertical bars.
• The height of a bar reflects the combined frequency of the values in that bin.
• A histogram uses a number line.
At this point students do not yet need to see the merits or limits of each graphical display; this work will be done in upcoming lessons. Students should recognize, however, how the structures of the
two displays are different (MP7) and start to see that the structural differences affect the insights we are able to glean from the displays.
Representation: Develop Language and Symbols. Create a display of important terms and vocabulary. Include the following terms and maintain the display for reference throughout the unit: histogram.
Supports accessibility for: Memory; Language
Representing, Listening, Speaking: MLR2 Collect and Display. Record and display the language students use as they discuss how dot plots and histograms are alike and different. Highlight language
related to individual specific data values and groups of data values, center, and spread. Keep the display visible for reference in upcoming lessons. This will help students connect unique and shared
characteristics of histograms and dot plots.
Design Principle(s): Support sense-making; Maximize meta-awareness
6.3: Population of States (20 minutes)
In this activity, students continue to develop their understanding of histograms. They begin to notice that a dot plot may not be best for representing a data set with a lot of variability (or where
few values are repeated) or when a data set has a large number of values. Histograms may help us visualize a distribution more clearly in these situations. Students organize a data set into “bins”
and draw a histogram to display the distribution.
As students work and discuss, listen for explanations for why certain questions might be easy, hard, or impossible to answer using each graphical display.
Give students a brief overview of census and population data, as some students may not be familiar with them. Refer to the dot plot of the population data and discuss questions such as:
• “How many total dots are there?” (51)
• “What's the population of the state with the largest population? Do you know what state that is?” (Between 37 and 38 million. It's California.)
• “Look at the leftmost dot. What state might it represent? Approximately what is its population?” (The leftmost dot represents Wyoming, with a population of around half a million.)
• “Do you know the approximate population of our state? Where do you think we are in the dot plot?”
Explain to students that they will now draw a histogram to represent the population data. Remind them that histograms organize data values into “bins” or groups. In this case, the bins sizes are
already decided for them. Then, arrange students in groups of 3–4. Provide access to straightedges. Give students 10–12 minutes to complete the activity. Encourage them to discuss their work within
their group as needed.
Action and Expression: Internalize Executive Functions. To support development of organizational skills, check in with students within the first 2-3 minutes of work time. Check to make sure students
have developed a way to keep track of counting the 2010 census data.
Supports accessibility for: Organization; Attention
Student Facing
Every ten years, the United States conducts a census, which is an effort to count the entire population. The dot plot shows the population data from the 2010 census for each of the fifty states
and the District of Columbia (DC).
1. Here are some statistical questions about the population of the fifty states and DC. How difficult would it be to answer the questions using the dot plot?
In the middle column, rate each question with an E (easy to answer), H (hard to answer), or I (impossible to answer). Be prepared to explain your reasoning.
│ statistical question │using the dot plot│using the histogram│
│a. How many states have populations greater than 15 million? │ │ │
│b. Which states have populations greater than 15 million? │ │ │
│c. How many states have populations less than 5 million? │ │ │
│d. What is a typical state population? │ │ │
│e. Are there more states with fewer than 5 million people or more states with between 5 and 10 million people?│ │ │
│f. How would you describe the distribution of state populations? │ │ │
2. Here are the population data for all states and the District of Columbia from the 2010 census. Use the information to complete the table.
│population (millions) │frequency │
│0–5 │ │
│5–10 │ │
│10–15 │ │
│15–20 │ │
│20–25 │ │
│25–30 │ │
│30–35 │ │
│35–40 │ │
3. Use the grid and the information in your table to create a histogram.
4. Return to the statistical questions at the beginning of the activity. Which ones are now easier to answer?
In the last column of the table, rate each question with an E (easy), H (hard), and I (impossible) based on how difficult it is to answer them. Be prepared to explain your reasoning.
Student Facing
Are you ready for more?
Think of two more statistical questions that can be answered using the data about populations of states. Then, decide whether each question can be answered using the dot plot, the histogram, or both.
Activity Synthesis
Much of the discussion about how to construct histograms should have happened in small groups. Address unresolved questions about drawing histograms if they are relatively simple. Otherwise, consider
waiting until students have more opportunities to draw histograms in upcoming lessons.
Focus the discussion on comparing the effectiveness of dot plots and histograms to help us answer statistical questions.
Select a few students or groups to share how their ratings of “easy,” “hard,” and “impossible,” changed when they transitioned from using dot plots to using histograms to answer statistical questions
about populations of states. Then, discuss and compare the two displays more generally. Solicit as many ideas and observations as possible to these questions:
• “What are some benefits of histograms?”
• “When might histograms be preferable to dot plots?”
• “When might dot plots be preferable to histograms?”
Students should walk away from the activity recognizing that in cases when a data set has many numerical values, especially if the values do not repeat, a histogram can give us a better visualization
of the distribution. In such a case, creating a dot plot can be very difficult to do including finding a scale that can meaningfully display the data while a histogram will be easier to create and
display the information in a way that is easier to understand at a glance.
Representing, Writing, Conversing: MLR8 Discussion Supports. Give students sentence frames during the discussion, such as: “Histograms are easy (or hard or impossible) to use when _____, because . .
.”. This will help students make decisions about the type of graph to use to display different types of data sets.
Design Principle(s): Optimize output (for generalization)
Lesson Synthesis
In this lesson, we learn about a different way to represent the distribution of numerical data—using a histogram. This histogram, for instance, represents the distribution for the weights of some
• “What could the smallest dog weigh? The largest?” (10 kilograms up to almost 40 kilograms)
• “What does the bar between 25 and 30 tell you?” (5 dogs weigh between 25 and just under 30 kilograms)
• “What can you say about the dogs who weigh between 10 and 20 kg?” (There are 16 total dogs in this range including 7 dogs between 10 and 15 kg and 9 between 15 and 20 kg)
• “In general, what information does a histogram allow us to see? How is it different from a dot plot?” (A bigger picture of the distribution is shown in the histogram, but some of the detail is
lost when compared to a dot plot. For example, this histogram does not show the weight of any individual dogs.)
• “When might it be more useful to use a histogram than a dot plot?” (When the data is very spread out, when there are not very many data points with the same value, or when an overall idea of the
distribution is more important than a detailed view.)
6.4: Cool-down - Rain in Miami (5 minutes)
Student Facing
In addition to using dot plots, we can also represent distributions of numerical data using histograms.
Here is a dot plot that shows the weights, in kilograms, of 30 dogs, followed by a histogram that shows the same distribution.
In a histogram, data values are placed in groups or “bins” of a certain size, and each group is represented with a bar. The height of the bar tells us the frequency for that group.
For example, the height of the tallest bar is 10, and the bar represents weights from 20 to less than 25 kilograms, so there are 10 dogs whose weights fall in that group. Similarly, there are 3 dogs
that weigh anywhere from 25 to less than 30 kilograms.
Notice that the histogram and the dot plot have a similar shape. The dot plot has the advantage of showing all of the data values, but the histogram is easier to draw and to interpret when there are
a lot of values or when the values are all different.
Here is a dot plot showing the weight distribution of 40 dogs. The weights were measured to the nearest 0.1 kilogram instead of the nearest kilogram.
Here is a histogram showing the same distribution.
In this case, it is difficult to make sense of the distribution from the dot plot because the dots are so close together and all in one line. The histogram of the same data set does a much better job
showing the distribution of weights, even though we can’t see the individual data values. | {"url":"https://curriculum.illustrativemathematics.org/MS/teachers/1/8/6/index.html","timestamp":"2024-11-14T10:15:39Z","content_type":"text/html","content_length":"126742","record_id":"<urn:uuid:f1475b6b-b7c9-4b33-90a7-a3a71e5459ef>","cc-path":"CC-MAIN-2024-46/segments/1730477028558.0/warc/CC-MAIN-20241114094851-20241114124851-00404.warc.gz"} |
[opentheory-users] Status of derived syntax
Rob Arthan rda at lemma-one.com
Sat Nov 23 15:53:19 UTC 2013
On 22 Nov 2013, at 18:58, Joe Leslie-Hurd wrote:
> Hi Rob,
> The derived syntax in OpenTheory is rather ad-hoc: I just hard-coded some common mathematical syntax such as numerals, pairs and sets which is triggered by specific names in the OpenTheory namespace.
> When exporting HOL Light theories to the OpenTheory library, one of the more involved things I had to do is remove as many HOL Light-specific tags as possible. So for example, the NUMERAL tag is easily removed, and OpenTheory numerals are instead recognized by sequences of bit0 and bit1 functions terminated by natural number zero.
> The OpenTheory version of GSPEC is called fromPredicate, and you can see the standard library renamings in the OpenTheory fork of HOL Light in the file
> opentheory/stdlib/stdlib.int
> This is how sets are recognized for printing purposes (assuming that the term argument to fromPredicate has the right shape).
> Finally, I'm rather pleased by my scheme for recognizing let expressions, although it sounds like it might be causing you problems displaying your theory.
I haven't got that far yet. I have just been looking at the article files and HTML pages in the repo.
> Lets are simply terms that can be beta-reduced, so
> (\v. t[v]) x
> is printed as
> let v = x in t[v]
> No need for any special term tags, and in all the theories I have so far encountered there are no subterms in theorems that can be beta-reduced (and thus accidentally print as let).
Nice idea. I don't think you would want that feature while doing interactive theorem-proving, but it makes sense in your context. I have just had a go with the ProofPower theorem finder and it supports your approach: there are no theorems saved in the ProofPower-HOL theory hierarchy containing beta-redexes.
However, some packages contain assumptions that do contain beta-redexes and hence have been printed out as let-expressions in the HTML. See:
The assumption in the last one of these is a real monster. They all look like intermediate lemmas generated by some proof procedure whose proofs have somehow gone missing.
> Hope that helps,
It does indeed. I have some other questions about using the article files in the repo, but I think I will start a separate thread for that.
> Joe
> On Fri, Nov 22, 2013 at 8:08 AM, Rob Arthan <rda at lemma-one.com> wrote:
> What is the status of things like let-terms and set comprehensions in the Gilith Open Theory Repo. I can't see the definitions of the magic constants that are used to represent these things (LET and GSPEC in HOL Light and HOL4)? A peculiar sequence of complicated assumptions involving let builds up in the HTML listings.
> Regards,
> Rob.
> _______________________________________________
> opentheory-users mailing list
> opentheory-users at gilith.com
> http://www.gilith.com/opentheory/mailing-list
> _______________________________________________
> opentheory-users mailing list
> opentheory-users at gilith.com
> http://www.gilith.com/opentheory/mailing-list
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Efficient Solutions for Merging Two Sorted Lists: A Guide
Mastering LeetCode: Merge Two Sorted Lists
An in-depth guide to merging sorted linked lists, featuring Python, TypeScript, and Java solutions.
The problem of merging two sorted linked (LeetCode 21) lists into a single sorted list is a classic algorithmic challenge often encountered in software engineering interviews. This task tests one's
understanding of linked list data structures, pointer manipulation, and algorithm efficiency.
Imagine you're given two lists: list1 = [1,2,4] and list2 = [1,3,4]. Your goal is to merge these lists into one sorted list, resulting in [1,1,2,3,4,4]. This seemingly straightforward task can reveal
deep insights into an engineer's problem-solving skills.
Problem Solving Strategy
To tackle this problem:
1. We start with a dummy node to simplify edge cases and maintain a current pointer to build the new list.
2. We compare the values of nodes from both lists, appending the smaller one to the current node, and moving the pointer of the appended list forward. This process continues until we reach the end
of one or both lists.
3. If one list is exhausted before the other, we link the remainder of the non-exhausted list to the end of the merged list. This ensures that all elements are included.
The time complexity of this algorithm is O(n + m), where n and m are the lengths of the two lists, as each element is visited exactly once.
The space complexity is O(1), as we only allocate a few pointers regardless of the input size.
Python Solution
class ListNode:
def __init__(self, val=0, next=None):
self.val = val
self.next = next
class Solution:
def mergeTwoLists(self, list1: Optional[ListNode], list2: Optional[ListNode]) -> Optional[ListNode]:
# Create a dummy node to act as the starting point
head = cur = ListNode(0)
# Traverse both lists
while list1 and list2:
# Link the smaller value to 'cur' and advance
if list1.val < list2.val:
cur.next = list1
list1 = list1.next
cur.next = list2
list2 = list2.next
cur = cur.next
# Attach any remaining elements
cur.next = list1 or list2
# Return the merged list, skipping the dummy node
return head.next
TypeScript Solution
class ListNode {
val: number;
next: ListNode | null;
constructor(val?: number, next?: ListNode | null) {
this.val = (val===undefined ? 0 : val);
this.next = (next===undefined ? null : next);
function mergeTwoLists(list1: ListNode | null, list2: ListNode | null): ListNode | null {
let cur = new ListNode(0);
const head = cur;
while (list1 && list2) {
if (list1.val < list2.val) {
cur.next = list1;
list1 = list1.next;
} else {
cur.next = list2;
list2 = list2.next;
cur = cur.next;
cur.next = list1 || list2;
return head.next;
Java Solution
public class ListNode {
int val;
ListNode next;
ListNode() {}
ListNode(int val) { this.val = val; }
ListNode(int val, ListNode next) { this.val = val; this.next = next; }
public class Solution {
public ListNode mergeTwoLists(ListNode list1, ListNode list2) {
ListNode head = new ListNode(0);
ListNode cur = head;
while (list1 != null && list2 != null) {
if (list1.val < list2.val) {
cur.next = list1;
list1 = list1.next;
} else {
cur.next = list2;
list2 = list2.next;
cur = cur.next;
cur.next = (list1 != null) ? list1 : list2;
return head.next;
Merging two sorted lists is an essential problem that showcases the importance of understanding data structures and algorithmic strategies.
Remember, the key to excelling in coding interviews is practice, understanding the underlying principles, and adapting to various problem-solving scenarios.
Did you find this article valuable?
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Equalization and specification of image gray histogram matlab
Gray Histogram
Gray histogram: Reflects the statistics of different gray levels in the image.
Examples are as follows:
Equalization: The process of transforming the gray-scale histogram of an original image from a certain gray-scale interval in a comparative set to a uniform distribution in all gray-scale ranges.
Histogram equalization is the process of stretching the image nonlinearly and reassigning the pixel values of the image so that the number of pixels in a certain gray-scale range is approximately the
same. Histogram equalization is the process of changing the histogram distribution of a given image to a uniform distribution"Uniform" distribution histogram distribution.
Gray histogram of an image is represented as a one-dimensional discrete function:
So each column of the histogram is the value of nk.
The normalized histogram is further obtained, expressed as the frequency of gray values:
Therefore, the operation of equalization is to establish a mapping function for each gray value of the original image to correspond to a new gray value, and all new gray values of the new image
should be evenly distributed, in other words, the mapping function needs to satisfy the following conditions:
1. Within 0 < r < 1, T® Monotonically increasing function; (This condition guarantees the same order of gray levels from black to white after equalization)
2. There is 0 < T in 0 < r < 1® Less than or equal to 1. (This condition guarantees that the gray level of the equalized image is within the allowable range)
The new gray values are evenly distributed between 0 and 1, that is, equalization.
Equalization steps:
The first step is to calculate the gray histogram of the original image.
The second step calculates the total number of pixels in the original image.
The third step is to calculate the gray distribution frequency of the original image.
The fourth step calculates the gray-scale cumulative distribution frequency of the original image.
The fifth step multiplies and rounds the normalized image so that the gray level of the equalized image is consistent with the original image before normalization.
The sixth step, based on the above mapping relationship, refers to the pixels in the original image to write the image after histogram equalization.
Code implementation:
close all;
% First read in the grayscale image, and extract the height and width of the image
image = imread('1.jpg');
if z>1
image = rgb2gray(image);
[height, width] = size(image);
% Then count the cumulative number of pixel values for each gray level
NumPixel = zeros(1,256); % Establish a 256-column row vector to count the number of pixels in each gray level
for i = 1 : height
for j = 1 : width
k = image(i,j); % k Is a pixel point(i,j)Gray value of
% because NumPixel The subscript of the array starts at 1, but the range of image pixels is 0~255
% So use NumPixel(k+1)
NumPixel(k+1) = NumPixel(k+1) + 1; % Add 1 to the number of pixels corresponding to the gray value
% Next, calculate the frequency value as frequency
ProbPixel = zeros(1,256); % Statistics the frequency of each gray level
for i = 1 : 256
ProbPixel(i) = NumPixel(i) / (height * width);
% Reuse function cumsum()To calculate the cumulative distribution function ( CDF)，And frequency (value range is 0)~1)Map to 0~255 Unsigned integer
CumPixel = cumsum(ProbPixel); % Array here CumPixel Size is also 1×256
CumPixel = uint8((256-1) .* CumPixel + 0.5);
% At the right of the assignment statement used as a histogram equalization implementation below, image(i,j)Used as CumPixel Index of
% For example, image(i,j)=120，Then from CumPixel Take out the 120 value as image(i,j)New Pixel Value
outImage = uint8(zeros(height, width)); % Preallocated Array
for i = 1 : height
for j = 1 : width
outImage(i,j) = CumPixel(image(i,j)+1);
subplot(3,3, 1);
subplot(3,3, [2 3])
subplot(3,3, 4);
subplot(3,3, [5 6]);
subplot(3,3, 7);
subplot(3,3, [8 9]);
Result comparison:
The result compares the output of the self-processed image with that of the standard library equalization code. The image display is basically the same, but the histogram display is slightly
Regularization operates on the basis of equalization, and the biggest difference from equalization is that the results of normalization are known.
Based on the principle of equalization, the relationship between the original image and the expected image (the histogram image to be matched) is established to match the histogram of the original
image to a specific shape, thus compensating for the non-interactive feature of histogram equalization.
1. Gray transformation functions can be automatically determined to obtain an output image with a uniform histogram.
2. Enhance image contrast with small dynamic range.
To achieve the spanning from the original image to the matched image, the equalized image is used as the middleware.
Generally speaking, A is mapped to A1 through s1 and C to C1 through s2. As shown in the figure above, A1=C1 can be mapped to C by A. In fact, elements in A1 do not necessarily exist in C1. When
element b is found to belong to A1 but not to C1, match the closest element to b in C1.
Main mapping methods:
Result comparison: | {"url":"https://www.fatalerrors.org/a/equalization-and-specification-of-image-gray-histogram-matlab.html","timestamp":"2024-11-09T14:01:10Z","content_type":"text/html","content_length":"15462","record_id":"<urn:uuid:85ab20c4-602a-46da-8dc1-8e5862844fda>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00895.warc.gz"} |
statistics answered |Statistics tutor online| My Mathlab answers |Ap Statistics Tutor | Suppose you wish to estimate a population mean correct to within 0.15 with a confidence level of 0.90.
data Genius
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Suppose you wish to estimate a population mean correct to within 0.15 with a confidence level of 0.90. You do not know sigma2, but you know that the observations will range in value between 33 and
41. Complete parts a and b.
a. Find the approximate sample size that will product the desired accuracy of the estimate. You wish to be conservative to ensure that the sample size will be ample for achieving the desired accurcay
of the estimaye (Hint: assume that the range of the observations will be equal 4sigma)
__ The approximate sample size is ___
b. Calculate the approximate sample size, making the less conservative assumption that the range of the observations is equal tp 6sigma.
The approximate sample size is ____ | {"url":"https://www.statisticsanswered.com/questions/112/suppose-you-wish-to-estimate-a-population-mean-correct-to-within-0-15-with-a-confidence-level-of-0-90","timestamp":"2024-11-11T14:50:20Z","content_type":"text/html","content_length":"75346","record_id":"<urn:uuid:72af09b7-f182-4a72-8c39-6006aa2cf2ba>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00117.warc.gz"} |
The Golden Ratio - Solution
The golden ratio is found from the limit, as n tends to infinity, of the ratio of two successive terms of the Fibonacci sequence: x(n)=x(n-1)+x(n-2).
Divide the recurrence relation by x(n-1) and setting r(n)=x(n)/x(n-1) we get r=1+1/r.
Solve this equation using the quadratic formula, using Newton-Raphson and using another fixed point method.
To solve this equation, where r!=0, we multiply both sides of the equation by r to get r^2=r+1
and subtracting r+1 from both sides we get
Solutions of a quadratic equation of the form a
are given by
x= (-b
Here a=1, b=-1 c=-1 giving
We only look for the positive solution in this case giving
(1+sqrt(5))/2 = 1.6180339887499 to 14 s.f.
To solve using Newton-Raphson we plot the graph of y=t^2-t-1 using plotXpose.
Graph of y=t^2-t-1
Tap on the 'Mode' icon and then on the submenu (⋮) and select 'Zero Find'. Change the Accuracy (No. of s.f.) to 14 and tap OK. You will get the message popup "Click on the t-axis" and after tapping
OK on this popup then click once on the t-axis in the vicinity of where the graph crosses the t-axis. After performing the zero-finding plotXpose will display the result and the sequence of values
found during the calculation. An example output is below, where Newton- has calculated the solution as 1.6180339887499 to 14 significant figures.
Solving t^2-t-1=0: Success. The Newton-Raphson method has converged to the value (1.6180339887499,0.00000000000000E+000).
The sequence of values found was
( n, t , y(t))
(0, 1.79738557338715, 4.332093260330936E-001)
(1, 1.63043083960585, 2.787388313198380E-002)
(2, 1.61810196367846, 1.520011816329436E-004)
(3, 1.61803399081616, 4.620309912439780E-009)
(4, 1.61803398874989, 0.000000000000000E+000)
(5, 1.61803398874989, 0.000000000000000E+000)
(6, 1.61803398874989, 0.000000000000000E+000)
To solve using an alternative fixed point method we can use the expression we started from above for the ratio, i.e. r=1+1/r.
Again we tap on the 'Mode' icon and then on the submenu (⋮) and select 'Zero Find' and this time select the Fixed Point method radio button and tap on 'fix(t)=?' to enter the fixed point function to
use. Enter fix(t) as 1+1/t and then tap on OK.
You will get the message popup "Tap on the t-axis" and after tapping OK on this popup then tap once on the t-axis in the vicinity of where the graph crosses the t-axis. After performing the
zero-finding plotXpose will display the result and the sequence of values found during the calculation. An example output is below, where the solution has been calculated once again to
1.618033988749925 to 14 significant figures.
Solving t^2-t-1=0: Success. The Fix point method, using function 1+1/t has converged to the value (1.6180339887499,-8.88178419700125E-016).
The sequence of values found was
( n, t , y(t))
(0, 1.64488017559052, 6.075061646016877E-002)
(1, 1.6079470193876, -2.245340223014103E-002)
(2, 1.6219110380769, 8.684377358790574E-003)
(3, 1.61655662765925, -3.301297230213773E-003)
(4, 1.61859880618472, 1.263289197872552E-003)
(5, 1.61781832297106, -4.821968301687019E-004)
(6, 1.61811637672859, 1.842319086773347E-004)
(7, 1.61800252094459, -7.036316153685718E-005)
(8, 1.61804600861573, 2.687738157280961E-005)
(9, 1.61802939760379, -1.026609370491372E-005)
(10, 1.61803574241664, 3.921321116706622E-006)
(11, 1.61803331890953, -1.497808139294676E-006)
(12, 1.61803424460625, 5.721122744439811E-007)
(13, 1.61803389102148, -2.185273741961424E-007)
(14, 1.61803402607883, 8.347003976894030E-008)
(15, 1.61803397449151, -3.188271668896903E-008)
(16, 1.61803399419611, 1.217811407272507E-008)
(17, 1.61803398666962, -4.651625307161567E-009)
(18, 1.61803398954449, 1.776762736938053E-009)
(19, 1.61803398844639, -6.786633477418036E-010)
(20, 1.61803398886582, 2.592264181089377E-010)
(21, 1.61803398870561, -9.901568454040444E-011)
(22, 1.61803398876681, 3.782063551227566E-011)
(23, 1.61803398874343, -1.444622199642254E-011)
(24, 1.61803398875236, 5.518252521596878E-012)
(25, 1.61803398874895, -2.108091479158247E-012)
(26, 1.61803398875026, 8.055778266680136E-013)
(27, 1.61803398874976, -3.077538224260934E-013)
(28, 1.61803398874995, 1.172395514004165E-013)
(29, 1.61803398874987, -4.440892098500626E-014)
(30, 1.6180339887499, 1.687538997430238E-014)
(31, 1.61803398874989, -6.661338147750939E-015)
(32, 1.6180339887499, 2.442490654175344E-015)
(33, 1.61803398874989, -8.881784197001252E-016)
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Find the Missing Numbers - Worksheet 2 - KidsPressMagazine.com
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9,352 research outputs found
Common wisdom states that teenage childbearing reduces schooling, labour market experience and adult wages. However, the decisions to be a teenage mother, to quit school, and be less attached to the
labour market might all stem from some personal or family characteristics. Using the National Child Development Study (NCDS), we find that in Britain teenage childbearing decreases the probability of
post-16 schooling by 12% to 24%. Employment experience is reduced by up to three years, and the adult pay differential ranges from 5% to 22%. The negative impact of teen motherhood on various adult
outcomes is not due to some pre-motherhood characteristics; hence policies aiming to encourage return to school and participation in the labour market may be an efficient way to reduce the long-term
consequences of teenage pregnancy.Teenage pregnancy, schooling decisions, wages
This paper presents a model of partial observability applied to the childcare market in Britain. We simultaneously estimate the demand and use and calculate the excess demand for childcare. We find a
large queue with nearly half of the mothers demanding childcare queuing for it. We also find that formal and informal care are not substitute, implying that policies increasing the supply of formal
care lead to an increase in the use of care rather than solely a shift from informal to formal care. This has implication on the efficiency of policies aiming at increasing the labour supply of
mothers.supply of childcare
We revisit the problem of the emission of gravitational waves from a test mass orbiting and thus perturbing a Kerr black hole. The source term of the Teukolsky perturbation equation contains a Dirac
delta function which represents a point particle. We present a technique to effectively model the delta function and its derivatives using as few as four points on a numerical grid. The source term
is then incorporated into a code that evolves the Teukolsky equation in the time domain as a (2+1) dimensional PDE. The waveforms and energy fluxes are extracted far from the black hole. Our
comparisons with earlier work show an order of magnitude gain in performance (speed) and numerical errors less than 1% for a large fraction of parameter space. As a first application of this code, we
analyze the effect of finite extraction radius on the energy fluxes. This paper is the first in a series whose goal is to develop adiabatic waveforms describing the inspiral of a small compact body
into a massive Kerr black hole.Comment: 21 pages, 6 figures, accepted by PRD. This version removes the appendix; that content will be subsumed into future wor
In the context of nuclear physics Pratt recently investigated noninteracting Fermi systems described by the microcanonical and canonical ensemble. As will be shown his discussion of the model of
equally spaced levels contains a flaw and a statement which is at least confusing.Comment: Comment on S. Pratt, Phys. Rev. Lett. 84, 4255 (2000) and nucl-th/990505
Systems containing few Fermions (e.g., electrons) are of great current interest. Fluorescence occurs when electrons drop from one level to another without changing spin. Only electron gases in a
state of equilibrium are considered. When the system may exchange electrons with a large reservoir, the electron-gas fluorescence is easily obtained from the well-known Fermi-Dirac distribution. But
this is not so when the number of electrons in the system is prevented from varying, as is the case for isolated systems and for systems that are in thermal contact with electrical insulators such as
diamond. Our accurate expressions rest on the assumption that single-electron energy levels are evenly spaced, and that energy coupling and spin coupling between electrons are small. These
assumptions are shown to be realistic for many systems. Fluorescence from short, nearly isolated, quantum wires is predicted to drop abruptly in the visible, a result not predicted by the Fermi-Dirac
distribution. Our exact formulas are based on restricted and unrestricted partitions of integers. The method is considerably simpler than the ones proposed earlier, which are based on second
quantization and contour integration.Comment: 10 pages, 3 figures, RevTe
Starting from the mean-field solution of a spin-orbital model of LiNiO$_2$, we derive an effective quantum dimer model (QDM) that lives on the triangular lattice and contains kinetic terms acting on
4-site plaquettes and 6-site loops. Using numerical exact diagonalizations and Green's function Monte Carlo simulations, we show that the competition between these kinetic terms leads to a resonating
valence bond (RVB) state for a finite range of parameters. We also show that this RVB phase is connected to the RVB phase identified in the Rokhsar-Kivelson model on the same lattice in the context
of a generalized model that contains both the 6--site loops and a nearest-neighbor dimer repulsion. These results suggest that the occurrence of an RVB phase is a generic feature of QDM with
competing interactions.Comment: 8 pages, 12 figure
As the most common environment in the universe, groups of galaxies are likely to contain a significant fraction of the missing baryons in the form of intergalactic gas. The density of this gas is an
important factor in whether ram pressure stripping and strangulation affect the evolution of galaxies in these systems. We present a method for measuring the density of intergalactic gas using
bent-double radio sources that is independent of temperature, making it complementary to current absorption line measurements. We use this method to probe intergalactic gas in two different
environments: inside a small group of galaxies as well as outside of a larger group at a 2 Mpc radius and measure total gas densities of $4 \pm 1_{-2}^{+6} \times 10^{-3}$ and $9 \pm 3_{-5}^{+10} \
times 10^{-4}$ per cubic centimeter (random and systematic errors) respectively. We use X-ray data to place an upper limit of $2 \times 10^6$ K on the temperature of the intragroup gas in the small
group.Comment: 6 pages, 1 figure, accepted for publication in Ap
The Mock LISA Data Challenges (MLDCs) have the dual purpose of fostering the development of LISA data analysis tools and capabilities, and demonstrating the technical readiness already achieved by
the gravitational-wave community in distilling a rich science payoff from the LISA data output. The first round of MLDCs has just been completed: nine challenges consisting of data sets containing
simulated gravitational-wave signals produced either by galactic binaries or massive black hole binaries embedded in simulated LISA instrumental noise were released in June 2006 with deadline for
submission of results at the beginning of December 2006. Ten groups have participated in this first round of challenges. All of the challenges had at least one entry which successfully characterized
the signal to better than 95% when assessed via a correlation with phasing ambiguities accounted for. Here, we describe the challenges, summarize the results and provide a first critical assessment
of the entries
The Mock LISA Data Challenge is a worldwide effort to solve the LISA data analysis problem. We present here our results for the Massive Black Hole Binary (BBH) section of Round 1. Our results cover
Challenge 1.2.1, where the coalescence of the binary is seen, and Challenge 1.2.2, where the coalescence occurs after the simulated observational period. The data stream is composed of Gaussian
instrumental noise plus an unknown BBH waveform. Our search algorithm is based on a variant of the Markov Chain Monte Carlo method that uses Metropolis-Hastings sampling and thermostated frequency
annealing. We present results from the training data sets and the blind data sets. We demonstrate that our algorithm is able to rapidly locate the sources, accurately recover the source parameters,
and provide error estimates for the recovered parameters.Comment: 11 pages, 6 figures, Submitted to CQG proceedings of GWDAW 11, AEI, Germany, Dec 200
Trying to detect the gravitational wave (GW) signal emitted by a type II supernova is a main challenge for the GW community. Indeed, the corresponding waveform is not accurately modeled as the
supernova physics is very complex; in addition, all the existing numerical simulations agree on the weakness of the GW emission, thus restraining the number of sources potentially detectable.
Consequently, triggering the GW signal with a confidence level high enough to conclude directly to a detection is very difficult, even with the use of a network of interferometric detectors. On the
other hand, one can hope to take benefit from the neutrino and optical emissions associated to the supernova explosion, in order to discover and study GW radiation in an event already detected
independently. This article aims at presenting some realistic scenarios for the search of the supernova GW bursts, based on the present knowledge of the emitted signals and on the results of network
data analysis simulations. Both the direct search and the confirmation of the supernova event are considered. In addition, some physical studies following the discovery of a supernova GW emission are
also mentioned: from the absolute neutrino mass to the supernova physics or the black hole signature, the potential spectrum of discoveries is wide.Comment: Revised version, accepted for publication
in Astroparticle Physic | {"url":"https://core.ac.uk/search/?q=authors%3A(Arnaud%20K.)","timestamp":"2024-11-03T14:08:34Z","content_type":"text/html","content_length":"187202","record_id":"<urn:uuid:feec9505-398f-45a7-a37e-91391ee2f775>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00035.warc.gz"} |
Calendar Math Kit
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Turbulence Modeling No Longer a Drag
A new theoretical framework allows scientists to accurately estimate the friction a surface experiences in a turbulent flow, such as an airplane wing flying through the sky.
When a fluid flows across a solid surface, friction arises between the fluid and the surface. This friction dictates the energy needed to pump oil through a pipe or the fuel needed to fly an aircraft
through the sky, for example. Previous models of such friction were accurate for only a limited range of turbulent flows. Now Shivsai Dixit and his colleagues at the Indian Institute of Tropical
Meteorology have developed a generalized model that is accurate for many more such flows [1].
To date, all models of friction in turbulent flows describe mathematically how the amount of friction varies as a quantity called the Reynolds number is changed. This number is the ratio between
inertial and viscous forces in the flow. Previously, these relationships were inferred from empirical data and were thus reliable only for flow types extensively studied in either experiments or
numerical simulations. These flows include idealized ones, such as those in wind tunnels, but not ones of practical interest, such as the airflows that whip around airplane wings or turbine blades.
In their new model, Dixit and his colleagues overcame this restriction by determining how the amount of friction varies with the Reynolds number directly from the equations that govern the dynamics
of turbulent flows. The key step was to carefully consider such dynamics and the boundary conditions associated with these equations. The researchers found that their model accurately predicts the
amount of friction measured in experiments and numerical simulations for a wide range of Reynolds numbers and flow types.
–Ryan Wilkinson
Ryan Wilkinson is a Corresponding Editor for Physics Magazine based in Durham, UK. | {"url":"https://physics.aps.org/articles/v17/s2","timestamp":"2024-11-04T22:07:45Z","content_type":"text/html","content_length":"22225","record_id":"<urn:uuid:630643d7-767f-42fd-907d-da6c332e1c7d>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.16/warc/CC-MAIN-20241104194528-20241104224528-00008.warc.gz"} |
Possible bug or I am blind about something??
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6 Replies
3 Total Likes
Possible bug or I am blind about something??
Just wanted to plot a circle to solve an analytic geometry problem: The equation of the circle is (x-2)^2 + (y+1)^2 == 6^2 I had mathematica solve for y so I could plot both sides of the circle.
Mathematica returned the following results:
{{y->-1-Sqrt[-4+4 x-x^2+36 ^2]},{y->-1+Sqrt[-4+4 x-x^2+36 ^2]}}
When the circle was plotted, the circle had a radius around 38 instead of 6 as indicated in the standard form of a circle.
What am I missing here???
6 Replies
Thankfully rare stumbling block in Mathematica: you can insert invisible operators in equations. I think I managed to find a shortcut for invisible addition (or something similar) once, and scratched
my head for half and hour trying to understand what made my results so strange.
Solve[(x - 2)^2 + (y + 1)^2 == 6^2 , y]
Solve[(x - 2)^2 + (y + 1)^2 == 6^2, y]
Plot[{-1 - Sqrt[-4 + 4 x - x^2 + 36^2], -1 +
Sqrt[-4 + 4 x - x^2 + 36^2]}, {x, -40, 40}]
Plot[{-1 - Sqrt[32 + 4 x - x^2], -1 + Sqrt[32 + 4 x - x^2]}, {x, -10,
{{y->-1-Sqrt[-4+4 x-x^2+36 ^2]},{y->-1+Sqrt[-4+4 x-x^2+36 ^2]}}
{{y->-1-Sqrt[32+4 x-x^2]},{y->-1+Sqrt[32+4 x-x^2]}}
Thanks Bruce - seems you were right. I retyped that equation and got the right results. it seems what I thought were just spaces, were something else. What can I do to tell what non-printing
characters slipped in?? again, thanks all for your help...
"What can I do to tell what non-printing characters slipped in?? "
May be this can help
It is about hidden chars that can slip in sometimes into the notebook cell.
There might have been a non-printing character that slipped into the input before the 6^2.
The corresponding output has 36 space ^2, which is suspicious.
You can select the cell bracket and go to Cell menu - Cell Expression. The problem might be apparent.
Usual fixes are
- re-type that section of the input, or the whole input, and
- select the input, use Edit menu - CopyAs - Plain Text, and paste it into a different place in the notebook.
Are you sure that the results you quote were returned by this Solve command?
Mathematica cannot have returned these solutions because they are not fully evaluated. Note the 36^2 - 4 part. Mathematica would always print 1292.
The actual solution returned by version 9.0.1 is correct.
I get this, what do you see?
Plot[y /. Solve[ (x - 2)^2 + (y + 1) ^2 == 6^2, y], {x, -10, 10},
AspectRatio -> 1, PlotRange -> {{-10, 10}, {-10, 10}}]
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Energy of a capacitor and an electric field
Electromagnetic Fields and Waves • Year 1
Energy of a capacitor and an electric field
Elementary work of external forces to move charge dq in electric field of a capacitor
$dA=dq*\left({\phi }_{1}–{\phi }_{2}\right)=dq\frac{q}{C}$
Total work is
$A={\int }_{0}^{Q}\frac{dqq}{C}=\frac{{Q}^{2}}{2C}$
this work determines total energy stored in a capacitor, Q is a total capacitor charge.
$Q=C\left({\phi }_{1}–{\phi }_{2}\right)$
and energy of a charged capacitor
$W=\frac{C\left({\phi }_{1}–{\phi }_{2}{\right)}^{2}}{2}$
Let’s express these characteristics through the electric field parameters. For flat capacitors
${\phi }_{1}–{\phi }_{1}=Ed$
$C=\frac{\epsilon {\epsilon }_{0}S}{d}$
$W=\frac{\epsilon {\epsilon }_{0}S}{d}\frac{\left(Ed{\right)}^{2}}{2}=\frac{\epsilon {\epsilon }_{0}{E}^{2}}{2Sd}=\frac{\epsilon {\epsilon }_{0}{E}^{2}}{2V}$
V – is is the volume between capacitor plates. Therefore, volume energy density can be found for a capacitor
${w}_{e}=\frac{W}{V}=\frac{\epsilon {\epsilon }_{0}{E}^{2}}{2}$
Volume energy density has local characteristics, and it corresponds to the piece of a capacitor where the electric field is uniform and equal to E. Let’s consider the term of volume energy density,
on the example of non-uniform electric field. Take a piece of space with volume dV, that characterises radius-vector r. Volume density of energy in general is the value expressed by the formula
Figure 22 dW[e ]is the energy of the small piece of electric field. So if we know the electric field E(r), we can calculate the energy of any piece of field with finite dimensions Ω. Therefore,
${w}_{e}=\frac{\epsilon {\epsilon }_{0}{E}^{2}\left(\overline{r}\right)}{2,{W}_{e}}={\int }_{\Omega }{w}_{e}dV={\int }_{\Omega }\frac{\epsilon {\epsilon }_{0}{E}^{2}\left(\overline{r}\right)}{2dV}$
These formulas work for the uniform capacitor material with permittivity ε=const.
Figure 22. Piece of space with calculated energy
<strong>Electric field in dielectrics. Electric dipole</strong>
Dipoles play an important role in describing dielectrics in the electric field. Electric dipole is a system of two equal charges q, with opposite signs, placed on the distance l. A Dipole moment is
the main dipole characteristic.
from negative to positive charge – called the dipole shoulder. Figure 23. For example, molecules
are dipoles. Let’s find out how dipoles behave in different kinds of electric fields.
Figure 23. Schematic picture of a dipole
<strong>Uniform field</strong>
Electric field is constant in any point of space, forces affecting the charges
are equal with an opposite sign. Resulting force is 0. The dipole of these forces is not 0, if the dipole is not oriented parallel to the electric field lines. Figure 24. Force momentum, according to
the axis normal to the figure plane:
This means a uniform field turns the dipole along the field lines.
Figure 24. Dipole in the uniform electric field
<strong>Non-uniform field</strong>
In non-uniform fields, the dipole is affected by turning the moment and electric force. Assuming that field changes only in one direction, Figure 25. Projection of force on the axis
${F}_{x}={F}_{x}^{–}+{F}_{x}^{+}=q{E}_{x}^{–}+q{E}_{x}^{+}=q*dE=q*\partial E/\partial xdx,dx=l*cos\alpha$
${F}_{x}=q*\frac{\partial E}{\partial x}*l*cos\alpha =q*l*\frac{\partial E}{\partial x}*cos\alpha =p*\frac{\partial E}{\partial x}*cos\alpha$
Dipoles are oriented along the field. F[x ]has a positive or negative sign depending on the electric field, and if it is increasing or decreasing along the 0X axis. Dipoles are drawn into the
stronger part of field. Electric field increment is
$\overline{dE}=\left(\frac{\partial E}{\partial x}dx\right)\overline{{e}_{x}}+\left(\frac{\partial E}{\partial y}dy\right)\overline{{e}_{y}}+\left(\frac{\partial E}{\partial z}dz\right)\overline{{e}_
then the force affecting the dipole is
$\overline{F}=\left(\frac{\partial E}{\partial x}{p}_{x}\right)\overline{{e}_{x}}+\left(\frac{\partial E}{\partial y}{p}_{y}\right)\overline{{e}_{y}}+\left(\frac{\partial E}{\partial z}{p}_{z}\right)
The conclusion is that dipoles are orienting along the electric field lines and drawing into the electric field with a bigger intensity.
Figure 25. Dipole in non-uniform field
Dipole energy in the electric field
$W=q{\phi }^{+}–q{\phi }^{–}.{\phi }^{+}–{\phi }^{–}=\frac{\partial \phi }{\partial x}dx=\frac{\partial \phi }{\partial x}l*cos\alpha ,{E}_{x}=–grad\phi$
$W=–p*E*\mathrm{cos}\alpha =–\left(p,E\right)$
Force of electric field
$\overline{F}=–gradW,{F}_{x}=–\frac{\partial W}{\partial x}=\frac{p\partial E}{\partial x}cos\alpha$
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New paper: "Proof-producing reflection for HOL" - Machine Intelligence Research Institute
New paper: “Proof-producing reflection for HOL”
MIRI Research Fellow Benya Fallenstein and Research Associate Ramana Kumar have co-authored a new paper on machine reflection, “Proof-producing reflection for HOL with an application to model
HOL stands for Higher Order Logic, here referring to a popular family of proof assistants based on Church’s type theory. Kumar and collaborators have previously formalized within HOL (specifically,
HOL4) what it means for something to be provable in HOL, and what it means for something to be a model of HOL.^1 In “Self-formalisation of higher-order logic,” Kumar, Arthan, Myreen, and Owens
demonstrated that if something is provable in HOL, then it is true in all models of HOL.
“Proof-producing reflection for HOL” builds on this result by demonstrating a formal correspondence between the model of HOL within HOL (“inner HOL”) and HOL itself (“outer HOL”). Informally
speaking, Fallenstein and Kumar show that one can always build an interpretation of terms in inner HOL such that they have the same meaning as terms in outer HOL. The authors then show that if
statements of a certain kind are provable in HOL’s model of itself, they are true in (outer) HOL. This correspondence enables the authors to use HOL to implement model polymorphism, the approach to
machine self-verification described in Section 6.3 of “Vingean reflection: Reliable reasoning for self-improving agents.”^2
This project is motivated by the fact that relatively little hands-on work has been done on modeling formal verification systems in formal verification systems, and especially on modeling them in
themselves. Fallenstein notes that focusing only on the mathematical theory of Vingean reflection might make us poorly calibrated about where the engineering difficulties lie for software
implementations. In the course of implementing model polymorphism, Fallenstein and Kumar indeed encountered difficulties that were not obvious from past theoretical work, the most important of which
arose from HOL’s polymorphism.
Fallenstein and Kumar’s paper was presented at ITP 2015 and can be found online or in the associated conference proceedings. Thanks to a grant by the Future of Life Institute, Kumar and Fallenstein
will be continuing their collaboration on this project. Following up on “Proof-producing reflection for HOL,” Kumar and Fallenstein’s next goal will be to develop toy models of agents within HOL
proof assistants that reason using model polymorphism.
1. Kumar showed that if there is a model of set theory in HOL, there is a model of HOL in HOL. Fallenstein and Kumar additionally show that there is a model of set theory in HOL if a simpler axiom
holds. ↩
2. For more on the role of logical reasoning in machine reflection, see Fallenstein’s 2013 conversation about self-modifying systems. ↩ | {"url":"https://intelligence.org/2015/12/04/new-paper-proof-producing-reflection-for-hol/","timestamp":"2024-11-15T04:26:40Z","content_type":"text/html","content_length":"54364","record_id":"<urn:uuid:8e50fd14-ffc2-4b4c-afa3-49a21ac957bc>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00085.warc.gz"} |
The complexity of finding temporal separators under waiting time constraints
In this work, we investigate the computational complexity of RESTLESS TEMPORAL (s,z)-SEPARATION, where we are asked whether it is possible to destroy all restless temporal paths between two distinct
vertices s and z by deleting at most k vertices from a temporal graph. A temporal graph has a fixed vertex set but the edges have (discrete) time stamps. A restless temporal path uses edges with
non-decreasing time stamps and the time spent at each vertex must not exceed a given duration Δ. RESTLESS TEMPORAL (s,z)-SEPARATION naturally generalizes the NP-hard TEMPORAL (s,z)-SEPARATION
problem. We show that RESTLESS TEMPORAL (s,z)-SEPARATION is complete for Σ[2]^P, a complexity class located in the second level of the polynomial time hierarchy. We further provide some insights in
the parameterized complexity of RESTLESS TEMPORAL (s,z)-SEPARATION parameterized by the separator size k.
• Computational complexity
• Parameterized complexity
• Restless temporal paths
• Temporal graphs
• Σ-completeness
ASJC Scopus subject areas
• Theoretical Computer Science
• Signal Processing
• Information Systems
• Computer Science Applications
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Variable Dimension Posterior
Next: Further Work Up: Bayesian Posterior Comprehension via Previous: Multimodal Likelihood Function
Variable Dimension Posterior
This section describes an algorithm suitable for variable dimension posterior distributions. Unlike the simple unimodal algorithm we must incorporate a Kullback-Leibler distance acceptance test, so
that regions contain only models that are similar. Such a constraint follows from Wallace's MMLA approximation (see, e.g. (Fitzgibbon, Dowe, and Allison, 2002a, section 2.2)). This also ensures that
the MMC instantaneous codebook corresponds to an epitome with BPC properties. We therefore augment the likelihood-based acceptance rule (Equation 12) to include the following requirement
where 11, and
In previous work the basic unimodal algorithm was modified to include this Kullback-Leibler distance acceptance rule and to make multiple passes through the sample (Fitzgibbon, Dowe, and Allison,
2002a, page 14). While this algorithm was found to be satisfactory for the univariate polynomial problem it was applied to, we found that the regions refused to grow for the change-point problem that
we consider later in this section. The problem was due to the discrete parameter space - the Kullback-Leibler acceptance rule stopped the regions from growing.
Therefore we have devised a slightly different algorithm that does not suffer from this problem. The algorithm consists of two phases. In the first phase we apply the unimodal algorithm to the sample
recursively. That is, we start the unimodal algorithm and keep track of which elements of the sample have been allocated. Once the first region has been formed we store the results and restart the
algorithm on the unallocated elements of the sample. This is repeated until all elements of the sample have been allocated to a region. We therefore end up with a set of regions: 40). We therefore
enter phase two of the algorithm where we recursively estimate the point estimate for each region and reassign elements between regions. The recursion stops when no reassignments were made in the
last iteration. The reassignment between regions is based on Kullback-Leibler distance. For each element of each region we test whether there is a region whose point estimate is closer in
Kullback-Leibler distance. If there is and the element passes the Kullback-Leibler distance acceptance rule (Equation 40) for the candidate region then the element is moved to the candidate region.
Phase two of the algorithm is given as pseudo-code in Figure 2. After phase two of the algorithm has completed we are left with an instantaneous MML codebook which defines an epitome having BPC
We now illustrate the use of the algorithm for a multiple change-point problem.
We now apply a multiple change-point model to synthetic data. In order to apply the MMC algorithm we require a sample from the posterior distribution of the parameters and a function for evaluation
of the Kullback-Leibler distance between any two models. The sampler that we use is a Reversible Jump Markov Chain Monte Carlo sampler (Green, 1995) that was devised for sampling piecewise polynomial
models by Denison, Mallick, and Smith (1998). The sampler is simple, fast and relatively easy to implement. The sampler can make one of three possible transitions each iteration:
• Add a change-point
• Remove a change-point
• Move an existing change-point
We fit constants in each segment and use a Gaussian distribution to model the noise. However, rather than include the Gaussian parameters in the sampler we use the maximum likelihood estimates. This
means that the only parameters to be simulated are the number of change-points and their locations. Use of the maximum likelihood estimates required us to use a Poisson prior over the number of
change-points with
The Kullback-Leibler distance is easily calculated for the piecewise constant change-point model and has time complexity that is linear in the number of change-points.
The function that we have used in the evaluation is the ``blocks" function from (Donoho and Johnstone, 1994). The function is illustrated in Figure 2 and consists of eleven change-points over the
The main results for the small (3. This figure shows the data sample from the blocks function with 4. In the figures a change-point is marked using a vertical bar, and for each segment the mean and
one and two standard deviation bars are shown allowing a change in the mean or standard deviation estimates for a segment to be easily seen.
With such little data we do not expect the true blocks function to be accurately estimated. The point estimates for regions 1-9 are all reasonable models of the data and represent a good degree of
variety. The maximum posterior model closely fits the data in the second segment and would be expected to have a very large Kullback-Leibler distance from the true model. We see that none of the
point estimates in the MMC epitome make this mistake.
The point estimates for regions 1 and 7 contain the same number of change-points, yet region 1 is given 42 times the weight of region 7 (
The main results for the large (5. In this MMC epitome there were 13 regions. Regions 4-14 are shown in Figure 6. In these results we see that the maximum posterior estimate is quite reasonable but
lacks one of the change-points that exists in the true function. The point estimate for region 1 is able to detect this change-point and looks almost identical to the true function. The point
estimates for the other regions (2-13) look reasonable and tend to increase in detail. Some of them contain superfluous change-points. This does not damage their predictive ability or
Kullback-Leibler distance to the true model, but can be distracting for human comprehension. This problem is discussed further in Section 7.2.
For these examples we have found that the MMC algorithm can produce reasonable epitomes of a variable dimension posterior distribution. For both examples,
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What is
What is fixed and random effect model?
A fixed-effects model supports prediction about only the levels/categories of features used for training. A random-effects model, by contrast, allows predicting something about the population from
which the sample is drawn.
What is a fixed effects model quizlet?
Fixed effect model. A parameter associated with a specific unit in a panel data model.
What is fixed effects in regression?
Fixed effects is a statistical regression model in which the intercept of the regression model is allowed to vary freely across individuals or groups. It is often applied to panel data in order to
control for any individual-specific attributes that do not vary across time.
What is the difference between fixed and random factors?
Here are the differences: Fixed effect factor: Data has been gathered from all the levels of the factor that are of interest. Random effect factor: The factor has many possible levels, interest is in
all possible levels, but only a random sample of levels is included in the data.
What are fixed effects?
Fixed effects models remove omitted variable bias by measuring changes within groups across time, usually by including dummy variables for the missing or unknown characteristics.
Which of the following is a difference between a fixed effects estimator and a first difference estimator quizlet?
Which of the following is a difference between a fixed effects estimator and a first-difference estimator? The fixed effects estimator is more efficient than the first-difference estimator when the
idiosyncratic errors are serially uncorrelated.
What is a fixed effects model meta-analysis?
The fixed-effects model assumes that all studies included in a meta-analysis are estimating a single true underlying effect. If there is statistical heterogeneity among the effect sizes, then the
fixed-effects model is not appropriate.
What is a fixed effect factor?
Fixed effect factor: Data has been gathered from all the levels of the factor that are of interest. Example: The purpose of an experiment is to compare the effects of three specific dosages of a drug
on the response.
What are fixed factors?
Fixed factors are those that do not change as output is increased or decreased, and typically include premises such as offices and factories, and capital equipment such as machinery and computer
What is the difference between random and fixed effects?
Age-group of the person (Below 18,18-30,30-50,50-70,70-90)
Gender of the person (Female,Male)
Whether the person is having prior health problems related to hypertension (blood pressure),diabetes (sugar) etc.
Country of the person
When to use fixed effects?
Fixed income investors are faced with a murky market outlook, with inflation, monetary policy and the pandemic all having differing effects across the yield curve. ETF.com surveyed several
The fixed effect assumption is that the individual-specific effects are correlated with the independent variables. If the random effects assumption holds, the random effects estimator is more
efficient than the fixed effects estimator. However, if this assumption does not hold, the random effects estimator is not consistent.
When should you use random effects model?
The random-effects model should be considered when it cannot be assumed that true homogeneity exists. Similarly, a fourth criterion refers to the likelihood of a common effect size. In fixed-effects
models, we assume that there is one common effect. | {"url":"https://musicofdavidbowie.com/what-is-fixed-and-random-effect-model/","timestamp":"2024-11-05T06:25:58Z","content_type":"text/html","content_length":"46685","record_id":"<urn:uuid:f59ea2c0-fee5-43d8-926c-b087954af6b5>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00807.warc.gz"} |
Kotze-Pereira transformation - electowikiKotze-Pereira transformationKotze-Pereira transformation
Visual representation of the KP-Tansform
The Kotze-Pereira transformation (KP transform) converts scored ballots into Approval ballots, which allows any Approval PR method to be run on scored ballots. Because the score winner will always
have the most approvals after the transformation, a PR method that elects the approval winner in the single-winner case will also elect the score winner in the single-winner case when converted to a
score method using the transformation. Score methods using this transformation are also generally satisfy Scale invariance (multiplying all score by a constant leaves the result unaffected), except
when the change in score causes differences in surplus handling due to quotas being met or not met. The transformation was independently invented by Kotze in 2011^[1] and Toby Pereira in 2015.^[2] It
was named by Forest Simmons in 2015.^[3]
A voter whose scores are (with the max score being 10) A=10 B=6 C=4, would have their 1 ballot transformed into 0.4 ABC, 0.2 AB, and 0.4 A Approval ballots. This is because the lowest score they gave
to any candidate is a 4 out of 10, which is 40% support, so a corresponding 40% portion of their ballot is treated as approving all candidates scored a 4/10 or higher (Candidates A, B, and C); the
next-lowest score they gave was a 6/10, 60% support, but because 40% of the ballot was already converted into Approval ballots, only the remaining 60% - 40% = 20% portion is converted, and all
candidates scored a 6 out of 10 or higher (Candidates A and B) are treated as approved in this 20% portion; finally, within the remaining 40% portion of the ballot, the next-lowest score the voter
gave is a 10/10, so all candidates scored a 10 or higher (Candidate A) are considered approved on this portion of the ballot.
To avoid having fractional approval ballots, some suggest that the KP transform should be done in such a way that one voter's score ballot always produces the smallest number of approval ballots such
that they all are integer amounts; with the above example, this would mean multiplying the number of Approval ballots in each set by 5, yielding 2 ABC, 1 AB, and 2 A Approval ballots.
The formal definition
Replace any ballot which rates the C candidates with scores S1≥S2≥S3≥...≥SC by these C weighted approval (meaning with {0,1}-scores only) ballots
(1,1,1,...,1,1) with weight SC
(1,1,1,...,1,0) with weight SC-1-SC
(1,1,0,...,0,0) with weight S2-S3
(1,0,0,...,0,0) with weight S1-S2
Note: the candidates were ordered by decreasing scores on the ballot under consideration. That assures that all the weights come out positive. For example, the score ballot (9,5,3) in a
three-candidate election would be replaced by 3×(1,1,1) + 2×(1,1,0) + 4×(1,0,0).
A "ballot with weight w" is to be interpreted the same as "w voters cast that ballot." This transform converts scores into approvals so that any method that uses approval ballots can be converted to
a method that uses score ballots without having to individually define how to do so for each method.
Note that the Approval ballots yielded by the KP transform can be converted into ranked ballots by considering all approved candidates on a ballot as ranked co-1st, and all disapproved candidates as
ranked co-equal last. With the above example of 0.4 ABC, 0.2 AB, and 0.4 A Approval ballots, this would converted into 0.4 A=B=C, 0.2 A=B(>C), and 0.4 A(>B=C) ranked ballots. This allows ranked PR
methods to be done on rated ballots.
Whole Ballot formulation
In the whole ballot formulation there is never any fractional approval ballots. All score ballots are turned into the same number of whole approval ballots. The number of ballots is the same as the
maximum score.
Example Ballot
An illustrative ballot for a max score 5 system is. A:1 B:3 C:4 D:0
The KP-Transform turns this into 5 approval ballots.
Ballot A B C D
Python Implementation
The whole ballot formulation is particularly simple computationally. Given a Pandas dataframe S with columns representing candidates and rows representing voters the entries would encode the score of
all the ballots. For a max score of K and an output set of approval ballots A.
import pandas as pd
import numpy as np
groups = []
for threshold in range(K):
groups.append(np.where(S.values > threshold, 1, 0))
A = pd.DataFrame(np.concatenate(groups), columns=S.columns)
Scale Invariance example for RRV
Reweighted Range Voting is one example of a method where the Kotze-Pereira Transformation will provide Scale invariance. This example will use Jefferson RRV because it's marginally easier to work
with, but it's basically the same with any divisors.
Let's say we have 5 candidates to elect and there are multiple A and B candidates, and we have the following approval ballots.
200 voters: A
100 voters: B
In this case A would win 3, B would win 1 out of the first 4, and there would be a tie for the final seat. After the initial 4 have been elected, the weighted approvals for A would be 200/4 = 50, and
for B it would be 100/2 = 50. So a tie. But now let's move onto score voting (out of 10)
200 voters: A1=10, A2=10, A3=9, A4=9
100 voters: B1=10, B2=9
A1, A2, and B1 would be elected as the first three. This gives us the proportional 2:1 ratio and this is the same as approval currently (because they had max scores). The A votes are now worth 1/3,
and the B votes worth 1/2. A3 would be the next elected. But because of the score of 9, there isn't a full deweighting. Each A vote would now be worth 1/(1 + 29/10) = 10/39 or about 0.256. Both A4
and B2 have been given a score of 9 by their voters. The total score for A4 is 200 x 0.9 x 10/39 = 46.15. For B2 it is 100 x 0.9 x 0.5 = 45. So A4 would be elected instead of there being a tie. OK,
so it's only a tie being broken, but if there were 101 B voters, then B2 should win the final seat but 101 x 0.9 x 0.5 = 45.45, so it wouldn't be enough to take the final seat. The wrong candidate
would win the final seat.
With KP, the initial ballots would be transformed to:
200 ballots: A1, A2
1800 ballots: A1, A2, A3, A4
100 ballots: B1
900 ballots: B1, B2
Imagine A1, A2, B1, A3 are already elected. The new weighted total approvals for A4 would be 1800 x 1/4 = 450. For B2 it would be 900 x 1/2 = 450. This is the exact tie we want. So Scale invariance
is preserved.
Using the "ranked KP transform" on Score ballots (converting them into Approval ballots which are then converted into ranked ballots, with approved candidates ranked 1st and all others last) and
running this through Smith-efficient Condorcet methods yields a Smith set with only the candidates who originally had the most points i.e. the Score voting winner.
Something similar to the KP transform can be done using randomness: if a voter approves a candidate with a probability proportional to their utility from that candidate, then with probability
approaching 1 with many voters, the candidate will have the same approval rating as they would if every voter had simply scored that candidate.
One way to visualize the KP transform is as follows: imagine that for each voter, 9 additional voters are added to the election, whose ballots are treated as "under the control of" that voter. If the
voter decided to make 8 of the 10 ballots under their control approve their favorite candidate, while not doing anything with the remaining 2, then this would be equivalent to them giving that
candidate an 8 out of 10 on a rated ballot. Thus, the KP transform helps with scale invariance.
The KP transform often improves or at least doesn't worsen a voting method that it is applied to, but this isn't always the case. For example, SMV depends on being able to spend an entire ballot even
if it didn't give full support to the winner.
The connection that the KP transform shows between Approval and Score ballots can most clearly be seen when the Score ballots are set to a scale of 0 to 1 (with in-between decimals allowed), because
a voter who gives a middling score to a candidate is seen to be giving them a fractional approval.
The online community SimDemocracy uses a form of KP-transform with SPAV (Jefferson), called "SPSV" (Sequential Proportional Score Voting) for its parliamentary elections^[4]. A custom build voting
tool is used to hold elections and requires a reddit account^[5].
Further Reading | {"url":"https://electowiki.org/wiki/KP_transform","timestamp":"2024-11-09T20:35:52Z","content_type":"text/html","content_length":"67579","record_id":"<urn:uuid:c59e8da1-a000-4daa-9cab-770dd910f77e>","cc-path":"CC-MAIN-2024-46/segments/1730477028142.18/warc/CC-MAIN-20241109182954-20241109212954-00372.warc.gz"} |
mimi tsuruga
I started writing a post about Homology Algorithms but realized that some discussion about homology should come first. And for those of you who already know what homology is, we should at least be on
the same page in terms of language and notation. Besides, there’s more than one way to think about homology.
The way that I understand homology—the way I describe it here—is influenced almost entirely by Carsten Lange. He helped me understand the basics of algebraic topology so that I can read about
discrete Morse theory without feeling like a total oaf. This also means that we are thinking about homology with the eventual goal of computing (i.e., with a computer) the homology of some explicit,
finite discrete topological space. In particular, we will work with simplicial complexes.
References used include the classics, Munkres and Hatcher, and for a speedy introduction, Zomorodian’s “Topology for Computing”. (And, as usual, a ton of help has been provided by my advisors Frank
and Bruno.)
And before I begin, let me just say: Big ups to Carsten and all dedicated educators like him who, for countless hours, over many days and weeks, sit together with students one-on-one until they have
the confidence to seek knowledge on their own.
Homology in topology
The big question in topology is whether or not two spaces are homeomorphic. To say that two spaces are homeomorphic, we need only find any homeomorphism between them. But to say that two spaces are
not homeomorphic, we would have to show that no function between them is a homeomorphism. One way to handle this latter (infinite) case is by using topological invariants.
A topological invariant is a property of a space that can be used to distinguish topological spaces. For example, say we have two sacks labeled $A$ and $B$. We can’t see inside them, but we can tell
that sack $A$ has one item in it while sack $B$ has two. Clearly the contents of sack $A$ and sack $B$ cannot be the same.
However, if $A$ and $B$ both have one item each, we can’t tell whether those two items are of the same type. Perhaps one is a steak and the other a rice crispy treat—clearly very different objects,
both nutritionally and topologically. (Can you tell I missed lunch?)
The topological invariant described in these examples is connectedness, which is only useful when the number of connected components is different. But if they have the same number of connected
components, we can’t say definitively whether the objects are of the same topological type. And that’s how it works with all topological invariants.
Definition (topological invariant) Given two topological spaces $K_1$ and $K_2$, a map $\tau$ is a topological invariant if whenever $\tau(K_1) \neq \tau(K_2)$, then $K_1 \ncong K_2$. But if $\tau
(K_1) = \tau(K_2)$, then $K_1 \overset{?}{\cong} K_2$ and we don’t know whether they are homeomorphic or not.
Homology groups are topological invariants. We take some geometric information about the topological space, form some algebraic object (specifically, a group) containing that information, then apply
some machinery from group theory to say something about the original space.
And if two topological spaces give rise to homology groups that are not isomorphic, then those two spaces are not homeomorphic.
Definition of Homology
Let’s jump right in with a definition, then explain it in detail using an example.
Definition (homology group) For a topological space $K$, we associate a chain complex $\mathcal{C}(K)$ that encodes information about $K$. This $\mathcal{C}(K)$ is a sequence of free abelian groups
$C_p$ connected by homomorphisms $\partial_p: C_p \to C_{p-1}$. \begin{align*}0 \overset{\partial_{d+1}}{\longrightarrow} C_d \overset{\partial_{d}}{\longrightarrow} C_{d-1} \overset{\partial_{d-1}}
{\longrightarrow} \cdots \overset{\partial_{2}}{\longrightarrow} C_1 \overset{\partial_{1}}{\longrightarrow} C_0 \overset{\partial_{0}}{\longrightarrow} 0\end{align*} Then the $p$-th homology group
is defined as \begin{align*}H_p(K):= \text{ker}(\partial_p) / \text{im}(\partial_{p+1}).\end{align*}
That’s a lot of terms we have yet to define.
Let’s start at the beginning with our input space $K$. We can assume that $K$ is a simplicial complex since we’ll only be dealing with simplicial complexes and they’re just nice to handle. So $K$ is
a simplicial complex of dimension $d$. This means that for any face $\sigma \in K$, its dimension is at most $|\sigma| \le d$.
Figure 1: Simplicial complex $K$ with vertices labeled 1,2,3,4.
In Figure 1, we have our simplicial complex $K$. I’ve labeled the vertices so it’s easier for me to refer to the many parts of $K$. Our complex $K$ consists of
• one 2-cell: $f_1 = [2 \, 3\, 4]$;
• five 1-cells: $e_1=[1 \, 2], e_2=[1 \, 4], e_3=[2 \, 3], e_4=[2 \, 4], e_5=[3 \, 4]$; and
• four 0-cells: $v_1=[1], v_2=[2], v_3=[3], v_4=[4]$.
This complex $K$ is of dimension $d=2$.
The group $C_p$ is a group of something called $p$-chains. Recall that a group is a set with a binary operation (in our case, addition) that is (i) associative, (ii) has an identity element, and
(iii) has inverses. And when we refer to any group, instead of listing all the elements of the group all the time, it’s much more economical to talk about groups with respect to its generators. For
the generators of $C_p$, we have to go back to $K$.
The $p$ in $C_p$ refers to the dimension of cells in $K$. A 1-chain of $K$, for example, is some strange abstract object that is generated by all the 1-cells of $K$, that is, we “add” up a bunch of
1-cells. So $e_1 + 2 e_5$ is an example of a 1-chain. It has no obvious geometric meaning (though, apparently, some people would disagree). If it helps, you can think of $C_p$ as a vectorspace whose
basis is the set of $p$-cells of $K$ and the scalars are in $\mathbb{Z}$.
Figure 2: Relabeled and oriented.
Notice that the labeling of the vertices from Figure 1 gives us a nice built-in orientation, i.e., ordering, for each cell, see Figure 2. For example, edge $e_1 = [1 \, 2]$ so the orientation we give
this edge is from $v_1$ to $v_2$ (because $1<2$). This orientation will help us define the additive inverse of a 1-cell. So $-e_1$ is what you would expect it to be; it’s just $e_1$ oriented in the
opposite direction, from $v_2$ to $v_1$.
Now the definition states that $C_p$ is not just any old group. It is a free Abelian group. This is not as bad as it sounds. It just means, again, that we can just go back to thinking of $C_p$ like
it’s a vectorspace.
So $C_p$ is free, which just means that every $p$-chain in $C_p$ can be uniquely written as $\sum_i n_i \sigma_i^p$, where $n_i \in \mathbb{Z}$ and $\sigma_p$ are $p$-cells. This idea should be
deeply ingrained into your understanding of basis vectors in a vectorspace.
And $C_p$ is also Abelian. So our addition is commutative $\sigma + \tau = \tau + \sigma$ and that also let’s us keep thinking about this vectorspace idea.
So now that we know what the $C_p$ are, let’s talk about the homomorphisms $\partial_p$ between them. These homomorphisms have a special name; they are called boundary maps. This is where the
geometry and combinatorics fit together very nicely. Let’s go back to our Figure 2.
The boundary of some $p$-cell $\sigma^p=[v_0 \, v_1 \, \dots \, v_p]$ is a $(p-1)$-chain with the alternating sum $$\partial_p(\sigma) = \sum_{i=0}^{p} (-1)^i [v_0 \, v_1 \, \cdots \, \hat{v_i} \, \
cdots \, v_p ]$$ where $\hat{v_i}$ means you leave that vertex out.
For example, $\partial_1(e_1) = v_2 – v_1$ or $\partial_2(f_1) = \partial_2([2 \, 3 \, 4]) = [3 \, 4] – [2 \, 4] + [2 \, 3] = e_3 + e_5 – e_4$. This corresponds nicely with the figure. Just follow
along the blue arrows on the boundary of $f_1$ from $v_2$ in a counter clockwise direction.
The boundary function works similarly for all dimensions $p$. For the higher dimensions, just keep in mind what I said earlier about orientation being determined by the labels. The orientation for an
edge is easy to visualize, but not so much for, say, a 6-simplex. Everything works out nicely combinatorially with the above definition.
And because $C_p$ is a free abelian group, we see that the boundary map $\partial_p$ is a homomorphism. Notice also that $\partial_p \circ \partial_{p+1} = 0$. Whenever this happens (to any sequence
of Abelian groups with homomorphisms between them with this property), we call such a sequence a chain complex.
Geometry of homology
Let’s introduce a few more important terms.
The set $B_p = \text{im}(\partial_{p+1})$ is the set of boundaries and the set $Z_p = \text{ker}(\partial_p)$ is the set of cycles (which, in German, is “Zyklus”).
So a cycle is any $p$-chain that sums up to zero. This is a very combinatorial/algebraic definition for a word that sounds very geometric. Intuitively, when you hear the word “cycle”, you might think
of some path that closes, maybe like a loop. It’s very important, however, to recognize that “cycle” and “loop” (from homotopy theory) are very very different objects.
Let me go off on a short tangent here. Have you noticed that the word “curve” is different depending on who you talk to? For a physicist, a curve is a trajectory–a function $\gamma: [0,1] \to X$ for
some space $X$. For a geometer, however, a curve is a subset of a space. Merely the image $\text{im}(\gamma)$. So it doesn’t matter whether a mosquito flies around in a circle 7 times or just once;
it still traces just a single circle.
Analogously, a cycle doesn’t care about how you “go around the cycle” as long as the algebraic sum of the corresponding $(p-1)$-chains in the cycle adds up to zero.
And remember: since we have an Abelian group, the order of the addition (or “direction” of the cycle) doesn’t matter at all. For a loop, however, it matters whether you go around the loop clockwise
or counterclockwise.
Let’s look again at the composition $\partial_p \circ \partial_{p+1} = 0$. It says, geometrically, that the boundary of $(p+1)$-chains are cycles. So the set of boundaries $B_p$ is contained in the
set of cycles $Z_p$, i.e., $B_p \subseteq Z_p$.
Not only that, because $\partial_{p+1}$ is a homomorphism, $B_p$ is even a subgroup of $Z_p$. Geometrically speaking, the quotient group $H_p = Z_p / B_p$ means we partition the set of cycles $Z_p$
to those that bound and those that don’t. This means we can isolate cycles that do not bound cells. That is, we can find holes!
Figure 2: Relabeled and oriented.
Let’s go back to our example and think about $H_1 (K)$. The “simplest” 1-chains that make a loop are $e_1 + e_4 – e_2 = \ell_1$ and $e_3 + e_5 – e_4=\ell_2$. All other loops are some linear
combination of $\ell_1, \ell_2$. For example $e_1 + e_3 + e_5 – e_2 + (e_4 – e_4) = \ell_1 + \ell_2$. So $Z_1$ is generated by $\ell_1, \ell_2$, i.e., $Z_1 = \{c\, \ell_1 + d\, \ell_2 \mid c,d \in \
mathbb{Z} \} \cong \mathbb{Z}^2$. But we also know that $\ell_2 = \partial_{2} (f_1)$ and since $f_1$ is the only 2-cell in $K$, we have $C_2 = \{ c \,f_1 \mid c \in \mathbb{Z} \}$ and therefore $B_1
= \text{im}(\partial_{2}) = \{ c \, \ell_2 \mid c \in \mathbb{Z}\} \cong \mathbb{Z}$.
So $H_1 = Z_1/B_1 \cong \mathbb{Z}^2 / \mathbb{Z} \cong \mathbb{Z}$.
Using homology
Now that the homology group has been defined, we should know what it means. There is an important theorem in group theory that we should recall at this point. It’s so important that it is a
fundamental theorem.
Theorem (Fundamental Theorem of Finitely Generated Abelian Groups) Every finitely generated Abelian group $G$ is isomorphic to a direct product \begin{align*}G \cong H \times T, \end{align*} where $H
= \mathbb{Z} \times \cdots \times \mathbb{Z} = \mathbb{Z}^{\beta}$ is a free Abelian subgroup of $G$ having finite rank $\beta$ and $T = \mathbb{Z}_{t_1} \times \mathbb{Z}_{t_2} \times \cdots \times
\mathbb{Z}_{t_k}$ consists of finite cyclic groups of rank $t_1, \dots, t_k$.
The $T$ is called the torsion subgroup of $G$; the $\beta$ is unique and called the Betti number of $G$; and the $t_1, \dots, t_k$ are also unique and called the torsion coefficients of $G$.
This fundamental theorem says that the homology group $H_p$ (which is a quotient group of a finitely generated Abelian group over a finitely generated Abelian group and therefore itself a finitely
generated Abelian group) will always be isomorphic to something that looks like \begin{align*}H_p \cong \mathbb{Z}_{t_1} \times \cdots \times \mathbb{Z}_{t_k} \times \mathbb{Z}^{\beta}\end{align*} a
finite direct sum of (finite and infinite) cyclic groups. So now all topological spaces can be nicely categorized by their homology groups according to their Betti numbers and torsion coefficients.
The homology groups $H_p$ are of dimension $p$ and each of them have an associated Betti number $\beta_p$. Since we figured out for our example that $H_1 (K) \cong \mathbb{Z} = \mathbb{Z}^1$, our
Betti number $\beta_1 = 1$.
These Betti numbers have a wonderful geometric meaning. This number $\beta_p$ counts the number of $p$-dimensional holes in $K$. Ah, yes. We finally get to something meaningful and even useful!
Betti numbers can count, as in our example, the number of holes.
Since $\beta_1(K)=1$, this means $K$ has one 1-dimensional hole. And there is indeed a hole in $K$ (because there is no face $[1 \, 2 \, 4]$).
But what is a “hole”?
Honestly, I don’t know how to draw a good picture of a general $p$-dimensional hole (though examples of specific cases are easy). Thinking of holes like in homotopy theory–as the empty spaces where
$p$-spheres $S^p$ cannot be shrunk down to a point–is a very nice and intuitive notion. But there are some spaces that, like the Poincare sphere for example, have $H_1 = 0$ and $\pi_1 \neq 0$. So
it’s not enough to think about holes as the holes we see when working with fundamental groups.
We saw from the definition that homology groups are equivalence classes of cycles, of those that bound and those that don’t. So we collect all the “simplest” cycles and check whether they are the
boundary of any faces. Any cycle that does not have a corresponding face which it bounds must bound a hole.
It turns out that counting holes has all kinds of industry applications. It can be used to analyze the structure of solid materials or proteins. It can be used in image processing or pattern
recognition. It can even be used in robotics and sensor networks.
So homology algorithms have been developed to help these industry people to count holes. And now we know exactly how to write such an algorithm, right?
The significance of adipra
I realize that anyone reading this post doesn’t know what adipra is. How can you? I made it up.
The sphere I constructed most recently was an idea of Karim Adiprasito. While building it, I needed to name it so that I can refer to it in my code. So I temporarily named it adipra and that’s what
I’m going to call it here.
Actually, in this post, I don’t want to talk about what adipra is. Not yet. Let’s talk just about what makes adipra special. For anyone who’s interested, all the specifics can be found here.
We need a little language to get started.
Def A combinatorial d-manifold is a triangulated d-manifold whose vertex links are PL spheres.
Let’s assume we know what a triangulated d-manifold means without hashing out the details here. If you need to, you can think of it as a simplicial complex.
The star of a vertex v in a triangulated d-manifold T is the collection of facets of T that contain v. The link of a vertex v is like the star of v, but take out all the v‘s. For example, let’s take
a simple hexagon with a vertex in the center.
Example of star and link of vertex
Let T=$\{[0\, 1 \,2],[0\, 1\, 6],[0\, 2\, 3],[0\, 3\, 4],[0\, 4\, 5],[0\, 5\, 6]\}$, then \begin{align*}\text{star}(3,T) &=\{[0\, 2 \,3],[0\, 3\, 4]\} \text{ (green triangles)}\\ \text{link}(3,T)&=\
{[0\, 2],[0\, 4]\} \text{ (red edges)} \end{align*}
That’s simple enough to understand even in higher dimensions. Just remember we’re always doing things combinatorially.
Next is the PL sphere. I said earlier that we can assume we know what a triangulated d-manifold is. What we’re actually talking about is a triangulated PL d-manifold. A PL-sphere is a PL manifold
that is bistellarly equivalent to the boundary of a d-simplex. I’ll write in more detail what bistellarly equivalent means in a later post. For now, just think of it as the discrete version of being
Putting all that together, we understand a combinatorial d-manifold to be a triangulated manifold, that is, some simplicial complex-like thing where we require that each of its verticies is sort of
covered by a ball. Naturally, the next question to ask is: are there triangulated manifolds that are not combinatorial?
For d=2,3, all triangulations (of the d-sphere) are combinatorial. For d=2, the vertex links should be homeomorphic to $S^1$ or bistellarly equivalent to the triangle (boundary of a 2-simplex).
Similarly, for d=3, vertex links are $S^2$. For d=4, all triangulated 4-manifolds are also combinatorial. This result is due to Perelman. The vertex links are 3-spheres, which, as you know, is what
Perelman worked on. [Remember that PL=DIFF in dim 4.] But it falls apart for d$\ge 5$ as there are non-PL triangulations of the d-sphere.
Ok, so here’s a spoiler about adipra: it’s a non-PL triangulation of the 5-sphere. But that’s not all.
There’s a nice theorem by Robin Forman, the father of discrete Morse theory.
Theorem (Forman) Every combinatorial d-manifold that admits a discrete Morse function with exactly two critical cells is a combinatorial d-sphere.
Actually, this is not really Forman’s theorem. This is a theorem by Whitehead which Forman reformulated using his language of discrete Morse theory. This is the original theorem.
Theorem (Whitehead) Any collapsible combinatorial d-manifold is a combinatorial d-ball.
What Forman did is take a sphere, take out one cell, call that guy a critical cell, then collapse the rest of it down (using Whitehead’s theorem) to get the other critical cell. Thus you have two
critical cells.
So the question you ask next is: can you have non-combinatorial spheres with 2 critical cells?
Yes, actually, you can! Karim showed that you can have a non-PL/non-combinatorial triangulation of the 5-sphere that has a discrete Morse function with exactly 2 critical cells! And then I built an
explicit example (with Karim’s instructions) and called it adipra.
KolKom 2012
My first conference talk since starting Phase II was at KolKom 2012, the colloquium on combinatorics in Berlin. But I don’t know anything about combinatorics or optimization. What am I doing at a
combinatorics conference!?
Actually, I don’t know much about anything. But it’s not hard to make me sound like I know about some things. The title of my talk was “constructing complicated spheres as test examples for homology
algorithms”. Yes, algorithms! That’s the key! It was Frank’s idea to add the word “algorithms” to my title so that I can at least pretend to belong to this conference. It made for a very long title,
but at least my abstract was accepted.
You can look at my slides, but I’ll give a short overview here.
We begin with a definition of homology and motivate some applications for homology algorithms. Betti numbers, in particular, are interesting for industrial applications. To compute the Betti numbers
of some given simplicial complex, we will be dealing with matrices. We want to find the Smith Normal Form of these matrices. But the bigger the complex, the bigger the matrix, and the longer it takes
to find the Smith normal form. So we throw the complex into a preprocess to reduce the size of the input. This preprocess uses discrete Morse theory.
Morse theory is used to distinguish topological spaces in differential topology. Forman came up with a nice discrete counterpart that conveniently preserves many of the main results from smooth Morse
theory including the Morse inequalities. The Morse inequality $\beta_i \le c_i \; \forall i$ means that the Betti number $\beta_i$ is bounded above by $c_i$, the number of critical points of
dimension $i$. Since our original goal was to compute Betti numbers, this result will definitely come in handy.
The critical points $c_i$ are the critical points of some discrete Morse function. These discrete Morse functions can be interpreted as a sort of collapse. By collapsing big complexes, we can reduce
big complexes (that have big matrices for which we need to find the Smith normal form) to smaller complexes.
Ideally, we want to find the discrete Morse function with the lowest $c_i$’s. But computing the optimal discrete Morse vector, a vector $c = (c_0,c_1,c_2, \dots)$, is $\mathcal{NP}$-hard! So Lutz and
Benedetti have come up with the idea of using random discrete Morse theory. Surprisingly, their algorithm is able to find the optimal discrete Morse vector most (but not all) of the time. To get a
better idea of when they’re better able to find the optimum, they came up with a complicatedness factor which measures how hard it is to find that optimum.
That’s where I come in. I constructed some complicated simplicial complexes. They are being used to test some homology algorithms that are currently under development.
The first of my complicated complexes is the Akbulut-Kirby sphere, one of a family of Cappell-Shaneson spheres. These spheres have an interesting history. The AK-sphere, in particular, was one
thought to be an exotic sphere. Exotic spheres are spheres that are homeomorphic, but not diffeomorphic to a standard sphere. In dimension $4$, this would mean it could be a counterexample to the
smooth Poincar\’e conjecture, or SPC4 for short. Unfortunately, some years after the AK-sphere was proposed, it was shown to be standard after all. You can learn more about it here.
So I built an AK-sphere. Actually, I wrote code that can build any of the whole family of the CS-spheres. Before talking about what we did, let’s start with how these spheres are built. To understand
the construction, you have to first accept two simple ideas:
• The boundary of a solid ball is a sphere.
That’s not too hard to accept, which is why I said it first even though it’s the last step.
• Given a donut, if you fill the “hole” (to make it whole, HA!), it will become a solid ball.
That’s also not too difficult to see. You can imagine the filling to be something like a solid cylinder that you “glue” onto the inside part of the donut.
Now for some language. This donut is actually a ball with a handle, which is known as a 1-handle. The filling for the hole is called a 2-handle. And we can specify how the 2-handle is glued in using
an attaching map.
So to build the AK-sphere here’s what we do:
1. Take a 5-ball.
2. Attach two 1-handles. Let’s call one of them $x$ and the other $y$ so we can tell them apart.
3. Glue in two 2-handles to close the “holes” but use attaching maps that have the following presentation: \begin{align*} xyx&= yxy & x^r = y^{r-1}\end{align*}This means that the attaching map goes
(on the boundary of the 5-ball) along $x$ first, then $y$, then $x$, then $y$ in the reverse direction, then $x$ in reverse, and finally $y$ in reverse. Similarly with $x^r = y^{r-1}$. [Note: For
the AK-sphere, $r = 5$. By varying $r$ you get the family of CS-spheres.]
4. Take its boundary.
Since the 2-handles have closed the holes created by the 1-handles, what we have left after step 3 is again a 5-ball, though a bit twisted. So its boundary in step 4 should be a sphere.
To construct these spheres, we decided to go in a different order. It would be hard to imagine the attaching maps in dimension 5. So we went down a smidge to dimension 3. And instead of drawing the
paths of the attaching maps on/in a ball, we built the ball around the paths.
As you might imagine, these paths come out a bit knotted. But by bringing them up in dimension, we can “untangle” it because there’s more space in higher dimensions. The way that we go up in
dimension is by “crossing by $I$” where $I$ is the usual unit interval. Crossing by $I$ has the convenient side effect that everything you “crossed” ends up on the boundary of the resulting complex.
So we start with these two paths (some triangulated torii). We fill in the space between them with cleverly placed tetrahedra so that it ends up being a ball with two 1-handles. We now have a genus 2
simplicial complex, where two sets of tetrahedra are labelled as the path representing $xyx = yxy$ and the other $x^r = x^{r-1}$. Cross $I$, cross $I$ so that we are in dimension 5 with the two paths
on the boundary of this ball with two 1-handles. Glue in the (5-dimensional) 2-handles (whatever that means). Take the boundary. Done!
The experiments we’ve run on them so far have shown some promise. The complexes are not too large (with f-vector$=(496,7990,27020,32540,13016)$) so the experiments don’t take too long to run. They’ve
already been used by Konstantin Mischaikow and Vidit Nanda to improve their on-going Perseus algorithm. [edit: In the original post, I mistakenly referenced CHomP, which has been static for some
The other complex I constructed is Mazur’s 4- manifold. We do a similar thing, but this time there’s only one path to deal with (that is, only one 1-handle to close up with one 2-handle). The results
from this one has some nice topological properties. I’ll get to that in a later post once we’re ready to publish. | {"url":"https://matsguru.com/?paged=2&page_id=85","timestamp":"2024-11-06T15:43:43Z","content_type":"text/html","content_length":"54650","record_id":"<urn:uuid:d18375e4-4419-4900-aec1-eb45b0960d51>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00314.warc.gz"} |
crevice volume
`Vol_c = "rD" * pi/4 * ( "bD" ^2 - "pD" ^2) `
Enter a value for all fields
The Piston Crevice Volume calculator computes the crevice volume (Vol[c]) of a Piston Ring Crevice combustion engine cylinder as a function the bore diameter(bD), piston diameter (pD) and the top
ring depth (rD).
INSTRUCTIONS: Choose your preferred units, the default is inches and mils, and enter the following:
Inch Equivalences
Fraction Decimal Mils
1/16^th 0.0625 62.5
1/32^th 0.03125 31.25
1/64^th 0.015625 15.625
Cylinder Crevice Volume: The volume is computed in cubic inches. However, the can be automatically converted to other volume units (e.g. cubic centimeters aka CCs) via the pull-down menu.
The Math
Engine Math (Baechtel) and I disagree on this equation. Engine Math takes the difference of the diameters (bore and piston), multiplies that difference by the circumference of the bore to create the
surface area of the crevice and then multiplies that surface area by the depth to the top ring to get the volume. The surface area of that equation is geometrically wrong. Fortunately these crevice
volumes don't affect much. But the mathematician in me can't let it sit. The correct formula for the crevice surface area is the difference between the two circle areas defined by the bore diameter
and the piston diameter, using PI*r^2 where r is half of the respective diameters. THEN multiply the correct surface area by the crevice depth from the deck to the top piston ring. I'm sure
Baechtel's answer is probably good enough, but this is more correct.
Ratios and Lengths
• Cylinder Bore Diameter: Computes the diameter (bore) based on the engine displacement, number of cylinders and the stroke length.
• Bore Stroke Ratio: Computes ratio based on the diameter of the bore and the length of the stroke.
• Combustion Ratio: Computes ratio base on the minimum and maximum displacements of the cylinder at the beginning (1-Induction) and compressed (3-Power) portions of the combustion cycle
• Displacement Ratio: Computes ratio based on the volumes at the beginning and end of the stroke.
• Rod and Stroke Length Ratio: Computes ratio base on the rod and stroke lengths.
• Stroke Length: Computes the required stroke length based on the total engine displacement, number of cylinders and the bore.
• Piston Position: Computes the piston position based on the crank angle, crank radius, and rod length.
• Piston Deck Height: Computes deck height based on Block Height, Rod Length, Stroke Length, and Pin Height.
Volumes and Displacements
• Total Volume (displacement) of a Combustion Engine: Computes volume based on the bore, stroke and number of cylinders.
• Volume (displacement) of a Engine Cylinder: Computes volume based on the bore and stroke.
• Volume (displacement) of an Engine with an Overbore: Computes volume based on the stroke, bore, overbore and number of cylinders.
• Equivalent Volume of a Rotary Engine: Estimates rotary engine volume based on the swept volume and number of pistons.
• Carburetor Air Flow: Estimates the volumetric flow of air through a carburetor based on a four-stroke engine's displacement, RPMs, and volumetric efficiency.
• Compressed Volume of a Cylinder: Compute volume when the piston is at the end of the stroke and the chamber is at its smallest (and most compressed) volume, based on the chamber, deck, crevice,
chamfer, gasket, valve relief and dome/dish volumes. This is the second volume (V2) in the Compression Ratio calculation.
• Volume of a Gasket: Computes gasket volume (displacement) based on the inner and outer diameters and the gasket's thickness.
• Volume of a Cylinder Deck: Computes the deck volume based on the deck height and the bore.
• Volume of a Cylinder Crevice: Computes crevice volume based on the piston diameter, cylinder bore and the crevice height.
• Volume of a Cylinder Chamfer: Computes chamfer volume based on the cylinder diameter and the chamfer height and width.
• Clearance Volume of a Piston: Computes the volume remaining in the combustion chamber when the piston is at its top dead center (TDC) which is the space above the piston crown when it's at its
highest point in the cylinder.
• Engine Compression Ratio: Computes the ratio of the volume of the combustion chamber with the piston at its bottom dead center (BDC) to the volume with the piston at its top dead center (TDC).
Speeds and RPMs
• Piston Speed (mean): Computes the mean (average) piston speed based on stroke length and RPMs.
• Max Piston Speed: Computes max speed based on stroke length and RPMs
• RPMs: Computes revolutions per minute based on piston speed and stroke length.
Other Automotive Calculators
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Change your browser options, then try again. | {"url":"https://www.vcalc.com/wiki/KurtHeckman/crevice-volume","timestamp":"2024-11-03T23:07:34Z","content_type":"text/html","content_length":"64340","record_id":"<urn:uuid:40fbce71-a9cd-4183-8f89-ebfe4db43d35>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00385.warc.gz"} |
Longdom Publishing SL | Open Access Journals
Research Article - (2014) Volume 3, Issue 6
Correlations of Remote Sensing Chlorophyll-a Data and Results of A Numerical Model of the Tropical and South Atlantic Ocean Circulation
^1MAG - Mar, Ambiente e Geologia, Rio de Janeiro, Brazil
^2Oceanographic Institute, University of Sao Paulo, Sao Paulo, Brazil
^3Institute of Astronomy, Geophysics and Atmospheric Sciences, University of Sao Paulo, Sao Paulo, Brazil
^*Corresponding Author:
Nair Emmanuela Da Silveira Pereira, MAG - Mar, Ambiente e Geologia, Rio de Janeiro, Brazil, Tel: 55-21-22537406/55-21- 982339612
The Tropical and South Atlantic Ocean are characterized by important large scale features that have seasonal character. The interactions between atmospheric and oceanic phenomena compose a complex
system where variations in physical parameters affect the distribution of primary production. Previous studies showed that the variability of physical parameters displays high values of
cross-correlation with chlorophyll-a, with strong dependence on latitude and variability in the biological response time. This study aims to correlate data of chlorophyll-a from MODIS with the
results of a hydrodynamic numerical model, in the period 2003 - 2009. The annual and semi-annual signals are predominant both in MODIS and model data but, even excluding these components, the
residual correlations are still high. On the other hand, annual and semi-annual signals have smaller standard deviation than the remaining (residual) frequencies. The cross-correlations between
chlorophyll-a and salinity, temperature and surface elevation showed spatial distribution patterns with well-defined latitudinal character, presenting higher modulus of correlation for temperature
and salinity, above +0.6 in the polar region and below -0.5 in the tropical area. A general pattern of negative correlations in the regions of low concentration and positive in regions of high
concentration was obtained, except the Equator (region of high chlorophyll concentration, which is characterized by a negative correlation for all variables, except the intensity of the currents).
The cross-correlations between chlorophyll and physical parameters corroborate the pattern found in the correlations considering lag zero, stressing aspects as the positive correlation with the
intensity of the currents in the equatorial region and the negative correlation with the surface elevation inside the South Atlantic Subtropical Gyre (SASG), both presenting immediate response. The
analysis of spatial distributions of the cross-covariance of Fourier spectra between chlorophyll and each of the physical variables, in the transect 20°W, showed that temperature and salinity
presented the best defined signals, especially in the periods of 3.5, 2.3, 0.7, and 1.7 years, with varying spatial distributions and time lags. These signals are found in the literature, being
associated with ENSO phenomena.
Keywords: Tropical; Circulation; Remote sensing
The use of remote sensing of ocean color in the study of biological phenomena has been increasingly frequent. Chlorophyll-a, for example, has a well-defined spectral response. These remote
measurements have shown good performance, as in study of Kampel et al. [1], where estimates of the concentrations of chlorophyll-a by remote sensing (SeaWiFS sensor algorithms), when compared to in
situ measurements in the southeastern Brazilian coastal region, showed good consistency.
The chlorophyll is present in all photosynthetic eukaryotic and cyanobacteria, and all photosystem pigments are capable of absorbing photons, however, only a specific pair of chlorophyll molecules
can utilize the energy of the photosynthetic reaction [2]. Primary production consists in fixing environmental carbon atoms by biological activity, whereas, in the marine environment, most of the
carbon is embedded in living tissues through photosynthesis [3].
According to Metsamaa et al. [4], the amount of phytoplankton, usually expressed as chlorophyll concentration, is one of the most important parameters in the description of water bodies, and
correlations between data from MODIS and in situ measurements have been high, as verified by Carder et al. [5]. Algorithms used in the inference of chlorophyll present relatively accurate results in
Case I waters (oceanic regions). However, often they fail in Case II waters (coastal regions), as shown in a study by Metsamaa et al. [4] with MODIS sensor data for the Baltic Sea Region.
The availability of nutrients, sunlight and temperature are the determinant factors in the development of phytoplankton in oceanic regions and physical processes (such as upwelling, turbulence and
subsidence) affect the transport and mixing of nutrients in the water columns [6]. Thus, variations in meteorological and oceanographic parameters influence the distribution of chlorophyll in the
ocean. As example, the upwelling of cold waters rich in nutrients, in some regions of the South American continental shelf, is mainly driven by seasonal winds. This is an important factor in the
platform circulation and its main impact is the increase of organic productivity, for example, in the Cabo Frio region [7].
Garcia et al. [8] demonstrated a good correlation between variations of chlorophyll and physical data, such as sea surface temperature and sea level, with a dominant annual signal. Another example is
found in the Black Sea region, with a 60% correlation between chlorophyll and surface temperature, so the understanding of this correlation is useful, for example, in studies about the impact of
climate changes on marine biota [9].
Chlorophyll is correlated with the sea level rise, suggesting that the increase in chlorophyll-concentrations would result from an uplift of the thermocline, which increases the supply of nutrients
to the surface [10]. As an example, negative anomalies in sea level (as those that occur in La Niña events or cyclonic eddies) are associated with the uplift of the isopicnals in the nutricline,
resulting an increase of chlorophyll-a [11]. The reverse occurs in El Niño and anticyclonic eddies and this relationship between sea level anomaly and chlorophyll-a depends on the spectral bands of
the time series analyzed, with the presence of signals related to El Niño - Southern Oscillation (ENSO).
Thus, the aim of this study is to determine patterns of spatial correlations between chlorophyll-a (measured by MODIS) and ocean physical parameters generated by numerical modeling (Princeton Ocean
Model - POM), discarding the influence of annual and semiannual signals, in the Tropical and South Atlantic Ocean.
Study area
The area under study in this project is the Tropical and South Atlantic Ocean, whose bathymetric distribution is shown in Figure 1. This region has a complex system of surface currents characterized
by the presence of an anticyclonic rotation forced mainly by wind [12,13] and strong associated features, as the Retroflection of the Agulhas Current [13] and the confluence of the Brazil and
Malvinas Currents along the South America continental shelf, forming a complex pattern of meanders and eddies [14]. In the Tropical and South Atlantic there is a large seasonal variability of surface
currents, upwelling regions, temperature and salinity [15], which can be related to variations in atmospheric forcings that also exhibit significant seasonal character [16,17]. Such variabilities may
induce variations of biological responses, which make its study highly relevant.
Pereira et al. [18] observed distributions of chlorophyll-a and net primary productivity (PPL) in the region (from MODIS sensor data), detecting high concentrations in coastal regions, especially
near the mouths of great rivers and regions of upwelling or confluence of currents, with marked seasonal variation. That study showed physical variables having high cross-correlation with
chlorophyll-a and PPL, but with strong latitudinal dependence and variability in the timing of the biological response, which is greatly influenced by the high seasonality of the physical parameters.
Data and Methods
Biological data
Presently, there is a range of sensors coupled to artificial satellites which are used for studies of terrestrial, atmospheric and ocean phenomena. One of these is the Moderate Resolution Imaging
Spectroradiometer (MODIS), which is a key instrument coupled to artificial satellites Terra (EOS AM) and Aqua (EOS PM), of the National Aeronautics and Space Administration (NASA). These satellites
form a complete coverage of the globe over a period of two days, capturing data in 36 spectral bands at moderate resolution (between 0.25 and 1 km) [19]. Biological data used in this work are
horizontal distributions of the concentration of chlorophyll-a based on the lognormal distribution algorithm described by Campbell et al. [20]. These data are derived from remote sensing through
MODIS with spatial resolution of 0.5 degrees in latitude and longitude, and temporal resolution of 8 days. Data for the entire period from 2003 to 2009 were used in this study (7 years), with
interpolations for missing data due to cloud cover. These data are the product of level 3, provided by Oregon State University (OSU).
The numerical model
The hydrodynamic model used is a version of Princeton Ocean Model (POM) written by Blumberg et al. [21], optimized by Harari et al. [22]. This implementation uses high-resolution grids for the
Tropical and South Atlantic and Brazilian Continental Shelf, considering simulations and predictions of circulations generated by tides, winds and density variations. This version has been used for
scientific and operational purposes, allowing the reproduction of the hydrodynamics in this area and any subdomain, through nested grids, especially in coastal and continental shelf [22,23].
The POM is a three-dimensional model that considers free surface and solves a set of three-dimensional nonlinear primitive equations of motion, discretized by the finite difference method and
considering modes separation. It is through this modes separation that volume transport (external mode) and velocity shear in the vertical (internal modes) are solved separately, saving computation
The complete hydrodynamic equations are written in the flux form, considering Boussinesq and hydrostatic approximations. The model also adopts a second order turbulent closure for coefficients of
vertical viscosity and diffusion; Smagorinsky parameterization for horizontal viscosity and diffusion; leapfrog scheme for time and horizontal space integration, and an implicit scheme for the
vertical integration. For the spatial differentiation, the model uses an alternating Arakawa C-type grid, suitable for high-resolution models (spacing less than 50 km).
The model was processed for 31 years (from 1979 to 2009), the first year being discarded (to avoid the influence of initial conditions at rest). The horizontal grid resolution was set to 0.5° in
longitude and latitude, and vertical resolution of 22 sigma (σ) levels of varying thickness, with the first 8 levels corresponding to 10% of the total depth.
The tidal potential was included in all runs and the results of the model were filtered each time step, averaging 5 points in space and 3 levels in time, in order to eliminate noise. The boundary
conditions for the currents were fixed as no-gradients and monthly climatological values of temperature and salinity were prescribed at the open limits of the grid. Harmonic constants of tidal
components were given at double boundaries, so that the elevations were partially clamped to harmonic oscillations with restoration period equivalent to the baroclinic time step. Table 1 gives some
of the conditions used in the model processing and Table 2 informs the input data.
Model parameters Values
Internal time step (integration) 1800 s (baroclinic)
External time step (integration) 30 s (barotropic)
Coefficient of relaxation to sea surface temperature climatology 100 W/m^2/K
Constant horizontal diffusivity (Smagorinsky) 0,08
Initial value of the Smagorinsky diffusion coefficient 100
Inverse diffusivity horizontal Prandtl number 1
Relaxation coefficient of TS model calculations to climatology (for all sigma levels) 10^-3
Weight assigned to the central point in the process of spatial averages 0,8
Table 1: Initial parameters used in the model.
Input data Source
Bathymetric data General bathymetric Chart of the Oceans (GEBCO)
Forcing of mean sea level at the boundaries is inserted at intervals of 24 h, with corresponding climatological averages. Ocean Circulation and Climate Advanced Model (OCCAM)
Harmonic tidal constants at the open contours Model TPXO7.1-version utilizes mission data TOPEX / POSEIDON [39]
Temperature and salinity annual climatology (inserted at the initial time, at each grid point); World Ocean Atlas in its 2008 version (WOA08)
Relaxation of model results to climatology Climate Forecast System Reanalysis (CFSR) [40]
Boundary conditions for winds and surface fluxes of heat and salt (every 6 hours); Reanalysis of atmospheric model the NCEP / NCAR [41]
Table 2: Input data used in the model.
Statistical and spectral analyses
As the outputs of the model were obtained at 6 h intervals, their results were reorganized on 8 days averages, for comparison with the time series of chlorophyll-a, in the same grid points positions
of the hydrodynamic model. We applied the method of least squares to obtain trends and annual and semiannual signals of the series [24].
The covariance and cross-correlation functions are used to examine the relationships between different series in the time domain. The mathematical basis for these calculations is provided by Spiegel
et al. [25]. By denoting the mean and standard deviation of a time series x as _ x and σ [x], and the mean and standard deviation of a time series y as
And the linear correlation coefficients between series x and y, with lags h, are given by:
The linear correlation coefficient indicates the linear dependence of the compared data, giving the degree of dispersion around an adjustment function that is equivalent to a straight line. The
values range between -1 (perfect inverse correlation) and +1 (perfect direct correlation), with values close to zero indicating that the variables in question are not correlated. This parameter
estimates then the linear interdependence of two series, where high positive values indicate a similar behavior of the variables, while strong negative values indicate an opposite behavior. For the
Fourier analysis of the time series, averages and trends of the series were removed, in order to avoid distortions in the low frequency components of the spectrum [24]. In frequencies domain, the
Fourier transform of a time series is defined as:
Finally, the cross-spectrum is defined by the Fourier transform of the cross-covariance function (6) whose amplitude peaks indicate the frequencies where there is a greater inter-relationship between
series and phases indicate their respective delays (in time).
Results and Discussion
Distribution of chlorophyll-a
The distribution of the mean values of chlorophyll-a (Figure 2a) showed behavior similar to that observed for long-term analysis with remote sensing data held by Deser et al. [26], Hardman et al. [27
] and Saraceno et al. [28], and optimal interpolation observational data by Reynolds et al. [29] For all the study area, this distribution had a mean of 0.31 ± 0.78 mg/m^3. One can observe a pattern
of high concentrations in regions of coastal upwelling, equatorial upwelling and high latitudes. These results corroborate those obtained by Wang et al. [30] for the climatology of SeaWiFS data in
the period 1997-2007, which also checks the seasonality of chlorophyll-a in the equatorial region, the Amazon and Congo River discharge areas and upwelling regions in North Africa, as can be observed
in the distribution of the standard deviation of the annual and semiannual signals.
The distribution of the chlorophyll trend (Figure 2b) showed an average of (0.22 ± 3.79).10^-2 mg/m^3/year. The largest variabilities (both positive and negative) concentrated in the coastal regions,
with the highlight of a region with negative trend near the mouth of the La Plata River.
The distribution of the standard deviation (after removal of annual and semiannual signal) had a mean of 0.22 ± 0.75 mg/m^3 (Figure 2c). This distribution pattern is similar to that of the mean
concentrations, with higher standard deviation in the regions of higher mean concentration. The spatial distribution of the standard deviation of annual and semiannual signals averaged 0.03 ± 0.10 mg
/m3 (Figure 2d), with values much lower than those found in the residual signals, indicating that although the seasonality is important, it does not represent the largest part of the chlorophyll
variability. Garcia et al. [31], by means of spectral analysis, found and highlighted the significance of the signal in the annual and semi-annual variability of chlorophyll-a in the region of the
Malvinas Current, with similar values to those of the residual distribution, but their signals were more evident in the region north of the belt of 40°S. Hardman et al. [27] verified the high values
of overall standard deviation in the Congo River Plume, probably contaminated by high loadings of suspended particulate matter and dissolved organic substances and strong biannual signature.
Correlations between chlorophyll-a and physical variables
Distributions of the linear correlation coefficients between chlorophyll-a and the physical variables computed by the model, considering zero time lag, are shown in Figure 3, for the Tropical and
South Atlantic. The distribution of the correlation coefficient (with zero lag) between the current intensity and chlorophyll-a (Figure 3a) has average of 0.00 ± 0.08, with a maximum of 0.38 and
minimum of -0.41, with no defined pattern. Anyway, it is possible to detect a belt of positive correlations in the equatorial region and a point of negative correlation near the mouth of the Amazon
The correlation between the surface elevation and chlorophyll (Figure 3b) has an average of -0.02 ± 0.13, with maximum of 0.37 and minimum of -0.48. Despite having values comparable to those found in
the correlation with the intensity of the currents, this distribution showed a more defined spatial pattern, so that in the area of the triangle of positive elevation (SASG) predominates negative
correlation coefficients. The equatorial region, south of the North Atlantic Subtropical Gyre (NASG) and a belt around 45°S also display negative correlations. For salinity, the correlations with
chlorophyll-a averaged -0.02 ± 0.12, with maximum and minimum of 0.44 and -0.58, respectively (Figure 3c). This distribution showed no defined spatial pattern, but is characterized by the
predominance of positive correlations at high latitudes (up to latitude 40°S) and a belt between the Equator and 10°N. The region of the mouth of the Amazon River showed negative correlations. This
fact would suggest that the lower salinity site (the larger river discharge), the greater supply of limiting nutrients in the process of photosynthesis. And for temperature, is seen spatial average
of 0.02 ± 0.17, with a maximum value of 0.68 and a minimum of -0.57, shown in Figure 3d, with a well-defined distribution pattern, having positive values at high latitudes (up to 40°S), and a region
with negative values constrained by Brazil Malvinas Confluence (BMC), the South Atlantic Current (SAC) and the Agulhas Current Retroflection (ACR). A high positive correlation in the region of the
Amazon and Congo River discharges and in the region near to the northeastern coast of Brazil can also be observed. Picaut et al. [32] presents a pattern of distribution for this seasonal signal of
sea surface temperature in the Atlantic equatorial region similar to that shown by the chlorophyll-a (Figure 2d), which justify this high and almost homogenous negative correlation in this region. In
medium and low latitudes, the temperature rise drives a stepping thermocline, forming a thermal barrier that hinders the entry of nutrients from the deeper layers into the mixed layer. Thus, there is
a reduction of primary productivity. On the other hand, at high latitudes, the low temperatures prevent the formation of a thermocline, so that the temperature rise is indicative of solar radiation
increase, resulting a deeper penetration of radiation into the water column, resulting then a primary productivity increase.
Next, Figure 4 presents the distributions of maximum crosscorrelations between chlorophyll-a and physical variables considering the respective time lags. Negative lags mean advance of the physical
variable with respect to the biological one, while positive lags mean delays in the occurrence of the physical variables. The time unit in the considered lags is equal to 8 days (approximately one
week). The distribution of the cross-correlation with the intensity of currents averaged 0.01 ± 0.15, with extremes of 0.41 and -0.41 (Figures 4a and 4b). Note a slight change in the distribution of
these correlations relative to zero lag, like a band of positive correlations in the equatorial region, with negative lag of up to 10 weeks, which means the physical variable precedence in relation
to the biological variable. There are regions of negative correlation, like near the mouth of the Amazon River, but there are regions with positive correlations, as in the southern and southeastern
Brazilian coast, lagged positive in approximately five weeks. This positive correlation at the equator is plausible when considering the increased intensity of the currents in the region as a result
of increased intensity of trade winds, since this system is responsible for upwelling in the region, due to the winds and consequent Ekman transport mechanism, which generates currents with southern
components. The greater intensity of winds the greater intensity of the currents, and the greater intensity of meridional currents. This would generate a greater vertical transport in the area,
increasing the supply of nutrients from deeper layers into the surface. The occurrence of high correlation values in module, with both positive and negative lags, highlights the existence of a factor
that influences the variability of both variables correlated, with no necessarily cause and effect relationship between these variables. The cross-correlations between chlorophyll-a and sea level
elevation showed a mean value of -0.02 ± 0.20, with maximum and minimum of 0.51 and -0.48, respectively (Figures 4c and 4d). Can be noted, in this distribution, a reinforcement of what was found
previously with zero lag. The triangle elevation keeps negative correlation, with virtually zero lag, this behavior followed in equatorial regions and NASG. In regions of positive correlation there
is predominance of positive lag, but with much variability. A strong negative correlation in the region inside the SASG may suggest that the increase in surface elevation causes an increase in the
thermocline depth, removing the region below this layer (more nutrients) in the photic zone, which would reduce the primary productivity. This type of event would have a biological response almost
With salinity, the cross-correlations averaged -0.04 ± 0.21, with maximum and minimum values of 0.48 and -0.64, respectively (Figures 4e and 4f). In this distribution there are similarities with the
distribution of the simple zero lag correlation, with a predominance of positive correlations at high latitudes and a belt above the equator, these ones being associated with negative lags of more
than 15 weeks. The negative correlations, especially the most intense in the region near the Antarctic continent, are associated, in general, to positive lags.
The same correlation with temperature showed a mean value of -0.02 ± 0.22, with 0.68 maximum and -0.57 minimum (Figures 4g and 4h). This distribution confirms the pattern already seen in the simple
zero lag correlation, highlighting the association of this correlation to a virtually instantaneous response time.
Analysis of transect 20°W
Next, latitudinal variabilities at the longitude of 20°W of the amplitudes of the Fourier spectrum of the cross-covariance and respective time lags (in years), between chlorophyll-a and physical
variables, are presented on Figure 5. For the intensity of currents and surface elevation there are greater amplitude for low frequency signals (more than 3 years) with great variability in response
time, oscillating positive and negative values over 1 year. These signals are concentrated in the belts of 10°N to 30°N, between 30°S and 40°S and around 60°S to 70°S, concentrating on the period of
7 years (equivalent to the extension of the series). Peaks in the spectrum with periods of 6 and 7.5 years have been cited in the analysis of IOS and ION [32], which emphasizes the shortness of the
series analyzed. There is a weak signal in the period of 2.3 years, with response almost instantaneous (lag zero) most salient between the Equator and 10°S. For surface elevation, across the transect
is evident a strong signal with period of 0.33 years and a less intense one located between 35°S and 60°S of 0.25 years (Figures 5c and 5d). Alvarez- Ramirez et al. [33] associate the signal of
period 0.33 years with the frequency spectrum of the Northern Oscillation Index (NOI), related to the North Atlantic Oscillation (NAO), with nonlinear resonant effects. The period of 0.25 years, as
it is a submultiple of the annual signal, can also be related to a resonant effect associated with seasonality. There is also a weak signal in the period of 1.4 years (at approximately 15°S) and 1.7
years (at approximately 35°S) with response almost instantaneous.The transects for salinity and temperature amplitudes present the most striking features for certain frequencies, with evident
signatures at low frequencies (Figures 5e and 5h). Lags have similar patterns to those found for other physical variables (with more immediate responses to higher frequencies), but with regions of
alternating signals better defined. These low-frequency signals are concentrated in the period of 7 years (extension of the time series itself). These transects, as well as surface elevation, clearly
feature the signal of period 0.33 years in much of the transect, intensified in the polar region. At high latitudes, similar to the other physical parameters, salinity and temperature have
highlighted signals of lower frequencies (period 2.3 years) concentrated from 30°S to 40°S and between the Equator and 10°S, both with positive lag about 1 year. Mainly for high latitudes, there are
signals with periods around 1 year which can be due to failure when removing the seasonality. It is also apparent, particularly in the region between 10°N and 20°N, a signal with period of
approximately 1.4 years, with a positive phase shift below 6 months. Note that this region is affected by coastal upwelling (North African coast).
For temperature, greater emphasis is given to the occurrence of a signal with high amplitude in the belt between 50°S and 60°S, also present but weaker in 5°N, with higher values around 3.5 years and
having a negative gap of less than 6 months. There is also the occurrence of weaker signals concentrated around the period of 2.3 years, at 10°N and 45°S with an almost immediate response (delay less
than 10 weeks). The occurrence of these same signals is verified for salinity, but with greater intensiy in the period of 2.3 years, concentrated between the Equator and 5°S and around 45°S (region
of higher salinity gradient).
Carton et al. [34] highlights variabilities in the equatorial mode of the sea surface temperature for a timescale of 2-5 years like being controlled by dynamical processes. Hardman-Mountford et al. [
27] observed cycles with periods between 3 and 5 years in spectral analysis of time series of surface temperature anomaly and wind components for the equatorial region, north of Angola and Benguela,
in the period from 1982 to 1999. The same period is highlighted in the analysis of wavelet transform (disregarding the annual signal) in 32-years integration of ROMS (Regional Ocean Modeling System)
for the same region by Colberg et al., [35], with the variability related to coastal Kelvin waves associated to ENSO events in the Tropical Pacific.
Schneider et al. [32] found that the variance spectrum of the Southern Oscillation Index (SOI) showed significant peaks with periods of 2.3, 2.9, 3.5 and 6 years. Phenomena of El Niño - Southern
Oscillation (ENSO) show interannual components ranging between 2.3 and 3.4 years and a low frequency component of 6 years or more, being the period of 2.3 years consistent with the average
periodicity of ENSO [36]. Thus, we can associate the signals of periods 3.5 and 2.3 years to ENSO phenomena. This periodicity is verified for Naujokat et al. [37] that found a quasi-biennial
oscillation with mean period of 27.7 months (2.308 years) in equatorial zonal winds data that can suggest the ocean atmosphere coupling. Other signals are observed with low intensity, as for the
period of 1.7 years evident at 35°S, with negative delay of about 6 months to temperature, and the period of 0.7 years and instantaneous response, concentrated at high latitudes in temperature and
salinity, but prominently in temperature. The signature of period 0.7 years closely resembles that analyzed for 0.33 years. Schneider et al. [27] found that the variance spectrum of the NOI had
significant peaks at periods of 1.7, 2.2 and 7.5 years, which allows to associate this signal of 1.7 years with the analyzed signal in NOI. In analyzing the Magnitude of the Squared Coherence for
temperature anomalies of the sea surface and the SOI, Campos et al. [38] found spectral peaks of frequencies 0.12 and 0.055 cycles per month (equivalent to the periods of 0.7 and 1.5 years,
respectively), relating them to the ENSO phenomena.
The distributions of mean values and standard deviations of chlorophyll-a exhibit behavior similar to that given in literature. The annual and semi-annual signals are predominant both in MODIS and
model data but, even excluding these components; the residual correlations are still high. On the other hand, annual and semi-annual signals have smaller standard deviation than the remaining
(residual) frequencies. The cross-correlations between chlorophyll-a and salinity, temperature and surface elevation showed spatial distribution patterns with well defined latitudinal character,
presenting higher modulus of correlation for temperature and salinity, above +0.6 in the polar region and below -0.5 in the tropical area. A general pattern of negative correlations in the regions of
low concentration and positive in regions of high concentration was obtained, except the Equator (region of high chlorophyll-a concentration, which is characterized by a negative correlation for all
variables, except the intensity of the currents). The cross-correlations between chlorophyll-a and physical parameters corroborate the pattern found in the correlations considering lag zero,
stressing aspects as the positive correlation with the intensity of the currents in the equatorial region and the negative correlation with the surface elevation inside the SASG, both presenting
immediate response. In the analysis of spectral distributions of Fourier spectra of the cross-covariance between chlorophyll-a and each of the physical variables, in the transect 20°W, the best
defined signals were found for temperature and salinity, especially the periods of 3.5, 2.3, 0.7, and 1.7 years, with varying spatial distribution and time lags. These signals are found in the
literature, being associated with ENSO phenomena. Thus, one can conclude from this study that there is a very characteristic spatial distribution of correlations between physical variables and
chlorophyll-a, even removing the annual and semiannual signals. Despite their importance, these signals are not responsible for much of the variability of chlorophyll-a concentrations. It should be
noted also that the spatial variability also occurs in the frequency domain, which denotes a probable influence of ENOS phenomena, both in the physical variables and chlorophyll-a. As a suggestion
for future work, further research should focus on the causes of the spatial variability of the correlations and phase delays, as well as a better understanding of how phenomena such as ENSO (El Niño
- Southern Oscillation) affect these correlations.
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Citation: da Silveira Pereira NE, Harari J, de Camargo R (2014) Correlations of Remote Sensing Chlorophyll-a Data and Results of A Numerical Model of the Tropical and South Atlantic Ocean
Circulation. J Geol Geosci 3:171.
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Range based Localization Scheme for 3D Wireless Sensor Network using Joint Distance and Angle Information:A Brief Review
Volume 03, Issue 06 (June 2014)
Range based Localization Scheme for 3D Wireless Sensor Network using Joint Distance and Angle Information:A Brief Review
DOI : 10.17577/IJERTV3IS061388
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Mr. S. P. Salgar, Mr. Sachidanand B. N. , Mr. Veeresh M. Metigoudar, Mr. Prashant P Zirmite, 2014, Range based Localization Scheme for 3D Wireless Sensor Network using Joint Distance and Angle
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Range based Localization Scheme for 3D Wireless Sensor Network using Joint Distance and Angle Information:A Brief Review
Mr. Santosh P. Salgar Electronics Engineering Dept DKTEs TEI, Ichalkaranji
Prof. B N Sachidanand Electronics Engineering Dept DKTEs TEI, Ichalkaranji
Prof. Veeresh M Metigoudar Electronics Engineering Dept DKTEs TEI, Ichalkaranji
Mr. Prashant P Zirmite Electronics Engineering Dept DKTEs TEI, Ichalkaranji
Abstract – This review paper describes range based 3D localization algorithm for wireless sensor network to satisfy the practical needs. The mentioned algorithm is anchor free, scalable and provides
accurate physical position. To estimate ranges between neighbors both distance and direction measurement technique is used. Based on this information a global network with wide coordinate system is
developed using local coordinate system leads to absolute position of nodes. Simulation results have shown that proposed algorithm achieves good tradeoff between localization percentage and precision
when node degree equals 12 or around.
Keywords – Localization, Wireless Sensor Network, adjacency transformation matrix
1. INTRODUCTION
Many localization schemes have been proposed for more precise localization of senor node in wireless sensor network. While working with these schemes a little attention is given towards their
application to the practical 3D environment. The existing 2D schemes not even found any deterministic algorithm to verify that uniquely localizable in 3D [1].
Among the existing range-based 3D localization mechanisms [2] tried to reduce the complexity and transform the 3D localization process into its 2D counterpart by employing sensor depth
information and additional hardware-upgradable modules. Many 3D localization schemes use trilateration method to calculate the desired position [3-5]. The basic idea is to use at least four
anchor nodes to implement the trilateration. This approach normally experiences accelerated error accumulation as more nodes are positioned iteratively, making it difficult to scale. Also
assumption is made that the sensor node density to be high in order to attain good localization coverage [6].
This review paper proposes the Three Dimensional Anchor free Localization (3DAFL) algorithm that tries to achieve accurate physical node positions for large-scale WSN in a3D scenario. This
algorithm makes use of both distance and angle of arrival (AOA)
information [7]. There are two phases in this algorithms. Initially, local coordinate system (LCS) is constructed at each individual node and relative neighbor positions can be calculated
accordingly. Later, the LCSs efficiently converge to form a global coordinate system by means of homogeneous coordinate transformation. This algorithm provides high accuracy, low communication
overhead, and is robust to node failure.
2. THREE DIMENTIONAL ANCHOR FREE LOCALIZATION(3DAFL)
The sensor nodes are randomly deployed in 3D scenarios rather than on pure 2D planes. From application (cost, power usage) point of view, minimizing the number of anchors in the network is highly
1. Local position computation
At the system initialization stage, every node starts exchanging beacon frames to detect its 1-hop neighbors and to build the LCS. The beacon response frame of node O contains sequence number
and a list of Os neighbors maintained by its neighbor list table. For a node O to build LCSO, it must have at least two non- collinear neighbors A and B. See Fig. 1. For the purpose of global
coordinate transformation, O needs another neighbor, Q, such that all the four nodes (O, A, B and Q) are non-collinear 1-hop neighbors of each other.
Without loss of generality, we choose A and B to setup LCSO. +X-axis is temporarily set as along with A lying on the +x-axis. Then XOY plane can be defined with
node B lying in the direction of +y-axis. For homogeneity
the positive direction of the z-axis is set to be extending out and conforms to the right handed rule.
Fig. 1 Local coordinate system and localposition computation
To avoid the flip ambiguity [8] while computing the node positions relative to LCSO, node O utilizes the bearing information OB to establish final +x-axis. If OB <
, +xaxis = . Or else, +x-axis= . This procedure guarantees that LCS at any node is always constructed such that +y-axis is /2 counterclockwise to +x-axis with +z axis pulling out when viewed
from the top. Now node O can use trilateration to compute the positions of its neighbors with respect to LCSO. The local position of node B, OPB = (Bx, By, Bz), is
= . , = . , = 0 (1) Where,
coordinate reference frame, then they are repositioned into a global coordinate scene as described in the next subsection.
2. Global position transformation
In this part we compute the position of a node to a fixed coordinate system (FCS) which acts as a physical location reference to all the nodes in the WSN. To transform position descriptions
from LCSs to the FCS, we need to develop an adjacency transformation matrix (ATM) that brings two LCSs into alignment. This process continues till all the nodes know their positions relative
to the FCS.
We illustrate global position computation by considering the example in Fig. 2. Node R can convert its
Fig.2 Global position computation
local position OPR to a position relative to Os neighor Q, and thus to a position relative to the FCS, FCSPR, as:
FCSPR=FCS[T]Q .Q[T]O .OPR (3)
2 + 2 2
(0, )
This process continues until all the nodes know
As shown in Fig. 1, the local position of another node C, OPC = (Cx, Cy, Cz), can be computed if it is the neighbor of both A and B such that dOC, dCA and dCB are explicitly known by ranging
2 2 + 2
their positions relative to the FCS. Such positions, which are consentaneous within the whole network, can easily be absolute positions once the FCS knows its physical position by means such
as GPS.
2 2 + 2 + 2 2
Since the z coordinate results from a square root calculation, it is possible to have one or two solutions for the trilateration problem. One solution predicates node C being in the same XOY
plane while the two alternative solutions can be determined on the AOA basis. Once a node estimates its position it becomes a beacon and assists other unknown nodes in estimating their
Similarly the local position of any node Q, OPQ, can be fixed if it is neighbors of at least three known nodes L, M and N such that they are all non-collinear neighbors of O [5]. In this way
sensor nodes are described in a local
Fig.3 Adjacency transformation matrix development
So the crucial problem is to acquire the ATM at any node. We can generate the ATMO, [] , by performing the following sequence of operations. See Fig. 3.
1. Rotate about the axis by the positive angle 1 to bring into the plane that is parallel to XQZ plane.
2. Rotate about the new axis (after step 1) by 2 to bring into the plane that is parallel to XQZ plane.
3. Rotate (after step 2) about the new axis (after step 2) by th positive angle 3 to bring + coincide with the axis that is parallel to + axis.
4. Translate the rotated LCSO (after step 3), which is already aligned with LCSQ, such that O moves to Q.
Accordingly, [] can be expressed in the homogeneous coordinate form [8] as a composite transformation involving combination of the four matrix multiplications.
such that with communication range r of 10m we obtain a node degree d varying from 6 to 16. The ranging error is Gaussian with a fixed standard deviation, for both distance and bearing
estimation.We assume a ranging error of 1% for all our simulations.
Fig.4 Localization percentage Vs Node degree
Figure 4 shows the ratio of deployed nodes that are able to localize themselves relative to the FCS at a
[] =
O O O
. 0 1 0 0 .
given node degree. We observe that with d 9 the localization percentage is low. First in a sparse WSN some nodes may not be able to find enough beaconing neighbors to calculate their positions
relative to a LCS. When the
node density gets higher, this happens mostly at the network borders. Second, as mentioned in section 2.2 a
node O may not be able to develop an ATM [] due to the absence of AQ or BQ. This problem is greatly alleviated as the node degree increases to 11 and more, which provides us with a valid
parameter in real WSN
1 = cos( , )
1 = ± 1 2 1
2 = cos ,
2 = ± 1 2 2
cos( , )
Localization error is the offset of the estimated node position from the actual node position. We express this metric relative to the communication range r, in terms of percentage. The average
localization error for different node density d is illustrated in Fig. 5. When the node
3 =
degree is low (d 9), the localization error is high due to
3 = ± 1 2 3
The plus or minus sign of 1 , 2 and 3 ( (0, 2)) is determined with the aid of AOA technique, respectively. Note that D is a virtual node whose relative position = ( , , ) can be calculated by
, , and DQAQ = /2. Thus one necessary
condition of obtaining [] is that the two nodes (AQ, BQ) determining XQY plane are also Os neighbors.
Iteratively all the nodes can know their positions relative to the FCS, which acts as a physical location reference. Due to the presence of the fixed data-sink in a WSN, it makes sense to select
it for defining the FCS.
3. SIMULATION
To evaluate through simulation the performance of 3DAFL, we deploy n sensors in a cubic region R such that the node positions are generated using a random uniform distribution.The values of n and
R are selected
the reason that enough neighbors are unavailable to reduce the size of the estimation region. Increasing d brings more rigid frameworks and more positive conectivity constraints to
Fig. 5 Mean errors of the x- y- and z- component sorted by the mean
position errors = ( )2 + ( )2 + ()2 of sensor nodes.
increase the accurate iterative calculation. One observation is that the errors of the z-component are always higher than those of the horizontal components. This is, however, an acceptable
deviation derived from the positioning formula where the z-component of depends more on the two
values of cos3 and sin3 .
4. Conclusion
In this paper we proposed a 3D distributed, distance and AOA information oriented, and anchor-free localization algorithm for practical WSN applications. Simulation results demonstrated the
effectiveness of our algorithm. For node degree equals 12 or around, the deployment wined good tradeoff between localization percentage and precision. One of our ongoing work is to verify 3DAFL
through quantitative comparisons to other 3D range-based localization algorithms.
1. R. Connelly, T. Jordan, and W. Whiteley, Generic global rigidity of body-bar frameworks, ERGCO Technical Report, TR-2009-13, 2009.
2. A. Y. Teymorian, W. Cheng, L. Ma, X. Cheng, X. Lu, and Z. Lu, 3D underwater sensor network localization, IEEE Trans. Mobile Comput., vol. 8, no. 12, pp. 16101621, Dec. 2009.
3. F. Thomas and L. Ros, Revisiting trilateration for robot localization,IEEE Trans. Robotics, vol. 21, no. 1, pp. 93101, Feb. 2005.
4. E. Doukhnitch, M. Salamah, and E. Ozen, An efficient approach for trilateration in 3D positioning, Computer Commun., vol. 31, no. 17, pp. 41244129, Nov. 2008.
5. G. S. Kuruoglu, M. Erol, and S. Oktug, Three dimensional localization in wireless sensor networks using the adapted multi- lateration technique considering range measurement errors, in Proc. IEEE
GLOBECOM 2009, pp. 15.
6. D. Moore, J. Leonard, D. Rus, and S. Teller, Robust distributed network localization with noisy range measurements, in Proc. SenSys 2004, pp. 5061.
7. G. D. Stefano and A. Petricola, A distributed AOA based localization algorithm for wireless sensor networks, J. Computers, vol. 3, no. 4, pp. 18, Apr. 2008.
8. D. Hearn and M. P. Baker, Computer Graphics: C Version, 2nd edition. Pearson Education, 2004.
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-16041 AN IMPLICIT OR EXPLICIT INVOCATION OF THE fn:boolean FUNCTION IN THE XQUERY EXPRESSION COULD NOT COMPUTE THE EFFECTIVE BOOLEAN VALUE OF THE SEQUENCE. ERROR QNAME=err:FORG0006
The effective boolean value of the argument passed to fn:boolean could not be computed. The effective boolean value can be computed only if the sequence operand is one of the following sequences:
• An empty sequence
• A sequence where the value of the first item is a node
• A singleton sequence with a value of type xs:string, xdt:untypedAtomic or a type derived from one of these types
• A singleton sequence with a value of any numeric type or derived from any numeric type
System action
The XQuery expression is not processed.
User response
Programmer response
Determine the possible expressions within the XQuery expression where an effective boolean value is calculated either implicitly or explicitly. An implicit invocation of the fn:boolean function can
occur when processing the following types of expressions:
• The logical expressions AND and OR
• An fn:not function invocation
• The WHERE clause of an FLWOR expression
• Certain types of predicates, such as a[b]
• Conditional expressions, such as IF
Ensure that the sequence operand of each effective boolean value calculation has a valid sequence operand, as described in the explanation. | {"url":"https://www.ibm.com/docs/en/db2-for-zos/10?topic=codes-16041","timestamp":"2024-11-09T13:23:23Z","content_type":"text/html","content_length":"6801","record_id":"<urn:uuid:ba4b4106-bffe-4c67-96eb-fd82a802eca4>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00466.warc.gz"} |
Zero Lower Bound and Yield Curve Distortions
It's crunch time, and the signals are not comforting.
Interest rates have taken a dive recently. Declines this week appear to be from lower inflation premiums in the first 10 years of the yield curve and from both lower real and nominal rates at the
long end of the curve. The declines are not coming from a delay in rising rates in 2015. The declines are coming entirely from lower rates at the long end and a lower slope during the rate recovery
This first graph shows the expected date of the first rate hike and the expected subsequent rate of rate increases (from Eurodollar futures). The expected date of the first hike has actually moved
back in time recently (blue line, inverted right scale). But the slope of rate hikes has dropped sharply down to 22 bps per quarter.
The next graph shows the Eurodollar curve at several points in time. In Sept. 2012, when QE3 was just getting under way, short term rates were very close to where they are now. Then, by the summer of
2013, both the date of the first hike and the rate of those hikes had taken bullish directions, with the first rate hike expected as early as 2014 and the slope hitting around 35 bps per quarter
(which is still low relative to previous recoveries). But, as QE3 has been tapered and then terminated, rates fell back to the previous level, and now long term rates are even lower than they had
been after QE3 began.
The main difference (comparing the white line to the green line) is that the slope in late 2012 had been as low as 15 bps per quarter. So, now, compared to late 2012, long term rates are lower, but
the slope of the recovery period is somewhat higher. My intuition here is that market participants are more confident that rates will rise at all than they were when QE3 began, but they now expect
the long term posture of the Fed to be more hawkish. But, I will address this further. I think this may be a difficult thing to read.
My model of the expected date of the first hike, the uncertainty about that date, and the slope of the curve during subsequent hikes has been based on the notion of the short end of the yield curve
at the zero lower bound (ZLB) as a sort of call option, where the ZLB is like the strike price, and the yield curve is the current "price" of forward rates, which includes an expectation of future
rates and a premium created from the ZLB and the level of uncertainty about future rate movements. A futures contract well into the future is like a call option far in-the-money. Its price generally
reflects the present value of future prices. But, a futures contract near the expected date of the future rate hike includes a premium because the ZLB limits losses, just like a strike price does for
a call option.
We are close enough to the expected rate hike that there isn't much "premium" curvature in the yield curve anymore. And, at the very short end of the curve, Fed discretion can be as important as
movements in natural rates. So, I think the ZLB could be causing distortions now in the yield curve in a slightly different way.
We should think of the price of each forward interest rate as the price settled by the marginal investor in each contract. The neutral final price that comes from a variety of different expectations.
So, each forward rate is the product, itself, of a distribution of expectations.
The recent decline in the slope of the yield curve has been roughly proportional to the decline in the expected long term interest rate. So, compared to 4 months ago, I think there has been an
increase in the expectation that we will, in fact, not escape the ZLB. I have been thinking of this in terms of a bifurcated set of expectations, where some portion of the market expects rates to
climb quickly to, say, 3% or 4%, and another portion expects us to remain at the ZLB or to leave it only temporarily. Thinking of it that way, the ZLB traders would be pulling the yield curve down
proportionately across the curve.
But, I am wondering if this is the right way to think about it. Here, I have graphed the yield curve as if short term rates were at 3%. We might imagine that at each point along the curve, there are
a range of relatively non-skewed expectations around the market rate. As the yield curve moves out into the future, the uncertainty about the expected rate would increase, but the market rate would
still be an unbiased reflection of a range of expectations.
But, if the rates are at the ZLB, then it is possible that the lower bound will affect the shape of expectations. At very short term rates, this is basically how I have been modeling the convexity of
the yield curve. I had assumed that this factor was not important at the longer end of the yield curve. But, as long rates have declined, the probability that there are some ZLB expectations at the
long end of the curve seems undeniable.
If instead of thinking of it as consisting of a two-humped distribution, as I had above, what if we think of it as a normal distribution where the ZLB acts as a lower limit, much as a strike price
does on a stock option? This reverses the effect on forward market rates.
Now, what we have is what would normally be the mean, median, and mode rate expectation. But, the low end of the expectations distributions would be truncated by the ZLB. We could presume that this
would lessen the speculative weight these market participants would place on taking a position on lower rates, since the potential profit is limited by the ZLB.
If this is the basic shape of market expectations, then the ZLB would push the mean expected rates above the median expected rates - and presumably to some degree the market rates would also be
pushed above the median expected rates.
In this scenario, the rate curve we are seeing now would actually be an overstatement of expected rates. After walking through this, I think this is very likely to be the case. If we take the
scenario to the extreme, where expected future short term rates are expected to be at 0% indefinitely, we would clearly expect the long end of the curve to rise above zero, even if there was no
maturity premium and we had a pure expectations context. Because, there would be an option premium there, just as is with a stock option at the money or the recent short end of the yield curve.
The scale of this distortion would be a product of the level of uncertainty. And this, ironically, might be good news. The long end of the curve is about 3/4% lower than it was in September 2012
after the start of QE3. While there are still some expectations of an enduring ZLB, there should be quite a bit less uncertainty about the near term economy than there was then. GDP growth has
remained steady. Employment has been strengthening. Commercial lending has continued to strengthen. Households have continued to deleverage.
So, we should expect the tendency of the ZLB to push up the market rates at the long end of the yield curve to be weaker now than it was then. Maybe the median expected long term rate level was, say,
2%, in September 2012 and is also at about that level now. Maybe then the market rate was biased up by 1 3/4% and now it's only biased up by 1%, because there is a tighter range of expectations now.
Even if this is the case, the most recent drop in rates is probably the result of falling mean and median rate expectations and a higher probability of being stuck at the ZLB.
Also, the slope in the 2016 period is a monkey wrench in this hypothesis. The expected date of the first rate increase has not changed. If recent rate changes were a result of lower economic
expectations, then I would have expected the date to move back in time. It seems odd that the expected slope has declined while the expected date of the first increase has remained bullish. This
suggests the bifurcated expectations distribution I originally described.
The fact that the slope is higher now than in September 2012 also points to bifurcated expectations, since lower variance of expectations would have lowered the slope if expectations were normally
There is also the possibility that expectations about the way the Fed manages the rate of hikes in 2015-2017 has changed, and the changing slope simply reflects these changing expectations. But, I'm
already having trouble keeping this all straight without bringing in another vague and moving variable.
The ramifications for speculation are problematic here, because I would have been looking for a flat yield curve as a signal for declining short term rates. This effect could mean that the next
downturn begins when short term rates are near zero and the yield curve is distorted so that even if median expectations are for a flat yield curve, the market prices of forward rates could have a
decent positive slope.
I hope that all of this becomes moot when we start to see mortgage credit flow and inflation expectations are re-invigorated. But, there has been no sign of that happening yet. | {"url":"https://www.idiosyncraticwhisk.com/2015/01/zero-lower-bound-and-yield-curve.html","timestamp":"2024-11-08T22:14:24Z","content_type":"application/xhtml+xml","content_length":"122281","record_id":"<urn:uuid:b1344be7-8cec-422b-bef2-033c60ea4688>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00327.warc.gz"} |
Introduction to Simple Interest, Formula and Examples for Class 10 WBBSE.
By Admin / Competitive Exam, Math, WBBSE / 8th May 2023
Introduction to Simple interest:
First we need to understand what is interest? Interest is the extra money that is paid to the lender of money for using his money, in another word, interest is the cost of borrowing money. Interest
may be either Simple interest or Compound interest.
Simple Interest is the interest which is calculated on the principal for a given period of time. In simple interest Interest remain same for every year.Formula to calculate simple interest is,
Principal is the money which is given or taken as loan or the money which is deposited into a bank.
For Example: Mr Karim opened a bank account in BANK OF INDIA and deposited Rs 10,000 in that bank. He got 10200 after 2 year
Here Rs 10,000 is the principal as this is money he deposited in the bank and 200 is interest,
Period for which loan is taken or time for which money is deposited in the bank.
In the above example, 2 years is the period for which money was deposited in the bank.
Rate of Simple Interest
The amount of interest on Rs100 for a given period of time is called Rate of Simple interest.
for Example : “Rate of simple interest 5% per annum” means that interest on Rs 100 in 1 year is Rs 5.
Principal along with interest is called amount.
For example: If I have deposited INR 10000 for 5 year at the rate of simple interest 5% per annum. After 5 year I get 12500. This 12500 will be called amount. Here Rs 12500 include principal i.e
10000 and 25000 simple interest.
Examples: 1
Rahim has taken Rs 5000 as loan from a bank at the rate of 4% p.a simple interest. How much will he pay after 6 years as interest?
In the above example
Rate of Interest (R)=4% p.a
Time(T)=6 years
In the above example
Rate of Interest (R)=4% p.a
Time(T)=6 years
Simple interest Introduction
West Bengal Madhyamik 2024 Routine pdf.
WBBSE, News / By Admin | {"url":"https://zedonline.in/simple-interest/","timestamp":"2024-11-05T22:08:16Z","content_type":"text/html","content_length":"157571","record_id":"<urn:uuid:440f4af7-34b8-4404-adc3-2aa51c5bdae4>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00650.warc.gz"} |
Programming Quantum Computers
Get Mark Richards’s Software Architecture Patterns ebook to better understand how to design components—and how they should interact.
Dive in for free with a 10-day trial of the O’Reilly learning platform—then explore all the other resources our members count on to build skills and solve problems every day. | {"url":"https://www.oreilly.com/library/view/programming-quantum-computers/9781492039679/","timestamp":"2024-11-13T08:42:51Z","content_type":"text/html","content_length":"96632","record_id":"<urn:uuid:8578c139-6fc7-4728-865c-b93e354b36ad>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00165.warc.gz"} |
Electrical conductivity of partially ionized Na-Cs plasma mixtures
The electrical conductivity of partially ionized sodium-cesium plasmas is calculated starting from the electron kinetic equation taking into account the electron-electron, electron-ion and the
elastic electron-atom scattering. The transport cross sections follow from the scattering phase shifts solving the corresponding Schrödinger's equation with effective interaction potentials. In order
to calculate the plasma composition a coupled law of mass action is used. The electrical conductivity of the plasma mixture is then discussed for densities 1016…1021 cm−3 and for temperatures 3000 K…
10000 K.
Contributions to Plasma Physics
Pub Date:
January 1989
□ Cesium Plasma;
□ Metallic Plasmas;
□ Plasma Composition;
□ Plasma Conductivity;
□ Plasma Diagnostics;
□ Mixtures;
□ Plasma Temperature;
□ Sodium;
□ Transport Theory;
□ Plasma Physics | {"url":"https://ui.adsabs.harvard.edu/abs/1989CoPP...29..413B/abstract","timestamp":"2024-11-02T16:18:33Z","content_type":"text/html","content_length":"36961","record_id":"<urn:uuid:9673836f-404c-44b2-bc67-9c6238b00307>","cc-path":"CC-MAIN-2024-46/segments/1730477027714.37/warc/CC-MAIN-20241102133748-20241102163748-00349.warc.gz"} |
Standard Error | Data and Metrics
In simple terms, the standard error is a measure of how much the sample mean (average) is likely to vary from the true population mean. It tells us how uncertain or "noisy" our estimate of the
population mean is based on a sample.
Formula for the standard error of the mean (SE):
SE = (σ / √n)
• SE is the standard error of the mean.
• σ (sigma) is the population standard deviation, which measures how spread out the values are in the entire population.
• √n represents the square root of the sample size (n), which tells us how many observations are in our sample.
Let's break it down with an example:
Imagine you want to find out the average height of all the students in a school, but it's impractical to measure every student. So, you randomly select 30 students and measure their heights.
• You find that the average height of these 30 students is 160 centimeters.
• You know from previous research (or a larger data set) that the population standard deviation (σ) of student heights in the school is 10 centimeters.
Now, you can use the formula for the standard error:
SE = (10 / √30) ≈ 1.83 centimeters
The standard error in this case is approximately 1.83 centimeters. It tells you that based on your sample of 30 students, the average height of all students in the school is likely to fall within
1.83 centimeters of 160 centimeters. In other words, you can be reasonably confident that the true average height of all students is somewhere between approximately 158.17 and 161.83 centimeters.
So, the standard error helps you understand how much your sample mean might differ from the actual population mean, providing a measure of the uncertainty associated with your estimate. Smaller
standard errors indicate a more precise estimate, while larger standard errors suggest more variability and less confidence in your estimate. | {"url":"https://www.dataandmetrics.com/home/statistics-concepts/standard-error","timestamp":"2024-11-05T12:53:36Z","content_type":"text/html","content_length":"913205","record_id":"<urn:uuid:f51c7779-da1d-4373-833a-c2b5ada2b779>","cc-path":"CC-MAIN-2024-46/segments/1730477027881.88/warc/CC-MAIN-20241105114407-20241105144407-00087.warc.gz"} |
Indices rules
There are a number of important rules of index numbers: ya × yb = ya+b. Examples. 24 × 28 Six rules of the Law of Indices. Rule 1: Any number, except 0, whose index is 0 is always equal to 1,
regardless of the value of the base. An Example: Simplify 20: Index laws are the rules for simplifying expressions involving powers of the same Examples: Simplify the following expressions, leaving
only positive indices in
Thorax. 2004 May;59(5):421-7. Validation of predictive rules and indices of severity for community acquired pneumonia. Ewig S(1), de Roux A, Bauer T, García E 24 Jun 2019 The ICE® U.S. 3000 Index is
a rules-based, float-adjusted market capitalization- weighted broad U.S. equity index that has the objective of The CBF benchmarks (TAB Nominal, TAB UF and TADO) and indices (ICP and ICP (Real))
were administered by the Chilean Banking Association, ABIF, prior to 1. The World Justice Project. Rule of Law Index. Juan Carlos Botero and Alejandro Ponce. World Justice Project. June 2010 The
index is rebalanced monthly to ensure the constituents' continued compliance with the index rules though bonds may be removed from the index during the BSE Index Calculation Rules · Methodology for
Calculation of Financial Ratios · Permissible day-count conventions on listing and trading in bond issues. implementation of the new Securities Law, continue to improve the basic rules and policies
of the bond market, increase market SZSE COMPONENT INDEX.
17 Nov 2019 Fund managers will therefore closely monitor the weight of particular companies in their benchmark index. Most benchmark indices use a market-
Examples, solutions and videos to help GCSE Maths students learn about the multiplication and division rules of indices. Maths : Indices : Multiplication Rule In this tutorial you are shown the
multiplication rule for indices. You are given a short test at the end. x m × x n = x m+n Indices rules (or laws of indices) are the rules governing mathematical operations with indices (or powers).
Indices rules (or laws of indices) are the rules governing mathematical operations with indices (or powers). tutor2u. Subjects Events Job board Shop Company Support Main menu. Cart . There are three
rules of indices (or laws of indices) which you have to know and be able to apply to problems involving both numbers and algebra.For any numbers, x, m, and n, those three rules are The multiplication
law – when you multiply terms, you add the powers:; x^m\times x^n=x^{m+n}. The division law – when you divide terms, you subtract the powers: Introduction The manipulation of powers, or indices or
exponents is a very crucial underlying skill to have in algebra. In essence there are just 3 laws and from those we can derive 3 other interesting/useful rules. Exponentiation is a mathematical
operation, written as b n, involving two numbers, the base b and the exponent or power n.When n is a positive integer, exponentiation corresponds to repeated multiplication of the base: that is, b n
is the product of multiplying n bases: = × ⋯ × ⏟. The exponent is usually shown as a superscript to the right of the base. In that case, b n is called "b Addition, Subtraction, Multiplication and
Division of Powers Addition and Subtraction of Powers. It is obvious that powers may be added, like other quantities, by uniting them one after another with their signs. Thus the sum of a 3 and b 2,
is a 3 + b. And the sum of a 3 - b n and h 5-d 4 is a 3 - b n + h 5 - d 4.. The same powers of the same letters are like quantities and their coefficients To add or subtract with powers, both the
variables and the exponents of the variables must be the same. You perform the required operations on the coefficients, leaving the variable and exponent as they are. When adding or subtracting with
powers, the terms that combine always have exactly the same variables with exactly the same […]
Index rules for Austrian, CEE & CIS and style indices - available for download as pdf-file.
Indices are a convenient tool in mathematics to compactly denote the process of taking a power or a root of a number. Therefore, it is important to clearly understand the concept as well as the laws
of indices to be able to apply them later in important applications. Indexes and indices are both accepted and widely used plurals of the noun index.Both appear throughout the English-speaking world,
but indices prevails in varieties of English from outside North America, while indexes is more common in American and Canadian English.Meanwhile, indices is generally preferred in mathematical,
financial, and technical contexts, while indexes is relatively common Examples, solutions and videos to help GCSE Maths students learn about the multiplication and division rules of indices. Maths :
Indices : Multiplication Rule In this tutorial you are shown the multiplication rule for indices. You are given a short test at the end. x m × x n = x m+n
Introduction The manipulation of powers, or indices or exponents is a very crucial underlying skill to have in algebra. In essence there are just 3 laws and from those we can derive 3 other
interesting/useful rules.
Indices are used to show numbers that have been multiplied by themselves. They can be used instead of the roots such as the square root. The rules make complex calculations that involve powers
easier. Six rules of the Law of Indices: To manipulate math expressions, we can consider using the Law of Indices. These laws only apply to expressions with the same base, for example, 3 4 and 3 2
can be manipulated using the Law of Indices, but we cannot use the Law of Indices to manipulate the expressions 4 5 and 9 7 as their base differs (their bases are 4 and 9, respectively). Indices are
a convenient tool in mathematics to compactly denote the process of taking a power or a root of a number. Therefore, it is important to clearly understand the concept as well as the laws of indices
to be able to apply them later in important applications. There are three rules of indices (or laws of indices) which you have to know and be able to apply to problems involving both numbers and
algebra.For any numbers, x, m, and n, those three rules are The multiplication law – when you multiply terms, you add the powers:; x^m\times x^n=x^{m+n}. The division law – when you divide terms, you
subtract the powers: The laws of indices Introduction A power, or an index, is used to write a product of numbers very compactly. The plural of index is indices. In this leaflet we remind you of how
this is done, and state a number of rules, or laws, which can be used to simplify expressions involving indices. 1. Powers, or indices We write the expression 3×3× 3 Indices revision for A-Level
Maths. This section covers Indices and includes examples. Indexes and indices are both accepted and widely used plurals of the noun index.Both appear throughout the English-speaking world, but
indices prevails in varieties of English from outside North America, while indexes is more common in American and Canadian English.Meanwhile, indices is generally preferred in mathematical,
financial, and technical contexts, while indexes is relatively common
17 Nov 2019 Fund managers will therefore closely monitor the weight of particular companies in their benchmark index. Most benchmark indices use a market-
Indexes and indices are both accepted and widely used plurals of the noun index.Both appear throughout the English-speaking world, but indices prevails in varieties of English from outside North
America, while indexes is more common in American and Canadian English.Meanwhile, indices is generally preferred in mathematical, financial, and technical contexts, while indexes is relatively common
rules of indices are discussed in school textbooks. I pay particular attention to how authors explain the shift from integer to rational exponents. The notation an is Thorax. 2004 May;59(5):421-7.
Validation of predictive rules and indices of severity for community acquired pneumonia. Ewig S(1), de Roux A, Bauer T, García E 24 Jun 2019 The ICE® U.S. 3000 Index is a rules-based, float-adjusted
market capitalization- weighted broad U.S. equity index that has the objective of The CBF benchmarks (TAB Nominal, TAB UF and TADO) and indices (ICP and ICP (Real)) were administered by the Chilean
Banking Association, ABIF, prior to 1. The World Justice Project. Rule of Law Index. Juan Carlos Botero and Alejandro Ponce. World Justice Project. June 2010 | {"url":"https://cryptongsjkn.netlify.app/wacker47975wu/indices-rules-262.html","timestamp":"2024-11-03T10:23:26Z","content_type":"text/html","content_length":"36405","record_id":"<urn:uuid:38a49c64-4928-4c5c-96ef-4c47dea7553b>","cc-path":"CC-MAIN-2024-46/segments/1730477027774.6/warc/CC-MAIN-20241103083929-20241103113929-00288.warc.gz"} |
Shear Panic: Seeking Stress Relief - Digital Engineering
Shear Panic: Seeking Stress Relief
Stress can be puzzling for engineers; FEA modeling can provide guidance.
Stress can be puzzling for engineers; FEA modeling can provide guidance.
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Earlier this year, I presented a short course on interpreting stresses at the NAFEMS World Congress in Quebec. The material was largely drawn from my series of articles in DE (March, May and July
2016; see links at end of article).
At the beginning of the session, I explained why I preferred to start a discussion on stress by looking at force vectors. I was interrupted by a gentleman who made it clear he did not understand—or
even believe—what I was talking about. It turns out he had a computational fluid dynamics (CFD) background, had never dealt with stress and considered it was trivial to deal with!
That got me thinking about revisiting the topic. I decided to build a few simple finite element analysis (FEA) models to explain stress in context, and also expand on the role of shear stress. Shear
stress is something that puzzles a lot of engineers and took me many years to figure out.
Force is a Simple Vector
Fig. 1 shows the end of a rod that is loaded with three separate load cases: Fz, axial force in z, Fx shear force in x and Fy shear force in y.
The relationship between the force vector and the plane it acts on defines the sense of the force. The common loaded plane is the XY plane (or in a contracted notation, referred to as the Z normal
plane). I have color-coded the forces with the format used in the FEA coordinate system symbol seen in the figure.
The forces can be combined to form new loadings, and a vector can describe the resultant. Fig. 2 shows some examples of load combinations. The contributions of the axial and two shear forces are very
easily seen.
In each case, the resultant force is shown as a black arrow.
The force notation used in Figs. 1 and 2 is a bit “wooly,” or arbitrary, and there is no clear distinction between direct and shear forces. You have to rely on the descriptions already noted!
In previous articles, I described the “force cube.” For our purposes, I have modified the diagram in the figure from those articles to align with the specific loading plane and loading direction
across the rod section. I have also labeled direct forces with blue and shear forces with red (not to be confused now with the previous colors.)
This is formalizing the force notation based on the plane the force is acting normal to, and the direction of the force. This allows us to clearly differentiate between direct and shear forces. The
first force index is the plane normal; the second index is the force direction.
In this case, the axial load in Figs. 1 and 2 is Fzz, simplified to Fz. The vertical shear force loosely described in Figs. 1 and 2 as Fx, is more precisely defined as Fzx. It is in the plane normal
to z and is pointing in the x direction. The lateral shear force, described as Fy, is properly defined now as Fzy. It is in the z normal plane, but points in the y direction.
Stress is Not a Simple Vector
As a non-mathematician, I can understand why this heading may leave you cold. What does it mean? Why is it important? This is why I like to use force to start the discussion on stress.
If we imagine the forces are now internally applied to a very tiny cube of material deep inside the rod, we can imagine the stress state does not change across any of the faces. A “stress cube” can
be drawn, where each stress is just the force/area in the limit. Fig. 3 shows the stress cube, which corresponds exactly to the force cube.
Using this diagram, imagine we have an axial stress Sz and a shear stress Szx. Both are being applied to the z normal face, but I cannot combine them as a vector with the simple chain rule I used
with forces. I have to describe the stress state in terms of the component stresses Sz and Szx. And therein lies the difficulty with stresses; it is difficult to describe, or indeed visualize, a
combined stress state.
We have to resort to other methods to try to understand the importance of the stress in terms of its effect on strength. One of the most common methods is to use the von Mises equivalent stress. This
applies weighting to shear terms and combines all the stress contributions. In the case already mentioned, we add (Sz)^2 and (3*Szx)^2 and then take the square root. The factor of 3 on shear stress
inside the square root is based on von Mises theory and ties up very well with the experimental evidence that yield strength in shear is less than yield strength in tension.
Examples Using FEA
First, let’s apply the axial force Fz to the rod. The value is 31,153 lbf. The cross-sectional area, A, is 3.14 inches ^ 2. The stress should be constant across the section at 9,921 psi. The FEA
result is 9,928 psi due to small errors in the mesh cross-sectional area.
It would seem that there is a simple relationship between the axial load and the axial stress. However, this is a stress state, not a vector, and we can look at it in different ways. For example, the
commonly used Tresca failure theory uses the maximum shear stress component from a uniaxial tensile test. How can this be? It is a tension test!
In fact, we can look at the small stress cube, but use it to develop force components, which can be balanced, as they are vectors. The procedure is shown in Fig. 4.
Fig. 4(a) describes the stress state, looking at the side of the cube. A section is cut at an angle theta. A force diagram is shown in Fig. 4(b). The forces are related to the stresses shown in Fig.
4(c) by the cut areas (assuming a unit depth in y).
Fz = Sz*AC
Fzx’= Szx’*AB
The prime symbol (‘) relates to the new coordinate system after rotation by theta.
Using force balances in Fig. 6(b):
Horizontally; Fz’=Fz cos(theta) - equation (1)
Vertically; Fzx’=Fz sin(theta) - equation (2)
From the geometry AC = AB cos(theta)
Hence in equation (1); Sz’*AB = Sz*AC cos(theta) = Sz* AB cos(theta) cos(theta)
Direct stress, Sz’ = Sz* cos^2(theta)
Hence in equation (2); Szx’*AB = Sz*AC sin(theta) = Sz* AB cos(theta) sin(theta)
Shear stress, Szx’ = Sz* cos(theta) sin(theta)
This relationship is shown in Fig. 5.
At zero cut plane angle, all the Sz’ stress is axial (being aligned with z direction). Rotating to 45º gives the maximum shear, Szx’. The balancing direct stress in the local x’ direction can be
calculated in the same way, but using a cut plane rotated 90º.
We can check out these results by using the model, as shown in Fig. 6.
I created two local coordinate systems in the model, rotated at 45º and 60º about the y axis. In the post-processor, I transformed the stresses to each local coordinate system in turn. The stresses
are constant across the section, with very small perturbations due to numerical drift. The stresses in Fig. 6 agree with the theory, using the graph shown in Fig. 5.
The shear stress is labeled as Sxz’ in Fig. 6(b) and 6(c). However, remembering the index notation, this should mean it is acting in the x (normal) plane shearing in the z’ direction. Surely the
stress should be labeled Szx’ as it is in the z’ plane and acts in the x’ direction! Check Fig. 3 for the answer; Szx is equal to Sxz. The post-processor rather lazily just uses one term to describe
both equivalent shear stresses. Though it would take an intensive and complicated effort to automate this.
For each of the coordinate system transformations used (0º, 45º, 90º), the von Mises stress was found to be identical. This is because we only have one stress state due to the axial loading. This is
reduced to a single scalar value with the von Mises equation. The cut plane—or transformation angle—just provides a different way of looking at the same stress state.
The 45º plane is interesting, because it provides the biggest shear component, Sxz’. It is numerically equal to half the axial tensile stress Sz at a zero cut plane. If a material is weaker in shear
compared to tension by a factor greater than 0.5, then a shear failure may occur in a tensile test. If the axial load is compressive, then some materials exhibit a shear failure on a 45º slip plane.
In mild steel Lüders lines or slip lines can sometimes be seen at 45º in a tensile test (see vimeo.com/4586024). In general, the material behavior can get very complex, but the point is that shear
components are lurking in the stress state under axial loading.
Shear Loading
In the next example, I applied a force of 31,153 lbf in the x direction at the end of the rod. This then acts like a cantilever beam with a constant shear force and a linearly increasing bending
moment. The stress state at a position along the rod is shown in Fig. 7.
The direct stress Sz, which is due to bending, is shown in Fig. 7(a). As expected, we have maximum at the top, tension surface and minimum at the bottom compression surface. The values are equal and
opposite with a small numerical drift.
The shear distribution through the depth of a solid section is parabolic. The shears at the extreme top and bottom surfaces must be zero, and build to a maximum at the neutral axis. Basic shear
theory calculates the maximum shear stress to be 4/3 of the nominal shear stress for a circular cross-section. The nominal value is 31,153/3.14 = 9,921 psi. The theoretical maximum is 13,228 psi. The
FEA maximum value is higher at 13,983 psi.
What’s interesting is that the basic theory assumes that shear variation is only a function of depth. This would imply non-zero tangential shears building up at the periphery of the rod, below the
top and bottom positions. The plot in Fig. 7(b) is not in fact a function of depth alone. Popov 1 suggests that the manual calculation gives the correct vertical component of shear stress, but that
there is another horizontal component missing. Fig. 7(c) shows the missing component Syz, which balances the shear state to give zero “out of surface” shear at the free edges. The maximum shear
stress is assumed to be within 5% of the simple theory, which seems to be the case here.
Fig. 8 shows a further attempt to describe the shear state, using a cylindrical coordinate system to look at radial and hoop components.
In Fig. 8(a) the radial stresses are zero at the outside wall—and this is the improvement over the manual calculation. Fig. 8(b) shows the shear flow around the outside wall, being a maximum at the
neutral axis and decaying to zero at the top and bottom. The shear stress sign change is due to left and right edges acting positive upwards, that is, anticlockwise, then clockwise.
Unfortunately, using von Mises stress will not help give a picture of the shear stress contributions. Remember Fig. 7(a)? This shows that the axial stress Sz dominates. This will be included in the
von Mises calculation and obscure the shear stress terms. It may be that a macro could be written to exclude the direct stress in producing a “shear only” pseudo von Mises stress.
Torsional Stresses
A torsional load of 100,000 lbf in. was applied at the end of the bar. From basic calculations, the torsional constant, J, is given by:
J = pi*D^4/32, where D is the diameter.
The torsional constant with D of 2.0 inches is 1.571 in^4. The maximum distance from the center of rotation, c is D/2. The peak shear stress due to torsion is given by:
S,shear,max = TJ/c
The maximum shear stress is 63,662 psi. The FEA results are shown in Fig. 9.
Fig. 9(a) shows the shear stress in the hoop direction, using the basic cylindrical coordinate system. This is the most logical way to plot torsional shear as it flows in this sense. The maximum
value from FEA is 64,319 psi, compared to 63,662 psi in the manual calculation. The error is probably a combination of errors in effective diameter due to meshing of the circular cross-section and
numerical drift in the FEA method. The stress distribution is also distorted away from purely concentric rings of constant stress at the center of the section.
Figs. 9(b) and 9(c) show the component shear stresses in the basic cartesian coordinate system. This is a similar issue to that found previously: shear stress components have to be combined to
achieve meaningful results. In this case the shear stresses in the cylindrical hoop direction, Szy’ completely define the stress state as shown in Fig. 9(a). The radial component, Szx’ is zero.
A further interesting reflection on manual calculation methods is that it is extremely difficult to predict shear stresses under combined loadings, particularly with arbitrary cross-sections. This is
where FEA can give a full distribution—we just have to figure out how to visualize it!
Stress is a difficult quantity to describe and interpret. Considering contributing force vectors in parallel may help visualize stress states. Using simple FEA models of configurations that have
known classical solutions can be a big help.
However, even then analysts need care to interpret the stress components, particularly in shear. We also have the issue that some problems such as shear stress in a circular section due to shear
loading are simplified in classical solutions.
I suggest trying a few other sections using this approach—perhaps an I-beam or thick box girder will show similar interesting results!
1 E. P. Popov. Mechanics of Materials. 2nd ed. Upper Saddle River, NJ: Prentice Hall, 1976.
Previous DE articles by Tony on FEA
About the Author
Tony Abbey
Tony Abbey is a consultant analyst with his own company, FETraining. He also works as training manager for NAFEMS, responsible for developing and implementing training classes, including e-learning
classes. Send e-mail about this article to [email protected].
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Numerical Simulation of Tunneling Current in an Anisotropic Metal-Oxide-Semiconductor Capacitor
In this paper, we have developed a model of the tunneling current through a high- dielectric stack in MOS capacitors with anisotropic masses. The transmittance was numerically calculated by
employing a transfer matrix method and including longitudinal-transverse kinetic energy coupling which is represented by an electron phase velocity in the gate. The transmittance was then applied to
calculate tunneling currents in TiN/HfSiOxN/SiO2/p-Si MOS capacitors. The calculated results show that as the gate electron velocity increases, the transmittance decreases and therefore the tunneling
current reduces. The tunneling current becomes lower as the equivalent oxide thickness (EOT) of HfSiOxN layer increases. When the incident electron passed through the barriers in the normal incident
to the interface, the electron tunneling process becomes easier. It was also shown that the tunneling current was independent of the substrate orientation. Moreover, the model could be used in
designing high speed MOS devices with low tunneling currents.
Govoreanu B, Blomme P, Rosmeulen M, Houdt J V, Meyer K D. A Model for Tunneling Current in Multi-layer Tunnel Dielectrics. Solid-State Electronics. 2003; 47(6): 1045-1053
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Khairurrijal, Noor F A, Abdullah M, Sukirno, Miyazaki S. Theoretical Study on Leakage Current in MOS with High-K Dielectric Stack: Effects of In-plane-Longitudinal Kinetic Energy Coupling and
Anisotropic Masses. Transactions of the Materials Research Society of Japan. 2009; 34(2): 291-295
Wilk G D, Wallace R M, Anthony J M. High-k Gate Dielectrics: Current Status and Materials Properties Considerations. Journal of Applied Physics. 2001; 89(10): 5243-5275
Green M L, Gusev E P, Degraeve R, Garfunkel E. Ultrathin (<4 nm) SiO2 and Si-O-N Gate Dielectric Layers for Silicon Microelectronics: Understanding the Processing, Structure, and Physical and
Electrical Limits. Journal of Applied Physics. 2001; 90(5): 2057-2121
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Workshop on Dielectric Thin Films for Future ULSI Devices – Science Technology. Kanagawa. 2006; 107-108
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Noor F A, Abdullah M, Sukirno, Khairurrijal. Comparison of Electron Direct Transmittance and Tunneling Time of Si(100)/HfO2/Si(100) and Si(110)/HfO2/Si(110) Structures with Ultra-thin Trapezoidal
Barrier. Indonesian Journal of Physics. 2007; 18(2): 41-45
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Noor F A, Abdullah M, Sukirno, Khairurrijal, Ohta A, Miyazaki S. Electron and hole components of tunneling currents through an interfacial oxide-high-k gate stack in metal-oxide-semiconductor
capacitors. Journal of Applied Physics. 2010; 108(9): 093711-1-093711-4
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97(5): 053702-1053702-12
Hasanah L, Abdullah M, Sukirno, Winata T, Khairurrijal. Model of a Tunneling Current in an Anisotropic Si/Si1−xGex/Si Heterostructure with a Nanometer-thick Barrier Including the Effect of
Parallel–Perpendicular Kinetic Energy Coupling. Semiconductor Science and Technology. 2008; 23(12): 125024-1-125024-6
Noor F A, Abdullah M, Sukirno, Khairurrijal. Comparison of Electron Transmittances and Tunneling Currents in an Anisotropic TiNx/HfO2/SiO2/p-Si(100) Metal–Oxide–Semiconductor (MOS) Capacitor. 2010.
Journal of Semiconductors. 31(12): 124002-1-124002-5
Noor F A, Darma Y, Abdullah M, Khairurrijal. 2010. The Effect of Electron Incident Angle on Transmittance and Tunneling Current in an Anisotropic Metal-Oxide-Semiconductor Capacitor with High-
Dielectric Gate Stacks. Asian Physics Symposium 2010 (published in American Institute of Physics (AIP) Conference Proceedings). Bandung. 2010; 1325: 206-209
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Shell quenching phenomena in nuclear charge radii are typically observed at the well-established neutron magic numbers. However, the recent discovery of potential new magic numbers at the neutron
numbers $N = 32$ and $N = 34$ has sparked renewed interest in this mass region. This work further inspects into the charge radii of nuclei around the $N = 28$ shell closure using the relativistic
Hartree-Bogoliubov model. We incorporate meson-exchange and point-coupling effective nucleon-nucleon interactions alongside the Bogoliubov transformation for pairing corrections. To accurately
capture the odd-even staggering and shell closure effects observed in charge radii, neutron-proton correlations around Fermi surface are explicitly considered. The charge radii of Ca and Ni isotopes
are used to test the theoretical model and show an improvement with neutron-proton pairing corrections, in particular for neutron-rich isotopes.Our calculations reveal a inverted parabolic-like trend
in the charge radii along the $N = 28$ isotones for proton numbers $Z$ between 20 and 28. Additionally, the shell closure effect of $Z = 28$ persists across the $N = 28$, $30$, $32$, and $34$
isotonic chains, albeit with a gradual weakening trend. Notably, the significantly abrupt changes in charge radii are observed across $Z = 22$ along both the $N = 32$ and $N = 34$ isotonic chains.
This kink at $Z = 22$ comes from the sudden decrease of the neuron-proton correlation around Fermi surfaces across $Z = 22$ for $N = 30$, 32, and 34 isotones, and might provide a signature for
identifying the emergence of neutron magic numbers $N=32$ and 34.Furthermore, the calculated charge radii for these isotonic chains ($N = 28$, 30, 32, and 34) can serve as reliable guidelines for
future experimental measurements.
This study investigates the tetrahedral structure in $^{80}$Zr and Lambda ($\Lambda$) impurity effect in $^{81}_{~\Lambda}$Zr using the multidimensionally constrained relativistic Hartree-Bogoliubov
model. The ground states of both $^{80}$Zr and $^{81}_{~\Lambda}$Zr exhibit a tetrahedral configuration, accompanied by prolate and axial-octupole shape isomers. Our calculations reveal there are
changes in the deformation parameters $\beta_{20}$, $\beta_{30}$, and $\beta_{32}$ upon $\Lambda$ binding to $^{80}$Zr, except for $\beta_{32}$ when $\Lambda$ occupies $p$-orbits. Compared to the two
shape isomers, the $\Lambda$ particle exhibits weaker binding energy in the tetrahedral state when occupying the $1/2^+[000](\Lambda_s)$ or $1/2^-[110]$ single-particle states. In contrast, the
strongest binding occurs for the $\Lambda$ particle in the $1/2^-[101]$ state with tetrahedral shape. Besides, a large $\Lambda$ separation energy may not necessarily correlate with a significant
overlap between the density distributions of the $\Lambda$ particle and the nuclear core, particularly for tetrahedral hypernuclei.
The equation of state (EOS) of extremely dense matter is crucial for understanding the properties of rotating neutron stars. Starting from the widely used realistic Bonn potentials rooted in a
relativistic framework, we derive EOSs by performing the state-of-the-art relativistic Brueckner-Hartree-Fock (RBHF) calculations in the full Dirac space. The self-consistent and simultaneous
consideration of both positive- and negative-energy states (NESs) of the Dirac equation allows us to avoid the uncertainties present in calculations where NESs are treated using approximations. To
manifest the impact of rotational dynamics, several structural properties of neutron stars across a wide range of rotation frequencies and up to the Keplerian limit are obtained, including the
gravitational and baryonic masses, the polar and equatorial radii, and the moments of inertia. Our theoretical predictions align well with the latest astrophysical constraints from the observations
on massive neutron stars and joint mass-radius measurements. The maximum mass for rotating configurations can reach up to $2.93M_{\odot}$ for Bonn A potential, while the radius of a $1.4M_\odot$
neutron star for non-rotating case can be extended to around 17 km through the constant baryonic mass sequences. Relations with good universalities between the Keplerian frequency and static mass as
well as radius are obtained, from which the radius of the black widow PSR J0952-0607 is predicted to be less than 19.58 km. Furthermore, to understand how rotation deforms the equilibrium shape of a
neutron star, the eccentricity is also calculated. The approximate universality between the eccentricity at the Keplerian frequency and the gravitational mass is found.
The subtraction function plays a pivotal role in calculations involving the forward Compton amplitude, which is crucial for predicting the Lamb shift in muonic atom, as well as the proton-neutron
mass difference. In this work, we present a lattice QCD calculation of the subtraction function using two domain wall fermion gauge ensembles at the physical pion mass. We utilize a recently proposed
subtraction point, demonstrating its advantage in mitigating statistical and systematic uncertainties by eliminating the need for ground-state subtraction. Our results reveal significant
contributions from $N\pi$ intermediate states to the subtraction function. Incorporating these contributions, we compute the proton, neutron and nucleon isovector subtraction functions at photon
momentum transfer $Q^2\in[0,2]$ GeV$^2$. For the proton subtraction function, we compare our lattice results with chiral perturbation theory prediction at low $Q^2$ and with the results from the
perturbative operator-product expansion at high $Q^2$. Finally, using these subtraction functions as input, we determine their contribution to two-photon exchange effects in the Lamb shift and
isovector nucleon electromagnetic self-energy.
We propose a mechanism which explains the masses of $\eta$ and $\eta'$ mesons without invoking the explicit violation of $U(1)_A$ symmetry by the chiral anomaly. It is shown that the U(1) problem,
the problem for which the prediction of $\eta$ and $\eta'$ masses in the simple chiral perturbation theory largely deviates from the experimental values, is actually resolved by considering the first
order contribution of the disconnected meson correlator with respect to the quark mass. The bound of Weinberg $m_\eta^2 \le 3 m_\pi^2$ is fulfilled by considering the negative squared mass of $\eta$
or $\eta'$ which is just the saddle point of the QCD effective potential, and 20% level agreements with experimental data are obtained by just fitting one low energy constant. We provide the leading
chiral Lagrangian due to the disconnected contribution in 3-flavor QCD, and also discuss the 2- and 4-flavor cases as well as the consistency of our mechanism with the chiral restoration at high
temperature found in lattice calculations.
Simulations of collisions of fundamental particles on a quantum computer are expected to have an exponential advantage over classical methods and promise to enhance searches for new physics.
Furthermore, scattering in scalar field theory has been shown to be BQP-complete, making it a representative problem for which quantum computation is efficient. As a step toward large-scale quantum
simulations of collision processes, scattering of wavepackets in one-dimensional scalar field theory is simulated using 120 qubits of IBM's Heron superconducting quantum computer ibm_fez. Variational
circuits compressing vacuum preparation, wavepacket initialization, and time evolution are determined using classical resources. By leveraging physical properties of states in the theory, such as
symmetries and locality, the variational quantum algorithm constructs scalable circuits that can be used to simulate arbitrarily-large system sizes. A new strategy is introduced to mitigate errors in
quantum simulations, which enables the extraction of meaningful results from circuits with up to 4924 two-qubit gates and two-qubit gate depths of 103. The effect of interactions is clearly seen, and
is found to be in agreement with classical Matrix Product State simulations. The developments that will be necessary to simulate high-energy inelastic collisions on a quantum computer are discussed. | {"url":"https://papers.cool/arxiv/nucl-th","timestamp":"2024-11-06T05:18:08Z","content_type":"text/html","content_length":"34257","record_id":"<urn:uuid:1e62b303-35f7-4ead-ac6c-581d6e9bfd80>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00119.warc.gz"} |
CS3 Data Structures & Algorithms - BC and Slides -
7.5.1. Binary Tree Traversals
Often we wish to process a binary tree by “visiting” each of its nodes, each time performing a specific action such as printing the contents of the node. Any process for visiting all of the nodes in
some order is called a traversal. Any traversal that lists every node in the tree exactly once is called an enumeration of the tree’s nodes. Some applications do not require that the nodes be visited
in any particular order as long as each node is visited precisely once. For other applications, nodes must be visited in an order that preserves some relationship.
7.5.1.1. Preorder Traversal
For example, we might wish to make sure that we visit any given node before we visit its children. This is called a preorder traversal.
Figure 7.5.1: A binary tree for traversal examples.
Example 7.5.1
The preorder enumeration for the tree of Figure 7.5.1 is A B D C E G F H I.
The first node printed is the root. Then all nodes of the left subtree are printed (in preorder) before any node of the right subtree.
Server Error
7.5.1.2. Postorder Traversal
Alternatively, we might wish to visit each node only after we visit its children (and their subtrees). For example, this would be necessary if we wish to return all nodes in the tree to free store.
We would like to delete the children of a node before deleting the node itself. But to do that requires that the children’s children be deleted first, and so on. This is called a postorder traversal.
Example 7.5.2
The postorder enumeration for the tree of Figure 7.5.1 is D B G E H I F C A.
Server Error
7.5.1.3. Inorder Traversal
An inorder traversal first visits the left child (including its entire subtree), then visits the node, and finally visits the right child (including its entire subtree). The binary search tree makes
use of this traversal to print all nodes in ascending order of value.
Example 7.5.3
The inorder enumeration for the tree of Figure 7.5.1 is B D A G E C H F I.
Server Error
7.5.1.4. Implementation
Now we will discuss some implementations for the traversals, but we need to define a node ADT to work with. Just as a linked list is composed of a collection of link objects, a tree is composed of a
collection of node objects. Here is an ADT for binary tree nodes, called BinNode. This class will be used by some of the binary tree structures presented later. Member functions are provided that set
or return the element value, return a pointer to the left child, return a pointer to the right child, or indicate whether the node is a leaf.
interface BinNode { // Binary tree node ADT
// Get and set the element value
public Object value();
public void setValue(Object v);
// return the children
public BinNode left();
public BinNode right();
// return TRUE if a leaf node, FALSE otherwise
public boolean isLeaf();
interface BinNode<E> { // Binary tree node ADT
// Get and set the element value
public E value();
public void setValue(E v);
// return the children
public BinNode<E> left();
public BinNode<E> right();
// return TRUE if a leaf node, FALSE otherwise
public boolean isLeaf();
A traversal routine is naturally written as a recursive function. Its input parameter is a pointer to a node which we will call rt because each node can be viewed as the root of a some subtree. The
initial call to the traversal function passes in a pointer to the root node of the tree. The traversal function visits rt and its children (if any) in the desired order. For example, a preorder
traversal specifies that rt be visited before its children. This can easily be implemented as follows.
static void preorder(BinNode rt) {
if (rt == null) return; // Empty subtree - do nothing
visit(rt); // Process root node
preorder(rt.left()); // Process all nodes in left
preorder(rt.right()); // Process all nodes in right
static <E> void preorder(BinNode<E> rt) {
if (rt == null) { return; } // Empty subtree - do nothing
visit(rt); // Process root node
preorder(rt.left()); // Process all nodes in left
preorder(rt.right()); // Process all nodes in right
Function preorder first checks that the tree is not empty (if it is, then the traversal is done and preorder simply returns). Otherwise, preorder makes a call to visit, which processes the root node
(i.e., prints the value or performs whatever computation as required by the application). Function preorder is then called recursively on the left subtree, which will visit all nodes in that subtree.
Finally, preorder is called on the right subtree, visiting all nodes in the right subtree. Postorder and inorder traversals are similar. They simply change the order in which the node and its
children are visited, as appropriate. | {"url":"https://opendsa-server.cs.vt.edu/ODSA/Books/usek/gin231-202310/fall-2022/TR_930am/html/BinaryTreeTraversal.html","timestamp":"2024-11-02T14:54:49Z","content_type":"text/html","content_length":"31019","record_id":"<urn:uuid:e5c0ec20-31d9-41be-97f5-f00dbf860b50>","cc-path":"CC-MAIN-2024-46/segments/1730477027714.37/warc/CC-MAIN-20241102133748-20241102163748-00130.warc.gz"} |
Overview of Hmisc Library
HmiscOverview {Hmisc} R Documentation
Overview of Hmisc Library
The Hmisc library contains many functions useful for data analysis, high-level graphics, utility operations, functions for computing sample size and power, translating SAS datasets into R, imputing
missing values, advanced table making, variable clustering, character string manipulation, conversion of R objects to LaTeX code, recoding variables, and bootstrap repeated measures analysis. Most of
these functions were written by F Harrell, but a few were collected from statlib and from s-news; other authors are indicated below. This collection of functions includes all of Harrell's submissions
to statlib other than the functions in the rms and display libraries. A few of the functions do not have “Help” documentation.
To make Hmisc load silently, issue options(Hverbose=FALSE) before library(Hmisc).
Function Name Purpose
abs.error.pred Computes various indexes of predictive accuracy based
on absolute errors, for linear models
addMarginal Add marginal observations over selected variables
all.is.numeric Check if character strings are legal numerics
approxExtrap Linear extrapolation
aregImpute Multiple imputation based on additive regression,
bootstrapping, and predictive mean matching
areg.boot Nonparametrically estimate transformations for both
sides of a multiple additive regression, and
bootstrap these estimates and R^2
ballocation Optimum sample allocations in 2-sample proportion test
binconf Exact confidence limits for a proportion and more accurate
(narrower!) score stat.-based Wilson interval
(Rollin Brant, mod. FEH)
bootkm Bootstrap Kaplan-Meier survival or quantile estimates
bpower Approximate power of 2-sided test for 2 proportions
Includes bpower.sim for exact power by simulation
bpplot Box-Percentile plot
(Jeffrey Banfield, umsfjban@bill.oscs.montana.edu)
bpplotM Chart extended box plots for multiple variables
bsamsize Sample size requirements for test of 2 proportions
bystats Statistics on a single variable by levels of >=1 factors
bystats2 2-way statistics
character.table Shows numeric equivalents of all latin characters
Useful for putting many special chars. in graph titles
(Pierre Joyet, pierre.joyet@bluewin.ch)
ciapower Power of Cox interaction test
cleanup.import More compactly store variables in a data frame, and clean up
problem data when e.g. Excel spreadsheet had a non-
numeric value in a numeric column
combine.levels Combine infrequent levels of a categorical variable
confbar Draws confidence bars on an existing plot using multiple
confidence levels distinguished using color or gray scale
contents Print the contents (variables, labels, etc.) of a data frame
cpower Power of Cox 2-sample test allowing for noncompliance
Cs Vector of character strings from list of unquoted names
csv.get Enhanced importing of comma separated files labels
cut2 Like cut with better endpoint label construction and allows
construction of quantile groups or groups with given n
datadensity Snapshot graph of distributions of all variables in
a data frame. For continuous variables uses scat1d.
dataRep Quantify representation of new observations in a database
ddmmmyy SAS “date7” output format for a chron object
deff Kish design effect and intra-cluster correlation
describe Function to describe different classes of objects.
Invoke by saying describe(object). It calls one of the
describe.data.frame Describe all variables in a data frame (generalization
of SAS UNIVARIATE)
describe.default Describe a variable (generalization of SAS UNIVARIATE)
dotplot3 A more flexible version of dotplot
Dotplot Enhancement of Trellis dotplot allowing for matrix
x-var., auto generation of Key function, superposition
drawPlot Simple mouse-driven drawing program, including a function
for fitting Bezier curves
Ecdf Empirical cumulative distribution function plot
errbar Plot with error bars (Charles Geyer, U. Chi., mod FEH)
event.chart Plot general event charts (Jack Lee, jjlee@mdanderson.org,
Ken Hess, Joel Dubin; Am Statistician 54:63-70,2000)
event.history Event history chart with time-dependent cov. status
(Joel Dubin, jdubin@uwaterloo.ca)
find.matches Find matches (with tolerances) between columns of 2 matrices
first.word Find the first word in an R expression (R Heiberger)
fit.mult.impute Fit most regression models over multiple transcan imputations,
compute imputation-adjusted variances and avg. betas
format.df Format a matrix or data frame with much user control
(R Heiberger and FE Harrell)
ftupwr Power of 2-sample binomial test using Fleiss, Tytun, Ury
ftuss Sample size for 2-sample binomial test using " " " "
(Both by Dan Heitjan, dheitjan@biostats.hmc.psu.edu)
gbayes Bayesian posterior and predictive distributions when both
the prior and the likelihood are Gaussian
getHdata Fetch and list datasets on our web site
hdquantile Harrell-Davis nonparametric quantile estimator with s.e.
histbackback Back-to-back histograms (Pat Burns, Salomon Smith
Barney, London, pburns@dorado.sbi.com)
hist.data.frame Matrix of histograms for all numeric vars. in data frame
Use hist.data.frame(data.frame.name)
histSpike Add high-resolution spike histograms or density estimates
to an existing plot
hoeffd Hoeffding's D test (omnibus test of independence of X and Y)
impute Impute missing data (generic method)
interaction More flexible version of builtin function
is.present Tests for non-blank character values or non-NA numeric values
james.stein James-Stein shrinkage estimates of cell means from raw data
labcurve Optimally label a set of curves that have been drawn on
an existing plot, on the basis of gaps between curves.
Also position legends automatically at emptiest rectangle.
label Set or fetch a label for an R-object
Lag Lag a vector, padding on the left with NA or ''
latex Convert an R object to LaTeX (R Heiberger & FE Harrell)
list.tree Pretty-print the structure of any data object
(Alan Zaslavsky, zaslavsk@hcp.med.harvard.edu)
Load Enhancement of load
mask 8-bit logical representation of a short integer value
(Rick Becker)
matchCases Match each case on one continuous variable
matxv Fast matrix * vector, handling intercept(s) and NAs
mgp.axis Version of axis() that uses appropriate mgp from
mgp.axis.labels and gets around bug in axis(2, ...)
that causes it to assume las=1
mgp.axis.labels Used by survplot and plot in rms library (and other
functions in the future) so that different spacing
between tick marks and axis tick mark labels may be
specified for x- and y-axes.
Use mgp.axis.labels('default') to set defaults.
Users can set values manually using
mgp.axis.labels(x,y) where x and y are 2nd value of
par('mgp') to use. Use mgp.axis.labels(type=w) to
retrieve values, where w='x', 'y', 'x and y', 'xy',
to get 3 mgp values (first 3 types) or 2 mgp.axis.labels.
minor.tick Add minor tick marks to an existing plot
mtitle Add outer titles and subtitles to a multiple plot layout
multLines Draw multiple vertical lines at each x
in a line plot
%nin% Opposite of %in%
nobsY Compute no. non-NA observations for left hand formula side
nomiss Return a matrix after excluding any row with an NA
panel.bpplot Panel function for trellis bwplot - box-percentile plots
panel.plsmo Panel function for trellis xyplot - uses plsmo
pBlock Block variables for certain lattice charts
pc1 Compute first prin. component and get coefficients on
original scale of variables
plotCorrPrecision Plot precision of estimate of correlation coefficient
plsmo Plot smoothed x vs. y with labeling and exclusion of NAs
Also allows a grouping variable and plots unsmoothed data
popower Power and sample size calculations for ordinal responses
(two treatments, proportional odds model)
prn prn(expression) does print(expression) but titles the
output with 'expression'. Do prn(expression,txt) to add
a heading (‘txt’) before the ‘expression’ title
pstamp Stamp a plot with date in lower right corner (pstamp())
Add ,pwd=T and/or ,time=T to add current directory
name or time
Put additional text for label as first argument, e.g.
pstamp('Figure 1') will draw 'Figure 1 date'
putKey Different way to use key()
putKeyEmpty Put key at most empty part of existing plot
rcorr Pearson or Spearman correlation matrix with pairwise deletion
of missing data
rcorr.cens Somers' Dxy rank correlation with censored data
rcorrp.cens Assess difference in concordance for paired predictors
rcspline.eval Evaluate restricted cubic spline design matrix
rcspline.plot Plot spline fit with nonparametric smooth and grouped estimates
rcspline.restate Restate restricted cubic spline in unrestricted form, and
create TeX expression to print the fitted function
reShape Reshape a matrix into 3 vectors, reshape serial data
rm.boot Bootstrap spline fit to repeated measurements model,
with simultaneous confidence region - least
squares using spline function in time
rMultinom Generate multinomial random variables with varying prob.
samplesize.bin Sample size for 2-sample binomial problem
(Rick Chappell, chappell@stat.wisc.edu)
sas.get Convert SAS dataset to S data frame
sasxport.get Enhanced importing of SAS transport dataset in R
Save Enhancement of save
scat1d Add 1-dimensional scatterplot to an axis of an existing plot
(like bar-codes, FEH/Martin Maechler,
maechler@stat.math.ethz.ch/Jens Oehlschlaegel-Akiyoshi,
score.binary Construct a score from a series of binary variables or
sedit A set of character handling functions written entirely
in R. sedit() does much of what the UNIX sed
program does. Other functions included are
substring.location, substring<-, replace.string.wild,
and functions to check if a string is numeric or
contains only the digits 0-9
setTrellis Set Trellis graphics to use blank conditioning panel strips,
line thickness 1 for dot plot reference lines:
setTrellis(); 3 optional arguments
show.col Show colors corresponding to col=0,1,...,99
show.pch Show all plotting characters specified by pch=.
Just type show.pch() to draw the table on the
current device.
showPsfrag Use LaTeX to compile, and dvips and ghostview to
display a postscript graphic containing psfrag strings
solvet Version of solve with argument tol passed to qr
somers2 Somers' rank correlation and c-index for binary y
spearman Spearman rank correlation coefficient spearman(x,y)
spearman.test Spearman 1 d.f. and 2 d.f. rank correlation test
spearman2 Spearman multiple d.f. \rho^2, adjusted \rho^2, Wilcoxon-Kruskal-
Wallis test, for multiple predictors
spower Simulate power of 2-sample test for survival under
complex conditions
Also contains the Gompertz2,Weibull2,Lognorm2 functions.
spss.get Enhanced importing of SPSS files using read.spss function
src src(name) = source("name.s") with memory
store store an object permanently (easy interface to assign function)
strmatch Shortest unique identifier match
(Terry Therneau, therneau@mayo.edu)
subset More easily subset a data frame
substi Substitute one var for another when observations NA
summarize Generate a data frame containing stratified summary
statistics. Useful for passing to trellis.
summary.formula General table making and plotting functions for summarizing
summaryD Summarizing using user-provided formula and dotchart3
summaryM Replacement for summary.formula(..., method='reverse')
summaryP Multi-panel dot chart for summarizing proportions
summaryS Summarize multiple response variables for multi-panel
dot chart or scatterplot
summaryRc Summary for continuous variables using lowess
symbol.freq X-Y Frequency plot with circles' area prop. to frequency
sys Execute unix() or dos() depending on what's running
tabulr Front-end to tabular function in the tables package
tex Enclose a string with the correct syntax for using
with the LaTeX psfrag package, for postscript graphics
transace ace() packaged for easily automatically transforming all
variables in a matrix
transcan automatic transformation and imputation of NAs for a
series of predictor variables
trap.rule Area under curve defined by arbitrary x and y vectors,
using trapezoidal rule
trellis.strip.blank To make the strip titles in trellis more visible, you can
make the backgrounds blank by saying trellis.strip.blank().
Use before opening the graphics device.
t.test.cluster 2-sample t-test for cluster-randomized observations
uncbind Form individual variables from a matrix
upData Update a data frame (change names, labels, remove vars, etc.)
units Set or fetch "units" attribute - units of measurement for var.
varclus Graph hierarchical clustering of variables using squared
Pearson or Spearman correlations or Hoeffding D as similarities
Also includes the naclus function for examining similarities in
patterns of missing values across variables.
num.denom.setup Set of function for obtaining weighted estimates
xy.group Compute mean x vs. function of y by groups of x
xYplot Like trellis xyplot but supports error bars and multiple
response variables that are connected as separate lines
ynbind Combine a series of yes/no true/false present/absent variables into a matrix
zoom Zoom in on any graphical display
(Bill Dunlap, bill@statsci.com)
Copyright Notice
This program is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 2, or (at your
option) any later version.
This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General
Public License for more details.
In short: You may use it any way you like, as long as you don't charge money for it, remove this notice, or hold anyone liable for its results. Also, please acknowledge the source and communicate
changes to the author.
If this software is used is work presented for publication, kindly reference it using for example:
Harrell FE (2014): Hmisc: A package of miscellaneous R functions. Programs available from https://hbiostat.org/R/Hmisc/.
Be sure to reference R itself and other libraries used.
Frank E Harrell Jr
Professor of Biostatistics
Vanderbilt University School of Medicine
Nashville, Tennessee
See Alzola CF, Harrell FE (2004): An Introduction to S and the Hmisc and Design Libraries at https://hbiostat.org/R/doc/sintro.pdf for extensive documentation and examples for the Hmisc package.
version 5.1-3 | {"url":"https://search.r-project.org/CRAN/refmans/Hmisc/html/Overview.html","timestamp":"2024-11-13T15:29:56Z","content_type":"text/html","content_length":"40361","record_id":"<urn:uuid:3fc2b27a-0082-4652-bcd4-69289b683052>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00808.warc.gz"} |
Mathematically Correct Breakfast -- Linked Bagel Halves
Jay Taylor's notes
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Mathematically Correct Breakfast -- Linked Bagel Halves
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Clipped on: 2018-01-26
Mathematically Correct Breakfast
How to Slice a Bagel into Two Linked Halves
George W. Hart
A is the highest point above the +X axis. B is where the +Y axis enters the bagel.
C is the lowest point below the -X axis. D is where the -Y axis exits the bagel.
and the points. You don't need to actually write on the bagel to cut it properly.
As it goes 360 degrees around the Z axis, it also goes 360 degrees around the bagel.
An ideal knife could enter on the black line and come out exactly opposite, on the red line.
But in practice, it is easier to cut in halfway on both the black line and the red line.
The cutting surface is a two-twist Mobius strip; it has two sides, one for each half.
the hole of the other. (So when you buy your bagels, pick ones with the biggest holes.)
not need to draw on the bagel. Here the two parts are pulled slightly apart.
(You can make both be the opposite handedness if you follow these instructions in a mirror.)
You can toast them in a toaster oven while linked together, but move them around every
minute or so, otherwise some parts will cook much more than others, as shown in this half.
the intellectual stimulation, you get more cream cheese, because there is slightly more surface area.
Topology problem: Modify the cut so the cutting surface is a one-twist Mobius strip.
(You can still get cream cheese into the cut, but it doesn't separate into two parts.)
Calculus problem: What is the ratio of the surface area of this linked cut
to the surface area of the usual planar bagel slice?
For future research: How to make Mobius lox...
Note: I have had my students do this activity in my Computers and Sculpture class. It is very successful if the students work in pairs, with two bagels per team. For the first bagel, I have them
draw the indicated lines with a "sharpie". Then they can do the second bagel without the lines. (We omit the schmear of cream cheese.) After doing this, one can better appreciate the stone carving
Keizo Ushio
, who makes analogous cuts in granite to produce monumental sculpture.
Addendum: I made a
showing how to do this. | {"url":"https://jaytaylor.com/notes/node/1517003130000.html","timestamp":"2024-11-04T15:11:27Z","content_type":"text/html","content_length":"6806","record_id":"<urn:uuid:abb30fff-aad5-4772-a792-7751c6c46f26>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00390.warc.gz"} |
Compositionality and the Mass Customization of Economic Models
I thank Oliver Beige for many helpful comments.
Fables or algorithms?
Economic theory formulates thoughts via what we call “models.” The word model sounds more scientific than the word fable or tale, but I think we are talking about the same thing. (Ariel
Are economic models useful for making decisions? One might expect that there is a clear answer to this simple question. But in fact opinions on the usefulness or non-usefulness of models as well as
what exactly makes models useful vary widely - within the economic profession and of course even more so beyond. Sometimes the question feels like a Rorschach test - telling more about the person
than about the subject.
In this post, I want to explore the question of usefulness. Even more so, I want to explore how the usefulness ties into the modelling process. The reason for doing so is simple: Part of our efforts
at CyberCat is to build software tools to improve and accelerate the modelling process.
The importance of this is also evident: If models are useful, and we improve the process generating useful models, we improve our decision-making. And in so far as these improvements tie into
computing technology, as they do in our opinion, improvements could be significant.
Economic models
My question, "are economic models useful", is quite lofty. So, let's first do some unpacking.
What do I mean by economic model? A mathematical, formal model which relates to the domain of decision-making at hand. A prototypical example is a model that tells us how to bid in an auction. Such
models are often classified as applied economic models.^2
Why do I emphasize "economic"? If my question was: Are mathematical models useful for decision-making, the answer would be a simple yes and we could call it a day. Operations research models are in
production for a multitude of tasks (job scheduling, inventory management, revenue management etc.). In fact, many of these models are so pervasive that it is easy to forget them. Just think about
the business models that have been built on the navigation and prediction functionalities of Google maps.
The distinction between operations research and economics is obviously blurry and more due to artificial academic barriers than fundamental differences (check out Oliver's post on this). I am making
the crude distinction that economic models are about several agents interacting - most often strategically - whereas traditional operations research models are focused on single decision-makers.
Now, this is crude because obviously operations research by now also includes auctions and other models that are interactive in this way. Moreover, as Oliver pointed out in another post several
leading economists who advanced the practical use of economic models (which we still come to) have an operations research background.
It is, I think, also not a coincidence that operations research has moved into the realm of interactive agents: Due to globalization and in particular the internet, companies have become more
interconnected and also have much more technical leverage. 50 years ago, the idea that a regular company could be designing their own market probably would have been quite a thing. Today, it is part
of the standard startup toolkit.
Technology and interconnectedness are driving the need for models that help decide in such a world as well as design the frameworks and protocols in which decisions take place. Economic models are
the natural candidate for this task.
Let's turn to the central part of my question. What do I mean by useful? Opinions on this vary widely. According to Rubinstein, the question how a model can be useful is already ill-posed. Models are
not useful. Models might carry a lesson and can transform our thinking. But they are of little value for concrete decisions.
In economics, Rubinstein's position is an extreme point. On the other side of the extreme, economists and even more importantly computer scientists are working on market design and mechanism design
models.^3 Models in this spirit are "very" practical: they do affect decisions in a concrete sense - they get implemented in the form of algorithms and are embedded in software systems.
We can think of fables and algorithms as two ends of a spectrum - from basically irrelevant to decisive for a choice we have to make. While it is hard to precisely locate a given model on this
"usefulness" line, we can consider how a model can become more useful when moving along the spectrum. Of course, what constitutes value and who benefits how from a model changes along this path as
well. The usefulness of a model is a matter of degree and not an absolute.
Let's begin at the fable end and start moving inroads. How can a model produce value? If we are faced with infinitely many ways to think about a situation, even a simplistic model can be valuable. It
helps to focus and to select a few key dimensions. This aspect becomes even more important in an organizational context where people have to collaborate and it is very easy to get lost in a myriad of
possibility and different interpretations.
Many classic games (in the game theory sense) like the Battle of the Sexes, Matching Pennies, and of course the Prisoners' Dilemma help to focus on key issues - for instance the interdependency
between actions and their consequences. To be clear, the connection how to map a model into a concrete decision is very loose in this case and the value of the model lies in the eyes of the analyst.
These games often focus on a few actions ("defect" or "cooperate"). Moreover, agents have perfect information about the consequences of their actions and the actions of others. In many situations,
e.g. in business contexts, choices are more fine-grained and information is not perfect. Models in Industrial Organization routinely incorporate these aspects, for instance analyzing competition
between companies. From a practical perspective, these models often resemble the following pattern: If we had information X, the model would help us make a decision. Consider strategic pricing: It is
standard in these models to assume demand to be known or at least drawn from a known distribution. The demand curve will then be typically a smooth, mathematically well behaved object. Such models
can produce insights - no doubt about it.
But they rarely help to make a concrete decision, e.g. what prices to charge. There are many reasons for this but let me just give an obvious one as a co-founder of a startup: I would love to
maximize a demand curve and price our services accordingly. But the reality is: I do not have a curve. Hell, if I am lucky I observe a handful of points (price-quantity combinations). But these
points might not even be on any actual demand curve in the model's sense. So, while useful for structuring discussions around pricing, in the actual decision to set prices, the model is only one
(possibly small) input. And this is very typical. Such models provide insights and do help to inform decisions. But they are only part of a collage of inputs into a decision.
There are economic models which do play a more dominant role in shaping decisions. Consider auctions. There is a rich theory that helps to choose a specific auction format to solve a given allocation
problem. Still, even in this case, there are gaps between the model and the actual implementation, for instance when it comes to multi-unit auctions.
The examples I gave are obviously not meant to be exhaustive. There are other ways how a model can be useful. But this is not so important. The main point is, that all along the usefulness line,
economic models can produce value. The question is not whether a model produces a choice but whether, at the margin, it helps us make better decisions. And this can happen all along the spectrum.
Moreover, ceteris paribus, the further we move along the path towards the algorithm end, the more influence the economic model gains relative to other inputs into a decision and the more value it
If we accept this, then an immediate question comes up: How can we push models from the fable side more towards the algorithm side? Let's explore this.
The process of modelling and the library of models
I first need to discuss how models get located on a specific point on the usefulness line in the first place. But this requires digging into the actual modelling process. Note again that I am only
interested in "instrumental" modelling - models that are useful for a specific decision at hand. My exposition will be simplistic and subjective. I will neither cover the full range of opinions nor
be grounded in any philosophical discussions of economics. This is just me describing how I see this (and also how I have used models in my work at 20squares).
Applied models in economics are a mixture of mathematical formalism and interpretative mapping connecting the internals of the model to the outside world. Mappings are not exclusive: The same formal
structure can be mapped to different domains. The Prisoner's dilemma is such an example. It has various interpretations from two prisoners in separate cells to nuclear powers facing each other.
The formal, inner workings of models are "closed" objects. What do I mean by that? Each model describes a typically isolated mechanism, e.g. connecting a specific market design with some desirable
properties. The formal model has no interfaces to the outside world. And therefore it cannot be connected to other models at the formal level. In that sense a model is a self-contained story.
Let me contrast this with a completely different domain: If one thinks about functional programming, then everything is about the composability of functions (modulo types). The whole point of
programming is that one program (which is a function) can be composed with another program (which is a function) to produce a new program (which is a function).^4
Back to economic models. When it comes to applications, the "right" model is not god given. So, how does the process of modelling real world phenomena look like?
As observed by Dani Rodrik^5, the evolution of applied models in economics is different from the evolution of theories in physics. In physics one theory regularly supersedes another theory. In
economics, the same rarely happens. The practice of modelling is rather about developing new models, like new stories, that then get added to the canon.
One can compare this to a library where each book stands for a model that has been added at some point. Applied modelling then means mapping a concrete problem into a model among the existing staple
or, if something is missing, develop a new model and add it to the canon.
Inherent in this process is the positioning of a model on a specific point in the spectrum between fables and algorithms. Models mostly take on a fixed position on the line and will stay there. There
are exogenous factors that influence the positioning and that can change over time. For instance, the domain matters. If you build a model of an intergallactic trading institution, it is safe to
assume that this model will not be directly useful. Of course, this might change.
Like stories, certain models do get less fashionable over time, others become prominent for a while, and a select few stay ever-greens. Economists studying financial crises in 2006 were not really
standing in the spotlight of attention. That changed radically one year later.^6
Let me emphasize another aspect. I depicted applied models as packages of internal, formal structure and interpretative maps connecting the internals with some outside phenomenon. This interpretative
mapping is subjective. And indeed discussions in economic policy often do not focus on the internal consistency of models but instead are more about the adequateness of the model's mapping (and its
assumptions) for the question at hand. Ultimately, this discourse is verbal and it is structurally not that different from deciding which story in the bible (or piece of literature, or movie) is the
best representation of a specific decision problem.
The more a model will lean towards the fable side, the more it will be just one piece in a larger puzzle and the more other sources of information a decision-maker will seek. This might include other
economic models but of course also sources outside. Different models and other sources of information need to be integrated.
As a consequence, whatever powers we gain through the formal model, a lot of it is lost the moment we move beyond the model's inner working and need to compare and select between different models as
well as integrate with other sources. A synthesis at the formal level is not feasible.
Let me summarize so far: A model's position on the spectrum of fable to algorithm is mostly given. There is not much we can do to push a single model along. Moreover, we have no systematic way of
synthesizing different models - which would be another possibility to advance along the spectrum.
We have been mostly concerned with the type of output the modelling process generates. Let's also briefly turn to the inputs. Modelling by and large today is not that different compared to 50 years
ago. Sure, co-authorships have increased, computers are used, and papers circulate online. But in the end, the modelling process is still a slow, labor-intensive craft and demands a lot from the
modeller. He or she needs knowledge in the domain, must be familiar with the canon of models, needs judgment to balance off the tradeoffs involved in different models, etc.
This makes the modelling process costly. And it means we cannot brute force our way to push models from fable to algorithm. In fact, in the context of policy questions many economists like Dani
Rodrik^7 criticize the fact that discussions focus on a single model whereas a discussion would be more robust if it could be grounded in a collage of different models. But generating an adequate
model is just very costly.^8
Taken together, the nature of the model generating process as well as its cost function, are bottlenecks that we need to overcome if we want to transform the modelling process.
Let's go back to our (functional) programming domain to see an alternative paradigm. Here, we are also relying on libraries. But the process of using them is markedly different. Sure, one can just
simply choose programs from a library an apply it. But one can also compose models and form new, more powerful programs. One can synthesize different programs; and one can find better abstractions
through the patterns of multiple programs which do similar things. Lastly, one can refine a program by adding details. And of course, if you consider statistical modelling, this modularity is already
present in many software packages.
It is modularity which gives computing scalability. And it is this missing modularity which severely limits the scalability of economic modelling.
Consider the startup pricing example I gave before. Say, I thought about using a pricing model to compute prices but I am lacking the demand information. What am I supposed to do? Right now, I am
most likely forced to abandon the model altogether and choose a different framework instead.
What I would like to do instead is to have my model in a modular shape so that I could add a "demand" module and combine it with my pricing optimization - maybe a sampling procedure or even just a
heuristic. The feature I want is that I have a coherent path from low to higher resolution.
The goal behind our research and engineering efforts is to lift economic modelling to this paradigm. Yet, we do not just want to compose software packages. We want an actual composition of economic
models AND the software built on top.
How to get there? Compositionality!
Say, we want to turn the manual modelling process, which mostly relies on craft, experience and judgement, into a software engineering process. But not only that. We are aiming for a framework of
synthesis in which formal mathematical models can be composed.
How should we go about this? This is totally unclear! Even more, the question does not even make sense. This is a bit like asking how do we multiply a story from Hemingway with a story by Marquez.^9
Similarly, models in economics are independent and closed objects and generally do not compose. It is here where the "Cat" in CyberCat comes in. Category theory gives us a way to consider open
systems and model them by default relative to an environment. It is this feature which allows us to even consider the composition of models - for instance the composition of game theoretic models we
Another central feature that is enabled through category theory is the following paradigm:
model == code
That is, the formalism can be seamlessly translated back and forth between model and an actual (software) implementation. Thereby, instead of modelling on pen and paper, modelling itself becomes
programming. It is important to note that we do not just want to translate mathematical models into simulations but code does actually symbolically represent mathematical statements.
To summarize, category theory gives us a formal language of composable economic models which can be directly implemented.
Equipped with this foundation, we can turn to the programming language design task to turn the modelling process into a process of software engineering.
Industrial mass customization of economic models
Modelling as programming enables the iterative refinement of models. Whereas in the traditional sense, models are not only closed but also dead wood (written on paper), under this paradigm models are
more like living objects which can be (automatically) updated over time.
Instead of building a library of books, in our case the models are part of a software library. Which means the overall environment becomes way more powerful over time, as the ecosystem grows.
Composition also means division of labor. We can build models where parts are treated superficially at first but then details get filled in later. This can mean more complexity but most importantly
means that we can build consistent models that are extended, refined, and updated over time.
These aspects resemble similar attempts in mathematics and the use of proof assistants and verification systems more generally. Here is Terence Tao on these efforts^10:
One thing that changed is the development of standard math libraries. Lean, in particular, has this massive project called mathlib. All the basic theorems of undergraduate mathematics, such as
calculus and topology, and so forth, have one by one been put in this library. So people have already put in the work to get from the axioms to a reasonably high level. And the dream is to
actually get [the libraries] to a graduate level of education. Then it will be much easier to formalize new fields [of mathematics]. There are also better ways to search because if you want to
prove something, you have to be able to find the things that it already has confirmed to be true. So also the development of really smart search engines has been a major new development.
It also means different forms of collaboration between field experts and across traditional boundaries. Need a financial component in that traditional IO model? No problem, get a finance expert to
write this part - a modern pin factory equivalent. See again Terence Tao^11:
With formalization projects, what we’ve noticed is that you can collaborate with people who don’t understand the entire mathematics of the entire project, but they understand one tiny little
piece. It’s like any modern device. No single person can build a computer on their own, mine all the metals and refine them, and then create the hardware and the software. We have all these
specialists, and we have a big logistics supply chain, and eventually we can create a smartphone or whatever. Right now, in a mathematical collaboration, everyone has to know pretty much all the
mathematics, and that is a stumbling block, as [Scholze] mentioned. But with these formalizations, it is possible to compartmentalize and contribute to a project only knowing a piece of it.
Lastly, the current developments of ML and AI favor the setup of our system. We can leverage the rapid development of ML and AI to improve the tooling on both ends of the pipeline: Users are
supported in the modelling setup and solving or analyses of models becomes easier.
The common thread behind all of our efforts is to boost the modelling process. The traditional process is manual, slow, and limited by domain expertise - in other words very expensive.
Our goal is to turn manual work into mass customizable production.
Closing remarks
What I described so far is narrowly limited to economic modelling. Where is the "Cybernetics"?
First, I focused on the composability of economic models. But the principles of the categorical approach extend beyond this domain. This includes the understanding how apparently distinct approaches
share commonality (e.g. game theory and learning) and how different structures can be composed (build game theoretic models on top of some underlying structure like networks). In short, we work
towards a whole "theory stack".
Second, the software engineering process depicted above focuses very narrowly on extending the economic modelling process itself. But the same approach will mirror the theory stack with software
enabling analyses along each level.
Third, once we are operating software, we open the ability towards leveraging other software to support the modelling process. This follows pragmatic needs and can range from data analytics to LLMs.
A general challenge to decision-making is the hyper-specialization of expert knowledge. But as decisions are more and more interconnected, what is lacking is the ability to synthesize this knowledge.
Just consider the decision-making of governments during the Covid epidemic. For instance, in the decision to close schools, one cannot simply rely on a single group of domain experts (say
physicians). One needs to synthesize the outcomes of different models following different methodologies from different domains. We want to develop frameworks in which these tradeoffs can be
1. Ariel Rubinstein. Economic fables. Open book publishers, 2012, p.16 ↩
2. I will focus on micro-economic models. They are simply closest to my home base and relevant for my daily work. ↩
3. The view on what economists do there is markedly different from Rubinstein's. Prominently Al Roth: The Economist as Engineer: Game Theory, Experimentation, and Computation as Tools for Design
Economics. ↩
4. And probably most importantly, functions themselves can be input to other functions. ↩
5. Economics Rules: The Rights and Wrongs of The Dismal Science. New York: W.W. Norton; 2015 ↩
6. Of course, the classification of practical and non-practical is not exclusive to economics. Mathematics is full of examples of domains that are initially seen as without any practical use and
then turned out to be important later on. ↩
7. Ibid. ↩
8. In addition, if the modelling falls to academics, then also their incentives kick in. The chances for publishing a model on a subject that has already been tackled by a prominent model can be
very low - in particular in the case of a null-result. ↩
9. We might of course come up with a way how these two stories can be combined or compared. But this requires extra work; there is no operation to achieve this generically. These days we might ask
an LLM to do so. And indeed this might be a useful direction for the future to support this process. ↩
10. Quoted from this interview ↩
11. Ibid. ↩ | {"url":"https://cybercat.institute/2024/07/15/usefulness-models/","timestamp":"2024-11-14T20:07:23Z","content_type":"text/html","content_length":"33768","record_id":"<urn:uuid:050f0f6d-8447-44e3-959d-b8cd9ec4de5d>","cc-path":"CC-MAIN-2024-46/segments/1730477395538.95/warc/CC-MAIN-20241114194152-20241114224152-00144.warc.gz"} |
How to find the GCD of two numbers using Python?
Here's one way to find the GCD of two numbers using Python:
1 def gcd(a, b):
2 if b == 0:
3 return a
4 return gcd(b, a % b)
6 # example usage
7 print(gcd(24, 36)) # output: 12
This function uses a recursive algorithm called the Euclidean algorithm to find the GCD. If b is zero, then a is the GCD. Otherwise, we call the function again with b and the remainder of a divided
by b, until b is zero. | {"url":"https://devhubby.com/thread/how-to-find-the-gcd-of-two-numbers-using-python","timestamp":"2024-11-08T21:20:51Z","content_type":"text/html","content_length":"112782","record_id":"<urn:uuid:d8f037a6-f1ff-46d1-ad97-cdb2b6e0e2a7>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00531.warc.gz"} |
Rotation Curve
A rotation curve is a plot showing how orbital velocity, V, varies with distance from the centre of the object, R. Rotation curves can be determined for any rotating object, and in astronomy are
generally used to show how mass is distributed in the Solar System (Keplerian Rotation curves) or in spiral galaxies (galactic rotation curves).
The rotation curves of galaxies can be measured using neutral hydrogen observations with radio telescopes. By equating the gravitational force to the centrifugal force we can estimate the mass inside
a certain radius.
where v is the rotation velocity, G is Newton’s gravitational constant, and M the mass inside a particular radius R. | {"url":"https://astronomy.swin.edu.au/cosmos/R/Rotation+Curve","timestamp":"2024-11-08T06:03:23Z","content_type":"application/xhtml+xml","content_length":"10133","record_id":"<urn:uuid:9e4f6deb-bcdf-4efc-b81a-5c66853ed690>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00848.warc.gz"} |
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How to Cite This Entry:
Measure in a topological vector space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Measure_in_a_topological_vector_space&oldid=47816
This article was adapted from an original article by R.A. Minlos (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098.
See original article | {"url":"https://encyclopediaofmath.org/index.php?title=Measure_in_a_topological_vector_space&action=info","timestamp":"2024-11-10T05:07:29Z","content_type":"text/html","content_length":"16260","record_id":"<urn:uuid:c064e007-c601-43e9-8075-49bc2f439110>","cc-path":"CC-MAIN-2024-46/segments/1730477028166.65/warc/CC-MAIN-20241110040813-20241110070813-00510.warc.gz"} |
This number is a prime.
The largest known prime number that cannot be represented as the sum of less than nineteen fourth powers.
The absolute value of n^2 - 47n + 479 isn't composite for 0 < n < 47. [Wesolowski]
The smallest prime that is the sum of first n consecutive Hoax numbers, (n=6), i.e., 22+58+84+85+94+136=479. [Loungrides]
In the 611 distinct-digit squares only the digit 6 appears a prime number of times, appearing 479 times in the 4358 total digits. [Gaydos]
The first and last digit of 479 is 7 squared. If we repeat with the digits of 479 we obtain the prime 146479891. Will this procedure work again with 146479891? [Petrov]
The pair of consecutive primes (479, 487) is the only pair, at least up to 10^9, of consecutive primes (p,q) such that the reverse of the digits of p is equal to 2*q. [Galliani]
An 11-smooth number has only prime factors 2, 3, 5, 7, or 11. 479 is the smallest prime which is not the sum of two 11-smooth numbers. [Drost]
(There are 5 curios for this number that have not yet been approved by an editor.)
Printed from the PrimePages <t5k.org> © G. L. Honaker and Chris K. Caldwell | {"url":"https://t5k.org/curios/page.php?number_id=273","timestamp":"2024-11-14T00:51:17Z","content_type":"text/html","content_length":"10607","record_id":"<urn:uuid:ced4acc3-bd7e-48af-b104-530030c43e7a>","cc-path":"CC-MAIN-2024-46/segments/1730477028516.72/warc/CC-MAIN-20241113235151-20241114025151-00827.warc.gz"} |
Irreducible Elements in a Commutative Ring
Irreducible Elements in a Commutative Ring
Recall from The Greatest Common Divisor of Elements in a Commutative Ring page that if $(R, +, \cdot)$ is a commutative ring and $a_1, a_2, ..., a_n \in R$ then a greatest common divisor of these
elements is an element $d \in R$ which satisfies the following properties:
• 1) $d | a_1, a_2, ..., a_n$.
• 2) If $c \in R$ is such that $c | a_1, a_2, ..., a_n$ then $c | d$.
We proved some very important results. First, if $d' \in R$ is an associate of $d$, that is, $d \sim d'$, then $d'$ is also a greatest common divisor of $a_1, a_2, ..., a_n$.
Furthermore, if $a, b, d \in R$ with $d \neq 0$ and $aR + bR = dR$ then $d$ is a greatest common divisor of $a$ and $b$.
Lastly, if $(R, +, \cdot)$ is a principal ideal domain and $a, b \in R$ with $a \neq 0$ and $b \neq 0$ then there exists a greatest common divisor of $a$ and $b$ of the form:
\quad d = as + bt
We have established the concepts of divisors and greatest common divisors to commutative rings, and now we will extend the concept of irreducible elements to commutative rings.
Definition: Let $(R, +, \cdot)$ be a commutative ring. An element $p \in R$ is said to be Irreducible in $R$ if $p$ satisfies the following properties:
1) $p$ is not a unit.
2) If $p = ab$ then either $a$ or $b$ is a unit.
Theorem 1: Let $(R, +, \cdot)$ be a principal ideal domain and let $a, b, p \in R$. If $p$ is irreducible and $p | ab$ then either $p | a$ or $p | b$.
• Proof: Let $p$ be irreducible and let $p | ab$ and suppose that $p$ does not divide $a$. Then $p$ is not a common divisor of $p$ and $a$. The only divisors of $p$ are units and $1$. In
particular, every unit is an associate with $1$, so $1$ is a greatest common divisor of $a$ and $p$. So there exists $s, t \in R$ such that:
\quad 1 = as + pt
• Multiply both sides of the equation above by $b$ to get:
\quad b = abs + bpt
• Since $p | abs$ and $p | bpt$ we have that $p | b$. $\blacksquare$
Theorem 2: Let $(R, +, \cdot)$ be a principal ideal domain and let $p \in R$ with $p \neq 0$. Then $p$ is irreducible if and only if $pR$ is a prime ideal.
Recall that an ideal $I$ in a ring $R$ is prime if whenever $ab \in R$ we have that either $a \in R$ or $b \in R$ and secondly, $pR \neq R$.
• Proof: $\Rightarrow$ Suppose that $p$ is irreducible and let $ab \in pR$. Then $p | ab$. By the previous theorem we have that either $p | a$ or $p | b$. So $pR$ is a prime ideal.
• $\Leftarrow$ Suppose that $pR$ is a prime ideal. Then if $ab \in pR$ we have that either $a \in pR$ or $b \in pR$. Suppose that $p$ is not irreducible. Then there exists elements $a, b \in R$
such that $a$ and $b$ are not units and:
\quad p = ab
• Clearly $ab = p \in pR$ so either $p | a$ or $p | b$. If $p | a$ then there exists an element $q \in R$ such that $a = pq$. Substituting this into the equation above yields $p = pqb$. So $1 = qb$
. So $b$ is a unit which is a contradiction.
• Similarly, if $p | b$ then there exists an element $q \in R$ such that $b = pq$. Substituting this into the equation above yields $p = paq$. So $1 = aq$. So $a$ is a unit which is a
• So if $p = ab$ then either $a$ or $b$ is a unit so (2) in the definition holds. Furthermore, $p$ itself is not a unit since $pR \neq R$ so (1) in the definition holds. Hence $p$ is irreducible. $ | {"url":"http://mathonline.wikidot.com/irreducible-elements-in-a-commutative-ring","timestamp":"2024-11-13T18:59:42Z","content_type":"application/xhtml+xml","content_length":"20833","record_id":"<urn:uuid:d554094a-5331-44d3-8760-72be09ad3910>","cc-path":"CC-MAIN-2024-46/segments/1730477028387.69/warc/CC-MAIN-20241113171551-20241113201551-00779.warc.gz"} |
seminars - Dimension theory of groups of diffeomorphisms of the circle.
Abstract : In the talk, consider the action of a finitely generated group on the circle by analytic diffeomorphisms. We will discuss some results concerning the dimensions of objects arising from
this action. More precisely, we will present connections among the dimension of minimal subsets, that of stationary measures, entropy of random walks, Lyapunov exponents and critical exponents. These
can be viewed as generalizations of well-known results in the situation of PSL(2,R) acting on the circle.
This talk is based on a joint work with Yuxiang Jiao and Disheng Xu | {"url":"https://www.math.snu.ac.kr/board/index.php?mid=seminars&l=en&sort_index=speaker&order_type=desc&page=56&document_srl=1026784","timestamp":"2024-11-05T04:30:15Z","content_type":"text/html","content_length":"49834","record_id":"<urn:uuid:0f0644ed-03f2-485a-9470-2ef67c6f4653>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00673.warc.gz"} |
Functional Alchemy
2021-03-17 This post is over 2 years old
While reading through the Pragmatic Programmer, I happened upon a section which discussed an alternative model for looking at our code. Rather than looking at it as a series of object interactions,
this passage advocated for thinking in terms of Data-transformations, and about how the data would flow through each transformation from input to output. I’ve been using a variation of this model for
a long while, though I did not know it. In fact I used to call it ‘Functional Alchemy’ because it reminded me of the hunt for the Philosopher’s stone, and the transmutation of the elements. I recall
leveraging it for mapping computations for data going into Redux store and from there into React Components so that only subsets would be used at a given moment.
I want to spend some time now to discuss how you too could apply this model by way of an example. I will start with notational structure. When you see (A, B) => c, you should read this as “There is a
function that given A and B inputs can produce/find C”.
For my part, I tend to enumerate the set of function starting at the Top, or with my inputs. But I always note my desired output as me second step. In order to get anyway, you need a goal right? It
usually looks like this:
1. I have A and B
2. I need E
3. (A,B) => C
4. (B) => D
5. (C, D) => E
This helps me create my eventual method signature: (A,B) => E, where E is my desired output, and A, B are the necessary inputs to whatever computation I am describing.
Now let us take a practical case:
I want to determine the Taxes Due on an Invoice, which has LineItems, and a Location.
The LineItems have a Unit, RatePerUnit, TotalBillAmount, DiscountAmount, IsExpedited, TotalAfterExpeditedRateApplied and an ExtendedBillAmount. LineItems in this case can have a surcharge if they are
The Taxes Due for the Invoice must be determined by Location of the Invoice, and the total before any Discounts are applied. Taxes are applied to ExpeditedAmount if a LineItem is Expedited.
With this in mind, let’s start with the DesiredOutput: TotalTaxes. Now by construction, we can assume that given an Invoice we can derive the LineItems, so the probable final signature may read:
(Invoice) => TotalTaxes
1. I have Invoice, which has LineItems
2. I need Total Taxes
Now, Per the problem description, Taxes are derived from Either ExpeditedTotals or from TotalBill Amount, depending on whether the the LineItem was Expedited. LineItems show us Expedited or No by a
Boolean so we can get the two lists rather simply from LineItems.
1. I have Invoice, which has LineItems
2. I need Total Taxes
3. Invoice => LineItems
4. LineItems => RegularLineItems
5. LineItems => ExpeditedLineItems
From what we have, and the basic model for LineItems, the Expedited Function (#5)would be equivalent to
1 lineItems.filter((li) => li.IsExpedited);
Thinking functionaly, the next step would be a reduce, since we want to Sum of the Taxable Amount for both LineItems and ExpeditedLineItems. We could handle this in a couple of ways, but let’s favor
simplicity of explanation for now.
1. I have Invoice, which has LineItems
2. I need Total Taxes
3. Invoice => LineItems
4. LineItems => RegularLineItems
5. LineItems => ExpeditedLineItems
6. RegularLineItems => RegularTaxableSubtotal
7. ExpeditedLineTimes => ExpeditedTaxableSubtotal
For Function #6, this might look like:
1 regularLineItems.map((rli) => rli.TotalBillAmount).reduce((a, b) => a + b, 0);
Now we’re within inches of our final transformation, but we’ll need to get the TaxRate for the Location of the Invoice. Presuming this is a simple lookup, we add two additional functions:
1. I have Invoice, which has LineItems
2. I need Total Taxes
3. Invoice => LineItems
4. LineItems => RegularLineItems
5. LineItems => ExpeditedLineItems
6. RegularLineItems => RegularTaxableSubtotal
7. ExpeditedLineTimes => ExpeditedTaxableSubtotal
8. Invoice => Location
9. Location => ApplicableTaxRate
With this last piece of data ready, and the TaxableSubTotals, we can finally transform into our TotalTaxes:
1. I have Invoice, which has LineItems
2. I need Total Taxes
3. Invoice => LineItems
4. LineItems => RegularLineItems
5. LineItems => ExpeditedLineItems
6. RegularLineItems => RegularTaxableSubtotal
7. ExpeditedLineTimes => ExpeditedTaxableSubtotal
8. Invoice => Location
9. Location => ApplicableTaxRate
10. (RegularTaxableSubtotal, ExpeditedTaxableSubtotal, ApplicableTaxRate) => TotalTaxes
I want to point out a couple facets of this approach, which only became obvious to me after extended use. First, this approach has you focusing solely on Data, transformations and flow. It provides a
simple and clean way to describe the operation, and it can be tempting to try to force all interactions into this model. Be cautious. Not all computing problems are created equal, and this model is
useful for some, but should never be the only model you apply.
With that said, this model is exceptionally useful for breaking down complex computational problems. The Example is admittedly contrived. But consider that at no time did we need to consider the
implementation details of Getting the Applicable TaxRate given a Location. Nor did we dive into the Total Tax computation, it might have been a simple multiplication. It might be some step-function
or any other thing, depending on what ‘ApplicableTax’ rate actually is. The model, and form of operational description allows us to hide significant information about our problem by abstracting it
into a named component, which we can pipe to the necessary place.
Another useful facet, which in part derives from the the abstraction mentioned above, is that this method allows you to jump around a bit. While in my example I derived the Taxable Subtotals first,
if that computation had been more difficult, the method could easily hide that complexity behind ‘LineItmes => TaxableSubtotal’, while you continued to design the rest of the flow around that
unknown. Like water navigating around a stone in it’s path, you are free to encounter unknowns and pivot around them, by wrapping them in a neat little bundle of inputs and outputs. Oh I need Y? I
know I can get them from X, but I don’t want to think about that for now… So given X I can get Y, (X) => Y
So, while I wouldn’t recommend this model for say… an Event-Driven Workflow, consider giving it a shot before you design your next computational method. Things like TaxCalculation or determining a
complex enable/disable behavior might be good candidates. Meanwhile, I hope this example proves useful, and if nothing else interesting. May it make for some delightful development in your near | {"url":"https://scheufler.io/2021/03/17/functional-alchemy/","timestamp":"2024-11-10T09:24:32Z","content_type":"text/html","content_length":"17782","record_id":"<urn:uuid:82be35ab-4c59-4ed7-8157-9b3519ec723a>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00319.warc.gz"} |
Wolfram|Alpha Examples: Common Core Math: Counting & Cardinality
Examples for
Common Core Math: Counting & Cardinality
The Counting and Cardinality domain is covered entirely in kindergarten, where students learn number names up to one hundred and use them to count the numbers of objects in groups. Students develop
an understanding of counting as pairing the counting numbers with the objects being counted. Students practice counting groups of up to 20 objects, reporting their results by writing numerals and
comparing groups based on their numbers of objects.
Common Core Standards
Get information about Common Core Standards.
Look up a specific standard:
Search for all standards in a domain:
Use counting to determine the number of objects.
Count the number of objects (CCSS.Math.Content.K.CC.B.5):
Compare the sizes of two groups (CCSS.Math.Content.K.CC.C.6):
Compare two numbers (CCSS.Math.Content.K.CC.C.7):
Know number names and the counting sequence.
Count to one hundred (CCSS.Math.Content.K.CC.A.1):
Count between any two numbers (CCSS.Math.Content.K.CC.A.2): | {"url":"https://www.wolframalpha.com/examples/mathematics/common-core-math/common-core-math-kindergarten/common-core-math-counting-and-cardinality","timestamp":"2024-11-04T22:05:03Z","content_type":"text/html","content_length":"85172","record_id":"<urn:uuid:f7ccae36-82a0-4487-9fbb-4766cf31fb71>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.16/warc/CC-MAIN-20241104194528-20241104224528-00112.warc.gz"} |
Academic Personnel
Differential Equations and Mathematical Physics Department
Differential Equations and Mathematical Physics Department
Bobeva, Galya
Ph.D. student
+359 2 979 2841
Acad. G. Bonchev St., building 8, office 520
Sofia, 1113
Research Interests:
Cellular Nonlinear Networks, Differential equations, Local Activity Theory, Dynamical Systems.
Differential Equations and Mathematical Physics Department
Boyadjiev, Georgi
Assoc. Prof.,Ph.D.
+359 2 979 2842
Acad. G. Bonchev St., building 8, office 514
Sofia, 1113
Research Interests:
The research interests of Dr. Boyadzhiev include: the comparison principle for systems of elliptic and parabolic differential equations; PDEs, including both cooperative, and non-cooperative systems;
movement of waves in a solid body - in particular, the use of characteristics to track the path of seismic waves from an earthquake source to reception stations; development of methods for choosing
the optimal solution of nonlinear vapor problem in mathematical modelling of Earth structure; deterministic models in geophysics and financial markets. | {"url":"http://old.math.bas.bg/en/en-about-mission-2/en-mnu-scientific-community?view=trombinoscopeextended&letter=B&category=169","timestamp":"2024-11-11T01:00:51Z","content_type":"application/xhtml+xml","content_length":"43572","record_id":"<urn:uuid:88e3303a-63fb-4ded-b207-d4b08103c8d0>","cc-path":"CC-MAIN-2024-46/segments/1730477028202.29/warc/CC-MAIN-20241110233206-20241111023206-00398.warc.gz"} |
Visual Fractions of Sets Worksheet - 15 Worksheets.com
Visual Fractions of Sets
Worksheet Description
This worksheet is designed to help students learn to calculate the fraction of a set of objects. Each problem presents a visual array of circles grouped to represent a whole set, with a fraction
beside it. The students are instructed to determine what part of the total group the fraction represents and write the number of circles that corresponds to the fraction in the provided space. One
example is already completed, serving as a guide for how to approach the remaining problems.
The worksheet is teaching students to visualize and understand fractions as parts of a whole in a concrete manner. By counting and shading the appropriate number of circles, students learn to
translate a fraction into a tangible number of objects from a set. This skill is fundamental for comprehending fractions in real-world scenarios, such as dividing items into groups or portions.
Moreover, it helps students to reinforce the concept of fractions and to develop their ability to perform basic calculations involving fractions and whole numbers. | {"url":"https://15worksheets.com/worksheet/fractions-of-a-whole-number-8/","timestamp":"2024-11-08T09:29:47Z","content_type":"text/html","content_length":"109037","record_id":"<urn:uuid:2133da07-0c28-4c88-969b-3f01fe40f958>","cc-path":"CC-MAIN-2024-46/segments/1730477028032.87/warc/CC-MAIN-20241108070606-20241108100606-00691.warc.gz"} |
UNILAG Post UTME Past Questions | PastQuestions.com.ng
UNILAG Post UTME Past Questions
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UNILAG Post UTME Past Questions Sample
In other to know that we are giving out the original UNILAG Post UTME Past Questions, we have decided to give you some free samples as proof of what we said earlier.
1. As a general rule, ___ is the best measure of central tendency because it is more precise
(a) Mean
(b) Median
(c) Mode
(d) Range
1. The most frequently occurring number in a set of values is called the
(a) Mean
(b) Median
(c) Mode
(d) Range
1. A rectangular box with a square base and no top has a volume of 500 cm, the dimensions of the box that require the least amount of material are
(a) 10x10x5cm
(b) 4x5x25cm
(c) 50x5x2cm
(d) 25x10x2
(e) 10x50x1
1. The roof of shelter is made from a piece of corrugated iron 2.3m long inclined at 18º to the horizontal. How far from the wall does the roof stick out?
(a) 0.7m
(b) 2.2m
(c) 1.1m
(d) 1.2m
(e) 2.1m
5. A ladder 20m long rests against a vertical wall so that the foot of the ladder 9m from the wall. The height (correct to 1 decimal place above the ground at which the upper end of the ladder
touches the wall is
(a) 19.7m
(b) 18.1m
(c) 18.7m
(d) 17.1m
(e) 17.9m
6. A chord 6.6m long is 5.6m from the center of a circle. The radius of the circle is
(a) 3.2m
(b) 6.3m
(c) 6.5m
(d) 1.6m
(e) 2.56m
7. The heights in cm, of 10 children are 145, 163, 159, 162, 167, 149, 150, 160, 170, and 155. The mean height of the children is
(a) 156cm
(b) 158cm
(c) 160cm
(d) 162cm
(e) 159cm
8. The heights in cm, of 10 children are 145, 163, 159, 162, 167, 149, 150, 160, 170, and 155. The standard deviation of the heights of the children is
(a) 5.5cm
(b) 5.7cm
(c) 6.5cm
(d) 6.7cm
(e) 7.7cm
9. A class of all possible subsets of space S is called
(a) Universal set
(b) alpha – field
(c) sample space
(d) probability space
(e) random space
10. One of these is a demerit of a sample
(a) It is cheaper to enumerate a sample
(b) It is faster to survey a sample
(c) Results obtained from the sample are oftentimes as informative as those from a sensor
(d) All of the above
(e) None of the above
1. One of these is not a desirable feature of a good statistical table
(a) A table must reveal salient features of data
(b) A table must clearly communicate information in a neat and concise form
(c) A table must be self-sufficient
(d) A table must be self-explanatory
(e) None of the above
1. Which of these is a measure of location
(a) Mean
(b) Standard deviation
(c) Variance
(d) All of the above
(e) None of the above
1. Statistics is a set of tools whose proper use will …….. the decision-maker
(a) Completely fill the needs of
(b) Encumber
(c) Aid
(d) Confuse
(e) None of the above
1. The graph of a cumulative frequency distribution is called
(a) Frequency polygon
(b) Frequency distribution curve
(c) Frequency curve
(d) Step function
(e) Ogive
1. A ……… variation is one whose values convey the concept of attribute rather than number
(a) Quantitative
(b) Qualitative
(c) Discrete
(d) Continuous
(e) None of the above
1. A company employs 100 people, 65 of whom are men. 60 people including all the women are paid weekly. The number of men that are paid weekly is
(a) 35
(b) 40
(c) 25
(d) 30
(e) None of the above
1. In a survey of villagers, it is found that 20% of the people have visited Kano and 25% have visited Port Harcourt. If 5% have been to both cities, then the percentage that have visited neither
Kano nor Port Harcourt is
(a) 75%
(b) 65%
(c) 50%
(d) 60%
(e) 70%
1. The length of a rectangle is three times its width. If the perimeter is 72 cm, the width of the rectangle is
(a) 6cm
(b) 8cm
(c) 9cm
(d) 10cm
(e) 11cm
1. A frustum of a pyramid is 3cm square at the top and 6cm square at the bottom and is 5cm high. The volume in cm3 of the frustum is
(a) 15
(b) 150
(c) 105
(d) 115
(e) 36
1. If x = 2, then (x – 1)(2x – 3) =
(a) 1
(b) 3
(c) -1
(d) 0
(e) -3
1. What is sin30?
(a) ½
(b) 3/2
(c) 0.866
(d) 1
(e) 2
1. Given the list of numbers {1, 6, 3, 9, 16, 11, 2, 9, 5, 7, 12, 13, 8}, what is the median?
(a) 7
(b) 8
(c) 9
(d) 11
(e) 6
1. What is the slope of the line that is perpendicular to y – 2x = 1?
(a) 2
(b) -2
(c) ½
(d) -½
(e) 1
1. What is the sum of the first 40 even positive integers?
(a) 1,600
(b) 1,560
(c) 820
(d) 1,640
(e) 400
1. What is the length of an arc of a circle with a radius of 5 if it subtends an angle of 60° at the center?
(a) 3.14
(b) 5.24
(c) 10.48
(d) 2.62
(e) 4.85
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7 Tips to Prepare for UNILAG Post UTME Exams
1. Don’t make reading your hobby: A lot of people put reading as a hobby in their CV, they might be right because they have finished schooling. But “You” are still schooling, so reading should be a
top priority and not a hobby. Read far and wide to enhance your level of aptitude
2. Get Exams Preparation Materials: These involve textbooks, dictionaries, Babcock University Post UTME Past Questions and Answers, mock questions, and others. These materials will enhance your
mastery of the scope of the exams you are expecting.
3. Attend Extramural Classes: Register and attend extramural classes at your location. This class will help you refresh your memory and boost your classroom understanding and discoveries of new
4. Sleep when you feel like: When you are preparing for any exams, sleeping is very important because it helps in the consolidation of memory. Caution: Only sleep when you feel like it and don’t
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Make two sets of objects have same angle between them
For those of you that have played Battlefield 1942, what I’m trying to build is the tank turret-to-body rotation indicator (at the bottom): http://www.realtimerendering.com/erich/bf1942/
For those of you that haven’t, I’m trying to show on the GUI what the angular difference is between the camera’s Forward and the “body” Forward, in a 2D space of course. What I have so far is:
if(turret) {
Vector2 reducedVectorTurret = new Vector2(transform.forward.x, transform.forward.z);
Vector2 reducedVectorBody = new Vector2(playerBody.transform.forward.x, playerBody.transform.forward.z);
float angleDifference = Vector2.Angle (reducedVectorCam, reducedVectorBody);
but I’m stumped as to where to go from there. I know using Vector2.Angle won’t work since the angle will always be positive, which won’t let me set the RectTransform.rotation on the UI indicator
correctly since it doesn’t discern between left and right. I did some research on similar questions and it seems the Mathf.Atan2 function is involved but couldn’t find any cases close enough to mine
to figure it out.
Thanks for taking the time to read this, and I hope I’ve been clear enough!
The following code returns the signed angle between two Vector3 considering the normal of the surface (here Vector3.up I suppose) :
float SignedAngleBetween( Vector3 a, Vector3 b, Vector3 n )
float angle = Vector3.Angle( a, b );
float sign = Mathf.Sign( Vector3.Dot( n, Vector3.Cross( a, b ) ) );
return angle * sign;
See this manual page to understand the mathematics principles involved :
To be quick : Cross product between two vectors will return a third vector so that it’s perpendicular to the two others and the three vectors make an “direct basis” (using the left hand rule because
Unity uses a left-handed coordinate system)
The dot operation will help determining if the vector calculated just above, and the normal are “in the same direction”. If yes, the angle between a and b is positive (anti clockwise).
With this angle, you will be able to rotate your 2D tank
tankBody.GetComponent<RectTransform>().rotation = Quaternion.Euler( 0, 0, signedAngle ) ;
Ended up figuring it out myself. I’m not sure if it’s the most elegant solution, but it’s 10 lines and works:
if(turret) {
//reduce 3D vectors to 2D since Y is irrelevant
reducedVectorTurret = new Vector2(transform.forward.x, transform.forward.z);
reducedVectorBody = new Vector2(playerBody.transform.forward.x, playerBody.transform.forward.z);
//calculate angular difference
angleDifference = Vector2.Angle (reducedVectorTurret, reducedVectorBody);
//x is + or - depending on whether angle is to left or right
//turretIcon is always pointing Vector2.up, so just apply the
//earlier angular difference to bodyIcon Z rotation directly every frame
if(transform.InverseTransformDirection(playerBody.transform.forward).x < 0)
bodyIcon.eulerAngles = new Vector3(0, 0, angleDifference);
bodyIcon.eulerAngles = new Vector3(0, 0, -angleDifference);
I have a feeling there’s a way to do this using InverseTransformDirection alone and not calculating the angular difference separately, but working with 2D vectors is much easier to understand and
being able to plug the angleDifference into eulerAngles.z directly is a nice touch. | {"url":"https://discussions.unity.com/t/make-two-sets-of-objects-have-same-angle-between-them/141406","timestamp":"2024-11-15T01:43:16Z","content_type":"text/html","content_length":"32549","record_id":"<urn:uuid:d740b544-c3e6-447c-ba44-5eaab605d8a2>","cc-path":"CC-MAIN-2024-46/segments/1730477397531.96/warc/CC-MAIN-20241114225955-20241115015955-00528.warc.gz"} |
Risk (Part 4) - Correlation Matrix & Portfolio Variance – Varsity by Zerodha
In the previous chapter, we successfully calculated the variance-covariance matrix. As we discussed, these numbers are too small for us to make any sense. Hence, as a practice, it always makes sense
to calculate the correlation matrix when we calculate the variance-covariance matrix.
So let us go ahead and do this.
How is the correlation between two stocks calculated? Well, hopefully from the previous chapter, you will recall the formula for correlation –
Cov (x,y) is the covariance between the two stocks
σ[x ]= Standard deviation of stock x
σ[y ]= Standard deviation of stock y
This works fine if we have 2 stocks in the portfolio, but since we have 5 stocks in the portfolio, we need to resort to matrix operation to find correlations. So, when we have multiple stocks in the
portfolio, the correlations between stocks are all stacked up in a n x n (read it as n by n) matrix. For example, if it is a 5 stock portfolio (5 being the n here), then we need to create a 5 x 5
The formula for calculating the correlation remains the same. Recall, from the previous chapter, we have the variance-covariance matrix. For the sake of convenience, I’ll paste the image again here –
This takes care of the numerator part of the formula. We need to now calculate the denominator, which is simply the product of the standard deviation of stock A with the standard deviation of stock
B. If the portfolio has 5 stock, then we need the product of the standard deviation of all possible combination between the stocks in the portfolio.
Let’s go ahead and set this up.
We first need to calculate the standard deviations of each of the stocks in the portfolio. I’m assuming you are familiar with how to do this. You just need to use the ‘=Stdev()’ function on the daily
returns array to get the standard deviations.
I’ve calculated the same on excel used in the previous chapter. Here is the image –
Given that we have the stock-specific standard deviations, we now need to get the product of the standard deviation of all possible portfolio combination. We resort to matrix multiplication for this.
This can be easily achieved by multiply the standard deviation array with the transpose of itself.
We first create the matrix skeleton and keep all the cells highlighted –
Now, without deselecting the cells, we apply the matrix multiplication function. Note, we are multiplying the standard deviation array with the transpose of itself. The image below should give you an
idea, do look at the formula used –
As I mentioned in the previous chapter, whenever you use matrix or array function in excel, always hold the ‘ctrl+shift+enter’ combo. The resulting matrix looks like this –
At this point let me paste the formula for the correlation again –
The numerator is the variance-covariance matrix as seen below, and the denominator is the product of the standard deviations which we have just calculated above –
Dividing the variance-covariance matrix by the product of the standard deviations should result in the correlation matrix. Do note, this is an element by element division, which is still an array
function, so the use of ‘ctrl+shift+enter’ is necessary.
The resulting correlation matrix looks like this –
The correlation matrix gives us the correlation between any two stocks. For example, if I have to know the correlation between Cipla and Alkem, I simply have to look under the intersecting cell
between Cipla and Alkem. There are two ways you can do this –
1. Look at the row belonging to Cipla and scroll till the Alkem column
2. Look at the row belonging to Alkem and scroll till the Cipla column
Both these should reflect the same result i.e 0.2285. This is quite obvious since the correlation between stock A with Stock B is similar to the correlation of Stock B with Stock A. For this reason,
the matrix displays symmetrically similar values above and below the diagonal. Check this image below, I have highlighted the correlation between Cipla and Alkem and Alkem and Cipla –
The correlations along the diagonal represent the correlation of certain stock with itself. Do note, the correlation numbers above the diagonal are symmetrically similar to the correlation numbers
below the diagonal.
Needless to say, correlation of Stock A with Stock A is always 1, which is what we have got in the diagonal and the same is highlighted in yellow boxes.
5.2 – Portfolio Variance
We are just a few steps away from calculating the Portfolio Variance. As I have discussed earlier, we need the portfolio variance to identify the extent of risk my portfolio is exposed to. With this
information, I’m no longer driving blind. One can develop many other insights based on this. Of course, we will talk about this going forward.
The first step in calculating portfolio variance is to assign weights to the stocks. Weights are simply the amount of cash we decide to invest in each stock. For example, if I have Rs.100, and I
decide to invest all of that money in Stock A, then the weight in stock A is 100%. Likewise, if I decide to invest Rs.50 in A, Rs.20 in B and Rs.30 in C, the weights in A, B, and C would be 50%, 20%,
and 30% respectively.
I have arbitrarily assigned weights to the 5 stocks in the portfolio –
• Cipla @ 7%
• Idea @ 16%
• Wonderla @ 25%
• PVR @ 30%
• Alkem @ 22%
There is no science to assigning weights at this stage. However, at a later point in the module, I will discuss more this part.
The next step is to calculate the weighted standard deviation. The Weighted standard deviation is simply the weight of a stock multiplied by its respective standard deviation. For example, Cipla’s
standard deviation is 1.49%, hence its weighted standard deviation would be 7% * 1.49% = 0.10%
Here are the weights and the weighted standard deviation of 5 stocks in the portfolio –
Do note, the total weight should add up to 100% i.e the sum of the individual weights in stocks should add up to 100%.
At this stage, we have all the individual components needed to calculate the ‘Portfolio Variance’. The formula to calculate the Portfolio Variance is as shown below –
Portfolio Variance = Sqrt (Transpose (Wt.SD) * Correlation Matrix * Wt. SD)
Wt.SD is the weights standard deviation array.
We will implement the above formula in 3 steps –
1. Calculate the product of Transpose of Wt.SD with correlation matrix. This will result in a row matrix with 5 elements
2. Multiply the result obtained above (row matrix) with the weighted standard deviation array. This will result in a single number
3. Take the square root of the result obtained above to get the portfolio variance
So, let’s jump straight ahead and solve for portfolio variance in the same order –
I will create a row matrix called ‘M1’ with 5 elements. This will contain the product of the Transpose of Wt.SD with correlation matrix.
Do note, you will have to select the empty array space and hold down the ctrl+shift+enter keys simultaneously.
We now create another value called ‘M2’, which contains the product of M1 and weighted standard deviation –
We obtain the value of M2 as 0.000123542, the square root of this value is the portfolio variance.
The result for the above operation yields a value of 1.11%, which is the portfolio variance of the 5 stocks portfolio.
I need a break at this. Let’s figure out the next steps in the next chapter J
Download the excel sheet used in this chapter.
Key takeaways from this chapter –
1. Correlation matrix gives out the correlation between any two stocks in a portfolio
2. Correlation between stock A with stock B is the same as the correlation between stock B with stock A
3. Correlation of stock with itself is always 1
4. The diagonals of a correlation matrix should represent the correlation of stock A with itself
5. The correlation matrix contains symmetrical values above and below the diagonals
105 comments
1. Hello sir
In next module would you teach us about algo trading. This just a guess by name “Trading strategies and systems”
□ As on now yes, but there is a lot of demand for a personal finance module as well 🙂
2. Thank you for the detailed steps , I have few questions
1. How do we add new stock to already existing portfolio , for example a stock from IPO since it doesn’t have enough data points
2. Do we wait for enough data points to calculate ? is there a minimum no of data points we should have for calculation ?
3. Can portfolio variance be used to decide to buy or sell a stock , and decide on the weight-age allocation of stock ?
□ 1) That would be tough. We need enough data points to do this
2) No minimum as such, I personally prefer at least 6 months of data
3) Yes you can. Watch out for chapter 7 for more details on this.
3. Karthik
Now that we have calculated the Portfolio Variance to be 1.11%, what do we understand from the same? Is it good or bad? What should be an ideal variance figure?
□ There is nothing like an ideal variance figure. However, you need to watch out for higher numbers like 2.5-3%. Estimating variance gives you a sense of the risk associated with it.
4. Hi Karthik,
Thanks a lot for all your great articles. It’s simple and very elegant. I have a slight clarification in the portfolio variance formula. Since there is Sqrt, shouldn’t it be portfolio standard
deviation instead of portfolio variance.
□ Interesting, I use both Portfolio variance and portfolio SD interchangeably. But yes, what you’ve said is technically correct.
5. Hi,
Can you please explain what is the meaning of Portfolio Variance value=1.11%? What does 1.11% indicate here? Does it mean that total portfolio value can move 1.11% up or down when the market goes
up or down?.
□ Yes, that is the risk of an overall portfolio.
6. Hi Karthik,
Thanks a lot for your explanation in detail. You have made the difficult subject of statistics very easy.
While calculating the correlation:
“The correlations along the diagonal represents the correlation of certain stock with itself ”
In your example, it is showing as “1”. However, when I do it, I am getting 0.99 and 0.98 values. Not exactly 1. Is it fine?
□ Glad you thought so, Ruban 🙂
The correlation of a stock with itself is 1, but 0.9999 is also an acceptable value 🙂
☆ Thanks Karthik,
I am getting the values not as ‘0.9999’ but as :
Is this fine too?
○ Yes, this should be ok.
7. Hi Karthik – How do we account for dividend / bonus share in the excel sheet while calculating daily return. For Ex. BPCL gave 1:2 bonus which resulted in price falling from 683.70 to 485.55 on
14th July. Should we take this (485.55) price / adjust 13th July price or just ignore this data as one day noise?
Would assume removing one day will not really affect the average but, need your guidance before doing the same.
□ Yes, you always take the price adjusted for corporate actions. Else, the data will be skewed.
8. Many thanks and much appreciated your response. With the risk of not bugging you would like to ask two more question for the time being
1) If, I use daily adjusted closed price as published on yahoo finance it should be ok?
2) The portfolio variance (for my portfolio) calculated by Equity curve comes to 0.99545% however, the one calculated by the variance / co-variance & correlation method comes to 0.95332% for same
set of data. Is it possible?
□ 1) Yes, last time when I was using Yahoo, I remember they were putting up clean data. Not sure if they still do that
2) Possible. I always take the EQ curve one – helps me a quick and dirty estimate and I’m absolutely fine with it 🙂
☆ Thank you very much sir… You guys are doing a great job. Will bother you when I read the next module 🙂
○ Yes, please! Looking forward to it 🙂
9. Dear K,
If portfolio variance is 1.11, means that portfolio Standard deviation is 10.54%? is this daily STDV of a portfolio???
□ It means the standard deviation is 1.11%…which is the measure of volatility. Yes, in this case, it is for the entire portfolio and not just a single stock.
☆ Many thanks
○ Welcome!
10. Do you have any part talking about correlation between portfolio and benchmark?
□ I’m not sure if I’ve touched that topic, but you can run a correlation between the portfolio’s equity curve and the desired benchmark. This will help you get started.
11. Dear Karthik,
I appriciate your effort for providing such a wonderful material on Risk Management. However, there is a thing I would like to clarify from you. As mentioned in the chapter, calculation of
portfolio variance can be calculated as,
“We will implement the above formula in 3 steps –
1- Calculate the product of Transpose of Wt.SD with correlation matrix. This will result in a row matrix with 5 elements
2- Multiply the result obtained above (row matrix) with the weighted standard deviation array. This will result in a single number
3- Take the square root of the result obtained above to get the portfolio variance”
As per my understanding, #Step 2 should give me the Portfolio variance. If I use the step 3, it will give Standard Deviation of Portfolio.
Please help me understanding the same.
With Regards,
Niteesh Sharma
□ Yes, that’s absolutely correct, Niteesh.
12. Why is portfolio variance a square root of the matrix multiplication? the matrix multiplication should yield a result in the unit of variance, so a square root of that result should yield
portfolio standard deviation. Did I miss something?
Reference to your article:
Portfolio Variance = Sqrt (Transpose (Wt.SD) * Correlation Matrix * Wt. SD)
Thanks for the detailed walkthrough!
□ Yeah, maybe I should have specified –
Portfolio Standard Deviation = Sqrt (Transpose (Wt.SD) * Correlation Matrix * Wt. SD)
Thanks for point out 🙂
13. The excel file is wrong. Correlation for same asset should be exactly 1 (even not 0.99999)
Average returns
number of data (N=126, not 127)
number of data for variance – co-variance should be N-1 N-1 = 125 to be used to compute variance – co-variance.
Then, correlation matrix will be correct and will give exactly 1 on the diagonal.
My friend, check your results before publishing.
□ Thanks for pointing that out, Fred. Will go through the sheet again.
14. Dear Karthik Sir,
I have been repeatedly trying following all steps very carefully, but unable to get “1” value along the diagonal of correlation matrix. Kindly help. what do u think, might be the possible issue
with this
□ Simranjeet, it would be hard to figure out what is going wrong with your excel as there are multiple steps involved. I’d suggest you retract it or best try rebuilding (if time permits).
☆ Just figured it out man . It was a very fundamental problem. Kindly tell me weather “MMULT(A,B)” is equal to “MMULT(B,A)” or not. U must be laughing but i was afraid of matrix algebra in
school. Anyway Cheers..
○ I’m still afraid of Matrix algebra 🙂
Anyway, these two are not the same. You can experiment on excel actually. Btw, check this – https://math.stackexchange.com/questions/2853239/
15. Hi, I’m trying this with a portfolio of over 100 holdings. My correlation matrix is only yielding perfect correlation for the first security, all others are in the ranges of 98%-103%. I have a
few outliers, like 106% and 183%. I’ve followed the guide the best I can and have tried recreating many times. Any idea why this is the case?
□ How are you doing this, Alyn? Is it on excel?
One of the best checks for the correlation matrix is to see the values at the diagonal. If its near to 1, then probably you are on the right track.
16. how to find portfolio variance if stock invested at different period for eg invested in stock A on 1st april and invested in B on 28th april.
□ It would still be the same procedure.
17. Sir if I have invested in stock at different period for eg. invested in stock A on 24thdec 2019 in stock B on 1st jan 19, Stock C on 20th feb 19 and stock D and E on 31st march 19.Then to find
portfolio variance how many data points should I considered.Should I considered from the date of investment of first stock A or past 1 year data?
□ Yes Rajat, the idea is to have the data for the stock for at least 1 year.
☆ So Sir inorder to find out portfolio variance that I have invested from 24th dec 2018 onwards I have to consider data of past 1 year from the date of first investment in stock that is on
24th dec 2018.
○ Yeah, also ensure you have the data for the same period for all the stocks.
18. shouldn’t it be a standard deviation from the portfolio instead of variance? = sqrt(portfolio variance)
□ Hmm, unable to get that. Can you please shed more context to this? Thanks.
19. Thanks a ton for such wonderful content. The real beauty is the ease with which the concepts have been explained. Truly a gem for all learners. Two quick queries please, if you could throw some
If my data range of the data is from 01 Jan 2017 till 31 Dec 2019, what happens when I add an IPO to my portfolio which launched on 01 Apr 2019. In this case the historical data (prior Apr 2019)
will not be there. If i put blank in the daily return column, the matrix throws up an error. Other option is to keep daily return at 0% till the date IPO was issued (avg return will be calculated
for the number of days since IPO only). Excess return matrix will also have to be manually set to 0% till IPO date. But will that affect he overall calculation?
Second query is that one of my stocks had a stock split six months back. the face value changed from Rs 10 to Re 1. As the share price reduced the daily return will be erroneous. So how do I
cater for that.
It would be great if you could please throw some light on these issue. Thanks a ton!
□ Agreed. Hence it is best to consider the portfolio without the IPO stock. I know this is not the right thing, but unfortunately, you can’t help it. Another practice is to only buy stocks
which have at least 2 years of trading history.
20. OK, guess we can include that in the portfolio once it’s built an year of history. But what about the stock split. How do we cater for that?
□ Yes.
Nothing really changes in a stock split, except that you will have to replace the data with the new adjusted values.
21. No but when the stock split happens as the per share cost reduces to 1/10th of the pre split amount, hence the daily return shows as -90%. How to cater for this error. Should I change the
previous price also by a factor of 1/10 to avoid this error?
Sorry for pestering, but I am really stuck at this stage. Thanks again.
□ True, hence you need to replace the entire column with the adjusted data. You won’t have this problem then.
22. Hey Karthik,
Did you maintain a trading journal when you started trading? If so can give some tips on how to keep one myself?
□ Unfortunately no, I never maintained one 🙁
23. Sir can you elaborate the meaning of 1.11% portfolio variance?
□ Think of the entire portfolio as a single stock. The volatility of that being 1.11%.
24. Is 1.1% daily variance? How to we find annualized standard deviation from this ?
□ Multiply this ith square root of time to get the annualized SD.
25. By calculating portfolio return of same cipla,idea,wonderl,pvr and alkem by,w1r1+w2r2+w3r3+w4r4+w5r5
and then portfolio variance on excel by; =var(number1………), (on portfolio returns)
portfolio variance is coming 0.01235%
why is it so?
□ Do you mean to say that it does not match with the calculations shown in the chapter? Can you try looking at the excel and matching the steps?
26. ive done it by different method,by calculating overall portfolio return , then applying [=var(……..)] on portfolio return column,
and variance is coming 0.01235% and by sqroot(variance) to calculate standard deviation SD is comin 1.11%.
□ Ok. Do the values match?
27. Hi sir , thank you for the streamlined content. Never found such clear explanation anywhere. I would like to point out a typo error here. In this chapter , in place of shift – it got printed as
shit. You might want to correct this.
□ Thanks, Yashwanth. That’s an embarrassing typo, thanks for pointing that out. Will fix it 🙂
28. Nice Articles, easy to understand for novice investor. Was wondering if these kind of calculation are readily available in Zerodha console/back office?
□ No Gaurav, you need to calculate these.
29. Hey Kartik,
Thanks a lot man for sharing your work with us, Actually after getting inspired of uhh I did created a portfolio for Crypto and the Portfolio variance turned out to be =4.859% and later when I
did the same thing with your 2nd alternate way then that portfolio variance was completely different which is = 0.344% I don’t know what mistake I made……can uhh help me out in this?
□ Very tough to analyze this without actually looking at the steps 🙂
30. Hi,
why does portfolio variance = sqrt (….)? If we are taking the sqrt(..) , it means the portfolio standard deviation, no?
□ Yes, the square root of the variance is the standard deviation.
31. Thank you for the excellent articles.
In the resulting correlation matrix the values in the diagonals representing correlation between the same stocks does not actually equal 1.0 as shown in the screenshot, but is close to one and
formatted to display to 1 decimal place in Excel.
Is this an artefact of rounding/truncation through the multiple steps or is something else at play?
Kind regards.
□ Its actually rounded off by 1/1000th, so its ok, I guess.
32. sir, how to find correlation between two set of values with different lengths. I am trying to find correlation between backtest values of two systems.
□ That would be tough. To correlate, one of the key assumptions is to ensure both series has the same number of data sets.
33. I was thinking to convert my returns to weekly and find correlation between them.
□ You can, Mani. Ensure the number of data points match.
34. I did a weekly correlation test on two of my systems and I got -0.22.
Is the negative correlation significant enough?
Is there any other metrics I have to look at?
□ -ve indicates that the assets move in opposite directions. Are these just stocks in the portfolio?
35. no sir, weekly pnls of intraday BNF trend following system and BNF intraday option writing system.
□ Sure, please do check the calculations again. Something seems off to me.
36. I am not getting it sir, what seems off?
37. Hi, I’m a finance student and I’m trying to implement this to make a variance covariance matrix. I think I completed it correctly; it it possible to review mine for me? If so, how can I send the
excel attachment? If it’s more comfortable for you, I can send a OneDrive link (no download required of the attachment) and it can be viewed via the web.
Thank you!
□ Jay, I’m really pressed for time, not sure if I can review individual worksheets. Sorry, but I hope you understand.
38. UNABLE TO DOWNLOAD EXCEL File
□ Can you try another browser, please?
39. Hello Sir!
Great work by you sir! I had doubt regarding the no. of observations we should take for calculation. What amount of historical data is considered to be considered good for all these calculations?
1 year, 2 years, or 5 years? I am trying to apply all these concepts to my own portfolio.
Thanks & Regards,
□ At least a year, I’d say 🙂
40. Hello Sir, This question was already answered earlier but I was not able to understand it….
So, we have calculated the portfolio variance to be 1.11%…. Let’s say, my portfolio has an expected return of 20%
Does that mean I will have a deviation of 1.11% in the expected return that is 20% in my portfolio on a yearly basis?
□ Yes, thats what it means. The variance from the expected return.
41. Hi
Unable to download excel sheet, please check is it sill available?
□ Can you please try another browser, Vicky?
42. I can see comments feed updating on Varsity home page. Is there any way to get a feed of changes happening to the module chapters so that we are updated for any corrections or latest updates?
□ Will share that feedback with the team, Amit. Thanks.
43. Simple, however maybe a silly question. How do I deal with the case where the product of the standard deviations is zero, introducing a zero denominator.
□ Hmm, its unlikely a zero, could be a small decimal number. Can you try expanding the decimals in the cells and check?
44. sir, is portfolio variance and portfolio SD same thing ?
□ Yup, thats right. They both are the same.
45. Excel sheet did not downloaded
□ Please try downloading from another browser?
46. Your division is wrong, if you use 125 you won’t need to force the yellow 1s. You should count the number of excess returns and subtract 1. 125
□ Thanks, let me recheck this.
47. Hello sir, thanks for the great content.I tried the calculations on excel myself(using different stocks) and while calculating the correlation matrix my diagonals are coming out to be equal to
0.99596774, is this acceptable or have i made some error on my end. I have used previous 1 year data of all the 5 stocks.
□ Thats ok. Its near 1 🙂
48. Thanks for the reply, but caught my mistake did not do (n-1) while calculating covariance matrix
□ Got it. Good luck and happy number crunching 🙂
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Theorem Proof Part 3. Stokes' Theorem Proof Part 4. Stokes' Theorem Proof Part 5. Stokes' Theorem Proof Part 6. Stokes' Theorem Proof Part 7. Introduction to Parametrizing a Surface with Two
Parameters. Determining a Position Vector-Valued Function for a Parametrization of Two Parameters. Partial Derivatives of Vector-Valued Functions. Introduction to the Surface Integral. Example of
calculating a surface integral part 1. Example of calculating a surface integral part 2. Example of calculating a surface integral part 3. Surface Integral Example Part 1 - Parameterizing the Unit
Sphere. Surface Integral Example Part 2 - Calculating the Surface Differential. Surface Integral Example Part 3 - The Home Stretch. Surface Integral Ex2 part 1 - Parameterizing the Surface. Surface
Integral Ex2 part 2 - Evaluating Integral. Surface Integral Ex3 part 1 - Parameterizing the Outside Surface. Surface Integral Ex3 part 2 - Evaluating the Outside Surface. Surface Integral Ex3 part 3
- Top surface. Surface Integral Ex3 part 4 - Home Stretch. Conceptual Understanding of Flux in Three Dimensions. Constructing a unit normal vector to a surface. Vector representation of a Surface
Integral. Stokes' Theorem Intuition. Green's and Stokes' Theorem Relationship. Orienting Boundary with Surface. Orientation and Stokes. Conditions for Stokes Theorem. Stokes Example Part 1. Part 2
Parameterizing the Surface. Stokes Example Part 3 - Surface to Double Integral. Stokes Example Part 4 - Curl and Final Answer. Evaluating Line Integral Directly - Part 1. Evaluating Line Integral
Directly - Part 2. Stokes' Theorem Proof Part 1. Stokes' Theorem Proof Part 2. Stokes' Theorem Proof Part 3. Stokes' Theorem Proof Part 4. Stokes' Theorem Proof Part 5. Stokes' Theorem Proof Part 6.
Stokes' Theorem Proof Part 7.
Why we do the same thing to both sides: simple equations. Representing a relationship with a simple equation. One-Step Equation Intuition. One step equation intuition exercise intro. One step
equation intuition. Adding and subtracting the same thing from both sides. Intuition why we divide both sides. Why we do the same thing to both sides: two-step equations. Why we do the same thing to
both sides: multi-step equations. Why we do the same thing to both sides basic systems. Super Yoga Plans- Basic Variables and Equations. Super Yoga Plans- Solving One-Step Equations. Constructing and
solving equations in the real world 1. Super Yoga Plans- Plotting Points. Super Yoga Plans- Solving Systems by Substitution. Super Yoga Plans- Solving Systems by Elimination. Constructing and solving
equations in the real world 1 exercise. Simple Equations of the form Ax=B. Example solving x/3 =14. One-step equations with multiplication. Example solving x+5=54. Examples of one-step equations like
Ax=B and x+A = B. One step equations. Solving Ax+B = C. Two-Step Equations. Example: Dimensions of a garden. Example: Two-step equation with x/4 term. 2-step equations. Basic linear equation word
problem. Linear equation word problems. Linear equation word problem example. Linear equation word problems 2. Variables on both sides. Example 1: Variables on both sides. Example 2: Variables on
both sides. Equation Special Cases. Equations with variables on both sides. Number of solutions to linear equations. Number of solutions to linear equations ex 2. Number of solutions to linear
equations ex 3. Solutions to linear equations. Another Percent Word Problem. Percent word problems. Percent word problems 1 example 2). Solving Percent Problems 2. Solving Percent Problems 3.
Percentage word problems 1. Percentage word problems 2. Rearrange formulas to isolate specific variables. Solving for a Variable. Solving for a Variable 2. Example: Solving for a variable. Solving
equations in terms of a variable. Converting Repeating Decimals to Fractions 1. Converting 1-digit repeating decimals to fractions. Converting Repeating Decimals to Fractions 2. Converting
multi-digit repeating decimals to fractions. Ex 1 Age word problem. Ex 2 Age word problem. Ex 3 Age word problem. Age word problems. Absolute Value Equations. Absolute Value Equations Example 1.
Absolute Value Equation Example 2. Absolute Value Equations. Absolute Value Equations 1. Absolute value equation example. Absolute value equation with no solution. Absolute value equations. Absolute
Value Inequalities. Absolute value inequalities Example 1. Absolute Inequalities 2. Absolute value inequalities example 3. Ex 2 Multi-step equation. Solving Equations with the Distributive Property.
Solving equations with the distributive property 2. Ex 2: Distributive property to simplify . Ex 1: Distributive property to simplify . Ex 3: Distributive property to simplify . Multistep equations
with distribution. Evaluating expressions where individual variable values are unknown. Evaluating expressions with unknown variables 2. Expressions with unknown variables. Expressions with unknown
variables 2. Mixture problems 2. Basic Rate Problem. Early Train Word Problem. Patterns in Sequences 1. Patterns in Sequences 2. Equations of Sequence Patterns. Finding the 100th Term in a Sequence.
Challenge example: Sum of integers. Integer sums. Integer sums. 2003 AIME II Problem 1. Bunch of examples. Mixture problems 3. Order of Operations examples. Algebra: Linear Equations 1. Algebra:
Linear Equations 2. Algebra: Linear Equations 3. Algebra: Linear Equations 4. Averages. Taking percentages. Growing by a percentage. More percent problems. Age word problems 1. Age word problems 2.
Age word problems 3. Why we do the same thing to both sides: simple equations. Representing a relationship with a simple equation. One-Step Equation Intuition. One step equation intuition exercise
intro. One step equation intuition. Adding and subtracting the same thing from both sides. Intuition why we divide both sides. Why we do the same thing to both sides: two-step equations. Why we do
the same thing to both sides: multi-step equations. Why we do the same thing to both sides basic systems. Super Yoga Plans- Basic Variables and Equations. Super Yoga Plans- Solving One-Step
Equations. Constructing and solving equations in the real world 1. Super Yoga Plans- Plotting Points. Super Yoga Plans- Solving Systems by Substitution. Super Yoga Plans- Solving Systems by
Elimination. Constructing and solving equations in the real world 1 exercise. Simple Equations of the form Ax=B. Example solving x/3 =14. One-step equations with multiplication. Example solving x+5=
54. Examples of one-step equations like Ax=B and x+A = B. One step equations. Solving Ax+B = C. Two-Step Equations. Example: Dimensions of a garden. Example: Two-step equation with x/4 term. 2-step
equations. Basic linear equation word problem. Linear equation word problems. Linear equation word problem example. Linear equation word problems 2. Variables on both sides. Example 1: Variables on
both sides. Example 2: Variables on both sides. Equation Special Cases. Equations with variables on both sides. Number of solutions to linear equations. Number of solutions to linear equations ex 2.
Number of solutions to linear equations ex 3. Solutions to linear equations. Another Percent Word Problem. Percent word problems. Percent word problems 1 example 2). Solving Percent Problems 2.
Solving Percent Problems 3. Percentage word problems 1. Percentage word problems 2. Rearrange formulas to isolate specific variables. Solving for a Variable. Solving for a Variable 2. Example:
Solving for a variable. Solving equations in terms of a variable. Converting Repeating Decimals to Fractions 1. Converting 1-digit repeating decimals to fractions. Converting Repeating Decimals to
Fractions 2. Converting multi-digit repeating decimals to fractions. Ex 1 Age word problem. Ex 2 Age word problem. Ex 3 Age word problem. Age word problems. Absolute Value Equations. Absolute Value
Equations Example 1. Absolute Value Equation Example 2. Absolute Value Equations. Absolute Value Equations 1. Absolute value equation example. Absolute value equation with no solution. Absolute value
equations. Absolute Value Inequalities. Absolute value inequalities Example 1. Absolute Inequalities 2. Absolute value inequalities example 3. Ex 2 Multi-step equation. Solving Equations with the
Distributive Property. Solving equations with the distributive property 2. Ex 2: Distributive property to simplify . Ex 1: Distributive property to simplify . Ex 3: Distributive property to simplify
. Multistep equations with distribution. Evaluating expressions where individual variable values are unknown. Evaluating expressions with unknown variables 2. Expressions with unknown variables.
Expressions with unknown variables 2. Mixture problems 2. Basic Rate Problem. Early Train Word Problem. Patterns in Sequences 1. Patterns in Sequences 2. Equations of Sequence Patterns. Finding the
100th Term in a Sequence. Challenge example: Sum of integers. Integer sums. Integer sums. 2003 AIME II Problem 1. Bunch of examples. Mixture problems 3. Order of Operations examples. Algebra: Linear
Equations 1. Algebra: Linear Equations 2. Algebra: Linear Equations 3. Algebra: Linear Equations 4. Averages. Taking percentages. Growing by a percentage. More percent problems. Age word problems 1.
Age word problems 2. Age word problems 3.
Solving exponential and radical expressions and equations. Using scientific notation and significant figures. Understanding Exponents. Understanding Exponents 2. Exponent Rules 1. Exponent Rules 2.
Level 1 Exponents. Level 2 Exponents. Negative Exponent Intuition. Zero, Negative, and Fractional Exponents. Level 3 exponents. Exponent Rules Part 1. Exponent Rules Part 2. Exponent Properties 1.
Exponent Properties 2. Exponent Properties 3. Exponent Properties 4. Exponent Properties 5. Exponent Properties 6. Exponent Properties 7. Exponent Properties Involving Products. Negative and Positive
Exponents. Exponent Properties Involving Quotients. Rational Exponents and Exponent Laws. More Rational Exponents and Exponent Laws. Simplifying Expressions with Exponents. Multiplying and Dividing
Rational Expressions 1. Multiplying and Dividing Rational Expressions 2. Multiplying and Dividing Rational Expressions 3. Simplifying Expressions with Exponents 2. Simplifying Expressions with
Exponents 3. Fractional Exponent Expressions 1. Fractional Exponent Expressions 2. Fractional Exponent Expressions 3. Pythagorean Theorem 1. Pythagorean Theorem 2. Pythagorean Theorem 3. Evaluating
exponential expressions. Evaluating exponential expressions 2. Evaluating exponential expressions 3. Scientific notation 1. Scientific notation 2. Scientific notation 3. Scientific Notation I.
Scientific Notation Example 2. Scientific Notation 3 (new). Scientific Notation Examples. Multiplying in Scientific Notation. Significant Figures. More on Significant Figures. Addition and
Subtraction with Significant Figures. Multiplying and Dividing with Significant Figures. Understanding Square Roots. Approximating Square Roots. Square Roots and Real Numbers. Simplifying Square
Roots. Simplifying Square Roots Comment Response. Finding Cube Roots. Simplifying Cube Roots. Radical Equivalent to Rational Exponents. Radical Equivalent to Rational Exponents 2. Simplifying
radicals. Simplifying Radical Expressions1. Simplifying Radical Expressions 2. Simplifying Radical Expressions 3. More Simplifying Radical Expressions. Radical Expressions with Higher Roots. Adding
and Simplifying Radicals. Subtracting and Simplifying Radicals. Adding and Subtracting Rational Expressions. Multiply and Simplify a Radical Expression 1. Multiply and Simplify a Radical Expression
2. How to Rationalize a Denominator. Solving Radical Equations. Extraneous Solutions to Radical Equations. Solving Radical Equations 1. Solving Radical Equations 2. Solving Radical Equations 3.
Applying Radical Equations 1. Applying Radical Equations 2. Applying Radical Equations 3.
Understanding and solving equations with imaginary numbers. Introduction to i and Imaginary Numbers. Calculating i Raised to Arbitrary Exponents. Imaginary unit powers. Imaginary Roots of Negative
Numbers. i as the Principal Root of -1 (a little technical). Complex numbers. Complex numbers (part 1). Complex numbers (part 2). Plotting complex numbers on the complex plane. The complex plane.
Adding Complex Numbers. Subtracting Complex Numbers. Adding and subtracting complex numbers. Multiplying Complex Numbers. Multiplying complex numbers. Complex Conjugates Example. Dividing Complex
Numbers. Dividing complex numbers. Absolute value of a complex number. Absolute value of complex numbers. Example: Complex roots for a quadratic. Algebra II: Imaginary and Complex Numbers.
Introduction to i and Imaginary Numbers. Calculating i Raised to Arbitrary Exponents. Imaginary unit powers. Imaginary Roots of Negative Numbers. i as the Principal Root of -1 (a little technical).
Complex numbers. Complex numbers (part 1). Complex numbers (part 2). Plotting complex numbers on the complex plane. The complex plane. Adding Complex Numbers. Subtracting Complex Numbers. Adding and
subtracting complex numbers. Multiplying Complex Numbers. Multiplying complex numbers. Complex Conjugates Example. Dividing Complex Numbers. Dividing complex numbers. Absolute value of a complex
number. Absolute value of complex numbers. Example: Complex roots for a quadratic. Algebra II: Imaginary and Complex Numbers.
Sal working through the 53 problems from the practice test available at http://www.cde.ca.gov/ta/tg/hs/documents/mathpractest.pdf for the CAHSEE (California High School Exit Examination). Clearly
useful if you're looking to take that exam. Probably still useful if you want to make sure you have a solid understanding of basic high school math. CAHSEE Practice: Problems 1-3. CAHSEE Practice:
Problems 4-9. CAHSEE Practice: Problems 10-12. CAHSEE Practice: Problems 13-14. CAHSEE Practice: Problems 15-16. CAHSEE Practice: Problems 17-19. CAHSEE Practice: Problems 20-22. CAHSEE Practice:
Problems 23-27. CAHSEE Practice: Problems 28-31. CAHSEE Practice: Problems 32-34. CAHSEE Practice: Problems 35-37. CAHSEE Practice: Problems 38-42. CAHSEE Practice: Problems 43-46. CAHSEE Practice:
Problems 47-51. CAHSEE Practice: Problems 52-53. CAHSEE Practice: Problems 1-3. CAHSEE Practice: Problems 4-9. CAHSEE Practice: Problems 10-12. CAHSEE Practice: Problems 13-14. CAHSEE Practice:
Problems 15-16. CAHSEE Practice: Problems 17-19. CAHSEE Practice: Problems 20-22. CAHSEE Practice: Problems 23-27. CAHSEE Practice: Problems 28-31. CAHSEE Practice: Problems 32-34. CAHSEE Practice:
Problems 35-37. CAHSEE Practice: Problems 38-42. CAHSEE Practice: Problems 43-46. CAHSEE Practice: Problems 47-51. CAHSEE Practice: Problems 52-53.
Select problems from ck12.org's Algebra 1 FlexBook (Open Source Textbook). This is a good playlist to review if you want to make sure you have a good understanding of all of the major topics in
Algebra I. Variable Expressions. Order of Operations Example. Patterns and Equations. Equations and Inequalities. Domain and Range of a Function. Functions as Graphs. Word Problem Solving Plan 1.
Word Problem Solving Strategies. Integers and Rational Numbers. Addition of Rational Numbers. Subtraction of Rational Numbers. Multiplication of Rational Numbers. Distributive Property Example 1.
Division of Rational Numbers. Square Roots and Real Numbers. Problem Solving Word Problems 2. One Step Equations. Two-Step Equations. Ex 1: Distributive property to simplify . Ex 3: Distributive
property to simplify . Ratio and Proportion. Scale and Indirect Measurement. Percent Problems. Another percent example. The Coordinate Plane. Graphing Using Intercepts. Graphs of Linear Equations.
Slope and Rate of Change. Graphs Using Slope-Intercept Form. Direct Variation Models. Function example problems. Word Problem Solving 4. Linear Equations in Slope Intercept Form. Linear Equations in
Point Slope Form. Linear Equations in Standard Form. Equations of Parallel and Perpendicular Lines. Fitting a Line to Data. Predicting with Linear Models. Using a Linear Model. Inequalities Using
Addition and Subtraction. Inequalities Using Multiplication and Division. Compound Inequalities. Absolute Value Equations. Absolute Value Inequalities. Graphing Inequalities. Solving Linear Systems
by Graphing. Solving Linear Systems by Substitution. Solving Systems of Equations by Elimination. Solving Systems of Equations by Multiplication. Special Types of Linear Systems. Systems of Linear
Inequalities. Exponent Properties Involving Products. Exponent Properties Involving Quotients. Zero, Negative, and Fractional Exponents. Scientific Notation. Exponential Growth Functions. Exponential
Decay Functions. Geometric Sequences (Introduction). Word Problem Solving- Exponential Growth and Decay. Addition and Subtraction of Polynomials. Multiplication of Polynomials. Special Products of
Binomials. Polynomial Equations in Factored Form. Factoring quadratic expressions. Factoring Special Products. Factor by Grouping and Factoring Completely. Graphs of Quadratic Functions. Solving
Quadratic Equations by Graphing. Solving Quadratic Equations by Square Roots. Solving Quadratic Equations by Completing the Square. How to Use the Quadratic Formula. Proof of Quadratic Formula.
Discriminant of Quadratic Equations. Linear, Quadratic, and Exponential Models. Identifying Quadratic Models. Identifying Exponential Models. Quadratic Regression. Shifting functions. Radical
Expressions with Higher Roots. More Simplifying Radical Expressions. How to Rationalize a Denominator. Extraneous Solutions to Radical Equations. Radical Equation Examples. More Involved Radical
Equation Example. Pythagorean Theorem. Distance Formula. Midpoint Formula. Visual Pythagorean Theorem Proof. Average or Central Tendency: Arithmetic Mean, Median, and Mode. Range, Variance and
Standard Deviation as Measures of Dispersion. Stem and Leaf Plots. Histograms. Box-and-whisker Plot. Proportionality. Asymptotes of Rational Functions. Another Rational Function Graph Example. A
Third Example of Graphing a Rational Function. Polynomial Division. Simplifying Rational Expressions Introduction. Multiplying and Dividing Rational Expressions. Adding Rational Expressions Example
1. Adding Rational Expressions Example 2. Adding Rational Expressions Example 3. Solving Rational Equations. Two more examples of solving rational equations. Surveys and Samples.
Understanding fractions conceptually, using operations with fractions, and converting fractions. Introduction to fractions. Identifying Fraction Parts. Recognizing fractions exercise. Recognizing
fractions 0.5. Numerator and Denominator of a Fraction. Identifying numerators and denominators. Plotting basic fractions on the number line. Fractions on the number line 1. Fraction word problems 1
exercise. Fraction word problems 1. Visualizing equivalent fractions. Equivalent fraction word problem example. Equivalent fraction word problem example 2. Equivalent fraction word problem example 3.
Visualizing equivalent fractions. Fractions cut and copy 1 exercise. Fractions cut and copy 1. Fractions in lowest terms. Simplifying fractions. Equivalent fractions. Equivalent Fractions Example.
Finding Common Denominators. Equivalent fractions. Equivalent fractions 2. Comparing Fractions. Comparing Fractions 2. Comparing fractions 1. Comparing fractions 2. Ordering Fractions. Ordering
fractions. Decomposing a fraction visually. Decomposing a mixed number. Adding up to a fraction drag and drop example. Decomposing fractions. Adding Fractions with Like Denominators. Adding fractions
with common denominators. Subtracting Fractions. Subtracting fractions with common denominators. Adding and subtracting fractions. Adding Fractions with Unlike Denominators. Adding fractions (ex 1).
Adding fractions. Subtracting fractions with unlike denominators. Subtracting fractions. Adding and subtracting fractions. Adding fractions with different signs. Adding fractions examples. What
fraction of spider eyes are looking at me?. How much more piano practice?. How long is this lizard?. Adding and subtracting fractions with like denominators word problems. Adding fractions with
unlike denominators word problem. Subtracting fractions with unlike denominators word problem. Adding and subtracting fractions with unlike denominators word problems. Multiplying fractions and whole
numbers. Multiplying fractions by integers. Multiplying a fraction by a fraction. Multiplying Fractions. Multiplying fractions 0.5. Multiplying negative and positive fractions. Multiplying fractions.
Multiplying fractions by whole numbers word problems. Multiplication as scaling. Fraction multiplication as scaling. Creating a fraction through division of whole numbers. My share of soap as a mixed
number on a number line. Understanding fractions as division. Putting out bowls of potpurri. Spending the weekend studying. How many t-shirts can I make?. Dividing fractions by whole numbers.
Dividing whole numbers by fractions. Division with fractions and whole numbers word problems. Examples of dividing negative fractions. Dividing fractions. Dividing Fractions Example. Dividing
positive and negative fractions. Dividing fractions word problems. Dividing fractions word problems 2. How long will three movies last?. My family loves milk. We must have eaten more pie!.
Multiplying fractions and whole numbers word problems. Making banana oat muffins...mmm. How much laundry detergent left?. Biking to a friend. Multiplying fractions by fractions word problems.
Comparing improper fractions and mixed numbers. Mixed numbers and improper fractions. Proper and Improper Fractions. Converting Mixed Numbers to Improper Fractions. Changing a Mixed Number to an
Improper Fraction. Changing an Improper Fraction to a Mixed Number. Converting mixed numbers and improper fractions. Positive improper fractions on the number line. Fractions on the number line 2.
Ordering improper fractions and mixed numbers. Comparing improper fractions and mixed numbers. Ordering improper fractions and mixed numbers. Fractions cut and copy 2 exercise example. Fractions cut
and copy 2. Points on a number line. Fractions on the number line 3. Adding Mixed Numbers. Adding Mixed Numbers with Unlike Denominators. Adding Mixed Numbers Word Problem. Subtracting Mixed Numbers.
Subtracting Mixed Numbers 2. Subtracting Mixed Numbers Word Problem. Adding and subtracting mixed numbers 0.5 (ex 1). Adding and subtracting mixed numbers 0.5 (ex 2). Adding and subtracting mixed
numbers 0.5. Adding and subtracting mixed numbers 1 (ex 1). Adding and subtracting mixed numbers 1 (ex 2). Adding and subtracting mixed numbers 1. Multiplying Fractions and Mixed Numbers. Multiplying
Mixed Numbers. Multiplying mixed numbers 1. Reciprocal of a Mixed Number. Dividing Mixed Numbers. Dividing Mixed Numbers and Fractions. Decimals and Fractions. Converting Fractions to Decimals
Example. Converting fractions to decimals (ex1). Converting fractions to decimals (ex2). Converting fractions to decimals. Converting fractions to decimals. Representing a number as a decimal,
percent, and fraction 2. Representing a number as a decimal, percent, and fraction. Converting decimals to fractions 1 (ex 1). Converting decimals to fractions 1 (ex 2). Converting decimals to
fractions 1 (ex 3). Converting decimals to fractions 1. Converting decimals to fractions 2 (ex 1). Converting decimals to fractions 2 (ex 2). Converting decimals to fractions 2. Ordering numeric
expressions. Ordering numbers. Converting a fraction to a repeating decimal. Writing fractions as repeating decimals. Number Sets. Number Sets 1. Number Sets 2. Number Sets 3. Introduction to
fractions. Identifying Fraction Parts. Recognizing fractions exercise. Recognizing fractions 0.5. Numerator and Denominator of a Fraction. Identifying numerators and denominators. Plotting basic
fractions on the number line. Fractions on the number line 1. Fraction word problems 1 exercise. Fraction word problems 1. Visualizing equivalent fractions. Equivalent fraction word problem example.
Equivalent fraction word problem example 2. Equivalent fraction word problem example 3. Visualizing equivalent fractions. Fractions cut and copy 1 exercise. Fractions cut and copy 1. Fractions in
lowest terms. Simplifying fractions. Equivalent fractions. Equivalent Fractions Example. Finding Common Denominators. Equivalent fractions. Equivalent fractions 2. Comparing Fractions. Comparing
Fractions 2. Comparing fractions 1. Comparing fractions 2. Ordering Fractions. Ordering fractions. Decomposing a fraction visually. Decomposing a mixed number. Adding up to a fraction drag and drop
example. Decomposing fractions. Adding Fractions with Like Denominators. Adding fractions with common denominators. Subtracting Fractions. Subtracting fractions with common denominators. Adding and
subtracting fractions. Adding Fractions with Unlike Denominators. Adding fractions (ex 1). Adding fractions. Subtracting fractions with unlike denominators. Subtracting fractions. Adding and
subtracting fractions. Adding fractions with different signs. Adding fractions examples. What fraction of spider eyes are looking at me?. How much more piano practice?. How long is this lizard?.
Adding and subtracting fractions with like denominators word problems. Adding fractions with unlike denominators word problem. Subtracting fractions with unlike denominators word problem. Adding and
subtracting fractions with unlike denominators word problems. Multiplying fractions and whole numbers. Multiplying fractions by integers. Multiplying a fraction by a fraction. Multiplying Fractions.
Multiplying fractions 0.5. Multiplying negative and positive fractions. Multiplying fractions. Multiplying fractions by whole numbers word problems. Multiplication as scaling. Fraction multiplication
as scaling. Creating a fraction through division of whole numbers. My share of soap as a mixed number on a number line. Understanding fractions as division. Putting out bowls of potpurri. Spending
the weekend studying. How many t-shirts can I make?. Dividing fractions by whole numbers. Dividing whole numbers by fractions. Division with fractions and whole numbers word problems. Examples of
dividing negative fractions. Dividing fractions. Dividing Fractions Example. Dividing positive and negative fractions. Dividing fractions word problems. Dividing fractions word problems 2. How long
will three movies last?. My family loves milk. We must have eaten more pie!. Multiplying fractions and whole numbers word problems. Making banana oat muffins...mmm. How much laundry detergent left?.
Biking to a friend. Multiplying fractions by fractions word problems. Comparing improper fractions and mixed numbers. Mixed numbers and improper fractions. Proper and Improper Fractions. Converting
Mixed Numbers to Improper Fractions. Changing a Mixed Number to an Improper Fraction. Changing an Improper Fraction to a Mixed Number. Converting mixed numbers and improper fractions. Positive
improper fractions on the number line. Fractions on the number line 2. Ordering improper fractions and mixed numbers. Comparing improper fractions and mixed numbers. Ordering improper fractions and
mixed numbers. Fractions cut and copy 2 exercise example. Fractions cut and copy 2. Points on a number line. Fractions on the number line 3. Adding Mixed Numbers. Adding Mixed Numbers with Unlike
Denominators. Adding Mixed Numbers Word Problem. Subtracting Mixed Numbers. Subtracting Mixed Numbers 2. Subtracting Mixed Numbers Word Problem. Adding and subtracting mixed numbers 0.5 (ex 1).
Adding and subtracting mixed numbers 0.5 (ex 2). Adding and subtracting mixed numbers 0.5. Adding and subtracting mixed numbers 1 (ex 1). Adding and subtracting mixed numbers 1 (ex 2). Adding and
subtracting mixed numbers 1. Multiplying Fractions and Mixed Numbers. Multiplying Mixed Numbers. Multiplying mixed numbers 1. Reciprocal of a Mixed Number. Dividing Mixed Numbers. Dividing Mixed
Numbers and Fractions. Decimals and Fractions. Converting Fractions to Decimals Example. Converting fractions to decimals (ex1). Converting fractions to decimals (ex2). Converting fractions to
decimals. Converting fractions to decimals. Representing a number as a decimal, percent, and fraction 2. Representing a number as a decimal, percent, and fraction. Converting decimals to fractions 1
(ex 1). Converting decimals to fractions 1 (ex 2). Converting decimals to fractions 1 (ex 3). Converting decimals to fractions 1. Converting decimals to fractions 2 (ex 1). Converting decimals to
fractions 2 (ex 2). Converting decimals to fractions 2. Ordering numeric expressions. Ordering numbers. Converting a fraction to a repeating decimal. Writing fractions as repeating decimals. Number
Sets. Number Sets 1. Number Sets 2. Number Sets 3.
Lectures based on the Singapore Math curriculum. You can follow along through the workbooks available at singaporemath.com. Singapore Math: Grade 3a, Unit 1 (part 1). Singapore Math: Grade 3a Unit 1
(part 2). Singapore Math: Grade 3a, Unit 1 (part 3). Singapore Math: Grade 3a, Unit 1 (part 4). Singapore Math: Grade 3a, Unit 1 (part 5). Singapore Math: Grade 3a, Unit 1 (part 6). Singapore Math:
Grade 3a, Unit 1 (part 7). Singapore Math: Grade 3a, Unit 1 (part 8). Singapore Math: Grade 3a, Unit 1 (part 9). Singapore Math: Grade 3a Unit 2 (part 1). Singapore Math: Grade 3a Unit 2 (part 2).
Singapore Math: Grade 3a Unit 2 (part 3). Singapore Math: Grade 3a Unit 2 (part 4). Singapore Math: Grade 3a Unit 2 (part 5). Singapore Math: Grade 3a Unit 2 (part 6). Singapore Math: Grade 3a Unit 2
(part 7). Singapore Math: Grade 3a Unit 2 (part 8). Singapore Math: Grade 3a Unit 2 (part 9). Singapore Math: Grade 3a Unit 2 (part 10). Singapore Math: Grade 3a Unit 2 (part 11). Singapore Math:
Grade 3a, Unit 1 (part 1). Singapore Math: Grade 3a Unit 1 (part 2). Singapore Math: Grade 3a, Unit 1 (part 3). Singapore Math: Grade 3a, Unit 1 (part 4). Singapore Math: Grade 3a, Unit 1 (part 5).
Singapore Math: Grade 3a, Unit 1 (part 6). Singapore Math: Grade 3a, Unit 1 (part 7). Singapore Math: Grade 3a, Unit 1 (part 8). Singapore Math: Grade 3a, Unit 1 (part 9). Singapore Math: Grade 3a
Unit 2 (part 1). Singapore Math: Grade 3a Unit 2 (part 2). Singapore Math: Grade 3a Unit 2 (part 3). Singapore Math: Grade 3a Unit 2 (part 4). Singapore Math: Grade 3a Unit 2 (part 5). Singapore
Math: Grade 3a Unit 2 (part 6). Singapore Math: Grade 3a Unit 2 (part 7). Singapore Math: Grade 3a Unit 2 (part 8). Singapore Math: Grade 3a Unit 2 (part 9). Singapore Math: Grade 3a Unit 2 (part
10). Singapore Math: Grade 3a Unit 2 (part 11).
Adding and subtracting positive and negative whole numbers. Starts with 1+1=2 and covers carrying, borrowing, and word problems. Basic Addition. 1-digit addition. Basic Subtraction. 1-digit
subtraction. Example: Adding two digit numbers (no carrying). 2-digit addition. Subtraction 2. Example: 2-digit subtraction (no borrowing). 2 and 3-digit subtraction. Level 2 Addition. Introduction
to carrying when adding. Addition 3. Addition with carrying. Addition 4. 4-digit addition with carrying. Subtraction 3: Introduction to Borrowing or Regrouping. Why borrowing works. Borrowing once
example 1. Borrowing once example 2. Subtraction with borrowing. Regrouping (borrowing) twice example. 4-digit subtraction with borrowing. Alternate mental subtraction method. Level 4 Subtraction.
Subtraction Word Problem. Addition and subtraction word problems. Basic Addition. 1-digit addition. Basic Subtraction. 1-digit subtraction. Example: Adding two digit numbers (no carrying). 2-digit
addition. Subtraction 2. Example: 2-digit subtraction (no borrowing). 2 and 3-digit subtraction. Level 2 Addition. Introduction to carrying when adding. Addition 3. Addition with carrying. Addition
4. 4-digit addition with carrying. Subtraction 3: Introduction to Borrowing or Regrouping. Why borrowing works. Borrowing once example 1. Borrowing once example 2. Subtraction with borrowing.
Regrouping (borrowing) twice example. 4-digit subtraction with borrowing. Alternate mental subtraction method. Level 4 Subtraction. Subtraction Word Problem. Addition and subtraction word problems.
Using polynomial expressions and factoring polynomials. Terms coefficients and exponents in a polynomial. Interesting Polynomial Coefficient Problem. Polynomials1. Polynomials 2. Evaluating a
polynomial at a given value. Simplify a polynomial. Adding Polynomials. Example: Adding polynomials with multiple variables. Addition and Subtraction of Polynomials. Adding and Subtracting
Polynomials 1. Adding and Subtracting Polynomials 2. Adding and Subtracting Polynomials 3. Subtracting Polynomials. Subtracting polynomials with multiple variables. Adding and subtracting
polynomials. Multiplying Monomials. Dividing Monomials. Multiplying and Dividing Monomials 1. Multiplying and Dividing Monomials 2. Multiplying and Dividing Monomials 3. Monomial Greatest Common
Factor. Factoring and the Distributive Property. Factoring and the Distributive Property 2. Factoring and the Distributive Property 3. Multiplying Binomials with Radicals. Multiplication of
Polynomials. Multiplying Binomials. Multiplying Polynomials1. Multiplying Polynomials 2. Square a Binomial. Special Products of Binomials. Special Polynomials Products 1. Factor polynomials using the
GCF. Special Products of Polynomials 1. Special Products of Polynomials 2. Multiplying expressions 0.5. Factoring linear binomials. Multiplying expressions 1. Multiplying Monomials by Polynomials.
Multiplying Polynomials. Multiplying Polynomials 3. More multiplying polynomials. Multiplying polynomials. Level 1 multiplying expressions. Polynomial Division. Polynomial divided by monomial.
Dividing multivariable polynomial with monomial. Dividing polynomials 1. Dividing polynomials with remainders. Synthetic Division. Synthetic Division Example 2. Why Synthetic Division Works.
Factoring Sum of Cubes. Difference of Cubes Factoring. Algebraic Long Division. Algebra II: Simplifying Polynomials. Terms coefficients and exponents in a polynomial. Interesting Polynomial
Coefficient Problem. Polynomials1. Polynomials 2. Evaluating a polynomial at a given value. Simplify a polynomial. Adding Polynomials. Example: Adding polynomials with multiple variables. Addition
and Subtraction of Polynomials. Adding and Subtracting Polynomials 1. Adding and Subtracting Polynomials 2. Adding and Subtracting Polynomials 3. Subtracting Polynomials. Subtracting polynomials with
multiple variables. Adding and subtracting polynomials. Multiplying Monomials. Dividing Monomials. Multiplying and Dividing Monomials 1. Multiplying and Dividing Monomials 2. Multiplying and Dividing
Monomials 3. Monomial Greatest Common Factor. Factoring and the Distributive Property. Factoring and the Distributive Property 2. Factoring and the Distributive Property 3. Multiplying Binomials with
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percentage. Representing a number as a decimal, percent, and fraction 2. Ordering numeric expressions.
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Graphically. Graphing systems of equations. King's Cupcakes: Solving Systems by Elimination. How many bags of potato chips do people eat?. Simple Elimination Practice. Systems of equations with
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equations with substitution. Systems of equations. Systems of equations word problems. Solving linear systems by graphing. Graphing systems of equations. Solving linear systems by substitution.
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with elimination. Systems of equations. Special types of linear systems. Solutions to systems of equations. Old video on systems of equations. Solving linear systems by graphing. Testing a solution
for a system of equations. Graphing Systems of Equations. Graphical Systems Application Problem. Example 2: Graphically Solving Systems. Example 3: Graphically Solving Systems. Solving Systems
Graphically. Graphing systems of equations. Inconsistent systems of equations. Infinite solutions to systems. Consistent and Inconsistent Systems. Independent and Dependent Systems. Practice thinking
about number of solutions to systems. Graphical solutions to systems. Solutions to systems of equations. Constructing solutions to systems of equations. Constructing consistent and inconsistent
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equations with simple elimination. Systems with Elimination Practice. Systems of equations with elimination. Using a system of equations to find the price of apples and oranges. Linear systems word
problem with substitution. Systems of equations word problems. Systems of equation to realize you are getting ripped off. Thinking about multiple solutions to a system of equations. Understanding
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Problem. System of equations for passing trains problem. Overtaking Word Problem. Multple examples of multiple constraint problems. Testing Solutions for a System of Inequalities. Visualizing the
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of Inequalities Application. CA Algebra I: Systems of Inequalities. Systems of Three Variables. Systems of Three Variables 2. Solutions to Three Variable System. Solutions to Three Variable System 2.
Three Equation Application Problem. Non-Linear Systems of Equations 3. Non-Linear Systems of Equations 1. Non-Linear Systems of Equations 2. Non-Linear Systems of Equations 3. Systems of nonlinear
equations 1. Systems of nonlinear equations 2. Systems of nonlinear equations 3. Systems of nonlinear equations. Trolls, Tolls, and Systems of Equations. Solving the Troll Riddle Visually. Solving
Systems Graphically. Graphing systems of equations. King's Cupcakes: Solving Systems by Elimination. How many bags of potato chips do people eat?. Simple Elimination Practice. Systems of equations
with simple elimination. Systems with Elimination Practice. Systems of equations with elimination. Talking bird solves systems with substitution. Practice using substitution for systems. Systems of
equations with substitution. Systems of equations. Systems of equations word problems. Solving linear systems by graphing. Graphing systems of equations. Solving linear systems by substitution.
Systems of equations with substitution. Solving systems of equations by elimination. Systems of equations with simple elimination. Solving systems of equations by multiplication. Systems of equations
with elimination. Systems of equations. Special types of linear systems. Solutions to systems of equations. Old video on systems of equations. Solving linear systems by graphing. Testing a solution
for a system of equations. Graphing Systems of Equations. Graphical Systems Application Problem. Example 2: Graphically Solving Systems. Example 3: Graphically Solving Systems. Solving Systems
Graphically. Graphing systems of equations. Inconsistent systems of equations. Infinite solutions to systems. Consistent and Inconsistent Systems. Independent and Dependent Systems. Practice thinking
about number of solutions to systems. Graphical solutions to systems. Solutions to systems of equations. Constructing solutions to systems of equations. Constructing consistent and inconsistent
systems. Example 1: Solving systems by substitution. Example 2: Solving systems by substitution. Example 3: Solving systems by substitution. The Substitution Method. Substitution Method 2.
Substitution Method 3. Practice using substitution for systems. Systems of equations with substitution. Example 1: Solving systems by elimination. Example 2: Solving systems by elimination. Addition
Elimination Method 1. Addition Elimination Method 2. Addition Elimination Method 3. Addition Elimination Method 4. Example 3: Solving systems by elimination. Simple Elimination Practice. Systems of
equations with simple elimination. Systems with Elimination Practice. Systems of equations with elimination. Using a system of equations to find the price of apples and oranges. Linear systems word
problem with substitution. Systems of equations word problems. Systems of equation to realize you are getting ripped off. Thinking about multiple solutions to a system of equations. Understanding
systems of equations word problems. Systems and rate problems. Systems and rate problems 2. Systems and rate problems 3. Officer on Horseback. Two Passing Bicycles Word Problem. Passed Bike Word
Problem. System of equations for passing trains problem. Overtaking Word Problem. Multple examples of multiple constraint problems. Testing Solutions for a System of Inequalities. Visualizing the
solution set for a system of inequalities. Graphing systems of inequalities. Graphing systems of inequalities 2. Graphing systems of inequalities. Graphing and solving systems of inequalities. System
of Inequalities Application. CA Algebra I: Systems of Inequalities. Systems of Three Variables. Systems of Three Variables 2. Solutions to Three Variable System. Solutions to Three Variable System 2.
Three Equation Application Problem. Non-Linear Systems of Equations 3. Non-Linear Systems of Equations 1. Non-Linear Systems of Equations 2. Non-Linear Systems of Equations 3. Systems of nonlinear
equations 1. Systems of nonlinear equations 2. Systems of nonlinear equations 3. Systems of nonlinear equations.
Questions from previous IIT JEEs. IIT JEE Trigonometry Problem 1. IIT JEE Perpendicular Planes (Part 1). IIT JEE Perpendicular Plane (part 2). IIT JEE Complex Root Probability (part 1). IIT JEE
Complex Root Probability (part 2). IIT JEE Position Vectors. IIT JEE Integral Limit. IIT JEE Algebraic Manipulation. IIT JEE Function Maxima. IIT JEE Diameter Slope. IIT JEE Hairy Trig and Algebra
(part 1). IIT JEE Hairy Trig and Algebra (Part 2). IIT JEE Hairy Trig and Algebra (Part 3). IIT JEE Complex Numbers (part 1). IIT JEE Complex Numbers (part 2). IIT JEE Complex Numbers (part 3). IIT
JEE Differentiability and Boundedness. IIT JEE Integral with Binomial Expansion. IIT JEE Symmetric and Skew-Symmetric Matrices. IIT JEE Trace and Determinant. IIT JEE Divisible Determinants. IIT JEE
Circle Hyperbola Intersection. IIT JEE Circle Hyperbola Common Tangent Part 1. IIT JEE Circle Hyperbola Common Tangent Part 2. IIT JEE Circle Hyperbola Common Tangent Part 3. IIT JEE Circle Hyperbola
Common Tangent Part 4. IIT JEE Circle Hyperbola Common Tangent Part 5. IIT JEE Trigonometric Constraints. IIT JEE Trigonometric Maximum. Vector Triple Product Expansion (very optional). IIT JEE
Lagrange's Formula. Tangent Line Hyperbola Relationship (very optional). 2010 IIT JEE Paper 1 Problem 50 Hyperbola Eccentricity. Normal vector from plane equation. Point distance to plane. Distance
Between Planes. Complex Determinant Example. Series Sum Example. Trigonometric System Example. Simple Differential Equation Example. IIT JEE Trigonometry Problem 1. IIT JEE Perpendicular Planes (Part
1). IIT JEE Perpendicular Plane (part 2). IIT JEE Complex Root Probability (part 1). IIT JEE Complex Root Probability (part 2). IIT JEE Position Vectors. IIT JEE Integral Limit. IIT JEE Algebraic
Manipulation. IIT JEE Function Maxima. IIT JEE Diameter Slope. IIT JEE Hairy Trig and Algebra (part 1). IIT JEE Hairy Trig and Algebra (Part 2). IIT JEE Hairy Trig and Algebra (Part 3). IIT JEE
Complex Numbers (part 1). IIT JEE Complex Numbers (part 2). IIT JEE Complex Numbers (part 3). IIT JEE Differentiability and Boundedness. IIT JEE Integral with Binomial Expansion. IIT JEE Symmetric
and Skew-Symmetric Matrices. IIT JEE Trace and Determinant. IIT JEE Divisible Determinants. IIT JEE Circle Hyperbola Intersection. IIT JEE Circle Hyperbola Common Tangent Part 1. IIT JEE Circle
Hyperbola Common Tangent Part 2. IIT JEE Circle Hyperbola Common Tangent Part 3. IIT JEE Circle Hyperbola Common Tangent Part 4. IIT JEE Circle Hyperbola Common Tangent Part 5. IIT JEE Trigonometric
Constraints. IIT JEE Trigonometric Maximum. Vector Triple Product Expansion (very optional). IIT JEE Lagrange's Formula. Tangent Line Hyperbola Relationship (very optional). 2010 IIT JEE Paper 1
Problem 50 Hyperbola Eccentricity. Normal vector from plane equation. Point distance to plane. Distance Between Planes. Complex Determinant Example. Series Sum Example. Trigonometric System Example.
Simple Differential Equation Example.
Videos exploring why algebra was developed and how it helps us explain our world. Origins of Algebra. Abstract-ness. The Beauty of Algebra. Descartes and Cartesian Coordinates. Why all the letters in
Algebra?. What is a variable?. Why aren't we using the multiplication sign. Example: evaluating an expression. Example: evaluate a formula using substitution. Evaluating exponential expressions 2.
Evaluating expressions in one variable. Expressions with two variables. Example: Evaluating expressions with 2 variables. Examples of evaluating variable expressions. Evaluating expressions in 2
variables. Example evaluating expressions in word problems. Evaluating expressions 3. Combining like terms. Adding like rational terms. Combining Like Terms 1. Combining Like Terms 2. Combining Like
Terms 3. Combining like terms. Combining like terms and the distributive property. Combining like terms with distribution. Factoring a linear expression with rational terms. Distributive property
with rational terms. Manipulating linear expressions with rational coefficients. Equivalent forms of expressions 1. Equivalent forms of expressions 1. Writing Expressions 1. Writing expressions.
Writing Expressions 2. Writing expressions 2 exercise example. Writing expressions 2. Interpreting linear expressions example. Interpreting linear expressions example 2. Interpreting linear
expressions. Writing expressions 3 exercise example 1. Writing expressions 3 exercise example 2. Writing expressions 3 exercise example 3. Writing expressions 3. Why we do the same thing to both
sides: simple equations. Representing a relationship with a simple equation. One-Step Equation Intuition. One step equation intuition exercise intro. One step equation intuition. Adding and
subtracting the same thing from both sides. Intuition why we divide both sides. Why we do the same thing to both sides: two-step equations. Why we do the same thing to both sides: multi-step
equations. Why we do the same thing to both sides basic systems. Why all the letters in Algebra?. Super Yoga Plans- Basic Variables and Equations. Super Yoga Plans- Solving One-Step Equations.
Constructing and solving equations in the real world 1. Super Yoga Plans- Plotting Points. Super Yoga Plans- Solving Systems by Substitution. Super Yoga Plans- Solving Systems by Elimination.
Variables Expressions and Equations. Solving equations and inequalities through substitution example 1. Solving equations and inequalities through substitution example 2. Solving equations and
inequalities through substitution example 3. Solving equations and inequalities through substitution example 4. Solving equations and inequalities through substitution. Dependent and independent
variables exercise example 1. Dependent and independent variables exercise example 2. Dependent and independent variables exercise example 3. Dependent and independent variables. Origins of Algebra.
Abstract-ness. The Beauty of Algebra. Descartes and Cartesian Coordinates. Why all the letters in Algebra?. What is a variable?. Why aren't we using the multiplication sign. Example: evaluating an
expression. Example: evaluate a formula using substitution. Evaluating exponential expressions 2. Evaluating expressions in one variable. Expressions with two variables. Example: Evaluating
expressions with 2 variables. Examples of evaluating variable expressions. Evaluating expressions in 2 variables. Example evaluating expressions in word problems. Evaluating expressions 3. Combining
like terms. Adding like rational terms. Combining Like Terms 1. Combining Like Terms 2. Combining Like Terms 3. Combining like terms. Combining like terms and the distributive property. Combining
like terms with distribution. Factoring a linear expression with rational terms. Distributive property with rational terms. Manipulating linear expressions with rational coefficients. Equivalent
forms of expressions 1. Equivalent forms of expressions 1. Writing Expressions 1. Writing expressions. Writing Expressions 2. Writing expressions 2 exercise example. Writing expressions 2.
Interpreting linear expressions example. Interpreting linear expressions example 2. Interpreting linear expressions. Writing expressions 3 exercise example 1. Writing expressions 3 exercise example
2. Writing expressions 3 exercise example 3. Writing expressions 3. Why we do the same thing to both sides: simple equations. Representing a relationship with a simple equation. One-Step Equation
Intuition. One step equation intuition exercise intro. One step equation intuition. Adding and subtracting the same thing from both sides. Intuition why we divide both sides. Why we do the same thing
to both sides: two-step equations. Why we do the same thing to both sides: multi-step equations. Why we do the same thing to both sides basic systems. Why all the letters in Algebra?. Super Yoga
Plans- Basic Variables and Equations. Super Yoga Plans- Solving One-Step Equations. Constructing and solving equations in the real world 1. Super Yoga Plans- Plotting Points. Super Yoga Plans-
Solving Systems by Substitution. Super Yoga Plans- Solving Systems by Elimination. Variables Expressions and Equations. Solving equations and inequalities through substitution example 1. Solving
equations and inequalities through substitution example 2. Solving equations and inequalities through substitution example 3. Solving equations and inequalities through substitution example 4.
Solving equations and inequalities through substitution. Dependent and independent variables exercise example 1. Dependent and independent variables exercise example 2. Dependent and independent
variables exercise example 3. Dependent and independent variables.
GMAT Math: 1. GMAT Math: 2. GMAT Math: 3. GMAT Math: 4. GMAT Math: 5. GMAT: Math 6. GMAT: Math 7. GMAT: Math 8. GMAT: Math 9. GMAT: Math 10. GMAT: Math 11. GMAT: Math 12. GMAT: Math 13. GMAT: Math
14. GMAT: Math 15. GMAT: Math 16. GMAT: Math 17. GMAT: Math 18. GMAT: Math 19. GMAT Math 20. GMAT Math 21. GMAT Math 22. GMAT Math 23. GMAT Math 24. GMAT Math 25. GMAT Math 26. GMAT Math 27. GMAT
Math 28. GMAT Math 29. GMAT Math 30. GMAT Math 31. GMAT Math 32. GMAT Math 33. GMAT Math 34. GMAT Math 35. GMAT Math 36. GMAT Math 37. GMAT Math 38. GMAT Math 39. GMAT Math 40. GMAT Math 41. GMAT
Math 42. GMAT Math 43. GMAT Math 44. GMAT Math 45. GMAT Math 46. GMAT Math 47. GMAT Math 48. GMAT Math 49. GMAT Math 50. GMAT Math 51. GMAT Math 52. GMAT Math 53. GMAT Math 54.
Types of Decay. Half-Life. Exponential Decay Formula Proof (can skip, involves Calculus). Introduction to Exponential Decay. More Exponential Decay Examples. Types of Decay. Half-Life. Exponential
Decay Formula Proof (can skip, involves Calculus). Introduction to Exponential Decay. More Exponential Decay Examples.
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Difference Of Means Calculator - Calculator Wow
Difference Of Means Calculator
The Difference Of Means Calculator is a valuable tool in statistical analysis, enabling researchers, analysts, and students to quantify the disparity between two sample means. This article delves
into the intricacies of this calculator, shedding light on its utility in various fields of study and research.
The importance of the Difference Of Means Calculator lies in its ability to provide insights into the comparative analysis of sample data sets. Key aspects of its significance include:
• Statistical Inference: By computing the difference between two means, researchers can infer whether there is a significant distinction between the populations from which the samples were drawn.
• Hypothesis Testing: In hypothesis testing scenarios, the calculator aids in determining whether observed differences in sample means are statistically significant or merely due to random
• Data-driven Decision Making: Organizations utilize the calculator to analyze performance metrics, customer feedback, and other key indicators, facilitating informed decision-making processes.
How to Use
Utilizing the Difference Of Means Calculator is straightforward:
1. Input Means: Enter the values for the first mean (Mean 1) and the second mean (Mean 2) into the designated fields.
2. Calculate Difference: Click the “Calculate” button to compute the difference of means (DM) using the provided inputs.
3. Interpret Results: Review the calculated difference of means to understand the degree of disparity between the two sample populations.
10 FAQs and Answers
1. What does the Difference Of Means Calculator measure?
• The calculator quantifies the difference between two sample means, providing a numerical value that indicates the extent of disparity between the sample populations.
2. How is the difference of means interpreted in statistical analysis?
• A positive difference of means suggests that the first sample mean is higher than the second, while a negative difference indicates the opposite. The magnitude of the difference signifies the
degree of separation between the two sample populations.
3. Can the Difference Of Means Calculator be used for paired data analysis?
• Yes, the calculator can analyze paired data by comparing the means of paired observations within the same sample group.
In conclusion, the Difference Of Means Calculator serves as a valuable tool in statistical analysis, enabling researchers and analysts to quantify the discrepancy between sample means and draw
meaningful insights from comparative data sets. By leveraging this calculator, users can enhance their understanding of statistical relationships, conduct hypothesis tests, and make data-driven
decisions with confidence. Embrace the power of the Difference Of Means Calculator in unraveling statistical mysteries and advancing research across various domains of study. | {"url":"https://calculatorwow.com/difference-of-means-calculator/","timestamp":"2024-11-12T15:29:12Z","content_type":"text/html","content_length":"63916","record_id":"<urn:uuid:5b08e418-f491-4322-9461-bc6de3043f1a>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.63/warc/CC-MAIN-20241112145015-20241112175015-00145.warc.gz"} |
Viewing posts by christian
Posted by: christian on 24 Mar 2020
PyValem is a Python package, for parsing, representing and manipulating the formulas of simple chemical species, their states and reactions. Its source is available on GitHub under a GPL licence, and
it can be installed from the Python Package Index, PyPI, using pip:
The Recamán sequence is a famous sequence invented by the Colombian mathematician, Bernardo Recamán Santos. It is defined by the following algorithm, starting at $a_0 = 0$:
Posted by: christian on 25 Jan 2020
This blog post was inspired by Holly Krieger's video for Numberphile.
Posted by: christian on 19 Dec 2019
Piphilology comprises the creation and use of mnemonic techniques to remember a span of digits of the mathematical constant $\pi$. One famous technique, attributed to the physicist James Jeans uses
the number of letters in each word of the sentence:
Posted by: christian on 15 Nov 2019
The quadtree data structure is a convenient way to store the location of arbitrarily-distributed points in two-dimensional space. Quadtrees are often used in image processing and collision detection. | {"url":"https://scipython.com/blog/author/christian/?page=17","timestamp":"2024-11-14T13:17:25Z","content_type":"text/html","content_length":"24105","record_id":"<urn:uuid:73865636-3112-476b-a643-5a21d81f9722>","cc-path":"CC-MAIN-2024-46/segments/1730477028657.76/warc/CC-MAIN-20241114130448-20241114160448-00237.warc.gz"} |
What are the main advantages of using W.D. Gann Arcs and Circles over other methods? | Practical GANN
What are the main advantages of using W.D. Gann Arcs and Circles over other methods?
What are the main advantages of using W.D. Gann Arcs and Circles over other methods? In today’s article, I’ll provide you with an insightful guide on how to effectively use W.D. Gann Arcs and Circles
when you are in the making of a chart for over at this website a future financial condition within the market. For such analysis, these method can be used together with each other and also be
integrated in other methods. With W.D. Gann Arcs and Circles integrated, it can provide you a complete understanding of the market. But if it’s for your own interests, it’s best to go for any one
method alone. Now, let’s begin with a quick about the main advantages of these 2 analysis methods! What are the main advantages of W.D. Gann Arcs analysis? For any financial position of any stock
traded, we can map it onto a 2D chart.
As mentioned in the previous article, we have 2 components which are direction change and time duration respectively. The direction is either one Your Domain Name the 4 types of pattern – Caudal,
Cusp, Piercing and Piercing Sun. The time duration is obviously stock price movement. The question arises whether these will be able to provide an exact and accurate picture of the stock price
movement direction and time duration. I think that it really depends on the type of chart and the trading approach being taken. There are several methods that can be used for analyzing these
patterns. Two major methods include the conventional Gann Arcs and Circles for describing the direction and relative strength of the stock, while the Fibonacci method is used to analyze the time
duration of the move. In this article, we’ll discuss about the advantage of 1st method only. As it stands, with the conventional method of Gann Arcs, only after the reversal is completed, it will
become clear whether the direction is bearish or bullish. So,What are the main advantages of using W.D. Gann Arcs and Circles over other methods? If you can find your money, you’ve got an easier time
planning for the long term and because arc can buy, you can purchase a number of pieces of equipment and sell those purchases near at later. This offers great go now and can add company website the
value of the stock you are building.
Gann’s Law of Vibration
Unless you are willing to dedicate as much of your time as we do with the Gann method. Finally and most important are the benefits you will receive in trading and/or building an income fund. When the
new, in-depth investor education courses, our newsletters, and trading and building lessons are completed you will be closer to the dream of building a respectable portfolio. We know, that when the
student is the expert and when the trader is the student they will have an edge in the markets we are teaching as each method uses different information and techniques. Who’s Who in Rolle Circle
Trading and W.D. Gann D.J. Gann and W.D. Rolle have both built very large collections of bull, bear and neutral trading and index trading systems in their well-known Rolle and Gann circles, as well
as other systems written by J. W. Gann and others.
Gann Hexagon
These all bear analysis and history dates and are well-documented and vetted. A: While not proven or verified (anyone can claim anything), it is very common for a given symbol/concept to be
attributed to a master or creator. While possible that many years ago there may have been one or two people who were responsible for the creation of the strategy, a new his explanation is usually the
creation of multiple people and this pattern of the past doesn’t cease to exist. It is due to this history of creators and proven usage, that we have chosen to attribute any of our own creations to
some of the most famous builders of trading and investing. For example, among dozens of papers we could cite, one by George Stathatos (founder of Portfolio Analysis) refers to his system as being
“incorporating the work of our founders, JW Gann and DJ Gann, as well as our own studies.” Another by Mr. Stathatson refers to George as the father of Portfolio Analysis. Whether people have been
given credit for the construction of their own work has no bearing on the ideas of the ideas behind the strategies, but it adds confidence that you have the method that is the best way online nursing
homework help purchase, and execute, the strategy. Q: Does Gann Theory or methods in general apply to everyone? A: No – Gann principles were developed from his investment experience – specifically to
sell high and often, at what he called “peak” prices, and then sell in smaller quantities where price moves could be used to make money. While these principles do work very well for most individual
investors, many different people are looking for different philosophies which allow them to minimize risk exposure, reduceWhat are the main advantages of using W.D. Gann Arcs and Circles over other
methods? I know that it is usually not difficult for people to understand the basic methodology, so please consider the following example: We want to learn the rate that any given country’s
population doubles in the next 50 years. We use the formula: {the population in 1977} = 2 \times {the population in 0} + {the rate of population growth} \times {the time in years} The rate of
population growth is 60, and we are interested in the population in 1977.
Astrological Charting
If you plug 1960 into the 2 here you will get the population in 0. Country X has an estimated population of 5,000 in 1960 so you plug 5,000 into the 2 and get 10,000. Now, the formula is used to
predict (2) is what will happen to the population in 2005. Country X was not originally included in the study though and so does not appear in the formula. The formula will predict a population of
15,000 for country X, should the population actually be 1,000,000. One aspect of this is that to be sure of our results, we will change the time in the formula to represent a 4-year period spanning
from 1996 to 2000. We have to force our new calculation by changing the time to represent 1996-2000, and because the year is inside the formula we also have to change the value of the population in
1960 to get 15,000 as we want it to be. _________ Obviously this was just an example. What I really wanted to do was to see if using Gann arcs and circles can be generalized and be applied to a whole
load of different problems. So if we were to apply this method to actually find the rate at which British incomes in 2012 have been growing, we would not have to know how incomes in 1988 were to be
estimated. Again, it would just be a matter of a simple formula. Again it would mean that the | {"url":"https://practicalgann.com/what-are-the-main-advantages-of-using-w-d-gann-arcs-and-circles-over-other-methods-2/","timestamp":"2024-11-03T10:46:09Z","content_type":"text/html","content_length":"168749","record_id":"<urn:uuid:2270c67b-67b9-4355-bd2c-4bb3c1fcb771>","cc-path":"CC-MAIN-2024-46/segments/1730477027774.6/warc/CC-MAIN-20241103083929-20241103113929-00005.warc.gz"} |
Recognize and represent 2-digit numbers as tens and ones (Part 1)
Curriculum>Grade 2> Module 3>Topic D: Modeling Numbers Within 1,000 with Place Value Disks
Decompose a 2-digit number into its place values. First, identify the number of tens and ones and put the place value label under each digit. Then, represent the number with coins labeled 10 and 1 | {"url":"https://happynumbers.com/demo/cards/294485/","timestamp":"2024-11-08T11:04:13Z","content_type":"text/html","content_length":"14373","record_id":"<urn:uuid:0729d83c-bb94-4637-ac81-e187e3f35649>","cc-path":"CC-MAIN-2024-46/segments/1730477028059.90/warc/CC-MAIN-20241108101914-20241108131914-00149.warc.gz"} |
How does Planck solve the blackbody radiation problem?
Since the problem of blackbody radiation directly led to the beginning of the old quantum theory, we also emphasized, why do people study blackbody radiation? And what is blackbody radiation?
Today’s topic is, how does Planck, the founder of quantum theory, solve the problem of black body radiation?
In the 1890s, at the Imperial College in the suburbs of Berlin, there was an optical laboratory. This laboratory was set up specifically to overcome the problem of black body radiation. The leader of
the laboratory was Otto Lummel, and he had one The 29-year-old young man named William Wien; Wien first made a crucial breakthrough in the issue of blackbody radiation.
At this time, the optical laboratory has been able to more accurately measure the energy distribution curve of the black body radiation.
The picture we see now is the curve of the blackbody radiation energy distribution. The abscissa represents the wavelength, the unit is nanometers, the wavelength range of visible light is between
400 and 700, and the vertical axis is the radiation density, which can also be called the radiation intensity.
The lines of different colors represent the curve of the blackbody energy distribution at different temperatures. It can be easily seen that the radiation intensity will increase with the increase of
the wavelength. After reaching a peak, then it will begin to decrease. The whole curve is like a bell, so the curve of the black body radiation intensity is also called a bell curve.
The higher the temperature, the more prominent the curve and the more like a bell.
Because Wien can obtain first-hand experimental data in the laboratory, he proposed a displacement law to describe the black body radiation curve at different temperatures in February 1893, and gave
a simple mathematical formula .
The law that Wien discovered is relatively simple. It is said that as the temperature of the black body increases, the peak of the black body radiation intensity will move toward the short wave, that
is, the top of the bell curve moves toward the short wave, and the wavelength of the peak is multiplied by the temperature. constant.
This constant is easy to calculate. You only need to measure the wavelength of the blackbody’s radiation peak at a certain temperature to calculate this constant. Through this constant, we can know
the wavelength of the radiation peak of an object at different temperatures.
Wien’s law of displacement explains well why the color of an object changes from red, to yellow, and then to blue-white as the temperature increases.
Three years later, in 1896, Wien made persistent efforts and proposed a mathematical formula to describe the blackbody temperature and radiation intensity distribution. This is the famous Wien
In June of the same year, when Wien published the law of energy distribution, he left Imperial College and went to RWTH Aachen University, where he was promoted from a laboratory assistant to an
assistant professor.
This is a career path that every physicist must go through. Follow your supervisor as an assistant for several years, complete your doctoral dissertation, get a doctorate, and then stay in school or
go to other universities to be a supernumerary lecturer, or you can be lucky. Be an assistant professor and then be promoted to a full-time professor. At the step of being a full-time professor, you
are considered a civil servant.
Closer to home, let’s talk about the issue of blackbody radiation.
As soon as Wien’s law of energy distribution came out, the next task was to measure whether the formula met the experimental data. At that time, the electric heating black bodies used in major
laboratories did not have a unified standard, they were all manufactured by themselves, and the temperature of each experiment The measured wavelength ranges are different, so it took four years to
verify Wien’s formula.
Some scientists, such as Paschen (the Paschen series in the hydrogen atom spectrum, are named after him). After four years of testing, they say that Wien’s formula is okay, while Lummer and Prinscher
of Imperial College London Mu also tested for four years, and always found that Wien’s formula was in the long wave range, and there was an error with the experimental data.
The person who settled this dispute was Planck’s friend Heinrich Rubens. In 1900, the 35-year-old Rubens had joined a full-time professor at the Technical University of Berlin. In late September,
Rubens and Kurbo In the temperature range of 200 to 1500 degrees Celsius, the Wien formula was tested for the wavelength range of 0.03 to 0.06 mm, which is 30,000 to 60,000 nanometers, and it was
found that the Wien formula did not conform to the long-wave radiation.
On October 7, Rubens brought his wife to Planck’s house for dinner, and by the way told Planck the results of the experiment, and told Planck that in the long wave range where the Wien formula
failed, the experimental results showed that, The intensity of radiation is directly proportional to temperature.
On the day Rubens left, it is estimated that Planck did not sleep well all night. Because he did a great thing that can make his name go down in history.
Planck was born in Kiel, Germany on April 23, 1858. His great-grandfather and grandfather were both famous theologians, and his father was a law professor at the University of Munich. He was
influenced by his childhood culture and a superior living environment. After many children from ordinary families, when they grow up, others like to go to bars, while Planck likes to go to theaters
and concert halls.
After graduating from high school, Planck once wandered between musicians and physics for a long time. After all, people played a good piano, at least it was more beautiful than Einstein’s violin. In
1974, Planck. Kr chose to study physics at the University of Munich, and then transferred to the University of Berlin in 1977.
At the University of Berlin, he had talked to two top German physicists Kirchhoff and Helmholtz at the time, but Planck didn’t like the two lecture styles, which were monotonous and boring. Planck’s
later years When I recalled, especially Helmholtz had never prepared a lesson well.
By chance, Planck came into contact with the 56-year-old Clausius. He was a professor at the University of Bonn in Germany. Clausius’ research field is thermodynamics. It was the first time that he
proposed the concept of entropy. Planck not only I was attracted by the professor’s lecture style, and after reading Clausius’s thesis on thermodynamics, I was also deeply attracted by the two laws
of thermodynamics.
Like Clausius, Planck also believes that the second law of thermodynamics, like the first law, is absolute and universal. In other words, Planck believes that entropy increases whenever, wherever,
and energy The law of conservation is absolutely valid.
After staying in Berlin for a year, Planck returned to the University of Munich. His doctoral thesis was about thermodynamics. Before studying blackbody radiation, Planck’s main direction was also
In 1880, Planck became a non-staff lecturer in Munich. He has been working for 5 years. The non-staff lecturer is a temporary position. Planck collapsed a bit and felt that his future was a little
In 1885, the 27-year-old Planck unexpectedly received an invitation to the position of assistant professor at Keele University, but he suspected that his father had asked him to find a job.
Planck accidentally won the grand prize in 1888, his former mentor Kirchhoff passed away, and the position of full-time professor at the University of Berlin was vacant. The University of Berlin
invited Hertz, but Hertz directly refused, and invited Boltzmann again. When Boltzmann saw that Hertz was not going, he didn’t go either. At this moment Helmholtz thought of his student Planck.
In fact, Planck did not have a deep friendship with Helmholtz, but Planck once publicly supported Helmholtz’s point of view at a debate at the University of Göttingen, which gave Helmholz I am
Since then, Plank has been on the same footing and his career has been smooth sailing, especially in 1894, when Helmholtz and Comte died one after another. Planck suddenly discovered that 36-year-old
himself had not done any amazing results yet, unexpectedly unexpectedly. Zhong became a senior physicist at a top university in Germany.
He also served as a theoretical physics consultant for the top academic journal “Physics Yearbook” in Germany.
It may be destined that Planck is not a general in this life. On the night when his friend Rubens left, Planck decided to try to figure out a correct formula first. Now he has three important
messages in front of him:
Wien’s displacement law is correct, Wien’s formula is correct in the short wave range, and Wien’s formula is incorrect in the long wave range where the radiant energy is proportional to temperature.
Based on this information, Planck came up with a formula that night and sent a letter to Rubens overnight, asking Rubens to help him test it. A few days later, Rubens came to Planck with the answer.
At home, tell Planck that your formula exactly matches the experimental data.
On October 19th, the regular meeting of German physicists was held as usual. At the meeting, Planck wrote his formula on the blackboard, saying that this formula is a slight improvement on the Wien
formula, at least it seems to be correct for now; Ke didn’t say much about the physical meaning of this formula, nor did he explain too much. His colleagues also nodded politely.
A few days later, all the laboratories confirmed that Planck’s formula was okay, but for Planck, the things that really bothered him had just begun.
In the next few weeks, Planck had trouble sleeping and eating. Planck recalled in his later years that those few weeks were the most stressful period of his life.
Because Planck, as a theoretical physicist, you only give formulas and do not explain physics. This inevitably makes people feel that you are a bit opportunistic, and even make people suspect that
you are relying on luck.
So Planck had to give his formula a meaning in physics, and he had to be ahead of others, but Planck found that his formula could not be explained in terms of classical electromagnetism and classical
thermodynamics. I tried many ways to no avail.
In desperation, Planck introduced Boltzmann’s concepts of statistical mechanics and entropy, which he had always disliked. Here is a little bit. Planck did not believe in the existence of atoms at
this time, and he did not like Boltzmann. Mann’s explanation of the probability of entropy increase has been mentioned above. He inherited Clausius’ idea that the entropy increase is absolute and
there is no probability.
Boltzmann’s statistical entropy means that there is a certain probability that the entropy will decrease. For example, if you heat up a rock on the ground, it may jump into the air again, and the
broken glass may recover. The air may suddenly all move to a corner and so on, but the probability of this situation is very low and basically impossible.
Planck thinks this statement is extremely absurd. Entropy increase does not require probability, but Planck can no longer take care of so much in order to solve his own dilemma. He can only try to
accept Boltzmann’s physics.
And imagine the inner wall of the black body cavity as a huge array of oscillators. Each oscillator is a simple harmonic oscillator with a fixed vibration frequency. The oscillator of each frequency
can only emit radiation of the corresponding frequency, for example, one The vibration frequency of the vibrator is V, so it can only emit radiation with frequency V.
Then the frequencies released by all the oscillators are added together to release the frequencies of the entire black body radiation.
When Planck allocated the energy of the black body to each oscillator, he found that only the oscillator releases energy one part at a time, and the energy that the oscillator has can only be this
one. His equation can only be established when the energy is an integer multiple.
The minimum energy of each part is the frequency V of the oscillator multiplied by a constant h. This constant is the Planck constant, which is 6.626 times 10 to the minus 37 power, erg seconds.
We know that when the oscillator frequency is constant, the energy it has is proportional to the square of the amplitude. That is to say, when the amplitude of the oscillator is 1, the energy of the
oscillator is 1, and when the amplitude is 1.2, the energy is 1.44. When the amplitude is 1.5, the energy is 2.25, and when the amplitude is 2, the energy of the oscillator is 4; it can be deduced by
In the classical world, we believe that the vibrator can have any amplitude, and the amplitude can be arbitrary, which is continuous. Of course, the energy that the oscillator can have is also
But Planck now says that amplitude 1.2 and amplitude 1.5 are not allowed, because at this amplitude, the energy of the oscillator is 1.44 and 2.25, which are not integers.
Planck discovered that the release and absorption of energy is discontinuous, just like the white sugar in the supermarket. It is not in bulk, but stacked in bags. You have to buy at least one bag at
a time, not half a bag. , You can’t buy one and a half bags, but you can buy two bags, three bags, or four bags.
Then why do we not see in real life that the simple harmonic oscillator can only have a specific amplitude because the energy is part of it? We found that the amplitude is continuous. When we buy
white sugar, we also feel that we can buy any amount? This is because Planck’s constant is very small, 37 zeros after the decimal point. Such a small constant cannot be felt by our macroscopic world
at all, and it gives us the illusion that the world is continuous.
This day was particularly important on December 14, 1900. It was considered the birth day of quantum theory. On this day, the regular German scientific meeting proceeded as usual, and Planck
elaborated on the physics meaning of the black body radiation formula.
What needs special attention here is that Planck only emphasized at the meeting that energy is absorbed and released piece by piece, and the smallest piece is hv, but it never said that the nature of
energy is discontinuous. Not to mention that the world is not continuous.
And he also told his colleagues at the meeting, don’t think about it, I never said anything deviant. So when Planck put forward the hypothesis of the amount of energy, all physicists believed that
Planck was just a technique used to allow his formula to have an explanation and to give himself a step down.
But when the whole world thought it was a joke, a young man took it seriously. He is Einstein who has graduated, but can’t find a job, and even worry about eating.
You must be logged in to post a comment. | {"url":"http://www.verycoldscience.com/how-does-planck-solve-the-blackbody-radiation-problem/","timestamp":"2024-11-03T09:48:15Z","content_type":"text/html","content_length":"78790","record_id":"<urn:uuid:47f08de1-4785-4e51-b56b-a8b42baba887>","cc-path":"CC-MAIN-2024-46/segments/1730477027774.6/warc/CC-MAIN-20241103083929-20241103113929-00057.warc.gz"} |
Maths Venns
I've been having so much fun sharing
Maths Venns
with teachers. I love doing Venns in class with students, but they don't smile and express as much joy as teachers do when they're working out a Venn. Out of all of the types of math problems out
there, I've found that math Venns generate the loudest conversations among students and adults. I might be as bold to say that no type of activity allows for "critiquing the reasoning of others" like
Math Venns do. This activity lends itself to speaking your ideas out loud and critically listening to other's answers to give appropriate feedback.
Recently I gave a presentation with Kathleen Williams at the
West Suburban Math Institute Conference
about Open Middle Problems. We threw some Math Venns into the presentation as another example of deeper thinking. The room was full of middle school teachers and a few high school teachers. Even
though this group of strangers didn't know each other, the volume in the room immediately got louder with the buzz of teachers discussing the problem we posed.
Last week Kathleen and I presented Math Venns to Pre Calc teachers in our district. Teachers had to come up with graphs to fill in the regions "
has a minimum point, does not cross the x-axis, and has a positive y-intercept.
" The conversation began with quadratics and some teachers were thinking that some regions were impossible. As time went on, teachers got more creative with their answers. They started talking about
trig graphs, circles, and rational functions.
The premise of the Math Venn is that you have to find an equation/graph/number that could fit in each of the 8 regions (of a 3 circle Venn) or 4 regions (of a 2 circle Venn). Because the problem is
open-ended, it's helpful to have a partner listening and checking your reasoning. The great thing about being "wrong" about a particular answer fitting into a certain region is that you can often
still use that answer somewhere else, you just need to find where it fits.
When I facilitate a Math Venn in my classroom, I have students work with peers to fill in each of the regions. When the class is ready to go over the Venn, I randomly call on students by using note
cards with their names on them. I'll ask the student I call on to tell me their answer for any of the regions. This is a pretty safe way for students participate since they've already worked with a
buddy and have a choice of what answer they want to give.
Another way I'll facilitate a Math Venn is by having small groups write the answer for one region anonymously on the wall. Then students can check the work that's on the wall, and if they disagree,
they can write what they believe to be a correct answer next to the work that's there. As a class we decide which answer is correct.
I observed a teacher use Venns in an Algebra class related to linear equations. She had students in groups and gave one sheet of paper per group with a two circle Venn. Once students completed that
one, they went on to do more complicated Venns. The really fun thing that happened organically in her class was that students were filling in a region with multiple right answers. We talked about how
this would be a great way to differentiate for students who get done early by asking them to come up with multiple answers for a region.
I've heard the comment from teachers "This is great, but I don't have time to do anything like this in my class." These are the same teachers who complain that students don't retain knowledge from
year to year, or even chapter to chapter. I would argue that we have to make time to do Maths Venns, Open Middle, and other types of problems that require higher order thinking in order to make the
learning stick.
For more great info. on facilitating a Math Venn, check out the
about section
by Craig Barton, creator of the Maths Venns website. | {"url":"http://classroomfruition.blogspot.com/2019/03/maths-venns.html","timestamp":"2024-11-06T16:58:20Z","content_type":"text/html","content_length":"95053","record_id":"<urn:uuid:2a0d046c-0ee9-4219-9d42-5456f07741dd>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00506.warc.gz"} |
equivariant Chern character
Might anyone have an electronic copy of
• Jolanta Słomińska, On the Equivariant Chern homomorphism, Bull. Acad. Polon. Sci., Sér. Sci. Math. Astronom. Phys, Vol. 24 (1976), 909-913.
I have !include-ed the table incarnations of rational equivariant topological K-theory – table.
This reminds me that rational equivariant K-theory should be a page on its own…
diff, v7, current
added pointer to:
• Alejandro Adem, Yongbin Ruan, Section 5 of: Twisted Orbifold K-Theory, Commun. Math. Phys. 237 (2003) 533-556 (arXiv:math/0107168)
diff, v7, current
added point to
• J. Słomińska, Equivariant Chern homomorphism, Bull. Acad. Polon. Sci., Sér. Sci. Math. Astronom. Phys, Vol. 24 (1976), 909-913.
(the original reference(?), though I haven’t yet actually seen it)
diff, v7, current
added this pointer:
• Alain Connes, Paul Baum, Chern character for discrete groups, A Fête of Topology, Papers Dedicated to Itiro Tamura 1988, Pages 163-232 (doi:10.1016/B978-0-12-480440-1.50015-0)
diff, v5, current
Added pointer to:
• Ulrich Bunke, Markus Spitzweck, Thomas Schick, Inertia and delocalized twisted cohomology, Homotopy, Homology and Applications, vol 10(1), pp 129-180 (2008) (arXiv:math/0609576)
diff, v5, current
added this pointer:
• Ulrich Bunke, Section 3.1 in: Orbifold index and equivariant K-homology, Math. Ann. 339, 175–194 (2007) (arXiv:math/0701768, doi:10.1007/s00208-007-0111-5)
diff, v4, current
starting something
v1, current
Wow, that journal seems to have missed the digitisation boat. That volume is on Google books, but with no preview available (I can see a few lines of the start of the article when searching inside,
but nothing useful). For reference, here’s the complete MathSciNet review:
For any finite group $G$ the author constructs an equivariant Chern homomorphism from $K_G$ to $H^{ev}( ,R_G)$ which is a rational isomorphism for compact $G$-CW-complexes $X$, and uses it to
express $K_G(X)\otimes \mathbf{Q}$ in terms of all $K(X^H)$ with $H$ a cyclic subgroup of $G$. The basic tool is the notion of a “split coefficient system” and its properties; this is a $G$
-coefficient system in G. E. Bredon’s sense [Equivariant cohomology theories, Lecture Notes in Math., Vol. 34, Springer, Berlin, 1967; MR0214062] admitting suitable transfer maps.
The zbmath one is similarly brief.
Thanks for double-checking. Maybe I should walk into an actual library, for a change.
I don’t expect to find anything in Słomińska’s article that I wouldn’t essentially have seen reproduced elsewhere; but Mislin 2003 writes (p. 22) that the ideas on splitting of the rationalized
representation ring (discussed and referenced here) due to Lueck & Oliver 1998 “have their root in Słomińska’s paper”. Since that splitting is a somewhat subtle business, I grew interested in seeing
what Słomińska actually wrote about it.
Re #10: The PDF file is available here: https://dmitripavlov.org/scans/słomińska.pdf. | {"url":"https://nforum.ncatlab.org/discussion/11751/equivariant-chern-character/?Focus=97459","timestamp":"2024-11-09T21:45:03Z","content_type":"application/xhtml+xml","content_length":"59851","record_id":"<urn:uuid:c9faeb82-b158-4ab5-ba2e-eb69e848d68d>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.10/warc/CC-MAIN-20241109214337-20241110004337-00093.warc.gz"} |
Quantum Acausal Determinism (2)
As noted by the late David Bohm, and B. J. Hiley [1]:
It is well known that Einstein did not accept the fundamental and irreducible indeterminism of the usual interpretation of the quantum theory, e.g. as revealed by his statement: 'God does not play
dice with the universe'. However, what is much more important is his rejection of another fundamental and irreducible feature of the quantum theory, i.e. nonlocality
Let's examine this further, before looking at the E-P-R or Einstein - Podolsky -Rosen thought experiment which sought to show nonlocality didn't float so QM had to be incomplete.
If I have always investigated nature from a reductionist viewpoint, taking some phenomenon 'X' and reducing it into localized sub-assemblies:
X C x1 + x2 + x3 + ...... xN
wherein the total function of X is no greater than its parts, then clearly I will be averse to approaching a natural phenomenon any other way. This is particularly applicable if I have been
accustomed to seeing interrelations (in other phenomena) of the form:
[x1 + x2] -> x3
x3 -> [x4 + x5]
[x5 + x6] -> x7
where each of the arrows denotes a
'causal flow'
from the component(s) on the left hand side, to the component(s) on the right hand side. Nowhere is this type of 'causal flow' seen more clearly than in examples of Newtonian motion. (For example,
tossing a baseball at a wall).
Not coincidentally, this converges where the premise of locality attains near sacramental status. By locality I mean extraneous factors can be ignored in examining the immediate physical situation.
If I am studying the dynamics of simple pendulum, I need only pay attention to its mass, length and period, not what the pendulum clock in the next room is doing.
Giving up absolute 'causal flow' via direct interactions (or locality) in models, is not unlike asking a junkie to give up whatever gives him some temporary satisfaction. The equivalent of 'cold
turkey' occurred when experiments to test quantum conformity to Bell's Theorem were performed by Alain Aspect and his colleagues at the University of Paris.
In these experiments, the detection of the polarizations
of photons was the key. These were observed with the photons emanating from a Krypton-Dye laser and arriving at two different analyzers, e.g.
A1 (+ ½ ) <-----------[D]----------->(- ½ )A2
The above scene captures the instant just before each detector intercepts an atomic magnet from the device. The quantum state observed is described by the spin number, which is (+½ ) for A1 and (-½)
for A2, corresponding to the spin up and spin down orientations, respectively. It is important to understand that these values can only be known definitely at the instant of observation.
In the orthodox Copenhagen (and most conservative) interpretation of quantum theory, there can be no separation of observed (e.g. spin) state until an observation or measurement is made. Until that
instant (of detection) the states are in a superposition, as described above. More importantly, the fact of superposition imposes on all quantum phenomena an inescapable ‘black box’. In other words,
no information other than statistical can be extracted before observation.
Meanwhile, in the Bohmian acausal deterministic setting [1]: "it
is supposed that each experimental result is determined completely by a set of hidden variables, φ. Thus, the result A of measurement of spin in the direction â depends only on φ, while the result C,
of measurement of spin in direction ĉ depends only on φ and ĉ
This was a brilliant tour-de -force which can be summarized (ibid.):
A = A (â, φ)
C = C(ĉ, φ)
where measuring
such as: A = A(â, ĉ, φ) and C = C(â, ĉ, φ) are specifically excluded.
Thus, as the authors note(ibid.):
In other words, while nothing is said about the general dynamical laws of the hidden variables, φ, which may be as nonlocally connected as we please, we are requiring that the response of each
particular observing instrument to the set φ, depends only on it own state and not the state of any other piece of apparatus that is far away
This is a critical distinction, because it eliminates the pure objection Einstein had to unlimited nonlocality. Thus, Bohmian quantum physics introduces a measure of locality by tying the action of
hidden variables to the particular observing device.
To fix ideas and show differences, in the Aspect experiment four (not two - as shown) different analyzer orientation 'sets' were obtained. These might be denoted: (A1,A2)I, (A1,A2)II, (A1,A2,)III,
and (A1,A2)IV. Each result is expressed as a mathematical (statistical) quantity known as a 'correlation coefficient'. Aspect's final result yielded:
S = (A1,A2)I + (A1,A2)II + (A1,A2,)III + (A1,A2)IV = 2.70 ± 0.05
What is the significance? In a landmark theoretical achievement in 1964, mathematician John S. Bell formulated a thought experiment based on a design similar to that shown. He made the basic
assumption of locality (i.e. that no communication could occur between A1 and A2 at any rate faster than light speed). In what is now widely recognized as a seminal work of mathematical physics, he
set out to show that a theory which upheld locality could reproduce the predictions of quantum mechanics. His result predicted that the above sum, S, had to be less than or equal to 2 (S
less than or equal
2). This is known as the
'Bell Inequality'.
In the case of Bohm's hidden variables, deterministic model, the above sum came to less than 2.
The E-P-R experiment:
This was probably conceived explicitly because Einstein wanted to show quantum theory was incomplete, and hence an unlimited nonlocality could not be accepted - since otherwise anything could occur
say at point (x1, y1, z1, t1) 20 million light years from Earth and affect events on Earth, say at point (x2, y2, z2, t2). If that was true, then both locality and determinism went right out the
The original form of the experiment invoked the quantum state of a two particle system in which position deifferential (x1 - x2) and momentum sum (p1 + p2)are both determined. Then the wave function
U(x1, x2) = f(x1 - x2 -a) = SIGMA_k c_k exp[ik(x1 - x2 -a)]
where SIGMA denotes a summation over the k elements; f(x1 - x2 -a) is a packlet function sharply peaked at (x1 - x2) = a (for an example of such a functional form check out the imposed image in the
graphic just below "God" playing dice with Einstein looking on. ) Also c_k is a Fourier coefficient. Thus, in this particular dynamic state, p1 + p2 = 0 and x1 - x2 can be as well-refined as one
pleases. (In the normally applied
Heisenberg Uncertainty Principle
, when nothing is known on the momentum then the position x can be precisely obtained. It's just that both position x and momentum p can't be known to the same exact precision simultaneously)
In the EPR experiment when one measures x1 then one immediately knows: x2 = x1 + a. Alternatively, if p is measured, then one knows immediately that p2 = -p1 since p1 + p2 = 0. In both cases the 1st
particle is "disturbed" by measurement and this accounts for the Heisenberg uncertainty relations applied in 1-d. However, the 2nd particle is taken not to interact with the 1st at all- so one can
obtain its properties minus the assumption of any disturbance. Still, the Heisenberg principle still applies so: dp2 dx2 >= h-bar.
Where h-bar = h/ 2 pi as before (h is Planck's constant or 6.62 x 10^-34 J-s) , and dp2, dx2 denote the uncertainties in the 2nd particle momentum and position. The key point here is that given the
above conditions and parameters Heisenberg's explanation of the uncertainty arising from a "disturbance" can no longer be used. It was exactly this which prompted Einstein, Podolsky and Rosen to
argue that since both x2 and p2 were measurable to "arbitrary accuracy"
without any disturbance
, then they already existed independently (in particle 2) as localized elements of reality with well-defined values before any measurement took place. Hence, they concluded that QM is merely a
mathematical abstraction which gives only an incomplete picture of reality - and hence QM itself is incomplete.
Bohr's Reponse to the EPR paradox.
[1]Bohm, D. and Hiley, B.J.:1981,
Foundations of Physics
, Vol. 11, Nos. 7/8, p. 529.
[2] Aspect, A., Grangier, P. and Roger, G.: 1982,
Physical Review Letters
, Volume 49, No. 2, p. 91.
[3] Polarization is the orientation in space of the electric field E, associated with light. This can be altered, subject to the imposition of different filters and devices. | {"url":"https://brane-space.blogspot.com/2010/12/quantum-acausal-determinism-2.html","timestamp":"2024-11-09T08:00:43Z","content_type":"text/html","content_length":"123286","record_id":"<urn:uuid:431ee3a9-ddaa-41ef-8066-8502f73d9673>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00161.warc.gz"} |
197400 Yottahertz to Frames Per Second (YHz to FPS) | JustinTOOLs.com
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Category: frequencyConversion: Yottahertz to Frames Per Second
The base unit for frequency is hertz (Non-SI/Derived Unit)
[Yottahertz] symbol/abbrevation: (YHz)
[Frames Per Second] symbol/abbrevation: (FPS)
How to convert Yottahertz to Frames Per Second (YHz to FPS)?
1 YHz = 1.0E+24 FPS.
197400 x 1.0E+24 FPS =
Frames Per Second.
Always check the results; rounding errors may occur.
In relation to the base unit of [frequency] => (hertz), 1 Yottahertz (YHz) is equal to 1.0E+24 hertz, while 1 Frames Per Second (FPS) = 1 hertz.
197400 Yottahertz to common frequency units
197400 YHz = 1.974E+29 hertz (Hz)
197400 YHz = 1.974E+26 kilohertz (kHz)
197400 YHz = 1.974E+23 megahertz (MHz)
197400 YHz = 1.974E+20 gigahertz (GHz)
197400 YHz = 1.974E+29 1 per second (1/s)
197400 YHz = 1.2403007803534E+30 radian per second (rad/s)
197400 YHz = 1.1844000004738E+31 revolutions per minute (rpm)
197400 YHz = 1.974E+29 frames per second (FPS)
197400 YHz = 4.2638672887506E+33 degree per minute (°/min)
197400 YHz = 1.974E+17 fresnels (fresnel) | {"url":"https://www.justintools.com/unit-conversion/frequency.php?k1=yottahertz&k2=frames-per-second&q=197400","timestamp":"2024-11-12T17:20:13Z","content_type":"text/html","content_length":"66339","record_id":"<urn:uuid:9b34205a-cbcc-400f-86f1-bba62649b669>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.63/warc/CC-MAIN-20241112145015-20241112175015-00121.warc.gz"} |
January 2015 News :: Transum Newsletter
January 2015 News
Monday 5th January 2015
Happy New Year! Best wishes for twenty-fifteen, or two thousand and fifteen or ‘two oh one five’. Which is the correct way to say the year? Well it certainly is not the third suggestion above. More
about that later. One of my Christmas presents this year was the book “Are you smart enough to work at Google”. It’s an excellent read for anyone interested in logic puzzles and a diverse range of
other topics which have been used as interview questions for prospective Google employees. I’ll use one of the mathematically related questions as the puzzle for this newsletter:
“You're playing football on a desert island and want to toss a coin to decide the advantage. Unfortunately, the only coin on the island is bent and is seriously biased. How can you use the biased
coin to make a fair decision?”
The answer is at the end of this newsletter. Thanks as always for the comments received last month. One in particular from Alan Ramm about the Starter for 16th December which asks pupils for a power
of three and a power of two which are consecutive numbers. The surprising result is that there are four different answers where both numbers are less than ten. Alan informs us that this leads to
Catalan’s conjecture, first stated in 1844 but not proved until 2002. It is explained nicely on Wikipedia but is a little beyond the scope of school Mathematics! December was a busy month for
updating pages on the Transum website there were also a number of new activities added including Christmas Consonants, Christmas Toggle Tree, Christmas Tree Trim, Set Notation Matching, Pu Wiang,
Pong Hau K’l and Cat Face. Questions in my mind at the moment are about Christmas Toggle Tree. I have never seen a puzzle like this before and I’m really interested to hear what strategies people
will use to do it. As yet I have not come up with an elegant method but sure someone out there can. I’m also wondering whether we have finally found the minimum number of moved required to do Pu
Wiang. I thought I had the answer until Colleen Young's (@ColleenYoung) and her colleague, Elaine, found a more efficient solution. The record currently stands at 19 moves but I need an insightful
mathematician to prove this is indeed the minimum number of moves required. Now to the pronunciation of the year question. The aspect I’d like to examine is the use of ‘oh’ for nought or zero. For
the whole of my teaching career I have told pupils that ‘oh’ is a letter that comes between n and p and it is not a number. But has common usage eroded this stance? For example how would you say the
number associated with James Bond (007)? The answer to these questions are clearly provided in a podcast I listen to called "Grammar Girl’s Quick and Dirty Tips" by Mignon Fogarty. She says the ‘oh’
is acceptable when saying phone numbers, hotel room numbers, post codes or credit card numbers. Zero or nought should be used in mathematical or scientific contexts. I will included the excerpt in
the Transum Podcast for this month as it is certainly something all Maths teachers should know. So here’s the answer to the puzzle as told by William Poundstone in his book “Are you smart enough to
work at Google”.
Firstly there is one method involving tossing the coin a large number of times to work out the value of the bias but the best answer comes from Google: "If you toss the coin twice. There are four
possible outcomes: HH, HT, TH, and TT. Since the coin favours one side, the chance of HH will not equal the chance of TT. But HT and TH must be equally probable, no matter what the bias. So toss
twice, after agreeing that HT means one team gets the advantage and TH means the other does. Should it come up HH or TT, ignore it and toss another two times. Repeat as necessary until you get HT or
That’s all for now. ps. Why is a dog with a bad foot like adding 6 and 7? A. Because he puts down three and carries the one.
Do you have any comments? It is always useful to receive feedback on this newsletter and the resources on this website so that they can be made even more useful for those learning Mathematics
anywhere in the world. Click here to enter your comments. | {"url":"https://transum.org/Newsletter/?p=99","timestamp":"2024-11-14T19:02:37Z","content_type":"text/html","content_length":"19661","record_id":"<urn:uuid:6da92ddf-25e2-4a5c-ad9b-35872198be24>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00607.warc.gz"} |
MUREM: A Multiplicative Regression Method for Software Development Effort Estimation
In this paper a multiplicative regression method to estimate software development effort is presented. This method, which we call MUREM, is a result of, on the one hand, a set of initial conditions
to frame the process of estimating software development effort and, on the other hand, a set of restrictions to be satisfied by the development effort as a function of software size. To evaluate the
performance of MUREM, it was compared with three regression models which are considered as important methods for estimating software development effort. In this comparison a battery of hypothesis and
standard statistical tests is applied to twelve samples taken from well-known public databases.These databases serve as benchmarks for comparing methods to estimate the software development effort.
In the experimentation it was found that MUREM generates more accurate point estimates of the development effort than those achieved by the other methods. MUREM corrects the heteroscedasticity and
increases the proportion of samples whose residuals show normality. MUREM thus generates more appropriate confidence and prediction intervals than those obtained by the other methods. An important
result is that residuals obtained by the regression model of MUREM satisfy the test for zero mean additive white gaussian noise which is proof that the estimation error of this model is random.
Software development effort estimation; method for estimating software development effort; estimating method; multiplicative method; regression model.
Full Text:
PDF (Spanish) | {"url":"https://cys.cic.ipn.mx/ojs/index.php/CyS/article/view/2378/0","timestamp":"2024-11-04T05:41:20Z","content_type":"application/xhtml+xml","content_length":"19680","record_id":"<urn:uuid:b765b857-e132-4b6c-ba72-1a2ef32c5356>","cc-path":"CC-MAIN-2024-46/segments/1730477027812.67/warc/CC-MAIN-20241104034319-20241104064319-00793.warc.gz"} |
Solving Equations With X On Both Sides Worksheet - Equations Worksheets
Solving Equations With X On Both Sides Worksheet
If you are looking for Solving Equations With X On Both Sides Worksheet you’ve come to the right place. We have 16 worksheets about Solving Equations With X On Both Sides Worksheet including images,
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Leave a Comment | {"url":"https://www.equationsworksheets.com/solving-equations-with-x-on-both-sides-worksheet/","timestamp":"2024-11-13T22:53:11Z","content_type":"text/html","content_length":"65166","record_id":"<urn:uuid:a01e7ce5-e84a-481e-8181-a5b2c1115ba8>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00326.warc.gz"} |
What is Principal Component Analysis?
Principal Component Analysis is a statistical technique used for dimensionality reduction in data analysis. It transforms a set of possibly correlated variables into a set of linearly uncorrelated
variables called principal components.
In Principal Component Analysis the first principal component accounts for the most variance in the data, and each succeeding component, in turn, has the highest variance possible under the
constraint that it is orthogonal to the preceding components. PCA is commonly used in exploratory data analysis and for making predictive models more efficient.
n Sklearn, a popular Python library for machine learning, Principal Component Analysis is implemented as a part of the decomposition module. It allows users to perform PCA on a dataset to reduce its
dimensionality. Sklearn's PCA class provides options to specify the number of components to retain, handle data centering, and compute explained variance. It's typically used in preprocessing steps
to improve model performance and reduce computational costs by focusing on the most informative features.
The key difference between Kernel PCA and standard PCA lies in their approach to handling non-linear relationships. Standard PCA is linear and works well when the data structure is linearly
separable. However, in cases where the data structure is non-linear, Kernel PCA becomes more effective. It uses kernel functions to map the original nonlinear observations into a higher-dimensional
space where they become linearly separable. This process allows for capturing more complex patterns in the data compared to standard PCA.
In R, a programming language and environment for statistical computing, Principal Component Analysis can be performed using various functions and packages. The most common method is the prcomp
function from the base stats package. This function computes PCA using singular value decomposition, which is efficient for large datasets. Users can specify the number of components, scale the data,
and access various outputs such as scores and loadings. R's rich ecosystem offers other packages like factoextra for enhanced visualization and interpretation of PCA results.
Principal Component Analysis in Python can be conducted using libraries such as Sklearn and SciPy. These libraries provide functions to perform PCA, allowing users to reduce the dimensionality of
large datasets. Python's implementation involves computing eigenvalues and eigenvectors of the covariance matrix of the data, or using singular value decomposition. Users can choose the number of
components to retain and visualize the results using libraries like Matplotlib and Seaborn to interpret the principal components and their contribution to variance in the data.
Here are some fascinating statistics and insights about Principal Component Analysis:
Diverse Applications: PCA is applied in a myriad of fields. In bioinformatics, it's used for genetic data analysis. In finance, PCA helps in risk management and portfolio optimization. In machine
learning, it's a fundamental tool for feature reduction and data visualization.
Growth in Data Science: The increasing importance of data science and big data analytics has led to a rise in PCA usage. As datasets grow larger and more complex, PCA becomes crucial for reducing
dimensionality and extracting meaningful patterns from data.
Advancements in Computing: The evolution of computing power and the advent of big data technologies have made PCA more accessible and efficient, even for extremely large datasets, enhancing its
utility in real-time data analysis and complex simulations. Common Segmentation Criteria: The average company uses about 3.5 different segmentation criteria, with demographics, psychographics, and
behavior being the most common.
Integration in Software Tools: PCA is integrated into various software tools and platforms used in academia and industry, such as MATLAB, R, Python’s Sklearn, and others, indicating its fundamental
role in data analysis and machine learning. | {"url":"https://forgility.com/dictionary/principal-component-analysis/","timestamp":"2024-11-10T11:48:55Z","content_type":"text/html","content_length":"38621","record_id":"<urn:uuid:af86f183-b315-45d2-b5d7-59be23129570>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00194.warc.gz"} |
seminars - Landis-type results for discrete equations
We prove Landis-type uniqueness results for both the semidiscrete heat and the stationary discrete Schrödinger equations. To establish a nomenclature, we refer to Landis-type results when we are
interested in the maximum vanishing rate of solutions to equations with potentials. The results are obtained through quantitative estimates within a spatial lattice which manifest an interpolation
phenomenon between continuum and discrete scales. In the case of the elliptic equation, these quantitative estimates exhibit a rate decay which, in the range close to continuum, coincides with the
same exponent as in the classical results of the Landis conjecture in the Euclidean setting.
Joint work with Aingeru Fern'andez-Bertolin and Diana Stan. | {"url":"http://www.math.snu.ac.kr/board/index.php?mid=seminars&page=72&sort_index=speaker&order_type=desc&l=en&document_srl=1221292","timestamp":"2024-11-14T23:48:03Z","content_type":"text/html","content_length":"46130","record_id":"<urn:uuid:f4571e0d-27a3-4564-b785-98840ec8ac65>","cc-path":"CC-MAIN-2024-46/segments/1730477397531.96/warc/CC-MAIN-20241114225955-20241115015955-00575.warc.gz"} |