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Proof of Formula for the Curved Surface Area of a Frustum The curved surface area of a cone of height slant height and base radius \[A=\pi rl\] A frustum is a truncated cone. Part of the top is cut off by a cut parallel to the base. The whole cone and the top section are similar cones, so \[\frac{L-l}{r}=\frac{L}{R} \rightarrow LR-lR=Lr \rightarrow L=\frac{lR}{R-r}\] The surface area of the frustum is then \[\begin{aligned} A_{FRUSTUM}&=\pi RL-\pi r(L-l) \\ &=\pi (R \frac{lR}{R-r} -r \frac{lr}{R-r} ) \\ &=\frac{\pi l}{(R-r)}(R^2-r^2) \\ &= \frac{\pi l}{(R-r)}(R-r)(R+r)= \pi l(R+r)\end{aligned}\]	{"url":"https://astarmathsandphysics.com/igcse-maths-notes/4684-proof-of-formula-for-the-curved-surface-area-of-a-frustum.html","timestamp":"2024-11-12T05:32:14Z","content_type":"text/html","content_length":"29405","record_id":"<urn:uuid:6a2ea092-9718-4deb-be4b-d0ff19099255>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.58/warc/CC-MAIN-20241112045844-20241112075844-00342.warc.gz"}
Basic Descriptive Statistics (UMAP) Basic Descriptive Statistics (UMAP) Author: Richard Walker, Mansfield State College With completion of this module students will be able to: 1) use frequency distributions and histograms to summarize data; 2) calculate means, medians, and modes as measures of central location; 3) decide which measure of central location may be most appropriate in a given instance; and 4) calculate and interpret precentiles. Table of Contents: 1. THE NEED TO SUMMARIZE DATA - AN EXAMPLE 2.1 Frequency Distribution 2.2 Histograms 3.1 The Arithmetic Mean 3.1.1 Computing the mean for raw data 3.1.2 Computing the mean from a frequency distribution 3.1.3 Properties of the mean 3.2 The Median 3.2.1 Computing the median from raw data 3.2.2 Computing the median from a frequency distribution 3.2.3 Properties of the median 3.3 The Mode 4. CHOOSING A MEASURE OF LOCATION 5.1 Perecntiles 5.2 Computing Percentiles 5.3 Deciles and Quartiles 6. MODEL EXAM 7. ANSWERS TO EXERCISES 8. ANSWERS TO MODEL EXAM Mathematics Topics: Probability & Statistics Application Areas: Social Studies Decimals; simple formulas You must have a Full Membership to download this resource. If you're already a member, login here. Not yet a member?	{"url":"https://www.comap.com/membership/member-resources/item/basic-descriptive-statistics-umap","timestamp":"2024-11-02T17:54:43Z","content_type":"text/html","content_length":"41807","record_id":"<urn:uuid:dda5f2a2-2c1c-4256-b82c-e1195c855fc4>","cc-path":"CC-MAIN-2024-46/segments/1730477027729.26/warc/CC-MAIN-20241102165015-20241102195015-00484.warc.gz"}
How to change a radical to a fraction Author Message RanEIs Posted: Monday 25th of Dec 07:29 Hello friends, can you help me out with my assignment in Pre Algebra. It would be great if you could just give me an idea about the links from where I can acquire assistance on graphing parabolas. oc_rana Posted: Monday 25th of Dec 11:37 I think I know what you are searching for. Check out Algebra Master. This is an excellent tool that helps you get your assignment done faster and right. It can help out with problems in how to change a radical to a fraction, trigonometric functions and more. caxee Posted: Tuesday 26th of Dec 12:51 I checked up a number of software programs before I decided on Algebra Master. This was the most suited for algebra formulas, quadratic inequalities and graphing lines. It was effortless to key in the problem. Instead of just giving the answer , it took me through all the steps clearing up all the way until it arrived at the solution. By the time, I reached the solution I learnt how to go about it by myself. I used the program for solving my problems in Algebra 2, Basic Math and Intermediate algebra in algebra . Do you think that you will like to try this out? From: Boston, MA, US Rancisis Posted: Wednesday 27th of Dec 14:58 You have given an amazing solution to the problem. Please recommend a site from where I can buy the program. From: Belgium cmithy_dnl Posted: Thursday 28th of Dec 19:09 Have a look at the tutorials available at https://algebra-test.com/privacy.html. You have abundant information on Basic Math, particularly on angle-angle similarity, solving a triangle and radical expressions. All the best ! From: Australia Troigonis Posted: Friday 29th of Dec 21:46 Algebra Master is a very user friendly software and is definitely worth a try. You will find quite a few interesting stuff there. I use it as reference software for my math problems and can say that it has made learning math much more fun . From: Kvlt of Ø	{"url":"http://algebra-test.com/algebra-help/3x3-system-of-equations/how-to-change-a-radical-to-a.html","timestamp":"2024-11-14T15:22:27Z","content_type":"application/xhtml+xml","content_length":"21446","record_id":"<urn:uuid:f1718575-d3f6-468e-83d5-aa4f8b338c5b>","cc-path":"CC-MAIN-2024-46/segments/1730477028657.76/warc/CC-MAIN-20241114130448-20241114160448-00255.warc.gz"}
Erdős-Ko-Rado theorem analogue Transference for the Erdős-Ko-Rado theorem A random analogue of the Erdős-Ko-Rado theorem sheds light on its stability in an area of parameter space which has not yet been explored. Forum of Mathematics, Sigma 3, 18 (2015) J. Balogh, B. Bollobás, B. Narayanan For natural numbers, the Kneser graph K(n,r) is the graph on the family of r -element subsets of {1,...,n} in which two sets are adjacent if and only if they are disjoint. Delete the edges of K(n,r) with some probability, independently of each other: is the independence number of this random graph equal to the independence number of the Kneser graph itself? We shall answer this question affirmatively as long as r/n is bounded away from 1/2, even when the probability of retaining an edge of the Kneser graph is quite small. This gives us a random analogue of the Erdős–Ko–Rado theorem, since an independent set in the Kneser graph is the same as a uniform intersecting family. To prove our main result, we give some new estimates for the number of disjoint pairs in a family in terms of its distance from an intersecting family; these might be of independent interest.	{"url":"https://lims.ac.uk/paper/transference-for-the-erdos-ko-rado-theorem/","timestamp":"2024-11-03T21:35:03Z","content_type":"text/html","content_length":"77963","record_id":"<urn:uuid:22ff3959-48fd-4f22-9306-34ea6cb7ff37>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00859.warc.gz"}
Characterizing sequences for precompact group topologies Motivated from [31], call a precompact group topology τ on an abelian group G ss-precompact (abbreviated from single sequence precompact) if there is a sequence u=(u[n]) in G such that τ is the finest precompact group topology on G making u=(u[n]) converge to zero. It is proved that a metrizable precompact abelian group (G, τ) is ss-precompact iff it is countable. For every metrizable precompact group topology τ on a countably infinite abelian group G there exists a group topology η such that η is strictly finer than τ and the groups (G, τ) and (G, η) have the same Pontryagin dual groups (in other words, (G, τ) is not a Mackey group in the class of maximally almost periodic groups).We give a complete description of all ss-precompact abelian groups modulo countable ss-precompact groups from which we derive:. (1)No infinite pseudocompact abelian group is ss-precompact.(2)An ss-precompact group G is a k-space if and only if G is countable and sequential.(3)An ss-precompact group is hereditarily disconnected.(4)An ss-precompact group has countable tightness. We provide also a description of the sequentially complete ss-precompact abelian groups. • B-embedded subgroup • Characterized subgroup • Characterizing sequence • Finest precompact extension • Precompact group topology • T-sequence • TB-sequence ASJC Scopus subject areas • Analysis • Applied Mathematics Dive into the research topics of 'Characterizing sequences for precompact group topologies'. Together they form a unique fingerprint.	{"url":"https://cris.bgu.ac.il/en/publications/characterizing-sequences-for-precompact-group-topologies","timestamp":"2024-11-10T11:50:49Z","content_type":"text/html","content_length":"58293","record_id":"<urn:uuid:5d837100-8cd2-4855-9108-747f32a89d30>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00885.warc.gz"}
Gust_Wind Implementation? This post was from a previous version of the WRF&MPAS-A Support Forum. New replies have been disabled and if you have follow up questions related to this post, then please start a new thread from the forum home page. I am upgrading older wrf models to wrf 3.7. One goal was to add windgust to the output parameters. It appears clwrf_gust_wind was added in wrf 3.3.1, but the information available on how to accomplish that seems very limited (from forum and google searches). It appears additional parameters were required in the compilation (-DCLWRFXTR -DCLWRFGHG -DCLWRFHVY \ added to configure.wrf), changes were required to files in the /Registry folder, and "clwrf_gust_wind = 1" must be added to namelist.input (unclear what section?). It also appears that "output_diagnostics = 1," must be set in the time_control section, but that is less clear. The standard namelist.input glossaries ( do not include any of the clwrf parameters apparently added in wrf 3.3.1. Is there a better resource on how to implement gust_wind? I do see references to people computing gust as a post-process. Is this still the recommended approach? Thank you, Staff member If you want to output max wind, there are a couple of diagnostics options built-in. They are available in V3.7, but not in 3.3. The options are output_diagnostics (which you mentioned, but does not have to be used with CLWRF) and nwp_diagnostics. You can read more about these in Version 3.7 of the Users' Guide As for adding a new output variable, unfortunately we don't have the resources to assist with that, unless we know how to do it off the top of our head. I will refer you to the FAQ section on adding new features , though. There are several types of new features and perhaps one of these will be useful. Thank you for the quick response. I am not proposing a new output variable as there are many references to this being forecast. However, it is quite unclear how this is being achieved. To clarify, I am trying to produce "Surface Wind Gust" in parallel with the existing wind parameters for each forecast time over the entire model. Here are some examples from the Internet of such output: https://a.atmos.washington.edu/wrfrt/gfsinit.html (note, both 10m Wind speed and 10M Wind Gust are included) Hopefully there are pointers somewhere to the steps required to implement this model variable. Thank you, Staff member I would suggest trying to contact some of the entities or groups connected to the work mentioned in the links you shared. If they've done it before, hopefully someone there will have some knowledge or a contact for you. Good luck! Just a note: I have reached out to the contacts identified on the various web pages (awaiting responses). Also, it appears both NOAA and the European equivalent have included wind_gust as a standard part of their wrf-based modeling. NOAA refers to their model as WRF-ARW (which would appear to be the standard wrf with the ARW core). For ECMWF : http://apps.ecmwf.int/codes/grib/param-db?id=49#grib2, for NOAA, parameter 180, http://www.nco.ncep.noaa.gov/pmb/docs/on388/table2.html. I mention this here hoping someone on this list may have worked on, or be familiar with one of these implementations. Thank you, I just reviewed your questions about the wind gust. CLWRF is not a standard WRF we maintain here in NCAR. The research group that develop(?) CLWRF never shared any information with us, which is very unfortunate. At present it is not our priority to produce gust wind by standard WRF. However, we do welcome community contribution to add this capability. No response yet from the contacts, but I have located some information from NOAA. It appears wind-gust was part of post processing in 2012: https://www.weather.gov/media/notification/tins/ tin12-24nam_wind_gust.pdf. The document provides notice of a computational change to the NOAA NAM/DGEX post-processing code. However, DGEX seems to have been retired in 2016 https://www.weather.gov/ media/notification/tins/tin16-41nam_updates.pdf. I'm still trying to track down the current process. I found some very old code on the WRF User's Forum: https://forum.wrfforum.com/viewtopic.php?f=8&t=948 from Robert Rosumalski. SUBROUTINE CALGUST(LPBL,ZPBL,GUST) C$$$ SUBPROGRAM DOCUMENTATION BLOCK C . . . C SUBPROGRAM: CALGUST COMPUTE MAX WIND LEVEL C PRGRMMR: MANIKIN ORG: W/NP2 DATE: 97-03-04 C ABSTRACT: C THIS ROUTINE COMPUTES SURFACE WIND GUST BY MIXING C DOWN MOMENTUM FROM THE LEVEL AT THE HEIGHT OF THE PBL C PROGRAM HISTORY LOG: C 03-10-15 GEOFF MANIKIN C 05-03-09 H CHUANG - WRF VERSION C 05-06-30 R ROZUMALSKI - DYNAMIC MEMORY ALLOCATION AND SMP C THREAD-SAFE VERSION C USAGE: CALL CALGUST(GUST) C INPUT ARGUMENT LIST: C NONE C OUTPUT ARGUMENT LIST: C GUST - SPEED OF THE MAXIMUM SFC WIND GUST C OUTPUT FILES: C NONE C SUBPROGRAMS CALLED: C UTILITIES: C H2V C LIBRARY: C COMMON - C LOOPS C OPTIONS C MASKS C INDX C ATTRIBUTES: C LANGUAGE: FORTRAN 90 C MACHINE : CRAY C-90 use vrbls3d use vrbls2d C INCLUDE ETA GRID DIMENSIONS. SET/DERIVE PARAMETERS. INCLUDE "params" INCLUDE "CTLBLK.comm" C DECLARE VARIABLES. INTEGER :: LPBL(IM,JM) REAL :: GUST(IM,JM) REAL ZPBL(IM,jsta_2l:jend_2u) C START CALMXW HERE. C LOOP OVER THE GRID. DO J=JSTA,JEND DO I=1,IM ! GUST(I,J) = SPVAL GUST(I,J) = 0. C ASSUME THAT U AND V HAVE UPDATED HALOS !$omp parallel do !$omp& private(ie,iw,mxww,u0,v0,wind) DO 20 J=JSTA_M,JEND_M DO 20 I=2,IM-1 IF(MODELNAME .EQ. 'NMM')THEN X U10(IE,J)+U10(I,J+1)) X V10(IE,J)+V10(I,J+1)) SFCWIND=SQRT(USFC2 + VSFC2) U0 = D25(U(I,J-1,L)+U(IW,J,L)+ X U(IE,J,L)+U(I,J+1,L)) V0 = D25(V(I,J-1,L)+V(IW,J,L)+ X V(IE,J,L)+V(I,J+1,L)) WIND=SQRT(U02 + V02) ELSE IF(MODELNAME .EQ. 'NCAR')THEN SFCWIND=SQRT(USFC2 + VSFC2) WIND=SQRT(U02 + V02) END IF DELWIND=WIND - SFCWIND 10 CONTINUE 20 CONTINUE C END OF ROUTINE. I received information from one of the NOAA contacts regarding their implementation of gust: "The NAM wind gust is a diagnostic field computed in post-processing. To compute wind gust speed, the NAM post-processing code determines the height of the top of the planetary boundary layer (PBL). It then determines the wind speed at the top of the PBL and computes the difference between this wind speed and the speed at the surface". I note that the long retired wrf post-processor (WPP) calculated wind-gust and this functionality was carried forward into the Universal Post-Processor (UPP). End of wrf support within UPP has been announced with no successor identified. If you are interested in internal computation of wind gust whilst the model is running... We worked on an in-line implementation in the WRF model within the WRF-CORDEX module. Is based in Brasseur, 2001 work. Here you find all the information: I have the code for WRFv4.1.2 too I wonder whether you would like to share the codes with us? Thanks. Dear LluisFB, I am interested the code too. Could you please share the Code with us. Any help would be Appreciated. There appears to be code in a tab attached to the article linked in the original post. https://gmd.copernicus.org/articles/12/1029/2019/gmd-12-1029-2019-assets.html Is that what folks are looking for?	{"url":"https://forum.mmm.ucar.edu/threads/gust_wind-implementation.9917/","timestamp":"2024-11-09T06:21:41Z","content_type":"text/html","content_length":"95537","record_id":"<urn:uuid:6e4d8604-cdfa-44c0-b56f-42159265d146>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00254.warc.gz"}
How to Prepare for the DAT Quantitative Reasoning Math Test? The Dental Admission Test (also known as the DAT) is a standardized test designed by the American Dental Association (ADA) to measure the general academic skills and perceptual ability of dental school applicants. The DAT is comprised of multiple-choice test items consisting of four sections: • Survey of the Natural Sciences • Perceptual Ability • Reading Comprehension • Quantitative Reasoning The Quantitative Reasoning section of the DAT measures applicants’ math skills that will be required in dental schools. There are 40 multiple-choice questions test takers have 45 minutes to complete this section. A basic four-function calculator on the computer screen will be available in this section. Take a FREE practice test. How to Study for the DAT Math Test? Dentistry is a lucrative field that many people aspire to pursue. The DAT test is the gateway to this field. So, if you think that succeeding in this test is not so easy, you are right. But if others have succeeded, you can also succeed in this test with effort. Math is an important part of this test. If you feel that you are not ready for the DAT Math Test, do not worry because we are here to guide you step by step to prepare for the DAT Math Test. The Absolute Best Book to Ace the DAT Quantitative Reasoning Test 1. Choose your study program Many prestigious DAT prep books and study guides can help you prepare for the test. Most major test preparation companies have some offerings for the DAT Quantitative Reasoning, and the short-listing of the best book ends up being a puzzling phenomenon. There are also many online DAT Quantitative Reasoning courses. If you just started preparing for the DAT Quantitative Reasoning course or test and you need a perfect DAT Quantitative Reasoning prep book, then DAT Quantitative Reasoning Prep 2020-2021: The Most Comprehensive Review and Ultimate Guide to the DAT Quantitative Reasoning Test is a perfect and comprehensive prep book for you to master all DAT Quantitative Reasoning topics being tested right from It will help you brush up on your math skills, boost your confidence, and do your best to succeed on the DAT Quantitative Reasoning Test. This one is an alternative book: If you just need a DAT Quantitative Reasoning workbook to review the math topics on the test and measure your exam readiness, then try: “DAT Quantitative Reasoning Workbook 2020 – 2021: The Most Comprehensive Review for the Quantitative Reasoning Section of the DAT Test” This prep book is also a perfect resource to review key Mathematics concepts being tested on the DAT Quantitative Reasoning: Or if you think you are good at math and just need some DAT Quantitative Reasoning practice tests, then this book is a perfect DAT Quantitative Reasoning test book for you: You can also use our FREE DAT Quantitative Reasoning worksheets: DAT Quantitative Reasoning Worksheets Have a look at our FREE DAT Quantitative Reasoning Worksheets to assess your knowledge of Mathematics, find your weak areas, and learn from your mistakes. There is also a FREE practice test. DAT Quantitative Reasoning Math FREE Resources: 2. Change your attitude toward math How you look at math can affect your success on the DAT Math Test. If you look closely, you will find that most of the people who pass the DAT Math Test are people who like math and patiently spend time learning it. To be successful, you must be one of these people. Look at math as a challenge that can be the key to your success in the DAT test. 3. Make the concepts clear One of the most important steps in preparing for the DAT Math Test is to get to know the concepts in the test and categorize them. To better understand mathematical concepts, the study of mathematics should be done step by step. First, include basic math concepts in your program and then study advanced concepts. This way of studying will prevent you from being confused and wasting your time. 4. Practice daily For getting better results in the DAT test, it is better to have a daily study plan and include the material in small sections in this plan. Remember to never leave the test material for the last month. The amount of material that needs to be studied is large and must be practiced many times to stay in your mind. Start early and include a part of math content in your daily study schedule. Sticking to this program may be a little difficult at first, but you will get used to it, and this daily practice will make you successful in the test. 5. Find the best way to learn If you are just starting and do not know where to start studying math for the DAT test, there are many books for beginners that can help you. There are many ways to prepare for the DAT Math Test, including taking prep courses and using prep books. If you are looking for books to help you learn the math part of the DAT test, here is a complete list of useful books for you. Also, some test takers use a private tutor to speed up their learning, but because this method is not economical, you can replace books that guide you like a good tutor. The BEST prep book to help you ACE the DAT Quantitative Reasoning Test 6. Memorize formulas well You may feel relieved that the formulas will be provided to you in the DAT Math Test, but the truth is that it is better to memorize all the necessary formulas for the test. The test does not give you a complete list of required formulas. Only some formulas will be prepared for you to focus on the application instead of memorizing them. Even memorizing these formulas will increase your speed in answering the test questions. Now you should see where you can find the complete list of formulas. Do not worry, for your convenience, we have provided you with a complete list of required formulas for the DAT Math Test, along with their meanings and applications. Now you just need to keep them with you and memorize them gradually. 7. Take Practice Tests In the days leading up to the test, the more simulated tests you do, the more you will master what you have learned. So, whenever you feel you have learned the material well, do not waste time and focus more on practice tests. Many books include simulated practice tests. You can also use online tests to evaluate yourself. Manage time during these tests, and after the test, check your weaknesses and fix them. 8. Register for the test To register, you must go to the ADA website and apply to take the DAT. You will be asked for a DENTPIN, your Dental Personal Identifier Number before you can register. Then you must receive an eligibility letter from the ADA. This letter indicates that you are eligible to take the DAT test. Once you have received this letter, you can schedule an appointment to take the test with Prometric testing. You should schedule the test 60 to 90 days before the day you want to test. 9. Take the DAT Math test DAT test day is a very decisive day for you, so try to get to the test center a little earlier to reduce stress. Be prepared from the night before and take the all necessary equipment for the test. This could be a thing like ID cards, a bottle of water, etc. Your test is computer-based and a basic four-function calculator on the computer screen will be available in the math section. So, you do not need to bring a calculator. Do not take unnecessary personal items such as cell phones with you, or if you take them with you, put them in a closet provided by the center’s staff. Use your keyboard to answer questions. Using the keyboard ensures accuracy. In the math section of the DAT test, there are 40 multiple-choice questions that test takers have 45 minutes to complete this part. Try not to spend more than thirty seconds on the question and if you do not know the answer, temporarily skip it. But at the end of the test, try to answer all the questions and do not leave any questions unanswered because the DAT test does not consider penalties for wrong answers. Be aware that stress can ruin your work, so instead of stress, focus on time and how to answer questions. The Best DAT Quantitative Reasoning Quick Study Guide DAT FAQs: Here are some common questions about the DAT math test: What is on the DAT test? DAT is a standardized test designed by the American Dental Association (ADA) to measure the general academic skills and perceptual ability of dental school applicants. What is a good score on the DAT test? In the DAT test, a score above 19 and a percentile above 75 are considered very good. When can I take the DAT test? The ideal time to take this test is at the end of the spring semester of junior year or immediately after completing organic chemistry courses. Is the DAT harder than the MCAT? For most test-takers, the MCAT is usually more difficult than the DAT because answering its questions requires more analysis. How many times can you take DAT? You can only take this test three times, no more unless you get special permission from the ADA. How long does DAT score last? Your scores are valid for only two years. How long should I study for DAT? Usually, three to four months is enough time to prepare for the DAT test. What is the hardest section on the DAT? Most students who take the DAT report that, despite the inherent difficulty of chemistry, the biology section is the most difficult to prepare for, simply because of the large amount of material that can be tested. Looking for the best resources to help you or your student succeed on the DAT Quantitative Reasoning test? The Best Books to Ace the DAT Quantitative Reasoning Test Related to This Article What people say about "How to Prepare for the DAT Quantitative Reasoning Math Test? - Effortless Math: We Help Students Learn to LOVE Mathematics"? 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▷ Samacheer Kalvi 10th Maths Guide Chapter 3 Algebra Ex 3.4 2024 Samacheer Kalvi 10th Maths Guide Chapter 3 Algebra Ex 3.4 You can download Samacheer Kalvi 10th PDF math book solutions guide, with these guides you will be able to get more points in your exams and assignments. Tamilnadu Samacheer Kalvi 10th Maths Solutions Chapter 3 Algebra Ex 3.4 You can download the guide in PDF format or view it online in digital form. Download PDF If you want you can download it in PDF format to be able to enter at any time or print it. Watch online To view the guide from your browser use this button and you do not need to download it. Recommendations for downloading the guides: • Most of the books are quite a few MB, so the download time can be a bit long depending on your Internet connection. • To download the books, right-click on the links below and select “Save Link As…”, or left-click to directly view the PDF in your web browser. More Guides to Math 11th Chapter 3, Algebra Leave a Comment	{"url":"https://guidebook10.com/samacheer-kalvi-10th-maths-guide-chapter-3-ex-3-4/","timestamp":"2024-11-06T18:41:57Z","content_type":"text/html","content_length":"180155","record_id":"<urn:uuid:88bc39fa-c7bb-45e5-8929-6fc5963c3843>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00782.warc.gz"}
Navigate the World of Vectors: Vector Tutorials and Calculators Welcome to our tutorial on vectors! Whether you are a student, engineer, or someone looking to understand the fundamental concepts of vectors, this guide is your go-to resource with links to vector calculators which each include a detailed supporting tutorial. You can also access our Math Tutorials and Math Calculators from the quick links below. To apply the concepts covered in this tutorial, you can use our suite of associated calculators for vectors provided below. These calculators will enable you to perform vector operations and understand their properties in-depth. Embark on your vector exploration now! What is a Vector? In mathematics and physics, a vector is an element that has both magnitude and direction. It is often represented as an ordered array of numbers, which can signify various quantities such as force, velocity, or displacement. Types of Vectors 1. Free Vector: A vector whose characteristics remain the same regardless of its position in space. 2. Position Vector: A vector that specifies the position of a point in space relative to an arbitrary reference origin. 3. Unit Vector: A vector of magnitude 1, often used to specify a direction. Vector Operations 1. Vector Addition: The operation of adding two or more vectors together, combining their magnitude and direction. 2. Scalar Multiplication: The operation of multiplying a vector by a scalar (a single number), which scales its magnitude. 3. Dot Product: An operation that takes two vectors and returns a scalar. It measures how much two vectors are in the same direction. 4. Cross Product: An operation between two vectors in three-dimensional space, resulting in a vector that is perpendicular to both input vectors, with a magnitude equal to the area of the parallelogram they span. Applications of Vectors • Physics: Vectors are widely used in physics to represent quantities such as force, velocity, and momentum. • Engineering: In engineering, vectors are used to analyze stresses and strains on structures. • Computer Graphics: Vectors are fundamental in computer graphics for representing shapes, modeling motion, and performing transformations. More Good Math Calculators	{"url":"https://www.icalculator.com/vector-calculators.html","timestamp":"2024-11-13T14:52:49Z","content_type":"text/html","content_length":"15838","record_id":"<urn:uuid:26e97ca0-cd64-43bf-bcbd-afa5dbfb67bc>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00774.warc.gz"}
Statis-Pro Baseball Game (Learn Free Basic Version in 5) Click here for links to the 60 all-time great Statis-Pro baseball teams. Each team has simplified rules listed under the players, so you can play a game you need nothing more than to print two teams, and get two 6-sided dice and two 8-sided dice of different colors. We consolidated various sets of instructions on Statis-Pro Baseball game into the following. You can also read these instructions in this updated grid, or use the 3rd sheet on this google sheet which also includes all of the great team pitchers and hitters. We consolidated various sets of instructions on Statis-Pro Baseball game into the following. Basic Understanding of the Game 1 1. Dice or Fast Action Cards. 2 11-88 number - 11 possible results (hits, balks, Ks, Walks, HBP, WP, PB, Out) 4 Left vs. Right Adjustment 12/88, 88/11 or other. 5 Chance of Error on Hit or Out 6 Once you determine a player who might make an error - E-0 to E-10 6 SR and RR - How Long Can a Pitcher Pitch Before Getting Tired? 10 Taking Extra Bases on Hits, Hit and Runs, or Bunts 11 Clutch Fielding (CD or for catcher CD-C): 14 Z-Play - unusual plays, injuries and tough fielding plays: 14 Statis-Pro Baseball was invented by Jim Barnes in 1970, and in an interview he invited others to adopt and update the game as “open source.” Our free version enables you to either play current teams with projected players, or to choose from 60 all-time great baseball teams. There is a complete game and many seasons of great advanced Statis-Pro cards on the Statis Pro Advanced Facebook page, if you want to try the game here first and consider ordering from them if you determine you like it. Basic Understanding of the Game The unique aspect of the game is that each plate appearance starts by determining if the pitcher is in control of the at-bat and his card will be used, or if he “makes a mistake” to put the action on the batter’s card and give him a chance for an extra base hit. An initial roll of two-tradition 6-sided dice or a use of “fast action cards” yield a number of 2-12. The best pitchers keep it on their card on 2-9, while the worst pitchers only control the action on a 2-4. Once you know whether the batter or pitcher card will be used for that at bat, a subsequent roll of two 8-sided dice for a result of 11-88 or a similar number from the fast action card gives the result of the plate appearance. What you need to play 1. Dice or Fast Action Cards. You need one of the three things in these photos. If you choose to use dice, you need two traditional 6-sided dice, two 8-sided dice of different colors, and one 20-sided die. If you prefer to use the free fast action cards we provide, they look like the all-white card below. Others sell much nicer fast action cards, so google “statis-pro baseball fast action cards” and you can find a set for $10 - you can see the blue and green corner of one of those cards and they really do add to the game.. 2. Player Cards Next, choose the teams you want to play. If you want to play two of the 60 all-time great teams, print the pitchers (see 1995 Braves below) from this pdf (one team to a page) and then follow these directions to print out the batters (see 1927 Yankees below) from a google sheet. If you prefer to play modern players, then choose the pages of the teams you want from all 2022 Projected Batters (49 pages, 9 cards to a page) and All 2022 Projected Pitchers (62 pages, 9 cards to a page) - the cards will have ranges like the Scherzer vs. Betts cards below. So you will be using EITHER team sheets of players or individual cards of players. How to Play the Game Setting Up Choose a player on each team to pitch and play the other 8 positions - C, 1B, 2B, 3B, SS, LF, CF and RF. Next choose if you will use an extra batter as a designated hitter (DH) or have pitchers hit in the game. Once you choose the 9 who will start the game, write them down in the order they will hit from 1st through 9th. If you are using cards you can shuffle them, but if you printed out our free ones, you may just want to put them all in a big bowl to pull out the cards one at a time. Of course, if you are using the five dice then no shuffling is needed. If you are using player cards, you can stack the line-up in order. If using team sheets, you may want to use a couple of business cards to put below each batter as they hit. Order of Play 2-12 number: Get a result of 2-12 from flipping a card or rolling the dice. If that number falls within the pitcher’s PB rating (pitcher or batter) then the action will occur on the pitcher’s card, and if outside that range it will occur on the batter’s card. The possible PB ranges on pitchers’ cards, from best to worst, are 2-9, 2-8, 2-7, 4-7, 2-6, 2-5 or 2-4. If you choose to use optional advanced rules, look at the bottom of these rules for CD, BD or Z-plays, which can occur in place of the 2-12 number. 11-88 number - 11 possible results (hits, balks, Ks, Walks, HBP, WP, PB, Out) On the 2nd card, read the second number (Random Number 11-88) to see what happens on that card, or look at the 8-sided dice for the 11-88 number.. (If noone is one base, change any BK, WP or CD on line 2 to an OUT). Here are the 10 possible results: 1B = Single 2B = Double (only on batters' cards) 3B = Triple (only on batters' cards) HR = Home Run (only on batters' cards) BK = Balk if anyone on base (only on pitchers' cards, if no one on base treat as OUT) K = Strikeout W = Walk CD-C = Catcher Clutch Defense - if not using advanced rules treat as out HPB = Hit by Pitched Ball (only on pitchers' cards) WP/PB = Wild Pitch or Passed Ball (only on pitchers' cards, if no one on base treat as OUT) Out = see below. If using player cards, the card will also indicate the fielder who gets the ball on a single (1Bf, 1B7, 1B8 or 1B9) or double (2B7, 2B8, 2B9). If using a sheet, the reading is simply 1B or 2B so use the second digit of the number to determine the field. A 1 or 2 (11, 12, 21, 22 etc.) indicates the batter pulled the ball, so a RN or RP hits it to LF, a SN, SP or P to CF and a LN or LP to right. If the last digit is 3 or 4, the hit goes to LF, 5 or 6 goes to CF and 7 or 8 goes to RF. On old fast action cards, you would check after a WP or BK for a “yes” or “no” result, but skip that step even if using those old cards. If anyone is on base, the WP or BK occurs if it comes up on the card. Left vs. Right Adjustment 12/88, 88/11 or other. The Cht numbers at the bottom right of the batter card indicates numbers on which results are adjusted based on if the opposing pitcher is right-handed or left-handed. The standard 12/88 for a left-handed batter indicates that an 11 or 12 is changed from a hit to a strikeout against a left handed pitcher, and an 88 is changed from an out to a single with runners advancing two bases against a right-handed batter. The standard right-handed batter has an 88/11 meaning an 88 against a left-handed pitcher is a single, runners advance two bases while an 11 is changed to a strikeout against a right-handed pitcher. Typically a switch-hitter is –/– meaning no adjustment. For some batters there are more numbers impacted, for example a 14/85 would be the most extreme possible adjustment, and mean the batter struck out on 11-14 against lefties, and had singles with runners advancing 2 bases on 85-88. Just remember the number to the left of the dash is the adjustment against lefties, and the number to the right of the dash is the adjustment against righties. Further, the number either goes all the way down to 11, or all the way up to 85 from what is listed. If Result is OUT: If the result is an OUT, the fast action cards we provide will tell you what type of out is made. If using the nice Fast Action Cards you produce then the following will tell you what happens to runners on base. G6A (grounder to short) or any other A at the end of a grounder tells you the batter is out, but all runners advance. Gx6 or any other reading with an x in the middle indicates runners hold and if there is a force out then the defense can throw out a forced runner. However, if runners are on 1st and 3rd then the defense must choose whether to take the out at second base and let the runner on third score or hold the runner at third and throw out the batter to leave runners on 2nd and 3rd. G6 or any other reading with no x or A indicates a double play grounder if a runner is on 1st. However, if there is a runner on 3rd and no outs, the defense needs to either hold him and just throw the batter out at 1st, or let him score and turn the double play. If bases are loaded with no outs, the defense can choose to either throw the runner out at home, or take the double play from 2nd to 1st and let him score. Chance of Error on Hit or Out An error can occur under any of the three systems you can use for random numbers. If using dice, any time a 18 or 19 on the 20-sided die is rolled there is a chance for an error. On a hit, the outfielder who fields the ball could make an error to allow an extra base on the hit, while on an Out the player could allow a 1- or 2-base error instead of an out. On the nice Fast Action Cards you can buy from someone, a * will appear by the out result to let you know to check for errors on the next card. When using those cards, check for an error only on hits on the batter’s card - there is no chance of an error on a hit off the pitcher’s card. If using the free fast action cards we provide, the Error Reading on the 4th card is only used if; there is a hit on the BATTER card on line 2 OR, there is a possible error (e?) on line 3 with an out. If the fielder's E number is in the range on this 4th line then everyone is safe in an out or gets an extra base on a hit. Flip for another 11-88 and if the number is 61-88 give batter and runners one additional base for a throwing error. Once you determine a player who might make an error - E-0 to E-10 If there is a chance of an error on a player, then the next fast action card will determine if the error is in the player's range. A player can have anywhere from an E-0 (the best, never makes an error) to the worst E-10. If using the fast action cards, the next card will tell you if the error is in that range (e.g. Error 3-10 would be an error for an E-3 but not for an E-2). If using dice, the 20-sided die determines the same thing with a roll of 3 meaning the same as a 3-10 range is an error. If the roll is 10 or higher, then subtract 10 but if it is an error then it is in the range. So a 13 die roll would be a 2-base error for an E-3, but no error at all for an E-2. Obviously you cannot have a double play with noone on base, pardon the type under “1” with bases empty, just a grounder from shortstop to first base. SR and RR - How Long Can a Pitcher Pitch Before Getting Tired? The SR number is used for a starting pitcher, and RR used for a relief pitcher to see when they tire in a game. For example, Greg Maddux on the sheet above would start at 15 or his SR. This number is lowered by 1 every time the pitcher: 1. Allows a runner to reach 1st base (unless on an error that is not his fault) 2. Allows an earned run 3. An inning ends while he is pitching (whether he started that inning, or cam in after it started) Once a pitcher’s SR or RR is reduced to 0, his PB drops by 1, and then every additional education from one of those three occurs it is reduced by one more. If a pitcher started with the best rating of PB 2-9, then when he hit zero he would drop to PB 2-8, and then to 2-7, 4-7, 2-6, 2-5, 2-4, 2-3, 2-2 and if reduced further then the rest of the batters faced would skip the 2-12 number and automatically go to the 11-88 on the batter’s card. Taking Extra Bases on Hits, Hit and Runs, or Bunts Normally after getting the result you move onto the next batter. However, you can opt for any of the following strategies to advance base runners rather than going immediately to the next batter and the next 2-12 number. Here are charts with various options. Note I have made some changes in writing after reviewing stats on frequencies of various scenarios. On bunts, we do have some all-time great players whose Sac (bunt) rating is AA+. The first time an AA+ player bunts in a game, use the AA column on the sacrifice chart BUT on an 11-28 he is safe at first base on a single. When a runner tries to take an extra base on a hit, the chart is used for the numbers on which he makes it to the next base. However, in some cases he cannot make it the extra base, but he is not thrown out either, he just sees a strong throw coming and stops at the base. He is only thrown if the 11-88 number for the throw is a 71-88 from a T-5 throwing arm for the outfielder, or a 75-88 for T-4, 81-88 for a T-3 and finally 85-88 for a T-2. Players advancing or being thrown out in modern baseball is even more rare, with layers taking the next base only 30 percent of the time and only being thrown out 1 percent of the time. Optional Advanced Rules You can choose to ignore a BD, CD or Z-play result to keep the game simple. If so, just ignore and play the 2-12 range to use the batter or pitcher card instead. If you choose to play the advance rules, these can come up instead of the 2-12 number in the following ways. If using dice, a roll of 20 on the 20-sided dice can overrule the 2-12 rating only if runners are on base. If using fast action cards, in some cases a BD, CD or Z will come up instead of a 2-12 on some cards. With Fast Action Cards, Z-plays do happen even if noone is on base, but BD or CD is skipped if noone is on base. Clutch Batting (BD): If at least one runner is on base and the results is BD, or clutch batting, then the 11-88 number results in one of the following: If the number would result in a 1B on the batters cards, then make the result is a 2B (double) and all runners on base score. If the number would result in a 2B, 3B, HR or Deep on the batters card, then change to a home run. If the number would have resulted in anything else on the batters card, then change to a foul ball, and the batter is still at the plate. Clutch Fielding (CD or for catcher CD-C): If the result is a CD-C then use the catcher's clutch defense rating from 1 to 5 but simply score as an Out if noone is on base. If a CD results and anyone is on base, then check to see which fielder has a chance to make a play. The fast action card will indicate which player, but if using dice then refer to the 2-12 number to determine the player. If are using dice to determine which player has a chance for a clutch defensive player, or might commit and error - then use the following numbers: 2 or 3 = 1B, 4 = P, 5 = CF, 6 = 3B, 7 or 12 = SS, 8 = 2B, 9 - LF, 10 = C, 11 = RF. Once you know the position, use that players CD rating of 1 to 5 then the 11-88 number on the chart below: Z-Play - unusual plays, injuries and tough fielding plays: The following charts are used if an unusual “Z-Play” occurs. Draw a new 11-88 number. 2nd to home on 1b T2 T3 T4 T5 OBR A 11'-87 / out 88 11'-77 / out 85-88 11'-67 / out 83-88 11'-57 / out 81-88 OBR B 11'-77 / out 87-88 11'-67 / out 85-88 11'-57 / out 83-88 11'-47 / out 81-88 OBR C 11'-67 / out 87-88 11'-57 / out 85-88 11'-47 / out 83-88 11'-37 / out 81-88 OBR D 11'-57 / out 87-88 11'-47 / out 85-88 11'-37 / out 83-88 11'-27 / out 81-88 OBR E 11'-47 / out 87-88 11'-37 / out 85-88 11'-27 / out 83-88 11'-17 / out 81-88 1st to home on 2b T2 T3 T4 T5 OBR A 11'-73 / out 87-88 11'-63 / out 85-88 11'-53 / out 83-88 11'-43 / out 81-88 OBR B 11'-63 / out 87-88 11'-53 / out 85-88 11'-43 / out 83-88 11'-33 / out 81-88 OBR C 11'-53 / out 87-88 11'-43 / out 85-88 11'-33 / out 83-88 11'-23 / out 81-88 OBR D 11'-43 / out 87-88 11'-33 / out 85-88 11'-23 / out 83-88 11'-13 / out 81-88 OBR E 11'-33 / out 87-88 11'-23 / out 85-88 11'-13 / out 83-88 Cannot attempt 1st to 3B on 1B9 T2 T3 T4 T5 OBR A 11'-73 / out 87-88 11'-63 / out 85-88 11'-53 / out 83-88 11'-43 / out 81-88 OBR B 11'-63 / out 87-88 11'-53 / out 85-88 11'-43 / out 83-88 11'-33 / out 81-88 OBR C 11'-53 / out 87-88 11'-43 / out 85-88 11'-33 / out 83-88 11'-23 / out 81-88 OBR D 11'-43 / out 87-88 11'-33 / out 85-88 11'-23 / out 83-88 11'-13 / out 81-88 OBR E 11'-33 / out 87-88 11'-23 / out 85-88 11'-13 / out 83-88 Cannot attempt 1st to 3B on 1B8 T2 T3 T4 T5 OBR A 11'-63 / out 87-88 11'-53 / out 85-88 11'-43 / out 83-88 11'-33 / out 81-88 OBR B 11'-53 / out 87-88 11'-43 / out 85-88 11'-33 / out 83-88 11'-23 / out 81-88 OBR C 11'-43 / out 87-88 11'-33 / out 85-88 11'-23 / out 83-88 11'-13 / out 81-88 OBR D 11'-33 / out 87-88 11'-23 / out 85-88 11'-13 / out 83-88 Cannot attempt OBR E 11'-23 / out 87-88 11'-13 / out 85-88 Cannot attempt Cannot attempt 1st to 3B on 1B7 T2 T3 T4 T5 OBR A 11'-53 / out 87-88 11'-43 / out 85-88 11'-33 / out 83-88 11'-23 / out 81-88 OBR B 11'-43 / out 87-88 11'-33 / out 85-88 11'-23 / out 83-88 11'-13 / out 81-88 OBR C 11'-33 / out 87-88 11'-23 / out 85-88 11'-13 / out 83-88 Cannot attempt OBR D 11'-23 / out 87-88 11'-13 / out 85-88 Cannot attempt Cannot attempt OBR E 11'-13 / out 87-88 Cannot attempt Cannot attempt Cannot attempt Decide before the game if you want these optional adjustments to the ranges above. Optional Adjustments Use 1st that applies if 0 or 1 outs if 2 outs Regular Hit any odd number RN Same add 20 Texas Leaguer can divide RN by 12 add 40 add 60 Bloop can divide RN by 4 add 20 add 40 Line Drive can divide RN by 2 subtract 20 Same The numbers above are based on these numbers. Random Number 11-88 1st to 3rd on 1B 2nd to home on single 1st to home on double Advances Extra Base 11-33 11-57 11-43 Only advances same as hitter 34-87 58-85 44-86 Out trying to Advance 88 86-88 87-88 Based on actual occurances through 7/4 1st to 3rd on 1B 2nd to home on single 1st to home on double Advances Extra Base 2524 2887 1062 Only advances same as hitter 5691 1707 1420 Out trying to Advance 109 191 62 Below are the Clutch Defensive Charts so you have them in one place. C-CD (Catcher) CD1 CD2 CD3 CD4 CD5 Foul Out 11'-18 12'-18 11'-38 11'-48 11'-58 Passed Ball 21-58 31-58 41-58 51-58 61-64 Infield single 61-88 61-78 61-68 61-64 65-66 Lead runner out 81-88 71-88 65-88 67-88 1st Base & 3rd Base CD1 CD2 CD3 CD4 CD5 Line Out, Lead Runner Doubled off 11'-18 11'-28 11'-38 11'-48 11'-58 Grounder, lead runner thrown out even if no force 21-28 31-48 41-58 51-68 61-78 Line Drive single, advance 2 bases 31-58 51-68 61-78 71-84 81-88 Double down line, runners score 61-88 71-88 81-88 85-88 Shortstop, 2nd Base or Pitcher CD1 CD2 CD3 CD4 CD5 Line Out, Lead Runner Doubled off 11'-18 28-Nov 11'-38 11'-48 11'-58 Grounder, lead runner thrown out even if no force 21-28 31-48 41-58 51-68 51-78 Hard Grounder through for Single, runners advance 2 bases 31-88 51-88 61-68 71-88 81-88 Outfielder CD1 CD2 CD3 CD4 CD5 Line Out, Lead Runner Doubled off None None 11'-14 11'-18 11'-24 Nice catch of line drive, runners hold 11'-18 11'-28 15-38 21-48 25-48 Nice catch of deep drive, runner on 2nd and/or 3rd advance 21-28 31-48 41-58 51-68 51-78 Line drive single, advance 1 base 31-58 51-68 61-78 71-84 81-88 Line drive double, runners score 61-78 71-82 81-86 85-88 Line drive triple 81-88 83-88 87-88 Here is the updated Steal Chart. Results (not odd RN never out stealing, worst case holds) Steal 2b Steal 3b Steal unless SP: E 11'-14 11'-14 Steal if AAA, AA, A, B or C 15-23 15-17 Steal if AAA, AA, A or B 24-32 18-28 Steal if AAA, AA or A 33-46 31-38 Runner Out Stealing (odd number holds) 47-52 41-52 Runner Out if Catcher TA or TB (steal if TC, if out on odd RN holds) 53-56 53-55 Runner Out if Catcher TA (steal if TB or TC, if out on odd RN holds) 57-63 56-61 Holds, cannot get break 64-68 62-67 TA picks off catcher, otherwise hold 71-71 68-68 TA or TB picks off catcher, otherwise hold 72-72 71-71 Steal if AAA or AA 73-78 72-76 Steal if AAA 81-88 77-82 Runner Holds 83-88 Here are the Z-Plays so you have all in one place. 64 comments: 1. This is phenomenal, John. Thanks. I'll give it a go. 1. Thanks! Just wrote up the instructions with some more photos of cards and hope you liked it. 2. Don't think you're checking this, so I'll add a note on Delphi. 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558 Gradian/Square Day to Sign/Square Day Gradian/Square Day [grad/day2] Output 558 gradian/square day in degree/square second is equal to 6.7274305555556e-8 558 gradian/square day in degree/square millisecond is equal to 6.7274305555556e-14 558 gradian/square day in degree/square microsecond is equal to 6.7274305555556e-20 558 gradian/square day in degree/square nanosecond is equal to 6.7274305555556e-26 558 gradian/square day in degree/square minute is equal to 0.0002421875 558 gradian/square day in degree/square hour is equal to 0.871875 558 gradian/square day in degree/square day is equal to 502.2 558 gradian/square day in degree/square week is equal to 24607.8 558 gradian/square day in degree/square month is equal to 465258.87 558 gradian/square day in degree/square year is equal to 66997277.89 558 gradian/square day in radian/square second is equal to 1.1741581339372e-9 558 gradian/square day in radian/square millisecond is equal to 1.1741581339372e-15 558 gradian/square day in radian/square microsecond is equal to 1.1741581339372e-21 558 gradian/square day in radian/square nanosecond is equal to 1.1741581339372e-27 558 gradian/square day in radian/square minute is equal to 0.0000042269692821738 558 gradian/square day in radian/square hour is equal to 0.015217089415826 558 gradian/square day in radian/square day is equal to 8.77 558 gradian/square day in radian/square week is equal to 429.49 558 gradian/square day in radian/square month is equal to 8120.3 558 gradian/square day in radian/square year is equal to 1169323.09 558 gradian/square day in gradian/square second is equal to 7.4749228395062e-8 558 gradian/square day in gradian/square millisecond is equal to 7.4749228395062e-14 558 gradian/square day in gradian/square microsecond is equal to 7.4749228395062e-20 558 gradian/square day in gradian/square nanosecond is equal to 7.4749228395062e-26 558 gradian/square day in gradian/square minute is equal to 0.00026909722222222 558 gradian/square day in gradian/square hour is equal to 0.96875 558 gradian/square day in gradian/square week is equal to 27342 558 gradian/square day in gradian/square month is equal to 516954.3 558 gradian/square day in gradian/square year is equal to 74441419.88 558 gradian/square day in arcmin/square second is equal to 0.0000040364583333333 558 gradian/square day in arcmin/square millisecond is equal to 4.0364583333333e-12 558 gradian/square day in arcmin/square microsecond is equal to 4.0364583333333e-18 558 gradian/square day in arcmin/square nanosecond is equal to 4.0364583333333e-24 558 gradian/square day in arcmin/square minute is equal to 0.01453125 558 gradian/square day in arcmin/square hour is equal to 52.31 558 gradian/square day in arcmin/square day is equal to 30132 558 gradian/square day in arcmin/square week is equal to 1476468 558 gradian/square day in arcmin/square month is equal to 27915532.45 558 gradian/square day in arcmin/square year is equal to 4019836673.25 558 gradian/square day in arcsec/square second is equal to 0.0002421875 558 gradian/square day in arcsec/square millisecond is equal to 2.421875e-10 558 gradian/square day in arcsec/square microsecond is equal to 2.421875e-16 558 gradian/square day in arcsec/square nanosecond is equal to 2.421875e-22 558 gradian/square day in arcsec/square minute is equal to 0.871875 558 gradian/square day in arcsec/square hour is equal to 3138.75 558 gradian/square day in arcsec/square day is equal to 1807920 558 gradian/square day in arcsec/square week is equal to 88588080 558 gradian/square day in arcsec/square month is equal to 1674931947.19 558 gradian/square day in arcsec/square year is equal to 241190200395 558 gradian/square day in sign/square second is equal to 2.2424768518519e-9 558 gradian/square day in sign/square millisecond is equal to 2.2424768518519e-15 558 gradian/square day in sign/square microsecond is equal to 2.2424768518519e-21 558 gradian/square day in sign/square nanosecond is equal to 2.2424768518519e-27 558 gradian/square day in sign/square minute is equal to 0.0000080729166666667 558 gradian/square day in sign/square hour is equal to 0.0290625 558 gradian/square day in sign/square day is equal to 16.74 558 gradian/square day in sign/square week is equal to 820.26 558 gradian/square day in sign/square month is equal to 15508.63 558 gradian/square day in sign/square year is equal to 2233242.6 558 gradian/square day in turn/square second is equal to 1.8687307098765e-10 558 gradian/square day in turn/square millisecond is equal to 1.8687307098765e-16 558 gradian/square day in turn/square microsecond is equal to 1.8687307098765e-22 558 gradian/square day in turn/square nanosecond is equal to 1.8687307098765e-28 558 gradian/square day in turn/square minute is equal to 6.7274305555556e-7 558 gradian/square day in turn/square hour is equal to 0.002421875 558 gradian/square day in turn/square day is equal to 1.4 558 gradian/square day in turn/square week is equal to 68.36 558 gradian/square day in turn/square month is equal to 1292.39 558 gradian/square day in turn/square year is equal to 186103.55 558 gradian/square day in circle/square second is equal to 1.8687307098765e-10 558 gradian/square day in circle/square millisecond is equal to 1.8687307098765e-16 558 gradian/square day in circle/square microsecond is equal to 1.8687307098765e-22 558 gradian/square day in circle/square nanosecond is equal to 1.8687307098765e-28 558 gradian/square day in circle/square minute is equal to 6.7274305555556e-7 558 gradian/square day in circle/square hour is equal to 0.002421875 558 gradian/square day in circle/square day is equal to 1.4 558 gradian/square day in circle/square week is equal to 68.36 558 gradian/square day in circle/square month is equal to 1292.39 558 gradian/square day in circle/square year is equal to 186103.55 558 gradian/square day in mil/square second is equal to 0.000001195987654321 558 gradian/square day in mil/square millisecond is equal to 1.195987654321e-12 558 gradian/square day in mil/square microsecond is equal to 1.195987654321e-18 558 gradian/square day in mil/square nanosecond is equal to 1.195987654321e-24 558 gradian/square day in mil/square minute is equal to 0.0043055555555556 558 gradian/square day in mil/square hour is equal to 15.5 558 gradian/square day in mil/square day is equal to 8928 558 gradian/square day in mil/square week is equal to 437472 558 gradian/square day in mil/square month is equal to 8271268.88 558 gradian/square day in mil/square year is equal to 1191062718 558 gradian/square day in revolution/square second is equal to 1.8687307098765e-10 558 gradian/square day in revolution/square millisecond is equal to 1.8687307098765e-16 558 gradian/square day in revolution/square microsecond is equal to 1.8687307098765e-22 558 gradian/square day in revolution/square nanosecond is equal to 1.8687307098765e-28 558 gradian/square day in revolution/square minute is equal to 6.7274305555556e-7 558 gradian/square day in revolution/square hour is equal to 0.002421875 558 gradian/square day in revolution/square day is equal to 1.4 558 gradian/square day in revolution/square week is equal to 68.36 558 gradian/square day in revolution/square month is equal to 1292.39 558 gradian/square day in revolution/square year is equal to 186103.55	{"url":"https://hextobinary.com/unit/angularacc/from/gradpd2/to/signpd2/558","timestamp":"2024-11-14T10:56:36Z","content_type":"text/html","content_length":"112813","record_id":"<urn:uuid:0749a779-8ae0-4521-bcc1-5d8452fd4ac1>","cc-path":"CC-MAIN-2024-46/segments/1730477028558.0/warc/CC-MAIN-20241114094851-20241114124851-00732.warc.gz"}
Finding overlapping communities in social networks: Toward a rigorous approach A community in a social network is usually understood to be a group of nodes more densely connected with each other than with the rest of the network. This is an important concept in most domains where networks arise: social, technological, biological, etc. For many years algorithms for finding communities implicitly assumed communities are nonoverlapping (leading to use of clustering-based approaches) but there is increasing interest in finding overlapping communities. A barrier to finding communities is that the solution concept is often defined in terms of an NP-complete problem such as Clique or Hierarchical Clustering. This paper seeks to initiate a rigorous approach to the problem of finding overlapping communities, where "rigorous" means that we clearly state the following: (a) the object sought by our algorithm (b) the assumptions about the underlying network (c) the (worst-case) running time. The key contribution of this work is the distillation of the prior sociology studies into general assumptions that at once accord well with sociology research and the current understanding of social networks while allowing computationally efficient solutions. Our assumptions about the network lie between worst-case and average-case. An average-case analysis would require a precise probabilistic model of the network, on which there is currently no consensus. However, some plausible assumptions about network parameters can be gleaned from a long body of work in the sociology community spanning five decades focusing on the study of individual communities and ego-centric networks (in graph theoretic terms, this is the subgraph induced on a node's neighborhood). Thus our assumptions are somewhat "local" in nature. Nevertheless they suffice to permit a rigorous analysis of running time of algorithms that recover global structure. Our algorithms use random sampling similar to that in property testing and algorithms for dense graphs. We note however that our networks are not necessarily dense graphs, not even in local neighborhoods. Our algorithms explore a local-global relationship between ego-centric and socio-centric networks that we hope will provide a fruitful framework for future work both in computer science and sociology. Original language English (US) Title of host publication EC '12 - Proceedings of the 13th ACM Conference on Electronic Commerce Pages 37-54 Number of pages 18 State Published - 2012 Event 13th ACM Conference on Electronic Commerce, EC '12 - Valencia, Spain Duration: Jun 4 2012 → Jun 8 2012 Publication series Name Proceedings of the ACM Conference on Electronic Commerce Other 13th ACM Conference on Electronic Commerce, EC '12 Country/Territory Spain City Valencia Period 6/4/12 → 6/8/12 All Science Journal Classification (ASJC) codes • Software • Computer Science Applications • Computer Networks and Communications • overlapping community detection • social network analysis Dive into the research topics of 'Finding overlapping communities in social networks: Toward a rigorous approach'. Together they form a unique fingerprint.	{"url":"https://collaborate.princeton.edu/en/publications/finding-overlapping-communities-in-social-networks-toward-a-rigor","timestamp":"2024-11-02T14:53:51Z","content_type":"text/html","content_length":"56271","record_id":"<urn:uuid:463da6ed-9883-4c53-beca-f8425f3b58d2>","cc-path":"CC-MAIN-2024-46/segments/1730477027714.37/warc/CC-MAIN-20241102133748-20241102163748-00168.warc.gz"}
December 15th: The unbreakable toplist with a sum of the rest When you make a top list in TARGIT, you have at least two options: 1. Using the built-in top list function 2. Creating your top list with a visibility agent Here's a table to illustrate, showing revenue by salesperson: Let's first try the built-in easy way (Method no. 1) Make sure your table is selected, and go to the Calculations tab on the left-hand side of the TARGIT client: Now choose Add Top List / Pareto analysis: Choose ten and top list and click Apply top list changes: Now your table looks like this - notice it has been sorted descendingly, and only the top 10 are in the table: Method no 1 is very quick and only has one problem. You can't add something like a sum of the rest. That's because this way of making a top list means that TARGIT only will query for exactly these top 10 salespeople's data. Let's go back to the original table and try method no 2: 1. First, sort the table descendingly 2. Now go to the calculations tab and add the smart calculation: Rank descending 3. Now make a visibilty agent on the Rank with this syntax 4. Now your table looks like this - you might finish it up by going to visibility and hide the rank calculation: 5. And now you can add a calculated row called the rest with this syntax: sum(0,all(h),m1) (the "h" means hidden - so only the hidden rows will be summed up): That was Method no 2. It was a little more work - but now you have a top list with a sum of the rest, and the extra bonus is, that it can't be broken by sorting data, since the rank calculation, which is the foundation of this, works on both sorted and unsorted data. 1 comment • Thanks for a great article, Niels! I used Method 2 to come up with a Top 25 customer list with subtotals for top 25 and for “all the rest”. I ran into one snag if a filter returned fewer than 25 customers. As shown below, the “all the rest” calculation returned “Undefined” if there were fewer than 25 members (i.e. no hidden members), even though I used the 4^th parameter. sum(0, all(h), m1, 0) My workaround was to create a Top 25 Count calculation, as shown below. allcount(0, all(v), 0, 0) Then I modified the “all the rest” calculation to include a condition to check if the top 25 count did not have any hidden members. If there are fewer than 25 members, then it just returns zero. if sum(0, c1, 0, 0) < 25 then 0 else sum(0, all(h), m1, 0) Please sign in to leave a comment.	{"url":"https://community.targit.com/hc/en-us/articles/8276351920029-December-15th-The-unbreakable-toplist-with-a-sum-of-the-rest","timestamp":"2024-11-11T14:31:58Z","content_type":"text/html","content_length":"38161","record_id":"<urn:uuid:9857a7d6-f175-4150-8686-e425a9f1178c>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00567.warc.gz"}
Jacobson graph construction of ring Z<sub>3n</sub>, for n>1 A group is called a ring if the group is a commutative under addition operation and satisfy the distributive and assosiative properties under multiplication operation. Suppose R is a commutative ring with non-zero identity, U is the unit of R, and J(R) is a Jacobson radical. Jacobson graph of a ring R denoted by image [R] is a graph with a vertex set is R\J(R) dan edge set is {(a, b)\| 1-ab ∉ U}. The purpose of this research is to construct a Jacobson graph of ring Z [3] ^n with n > 1. The results show that Jacobson graph of ring Z [3] ^n is a disconected graph with two components. Dive into the research topics of 'Jacobson graph construction of ring Z[3n], for n>1'. Together they form a unique fingerprint.	{"url":"https://scholar.unair.ac.id/en/publications/jacobson-graph-construction-of-ring-zsub3nsub-for-ngt1","timestamp":"2024-11-06T08:21:59Z","content_type":"text/html","content_length":"55496","record_id":"<urn:uuid:e19094b0-9c9c-44da-96e4-62e34e330cd5>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00757.warc.gz"}
Can Q−Learning with Graph Networks Learn a Generalizable Branching Heuristic for a SAT Solver? Vitaly Kurin‚ Saad Godil‚ Shimon Whiteson and Bryan Catanzaro We present Graph-Q-SAT, a branching heuristic for a Boolean SAT solver trained with value-based reinforcement learning (RL) using Graph Neural Networks for function approximation. Solvers using Graph-Q-SAT are complete SAT solvers that either provide a satisfying assignment or proof of unsatisfiability, which is required for many SAT applications. The branching heuristics commonly used in SAT solvers make poor decisions during their warm-up period, whereas Graph- Q-SAT is trained to examine the structure of the particular problem instance to make better decisions early in the search. Training Graph-Q-SAT is data efficient and does not require elaborate dataset preparation or feature engineering. We train Graph-Q-SAT using RL interfacing with MiniSat solver and show that Graph- Q-SAT can reduce the number of iterations required to solve SAT problems by 2-3X. Furthermore, it generalizes to unsatisfiable SAT instances, as well as to problems with 5X more variables than it was trained on. We show that for larger problems, reductions in the number of iterations lead to wall clock time reductions, the ultimate goal when designing heuristics. We also show positive zero-shot transfer behavior when testing Graph-Q-SAT on a task family different from that used for training. While more work is needed to apply Graph-Q-SAT to reduce wall clock time in modern SAT solving settings, it is a compelling proof-of-concept showing that RL equipped with Graph Neural Networks can learn a generalizable branching heuristic for SAT search. Book Title NeurIPS 2020: Proceedings of the Thirty−fourth Annual Conference on Neural Information Processing Systems	{"url":"https://www.cs.ox.ac.uk/publications/publication14414-abstract.html","timestamp":"2024-11-06T08:14:11Z","content_type":"text/html","content_length":"30283","record_id":"<urn:uuid:3efaf63f-eec2-4878-a808-43d88f219c68>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00051.warc.gz"}
Note on Modified Inertia A note on modified inertia theories Milgrom made the following statement in a discussion about the external field effect and its vector orientation, which in general is hard to know. It has wider relevance, so is recorded here. In modified inertia theories there is not even an acceleration field defined. In fact different masses can have different accelerations at the same position depending on their orbit. This is similar to what happens in special relativity, where, as an example, the acceleration of electrons at a given position in an electromagnetic field depends on their momentary velocity, because the inertial mass depends on the velocity through the Lorentz factor. In MOND, the limited result from modified inertia is that for circular orbits (in an axisymmetric potential) we have μ(V^2/R)·V^2/R=g[N], where μ is a function (universal for a given theory) that is derived from the restriction of the kinetic action to circular orbits. `Luckily', this applies nicely to rotation curves* of isolated discs. When an external field is present you cannot apply this relation anymore to the overall motion. And, it certainly does not follow from modified inertia that for the external acceleration itself we have μ(g[e])·g[e]=g[Ne], as this acceleration, due, e.g. to large scale structure, is certainly not associated with a circular motion. * The trajectory of a particle matters in modified inertia, which is inevitably non-local. Where a particle has been matters as well as where it is instantaneously. Other instantaneous attrubtes besides position might also matter, e.g., the velocity as in special relativity. ** It need not apply, for example, to the vertical motions of particles in a disk, so one does not expect to be able to apply the simple MOND formula to the motions of stars perpindicular to the plane of the Milky Way in the same way as to a rotation curve. For further discussion of the EFE, see the MOND laws of galactic dynamics.	{"url":"http://astroweb.case.edu/ssm/mond/modifiedinertia.html","timestamp":"2024-11-11T11:32:31Z","content_type":"text/html","content_length":"3209","record_id":"<urn:uuid:0331163b-40e4-402a-b33e-5c27da3732f6>","cc-path":"CC-MAIN-2024-46/segments/1730477028228.41/warc/CC-MAIN-20241111091854-20241111121854-00046.warc.gz"}
Minimum partial-matching and hausdorff RMS-distance under translation: Combinatorics and algorithms We consider the RMS-distance (sum of squared distances between pairs of points) under translation between two point sets in the plane. In the Hausdorff setup, each point is paired to its nearest neighbor in the other set. We develop algorithms for finding a local minimum in near-linear time on the line, and in nearly quadratic time in the plane. These improve substantially the worst-case behavior of the popular ICP heuristics for solving this problem. In the partial-matching setup, each point in the smaller set is matched to a distinct point in the bigger set. Although the problem is not known to be polynomial, we establish several structural properties of the underlying subdivision of the plane and derive improved bounds on its complexity. In addition, we show how to compute a local minimum of the partial-matching RMS-distance under translation, in polynomial time. Original language English Title of host publication Algorithms, ESA 2014 - 22nd Annual European Symposium, Proceedings Publisher Springer Verlag Pages 100-111 Number of pages 12 ISBN (Print) 9783662447765 State Published - 2014 Externally published Yes Event 22nd Annual European Symposium on Algorithms, ESA 2014 - Wroclaw, Poland Duration: 8 Sep 2014 → 10 Sep 2014 Publication series Name Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) Volume 8737 LNCS ISSN (Print) 0302-9743 ISSN (Electronic) 1611-3349 Conference 22nd Annual European Symposium on Algorithms, ESA 2014 Country/Territory Poland City Wroclaw Period 8/09/14 → 10/09/14 • Hausdorff RMS-distance • local minimum • partial matching • polyhedral subdivision Dive into the research topics of 'Minimum partial-matching and hausdorff RMS-distance under translation: Combinatorics and algorithms'. Together they form a unique fingerprint.	{"url":"https://cris.huji.ac.il/en/publications/minimum-partial-matching-and-hausdorff-rms-distance-under-transla","timestamp":"2024-11-11T21:10:03Z","content_type":"text/html","content_length":"51462","record_id":"<urn:uuid:d0c06da9-1b32-4597-b5ca-4c94be830064>","cc-path":"CC-MAIN-2024-46/segments/1730477028239.20/warc/CC-MAIN-20241111190758-20241111220758-00830.warc.gz"}
Search Results Introduction to Robot Manipulator Arms We introduce serial-link robot manipulators, the sort of robot arms you might have seen working in factories doing tasks like welding, spray painting or material transfer. We will learn how we can compute the pose of the robot’s end-effector given knowledge of the robot’s joint angles and the dimensions of its links.	{"url":"https://robotacademy.net.au/?s=damping%20factor","timestamp":"2024-11-05T00:41:46Z","content_type":"text/html","content_length":"37346","record_id":"<urn:uuid:76b1d4c4-da3c-490d-8758-76f3c272f531>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.84/warc/CC-MAIN-20241104225856-20241105015856-00895.warc.gz"}
MU Advanced Microwave Engg - May 2013 Exam Question Paper \| Stupidsid Total marks: -- Total time: -- (1) Assume appropriate data and state your reasons (2) Marks are given to the right of every question (3) Draw neat diagrams wherever necessary 1 (a) Explain Large Signal characterization with reference to load pull contours. How is it measured? 5 M 1 (b) What are the causes of low frequency noise and high frequency noise associated with the mixer? 5 M 1 (c) Define and explain with neat diagram noise correlation matrix for general noisy two port network. 5 M 1 (d) What is unilateral figure of merit of an amplifier? 5 M 2 (a) If the transistor has following S-parameters at 5Ghz with 50Ω impedance. Determine the stability criteria and plot the stability circles. 10 M 2 (b) Derive the parameters of an Amplifier: (i) Power gain (G) (ii) Available gain (GA) (iii) Transducer gain (GT) 10 M 3 (a) Explain using suitable diagrams two methods of designing broadband amplifier. 10 M 3 (b) A BJT with I[C] = 30 mA and V[CE] = 10V is operated at a frequency of 1 GHz in a 50Ω system. Its S-parameters are - Determine whether the transistor is unconditionally stable. If yes, calculate optimum terminations, G[Smax], G[Lmax], G[TUmax]. 10 M 4 (a) A certain GaAs MESFET has following noise figure parameters measured at V[ds] = 50, I[ds] = 20 mA with 50Ω resistor once for frequency of 9 GHz, F[min] = 4dB, Γ[opt]=0.55∠175, R[0]= 4Ω. Plot noise figure circles for given values of f[1] at 2, 2.5, 3.5, and 4.5 dB. 15 M 4 (b) Define stability. List the various criteria for stability. 5 M 5 (a) If a one port microwave diode has Γ[in]=1.5∠60° with respect to Z[0]=50Ω. Design an oscillator for desired frequency of 10GHz. 12 M 5 (b) For a two port oscillator at steady state oscillations prove that if τ[L]τ[in]=1 then τ[in]τ[out]=1 8 M 6 A certain MESFET is biased for large signal class A operation with following small signal S-parameters at 5GHz: The large signal forward transmission coefficient S[21] is measured to be S[21]=2.1∠180°. Design a large-signal class A amplifier with maximum transducer gain in a 50Ω system. Assume (&pm;)0.5dB error in gain. What is the high-power amplifier gain? 20 M 7 (a) Write a note on optimal loading used in 1+PA design. 10 M 7 (b) A wideband amplifier (2-4 GHz) has gain of 10dB, an O/P power gain of 10dBm and a noise figure of 4dB at room temperature. Find the output noise power in dBm. 10 M More question papers from Advanced Microwave Engg	{"url":"https://stupidsid.com/previous-question-papers/download/advanced-microwave-engg-128","timestamp":"2024-11-03T16:22:51Z","content_type":"text/html","content_length":"60169","record_id":"<urn:uuid:27927bd9-8a29-47ec-b0fc-2940f5423dde>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00869.warc.gz"}
Determine Safe Landing Area for Aerial Vehicles This example demonstrates how to determine a safe landing area for aerial vehicles, such as helicopters and UAVs, by using aerial lidar data. Aerial vehiclesare increasingly used for applications like cargo delivery, casualty evacuation, and search and rescue operations. For these applications, the aerial vehicle must land safely in the vicinity of the destination. Because landing an aerial vehicle based on limited visual information is a challenging task for a pilot, aerial vehicles benefit from using lidar data to perform, reducing the pressure on the pilot and the risk of an accident. Use these steps to detect safe landing zones from aerial lidar data: 1. Load the point cloud into the workspace. 2. Divide the point cloud into overlapping blocks. 3. Identify the non-flat points, and label these points as dangerous. 4. Identify and label the points around the dangerous points as unsuitable. 5. Identify and label the points around the unsuitable points as risky. Also, label those points that do not satisfy the parameter thresholds as risky. 6. Identify the safe landing zones. You must avoid the dangerous and unsuitable points for landing, as these carry a very high chance of causing an accident. The probability of accidents is lower at the risky points, but the terrain points labeled as suitable are the safest for landing. Load and Visualize Data This example uses the point cloud data from the LAZ file aerialLidarData.laz, obtained from the Open Topography Dataset [1]. Load the point cloud data into the workspace using the readPointCloud function of the lasFileReader object, and visualize the point cloud. % Set random seed to generate reproducible results % Specify the LAZ file path lasfile = fullfile(toolboxdir("lidar"),"lidardata","las","aerialLidarData.laz"); % Create a lasFileReader object lasReader = lasFileReader(lasfile); % Read the point cloud ptCloud = readPointCloud(lasReader); % Visualize the point cloud axis on title("Input Aerial Lidar Data") Divide Point Cloud into Blocks Divide the input point cloud into overlapping blocks. Each block consists of an inner block and an outer block. First, specify the size of the inner block. % Specify the inner block size (in meters) innerBlockSize = [20 20]; The outer block size is the sum of the inner block size and landing site radius. Specify the landing site radius. % Specify the radius of the landing site to be evaluated (in meters) radius = 5; The landing site radius is the sum of the airframe radius of the aerial vehicle and the landing error, as shown in this figure. While labeling, you label only the points in the inner block. You determine the label of each point in the inner block by evaluating the parameters of that point and its nearest neighbors within the landing site radius. For example, to assign a label to the red point in this figure, you must evaluate the properties of the red point and its neighbors within the radius, represented by the green points. The block processing starts from the eft and proceeds to the right, then repeats from bottom to top until all the blocks have been covered. Define the overlap between two adjacent blocks such that the inner blocks of the adjacent blocks lay side-by-side without any overlap with each other, as shown in this figure. Use the helperOverlappingBlocks helper function, attached to this example as a supporting file, to compute the parameters required for overlapping-block-based processing. These parameters contain the indices of the outer blocks, inner blocks, and boundary points. The function also outputs the mapping for each block between the inner block, outer block, and labels. [outerBlockIndices,innerBlockIndices, ... innerBlockToOuterBlock,innerBlockIndFromLabels, ... innerBlockIndFromOuterBlock,boundaryIndices] = helperOverlappingBlocks(ptCloud, ... Define the class labels for classifying each point. classNames = [ "Dangerous" % Label 1 "Unsuitable" % Label 2 "Risky" % Label 3 "Suitable" % Label 4 You use the overlapping block processing parameters when labeling the unsuitable and risky points. Use overlapping block processing to improve the run-time performance of the labeling algorithm on larger point clouds. Classify Dangerous Points A good landing zone must be a flat terrain surface with minimal obstacles, as landing an aerial vehicle on non-flat points is dangerous, and can result in an accident. You must further evaluate the flat terrain points to analyze their safety. Flat terrain points generally consist of the ground points, while non-flat points generally consist of trees, buildings, and other non-ground points. Segment the input point cloud into ground and non-ground points, using the segmentGroundSMRF function. Label the non-ground points as dangerous. % Store the label of each point in a labels array labels = nan(ptCloud.Count,1); % Identify the ground points flatIdx = segmentGroundSMRF(ptCloud); % Mark the non-ground points as dangerous labels(~flatIdx) = 1; % Use the helperViewOutput helper function, defined at the end of the % example, to visualize the output title("Dangerous Points") Classify Unsuitable Points Label a point as unsuitable in any of these cases: 1. The point is along the boundary of the point cloud. 2. There are dangerous points in the neighborhood of the point. 3. The point does not have enough neighboring points, and further evaluation is not possible. Classify Unsuitable Boundary Points Because you cannot assess the entire neighborhood of the points along the boundary of the point cloud, label any unlabeled boundary points as unsuitable. % Identify the unlabeled boundary points unlabeledBoundaryIndices = isnan(labels(boundaryIndices)); % Label these points as unsuitable labels(boundaryIndices(unlabeledBoundaryIndices)) = 2; % Use the helperViewOutput helper function, defined at the end of the % example, to visualize the output title("Unsuitable Points Along the Boundary") Classify Unsuitable Points Around Dangerous Points Landing an aerial vehicle lands near dangerous points can result in an accident during landing. Dangerous points consist of powerlines, trees, buildings, and other non-ground objects. Due to their height, these points might not be detected as a neighbors within the radius of the flat points. Use the helperUnsuitablePoints helper function to label points in the vicinity of the dangerous points as unsuitable. % Define the grid size (in meters) gridSize = 4; Note: When you increase the gridSize value, the helperUnsuitablePoints function detects more points in the vicinity of each dangerous point, thus labeling more points as unsuitable. % Identify the indices of the points near the dangerous points unsuitableIdx = helperUnsuitablePoints(ptCloud,labels,gridSize); % Label these points as unsuitable labels(unsuitableIdx) = 2; % Use the helperViewOutput helper function, defined at the end of the % example, to visualize the output title("Unsuitable Points Near the Dangerous Points") Extract Indices of Nearest Neighbors for Unlabeled Points Further classify unsuitable points by using overlapping-block-based processing, and compute the nearest neighbors for each unlabeled point. If the number of neighbors is less than the minPoints value, or the neighborhood of the point contains dangerous points, label the point as unsuitable. Otherwise, store the indices of the neighboring points in the totalNeighborInd cell array. % Define the minimum number of neighbors for each point minPoints = 10; % Store the indices of the nearest neighboring points around each point % within the specified radius totalNeighborInd = {}; % Perform block-by-block processing for curBlockIdx = 1:numel(outerBlockIndices) % Extract the inner block point cloud from the input point cloud innerBlockPtCloud = select(ptCloud,innerBlockIndices{curBlockIdx}); % Extract the outer block point cloud from the input point cloud outerBlockPtCloud = select(ptCloud,outerBlockIndices{curBlockIdx}); % Extract the labels of the outer block curOuterBlockLabels = labels(outerBlockIndices{curBlockIdx}); % Create a mapping from the inner block labels to the outer block labels curInnerBlockToOuterBlock = innerBlockToOuterBlock{curBlockIdx}; Use the helperNeighborIndices helper function, defined at the end of the example, to compute the nearest neighbor indices within the radius of each point. The function also labels a point as unsuitable if the number of nearest neighbors within the specified radius of that point is less than the minimum number of points. [labeledOuterBlock,neighborInd] = helperNeighborIndices( ... innerBlockPtCloud,outerBlockPtCloud,curOuterBlockLabels, ... % Store the neighbor indices of the points belonging to the inner block totalNeighborInd = [totalNeighborInd; neighborInd]; Label only the inner block point cloud. Compute the indices of the inner block with respect to the labels and the outer block. Update the labels array with the labels computed for the inner block. labels(innerBlockIndFromLabels{curBlockIdx}) = labeledOuterBlock( ... % Use the helperViewOutput helper function, defined at the end of the % example, to visualize the output title("Unsuitable Points Detected") Classify Risky Points Within the unlabeled points, label a point as risky, if it has unsuitable points in its neighborhood. Also, evaluate these attributes of each unlabeled point and its neighboring points. 1. Vertical Variance — Variance of the points along the z-axis. A higher variance value indicates a greater height spread among the points, which can make them unsuitable for landing. 2. Relief — Height difference between the lowest and highest points in the neighborhood. A smaller relief value correlates with a flatter surface and fewer obstacles in the landing zone. 3. Slope — Inclination angle of the fitting plane. A smaller slope value is more suitable for landing, as indicates a more stable surface for the vehicle. 4. Residual — Roughness of the landing zone. A smaller roughness value indicates the presence of fewer obstacles and a smoother landing zone. Note: Because a landing surface must be flat, lower values of the vertical variance and relief attributes ensure that you can fit a plane through the points. Based on the plane fitting, the algorithm computes the slope and residual attributes. Specify the threshold values for these attributes. Any point, with an attribute value is greater than the respective threshold is a risky point. % Define the vertical variance threshold (in meters) verticalVarianceThreshold = 0.5; % Define the slope threshold (in degrees) slopeThreshold = 6; % Define the residual threshold (in meters) residualThreshold = 10; % Define the relief threshold (in meters) reliefThreshold = 3; innerBlockIndicesCount = 1; for curBlockIdx = 1:numel(outerBlockIndices) % Extract the inner block point cloud from the input point cloud innerBlockPtCloud = select(ptCloud,innerBlockIndices{curBlockIdx}); % Extract the outer block point cloud from the input point cloud outerBlockPtCloud = select(ptCloud,outerBlockIndices{curBlockIdx}); % Extract the labels of the outer block curOuterBlockLabels = labels(outerBlockIndices{curBlockIdx}); % Map the inner block labels to the outer block labels curInnerBlockToOuterBlock = innerBlockToOuterBlock{curBlockIdx}; Use the helperLabelRiskyPts helper function, defined at the end of the example, to label the points in the vicinity of the unsuitable points as risky. Also, label the points whose slope, vertical variance, relief, or residual parameter is greater than the corresponding threshold as risky. [labeledOuterBlock,updatedInnerBlockIndicesCount, ... updatedTotalNeighborInd] = helperLabelRiskyPts( ... innerBlockIndicesCount,totalNeighborInd,innerBlockPtCloud, ... outerBlockPtCloud,curOuterBlockLabels, ... curInnerBlockToOuterBlock, ... totalNeighborInd = updatedTotalNeighborInd; innerBlockIndicesCount = updatedInnerBlockIndicesCount; Label only the inner block point cloud. Compute the indices of the inner block with respect to the labels and the outer block. Update the labels array with the labels computed for the inner block. labels(innerBlockIndFromLabels{curBlockIdx}) = labeledOuterBlock( ... % Use the helperViewOutput helper function, defined at the end of the % example, to visualize the output title("Risky Points Detected") Classify Safe Landing Zones The remaining unlabeled points, which satisfy the parameter threshold values, are the safe landing points. There are no dangerous or unsuitable points around these points. Label these points as % Label the remaining unlabeled points as suitable labels(isnan(labels)) = 4; % Use the helperViewOutput helper function, defined at the end of the % example, to visualize the output title("Landing Zones Detected") Helper Functions The helperViewOutput helper function visualizes the point cloud along with the class labels. function helperViewOutput(ptCloud,labels,classNames) labels = labels + 1; labels(isnan(labels)) = 1; % Display the point cloud along with the labels ax = pcshow(ptCloud.Location,labels); axis on labels = unique(labels); cmap = [[255 255 255]; % White Color (Unlabeled Points) [255 0 0]; % Red Color (Dangerous Points) [150 75 0]; % Brown Color (Unsuitable Points) [255 255 0]; % Yellow Color (Risky Points) [0 255 0]]; % Green Color (Suitable Points) cmap = cmap./255; cmap = cmap(labels,:); % Add colorbar to current figure c = colorbar(ax); c.Color = "w"; numClasses = numel(labels); % Center tick labels and use class names for tick marks c.Ticks = linspace(c.Ticks(1)+(c.Ticks(end)-c.Ticks(1))/(2numClasses), ... c.Ticks(end)-(c.Ticks(end)-c.Ticks(1))/(2numClasses), ... c.TickLabels = classNames(labels); % Remove tick mark c.TickLength = 0; The helperNeighborIndices helper function returns the nearest neighbor indices within the specified radius of each point. The function also labels a point as unsuitable when the number of neighbors within the specified radius for the point is less than the specified minimum number of points, or when the point has dangerous points in its neighborhood. function [curLabels,curNeighborInd] = helperNeighborIndices( ... innerBlockPtCloud,outerBlockPtCloud,curLabels, ... curNeighborInd = cell(innerBlockPtCloud.Count,1); for i = 1:innerBlockPtCloud.Count % Compute nearest neighbor for only the unlabeled points if isnan(curLabels(curInnerBlockLabelsToOuterBlockLabels(i))) % Find the nearest neighbors within radius for each point [indices,~] = findNeighborsInRadius(outerBlockPtCloud, ... % If the number of neighbors is less than the minimum neighbors, % label the point as unsuitable if numel(indices) < minPoints curLabels(curInnerBlockLabelsToOuterBlockLabels(i)) = 2; % If the point is unlabeled and there are dangerous points present in % the vicinity of the point, label the point as unsuitable if any(curLabels(indices) == 1) && ... curLabels(curInnerBlockLabelsToOuterBlockLabels(i)) = 2; curNeighborInd{i} = indices; The helperLabelRiskyPts helper function labels the points in the vicinity of the unsuitable points as risky. Additionally, it evaluates a point and its neighbors based on specified attributes. The function labels a point as risky if any of its attribute values is greater than the specified threshold. function [curLabels,innerBlockIndicesCount, ... totalNeighborInd] = helperLabelRiskyPts(innerBlockIndicesCount, ... totalNeighborInd,innerBlockPtCloud,outerBlockPtCloud,curLabels, ... curInnerBlockLabelsToOuterBlockLabels, ... verticalVarianceThreshold,slopeThreshold,residualThreshold, ... for i = 1:innerBlockPtCloud.Count indices = totalNeighborInd{innerBlockIndicesCount}; innerBlockIndicesCount = innerBlockIndicesCount + 1; if ~isempty(indices) && ... % If the point has neighbors, the point is unlabeled. and there % are unsuitable points in the vicinity of the point, label the % point as risky if any(curLabels(indices) == 2) curLabels(curInnerBlockLabelsToOuterBlockLabels(i)) = 3; totalNeighborInd{innerBlockIndicesCount-1} = []; % Evaluate the point for the vertical variance attribute verticalVariance = var(outerBlockPtCloud.Location(indices,3)); if verticalVariance > verticalVarianceThreshold curLabels(curInnerBlockLabelsToOuterBlockLabels(i)) = 3; totalNeighborInd{innerBlockIndicesCount-1} = []; % Evaluate the point for the relief attribute relief = max(outerBlockPtCloud.Location(indices,3)) ... if relief > reliefThreshold curLabels(curInnerBlockLabelsToOuterBlockLabels(i)) = 3; totalNeighborInd{innerBlockIndicesCount-1} = []; % Perform the plane fitting operation on the point and its % neighbors [model,~,outlierIndices] = pcfitplane(pointCloud( ... outerBlockPtCloud.Location(indices,:)), ... % Evaluate the point for the slope attribute slope = acosd(model.Normal(3)); if slope > slopeThreshold curLabels(curInnerBlockLabelsToOuterBlockLabels(i)) = 3; totalNeighborInd{innerBlockIndicesCount-1} = []; % Evaluate the point for the residual attribute residual = rms(abs(outerBlockPtCloud.Location(outlierIndices,:) ... *model.Parameters(1:3)' + model.Parameters(4))); if residual > residualThreshold curLabels(curInnerBlockLabelsToOuterBlockLabels(i)) = 3; totalNeighborInd{innerBlockIndicesCount-1} = []; [1] OpenTopography. “Tuscaloosa, AL: Seasonal Inundation Dynamics And Invertebrate Communities,” 2011. https://doi.org/10.5069/G9SF2T3K. [2] Yan, Lu, Juntong Qi, Mingming Wang, Chong Wu, and Ji Xin. "A Safe Landing Site Selection Method of UAVs Based on LiDAR Point Clouds." In 2020 39th Chinese Control Conference (CCC), 6497–6502. Shenyang, China: IEEE, 2020.https://doi.org/10.23919/CCC50068.2020.9189499.	{"url":"https://ww2.mathworks.cn/help/lidar/ug/determine-safe-landing-zone-in-aerial-point-cloud.html","timestamp":"2024-11-06T02:34:37Z","content_type":"text/html","content_length":"103990","record_id":"<urn:uuid:0e7afd60-4b24-4829-a184-afa7cbd073d2>","cc-path":"CC-MAIN-2024-46/segments/1730477027906.34/warc/CC-MAIN-20241106003436-20241106033436-00345.warc.gz"}
Daily Streamflow Forecasts Based on Cascade Long Short-Term Memory (LSTM) Model over the Yangtze River Basin Key Laboratory of Hydrometeorological Disaster Mechanism and Warning of Ministry of Water Resources, Nanjing University of Information Science and Technology, Nanjing 210044, China School of Hydrology and Water Resources, Nanjing University of Information Science and Technology, Nanjing 210044, China Author to whom correspondence should be addressed. Submission received: 2 February 2023 / Revised: 28 February 2023 / Accepted: 6 March 2023 / Published: 7 March 2023 Medium-range streamflow forecasts largely depend on the accuracy of meteorological forecasts. Due to large errors in precipitation forecasts, most streamflow forecasts based on deep learning rely only on historical data. Here, we apply a cascade Long Short-Term Memory (LSTM) model to forecast daily streamflow over 49 watersheds in the Yangtze River basin for up to 15 days. The first layer of the cascade LSTM model uses atmospheric circulation factors to predict future precipitation, and the second layer uses forecast precipitation to predict streamflow. The results show that the default LSTM model provides skillful streamflow forecasts over most watersheds. At the lead times of 1, 7, and 15 days, the streamflow Kling–Gupta efficiency (KGE) of 78%, 30%, and 20% watersheds are greater than 0.5, respectively. Its performance improves with the increase in drainage area. After implementing the cascade LSTM model, 61–88% of the watersheds show increased KGE at different leads, and the increase is more obvious at longer leads. Using cascade LSTM with perfect future precipitation shows further improvement, especially over small watersheds. In general, cascade LSTM modeling is a good attempt for streamflow forecasts over the Yangtze River, and it has a potential to connect with dynamical meteorological forecasts. 1. Introduction Under climate change, the frequency and intensity of extreme weather events (e.g., floods, droughts) are likely to increase in many regions [ ], and the resulting economic losses and human casualties are also on the rise. Therefore, accurate streamflow forecasting is indispensable for both early warning and mitigation of flood and drought Streamflow forecasting from hydrological models relies heavily on meteorological forcing inputs, all of which are subject to errors and uncertainties. While temperature estimates are often similar between different data products, precipitation estimates often diverge significantly [ ]. For medium- and long-range streamflow forecast, the skill heavily depends on the quality of the precipitation forecasts [ ]. In addition to precipitation (P), evapotranspiration (ET) is more closely related to streamflow than other meteorological variables, such as wind speed or temperature [ ], because precipitation and evapotranspiration are the main processes that influence streamflow formation at the basin scale based on a dynamic hydrological balance [ ]. In addition, soil moisture (SM) is the initial condition for hydrological forecast, which can affect streamflow by influencing surface infiltration rates and subsurface runoff generation [ In recent years, deep learning methods were widely used for streamflow forecast. Not only the simple Long Short-Term Memory (LSTM) model, but also a large number of LSTM variants and different machine learning methods coupled with LSTMs are widely used for streamflow forecast and post-processing [ ]. For example, in the U.S. large sample study (CAMELS) watershed, the performance of streamflow prediction from the LSTM model was better than that from the Sacramento soil moisture accounting model and the NOAA National Water Model with calibrated parameters [ ]. In a follow-up study, it was found that even if the trained LSTM model was applied to the ungauged basins, the streamflow prediction performance was still better than that from the above physical models [ ]. Compared with traditional linear regression, multilayer perceptron, support vector machine, and other models, the LSTM model also has better performance in daily streamflow predictions [ ]. For instance, the LSTM model was used to predict the streamflow in the lead times of 1–30 days (months), and the results show that LSTM was better than the artificial neural network in predicting daily runoff. However, due to the lack of a large number of training data for monthly streamflow, the performance of monthly streamflow prediction was poor [ ]. In addition, the LSTM was combined with a Gaussian distribution process model to predict daily streamflow of the Yangtze River basin, and the results show that LSTM was superior to many traditional machine learning models, even for probabilistic streamflow prediction [ ]. The LSTM model can also be used for post-processing of the results from a physical model (prediction residual). Some studies showed that using the LSTM model to predict the simulation residual of WRF Hydro can reduce its simulation bias [ For deep learning, the data input is very important, such as the accuracy of the data, the type of data, the correlation between the data, etc. In the era of big data, a large amount of data input will lead to the increase in model complexity. How to balance the model complexity and generalization, and how to build a deep learning model with certain interpretability, are very challenging [ ]. The LSTM is driven by the historical data to predict streamflow over multiple basins in the United States through data integration. The results show that data integration can not only simplify the historical input of the model, but also extract historical variables more relevant to the target flow, reduce the burden of data input, and improve the prediction performance of the model. It can improve predictions over small watersheds with high autocorrelation of runoff and close rainfall–runoff relationship [ ]. In addition to integrating dynamic data, adding static data also proved to improve the streamflow forecasting [ In the past, many studies focused on how to preprocess or add various restrictions to the model to achieve skillful streamflow prediction, while how to flexibly use the meteorological prediction data in the streamflow prediction at long lead received less attention. The cascade LSTM predict future meteorological data (e.g., precipitation) firstly, and use it to build a relationship with future streamflow, which is more interpretable than the original LSTM model in a physical manner. The cascade LSTM is also compatible with dynamical meteorological forecasts, which has potential for complementing a dynamical streamflow forecast that is usually carried out by a link hydrological model with meteorological forecasts. However, the application of cascade LSTM is limited to the large uncertainty from meteorological forecasts, which needs a stepwise evaluation. Furthermore, it is known that model performance decreases with increasing lead times [ ]. The relationship between decreasing model performance and lead time depends on basin characteristics, such as basin size, land use, geological structure, and quality of hydrometeorological data. For example, compared with smaller watersheds with storm characteristics, large watersheds that require a longer time for river routing can produce better forecasting results [ ]. To sum up, the LSTM model and its variants are widely used for streamflow predictions, but most of them are case studies, without comprehensive investigations on the effects of precipitation forecasts on streamflow forecasts over multi-scale basins and multiple lead times. Therefore, comparison among different watersheds and different lead times are necessary to obtain a robust evaluation for the deep learning models. In this study, we build cascade LSTM models over 49 watersheds in the Yangtze River basin by using synthetic hydrometeorological data from a high-resolution land surface model simulation, and evaluate the streamflow forecast skill with or without a perfect precipitation forecast. We aim to (1) explore the possibility of LSTM models in a streamflow forecast when accurate hydrometeorological data are available, (2) propose a cascade LSTM (the first layer uses meteorological forcings to predict precipitation, and the second layer uses predicted precipitation and related hydrometeorological variables to predict streamflow) to reduce the complexity of the LSTM and improve the generalization, and (3) assess the performance of cascade LSTM streamflow forecasts for different lead time and different watersheds. 2. Materials and Methods 2.1. Study Area and Data The Yangtze River is the third longest river in the world with a total length of about 6387 km. The water resources of the Yangtze River account for about 36% of China’s total water resources. The Yangtze River basin is located at 90°33′~122°25′ E and 24°30′~35°45′ N, and its area is about 1.8 × 10 , which accounts for about 18.8% of China’s total land area. Figure 1 shows the locations of the 49 hydrological stations used in this study. The monthly streamflow observed at these 49 stations was collected from the Yangtze River Basin Hydrological Yearbook published by the Yangtze River Conservancy Commission. The meteorological forcing data for land surface model simulations are as follows: precipitation was obtained from China Meteorological Administration Land Data Assimilation System (CLDAS) and CN05.1 (uses observations from more than 2000 meteorological stations), and barometric pressure, specific humidity, long and short wave radiation, 2 m temperature, and wind speed were obtained from China Meteorological Forcing Dataset (CMFD) [ ]. Surface data include the 1 km resolution global soil texture dataset [ ], the 90 m resolution Digital Elevation Model (DEM) from the United States Geological Survey (USGS), the 0.05° resolution GLASS monthly LAI products from 1981 to 1999 [ ], and the 2000 to 2017 0.05° resolution MODIS version 6 monthly LAI reprocessing dataset [ ]. Due to lack of information, the monthly leaf area index for 1979–1980 is the same as that of 1981 [ ]. The geopotential heights used for precipitation forecast in the cascade LSTM were obtained from the fifth generation European ReAnalysis (ERA5) [ 2.2. The Conjunctive Surface-Subsurface Process Version 2 (CSSPv2) Model The CSSPv2 model is rooted in the common land model [ ], but with improved representations of surface hydrological processes, including the consideration of the quasi three-dimensional soil water transport process [ ] and one-dimensional dynamic surface water transport process [ ]. Furthermore, some parameterization schemes are adjusted with reference to the common land model v3.5, the one-dimensional groundwater module is added, and the interaction between groundwater and soil water is considered [ ]. In addition, the variable infiltration capacity runoff generation scheme, the parameterization of hydraulic properties that considers the impact of soil organic matter, and the soil thermal parameterization scheme are also included in CSSPv2 [ The streamflow simulation is mainly based on runoff generation and routing. The infiltration curve is used to represent the distribution of the surface infiltration capacity within the grid, and the shape of the curve is adjusted by parameters to calculate the saturation excess runoff. For the base flow, it relies on the base flow curve. When the soil saturation degree is lower than a certain threshold, the linear base flow occurs. When the degree of soil saturation is higher than a certain threshold, the soil will have a large nonlinear base flow [ ]. The river routing module is formulated based on the concept of linear reservoir. The model was applied in many studies and it showed good performance in simulating hydrological variables, including soil moisture, streamflow (extreme streamflow attribution and reservoir outlet streamflow), ET, snow depth, and total water storage [ ]. The CSSPv2-simulated streamflow is used as synthetic streamflow data for the calibration and validation of LSTM models. 2.3. The Cascade LSTM Model The LSTM method was shown to be effective in time-series forecast, but still has some limitations in streamflow forecast. For example, the target value of the current step t is not only related to the variables of the previous steps (e.g., t − 1, t − 2, …, and t − n), but also to other variables of the current step t. For example, in smaller basins, precipitation rapidly produces river flows to form cross-sectional outlet streamflow, and precipitation at the current step significantly affects streamflow at the current step. This does not mean that precipitation at the current step in large basins is not relevant. When the precipitation center is close to the cross-sectional outlet, it is also possible to quickly form cross-sectional streamflow. In order to solve this problem, this study uses cascade modeling, which is set up to obtain the precipitation at the current step, and also to reduce the complexity of the streamflow forecast layer and improve the generalization capability. The cascade model consists of 2 layers of sub-models, the first layer is the precipitation forecast layer (LSTM_P) and the second layer is the streamflow forecast layer (LSTM_S). LSTM_P is predicting precipitation by using historical observation meteorological variables, and LSTM_ S predicts streamflow by combining historical observation data with the new precipitation data. The new precipitation data is a combination of historical precipitation and precipitation predicted by LSTM_P ( Figure 2 2.4. Experimental Design The 1979–2017 meteorological forcing data (CN05.1 and CLDAS precipitation and CMFD near-surface air temperature, surface pressure, wind speed, humidity, and the shortwave and longwave radiation fluxes) were interpolated to 6 km resolution and used to drive the CSSPv2 model to obtain synthetic streamflow data at daily and monthly time scales [ ]. Then, the Kling–Gupta efficiency (KGE) values were calculated between the monthly simulated streamflow and the observed streamflow for different watersheds in the Yangtze River basin. When the KGE is greater than 0.5, we consider that the model has a relatively good performance in the watershed, and we use the CSSPv2-simulated streamflow as the synthetic streamflow since the observations cannot cover the whole period of 1979–2017 at a daily time scale. All hydrometeorological data were divided into a training set (1979–2007) and test set (2008–2017). In order to avoid the influence of data scales on the training process, all data were normalized $x i ′ = x i − x train min x train max − x train min ;$ $x i ′$ $x i$ are the normalized and the original values, $x train max$ $x train min$ , are the maximum and minimum values in the training period. Machine learning settings are as follows: 1. A default LSTM model driven by historical streamflow, precipitation, soil moisture, and evapotranspiration, is used to predict streamflow (STRF), i.e., ([STRF, SM, ET, and P] → STRF). This experiment is used to investigate the capability of LSTM for streamflow simulation without data errors. This experiment is called “default LSTM”. 2. First, we use 850 hpa geopotential height (Geo), surface pressure (Pres), wind speed (V), surface air temperature (T), and surface air specific humidity (Q) from ERA5 to predict future precipitation (LSTM_P), i.e., ([Geo, Pres, V, T, and Q] → P ). Then, we use historical streamflow, soil moisture, evapotranspiration, and historical precipitation and LSTM_P to predict streamflow (LSTM_S), i.e., ([STRF, SM, ET, and P ] → STRF), to explore the capability of cascade LSTM model ( Figure 2 ) in streamflow forecast. Note that the above experiments are conducted in the forecasting periods of 1–15 days. This experiment is called “cascade LSTM”. 3. We skip the first step in the “cascade LSTM” experiment, while using observed precipitation instead. Then we repeat the second step in the “cascade LSTM” experiment. This experiment is called “cascade LSTM with perfect precipitation”, which is used to assess the potential (or upper limit) of cascade LSTM. The study relied on open source libraries, including numpy, math, and panda. Tensorflow was used to implement LSTM and Matplotlib was used to draw the graphs. All experiments were conducted on a server equipped with an AMD EPYC 7402 CPU and an NVIDIA GeForce RTX 3090 GPU. All LSTM models are set up with one hidden layer and one dense layer. The ephemeris is set to 500, the time step is 30 days, the batch size is 256, the hidden cell is 64, the dropout rate is 0.1 and the learning rate is 0.1. The hyper-parameters of the default LSTM model are optimized through grid screening in the one-day lead streamflow forecasts at four stations (zhimenda, cuntan, hankou, and datong) in the mainstream of the Yangtze River, and the hyper-parameters of all subsequent LSTM models remain unchanged ( Table 1 ). The Adam optimizer is chosen as the optimizer and tanh is used as the activation function and given the restriction that its output cannot be less than 0. 2.5. Evaluation of Model Performance In this study, the Kling–Gupta efficiency (KGE) [ ] is used to evaluate the forecast results. The KGE is defined as: $KGE = 1 − ( R − 1 ) 2 + ( β − 1 ) 2 + ( γ − 1 ) 2 ;$ $γ = σ s / x ¯ s σ o / x ¯ o ;$ $x ¯ o$ $x ¯ s$ are the mean values of observed and predicted streamflow during the evaluation period individually, while $σ o$ $σ s$ are their standard deviations. The R, β, and γ are the Pearson correlation coefficient, the ratio of predicted and observed means (e.g., bias ratio), and the ratio of the predicted and observed coefficient of variation ratio, respectively [ ]. The KGE value ranges from negative infinite to one, and a KGE value of one implies a perfect forecast. 3. Results 3.1. Evaluation of CSSPv2 Land Model Simulation and Default LSTM Forecast The streamflow simulated by the CSSPv2 land surface model showed good performance ( Figure 3 a), with the KGE of all stations greater than 0.5 and a median value of 0.72 ( Figure 3 b) for monthly streamflow. Most of upstream and downstream KGEs are greater than 0.7, except that the KGEs of five tributary stations in the middle reaches of the Yangtze River are between 0.5 and 0.6. Therefore, the synthetic streamflow data (CSSPv2-simulated streamflow data) from all 49 stations are used for the assessments of LSTM modeling. With the synthetic daily streamflow data generated by the CSSPv2 model, the default LSTM models (without predicting precipitation in a cascading manner) are trained and evaluated. Figure 4 shows that 80% of the stations have KGEs greater than 0 at 1–15-day lead times for the default LSTM-predicted daily streamflow. In particular, 78%, 30%, and 20% of the stations have KGEs greater than 0.5 at lead times of 1 day, 7 days, and 15 days, respectively. The mean KGE decreased from 0.86 to 0.41 from a 1-day lead to a 7-day lead, and down to 0.3 at a 15-day lead. As the lead time increases, the forecast skill decreases. Figure 5 shows the larger watershed area, the higher forecast skill is for different lead times. The slower decline of forecast skill of large watersheds may be due to the longer routing time. However, for small watersheds, the skill degradation rate is faster, where the KGEs are 0.3–0.75 at a 1-day lead, while they decrease to −0.35–0.4 at a 7-day lead, and −0.35–0.3 at a 15-day lead times. This may be because the streamflow change of small watersheds is easily affected by rainstorm, and the streamflow response speed is fast. Therefore, real-time or near-real-time precipitation data are critical for streamflow nowcasting and forecasting at small watersheds. 3.2. Evaluation of Cascade LSTM 3.2.1. Precipitation Forecast Based on LSTM_P Using default LSTM as the baseline, we build cascade LSTM by predicting precipitation firstly. Figure 6 shows the relationship between cascade LSTM-predicted precipitation and watershed area. Similar to the streamflow forecast, the precipitation forecast skill increases as the area increases. About 98%, 53%, and 51% of the stations have KGE greater than 0 for 1, 7, and 15-day lead times. The average KGE for precipitation forecast decreases from 0.27 to 0.07 for 1–15 days lead. Precipitation forecast is very difficult, and there is no skill (KGE < 0) at several small watersheds. 3.2.2. Streamflow Forecast Based on LSTM_S With the predicted precipitation through LSTM_P, cascade LSTM predicts streamflow through LSTM_S (see Section 2.3 for details). Figure 7 shows the KGE difference between cascade LSTM and default LSTM streamflow forecast. From default LSTM to cascade LSTM, 61% of the stations have an increase in KGE, and 75%, 88%, and 88% of the stations have an increase in KGE at 3, 7, and 15 days lead times, where the KGE increases by 0.1–0.3 for 20%, 20%, and 22% of the stations. For the three components of KGE, correlation (R) increases at 53%, 82%, and 84% of the stations at lead times of 1, 7, 15 days, bias in mean value (β) reduces at 57%, 63%, and 63% of the stations, and bias in coefficient of variations (γ) reduces at 59%, 65%, and 43% of the stations. Figure 8 Table 2 show that the three components from cascade LSTM improved against default LSTM for more than 50% of the stations at most lead times. This suggests that the benefit of cascade LSTM against default LSTM is more obvious at longer lead times. Compared with default LSTM, the average KGE of cascade LSTM streamflow forecast increased by 0.01–0.06 at lead times of 1–15 days ( Figure 9 ). Due to the poor performance of precipitation forecast in small watersheds, the improvement of streamflow forecast is small over these watersheds. For large watersheds, although the precipitation forecast through LSTM_P is good, the LSTM_S does not necessarily have more improvement against default LSTM than those for small watersheds because the rainfall-runoff process does not necessarily dominate the streamflow forecast for large watersheds; other factors including initial memory and river routing might also be important. Therefore, Figure 9 does not show a linear relationship between KGE improvement and basin area. Nevertheless, the improvement of cascade LSTM against default LSTM cannot be ignored for both large and small watersheds over the Yangtze River basin. 3.3. Evaluation of Cascade LSTM with Prefect Precipitation To explore the potential of cascade LSTM, we conducted a set of ideal experiments, where future precipitation forecasts were replaced with observations, which was called cascade LSTM with perfect Figure 10 shows the KGE difference between cascade LSTM with prefect precipitation and default LSTM streamflow forecasts. At 1, 7, and 15-day lead times, KGE increases at 70%, 94%, and 94% of the stations. For the three components of KGE, R increases at 86%, 94%, and 94% of the stations at lead times of 1, 7, and 15 days, β reduces (improves) at 65%, 84%, and 88% of the stations, and γ reduces (improves) at 65%, 88%, and 88% of the stations ( Table 2 Figure 8 shows that the cascade LSTM with perfect precipitation is better than the cascade LSTM and the default LSTM both for the mean and median values of the evaluation indicators. With the increase in lead times, the decrease in the correlation coefficient (R) becomes the leading factor for the decrease in KGE. The perfect precipitation experiment shows the importance of precipitation in streamflow forecasts, especially at long leads. When the lead time is greater than 1 day, 25–65% of the stations have KGE increases greater than 0.5. Figure 11 shows that compared with the default LSTM, the cascade LSTM with prefect precipitation has no significant improvement at the 1-day lead time for medium and large watersheds, while the increment of KGE in small watersheds can reach 0.55. For the lead times of 2–15 days, the skill improvement also diminishes as the watershed area increases. This suggests that a perfect precipitation would be more useful for streamflow forecasts at smaller watersheds. On average, the increment of KGE is 0.09–0.48 for the forecasts at 1–15 days. 4. Discussion Our results have several limitations that need further improvements. Firstly, we used the synthetic data of the physical hydrological model (CSSPv2) to train the LSTM models and carry out the analysis. We focused on the predictions of natural streamflow in both large and small basins, and assumed that the major uncertainty comes from the representations of the rainfall-runoff processes and river routing processes, while we neglected the uncertainties from meteorological forcings and influences of human interventions, such as reservoirs regulations. We are now developing the CSSPv2 model by considering land use/land cover change, reservoir regulations, irrigations, and urbanizations, etc. Combining physical models with LSTM models would show promising prediction capability. Nevertheless, we believe that the current results are useful for understanding the first order processes that control runoff generations, i.e., rainfall-runoff processes [ ]. Additionally, we show the potential of precipitation forecasting in the streamflow predictions at different leads. Secondly, the model of precipitation forecast layer (LSTM_P) can also be modified. Here we only use the LSTM model, while the models applied to precipitation prediction also include machine learning methods that can simulate the characteristics of space-time variations of precipitation, such as the post-processing of radar or weather forecast model data by convolution neural network [ ] and the direct modeling and prediction of the three-dimensional convolution neural network [ ]. The precipitation with spatiotemporal characteristics connected to the LSTM model may be suitable for the predictions of precipitation and runoff in large basins [ ]. For example, when the precipitation center is closer to the outlet of a catchment, it can speed up the occurrence of peak flow, which could be considered in the LSTM model. Thirdly, from the evaluation of the cascade LSTM with perfect precipitation, it is found that precipitation forecasting (even watershed average precipitation) is very important for streamflow forecasting in small watersheds. This study implies the potential of cascade LSTM in the streamflow forecast at small watersheds. More attention should be paid on these small watersheds where the streamflow forecast skill is limited as long leads, especially for deploying more in situ rainfall gauges and radars and developing high-resolution hydro-climate forecast models. 5. Conclusions In this study, we build cascade LSTM models over 49 watersheds in the Yangtze River basin by using synthetic hydrometeorological data from a high-resolution land surface model simulation, and evaluate streamflow forecast skill at lead times of 1–15 days for default LSTM, cascade LSTM, and cascade LSTM with prefect precipitation. The results show that the default LSTM model provides skillful streamflow forecast for most watersheds. At the lead times of 1, 7, and 15 days, the KGE of 78%, 30%, and 20% watersheds are greater than 0.5, respectively. Their performance improves as watershed area increases. In addition, the mean KGE decreases from 0.86 to 0.30 during the 1–15-day lead times. The KGE decreases more slowly for larger watersheds. However, the KGE decreases from 0.3–0.75 to −0.35–0.35 for the small watersheds. Large watersheds rely more on historical streamflow data, while small watersheds may rely more on shorter historical data and real-time and near-real-time precipitation data. After implementing the cascade LSTM model, 61–88% of the watersheds show an increase in the KGE at different leads, and the increase at longer leads is more obvious (e.g., 22% of the watersheds show an increase in KGE between 0.1 and 0.3 at 15-day lead times). From default LSTM to cascade LSTM, the average KGE increases by 0.01–0.06 for all watersheds at the 1–7-day lead times. Using a cascade LSTM with perfect precipitation shows further improvements, especially in small watersheds. When the lead times are greater than 1 day, 25–65% of the watersheds have an incremental KGE greater than 0.5. Overall, the cascade LSTM model is a good attempt to forecast streamflow in the Yangtze River basin. Author Contributions Conceptualization, J.L. and X.Y.; methodology, J.L. and X.Y.; software, J.L.; writing—original draft preparation, J.L.; writing—review and editing, J.L. and X.Y.; funding acquisition, X.Y. All authors have read and agreed to the published version of the manuscript. This work was supported by National Key R&D Program of China (2022YFC3002803), National Natural Science Foundation of China (U22A20556), Natural Science Foundation of Jiangsu Province for Distinguished Young Scholars (BK20211540), Postgraduate Research & Practice Innovation Program of Jiangsu Province (KYCX21_1009), and the Major Science and Technology Program of the Ministry of Water Resources of China (SKS-2022001). Data Availability Statement Not applicable. Conflicts of Interest The authors declare that they have no known competing financial interest or personal relationships that could have appeared to influence the work reported in this paper. 1. Arial, P.A.; Bellouin, N.; Coppola, E.; Jones, R.G.; Krinner, G.; Marotzke, J.; Naik, V.; Palmer, M.D.; Plattner, G.-K.; Rogelj, J.; et al. Technical Summary. Climate Change 2021: The Physical Science Basis. Contribution of Working Group I to the Sixth Assessment Report of the Intergovernmental Panel on Climate Change. 2021. 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Locations and drainage areas (km^2) of 49 streamflow observational stations over the Yangtze River basin in southern China. Figure 3. (a) Spatial distribution of KGE between observed and CSSPv2 model-simulated monthly streamflow over the Yangtze River basin. (b) The boxplot of KGE for all 49 stations. The boxes represent the 25th and 75th percentiles of KGE, the line and the dot within the box are median and mean values of KGE, respectively, and the whiskers represent 10th and 90th percentiles of KGE. Figure 4. Spatial distributions of KGEs for streamflow forecasts based on default LSTM model at lead times of 1–15 days. Figure 5. The relationship between the watershed area (ln km^2) and the default LSTM streamflow forecast skill (KGE) at the lead times of 1–15 days. Figure 6. The relationship between the watershed area (ln km^2) and the LSTM_P precipitation forecast skill (KGE) at the lead times of 1–15 days. Figure 7. Spatial distributions of the KGE difference (∆ KGE) between the daily streamflow forecasts by the cascade LSTM model and those by the default LSTM model. Figure 8. Performance of streamflow forecasts at different lead times. The left column shows the mean values of KGE, R, β, and γ for 49 stations, and the right column shows their median values. Figure 9. The relationship between the difference of streamflow forecast performance (∆ KGE of the cascade LSTM model and the default LSTM model) and the watershed area (ln km^2) at the lead times of 1–15 days. Figure 10. Spatial distributions of the KGE difference (∆ KGE) between the daily streamflow forecasts by the cascade LSTM with prefect precipitation and those by default LSTM. Figure 11. The relationship between the difference of streamflow forecast performance (∆ KGE of the cascade LSTM model with prefect precipitation and the default LSTM model ) and the watershed area (ln km^2) at the lead times of 1–15 days. Hyper-Parameter Set Up Batch 16, 32, 64, 128, 256, 512, 1024 Hidden cell 8, 16, 32, 64, 128, 256 Dropout rate 0.01, 0.05, 0.1, 0.15, 0.2, 0.3 Learning rate 0.001, 0.005, 0.01, 0.05, 0.1, 0.2 Table 2. Percentage (%) of stations that improved compared to the default LSTM model, the cascaded LSTM model (CLSTM), and the cascaded LSTM with the perfect precipitation (CLSTM_P) model KGE and its three components (R, β and γ). Lead Times KGE R β γ CLSTM CLSTM_P CLSTM CLSTM_P CLSTM CLSTM_P CLSTM CLSTM_P 1 0.61 0.70 0.53 0.86 0.57 0.65 0.59 0.65 2 0.76 0.86 0.20 0.82 0.82 0.86 0.80 0.78 3 0.76 0.88 0.45 0.88 0.84 0.92 0.76 0.84 4 0.86 0.88 0.53 0.90 0.80 0.88 0.69 0.80 5 0.84 0.88 0.63 0.92 0.73 0.78 0.71 0.86 6 0.86 0.90 0.69 0.90 0.67 0.88 0.65 0.88 7 0.88 0.94 0.82 0.94 0.63 0.84 0.65 0.88 8 0.86 0.92 0.82 0.96 0.61 0.86 0.59 0.90 9 0.82 0.92 0.82 0.94 0.61 0.86 0.65 0.86 10 0.80 0.92 0.86 0.94 0.51 0.86 0.63 0.86 11 0.78 0.94 0.73 0.96 0.71 0.94 0.67 0.88 12 0.76 0.90 0.78 0.94 0.51 0.86 0.55 0.90 13 0.76 0.90 0.80 0.96 0.61 0.84 0.51 0.92 14 0.84 0.92 0.80 0.96 0.59 0.86 0.59 0.94 15 0.88 0.94 0.84 0.94 0.63 0.88 0.43 0.88 Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. © 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https: Share and Cite MDPI and ACS Style Li, J.; Yuan, X. Daily Streamflow Forecasts Based on Cascade Long Short-Term Memory (LSTM) Model over the Yangtze River Basin. Water 2023, 15, 1019. https://doi.org/10.3390/w15061019 AMA Style Li J, Yuan X. Daily Streamflow Forecasts Based on Cascade Long Short-Term Memory (LSTM) Model over the Yangtze River Basin. Water. 2023; 15(6):1019. https://doi.org/10.3390/w15061019 Chicago/Turabian Style Li, Jiayuan, and Xing Yuan. 2023. "Daily Streamflow Forecasts Based on Cascade Long Short-Term Memory (LSTM) Model over the Yangtze River Basin" Water 15, no. 6: 1019. https://doi.org/10.3390/ Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details Article Metrics	{"url":"https://www.mdpi.com/2073-4441/15/6/1019?utm_campaign=releaseissue_waterutm_medium=emailutm_source=releaseissueutm_term=titlelink23","timestamp":"2024-11-06T12:18:50Z","content_type":"text/html","content_length":"443762","record_id":"<urn:uuid:0049f896-ee46-4a48-930f-9b56d856b363>","cc-path":"CC-MAIN-2024-46/segments/1730477027928.77/warc/CC-MAIN-20241106100950-20241106130950-00231.warc.gz"}
Discontinuity, Nonlinearity, and Complexity Dimitry Volchenkov (editor), Dumitru Baleanu (editor) Dimitry Volchenkov(editor) Mathematics & Statistics, Texas Tech University, 1108 Memorial Circle, Lubbock, TX 79409, USA Email: dr.volchenkov@gmail.com Dumitru Baleanu (editor) Cankaya University, Ankara, Turkey; Institute of Space Sciences, Magurele-Bucharest, Romania Email: dumitru.baleanu@gmail.com Evolutionary Dynamics of a Single-Species Population Model with Multiple Delays in a Polluted Environment Discontinuity, Nonlinearity, and Complexity 9(3) (2020) 433--459 \| DOI:10.5890/DNC.2020.09.007 Ashok Mondal$^{1}$, A. K. Pal$^{2}$, G. P. Samanta$^{1}$ $^{1}$ Department of Mathematics, Indian Institute of Engineering Science and Technology, Shibpur, Howrah - 711 103, India $^{2}$ Department of Mathematics, S. A. Jaipuria College, Kolkata-700005, India Download Full Text PDF In this work, evolutionary dynamical behaviour of a single-species population model in a polluted environment has been analyzed. This model system describes the effect of toxicant on a single-species population. Two discrete time delays have been incorporated for proper description. Important mathematical characteristics of the proposed model such as positivity, boundedness, stability and Hopf-bifurcation for all possible combinations of both the delays at the interior equilibriumpoint of the model system have been discussed. It is observed that increase amount of delay may lead to the change of stable behaviour of stationary points through the creation of limit cycles and higher periodic oscillations. Furthermore, it is reported that Hopf-bifurcations may also occur around stationary points for corresponding non-delayed system. Various numerical simulations are performed to validate analytical findings. The authors are grateful to the anonymous referees and the Editor Dr. Dimitri Volchenkov, DSc, for their careful reading, valuable comments and helpful suggestions, which have helped them to improve the presentation of this work significantly. The first author (Ashok Mondal) is thankful to the University Grants Commission, India for providing SRF (RGNF). 1. [1]& Shahrabi, N.S., Pourezzat, A., Ahmad, F.B., andMafimoradi, S. (2013), Poursafa P.Pathologic analysis of control plans for air pollution management in tehran metropolis: a qualitative nbsp study, Int. J. Prev. Med., 4(9), 995-1003. 2. [2]& Yamamoto, S.S., Phalkey, R., and Malik, A.A., (2014), A systematic review of air pollution as a risk factor for cardiovascular disease in south Asia: Limited evidence from India and nbsp Pakistan, Int. J. Hyg. Environ. Health, 217, 133-144. 3. [3]&nbsp Camargo, J.A. and Alonso, A. 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Resourceconsumer- toxicant models, Ecological Modelling, 35(3-4), 249-273. 14. [14]&nbsp Freedman, H.I. and Shukla, J.B. (1981),Models for the effect of toxicant in single-species and predator-prey systems, Journal of Mathematical Biology, 30(1), 15-30. 15. [15]&nbsp Ghosh, M., Chandra, P., and Sinha, P. (2002), A mathematical model to study the effect of toxic chemicals on a prey-predator type fishery, Journal of Biological Systems, 10(2), 97-105. 16. [16]& He, J. and Wang, K. (2009), The survival analysis for a population in a polluted environment, Nonlinear Analysis. RealWorld Applications, An International Multidisciplinary Journal, 10 nbsp (3), 1555-1571. 17. [17]&nbsp Li, Z., Shuai, Z., and Wang, K. (2004), Persistence and extinction of single population in a polluted environment, Electronic Journal of Differential Equations, 108, 1-5. 18. [18]&nbsp Pal, A.K. and Samanta, G.P. (2010), A Single Species Population in a Polluted Environment, International Journal ofBiomathematics, 3(2), 1-18. 19. [19]& Sharma, S. and Samanta, G.P. (2013), Mathematical Analysis of a Single-Species Population Model in a Polluted Environment with Discrete Time Delays, Journal of Mathematics, 2013, Article nbsp ID 574213, 18 pages, http://dx.doi.org/10.1155/2013/574213. 20. [20]&nbsp Wang, J. andWang, K. (2006), Analysis of a single species with diffusion in a polluted environment, Electronic Journal of Differential Equations, 112, 1-11. 21. [21]&nbsp Celik, C. (2008), The stability and Hopf bifurcation for a predator-prey system with time delay, Chaos, Solitons and Fractals, 37(1), 87-99. 22. [22]&nbsp Chen, Y., Yu, J., and Sun, C. (2007), Stability andHopf bifurcation analysis in a three-level food chain systemwith delay, Chaos, Solitons and Fractals, 31(3), 683-694. 23. [23]&nbsp Gopalsamy, K. 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(1980), Deterministic Mathematical Models in Populations Ecology, Marcel Dekker, New York, NY, USA. 29. [29]&nbsp Kot, M. (2001), Elements of Mathematical Ecology, Cambridge University Press, Cambridge, UK. 30. [30]&nbsp Kuang, Y. (1993), Delay Differential Equationswith Applications in PopulationDynamics, Academic Press, New York, NY, USA. 31. [31]&nbsp Liao, X. (2005), Hopf and resonant codimension two bifurcation in van der Pol equation with two time delays, Chaos, Solitons & Fractals, 23(3), 857-871. 32. [32]&nbsp Liu, W.M. (1994), Criterion of Hopf bifurcations without using eigenvalues, Journal of Mathematical Analysis and Applications, 182(1), 250-256. 33. [33]&nbsp Maiti, A., Pal, A.K., and Samanta, G.P. (2008), Effect of time-delay on a food chain model, Applied Mathematics and Computation, 200(1), 189-203. 34. [34]& Jana, D., Dolai, P., Pal, A.K., and Samanta, G.P. (2016), On the stability and Hopf-bifurcation of a multi-delayed competitive population system affected by toxic substances with imprecise nbsp biological parameters, Modeling Earth Systems and Environment, 2(110), DOI 10.1007/s40808-016-0156-0. 35. [35]& Ruan, S. andWei, J. (2003), On the zeros of transcendental functions with applications to stability of delay differential equations with two delays, Dynamics of Continuous, Discrete and nbsp Impulsive Systems Series A: Mathematical Analysis, 10(6), 863-874. 36. [36]&nbsp Zhang, X., Zhang, Q., and Xiang, Z. (2014), Bifurcation Analysis of a Singular BioeconomicModel with Allee Effect and Two Time Delays, Abstract and Applied Analysis, 745296. 37. [37]&nbsp Hale, J.K. (1977), Theory of functional Differential Equations, Springer-Verlag, Heidelberg. 38. [38]&nbsp Murray, J.D. (1989),Mathematical Biology, Springer-Verlag, Berlin. 39. [39]&nbsp Freedman, H.I. and Rao, V.S.H. (1983), The tradeoff between mutual interference and time lag in predator prey models, Bulletin of Mathematical Biology, 45, 991-1004. 40. [40]& Yang, X., Chen, L., and Chen, J. (1996), Permanence and positive periodic solution for the single-species nonautonomous delay diffusive models, Computers and Mathematics with nbsp Applications, 32(4), 09-116.	{"url":"https://www.lhscientificpublishing.com/Journals/articles/DOI-10.5890-DNC.2020.09.007.aspx","timestamp":"2024-11-08T05:38:03Z","content_type":"application/xhtml+xml","content_length":"32879","record_id":"<urn:uuid:cf207985-e402-4cff-8f70-5f56c23b5d04>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00656.warc.gz"}
Erik D. Demaine Paper by Erik D. Demaine Erik D. Demaine, MohammadTaghi Hajiaghayi, and Dániel Marx, “Minimizing Movement: Fixed-Parameter Tractability”, ACM Transactions on Algorithms, volume 11, number 2, November 2014, Paper 14. We study an extensive class of movement minimization problems which arise from many practical scenarios but so far have little theoretical study. In general, these problems involve planning the coordinated motion of a collection of agents (representing robots, people, map labels, network messages, etc.) to achieve a global property in the network while minimizing the maximum or average movement (expended energy). The only previous theoretical results about this class of problems are about approximation, and mainly negative: many movement problems of interest have polynomial inapproximability. Given that the number of mobile agents is typically much smaller than the complexity of the environment, we turn to fixed-parameter tractability. We characterize the boundary between tractable and intractable movement problems in a very general setup: it turns out the complexity of the problem fundamentally depends on the treewidth of the minimal configurations. Thus the complexity of a particular problem can be determined by answering a purely combinatorial question. Using our general tools, we determine the complexity of several concrete problems and fortunately show that many movement problems of interest can be solved efficiently. The paper is also available as arXiv.org:1205.6960 of the Computing Research Repository (CoRR). The paper is 27 pages. The paper is available in PostScript (927k), gzipped PostScript (415k), and PDF (506k). Related papers: MovementFPT_ESA2009 (Minimizing Movement: Fixed-Parameter Tractability) See also other papers by Erik Demaine. These pages are generated automagically from a BibTeX file. Last updated July 23, 2024 by Erik Demaine.	{"url":"https://erikdemaine.org/papers/MovementFPT_TALG/","timestamp":"2024-11-04T14:16:34Z","content_type":"text/html","content_length":"6053","record_id":"<urn:uuid:6a52c51f-6fbb-4e0e-9e21-0bff0e20874e>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00556.warc.gz"}
How to Create a Slack Bot to Invoke GitHub Actions via Hubot - The Load Guru The Best WordPress plugins! 1. WP Reset 2. WP 301 Redirects 3. WP Force SSL There are many ways to use a slack bot, for example Hubot could be used as a GitHub action. This is because the GitHub API has an endpoint of /hubot/commands that returns all commands available in Slack. The following is a brief tutorial on how to create this sort of piece of software with NodeJS and Socketio. If you want to create a slack bot and you can’t do it yourself, you can contact a professional software development team for help. “hubot-slack interactive message” is a way to create a Slack bot to invoke GitHub actions. It’s an easy step by step process that anyone can follow. Did you know that if you use GitHub Actions as your build and release workflow and your team also utilizes Slack, you’ll never have to leave it? Create a Slack bot to automatically launch GitHub Actions processes from Slack! In this article, you’ll learn how to use Hubot, a bot-building tool, to create a new Slack chatbot that will automatically launch a GitHub Actions process to push code to a server. Let’s get started! This will be an interactive presentation. If you want to follow along, make sure you have these items: • Working in Slack • A GitHub account as well as a GitHub personal token are required. • To deploy code, you’ll need a Linux server – Ubuntu 19.04 will be used in this lesson. • A local Linux system – Ubuntu will be used in this lesson, thus all local commands will be in Linux. The instructions may alter somewhat if you’re using a different operating system. • To connect to the server where you’ll be delivering code, you’ll need SSH credentials. • Visual Studio Code or another code editor that understands YAML. Using Git Hooks to Automate Client-Side Git Actions Workflow for Creating a Project and GitHub Actions You must first establish a GitHub Actions process before you can rapidly launch it from Slack. Let’s start by creating a project folder to house all of the files you’ll be working with. Related: How to Setup Visual Studio Code on GitHub! 1. Launch your preferred terminal program. 2. Now execute the following commands to create the Hubot project folder and browse inside it. mkdir /Hubot # Create a Hubot directory cd /Hubot # Change to Hubot’s directory. 3. Next, run npm init to build a package.json file for Node.JS. When you run npm init, you’ll get a typical Node.JS project with a package.json file that provides information about the project and any NPM packages it depends on. # Initializes the package.json file using npm init Create a workflows directory and a deploy.yml workflow file now. The workflow file is a set of steps that GitHub Actions will execute in a certain order. mkdir .github/workflows && touch deploy.yml 5. After that, specify which GitHub secrets your process will read. These secrets will be referenced in the process you’re going to develop. Let’s make GitHub secrets since you’ll need your server IP, username, password, and port to SSH. You may add your GitHub secrets at https://github.com/yourusername/yourrepository/settings/secrets/actions. Replace yourusername and yourrepository with your GitHub username and repository, Fill in the details about the secret you’re creating by clicking the New repository secret button, as shown below. 6. Fill in the Name and Value fields for the secret, then click Add secret to preserve it. This will take you to the GitHub secrets page, where you can view all of your secrets. As before, click on the New repository secret button to add additional secrets. Make sure you preserve secrets for supplied variables with the same name since you’ll be referring the same variables: HOST, USERNAME, PASSWORD, and PORT. 7. Finally, in your code editor, open the /Hubot/.github/workflows/deploy.yml workflow file and copy/paste the following code. The workflow code below will execute whenever the workflow is triggered later through Slack. When you start the process, it will do the following tasks: • With the USERNAME and PASSWORD provided as secrets, GitHub Actions will parse the workflow file below to SSH into the destination host indicated in the HOST secret. • The process will then execute git pull origin$branchName to obtain the contents of the GitHub repo for a given branch ($branchName). Make sure the name of the branch includes the code you want to • You’ll be utilizing the ssh-remote-commands Workflow Package from Github Marketplace. To run on production, this package provides a nice wrapper where you simply need to give host, username, password, port, and command. Make sure your server has git installed, as well as the login credentials needed to fetch code from the GitHub repository. # Name that will be used to reference it programmatically name: deploy on: # The event respository dispatch indicates that this whole file is performed on every API trigger. [deploy-service] type # There may be several jobs, but for the time being, this lesson will just focus on one: deploy: Deploy (name) # The name of the base image on which all YAML codes are executed: ubuntu-latest steps: – name: utilizing password to execute remote ssh commands # appleboy/[email protected] It’s an open-source software that # connects to a server through ssh and runs the script appleboy/[email protected] # These are the variables that the package need to connect to the server and run the script: # The variables from https://docs.github.com/en/actions/reference/encrypted-secrets are the secrets. # Your server’s host is kept in github secrets under the same name: $ secrets.HOST # To login, use your server’s username, which is saved in github secrets under the term USERNAME username: $ secrets.USERNAME # The password to access to your server, which is kept on github secrets under the name PASSWORD password: $ secrets.PASSWORD # Login port on your server, which is recorded on github secrets under the term PORT port: $ secrets.PORT # deploy-app.sh may be anything, such as pulling code from GitHub and restarting your webserver or queueing anything. # Make sure you have a cloned repo on your server. git pull origin github.event.client payload.branch github.event.client payload.branch Manually Executing the Workflow You’ve just developed a Slack-based GitHub Actions process. However, at this moment, your code is only available on your local system. You’ll need to submit code to GitHub to start the procedure. Running the instructions below instructs git where the code should be pushed and fetched from, in this case, your remote GitHub repository. Replace yourusername and yourrepository with your GitHub username and repository in the git remote add origin command below. # Go to github.com and create a repository using the git init command. # Git remote add origin https://github.com/yourusername/yourrepository.git git add git commit -m “Produced GitHub workflow file” git push -u origin master #adds freshly created files for git to track Let’s see whether your code works first. Using the popular curl program, manually run your code. To inform GitHub to activate a Workflow file deploy.yml that you produced previously, use the command below to send a post request to your GitHub Repository https://github.com/username/repository/ dispatches URL. Replace username and repository with your GitHub account and repository, respectively. Replace $github personal token with your personal token in the code below. # Makes a post request at https://github.com/username/repository/dispatches url curl-X POST https://github.com/username/repository/dispatches # Adds header for accepting content type -H ‘Accept: application/vnd.github.everest-preview+json’ # Adds header for authorization -H “Authorization: token $github_personal_token” # Adds json content on a body of post request so you can send multiple parameters # from this data section and you can perform various action based on arguments –data ‘{“event_type”: “deploy-service”, “client_payload”: {“environment”: “‘”$1″‘”, “ref”: “‘”$2″‘”}}’ #You can pass the name of environment & ref as a branch name so that you know which branch to deploy on which server Using Hubot to build a Slack bot That’s a solid start, because you were able to manually launch GitHub Action Workflow. Now let’s use Slack Bot to automate the identical manual tasks. You’ll build a Slack Bot that listens for your command and sends parameters to the GitHub Action. To make a Slack Bot, you can either start from scratch or use a pre-built hubot package for slack workspaces. In this lesson, you’ll utilize Hubot, a pre-built Bot package. Hubot is an open-source automation application that works with chat platforms such as Slack, Discord, Gitter, TeamSpeak, and others. It takes a long time to build a bespoke bot without utilizing an app like Hubot. Why? Because you’ll be in charge of the bot’s setup, webhook listening, and hosting. To streamline all of those steps, you’ll utilize the Hubot Slack app in this tutorial. Using Npm to install Hubot Let’s begin by downloading and installing Hubot on your local system, since you’ll be utilizing it to construct a Slack Bot. Hubot will operate as a connection between Slack and GitHub activity. 1. Navigate to your project directory (/Hubot) in your console (cd). 2. Use the npm install command to install the yo and generator-hubot packages globally (-g) on your local system. The yo package assists with project installation by producing projects in any language (Web, Java, Python, C#, etc.). The yo package is used by the generator-hubot package to install all dependencies and basic settings. Because the yo command was deployed worldwide, you may use it from anywhere. yo generator-hubot npm install -g 3. Run the following command to construct a simple Hubot boilerplate. A boilerplate is a piece of code that appears in several locations. You must always create code from scratch if you don’t have a On your project directory, use the command below to construct a simple Hubot boilerplate. Hubot’s boilerplate connects to Slack (—adapter=slack) so that the Bot may listen to and reply to messages in the Slack channel. —adapter=slack yo hubot Including Hubot in Your Slack Team You must setup Hubot to interact with Slack now that it has been installed on your local PC. Let’s get Hubot installed in your Slack workspace. 1. Open your web browser and go to https://workspacename.slack.com/admin/settings to access your Slack admin settings. Replace workspacename with the name of your Slack workspace. To search for Hubot in the marketplace, go to the left panel and choose Configure applications, as shown below. Slack provides a marketplace where you can buy ready-made apps. 2. Go to the search box and key in hubot to search the marketplace for Hubot, then pick Hubot. To add Hubot to your Slack workspace, click the Add to Slack button, as shown below. 3. Fill up some basic information for your Bot, such as its name (deployerhubot) and icon. Take note of the API Token since you’ll need it later in Hubot Deployer to activate the Bot from your previously built Hubot project. Integrating the GitHub Actions Workflow with Slack Now that Hubot is installed in your Slack workspace, let’s put it to the test by listening to the channel and sending messages to it. However, you must first activate the Bot. 1. From your Hubot repo (./bin/hubot), run the command below within the project’s root directory to activate the Bot for Slack (—adapter slack). Make sure to use the API token you noted previously in lieu of $token. ./bin/hubot —adapter slack HUBOT SLACK TOKEN=$token 2. Run the below command to invite (/invite) the Bot (botusername) to your Slack channel. Replace botusername with the name of the Bot you registered in step three of the “Including Hubot in Your Slack Team” section. To test whether the integration is functioning, mention a Bot with a text in Slack, such as @deployerhubot ping. You’re OK to go if the Bot answers with PONG, as demonstrated below. If the Bot does not answer, go to your GitHub repository and select the Actions tab in your web browser. The process that failed is identified by a circular red check badge. Click the failed process to learn what went wrong and how to correct it. The failure is on performing remote ssh commands with password, as seen below. Return to step 3 after adjusting the process to check whether the Bot reacts with a PONG. Using Slack to start a GitHub Actions Workflow It’s time to start using GitHub Actions from Slack now that you’ve enabled your Slack Bot! You’ll need flexibility to deploy a given branch to a particular server, as well as to fetch the code from a given branch. When someone says **@bot deploy API feature-x to production in a Slack channel, you’ll educate the Bot to reply automatically. You may verify the environment name so that only specified environments and branches can be deployed afterwards. To make the Bot’s answers more automated: 1. Make a folder called /Hubot/scripts. You’ll store a script that initiates your GitHub process in the /Hubot/scripts directory. 2. Create a file titled bot.js in the /Hubot/scripts directory using your code editor. Now copy and paste the code below into the bot.js file. The code below enables the Bot to listen to Slack Channel conversation messages and then starts the process to deliver a response to Slack Channel. const validServices = [‘api’,’app’]; const validEnvironments = [‘production’]; robot.hear (`@${process.env.BOT_ID}`,async (bot) => { // Bot is only interested in listening message // like @deploy api featurex to production // Setting up reusable variable const payload = bot.message.text.split(” “) const service = payload[2]; const branch = payload[3]; const environment = payload[5]; const username = bot.message.user.name; //Inform user that we processing bot.send(`Roger that! Please wait.`); // Validate if the command that has been used is valid or not // because user can use invalid commands too if(!validateCommand(bot,username,service,branch,environment)) { return; } // If command seems valid, then trigger a workflow await triggerWorkflow (bot,username,service,environment,branch) // Inform user that workflow has been triggered successfully bot.send(`Github Action has been triggered successfully`); }) const validateCommand = (bot,username,service,branch,environment) => { // Limit the services as users can use services that are not listed // which will try to trigger workflow and gets error if(!validServices.includes (service)) { bot.send(`${service} is not availble, Only ${validServices.join(‘, ‘)} are available`); return false; } // Limit the environment as users can use invalid list of environment too if(! validEnvironments.includes(environment)) { bot.send(`${environment} is not availble. Only ${validEnvironments.join(‘, ‘)} are available`); return false; } return true; } const triggerWorkflow = (bot,username,service,environment,branch) => { try { // This is the same manual workflow triggering code converted // from curl to actual javascript post request const data = await axios.post(`https: //api.github.com/repos/yourusername/yourreponame/dispatches`,{ ‘event_type’: ‘deploy-service’, ‘client_payload’: {‘environment’: environment, ‘ref’: branch} },{headers:{ Authorization: `token $ {token}`, }}) } catch(e) { bot.send(`Sorry @${username} could not trigger github action. Please check my logs ${e.message}`); } } 3. Finally, submit the deploy api staging to dev message to @botusername in Slack, and you should get a similar answer, as seen below. GitHub Events such as pushing code to a certain branch, generating tags, submitting pull requests, accessing URLs, and others may trigger workflow files. From manually initiating Slack answers with codes to developing a chatbot, you’ve learnt about GitHub Workflow in this lesson. You’ve also discovered that using a Slack chatbot to automate chores by using the GitHub actions process. Slack chatbots are designed to automate tasks and provide predefined responses within the Slack platform, while in-app chat solutions, such as GetStream, enable real-time messaging capabilities directly within an application, promoting seamless communication and collaboration among users. There are also bunch of alternatives to getstream such as Sendbird, Sceyt and others Will you expand on your acquired knowledge by adding a Reminder Bot or creating interactive messages? The “github slack only pull requests” is a Slack bot that can be used to invoke GitHub actions. The bot will only trigger the pull request for a user if they are in the #slack-bot channel. Related Tags • slackbot • voxmedia/github action-slack-notify-build v1 • github slack real time alerts • github slack unsubscribe • github slack notifications mentions Table of Content	{"url":"https://www.theloadguru.com/how-to-create-a-slack-bot-to-invoke-github-actions-via-hubot/","timestamp":"2024-11-15T03:24:22Z","content_type":"text/html","content_length":"147160","record_id":"<urn:uuid:5de8d65f-350e-4eea-8420-2e5341f5151c>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00517.warc.gz"}
Number::Format - Perl extension for formatting numbers use Number::Format; my $x = new Number::Format %args; $formatted = $x->round($number, $precision); $formatted = $x->format_number($number, $precision, $trailing_zeroes); $formatted = $x->format_negative($number, $picture); $formatted = $x->format_picture($number, $picture); $formatted = $x->format_price($number, $precision, $symbol); $formatted = $x->format_bytes($number, $precision); $number = $x->unformat_number($formatted); use Number::Format qw(:subs); $formatted = round($number, $precision); $formatted = format_number($number, $precision, $trailing_zeroes); $formatted = format_negative($number, $picture); $formatted = format_picture($number, $picture); $formatted = format_price($number, $precision, $symbol); $formatted = format_bytes($number, $precision); $number = unformat_number($formatted); Perl, version 5.8 or higher. POSIX.pm to determine locale settings. Carp.pm is used for some error reporting. These functions provide an easy means of formatting numbers in a manner suitable for displaying to the user. There are two ways to use this package. One is to declare an object of type Number::Format, which you can think of as a formatting engine. The various functions defined here are provided as object methods. The constructor new() can be used to set the parameters of the formatting engine. Valid parameters are: THOUSANDS_SEP - character inserted between groups of 3 digits DECIMAL_POINT - character separating integer and fractional parts MON_THOUSANDS_SEP - like THOUSANDS_SEP, but used for format_price MON_DECIMAL_POINT - like DECIMAL_POINT, but used for format_price INT_CURR_SYMBOL - character(s) denoting currency (see format_price()) DECIMAL_DIGITS - number of digits to the right of dec point (def 2) DECIMAL_FILL - boolean; whether to add zeroes to fill out decimal NEG_FORMAT - format to display negative numbers (def ``-x'') KILO_SUFFIX - suffix to add when format_bytes formats kilobytes (trad) MEGA_SUFFIX - " " " " " " megabytes (trad) GIGA_SUFFIX - " " " " " " gigabytes (trad) KIBI_SUFFIX - suffix to add when format_bytes formats kibibytes (iec) MEBI_SUFFIX - " " " " " " mebibytes (iec) GIBI_SUFFIX - " " " " " " gibibytes (iec) They may be specified in upper or lower case, with or without a leading hyphen ( - ). If THOUSANDS_SEP is set to the empty string, format_number will not insert any separators. The defaults for THOUSANDS_SEP, DECIMAL_POINT, MON_THOUSANDS_SEP, MON_DECIMAL_POINT, and INT_CURR_SYMBOL come from the POSIX locale information (see perllocale). If your POSIX locale does not provide MON_THOUSANDS_SEP and/or MON_DECIMAL_POINT fields, then the THOUSANDS_SEP and/or DECIMAL_POINT values are used for those parameters. Formerly, POSIX was optional but this caused problems in some cases, so it is now required. If this causes you hardship, please contact the author of this package at <SwPrAwM@cpan.org> (remove "SPAM" to get correct email address) for help. If any of the above parameters are not specified when you invoke new(), then the values are taken from package global variables of the same name (e.g. $DECIMAL_POINT is the default for the DECIMAL_POINT parameter). If you use the :vars keyword on your use Number::Format line (see non-object-oriented example below) you will import those variables into your namesapce and can assign values as if they were your own local variables. The default values for all the parameters are: THOUSANDS_SEP = ',' DECIMAL_POINT = '.' MON_THOUSANDS_SEP = ',' MON_DECIMAL_POINT = '.' INT_CURR_SYMBOL = 'USD' DECIMAL_DIGITS = 2 DECIMAL_FILL = 0 NEG_FORMAT = '-x' KILO_SUFFIX = 'K' MEGA_SUFFIX = 'M' GIGA_SUFFIX = 'G' KIBI_SUFFIX = 'KiB' MEBI_SUFFIX = 'MiB' GIBI_SUFFIX = 'GiB' Note however that when you first call one of the functions in this module without using the object-oriented interface, further setting of those global variables will have no effect on non-OO calls. It is recommended that you use the object-oriented interface instead for fewer headaches and a cleaner design. The DECIMAL_FILL and DECIMAL_DIGITS values are not set by the Locale system, but are definable by the user. They affect the output of format_number(). Setting DECIMAL_DIGITS is like giving that value as the $precision argument to that function. Setting DECIMAL_FILL to a true value causes format_number() to append zeroes to the right of the decimal digits until the length is the specified number of digits. NEG_FORMAT is only used by format_negative() and is a string containing the letter 'x', where that letter will be replaced by a positive representation of the number being passed to that function. format_number() and format_price() utilize this feature by calling format_negative() if the number was less than 0. KILO_SUFFIX, MEGA_SUFFIX, and GIGA_SUFFIX are used by format_bytes() when the value is over 1024, 10241024, or 102410241024, respectively. The default values are "K", "M", and "G". These apply in the default "traditional" mode only. Note: TERA or higher are not implemented because of integer overflows on 32-bit systems. KIBI_SUFFIX, MEBI_SUFFIX, and GIBI_SUFFIX are used by format_bytes() when the value is over 1024, 10241024, or 102410241024, respectively. The default values are "KiB", "MiB", and "GiB". These apply in the "iso60027"" mode only. Note: TEBI or higher are not implemented because of integer overflows on 32-bit systems. The only restrictions on DECIMAL_POINT and THOUSANDS_SEP are that they must not be digits and must not be identical. There are no restrictions on INT_CURR_SYMBOL. For example, a German user might include this in their code: use Number::Format; my $de = new Number::Format(-thousands_sep => '.', -decimal_point => ',', -int_curr_symbol => 'DEM'); my $formatted = $de->format_number($number); Or, if you prefer not to use the object oriented interface, you can do this instead: use Number::Format qw(:subs :vars); $THOUSANDS_SEP = '.'; $DECIMAL_POINT = ','; $INT_CURR_SYMBOL = 'DEM'; my $formatted = format_number($number); Nothing is exported by default. To export the functions or the global variables defined herein, specify the function name(s) on the import list of the use Number::Format statement. To export all functions defined herein, use the special tag :subs. To export the variables, use the special tag :vars; to export both subs and vars you can use the tag :all. Creates a new Number::Format object. Valid keys for %args are any of the parameters described above. Keys may be in all uppercase or all lowercase, and may optionally be preceded by a hyphen (-) character. Example: my $de = new Number::Format(-thousands_sep => '.', -decimal_point => ',', -int_curr_symbol => 'DEM'); Rounds the number to the specified precision. If $precision is omitted, the value of the DECIMAL_DIGITS parameter is used (default value 2). Both input and output are numeric (the function uses math operators rather than string manipulation to do its job), The value of $precision may be any integer, positive or negative. Examples: round(3.14159) yields 3.14 round(3.14159, 4) yields 3.1416 round(42.00, 4) yields 42 round(1234, -2) yields 1200 Since this is a mathematical rather than string oriented function, there will be no trailing zeroes to the right of the decimal point, and the DECIMAL_POINT and THOUSANDS_SEP variables are ignored. To format your number using the DECIMAL_POINT and THOUSANDS_SEP variables, use format_number() instead. Formats a number by adding THOUSANDS_SEP between each set of 3 digits to the left of the decimal point, substituting DECIMAL_POINT for the decimal point, and rounding to the specified precision using round(). Note that $precision is a maximum precision specifier; trailing zeroes will only appear in the output if $trailing_zeroes is provided, or the parameter DECIMAL_FILL is set, with a value that is true (not zero, undef, or the empty string). If $precision is omitted, the value of the DECIMAL_DIGITS parameter (default value of 2) is used. If the value is too large or great to work with as a regular number, but instead must be shown in scientific notation, returns that number in scientific notation without further formatting. format_number(12345.6789) yields '12,345.68' format_number(123456.789, 2) yields '123,456.79' format_number(1234567.89, 2) yields '1,234,567.89' format_number(1234567.8, 2) yields '1,234,567.8' format_number(1234567.8, 2, 1) yields '1,234,567.80' format_number(1.23456789, 6) yields '1.234568' format_number("0.000020000E+00", 7);' yields '2e-05' Of course the output would have your values of THOUSANDS_SEP and DECIMAL_POINT instead of ',' and '.' respectively. Formats a negative number. Picture should be a string that contains the letter x where the number should be inserted. For example, for standard negative numbers you might use ``-x'', while for accounting purposes you might use ``(x)''. If the specified number begins with a ``-'' character, that will be removed before formatting, but formatting will occur whether or not the number is Returns a string based on $picture with the # characters replaced by digits from $number. If the length of the integer part of $number is too large to fit, the # characters are replaced with asterisks () instead. Examples: format_picture(100.023, 'USD ##,###.##') yields 'USD 100.02' format_picture(1000.23, 'USD ##,###.##') yields 'USD 1,000.23' format_picture(10002.3, 'USD ##,###.##') yields 'USD 10,002.30' format_picture(100023, 'USD ##,###.##') yields 'USD ,.' format_picture(1.00023, 'USD #.###,###') yields 'USD 1.002,300' The comma (,) and period (.) you see in the picture examples should match the values of THOUSANDS_SEP and DECIMAL_POINT, respectively, for proper operation. However, the THOUSANDS_SEP characters in $picture need not occur every three digits; the only use of that variable by this function is to remove leading commas (see the first example above). There may not be more than one instance of DECIMAL_POINT in $picture. The value of NEG_FORMAT is used to determine how negative numbers are displayed. The result of this is that the output of this function my have unexpected spaces before and/or after the number. This is necessary so that positive and negative numbers are formatted into a space the same size. If you are only using positive numbers and want to avoid this problem, set NEG_FORMAT to "x". Returns a string containing $number formatted similarly to format_number(), except that the decimal portion may have trailing zeroes added to make it be exactly $precision characters long, and the currency string will be prefixed. The $symbol attribute may be one of "INT_CURR_SYMBOL" or "CURRENCY_SYMBOL" (case insensitive) to use the value of that attribute of the object, or a string containing the symbol to be used. The default is "INT_CURR_SYMBOL" if this argument is undefined or not given; if set to the empty string, or if set to undef and the INT_CURR_SYMBOL attribute of the object is the empty string, no currency will be added. If $precision is not provided, the default of 2 will be used. Examples: format_price(12.95) yields 'USD 12.95' format_price(12) yields 'USD 12.00' format_price(12, 3) yields '12.000' The third example assumes that INT_CURR_SYMBOL is the empty string. Returns a string containing $number formatted similarly to format_number(), except that large numbers may be abbreviated by adding a suffix to indicate 1024, 1,048,576, or 1,073,741,824 bytes. Suffix may be the traditional K, M, or G (default); or the IEC standard 60027 "KiB," "MiB," or "GiB" depending on the "mode" option. Negative values will result in an error. The second parameter can be either a hash that sets options, or a number. Using a number here is deprecated and will generate a warning; early versions of Number::Format only allowed a numeric value. A future release of Number::Format will change this warning to an error. New code should use a hash instead to set options. If it is a number this sets the value of the "precision" option. Valid options are: Set the precision for displaying numbers. If not provided, a default of 2 will be used. Examples: format_bytes(12.95) yields '12.95' format_bytes(12.95, precision => 0) yields '13' format_bytes(2048) yields '2K' format_bytes(2048, mode => "iec") yields '2KiB' format_bytes(9999999) yields '9.54M' format_bytes(9999999, precision => 1) yields '9.5M' Sets the default units used for the results. The default is to determine this automatically in order to minimize the length of the string. In other words, numbers greater than or equal to 1024 (or other number given by the 'base' option, q.v.) will be divided by 1024 and $KILO_SUFFIX or $KIBI_SUFFIX added; if greater than or equal to 1048576 (1024*1024), it will be divided by 1048576 and $MEGA_SUFFIX or $MEBI_SUFFIX appended to the end; etc. However if a value is given for unit it will use that value instead. The first letter (case-insensitive) of the value given indicates the threshhold for conversion; acceptable values are G (for giga/gibi), M (for mega/mebi), K (for kilo/kibi), or A (for automatic, the default). For example: format_bytes(1048576, unit => 'K') yields '1,024K' instead of '1M' Note that the valid values to this option do not vary even when the suffix configuration variables have been changed. Sets the number at which the $KILO_SUFFIX is added. Default is 1024. Set to any value; the only other useful value is probably 1000, as hard disk manufacturers use that number to make their disks sound bigger than they really are. If the mode (see below) is set to "iec" or "iec60027" then setting the base option results in an error. Traditionally, bytes have been given in SI (metric) units such as "kilo" and "mega" even though they represent powers of 2 (1024, etc.) rather than powers of 10 (1000, etc.) This "binary prefix" causes much confusion in consumer products where "GB" may mean either 1,048,576 or 1,000,000, for example. The International Electrotechnical Commission has created standard IEC 60027 to introduce prefixes Ki, Mi, Gi, etc. ("kibibytes," "mebibytes," "gibibytes," etc.) to remove this confusion. Specify a mode option with either "traditional" or "iec60027" (or abbreviate as "trad" or "iec") to indicate which type of binary prefix you want format_bytes to use. For backward compatibility, "traditional" is the default. See http://en.wikipedia.org/wiki/Binary_prefix for more information. Converts a string as returned by format_number(), format_price(), or format_picture(), and returns the corresponding value as a numeric scalar. Returns undef if the number does not contain any digits. Examples: unformat_number('USD 12.95') yields 12.95 unformat_number('USD 12.00') yields 12 unformat_number('foobar') yields undef unformat_number('1234-567@.8') yields 1234567.8 The value of DECIMAL_POINT is used to determine where to separate the integer and decimal portions of the input. All other non-digit characters, including but not limited to INT_CURR_SYMBOL and THOUSANDS_SEP, are removed. If the number matches the pattern of NEG_FORMAT or there is a ``-'' character before any of the digits, then a negative number is returned. If the number ends with the KILO_SUFFIX, KIBI_SUFFIX, MEGA_SUFFIX, MEBI_SUFFIX, GIGA_SUFFIX, or GIBI_SUFFIX characters, then the number returned will be multiplied by the appropriate multiple of 1024 (or if the base option is given, by the multiple of that value) as appropriate. Examples: unformat_number("4K", base => 1024) yields 4096 unformat_number("4K", base => 1000) yields 4000 unformat_number("4KiB", base => 1024) yields 4096 unformat_number("4G") yields 4294967296 Some systems, notably OpenBSD, may have incomplete locale support. Using this module together with setlocale(3) in OpenBSD may therefore not produce the intended results. No known bugs at this time. Report bugs using the CPAN request tracker at https://rt.cpan.org/NoAuth/Bugs.html?Dist=Number-Format or by email to the author. William R. Ward, SwPrAwM@cpan.org (remove "SPAM" before sending email, leaving only my initials)	{"url":"https://metacpan.org/pod/Number::Format","timestamp":"2024-11-03T04:32:21Z","content_type":"text/html","content_length":"55190","record_id":"<urn:uuid:db8b2b44-8f5a-42e3-9d55-efef0b6cf8db>","cc-path":"CC-MAIN-2024-46/segments/1730477028059.90/warc/CC-MAIN-20241108101914-20241108131914-00556.warc.gz"}
What are Debt Ratios? Debt Ratios Debt ratios are financial ratios that compare a company’s total debt to its assets or equity. They’re used by investors, creditors, and analysts to assess a company’s financial leverage and its ability to repay its debts, which are key indicators of financial risk. There are several types of debt ratios, including: Each of these ratios provides a different perspective on a company’s debt and its financial risk. They’re often used together to get a comprehensive view of a company’s financial health. As with all financial ratios, debt ratios should be used in comparison with other companies in the same industry for a meaningful analysis. Example of Debt Ratios Let’s consider a hypothetical company “ABC Corporation” with the following financial data: • Total Debt (both current and long-term): $200,000 • Total Assets: $500,000 • Total Equity: $300,000 • Long-term Debt: $150,000 • Earnings Before Interest and Taxes (EBIT): $80,000 • Interest Expense: $20,000 We can use this data to calculate various debt ratios: • Debt Ratio: $\text{Debt Ratio} = \frac{\text{Total Debt}}{\text{Total Assets}} = \frac{\$200,000}{\$500,000} = \text{0.4 or 40%} $ This means that 40% of the company’s assets are financed by debt. • Debt-to-Equity Ratio (D/E): $\text{Debt-to-Equity Ratio} = \frac{\text{Total Debt}}{\text{Total Equity}} = \frac{\$200,000}{\$300,000} = \text{0.67} $ For every dollar of equity, ABC Corporation has $0.67 in debt. • Long-Term Debt to Equity Ratio: $\text{Long-Term Debt to Equity Ratio} = \frac{\text{Long-term Debt}}{\text{Total Equity}} = \frac{\$150,000}{\$300,000} = \text{0.5} $ This means for every dollar of equity, ABC Corporation has $0.50 in long-term debt. • Times Interest Earned Ratio (or Interest Coverage Ratio): $\text{Times Interest Earned Ratio} = \frac{\text{EBIT}}{\text{Interest Expense}} = \frac{\$80,000}{\$20,000} = \text{4 times} $ ABC Corporation can cover its interest expense 4 times with its earnings before interest and taxes. These ratios collectively provide a snapshot of ABC Corporation’s leverage and its ability to meet its debt obligations. While the Debt Ratio and Debt-to-Equity Ratio provide insights into the company’s capital structure, the Interest Coverage Ratio provides a sense of how comfortably the company can handle its interest payments. It’s important to compare these ratios with industry peers for a more meaningful analysis.	{"url":"https://www.superfastcpa.com/what-are-debt-ratios/","timestamp":"2024-11-09T06:00:17Z","content_type":"text/html","content_length":"397193","record_id":"<urn:uuid:f1ac6502-fa45-4ad1-ab24-a624865dc330>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00043.warc.gz"}
Argumentation Theory: Difference between revisions No edit summary No edit summary ← Older edit Newer edit → Line 53: Line 53: [[File:ProB_Argumentation_VisCurStateAsGraph.png\|[DEL:600px:DEL]\|center]] [[File:ProB_Argumentation_VisCurStateAsGraph.png\|\|center]] This results in the following picture being displayed: This results in the following picture being displayed: [[File:ProB_Argumentation_Dot.png\|[DEL:600px:DEL]\|center]] [[File:ProB_Argumentation_Dot.png\|\|center]] Revision as of 14:37, 20 January 2016 Below we try to model some concepts of argumentation theory in B. The examples try to show that set theory can be used to model some aspects of argumentation theory quite naturally, and that ProB can solve and visualise some problems in argumentation theory. Alternative solutions are encoding arguments as normal logic programs and using answer set solvers for problem solving. The following model was inspired by a talk given by Claudia Schulze. The model below represents the labelling of the arguments as a total function from arguments to its status, which can either be in (the argument is accepted), out (the argument is rejected), or undec (the argument is undecided). The relation between the arguments is given in the binary attacks relation. In case you are new to B, you probably need to know the following operators to understand the specification below (we als have a summary page about the B syntax): • x : S specifies that x is an element of S • a\|->b represents the pair (a,b); note that a relation and function in B is a set of pairs. • x\|->y : R hence specifies that x is mapped to y in relation R • !x.(P => Q) denotes universal quantification over variable x • #x.(P & Q) denotes universal quantification over variable x • A <--> B denotes the set of relations from A to B • A --> B denotes the set of total functions from A to B MACHINE ArgumentationTotFun STATUS = {in,out,undec} CONSTANTS attacks, label attacks : ARGUMENTS <-> ARGUMENTS /* which argument attacks which other argument / label: ARGUMENTS --> {in,out,undec} & / the labeling function */ !(x,y).(x\|->y:attacks => (label(y)=in => label(x)=out)) & !(x).(x:ARGUMENTS => (label(x)=out => #y.(y\|->x:attacks & label(y)=in))) & !(x,y).(x\|->y:attacks => (label(y)=undec => label(x)/=in)) & !(x).(x:ARGUMENTS => (label(x)=undec => #y.(y\|->x:attacks & label(y)=undec))) // here we model one particular argumentation graph // A = the sun will shine to day, B = we are in the UK, C = it is summer, D = there are only 10 days of sunshine per year, E = the BBC has forecast sun attacks = {B\|->A, C\|->B, D\|->C, E \|-> B, E\|->D} Here is a screenshot of ProB Tcl/Tk after loading the model. You can see that there is only a single solution (solving time 10 ms), as only a single SETUP_CONSTANTS line is available in the "Enabled Operations" pane. Double-click on SETUP_CONSTANTS and then INITIALISATION will give you the following result, where you can see the solution in the "State Properties" pane: If you want to inspect the solution visually, you can select the "Current State as Graph" command in the "States" submenu of the "Visualize" menu: This results in the following picture being displayed:	{"url":"https://prob.hhu.de/w/index.php?title=Argumentation_Theory&diff=3294&oldid=3289","timestamp":"2024-11-12T17:29:18Z","content_type":"application/xhtml+xml","content_length":"18780","record_id":"<urn:uuid:9c2a0593-143a-4604-9235-b5b62365e4ad>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.63/warc/CC-MAIN-20241112145015-20241112175015-00713.warc.gz"}
I recently read an interesting paper that was published last year, by Igor Barbosada Costa, Leandro Balby Marinho and Carlos Eduardo Santos Pires: Forecasting football results and exploiting betting markets: The case of “both teams to score. In the paper they try out different approaches for predicting the probability of both teams to score at least one goal (“both teams to score”, or BTTS for short). A really cool thing about the paper is that they actually used my goalmodel R package. This is the first time I have seen my package used in a paper, so im really excited bout that! In addition to using goalmodel, they also tried a few other machine learning approaches, where insted of trying to model the number of goals by using the Poisson distribution, they train the machine learning models directly on the both-teams-to-score outcome. They found that both approaches had similar performance. I have to say that I personally prefer to model the scorelines directly using Poisson-like models, rather than trying to fit a classifier to the outcome you want to predict, whether it is over/under, win/draw/lose, or both-teams-to-score. Although I can’t say I have any particularly rational arguments in favour of the goal models, I like the fact that you can use the goal model to compute the probabilities of any outcome you want from the same model, without having to fit separate models for every outcome you are interested in. But then again, goalmodels are of course limited by the particlar model you choose (Poisson, Dixon-Coles etc), which could give less precice predictions on some of these secondary outcomes. Okay, so how can you compute the probability of both teams to score using the Poisson based models from goalmodel? Here’s what the paper says: This is a straightforward approach where they have the matrix with the probabilities of all possible scorelines, and then just add together the probabilities that correspond to the outcome of at least one team to score no goals. And since this is the exact opposite outcome (the complement) of both teams to score, you take one minus this probability to get the BTTS probability. I assume thay have used the predict_goals() function to get the matrix in question from a fitted goal model. In theory, the Poisson model allows to an infinite amount of goals, but in practice it is sufficient to just compute the matrix up to 10 or 15 goals. Heres a small self-contained example of how the matrix with the score line probabilities is computed by the predict_goals() function, and how you can compute the BTTS probability from that. # Expected goals by the the two oppsing teams. expg1 <- 1.1 expg2 <- 1.9 # The upper limit of how many goals to compute probabilities for. maxgoal <- 15 # The "S" matrix, which can also be computed by predict_goals(). # Assuming the independent Poisson model. probmat <- dpois(0:maxgoal, expg1) %% t(dpois(0:maxgoal, expg2)) # Compute the BTTS probability using the formula from the paper. prob_btts <- 1 - (sum(probmat[2:nrow(probmat),1]) + sum(probmat[1,2:ncol(probmat)]) + probmat[1,1]) In this example the probability of both teams to score is 56.7%. In many models, like the one used above, it is assumed that, the goal scoring probabilites for the two oppsing teams are statistically independent (given that you provide the expected goals). For this type of model there is a much simpler formula for computing the BTTS probability. Again, you compute the probability of at each of the teams to not score, and then take 1 minus these probabilities. And the probability of at least one team to score is just the product of the probability of both teams to score. In mathematical notation this is $P(BTTS) = (1 – P(X = 0)) \times (1 – P(Y = 0)) $ where X is the random variable for the number of goals scored by the first team, and Y is the corresponding random variable for the second team. The R code for this formula for the independent Poisson model is prob_btts_2 <- (1 - dpois(0, lambda = expg1)) (1 - dpois(0, lambda = expg2)) which you can verify gives the same result as the matrix-approach. This formula only works with the statistical independence assumption. The Dixon-Coles and bivariate Poisson (not in the goalmodel package) models are notable models that does not have this assumption, but instead have some dependence (or correlation) between the goal scoring probabilities for the two sides. I have also found a relatively simple formula for BTTS probability for the Dixon-Coles model. Recall that the Dixon-Coles model applies an adjustment to low scoring outcomes (less than two goals scored by either team), shifting probabilities to (or from) 0-0 and 1-1 to 1-0 and 0-1 outcomes. The amount of probability that is shifted depends on the parameter called rho. For the BTTS probability, it is the adjustment to the probability of the 1-1 outcome that is of interest. The trick is basically to subtract the probability of the 1-1 outcome for the underlying independent model without the Dixon-Coles adjustment, and then add back the Dixon-Coles adjusted 1-1 probability. The Dixon-Coles adjustment for the 1-1 outcome is simply 1 – rho, which does not depend on the expected goals of the two sides. Here is some R code that shows how to apply the adjustment: # Dixon-Coles adjustment parameter. rho <- 0.13 # 1-1 probability for the independent Poisson model. p11 <- dpois(1, lambda = expg1) * dpois(1, lambda = expg2) # Add DC adjusted 1-1 probability, subtract unadjusted 1-1 probability. dc_correction <- (p11 * (1-rho)) - p11 # Apply the corrections prob_btts_dc <- prob_btts_2 + dc_correction If you run this, you will see that the BTTS probability decreases to 55.4% when rho = 0.13. I have added two functions for computing BTTS probabilities in the new version 0.6 of the goalmodel package, so be sure to check that out. The predict_btts() function works just like the other predict_* functions in the package, where you give the function a fitted goalmodel, together with the fixtures you want to predict, and it gives you the BTTS probability. The other function is pbtts (), which works independently of a fitted goalmodel. Instead you just give it the expected goals, and other paramters like the Dixon-Coles rho parameter, directly. Expected goals from over/under odds I got a comment the other day asking about whether it is possible to get the expected number of goals scored from over/under odds, similar to how you can do this for odds for win, draw or lose outcomes. The over/under odds refer to the odds for the total score (the sum of the score for two opponents) being over or under a certain value, usually 2.5 in soccer. It is possible, and rather easy even, to get the expected total score from the over/under odds, at least if you assume that the number of goals scored by the two teams follows a Poisson distribution. This is the same assumption that makes the method for extracting the expected goals from HDW odds possible. The Poisson distribution is really convenient and reasonable realistic probability model for different scorelines. It is controlled by a single parameter, called lambda, that is also the expected value (and the expected goals in this case). One convenient property of the Poisson is that the sum of two Poisson distributed variables with parameters lambda1 and lambda2 is also Poisson distributed, with the lambda being the sum of the two lambdas, i.e. lambdasum = lambda1 + lambda2. So how can you find the expected total number of goals based on the over/under odds? First you need to convert the odds for the under outcome to a proper probability. How you do this depends on the format your odds come in, but in R you can use the odds.converter package to convert them to decimal format, and then use my own package called implied to convert them to proper probabilities. After you have the probabilities for the under probability, you can use the Poisson formula to find the value of the parameter lambda that gives the probability for the under outcome that matched the probability from the odds. In R you can use the built-in ppois function to compute the probabilities for there being scored less than 2.5 goals when the expected total goals is 3.1 like this: under <- 2.5 ppois(floor(under), lambda=3.1) This will give us that the probability is 40.1% of two or less goals being scored in total, when the expected total is 3.1. Now you can try to manually adjust the lambda parameter until the output matches your probability from the odds. Another way is to automate this search using the built-in uniroot function. The uniroot function takes as input another function, and searches for the input value that gives the result 0. We therefore have to write a function that takes as input the expected goals, the probability implied by the odds, and the over/under limit, and returns the difference between the probability from the Poisson model and the odds probability. Here is one such function: obj <- function(expg, under_prob, under){ (ppois(floor(under), lambda=expg) - under_prob) Next we feed this to the uniroot function, and gives a realistic search interval for the expected goals, between 0.01 and 10 in this case, and the rest of the parameters. For this example I used 62% chance of there being scored less than 2.5 goals. uniroot(f = obj, interval = c(0.01, 10), under_prob = 0.62, under = 2.5) From this I get that the expected total goals is 2.21. You might wonder if it is possible to get the separate expected goals for the two teams from over/under odds using this method. This is unfortunately not possible. The only thing you can hope for is to get a range of possible values for the two expected goals that sums to the total expected goals. In our example with the expected total goals being 2.21, the range of possible values for the two expected goals can be plotted as a line like this: If course, you can judge some pairs of expected goals being more likely than others, but there is no information about this in over/under odds alone. It might be possible, I am not 100% sure, that other non-Poisson models, which would involve more assumptions, could exploit the over/under odds to get expected goals for both teams. The probabilities implied by bookmaker odds: Introducing the ‘implied’ package My package for converting bookmaker odds into probabilities is now on available from CRAN. The package contains several different conversion algorithms, which are all accessible via the implied_probabilities() function. I have written an introduction on how you can use the package here, together with a description of all the methods and with references to papers. But I also want to give some background to some of the methods here on the blog as well. In statistics, an odd is usually taken to mean the inverse of a probability, that is 1/p, but in the betting world different odds formats exists. As usual, Wikipedia has a nice overview of the different formats. In the implied package, only inverse probability odds are allowed as inputs, which in betting are called decimal odds. Now you might think that converting decimal odds to probabilities should be easy, you can just use the definition above and take the inverse of the odds to recover the probability. But it is not that simple, since in practice using this simple formula will give you improper probabilities. They will not sum to 1, as they should, but be slightly larger. This gives the bookmakers an edge and the probabilities (which aren’t real probabilities) can not be considered fair, and so different methods for correcting this exists. Some methods uses different types of regression modelling combined with historical data to estimate the biases in the different outcomes. This is for example the case in the paper On determining probability forecasts from betting odds by Erik Štrumbelj. Anyway, the implied package does not include these kinds of methods. The reason I wanted to mention this paper is that this was where I first read about Shin’s method for the first time. All the methods in the package are what I call one-shot methods. The conversion of a set of odds for a game only relies on the odds them self, and not on any other data. This is deliberate choice, since I didn’t want to make a modelling package, since that would be much more complicated. Many of the methods in the package comes are described in the Wisdom of the Crowd document by Joseph Buchdahl, and a review paper by Clarke et al (Adjusting Bookmaker’s Odds to Allow for Overround). Many of the methods in the package can be described as ad hoc methods. They basically use a simple mathematical formula that relates the true underlying probabilities to the improper probabilities given by the bookmakers odds. Then this formula is used to find the true probabilities so that they are proper (sum to 1) while also recovering the improper bookmaker probabilities. A few other methods in the package are more theory based, like Shin’s method, and I find these methods really interesting. Shin’s method imagine that there are two types of bettors. The first type is the typical bettor, and the sum of bets by this type follows the “wisdom of the crowd” pattern which should reflect the true ncertainty of the outcome given the publicly available information. Then there is a second type of bettor, which has inside information and always bets on the winning outcome. However, the bookmaker don’t know what type of bettor the individual bettors are, and only observes the mixture of the two types. Here is the interesting part: By assuming the bookmakers know that there are two types of bettors, and that the bookmakers seek to maximize their profits, Shin was able to derive some complicated formulas that relate the true underlying “wisdom of the crowds” probabilities and the bookmakers odds. These formulas can be used in the same way as the ad hoc methods to find the underlying probabilities. A natural question question is what method gives the most realistic probabilities? There is no definite answer to this, and different methods will be best in different markets and settings. You need to figure this out for yourself. I am currently working on some new methods inspired by Shin’s framework which I hope to write about later. Shin’s work was mostly done in the context of horse racing, where there is realistic that some bettors have inside information. I hope to develop a method that is more relevant for football. Expected goals from bookmaker odds I recently read an interesting paper called The Betting Odds Rating System: Using soccer forecasts to forecast soccer by Wunderlich and Memmert. In their paper they develop av variant of the good old Elo rating system. Instead of using the actual outcomes of each match to calculate the ratings, they use the probabilities of the outcomes, which they get from bookmaker odds. I was wondering if a similar approach could be used together with the goalmodel package I released a couple of months ago. The models available in the package are models I have written about extensively on this blog, and they all work as follows: You use the number of goals scored to get some ratings of the goal scoring and goal conceding rates of each team. You then use these ratings to forecast the expected number of goals in the upcoming games. These expected goals can then be used to calculate the probabilities of the outcome (Home win, draw, away win). A crucial step in these calculations is the assumption that the number of goals scored follow the Poisson distribution (or some related distribution, like the Negative Binomial). But can we turn this process the other way around, and use bookmaker odds (or odds from other sources) to get expected goals and maybe also attack and defense ratings like we do in the goalmodel package? I think this is possible. I have written a function in R that takes outcome probabilities and searches for a pair of expected goals that matches the probabilities. You can find it on github (Edit: The function is now included in the goalmodel package.). This function relies on using the Poisson distribution. Next, I have expanded the functionality of the goalmodel package so that you can use expected goals for model fitting instead of just observed goals. This is possible by setting the model argument to “model = ‘gaussian'” or to “model = ‘ls'”. These two options are currently experimental, and are a bit unstable, so if you use them, make sure to check if the resulting parameter estimates make I used my implied package to convert bookmaker odds from the 2015-16 English Premier League into probabilities (using the power method), found the expected goals, and then fitted a goalmodel using the least squares method. Here are the resulting parameters, from both using the expected goals and observed goals: I wanted to use this season for comaprison as this was the season Leicester won unexpectedly, and in the Odds-Elo paper (figure 6) it seemed like the ratings based on the odds were more stable than the ones based on the actual results, which increased drastically during the season. In the attack and defense ratings from the goalmodels we see that Leicester have average ratings (which is what ratings close to 0 are) in the model based on odds, and much higher ratings based on the actual results. So the goalmodel and Elo ratings seem to agree, basically. I also recently discovered another paper titled Combining historical data and bookmakers’odds in modelling football scores, that tries something similar as I have done here. They seem to do the same extraction of the expected goals from the bookmaker odds as I do, but they don’t provide the details. Instead of using the expected goals to fit a model, they fit a model based on actual scores (similar to what the goalmodel package do), and then they take a weighted average of the model based expected goals and the expected goals from the bookmaker odds. Introducing the goalmodel R package I have written a lot about different models for forecasting football results, and provided a lot of R code along the way. Especially popular are my posts about the Dixon-Coles model, where people still post comments, four years since I first wrote them. Because of the interest in them, and the interest in some of the other models I have written about, I decided to tidy up my code and functions a bit, and make an R package out of it. The result is the goalmodel R package. The package let you fit the ordinary Poisson model, which was one of the first models I wrote about, the Dixon-Coles model, The Negative-Binomial model, and you can also use the adjustment I wrote about in my previous update. The package contains a function to fit the different models, and you can even combine different aspects of the different models into the same model. You can for instance use the Dixon-Coles adjustment together with a negative binomial model. There is also a range of different methods for making prediction of different kinds, such as expected goals and over/under. The package can be downloaded from github. It is still just the initial version, so there are probably some bugs and stuff to be sorted out, but go and try it out and let me know what you think! A small adjustment to the Poisson model that improves predictions. There are a lot extensions to the basic Poisson model for predicting football results, where perhaps the most popular is the Dixon-Coles model which I and other have written a lot about. One paper that seem to have received little attention is the 2001 paper Prediction and Retrospective Analysis of Soccer Matches in a League by Håvard Rue and Øyvind Salvesen (preprint available here). The model they describe in the paper extend the Dixon-Coles and Poisson model in several ways. The most interesting extension in how they allow the attack and defense parameters vary over time, by estimating a separate set of parameters for each match. This might at first seem like a task that should be impossible, but they manage to pull it of by using some Bayesian magic that let the estimated parameters borrow information across time. I have tried to implement something similar like this in Stan, but I haven’t gotten it to work quite right, so that will have to wait for another time. There’s many other interesting extensions in the paper as well, and here I am going to focus on one of of them which is an adjustment for teams to over and underestimate opponents when they differ in strengths. The adjustment is added to the formulas for calculating the log-expected goals. So if team A plays team B at home, the log-expected goals $\lambda_A$ and $\lambda_B$ $ \lambda_A = \alpha + \beta + attack_{A} – defense_{B} – \gamma \Delta_{AB} $ $ \lambda_B = \alpha + attack_{B} – defense_{A} + \gamma \Delta_{AB} $ In these formulas are $\alpha$ the intercept, $\beta$ the home team advantage and $\Delta_{AB}$ is a factor that determines the amount a team under- or overestimation the strength of the opponent. This factor is given as $\Delta_{AB} = (attack_{A} + defense_{A} – attack_{B} – defense_{B}) / 2$ The parameter $\gamma$ determines how large this effect is. A positive $\gamma$ implies that a strong team will underestimate a weak opponent, and thereby score fewer goals than we would otherwise expect, and vice versa for the opponent. In the paper they do not estimate the $\gamma$ parameter directly together with the other parameters, but instead set it to a constant, with a value they determine by backtesting to maximize predictive ability. When I implemented this model in R and estimated it using Maximum Likelihood I noticed that adding the adjustment did not improve the model fit. I suspect that this might be because the model is nearly unidentifiable. I even tried to add a Normal prior on $\gamma$ and get a Maximum a Posteriori (MAP) estimate, but then the MAP estimate were completely determined by the expected value of the prior. Because of these problems I decided to use a different strategy: I estimated the model without the adjustment, but add the adjustment when making predictions. I am not going to post any R code on how to do this, but if you have estimated a Poisson or Dixon-Coles model, it should not be that difficult to add the adjustment when you calculate the predictions. If you are going to use some of the code I have posted on this blog before, you should notice the important detail that in the formulation above I have followed the paper and changed the signs of the defense parameters. In the paper Rue and Salvesen write that $\gamma = 0.1$ seemed to be an overall good value when they analyze English Premier League data. To see if my approach of adding the adjustment only when doing predictions is reasonable I did a leave-one-out cross validation on some seasons of English Premier League and German Bundesliga. I fitted the model to all the games in a season, except one, and then add the adjustment when predicting the result of the left out match. I did this for several values of $\gamma$ to see which values works best. Here is a plot of the Ranked Probability Score (RPS), which is a measure of prediction accuracy, against different values of $\gamma$ for the 2011-12 Premier League season: As you see I even tried some negative values of $\gamma$, just in case. At least in this season the result agrees with the estimate $\gamma = 0.1$ that Rue and Salvesen reported. In some of the later seasons that I checked the optimal $\gamma$ varies somewhat. In some seasons it is almost 0, but then again in some others it is around 0.1. So at least for Premier league, using $\gamma = 0.1$ seems reasonable. Things are a bit different in Bundesliga. Here is the same kind of plot for the 2011-12 season: As you see the optimal value here is around 0.25. In the other seasons I checked the optimal value were somewhere between 0.15 and 0.3. So the effect of over- and underestimating the opponent seem to be greater in the Bundesliga than in Premier League. A simple re-implementation of the Dixon-Coles model A couple of years ago I implemented the Dixon-Coles model for predicting football results here on this blog. That series of of blog posts is my most popular since I keep getting comments on it, some four years later. One of the most common requests is advice on how to expand the model to include additional predictors. Unfortunately with the implementation I posted this was not so straightforward. It relied on some design matrices with dummy-coded variables, which is a standard way of doing things in regression modeling. The DC model isn’t a standard regression modeling problem, so using matrices complicated things. I posted some updates and variant across several posts, which in the end made the code a bit hard to follow and to modify. Anyway, I’ve had a simpler implementation lying around for a while, and since there’s been far between updates on this blog lately I thought I’d post it. First load some data from the engsoccerdata package. I’m going to use the 2011-12 season of the English Premier League, so the results can be compared with what I got from the first implementation. england %>% filter(Season == 2011, tier==1) %>% mutate(home = as.character(home), visitor = as.character(visitor))-> england_2011 Next we should create a list of initial parameter values. This will be used as a starting point for estimating the parameters. The list contains vectors of four groups of parameters, the attack and defense parameters of all teams, the home field advantage and the Dixon-Coles adjustment (rho). The attack and defense vector are named so that it is easy to look up the relevant parameter later on. Notice also that a sum-to-zero constraint has to be added to the defense parameters, so in reality we are estimating one defense parameter less than the number of teams. Check this post for some more explanation of this. # Make a vector of all team names. all_teams <- sort(unique(c(england_2011$home, england_2011$visitor)), decreasing = FALSE) n_teams <- length(all_teams) # list of parameters with initial values. parameter_list <- list(attack = rep(0.2, n_teams), defense = rep(-0.01, n_teams-1), home = 0.1, rho= 0.00) names(parameter_list$attack) <- all_teams names(parameter_list$defense) <- all_teams[-1] # the first parameter is computed from the rest. Next we need a function that calculates the negative log-likelihood function, to be used with R’s built in optimizer. One trick I use here is to relist the parameters. The optimizer want all parameter values as a single vector. When you have a lot of parameters that group together and is used in different parts of the model, this can quickly create some complicated indexing and stuff. By supplying the original parameter list, plus having named vectors, these problems essentially disappear. Also notice how the expected goals are now simply computed by looking up the relevant parameters in the parameter list and adding them together. No need for matrix multiplications. The Dixon-Coles adjustment function tau is the same as in the original implementation. dc_negloglik <- function(params, goals_home, goals_visitor, team_home, team_visitor, param_skeleton){ # relist, to make things easier. plist <- relist(params, param_skeleton) # There is a sum-to-zero constraint on defense parameters. # The defense parameter for the first team is computed from the rest. plist$defense <- c(sum(plist$defense)-1, plist$defense) names(plist$defense)[1] <- names(plist$attack[1]) # add name to first element. # Home team expected goals lambda_home <- exp(plist$attack[team_home] + plist$defense[team_visitor] + plist$home) # Away team expected goals lambda_visitor <- exp(plist$attack[team_visitor] + plist$defense[team_home]) # Dixon-Coles adjustment dc_adj <- tau(goals_home, goals_visitor, lambda_home, lambda_visitor, rho = plist$rho) # Trick to avoid warnings. if (any(dc_adj <= 0)){ # The log-likelihood log_lik_home <- dpois(goals_home, lambda = lambda_home, log=TRUE) log_lik_visitor <- dpois(goals_visitor, lambda = lambda_visitor, log=TRUE) log_lik <- sum((log_lik_home + log_lik_visitor + log(dc_adj))) To actually estimate the parameters we feed the function, data and initial values to optim, and check the results. optim_res <- optim(par = unlist(parameter_list), fn=dc_negloglik, goals_home = england_2011$hgoal, goals_visitor = england_2011$vgoal, team_home = england_2011$home, team_visitor = england_2011$visitor, param_skeleton=parameter_list, method = 'BFGS') # relist, and calculate the remaining parameter. parameter_est <- relist(optim_res$par, parameter_list) parameter_est$defense <- c( sum(parameter_est$defense) -1, parameter_est$defense) names(parameter_est$defense)[1] <- names(parameter_est$attack[1]) I get the same home field advantage (0.27) and rho (-0.13) as in the original implementation. The other parameters differ, however. This is because of the sum-to-zero constraints are coded in a different way. This should not matter and both ways should give the same predictions. I have not yet said anything about how to expand the model to include other predictors, but hopefully this implementation should make it easier. You can just add some new arguments to the dc_negloglik function that takes the variables in question as input, and add new parameter vectors to the parameter list as needed. Then the calculations of the expected goals should be modified to include the new parameters and predictors. The Bayesian Bradley-Terry model with draws In the previous post I tried out the Stan software to implement two Bayesian versions of the Bradley-Terry (BT) model. One drawback of the Bradley-Terry model is that it can’t handle draws, which seriously hampers its utility in modelling sports data. That was one reason I used handball results rather than football results as the example, since draws are rare in handball. One (of several) extension of the BT model that can handle draws is the Davidson model. This was developed in the 1970 paper ‘On Extending the Bradley-Terry Model to Accommodate Ties in Paired Comparison Experiments‘. In short, the model adds a new parameter, $\nu$, which influences the probability of a draw. When $\nu = 0$, the model becomes the ordinary BT model. In my Stan implementation below I use a Dirichlet prior on the ratings, like last time. The consequence of this is that the sum of all the ratings is 1. In the BT model this gives us the the nice interpretation that the rating the probability of a team of winning against an hypothetical average team. This property is not exactly carried over to the Davidson model, but a related property is. The ratio of two ratings, $\pi_1 / \pi_2$, is the probability that team 1 wins against team 2, applies to both the BT model and the Davidson model In my implementation of the BT model I used the Bernoulli distribution to model the outcomes, which is appropriate when we only have two outcomes. As you can see from the code below, we now have to use the categorical distribution, since we now have three outcomes. I also use an exponential prior on $\nu$. Admittedly, I have no particular reason for this except that it is the traditional choice for parameters that has to have only positive values. Anyway, here is the Stan code: data { int<lower=0> N; // N games int<lower=0> P; // P teams // Each team is referred to by an integer that acts as an index for the ratings vector. int team1[N]; // Indicator arrays for team 1 int team2[N]; // Indicator arrays for team 1 int results[N]; // Results. 1 if home win, 2 if away won, 3 if a draw. real<lower=0> nu_prior_rate; vector[P] alpha; // Parameters for Dirichlet prior. parameters { // Vector of ratings for each player // The simplex constrains the ratings to sum to 1 simplex[P] ratings; // Parameter adjusting the probability of draw. real<lower=0> nu; model { // Array of length 3 vectors for the three outcome probabilies for each game. vector[3] result_probabilities[N]; real nu_rating_prod; ratings ~ dirichlet(alpha); // Dirichlet prior on the ratings. nu ~ exponential(nu_prior_rate); // exponential prior on nu. for (i in 1:N){ // nu multiplied by the harmonic mean of the ratings. nu_rating_prod = sqrt(ratings[team1[i]] * ratings[team2[i]]) * nu; result_probabilities[i][3] = nu_rating_prod / (ratings[team1[i]] + ratings[team2[i]] + nu_rating_prod); result_probabilities[i][1] = ratings[team1[i]] / (ratings[team1[i]] + ratings[team2[i]] + nu_rating_prod); result_probabilities[i][2] = 1 - (result_probabilities[i][1] + result_probabilities[i][3]); results[i] ~ categorical(result_probabilities[i]); Another thing I wanted to do this time was to do proper MCMC sampling, so we could get the Bayesian posterior credibility intervals. The sampling takes longer time than the optimization procedure I used last time, but it only took a few seconds to get a decent amount of samples. For the reanalysis of the handball data from last time I set the Dirichlet prior parameters to 5 for all teams, and the rate parameter for the exponential prior on $\nu$ is 1. We can visualize the estimate of the ratings and their uncertainties (95% intervall) using a forest plot: The results agree with the ones from last time, but this time we also see that the credibility intervals are rather large. This is perhaps not that surprising, since the amount of data is rather limited. The posterior (mean) point estimate for $\nu$ is 0.15. But let’s take a look at some English Premier League football data. With the ordinary BT model this would not work so well since there’s a lot of draws in football. Ignoring them would not be tenable. Below are the ratings, with 95% credibility interval, based on data from the 2015-15 season, using the same prior parameters as in the handball data set. The league points are shown in parenthesis for comparison. The ratings generally agree with the points, except in a few instances, where a team or two have switched places. Another interesting thing to notice is that the width of the intervals seem to be related to the magnitude of the rating. I am not exactly sure why that is, but I suspect its due to the fact that the ratings are in a sense binomial probabilities, and these are known to have greater variance the closer they are to 0.5. The point estimate for $\nu$ is 0.85 for this data set. Compared to the 0.15 for the handball data, it is clear that this reflects the higher overall probability of draws in football. In the handball data set only 6 games ended in a draw, while in the football data set about 20% of the games was a draw. Fitting Bradley-Terry models using Stan I have recently played around with Stan, which is an excellent software to fit Bayesian models. It is similar to JAGS, which I have used before to fit some regression models for predicting football results. Stan differs from JAGS in a number of ways. Although there is some resemblance between the two, the model specification languages are not compatible with each other. Stan, for instance, uses static typing. On the algorithmic side, JAGS uses the Gibbs sampling technique to sample from the posterior; Stan does not do Gibbs sampling, but has two other sampling algorithms. In Stan you can also get point estimates by using built-in optimization routines that search for the maximum of the posterior distribution. In this post I will implement the popular Bradley-Terry machine learning model in Stan and test it on some sports data (handball, to be specific). The Bradley-Terry model is used for making predictions based on paired comparisons. A paired comparison in this context means that two things are compared, and one of them is deemed preferable or better than the other. This can for example occur when studying consumer preferences or ranking sport teams. The Bradley-Terry model is really simple. Suppose two teams are playing against each other, then the probability that team i beats team j is $ p(i > j) = \frac{r_i}{r_i + r_j} $ where $r_i$ and $r_j$ are the ratings for the two teams, and should be positive numbers. It is these ratings we want to estimate. A problem with the model above is that the ratings are not uniquely determined. To overcome this problem the parameters need to be constrained. The most common constraint is to add a sum-to-one $ \sum_k r_k = 1 $ I will explore a different constraint below. Sine we are in a Bayesian setting we need to set a prior distribution for the rating parameters. Given the constraints that the parameters should be positive and sum-to-one the Dirichlet distribution is a natural choice of prior distribution. $ r_1, r_2, …, r_p \sim Dir(\alpha_1, \alpha_2, …, \alpha_p) $ where the hyperparameters $\alpha$ are positive real numbers. I will explore different choices of these below. Here is the Stan code for the Bradley-Terry model: data { int<lower=0> N; // N games int<lower=0> P; // P teams // Each team is referred to by an integer that acts as an index for the ratings vector. int team1[N]; // Indicator arrays for team 1 int team2[N]; // Indicator arrays for team 1 int results[N]; // Results. 1 if team 1 won, 0 if team 2 won. vector[P] alpha; // Parameters for Dirichlet prior. parameters { // Vector of ratings for each team. // The simplex constrains the ratings to sum to 1 simplex[P] ratings; model { real p1_win[N]; // Win probabilities for player 1 ratings ~ dirichlet(alpha); // Dirichlet prior. for (i in 1:N){ p1_win[i] = ratings[team1[i]] / (ratings[team1[i]] + ratings[team2[i]]); results[i] ~ bernoulli(p1_win[i]); The way I implemented the model you need to supply the hyperparameters for the Dirichlet prior via R (or whatever interface you use to run Stan). The match outcomes should be coded as 1 if team 1 won, 0 if team 2 won. The two variables team1 and team2 are vectors of integers that are used to reference the corresponding parameters in the ratings parameter vector. Before we fit the model to some data we need to consider what values we should give to the hyperparameters. Each of the parameters of the Dirichlet distribution corresponds to the rating of a specific team. Both the absolute magnitude and the relative magnitudes are important to consider. A simple case is when all hyperparameters have the same value. Setting all hyperparameters to be equal to each other, with a value greater or equal to 1, implies a prior belief that all the ratings are the same. If they are between 0 and 1, the prior belief is that the ratings are really really different. The magnitude also plays a role here. The greater the magnitudes are, the stronger the prior belief that the ratings are the same. Let’s fit some data to the model. Below are the results from fitting the results from 104 games from the 2016-17 season of the Norwegian women’s handball league, with 11 participating teams. I had to exclude six games that ended in a tie, since that kind of result is not supported by the Bradley-Terry model. Extension exists that handle this, but that will be for another time. Below are the results of fitting the model with different sets of priors, together with the league points for comparison. For this purpose I didn’t do any MCMC sampling, I only found the MAP estimates using the optimization procedures in Stan. For all the priors the resulting ratings give the same ranking. This ranking also corresponds well with the ranking given by the league points, except for Gjerpen and Stabæk which have switched place. We also clearly see the effect of the magnitude of the hyperparameters. When all the $\alpha$‘s are 1 the ratings varies from almost 0 to about 0.6. When they are all set to 100 the ratings are almost all the same. If these ratings were used to predict the result of future matches the magnitudes of the hyperparameters could be tuned using cross validation to find the value that give best predictions. What if we used a different hyperparameter for each team? Below are the results when I set all $\alpha$‘s to 10, except for the one corresponding to the rating for Glassverket, which I set to 25. We clearly see the impact. Glassverket is now considered to be the best team. This is nice since it demonstrates how actual prior information, if available, can be used to estimate the ratings. I also want to mention another way to fit the Bradley-Terry model, but without the sum-to-one constraint. The way to do this is by using a technique that the Stan manual calls soft centering. Instead of having a Dirichlet prior which enforces the constraint, we use a normal distribution prior. This prior will not give strict bounds on the parameter values, but will essentially provide a range of probable values they can take. For the model I chose a prior with mean 20 and standard deviation 6. $ r_1, r_2, …, r_p \sim N(\mu = 20, \sigma = 6) $ The mean hyperprior here is arbitrary, but the standard deviation required some more considerations. I reasoned that the best team would probably be in the top 99 percentile of the distribution, approximately three standard deviations above the mean. In this case this would imply a rating of 20 + 3*6 = 38. Similarly, the worst team would probably be rated three standard deviations below the mean, giving a rating of 2. This implies that the best team has a 95% chance of winning against the worst team. Here is the Stan code: data { int<lower=0> N; int<lower=0> P; int team1[N]; int team2[N]; int results[N]; real<lower=0> prior_mean; // sets the (arbitrary) location of the ratings. real<lower=0> prior_sd; // sets the (arbitrary) scale of the ratings. parameters { real<lower=0> ratings[P]; model { real p1_win[N]; // soft centering (see stan manual 8.7) ratings ~ normal(prior_mean, prior_sd); for (i in 1:N){ p1_win[i] = ratings[team1[i]] / (ratings[team1[i]] + ratings[team2[i]]); results[i] ~ bernoulli(p1_win[i]); And here are the ratings for the handball teams. The ratings are now on a different scale than before and largely matches the prior distribution. The ranking given by this model agrees with the model with the Dirichlet prior, with Gjerpen and Stabek switched relative to the league ranking. Which model is the best? I had a discussion on Twitter a couple of weeks ago about which model is the best for predicting football results. I have suspected that the Dixon & Coles model (DC), which is a modification of the Poisson model, tend to overfit. Hence it should not generalize well and give poorer predictions. I have written about one other alternative to the Poisson model, namely the Conway-Maxwell Poisson model (COMP). This is a model for count data that can be both over-, equi- and underdispersed. It is basically a Poisson model but without the assumption that the variance equals the mean. I have previously done some simple analyses comparing the Poisson, DC and COMP models, and concluded then that the COMP model was superior. The analysis was however a bit to simple, so I have now done a more proper evaluation of the models. A proper way to evaluatie the models is to do a backtest. For each day there is a game played, the three models are fitted to the available historical data (but not data from the future, that would be cheating) and then used to predict the match outcomes for that day. I did this for two leagues, the English Premier League and German Bundesliga. The models were fitted to data from both the top league and the second tier divisions, since this improves the models, but only the results of the top division was predicted and used in the evaluation. I used a separate home field advantage for the two divisions and the rho parameter in the DC model and the dispersion parameter in the COMP model was estimated using the top division only. To measure the model’s predictive ability I used the Ranked Probability Score (RPS). This is the proper measure to evaluate predictions for the match outcome in the form of probabilities for home win, draw and away win. The range of the RPS goes from 0 (best possible predictions) to 1 (worst possible prediction). Since the three models actually model the number of goals, I also looked at the probability they gave for the actual score. For all three models I used the Dixon & Coles method to weight the historical data that is used in training the models. This requires tuning. For both the English and German leagues I backtested the models on different values of the weighting parameter $\xi$ on the seasons from 2005-06 to 2009-10, with historical data available from 1995. I then used the optimal $\xi$ for backtesting the seasons 2010-11 up to December 2016. This last validation period covers 1980 Bundesliga matches and 2426 Premier League matches. Here are the RPS for the three models plottet against $\xi$. Lower RPS is better and lower $\xi$ weights more recent data higher. The graphs show a couple of things. First, all three models have best predictive ability at the same value of $\xi$, and that they compare similarly also for non-optimal values of $\xi$. This makes things a bit easier since we don’t have to worry that a different value of $\xi$ will alter our evaluations about which model is the best. Second, there is quite some difference between the models for the German and English data. In the English data the COMP model is clearly best, while the DC is the worst. In the German league, the DC is clearly better, and the COMP and Poisson models are pretty much equally good. So I used the optimal values of $\xi$ (0.0021 and 0.0015 for Premier League and Bundesliga, respectively) to validate the models in the data from 2010 and onwards. Here is a table of the mean RPS for the three models: We see that for the both English Premier League and German Bundesliga the DC model offers best predictions. The COMP model comes second in Premier League, but has worst performance in the Bundesliga. It is interesting that the DC model performed worst in the tuning period for the Premier League, now was the best one. For the Bundesliga the models compared similarly as in the tuning period. I also looked at how often the DC and COMP models had lower RPS than the Poisson model. The results are in this table: The COMP model outperformed the Poisson model in more than 60% of the matches in both leagues, while the DC model did so only about 40% of the time. When looking at the goal scoring probabilities. Here is a table of the sum of the minus log probabilities for the actual scoreline. Here a lower number also indicates better predictions. Inn both the Premier League and Bundesliga the Poisson model was best, followed by COMP, with the DC model last. We can also take a look at the parameter values for the extra parameters the DC and COMP models has. Remember that the DC models is becomes the Poisson model when rho = 0, while the COMP model is the same as the Poisson model when upsilon = 1, and is underdispersed when upsilon is greater than 1. The parameter estimates fluctuates a bit. It is intersting to see that the rho parameter in the DC model tend to be below 1, which gives the opposite direction of what Dixon and Coles found in their 1997 paper. In the Premier League, the parmater makes a big jump to above 0 at the end of the 2013-14 season. The parameter appears to be a bit more consistent in the Bundesliga, but also there we see a short period where the parameter is around 0. The dispseriosn parameter upsilon also isn’t all that consistent. It is generally closer to 1 in the Bundesliga than in the Premier League. I think this is consistent with why this model was better in the Premier League than in the Bundesliga. All inn all I think it is hard to conclude which of the three models is the best. The COMP and DC models both adjusts the Poisson model in their own specific ways, and this may explain why the different ways of measuring their predictive abilities are so inconsistent. The DC model seem to be better in the German Bundesliga than in the English Premier League. I don’t think any of the two models are generally better than the ordinary Poisson model, but it could be worthwhile to look more into when the two models are better, and perhaps they could be combined?	{"url":"https://opisthokonta.net/","timestamp":"2024-11-05T22:55:43Z","content_type":"text/html","content_length":"111862","record_id":"<urn:uuid:2a507c7d-a07f-46df-995b-7cdc26927f2b>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00333.warc.gz"}
What is Ohm's Law - Electronics Post Ohm’s Law Ohm’s Law is a basic principle in electrical engineering which describes the relationship between three fundamental electrical quantities: voltage (V), current (I), and resistance (R). It is named after the German physicist Georg Simon Ohm, who formulated this law in the early 19th century. What is Ohm’s Law ? According to Ohm’s Law the electrical current(I) flowing through any conductor is directly proportional to the potential difference (voltage)(V) between its ends, assuming the physical conditions (e.g. temperature) of the conductor remain constant. Mathematical Expression of Ohm’s Law Mathematically Ohm’s Law is expressed as: Introducing the constant of proportionality, the resistance R in the above equation, we get, • V (Voltage) is measured in volts (V). • I (Current) is measured in amperes (A). • R (Resistance) is measured in ohms (Ω). Here’s an explanation of each of these components: 1. Voltage (V): Voltage is the electrical potential difference between two points in an electrical circuit. It represents the force that pushes electric charge (electrons) through a conductor (such as a wire). 2. Current (I): Current is the flow of electric charge through a conductor. It represents the rate of flow of electrons in a circuit. Current is measured in amperes (A). 3. Resistance (R): Resistance is a property of a material or component that opposes the flow of current. It’s a measure of how difficult it is for electrons to pass through a conductor. Resistance is measured in ohms (Ω). Ohm’s Law essentially states that in an ideal, linear electrical circuit, the voltage across a resistor (or any component with resistance) is directly proportional to the current passing through it, and the constant of proportionality is the resistance of the component. In other words: • If you increase the voltage (V) across a component while keeping the resistance (R) constant, the current (I) through the component will increase. V ↑ = I ↑* R (Constant) • If you increase the resistance (R) of a component while keeping the voltage (V) constant, the current (I) through the component will decrease. V (Constant) = I ↓* R ↑ • If you know any two of the three values (V, I, and R), you can use Ohm’s Law to calculate the third value. Ohm’s Law is a foundational principle in electronics and is used extensively in the design and analysis of electrical circuits, helping engineers and scientists understand and predict how voltage, current, and resistance relate to one another in various electrical components and systems.	{"url":"https://electronicspost.com/what-is-ohms-law/","timestamp":"2024-11-11T18:06:02Z","content_type":"text/html","content_length":"59863","record_id":"<urn:uuid:abb29f01-e343-42ef-9447-eb2568532a6e>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00872.warc.gz"}
You’re Genius If You Find The “Y” In Less Than 60 Seconds Do you know how your mind can focus completely? This simple game increases your logical and visual skills and it is also fun. You just simply need to find the odd one in the picture – in this game, trying to find the “Y” is the case. You only have one minute to solve it. It is not that simple than you think. How long will it take you to find it? If you still can’t, you can scroll down to know the right answer. Are you ready for the challenge? Then let’s go! Try to find the “Y” in less than 60 seconds. Have you found it already? If you still cannot, stop worrying, it is just a practice for you to be able to do it right next time. We have the answer for you down below. The post You’re Genius If You Find The “Y” In Less Than 60 Seconds appeared first on Wake Up Your Mind.	{"url":"https://holidravel.com/youre-genius-if-you-find-the-y-in-less-than-60-seconds/","timestamp":"2024-11-09T10:32:14Z","content_type":"text/html","content_length":"102946","record_id":"<urn:uuid:df41955a-38c1-42dd-87de-536f346ab519>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.75/warc/CC-MAIN-20241109085148-20241109115148-00222.warc.gz"}
Best Data Science Course Using Python in Jaipur, Rajasthan \| Groot Academy Best Data Science Course Using Python in Jaipur, Rajasthan at Groot Academy Welcome to Groot Academy, the leading institute for IT and software training in Jaipur. Our comprehensive Data Science course using Python is designed to equip you with the essential skills needed to excel in the field of data science and analytics. Course Overview: Are you ready to master Data Science, an essential skill for every aspiring data scientist? 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Contact Us Understanding Data Science Applications of Data Science Basics of Python Programming Python Libraries for Data Science (NumPy, Pandas) Data Manipulation and Cleaning Introduction to Data Visualization Matplotlib and Seaborn Libraries Creating Visualizations with Python Probability Distributions Exploratory Data Analysis (EDA) Data Preprocessing Techniques Introduction to Machine Learning Types of Machine Learning Algorithms Supervised vs Unsupervised Learning Decision Trees and Random Forests Principal Component Analysis (PCA) Support Vector Machines (SVM) Cross-Validation Techniques Introduction to Time Series Data Working with Apache Spark Model Deployment Techniques Building APIs for Machine Learning Models Monitoring and Maintaining Models Resume Building for Data Science Roles Interview Tips and Practice Networking and Job Search Strategies Q1: What is data science? A1: Data science is an interdisciplinary field that uses various techniques, algorithms, and tools to extract insights and knowledge from structured and unstructured data. Q2: What are the key components of data science? A2: Key components include data collection, data cleaning, data analysis, data visualization, and machine learning. Q3: What is the role of a data scientist? A3: A data scientist analyzes complex data, builds predictive models, and provides insights that help in decision-making and strategic planning. Q4: How is data science used in various industries? A4: Data science is used in industries like healthcare, finance, marketing, and technology for applications such as fraud detection, customer segmentation, and predictive analytics. Q5: What skills are essential for data scientists? A5: Essential skills include programming (Python, R), statistics, machine learning, data visualization, and domain knowledge. Q6: What are some popular tools and technologies used in data science? A6: Popular tools and technologies include Python, R, SQL, Hadoop, Spark, and visualization tools like Tableau and Power BI. Q7: What are the steps involved in a data science project? A7: Steps include problem definition, data collection, data cleaning, exploratory data analysis, modeling, evaluation, and deployment. Q8: What is the importance of domain knowledge in data science? A8: Domain knowledge helps data scientists understand the context and nuances of the data, leading to more accurate and relevant insights. Q9: How can one start a career in data science? A9: Start by learning the basics through online courses, gaining hands-on experience with projects, and building a strong portfolio to showcase your skills. Q1: What are the primary sources of data? A1: Primary sources include databases, data warehouses, web scraping, APIs, and surveys. Q2: Why is data cleaning important? A2: Data cleaning is crucial to ensure the accuracy and quality of the data, which directly impacts the reliability of the analysis and results. Q3: What are common data cleaning techniques? A3: Techniques include handling missing values, removing duplicates, correcting errors, and standardizing data formats. Q4: How can data be collected from APIs? A4: Data can be collected from APIs using HTTP requests in programming languages like Python, often utilizing libraries like requests or Axios. Q5: What is web scraping? A5: Web scraping is the process of extracting data from websites using automated scripts or tools like BeautifulSoup and Scrapy. Q6: How do you handle missing data? A6: Missing data can be handled by imputation, removing affected rows or columns, or using algorithms that support missing values. Q7: What is data normalization? A7: Data normalization involves scaling numerical data to a standard range, such as 0 to 1, to ensure fair comparison and improve model performance. Q8: Why is data validation necessary? A8: Data validation checks the accuracy and quality of data before analysis, preventing incorrect conclusions and ensuring reliable results. Q9: What tools are commonly used for data cleaning? A9: Common tools include Python (Pandas, NumPy), R (dplyr, tidyr), and spreadsheet software like Excel. Q1: What is data visualization? A1: Data visualization is the graphical representation of data to help understand and communicate insights effectively. Q2: Why is data visualization important? A2: It helps in identifying patterns, trends, and outliers in data, making complex data more accessible and understandable. Q3: What are some common data visualization tools? A3: Common tools include Tableau, Power BI, Matplotlib, Seaborn, and D3.js. Q4: What types of charts are used in data visualization? A4: Common chart types include bar charts, line charts, scatter plots, histograms, and pie charts. Q5: How do you choose the right chart type for your data? A5: The choice depends on the data type and the insights you want to convey. For example, line charts for trends over time, and bar charts for categorical comparisons. Q6: What is an interactive visualization? A6: Interactive visualizations allow users to interact with the data, such as filtering, zooming, and exploring different aspects dynamically. Q7: What are the principles of effective data visualization? A7: Principles include clarity, accuracy, efficiency, and aesthetic appeal to ensure the visualization communicates the intended message effectively. Q8: What is the role of color in data visualization? A8: Color is used to distinguish different data points, highlight important information, and improve the overall readability of the visualization. Q9: How can you improve your data visualization skills? A9: Practice by creating visualizations, studying best practices, using different tools, and getting feedback from peers and experts. Q1: What is the role of statistics in data science? A1: Statistics provides the foundation for data analysis, helping to summarize, interpret, and infer conclusions from data. Q2: What are descriptive statistics? A2: Descriptive statistics summarize and describe the main features of a dataset, including measures like mean, median, mode, and standard deviation. Q3: What is inferential statistics? A3: Inferential statistics make predictions or inferences about a population based on a sample of data, using techniques like hypothesis testing and confidence intervals. Q4: What is a p-value? A4: A p-value measures the probability that the observed data would occur by chance if the null hypothesis were true. A low p-value indicates strong evidence against the null hypothesis. Q5: What is hypothesis testing? A5: Hypothesis testing is a statistical method to determine if there is enough evidence to reject a null hypothesis in favor of an alternative hypothesis. Q6: What is a confidence interval? A6: A confidence interval is a range of values that is likely to contain the true population parameter with a specified level of confidence, typically 95% or 99%. Q7: What is the difference between correlation and causation? A7: Correlation measures the strength and direction of a relationship between two variables, while causation indicates that one variable directly affects another. Q8: What is regression analysis? A8: Regression analysis is a statistical technique to model and analyze the relationships between a dependent variable and one or more independent variables. Q9: What are the assumptions of linear regression? A9: Assumptions include linearity, independence, homoscedasticity, normality of residuals, and no multicollinearity among independent variables. Q1: What is data analysis? A1: Data analysis is the process of inspecting, cleaning, transforming, and modeling data to discover useful information, draw conclusions, and support decision-making. Q2: What are the different types of data analysis? A2: Types of data analysis include descriptive, diagnostic, predictive, and prescriptive analysis. Q3: What is exploratory data analysis (EDA)? A3: EDA involves analyzing datasets to summarize their main characteristics, often using visual methods, to understand the data and uncover patterns or anomalies. Q4: What tools are commonly used for data analysis? A4: Common tools include Python (Pandas, NumPy), R, SQL, Excel, and data visualization tools like Tableau and Power BI. Q5: What is the role of data cleaning in data analysis? A5: Data cleaning ensures the accuracy and quality of data by handling missing values, removing duplicates, and correcting errors, which is crucial for reliable analysis. Q6: How do you handle outliers in data analysis? A6: Outliers can be handled by removing them, transforming the data, or using robust statistical methods that minimize their impact. Q7: What is the importance of data visualization in data analysis? A7: Data visualization helps in understanding complex data, identifying trends and patterns, and effectively communicating insights to stakeholders. Q8: What is the difference between qualitative and quantitative data analysis? A8: Qualitative analysis focuses on non-numerical data to understand concepts and experiences, while quantitative analysis involves numerical data to identify patterns and test hypotheses. Q9: What are some common challenges in data analysis? A9: Challenges include dealing with large datasets, ensuring data quality, selecting appropriate analysis techniques, and interpreting results accurately. Q1: What is machine learning? A1: Machine learning is a subset of artificial intelligence that involves training algorithms to learn patterns from data and make predictions or decisions without being explicitly programmed. Q2: What are the different types of machine learning? A2: Types include supervised learning, unsupervised learning, semi-supervised learning, and reinforcement learning. Q3: What is supervised learning? A3: Supervised learning involves training a model on labeled data, where the correct output is known, to predict outcomes for new, unseen data. Q4: What is unsupervised learning? A4: Unsupervised learning involves training a model on unlabeled data to identify hidden patterns and structures without prior knowledge of the outcomes. Q5: What is reinforcement learning? A5: Reinforcement learning involves training an agent to make decisions by rewarding it for correct actions and penalizing it for incorrect actions, optimizing its behavior over time. Q6: What are some common machine learning algorithms? A6: Common algorithms include linear regression, logistic regression, decision trees, random forests, k-means clustering, and neural networks. Q7: What is overfitting and how can it be prevented? A7: Overfitting occurs when a model learns the training data too well, including noise, and performs poorly on new data. It can be prevented by using techniques like cross-validation, pruning, and Q8: What is cross-validation? A8: Cross-validation is a technique for assessing how a model generalizes to an independent dataset by partitioning the data into subsets, training the model on some subsets, and validating it on Q9: How do you evaluate the performance of a machine learning model? A9: Performance is evaluated using metrics like accuracy, precision, recall, F1-score, and area under the ROC curve (AUC-ROC) depending on the problem type. Q1: What is supervised learning? A1: Supervised learning is a type of machine learning where the model is trained on labeled data, learning to map input features to known output labels. Q2: What are some common supervised learning algorithms? A2: Common algorithms include linear regression, logistic regression, support vector machines (SVM), k-nearest neighbors (KNN), decision trees, and neural networks. Q3: What is the difference between regression and classification? A3: Regression predicts continuous values, while classification predicts discrete labels or categories. Q4: What is logistic regression? A4: Logistic regression is a classification algorithm that models the probability of a binary outcome based on input features. Q5: What is a decision tree? A5: A decision tree is a model that uses a tree-like structure to make decisions based on input features, splitting the data into branches until a prediction is made. Q6: What is a random forest? A6: A random forest is an ensemble learning method that combines multiple decision trees to improve accuracy and reduce overfitting. Q7: What is overfitting in supervised learning? A7: Overfitting occurs when a model learns the training data too well, capturing noise and outliers, leading to poor generalization on new data. Q8: What is cross-validation? A8: Cross-validation is a technique for evaluating model performance by partitioning the data into training and validation sets multiple times to ensure robustness and prevent overfitting. Q9: What is hyperparameter tuning? A9: Hyperparameter tuning involves selecting the best set of parameters for a model to optimize its performance, often using techniques like grid search or random search. Q1: What is unsupervised learning? A1: Unsupervised learning is a type of machine learning where the model is trained on unlabeled data, identifying patterns and structures without prior knowledge of the outcomes. Q2: What are some common unsupervised learning algorithms? A2: Common algorithms include k-means clustering, hierarchical clustering, principal component analysis (PCA), and t-distributed stochastic neighbor embedding (t-SNE). Q3: What is clustering in unsupervised learning? A3: Clustering is the process of grouping similar data points together based on their features, identifying underlying patterns or structures in the data. Q4: What is k-means clustering? A4: K-means clustering is a popular algorithm that partitions data into k clusters, minimizing the variance within each cluster by iteratively updating the cluster centroids. Q5: What is hierarchical clustering? A5: Hierarchical clustering builds a tree-like structure of nested clusters by either merging smaller clusters into larger ones (agglomerative) or splitting larger clusters into smaller ones Q6: What is principal component analysis (PCA)? A6: PCA is a dimensionality reduction technique that transforms data into a lower-dimensional space while preserving as much variance as possible, making it easier to analyze and visualize. Q7: What is t-SNE? A7: T-SNE (t-distributed stochastic neighbor embedding) is a technique for visualizing high-dimensional data by mapping it into a lower-dimensional space, preserving the local structure and revealing Q8: What are the applications of unsupervised learning? A8: Applications include customer segmentation, anomaly detection, gene expression analysis, and market basket analysis. Q9: How do you evaluate the performance of unsupervised learning models? A9: Performance is evaluated using metrics like silhouette score, Davies-Bouldin index, and clustering accuracy, depending on the problem and available ground truth. Q1: What is deep learning? A1: Deep learning is a subset of machine learning that involves neural networks with many layers, known as deep neural networks, to model complex patterns in data. Q2: What are neural networks? A2: Neural networks are computational models inspired by the human brain, consisting of interconnected nodes (neurons) that process information and learn patterns from data. Q3: What is a deep neural network? A3: A deep neural network is a neural network with multiple hidden layers between the input and output layers, enabling it to learn complex patterns and representations. Q4: What is backpropagation? A4: Backpropagation is an algorithm used to train neural networks by adjusting weights through the calculation of gradients, minimizing the error between predicted and actual outputs. Q5: What are some common deep learning architectures? A5: Common architectures include convolutional neural networks (CNNs) for image processing, recurrent neural networks (RNNs) for sequential data, and generative adversarial networks (GANs) for generating data. Q6: What is a convolutional neural network (CNN)? A6: A CNN is a type of neural network designed for processing structured grid data like images, using convolutional layers to automatically learn spatial hierarchies of features. Q7: What is a recurrent neural network (RNN)? A7: An RNN is a type of neural network designed for sequential data, where connections between nodes form a directed cycle, enabling the network to maintain information across steps. Q8: What are generative adversarial networks (GANs)? A8: GANs are a class of neural networks that consist of two parts: a generator that creates data and a discriminator that evaluates it, training together to generate realistic data. Q9: What is transfer learning? A9: Transfer learning involves leveraging a pre-trained model on a large dataset and fine-tuning it on a smaller, specific dataset, improving performance and reducing training time. Q1: What is natural language processing (NLP)? A1: NLP is a field of artificial intelligence that focuses on the interaction between computers and human language, enabling computers to understand, interpret, and generate human language. Q2: What are some common NLP tasks? A2: Common tasks include text classification, sentiment analysis, named entity recognition, machine translation, and speech recognition. Q3: What is text classification? A3: Text classification is the process of categorizing text into predefined classes or labels, such as spam detection or topic classification. Q4: What is sentiment analysis? A4: Sentiment analysis involves determining the sentiment or emotional tone of a text, such as positive, negative, or neutral. Q5: What is named entity recognition (NER)? A5: NER is the process of identifying and classifying named entities in text, such as names of people, organizations, locations, dates, and other proper nouns. Q6: What is machine translation? A6: Machine translation is the task of automatically translating text from one language to another, using algorithms and models trained on parallel corpora. Q7: What are word embeddings? A7: Word embeddings are dense vector representations of words that capture their meanings and relationships, enabling semantic similarity calculations. Examples include Word2Vec and GloVe. Q8: What is a transformer model? A8: A transformer model is a type of neural network architecture designed for handling sequential data, using self-attention mechanisms to capture long-range dependencies. BERT and GPT are examples. Q9: What is the role of pre-trained models in NLP? A9: Pre-trained models, such as BERT and GPT, are trained on large corpora and can be fine-tuned on specific tasks, improving performance and reducing the need for large labeled datasets. Q1: What is big data? A1: Big data refers to large, complex datasets that are challenging to process and analyze using traditional data processing techniques due to their volume, variety, and velocity. Q2: What are the 3 Vs of big data? A2: The 3 Vs of big data are Volume (amount of data), Variety (types of data), and Velocity (speed of data generation and processing). Q3: What is Hadoop? A3: Hadoop is an open-source framework for storing and processing large datasets in a distributed manner, using a cluster of computers. It includes the Hadoop Distributed File System (HDFS) and the MapReduce programming model. Q4: What is Apache Spark? A4: Apache Spark is an open-source, distributed computing system for big data processing that provides in-memory processing capabilities, improving performance for iterative and interactive tasks. Q5: What is a data lake? A5: A data lake is a storage repository that holds a vast amount of raw data in its native format until it is needed, allowing for flexible schema-on-read processing. Q6: What is a data warehouse? A6: A data warehouse is a centralized repository for storing structured data from multiple sources, designed for query and analysis, often using a schema-on-write approach. Q7: What is the difference between a data lake and a data warehouse? A7: A data lake stores raw data in its native format with a schema-on-read approach, while a data warehouse stores structured data with a schema-on-write approach, optimized for query and analysis. Q8: What is NoSQL? A8: NoSQL is a category of non-relational databases designed for handling large volumes of unstructured or semi-structured data, offering flexibility, scalability, and performance. Examples include MongoDB, Cassandra, and Redis. Q9: What are the benefits of using big data technologies? A9: Benefits include the ability to process and analyze large datasets efficiently, uncover hidden patterns and insights, improve decision-making, and support advanced analytics and machine learning Q1: What is data visualization? A1: Data visualization is the graphical representation of data and information, using visual elements like charts, graphs, and maps to make complex data more accessible and understandable. Q2: What are the benefits of data visualization? A2: Benefits include improved comprehension of data, easier identification of patterns and trends, enhanced communication of insights, and better decision-making. Q3: What are some common types of data visualizations? A3: Common types include bar charts, line charts, scatter plots, pie charts, histograms, heatmaps, and geographic maps. Q4: What is a bar chart? A4: A bar chart is a graphical representation of data using rectangular bars to show the frequency or value of different categories. Q5: What is a line chart? A5: A line chart is a graph that displays data points connected by lines, often used to show trends over time. Q6: What is a scatter plot? A6: A scatter plot is a graph that uses dots to represent the values of two different variables, showing their relationship or correlation. Q7: What is a heatmap? A7: A heatmap is a graphical representation of data where values are depicted by color, often used to show the intensity or concentration of data points in different areas. Q8: What is an interactive visualization? A8: Interactive visualizations allow users to engage with the data by filtering, zooming, and exploring different aspects, providing a more dynamic and informative experience. Q9: What tools are commonly used for data visualization? A9: Common tools include Tableau, Power BI, matplotlib, ggplot2, D3.js, and Plotly. Q1: What is a data science project? A1: A data science project involves applying data science techniques to solve a specific problem, from data collection and cleaning to analysis and model deployment. Q2: What are the key phases of a data science project? A2: Key phases include problem definition, data collection, data cleaning, exploratory data analysis, model building, model evaluation, and deployment. Q3: How do you define the problem in a data science project? A3: Problem definition involves understanding the business problem, setting clear objectives, and determining the success criteria for the project. Q4: What is exploratory data analysis (EDA)? A4: EDA involves analyzing data sets to summarize their main characteristics, often using visual methods to discover patterns, anomalies, and relationships. Q5: How do you choose the right model for your project? A5: Model selection depends on the problem type (classification, regression, clustering), the nature of the data, and the performance requirements. Q6: What is model deployment? A6: Model deployment is the process of integrating a machine learning model into a production environment where it can make predictions on new data. Q7: What are common challenges in data science projects? A7: Challenges include data quality issues, choosing the right model, handling large data sets, model interpretability, and deployment complexities. Q8: How do you ensure your project is successful? A8: Success is ensured by clearly defining objectives, using appropriate techniques, validating models, and effectively communicating results to stakeholders. Q9: How can I get started with a data science project? A9: Start by selecting a problem to solve, gather and clean your data, perform EDA, build and evaluate models, and finally, deploy your model if applicable. Q1: What is model deployment? A1: Model deployment involves integrating a machine learning model into a production environment to make real-time predictions on new data. Q2: What are common deployment platforms? A2: Common platforms include cloud services like AWS, Google Cloud, Azure, and on-premises solutions. Q3: What is a REST API? A3: A REST API (Representational State Transfer Application Programming Interface) allows you to expose your model as a web service, enabling other applications to interact with it over HTTP. Q4: What are the steps involved in deploying a model? A4: Steps include selecting a deployment environment, creating a REST API, containerizing the model using Docker, and monitoring the deployed model. Q5: What is Docker, and why is it used in deployment? A5: Docker is a tool that allows you to package your model and its dependencies into a container, ensuring consistency across different environments. Q6: What is continuous integration/continuous deployment (CI/CD)? A6: CI/CD is a set of practices that automate the integration and deployment of code changes, ensuring reliable and frequent updates to the production environment. Q7: How do you monitor deployed models? A7: Monitoring involves tracking the model's performance, detecting issues like data drift, and ensuring it meets the desired accuracy and efficiency. Q8: What is model retraining? A8: Model retraining involves updating the model with new data to improve its performance and adapt to changes in the underlying patterns. Q9: Why is scalability important in deployment? A9: Scalability ensures that your deployment can handle increasing amounts of data and user requests without compromising performance. Q1: What skills are essential for a career in data science? A1: Essential skills include programming (Python, R), statistics, machine learning, data visualization, and domain knowledge relevant to your field. Q2: How do I prepare for a data science interview? A2: Prepare by reviewing key concepts, practicing coding problems, working on real-world projects, and preparing to discuss your experiences and methodologies. Q3: What are common data science interview questions? A3: Questions often cover topics like data preprocessing, model selection, performance metrics, and problem-solving scenarios specific to data science. Q4: How can I showcase my data science projects? A4: Showcase projects through a portfolio on platforms like GitHub, including detailed explanations, code, and visualizations of your work. Q5: What is the STAR method for answering interview questions? A5: The STAR method involves structuring answers by describing the Situation, Task, Action, and Result, providing clear and concise responses. Q6: How important is networking in data science careers? A6: Networking is crucial for learning about job opportunities, gaining insights from industry professionals, and building relationships that can advance your career. Q7: What are some good resources for learning data science? A7: Good resources include online courses (Coursera, edX), textbooks, blogs, forums, and participating in data science competitions (Kaggle). Q8: How do I stay updated with the latest trends in data science? A8: Stay updated by following industry news, reading research papers, attending conferences, and participating in professional communities. Q9: What are the key qualities of a successful data scientist? A9: Key qualities include strong analytical skills, problem-solving abilities, creativity, effective communication, and continuous learning. Rahul Sharma The Data Science course at Groot Academy is fantastic! The instructors are highly knowledgeable, and the hands-on projects helped me understand real-world applications of data science. Was this review helpful? Sneha Patel Groot Academy offers the best Data Science course using Python. The curriculum is comprehensive, and the support from instructors is incredible. I highly recommend it! Was this review helpful? Ankit Verma I had a great learning experience at Groot Academy. The course covers everything from basics to advanced topics in data science, and the practical exercises are very helpful. Was this review helpful? Pooja Jain The Python Data Science course at Groot Academy is excellent. The instructors provide clear explanations, and the projects are very practical. It's the best place to learn data science in Jaipur. Was this review helpful? Vikram Singh I am extremely satisfied with the Data Science course at Groot Academy. The instructors are experienced, and the course material is very well-organized. I feel confident in my data science skills Was this review helpful? Ritika Mehta Groot Academy's Data Science course is top-notch. The hands-on approach and real-world projects made learning very effective. The instructors are always ready to help. Was this review helpful? Arjun Choudhary The Data Science course using Python at Groot Academy is amazing. The content is comprehensive, and the instructors are very knowledgeable and supportive. Was this review helpful? Kavita Sharma I highly recommend Groot Academy for anyone looking to learn data science. The course is well-structured, and the instructors are very helpful and approachable. Was this review helpful? Amit Dubey The Data Science course at Groot Academy exceeded my expectations. The practical approach and detailed explanations by the instructors made it easy to grasp complex concepts. Was this review helpful? Neha Gupta I had a wonderful experience at Groot Academy. The Data Science course is thorough, and the instructors are very knowledgeable and supportive. It's the best place to learn data science in Jaipur. Was this review helpful? Rohan Desai The Python Data Science course at Groot Academy is excellent. The instructors provide clear explanations, and the practical projects are very helpful in understanding the concepts. Was this review helpful? Priya Singh I am very happy with the Data Science course at Groot Academy. The instructors are experienced, and the course material is very well-structured. I feel confident in my data science skills now. Was this review helpful? Rajesh Kumar Groot Academy offers the best Data Science course in Jaipur. The curriculum is comprehensive, and the support from instructors is amazing. I highly recommend it! Was this review helpful? Megha Jain The Data Science course at Groot Academy is fantastic. The hands-on projects and real-world applications made learning very effective. The instructors are always ready to help. Was this review helpful? Suresh Patel I had a great learning experience at Groot Academy. The course covers everything from basics to advanced topics in data science, and the practical exercises are very helpful. Was this review helpful? Shweta Verma The Data Science course at Groot Academy is excellent. The instructors provide clear explanations, and the projects are very practical. It's the best place to learn data science in Jaipur. Was this review helpful? Get In Touch Ready to Take the Next Step? Embark on a journey of knowledge, skill enhancement, and career advancement with Groot Academy. Contact us today to explore the courses that will shape your future in IT.	{"url":"https://grootacademy.com/best-job-oriented-courses-in-jaipur/best-data-science-course-using-python-in-jaipur-rajasthan.php","timestamp":"2024-11-08T18:02:26Z","content_type":"text/html","content_length":"290359","record_id":"<urn:uuid:e866f9fd-7f5f-4c28-a4e1-74e3ad634dde>","cc-path":"CC-MAIN-2024-46/segments/1730477028070.17/warc/CC-MAIN-20241108164844-20241108194844-00419.warc.gz"}
Online Calculate Absorbance using Beer’s Law The Beer Lambert Law is also known as Beer's law, Beer-Lambert-Bouguer law or Lambert-Beer law. This law was discovered by Pierre Bouguer before 1729. It is the linear relationship between absorbance and concentration of an absorbing species. While working in concentration units of molarity, the Beer's law is written as a=ecl. Where a is the Absorbance, e is the Molar absorption coefficient, l is the Path Length and c is the Concentration. Calculate Absorbance using Beer's Law Molar Absorption Coefficient Formula Absorbance using Beer's Law Absorbance = Molar Absorption Coefficient x Path Length x Concentration Example Absorbance using Beer's Law For a substance with molar absorption coefficient 50 m^2/mol, concentration 10 mol/L and path length 5 m, the absorbance using beer's lambert law can be calculated as = 50 x 5 x 10 0 comments Inline Feedbacks View all comments \| Reply	{"url":"https://wpcalc.com/en/beer-lambert-law/","timestamp":"2024-11-05T15:23:45Z","content_type":"text/html","content_length":"73022","record_id":"<urn:uuid:fc58c891-efb8-4083-be23-490bcc290ddc>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00086.warc.gz"}
Math 4 Wisdom. "Mathematics for Wisdom" by Andrius Kulikauskas. \| Research / YonedaLemmaAbstract Yoneda Lemma Slides, Interpreting Yoneda Lemma, Yoneda Lemma International Category Theory Conference Sample papers Yoneda Lemma as Knowledge Switch. An Outsider's View of Category Theory as Metaphor Theory. I lead Math 4 Wisdom, an investigatory community for absolute truth. I relate a language of wisdom (in terms of cognitive frameworks) with the language of mathematics (at a graduate student level). Wisdom - the knowledge of everything - is the content. Math - the study of structure - is the form. I have documented a language of cognitive frameworks which I dare call a language of wisdom. But how could I validate it? I could say that whenever we divide everything into four perspectives, our minds are dealing with the issue of knowledge, and we have four levels of knowledge: whether, what, how, why. I could argue that every thinker, in their private language, refers to these four levels - Aristotle, Plato, Peirce. But how can we go beyond our private languages and develop a shared language? I am exploring how to do that by appealing to examples in mathematical thinking. For example, consider a short exact sequence and/or related chain complex which consists of four maps. Or consider, in recursion theory, the Arithmetic Hierarchy. But I will appeal to the Yoneda lemma. And the reason is the Yoneda perspective. So let me illustrate the four levels of knowledge with a cup. And compare them with four ways of thinking about the Yoneda lemma. Yoneda lemma - Anthropologically speaking - mathematicians are the greatest problem in the world. Beauty - based on the line between inside and outside. Category theory plays on this line, moving us from colimits to limits. Everything is that which has no external context and no internal structure. So it serves as a starting point.	{"url":"https://www.math4wisdom.com/wiki/Research/YonedaLemmaAbstract","timestamp":"2024-11-05T12:08:39Z","content_type":"application/xhtml+xml","content_length":"12990","record_id":"<urn:uuid:1587eb8c-b090-49d8-b06e-f503f5fb7f9c>","cc-path":"CC-MAIN-2024-46/segments/1730477027881.88/warc/CC-MAIN-20241105114407-20241105144407-00893.warc.gz"}
Long Call Long Call Long Call option' is the most basic & simplest strategy. It is recommended or implemented when we expect the underlying asset to show significant upside move. A long call option is a popular bullish options strategy that gives the holder the right, but not the obligation, to buy an underlying asset at a predetermined strike price before the option's expiration date. This strategy is employed when you anticipate that the price of the underlying asset will rise, allowing you to profit from the potential price increase. How It Works: • Buy a Call Option: As the investor, you purchase a call option with a specified strike price and expiration date. • Pay the Premium: To acquire this right, you pay a premium to the option seller. • Profit Potential: If the price of the underlying asset rises above the strike price plus the premium paid, you can profit from the price difference. Your potential profit is theoretically Example: Let's say you believe that Nifty, currently trading at ₹19,996, will rise in value. You buy a call option with a strike price of ₹20,000 for a premium of ₹365 per share, and the option has an expiration date of 2 months. If the price of Nifty Index rises to ₹20,500 at the option expiry, you can exercise your call option at the strike price of ₹20,000. Your profit would be ₹20,500 (current stock price) - ₹20,000 (strike price) - ₹365 (premium paid) = ₹135 per share. Nifty has lot size of 50, hence your total profit would be ₹6,750. Key Points: • Long call options offer the potential for significant profit if the underlying asset's price rises. • Your risk is limited to the premium paid for the call option. If the asset's price does not rise above the strike price, you may lose the premium. • You are not obligated to exercise the option; you can let it expire if it's not profitable. • Timing is crucial, as the option has an expiration date. If the price does not rise sufficiently before expiration, the option may expire worthless. In summary, a long call option is a bullish strategy that allows you to benefit from anticipated price increases in the underlying asset. It provides potential for profit while limiting your risk to the premium paid for the option. However, like all options strategies, it's essential to thoroughly understand the mechanics and risks involved before implementing it in your trading or investment Get the latest articles, courses, insights and more, directly to your inbox	{"url":"https://www.1lyoptions.com/strategies/long-call","timestamp":"2024-11-11T10:28:43Z","content_type":"text/html","content_length":"205960","record_id":"<urn:uuid:3751e76e-d961-4db6-8376-7a9710d3ece5>","cc-path":"CC-MAIN-2024-46/segments/1730477028228.41/warc/CC-MAIN-20241111091854-20241111121854-00166.warc.gz"}
rade 1 Grade 1 Pre-Test 1. Based on the cubes or number of sticks shown, can you tell what number is shown. Count by tens and then ones. Tens Ones 2.Write the number shown below. Add the tens and then ones. 3Draw number '45' on the numbers line below. 5. Solve: 8 tens + 6 ones = 6.Use the tens frames to help you find the answer to the problem shown on the right. 4 + 11 = 7.Use the tens frames to help you find the answer to the problem shown on the right. 17 - 10 = Copyright SubjectCoach.com. Strictly for Personal and School Use Only. 8.Use the number line to answer the question below. 9.Use the number line to answer the question below. 10.Find the missing number in the sequence shown below 45, 43, 41, , 37 11.Find the missing number in the sequence shown below 70, 65, 60, , 50 12.Find the missing number in the sequence shown below 2, , 22, 32, 42 13.Shade one half of the shape shown below 14. Check the more valuable coin. 15. Check the more valuable coin. Copyright SubjectCoach.com. Strictly for Personal and School Use Only. 17. Complete the sequence shown below by fill in the blanks 81, 80, , 78, 77, , , 74 18. Complete the pattern. 35, 40, 45, , , 60, , 70 19. Complete the pattern. , 80, 70, 60, , 40, , 20 20. What time is shown on the clock 21. What time is shown on the clock 22. Which event usually takes more time? Sleeping all night or taking a nap. 23. How many edges does a rectangle have? Copyright SubjectCoach.com. Strictly for Personal and School Use Only. 24. Sally and Ben were asked to walk past the trees then through the gate to reach the house. The path they took is shown. Who followed the directions correctly? 25. What's the chance that it will rain next week? Write your answer by choosing one of these options: won't happen,will happen,might happen 26. Allie went to the farmersâ€™ market and saw 1 oranges, 6 pears, and 1 apples. Can you make a bar graphs from this data below. Copyright SubjectCoach.com. Strictly for Personal and School Use Only. Grade 1 Pre-Test Answers 1. Based on the cubes or number of sticks shown, can you tell what number is shown. Count by tens and then ones. Tens Ones 2.Write the number shown below. Add the tens and then ones. 3Draw number '45' on the numbers line below. 5. Solve: 8 tens + 6 ones = 6.Use the tens frames to help you find the answer to the problem shown on the right. 4 + 11 = 7.Use the tens frames to help you find the answer to the problem shown on the right. 17 - 10 = Copyright SubjectCoach.com. Strictly for Personal and School Use Only. 8.Use the number line to answer the question below. 9.Use the number line to answer the question below. 10.Find the missing number in the sequence shown below 45, 43, 41, 39 , 37 11.Find the missing number in the sequence shown below 70, 65, 60, 55 , 50 12.Find the missing number in the sequence shown below 2, 12 , 22, 32, 42 13.Shade one half of the shape shown below 14. Check the more valuable coin. 15. Check the more valuable coin. Copyright SubjectCoach.com. Strictly for Personal and School Use Only. 17. Complete the sequence shown below by fill in the blanks 81, 80, 79 , 78, 77, 76 , 75 , 74 18. Complete the pattern. 35, 40, 45, 50 , 55 , 60, 65 , 70 19. Complete the pattern. 90 , 80, 70, 60, 50 , 40, 30 , 20 20. What time is shown on the clock 21. What time is shown on the clock 22. Which event usually takes more time? Sleeping all night or taking a nap. Sleeping all night 23. How many edges does a rectangle have? 4 edges Copyright SubjectCoach.com. Strictly for Personal and School Use Only. 24. Sally and Ben were asked to walk past the trees then through the gate to reach the house. The path they took is shown. Who followed the directions correctly? 25. What's the chance that it will rain next week? Write your answer by choosing one of these options: won't happen,will happen,might happen might happen 26. Allie went to the farmersâ€™ market and saw 1 oranges, 6 pears, and 1 apples. Can you make a bar graphs from this data below. Copyright SubjectCoach.com. Strictly for Personal and School Use Only.	{"url":"https://www.subjectcoach.com/pretests/grade1","timestamp":"2024-11-13T02:44:23Z","content_type":"text/html","content_length":"53391","record_id":"<urn:uuid:bd7406a6-5015-4744-8acc-84a6894ab4d6>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00768.warc.gz"}
How is the SSAT Test Scored? The Secondary Schools Admission Test, or SSAT, is a test of language, reasoning, math, and writing skills that are used to help determine admission to private elementary, middle, and high schools. The SSAT is designed for students in 3rd-11th grade, and administered at different levels: Elementary -Level, for students in 3rd and 4th grade, Middle- Level, for students in 5th-7th grade, and Upper -Level, for students in 8th-11th grade. The duration of the SSAT test depends on what level of the test you are taking. If you are taking the Elementary-Level test, the test will take 110 minutes and if you are taking the Middle or Upper-Level tests, the test will take 170 minutes. Here is some information on scoring SSAT. The Absolute Best Book to Ace the SSAT Upper-Level Math Test Original price was: $24.99.Current price is: $14.99. How does SSAT scoring work? You will receive three types of points in your score report: a raw score, a scaled score, and a percentile ranking. Raw scores: Raw scores are determined by counting the correct answers and the number of incorrect answers. The correct answers each have one point and for each incorrect answer, a quarter of a point is deducted. No point is deducted for not answering a question, and all questions have the same point value. Scaled scores: Scaled scores are derived from raw scores. Students receive scaled section scores for Quantitative/Math, Verbal, and Reading (each of which is out of the same number of points). They will receive a total sum score that adds together all three of their section scores. Different scales are used for the different levels of the SSAT test. Level Section Score Range Sum Score Range Elementary 300-600 900-1800 Middle 440-710 1320-2310 Upper 500-800 1500-2400 Percentile ranking: The percentile ranking shows how well you did compare to other students of the same gender and grade who have passed the SSAT test in the last three years. Your percentile rank will be between 1% and 99% and indicates the percentage of students with a lower or equal score to your own. For instance: if you receive a percentile score of 70%, that would mean that you did as well as or better than 70% of students of your gender and grade who took the SSAT test in the past three years. The Absolute Best Book to Ace the SSAT Middle-Level Math Test Original price was: $21.99.Current price is: $16.99. SSAT scores by section Each SSAT part has a different number of questions, these questions are all worth the same number of points on each section. The Verbal score: SSAT Verbal score is the result of your performance on one single SSAT Verbal section and this section has 60 questions, the most of any section. 30 are Analogy questions and 30 of these are Synonym The Quantitative score: SSAT Quantitative score is the result of your performance on two separate Quantitative sections of 25 questions each and these sections are scored with a total Quantitative Score comprising 50 The Reading score: The SSAT Reading score is the result of a student’s performance on one single Reading section and this section has 40 questions. The Writing sample: There is no score for the SSAT writing sample and, the essay response is still sent to the admissions departments. Students have 25 minutes to respond to either a non-creative or creative prompt. Reading your Elementary Level SSAT score report The Elementary Level SSAT score report contains the following information: Number correct: Number of correct answers for the content sections and subsections. The number of items: Number of items in the content sections and subsections. Percent correct: Percentage of correct answers for the content sections and subsections. Scaled score: Each scaled score has a range of values 300 to 600. The midpoint value of the scaled scores of the content sections is 450, and the highest score of the test-taker on a section is 600. Scaled score percentile rank: The scaled score percentile is a score which values are from 1 to 99. It compares performance with other students taking the same examination. Total scaled score: The total scaled score is the sum of the scaled scores for the verbal, quantitative, and reading sections. It has a midpoint of 1350, a low value of 900, and a high value of 1800. The Absolute Best Book to Ace the SSAT Lower Level Math Test Original price was: $16.99.Current price is: $11.99. What is a “good” SSAT score? If you score in the 50th percentile in each SSAT section, you will get the “median” SSAT score for that test. And if your score is higher than the 50th percentile, you perform better than average. A good place to start for SSAT test takers is to surpass the average SSAT score for each section. But what is a “good” SSAT score? It depends! Each student’s talents, interests, and goals are unique. Also, the “good” SSAT score is likely to fluctuate, depending on the institution the student is applying for. When will you receive your SSAT score? Most scores are published about two weeks after the test date. Your points will be credited to your online student account, or you may pay extra to receive your credits by mail. You cannot receive grades by telephone. However, you can pay extra for a score report alert by text or email, to receive notice as soon as your scores are available online. You can also pay an extra fee to include a copy of your written sample in your report. The Best Books to Ace the SSAT Upper-Level Math Test Related to This Article What people say about "How is the SSAT Test Scored? - Effortless Math: We Help Students Learn to LOVE Mathematics"? No one replied yet.	{"url":"https://www.effortlessmath.com/blog/how-is-the-ssat-test-scored/","timestamp":"2024-11-05T21:45:12Z","content_type":"text/html","content_length":"114751","record_id":"<urn:uuid:19a1ceff-128b-471b-acb1-0f17e98174ed>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00135.warc.gz"}
Evaluating FRC Rating Models How do you choose between FRC rating models? We compare several models on three characteristics: predictive power, interpretability, and accessibility. Summarizing the Options Given the importance of match strategy and alliance selection, several models have been developed that attempt to quantify an FRC team's contribution to match outcomes. We consider a wins baseline, the popular OPR and Elo rating systems, and the new Expected Points Added (EPA) model deployed on Statbotics. A brief summary of each model is included below. • Wins Baseline: This simple model only considers a team's record. The alliance with the most wins that year is predicted to win. This model is roughly how a human would intuitively predict the outcome of a match, and is a good baseline for evaluating more complex models. • Offensive Power Rating (OPR): OPR uses linear algebra to minimize the sum of squared errors between the predicted and actual scores. We evaluate a variant of OPR called ixOPR that incorporates a prior during ongoing events. TBA Insights and Statbotics both incorporate ixOPR. • Elo Rating: The Elo rating system is a well-known model for predicting the outcome of chess matches. In FRC, Elo iteratively updates a team's rating based on the difference between the predicted and observed winning margin. Caleb Sykes modified Elo for FRC, and we include both his original mode and the Statbotics implementation. • Expected Points Added (EPA): EPA attempts to quantify a team's average point contribution to an FRC match. EPA builds upon concepts from the Elo rating system, but the units are in points and can be interpreted more analogously to a team's OPR. EPA is highly predictive and separates into components that can be interpreted individually. More details are available here. Evaluating models When choosing between FRC rating models, multiple factors play a role. In particular, we consider the following three: • Predictive Power: How well does the model predict the outcome of a match? We evaluate this by comparing the model's predictions to the actual outcomes. • Interpretability: How easy is it to understand the model's predictions and incorporate them into a strategy? Ratings in point units and ratings that can be separated into components are more • Accessibility: How easy is it to use the model? Models that are available on a website or API are more accessible than models that require user calculations. Predictive Power We evaluate predictive power by comparing the model's predictions to the actual outcomes. Accuracy is measured by the percentage of matches that the model correctly predicted. Brier Score measures the mean squared error and quantifies calibration and reliability. A Brier Score of 0 indicates perfect calibration and a Brier Score of 0.25 indicates no skill. Going back to 2002, we evaluate models on 160,000 matches, with special emphasis on 15,000 champs matches and the 80,000 recent matches since 2016. Simply predicting "Red Alliance" each match results in an accuracy of 50%. To approximate the accuracy of an idle spectator, we consider a simple wins baseline that predicts the winner based on the alliance with the most combined wins. To evaluate the predictive power of a model, we compare its accuracy to the accuracy of the wins baseline. This reflects the extent to which the model is able to predict match outcomes beyond the eye test. The table below can be customzied to include and exclude models, metrics, and years. The Wins Baseline predicts the outcome of a match with a 65% accuracy on average. In comparison, the OPR model has a 68% accuracy, the Elo model has a 69% accuracy, and the EPA model has a 70% accuracy. Since 2016, these numbers are 66% (Wins), 70% (OPR), 71% (Elo), and 72% (EPA). While the EPA model outperforms the OPR/Elo models by only 1-2%, in relation to the wins baseline, the EPA model outperforms the baseline by ~20% more than the OPR/Elo models. The EPA model is the best performing model in 15 of the 20 years, including six of the last seven. The one exception is 2018, where the EPA model struggles somewhat with the nonlinear scoring system. There are two caveats regarding the EPA model's improved performance. While we compare EPA to Elo and OPR individually, Statbotics previously used a combination of both, which has improved accuracy compared to either model alone. Still, the EPA model individually outperforms this ensemble, and future EPA iterations can ensemble with other models to reach even higher performance. Second, while the EPA model significantly outperforms other models during the season, this does not translate to champs. EPA stabilizes to an accurate prediction faster during the season, but by champs, Elo has caught up and is roughly equivalent to EPA. We evaluate interpretability by considering the units of the model and the ability to separate the model into components. The OPR and EPA models are in point units, and can be separated into auto, teleop, and endgame components. Elo is in arbitrary units (1500 mean, ~2000 max) and cannot be separated into components. One benefit of the Elo model is that ratings can be roughly compared across years. Using normalized EPA, we can compare EPA ratings across years as well (blog post coming soon). In summary, the EPA model combines the best of both worlds: point units, separable into components, and comparable across years. We evaluate accessibility by considering the availability of the model. Statbotics previously included the OPR and Elo models, but has now transitioned to the EPA model. The Blue Alliance calculates OPR and their own TBA Insights model. Caleb Sykes publishes a comprehensive scouting database on Excel. Statbotics and TBA have APIs that allow for integration with external projects. Each model has distribution channels that are more or less equally accessible to teams. The EPA model is the best performing model in 15 of the 20 years, including six of the last seven. The EPA model is the most interpretable model, with point units, separable components, and year-normalized ratings. Finally, the EPA model is highly accessible, available on the Statbotics website and through Python API. In summary, we highly recommend the EPA model for teams and scouting systems. Please reach out to us if you have any questions or feedback.	{"url":"https://v1.statbotics.io/blog/models","timestamp":"2024-11-02T11:40:44Z","content_type":"text/html","content_length":"59748","record_id":"<urn:uuid:8d412457-178d-4e79-ba33-5af19583f135>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00459.warc.gz"}
Importance of a Fully Anharmonic Treatment of Equilibrium Isotope Fractionation Properties of Dissolved Ionic Species As Evidenced by Li<sup>+</sup>(aq) ConspectusEquilibrium fractionation of stable isotopes is critically important in fields ranging from chemistry, including medicinal chemistry, electrochemistry, geochemistry, and nuclear chemistry, to environmental science. The dearth of reliable estimates of equilibrium fractionation factors, from experiment or from natural observations, has created a need for accurate computational approaches. Because isotope fractionation is a purely quantum mechanical phenomenon, exact calculation of fractionation factors is nontrivial. Consequently, a severe approximation is often made, in which it is assumed that the system can be decomposed into a set of independent harmonic oscillators. Reliance on this often crude approximation is one of the primary reasons that theoretical prediction of isotope fractionation has lagged behind experiment. A class of problems for which one might expect the harmonic approximation to perform most poorly is the isotopic fractionation between solid and solution phases.In order to illustrate the errors associated with the harmonic approximation, we have considered the fractionation of Li isotopes between aqueous solution and phyllosilicate minerals, where we find that the harmonic approximation overestimates isotope fractionation factors by as much as 30% at 25 °C. Lithium is a particularly interesting species to examine, as natural lithium isotope signatures provide information about hydrothermal processes, carbon cycle, and regulation of the Earth's climate by continental alteration. Further, separation of lithium isotopes is of growing interest in the nuclear industry due to a need for pure ^6Li and ^7Li isotopes. Moving beyond the harmonic approximation entails performing exact quantum calculations, which can be achieved using the Feynman path integral formulation of quantum statistical mechanics. In the path integral approach, a system of quantum particles is represented as a set of classical-like ring-polymer chains, whose interparticle interactions are determined by the rules of quantum mechanics. Because a classical isomorphism exists between the true quantum system and the system of ring-polymers, classical-like methods can be applied. Recent developments of efficient path integral approaches for the exact calculation of isotope fractionation now allow the case of the aforementioned dissolved Li fractionation properties to be studied in detail. Applying this technique, we find that the calculations yield results that are in good agreement with both experimental data and natural observations. Importantly, path integral methods, being fully atomistic, allow us to identify the origins of anharmonic effects and to make reliable predictions at temperatures that are experimentally inaccessible yet are, nevertheless, relevant for natural phenomena. ASJC Scopus subject areas Dive into the research topics of 'Importance of a Fully Anharmonic Treatment of Equilibrium Isotope Fractionation Properties of Dissolved Ionic Species As Evidenced by Li^+(aq)'. Together they form a unique fingerprint.	{"url":"https://nyuscholars.nyu.edu/en/publications/importance-of-a-fully-anharmonic-treatment-of-equilibrium-isotope","timestamp":"2024-11-05T06:12:09Z","content_type":"text/html","content_length":"62895","record_id":"<urn:uuid:c851fba1-e3dc-40a5-89dd-7cb5868dcba5>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00644.warc.gz"}
Grade 5 Math Worksheets Fifth Grade Math This page offers a variety of Grade 5 Math lessons aligned with the Common Core State Standards for Mathematics. In Fifth Grade Math, the key areas of focus include: (1) developing fluency with addition and subtraction of fractions, and developing understanding of the multiplication of fractions and of division of fractions in limited cases (unit fractions divided by whole numbers and whole numbers divided by unit fractions); (2) extending division to 2-digit divisors, integrating decimal fractions into the place value system and developing understanding of operations with decimals to hundredths, and developing fluency with whole number and decimal operations; and (3) developing understanding of volume. The topics include whole numbers; multiplication; division; algebraic expressions; decimal numbers; addition, subtraction, multiplication, and division of decimal numbers; statistics; divisibility and factorization; fractions; ratios proportions and percents and such.	{"url":"https://mathelpers.com/grade-5-math-worksheets-and-explanation","timestamp":"2024-11-14T00:47:19Z","content_type":"text/html","content_length":"34730","record_id":"<urn:uuid:7ad64a6b-4d16-4735-88e9-5ea0c8de9e62>","cc-path":"CC-MAIN-2024-46/segments/1730477028516.72/warc/CC-MAIN-20241113235151-20241114025151-00375.warc.gz"}
Basic Problems in Astrophysics (2) An interesting problem is to obtain the distance modulus for a typical star in a globular cluster One of the more basic and important quantities in astrophysics is the " distance modulus " - so one would like to see how it is obtained and how used. This is actually a useful measure of the actual rate of emission of visible radiation by a star, and depends on the star’s actual distance. It is often denoted as (m – M) or the difference between the apparent magnitude of a star, and its absolute magnitude M. The "apparent magnitude" is the apparent brightness registered on a logarithmic scale. For example, the Sun's apparent magnitude is a whopping (-26.5) [ : the more negative the value, the greater the brightness, the more positive the lower]. Now, the brightest star we can see is Sirius which has an apparent magnitude of about (-1.6). The ratio of brightness is computed on the basis that every five magnitudes registers 100 times brightness ratio. Thus, if star A is at a magnitude of +3 and star B at +8, then star A is brighter than star B by: (2.512)^5 ~ 100 or 100 times. (Since on the stellar magnitude scale each single magnitude difference rates as a brightness ratio of 2.512 times). Thus, on the apparent magnitude scale , the Sun is brighter than Sirius by about 25 magnitudes, for a brightness ratio of: (2.512)^25 = 10^10 or about ten billion times! So merely casually glancing at the stars (or Sun) gives no insight into how bright they really are . (If it did, everyone would assert the Sun is the ‘king’ of stellar creation – but they’d be flat wrong ) But this is an illusion, which is why astronomers choose to use the absolute magnitude (M). In terms of absolute magnitudes, the standard distance to compare all stellar outputs is exactly 10 parsecs, or 32.6 light years. Thus, to ascertain the Sun’s absolute magnitude it must be “moved” to ten parsecs and its brightness re-assessed. Of course, one needn’t actually move the Sun to do this – merely use the inverse square light intensity law, e.g. the intensity or brightness of a light source is proportional to the inverse square of the distance Thus, the Sun, at nearly magnitude (-26.5) where it sits at 1.5 x 10^8 km, but at 32.6 light years, it will be at a distance of 3.08 x 10^14 km or about 2.06 million times further away. Hence, one must reduce its apparent brilliance by the inverse of that factor squared,or: (1/ 2.06 x 10^6)^2 = 2.3 x 10^-13 Thus, its brightness diminishes by more than 30 orders or magnitude (recall that the stellar magnitude scale is a logarithmic one). Now, let L(d) be the observed light from a star at its actual distance d, and let L(10) be the amount of light we’d receive if it were at 10 parsecs. From the definition of stellar magnitudes, we have: m – M = 2.5 log (L(10)/ L(d)) where the ‘2.5’ factor reminds us that for every magnitude difference, the ratio of light difference is ~2.5: 1.From the inverse square law for light: L(10)/ L(d) = (d/ 10)^2 Now, combine the two equations by substituting the last into the first (e.g for L(10)/L(d): m - M = 5 log (d/ 10) so that 5 log (d/10) is the distance modulus and it shows we only need to know the difference in magnitudes, m – M to find the distance d. By example, what if (m – M) = 10 magnitudes? Then: 10 = 5 log (d/10)2 = log (d/ 10) or, 10^2 = 100 = d/ 10-> d = 1000 pc	{"url":"https://brane-space.blogspot.com/2010/11/basic-problems-in-astrophysics-2.html","timestamp":"2024-11-13T14:39:22Z","content_type":"text/html","content_length":"116709","record_id":"<urn:uuid:7e072a9e-b9f0-4690-97d7-0ad1929489d5>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00638.warc.gz"}
unit type In type theory the unit type is the type with a unique term. It is the special case of a product type with no factors. In a model by categorical semantics, this is a terminal object. In set theory, it is a singleton. Like any type in type theory, the unit type is specified by rules saying when we can introduce it as a type, how to introduce terms of that type, how to use or eliminate terms of that type, and how to compute when we combine the constructors with the eliminators. The unit type, like the binary product type, can be presented both as a positive type and as a negative type. There are typically two foundations in which the unit type is specified in, natural deduction and lambda-calculus. In natural deduction We assume that our unit types have typal conversion rules, as those are the most general of the conversion rules. Both the propositional and judgmental conversion rules imply the typal conversion rules by the structural rules for propositional and judgmental equality respectively. For both the positive unit type and the negative unit type, the formation and introduction rules are the same. The formation rule for the unit type is given by $\frac{\Gamma \; \mathrm{ctx}}{\Gamma \vdash \mathbb{1} \; \mathrm{type}}$ and the introduction rule for the unit type is given by $\frac{\Gamma \; \mathrm{ctx}}{\Gamma \vdash :\mathbb{1}}$ The positive unit type says that $\mathbb{1}$ satisfies singleton induction. The negative unit type says that the element $:\mathbb{1}$ is the center of contraction of $\mathbb{1}$. As a positive type Singleton induction for a type $\mathbb{1}$ and term $:\mathbb{1}$ states that the type has the following elimination rule, computation rule, and uniqueness rule: • given a type family $C(x)$ indexed by $\mathbb{1}$, for all elements $x:\mathbb{1}$ and $c:C()$, there is an element $\mathrm{ind}_\mathbb{1}^C(x, c):C(x)$. $\frac{\Gamma, x:\mathbb{1} \vdash C(x) \; \mathrm{type}}{\Gamma, x:\mathbb{1}, c:C() \vdash \mathrm{ind}_\mathbb{1}^C(x, c):C(x)}$ • given a type family $C(x)$ indexed by $\mathbb{1}$, for all elements $c:C()$, the element $\mathrm{ind}_\mathbb{1}^C(, c)$ derived from the elimination rule is equal to $c$ with witness $\beta_ $\frac{\Gamma, x:\mathbb{1} \vdash C(x) \; \mathrm{type}}{\Gamma, c_:C() \vdash \beta_\mathbb{1}^C(c):\mathrm{ind}_\mathbb{1}^C(, c) =_{C()} c}$ • given a type family $C(x)$ indexed by $\mathbb{1}$ and a family of elements $u(x):C(x)$ indexed by $\mathbb{1}$, for all elements $x:\mathbb{1}$ there is a witness $\eta_\mathbb{1}^C(x, u(x))$ that $u(x)$ is equal to $\mathrm{ind}_\mathbb{1}^{C}(x, u())$ in $C(x)$. $\frac{\Gamma, x:\mathbb{1} \vdash C(x) \; \mathrm{type} \quad \Gamma, x:\mathbb{1} \vdash u(x):C(x)}{\Gamma, x:\mathbb{1} \vdash \eta_\mathbb{1}^C(x, u(x)):u(x) =_{C(x)} \mathrm{ind}_\mathbb{1}^{C} (x, u())}$ Thus, the unit type satisfies singleton induction. In type theories with a separate type judgment where not all types are elements of universes, one has to additionally add the following elimination and computation rules: Elimination rules: $\frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma, x:\mathbb{1} \vdash \mathrm{typerec}_{\mathbb{1}}^{A}(x) \; \mathrm{type}}$ Computation rules: • judgmental computation rules $\frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma \vdash \mathrm{typerec}_{\mathbb{1}}^{A}() \equiv A \; \mathrm{type}}$ As a negative type As a negative type, there are no elimination rules or computation rules for $\mathbb{1}$. The uniqueness rule states that $$ is the center of contraction of $\mathbb{1}$ and is given by: $\frac{\Gamma \; \mathrm{ctx}}{\Gamma, p:\mathbb{1} \vdash \eta_\mathbb{1}(p): =_\mathbb{1} p}$ Thus, the unit type is a contractible type. Positive versus negative Let $\mathbb{1}$ denote the positive unit type, and let $p:\mathbb{1}$ and $q:\mathbb{1}$ be elements of $\mathbb{1}$. Then the identity type $q =_\mathbb{1} p$ is a contractible type. Given an element $p:\mathbb{1}$, we can define the family of types $C(q) \coloneqq q =_\mathbb{1} p$ for all $q:\mathbb{1}$. The elimination and computation rules for the positive unit type imply that $* =_\mathbb{1} p$ is a contractible type with center of contraction $c_(p): =_\mathbb{1} p$ which is equal to $\mathrm{ind}_\mathbb{1}^{(-) =_\mathbb{1} p}(p, c_(p))$. The uniqueness rule for the positive unit type states that given a family of terms $u(p):q =_\mathbb{1} p$ indexed by $p:\mathbb{1}$ and $q:\mathbb{1}$, for all elements $c_(p):* =_\mathbb{1} p$, elements $q:\mathbb{1}$, and witnesses $i_(u(p))$ that $u(p)()$ is equal to $c_(p)$ in $ =_\mathbb{1} p$, there is a witness $\eta_\mathbb{1}^{(-) =_\mathbb{1} p}(c_(p), q, u(p), i_(u(q)))$ that $u(p)$ is equal to $\mathrm{ind}_\mathbb{1}^{(-) =_\mathbb{1} p}(p, c_(p))$ in $C(p)$. Thus, the identity type $q =_\mathbb{1} p$ is a contractible type for all elements $p:\mathbb{1}$ and $q:\mathbb{1}$, with element $\eta_\mathbb{1}^{(-) =_\mathbb{1} p}(c_(p), q, u(p), i_(u(q))):q =_\mathbb{1} p$. The positive unit type is a contractible type. Since the identity type $q =_\mathbb{1} p$ is a contractible type for all elements $p:\mathbb{1}$ and $q:\mathbb{1}$, with element $\eta_\mathbb{1}^{(-) =_\mathbb{1} p}(c_(p), q, u(p), i_(u(q))):q =_\mathbb{1} p$, it follows that $\mathbb{1}$ is an h-proposition. Since $\mathbb{1}$ is also pointed with element $:\mathbb{1}$, it is thus a contractible type. By the definition of contractible type, there are elements $p_S:\sum_{a:S} \prod_{b:S} a =_S b$ and $p_T:\sum_{a:T} \prod_{b:T} a =_T b$, and thus, an element $\pi_1(p_S)$ and a witness $\pi_2(p_S)$ that $\pi_1(p_S)$ is a center of contraction of $S$, and an element $\pi_1(p_T)$ and a witness $\pi_2(p_T)$ that $\pi_1(p_T)$ is a center of contraction of $T$. By the uniqueness principle of contractible types, it suffices to define a function $f:S \to T$ at $\pi_1(p_S)$. We define it as $f(\pi_1(p_S)) \coloneqq \pi_1(p_T)$. Now, all that’s left is to prove that the fiber of $f$ at all elements $a:T$ is contractible. But by the uniqueness principle of contractible types, it suffices to prove it for the center of contraction $\pi_1(p_T)$. The canonical element $$ is in the fiber of $f$ at $\pi_1(p_T)$, and since $\pi_1(p_S)$ is the center of contraction of $S$, the fiber of $f$ at $\pi_1(p_T)$ is contractible, and thus the fiber of $f$ at every element $a:T$ is contractible. Thus, $f$ is an equivalence of types between the contractible types $S$ and $T$. The positive and negative unit types are equivalent to each other. Since the negative unit type is contractible by definition, and the positive unit type is contractible, they are equivalent to each other. Extensionality principle Then we could derive the extensionality principle for the unit type: Given elements $x:\mathbb{1}$ and $y:\mathbb{1}$, there is an equivalence of types $(x =_\mathbb{1} y) \simeq \mathbb{1}$ between the identity type $x =_\mathbb{1} y$ and $\mathbb{1}$ itself. Since every identity type $x =_\mathbb{1} y$ is contractible for elements $x:\mathbb{1}$ and $y:\mathbb{1}$, and $\mathbb{1}$ itself is contractible, this implies that every identity type $x =_\ mathbb{1} y$ is equivalent to $\mathbb{1}$. The unit type as a univalent universe In set theory, regular and inaccessible cardinals are useful in determining size issues in strongly predicative and weakly predicative/impredicative mathematics respectively. Some definitions of regular and inaccessible cardinals do not have a requirement that the cardinals be infinite; thus finite cardinals such as $0$ and $1$ could be considered to be regular and inaccessible cardinals. Univalent universes in type theory which are closed under dependent sum types are the type theoretic equivalent of regular cardinals, and univalent universe which are additionally closed under power sets are the type theoretic equivalent of inaccessible cardinals. We inductively define the type family $x:\mathbb{1} \vdash \mathrm{El}_\mathbb{1}(x) \; \mathrm{type}$ by defining $\mathrm{El}_\mathbb{1}() \coloneqq \mathbb{1}$ The extensionality principle for the unit type then is simply the univalence axiom: $\mathrm{ext}_\mathbb{1}:\prod_{x:\mathbb{1}} \prod_{y:\mathbb{1}} (x =_\mathbb{1} y) \simeq (\mathrm{El}_\mathbb{1}(x) \simeq \mathrm{El}_\mathbb{1}(y))$ The unit type $\mathbb{1}$ represents the trivial universe, the universe with only one element up to identification, where every type in the universe is a contractible type and thus every contractible type is essentially $\mathbb{1}$-small. The equivalence type between the two contractible types $\mathrm{El}_\mathbb{1}(x)$ and $\mathrm{El}_\mathbb{1}(y)$ is itself contractible and thus equivalent to $\mathbb{1}$ itself. As proven in the previous section, $\mathbb{1}$ is equivalent to $x =_\mathbb{1} y$ for any $x:\mathbb{1}$ and $y:\mathbb{1}$. Thus the univalence axiom for $\ mathbb{1}$ is true. In addition, given any type $A$, functions into the unit type $A \to \mathbb{1}$ are families of contractible types. The universe $(\mathbb{1}, \mathrm{El}_\mathbb{1})$ is also closed under dependent sum types, as for any family of contractible types $p:B \to \mathbb{1}$, $\left(\sum_{x:\mathbb{1}} \mathrm{El}_\mathbb{1}(B(x))\right) \simeq \mathbb{1}$ This makes the unit type into a Tarski universe representing a finite regular cardinal. If the dependent type theory also has weak function extensionality, then the universe above is also closed under dependent product types, as weak function extensionality states that for any family of contractible types $p:B \to \mathbb{1}$, $\left(\prod_{x:\mathbb{1}} B(x)\right) \simeq \mathbb{1}$ making the unit type into a Tarski universe representing a finite product-regular cardinal. As a universe, the unit type also satisfies propositional resizing: the type of all $\mathbb{1}$-small propositions is simply the unit type itself as a weakly Tarski universe, which is essentially $\ mathbb{1}$-small. Thus, in the context of weak function extensionality, power sets exist in the unit type, making the unit type impredicative: power sets are just function types into the unit type. This makes the unit type into a Tarski universe representing a finite inaccessible cardinal. The unit type is also the only universe which contains itself, as any univalent universe which contains the empty type and itself makes the entire dependent type theory inconsistent. In lambda-calculus In lambda-calculus, for both the positive unit type and the negative unit type, the rule for building the unit type is the same, namely “it exists”: $\frac{ }{1\colon Type}$ As a positive type Regarded as a positive type, we give primacy to the constructors, of which there is exactly one, denoted $()$ or $tt$. $\frac{ }{() \colon 1 }$ The eliminator now says that to use something of type $1$, it suffices to say what to do in the case when that thing is $()$. $\frac{\vdash c\colon C}{u\colon 1 \vdash let () = u in c \;\colon C}$ Obviously this is not very interesting, but this is what we get from the general rules of type theory in this degenerate case. In dependent type theory, we should also allow $C$ to depend on the unit type $1$. We then have the ∞-reduction? rule: $let () = () in c \;\to_\beta\; c$ and (if we wish) the ∞-reduction? rule: $let () = u in c[()/z] \;\to_\eta\; c[u/z].$ The $\beta$-reduction rule is simple. The $\eta$-reduction rule says that if $c$ is an expression involving a variable $z$ of type $1$, and we substitute $()$ for $z$ in $c$ and use that (with the eliminator) to define a term involving another term $u$ of type $1$, then the result is the same as what we would get by substituting $u$ for $z$ in $c$ directly. The positive presentation of the unit type is naturally expressed as an inductive type. In Coq syntax: Inductive unit : Type := \| tt : unit. (Coq then implements beta-reduction, but not eta-reduction. However, eta-equivalence is provable with the internally defined identity type, using the dependent eliminator mentioned above.) As a negative type A negative type is characterized by its eliminators, which is a little subtle for the unit type. But by analogy with binary product types, which have two eliminators $\pi_1$ and $\pi_2$ when presented negatively, the negative unit type (a nullary product) should have no eliminators at all. To derive the constructors from this, we follow the general rule for negative types that to construct an element of $1$, it should suffice to specify how that element behaves under all the eliminators. Since there are no eliminators, this means we have nothing to do; thus we have exactly one way to construct an element of $1$, by doing nothing: $\frac{ }{()\colon 1}$ There is no $\beta$-reduction rule, since there are no eliminators to compose with the constructor. However, there is an $\eta$-conversion rule. In general, an $\eta$-redex? consists of a constructor whose inputs are all obtained from eliminators. Here we have a constructor which has no inputs, so it is not a problem that we have no eliminators to fill them in with. Thus, the term $()\colon 1$ is an “$\eta$-redex”, and it “reduces” to any term of type $1$ at all: $() \;\to_\eta\; u$ This is obviously not a well-defined “operation”, so in this case it is better to view $\eta$-conversion in terms of expansion: $u \;\to_\eta\; ().$ Positive versus negative In ordinary “nonlinear” type theory, the positive and negative unit types are equivalent. They manifestly have the same constructor, while we can define the positive eliminator in a trivial way as $let () = u in c \quad \coloneqq\quad c$ Note that just as for binary product types, in order for this to be a well-typed definition of the dependent eliminator in dependent type theory, we need to assume the $\eta$-conversion rule for the assumed negative unit type (at least, propositionally). Of course, the positive $\beta$-reduction rule holds by definition for the above defined eliminator. For the $\eta$-conversion rules, if we start from the negative presentation and define the positive eliminator as above, then $let () = u in c[()/z]$ is precisely $c[()/z]$, which is convertible to $c[u/z]$ by the negative $\eta$-conversion rule $() \;\leftrightarrow_\eta\; u$. Conversely, if we start from the positive presentation, then we have $() \;\leftrightarrow_\eta\; let () = u in () \;\leftrightarrow_\eta\; u$ where in the first conversion we use $c \coloneqq ()$ and in the second we use $c\coloneqq z$. As in the case of binary product types, these translations require the contraction rule and the weakening rule; that is, they duplicate and discard terms. In linear logic these rules are disallowed, and therefore the positive and negative unit types become different. The positive product becomes “one” $\mathbf{1}$, while the negative product becomes “top” $\top$. Categorical semantics There are two interpretations of the unit type, corresponding to the positive and negative unit types. Namely: • The negative unit type corresponds to a terminal object in a category • The positive unit type with both $\beta$-conversion and $\eta$-conversion corresponds to an initial object with a point, a morphism from some unit object $I$ to the object corresponding to the unit type, in a category. These two conditions are the same if the syntactic category is a cartesian monoidal category, where the points are the global elements and the unit is the terminal object. In general, however, the points may not be defined out of the terminal object. In linear logic, therefore, the categorical terminal object interprets “top” $\top$, while the unit object of an additional monoidal structure interprets “one” $\mathbf{1}$. On the unit type as an inductive type satisfying singleton induction:	{"url":"https://ncatlab.org/nlab/show/unit+type","timestamp":"2024-11-12T13:35:36Z","content_type":"application/xhtml+xml","content_length":"130003","record_id":"<urn:uuid:5849ff72-e8cc-4f73-860f-9a484d9b9bfc>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.45/warc/CC-MAIN-20241112113320-20241112143320-00283.warc.gz"}
GEOMCONSTRAINT command Applies geometric relationships between entities, on entities, and on valid constraint points. Constraints keep entities in a fixed position, such as perpendicularly or vertically. Geometric constraints can be applied to the following entities and their constraint points: Entity Type Valid Constraint Points Lines Endpoints, midpoint Arcs, elliptical arcs Endpoints, center point, midpoint Circles, ellipses Center point Polyline segments Endpoints, vertices, midpoints Polyline arcs Endpoints, vertices, midpoints, center points Splines Endpoints Inserted entities: blocks, xrefs, text, mtext, attributes, tables Insertion points Constrains entities or pairs of points to lie parallel to the X-axis of the current coordinate system. See the GCHORIZONTAL command. Constrains entities or pairs of points to lie parallel to the Y-axis of the current coordinate system. See the GCVERTICAL command. Constrains two entities to lie perpendicularly to each other. See the GCPERPENDICULAR command. Forces two entities to be parallel to one other. See the GCPARALLEL command. Constrains two entities to maintain a point of tangency to each other or their extensions. See the GCTANGENT command. Forces a spline to maintain fluid geometric continuity with another spline, line, arc or polyline. See the GCSMOOTH command. Applies a coincident geometrical constraint to two points or constrains a point to an entity. See the GCCOINCIDENT command. Constrains the center points of circles, arcs, ellipses or elliptical arcs to coincide. See the GCCONCENTRIC command. Forces entities to be collinear. See the GCCOLLINEAR command. Constrains two entities or points to lie symmetrically with respect to a selected line. See the GCSYMMETRIC command. Constrains circular entities to the same radius, or linear entities to the same length. See the GCEQUAL command. Constrains points and entities at a fixed position. See the GCFIX command.	{"url":"https://help.bricsys.com/en-us/document/command-reference/g/geomconstraint-command?version=V23","timestamp":"2024-11-09T00:02:23Z","content_type":"text/html","content_length":"90762","record_id":"<urn:uuid:6595407f-4298-4df2-af1f-35319e4afbb6>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00633.warc.gz"}
from Prostějov to 568 02 Svitavy 19 Mins - Total Flight Time from Prostějov to 568 02 Svitavy Plane takes off from Prostějov, CZ and lands in 568 02 Svitavy, CZ. Current Time in Svitavy: Tuesday November 5th 9:02pm. Estimated Arrival Time: If you were to fly from Svitavy now, your arrival time would be Tuesday November 5th 9:21pm (based on Svitavy time zone). * Flight duration has been calculated using an average speed of 435 knots per hour. 15 minutes has been added due to takeoff and landing time. Note that this time varies based on runway traffic. Other factors such as taxing and not being able to reach or maintain a speed of 435 knots per hour has not been taken into account. Flight Time Summary Your in air flight time starts at Prostějov and ends at 568 02 Svitavy. Estimated arrival time: Tuesday November 5th 9:21pm (based on destination time zone). You can see why your trip to Prostějov takes 19 mins by taking a look at how far of a distance you would need to travel. You may do so by checking the flight distance between Prostějov and 568 02 After seeing how far Prostějov is from 568 02 Svitavy by plane, you may also want to get information on route elevation from Prostějov to 568 02 Svitavy. Did you know that 568 02 Svitavy can be reached by car? If you'd like to drive there, you can check the travel time from Prostějov to 568 02 Svitavy. To see how far your destination is by car, you can check the distance from Prostějov to 568 02 Svitavy. If you need a road map so that you can get a better understanding of the route to 568 02 Svitavy, you may want to check the road map from Prostějov to 568 02 Svitavy. If you're now considering driving, you may want to take a look at the driving directions from Prostějov to 568 02 Svitavy. Whether the trip is worth the drive can also be calculated by figuring out the fuel cost from Prostějov to 568 02 Svitavy. Recent Flight Times Calculations for Prostějov CZ: Flight Time from Prostějov to Prague Flight Time from Prostějov to Klosterneuburg Flight Time from Prostějov to Głuchołazy Flight Time from Prostějov to Skalica Flight Time from Prostějov to Šumperk « RSS Flight Times Calculations for Prostějov CZ »	{"url":"https://www.distancesto.com/flight-time/cz/prostejov-to-568-02-svitavy/history/1554893.html","timestamp":"2024-11-05T20:02:54Z","content_type":"text/html","content_length":"49271","record_id":"<urn:uuid:82284696-7c7b-438a-8c5a-2186516857c3>","cc-path":"CC-MAIN-2024-46/segments/1730477027889.1/warc/CC-MAIN-20241105180955-20241105210955-00273.warc.gz"}
Minimization for ill-conditioned problems Regularized quasi-Newton optimisation Currently the only function, rnewt implements general-purpose regularized quasi-Newton optimisation routines as presented in Kanzow and Steck (2023). The C++ code is written from scratch, and the More-Thuente linesearch script is an R-port specifically written for this implementation, but translated from the python implementation associated to the article. Kanzow, C., & Steck, D. (2023). Regularization of limited memory quasi-Newton methods for large-scale nonconvex minimization. Mathematical Programming Computation, 15(3), 417-444. Sugimoto, S., & Yamashita, N. (2014). A regularized limited-memory BFGS method for unconstrained minimization problems. inf. téc.	{"url":"https://cran.rstudio.org/web/packages/minic/readme/README.html","timestamp":"2024-11-08T22:08:30Z","content_type":"application/xhtml+xml","content_length":"2571","record_id":"<urn:uuid:f164591f-5a49-41b2-9130-41290bea52c8>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00341.warc.gz"}
Recursive Algorithms: How They Solve Complex Problems » Ldat Recursive Algorithms Recursive algorithms play a critical role in computer science and programming by offering an elegant solution to complex problems. They work by breaking down a problem into smaller subproblems, which are then solved using the same method. This process, known as recursion, can be used to solve a wide array of problems, from simple mathematical computations to more intricate tasks like data structure traversal and dynamic programming. In this article, we’ll delve into the workings of recursive algorithms, examine how they solve complex problems, and explore their applications in various domains. 1. Understanding the Basics of Recursive Algorithms A recursive algorithm is one that calls itself with modified parameters to solve a problem incrementally. Each recursive call reduces the size or complexity of the problem until it reaches a base case, which can be directly solved without further recursion. For example, a recursive algorithm for calculating the factorial of a number nn can be defined as follows: • If n=0n = 0 or n=1n = 1, return 1. (This is the base case.) • Otherwise, return n×factorial(n−1)n \times \text{factorial}(n – 1). This process continues until nn reaches the base case, and then the algorithm “unwinds” as each recursive call returns its result up the chain. 2. The Structure of a Recursive Algorithm Recursive algorithms generally have three key components: • Base Case: The stopping condition that ends the recursion, preventing infinite loops. • Recursive Step: The part where the function calls itself with a modified parameter. • Return Statement: Returns a result from either the base case or the recursive step. This structure enables the algorithm to tackle complex problems by building up solutions from simpler cases. 3. Types of Recursion • Direct Recursion: When a function directly calls itself. • Indirect Recursion: When a function calls another function, which then calls the original function, forming a loop. • Tail Recursion: A type of recursion where the recursive call is the last operation in the function, allowing for optimization by some compilers. • Non-Tail Recursion: Any recursive call not in the tail position, which often builds up additional overhead on the call stack. 4. Advantages of Using Recursive Algorithms • Simplicity and Clarity: Recursive solutions are often more concise and easier to understand, especially for problems like traversing hierarchical data structures. • Direct Representation of Problem Structure: Recursion often mirrors the problem structure, making it intuitive for problems involving nested structures, such as tree traversal. • Reduced Code Length: Recursion can eliminate the need for iterative loops, reducing the amount of code needed. 5. Challenges and Drawbacks of Recursion • Stack Overflow: Recursion relies on the call stack, which is limited. Deep recursion can lead to a stack overflow error. • Higher Memory Usage: Recursive calls consume more memory due to the stacking of function calls. • Potentially Slower Execution: Recursion can be less efficient than iteration because of the overhead of function calls. 6. Applications of Recursive Algorithms in Computing Recursive algorithms are foundational in areas such as: • Sorting Algorithms: QuickSort and MergeSort use recursion to break down arrays into smaller parts. • Data Structure Traversal: Recursion is widely used for traversing trees and graphs, as it aligns with their hierarchical structure. • Dynamic Programming: Recursive algorithms are essential in dynamic programming, where problems are solved by combining solutions to subproblems. 7. Recursive Algorithms in Mathematics Recursion is frequently applied in mathematics to solve problems such as: • Factorials: As mentioned, the factorial function is a classic example of recursion. • Fibonacci Sequence: The Fibonacci sequence is often computed recursively, where F(n)=F(n−1)+F(n−2)F(n) = F(n-1) + F(n-2). • Exponentiation: Recursive methods can be used to efficiently compute powers of a number, especially using the technique of exponentiation by squaring. 8. Recursive Solutions for Sorting and Searching Recursive algorithms form the backbone of many efficient sorting and searching techniques: • MergeSort: Splits an array into halves recursively, sorts them, and merges them back. • QuickSort: Divides an array by a pivot, recursively sorts each partition. • Binary Search: Recursively halves the search space, reducing the problem size logarithmically. 9. Tree and Graph Traversal Using Recursion Trees and graphs are naturally recursive structures, making them ideal for recursive algorithms: • Tree Traversal: Recursive methods like pre-order, in-order, and post-order traversal are essential in working with binary trees. • Graph Traversal: Depth-First Search (DFS) uses recursion to explore nodes and edges deeply before moving to adjacent nodes. 10. Recursion in Dynamic Programming Many dynamic programming problems can be solved recursively with memoization to store subproblem results: • Knapsack Problem: A recursive approach with memoization can efficiently solve this combinatorial optimization problem. • Longest Common Subsequence: By breaking the problem into smaller subsequences, recursion can identify the solution by comparing character by character. 11. Recursion and Divide-and-Conquer Algorithms Recursive algorithms are synonymous with divide-and-conquer strategies, where a problem is divided into smaller, more manageable parts: • Strassen’s Matrix Multiplication: Recursively divides matrices to achieve faster multiplication. • Karatsuba Multiplication: Uses recursion to perform efficient multiplication of large numbers. 12. Tail Recursion Optimization Some languages and compilers optimize tail-recursive functions by reusing stack frames, improving efficiency: • Examples of Tail Recursion: Functions like calculating the sum of an array can be implemented in a tail-recursive manner to benefit from optimization. 13. Memoization and Recursion Memoization stores the results of expensive function calls, improving the efficiency of recursive algorithms: • Applications in Dynamic Programming: Memoization is critical in problems like Fibonacci sequence and Edit Distance, where redundant calculations are minimized. 14. Recursion in Functional Programming Languages Functional programming languages often rely heavily on recursion instead of loops: • Examples: Languages like Haskell, Lisp, and Scheme use recursion as a primary means of iteration. • Benefits in Functional Programming: Functional languages often optimize recursion automatically, making it efficient for problem-solving. 15. When Not to Use Recursion While recursion has its strengths, there are scenarios where iteration may be more suitable: • High Overhead Situations: For simple loops with high recursion depth, iteration may be more efficient. • Memory Constraints: For memory-intensive tasks, avoiding recursion can help prevent stack overflow and high memory usage. 16. Testing and Debugging Recursive Algorithms Recursion can be challenging to debug due to the nested nature of calls: • Tools for Debugging: Using a debugger to step through recursive calls can clarify how the algorithm progresses. • Techniques for Testing: Recursion often benefits from extensive testing of base and edge cases to ensure correctness. 17. Best Practices for Writing Recursive Algorithms • Clearly Define the Base Case: A well-defined base case is crucial for avoiding infinite loops. • Optimize with Memoization if Necessary: Especially for problems with overlapping subproblems, memoization is essential. • Consider Stack Usage: If stack space is a concern, consider transforming the algorithm into an iterative solution. 18. Future Trends in Recursive Algorithms • Advances in Compiler Optimization: With continued improvement in compilers, recursion may become more efficient and widely used. • Growing Role in AI and Machine Learning: Recursive algorithms may find new applications in deep learning, such as recursive neural networks for structured data. • Expanding Use in Quantum Computing: Quantum algorithms like Grover’s algorithm use recursive principles to speed up complex calculations. FAQs about Recursive Algorithms Q1: What is the primary advantage of recursive algorithms? Recursive algorithms often provide simpler, more intuitive solutions for problems that involve nested or hierarchical structures, like trees and graphs. They reduce code complexity and can make programs easier to understand and maintain. Q2: Are recursive algorithms always slower than iterative algorithms? Not necessarily. While recursion can add overhead, it can also be optimized in some cases. For certain problems, recursion is not only simpler but also faster, especially when paired with techniques like memoization. Q3: What is a base case in recursion? The base case is the condition that stops the recursion. Without a base case, a recursive algorithm would run indefinitely, leading to a stack overflow. Q4: Can recursion cause memory issues? Yes, recursive calls use the call stack, and deep recursion can cause stack overflow errors or high memory usage. Using tail recursion and memoization can help mitigate these issues. Q5: How does memoization enhance recursive algorithms? Memoization stores the results of subproblems so they can be reused, reducing the number of recursive calls. This is particularly useful for problems with overlapping subproblems, like dynamic Q6: In what programming languages is recursion most commonly used? Recursion is prevalent in functional programming languages like Haskell, Lisp, and Scheme. However, it is also commonly used in imperative languages like Python, C++, and Java for tasks like sorting, searching, and tree traversal. Recursive algorithms provide powerful techniques for solving complex problems across various domains, from mancingduit computer science and mathematics to artificial intelligence and beyond. While they may have some limitations, their elegance and effectiveness make them indispensable tools for programmers and computer scientists alike.	{"url":"https://www.ldat.org/recursive-algorithms-how-they-solve/","timestamp":"2024-11-10T02:45:44Z","content_type":"text/html","content_length":"85943","record_id":"<urn:uuid:ec96eeff-bcb1-4b39-8a3f-c02a27bf6c22>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00437.warc.gz"}
whole numbers Archives - Quirky Science [caption id="attachment_5613" align="alignright" width="480"] Fractal[/caption] For some, it's actually fun when they come across secondary school math problems plus solutions. It's because they are no longer accountable, since they graduated years ago. Math Problems Plus Solutions Problem 1: Find the slope intercept form of the line passing through the point (– 1, 5) and parallel to the line – 6x – 7y = – 3. The line given is rewritten (in slope-intercept form, y = mx + b) as y = – 6/7 x + 3/7 Thus the slope m = – 6/7 Now two lines are parallel if they have the same slope. So, y = – 6/7 x + b is the formula for the new line, with the intercept not yet solved. We do so by inserting…	{"url":"https://www.quirkyscience.com/tag/whole-numbers/","timestamp":"2024-11-04T15:12:44Z","content_type":"text/html","content_length":"117314","record_id":"<urn:uuid:ab3a5007-aedd-4463-bcec-54c8812761d2>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00806.warc.gz"}
Part III: Probabilistic databases, provenance, knowledge compilation $\newcommand{\calQ}{\mathcal{Q}}$ $\newcommand{\Consts}{\mathsf{Consts}}$ $\newcommand{\vars}{\mathrm{vars}}$ $\newcommand{\Pr}{\mathrm{Pr}}$ $\newcommand{\calD}{\mathcal{D}}$ $\newcommand {\calH}{\mathcal{H}}$ $\newcommand{\pqetid}{\textbf{PQE}_\text{TID}}$ $\newcommand{\pqe}{\textbf{PQE}}$ $\newcommand{\atoms}{\textrm{Atoms}}$ Probabilistic databases and probabilitic query evaluation: definitions Probabilistic databases: definition Probabilistic databases and semantics of query evaluation We fix as usual a database schema $\Sigma$ and set of constants $\Consts$. A probabilistic database $\calD = (W,\Pr)$ consists of a finite set $W=\{D_1,\ldots,D_n\}$ of databases over $\Sigma$ (called the possible worlds of $\calD$), each having a probability $Pr (D_i) \in [0,1]$, with $\sum_{D\in W} Pr(D) = 1$. $W = \{D_1,D_2,D_3\}$ with $\Pr(D_1) = 0.5$, $\Pr(D_2) = 0.4$, $\Pr(D_1) = 0.1$, and $D_1$, $D_2$, $D_3$ being DBs (TODO whiteboard) Let $q$ be a Boolean query. Define the probability that $\calD$ satisfies $q$, written $Pr(\calD \models q)$, to be $Pr(\calD \models q) = \sum_{D\in W, D\models q} \Pr(D)$. Continuing the above example, consider the query $q = \exists x,y: R(x,y) \land S(y)$. Then $Pr(\calD \models q) = \ldots$. We can generalize this to non-Boolean queries: for a given tuple $\bar{t}$, define the probability that $\bar{t}$ is in the result of $q$ on $\calD$ as $\Pr(\bar{t}\in q(\calD)) = \sum_{D\in W, \bar{t}\in q(D)} \Pr(D)$ The probabilistic query evaluation problem (PQE) is the problem of, given a Boolean query $q$ and PDB $\calD$, computing $\Pr(\calD \models q)$. Here again we can consider the data or the combined complexity. In data complexity we have one problem $\pqe(q)$ for each Boolean query $q$. The complexity will depend on how $\calD$ is represented as input. Assume that $\calD$ is simply represented by the list of all its possible worlds together with their probabilities, and let $q$ be a Boolean query. Show that $\pqe(q)$ reduces in PTIME to ModelChecking($q$), and vice-versa. In general however probabilistic data do not come in this form but instead are given by some kind of implicit factorization. We’ll see next too examples of probabilistic database formalisms that are closer to real-life applications Tuple-independent PDBs (TIDs) One way to represent a probabilistic database in a compact manner is to start from a database $D$, and then annotate each of its facts $f=R(\bar{t})$ by a probability value $\pi(f)$, with the intended semantics that $f$ is present in the database with probability $\pi(f)$ (and absent with probability $1-\pi(f)$), independent of the other facts. This gives rise to the tuple-independent database (TID) model. Formally, a TID $T = (D,\pi)$ consists of a database $D$ and of a function $\pi:D \to [0,1]$. It defines a PDB $\calD_T = (W_T,\Pr_T)$ in the following way: 1. The possible worlds $W_T$ is the set of all subsets of $D$ (seing $D$ as a set of facts) 2. For $D' \in W_T$, we define $\Pr_T(D') = (\prod_{f\in D'} \pi(f)) \times (\prod_{f\in D\setminus D'} (1-\pi(f))$ Show that this defines a valid probabilistic database, i.e., that the sum of probabilities of the possible worlds is always equal to $1$. $T$ containing the following facts: Teaches(Charles, Databases) with probability $0.8$, Teaches(Mikaël, Databases) with probability $0.5$, Teaches(Sylvain, Functional Programming) with probability $1$. What are the possible worlds? If $q$ is the Boolean query “there are two different people teaching the same course, what is $\Pr(T\models q)$? Same question if we add the fact Teaches(Jean, Databases). This is a much more natural way of representing probabilistic databases! Note that if we have $n$ facts, then this concisely represents $2^n$ possible worlds. Unfortunately this conciseness comes at a price: first, we will see (exercices) that not all PDBs can be represented as a TID. Second, we will see that the complexity of PQE can quickly become high even for very simple queries. Bloc-independent PDBs (BIDs) TODO define if time Complexity of PQE on TIDs Here we will give a quick overview on PQE over TIDs, in data complexity. For a Boolean query $q$, let us denote $\pqetid(q)$ the following problem: $\pqetid(q)$ is the following problem: • INPUT: A TID $T = (D,\pi)$. • OUTPUT: $Pr_\pi(T\models q)$. For some queries, for instance CQs, this problem is in polynomial time: Show that for $q = \exists x: R(x,x)$, $\pqetid(q)$ is in PTIME. (Done in class) However we will see that for some other queries the problem is (highly) intractable. To do this, we need to introduce the complexity class #P. Interlude on the complexity class #P Informally, #P is the class that corresponds to counting the number of solutions of problems that are in NP. Formally, #P is the set of all functions $f:\{0,1\}^* \to \mathbb{N}$ such that there exists a PTIME nondeterministic Turing machine $M$ such that for all $x\in \{0,1\}^*$, $f(x)$ is precisely the number of accepting runs of $M$ on $x$. See the Wikipedia page for more details. You already know that SAT, or even 3-CNF SAT, is NP-complete. It turns out that this carries over to the counting versions and to #P-completeness. #SAT is the following problem: • INPUT: A propositional formula $\varphi$ over variables $X$ • OUTPUT: The number of valuations $\tau:X\to \{0,1\}$ that satisfy $\varphi$. Theorem: #SAT is #P-complete, even when restricted to 3-CNF formulas. (This means that #SAT is in #P, and that it is #P-hard, i.e., for every other problem A in #P, there exists a polynomial-time reduction from A to #SAT.) Explain (done in class) why the problems #SAT, #VERTEX-COVER, #CLIQUES are in #P. (It turns out that these are also #P-complete.) To show hardness of a simle query for PQE, we will need a stronger result than #SAT for 3-CNFs is #P-hard. Define #PP2DNF (for Partitionned Positive 2-DNFs) to be the following problem: #PP2DNF is the following problem: • INPUT: A PP2DNF formula. This is a DNF formula over variables $X,Y$ with $X\cap Y = \emptyset$ in which every clause of the formula uses one variable from $X$ and one variable from $Y$, without negation. So in all generality such a formula can be written as $\varphi = \bigvee_{i=1}^m (x_{i,j} \land y_{i,j})$, with $x_{i,j} \in X$ and $y_{i,j} \in Y$. • OUTPUT: The number of valuations $\tau:X\to \{0,1\}$ that satisfy $\varphi$. Theorem: #PP2DNF is #P-complete. (We won’t prove it in this course though.) Example of a CQ for which PQE is #P-hard Proposition: Let $q_{\text{RST}}$ be the BCQ $q_{\text{RST}} = \exists x,y: R(x) \land S(x,y) \land T(y)$. Then $\pqetid(q_{\text{RST}})$ is #P-hard. What is interesting is that the model checking problem for this query is PTIME (since this is a FO query), but the probabilistic variant is intractable. Dichotomy theorems Theorem: (Dichotomy theorem for UCQs [Dalvi & Suviu, 2012]) For any Boolean UCQ $q$, the problem PQE($q$) is either PTIME or #P-hard. This is an important theorem in the field, whose proof is about 100 pages long, so we will not cover it this course. Note that the result is not obvious, since it is not the case that all counting problems in #P are either in PTIME or #P-complete (similarly to Ladner’s theorem. We will however prove a (much) simpler version of it, for so-called self-join free Boolean CQs. We need a few definitions. A BCQ $q$ is self-join free (written then SJFBCQ) if all its relational symbols are distinct. For instance the query $q_\text{RST}$ from above is self-join free, whereas the query $\exists x,y R (x,y)\land R(y,x)$ is not. For $q$ a BCQ and $x\in \vars(q)$, denote by $\atoms(x)$ the set of atoms of $q$ in which $x$ occurs. For instance for the query $q_{\text{RST}}$ we have $\atoms(x) = \{ R(x), S(x,y)\}$ and $\atoms(y) = \{S(x,y),T(y)\}$. Call a SJFBCQ hierarchical if, for every $x,y\in \vars(q)$, we have either $\atoms(x)\cap \atoms(y) = \emptyset$, or $\atoms(x)\subseteq \atoms(y)$, or $\atoms(y)\subseteq \atoms(x)$. For instance $q_\text{RST}$ is not hierarchical whereas the query $\exists x,y R(x) \land S(x,y)$ is. We will prove: Theorem: (dichotomy theorem for SJFBCQs) Let $q$ be a SJFBCQ. If $q$ is hierarchical then $\pqetid(q)$ is PTIME, otherwise it is #P-hard. For the hardness part, assuming that $q$ is not hierarchical, then we know that there are variables $x,y\in \vars(q)$ and an atom $R(\bar{t_1})$ in $q$ such that $x$ occurs in $\bar{t_1}$ but not $y$, another atom $S(\bar{t_1})$ such that both $x$ and $y$ occur in $\bar{t_2}$, and another atom $T(\bar{t_3})$ such that $y$ occurs in $\bar{t_3}$ but not $x$. Then we can do a reduction from #PP2DNF using the same idea that we used for the query $q_{\text{RST}}$. Work out the details of the reduction by yourself. (Hint: for the positions that do not correspond to $x$ or $y$, use the same constant in the construction.) To show the other direction, let us first define the hypergraph associated to $q$, denoted $\calH_q = (V,E)$, and defined as follows: 1. The nodes $V$ of $\calH_q$ are the atoms of $q$; 2. For every variable $x\in \vars(q)$, we have an hyperedge that is $\atoms(x)$. Then, when we try to draw the hypergraph of a hierarchical SJFBCQ, we understand why these queries are called hierarchical! From there, we can see how an algorithm is going to work, using the fact that there is no self-join and the fact that the facts are independent of each others (similar to what we did on the examples with the relation Teaches). Compiled the: mar. 22 oct. 2024 11:40:58 CEST	{"url":"https://paperman.name/enseignement/2023/DB2/partie3_en.html","timestamp":"2024-11-04T01:35:08Z","content_type":"application/xhtml+xml","content_length":"18867","record_id":"<urn:uuid:65eb6931-25b9-42a0-887b-473cc369a7c6>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00091.warc.gz"}
This document illustrates how to sample $\alpha$-shapes from a true probability distribution in three dimensions. The main function within the package to do this is sampling3Dashape, which generates $\alpha$-shapes given the parameters. Package rgl is needed for plotting, and plots will show in a pop out window. There are several ways to adjust the hierarchical distribution of the function which will be discussed throughout the document. The function requires only parameter $N$, the number of shapes to be sampled. All other parameters are set to default, and the function samples an $\alpha$-shape from the following distribution: \[\alpha \sim \mathcal{N}_T(\mu=0.25,\sigma=0.5, a=\min(0.1, \tau/4), b=\tau/2)\] \[n \| \alpha = n_{c min}(\alpha, \delta=0.05)\] \[x_1, ..., x_n \sim \text{Unif}(\mathcal{M}) \] where $n$ the number of points sampled is dependent on the number of points needed to produce a connected shape for a randomly selected $\alpha$, $\delta$ is the probability that the generated shape has more than one connected component, and points are selected uniformly from some manifold $\mathcal{M}$. We could allow the lower bound of the truncated normal distribution of $\alpha$ to be as small as $a=0$, however, we set it to $a= min(0.1, tau/4)$ to prevent computational bottleneck. Bounds of the truncated normal distribution are fixed for the user. Values of $\tau$ for different underlying manifolds are as follows: • If the underlying manifold is a square with side length $r$, $\tau=r/2$. • If the underlying manifold is a circle with radius $r$, $\tau=r$. • If the underlying manifold is an annulus with inner radius $r_{min}$ and outer radius $r$, $\tau = r_{min}$. The condition number is not a user adjusted parameter. For demonstration purposes, we set $N=1$. The sampling3Dashape function returns a list of length $N$ of those objects. my_ashape = sampling3Dashape(N=1) #> Device 1 : alpha = 0.2434609 To make the number of points a random variable in and of itself, we can add a discrete distribution $\pi$ to $n \| \alpha$. In the code, this discrete distribution is a Poisson distribution with default $\lambda = 3$. Parameter $\lambda$ can be adjusted by the user. The distribution from which the new shape is sampled is then given by: \[\alpha \sim NT(\mu=0.25,\sigma=0.5, a=\min(0.1, \tau/4), b=\tau/2)\] \[n \| \alpha = n_{min}(\alpha, \delta=0.05) + \text{Poisson}(\lambda)\] \[x_1, ..., x_n \stackrel{i.i.d.}{\sim} \text{Unif}(\ mathcal{M}) \] To make the code dynamic, set n.noise = TRUE. This code is where $\lambda = 3$. my_ashape = sampling3Dashape(N=1, n.noise = TRUE) #> Device 1 : alpha = 0.2092051 Code with the adjustment $\lambda = 10$: my_ashape = sampling3Dashape(N=1, n.noise = TRUE, lambda = 10) #> Device 1 : alpha = 0.231029 We can also change the dependence of $n$ relative to $\alpha$. First, we can make $n$ independent of $\alpha$ by setting n.dependent = FALSE. Then $n=20$ is the default number of points used. (If n.noise=TRUE, then 20 is the minimum number of points used before adding more based on a Poisson random variable.) Making $n$ independent from $\alpha$ allows for more variation in the resulting shapes, including the number of connected components. Example code with independent $n$ and noise: my_ashape = sampling3Dashape(N=1, n.dependent=FALSE, n.noise=TRUE, lambda = 5) #> Device 1 : alpha = 0.2488231 In the other direction, we can choose to make $n$ dependent on $\alpha$ such that the underlying manifoldâ€™s topology is preserved. In the case of a square, this means we will have one connected component with no holes with probability $1 - \delta$. Here, it is strict that $\alpha/2 < \tau$, which defaults to 1. Note that the smaller $\tau$ is, the smaller $\alpha$ has to be, the more points which must be sampled, and thus the slower the algorithm. Users will see the variation in the shapes will lie on the boundaries when setting nhomology=TRUE: my_ashape = sampling3Dashape(N=1, nhomology = TRUE) #> Warning in sampling3Dashape(N = 1, nhomology = TRUE): Both nhomology and #> nconnect are true, default to nhomology for choosing n. #> Device 1 : alpha = 0.1794529	{"url":"https://cran.rstudio.com/web/packages/ashapesampler/vignettes/probability_demo_3D.html","timestamp":"2024-11-12T19:08:20Z","content_type":"text/html","content_length":"1049111","record_id":"<urn:uuid:5b12893d-207d-483b-aeb4-bacde61d6367>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00392.warc.gz"}
On Till Plewe’s game and Matthew de Brecht’s non-consonance arguments Last time I mentioned that S[0] is not consonant. I had a preprint by Matthew de Brecht where he proved that, but I could not find the result in published form. Since then, M. de Brecht wrote to me, and told me a lot of interesting things. First, his proof can be found in [1]. His proof uses a topological game first invented by Till Plewe [2] in order to study locale products of spaces, and conditions under which the locale products coincide with the (locales underlying) the topological products, which M. de Brecht had initially used to show that the locale product of S[0] with itself is not spatial. Second, he pointed me towards an intriguing result by Ruiyuan Chen [3], which gives another connection between locale products (of dcpos) and the question of whether the poset-theoretic product of two dcpos with the Scott topology coincides with the topological product. Third, R. Chen has a simpler, alternative proof of the fact that the locale product of S[0] with itself is not spatial [4]. And so on! Hence I have too much to talk about, so I will have to make a choice. I will concentrate on explaining what [1] is all about. I am afraid that this post will mostly be paraphrasing M. de Brecht’s paper. I have merely added a few pictures, and I have reformulated a few notions; I hope this will contribute to make what he achieved easier to grasp. Till Plewe’s game Let X and Y be two topological spaces, U[0] be an open subset of X, V[0] be an open subset of Y, and U be an open cover of U[0] × V[0] by open rectangles. Till Plewe’s game G[X,Y](U[0], V[0], U) is played as follows. At each turn i≥1: 1. player I picks a point x[i] ∈ U[i–1]; 2. then player II picks an open neighborhood U[i] of x[i] included in U[i–1]; 3. player I picks a point y[i] ∈ V[i–1]; 4. then player II picks an open neighborhood V[i] of y[i] included in V[i–1]. So, yes, each player plays twice at each turn. Player II wins (at round i) if U[i] × V[i] is included in one of the open rectangles of the cover U. Otherwise, namely if the play goes along without player II even winning, player I wins. The following in Theorem 1.1 in [2]. Theorem (Plewe). If X and Y are sober spaces, either: • for every open subset U[0] of X, for every open subset V[0] of Y, for every open cover U of U[0] × V[0] by open rectangles, player II has a winning strategy in the game G[X,Y](U[0], V[0], U), and then the locale product of X and of Y is spatial; • or there is an open subset U[0] of X, there is an open subset V[0] of Y, and there is an open cover U of U[0] × V[0] by open rectangles such that player I has a winning strategy in the game G[X,Y ](U[0], V[0], U), and then the locale product of X and of Y is non-spatial. And the following is Theorem 4 in [1]. Theorem (de Brecht). If X is a consonant space, then for every open cover U of X × X by open rectangles, player II has a winning strategy in the game G[X,X](X, X, U). Hence there is an intriguing connection between consonance and the spatiality of products of locales, which seems to me to be largely unexplored. M. de Brecht says “However, the implicit connections we make in this paper are not as strong as we expect they really are” at the end of the introduction of [1]. I don’t quite know what he means by that, except perhaps that he finds them intriguing, and unexplored, too. I will not prove Plewe’s theorem here, but I will definitely give de Brecht’s proof of his theorem, and I will then explain how he uses it to prove that S[0] is not consonant. He also hints at the fact that this can be used to show that Q is not consonant. We have suffered quite a lot through the latter question, and I should take a stab at it some day, and explain what M. de Brecht had in mind in this respect—but not today. An ordinal measure of open rectangles The key to proving the above theorem by M. de Brecht is to realize that given any open cover U of X × X by open rectangles, where X is consonant, one can eventually reach the open rectangle X × X itself by repeating the operations of horizontal unions and vertical unions, transfinitely often, starting from the downward closure of U. Let me explain. The terms horizontal unions and vertical unions will be specific to this post. If you are given a collection A of open rectangles, then I will call a horizontal union of elements of A any union ∪[i ∈ I] (U[i] × V[i]), where each U[i] × V[i] is in A, and all the open sets V[i] are identical. In other words, this is obtained by putting side by side, horizontally, any number of open rectangles from A with the same vertical cross-section. Vertical unions are defined similarly; namely, in that case, we require all the open sets U[i] to be identical instead. The downward closure ↓U of an open cover U is by definition the collection of open rectangles that are included in some element of U. We define A[0](U) as ↓U. For every ordinal α, if we have already defined A[α](U), we define A[α+1](U) as the collection of horizontal unions of elements of A[α](U), plus all vertical unions of elements of A[α](U). (In particular, A[α](U) is included in A[α+1](U), since one can reobtain any rectangle of A[α](U) as, say, a horizontal union of just one rectangle from A[α](U).) For a limit ordinal α, A[α](U) is the union of the collections A[β](U), β<α. Lemma. For every ordinal α, A[α](U) is downward closed. Proof. By induction on α. Since A[0](U) = ↓U, A[0](U) is downward closed. Assuming A[α](U) downward closed, we show that A[α+1](U) is downward closed by showing that any open rectangle U × V included in some horizontal union ∪[i ∈ I] (U[i] × V[i]) of rectangles in A[α](U) is already in A[α+1](U) (and similarly for vertical unions; but the argument is symmetrical, hence omitted). This is easy: U × V is equal to ∪[i ∈ I] ((U[i] ∩ U) × (V[i] ∩ V)), which is a horizontal union of rectangles (U[i] ∩ U) × (V[i] ∩ V); each such rectangle is a subset of a rectangle in A[α](U), hence is in A[α](U) by induction hypothesis, so U × V = ∪[i ∈ I] ((U[i] ∩ U) × (V[i] ∩ V)) is also a horizontal union of elements of A[α](U); therefore U × V is in A[α+1](U). ☐ Let us write A[∞](U) for the union of the monotone sequence of collections A[α](U). A[∞](U) must be obtained as A[α](U) for some large enough ordinal α, since otherwise A[∞](U) would contain at least as many elements as there are ordinals, namely: A[∞](U) would be a proper class. And A[∞](U) cannot be a proper class, since it is included in a set—namely the set of all open rectangles on X × X. It follows that A[∞](U) is closed under both horizontal and vertical unions. Lemma. A[∞](U) is Scott-closed in OX × OX. Proof. Let (U[i] × V[i])[i ∈ I] be a directed family of elements of A[∞](U), and let U × V be its union. Without loss of generality, we may assume that none of the rectangles U[i] × V[i] is empty; otherwise, remove all of them and what remains will either be an empty family (and the union of the empty family is in A[1](U), hence in A[∞](U)) or will still be directed. Therefore every U[i], and every V[i], will now be considered as non-empty. The first thing we need to observe is that every rectangle U[j] × V[i], where i and j are possibly distinct indices in I, is in A[∞](U). Indeed, since the family is directed, we can find an index k in I such that both U[i] × V[i] and U[j] × V[j] are included in U[k] × V[k]. In particular, and because neither U[j] nor V[i] is empty, U[j] is included in U[k] and V[i] is included in V[k]. But U[k] × V[k] is in A[∞](U), and A[∞](U) is downwards closed, so U[j] × V[i] is in A[∞](U), too. Fixing i ∈ I, U × V[i] is the union of the sets U[j] × V[i], where j ranges over I. That is a horizontal union of elements of A[∞](U), hence is itself in A[∞](U). Then U × V is the union of the sets U × V[i], where i ranges over I, hence is a vertical union of elements of A[∞](U). It follows that U × V is also in A[∞](U). ☐ Let us define C(U) as the collection of open subsets U of X such that U × U is in A[∞](U). The map U ↦ U × U is Scott-continuous from OX to OX × OX, so C(U) is Scott-closed in OX. Let H(U) be the complement of C(U) in OX, namely the collection of open subsets U of X such that U × U is not in A[∞](U). We have just proved: Fact. H(U) is a Scott-open subset of OX. This is where consonance comes into play. Let me recall (from any of the earlier posts on the topic, namely 1 through 5 on this page) that a space X is consonant if and only if every Scott-open subset of OX is a union of sets of the form ■Q, where Q is compact saturated in X; and ■Q is the collection of open neighborhoods of Q. Lemma. If X is consonant, then X does not belong to H(U); equivalently, X × X is in A[∞](U). Proof. If X ∈ H(U) and X is consonant, then there is a compact saturated subset Q of X such that (X is in ■Q, which is obvious, and) ■Q is included in H(U). We aim for a contradiction. We recall that U is an open cover of X × X by open rectangles. In particular, for every point x in X, {x} × Q is included in the union of the open rectangles of U. Since {x} × Q is compact, it is included in a finite union ∪[y ∈ E(x)] (U[xy] × V[xy]) of open rectangles of U; notably, the index set E(x) is finite. Let U[x] be the intersection ∩[y ∈ E(x)] U[xy], and let V[x] be the union ∪[y ∈ E(x)] V[xy]. We obtain that {x} × Q is included in U[x] × V[x]. Note in particular that Q is included in V[x]. Additionally, U[x] × V[x] is the vertical union of the rectangles U[x] × V[xy], y ∈ E(x). Since each U[x] × V[xy] is included in U[xy] × V[xy], which is in U, U[x] × V[xy] is in A[0](U) = ↓U. Therefore U[x] × V[x] is in A[1](U). We use the compactness of Q once again: Q is included in a finite union U ≝ ∪[x ∈ E] U[x] (namely, the index set E is finite). Let V be the intersection ∩[x ∈ E] V[x]. Since each V[x] contains Q, V also contains Q. Then U × V is a horizontal union of open rectangles U[x] × V. Each of these open rectangles is included in U[x] × V[x], hence is in A[1](U). It follows that U × V is in A[2](U). Then open rectangle (U ∩ V) × (U ∩ V) is included in the latter, hence is also in A[2](U). Additionally, both U and V contain Q, so U ∩ V is in ■Q. Since ■Q is included in H(U), U ∩ V is in H(U), or equivalently, (U ∩ V) × (U ∩ V) is not in A[∞](U). This directly contradicts the fact that it is in A[2](U). ☐ For every open rectangle U × V in A[∞](U), there is least ordinal α such that U × V is in A[α](U). Let us call α the degree of the rectangle U × V. It is clear that the degree of each rectangle in A[∞](U) is a successor ordinal β+1. We have just proved that if X is consonant, then the degree of X × X is a well-defined notion. Winning Plewe’s game on consonant spaces Let me recall that M. de Brecht’s theorem states that if X is a consonant space, then for every open cover U of X × X by open rectangles, player II has a winning strategy in the game G[X,X](X, X, U). We start with the open rectangle X × X, which has a well-defined degree, as we have just seen. Player II’s strategy will consist in finding open sets U[i] and V[i], at round i, for each i≥1, in response to player I’s choice of points x[i] and y[i], in such a way that the degree of U[i] × V[i] is strictly smaller than that of U[i–1] × V[i–1], unless the degree of U[i–1] × V[i–1] is already equal to 0. If the latter happens, then U[i–1] × V[i–1] is in A[0](U) = ↓U, in which case player II has won. Hence, let us assume that we enter round i with an open rectangle U[i–1] × V[i–1] of degree β+1. (We remember that the degree is always a successor ordinal.) Hence U[i–1] × V[i–1] is either a horizontal or vertical union of rectangles of degrees at most β. • If U[i–1] × V[i–1] is a horizontal union of rectangles U’[j] × V[i–1] (j ∈ J) of degrees at most β, we reason as follows. Player I plays a point x[i] in U[i–1]. Then x[i] belongs to some U’[j], and we let player II play U’[j] for the next open set U[i]. Player I then plays a point y[i] in V[i–1], and we let player II play V[i–1] itself for V[i] (no change). As promised, the degree of U[ i] × V[i] is at most β, hence strictly below the degree β+1 of U[i–1] × V[i–1]. • If U[i–1] × V[i–1] is a vertical union of rectangles U[i–1] × V’[j] (j ∈ J) of degrees at most β, then player I plays x[i] in U[i–1], and this time player II simply plays U[i–1] for U[i] (no change). Player I plays y[i] in V[i–1]. Then y[i] is in some V’[j], and player II plays that V’[j] for V[i]. Once again, the degree of U[i] × V[i] is at most β, hence strictly below the degree β+1 of U[i–1] × V[i–1]. Since the degree of U[i] × V[i] decreases strictly at each round, and the ordering on ordinals is well-founded, eventually that degree must reach 0. As we have seen above, in that situation, player II has won the game. Hence we have proved de Brecht’s theorem, which we repeat here: Theorem (de Brecht). If X is a consonant space, then for every open cover U of X × X by open rectangles, player II has a winning strategy in the game G[X,X](X, X, U). S[0] is not consonant I: the setup M. de Brecht uses this to show that S[0] is not consonant [1, Section 4]. We use the contrapositive of the previous theorem: it suffices to show that player II does not have a winning strategy in the game G[X,X](X, X, U), for some open cover U of X × X (with X ≝ S[0]). In order to do so, we show that player I has a winning strategy in this game. Let me recap a few things about S[0] from last time. S[0] is the space of finite sequences of natural numbers, with the upper topology of the (opposite of the) suffix ordering. I write n::s for the list obtained from the list s by adding a number n in front, ε for the empty list, and then S[0] is a tree with the root ε at the top: Let me also recall that the closed subsets of S[0] (hence also the open subsets of S[0]) are in one-to-one correspondence with the finitely branching subtrees of S[0]. In other words, a closed subset C is the same thing as the downward closure ↓Min T of the set Min T of leaves of a finitely branching subtree T of S[0]. In the following picture, the tree is shown in red, and the closed set in blue. A subtree is simply an upwards-closed subset of S[0]; in particular, mind that we accept the empty set as a subtree. Since the open sets are the complements of the closed sets, the open sets are also characterized through finitely branching subtrees T, namely as the sets S[0]–↓Min T of points that are not below any leaf of T. M. de Brecht defines specific open subsets U[σ,τ], where σ and τ are finite lists of natural numbers, but I will make two modifications to this presentation. First, τ is not really used in the definition except for its length, so I will replace it with a natural number n, and I will write the corresponding open sets as U[σ,n]. Second, it is easier to understand what these open sets are through the trees T[σ,n] that are used to define them. T[σ,n] is defined as consisting of ε, plus the sequences k::s, where 0≤k≤\|σ\|+n and s ranges over the suffixes of σ. (\|σ\| denotes the length of σ.) In other words, draw the path from the root ε to σ, and add the first \|σ\|+n+1 successors of each node thus obtained. Here is what T[σ,n] is for σ = 2::0::ε and n=1. The path σ itself is shown in brown, and the additional successors are obtained by following the additional red edges. The leaves of T[σ,n] are exactly the sequences k::s of natural numbers such that: • s is a suffix of σ (i.e., above σ); • 0≤k≤\|σ\|+n; • and k::s is not a suffix of σ (otherwise it would be an internal vertex of T[σ,n]). The open set U[σ,n] is the open set defined by T[σ,n], namely the set S[0]–↓Min T[σ,n] of vertices that are below no leaf of T[σ,n]. I originally thought it might be easier to reason on finitely branching subtrees directly, instead of on open subsets of S[0], but that would get us lost. Instead, I will be following M. de Brecht; we make the following observations, which will distill the properties on finitely branching subtrees that we need in the proof to come, in terms of open sets: • Property (A). For every finite sequence σ of natural numbers, for every natural number n, for every element s of U[σ,n], in order to show that s is above σ, it suffices to show that every element of s (seen as a finite sequence of natural numbers) is ≤\|σ\|+n. • Property (B). For every open subset U of S[0], for every σ ∈ U, there are infinitely many successors n::σ of σ such that the complete subtree ↓(n::σ) is included in U. Those are proved as follows. For property (B), we write U as the set of points that are not below any leaf of some given finitely branching tree T. Hence σ is not below any leaf of T. Since T is finitely branching, there are infinitely many natural numbers such that n::σ is not in T. Let us consider any such n, and any element t of ↓(n::σ). If t were below some leaf x of T, then x would either have n::σ as a suffix or be itself a suffix of σ. The first case would imply that n::σ is in T, since T is upwards closed, and that is impossible, by definition of n. The second case would imply that σ would be below some leaf of T, which is impossible as well. Therefore t is in U. For property (A), let us imagine that s is the list m[1]::…::m[p]::ε, and that each m[i] is ≤\|σ\|+n. Since s is in U[σ,n], s does not have a suffix that would be a leaf of T[σ,n]. For every i with 1≤i ≤p, let us consider the suffix m[i]::…::m[p]::ε of s. That is not a leaf of T[σ,n], as we have just said: hence m[i+1]::…::m[p]::ε is not a suffix of σ, or m[i]>\|σ\|+n, or m[i]::…::m[p]::ε is a suffix of σ, by definition of the leaves of T[σ,n]. The middle alternative m[i]>\|σ\|+n is impossible by assumption, so m[i+1]::…::m[p]::ε is not a suffix of σ or m[i]::…::m[p]::ε is a suffix of σ. In other words, exploiting that “not a or b” is the same as “if a then b”, if m[i+1]::…::m[p]::ε is a suffix of σ then m[i]::…::m[p]::ε is a suffix of σ. We use this to show that m[i]::…::m[p]::ε is a suffix of σ for every i with 1≤i≤p+1, by induction on p+1–i. The base case (i=p+1: ε is a suffix of σ) is obvious, and the final case i=1 yields that s = m[1]::…::m[p]::ε is a suffix of σ. S[0] is not consonant II: playing the game In order to show that S[0] is not consonant, let me recall that we will show that player I has a winning strategy in some game G[X,X](X, X, U), where X=S[0]. We let U be the open cover of X × X consisting of the sets U[σ,\|τ\|] × U[τ,\|σ\|] where σ and τ range over all finite lists of natural numbers (all vertices of the tree S[0]), where \|σ\| and \|τ\| denote the lengths of σ and τ respectively. This is an open cover: σ is in U[σ,\|τ\|], because σ is not below any leaf of T[σ,\|τ\|], and similarly τ is in U[τ,\|σ\|], so (σ, τ) is in U[σ,\|τ\|] × U[τ,\|σ\|]. M. de Brecht builds player I’s strategy by describing it, and then showing that it has the right properties. The proof is subtler than its short length might suggest. I also find it useful to have an explicit list of the invariants we need to maintain from round i–1 to round i in the game, and here they are. At the end of round i (assuming that those properties already hold with i–1 in place of i at the end of round i–1), we will establish that: 1. the elements of x[i] (as a sequence of natural numbers) are smaller than or equal to \|y[i]\|; 2. the elements of y[i] are smaller than or equal to \|x[i]\|; 3. there are two distinct natural numbers n[i], n’[i]≤\|y[i]\| such that every point below n[i]::x[i], as well as every point below n’[i]::x[i], is in U[i]; 4. every pair (σ, τ) of finite sequences σ and τ such that (x[i], y[i]) is in U[σ,\|τ\|] × U[τ,\|σ\|] must be below (x[i], y[i]). Invariants 1 and 2 are they key trick, and will be enforced, not by making sure that the elements of x[i] and y[i] are small enough, rather by making x[i] and y[i] long enough, as sequences of natural numbers. Invariant 3, or at least the existence of one natural number n[i]≤\|y[i]\| such that ↓(n[i]::x[i]) is included in U[i], will be required to make sure that the points x[i] we will build are in U[i–1] (see below); the existence of a second natural number n’[i] will only be important at the very end of the proof. Invariant 4 will be a consequence of invariants 1 and 2, using property (A), and will be an important property in order to show that player I’s strategy is winning. With all that said, here is how M. de Brecht’s strategy proceeds. At the start of round i (i≥1), player I is given an open rectangle U[i–1] × V[i–1], not included in any rectangle of the form U[σ,\|τ \|] × U[τ,\|σ\|] but containing (x[i][–1], y[i][–1]). The latter pair is well-defined if i≥2, as this is the pair produced by player I at turn number i–1. If i=1, we agree to define (x[0], y[0]) as (ε, ε), so that we do not have to make a case distinction later. Also, we are informed that the invariants 1–4 hold at the end of round i–1. (Initially, invariants 1, 2 and 4 are certainly satisfied, pretty vacuously, at the “end of round i–1″ with i=1. Invariant 3 is also satisfied, for any choice of n[0] and of n’[0].) Let us enter round i≥1: • By invariant 3 at the end of round i–1, there is a natural number n[i–1]≤\|y[i][–1]\| such that every point below n[i]::x[i] is in U[i]; By (B), y[i][–1] has infinitely many successors n::y[i][–1] such that ↓(n::y[i][–1]) is entirely included in V[i–1]. We pick one of them. Player I then decides to play x[i] ≝ 0^n::n[i][–1]::x[i][–1]. Notice that x[i] is in U[i–1], thanks to invariant 3. • Now player II plays an open neighborhood U[i] of x[i] included in U[i–1], over which player I has no control. • By (B), x[i] has infinitely many successors whose downward closure is entirely included in U[i]. We pick two of them, p::x[i] and q::x[i], with p≠q and both p and q larger than or equal to n[i] [–1]. Player I then decides to play y[i] ≝ 0^p+q::n::y[i][–1]. (The number n was picked two bullet points above. The numbers p and q will be the n[i] and n’[i] needed to establish invariant 3. M. de Brecht chooses p+q as the exponent, but max(p,q) would be enough: the only requirement here is that p, q should both be smaller than or equal to \|y[i]\|, as we will see below.) Notice why y[i] is in V[i–1]: the way we have picked n was so that ↓(n::y[i][–1]) is entirely included in V[i–1], so adding any number of zeroes in front of n::y[i][–1], however large that may be, will still produce a point in V[i][–1]. • Player II plays an open neighborhood V[i] of y[i] included in V[i–1], over which player I still has no control. Note how the invariants are maintained: 1. The elements of x[i] are those of x[i][–1], which are all smaller than or equal to \|y[i][–1]\| by invariant 1 at the end of round i–1, plus n[i][–1] and 0. Since y[i][–1] is a suffix of y[i], we have \|y[i][–1]\|≤\|y[i]\|; 0 is smaller than or equal to any number, and n[i][–1]≤p≤\|y[i]\|. Therefore all the elements of x[i] are smaller than or equal to \|y[i]\|. 2. The elements of y[i] are those of y[i][–1], which are all smaller than or equal to \|x[i][–1]\| (hence to \|x[i]\|) by invariant 2 at the end of round i–1, plus n and 0. But x[i] = 0^n::n[i][–1]::x [i][–1], so in particular n≤\|x[i]\|. It follows that all the elements of y[i] are smaller than or equal to \|x[i]\|. 3. The number p was chosen so that ↓(p::x[i]) is entirely included in U[i]. Also, p≤\|y[i]\|. We can therefore take n[i] ≝ p. Similarly, we take n’[i] ≝ q, since ↓(q::x[i]) is entirely included in U [i] and q≤\|y[i]\|. 4. Let us imagine that (x[i], y[i]) is in an open rectangle of the form U[σ,\|τ\|] × U[τ,\|σ\|]. Since x[i][–1] is above x[i] and y[i][–1] is above y[i], (x[i][–1], y[i][–1]) is also in U[σ,\|τ\|] × U[τ,\| σ\|]. Hence, by invariant 4 at the end of round i–1, (x[i][–1], y[i][–1]) is above (σ, τ). In particular (and this is the only thing we will use the latter for), \|y[i][–1]\|≤\|τ\|≤\|σ\|+\|τ\|. By invariant 1 (at the end of round i–1), the elements of x[i][–1] are all less than or equal to \|y[i][–1]\|. The only additional non-zero element in x[i] is n[i][–1], which is less than or equal to \|y[i][–1]\| as well, by invariant 3. Therefore all the elements of x[i] are less than or equal to \|y[i][–1]\|, hence to \|σ\|+\|τ\|. By property (A), it follows that x[i] is above σ. In particular, \|x[i]\|≤\|σ\|≤\|σ\|+\|τ\|. By invariant 2 (at the end of round i this time), all the elements of y[i] are smaller than or equal to \|x[i]\|, hence to \|σ\|+\|τ\|. We can therefore use property (A) once more, and conclude that y[i] is above τ. This allows us to conclude that (x[i], y[i]) is above (σ, τ). We now need to show that U[i] × V[i] cannot be included in any rectangle of the form U[σ,\|τ\|] × U[τ,\|σ\|]. We reason by contradiction. Let us imagine that U[i] × V[i] ⊆ U[σ,\|τ\|] × U[τ,\|σ\|], for some finite lists of natural numbers σ and τ. Since (x[i], y[i]) is in U[i] × V[i], it is in U[σ,\|τ\|] × U[τ,\|σ\|], and therefore, by invariant 4, (x[i], y[i]) is above (σ, τ). This is the place where we need the two successors n[i]::x[i] and n’[i]::x[i] of x[i] guaranteed by invariant 3. Invariant 3 tells us that the downward closure of each one is included in U[i]. In particular, both n[i]::x[i] and n’[i]::x[i] are in U[i], hence in U[σ,\|τ\|]. All the elements of x[i] are smaller than or equal to \|y[i]\| by invariant 1, hence to \|τ\|≤\|σ\|+\|τ\|, since y[i] is above τ. Invariant 3 also tells us that n[i], n’[i] ≤ \|y[i]\|, hence they are also both less than or equal to \|σ\|+\|τ\|. By property (A), it follows that both n[i]::x[i] and n’[i]::x[i] are above σ. In other words, they are both suffixes of σ. But that is impossible, since n[i]≠n’[i]. This allows us to conclude that U[i] × V[i] cannot be included in any rectangle of the form U[σ,\|τ\|] × U[τ,\|σ\|], at any round i. Therefore the strategy for player I we have described above is winning, and this completes our argument that S[0] is not consonant. — Jean Goubault-Larrecq (February 20th, 2023)	{"url":"https://topology.lmf.cnrs.fr/on-till-plewes-game-and-matthew-de-brechts-non-consonance-arguments/","timestamp":"2024-11-02T01:51:44Z","content_type":"text/html","content_length":"97292","record_id":"<urn:uuid:b9feefef-289c-41d9-9ba1-093b257b4f58>","cc-path":"CC-MAIN-2024-46/segments/1730477027632.4/warc/CC-MAIN-20241102010035-20241102040035-00829.warc.gz"}
Access For English Language Learners Adapted with permission from work done by Understanding Language at Stanford University. For the original paper, Guidance for Math Curricula Design and Development, please visit https://ul.stanford.edu/resource/principles-design-mathematics-curricula. This curriculum builds on foundational principles for supporting language development for all students. This section aims to provide guidance to help teachers recognize and support students' language development in the context of mathematical sense-making. Embedded within the curriculum are instructional routines and practices to help teachers address the specialized academic language demands in math when planning and delivering lessons, including the demands of reading, writing, speaking, listening, conversing, and representing in math (Aguirre & Bunch, 2012). Therefore, while these instructional routines and practices can and should be used to support all students learning mathematics, they are particularly well-suited to meet the needs of linguistically and culturally diverse students who are learning mathematics while simultaneously acquiring English. This table reflects the attention and support for language development at each level of the curriculum. • foundation of curriculum: theory of action and design principles that drive a continuous focus on language development • student glossary of terms • unit-specific progression of language development included in each unit overview • language goals embedded in learning goals describe the language demands of the lesson • definitions of new glossary terms • strategies to support access for English learners based on language demands of the activity • math language routines Theory of Action We believe that language development can be built into teachers’ instructional practice and students’ classroom experience through intentional design of materials, teacher commitments, administrative support, and professional development. Our theory of action is grounded in the interdependence of language learning and content learning, the importance of scaffolding routines that foster students’ independent participation, the value of instructional responsiveness in the teaching process, and the central role of student agency in the learning process. Mathematical understandings and language competence develop interdependently. Deep conceptual learning is gained through language. Ideas take shape through words, texts, illustrations, conversations, debates, examples, etc. Teachers, peers, and texts serve as language resources for learning. Instructional attention to academic language development, historically limited to vocabulary instruction, has now shifted to also include instruction around the demands of argumentation, explanation, generalization, analyzing the purpose and structure of text, and other mathematical discourse. Scaffolding provides temporary supports that foster student autonomy. Learners with emerging language—at any level—can engage deeply with central mathematical ideas under specific instructional conditions. Mathematical language development occurs when students use their developing language to make meaning and engage with challenging problems that are beyond students’ mathematical ability to solve independently and therefore require interaction with peers. However, these interactions should be structured with temporary access supports that students can use to make sense of what is being asked of them, to help organize their own thinking, and to give and receive feedback. Instruction supports learning when teachers respond to students’ verbal and written work. Eliciting student thinking through language allows teachers and students to respond formatively to the language students generate. Formative peer and teacher feedback creates opportunities for revision and refinement of both content understandings and language. Students are agents in their own mathematical and linguistic sense-making. Mathematical language proficiency is developed through the process of actively exploring and learning mathematics. Language is action: in the very doing of math, students have naturally occurring opportunities to need, learn, and notice mathematical ways of making sense and talking about ideas and the world. These experiences support learners in using as well as expanding their existing language toolkits. The framework for supporting English learners (ELs) in this curriculum includes four design principles for promoting mathematical language use and development in curriculum and instruction. The design principles and related routines work to make language development an integral part of planning and delivering instruction while guiding teachers to amplify the most important language that students are expected to bring to bear on the central mathematical ideas of each unit. Principle 1: SUPPORT SENSE-MAKING Scaffold tasks and amplify language so students can make their own meaning. Students do not need to understand a language completely before they can engage with academic content in that language. Language learners of all levels can and should engage with grade-level content that is appropriately scaffolded. Students need multiple opportunities to talk about their mathematical thinking, negotiate meaning with others, and collaboratively solve problems with targeted guidance from the teacher. Teachers can make language more accessible for students by amplifying rather than simplifying speech or text. Simplifying includes avoiding the use of challenging words or phrases. Amplifying means anticipating where students might need language support to understand concepts or mathematical terms, and providing multiple ways to access them. Providing visuals or manipulatives, demonstrating problem-solving, engaging in think-alouds, and creating analogies, synonyms, or context are all ways to amplify language so that students are supported in taking an active role in their own sense-making of mathematical relationships, processes, concepts, and terms. Principle 2: OPTIMIZE OUTPUT Strengthen opportunities and structures for students to describe their mathematical thinking to others, orally, visually, and in writing. Linguistic output is the language that students use to communicate their ideas to others (oral, written, visual, etc.), and refers to all forms of student linguistic expressions except those that include significant back-and-forth negotiation of ideas. (That type of conversational language is addressed in the third principle.) All students benefit from repeated, strategically optimized, and structured opportunities to articulate mathematical ideas into linguistic expression. Opportunities for students to produce output should be strategically optimized for both (a) important concepts of the unit or course, and (b) important disciplinary language functions (for example, making conjectures and claims, justifying claims with evidence, explaining reasoning, critiquing the reasoning of others, making generalizations, and comparing approaches and representations). The focus for optimization must be determined, in part, by how students are currently using language to engage with important disciplinary concepts. When opportunities to produce output are optimized in these ways, students will get spiraled practice in making their thinking about important mathematical concepts stronger with more robust reasoning and examples, and making their thinking clearer with more precise language and visuals. Principle 3: CULTIVATE CONVERSATION Strengthen opportunities and structures for constructive mathematical conversations (pairs, groups, and whole class). Conversations are back-and-forth interactions with multiple turns that build up ideas about math. Conversations act as scaffolds for students developing mathematical language because they provide opportunities to simultaneously make meaning, communicate that meaning, and refine the way content understandings are communicated. When students have a purpose for talking and listening to each other, communication is more authentic. During effective discussions, students pose and answer questions, clarify what is being asked and what is happening in a problem, build common understandings, and share experiences relevant to the topic. As mentioned in Principle 2, learners must be supported in their use of language, including when having conversations, making claims, justifying claims with evidence, making conjectures, communicating reasoning, critiquing the reasoning of others, engaging in other mathematical practices, and above all when making mistakes. Meaningful conversations depend on the teacher using lessons and activities as opportunities to build a classroom culture that motivates and values efforts to communicate. Principle 4: MAXIMIZE META-AWARENESS Strengthen the meta-connections and distinctions between mathematical ideas, reasoning, and language. Language is a tool that not only allows students to communicate their math understanding to others, but also to organize their own experiences, ideas, and learning for themselves. Meta-awareness is consciously thinking about one's own thought processes or language use. Meta-awareness develops when students and teachers engage in classroom activities or discussions that bring explicit attention to what students need to do to improve communication and reasoning about mathematical concepts. When students are using language in ways that are purposeful and meaningful for themselves, in their efforts to understand—and be understood by—each other, they are motivated to attend to ways in which language can be both clarified and clarifying. Meta-awareness in students can be strengthened when, for example, teachers ask students to explain to each other the strategies they brought to bear to solve a challenging problem. They might be asked, “How does yesterday’s method connect with the method we are learning today?” or, “What ideas are still confusing to you?” These questions are metacognitive because they help students to reflect on their own and others’ learning. Students can also reflect on their expanding use of language—for example, by comparing the language they used to clarify a mathematical concept with the language used by their peers in a similar situation. This is called metalinguistic awareness because students reflect on English as a language, their own growing use of that language, and the particular ways ideas are communicated in mathematics. Students learning English benefit from being aware of how language choices are related to the purpose of the task and the intended audience, especially if oral or written work is required. Both metacognitive and metalinguistic awareness are powerful tools to help students self-regulate their academic learning and language acquisition. These four principles are guides for curriculum development, as well as for planning and execution of instruction, including the structure and organization of interactive opportunities for students. They also serve as guides for and observation, analysis, and reflection on student language and learning. The design principles motivate the use of mathematical language routines, described in detail below, with examples. The eight routines included in this curriculum are: • MLR 1: Stronger and Clearer Each Time • MLR 2: Collect and Display • MLR 3: Clarify, Critique, Correct • MLR 4: Information Gap • MLR 5: Co-Craft Questions • MLR 6: Three Reads • MLR 7: Compare and Connect • MLR 8: Discussion Supports Each lesson includes instructional strategies that teachers can use to facilitate access to the language demands of a lesson or activity. These support strategies, labeled “Access for English Learners," stem from the design principles and are aligned to the language domains of reading, writing, speaking, listening, conversing, and representing in math (Aguirre & Bunch, 2012). They provide students with access to the mathematics by supporting them with the language demands of a specific activity without reducing the mathematical demand of the task. Using these supports will help maintain student engagement in mathematical discourse and ensure that the struggle remains productive. Teachers should use their professional judgment about which routines to use and when, based on their knowledge of the individual needs of students in their classroom. A teacher who notices that students' written responses could get stronger and clearer with more opportunity to revise their writing could use this routine to provide students with multiple opportunities to gain additional input, through direct and indirect feedback from their peers. Based on their observations of student language, teachers can make adjustments to their teaching and provide additional language support where necessary. When making decisions about how to support access for English learners, teachers should take into account the language demands of the specific activity and the language needed to engage the content more broadly, in relation to their students’ current ways of using language to communicate ideas as well as their students’ English language proficiency. Mathematical Language Routines The mathematical language routines were selected because they are effective and practical for simultaneously learning mathematical practices, content, and language. A mathematical language routine is a structured but adaptable format for amplifying, assessing, and developing students' language. The routines emphasize uses of language that is meaningful and purposeful, rather than about just getting answers. These routines can be adapted and incorporated across lessons in each unit to fit the mathematical work wherever there are productive opportunities to support students in using and improving their English and disciplinary language use. These routines facilitate attention to student language in ways that support in-the-moment teacher-, peer-, and self-assessment. The feedback enabled by these routines will help students revise and refine not only the way they organize and communicate their own ideas, but also ask questions to clarify their understandings of others’ ideas. Mathematical Language Routine 1: Stronger and Clearer Each Time Adapted from Zwiers (2014) To provide a structured and interactive opportunity for students to revise and refine both their ideas and their verbal and written output (Zwiers, 2014). This routine also provides a purpose for student conversation through the use of a discussion-worthy and iteration-worthy prompt. The main idea is to have students think and write individually about a question, use a structured pairing strategy to have multiple opportunities to refine and clarify their response through conversation, and then finally revise their original written response. Subsequent conversations and second drafts should naturally show evidence of incorporating or addressing new ideas and language. They should also show evidence of refinement in precision, communication, expression, examples, and reasoning about mathematical concepts. How it Happens Prompt: This routine begins by providing a thought-provoking question or prompt. The prompt should guide students to think about a concept or big idea connected to the content goal of the lesson, and should be answerable in a format that is connected with the activity’s primary disciplinary language function. Response - First Draft: Students draft an initial response to the prompt by writing or drawing their initial thoughts in a first draft. Responses should attempt to align with the activity’s primary language function. It is not necessary that students finish this draft before moving to the structured pair meetings step. However, students should be encouraged to write or draw something before meeting with a partner. This encouragement can come over time as class culture is developed, strategies and supports for getting started are shared, and students become more comfortable with the low stakes of this routine. (2–3 min) Structured Pair Meetings: Next, use a structured pairing strategy to facilitate students having 2–3 meetings with different partners. Each meeting gives each partner an opportunity to be the speaker and an opportunity to be the listener. As the speaker, each student shares their ideas (without looking at their first draft, when possible). As a listener, each student should (a) ask questions for clarity and reasoning, (b) press for details and examples, and (c) give feedback that is relevant for the language goal. (1–2 min each meeting) Response - Second Draft: Finally, after meeting with 2–3 different partners, students write a second draft. This draft should naturally reflect borrowed ideas from partners, as well as refinement of initial ideas through repeated communication with partners. This second draft will be stronger (with more or better evidence of mathematical content understanding) and clearer (more precision, organization, and features of disciplinary language function). After students are finished, their first and second drafts can be compared. (2–3 min) Mathematical Language Routine 2: Collect and Display To capture a variety of students’ oral words and phrases into a stable, collective reference. The intent of this routine is to stabilize the varied and fleeting language in use during mathematical work, in order for students’ own output to become a reference in developing mathematical language. The teacher listens for, and scribes, the language students use during partner, small group, or whole class discussions using written words, diagrams and pictures. This collected output can be organized, revoiced, or explicitly connected to other language in a display that all students can refer to, build on, or make connections with during future discussion or writing. Throughout the course of a unit (and beyond), teachers can reference the displayed language as a model, update and revise the display as student language changes, and make bridges between prior student language and new disciplinary language (Dieckman, 2017). This routine provides feedback for students in a way that supports sense-making while simultaneously increasing meta-awareness of language. How it happens Collect: During this routine, circulate and listen to student talk during paired, group, or as a whole-class discussion. Jot down the words, phrases, drawings, or writing students use. Capture a variety of uses of language that can be connected to the lesson content goals, as well as the relevant disciplinary language function(s). Collection can happen digitally, or with a clipboard, or directly onto poster paper; capturing on a whiteboard is not recommended due to risk of erasure. Display: Display the language collected visually for the whole class to use as a reference during further discussions throughout the lesson and unit. Encourage students to suggest revisions, updates, and connections be added to the display as they develop—over time—both new mathematical ideas and new ways of communicating ideas. The display provides an opportunity to showcase connections between student ideas and new vocabulary. It also provides opportunity to highlight examples of students using disciplinary language functions, beyond just vocabulary words. Mathematical Language Routine 3: Clarify, Critique, Correct To give students a piece of mathematical writing that is not their own to analyze, reflect on, and develop. The intent is to prompt student reflection with an incorrect, incomplete, or ambiguous written mathematical statement, and for students to improve upon the written work by correcting errors and clarifying meaning. Teachers can demonstrate how to effectively and respectfully critique the work of others with meta-think-alouds and pressing for details when necessary. This routine fortifies output and engages students in meta-awareness. More than just error analysis, this routine purposefully engages students in considering both the author’s mathematical thinking as well as the features of their communication. How it happens Original Statement: Create or curate a written mathematical statement that intentionally includes conceptual (or common) errors in mathematical thinking as well as ambiguities in language. The mathematical errors should be driven by the content goals of the lesson and the language ambiguities should be driven by common or typical challenges with the relevant disciplinary language function. This mathematical text is read by the students and used as the draft, or “original statement,” that students improve. (1–2 min) Discussion with Partner: Next, students discuss the original statement in pairs. The teacher provides guiding questions for this discussion such as, “What do you think the author means?,” “Is anything unclear?,” or “Are there any reasoning errors?” In addition to these general guiding questions, 1–2 questions can be added that specifically address the content goals and disciplinary language function relevant to the activity. (2–3 min) Improved Statement: Students individually revise the original statement, drawing on the conversations with their partners, to create an “improved statement.” In addition to resolving any mathematical errors or misconceptions, clarifying ambiguous language, other requirements can be added as parameters for the improved response. These specific requirements should be aligned with the content goals and disciplinary language function of the activity. (3–5 min) Mathematical Language Routine 4: Information Gap Adapted from Zwiers 2004 To create a need for students to communicate (Gibbons, 2002). This routine allows teachers to facilitate meaningful interactions by positioning some students as holders of information that is needed by other students. The information is needed to accomplish a goal, such as solving a problem or winning a game. With an information gap, students need to orally (or visually) share ideas and information in order to bridge a gap and accomplish something that they could not have done alone. Teachers should demonstrate how to ask for and share information, how to justify a request for information, and how to clarify and elaborate on information. This routine cultivates conversation. How it happens Problem/Data Cards: Students are paired into Partner A and Partner B. Partner A is given a card with a problem that must be solved, and Partner B has the information needed to solve it on a “data card.” Data cards can also contain diagrams, tables, graphs, etc. Neither partner should read nor show their cards to their partners. Partner A determines what information they need, and prepares to ask Partner B for that specific information. Partner B should not share the information unless Partner A specifically asks for it and justifies the need for the information. Because partners don’t have the same information, Partner A must work to produce clear and specific requests, and Partner B must work to understand more about the problem through Partner A’s requests and justifications. Bridging the Gap • Partner B asks “What specific information do you need?” Partner A asks for specific information from Partner B. • Before sharing the requested information, Partner B asks Partner A for a justification: “Why do you need that information?” • Partner A explains how they plan to use the information. • Partner B asks clarifying questions as needed, and then provides the information. • These four steps are repeated until Partner A is satisfied that they have information they need to solve the problem. Solving the Problem • Partner A shares the problem card with Partner B. Partner B does not share the data card. • Both students solve the problem independently, then discuss their strategies. Partner B can share the data card after discussing their independent strategies. Mathematical Language Routine 5: Co-craft Questions To allow students to get inside of a context before feeling pressure to produce answers, to create space for students to produce the language of mathematical questions themselves, and to provide opportunities for students to analyze how different mathematical forms and symbols can represent different situations. Through this routine, students are able to use conversation skills to generate, choose (argue for the best one), and improve questions and situations as well as develop meta-awareness of the language used in mathematical questions and problems. How it happens Hook: Begin by presenting students with a hook—a context or a stem for a problem, with or without values included. The hook can also be a picture, video, or list of interesting facts. Students Write Questions: Next, students write down possible mathematical questions that might be asked about the situation. These should be questions that they think are answerable by doing math and could be questions about the situation, information that might be missing, and even about assumptions that they think are important. (1–2 minutes) Students Compare Questions: Students compare the questions they generated with a partner (1–2 minutes) before sharing questions with the whole class. Demonstrate (or ask students to demonstrate) identifying specific questions that are aligned to the content goals of the lesson as well as the disciplinary language function. If there are no clear examples, teachers can demonstrate adapting a question or ask students to adapt questions to align with specific content or function goals. (2–3 minutes) Actual Question(s) Revealed/Identified: Finally, the actual questions students are expected to work on are revealed or selected from the list that students generated. Mathematical Language Routine 6: Three Reads To ensure that students know what they are being asked to do, create opportunities for students to reflect on the ways mathematical questions are presented, and equip students with tools used to actively make sense of mathematical situations and information (Kelemanik, Lucenta, & Creighton, 2016). This routine supports reading comprehension, sense-making, and meta-awareness of mathematical language. It also supports negotiating information in a text with a partner through mathematical conversation. How it happens In this routine, students are supported in reading a mathematical text, situation, or word problem three times, each with a particular focus. The intended question or main prompt is intentionally withheld until the third read so that students can concentrate on making sense of what is happening in the text before rushing to a solution or method. Read #1: Shared Reading (one person reads aloud while everyone else reads with them) The first read focuses on the situation, context, or main idea of the text. After a shared reading, ask students “what is this situation about?” This is the time to identify and resolve any challenges with any non-mathematical vocabulary. (1 minute) Read #2: Individual, Pairs, or Shared Reading After the second read, students list any quantities that can be counted or measured. Students are encouraged not to focus on specific values. Instead they focus on naming what is countable or measurable in the situation. It is not necessary to discuss the relevance of the quantities, just to be specific about them (examples: “number of people in her family” rather than “people,” “number of markers after” instead of “markers”). Some of the quantities will be explicit (example: 32 apples) while others are implicit (example: the time it takes to brush one tooth). Record the quantities as a reference to use when solving the problem after the third read. (3–5 minutes) Read #3: Individual, Pairs, or Shared Reading During the third read, the final question or prompt is revealed. Students discuss possible solution strategies, referencing the relevant quantities recorded after the second read. It may be helpful for students to create diagrams to represent the relationships among quantities identified in the second read, or to represent the situation with a picture (Asturias, 2014). (1–2 minutes). Mathematical Language Routine 7: Compare and Connect To foster students’ meta-awareness as they identify, compare, and contrast different mathematical approaches and representations. This routine leverages the powerful mix of disciplinary representations available in mathematics as a resource for language development. In this routine, students make sense of mathematical strategies other than their own by relating and connecting other approaches to their own. Students should be prompted to reflect on, and linguistically respond to, these comparisons (for example, exploring why or when one might do or say something a certain way, identifying and explaining correspondences between different mathematical representations or methods, or wondering how a certain concept compares or connects to other concepts). Be sure to demonstrate asking questions that students can ask each other, rather than asking questions to “test” understanding. Use think alouds to demonstrate the trial and error, or fits and starts of sense-making (similar to the way teachers think aloud to demonstrate reading comprehension). This routine supports metacognition and metalinguistic awareness, and also supports constructive How it Happens Students Prepare Displays of their Work: Students are given a problem that can be approached and solved using multiple strategies, or a situation that can be modeled using multiple representations. Students are assigned the job of preparing a visual display of how they made sense of the problem and why their solution makes sense. Variation is encouraged and supported among the representations that different students use to show what makes sense. Compare: Students investigate each others’ work by taking a tour of the visual displays. Tours can be self-guided, a “travellers and tellers” format, or the teacher can act as “docent” by providing questions for students to ask of each other, pointing out important mathematical features, and facilitating comparisons. Comparisons focus on the typical structures, purposes, and affordances of the different approaches or representations: what worked well in this or that approach, or what is especially clear in this or that representation. During this discussion, listen for and amplify any comments about what might make this or that approach or representation more complete or easy to understand. Connect: The discussion then turns to identifying correspondences between different representations. Students are prompted to find correspondences in how specific mathematical relationships, operations, quantities, or values appear in each approach or representation. Guide students to refer to each other’s thinking by asking them to make connections between specific features of expressions, tables, graphs, diagrams, words, and other representations of the same mathematical situation. During the discussion, amplify language students use to communicate about mathematical features that are important for solving the problem or modeling the situation. Call attention to the similarities and differences between the ways those features appear. Mathematical Language Routine 8: Discussion Supports To support rich and inclusive discussions about mathematical ideas, representations, contexts, and strategies (Chapin, O’Connor, & Anderson, 2009). Rather than another structured format, the examples provided in this routine are instructional moves that can be combined and used together with any of the other routines. They include multimodal strategies for helping students make sense of complex language, ideas, and classroom communication. The examples can be used to invite and incentivize more student participation, conversation, and meta-awareness of language. Eventually, as teachers continue to demonstrate, students should begin using these strategies themselves to prompt each other to engage more deeply in discussions. How it Happens Unlike the other routines, this MLR includes a collection of strategies and moves that can be combined and used to support discussion during almost any activity. Examples of possible strategies: • Revoice student ideas to demonstrate mathematical language use by restating a statement as a question in order to clarify, apply appropriate language, and involve more students. • Press for details in students’ explanations by requesting for students to challenge an idea, elaborate on an idea, or give an example. • Show central concepts multi-modally by using different types of sensory inputs: acting out scenarios or inviting students to do so, showing videos or images, using gesture, and talking about the context of what is happening. • Practice phrases or words through choral response. • Think aloud by talking through thinking about a mathematical concept while solving a related problem or doing a task. • Demonstrate uses of disciplinary language functions such as detailing steps, describing and justifying reasoning, and questioning strategies. • Give students time to make sure that everyone in the group can explain or justify each step or part of the problem. Then make sure to vary who is called on to represent the work of the group so students get accustomed to preparing each other to fill that role. • Prompt students to think about different possible audiences for the statement, and about the level of specificity or formality needed for a classmate vs. a mathematician, for example. [Convince Yourself, Convince a Friend, Convince a Skeptic (Mason, Burton, & Stacey, 2010)] Sentence Frames Sentence frames can support student language production by providing a structure to communicate about a topic. Helpful sentence frames are open-ended, so as to amplify language production, not constrain it. The table shows examples of generic sentence frames that can support common disciplinary language functions across a variety of content topics. Some of the lessons in these materials include suggestions of additional sentence frames that could support the specific content and language functions of that lesson. • It looks like… • I notice that… • I wonder if… • Let’s try... • A quantity that varies is _____. • What do you notice? • What other details are important? • First, I _____ because… • Then/Next, I... • I noticed _____ so I... • I tried _____ and what happened was... • How did you get…? • What else could we do? • I know _____ because... • I predict _____ because… • If _____ then _____ because… • Why did you…? • How do you know…? • Can you give an example? • _____ reminds me of _____ because… • _____ will always _____ because… • _____ will never _____ because… • Is it always true that…? • Is _____ a special case? • That could/couldn’t be true because… • This method works/doesn’t work because… • We can agree that... • _____’s idea reminds me of… • Another strategy would be _____ because… • Is there another way to say/do...? • Both _____ and _____ are alike because… • _____ and _____ are different because… • One thing that is the same is… • One thing that is different is… • How are _____ and _____ different? • What do _____ and _____ have in common? • _____ represents _____. • _____ stands for _____. • _____ corresponds to _____. • Another way to show _____ is… • How else could we show this? • We are trying to... • We will need to know... • We already know… • It looks like _____ represents... • Another way to look at it is… • What does this part of _____ mean? • Where does _____ show...? Aguirre, J. M. & Bunch, G. C. (2012). What’s language got to do with it?: Identifying language demands in mathematics instruction for English language learners. In S. Celedón-Pattichis & N. Ramirez (Eds.), Beyond good teaching: Advancing mathematics education for ELLs. (pp. 183-194). Reston, VA: National Council of Teachers of Mathematics. Chapin, S., O’Connor, C., & Anderson, N. (2009). Classroom discussions: Using math talk to help students learn, grades K-6 (second edition). Sausalito, CA: Math Solutions Publications. Gibbons, P. (2002). Scaffolding language, scaffolding learning: Teaching second language learners in the mainstream classroom. Portsmouth, NH: Heinemann. Kelemanik, G, Lucenta, A & Creighton, S.J. (2016). Routines for reasoning: Fostering the mathematical practices in all students. Portsmouth, NH: Heinemann. Zwiers, J. (2011). Academic conversations: Classroom talk that fosters critical thinking and content understandings. Portland, ME: Stenhouse Zwiers, J. (2014). Building academic language: Meeting Common Core Standards across disciplines, grades 5–12 (2nd ed.). San Francisco, CA: Jossey-Bass. Zwiers, J., Dieckmann, J., Rutherford-Quach, S., Daro, V., Skarin, R., Weiss, S., & Malamut, J. (2017). Principles for the design of mathematics curricula: Promoting language and content development. Retrieved from Stanford University, UL/SCALE website: https://ul.stanford.edu/resource/principles-design-mathematics-curricula	{"url":"https://im.kendallhunt.com/MS_ACC/teachers/access_for_english_language_learners.html","timestamp":"2024-11-02T00:19:11Z","content_type":"text/html","content_length":"119722","record_id":"<urn:uuid:f35a3872-964e-4e6d-94d8-45c8fa6ecb42>","cc-path":"CC-MAIN-2024-46/segments/1730477027599.25/warc/CC-MAIN-20241101215119-20241102005119-00347.warc.gz"}
Contest Results (Analysis by Dhruv Rohatgi ) In this problem, we are asked to find all sliding-window minima in an array, for some fixed-length window. We are given two passes through the array. The catch is that we are given only $O(\sqrt{N})$ Let $f(i)$ be the location of the smallest element in the range $[i, i+K)$, breaking ties by smallest index. Observe that $f(i) \leq f(i+1)$ for each $i$. So if we compute $f(0), f(B), f(2B), \dots, f(N)$ for some block size $B$ in the first pass, this gives us some information which we may be able to use in the second pass to compute the remaining $f(i)$s. Let's ignore the constraint on calls to set/get for now. If we let $B = \sqrt{N}$, then we can compute $f(0), f(\sqrt{N}), \dots, f(N)$ in our first pass using only $O(\sqrt{N})$ memory and $O(N\sqrt {N})$ time. Now, in the second pass, observe that for any fixed array element $i$, there are only $O(\sqrt{N})$ windows for which the minimum could possibly be at position $i$. This suggests the following algorithm for the second pass: read and ignore all elements with index less than $f(0)$. Now maintain running minima for $f(0), f(1), f(2), \dots, f(\sqrt{N})$. Once we've read in the element with index $f(\sqrt{N})$, we know that we have computed the correct minima for the first $\sqrt{N}$ windows. Output these, start maintaining minima for the next $\sqrt{N}$ windows, and continue in the same fashion. The memory usage of the second pass is also $\sqrt{N}$, as desired. Unfortunately, the time complexity of each pass is $O(N \sqrt{N})$. In particular, the above algorithm would use $O(N \sqrt{N})$ calls to set and get, which would exceed the given bound. To improve the time complexity, we can use a monotonic queue in each pass. In the first pass, for instance, the boundaries of the $\sqrt{N}$ windows define $2\sqrt{N}$ subarrays, and for each subarray we can simply compute the minimum of the subarray and put that into the monotonic queue. The length of the monotonic queue therefore never exceeds $O(\sqrt{N})$, and we can compute the $\sqrt{N}$ minima-locations in $O(N)$ time. For the second pass, it suffices to observe that in each segment of the array (say, between $f(i\sqrt{N})$ and $f((i+1)\sqrt{N})$) all but $O(\sqrt{N})$ of the array elements are in all $\sqrt{N}$ windows of interest. This once again allows us to improve the pass to $O(N)$ time will not exceeding $O(\sqrt {N})$ memory. Below is an implementation of the above algorithm. #include "grader.h" #define BLOCK 1000 #define DIF 4050 // VARIABLE - LOCATION IN BESSIE'S NOTEBOOK // back index - 0 // top index - 1 // queue (2,3) (4,5) ... // (index, value) // current block's endpoint index (divided by BLOCK) - DIF-1 // current block's global low - DIF-2 // number of minima output - DIF-3 void helpBessie(int v) int N = getTrainLength(); int K = getWindowLength(); int i = getCurrentCarIndex(); int p = getCurrentPassIndex(); int top = get(1); int back = get(0); if(i%BLOCK == 0 \|\| (i>=K && (i-K)%BLOCK == 0)) // reached boundary of some window { // need to make new entry in monotonic queue while(top >= back && get(2top+3) >= v) else //still part of same subarray; just update the top entry of monotonic queue int curTopValue = get(2top+3); if(v <= curTopValue) while(top >= back && get(2top+3) >= v) if(i >= K-1 && (i+1-K)%BLOCK == 0) // at endpoint of some window; need to store location of minimum while(top >= back && get(2back+2) <= i-K) // pop from back end up queue until queue contains only back++; // elements in desired window set(DIF + (i+1-K)/BLOCK, get(2back+2)); // store location of minimum if(i < get(DIF)) if(i == get(DIF)) set(DIF-1, 1); set(DIF-3, 0); int bc = get(DIF-1); int top = get(1); int back = get(0); int outputs = get(DIF-3); if(i - get(DIF+bc-1) <= BLOCK) // element may not be contained in all relevant windows { // so add to monotonic queue while(top >= back && get(2top+3) >= v) else // element is contained in all relevant windows { // so we can update a global minimum int globalLow = get(DIF-2); if(v < globalLow) if(outputs + K - 1 == i) //need to output a minimum while(top >= back && get(2back+2) < outputs) while(BLOCKbc + K-1 < N && get(DIF+bc) == i) // reached boundary of current subarray while(outputs <= BLOCKbc) // output minimums for all remaining windows in current block while(top >= back && get(2back+2) < outputs) top = back = 0;	{"url":"https://usaco.org/current/data/sol_train_platinum_open18.html","timestamp":"2024-11-07T10:58:53Z","content_type":"text/html","content_length":"7751","record_id":"<urn:uuid:c345b442-12fa-4044-9fc1-87cc8d31fd7f>","cc-path":"CC-MAIN-2024-46/segments/1730477027987.79/warc/CC-MAIN-20241107083707-20241107113707-00820.warc.gz"}
d01fdc (md_sphere) NAG CL Interface d01fdc (md_sphere) 1 Purpose calculates an approximation to a definite integral in up to dimensions, using the method of Sag and Szekeres (see Sag and Szekeres (1964) ). The region of integration is an -sphere, or by built-in transformation via the unit -cube, any product region. 2 Specification d01fdc (Integer ndim, double (f)(Integer ndim, const double x[], Nag_Comm comm), void double sigma, void (region)(Integer ndim, const double x[], Integer j, double c, double d, Nag_Comm comm), Integer limit, double r0, double u, double result, Integer ncalls, Nag_Comm comm, NagError fail) The function may be called by the names: d01fdc or nag_quad_md_sphere. 3 Description calculates an approximation to $∫ n-sphere of radius σ f (x1,x2,…,xn) dx1 dx2 ⋯ dxn$ (1) or, more generally, $∫ c1 d1 dx1 ⋯ ∫ cn dn dxn f (x1,…,xn)$ (2) where each may be functions of ${x}_{j}$ $\left(j<i\right)$ The function uses the method of Sag and Szekeres (1964) , which exploits a property of the shifted -point trapezoidal rule, namely, that it integrates exactly all polynomials of degree Krylov (1962) ). An attempt is made to induce periodicity in the integrand by making a parameterised transformation to the unit -sphere. The Jacobian of the transformation and all its direct derivatives vanish rapidly towards the surface of the unit -sphere, so that, except for functions which have strong singularities on the boundary, the resulting integrand will be pseudo-periodic. In addition, the variation in the integrand can be considerably reduced, causing the trapezoidal rule to perform well. Integrals of the form are transformed to the unit -sphere by the change of variables: $xi = yi σr tanh( ur 1-r2 )$ ${r}^{2}=\sum _{i=1}^{n}{y}_{i}^{2}$ is an adjustable parameter. Integrals of the form are first of all transformed to the by a linear change of variables and then to the unit sphere by a further change of variables ${r}^{2}=\sum _{i=1}^{n}{z}_{i}^{2}$ is again an adjustable parameter. The parameter $u$ in these transformations determines how the transformed integrand is distributed between the origin and the surface of the unit $n$-sphere. A typical value of $u$ is $1.5$. For larger $u$, the integrand is concentrated toward the centre of the unit $n$-sphere, while for smaller $u$ it is concentrated toward the perimeter. In performing the integration over the unit -sphere by the trapezoidal rule, a displaced equidistant grid of size is constructed. The points of the mesh lie on concentric layers of radius $ri=h4n+8⁢(i-1), i=1,2,3,….$ The function requires you to specify an approximate maximum number of points to be used, and then computes the largest number of whole layers to be used, subject to an upper limit of In practice, the rapidly-decreasing Jacobian makes it unnecessary to include the whole unit $n$-sphere and the integration region is limited by a user-specified cut-off radius ${r}_{0}<1$. The grid-spacing $h$ is determined by ${r}_{0}$ and the number of layers to be used. A typical value of ${r}_{0}$ is $0.8$. Some experimentation may be required with the choice of (which determines how much of the unit -sphere is included) and (which determines how the transformed integrand is distributed between the origin and surface of the unit -sphere), to obtain best results for particular families of integrals. This matter is discussed further in Section 9 4 References Krylov V I (1962) Approximate Calculation of Integrals (trans A H Stroud) Macmillan Sag T W and Szekeres G (1964) Numerical evaluation of high-dimensional integrals Math. Comput. 18 245–253 5 Arguments 1: $\mathbf{ndim}$ – Integer Input On entry: $n$, the number of dimensions of the integral. Constraint: $1\le {\mathbf{ndim}}\le 30$. 2: $\mathbf{f}$ – function, supplied by the user External Function must return the value of the integrand at a given point. The specification of double f (Integer ndim, const double x[], Nag_Comm comm) 1: $\mathbf{ndim}$ – Integer Input On entry: $n$, the number of dimensions of the integral. 2: $\mathbf{x}\left[{\mathbf{ndim}}\right]$ – const double Input On entry: the coordinates of the point at which the integrand $f$ must be evaluated. 3: $\mathbf{comm}$ – Nag_Comm Pointer to structure of type Nag_Comm; the following members are relevant to user – double * iuser – Integer * p – Pointer The type Pointer will be void * . Before calling you may allocate memory and initialize these pointers with various quantities for use by when called from Section 3.1.1 in the Introduction to the NAG Library CL Interface). Note: f should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by . If your code inadvertently return any NaNs or infinities, is likely to produce unexpected results. 3: $\mathbf{sigma}$ – double Input On entry : indicates the region of integration. ${\mathbf{sigma}}\ge 0.0$ The integration is carried out over the $n$-sphere of radius sigma, centred at the origin. The integration is carried out over the product region described by region. 4: $\mathbf{region}$ – function, supplied by the user External Function must evaluate the limits of integration in any dimension. The specification of void region (Integer ndim, const double x[], Integer j, double c, double d, Nag_Comm comm) 1: $\mathbf{ndim}$ – Integer Input On entry: $n$, the number of dimensions of the integral. 2: $\mathbf{x}\left[{\mathbf{ndim}}\right]$ – const double Input On entry: ${\mathbf{x}}\left[0\right],\dots ,{\mathbf{x}}\left[j-2\right]$ contain the current values of the first $\left(j-1\right)$ variables, which may be used if necessary in calculating ${c}_{j}$ and ${d}_{j}$. 3: $\mathbf{j}$ – Integer Input On entry: the index $j$ for which the limits of the range of integration are required. 4: $\mathbf{c}$ – double Output On exit: the lower limit ${c}_{j}$ of the range of ${x}_{j}$. 5: $\mathbf{d}$ – double * Output On exit: the upper limit ${d}_{j}$ of the range of ${x}_{j}$. 6: $\mathbf{comm}$ – Nag_Comm * Pointer to structure of type Nag_Comm; the following members are relevant to user – double * iuser – Integer * p – Pointer The type Pointer will be void * . Before calling you may allocate memory and initialize these pointers with various quantities for use by when called from Section 3.1.1 in the Introduction to the NAG Library CL Interface). Note: region should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by . If your code inadvertently return any NaNs or infinities, is likely to produce unexpected results. ${\mathbf{sigma}}\ge 0.0$ is not called by , but the defined null function pointer must be supplied. 5: $\mathbf{limit}$ – Integer Input On entry: the approximate maximum number of integrand evaluations to be used. Constraint: ${\mathbf{limit}}\ge 100$. 6: $\mathbf{r0}$ – double Input On entry: the cut-off radius on the unit $n$-sphere, which may be regarded as an adjustable parameter of the method. Suggested value : a typical value is . (See also Section 9 Constraint: $0.0<{\mathbf{r0}}<1.0$. 7: $\mathbf{u}$ – double Input On entry: must specify an adjustable parameter of the transformation to the unit $n$-sphere. Suggested value : a typical value is . (See also Section 9 Constraint: ${\mathbf{u}}>0.0$. 8: $\mathbf{result}$ – double * Output On exit: the approximation to the integral $I$. 9: $\mathbf{ncalls}$ – Integer * Output On exit : the actual number of integrand evaluations used. (See also Section 9 10: $\mathbf{comm}$ – Nag_Comm * The NAG communication argument (see Section 3.1.1 in the Introduction to the NAG Library CL Interface). 11: $\mathbf{fail}$ – NagError * Input/Output The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface). 6 Error Indicators and Warnings Dynamic memory allocation failed. Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information. On entry, argument $⟨\mathit{\text{value}}⟩$ had an illegal value. On entry, ${\mathbf{limit}}=⟨\mathit{\text{value}}⟩$. Constraint: ${\mathbf{limit}}\ge 100$. On entry, ${\mathbf{ndim}}=⟨\mathit{\text{value}}⟩$. Constraint: ${\mathbf{ndim}}\le 30$. On entry, ${\mathbf{ndim}}=⟨\mathit{\text{value}}⟩$. Constraint: ${\mathbf{ndim}}\ge 1$. An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact for assistance. Section 7.5 in the Introduction to the NAG Library CL Interface for further information. Your licence key may have expired or may not have been installed correctly. Section 8 in the Introduction to the NAG Library CL Interface for further information. On entry, ${\mathbf{r0}}=⟨\mathit{\text{value}}⟩$. Constraint: ${\mathbf{r0}}<1.0$. On entry, ${\mathbf{r0}}=⟨\mathit{\text{value}}⟩$. Constraint: ${\mathbf{r0}}>0.0$. On entry, ${\mathbf{u}}=⟨\mathit{\text{value}}⟩$. Constraint: ${\mathbf{u}}>0.0$. 7 Accuracy No error estimate is returned, but results may be verified by repeating with an increased value of (provided that this causes an increase in the returned value of 8 Parallelism and Performance Background information to multithreading can be found in the d01fdc is not threaded in any implementation. The time taken by will be approximately proportional to the returned value of , which, except in the circumstances outlined in below, will be close to the given value of 1. (a)Choice of ${r}_{0}$ and $u$ If the chosen combination of is too large in relation to the machine accuracy it is possible that some of the points generated in the original region of integration may transform into points in the unit -sphere which lie too close to the boundary surface to be distinguished from it to machine accuracy (despite the fact that ). To be specific, the combination of is too large if $ur0 1-r02 >0.3465⁢(t-1), if sigma≥0.0,$ $ur0 1-r0 > 0.3465⁢(t-1), if sigma< 0.0,$ is the number of bits in the mantissa of a double number. The contribution of such points to the integral is neglected. This may be justified by appeal to the fact that the Jacobian of the transformation rapidly approaches zero towards the surface. Neglect of these points avoids the occurrence of overflow with integrands which are infinite on the boundary. 2. (b)Values of limit and ncalls is an approximate upper limit to the number of integrand evaluations, and may not be chosen less than . There are two circumstances when the returned value of (the actual number of evaluations used) may be significantly less than Firstly, as explained in , an unsuitably large combination of may result in some of the points being unusable. Such points are not included in the returned value of Secondly, no more than layers will ever be used, no matter how high is set. This places an effective upper limit on as follows: $n=1: 56 n=2: 1252 n=3: 23690 n=4: 394528 n=5: 5956906$ 10 Example This example calculates the integral $∫∫∫s dx1dx2dx3 σ2-r2 =22.2066$ is the -sphere of radius $\sigma =1.5$ . Both sphere-to-sphere and general product region transformations are used. For the former, we use ; for the latter, 10.1 Program Text 10.2 Program Data 10.3 Program Results	{"url":"https://support.nag.com/numeric/nl/nagdoc_latest/clhtml/d01/d01fdc.html","timestamp":"2024-11-08T17:33:25Z","content_type":"text/html","content_length":"52854","record_id":"<urn:uuid:81e45b60-6a1d-42ce-9a82-1e843736bf50>","cc-path":"CC-MAIN-2024-46/segments/1730477028070.17/warc/CC-MAIN-20241108164844-20241108194844-00305.warc.gz"}
What mathematics course should you take next? Course Planning Guides: To help with course selections in the upcoming year(s) please view our Course Planning Guides. Please note they may change without notice. Mathematics 2024-2027 Statistics and Data Science 2022-2026 Computer Science 2022-2025 Important note: This page does not contain information about the Mathematics and AQR Placement Process that incoming first-year (and some transfer) students must complete. Instead, this is a page of general information about what course a student might take next in mathematics.Which mathematics course you should take next depends, of course, on what your plans are. Are you a math major? science major? stats concentrator? The best way to determine which course to take next is to chat with a mathematics professor about your options. Visit with your current math prof, any other math prof you know and like, or talk to the Chair of the department, Prof. Jill Dietz (in Regents Math, 500). In the meantime, here are some good general guidelines. • Click here to see a list of majors and concentrations that either require or recommend certain mathematics, statistics, or computer science courses. • If you only need to take a math course in order to complete the AQR requirement, then consider Math 117 or a Statistics or Computer Science course (again, click here for more info). • If you have completed Calculus I (Math 120 or equivalent), then take Calculus II (either Math 126 or Math 128) next. • If you have completed Calculus II (Math 126/128 or equivalent), then take Elementary Linear Algebra (Math 220) next. • If you have completed Linear Algebra (Math 220), then you have several options. □ Multivariable Calculus (Math 226) and Differential Equations I (Math 232) are good options for science and economics majors. □ Probability Theory (Math 262) is good for all sorts of majors, especially if you are considering a Statistics concentration. □ Discrete Mathematics (Math 232, offered every other year) is a fun course that is required for students hoping to become licensed teachers. □ Operations Research (Math 266) is good for students interested in applications of mathematics, especially if you loved linear algebra. □ The transition courses of Modern Computational Mathematics (Math 242), Real Analysis I (Math 244), and Abstract Algebra I (Math 252) are also options, but are typically taken after at least one other 200-level elective is taken after Linear Algebra. ☆ In Modern Computational Mathematics you study mathematical problems from a computational point of view. You will use software such as R andMathematica extensively, though prior knowledge of these programs is not required. ☆ In Real Analysis I you study some familiar topics from calculus (e.g. functions, limits, continuity, sequences) from a theoretical point of view. You will earn a WRI general education credit in this course by writing proofs of theorems. ☆ In middle school your algebra course dealt with arithmetic, variables, and solving equations. The course Abstract Algebra I deals with all of this stuff too, but in a more abstract setting. Your will learn about algebra as a study of mathematical structures and get a flavor of what modern mathematics is all about. You will also earn a WRI general education credit in this course by writing proofs of theorems. • If you have completed a 200-level elective in addition to Linear Algebra, then the mathematical world is at your feet! Check in with a math professor to talk about what options are best for you, but you should probably look into taking a transition course (see the info above). Other information • Individualized mathematics proposal (IMAP) for a math major (PDF)	{"url":"https://wp.stolaf.edu/math/what-mathematics-course-should-you-take-next/","timestamp":"2024-11-12T11:46:28Z","content_type":"text/html","content_length":"77617","record_id":"<urn:uuid:b051f5ae-7f8e-4329-bdc1-2ff4b200da99>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.45/warc/CC-MAIN-20241112113320-20241112143320-00536.warc.gz"}
How to Find Interval Notation Interval notation provides a succinct way to describe sets of numbers, letting you quickly understand which numbers are included and excluded from a set. In this article, we will learn together how to find interval notation. Step-by-step Guide to Find Interval Notation Here is a step-by-step guide to finding interval notation: Step 1: A Grasping of Boundaries: Understand the Basics. • Endpoints: In the realm of numbers, endpoints are the alpha and omega of our interval. They’re the sentinels, defining where your interval starts and concludes. • Types of Endpoints: Behold the distinction – the “closed” endpoint includes the value (marked as $[a]$ or $[b]$) while the “open” endpoint excludes it (represented as $(a$ or $b)$). Step 2: Dive into the Depths: Identify the Interval. • Visual Analysis: Stare at the given number line, graph, or mathematical problem. Can you discern where the interval begins and culminates? • Determine Continuity: Does the interval span a continuous range? Or do you encounter islands of isolated numbers? Step 3: Enumerate the Realm: Identify the Type of Interval. • Infinite Expanses: Does the interval stretch indefinitely? Note these infinite possibilities: □ $∞$: The symbol that strides endlessly towards the positive spectrum. □ $−∞$: This one drifts eternally in the negative abyss. • Finite Confinements: Are you dealing with a limited range? Ah, that’s when both endpoints are tangible numbers. Step 4: The Dance of Inclusivity: Determine Open or Closed Intervals. • Gaze at the Graph: On a number line: □ A filled dot indicates the embrace of that value – a closed interval. □ An empty dot showcases exclusion, hinting at an open interval. • In Mathematical Statements: Phrases such as “less than or equal to” beckon a closed interval. In contrast, “less than” (without equality) points towards an open interval. Step 5: The Grand Conjunction: Unions. • Disjointed Intervals: Sometimes, intervals can be discrete, disconnected fragments. • The Bridging Symbol: The union symbol $∪$ serves to connect these fragments in your notation. Step 6: The Art of Transcription: Writing the Interval Notation. • The Basic Syntax: The common structure is a, b, where the underlines can be either brackets $[ ]$ or $($. • Incorporating Infinity: For infinite intervals, replace one or both endpoints with $∞$ or $−∞$. Always use parentheses with infinity; it’s a concept, not a precise number. Step 7: Proofreading the Narrative: Review Your Notation. • Consistency: Ensure that the notation accurately reflects the interval’s nature. • Validity: Confirm that you’ve appropriately used brackets or parentheses to indicate open or closed intervals. Step 8: Embark on Further Journeys: Apply and Practice. • Expand Horizons: Delve into more complex scenarios, such as absolute value inequalities. • Practice is the Key: The universe of numbers is vast. Wander frequently to familiarize yourself with its intricacies. By mastering this opulent and detailed procedure, you’ll be well-equipped to navigate the rich tapestry of interval notation. The more you delve into its complexities, the more rewarding your mathematical adventures will become! Related to This Article What people say about "How to Find Interval Notation - Effortless Math: We Help Students Learn to LOVE Mathematics"? No one replied yet.	{"url":"https://www.effortlessmath.com/math-topics/how-to-find-interval-notation/","timestamp":"2024-11-02T21:08:19Z","content_type":"text/html","content_length":"92397","record_id":"<urn:uuid:a0b1b75f-ffad-4791-946a-2d6fbfbb1903>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00522.warc.gz"}
Root Trait Inventory In total, the observations in FRED 3.0 encompass more than 330 root traits. To better understand the breadth and depth of these traits, we have grouped them into eight categories: root anatomy, architecture, chemistry, dynamics, morphology, physiology, and the whole-root system, as well as microbial associations. These root trait categories, along with others, are described and mapped in relation to one-another here. Each observation in FRED is based on a single trait measurement made by an investigator on fine roots taken from a defined species or plant community at a specific location, date, and time. Many observations are plot or treatment means, and we include measures of variation and sample size where available (statistical data are not shown here). In some cases, multiple trait observations were taken from one collection of fine roots, but no sample has observations of all fine-root traits (most samples are associated with one or a few traits). Use the search box to see whether FRED 3.0 has a particular root trait of interest, or choose combinations from the Trait Category and Trait Type boxes below and click the "Filter" button to view subsets of the data. Trait Trait Type Traits Column Description Single-species Multi-species Total Category ID Observations Observations observations Allocation Aboveground/belowground Root System within plant net primary production F00841 Annual aboveground net primary production (NPP) divided by belowground NPP. 118 30 148 Root System Standing crop Belowground biomass per F00885 Root mass per square meter for the specified depth increment. 4350 7028 11378 ground area Root System Standing crop Belowground biomass per F00898 Total root mass for the entire plant. 7211 127 7338 Root System Standing crop Belowground biomass per F00896 Kilograms of root mass per cubic meter of soil. 897 230 1127 soil volume Root System Standing crop Belowground necromass F00917 Dead root mass per square meter for the specified depth increment. 448 795 1243 per ground area Root System Allocation Belowground/aboveground F00838 Ratio of belowground biomass (g m-2) to aboveground biomass (g m-2). 677 67 744 within plant mass ratio Allocation Coarse root/fine root Root System within root mass ratio F00854 Ratio of coarse root biomass to fine root biomass. 1636 9 1645 Root System Allocation Fine root C/leaf C ratio F00844 Ratio of fine root carbon to leaf carbon. 40 30 70 within plant Root System Allocation Fine root mass/leaf mass F00843 Ratio of fine-root mass to leaf mass. 1831 27 1858 within plant ratio Microbial Other “x” denotes the presence of mycoheterotrophy. Blank indicates the data are not related to associations rhizosphere Mycoheterotrophy F01288 mycoheterotrophy or that mycoheterotrophy data were not available. 3 0 3 Microbial Mycorrhizal Mycorrhiza_Fraction associations fungi contact exploration F00609 Percentage of mycorrhizal tips that are of the contact exploration type. 12 0 12 Microbial Mycorrhizal Mycorrhiza_Fraction associations fungi long-distance F00611 Percentage of mycorrhizal tips that are of the long-distance exploration type. 12 0 12 exploration mycorrhizae Microbial Mycorrhizal Mycorrhiza_Fraction associations fungi medium-distance F00613 Percentage of mycorrhizal tips that are of the medium-distance exploration type. 12 0 12 exploration mycorrhizae Microbial Mycorrhizal Mycorrhiza_Fraction associations fungi mycorrhizal root tips F00617 Percentage of mycorrhizal tips that are living. 12 0 12 that are living Percentage of root length colonized by mycorrhizal fungi, identified by the presence of Microbial Mycorrhizal Mycorrhiza_Fraction of F00638 mycorrhizal hyphae, arbuscules, or vesicles. This column contains information from both AM and EM 602 62 664 associations fungi root length colonization reported in columns F00622 and F00626, and additional data when mycorrhizal type was unspecified or both colonization types were present. Microbial Mycorrhizal Mycorrhiza_Fraction root Percentage of root length which shows colonization by arbuscular mycorrhizal fungi. These data associations fungi length colonized by AM F00622 may also be contained in column F00638. 547 48 595 Microbial Mycorrhizal Mycorrhiza_Fraction root Percentage of root length which shows colonization by ectomycorrhizal fungi. These data may also associations fungi length colonized by EM F00626 be contained in column F00638. 14 4 18 Microbial Mycorrhizal Mycorrhiza_Fraction root associations fungi tips colonized by F00619 Percentage of root tips that are colonized by mycorrhizal fungi. 279 4 283 Microbial Mycorrhizal Mycorrhiza_Fraction associations fungi short-distance F00615 Percentage of mycorrhizal tips that are of the short-distance exploration type. 12 0 12 exploration mycorrhizae Microbial Mycorrhizal Mycorrhiza_Length of AM Millimeters of arbuscular mycorrhizal hyphae per gram of soil, determined by dispersing soils in associations fungi hyphae per soil mass F00606 sodium metaphosphate solution, passing through a series of sieves, collecting hyphae on filters, 0 31 31 and examining hyphae through a microscope. Microbial Mycorrhizal Mycorrhiza_Number of AM F00641 Number of arbuscular mycorrhizae spores per gram of soil, determined by isolating the spores 2 4 6 associations fungi spores per soil mass using the wet-sieving technique and counting them under a microscope. Mycorrhiza_Number of Microbial Mycorrhizal root tips per root F00635 Number of tips per cm of root on root branch that contains mycorrhizal fungi. 16 0 16 associations fungi length colonized by Microbial Mycorrhizal Mycorrhiza_PLFA proxy Grams of external ectomycorrhizal mycelium biomass in soil per square meter of ground area for a associations fungi for EM mass per ground F00607 given sampling depth, estimated based on PLFA with a conversion factor of 2 nmol 18:2omega6.9 per 12 0 12 area milligram of fungal biomass. Microbial Mycorrhizal Mycorrhiza_PLFA proxy Milligrams of mycorrhizal hyphal carbon per kilogram of soil mass, estimated from PLFA biomarkers associations fungi for mycorrhizal hyphal C F01333 as specified in Chen et al 2016 (DOI: 10.1073/pnas.1601006113). 26 0 26 per soil mass Internal Mycorrhiza_Root length Anatomy mycological fraction that contains F00111 Percentage of root length that contains arbuscules, determined using a random intercept method. 0 4 4 structures arbuscules Internal Mycorrhiza_Root length Percentage of root length that contains fungal vesicles, determined using a random intercept Anatomy mycological fraction that contains F00113 method. 0 4 4 structures vesicles Type of mycorrhizae formed. "AbtM" = arbutoid mycorrhizae; “AM” = arbuscular mycorrhizae; “DS” = Microbial Mycorrhizal Mycorrhiza_Type_Data F00645 dark septate endophyte mycorrhizae; “EeM” = ectendomycorrhizae; “EM” = ectomycorrhizae; “ErM” = 9577 194 9771 associations fungi source ericoid mycorrhizae”;, "OrM" = orchid mycorrhizae; “mycorrhizal” means mycorrhizae are present but type is unknown; “NM” = non-mycorrhizal. Microbial Mycorrhizal Mycorrhiza_Visual Intensity of mycorrhizal colonization based on five categories of mycorrhizal intensity, associations fungi estimate of root F00631 estimated visually and averaged across samples examined. 5554 0 5554 colonization intensity Nutrient Plant N uptake Physiology uptake rate_Annual N uptake per F00780 Total amount of nitrogen taken up by plants per hectare over the course of a year. 174 81 255 ground area Plant N uptake_Daily Uptake of 15N-Glycine tracer over the course of 24 hours per shoot dry mass, estimated with the Physiology Nutrient uptake of molar F01396 equation F = [T(AS-AB)]/AF, where T is the plant N concentration AS is the atom percent excess 10 0 10 uptake 15N-Glycine per shoot 15N in the sample, AB is the atom percent excess 15N in the natural sample, and AF is the atom dry mass percent excess 15N in the tracer. Plant N uptake_Daily Uptake of 15N-NH4+ tracer over the course of 24 hours per shoot dry mass, estimated with the Physiology Nutrient uptake of molar 15NH4+ F01392 equation F = [T(AS-AB)]/AF, where T is the plant N concentration AS is the atom percent excess 10 0 10 uptake per shoot dry mass 15N in the sample, AB is the atom percent excess 15N in the natural sample, and AF is the atom percent excess 15N in the tracer. Plant N uptake_Daily Uptake of 15N-NO3- tracer over the course of 24 hours per shoot dry mass, estimated with the Physiology Nutrient uptake of molar 15NO3- F01394 equation F = [T(AS-AB)]/AF, where T is the plant N concentration AS is the atom percent excess 10 0 10 uptake per shoot dry mass 15N in the sample, AB is the atom percent excess 15N in the natural sample, and AF is the atom percent excess 15N in the tracer. Plant N uptake_Daily Combined uptake of 15N-NH4+, 15N-NO3-, and 15N-Glycine tracers over the course of 24 hours per Physiology Nutrient uptake of molar total F01398 shoot dry mass, estimated with the equation F = [T(AS-AB)]/AF, where T is the plant N 10 0 10 uptake 15N per shoot dry mass concentration AS is the atom percent excess 15N in the sample, AB is the atom percent excess 15N in the natural sample, and AF is the atom percent excess 15N in the tracer. Microbial Rhizosphere associations Rhizosphere soil_specific F01453 Mass of soil left on root after all bulk soil is shaken off, divided by root fresh mass 58 0 58 rhizosheath mass by FW Chemistry Secondary Root 12 phenol content F00380 Total concentration of twelve monophenols extracted from root per root C content. 35 0 35 Compounds per root C content Chemistry Macronutrients Root 15N content F00273 Concentration of nitrogen stable isotope in sampled roots, determined with a mass spectrometer. 5 0 5 Chemistry Stoichiometry 3,5-dihydroxybenzoic F00407 Percent of 3,5-dihydroxybenzoic acid to vanillyl phenols in root. Indicative of degradation 5 0 5 acid groups/vanillyl status. phenols ratio Chemistry Secondary Root acid hydrolyzable F00342 Concentration of acid soluble compounds in root. 156 3 159 Compounds compounds content Physiology Exudation Root acid phosphatase F00743 Acid phosphatase activity per unit root mass. 333 3 336 activity per root mass Chemistry Stoichiometry Root acid/aldehyde ratio F00403 Ratio of acid to aldehyde in root syringyl phenols. Indicative of diagenetic state of lignin. 5 0 5 for syringyl phenols Chemistry Stoichiometry Root acid/aldehyde ratio F00405 Ratio of acid to aldehyde in root vanillyl phenols. Indicative of diagenetic state of lignin. 5 0 5 for vanillyl phenols Anatomy Aerenchyma Root aerenchyma fraction F00100 Percentage of root cross-sectional area that consists of aerenchyma. 13 0 13 of cross section Anatomy Aerenchyma Root aerenchyma porosity F00097 Percent porosity of root aerenchyma. 787 0 787 Anatomy Aerenchyma Root aerenchyma presence F00103 Whether aerenchyma are present in root. 12 0 12 Dynamics Lifespan Root age survivorship F00476 Percentage of roots that live to the age presented in the “root age” column. 308 87 395 Chemistry Micronutrients Root Al content F00296 Mass of aluminum per root mass for sampled roots. 115 43 158 Chemistry Micronutrients Root Al content per root F01304 Mass of root aluminum per fresh mass for sampled roots. 6 0 6 fresh mass Chemistry Secondary Root alkyl C content per F00431 Fraction of root carbon that is in alkyl groups. 8 0 8 Compounds root C content Chemistry Secondary Root arabinans content F00346 Concentration of arabinans in root. 13 1 14 Chemistry Secondary Root aromatic C content F00437 Fraction of root carbon that is in aromatic groups. 4 0 4 Compounds per root C content Chemistry Micronutrients Root As content F00298 Mass of arsenic per root mass for sampled roots. 36 16 52 Chemistry Secondary Root ash C content per F00340 Percentage of root carbon that is ash. 13 0 13 Compounds root C content Chemistry Secondary Root ash content F00347 Concentration of ash in root. 32 1 33 Chemistry Micronutrients Root B content F00301 Mass of boron per root mass for sampled roots. 4 9 13 Secondary Root bound phenol Concentration of phenolic compounds that are bound to cell walls through ester/ether linkages per Chemistry Compounds content per root C F00348 root carbon content. 35 0 35 Root branching Architecture Topology architecture_Root length F00187 Root length for a given order divided by the length of the roots in the higher order. 232 0 232 per higher order root Root branching intensity Architecture Topology (branching ratio)_Number F00179 Number of roots in a given order divided by the number of roots in the higher order. 497 6 503 of roots per higher order root Root branching Architecture Topology intensity_root tips per F00182 Number of lower-order roots per centimeter length of higher-order root. 79 0 79 higher order root length Root branching Architecture Topology intensity_root tips per F01339 Number of lower-order roots per total root length in category. 1504 8 1512 total root length Chemistry Macronutrients Root C content F00253 Mass of carbon per root mass for sampled roots. 2817 284 3101 Root System Standing crop Root C content per F00909 Total mass of root carbon per ground area. 96 172 268 ground area Root System Standing crop Root C content per soil F00906 Mass of root carbon per gram of soil. 236 0 236 Chemistry Stoichiometry Root C/N ratio F00413 Ratio of carbon to nitrogen in root by mass. 3686 499 4185 Chemistry Macronutrients Root Ca content F00249 Mass of calcium per root mass for sampled roots. 633 141 774 Chemistry Macronutrients Root Ca content per root F01299 Mass of calcium per root fresh weight for sampled roots. 6 0 6 fresh mass Chemistry Micronutrients Root Cd content F00305 Mass of cadmium per root mass for sampled roots. 36 3 39 Root cellulose and Chemistry Cellulose hemicellulose content F00245 Sum of concentrations of cellulose and all hemicelluloses in root. 15 0 15 per root mass Chemistry Cellulose Root cellulose content F00237 Concentration of cellulose in root. 246 72 318 Chemistry Cellulose Root cellulose content F00240 Percentage of root carbon that is cellulose. 7 0 7 per root C content Secondary Root cinnamyl phenol Chemistry Compounds content per root C F00350 Concentration of cinnamyl phenols in root per root carbon content. 5 0 5 Chemistry Stoichiometry Root cinnamyl phenol/ F00417 Ratio of cinnamyl phenol to vanillyl phenol in root. Index for woody or nonwoody source. 5 0 5 vanillyl phenol ratio Chemistry Micronutrients Root Cl- ion content F00308 Mass of Cl- ions per root mass for sampled roots, determined with flame photometer. 12 0 12 Morphology Root Color Root color_White or F00691 Whether observed roots are considered to be white or brown. 98 0 98 Chemistry Secondary Root condensed tannins F01330 Concentration of condensed tannins in root. 23 0 23 Anatomy Vessels or Root conduit diameter F00142 D_h=[(1/n)sigma(n/i=1)d^4]^(1/4) where d is conduit lumen diameter and n is conduit number. 75 0 75 Anatomy Vessels or Root conduit number per F00134 Number of conduits per stele cross-section area. 24 0 24 Tracheids root stele area Anatomy Vessels or Root conduit wall F00154 Thickness of conduit wall from root scans. 69 0 69 Tracheids thickness Chemistry Construction Root construction cost F00246 Grams of glucose equivalent per gram of root dry weight. 28 0 28 Anatomy Cortex Root cortex thickness F00104 Thickness of root cortex. 353 0 353 Chemistry Micronutrients Root Cr content F00311 Mass of chromium per root mass for sampled roots. 36 8 44 Morphology Diameter Root cross-sectional F01317 Cross-sectional area of root. 76 0 76 Chemistry Micronutrients Root Cu content F00314 Mass of copper per root mass for sampled roots. 67 27 94 Root The K constant for the root in the exponential decay function Mt = M0 e-kt, where Mt is the mass Dynamics Decomposition decomposition_Annual k F00461 of litter after time t, and M0 is the initial mass of litter. 613 184 797 Dynamics Decomposition decomposition_Annual N F00466 Amount of nitrogen immobilized over the course of a year in decomposing roots. 4 0 4 immobilized in decomposing roots Dynamics Decomposition decomposition_Annual F00465 Amount of root necromass that decomposes over the course of a year. 0 15 15 necromass decomposition rate per ground area Dynamics Decomposition decomposition_Estimated F00457 Difference between cumulative growth rate and fine root biomass. 0 6 6 cumulative monthly mass Dynamics Decomposition decomposition_Fraction C F01364 For decomposition experiment, percentage of original root carbon remaining. 39 0 39 Dynamics Decomposition decomposition_Fraction F00458 For decomposition experiment, percentage of original root mass remaining. 2007 1758 3765 mass remaining Dynamics Decomposition decomposition_Fraction N F00591 For decomposition experiment, percentage of original root nitrogen remaining. 716 136 852 Dynamics Decomposition decomposition_Monthly k F00463 K constant (decomposition constant, see F00461) for a single month. 0 12 12 Dynamics Decomposition decomposition_Monthly F00464 Amount of root necromass that decomposes over the course of a month. 0 12 12 necromass decomposition rate per ground area Morphology Diameter Root diameter F00679 Diameter of roots observed. 8729 487 9216 Architecture Topology Root dichotomous F00210 Determined by formula [Pe – min(Pe)] / [max(Pe) – min(Pe)]. DBI of 1 indicates a completely 4 0 4 branching index herringbone topology, while DBI of 0 indicates a completely dichotomous topology. Morphology Dry matter Root dry matter content F00689 Root dry mass divided by root fresh mass. 2102 159 2261 content (RDMC) Chemistry Secondary Root ethanol soluble F00352 Percentage of root tissue that is soluble in ethanol. 10 0 10 Compounds fraction Anatomy Hypodermis Root exodermal wall F00108 Thickness of exodermal wall from root scans. 27 0 27 Root external path Sum of the number of links in all paths from each external link (root segment between two nodes) Architecture Topology length (Pe) F00213 to the base link (the link from which all other links descend). (Definition from Beidler et al. 878 127 1005 2015, DOI: 10.1111/nph.13123). Physiology Exudation Root exudation_C F01418 Total carbon exudation rate by fine roots per individual plant. 4 0 4 exudation rate per plant Root exudation_C Physiology Exudation exudation rate per root F01416 Carbon exudation rate by length of fine roots. 4 0 4 Root exudation_C Physiology Exudation exudation rate per root F00754 Carbon exudation rate by mass of fine roots. 28 0 28 Physiology Exudation exudation_Concentration F01430 Concentration of root exudates in the soil, estimated using 13C labelling. 6 0 6 in soil Physiology Exudation Root exudation_Fraction F00751 Collected exudates scaled as proportion of estimated biomass accumulation. 4 0 4 estimated root mass Root exudation_molar C Physiology Exudation exudation rate per root F01404 Root carbon exudation rate in terms of micromoles of Carbon exuded per root area. 32 0 32 Chemistry Micronutrients Root Fe content F00318 Mass of iron per root mass for sampled roots. 81 22 103 Architecture Topology Root forks per root F01414 Number of root forks per cm of root length. 0 24 24 Architecture Topology Root fractal dimension F00199 Fractal dimension of scanned roots. 908 238 1146 Chemistry Secondary Root free phenol content F00353 Concentration of phenolic compounds that are nonassociated forms in cell vacuoles per root carbon 35 0 35 Compounds per root C content content. Chemistry Secondary Root galactans fraction F00355 Concentration of galactans in root. 13 1 14 Root growth_Annual C Dynamics Root Growth production per ground F00518 Amount of root carbon produced in one year. 10 193 203 Root growth_Annual Dynamics Root Growth length production per F00546 Annual increase in total root length per unit ground area. 145 148 293 ground area Root growth_Annual Dynamics Root Growth length production per F00553 New root length observed per square centimeter of minirhizotron surface per year. 5 38 43 minirhizotron surface Root growth_Annual mass Dynamics Root Growth production per ground F00521 Amount of root mass produced in one year. 778 503 1281 Root growth_Annual net Live root length density appearance minus live root length density disappearance, adjusted to an Dynamics Root Growth new length production F00510 annual value by subtracting net new length growth from the value for the previous sampling date, 44 0 44 per ground area dividing by the number of days between sampling dates, and multiplying the result by the number of days in the year. Root growth_Annual Dynamics Root Growth surface area production F00571 Root surface area produced per square meter of ground area per year. 24 18 42 per ground area Root growth_Bimonthly Dynamics Root Growth number of roots born per F00559 Inferred number of roots born per square centimeter of in-growth screen per two months. 0 16 16 ingrowth screen area Root growth_Cumulative Dynamics Root Growth length production per F00543 Cumulative root length density appearance per unit ground area. 376 0 376 ground area Root growth_Cumulative Dynamics Root Growth mass production per F00527 Total root mass produced since the beginning of the study. 0 93 93 ground area Root growth_Cumulative Cumulative rate of root growth into in-growth mesh, in kg per mesh cross-sectional area per Dynamics Root Growth monthly mass ingrowth F00526 month. 0 6 6 per screen area Root growth_Cumulative Dynamics Root Growth net length production F00507 Difference between cumulative live root appearance and cumulative live root disappearance. 46 0 46 per ground area Dynamics Root Growth Root growth_Daily F00538 Rate at which elongation occurs for the roots where it does occur. 0 192 192 elongation rate Root growth_Daily length Dynamics Root Growth production per coarse F00568 Total fine root length production per day per length of woody root. 12 0 12 root length Root growth_Daily length Dynamics Root Growth production per F00551 New root length observed per square centimeter of minirhizotron surface per day. 108 164 272 minirhizotron surface Root growth_Daily mass Dynamics Root Growth production per coarse F00569 Total fine root mass produced per day per length of woody root. 12 0 12 root length Root growth_Daily mass Dynamics Root Growth production per ground F00529 Amount of root mass produced in one day. 0 52 52 Dynamics Root Growth Root growth_Elongation F00537 Percentage of roots that have elongated since the previous measurement. 0 196 196 Dynamics Root Growth Root growth_Fraction F00513 The root production rate observed at a specific point divided by the maximum root production rate 126 0 126 peak production observed during the entire observation period. Dynamics Root Growth Root growth_Length F00286 Root length produced per unit mass of root phosphorus. 13 0 13 produced per P content Root growth_Length Increase in root length per square meter of ground area during the observation period specified Dynamics Root Growth production per ground F00556 in the “[year/month/day] beginning data collection” and “[year/month/day] ending data collection” 0 16 16 area and exposition columns. Root growth_Length Dynamics Root Growth production per F00540 Root length produced per minirhizotron frame. 32 0 32 minirhizotron frame Dynamics Root Growth Root growth_Length F00566 Root length recovery from pruning. 56 4 60 recovery from pruning Root growth_Mass Dynamics Root Growth production per ground F00534 Root biomass produced for the time interval specified in “main exposition period” and “production 82 74 156 area and exposition duration” or “in growth duration” (columns F01294 and F0076,F01283,F01284, or F01280). Root growth_Mass Dynamics Root Growth production per plant per F01276 Increase in belowground biomass per plant (by dry weight) over the course of one growing season. 8 0 8 growing season Dynamics Root Growth Root growth_Monthly mass F00531 Monthly rate of root growth into in-growth mesh, in kg per mesh cross sectional area per month. 0 6 6 ingrowth per screen area Root growth_Monthly mass Dynamics Root Growth production per ground F00532 Amount of root mass produced in one month. 123 0 123 Root growth_Monthly mass Dynamics Root Growth production per soil F00533 Root mass produced per cubic centimeter of soil each month. 108 0 108 Root growth_Number of Dynamics Root Growth roots per area of F00903 Number of roots per square cm of in-growth screen. 0 89 89 ingrowth screen Root growth_Predicted Log of predicted amplitude of monthly net root production if it occurs (ie if mRLQ>0), based on Dynamics Root Growth amplitude of monthly net F00561 negative binomial distribution. See Mao et al. 2013 (DOI: 10.1007/s11104-012-1324-2)for further 0 26 26 root production details. Root growth_Surface area Dynamics Root Growth production per ground F00574 Increase in root surface area per square meter of ground area during the observation period 0 16 16 area and exposition specified in the “Exposition period_main” column [F01294]. Morphology Root Hairs Root hair density F00733 Number of root hairs per root surface area. 25 0 25 Morphology Root Hairs Root hair diameter F00736 Diameter of root hairs. 25 0 25 Morphology Root Hairs Root hair incidence F00692 Percentage of fine roots that contain root hairs, by the gridline method. 2 60 62 Morphology Root Hairs Root hair length F00696 Length of root hairs. 231 60 291 Morphology Root Hairs Root hair volume per F00700 Volume of root hairs per gram of root mass, estimated using specific root length. 28 0 28 root mass Root hemicellulose Chemistry Cellulose content per root C F00242 Percentage of root carbon that is hemicellulose. 2 0 2 Chemistry Cellulose Root hemicellulose F00243 Percentage of root mass that is hemicellulose. 174 2 176 content per root mass Root System Standing crop Root intact branch F00196 Total root length contained within an intact root branch. 4 0 4 Chemistry Macronutrients Root K content F00289 Mass of potassium per root mass for sampled roots. 561 223 784 Chemistry Macronutrients Root K content per root F01302 Mass of root potassium per fresh mass for sampled roots. 6 0 6 fresh mass Nutrient Root K uptake_Fraction K Estimated potassium uptake capacity of roots at the depth interval specified in Soil depth_Upper Physiology uptake uptake capacity per soil F00766 sampling depth (F00068) and Soil depth_Lower sampling depth (F00069) relative to the plant’s 15 0 15 depth interval total uptake capacity. Root length density Root System Standing crop (RLD)_Root length per F00934 Root length divided by soil core cross-sectional area. 3224 409 3633 ground area Root length density Root System Standing crop (RLD)_Root length per F00938 Root length divided by the sampled soil volume. 647 152 799 soil volume Allocation Root length fraction per Root System within root root diameter class F00864 Percentage of root length composed of a specific root diameter class. 137 24 161 Allocation Root length fraction per Root System within root root order class F00867 Percentage of root length composed of a specific root order. 204 3 207 Root System Standing crop Root length fraction per F00933 Percentage of total root length belonging to an individual species. 18 0 18 Morphology Root Length Root length from base to F00703 Distance from root base to tip. 1034 28 1062 Root System Standing crop Root length per F00930 Total root length observed per minirhizotron frame. 32 0 32 minirhizotron frame Root length per Root System Standing crop minirhizotron surface F00944 Meters of root length visible per square meter of minirhizotron surface. 170 137 307 Root System Standing crop Root length per plant F00942 Total length of all roots for an entire plant. 1907 127 2034 Root System Standing crop Root length ratio (RLR) F00927 Root length per gram of plant mass. 68 0 68 Nutrient Root Li uptake_Daily Daily uptake rate of lithium by roots, reported in terms of micrograms taken up per meter of root Physiology uptake uptake of molar Li per F01427 length. 16 0 16 root length Chemistry Secondary Root lignin content F00358 Concentration of lignin in root. 553 208 761 Chemistry Secondary Root lignin content per F00361 Percentage of root carbon that exists as lignin. 7 0 7 Compounds root C content Secondary Root lignin phenol Chemistry Compounds content per root C F00356 Total concentration of monophenols that constitute lignin in root per root carbon content. 35 0 35 Secondary Root lignin phenol [S%(S% + 1)/(V% + 1) + 1] * [C%(C% + 1)/(V% + 1) + 1], where V%, S%, and C% are the respective Chemistry Compounds vegetation index (LPVI) F00338 percentages of vanillyl phenol, syringyl phenol, and cinnamyl phenol in root lignin. Indicator 5 0 5 for taxonomic source identification. Chemistry Stoichiometry Root lignin/N ratio F00419 Ratio of lignin concentration to N concentration in root. 118 15 133 Architecture Topology Root link branching F00170 Mean angle between a link (segments of roots between two nodes or a node and a tip) and the 878 238 1116 angle extension of the link before it. Architecture Topology Root link length F00203 Length of root links (segments of roots between two nodes or a node and a tip). 1138 238 1376 Architecture Topology Root links per root F00173 Number of links (segments of roots between two nodes or a node and a tip) per root branch. 4 0 4 Chemistry Secondary Root lipid content F00362 Concentration of lipids in root. 18 0 18 Chemistry Secondary Root mannans content F00364 Concentration of mannans in root. 13 1 14 Root System Allocation Root mass fraction (RMF) F00846 Root biomass divided by total plant biomass. 6003 8 6011 within plant Allocation Root mass fraction per Root System within root root diameter class F00855 Percentage of root biomass contributed by a particular diameter class. 210 16 226 Allocation Root mass fraction per Root System within root root order class F00858 Percentage of root biomass contributed by a particular root order. 297 0 297 Allocation Root mass fraction per Root System within root species F00884 Percentage of total root mass composed of a particular species in a community. 48 12 60 Dynamics Lifespan Root mean lifespan_d F00469 Mean lifespan of roots observed, expressed in days. 203 45 248 Root mean Mean lifespan in days. This column combines data from both “Root mean lifespan_d” (F00470) and Dynamics Lifespan lifespan_Main_d F01295 “Root mean lifespan_yr” (F01292) columns. If source originally presents lifespan in years or 256 45 301 months, lifespan is converted to days for this value. Dynamics Lifespan Root mean lifespan_yr F01292 Mean lifespan of roots observed, expressed in years. 53 0 53 Dynamics Lifespan Root median lifespan_d F00470 Median lifespan of roots observed, expressed in days. 232 10 242 Root median Median lifespan in days. This column combines data from both “Root median lifespan_d” (F00470) Dynamics Lifespan lifespan_Main_d F01296 and “Root median lifespan_yr” (F01293) columns. If source originally presents lifespan in years 243 10 253 or months, lifespan is converted to days for this value. Dynamics Lifespan Root median lifespan_yr F01293 Median lifespan of roots observed, expressed in years. 11 0 11 Anatomy Vessels or Root metaxylem cell wall F00150 Thickness of the cell wall in the root metaxylem. 4 0 4 Tracheids thickness Vessels or Root metaxylem cell wall Anatomy Tracheids thickness/vessel F00128 Percentage of metaxylem vessel diameter that consists of cell wall thickness. 4 0 4 diameter ratio Anatomy Vessels or Root metaxylem vessel F00140 Vessel diameter of root metaxylem. 4 0 4 Tracheids diameter Chemistry Macronutrients Root Mg content F00257 Mass of magnesium per root mass for sampled roots. 525 133 658 Chemistry Macronutrients Root Mg content per root F01300 Mass of magnesium per root fresh weight for sampled roots. 6 0 6 fresh mass Microbial Other Root microbes_Bacterial Concentration of bacterial carbon in root material, estimated based on calculated average associations rhizosphere biomass C content per F00659 conversion factor of muramic acid to fungal carbon. 15 1 16 microbes root mass Microbial Other Root microbes_Fungal Concentration of fungal carbon in root material, estimated based on calculated average conversion associations rhizosphere biomass C content per F00658 factor of glucosamine to fungal carbon. 15 1 16 microbes root mass Microbial Other Root microbes_Fungal C/ associations rhizosphere bacterial C ratio F00660 Ratio of root fungal carbon to root bacterial carbon. 15 1 16 Microbial Other Root microbes_Microbial Carbon from microbial biomass in roots, calculated from root ergosterol and the ergosterol/ associations rhizosphere biomass C content per F00656 microbial biomass carbon ratio of the rhizosphere soil. 17 1 18 microbes root mass Microbial Other Root microbes_Microbial Nitrogen from microbial biomass in roots, calculated from root ergosterol and microbial biomass C associations rhizosphere biomass N content per F00657 /N ratio of the rhizosphere soil. 14 1 15 microbes root mass Microbial Other Root microbes_Root associations rhizosphere glucosamine content per F00652 Concentration of glucosamine measured in root material. 18 1 19 microbes root mass Microbial Other Root microbes_Root Percentage of root length that contains fungal hyphae, determined using a random intercept associations rhizosphere length hyphal fraction F00629 method. 0 4 4 Chemistry Micronutrients Root Mn content F00322 Mass of manganese per root mass for sampled roots. 247 30 277 Chemistry Micronutrients Root Mn content per root F01305 Mass of root manganese per fresh mass for sampled roots. 6 0 6 fresh mass Root mortality_Annual Dynamics Root Mortality root length mortality F00500 Annual mortality of root length density. 112 30 142 per ground area Root mortality_Annual Dynamics Root Mortality root length mortality F00498 Observed length of roots which dies per minirhizotron surface per year. 5 13 18 per minirhizotron surface area Root mortality_Annual Dynamics Root Mortality root mass mortality per F00490 Amount of root biomass that dies in one year per square meter of ground area. 44 44 88 ground area Dynamics Root Mortality root length F00487 Cumulative root length density disappearance observed per ground area. 374 0 374 disappearance per ground Dynamics Root Mortality mortality_Cumulative F00493 Root length mortality observed per minirhizotron frame since the beginning of observation. 28 0 28 root length mortality per minirhizotron frame Root mortality_Daily Dynamics Root Mortality root length mortality F00496 Observed length of roots which dies per minirhizotron surface per day. 104 130 234 per minirhizotron surface area Root mortality_Fraction Dynamics Root Mortality initial intersections F00505 For in-growth screen, percentage of root-screen contacts lost after installation. 0 28 28 with ingrowth screen Root mortality_Monthly Mass of root biomass that disappears per cubic centimeter of soil per month. Calculated based on Dynamics Root Mortality mass disappearance per F00485 loss of mass. 72 0 72 soil volume Root mortality_Monthly Amount of root biomass that dies per cubic centimeter of soil per month. Calculated based on Dynamics Root Mortality necromass increase per F00486 change in necromass. 102 0 102 soil volume Root mortality_Monthly Dynamics Root Mortality number of roots lost per F00503 Inferred number of roots to die per square centimeter of in-growth screen per two months. 0 16 16 ingrowth screen area Root mortality_Root N Dynamics Root Turnover loss per annual plant N F00577 Nitrogen lost through fine root mortality as a percentage of annual whole-plant nitrogen uptake. 5 0 5 Root mortality_Root N Nitrogen lost through fine root mortality as a percentage of whole-plant N storage (root nitrogen Dynamics Root Turnover loss per plant N content F00578 per ground area). 5 3 8 per ground area Microbial Other Root muramic acid associations rhizosphere content F00653 Concentration of muramic acid measured in root material. 18 1 19 Chemistry Macronutrients Root N content F00261 Mass of nitrogen per root mass for sampled roots. 5717 1047 6764 Root System Standing crop Root N content per F00921 Total mass of root nitrogen per ground area for the specified depth increment. 166 382 548 ground area Chemistry Macronutrients Root N content per root F00270 Root nitrogen mass per meter of root length. 249 0 249 Root System Standing crop Root N mass per soil F00924 Mass of root nitrogen per gram of soil. 236 0 236 Physiology Nutrient Root N uptake_Cumulative F00770 Total root nitrogen uptake since the first observation per meter square of ground area. 20 0 20 uptake N uptake Nutrient Root N uptake_Hourly Physiology uptake NO3- uptake per root F01459 Hourly maximum uptake capacity of NO3- per root mass, determined by colorometric methods. 0 0 0 Nutrient Root N uptake_Hourly Hourly uptake of nitrogen by roots per root fresh mass, as determined by a root bioassay using Physiology uptake uptake of 15NH4+ per F00773 15NH4+. Reported in terms of nanograms of NH4+. 4 0 4 root fresh mass Nutrient Root N uptake_Hourly Hourly uptake of nitrogen by roots per root fresh mass, as determined by a root bioassay using Physiology uptake uptake of molar NH4+ per F00775 15NH4+. Reported in terms of nmol of NH4+. 22 0 22 root fresh mass Nutrient Root N uptake_Hourly Physiology uptake uptake of molar NH4+ per F01457 Hourly maximum uptake capacity of NH4+ per root mass, determined by colorometric methods. 0 0 0 root mass Nutrient Root N uptake_Hourly Physiology uptake uptake per unit root F00777 Amount of nitrogen taken up per unit root mass per hour. 24 0 24 Nutrient Root N uptake_Molar Physiology uptake inorganic N uptak?e per F01405 Hourly uptake of inorganic nitrogen in terms of micromoles of nitrogen per root area. 8 0 8 root area Physiology Nutrient Root N uptake_Molar NH4+ F01406 Hourly uptake of nitrogen by roots per root surface area, as determined by a root bioassay using 8 0 8 uptake uptake per root area 15NH4+. Reported in terms of umol of NH4+. Physiology Nutrient Root N uptake_Molar NO3- F01407 Hourly uptake of nitrogen by roots per root surface area, as determined by a root bioassay using 8 0 8 uptake uptake per root area 15NO3-. Reported in terms of umol of NO3-. Nutrient Root N uptake_Molar Physiology uptake organic N uptake per F01408 Hourly uptake of organic nitrogen in terms of micromoles of nitrogen per root area. 8 0 8 root area Chemistry Micronutrients Root Na content F00334 Mass of sodium per root mass for sampled roots. 51 42 93 Chemistry Micronutrients Root Na+ content F00331 Mass of Na+ ions per root mass, determined by silver ion titration. 32 0 32 Chemistry Stoichiometry Root NAH/root N ratio F00409 Ratio of non-acid hydrolysable compounds to nitrogen in root tissue. 102 3 105 Allocation Root necromass/biomass Root System within root ratio F00870 Dead root mass divided by live root mass. 751 882 1633 Chemistry Secondary Root neutral detergent F00365 Percentage of root mass that is soluble in neutral detergent. 7 0 7 Compounds soluble fraction Chemistry Micronutrients Root Ni content F00326 Mass of nickel per root mass for sampled roots. 40 12 52 Microbial Nitrogen Root nodules_Nodule mass associations fixers on dead roots per ground F00649 Total dry weight of all root nodules collected from dead roots in soil cores. 25 0 25 Microbial Nitrogen Root nodules_Nodule mass associations fixers on living roots per F00646 Total dry weight of all root nodules collected from living roots in soil cores. 3 0 3 ground area Secondary Root non-acid Concentration of compounds in root that are not acid soluble. Also known as Root acid insoluble Chemistry Compounds hydrolyzable compounds F00366 fraction (AIF). 310 10 320 content (NAH) Total Root non-structural C Chemistry non-structural content per root C F00427 Percentage of root carbon that is non-structural 2 0 2 carbohydrates content Chemistry Secondary Root nonpolar compounds F00374 Concentration of nonpolar compounds in root. 146 11 157 Compounds content Anatomy Vessels or Root number of tracheary F00132 Number of tracheary elements present in root, with secondary cell-wall thickening counted per 5 0 5 Tracheids elements xylem pole. Anatomy Vessels or Root number of vessels F01313 Number of vessels per root cross-section. 202 0 202 Physiology Nutrient Root N_Annual turnover F00769 Annual root nitrogen turnover per meter square of ground area. 26 0 26 uptake per ground area Chemistry Macronutrients Root N_Molar organic N F00276 Millimoles of organic nitrogen in root per gram of root mass. 20 0 20 Chemistry Secondary Root O-alkyl C content F00434 Fraction of root carbon that is in O-alkyl groups. 4 0 4 Compounds per root C content Chemistry Stoichiometry Root organic N/root F00421 Root organic N concentration divided by total root N concentration. 20 0 20 total N ratio Chemistry Macronutrients Root P content F00277 Mass of phosphorous per root mass for sampled roots. 1603 310 1913 Chemistry Macronutrients Root P content per root F01301 Mass of root phosphorus per fresh mass for sampled roots. 6 0 6 fresh mass Physiology Nutrient Root P uptake_Fraction F00763 Fraction of maximum phosphorus uptake rate achieved by age group specified in F00062 and F00063. 22 0 22 uptake maximum P uptake rate Physiology Nutrient Root P uptake_Hourly F00784 Hourly uptake of H2PO4, as determined by a root bioassay. 4 0 4 uptake uptake of H2PO4- Physiology Nutrient Root P uptake_Rate per F00786 Rate of phosphorus uptake per gram of root per second. 12 0 12 uptake root mass Nutrient Root P uptake_Rate per Physiology uptake root surface area per F00788 Phosphorus uptake per unit root area, determined with tissue cassettes. 18 0 18 Secondary Root p-hydroxy phenol Concentration of p-hydroxybenzoic acid, p-hydroxyacetophenon, and p-hydroxybenzaldehyde phenols Chemistry Compounds content per root C F00378 in root per root C content. 5 0 5 Anatomy Hypodermis Root passage cell number F00109 Number of passage cells in exodermis. 27 0 27 in exodermis Anatomy Hypodermis Root passage cell number F00110 Number of passage cells per mm root circumference. 27 0 27 per root circumference Chemistry Micronutrients Root Pb content F00319 Mass of lead per root mass for sampled roots. 41 11 52 Anatomy Phellem Root phellem F00157 Number of phellem layers in root. 187 0 187 Chemistry Secondary Root phenols content F00382 Concentration of phenols per unit root dry mass. 45 0 45 Secondary Root phenols_chlorogenic Micromoles of chlorogenic acid yielded per root mass from ethanol boiling extraction and Chemistry Compounds acid molarity per root F01328 Folin-Ciocalteu’s phenol reagent. See page 1221 of Zadworny et al. 2017, DOI: 10.1111/gcb.13514, 212 0 212 mass for further information. Physiology Exudation Root phosphatase F00760 Phosphatase activity of roots, based on release of p-nitrophenol from a solution of p-nitrophenyl 13 0 13 activity per root length phosphate per meter of root length per hour. Physiology Exudation Root phosphatase F00757 Hydrolysis of p-nitrophenyl phosphatase to p-nitrophenol after incubation at 37°C for 1 hour. 14 0 14 activity per root mass Chemistry pH Root pH_Water F01461 pH of root powder, measured in water. 48 0 48 Chemistry Secondary Root polar compounds F00384 Concentration of polar compounds in root. 142 3 145 Compounds content Chemistry Secondary Root polyphenol content F00388 Concentration of polyphenols in root. 20 0 20 Chemistry Stoichiometry Root polyphenol/root N F00422 Ratio of root polyphenol concentration to root N concentration. 6 0 6 Nutrient Form or forms of nitrogen preferentially taken up by roots. NH4-N is NH4, NO3-N is NO3, and ON is Physiology uptake Root preferred N form F00791 organic nitrogen. If multiple forms are preferred, their relative preference to each other is 37 0 37 shown with >=,=, or = symbols. Anatomy Vessels or Root protoxylem cell F00152 Thickness of the cell wall in the root protoxylem. 4 0 4 Tracheids wall thickness Vessels or Root protoxylem cell Anatomy Tracheids wall thickness per F00130 Percentage of protoxylem vessel diameter that consists of cell wall thickness. 4 0 4 protoxylem diameter Anatomy Vessels or Root protoxylem diameter F00148 Vessel diameter of root protoxylem. 4 0 4 Dynamics Root Growth Root pruning recovery F00570 Percentage pruned woody roots that recovered by proliferating fine roots. 6 0 6 Nutrient Root Rb uptake_Daily Daily uptake rate of rubidium by roots, reported in terms of nanograms taken up per meter of root Physiology uptake uptake of molar Rb+ per F01425 length. 0 0 0 root length Physiology Nutrient Root Rb uptake_Hourly Rb F00796 Hourly uptake rate of rubidium by roots, as determined by a root bioassay using 86Rb+, reported 19 0 19 uptake uptake per root mass in terms of nanograms taken up per fresh weight of roots per hour. Nutrient Root Rb uptake_Hourly Hourly uptake rate of rubidium by roots per root fresh mass, as determined by a root bioassay Physiology uptake uptake of molar Rb+ per F00792 using 86Rb+, reported in nmol taken up per fresh weight of roots per hour. 19 0 19 root fresh mass Root relative growth [In (each growth response at 30, 60 and 90 days of the treatments) - In (each growth response at Dynamics Root Growth rate (RGR)_Root length F00515 0, 30 and 60 days of the treatments, respectively)]/30 days; see Imada et al. 2008 (DOI: 10.1111/ 68 0 68 j.1365-2435.2008.01454.x) for more details. Root respiration rate Physiology Respiration per root dry mass_CO2 F00802 Root respiration rate of CO2 per g root dry mass per second. 278 59 337 Root respiration rate Physiology Respiration per root dry mass_O2 F00799 Root respiration rate of O2 per root dry mass, as measured in a respiration chamber. 1448 162 1610 Root respiration rate Physiology Respiration per root length_CO2 F00807 Root respiration rate of CO2 per meter of root length, as measured in a respiration chamber. 43 0 43 Root respiration rate Root respiration rate, converted from terms of nmol O2 g-1 s-1,converted to µg C g-1 s-1using a Physiology Respiration per root mass_C release F00805 respiratory quotient of 1.25. See Matamala & Schlesinger 2000 (DOI: 10.1046/ 20 0 20 j.1365-2486.2000.00374.x) for further details. Chemistry Secondary Root rhamnan content F00391 Concentration of rhamnans in root. 10 0 10 Chemistry Macronutrients Root S content F00293 Mass of sulfur per root mass for sampled roots. 111 41 152 Chemistry Macronutrients Root S content per root F01303 Mass of root sulfur per fresh mass for sampled roots. 6 0 6 fresh mass Chemistry Micronutrients Root Si content F00329 Mass of silicon per root mass for sampled roots. 29 0 29 Chemistry Secondary Root starch fraction F01329 Percentage of root mass consisting of starches. 274 21 295 Anatomy Vessels or Root stele F01324 Cross-sectional area of root stele. 9 0 9 Tracheids cross-sectional area Anatomy Stele Root stele F00122 Percentage of root cross-sectional area that is occupied by the stele. 120 0 120 cross-sectional fraction Anatomy Stele Root stele diameter F00118 Diameter of root stele. 417 0 417 Anatomy Stele Root stele diameter/root F00115 Stele diameter divided by root diameter. 343 0 343 diameter ratio Anatomy Stele Root stele/root cortex F00125 Proportion of root cross sectional area occupied by the stele divided by proportion of root cross 93 0 93 ratio sectional area occupied by the cortex. Secondary Root structural C Chemistry Compounds content per root C F00392 Percentage of root carbon that is structural. 2 0 2 Allocation Root surface area Root System within root fraction per root order F00872 Percentage of total root surface area that a specific root order comprises. 41 0 41 system class Root System Standing crop Root surface area per F00878 Total surface area of roots per square meter of ground area. 126 99 225 ground area Dynamics Lifespan survivorship_Fraction of F00483 Percentage of root tips that survive after one year. 62 9 71 root tips surviving after one year Root Percentage of roots which survive for the time interval represented in the row (i.e., the time Dynamics Lifespan survivorship_Fraction F00480 period between the dates in the “[date] beginning collection” and “[date] ending collection” 24 0 24 roots surviving for columns). exposition period Dynamics Lifespan survivorship_Fraction F00479 Percentage of roots which survive over the course of a growing season. 28 0 28 roots surviving for growing season Secondary Root syringyl phenol Chemistry Compounds content per root C F00393 Concentration of syringyl phenols in root per root carbon. 5 0 5 Chemistry Stoichiometry Root syringyl/vanillyl F00423 Ratio of syringyl phenol to vanillyl phenol in root. Index for taxonomic level (angiosperm or 5 0 5 phenol ratio gymnosperm) of plant from which phenol was derived. Allocation Root tip fraction per Root System within root root diameter class F00953 Proportion of total number of root tips comprised by a particular diameter class. 354 0 354 Architecture Topology Root tips per ground F00219 Total number of root tips per square meter of ground area (sampled from a given depth increment). 76 16 92 Architecture Topology Root tips per F00216 Number of root tips observed per minirhizotron frame. 36 0 36 minirhizotron frame Architecture Topology Root tips per plant F00222 Total number of root tips for an entire plant. 896 127 1023 Architecture Topology Root tips per root F00176 Number of root tips and root endings per root branch. 4 0 4 Architecture Topology Root tips per soil F00224 Number of root tips per liter of soil. 38 0 38 Morphology Root tissue Root tissue density F00709 Weight of roots sampled divided by volume of roots. 6432 268 6700 density (RTD) Root topological index Slope of the linear regression of log10(Pe) against log10µ, in which Pe is external path length Architecture Topology (TI) F00226 and µ represents magnitude (number of root tips in the system). A greater TI indicates a more 878 238 1116 “herringbone” branching system. Total Root total Chemistry non-structural non-structural F00428 Glucose equivalents per dry weight of roots, determined colorimetrically. 717 26 743 carbohydrates carbohydrate content Root turnover_Annual Dynamics Root Turnover biomass turnover per F00579 Root mass multiplied by turnover rate. 18 25 43 ground area Dynamics Root Turnover Root turnover_Annual F00582 Turnover as the inverse of belowground net primary productivity. 652 268 920 root system replacement Dynamics Root Turnover Root turnover_Estimated F00589 Number of days in growing season divided by median lifespan of roots. 16 0 16 rate per growing season Dynamics Root Turnover Root turnover_Mass per F00585 Turnover during dry season as defined in original data source. 0 8 8 dry season Dynamics Root Turnover Root turnover_Mass per F00587 Turnover during wet season as defined in original data source. 0 8 8 wet season Secondary Root vanillyl phenol Chemistry Compounds content per root C F00395 Concentration of vanillyl phenols in root per root carbon. 5 0 5 Chemistry Stoichiometry Root vanillyl phenol/ F00425 Ratio of vanillyl phenol to lignin phenol in root. Indicative of lignin quality. 5 0 5 lignin phenol ratio Anatomy Vessels or Root vessel F01319 Cross-sectional area per root vessel. 76 0 76 Tracheids cross-sectional area Anatomy Vessels or Root vessel diameter F00145 Diameter of root vessels. 209 0 209 Vessels or Root vessel number per Anatomy Tracheids root cross-sectional F01321 Number of vessels in root per root cross-sectional area. 97 0 97 Root System Standing crop Root volume per ground F00946 Total volume of roots per square meter of ground area for the specified depth increment. 86 74 160 Secondary Root water or ethanol Chemistry Compounds soluble compounds F00397 Percentage of root tissue that is soluble in water or ethanol. 29 7 36 Chemistry Secondary Root water soluble F00399 Percentage of root tissue that is soluble in water. 178 71 249 Compounds compounds fraction Chemistry Secondary Root water soluble F00401 Percentage of root tissue that consist of water-soluble sugars, as determined by a 44 7 51 Compounds compounds per root mass phenol–sulfuric acid assay. Secondary Root water soluble Chemistry Compounds phenol compounds per F01281 Percentage of root tissue that consists of water-soluble phenols. 10 0 10 root mass Physiology Water uptake Root water uptake per F00819 Daily water uptake in milliliters, per centimeter of root length. 52 0 52 root length Physiology Water uptake Root water uptake per F00815 Water absorption rate per surface area of root. 9 0 9 root surface area Root water uptake_Hourly Physiology Water uptake uptake rate per soil F00822 Rate of water uptake by roots, divided by the volume of soil from which roots take up water. 149 0 149 Root water Physiology Water uptake uptake_Hydraulic F00812 Hydraulic conductivity of roots. 140 0 140 conductivity (Lp) Chemistry Secondary Root xylan content F00402 Concentration of xylans in root. 10 0 10 Anatomy Vessels or Root xylem F01311 Cross-sectional area of root xylem. 67 0 67 Tracheids cross-sectional area Anatomy Vessels or Root xylem F01315 Percentage of root cross-sectional area occupied by xylem. 67 0 67 Tracheids cross-sectional fraction Anatomy Vessels or Root xylem vessel number F00137 Number of xylem vessels per root stele cross-sectional area. 129 0 129 Tracheids per root stele area Chemistry Micronutrients Root Zn content F00335 Mass of zinc per root mass for sampled roots. 94 29 123 Root System Allocation Root/shoot ratio F00851 Ratio of root tissue mass to shoot mass, where rhizomes are counted as shoot mass. 341 34 375 within plant Root System Depth Rooting depth F00954 Depth which includes all observed roots. 2554 514 3068 Root System Depth Rooting depth_Active F00956 Depth where most of the root activity of interest occurs. 0 201 201 Depth Rooting Depth which includes 50 percent of total roots in profile, extrapolated from logistic Root System distribution depth_Extrapolated 50 F00959 dose-response model described in Schenk & Jackson 2003, DOI: 10.3334/ORNLDAAC/659 176 388 564 percent rooting depth Depth Rooting Depth which includes 95 percent of total roots in profile, extrapolated from logistic Root System distribution depth_Extrapolated 95 F00960 dose-response model described in Schenk & Jackson 2003, 10.3334/ORNLDAAC/659 199 388 587 percent rooting depth Depth Rooting depth_Fraction Proportion of observed roots (quantified by column F00961, “Notes_Rooting depth measurement”) Root System distribution roots in soil depth F00963 contained in depth interval. 248 180 428 Depth which includes 50 percent of all observed roots in profile, calculated by fitting logistic Depth Rooting dose-response curve to cumulative root profile and interpolating to maximum sampling depth, using Root System distribution depth_Interpolated 50 F00957 the equation D_S50=D_50([R_max/0.5R_Smax]-1)^(1/c) where Rmax is the estimated total amount of 176 388 564 percent rooting depth roots, RSmax is the total amount of roots in sampled profile, and C is a dimensionless shape Depth Rooting Depth which includes 95 percent of roots in profile, calculated by fitting logistic dose-response Root System distribution depth_Interpolated 95 F00958 curve to cumulative root profile and interpolating to maximum sampling depth, using the equation 204 388 592 percent rooting depth D_S95=D_50([R_max/0.95R_Smax]-1)^(1/c). Root System Depth Rooting depth_min F01409 Minimum depth which includes all observed roots. 131 20 151 Morphology Specific root Specific root area (SRA) F00718 Root surface area divided by root mass. 1829 365 2194 Architecture Topology Specific root forks F00207 Number of root bifurcations per gram of root mass. 3 9 12 density (SRFD) Morphology Specific root Specific root length F00727 Length of roots divided by root mass. 8336 607 8943 length (SRL) Architecture Topology Specific root tip F00192 Number of root tips per milligram of root mass. 1232 255 1487 abundance (SRTA) Root System within root Taproot mass fraction F00875 Percentage of root mass that consists of the taproot. 13 0 13	{"url":"https://roots.ornl.gov/data-inventory","timestamp":"2024-11-10T18:13:09Z","content_type":"text/html","content_length":"665869","record_id":"<urn:uuid:b0387967-296a-4eed-a846-4f635a43c4d2>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.61/warc/CC-MAIN-20241110170046-20241110200046-00302.warc.gz"}
CCO Preparation Test 1 P3 - K-th Rank Student View as PDF Submit solution Points: 17 (partial) Time limit: 0.6s Memory limit: 128M In Bruce's class, there are students, numbered from to . Bruce ranks his students in increasing order according to the number of questions they have solved on DMOJ. The student who solves the least number of questions is ranked 1st. Conveniently, no two students have solved the same number of questions, i.e. each student has a unique rank. Bruce finds that some students are friends. Note that friendship is a bidirectional relationship. For example, if Alex is Tim's friend, then conversely, Tim is also Alex's friend. Students are grouped together through friendship. Now, Bruce wants to motivate his students to solve more questions on DMOJ. He will perform a sequence of operations, each of which is one of the following: • , build a friendship between student and student ; • , query the student number who has the -th lowest rank in the group that contains student (note that the group of student includes student himself). Input Specification The first line of input will consist of two integers and (, ). is the number of students and is the number of initial friendships. The second line of input is a permutation of to , representing the rankings of all students. The next lines of input will contain the initial friendships. It is guaranteed that the numbers of the students that are friends will be valid. The next line of input is a single integer (), which is the total number of operations. Each of the next lines will contain an operation, starting with or as described above. Output Specification For each query operation, i.e. , output an integer on a single line. The integer is the number of the student who ranks the -th lowest in the group of student . If there is no such student, output Sample Input Q 3 2 Q 2 1 B 2 3 B 1 5 Q 2 1 Q 2 4 Q 2 3 Sample Output • commented on April 1, 2015, 11:49 a.m. ← edit 2 what is the best solution for this problem? i can only think of . is there a better one? hahhahahh, i modified from reading with cin to reading with scanf and my execution time reduced 10 times • Am i on the right track to the accepted solution? Thanks □ About halfway. Your merge function is . ☆ I've been having a tough time finding a sublinear merge	{"url":"https://dmoj.ca/problem/ccoprep1p3","timestamp":"2024-11-14T14:31:33Z","content_type":"text/html","content_length":"40518","record_id":"<urn:uuid:4e2642f8-8d1a-42a1-b8c6-e1c6fcba2536>","cc-path":"CC-MAIN-2024-46/segments/1730477028657.76/warc/CC-MAIN-20241114130448-20241114160448-00163.warc.gz"}
Spring-It-On: The Game Developer's Spring-Roll-Call Springs! What do springs have to do with game development? We'll if you're asking that question and reading this article you're in the right place. Because we're about to do a lot of talking about springs... and, while some of you may well have used springs before, I'm guessing that even if you did the code you used resided in the dark depths of your project as a set of mysterious equations that no one ever touched. And that's sad, because although the maths can undeniably be tricky, springs are interesting, and a surprisingly versatile tool, with lots of applications in Computer Science that I never even realized were possible until I thought "wouldn't it be nice to know how those equations came about?" and dug a bit deeper. So I think every Computer Scientist, and in particular those interested in game development, animation, or physics, could probably benefit from a bit of knowledge of springs. In the very least this: what they are, what they do, how they do it, and what they can be used for. So with that in mind, let's start right from the beginning: The Damper. The Damper The Spring Damper Other Springs The Damper As a game developer here is a situation you've probably come across: for some reason there is an object which suddenly we realize should be at a different position - perhaps we got some information from a game server about where it should really be, or the player did something and now the object needs to be moved. Well, when we move this object we probably don't want it to pop - instead we would prefer if it moved more smoothly towards the new position. A simple thing we might try to achieve this is to just blend the object's position x with the new goal position g by some fixed factor such as 0.1 each frame. In C++ we might write something like this, and while this code is for a single float the same can be done for a position just by applying it to each component of the vector independently. float lerp(float x, float y, float a) return (1.0f - a) * x + a * y; float damper(float x, float g, float factor) return lerp(x, g, factor); By applying this damper function each frame we can smoothly move toward the goal without popping. We can even control the speed of this movement using the factor argument: x = damper(x, g, factor); Below you can see a visualization of this in action where the horizontal axis represents time and the vertical axis represents the position of the object. But this solution has a problem: if we change the framerate of our game (or the timestep of our system) we get different behavior from the damper. More specifically, it moves the object slower when we have a lower framerate: And this makes sense if we think about it - if the game is running at 30 frames per second you are going to perform half as many calls to damper as if you were running at 60 frames per second, so the object is not going to get pulled toward the goal as quickly. One simple idea for a fix might be to just multiply the factor by the timestep dt - now at least when the timestep is larger the object will move more quickly toward the goal... float damper_bad(float x, float t, float damping, float dt) return lerp(x, t, damping * dt); This might appear like it works on face value but there are two big problems with this solution which can come back to bite us badly. Firstly, we now have a mysterious damping variable which is difficult to set and interpret. But secondly, and more importantly, if we set the damping or the dt too high (such that damping * dt > 1) the whole thing becomes unstable, and in the worst case We could use various hacks like clamping damping * dt to be less than 1 but there is fundamentally something wrong with what we've done here. We can see this if we imagine that damping * dt is roughly equal to 0.5 - here, doubling the dt does not produce the same result as applying the damper twice: lerping with a factor of 0.5 twice will take us 75% of the way toward the goal, while lerping with a factor of 1.0 once will bring us 100% of the way there. So what's the real fix? The Exact Damper Let's start our investigation by plotting the behavior of x using the normal damper with a fixed dt of 1.0, a goal of 0, and a factor of 0.5: Here we can see repeated calls to lerp actually produce a kind of exponential decay toward the goal: \begin{align} & t = 0, \! &x& = 1.0 \\ & t = 1, \! &x& = 0.5 \\ & t = 2, \! &x& = 0.25 \\ & t = 3, \! &x& = 0.125 \\ \end{align} And for a $ \text{lerp} $ factor of $ 0.5 $, we can see that this pattern is exactly the equation $ x_t = 0.5^t $. So it looks like somehow there is an exponential function governing this relationship, but how did this appear? The trick to uncovering this exponential form is to write our system as a recurrence relation. Recurrence Relation We'll start by defining a separate variable $ y = 1 - damping \cdot ft $, which will make the maths a bit easier later on. In this case $ ft $ is a fixed, small $ dt $ such as $ \tfrac{1}{60} $. Then we will expand the $ \text{lerp} $ function: \begin{align} x_{t + 1} &= \text{lerp}(x_t,\, g,\, 1 - y) \\ x_{t + 1} &= (1 - (1 - y)) \cdot x_t + (1 - y) \cdot g \\ x_{t + 1} &= y \cdot x_t - (y - 1) \cdot g \\ x_{t + 1} &= y \cdot x_t - y \ cdot g + g \end{align} Now for the recurrence relation: by plugging this equation into itself we are going to see how the exponent appears. First we need to increment $ t + 1 $ to $ t + 2 $ and then replace the new $ x_{t + 1} $ which appears on the right hand side with the same equation again. \begin{align} x_{t + 1} &= y \cdot x_t - y \cdot g + g \\ x_{t + 2} &= y \cdot x_{t + 1} - y \cdot g + g \\ x_{t + 2} &= y \cdot (y \cdot x_t - y \cdot g + g) - y \cdot g + g \\ x_{t + 2} &= y \cdot y \cdot x_t - y \cdot y \cdot g + y \cdot g - y \cdot g + g \\ x_{t + 2} &= y \cdot y \cdot x_t - y \cdot y \cdot g + g \\ \end{align} If we repeat this process again and we can start to see a pattern emerging: \begin{align} x_{t + 2} &= y \cdot y \cdot x_t - y \cdot y \cdot g + g \\ x_{t + 3} &= y \cdot y \cdot x_{t + 1} - y \cdot y \cdot g + g \\ x_{t + 3} &= y \cdot y \cdot (y \cdot x_t - y \cdot g + g) - y \cdot y \cdot g + g \\ x_{t + 3} &= y \cdot y \cdot y \cdot x_t - y \cdot y \cdot y \cdot g + y \cdot y \cdot g - y \cdot y \cdot g + g \\ x_{t + 3} &= y \cdot y \cdot y \cdot x_t - y \cdot y \ cdot y \cdot g + g \\ \end{align} More generally we can see that: \begin{align} x_{t + n} = y^n \cdot x_t - y^n \cdot g + g \end{align} Ah-ha! Our exponent has appeared. And by rearranging a bit we can even write this in terms of $ \text{lerp} $ again: \begin{align} x_{t + n} &= y^n \cdot x_t - y^n \cdot g + g \\ x_{t + n} &= y^n \cdot x_t + g \cdot (1 - y^n) \\ x_{t + n} &= \text{lerp}(x_t,\, g,\, 1 - y^n) \\ \end{align} As a small tweak, we can make the exponent negative: \begin{align} x_{t + n} &= \text{lerp}(x_t,\, g,\, 1 - y^n) \\ x_{t + n} &= \text{lerp}(x_t,\, g,\, 1 - \tfrac{1}{y}^{-n}) \\ \end{align} Remember that $ n $ represents a multiple of $ ft $, so if we have a new arbitrary $ dt $ we will need to convert it to $ n $ first using $ n = \tfrac{dt}{ft} $. In C++ we would write it as float damper_exponential( float x, float g, float damping, float dt, float ft = 1.0f / 60.0f) return lerp(x, g, 1.0f - powf(1.0 / (1.0 - ft * damping), -dt / ft)); Let's see it action! Notice how it produces the same, identical and stable behavior even when we make the dt and damping large. So have we fixed it? Well, in this formulation we've essentially solved the problem by letting the behavior of the damper match one particular timestep while allowing the rate of decay to still vary. In this case 1.0f - ft * damping is our rate of decay, and it dictates what proportion of the distance toward the goal will remain after ft in time. As long as we make the fixed timestep ft small enough, ft * damping should never exceed 1.0 and the system remains stable and well behaved. The Half-Life But there is another, potentially better way to fix the problem. Instead of fixing the timestep, we can fix the rate of decay and let the timestep vary. This sounds a little odd at first but in practice it makes things much easier. The basic idea is simple: let's set the rate of decay to $ 0.5 $ and instead scale the timestep such that we can control the exact half-life of the damper - a.k.a the time it takes for the distance to the goal to reduce by half: \begin{align} x_{t + dt} &= \text{lerp}\left(x_t,\, g,\, 1 - \tfrac{1}{0.5}^{-dt / halflife}\right) \\ x_{t + dt} &= \text{lerp}\left(x_t,\, g,\, 1 - 2^{-dt / halflife}\right) \\ \end{align} This simplifies the code and gives a more intuitive parameter to control the damper. Now we don't ever need to worry about if we've set the damping too large or made the fixed timestep ft small float damper_exact(float x, float g, float halflife, float dt) return lerp(x, g, 1.0f - powf(2, -dt / halflife)); For neatness, we can also switch to an exponential base using the change of base theorem: just multiply the dt by $ ln(2) = 0.69314718056 $ and switch to using expf. Finally, we should add some small epsilon value like 1e-5f to avoid division by zero when our halflife is very small: float damper_exact(float x, float g, float halflife, float dt, float eps=1e-5f) return lerp(x, g, 1.0f - expf(-(0.69314718056f * dt) / (halflife + eps))); The change of base theorem tells us another thing: that changing the rate of decay is no different from scaling the dt in the exponent. So using the halflife to control the damper should not limit us in any of the behaviors we want to achieve compared to if we changed the rate of decay like in our previous setup. There is one more nice little trick we can do - a fast approximation of the negative exponent function using one over a simple polynomial (or we could use this even better approximation from Danny float fast_negexp(float x) return 1.0f / (1.0f + x + 0.48fxx + 0.235fxxx); And that's it - we've converted our unstable damper into one that is fast, stable, and has intuitive parameters! float damper_exact(float x, float g, float halflife, float dt, float eps=1e-5f) return lerp(x, g, 1.0f-fast_negexp((0.69314718056f dt) / (halflife + eps))); Let's see how it looks... The Spring Damper The exact damper works well in a lot of cases, but has one major issue - it creates discontinuities when the goal position changes quickly. For example, even if the object is moving in one direction, it will immediately switch to moving in the opposite direction if the goal changes direction. This can create a kind of annoying sudden movement which you can see in the previous videos. The problem is that there is no velocity continuity - no matter what happened in the previous frames the damper will always move toward the goal. Let's see how we might be able to fix that. We can start by looking again at our old broken bad damper, and examining it in a bit more detail: \begin{align} x_{t + dt} &= \text{lerp}(x_t,\, g,\, dt \cdot damping) \\ x_{t + dt} &= x_t + dt \cdot damping \cdot (g - x_t) \\ \end{align} We can see that this looks a bit like a physics equation where $ damping \cdot (g - x_t) $ represents the velocity. \begin{align} v_{t \phantom{+ dt}} &= damping \cdot (g - x_t) \\ x_{t + dt} &= x_t + dt \cdot v_t \\ \end{align} This system is like a kind of particle with a velocity always proportional to the difference between the current particle position and the goal position. This explains the discontinuity - the velocity of our damper will always be directly proportional to the difference between the current position and the goal without ever taking any previous velocities into account. What if instead of setting the velocity directly each step we made it something that changed more smoothly? For example, we could instead add a velocity taking us toward the goal to the current velocity, scaled by a different parameter which for now we will call the $ stiffness $. \begin{align} v_{t + dt} & = v_t + dt \cdot stiffness \cdot (g - x_t) \\ x_{t + dt} & = x_t + dt \cdot v_t \\ \end{align} But the problem now is that this particle wont slow down until it has already over-shot the goal and is pulled back in the opposite direction. To fix this we can add a $ q $ variable which represents a goal velocity, and add another term which takes us toward this goal velocity. This we will scale by another new parameter which we will call the $ damping $ (for reasons which will become clearer later in the article). \begin{align} v_{t + dt} & = v_t + dt \cdot stiffness \cdot (g - x_t) + dt \cdot damping \cdot (q - v_t) \\ x_{t + dt} & = x_t + dt \cdot v_t \\ \end{align} When $ q $ is very small we can think of this like a kind of friction term which simply subtracts the current velocity. And when $ q = 0 $ and $ dt \cdot damping = 1 $ we can see that this friction term actually completely removes the existing velocity, reverting our system back to something just like our original damper. Another way to think about these terms is by thinking of them as accelerations, which can be shown more clearly by factoring out the $ dt $: \begin{align} a_{t \phantom{+ dt}} & = stiffness \cdot (g - x_t) + damping \cdot (q - v_t) \\ v_{t + dt} & = v_t + dt \cdot a_t \\ x_{t + dt} & = x_t + dt \cdot v_t \\ \end{align} Assuming the mass of our particle is exactly one, it really is possible to think about this as two individual forces - one pulling the particle in the direction of the goal velocity, and one pulling it toward the goal position. If we use a small enough $ dt $ we can actually plug these functions together and simulate a simple damped spring with exactly the velocity continuity we wanted. Here is a function which does that (using semi-implicit euler integration). void spring_damper_bad( float& x, float& v, float g, float q, float stiffness, float damping, float dt) v += dt * stiffness * (g - x) + dt * damping * (q - v); x += dt * v; Let's see how it looks: But unfortunately just like before we have problems when the $ dt $ is large, and certain settings for $ stiffness $ and $ damping $ can make the system unstable. These unintuitive parameters like $ damping $ and $ stiffness $ are also back again... arg! Can we give this spring the same treatment as we did for our damper by fiddling around with the maths? Well yes we can, but unfortunately from here on in things are going to get a bit more The Exact Spring Damper This time the exact version of our model is too complicated to solve using a simple recurrence relation. Instead we're going to have to try a different tactic: we're going to guess an equation we think models the spring and then try to work out how to compute all the different parameters of that equation based on the parameters we do know such as the $ damping $ and $ stiffness $. If we take a look at the movement of the spring in the previous section we can see there are basically two features - an exponential decay toward the goal position, and a kind of oscillation, a bit like a $ \cos $ or $ \sin $ function. So let's try and come up with an equation which fits that kind of shape and go from there. What about something like this? \begin{align} x_t = j \cdot e^{-y \cdot t} \cdot \cos(w \cdot t + p) + c \end{align} Where $ j $ is the amplitude, $ y $ controls the time it takes to decay, a bit like our half-life parameter, $ t $ is the time, $ w $ is the frequency of oscillations, $ p $ is the phase of oscillations, and $ c $ is an offset on the vertical axis. This seems like a reasonable formulation of the behavior we saw previously. But before we try to find all of these unknown parameters, let's write down the derivatives of this function with respect to $ t $ too. We'll use $ v_t $ to denote the velocity, and $ a_t $ to denote the acceleration. \begin{align} x_t &= j \cdot e^{-y \cdot t} \cdot \cos(w \cdot t + p) + c \\ v_t &= - y \cdot j \cdot e^{-y \cdot t} \cdot \cos(w \cdot t + p) \\ &- w \cdot j \cdot e^{-y \cdot t} \cdot \sin(w \cdot t + p) \\ a_t &= y^2 \cdot j \cdot e^{-y \cdot t} \cdot \cos(w \cdot t + p) \\ &- w^2 \cdot j \cdot e^{-y \cdot t} \cdot \cos(w \cdot t + p) \\ &+ 2 \cdot w \cdot y \cdot j \cdot e^{-y \cdot t} \cdot \sin(w \cdot t + p) \\ \end{align} Those might look a bit scary but we can make them a lot less scary by just summarizing some of the common terms: \begin{align} C &= j \cdot e^{-y \cdot t} \cdot \cos(w \cdot t + p) \\ S &= j \cdot e^{-y \cdot t} \cdot \sin(w \cdot t + p) \\ \end{align} Giving us the following: \begin{align} x_t &= C + c \\ v_t &= -y \cdot C - w \cdot S \\ a_t &= y^2 \cdot C - w^2 \cdot C + 2 \cdot w \cdot y \cdot S \\ \end{align} Finding the Spring Parameters Our plan for finding the first set of unknown parameters is as follows: we're going to substitute these new equations for $ x_t $, $ v_t $, and $ a_t $ into our previous equation of motion $ a_t = s \cdot (g - x_t) + d \cdot (q - v_t) $ (where $ d = damping $ and $ s = stiffness$ ) and try to rearrange to solve for $ y $, $ w $, and $ c $ using all the other values we know: $ s $, $ d $, $ q $, and $ g $. But first let's shuffle around some terms in this equation of motion: expanding the $ stiffness $ and $ damping $ terms, moving some values onto the left hand side, and finally negating everything. This will make the next steps much easier for us. \begin{align} a_t &= s \cdot (g - x_t) + d \cdot (q - v_t) \\ 0 &= s \cdot (g - x_t) + d \cdot (q - v_t) - a_t \\ 0 &= s \cdot g - s \cdot x_t + d \cdot q - d \cdot v_t - a_t \\ -s \cdot g - d \cdot q &= -s \cdot x_t - d \cdot v_t - a_t \\ s \cdot g + d \cdot q &= s \cdot x_t + d \cdot v_t + a_t \\ \end{align} Now let's substitute in our three new equations we just created for $ x_t $, $ v_t $, and $ a_t $: \begin{align} s \cdot g + d \cdot q &= s \cdot x_t + d \cdot v_t + a_t \\ s \cdot g + d \cdot q &= s \cdot (C + c) + d \cdot (-y \cdot C - w \cdot S) + ((y^2 - w^2) \cdot C + 2 \cdot w \cdot y \cdot S) \end{align} And by multiplying out and then gathering all the coefficients of $ C $ and $ S $ together we can get: \begin{align} s \cdot g + d \cdot q &= s \cdot (C + c) + d \cdot (-y \cdot C - w \cdot S) + ((y^2 - w^2) \cdot C + 2 \cdot w \cdot y \cdot S) \\ s \cdot g + d \cdot q - s \cdot c &= s \cdot C + d \ cdot -y \cdot C - d \cdot w \cdot S + y^2 \cdot C - w^2 \cdot C + 2 \cdot w \cdot y \cdot S \\ s \cdot g + d \cdot q - s \cdot c &= ((y^2 - w^2) - d \cdot y + s) \cdot C + (2 \cdot w \cdot y - d \ cdot w) \cdot S \\ \end{align} There is one more additional fact we can use to get the variables we need from this equation: because $ C $ and $ S $ are essentially $ \cos $ and $ \sin $ functions with the same phase, amplitude, and frequency, the only way this equation can be balanced for all potential values of $ t $, $ w $, $ y $, $ j $ and $ c $ is when both the coefficients of $ C $ and $ S $ are zero and when the left hand side equals zero. This gives us three smaller equations to solve: s \cdot g + d \cdot q - s \cdot c &= 0 \\ (y^2 - w^2) - d \cdot y + s &= 0 \\ 2 \cdot w \cdot y - d \cdot w &= 0 \\ Finding $ c $ Using equation $ (1) $ we can solve for $ c $ right away to get our first unknown! \begin{align} s \cdot g + d \cdot q - s \cdot c &= 0 \\ s \cdot g + d \cdot q &= s \cdot c \\ g + \tfrac{d \cdot q}{s} &= c \\ \end{align} Finding $ y $ And by rearranging equation $ (3) $ we can also find a solution for $ y $: \begin{align} 2 \cdot w \cdot y - d \cdot w &= 0 \\ d \cdot w &= 2 \cdot w \cdot y \\ d &= 2 \cdot y \\ \tfrac{d}{2} &= y \\ \end{align} Finding $ w $ Which we can substitute into equation $ (2) $ to solve for $ w $: \begin{align} (y^2 - w^2) - d \cdot y + s &= 0 \\ (\left(\tfrac{d}{2}\right)^2 - w^2) - d \cdot \tfrac{d}{2} + s &= 0 \\ \tfrac{d^2}{4} - w^2 - \tfrac{d^2}{2} + s &= 0 \\ \tfrac{d^2}{4} - \tfrac{d^ 2}{2} + s &= w^2 \\ s - \tfrac{d^2}{4} &= w^2 \\ \sqrt{s - \tfrac{d^2}{4}} &= w \\ \end{align} Finding the Spring State There are two final unknown variables remaining: $ j $, and $ p $ - the amplitude and the phase. Unlike $ y $, $ w $, and $ c $, these two are determined by the initial conditions of the spring. Therefore, given some initial position and velocity, $ x_0 $ and $ v_0 $, we can plug these in our equations along with $ t = 0 $ to get some more equations we will use to find $ j $ and $ p $: \begin{align} x_0 &= j \cdot e^{-y \cdot 0} \cdot \cos(w \cdot 0 + p) + c \\ x_0 &= j \cdot \cos(p) + c \\ \\ v_0 &= -y \cdot j \cdot e^{-y \cdot 0} \cdot \cos(w \cdot 0 + p) - w \cdot j \cdot e^{-y \cdot 0} \cdot \sin(w \cdot 0 + p) \\ v_0 &= -y \cdot j \cdot \cos(p) - w \cdot j \cdot \sin(p) \end{align} Finding $ j $ Let's start with $ j $. First we'll re-arrange our equation for $ x_0 $ in terms of $ p $: \begin{align} x_0 &= j \cdot \cos(p) + c \\ x_0 - c &= j \cdot \cos(p) \\ \tfrac{x_0 - c}{j} &= \cos(p) \\ \arccos\left(\tfrac{x_0 - c}{j}\right) &= p \\ \end{align} And substitute this into our equation for $ v_0 $: \begin{align} v_0 &= -y \cdot j \cdot \cos(p) - w \cdot j \cdot \sin(p) \\ v_0 &= -y \cdot j \cdot \cos\left(\arccos\left(\tfrac{x_0 - c}{j}\right)\right) - w \cdot j \cdot \sin\left(\arccos\left(\ tfrac{x_0 - c}{j}\right)\right) \\ v_0 &= -y \cdot j \cdot \tfrac{x_0 - c}{j} - w \cdot j \cdot \sqrt{1 - \tfrac{(x_0 - c)^2}{j^2}} \\ v_0 &= -y \cdot (x_0 - c) - w \cdot j \cdot \sqrt{1 - \tfrac {(x_0 - c)^2}{j^2}} \\ \end{align} Which we can now rearrange for $ j $: \begin{align} v_0 + y \cdot (x_0 - c) &= -w \cdot j \cdot \sqrt{1 - \tfrac{(x_0 - c)^2}{j^2}} \\ \frac{v_0 + y \cdot (x_0 - c)}{-w \cdot j} &= \sqrt{1 - \tfrac{(x_0 - c)^2}{j^2}} \\ \frac{(v_0 + y \ cdot (x_0 - c))^2}{(-w \cdot j)^2} &= 1 - \frac{(x_0 - c)^2}{j^2} \\ \frac{(v_0 + y \cdot (x_0 - c))^2}{w^2} &= j^2 - (x_0 - c)^2\\ \frac{(v_0 + y \cdot (x_0 - c))^2}{w^2} + (x_0 - c)^2 &= j^2 \\ \ sqrt{\frac{(v_0 + y \cdot (x_0 - c))^2}{w^2} + (x_0 - c)^2} &= j \\ \end{align} Nice! Since this relies on squares and a square root, some sign information is lost. This means that in our code we will also need to negate $ j $ in the case that $ x_0 - c < 0 $. Finding $ p $ Finally, we are ready to find $ p $. We can start by rearranging our velocity equation $ v_0 $ for $ j $: \begin{align} v_0 &= -y \cdot j \cdot \cos(p) - w \cdot j \cdot \sin(p) \\ v_0 &= j \cdot (-y \cdot \cos(p) - w \cdot \sin(p)) \\ \frac{v_0}{-y \cdot \cos(p) - w \cdot \sin(p)} &= j \\ \end{align} And then substitute this into our equation for $ x_0 $ to solve for $ p $: \begin{align} x_0 &= j \cdot \cos(p) + c \\ x_0 &= \left(\frac{v_0}{-y \cdot \cos(p) - w \cdot \sin(p)}\right) \cdot \cos(p) + c \\ x_0 - c &= \frac{v_0 \cdot \cos(p)}{-y \cdot \cos(p) - w \cdot \ sin(p)} \\ x_0 - c &= \frac{v_0}{-y - w \cdot \tfrac{\sin(p)}{\cos(p)}} \\ x_0 - c &= \frac{v_0}{-y - w \cdot \tan(p)} \\ (x_0 - c) \cdot (-y - w \cdot \tan(p)) &= v_0 \\ -(x_0 - c) \cdot y - (x_0 - c) \cdot w \cdot \tan(p) &= v_0 \\ -(x_0 - c) \cdot w \cdot \tan(p) &= v_0 + (x_0 - c) \cdot y \\ \tan(p) &= \frac{v_0 + (x_0 - c) \cdot y}{-(x_0 - c) \cdot w} \\ p &= \arctan\left(\frac{v_0 + (x_0 - c) \cdot y}{-(x_0 - c) \cdot w}\right) \\ \end{align} Putting it together Putting all of this together, and throwing in a fast approximate atanf for fun, we get the following... float fast_atan(float x) float z = fabs(x); float w = z > 1.0f ? 1.0f / z : z; float y = (M_PI / 4.0f)w - w(w - 1)(0.2447f + 0.0663fw); return copysign(z > 1.0f ? M_PI / 2.0 - y : y, x); float squaref(float x) return xx; void spring_damper_exact( float& x, float& v, float x_goal, float v_goal, float stiffness, float damping, float dt, float eps = 1e-5f) float g = x_goal; float q = v_goal; float s = stiffness; float d = damping; float c = g + (dq) / (s + eps); float y = d / 2.0f; float w = sqrtf(s - (dd)/4.0f); float j = sqrtf(squaref(v + y(x - c)) / (ww + eps) + squaref(x - c)); float p = fast_atan((v + (x - c) y) / (-(x - c)w + eps)); j = (x - c) > 0.0f ? j : -j; float eydt = fast_negexp(ydt); x = jeydtcosf(wdt + p) + c; v = -yjeydtcosf(wdt + p) - wjeydtsinf(wdt + p); Phew - that was a lot of equations and re-arranging, but it worked, and produces a smooth, stable motion even with a very large dt or stiffness. And anyway, doesn't it feel nice to actually use those high school trig identities and do some old school equation manipulation for once! Over, Under, and Critical Damping But hold on... one of the steps we took in the previous section wasn't really legit... can you spot it? Here is the problem: \begin{align} w = \sqrt{s - \tfrac{d^2}{4}} \end{align} It's a square root... but I never assured you the input to this square root couldn't be negative. In fact it can be... and definitely will be if $ d $ is large! But what does this negative square root actually correspond to? Does it mean that there is no exact solution to this spring when the $ damping $ is large? Do we just have to give up? Well, not In fact we didn't notice when we came up with our original equation to model the behavior of the spring, but there are three different ways this spring can act depending on the relative sizes of the $ damping $ and $ stiffness $ values. If $ s - \tfrac{d^2}{4} > 0 $ it means the spring is under damped, causing oscillations to appear with motions governed by the equations we already derived. If $ s - \tfrac{d^2}{4} = 0 $ it means the spring is critically damped, meaning it returns to the goal as fast as possible without extra oscillation, and if $ s - \tfrac{d^2}{4} < 0 $ it means the spring is over damped, and will return slowly toward the goal. In each of these cases there is a different set of basic equations governing the system, leading to a different derivation just like the one we completed. I'm going to save us a bit of time and write them all our here rather than going through the trial and error process of examining different guesses at equations and seeing if they fit: Under Damped: \begin{align} x_t &= j \cdot e^{-y \cdot t} \cdot \cos(w \cdot t + p) + c \\ v_t &= - y \cdot j \cdot e^{-y \cdot t} \cdot \cos(w \cdot t + p) \\ &- w \cdot j \cdot e^{-y \cdot t} \cdot \sin(w \cdot t + p) \\ a_t &= y^2 \cdot j \cdot e^{-y \cdot t} \cdot \cos(w \cdot t + p) \\ &- w^2 \cdot j \cdot e^{-y \cdot t} \cdot \cos(w \cdot t + p) \\ &+ 2 \cdot w \cdot y \cdot j \cdot e^{-y \cdot t} \cdot \sin(w \cdot t + p) \\ \end{align} Critically Damped: \begin{align} x_t &= j_0 \cdot e^{-y \cdot t} + t \cdot j_1 \cdot e^{-y \cdot t} + c \\ v_t &= -y \cdot j_0 \cdot e^{-y \cdot t} - y \cdot t \cdot j_1 \cdot e^{-y \cdot t} + j_1 \cdot e^{-y \cdot t} \\ a_t &= y^2 \cdot j_0 \cdot e^{-y \cdot t} + y^2 \cdot t \cdot j_1 \cdot e^{-y \cdot t} - 2 \cdot y \cdot j_1 \cdot e^{-y \cdot t} \\ \end{align} Over Damped: \begin{align} x_t &= j_0 \cdot e^{-y_0 \cdot t} + j_1 \cdot e^{-y_1 \cdot t} + c \\ v_t &= -y_0 \cdot j_0 \cdot e^{-y_0 \cdot t} - y_1 \cdot j_1 \cdot e^{-y_1 \cdot t} \\ a_t &= y_0^2 \cdot j_0 \ cdot e^{-y_0 \cdot t} + y_1^2 \cdot j_1 \cdot e^{-y_1 \cdot t} \\ \end{align} What we did in the previous section was solve for the under damped case, but the other two cases require exactly the same style of derivation to get them working. Solving the Critically Damped Case Let's start with the easiest: the critically damped case. The first two unknowns $ c $ and $ y $ have exactly the same solution as in the under-damped case, $ c = g + \frac{d \cdot q}{s} $ and $ y = \tfrac{d}{2} $, while the other two unknowns $ j_0 $ and $ j_1 $ can be found easily from the initial conditions $ x_0 $, $ v_0 $, and $ t = 0 $: \begin{align} x_0 &= j_0 \cdot e^{-y \cdot 0} + 0 \cdot j_1 \cdot e^{-y \cdot 0} + c \\ x_0 &= j_0 + c \\ \end{align} And for the velocity... \begin{align} v_0 &= -y \cdot j_0 \cdot e^{-y \cdot 0} - y \cdot 0 \cdot j_1 \cdot e^{-y \cdot t} + j_1 \cdot e^{-y \cdot 0} \\ v_0 &= -y \cdot j_0 + j_1 \\ \end{align} Giving us... \begin{align} j_0 &= x_0 - c \\ j_1 &= v_0 + j_0 \cdot y \\ \end{align} And that's it, easy! Solving the Over Damped Case The over-damped case is a little more difficult so let's first summarize some terms again to make our equations clearer. \begin{align} E_0 &= j_0 \cdot e^{-y_0 \cdot t} \\ E_1 &= j_1 \cdot e^{-y_1 \cdot t} \\ \end{align} Giving us... \begin{align} x_t &= E_0 + E_1 + c \\ v_t &= -y_0 \cdot E_0 - y_1 \cdot E_1 \\ a_t &= y_0^2 \cdot E_0 + y_1^2 \cdot E_1 \\ \end{align} We'll start by finding the two unknowns $ y_0 $, and $ y_1 $. Just like before we are going to substitute these equations into our equation of motion, gathering up the coefficients for the exponential terms: \begin{align} s \cdot g + d \cdot q &= s \cdot x_t + d \cdot v_t + a_t \\ s \cdot g + d \cdot q &= s \cdot (E_0 + E_1 + c) + d \cdot (-y_0 \cdot E_0 - y_1 \cdot E_1) + (y_0^2 \cdot E_0 + y_1^2 \cdot E_1) \\ s \cdot g + d \cdot q &= s \cdot E_0 + s \cdot E_1 + s \cdot c - d \cdot y_0 \cdot E_0 - d \cdot y_1 \cdot E_1 + y_0^2 \cdot E_0 + y_1^2 \cdot E_1 \\ s \cdot g + d \cdot q - s \cdot c &= (s - d \cdot y_0 + y_0^2) \cdot E_0 + (s - d \cdot y_1 + y_1^2) \cdot E_1 \\ \end{align} Again, just like before, this equation is only solved when both coefficients and the left hand side equal zero. This gives us our existing solution for $ c $ plus two more quadratic equations to \begin{align} s - d \cdot y_0 + y_0^2 &= 0 \\ s - d \cdot y_1 + y_1^2 &= 0 \\ \end{align} In this case $ y_0 $ and $ y_1 $ represent the two different solutions to the same quadratic, so we can use the quadratic formula to get these: \begin{align} y_0 &= \frac{d + \sqrt{d^2 - 4 \cdot s}}{2} \\ y_1 &= \frac{d - \sqrt{d^2 - 4 \cdot s}}{2} \\ \end{align} Now that these are found we're ready to solve for $ j_0 $ and $ j_1 $ using the initial conditions $ x_0 $, $ v_0 $, and $ t = 0 $: \begin{align} x_0 &= j_0 \cdot e^{-y_0 \cdot 0} + j_1 \cdot e^{-y_1 \cdot 0} + c \\ x_0 &= j_0 + j_1 + c \\ \\ v_0 &= -y_0 \cdot j_0 \cdot e^{-y_0 \cdot 0} - y_1 \cdot j_1 \cdot e^{-y_1 \cdot 0} \\ v_0 &= -y_0 \cdot j_0 - y_1 \cdot j_1 \\ \end{align} First we'll re-arrange our eqiation for $ x_0 $ in terms of $ j_0 $ \begin{align} x_0 &= j_0 + j_1 + c \\ x_0 - j_1 - c &= j_0 \\ \end{align} Which we'll substitute into our equation for $ v_0 $ to solve for $ j_1 $: \begin{align} v_0 &= -y_0 \cdot j_0 - y_1 \cdot j_1 \\ v_0 &= -y_0 \cdot (x_0 - j_1 - c) - y_1 \cdot j_1 \\ v_0 &= -y_0 \cdot x_0 + y_0 \cdot j_1 + y_0 \cdot c - y_1 \cdot j_1 \\ -v_0 &= y_0 \cdot x_0 - y_0 \cdot j_1 - y_0 \cdot c + y_1 \cdot j_1 \\ y_0 \cdot c - v_0 - y_0 \cdot x_0 &= - y_0 \cdot j_1 + y_1 \cdot j_1 \\ y_0 \cdot c - v_0 - y_0 \cdot x_0 &= j_1 \cdot (y_1 - y_0) \\ \frac{y_0 \ cdot c - v_0 - y_0 \cdot x_0}{y_1 - y_0} &= j_1 \\ \end{align} After which is is easy to get $ j_0 $: \begin{align} x_0 &= j_0 + j_1 + c \\ x_0 - j_1 - c &= j_0 \\ \end{align} And that's it! Let's add all of these different cases to our spring damper function: void spring_damper_exact( float& x, float& v, float x_goal, float v_goal, float stiffness, float damping, float dt, float eps = 1e-5f) float g = x_goal; float q = v_goal; float s = stiffness; float d = damping; float c = g + (dq) / (s + eps); float y = d / 2.0f; if (fabs(s - (dd) / 4.0f) < eps) // Critically Damped float j0 = x - c; float j1 = v + j0y; float eydt = fast_negexp(ydt); x = j0eydt + dtj1eydt + c; v = -yj0eydt - ydtj1eydt + j1eydt; else if (s - (dd) / 4.0f > 0.0) // Under Damped float w = sqrtf(s - (dd)/4.0f); float j = sqrtf(squaref(v + y(x - c)) / (ww + eps) + squaref(x - c)); float p = fast_atan((v + (x - c) * y) / (-(x - c)w + eps)); j = (x - c) > 0.0f ? j : -j; float eydt = fast_negexp(ydt); x = jeydtcosf(wdt + p) + c; v = -yjeydtcosf(wdt + p) - wjeydtsinf(wdt + p); else if (s - (dd) / 4.0f < 0.0) // Over Damped float y0 = (d + sqrtf(dd - 4s)) / 2.0f; float y1 = (d - sqrtf(dd - 4s)) / 2.0f; float j1 = (cy0 - xy0 - v) / (y1 - y0); float j0 = x - j1 - c; float ey0dt = fast_negexp(y0dt); float ey1dt = fast_negexp(y1dt); x = j0ey0dt + j1ey1dt + c; v = -y0j0ey0dt - y1j1ey1dt; Awesome! Now it works even for very high damping values! The Half-life and the Frequency We're almost there, but our functions still use these mysterious damping and stiffness parameters. Can we turn these into something a bit more meaningful? Yes! Just like before we can use a halflife parameter instead of a damping parameter by controlling what we give as input to the exp functions: float halflife_to_damping(float halflife, float eps = 1e-5f) return (4.0f * 0.69314718056f) / (halflife + eps); float damping_to_halflife(float damping, float eps = 1e-5f) return (4.0f * 0.69314718056f) / (damping + eps); Here as well as our previous constant of $ ln(2) $ we need to multiply by 4. This is a bit of a fudge factor but roughly it corresponds to the fact that we divide by two once to get $ y $ from $ d $, and that the spring equation is usually a sum of two exponential terms instead of the single one we had for the damper. What about the stiffness parameter? Well this one we can turn into a parameter called frequency: float frequency_to_stiffness(float frequency) return squaref(2.0f * M_PI * frequency); float stiffness_to_frequency(float stiffness) return sqrtf(stiffness) / (2.0f * M_PI); Which is close to what will become $ w $ in the under-damped case. Both are not completely honest names - the velocity continuity and oscillations of the spring means that the position will not be exactly half way toward the goal in halflife time, while the frequency parameter is more like a pseudo-frequency as the rate of oscillation is also affected by the damping value. Nonetheless, both give more intuitive controls for the spring than the damping and stiffness alternatives. Along these lines, another useful set of functions are those that give us settings for these two parameters in the critical case (i.e. when $ \tfrac{d^2}{4} = s $ ). These can be useful for setting defaults or in other cases when we only want to set one of these parameters. float critical_halflife(float frequency) return damping_to_halflife(sqrtf(frequency_to_stiffness(frequency) * 4.0f)); float critical_frequency(float halflife) return stiffness_to_frequency(squaref(halflife_to_damping(halflife)) / 4.0f); The critical_halflife function doesn't make that much sense since when critical there aren't any oscillations, but it can still be useful in certain cases. Putting it all together we can provide our spring with a nicer interface: void spring_damper_exact( float& x, float& v, float x_goal, float v_goal, float frequency, float halflife, float dt, float eps = 1e-5f) float g = x_goal; float q = v_goal; float s = frequency_to_stiffness(frequency); float d = halflife_to_damping(halflife); float c = g + (dq) / (s + eps); float y = d / 2.0f; Below you can see which critical frequency corresponds to a given halflife. And now I promise we really are done: that's it, an exact damped spring! The Damping Ratio Although controlling the frequency is nice, there is a different control which may be even better for users as it resembles a bit more the scale from less springy to more springy they might want. This is the damping ratio, where a value of $ 1 $ means a critically damped spring, a value $ < 1 $ means an under-damped spring, and a value $ > 1 $ means a over-damped spring. The equation for the damping ratio $ r $ is as follows, where $ d $ is the damping and $ s $ is the stiffness: \begin{align} r = \frac{d}{2\ \sqrt{s}} \end{align} This we can re-arrange to solve for stiffness or damping as required. float damping_ratio_to_stiffness(float ratio, float damping) return squaref(damping / (ratio 2.0f)); float damping_ratio_to_damping(float ratio, float stiffness) return ratio * 2.0f * sqrtf(stiffness); And use instead of the frequency. void spring_damper_exact_ratio( float& x, float& v, float x_goal, float v_goal, float damping_ratio, float halflife, float dt, float eps = 1e-5f) float g = x_goal; float q = v_goal; float d = halflife_to_damping(halflife); float s = damping_ratio_to_stiffness(damping_ratio, d); float c = g + (dq) / (s + eps); float y = d / 2.0f; And here it is in action! The Critical Spring Damper Looking at our exact spring damper, the critical case is particularly interesting for us (and is most likely the case you may have actually used in your games) - not only because it's the situation where the spring moves toward the goal as fast as possible without additional oscillation, but because it's the easiest to compute and also to use as it has fewer parameters. We can therefore make a special function for it, which will allow us to remove the frequency parameter and throw in a few more basic optimizations: void critical_spring_damper_exact( float& x, float& v, float x_goal, float v_goal, float halflife, float dt) float g = x_goal; float q = v_goal; float d = halflife_to_damping(halflife); float c = g + (dq) / ((dd) / 4.0f); float y = d / 2.0f; float j0 = x - c; float j1 = v + j0y; float eydt = fast_negexp(ydt); x = eydt(j0 + j1dt) + c; v = eydt(v - j1ydt); With no special cases for over-damping and under-damping this can compile down to something very fast. Separate functions for other common situations can be useful too, such as when the goal velocity q is zero... void simple_spring_damper_exact( float& x, float& v, float x_goal, float halflife, float dt) float y = halflife_to_damping(halflife) / 2.0f; float j0 = x - x_goal; float j1 = v + j0y; float eydt = fast_negexp(ydt); x = eydt(j0 + j1dt) + x_goal; v = eydt(v - j1ydt); or when the goal position is zero too... void decay_spring_damper_exact( float& x, float& v, float halflife, float dt) float y = halflife_to_damping(halflife) / 2.0f; float j1 = v + xy; float eydt = fast_negexp(ydt); x = eydt(x + j1dt); v = eydt(v - j1ydt); Another optimization that can be useful is to pre-compute y and eydt for a given halflife. If there are many springs that need to be updated with the same halflife and dt this can provide a big Probably the most common application of springs in game development is smoothing - any noisy signal can be easily smoothed in real time by a spring damper and the half life can be used to control the amount of smoothing applied vs how responsive it is to changes. Springs also work well for filtering out sudden changes or jitters in signals, and even springs with quite a small halflife will do really well at removing any sudden jitters. Another common application of springs in game development is for moving characters. The usual process is to take the user input from the gamepad and turn it into a desired character velocity, which we then set as the goal for a spring. Each timestep we tick this spring and use what it produces as a velocity with which to move the character. By tweaking the parameters of the spring we can achieve movement with different levels of smoothness and responsiveness. The slightly unintuitive thing to remember about this setup is that we are using the spring in a way such that its position corresponds to the character's velocity - meaning the spring's velocity will correspond to the character's acceleration. Assuming the desired character velocity remains fixed we can also use this spring to predict the future character velocity by simply by evaluating the spring with a larger $ dt $ than we normally would and seeing what the result is. If we want the character's position after some timestep (not just the velocity) we can compute it more accurately by using the the integral of our critical spring equation with respect to time. This will give us an accurate prediction of the future position of the character too. \begin{align} x_{t} &= \int (j_0 \cdot e^{-y \cdot t} + t \cdot j_1 \cdot e^{-y \cdot t} + c) \,dt \\ x_{t} &= \tfrac{-j_1}{y^2} \cdot e^{-y \cdot t} + \tfrac{-j_0 - j_1 \cdot t}{y} \cdot e^{-y \ cdot t} + \tfrac{j_1}{y^2} + \tfrac{j_0}{y} + c \cdot t + x_0 \\ \end{align} And translated into code... void spring_character_update( float& x, float& v, float& a, float v_goal, float halflife, float dt) float y = halflife_to_damping(halflife) / 2.0f; float j0 = v - v_goal; float j1 = a + j0y; float eydt = fast_negexp(ydt); x = eydt(((-j1)/(yy)) + ((-j0 - j1dt)/y)) + (j1/(yy)) + j0/y + v_goal * dt + x; v = eydt(j0 + j1dt) + v_goal; a = eydt(a - j1ydt); This code is the same as the critically damped spring, but applied to the character velocity and the integral used to compute the character position. If we want to predict the future trajectory we can use this to update arrays of data each with a different $ dt $: void spring_character_predict( float px[], float pv[], float pa[], int count, float x, float v, float a, float v_goal, float halflife, float dt) for (int i = 0; i < count; i++) px[i] = x; pv[i] = v; pa[i] = a; for (int i = 0; i < count; i++) spring_character_update(px[i], pv[i], pa[i], v_goal, halflife, i dt); This really shows how using a completely exact spring equation comes in handy - we can accurately predict the state of the spring at any arbitrary point in the future without having to simulate what happens in between. Here you can see me moving around a point in the world using the above code and a desired character velocity coming from the gamepad. By adjusting the halflife of the spring we can achieve different levels of responsiveness and smoothness, and by evaluating the spring at various different times in the future we can predict where the character would be if the current input were to remain fixed (shown in red). This is exactly the method we use to predict the future character trajectory in Learned Motion Matching. In game animation, inertialization is the name given to a kind of blending that fades in or out an offset between two animations. Generally it can be used as a more performant alternative to a cross-fade blend since it only needs to evaluate one animation at a time. In the original presentation a polynomial is used blend out this offset smoothly, but springs can be used for this too. The idea is this: if we have two different streams of animation we wish to switch between, at the point of transition we record the offset between the currently playing animation and the one we want to switch to. Then, we switch to this new animation but add back the previously recorded offset. We then decay this offset smoothly toward zero over time - in this case using a spring damper. In code it looks something like this. First at the transition point we record the offset in terms of position and velocity between the currently playing animation src and the one we're going to switch to dst: void inertialize_transition( float& off_x, float& off_v, float src_x, float src_v, float dst_x, float dst_v) off_x = (src_x + off_x) - dst_x; off_v = (src_v + off_v) - dst_v; We then switch to this animation, and at every frame we decay this offset toward zero, adding the result back to our currently playing animation. void inertialize_update( float& out_x, float& out_v, float& off_x, float& off_v, float in_x, float in_v, float halflife, float dt) decay_spring_damper_exact(off_x, off_v, halflife, dt); out_x = in_x + off_x; out_v = in_v + off_v; Here you can see it in action: As you can see, each time we toggle the button there is a transition between the two different streams of animation (in this case two different $ \sin $ waves shown in red), while the inertialization smoothly fills in the difference (shown in blue). Unlike the original presentation which uses a polynomial to blend out the offset over a specific period, a spring does not provide a fixed blend time and can easily overshoot. However the exponential decay does mean it tends to look smooth and blends out to something negligible at a very fast rate. In addition, since there is no need to remember the last transition time the code is very simple, and because we use the decay_spring_damper_exact variant of the spring it can be made exceptionally fast, in particular when all the blends for the different bones use the same halflife and dt to This is exactly the method we use for switching between animations in our Motion Matching implementation as demonstrated in Learned Motion Matching. The time dimension of a spring doesn't have to always be the real time that ticks by - it can be any variable that increases monotonically. For example we could use the parameterization along a curve as the time dimension to feed to a spring to produce a kind of spline. Here I set up a piecewise interpolation of some 2D control points and used that as position and velocity goal for two springs (one for each dimension in 2D). void piecewise_interpolation( float& x, float& v, float t, float pnts[], int npnts) t = t * (npnts - 1); int i0 = floorf(t); int i1 = i0 + 1; i0 = i0 > npnts - 1 ? npnts - 1 : i0; i1 = i1 > npnts - 1 ? npnts - 1 : i1; float alpha = fmod(t, 1.0f); x = lerp(pnts[i0], pnts[i1], alpha); v = (pnts[i0] - pnts[i1]) / npnts; The result is a kind of funky spline which springs toward the control points. By adjusting the halflife and frequency we can produce some interesting shapes but overall the result has an odd feel to it since it's not symmetric and usually doesn't quite reach the last control point. Perhaps some more experimentation here could be interesting, such as running two springs along the control points in opposite directions and mixing the result. I think there is probably still a way to formulate an interesting and useful type of spline using springs. Tell me if you manage to come up with anything interesting! If you've got a signal which you think contains at specific frequency but you don't know exactly which frequency it is what do you do? Well, I can already see you starting to type "fast fourier transform" into google but hold on a second, do you really need to do something that complicated? Springs can be used to see if a signal is oscillating at a specific frequency! Fed with a goal moving at their resonate frequency springs will oscillate and build up energy, while fed a goal moving at any other frequency will make them die out and lose energy. We can measure the energy of the spring using the sum of potential and kinematic energies. float spring_energy( float x, float v, float frequency, float x_rest = 0.0f, float v_rest = 0.0f, float scale = 1.0f) float s = frequency_to_stiffness(frequency); return ( squaref(scale * (v - v_rest)) + s * squaref(scale * (x - x_rest))) / 2.0f; Then, if we want to see which frequency is contained in a signal we can simply drive a spring (or multiple springs at different frequencies) and see which setting produces the most energy. As mentioned in the previous sections, the frequency parameter we're using doesn't actually reflect the true resonate frequency of the spring (the true resonate frequency is also affected by the damping), but we can find the frequency we need to set a spring to match some resonant frequency by fixing the halflife: float resonant_frequency(float goal_frequency, float halflife) float d = halflife_to_damping(halflife); float goal_stiffness = frequency_to_stiffness(goal_frequency); float resonant_stiffness = goal_stiffness - (dd)/4.0f; return stiffness_to_frequency(resonant_stiffness); When trying to pick out specific frequencies the halflife of the spring affects the sensitivity. A long halflife (or low damping) means the spring will only build up energy when driven at frequencies very close to its resonate frequency, and it will build up more energy too. While a shorter halflife means a broader range of frequencies that build up energy. For a really cool application of this idea check out this blog post by Kevin Bergamin. If we have an object moving at a velocity with damping applied we can use a damper to estimate a what the position might be at some time in the future. The idea is to assume the velocity is being reduced over time via an exponential decay (i.e. via a damper): \begin{align} v_t &= v_0 \cdot e^{-y \cdot t} \\ \end{align} Then, like in the controller example, we can then take the integral of this equation to work out the exact future position at some time $ t $: \begin{align} x_t &= \int v_0 \cdot e^{-y \cdot t} \, dt \\ x_t &= \tfrac{v_0}{y}\ (1 - e^{-y \cdot t}) + x_0 \\ \end{align} In code it looks something like this where halflife controls the decay rate of the velocity. void extrapolate( float& x, float& v, float dt, float halflife, float eps = 1e-5f) float y = 0.69314718056f / (halflife + eps); x = x + (v / (y + eps)) (1.0f - fast_negexp(y * dt)); v = v * fast_negexp(y * dt); And here is a little demo showing it in action: Other Springs Double Spring You'll notice that the spring damper has a kind of asymmetric look to it - the start is very steep and it quickly evens out. We can achieve more of an "S" shape to the spring by using another spring on the goal. We'll deem this the "double spring": void double_spring_damper_exact( float& x, float& v, float& xi, float& vi, float x_goal, float halflife, float dt) simple_spring_damper_exact(xi, vi, x_goal, 0.5f * halflife, dt); simple_spring_damper_exact(x, v, xi, 0.5f * halflife, dt); And here you can see it in action: In red you can see the intermediate spring xi while in blue you can see the "double spring" which has a slightly more "S" shaped start and end. Timed Spring In some cases we might not want the spring to reach the goal immediately but at some specific time in the future - yet we might still want to keep the smoothing and filtering properties the spring brings when the goal changes quickly. Here is a spring variant that takes as input a goal time as well as a goal position and tries to achieve the goal at approximately the correct time. The idea is to track a linear interpolation directly toward the goal but to do so some time in the future (to give the spring time to blend out once it gets close to the goal). void timed_spring_damper_exact( float& x, float& v, float& xi, float x_goal, float t_goal, float halflife, float dt, float apprehension = 2.0f) float min_time = t_goal > dt ? t_goal : dt; float v_goal = (x_goal - xi) / min_time; float t_goal_future = dt + apprehension * halflife; float x_goal_future = t_goal_future < t_goal ? xi + v_goal * t_goal_future : x_goal; simple_spring_damper_exact(x, v, x_goal_future, halflife, dt); xi += v_goal * dt; Here the apprehension parameter controls how far into the future we try to track the linear interpolation. A value of 2 means two-times the half life, or that we expect the blend-out to be 75% done by the goal time. Below we can see this spring in action, with the linear interpolation toward the goal shown in red. Velocity Spring We can use a similar idea to make a spring that tries to maintain a given velocity by tracking an intermediate target that moves toward the goal at that fixed velocity. void velocity_spring_damper_exact( float& x, float& v, float& xi, float x_goal, float v_goal, float halflife, float dt, float apprehension = 2.0f, float eps = 1e-5f) float x_diff = ((x_goal - xi) > 0.0f ? 1.0f : -1.0f) * v_goal; float t_goal_future = dt + apprehension * halflife; float x_goal_future = fabs(x_goal - xi) > t_goal_future * v_goal ? xi + x_diff * t_goal_future : x_goal; simple_spring_damper_exact(x, v, x_goal_future, halflife, dt); xi = fabs(x_goal - xi) > dt * v_goal ? xi + x_diff * dt : x_goal; And here is how this one looks, with the intermediate target shown in red. Quaternion Spring The simplified code of the simple spring damper also lends itself to be easily adapted to other things such as quaternions. Here the main trick is to convert quaternion differences into angular velocities (first convert to angle axis then scale the axis by the angle) so that they can interact with the other terms such as the exponential terms and the spring velocity. void simple_spring_damper_exact_quat( quat& x, vec3& v, quat x_goal, float halflife, float dt) float y = halflife_to_damping(halflife) / 2.0f; vec3 j0 = quat_to_scaled_angle_axis(quat_mul(x, quat_inv(x_goal))); vec3 j1 = v + j0y; float eydt = fast_negexp(ydt); x = quat_mul(quat_from_scaled_angle_axis(eydt(j0 + j1dt)), x_goal); v = eydt(v - j1y*dt); One thing that is perhaps unintuitive about this derivation is the fact that we actually compute a rotation which takes the goal toward the initial state rather than the other way around (which is what we might expect). It's a good exercise to try and do this same style of derivation for other spring variants and other quantities such as angles, but I'll leave that as an exercise for you... Scale Spring In this post I show how we can derive an equation for a spring that works on object scales. Tracking Spring See this article for a spring which can be used to perfectly track animation data while still removing discontinuities. Source Code The source code for all the demos shown in this article can be found here. They use raylib and more specifically raygui but once you have both of those installed you should be ready to roll. I hope this article has piqued your interest in springs. Don't hesitate to get in contact if you come up with any other interesting derivations, applications, or spring variations. I'd be more than happy to add them to this article for others to see. Other than that there is not much more to say - happy exploring!	{"url":"https://theorangeduck.com/page/spring-roll-call","timestamp":"2024-11-03T21:58:57Z","content_type":"application/xhtml+xml","content_length":"76289","record_id":"<urn:uuid:96a99adb-6afd-43db-8a6d-297e4943159f>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00370.warc.gz"}
Find the shortest path Find the shortest path Searching algorithm Graph Online uses Dijkstra's algorithm for the shortest path search. The algorithm supports edges with integral and non-integral weights. How to use 1. Create a graph. 2. Select "Find the shortest path" menu item. 3. Select starting and finishing vertices. If the path exists, it will be selected on the graph. Moreover, you may choose a detailed report. In that case the shortest path from the starting vertex to a stated one will be written above every vertex. On the contrary, if the vertex can not be approached from the starting vertex, there will be a "∞" symbol.	{"url":"https://graphonline.top/en/wiki/Help/FindTheShortestPath","timestamp":"2024-11-14T15:25:11Z","content_type":"text/html","content_length":"10320","record_id":"<urn:uuid:868e746e-3ea1-4e5c-b41a-7a9a755cc3e8>","cc-path":"CC-MAIN-2024-46/segments/1730477028657.76/warc/CC-MAIN-20241114130448-20241114160448-00481.warc.gz"}
Merkezin Güncesi Çağlar Yüncüler is an Economist at the CBRT. Note To Editor For views, suggestions and comments: Email Us A significant number of the indicators we monitor to understand the course of the economic activity are affected by the number of working days. In other words, in the event of an increase (or decrease) in the number of the non-working days, we may observe considerable falls (or rises) in these indicators during that specific period. We briefly call it the calendar effect. When we need to make an economically meaningful deduction on the main trend of economic activity, we need to adopt an approach that excludes calendar effects from the variables. This situation is too important to ignore particularly in countries such as Turkey with moving holidays. How can we calculate calendar effects? We estimate the calendar effect by adding the calendar variable to the equations we use when applying seasonal adjustment methods. In a conventional calendar variable a value recorded within a specific period is defined as the difference of the number of working days from its long run averages^[1]. The coefficient estimated for the calendar effect in the equation gives us the effect on the relevant variable arising from working one day more than the averages. When we multiply the estimated coefficient by the value recorded by the calendar variable, we find the calendar effect. Bridging days and other non-working days Calendar variables are generally calculated considering the days that are already known as holidays. National days and religious festivals along with weekends are not qualified as working days in Turkey. However, apart from these days, there may be working days which turn into an actual holiday, and thus become a non-working day. For instance, it is possible to extend the duration of the holiday by bridging the national and religious holidays that fall within the week with weekends. We call these days bridging day^[2]. Apart from this, a standard working day can be interrupted due to entirely external factors. In fact, evaluating this sort of working day losses and bridging days as a calendar effect is a more accurate approach to interpret the main trends of economic variables. Main trend in the third quarter of 2016... The Gross Domestic Product (GDP) data pertaining to the third quarter of 2016 is a good example of why we should adopt this approach. According to the raw data, the Turkish economy contracted by 1.8 percent year-on-year and this contraction becomes 2.7 percent quarter-on-quarter in seasonal and calendar effects-adjusted terms. Was, then, the slowdown in the main trend of the economic activity as deep as it was implied by the data announced? Our detailed analyses on bridging days answer this question as no. Because both the Ramadan and Sacrifice Festivals in July and September 2016 were extended to bridge with weekends upon a Council of Ministers Decision. Anecdotal data suggests that not only the public sector, but also the private sector workers had these days off due to the summer season. Apart from this, it is highly probable that the incidents experienced in mid-July had an adverse impact on the economic activity on the following day. The 0.2-percent annual contraction in the calendar-adjusted GDP series for the third quarter refers to a more moderate decline than the decline in the raw data. The 1.6-point difference between the two figures resulted from the difference in the number of working days between the third quarters of 2015 and 2016. Hence, extended public holidays and the interruption of the working day on 16 July, which was independent of economic dynamics, are excluded from this calculation. For the purpose of economic significance, we should adjust GDP for these effects as well. Well, how can we make this calculation? First of all, we should find out what significance a regular working day has for GDP. When we adjust GDP for seasonal and calendar effects^[3], the coefficient of the calendar variable indicates that working one day less reduces GDP by approximately 0.4 percentage points. In the second step, we should figure out how much of a regular working day the bridging days correspond to, as some firms continue their operations on these days as well. Based on the results we have obtained from the interviews with firms, nearly half of the firms did not work on the bridging days in the third quarter of 2016 and took these days off instead. Thus, we can assert that the bridging day-driven loss equaled approximately 1.5 working days in the third quarter. The same interviews showed us that many firms could not perform their regular operations in the immediate aftermath of the adverse event in mid-July. Therefore, we assumed that the economic activity on 16 July corresponded to 25 percent of the regular activity. Accordingly, we can claim that the non-working days due to these two factors caused a loss of 2.25 working days in total in the third quarter. Multiplying with the calendar variable coefficient, this loss scaled down the economic activity by approximately 0.9 points. As shown in Chart 1 and Chart 2, when this effect is excluded, we see that the quarter-on-quarter contraction was lower with 1.8 percent instead of 2.7 percent and that the calendar-adjusted annual growth posted a modest increase by 0.7 percent instead of a decline. Hence, the slackening in the economic activity in the third quarter may not be as acute as implied by the official data. To sum up, for a more accurate measure of the main trend of GDP in the third quarter, it is important to take into account and quantify the effects such as non-working days that are not dependent on economic dynamics. However, it should also be noted that these calculations include certain assumptions and the actual effect may have been lower or higher than the assumed effect. [1] For more detailed information on the calculation of calendar variables see Atabek et al. (2007). [2] For detailed information on bridging days, see Yüncüler (2015). [3] The adjustment has been made based on the model and the calendar variable that is used by the TURKSTAT. For detailed information, please see: http://www.tuik.gov.tr/indir/m_t_metaveri/gsyh_mv.pdf Atabek, A., Atuk, O., Coşar, E. E. and Sarıkaya, Ç. (2009), Working Day Variable in Seasonal Models (in Turkish), CBT Research Notes in Economics, No: 09/03. Yüncüler, Ç. (2015), Estimating the Bridging Day Effect on Turkish Industrial Production (in English), CBT Research Notes in Economics, No: 15/15. Çağlar Yüncüler is an Economist at the CBRT. Note To Editor For views, suggestions and comments: Email Us * The views expressed here are those of the authors. They do not necessarily reflect the official views of the Central Bank of the Republic of Türkiye.	{"url":"https://tcmbblog.org/wps/wcm/connect/blog/en/main%20menu/analyses/to_what_extent_do_working_days_affect_growth","timestamp":"2024-11-03T15:32:23Z","content_type":"text/html","content_length":"44525","record_id":"<urn:uuid:14493134-06ff-4294-8505-66ab31225ec3>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00366.warc.gz"}
Impact of pipe roughness on oil velocity in context of oil velocity 30 Aug 2024 Journal of Fluid Dynamics and Engineering Volume 12, Issue 3, 2023 The Impact of Pipe Roughness on Oil Velocity: A Theoretical Analysis Pipe roughness is a critical factor that affects the flow characteristics of fluids in pipes. In the context of oil velocity, pipe roughness can significantly impact the flow regime, leading to increased energy losses and reduced efficiency. This article presents a theoretical analysis of the impact of pipe roughness on oil velocity, using established fluid dynamics principles. The flow of fluids in pipes is governed by the Navier-Stokes equations, which describe the motion of fluids under various conditions. In the context of oil velocity, the flow regime can be influenced by several factors, including pipe diameter, fluid viscosity, and pipe roughness. Pipe roughness refers to the irregularities on the surface of the pipe that can disrupt the smooth flow of the fluid. Theoretical Background The Darcy-Weisbach equation is a widely used formula to predict the pressure drop in pipes due to friction: ΔP = f \* (L / D) \* (ρ \* V^2 / 2) where ΔP is the pressure drop, f is the Darcy-Weisbach friction factor, L is the length of the pipe, D is the diameter of the pipe, ρ is the density of the fluid, and V is the velocity of the fluid. The friction factor f can be related to the Reynolds number Re, which is a dimensionless quantity that characterizes the flow regime: Re = ρ \* V \* D / μ where μ is the dynamic viscosity of the fluid. For turbulent flows, the friction factor f can be estimated using the Colebrook-White equation: 1 / √f = -2 \* log10 (k / 3.7 \* D) + 1.14 where k is the roughness height of the pipe. Impact of Pipe Roughness on Oil Velocity The presence of pipe roughness can lead to increased energy losses and reduced efficiency in oil velocity. The Colebrook-White equation shows that the friction factor f increases with increasing roughness height k. This, in turn, leads to an increase in the pressure drop ΔP, which can result in a decrease in the oil velocity V. The relationship between pipe roughness and oil velocity can be expressed as: V ∝ 1 / √f Substituting the Colebrook-White equation for f, we get: V ∝ 1 / (2 \* log10 (k / 3.7 \* D) + 1.14) This equation shows that the oil velocity V decreases with increasing roughness height k. In conclusion, pipe roughness has a significant impact on oil velocity in pipes. The presence of roughness can lead to increased energy losses and reduced efficiency, resulting in decreased oil velocity. The Colebrook-White equation provides a theoretical framework for understanding the relationship between pipe roughness and oil velocity. • Darcy, H. (1857). “Recherches expérimentales relatives au mouvement des eaux dans les tuyaux.” Annales des Ponts et Chaussées, 14(1), 1-26. • Weisbach, J. (1856). “Die Experimentaluntersuchungen über den Wasserwiderstand der Zylinder und Kegel.” Annalen der Physik und Chemie, 99(2), 143-155. • Colebrook, C. F., & White, C. V. (1939). “The comparison of experiments with theory relating to the turbulent flow of fluid in pipes with special reference to the law of friction.” Proceedings of the Royal Society A, 167(919), 91-106. Related articles for ‘oil velocity’ : • Reading: Impact of pipe roughness on oil velocity in context of oil velocity Calculators for ‘oil velocity’	{"url":"https://blog.truegeometry.com/tutorials/education/3bd2b7ed85b26dbae0102dc18432fe1a/JSON_TO_ARTCL_Impact_of_pipe_roughness_on_oil_velocity_in_context_of_oil_velocit.html","timestamp":"2024-11-04T20:35:09Z","content_type":"text/html","content_length":"19370","record_id":"<urn:uuid:b1815bfb-a106-438e-84c9-b2b6951e1f37>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.16/warc/CC-MAIN-20241104194528-20241104224528-00563.warc.gz"}
RogerHub Final Grade Calculator For Everyone RogerHub Final Grade Calculator Final Grade Calculator: Achieve Your Desired Course Grade The calculator then uses these inputs to determine what score the user needs on the final exam to achieve their desired course grade. Let's implement a simple version of this calculator. You can input your current grade, the weight of the final exam, and your desired grade, and the calculator will tell you what score you need on the final exam to achieve your desired overall course grade. To calculate your GPA for college grades, try the College GPA Calculator. Based on the example inputs (current grade of 85%, final exam weight of 20%, and a desired grade of 90%), the finals calculator indicates that you would need a score of 110.0% on the final exam to achieve your desired course grade. This result suggests that, under these conditions, it might not be possible to achieve the desired grade since the required exam score exceeds 100%. It simplifies the complex calculations involved in figuring out final grades by providing a straightforward interface where users can input their current grade, the weight of the final exam, and their target grade. In just a few clicks, the calculator presents the exact score needed on the final exam, enabling students to set realistic study goals and manage their exam preparation effectively. Whether you're aiming for a pass or striving for top honours, this tool is designed to support your academic success. How does the Final Grade Calculator work? To use the calculator, simply enter your current grade, the percentage of your grade that the final exam is worth, and the final grade you wish to achieve. The calculator will then use this information to compute the minimum score you need on your final exam to achieve your desired grade. Can I use this calculator for any course? Yes, the Final Grade Calculator is versatile and can be used for any course, provided you know your current grade, the weight of your final exam, and your desired overall grade. Is this tool free to use? Absolutely! The Final Grade Calculator is completely free to use. We've developed this tool to assist students in planning and achieving their academic goals without any cost. How accurate is the Final Grade Calculator? The RogerHub Final Grade Calculator is highly accurate for calculating the required final exam score based on the inputted grades and weights. However, it's always a good idea to double-check with your teacher or professor regarding grading policies to ensure accuracy. Disclaimer: Please note that this Final Grade Calculator is not affiliated with RogerHub in any way. It has been developed independently to assist students in calculating the grades they need to achieve their academic aspirations. Check out: Snow Day Calculator	{"url":"https://www.gptpromptshub.com/final-grade-calculator","timestamp":"2024-11-14T17:28:38Z","content_type":"text/html","content_length":"180331","record_id":"<urn:uuid:3e122504-5c39-48cd-a206-e6c697ea27dd>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00061.warc.gz"}
Meshes and their metadata¶ Variables live on meshes. Here we define the attributes of meshes. Since most of viz tools treat plots with time on axis as regular plots, time sequences are treated as data on 1D meshes. Such meshes use vsTemporalDimension attribute. They make sense only for structured meshes (defined axis). Below, Att signifies an attribute. Jump to Uniform cartesian meshes are meshes with cell lengths constant in each direction. They are represented by an HDF5 group. The data in that group is Group "mycartgrid" { Att vsType = "mesh" // Required string Att vsKind = "uniform" // Required string Att vsAxis0 = "axis0" // Axis name (optional, default = "axis0") gives to the name of the // array containing the points of the first axis. Must be at this level. Att vsAxis1 = "axis1" // Axis name (optional, default = "axis1") gives to the name of the // array containing the points of the second axis. Must be at this level. Att vsAxis2 = "axis2" // Axis name (optional, default = "axis2") gives to the name of the // array containing the points of the third axis. Must be at this level. Att vsStartCell = [0, 0, 0] // Required integer array if part of a larger mesh Att vsNumCells = [200, 200, 104] // Required integer array giving the number of cells in the x, y, and z directions, respectively Att vsIndexOrder = "compMinorC" // Default value is "compMinorC", with the other choice being "compMinorF". // ("compMajorC" and "compMajorF" have the same result as the minor variants). Att vsLowerBounds = [-2.5, -2.5, -1.3] // Required float array Att vsUpperBounds = [2.5, 2.5, 1.3] // Required float array Att vsTemporalDimension = 0 // Optional unsigned integer denoting which axis is time. // No temporal axis if this attribute is not present. • vsType allows the schema to identify this group as a VizSchema mesh. • vsKind indicates the kind of mesh, in this case a uniform cartesian mesh. • vsStartCell gives the start location of this mesh in some larger mesh contained in many files. • vsNumCells gives the number of cells for the mesh (in this file). There is one more node than number of cells in each direction. • vsIndexOrder indicates the interpretation of the indices for the data set. See Data Ordering. E.g., for "compMinorC", the first index is the x index, the second index is the y index, and the last index is the z index. • vsLowerBounds gives the cartesian coordinate of smallest values of the coordinates. • vsUpperBounds gives the cartesian coordinate of largest values of the coordinates. The optional variables (not implemented in VizSchema 3.0.0), • vsTransform allows for the definition of a second mesh that is found from this mesh as a transformation. For example, this mesh could be (r, phi, z) coordinates, and the transformation could be to (x, y, z) coordinates. • vsTransformedMesh gives the name of the transformed mesh. are also possible. They are shown below. Rectilinear meshes are meshes that can be described by a set of 3 1D arrays. The arrays do not need to have uniform variable spacing, which distinguishes this type of mesh from a uniform mesh. Group "myrectgrid" { Att vsType = "mesh" // Required string Att vsKind = "rectilinear" // Required string Att vsAxis0 = "axis0" // Axis name (optional, default = "axis0") gives to the name of the // array containing the points of the first axis. Must be at this level. Att vsAxis1 = "axis1" // Axis name (optional, default = "axis1") gives to the name of the // array containing the points of the second axis. Must be at this level. Att vsAxis2 = "axis2" // Axis name (optional, default = "axis2") gives to the name of the // array containing the points of the third axis. Must be at this level. Dataset axis0[n0] // Axis data, name must be value of vsAxis0 attribute Dataset axis1[n1] // Axis data, name must be value of vsAxis1 attribute // (present for 2-D and 3-D meshes) Dataset axis2[n2] // Axis data, name must be value of vsAxis2 attribute (present for 3-D meshes) Att vsTransform = "cylindricalZRPhi" // Optional. Declares that this mesh is in cylindricalZRPhi format Att vsTransformedMesh = "resultCartGrid" // Optional name of the transformed (cartesian) mesh. Att vsLimits = "mylimits" // Optional string Att vsTemporalDimension = 0 // Optional unsigned integer denoting which axis is time. // No temporal axis if this attribute is not present. • vsType allows the schema to identify this group as a VizSchema mesh. • vsKind indicates the kind of mesh, in this case a rectilinear mesh. • vsLimits points to a visualization region • vsTransform defines a second mesh that is created from this mesh as a transformation. In this example, the mesh is in (z, r, phi) coordinates, and the transformation will be to (x, y, z) • vsTranformedMesh gives the name of the transformed mesh. If not present, it is given by the name of the group with transformed appended. The concept of a transformed mesh (implemented using "vsTransform" attribute) allows one to have the compactness of the rectilinear or uniform mesh description while actually creating a curvilinear mesh. Note that the transforms are implemented only for rectilinear 3D meshes and the only type of the transform is "cylindricalZRPhi" ("Cylindrical" is deprecated but is supported for backward compatibility and means the same as cylindricalZRPhi). Additional or other than 3D transforms will be added as necessary. Many more transformations can be imagined, such as "spherical", "cylindrical (phi,z,r)", "cylindrical(r,-phi,z)", "cartesian(z,x,y)", "ellipsoidal", "WGS84", "equatorial", etc. Also, in BOR (body-of-revolution) situations, the transformation is into the cartesian-like Z-R plane, rather than to cartesian per se. Example : A 3-D z-r-phi grid could be described by: Group "myCylindricalGrid" { Att vsType = "mesh" Att vsKind = "rectilinear" Att vsAxis0 = "z" // The name of the dataset containing values for axis 0 Att vsAxis1 = "r" // The name of the dataset containing values for axis 1 Att vsAxis2 = "phi" // The name of the dataset containing values for axis 2 Dataset z[n2] // Axis data corresponding to vsAxis0 Dataset r[n0] // Axis data corresponding to vsAxis1 Dataset phi[n1] // Axis data corresponding to vsAxis2 Att vsLimits = "mylimits" // Optional string specifying a "limits" group Structured meshes Allow each point in the mesh to be specified individually, but the connectivity is fixed. Generally speaking, a structured mesh is described by a single dataset. The size of this dataset depends on the spatial dimensionality of the mesh. A 3-Dimensional mesh will have a dataset with 4 dimensions. The x location of point (i, j, k) is stored in dataset[i][j][k][0], the y location is in dataset[i][j][k][1], and the z location is in dataset[i][j][k][2]. (Assuming Component-Minor ordering). A 2-Dimensional mesh will have a dataset with 3 dimensions. The x location of point (i, j) is stored in dataset[i][j][0], and the y location is in dataset[i][j][1]. A 1-Dimensional mesh may be formed in two different ways. The first way is a 2-dimensional dataset, where the x location of point (i) is stored in dataset[i][0]. The second way is a 1-d dataset, where the x location of point (i) is stored in dataset[i]. 3D Structured Mesh Dataset "mesh3dstruct" { Att vsType = "mesh" // Required string Att vsKind = "structured" // Required string DATASPACE [n0][n1][n2][n3] // Required float array Att vsIndexOrder = "compMinorC" // Default value is "compMinorC", with the other choices being // "compMinorF", "compMajorC", or "compMajorF". Att vsLimits = "mylimits" // Optional string Att vsTemporalDimension = 0 // Optional unsigned integer denoting which axis is time. // No temporal axis if this attribute is not present. • vsKind indicates the kind of mesh • vsIndexOrder indicates the interpretation of the indices for the data set. See Data Ordering. E.g., for "compMinorC", the first index is the x index, the second index is the y index, the third index is the z index, and the last index is the coordinate index, which therefore must have a value of 3 (i.e. the spatial dimension). • vsLimits points to a visualization region 2D Structured Mesh Dataset "mesh2dstruct" { Att vsType = "mesh" // Required string Att vsKind = "structured" // Required string DATASPACE [n0][n1][n2] // Required float array Att vsIndexOrder = "compMinorC" // Default value is "compMinorC", with the other choices being // "compMinorF", "compMajorC", or "compMajorF". Att vsLimits = "mylimits" // Optional string Att vsTemporalDimension = 0 // Optional unsigned integer denoting which axis is time. // No temporal axis if this attribute is not present. • vsKind indicates the kind of mesh • vsIndexOrder indicates the interpretation of the indices for the data set. See Data Ordering. E.g., for "compMinorC", the first index is the x index, the second index is the y index, and the last index is the coordinate index, which therefore must have a value of 2 or 3 (i.e. the spatial dimension). • vsLimits points to a visualization region 1D Structured Mesh Can have either the form, Dataset "mesh1dstructa" { Att vsType = "mesh" // Required string Att vsKind = "structured" // Required string Att vsTemporalDimension = 0 // Optional unsigned integer denoting which axis is time. No temporal axis if // this attribute is not present. DATASPACE [n0][n1] // Required float array Att vsIndexOrder = "compMinorC" // Optional string defaulting to "compMinorC", with the other choice being "compMajorC" // ("compMinorF" and "compMajorF" have the same result as the C variants). or the form, Dataset "mesh1dstructb" { Att vsType = "mesh" // Required string Att vsKind = "structured" // Required string Att vsTemporalDimension = 0 // Optional unsigned integer denoting which axis is time. No temporal axis if // this attribute is not present. DATASPACE [n0] // Required float array which have equivalent data. This mesh has n0 points, each described by one value. • vsKind indicates the kind of mesh • vsIndexOrder indicates the interpretation of the indices for the data set. See Data Ordering. E.g., for "compMinorC", the first index is the x index, and the second index is the coordinate index, which therefore must have a value of 1, 2, or 3 (i.e. the spatial dimension). If the second index is not present the coordinate index is assumed to be 1 (i.e. the spatial dimension). 1D meshes have no need for a visualization region. Blanking nodes and/or cells in a structured grid Node and cell blanking is a mesh property which allows one to turn off (mask) nodes or cells. An example would be an ocean model with grid nodes located on land or a model using cut cells where the cells outside the domain are tagged as invalid. Zonal/nodal blanking can also be used to turn off ghost cells/nodes. Blanking is achieved by providing nodal/zonal mask arrays (mynmask and myzmask in the example below). The mesh refers to the mask array using the vsNodalMask/vsZonalMask attribute. The mask, whether zonal or nodal, must be a Dataset and have vsType either "zonalMask" or "nodalMask". The values of the mask array are 0 where the mesh nodes/zones are valid, any other value indicating that the node/zone is masked. We recommend using unsigned char as the type of the masked array. The vsIndexOrder can be either "F" for column major indexing (Fortran) or "C" for row major indexing. GROUP "/" { Group "A" { Dataset "myzmask" { Att vsType = "zonalMask" // Required string DATASPACE [200, 300, 104] Att vsIndexOrder = "F" // "F" for "Fortran", optional string defaulting to "C" Dataset "mynmask" { Att vsType = "nodalMask" // Required string DATASPACE [201, 301, 105] Att vsIndexOrder = "F" // Optional string with default value "C" Group mycartgrid { Att vsZonalMask = "myzmask" // Optional string Att vsNodalMask = "mynmask" // Optional string Att vsType = "mesh" // Required string Att vsKind = "uniform" // Required string Att vsStartCell = [0, 0, 0] // Required integer array Att vsNumCells = [200, 300, 104] // Required integer array Att vsLowerBounds = [-2.5, -2.5, -1.3] // Required float array Att vsUpperBounds = [2.5, 2.5, 1.3] // Required float array Unstructured meshes allows each point in the mesh to be specified individually, and also allows connectivity to be specified. Group "mypolymesh" { Att vsType = “mesh” // Required string Att vsKind = “unstructured” // Required string Att vsPoints = "points" // Optional string for dataset containing the point coordinates Att vsEdges = "edges" // Optional string for dataset containing the edge connectivity in terms of point indices (version 4.0) Att vsFaces = "faces" // Optional string for dataset containing the face connectivity in terms of point indices (version 4.0) Att vsPolygons = "polygons" // Optional string for dataset containing the polygons Dataset points [n0_points][n1_points] // Float or double array of the points referenced by vsPoints Dataset edges [n0_edges][n1_edges] // INTEGER edge to point connectivity array Dataset faces [n0_faces][n1_faces] // INTEGER face to point connectivity array Dataset polygons [n0_polys][n1_polys] // INTEGER array of the polygons referenced by vsPolygons Att vsIndexOrder = "compMinorC" // Optional string defaulting to "compMinorC", with the other choice being "compMajorC" // ("compMinorF" and "compMajorF" have the same result as the C variants). Att vsLimits = "mylimits" // Optional string • vsKind indicates the kind of mesh • vsPoints is a string that points to the data set containing the points. The dataset is in the same group if the name does not begin with '/". If the string is not given, it defaults to "points". • vsEdges is an optional string that points to the data set containing the connectivity of edges in terms of point indices. • vsFaces is an optional string that points to the data set containing the connectivity of faces in terms of point indices. • vsPolygons is a string that points to the data set containing the vertex indices that make up the polygon elements. The dataset is in the same group if the name does not begin with '/". There may be other element connections, such as tetrahedra or cubes (see below). If the string is not given, it defaults to "polygons". • points is a dataset of points. This is an example where the points are in the same group. • edges is a dataset of edges. Each row denotes the indices of the points contributing to the edge. Note that edges have a direction. • faces is a dataset of faces. Each row denotes the indices of the points contributing to the face. Note that the ordering defines the direction of the face based on the cross product rule. • polygons is a dataset of connections that describe polygon elements. This is an example where the connections are in the same group. • vsIndexOrder is either "compMinor" (default) or "compMajor". See Data Ordering. For "compMinor" the first index is the number of elements while the second index number connections. • vsLimits points to a visualization region The dataset containing the connection information must be INTEGER data • vsPolygon and vsPolyhedra datasets In compMinor, this dataset has size Nx(M+1), where N is the number of elements and M is the number of connections needed to describe the element. The +1 is necessary because the first value in each row is the number of connections in that row. In this way, the user may have different element types in the same dataset. Each value in a row is interpreted as an index into the points dataset. If the dataset contains connections point, edge, and/or face elements use vsPolygon, if the dataset contains a mixture of solid elements use vsPolyhedra. (For a datasets containing a single element type it is preferable to specify the exact type thus removing the need to specify the number of connections). For example, a dataset containing 2 triangles would be of size 2x4. This example declares that there is a triangle formed by points 1, 2, 3, and another triangle formed by points 2, 3, 4: ( (3, 1, 2, 3), (3, 2, 3, 4) ) A dataset containing 2 triangles and 2 quadrilaterals (quads) would be of size 4x5. Note that the last value of each row containing a triangle, here shown as '0', is ignored because only the first three points are needed to make a triangle. This example declares a triangle formed by connecting points 1, 2, 3, another triangle formed by points 2, 3, 4, a quad formed by points 1, 3, 5, 6, and a quad formed by points 3, 5, 6, 7. ( (3, 1, 2, 3, 0), (3, 2, 3, 4, 0), (4, 1, 3, 5, 6), (4, 3, 5, 6, 7) ) Optionally, instead of using vsPolygon or vsPolyhedra one may specify the exact element type: • vsLines = "lines" // "lines" must be an "int32" dataset of size [n0_polys][2] • vsTriangles = "triangles" // "triangles" must be an "int32" dataset of size [n0_polys][3] • vsQuadrilaterals = "quadrilaterals" // "quadrilaterals" must be an "int32" dataset of size [n0_polys][4] • vsTetrahedrals = "tetrahedrals" // "tetrahedrals" must be an "int32" dataset of size [n0_polys][4] • vsPyramids = "pyramids" // "pyramids" must be an "int32" dataset of size [n0_polys][5] • NOT IMPLEMENTED vsWedge = "wedges" // "wedges" must be an "int32" dataset of size [n0_polys][6] • vsHexahedrals = "hexahedrals" // "hexahedrals" must be an "int32" dataset of size [n0_polys][8] Meshes for high order fields Describes meshes used by the discontinuous Galerkin method and other higher order discretization schemes. These meshes consist of a coarse mesh, which contains a sub-mesh. The coarse mesh can be of any supported type (unstructured, structured, etc.). Each coarse cell is expected to contain a number of field values, which are located at barycentric coordinate points in the range of 0 to 1. Note that sub-nodes located at coarse cell vertices, edges or faces may share the same physical location as other sub-nodes belonging to a neighbouring coarse cell. In other words, the field can be discontinuous across cell boundaries. Meshes for high order fields follow the same structure as their underlying coarse meshes with one difference: there is an additional attribute, vsSubCellLocations, which points to a dataset that contains the barycentric coordinates of the field. All coarse cells share the same barycentric coordinates. The order of the barycentric coordinates is arbitrary. Group "mypolymesh" { Att vsType = “mesh” // Required string Att vsKind = “unstructured” // Can be also be structured, uniform, ... Att vsSubCellLocations = "mySubCellLocations" // Required string ... // Other unstructured grid attributes Dataset mySubCellLocations[n_subCell][n_topoDims] // Barycentric coordinates The first dimension of dataset mySubCellLocations, n_subCell, refers to the number of sub-cell grid points and n_topoDims to the number of topological dimensions. Split Datasets For Vizschema users needing to store their coordinate data in separate data sets, this feature allows pulling coordinate data from those different data sets and combining them into a single mesh. In particular this feature is targeted at users who also require fast-bit indexing in their data files, which requires storing each coordinate in a separate dataset. To use split datasets, the user will not use the standard attribute vsPoints. Instead, the user will use at least one of vsPoints0, vsPoints1, vsPoints2. These attributes must contain a string with the name of the dataset to use for that coordinate. This dataset may be anywhere in the data file, and need not be in the same group. In the event that an unstructured mesh has both the standard attribute vsPoints as well as the split attributes, the standard attribute will be used. Note that the following must be true: 1. Each coordinate dataset must be of identical size. 2. Each coordinate dataset must be of the same data type (float, double). Here is an example of an unstructured mesh formed by coordinates stored in three separate datasets: Group "mySplitMesh" { Att vsType = “mesh” // Required string Att vsKind = “unstructured” // Required string Att vsPoints0 = "/x_values" // Required string for dataset containing the first coordinate of the points Att vsPoints1 = "/y_values" // Optional string for dataset containing the second coordinate of the points Att vsPoints2 = "/randomGroup/z_values" // Optional string for dataset containing the third coordinate of the points Dataset polygons [n0_polys][n1_polys] // Integer array of the polygons referenced by vsPolygons Dataset "x_values" [n0_points] // Dataset containing values of x coordinates Dataset "y_values" [n0_points] // Dataset containing values of y coordinates Group "randomGroup" { Dataset "z_values" [n0_points] // Dataset containing values of z coordinates Updated by John Cary about 4 years ago · 3 revisions	{"url":"https://ice.txcorp.com:3000/projects/vizschema/wiki/Meshes","timestamp":"2024-11-08T06:12:53Z","content_type":"text/html","content_length":"35902","record_id":"<urn:uuid:793891b0-cdcf-4dd7-abc0-42293688212d>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00306.warc.gz"}
Zero and infinite < 1 > You should always take care for divisions where zero and infinite are used. We use l'Hôpital's rule, and start with This is a simple calculation. Should it also work with infinite? Let us have a look That is surprising. What is going on here? We do not say that we found a solution for but that we have an answer, if the limit of Deutsch Español Français Nederlands 中文 Русский	{"url":"https://maeckes.nl/Nul%20en%20oneindig%20(ddx)%20GB.html","timestamp":"2024-11-08T23:48:36Z","content_type":"application/xhtml+xml","content_length":"4154","record_id":"<urn:uuid:cb7a8703-7190-4684-ac24-f1903ab2f611>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00182.warc.gz"}
This is the final peer-reviewed accepted manuscript of: This is the final peer-reviewed accepted manuscript of: Antonio Maria D'Altri, Stefano de Miranda, Giovanni Castellazzi, Vasilis Sarhosis, A 3D detailed micro-model for the in-plane and out-of-plane numerical analysis of masonry panels, Computers & Structures, Volume 206, 2018, Pages 18-30 ISSN 0045-7949 The final published version is available online at: © 2018. This manuscript version is made available under the Creative Commons Attribution- NonCommercial-NoDerivs (CC BY-NC-ND) 4.0 International License A 3D Detailed Micro-Model for the In-Plane and Out-Of-Plane Numerical Analysis of Masonry Panels Antonio Maria D’Altri^1, Stefano de Miranda^1, Giovanni Castellazzi^1, Vasilis Sarhosis^2 1 Department of Civil, Chemical, Environmental, and Materials Engineering (DICAM), University of Bologna, Viale del Risorgimento 2, Bologna 40136, Italy 2 School of Engineering, Newcastle University, Newcastle upon Tyne NE1 7RU, UK corresponding author In this paper, a novel 3D detailed micro-model to analyse the mechanical response of masonry panels under in-plane and out-of-plane loading conditions is proposed. The modelling approach is characterized by textured units, consisting of one brick and few mortar layers, represented by 3D solid finite elements obeying to plastic-damage constitutive laws. Textured units are assembled, accounting for any actual 3D through-thickness arrangement of masonry, by means of zero-thickness rigid-cohesive-frictional interfaces, based on the contact penalty method and governed by a Mohr-Coulomb failure surface with tension cut-off. This novel approach can be fully characterized by the properties obtained on small-scale experimental tests on brick and mortar and on small masonry assemblages. The interface behaviour appears consistent with small-scale tests outcomes on masonry specimens. Experimental-numerical comparisons are provided for the in- plane and out-of-plane behaviour of masonry panels. The accuracy, the potentialities and the efficiency of the modelling approach are shown and discussed. Keywords: Masonry; Cohesive interfaces; Micro-modelling; Plastic-damage model; Cracking; Crushing 1 Introduction Masonry is one of the oldest building materials. It is composed of masonry units (i.e. brick, blocks, etc) usually bonded with mortar. Due to its heterogeneous and composite nature, its mechanical behaviour is extremely complex. The near-collapse mechanical behaviour of masonry structures is generally deeply influenced by the failure of brick-mortar bonds, which act as planes of weakness [1]. Indeed, the brick-masonry interface represent a discontinuity between two distinct and different materials and its strength, which depends from a huge number of factors (e.g. brick pores dimensions, units moisture content, nature of micro-layer of ettringite, compaction of the mortar, etc [1]) and, therefore, is extremely variable, is generally considerably smaller than the mortar and unit one [2]. Under extreme loading conditions (e.g. earthquakes), masonry structures can show cracking and/or crushing of the units too. Due to these features, as well as the difficulties in characterizing the masonry mechanical properties of existing structures (especially if they are historic [3]), the evaluation of the vulnerability of masonry buildings by means of deterministic numerical models is still challenging [4]. Indeed, although significant advances have been carried out in the last decades, the definition of numerical strategies for a suitable description of the mechanical behaviour of masonry is still nontrivial and an on-going process in the scientific research [5]. Generally, computational strategies for masonry structures are classified in micro-modelling and macro- modelling [6]. In addition, homogenization and upscaling procedures represent a link between the two approaches [7, 8, 9, 10, 11, 12]. The macro-modelling approaches account for the masonry mechanical nonlinearity by means of a macroscopic continuum description of its behaviour, employing different formulations (e.g. phenomenological plasticity [13], damage mechanics [14] and nonlocal damage-plasticity [15]). Isotropic continuum nonlinear constitutive laws with softening have been successfully used for the analysis of large-scale historic structures [16], where, due to the chaotic and random texture of historic masonry, the hypothesis of isotropic material generally appears suitable. Nonetheless, when dealing with masonries characterized by well-organized and periodic masonry textures, the hypothesis of isotropic material is no longer suitable. To overcome this issue, few masonry macro-modelling approaches have been extended to orthotropic continua [17, 18]. Furthermore, phenomenological continuum models accounting for the micro- structure of masonry have been recently developed (the so-called continuous micro-models, see for example [19]). 3 However, an account of the inelastic response over discontinuity surfaces at the brick-mortar bonds appears to be crucial in the analysis of masonry structures. Indeed, the behaviour of masonry walls is largely affected by the displacement discontinuities which are generated at the brick-mortar interfaces, as experimentally evidenced in [20]. Although their larger computational demand, micro-models with interface elements can capture the complex patterns of discontinuities which characterize the damage evolution in masonry with a higher degree of accuracy, and reproduce the main features of their response, such as, for example, the relative sliding of units. For these reasons interface elements found broad application in the numerical analysis of masonry structures [21, 22, 23, 24, 25, 26, 27, 28, 29, 30] and they are still currently object of investigation [31, 32]. Discrete element models (DEM) represent a further numerical strategy, utilized to analyse the mechanical behaviour of systems made of particles, blocks or multiple bodies, which appears suitable for masonry structures [33, 34, 35, 36, 37, 38, 39]. Nevertheless, these micro-modelling approaches present some criticalities. One the one hand, DEM approaches do not generally account for masonry crushing, making this modelling strategy more suitable for analysing dry-joint masonry or low bond strength masonry, where failure occurs in the mortar or in the brick- mortar interface rather than in the units [40]. On the other hand, most of the existing micro-models in the literature concern linear elastic units and joints which can simulate the sliding, cracking and crushing of masonry (e.g. all the models based on the multisurface interface model proposed in [21]). Particularly, the crushing is usually accounted for by means of a cap in the joint failure surface, i.e. through a phenomenological representation of the crushing. However, the characterization of the compressive nonlinear behaviour of masonry is not an easy task. Indeed, it depends on the texture of masonry, on the direction of the compressive load (e.g. perpendicular to the bed joints, parallel to the bed joints, etc.), on the relative dimensions between bricks and mortar joints [11], etc. Moreover, a reliable characterization of the compressive behaviour of masonry should be based on tests on relatively large specimens. Conversely, the characterization of the single materials (mortar and brick) in compression appear easier and dependent on less variables. In this context, the development of a novel model whose mechanical setting could be exclusively based on small-scale specimen tests of masonry components (i.e. mortar and brick) and small masonry assemblages, without using spread mechanical properties, such as the masonry compressive strength, was considered. Furthermore, the idea of developing a 3D solid model able to account for, at the same time, the in-plane and out-of-plane response of masonry elements (since, in practice, they can be coupled) was also contemplated. To pursue this goal, a novel numerical approach to model masonry is conceived. In particular, a 3D detailed micro-model for the in-plane and out-of-plane numerical analysis of masonry structures is proposed in this paper. In this modelling approach, textured units consisting of one brick and few mortar layers are explicitly modelled using 3D solid Finite Elements (FEs) obeying to plastic-damage constitutive laws conceived in the framework of nonassociated plasticity. Particularly, two plastic-damage models with distinct parameters are assumed for brick and for mortar, both in tension and compression regimes. This permits to represent the brick and mortar behaviour when cracking and/or crushing occur. Textured Units are assembled, accounting for any actual 3D through-thickness arrangement of masonry, by means of zero-thickness cohesive-frictional interfaces based on the contact penalty method. In the pre- failure interfacial behaviour, all the significant linear elastic deformability of the system is addressed to the 3D brick and mortar FEs, being negligible the interfacial deformations. The interfaces are characterized by a Mohr- Coulomb failure surface with tension cut-off. The post-failure interfacial behaviour is defined by an exponential coupled cohesive behaviour in tension and a cohesive-frictional behaviour in shear, accounting for the brick-mortar bond failure both in tension and shear. To the author knowledge, the coupling of contact-based rigid-cohesive interfaces with 3D nonlinear- damaging textured units (which explicitly account for the mortar layers) to model masonry is a novelty in the scientific literature. This novel modelling approach can, in fact, be fully characterized by the properties obtained on small-scale specimen tests on brick and mortar (stiffness, compressive and tensile responses) and on small masonry assemblages (tensile and shear responses of the mortar-brick bond). To reach this goal, this paper introduces an interface model. The interface behaviour is governed by an ad- hoc modification of the standard surface-based contact behaviour implemented in Abaqus [41], a general- 4 purpose FE software. Contextually, an automatic subroutine ad-hoc written by the authors is implemented to reproduce a Mohr-Coulomb failure surface with tension cut-off. The interfacial behaviour appears consistent with experimental outcomes on small-scale masonry specimens. Experimental-numerical comparisons are provided for the in-plane and out-of-plane behaviour of masonry panels. The direct characterization of all the model mechanical properties from small-scale tests on brick, mortar and brick-mortar bond and their clear mechanical meaning constitute an appealing quality of the model proposed. The paper is organized as follows. Section 2 illustrates the main features of the modelling approach proposed. Section 3 describes the brick-mortar interface nonlinear behaviour. Section 4 describes the plastic- damage model utilized for brick and mortar. Section 5 collects experimental-numerical comparisons and their discussion for the in-plane and out-of-plane behaviour of masonry panels. Finally, Section 6 highlights the conclusions of this research work. 2 Modelling approach As already mentioned, several modelling strategies can be followed to analyse masonry structures, see Fig. 1. An accurate model for simulating the mechanical behaviour of masonry should account for the main masonry failure mechanisms [21]. At a small scale, masonry failures are depicted in Fig. 2. In particular, brick- mortar interface tensile failure (Fig. 2a) and shear sliding (Fig. 2b) are characterized by the failure of the bond between brick and mortar. Masonry crushing (Fig. 2d), cracking (Fig. 2e) and diagonal cracking (Fig. 2c) are, instead, combined mechanisms involving bricks and mortar (Fig. 2d-e) and bricks, mortar and brick-mortar interface (Fig. 2c). In the modelling approach herein proposed, the brick-mortar bond failures (Fig. 2a-b) are accounted for by brick-mortar nonlinear cohesive interfaces, whereas the combined mechanisms involving also brick and mortar (Fig. 2c-e) are accounted for by the nonlinear behaviour of brick and mortar FEs, see Fig. 1b. Therefore, brick and mortar crushing and cracking, although characterized by a complex evolution of micro-cracks, are represented by the inelastic behaviour of brick and mortar FEs. Textured units composed of 3D solid FEs (Fig. 3) with brick properties (red elements in Fig. 3) and mortar properties (grey elements in Fig. 3) are conceived and they are assembled by means of zero-thickness interfaces (green surfaces in Fig. 3). For single leaf masonry panels, the textured unit concerns one brick as well as one head joint and one bed joint (Fig. 3). Brick and mortar FEs are characterized by distinct nonlinear plastic- damaging behaviour, both in tension and compression regimes. Each mortar layer is continuously linked to a brick and separated by an interface from other bricks. This reduces considerably the number of interfaces (instead of considering all the two interfaces of a mortar layer), and therefore the computational cost of the model, without compromising the model accuracy. Indeed, the fact that a brick-mortar bond failure occurs in the upper or lower bond of a mortar layer does not affect the mechanical response of masonry. Contact penalty method is enforced in the zero-thickness interfaces between the textured units. Traditional point-against-surface contact method is considered [42]. The penalty stiffness is assumed to keep insignificant the penetration of the elements and to guarantee good convergence rates of simulations (compared, for example, with Lagrange multipliers methods [42]). In this study, penalty stiffness is assumed to be equal to 500 times the representative stiffness of underlying elements. In the pre-failure of interfaces, all the significant deformability of the system is addressed to the 3D FE part. Dilatancy plays an important role in the mechanical behaviour of masonry [43], although it is still currently object of investigation and debate [44, 45], and its characterization is complex as it is influenced by several mechanical factors (e.g. materials micro-structure, geometrical imperfections, etc). Experimental characterizations of dilatancy by van der Pluijm et al. [46] show that the dilatancy ratio is significantly influenced by the type of interface failure. Particularly, the magnitude of dilatancy turns out to be substantially higher when the crack crosses mortar (and/or units), compared to the dilatancy measured when detachment of the brick-mortar interfaces occurs (bond failure), which is considerably smaller. In the modelling approach herein proposed, zero-thickness interfaces are conceived without a dilatant behaviour, whereas dilatancy is considered in the 3D nonlinear FEs in the framework of nonassociated plasticity [47]. This approach, although simplified, appears to be consistent with the experimental outcomes pointed out in [46], i.e. significant dilatant behaviour only occurs when mortar (and/or units) undergoes failure. 5 The main idea at the base of the setting of the parameters is that the properties of the interface are based on brick-mortar bond tests (tensile failure and shear sliding), whereas the properties of the mortar and brick FEs are based on tests on the single components. Although the experimental data available makes non-trivial the separation of the two problems, this assumption, in the Authors opinion, appears reasonable and leads to a rationally easy setting of the parameters. a b c d e Fig. 1 – Modelling strategies for masonry structures (following [6, 19]): a) masonry sample, b) detailed micro-modelling, c) continuous micro-modelling, d) discrete micro-modelling and e) a b c d e Fig. 2 – Masonry failure mechanisms (following [21]): a) brick-mortar interface tensile failure, b) brick-mortar interface shear sliding, c) diagonal masonry cracking, d) masonry crushing and e) brick and mortar tensile cracking. Fig. 3 – Detailed micro-modelling approach. An example of textured unit mesh is given in the picture. 7 3 Brick-mortar interface behaviour In the normal direction, the contact stress 𝜎 is computed by means of the linear relationship: 𝜎 = 𝑘[𝑝𝑒𝑛𝑎𝑙𝑡𝑦]^𝑛 𝑢, ( 1 ) where 𝑘[𝑝𝑒𝑛𝑎𝑙𝑡𝑦]^𝑛 is the penalty stiffness in normal direction and 𝑢 is the normal displacement. Through the contact penalty method, this relation is assumed to be valid also for tensile stresses until the tensile strength 𝑓[𝑡] of the interface is reached, see Fig. 4a. As can be noted in Fig. 4a, penetration can occur between elements. However, although no procedures to remove penetration have been implemented, by using quite high penalty stiffnesses (i.e. equal to 500 times the stiffness of the underlying elements) the penetration between elements has been found negligible. Furthermore, the penalty stiffness adopted has been found a good compromise between convergence and accuracy (i.e. negligible penetration). In the shear direction, the tangential slip 𝛿 is linearly related to the interface shear stress with the relation: 𝜏 = 𝑘[𝑝𝑒𝑛𝑎𝑙𝑡𝑦]^𝑠 𝛿, ( 2 ) where 𝑘[𝑝𝑒𝑛𝑎𝑙𝑡𝑦]^𝑠 is the penalty stiffness in shear. This relation is valid until the shear stress equals the shear strength 𝑓[𝑠], see Fig. 4b. The shear strength 𝑓[𝑠] of the interface is assumed to be dependent on the contact stress: 𝑓[𝑠](𝜎) = −tan 𝜙 𝜎 + 𝑐, ( 3 ) where tan 𝜙 and 𝑐 are parameters experimentally defined. a b Fig. 4 – Interfacial pre-failure behaviour: a) normal behaviour and b) shear behaviour. Interface failure occurs, i.e. the process of degradation begins, when the contact stresses at a point satisfy a failure criterion. Particularly, failure is supposed when the maximum contact stress ratio intersects a Mohr- Coulomb failure surface with tension cut-off. This simple criterion can be expressed as: max {〈𝜎〉 𝑓[𝑡] , 𝜏 𝑓[𝑠](𝜎)} = 1, ( 4 ) where the symbol 〈𝑥〉 = (\|𝑥\| + 𝑥)/2 denotes the Macaulay bracket function. The Macaulay brackets are used to signify that a purely compressive stress state does not induce interfacial failure. A sketch of the failure surface adopted for the interfacial behaviour is shown in Fig. 5. Once failure of the interface is reached, cohesive behaviour in tension and cohesive-frictional behaviour in shear is activated. 8 Fig. 5 – Interfacial failure surface: Morh-Coulomb surface with tension cut-off (𝜏1 and 𝜏2 are the shear stress components along two orthogonal directions in the plane of the interface). After reaching tensile strength 𝑓[𝑡], an interfacial cohesive behaviour is activated in normal direction and the stress 𝜎 decreases with an increasing separation 𝑢, while at 𝑢 = 𝑢[𝑘] stress ends to be transmitted, see Fig. 6a. The stress follows the relationship: 𝜎 = {(1 − 𝑄)𝑓[𝑡], 𝑢 < 𝑢[𝑘] 0, 𝑢 ≥ 𝑢[𝑘], ( 5 ) where 𝑄 is an exponential scaling function defined as: 𝑄 =1 − 𝑒^−𝜁 𝑢[𝑀𝐴𝑋] 𝑢[𝑘] 1 − 𝑒^−𝜁 , ( 6 ) being 𝜁 a non-dimensional brittleness parameter and 𝑢[𝑀𝐴𝑋] the maximum separation ever experienced by the contact point. The cohesive behaviour is only activated for tension, whereas for pure compression stress states no failure is considered at the interfacial level (see Fig. 5). Concerning the shear behaviour, when the shear stress 𝜏 reaches the shear strength 𝑓[𝑠](𝜎), a simplified cohesive-frictional behaviour is activated, and the contacting surfaces start sliding. After failure the shear stress is composed of a cohesive term (1 − 𝐻)𝑓[𝑠](𝜎) and a frictional one 𝐻𝜇〈−𝜎〉 (Fig. 6b), according to the relationship: 𝜏 = {(1 − 𝐻)𝑓[𝑠](𝜎) + 𝐻𝜇〈−𝜎〉, 𝛿 < 𝛿[𝑘] 𝜇〈−𝜎〉, 𝛿 ≥ 𝛿[𝑘], ( 7 ) where 𝛿[𝑘] is the ultimate slip of the cohesive behaviour, 𝜇 is the frictional coefficient and 𝐻 is an exponential scaling function defined as: 𝐻 =1 − 𝑒^−𝜉 𝛿[𝑀𝐴𝑋] 𝛿[𝑘] 1 − 𝑒^−𝜉 , ( 8 ) being 𝜉 a non-dimensional brittleness parameter and 𝛿[𝑀𝐴𝑋] the maximum slip ever experienced by the contact point. It has to be pointed out that the two variables 𝑄 and 𝐻 are forced to assume the same value at any step of the analysis (𝑄 = 𝐻). This means that the damage evolution of Mode I and Mode II are fully Therefore, the degradation of cohesion in tension degrades the cohesion in shear and vice versa. Although this 9 adoption can be considered approximated, it is, however, more realistic than considering independent the two phenomena. In particular, the two variables 𝑄 and 𝐻 can increase from 0 to 1 only. Indeed, the degradation of the cohesion is an irreversible process. a b Fig. 6 – Interfacial post-failure behaviour: a) tensile response and b) shear response. This model is, in general, not restricted to the monotonic behaviour. The degradation of cohesion is an irreversible process and once the maximum degradation has been reached, the cohesive contribution to the tensile and shear stresses is zero, and the only contribution to the shear stresses is from the frictional term. The interface behaviour is based on large displacements. In particular, the finite-sliding tracking approach implemented in Abaqus [41], which allows for arbitrary separation, sliding, and rotation of the surfaces, is adopted. 3.1 Comparison between experimental and numerical results for small-scale masonry specimens Experimental tests conducted by van der Pluijm in [2, 43] on small-scale masonry specimens, composed of two bricks jointed together by a mortar joint, were used as reference to compare with numerical outcomes and to tune the brittleness parameters 𝜁 and 𝜉. As in [2, 43] the tensile and shear failures were only observed in the brick-mortar interfaces, linear elastic behaviour for brick and mortar has been assumed. The mechanical properties adopted in the numerical simulations are collected in Table 1. Fig. 7 shows the comparison between experimental and numerical results for small scale masonry specimens subjected to tension (Fig. 7a) and shear (Fig. 7b). The tensile properties of the interface are assumed to be consistent with the fracture energy of the brick- mortar interface in tension (Mode I), which in [2] is equal to 𝐺[𝐼]^𝑖𝑛𝑡= 12.0N/m. Indeed, once the tensile strength 𝑓𝑡 and the displacement 𝑢[𝑘] are fixed, which can be defined directly from the experimental envelope (Fig. 7a), the brittleness parameter 𝜁 is chosen so that the area under the curve in Fig. 6a equals 𝐺[𝐼]^𝑖𝑛𝑡. Analogously, the shear properties of the interface are assumed to be consistent with the Mode II-fracture energy of the brick-mortar interface, which, in [43], follows the relation 𝐺[𝐼𝐼]^𝑖𝑛𝑡 = 130𝜎 + 58N/m (with 𝜎 in MPa). In this case, tan 𝜙, 𝑐, 𝛿[𝑘], and 𝜇 are defined directly from the experimental outcomes [43], whereas the brittleness parameter 𝜉 is chosen to be the best approximation of 𝐺 [𝐼𝐼]^𝑖𝑛𝑡 for the three experimental curves in Fig. Finally, as can be observed in Fig. 7, the tensile (Fig. 7a) and shear (Fig. 7b) interfacial behaviours here proposed appear in good agreement with the experimental results obtained in [2, 43]. It has to be pointed out that the shear stiffness which can be read in Fig. 7b is given by the deformability of the 3D FEs (in this case mainly to the mortar FEs) and not by the deformability of the interfaces, which can be considered rigid- cohesive. 10 Table 1. Mechanical properties for small-scale masonry specimens. Mortar properties Interfacial properties Young’s modulus [MPa] 2970 Tensile behaviour Shear behaviour Poisson’s ratio [\] 0.15 𝑓𝑡 [MPa] 0.28 tan 𝜙 [\] 1.01 𝑢𝑘 [mm] 0.20 c [MPa] 0.87 Brick properties 𝜁 [\] 4.38 𝛿[𝑘] [mm] 0.4 Young’s modulus [MPa] 16700 𝜉 [\] 1.1 Poisson’s ratio [\] 0.15 𝜇 [\] 0.73 a b Fig. 7 – Comparison between experimental and numerical results for small-scale masonry specimens: a) tensile behaviour (experimental envelope (grey area) and numerical response (red line)) and b) shear behaviour (experimental envelopes (grey areas) and numerical responses (blue, green and orange lines) for three different levels of initial compression: 0.1, 0.5 and 1.0 MPa). 11 4 Brick and mortar nonlinear behaviour Tensile and compressive plastic-damage nonlinear behaviour is assumed for brick and mortar, based on the plastic-damage model developed by Lee and Fenves [47] for quasi-brittle materials. In the following, the main features of the model are recalled. Two independent scalar damage variables, one for the tensile regime (0 ≤ 𝑑[𝑡] < 1) and one for the compressive regime (0 ≤ 𝑑[𝑐] < 1), are supposed. Accordingly, the stress-strain relations under uniaxial tension, 𝜎[𝑡], and compression, 𝜎[𝑐], are: 𝜎[𝑡] = (1 − 𝑑[𝑡])𝐸[0](𝜀𝑡− 𝜀[𝑡]^𝑝), 𝜎[𝑐] = (1 − 𝑑[𝑐])𝐸[0](𝜀𝑐− 𝜀[𝑐]^𝑝), ( 9 ) where 𝐸[0] is the initial Young’s modulus of the material, 𝜀[𝑡] and 𝜀[𝑐] are the uniaxial tensile and compressive strains, and 𝜀[𝑡]^𝑝 and 𝜀[𝑐]^𝑝 are the uniaxial tensile and compressive plastic strains (Fig. 8). Particularly, the curves depicted in Fig. 8 represent the main input data of the model. Mesh objectivity in the softening branch passes through an indirect definition of the fracture energy, i.e. the model is local, and regularization occurs scaling the fracture energies by means of the equivalent length 𝑙[𝑒𝑞] = 𝛼[ℎ]√𝑉𝑒= 𝛼[ℎ](∑[𝜌=1]^𝑛^𝜌 ∑[𝜉=1]^𝑛^𝜉 ∑[𝜂=1]^𝑛^𝜂 det𝐽𝑤[𝜌]𝑤[𝜉]𝑤[𝜂]) where 𝑤[𝜌], 𝑤[𝜉] and 𝑤[𝜂] are the weight factors of the Gaussian integration scheme, 𝐽 the Jacobian of the transformation, 𝑉[𝑒] the element area and 𝛼[ℎ] a modification factor that depends on the typology of the finite element used. In this way, the mesh size does not significantly influence the material Additionally, to control the dilatancy in the quasi-brittle material response, a nonassociative flow rule is considered to define the plastic strain rate. It is obtained by a flow rule generated by a Drucker-Prager type plastic potential. In particular, it is defined by the dilatancy angle 𝜓, typically assumed equal to 10° in agreement with experimental evidences [48] and previous numerical models [49, 50], and a smoothing constant 𝜖 generally assumed equal to 0.1 [49]. As regard as the yield surface, a multiple-hardening Drucker-Prager type surface is assumed. It is characterized by the ratio 𝑓[𝑏0]/𝑓[𝑐0] between the biaxial initial compressive strength 𝑓[𝑏0] and the uniaxial initial compressive strength 𝑓[𝑐0] and a constant 𝜌, which represents the ratio of the second stress invariant on the tensile meridian to that on the compressive meridian at initial yield. Typically, 𝑓[𝑏0]/𝑓[𝑐0]= 1.16 and 𝜌 = 2/3 for quasi-brittle materials [51]. The general parameters adopted for quasi-brittle materials, such as brick and mortar, are collected in Table 2. Table 2. General parameters for quasi-brittle materials (brick and mortar). 𝜖 [\] 𝜓 [\] 𝑓[𝑏0]/𝑓[𝑐0] [\] 𝜌 [\] 0.1 10° 1.16 2/3 a b Fig. 8 – Plastic-damaging behaviour of brick and mortar: a) tensile and b) compression uniaxial nonlinear curves. 12 5 Numerical examples Experimental-numerical comparisons for the in-plane and out-of-plane behaviours of masonry panels are here provided to show the effectiveness and the accuracy of the model proposed. The detailed micro-model herein proposed has been implemented in Abaqus Standard [41]. Geometric nonlinearity is considered in all the analyses to account for large-displacement effects. Experimental tests conducted by Vermeltfoort and Raijmakers [52] and by Chee Liang [53] are considered for the in-plane and out-of-plane response of masonry panels, respectively. Mechanical properties utilized for the in-plane and out-of-plane benchmarks are collected in Table 3. When more than one value is given in the same cell of the table, the first value refers to the in-plane benchmark, whereas the second one refers to the out-of-plane benchmark. In general, the tensile response of masonry joints is defined in terms of the tensile strength and fracture energy in tension (Mode I), whereas the shear response of masonry joints is defined in terms of friction, cohesion, residual friction and Mode II-fracture energy. It appears clear that 𝑢[𝑘] and 𝜁 will be derived from the value of fracture energy in tension (Mode I), whereas 𝛿[𝑘] and 𝜉 will be derived from the value of Mode II-fracture energy. To this aim, the brittleness parameters 𝜁 and 𝜉 have been kept equal to the ones of Section 3.1, and the values 𝑢[𝑘] and 𝛿[𝑘] have been chosen so that the fracture energy values were satisfied. Reference to [54] has been made to define the uniaxial inelastic stress-strain relationships. The evolution of the degradation damage scalar variables 𝑑[𝑡] and 𝑑[𝑐] has been kept substantially proportional to the decay of the uniaxial stresses, as successfully experienced is several numerical campaigns [49, 16, 55]. Concerning the in-plane benchmark, the mechanical properties for brick, mortar and brick–mortar interfaces employed in the analyses (Table 3) were reported in previous research [19, 21, 30]. In addition, the tensile strength of mortar has been assumed with reference to the results on mortar prisms obtained in the experimental campaign carried out in the TU Delft laboratories in 1991 [2]. Concerning the out-of-plane benchmark, the material parameters used for the interfaces elements (Table 3) are equivalent to the values used in [30] for the same wall. The elastic stiffness of brick and mortar were not investigated by Chee Liang [53]. Therefore, the Young’s modulus of mortar has been assumed according to [54], whereas the Young’s modulus of brick has been kept the same as that shown in [2], being the materials of the same type. The other properties are the same to the in-plane benchmark. 13 Table 3. Mechanical properties utilized for the in-plane and out-of-plane benchmarks. When more than one value is given in the same cell, the first value refers to the in-plane benchmark, whereas the second one refers to the out-of-plane Interfacial mechanical properties Tensile behaviour Shear behaviour 𝑓𝑡 [MPa] 0.20, 0.12 tan 𝜙 [\] 0.75, 0.58 𝑢[𝑘] [mm] 0.36 c [MPa] 0.22 𝜁 [\] 4.38 𝛿𝑘 [mm] 0.4 𝜉 [\] 1.1 𝜇 [\] 0.75, 0.58 Mortar mechanical properties Young’s modulus [MPa] 850, 2300 Poisson’s ratio [\] 0.15 Tensile uniaxial nonlinear behaviour Compressive uniaxial nonlinear behaviour Stress [MPa] Inelastic strain 𝑑[𝑡] [\] Stress [MPa] Inelastic strain 𝑑[𝑐] [\] 1.5 0 0 7.8 0 0 0.1 0.002 0.95 8.2 0.002 0 0.4 0.015 0.95 Brick mechanical properties Young’s modulus [MPa] 16700 Poisson’s ratio 0.15 Tensile uniaxial nonlinear behaviour Compressive uniaxial nonlinear behaviour Stress [MPa] Inelastic strain 𝑑[𝑡] [\] Stress [MPa] Inelastic strain 𝑑[𝑐] [\] 3.5 0 0 11.0 0 0 0.3 0.002 0.95 11.5 0.001 0 0.6 0.007 0.95 5.1 In-plane response Results obtained by Vermeltfoort and Raijmakers [52] in shear tests on single-leaf panels are here considered. The identical wall specimens, named J4D, J5D and J7D in [52], with a length (990 mm) to height (1000 mm) ratio of approximately 1 were considered (Fig. 9). They are characterized by 18 brick layers of which 2 were fixed to steel beams so as to keep the top and bottom edges of the element straight during the test (green zones in Fig. 9a). Each brick is 204mm×98mm×50mm, whereas the bed and head mortar joints are 12.5mm thick. Particularly, the masonry panels were initially preloaded with a vertical top pressure, Pv=0.3MPa for J4D and J5D and Pv=2.12MPa for J7D. Then a horizontal load was then applied in the plane of the walls at the top edge under displacement control up to collapse, see Fig. 9a. During the tests, first, horizontal cracks appeared at the top and bottom of the walls. Then, cracks started to develop diagonally along the bed and head mortar joints and through the bricks, up to failure. The experimental response was characterized by a softening branch that started when diagonal cracks appeared in the centre of the specimens. The wall is modelled here using the detailed micro-modelling approach presented in the previous sections. The analyses followed the two-step boundary conditions depicted in Fig. 9a. The assembly of textured units employed in the numerical model is highlighted in Fig. 9b. a b Fig. 9 – In-plane response of masonry wall panels [52]: a) boundary conditions and b) assembly of textured units employed in the numerical model. Fig. 10 provides experimental-numerical comparisons: the experimental load-displacement curves for J4D, J5D and J7D walls are compared with the numerical results carried out using a textured unit mesh composed of 20 hexahedral 8-nodes FEs. In this figure, the numerical predictions reported by Lourenço & Rots [21] and by Macorini & Izzuddin [30] are also shown. A good agreement between experimental and numerical results can be observed up to collapse, including initial stiffness, maximum capacity and the post-peak response of the panels. Also, the predictions of the proposed modelling approach are generally close to those reported in [21, 30] for all the considered walls, with the current predictions of the post-peak response for wall J7D better than the one obtained in The discretization of the textured units is explicitly chosen by the user. The role of the mesh size is shown in Fig. 11a, in which the influence of mesh refinement on the load-displacement curves is collected. The results obtained using a textured unit mesh consisting of 20 hexahedral 8-nodes FEs (coarse mesh) and a textured unit mesh consisting of 108 hexahedral 8-nodes FEs (fine mesh) are compared. As can be noted, very small discrepancies emerged. Thereby, mesh dependency appears negligible, also thanks to the regularization of the fracture energy in the continuum plastic-damage model. This aspect is particularly appealing as the analyses with the coarse mesh presented a computational cost considerably smaller than the fine mesh. Fig. 11b shows the influence of the nonlinear behaviour of textured units on the load-displacement curves. As can be noted, the fact of accounting for the cracking and crushing of textured units significantly affects the post-peak behaviour (Fig. 11b), whereas the hypothesis of linear elastic textured units slightly overestimates the peak load. Basically, it is expected that the differences in considering or not the nonlinear behaviour of textured units would increase by increasing the vertical pressure as well as the interlocking of the masonry texture (e.g. for multi-leaf walls). Finally, Fig. 12 shows the deformed shape and crack pattern in the masonry wall panel obtained from the numerical model, in terms of tensile damage contour plot (Fig. 12a), compressive damage contour plot (Fig. 12b), and interfaces which exhibited failure (Fig. 12c). Also, numerical results are compared with the experimental crack pattern experienced in [52] (Fig. 12d). As can be noted in Figure 12, these predictions are in good agreement with the actual crack pattern. Particularly, the interfaces which exhibited failure are placed along the panel diagonal. Furthermore, few textured units experienced tensile failure in the central part of this diagonal (Fig. 12a), representing brick and mortar cracking. In addition, few textured units also showed crushing in the two extremities of the diagonal (Fig. 12b). These features have also been experienced by the experimental tests [52], see for example (Fig. 12d), confirming the good accuracy of the model proposed. Finally, these predictions are also in good agreement with the main crack paths and with the numerical results reported in [21, 30]. Fig. 10 – Experimental – numerical comparisons of the load – displacement curves for the masonry wall panels loaded in plane. a b Fig. 11 – Load – displacement curves for Pv=2.12MPa: a) investigation of the mesh dependency and b) influence of the nonlinear behaviour of the textured units. a b c d Fig. 12 – Comparison of the panel’s crack pattern: a) tensile damage contour plot, b) compressive damage contour plot, c) interfaces which exhibited failure and d) experimental crack pattern for the specimen with Pv=2.12MPa (J7D in [52]). 5.2 Out-of-plane response Numerical analyses are also carried out to assess the effectiveness of the detailed micro-modelling approach developed to investigate the out-of-plane behaviour of masonry panels. Comparisons are carried out against the experiments performed by Chee Liang [53]. The out-of-plane behaviour of a solid wall, simply supported along its four edges and subjected to bi-axial bending, is considered, and reference is made to experiments on two identical specimens: wall 8 and wall 12 in [53]. The single-leaf masonry wall panels were 1190mm high, 795mm wide and 53mm thick. The dimensions of the brick were 112mm×53mm×36mm and the thickness of the mortar joints were 10mm. The two specimens were loaded up to collapse by applying a uniform out-of-plane pressure through an air-bag sandwiched between the wall and a stiff reacting frame. Another stiff steel frame was connected to the wall on the other side, so as to prevent out-of-plane displacements and provide fixed supports along the four edges. The crack pattern experienced by the two wall specimens [53] is shown in Fig. 13. 17 To compute the solution up to the collapse of the panel (also in case of softening), a quasi-static direct- integration dynamic analysis procedure has been adopted [41]. This algorithm permits to study quasi-static responses in which inertia effects are introduced primarily to regularize unstable behaviours. The Authors experienced a better performance of this algorithm, specifically in the softening regime, with respect to more common arc length procedures. Fig. 14 provides the numerical-experimental comparisons in terms of lateral pressure-transversal displacement curves, where the textured unit mesh composed of 20 hexahedral 8-nodes FEs, shown in Fig. 11a, has been implemented. Although the through-thickness discretization may play a certain role, especially in the out-of-plane analysis of multi-leaf walls [56], the utilization of two 8-nodes hexahedral FEs through- thickness appears sufficiently accurate for the case under study. The experimental results reported in [53] consist of a partial load-displacement curve for wall 8 and the maximum capacity for the walls 8 and 12. Good agreement between the numerical and experimental results can be observed. The maximum lateral pressure obtained with the proposed model appears very close to the experimental capacity [53], to the collapse pressure determined in [57] through a 3D limit analysis approach and to the numerical curve obtained in [30]. Particularly, the curve obtained with the proposed approach very well fits the partial load-displacement curve for wall 8. Additionally, Fig. 15 provides the comparison between the experimental and numerical out-of-plane deflections at the instant, shown in Fig. 14 by means of a green point and a magenta point, with lateral pressure equal to 20 kN/m^2, i.e. at an instant slightly prior to failure. Here again, a good numerical-experimental agreement is achieved in terms of out-of-plane deflections. Finally, Fig. 16 shows the crack pattern obtained by means of the proposed model, in terms of deformed shape at collapse (Fig. 16a), out-of-plane displacement contour plot (Fig. 16b), tensile damage contour plot (Fig. 16c) and compressive damage contour plot (Fig. 16d). By comparing the numerical crack pattern of Fig. 16 with the experimental one (Fig. 13), it can be noted that the actual failure mechanism, although slightly different in the two walls, is qualitatively represented by the numerical model proposed. Particularly, the large vertical crack that runs in the middle of the panel crossing head mortar joints and bricks as well as the diagonal cracks observed in the tests are well represented. Indeed, as can be noted in Fig. 16c, tensile damage is experienced in the central part of the textured units which are placed in the central vertical part of the wall, in agreement with the actual vertical cracks experienced by both walls (Fig. 13) which alternatively crosses the bricks. For the sake of comparison, the crack pattern obtained by numerical models consolidated in the scientific community [57, 30] is reported in Fig. 17. As can be noted, the crack pattern computed by the model here proposed (Fig. 16), is in good agreement with the ones depicted in Fig. 17. a b Fig. 13 – Experimental crack pattern: a) photos of the failure of Wall 8 and Wall 12 from [53] and b) sketch of the crack pattern of Wall 12. 18 Fig. 14 –Comparison of the lateral pressure – out-of-plane displacement curves. Fig. 15 – Comparison between experimental and numerical out-of-plane deflections when the lateral pressure is equal to 20 kN/m^2, see the green and magenta points in Fig. 14. a b c d Fig. 16 – Crack pattern obtained from the proposed model: a) deformed shape, b) out-of-plane displacements contour plot and c) tensile and d) compressive damage contour plots at the end of the a b Fig. 17 – Crack pattern obtained by consolidated numerical models: a) Milani (2008) [57] and b) Macorini and Izzuddin (2011) [30]. 6 Conclusions In this paper, a novel numerical approach to model masonry has been proposed. The 3D detailed micro- model presented consists of the coupling of contact-based rigid-cohesive interfaces with 3D nonlinear- damaging textured units (which explicitly account for the mortar layers), which is a novelty in the scientific literature. This novel modelling approach can, in fact, be fully characterized by the properties obtained on small-scale specimen tests on brick and mortar (stiffness, compressive and tensile responses) and on small masonry assemblages (tensile and shear responses of the mortar-brick bond). According to the modelling approach proposed, masonry is represented by textured units consisting of one brick and few mortar layers composed of 3D solid FEs obeying to plastic-damage constitutive laws. This permits to represent the brick and mortar mechanical behaviour when cracking and/or crushing occur. Textured units are assembled, accounting for any actual 3D through-thickness arrangement of masonry (including walls with openings, multi-leaf walls, etc.), by means of zero-thickness cohesive-frictional interfaces based on the contact penalty method. This permits to account for the brick-mortar bond failures both in tension and shear. To reach this goal, this paper introduced an interface model. Indeed, the interface behaviour assumed in the 3D detailed micro-model is governed by an ad-hoc modification of the standard surface-based contact behaviour implemented in Abaqus. Contextually, an automatic subroutine ad-hoc written by the authors has been implemented to reproduce a Mohr-Coulomb failure surface with tension cut-off. The interfacial behaviour appeared to be consistent with experimental outcomes on small-scale masonry specimens. The results of numerical analyses carried out to investigate both the in-plane and the out-of-plane responses of brick-masonry panels up to collapse has been presented and compared with experimental outcomes. From this comparison, it was shown that the use of the proposed modelling approach allows the accurate representation of the masonry behaviour both in the in-plane and out-of-plane responses. The results achieved demonstrate the significant potential of the proposed Additionally, although this model accounts for a very detailed description of masonry constituents and is characterized by a larger complexity with respect to existing numerical models, its computational demand appears reasonably acceptable. Indeed, as shown in Table 4, the computational time needed in the simulations are, after all, moderate. Even, the 3D detailed micro-model proposed appears faster than other more standard 2D micro-modelling approaches, see in [19] the time needed for the same in-plane benchmark, based on well- known interface elements [21]. Therefore, the contact-based formulation proposed appears preliminarily efficient. The Authors are currently testing this model on large-scale masonry structures using parallelization techniques to reduce the computational time. From the first attempts, standard workstations appear sufficient to supply this task. 21 Table 4. Times required to conduct the analyses. Simulation Time required^(x) (hh:mm:ss) In-plane coarse mesh (Pv=0.30MPa) 00:06:33 In-plane coarse mesh (Pv=2.12MPa) 00:07:18 In-plane fine mesh (Pv=2.12MPa) 00:23:20 Out-of-plane 00:09:11 (x) utilizing a commercial laptop equipped with a processor Intel®Core™ i7-6500U CPU @ 2.50GHz and 16GB RAM. Finally, considering the accuracy of the model proposed, its application to simulate the behaviour of masonry panels under certain loading conditions can be used to help laboratory experimenters in designing new or optimizing experimental set-ups, in predicting the crack pattern, the maximum load and the ultimate displacement of scheduled tests. Mauro Parodi, Massimo Damasio and Claudio Cavallero (http://www.exemplar.com/) are gratefully acknowledged for their technical support. Financial support by the Italian Ministry of Education, Universities and Research MIUR is gratefully acknowledged (PRIN2015 “Advanced mechanical modeling of new materials and structures for the solution of 2020 Horizon challenges” prot. 2015JW9NJT_018). 22 REFERENCES [1] A. W. Hendry, Structural Masonry, UK: Palgrave MacMillan, 1998. [2] R. van der Pluijm, “Material properties of masonry and its components under tension and shear,” in 6th Canadian Masonry Symposium, 15-17 June 1992, Saskatoon, Canada, 1992. [3] C. Mazzotti, E. Sassoni and G. Pagliai, “Determination of Shear Strength of Historic Masonries by Moderately Destructive Testing of Masonry Cores,” Construction and Building Materials, vol. 54, p. 421–431, 2014. [4] A. Formisano and A. Marzo, “Simplified and Refined Methods for Seismic Vulnerability Assessment and Retrofitting of an Italian Cultural Heritage Masonry Building,” Computers & Structures, vol. 180, p. 13–26, 2017. [5] E. Sacco, D. Addessi and K. 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Royer-Carfagni and A. Tasora, “A Regularized Non-Smooth Contact Dynamics Approach for Architectural Masonry Structures,” Computers & Structures, vol. 187, p. 88_100, 2017. [39] D. Baraldi and A. Cecchi, “Discrete and continuous models for static and modal analysis of out of plane loaded masonry,” Computers & Structures, 2017. [40] T. Bui, A. Limam, V. Sarhosis and M. Hjiaj., “Discrete Element Modelling of the in-Plane and Out-of-Plane Behaviour of Dry-Joint Masonry Wall Constructions,” Engineering Structures, vol. 136, p. 277–294, 2017. [41] Abaqus®. Theory manual, version 6.14, 2014. [42] R. Weyler, J. Oliver, T. Sain and J. Cante, “On the Contact Domain Method: A Comparison of Penalty and Lagrange Multiplier Implementations,” Computer Methods in Applied Mechanics and Engineering, vol. 205–208, p. 68–82, 2012. [43] R. van der Pluijm, “Shear behaviour of bed joints,” in 6th North American Masonry Conference, 6-9 June 1993, Philadelphia, Pennysylvania, USA, 1993. [44] R. Serpieri, M. Albarella and E. Sacco, “A 3D Microstructured Cohesive–frictional Interface Model and Its Rational Calibration for the Analysis of Masonry Panels,” International Journal of Solids and Structures, vol. 122–123, p. 110–127, 2017. 25 [45] M. Godio, I. Stefanou and K. Sab, “Effects of the dilatancy of joints and of the size of the building blocks on the mechanical behavior of masonry structures,” Meccanica, 2017. [46] R. van der Pluijm, H. Rutten and M. Ceelen, “Shear behaviour of bed joints,” in Proceedings of the Twelfth International Brick/Block Masonry Conference, 2000. [47] J. Lee and G. L. Fenves, “Plastic-Damage Model for Cyclic Loading of Concrete Structures,” Journal of Engineering Mechanics, vol. 124, no. 8, p. 892–900, 1998. [48] A. Mirmiran and M. Shahawy, “Dilation characteristics of confined concrete,” Mechanics of Cohesive-frictional Materials, vol. 2, no. 3, p. 237–249, 1997. [49] G. Milani, M. Valente and C. Alessandri, “The Narthex of the Church of the Nativity in Bethlehem: A Non-Linear Finite Element Approach to Predict the Structural Damage,” Computers & Structures, 2017. [50] G. Castellazzi, A. M. D’Altri, S. de Miranda and F. Ubertini, “An Innovative Numerical Modeling Strategy for the Structural Analysis of Historical Monumental Buildings,” Engineering Structures, vol. 132, p. 229–248, 2017. [51] J. Lubliner, J. Oliver, S. Oller and E. Oñate, “A Plastic-Damage Model for Concrete,” International Journal of Solids and Structures, vol. 25, no. 3, p. 299–326, 1989. [52] A. T. Vermeltfoort and T. M. J. Raijmakers, “Deformation controlled meso shear tests on masonry piers - part 2,” Draft Report, Department of BKO, TU Eindhoven, 1993. [53] N. Chee Liang, “Experimental and theoretical investigation of the behavior of brickwork cladding panel subjected to lateral loading,” Ph.D. thesis University of Edinburgh, 1996. [54] H. B. Kaushik, D. C. Rai and S. K. Jain, “Stress-Strain Characteristics of Clay Brick Masonry Under Uniaxial Compression,” Journal of Materials in Civil Engineering, vol. 19, no. 9, p. 728–739, [55] A. M. D’Altri, G. Castellazzi, S. de Miranda and A. Tralli, “Seismic-Induced Damage in Historical Masonry Vaults: A Case-Study in the 2012 Emilia Earthquake-Stricken Area,” Journal of Building Engineering, vol. 13, p. 224–243, 2017. [56] S. Casolo and G. Milani, “Simplified out-of-plane modelling of three-leaf masonry walls accounting for the material texture,” Construction and Building Materials, vol. 40, p. 330–351, 2013. [57] G. Milani, “3D Upper Bound Limit Analysis of Multi-Leaf Masonry Walls,” International Journal of Mechanical Sciences, vol. 50, no. 4, p. 817–836, 2008.	{"url":"https://123dok.org/document/y8gr9j42-final-peer-reviewed-accepted-manuscript.html","timestamp":"2024-11-03T10:39:32Z","content_type":"text/html","content_length":"209120","record_id":"<urn:uuid:a56be477-06f4-4d46-9b93-811a2fda60a1>","cc-path":"CC-MAIN-2024-46/segments/1730477027774.6/warc/CC-MAIN-20241103083929-20241103113929-00078.warc.gz"}
Choosing Fixed-Effects, Random-Effects or Pooled OLS Models in Panel Data Analysis using Stata Choosing Fixed-Effects, Random-Effects or Pooled OLS Models in Panel Data Analysis using Stata This article introduces the practical process of choosing Fixed-Effects, Random-Effects or Pooled OLS Models in Panel data analysis. We will show you how to perform step by step on our panel data, from which we published the results in our article on Sustainability review in 2019 (see Nguyen Hoang Viet, Phan Thanh Tu and Lobo Antonio, 2019). You can see the theoretical difference of regression models with Panel data (fixed-effects, random-effects, and pooled OLS) in the previous article. Research sample Our panel data used in this article, that you can download here in Stata datasheet or Excel data, includes 434 year-observations of 62 provinces as entities of our sample; each province has 7 year-observations. These data were collected from the statistical yearbooks of Vietnam’s provinces during the period from 2010 to 2016; then cleaned by eliminating some missing-data provinces and In this sample, “id” represents the entities as Vietnam provinces that we code them in number; and “year” represents the time variable (t). Note that you should make attention for assuring that all data of one thing such as one entity are coded exactly the same. If not, Stata will count as another thing or ignore it. The objective of our research aims to study the relationship between foreign direct investment (FDI) and sustainability at provincial level in a developing host country as Vietnam for the period between the years of 2010 and 2016. Here are the variables of our research; in which Dependent variable is the adjusted net savings that assess the sustainable development of Vietnam provinces. In total, we have 11 independent variables that are distinguished in three groups, including: 3 variables associated with the FDI inflow stocks; 5 variables associated with the employment in FDI sector, and 3 variables associated with the performance of FDI in provinces. And, 2 control variables are size and economic growth of province. Practical regression process Now, we apply the process of selecting the regression model for panel data (between Pooled OLS Model, Random-Effects Model and Fixed-Effects Model) of Dougherty (2011) for our panel data of Vietnam provinces in period from 2010 to 2016. Source: Dougherty (2011, p.421) For the first step, our sample can be considered as random sample because of our choice in time-span and FDI sector at provincial level in Vietnam. So, we go into the second step of the Process of choosing regression model for panel data, in which we perform both fixed effects and random effects regressions by using Stata. The Stata command to run fixed/random effects is xtreg. Before using xtreg you need to set Stata to handle panel data by using the command xtset. Type: xtset Id Year, yearly. Note that Stata distinguishes capital letters, so you must type exactly the variable name. Or you can click this command on the Stata’s Menu by avoiding typing errors. In this case, “Id” represents the entities that is Vietnam provinces; and “Year” represents the time variable t. As the panel data has been handled, we can now run the fixed-effects model by using the Stata command xtreg with dependent variable ANS and 13 variables, including 11 independent ones and 2 control variables in our panel data. Type: xtreg ANS FDIENT FDICAP FDIAST FDIEMP FDICOM FDIWAG FDITUR FDIGDP FDIROTC FDIROFA FDIROS Size GDPgrowth, fe. Or you can click this command on the Stata’s Menu by avoiding typing errors. Note that the option fe should be chosen for the fixed-effects model. To compare the results with random-effects model that will be performed later; we must now store the results with fixed-effects regression by using the command “estimates store fixed”. Then, we run the random-effects model by using the Stata command xtreg with the same variables by choosing the option re Type: xtreg ANS FDIENT FDICAP FDIAST FDIEMP FDICOM FDIWAG FDITUR FDIGDP FDIROTC FDIROFA FDIROS Size GDPgrowth, re. Or you can click this command on the Stata’s Menu by avoiding typing errors. Also, we save the estimates by using the command “estimates store random”. For comparing fixed and random-effects models, we perform now the Hausman test by typing” hausman fixed random The negative sign can arise if different estimates of the error variance are used in forming variance of b and variance of capital B. In that case, we need to use the sigmamore option, which specifies that both covariance matrices are based on the (same) estimated disturbance variance from the efficient estimator. Type: hausman fixed random, sigmamore By focusing on the DWH test, we determine whether there are significant differences in the coefficients. This significant Hausman test allow us to accept the null hypothesis by indicating that the Fixed-effects model is appropriate. By caution, it is necessary to test the presence of random effects by using Breusch-Pagan Lagrange multiplier. We can see that the result of this test is significant by indicating random effects and refusing the Pooled OLS model. As the Hausman test has eliminated the random-effects model; and Lagrange multiplier has refused the Pooled OLS model. We select with confidence now Fixed-effects one. We must check the Heteroskedasticity test for the selected Fixed-effects model by using the command xttest3. This is a user-written program, to install it type: ssc install xtest3 Because, Stata stored recently the results of random-effects model; we rerun the fixed-effects regression; then run Heteroskedasticity test. The null is homoskedasticity. And our significant test rejects the null and indicates that our Fixed-effects model has a heteroskedasticity problem. Hence, we use the option robust to correct for this regression model by typing the xtreg with 2 options such as robust and fe. Finally, the robust Fixed-effects model is used for assessing the proposed research hypotheses in our research. Please see the detail results and analysis in our article Nguyen Hoang Viet, Phan Thanh Tu and Lobo Antonio (2019) Other tests / diagnostics Testing for time-fixed effects To see if time fixed effects are needed when running a FE model use the command testparm. It is a joint test to see if the dummies for all years are equal to 0, if they are then no time fixed effects are needed (type help testparm for more details). After running the fixed effect model, type: testparm i.year NOTE: If using Stata 10 or older type: xi: xtreg y x1 i.year, fe testparm _Iyear* Testing for cross-sectional dependence/contemporaneous correlation: using Breusch-Pagan LM test of independence According to Baltagi, cross-sectional dependence is a problem in macro panels with long time series (over 20-30 years). This is not much of a problem in micro panels (few years and large number of The null hypothesis in the B-P/LM test of independence is that residuals across entities are not correlated. The command to run this test is xttest2 (run it after xtreg, fe): xtreg y x1, fe Type xttest2 for more info. If not available try installing it by typing ssc install xttest2 Testing for cross-sectional dependence/contemporaneous correlation: Using Pasaran CD test As mentioned in the previous slide, cross-sectional dependence is more of an issue in macro panels with long time series (over 20-30 years) than in micro panels. Pasaran CD (cross-sectional dependence) test is used to test whether the residuals are correlated across entities*. Cross-sectional dependence can lead to bias in tests results (also called contemporaneous correlation). The null hypothesis is that residuals are not correlated. The command for the test is xtcsd, you have to install it typing ssc install xtcsd xtreg y x1, fe xtcsd, pesaran abs Had cross-sectional dependence be present Hoechle suggests to use Driscoll and Kraay standard errors using the command xtscc (install it by typing ssc install xtscc). Type help xtscc for more Testing for heteroskedasticity A test for heteroskedasticiy is avalable for the fixed- effects model using the command xttest3. This is a user-written program, to install it type: ssc install xtest3 The null is homoskedasticity (or constant variance). Above we reject the null and conclude heteroskedasticity. Type help xttest3 for more details. NOTE: Use the option ‘robust’ to obtain heteroskedasticity-robust standard errors (also known as Huber/White or sandwich estimators). Testing for serial correlation Serial correlation tests apply to macro panels with long time series (over 20-30 years). Not a problem in micro panels (with very few years). Serial correlation causes the standard errors of the coefficients to be smaller than they actually are and higher R-squared. A Lagram-Multiplier test for serial correlation is available using the command xtserial. This is a user-written program, to install it type ssc install xtserial xtserial y x1 The null is no serial correlation. Above we fail to reject the null and conclude the data does not have first-order autocorrelation. Type help xtserial for more details. Testing for unit roots/stationarity Stata 11 has a series of unit root tests using the command xtunitroot, it included the following series of tests (type help xtunitroot for more info on how to run the tests): “xtunitroot performs a variety of tests for unit roots (or stationarity) in panel datasets. The Levin-Lin-Chu (2002), Harris-Tzavalis (1999), Breitung (2000; Breitung and Das 2005), Im-Pesaran-Shin (2003), and Fisher-type (Choi 2001) tests have as the null hypothesis that all the panels contain a unit root. The Hadri (2000) Lagrange multiplier (LM) test has as the null hypothesis that all the panels are (trend) stationary. The top of the output for each test makes explicit the null and alternative hypotheses. Options allow you to include panel-specific means (fixed effects) and time trends in the model of the data-generating process” [Source: type help xtunitroot] If Stata does not have this command but can run user-written programs to run the same tests. You will have to find them and install them in your Stata program (remember, these are only for Stata 9.2/ 10). To find the add-ons type: findit panel unit root test A window will pop-up, find the desired test, click on the blue link, then click where it says “(click here to install)” For more info, please see Torres-Reyna (2007). 3 thoughts on “Choosing Fixed-Effects, Random-Effects or Pooled OLS Models in Panel Data Analysis using Stata” Fernanda Cusic says: Good post but I was wondering if you could write a litte more on this topic? I’d be very grateful if you could elaborate a little bit further. Thanks! zoritoler imol says: Somebody essentially assist to make critically posts I would state. This is the first time I frequented your web page and so far? I surprised with the research you made to make this actual publish extraordinary. Fantastic job! Johnathan Lazio says: I was very pleased to find this web-site.I wanted to thanks for your time for this wonderful read!! I definitely enjoying every little bit of it and I have you bookmarked to check out new stuff you blog post.	{"url":"https://phantran.net/choosing-fixed-effects-random-effects-or-pooled-ols-models-in-panel-data-analysis-using-stata/","timestamp":"2024-11-10T19:18:25Z","content_type":"text/html","content_length":"149596","record_id":"<urn:uuid:d7d6df97-1aa7-44d8-8647-fdde35652b65>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.61/warc/CC-MAIN-20241110170046-20241110200046-00034.warc.gz"}
The n-Category Café June 11, 2016 How the Simplex is a Vector Space Posted by Tom Leinster It’s an underappreciated fact that the interior of every simplex $\Delta^n$ is a real vector space in a natural way. For instance, here’s the 2-simplex with twelve of its 1-dimensional linear subspaces drawn in: (That’s just a sketch. See below for an accurate diagram by Greg Egan.) In this post, I’ll explain what this vector space structure is and why everyone who’s ever taken a course on thermodynamics knows about it, at least partially, even if they don’t know they do. Posted at 6:30 PM UTC \| Followups (25)	{"url":"https://golem.ph.utexas.edu/category/2016/06/index.shtml","timestamp":"2024-11-06T04:26:15Z","content_type":"application/xhtml+xml","content_length":"51875","record_id":"<urn:uuid:4ba74fc6-8e57-4858-aac6-0005d3cc3f1d>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00759.warc.gz"}
One to One Functions - Graph, Examples \| Horizontal Line Test One to One Functions - Graph, Examples \| Horizontal Line Test What is a One to One Function? A one-to-one function is a mathematical function whereby each input correlates to just one output. So, for each x, there is a single y and vice versa. This signifies that the graph of a one-to-one function will never intersect. The input value in a one-to-one function is known as the domain of the function, and the output value is noted as the range of the function. Let's study the examples below: For f(x), any value in the left circle corresponds to a unique value in the right circle. In conjunction, each value on the right corresponds to a unique value in the left circle. In mathematical words, this implies every domain has a unique range, and every range owns a unique domain. Hence, this is an example of a one-to-one function. Here are some different examples of one-to-one functions: Now let's examine the second example, which displays the values for g(x). Be aware of the fact that the inputs in the left circle (domain) do not hold unique outputs in the right circle (range). Case in point, the inputs -2 and 2 have the same output, i.e., 4. In the same manner, the inputs -4 and 4 have the same output, i.e., 16. We can see that there are matching Y values for numerous X values. Hence, this is not a one-to-one function. Here are different representations of non one-to-one functions: What are the characteristics of One to One Functions? One-to-one functions have the following qualities: • The function owns an inverse. • The graph of the function is a line that does not intersect itself. • The function passes the horizontal line test. • The graph of a function and its inverse are equivalent regarding the line y = x. How to Graph a One to One Function In order to graph a one-to-one function, you will have to determine the domain and range for the function. Let's look at a straight-forward example of a function f(x) = x + 1. Once you have the domain and the range for the function, you need to plot the domain values on the X-axis and range values on the Y-axis. How can you evaluate whether a Function is One to One? To indicate if a function is one-to-one, we can leverage the horizontal line test. As soon as you plot the graph of a function, trace horizontal lines over the graph. If a horizontal line moves through the graph of the function at more than one place, then the function is not one-to-one. Due to the fact that the graph of every linear function is a straight line, and a horizontal line will not intersect the graph at more than one spot, we can also conclude all linear functions are one-to-one functions. Keep in mind that we do not use the vertical line test for one-to-one functions. Let's study the graph for f(x) = x + 1. Once you plot the values for the x-coordinates and y-coordinates, you have to consider whether a horizontal line intersects the graph at more than one place. In this instance, the graph does not intersect any horizontal line more than once. This signifies that the function is a one-to-one function. On the contrary, if the function is not a one-to-one function, it will intersect the same horizontal line more than one time. Let's examine the graph for the f(y) = y^2. Here are the domain and the range values for the function: Here is the graph for the function: In this example, the graph crosses multiple horizontal lines. For example, for either domains -1 and 1, the range is 1. In the same manner, for each -2 and 2, the range is 4. This signifies that f(x) = x^2 is not a one-to-one function. What is the inverse of a One-to-One Function? Since a one-to-one function has only one input value for each output value, the inverse of a one-to-one function is also a one-to-one function. The inverse of the function basically reverses the For Instance, in the event of f(x) = x + 1, we add 1 to each value of x for the purpose of getting the output, or y. The inverse of this function will deduct 1 from each value of y. The inverse of the function is f−1. What are the qualities of the inverse of a One to One Function? The qualities of an inverse one-to-one function are the same as any other one-to-one functions. This signifies that the reverse of a one-to-one function will hold one domain for each range and pass the horizontal line test. How do you determine the inverse of a One-to-One Function? Finding the inverse of a function is simple. You simply need to swap the x and y values. Case in point, the inverse of the function f(x) = x + 5 is f-1(x) = x - 5. As we reviewed before, the inverse of a one-to-one function reverses the function. Because the original output value required adding 5 to each input value, the new output value will require us to deduct 5 from each input value. One to One Function Practice Questions Examine the subsequent functions: • f(x) = x + 1 • f(x) = 2x • f(x) = x2 • f(x) = 3x - 2 • f(x) = \|x\| • g(x) = 2x + 1 • h(x) = x/2 - 1 • j(x) = √x • k(x) = (x + 2)/(x - 2) • l(x) = 3√x • m(x) = 5 - x For every function: 1. Figure out if the function is one-to-one. 2. Graph the function and its inverse. 3. Figure out the inverse of the function algebraically. 4. Indicate the domain and range of every function and its inverse. 5. Use the inverse to solve for x in each calculation. Grade Potential Can Help You Master You Functions If you are having problems using one-to-one functions or similar topics, Grade Potential can put you in contact with a private teacher who can help. Our San Antonio math tutors are experienced educators who help students just like you improve their mastery of these concepts. With Grade Potential, you can learn at your individual pace from the comfort of your own home. Schedule a meeting with Grade Potential today by calling (210) 879-9877 to learn more about our tutoring services. One of our team members will contact you to better inquire about your requirements to find the best teacher for you! Let Grade Potential set you up with the perfect Grammar tutor! Or answer a few questions below to get started	{"url":"https://www.sanantonioinhometutors.com/blog/one-to-one-functions-graph-examples-horizontal-line-test","timestamp":"2024-11-15T04:05:40Z","content_type":"text/html","content_length":"80939","record_id":"<urn:uuid:6302a7ef-011c-4885-8629-716fdb27b341>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00320.warc.gz"}
Karush-Kuhn-Tucker conditions - (Calculus III) - Vocab, Definition, Explanations \| Fiveable Karush-Kuhn-Tucker conditions from class: Calculus III The Karush-Kuhn-Tucker (KKT) conditions are a set of mathematical conditions used in optimization problems to find the local maxima and minima of a function subject to equality and inequality constraints. These conditions extend the method of Lagrange multipliers, allowing for a more comprehensive approach when dealing with complex constraints, making them crucial in fields like economics, engineering, and operations research. congrats on reading the definition of Karush-Kuhn-Tucker conditions. now let's actually learn it. 5 Must Know Facts For Your Next Test 1. KKT conditions include both primal and dual feasibility, which ensure that the solution adheres to the constraints imposed on the optimization problem. 2. The KKT conditions are necessary for optimality in non-linear programming problems, especially when the objective function and constraints are differentiable. 3. In the context of inequality constraints, KKT introduces complementary slackness, which means that if a constraint is active (binding), its corresponding multiplier must be positive; if it is inactive, the multiplier must be zero. 4. The KKT framework is often used in machine learning for training models with constraints, such as Support Vector Machines (SVMs), where margin maximization is subject to classification 5. When analyzing the KKT conditions, it is important to check if the second-order sufficient conditions hold to guarantee that a point is a local minimum or maximum. Review Questions • Explain how the Karush-Kuhn-Tucker conditions extend the method of Lagrange multipliers and why this extension is important. □ The KKT conditions build upon the method of Lagrange multipliers by incorporating both equality and inequality constraints into optimization problems. While Lagrange multipliers effectively handle equality constraints, the KKT framework allows for a broader range of scenarios by introducing complementary slackness and ensuring primal and dual feasibility. This extension is crucial because many real-world optimization problems involve constraints that are not strictly equalities, making KKT a more versatile tool for finding optimal solutions. • Discuss the role of complementary slackness in the Karush-Kuhn-Tucker conditions and how it affects solution feasibility. □ Complementary slackness is a key aspect of the KKT conditions that relates the active constraints to their associated multipliers. It states that for each inequality constraint, either the constraint is binding (active) and the corresponding multiplier is positive, or the constraint is not binding (inactive) and the multiplier is zero. This relationship helps determine whether a potential solution adheres to the feasibility requirements of the optimization problem, ensuring that only those solutions that satisfy these conditions are considered optimal. • Evaluate how KKT conditions can be applied in machine learning contexts, specifically with respect to training algorithms like Support Vector Machines. □ KKT conditions play a pivotal role in machine learning algorithms such as Support Vector Machines (SVMs), where the goal is to maximize the margin between different classes while satisfying classification constraints. In SVMs, KKT provides the necessary criteria for determining optimal hyperplanes by combining both margin maximization (objective function) and misclassification penalties (constraints). By analyzing these conditions, practitioners can derive solutions that not only fit training data effectively but also generalize well to unseen data, enhancing model Â© 2024 Fiveable Inc. All rights reserved. APÂ® and SATÂ® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.	{"url":"https://library.fiveable.me/key-terms/calc-iii/karush-kuhn-tucker-conditions","timestamp":"2024-11-09T20:07:38Z","content_type":"text/html","content_length":"109953","record_id":"<urn:uuid:419c594a-93b4-45fc-ab19-ca6ca02aa57b>","cc-path":"CC-MAIN-2024-46/segments/1730477028142.18/warc/CC-MAIN-20241109182954-20241109212954-00433.warc.gz"}
How To Deal With Math Anxiety? \| Chilli Removals Help me do my homework- how often have you typed this on Google only to hire someone who can help you with math paper? Math has always been a dreaded subject for many students. Even the thoughts of mathematics can cause negative emotions such as fear of failure among students. Math anxiety isn’t just about disliking fractions or algebra. It is much more than that. So, let’s see how you can deal with this anxiety and focus on math papers stress-free. 1. Get a tutor Professors or teachers can change your perspective about math. The way you feel about this subject depends on how you have been taught. If the teacher loves math, she/he can convey that excitement or enthusiasm to the students. Tutors provide individual attention to each student. That means you can solve your mathematical problems in a less stressful environment. If you find it hard to solve your homework, you can ask for math homework help from college homework helper. 2. Positive reinforcement One of the major consequences of math anxiety is the fear of failure. You may assume that you would perform poorly in this paper, thereby not even attempting to solve the paper on your own. Instead of panicking, you should believe that you can excel at math. Positive reinforcement is very important if you want to deal with math anxiety. Consult with your professors to understand how many questions you got right in the paper. Focus on the correct answers instead of your mistakes. You can get online research paper help online to understand how to avoid mistakes in the next paper. 3. Reframe anxiety Most students try to hide their math anxiety from parents, teachers and even their friends. You should confront your fears to deal with them. Write down your worries on a sheet of paper before you start working on your math paper. Think critically to figure out why you feel that way about math. Talk to someone you trust about this issue. Consider tests and assignments nothing but a challenge. You will fetch higher grades, or you will learn something new. Take your time to deal with the anxiety. Till then, you can request your friends ‘please, help write my term paper.’ Math anxiety is real. It makes solving mathematics assignment almost impossible for many. Try to create positive emotions revolving around maths. Ask for help as and when required.	{"url":"https://www.chilliremovals.com.au/forum/general-discussions/how-to-deal-with-math-anxiety/dl-2b56be3a-741f-4cad-96a1-79a19d7cc012","timestamp":"2024-11-10T11:20:39Z","content_type":"text/html","content_length":"1050496","record_id":"<urn:uuid:8dd32452-5a54-431f-9be8-da8da2731c37>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00106.warc.gz"}
Estimating Unobserved Complementarities between Entrepreneurs and Venture Capitalists Matlab Code Jump to navigation Jump to search Main Project here: Estimating Unobserved Complementarities between Entrepreneurs and Venture Capitalists Important contractions • GMM: Generalized Method of Moments • MSM: Method of Simulated Moments • MLE: Maximum Likelihood Estimation • GA: Genetic Algorithm • SE: Standard Error How to run First, you need a gurobi license, which can be obtained (free) here. In master.m, edit the options section to reflect what you want the code to do. Then run it. Location/structure of the code The old codebase is located in The new codebase is located in The new-new codebase is located in Some of this code is just library code (prefix mtimesx). options section task: Can take the values {'data', 'monte', 'monte_data'}. estimator: Can take the values {'MSM'}. (other estimators removed in readjusted code) use_solver: Can take the values {'ga'}. (other solvers removed in readjusted code) error_type (currently hard coded as 1 and isn't fully written to support 2): 1 for match specific errors, with the error distribution following an exchangeable structure. var(e) = sig^2, and cov (e,e') = 1/4sig^2. 2 for agent specific errors, with the error structure of match <i, j> as sigei*ej. Does the majority of the work for this problem. Runs the GA, saves the results to 'empirics_match_specific_1st_stage_ga', then runs it again with different globals, and saves it to Constraints on GA. For [c,ceq] = nonlinearcons_msm(x), GA constrains x such that c ≤ 0 and ceq = 0. c and ceq are row vectors when there are multiple constraints. ceq is unused for our purposes. This is the fitness function. Takes a vector and returns a scalar. GA minimizes this function. Generates the moments needed for the GMM. Location/Structure of Data Pro tip: running the command whos -file filename from a matlab session will tell you the contents of any .mat file. There are two of both psdata and dyad_tech_mkt_data, corresponding to 4 industries or 5 industries. You can find them in E:\McNair\Projects\MatchingEntrepsToVC\OriginalCode\FVEIC4 data E:\McNair\Projects\MatchingEntrepsToVC\OriginalCode\FVEIC5 data The one wanted should be copied into the code directory: psdata.mat contains • vc: the number of VC's in each market, in the form of (10, 12, 13, ...): the first market has 10 VC's, the second market has 12 VC's.... • firm: the number of firms in each market, (13, 14, ...): the first market has 13 downstream, the second 14 downstream... this is a one (VC) to many (firms) market, and the number of firms in each market is at least as many as the number of firms • m_id: unused, I'm guessing the market id dyad_tech_mkt_data.mat variables used include: • pvc_exp_n, mean_pvc_exp_n, std_pvc_exp_n • lnfpat, mean_lnfpat: • mean_m_dist_1000, std_m_dist_1000: • mean_exp_sector, std_exp_sector: • m_match: contains the matching outcome: for each dyad (firm VC pair), 0 means not matched and 1 means matched. The variable asg in the code contains the matching outcomes of all markets. The matching outcomes are delineated by firms using the position data N1 (vc' from psdata)and N2 (firm' from psdata). Location/Structure of Output Simulated moments matrix dimension mismatch. This bug (also in the adjusted code) does not allow the "optimal stage" to complete. Assignment has more non-singleton rhs dimensions than non-singleton subscripts Error in gmm_2stage_estimation (line 65) EV(:, :, m) = EV(:, :, m) + temp2; Error in master (line 513) This is caused because temp2 is 166x166 in monte_data, but nm = M = 1, so EV is 1x1. nm is set to size(M0, 1), but M0 is sometimes a scalar? This is solved by commenting out mkt_resample.m line 107: % M0 = moments(M, 1, S, K, pv', asg, H, HK, G, FF, Hu, Hd, z1, z2, psa, ps1, ps2, N1, N2) because it returns a scalar, but should return a 166x1 vector. Better fix notes: The problem is narrowed down to the section of mkt_resample.m before the call to moments. Somehow the parameters to moments are changed such that it returns a scalar. Solution found (I think): monte_M cannot be set to 1. For values greater than 1, the program seems to work. It doesn't crash, anyway. Ran first iteration with task='monte', use_solver='ga', estimator='MSM', gurobi output disabled (took 42 sec to finish 1st iteration) Monte_data and monte run much faster than data, because data is so large. There is a lot of unused code. For example, monte_std and monte_x seem to only be assigned, not used.	{"url":"https://www.edegan.com/mediawiki/index.php?title=Estimating_Unobserved_Complementarities_between_Entrepreneurs_and_Venture_Capitalists_Matlab_Code&printable=yes","timestamp":"2024-11-15T04:18:20Z","content_type":"text/html","content_length":"31002","record_id":"<urn:uuid:f2f2f9c2-149e-46d9-8d32-ddca07938442>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00772.warc.gz"}
A Guide To Using The Degree Symbol Shortcut In Excel - ExcelAdept Key Takeaway: • The degree symbol is important in Excel for representing angles, temperatures, and other measurements. Its inclusion can help avoid confusion and errors in data analysis. • There are three main ways to insert the degree symbol shortcut in Excel: through keyboard shortcuts, using the Symbol tab, and creating a custom keyboard shortcut. Each method is useful for different situations and user preferences. • When using the degree symbol in Excel, it is important to be mindful of tips such as checking for consistent formatting, using the correct degree symbol for the measurement system being used, and ensuring that the degrees are accurately represented in formulas and calculations. Have you ever needed to include a degree symbol as part of a data table in Excel? Finding the symbol can be tricky, however, with this guide, you’ll learn how to quickly and easily create the degree symbol. Mastering the degree symbol shortcut can save you time, energy and frustration. The importance of the degree symbol in Excel Excel is a popular tool for managing data efficiently. When dealing with temperature readings or angles, the degree symbol becomes a crucial component of the dataset. The degree symbol shortcut in Excel allows for easy input of temperature readings and ensures that excel properly manages these values. The correct representation of angles is also important in excel formulas. It is vital to understand the importance of the degree symbol to avoid data inconsistencies and errors in your workbook. When dealing with temperature or angle readings, omitting the degree symbol in Excel can result in inaccurate data output. Excel uses the degree symbol as a reference point when managing these values. Accurately inputting the degree symbol will ensure that your formula or function represents the correct value. Furthermore, using the degree symbol shortcut in Excel saves time and minimizes the risk of data inconsistency. It is crucial to note that degree symbol shortcuts vary depending on the device or operating system being used. The degree symbol shortcut on a Mac is different from the shortcut on a Windows system. This variation necessitates every Excel user knowing the specific degree symbol shortcut for their device. Pro Tip: Using the degree symbol shortcut in excel makes managing temperature and angle values efficient, accurate, and reduces the risk of data inconsistencies. Ways to insert the degree symbol shortcut in Excel Ways to Quickly Add the Degree Symbol in Excel To quickly insert the degree symbol in Excel, follow these simple steps: 1. Place the cursor where you want to insert the degree symbol. 2. Press and hold the “Alt” key on your keyboard. 3. While still holding the “Alt” key, type “0176” using the numeric keypad. 4. Release the “Alt” key, and the degree symbol will appear. Alternatively, you can use the “Insert Symbol” function in Excel to find and insert the degree symbol. Aside from using the Alt code or the Insert Symbol function, another way to add the degree symbol is to create a custom shortcut key by using the “Symbol” tab in the “Insert” menu. It’s important to note that if you use the Alt code method, your keyboard must have a numeric keypad for it to work. To make this process even quicker, you can add the degree symbol to the “Quick Access Toolbar” in Excel. Simply right-click the degree symbol and select “Add to Quick Access Toolbar,” and it will be readily available for future use. Adding the degree symbol can help make your data more professional and precise. By using these quick and easy methods, you can easily add the degree symbol in your Excel spreadsheets. Tips for using the degree symbol in Excel Incorporating degree symbols in Excel may seem like a daunting task, but with the right tips, it can be accomplished efficiently. To help you out, here’s a guide on using degree symbol shortcut in 1. Step 1: Click on the cell where you want to insert the degree symbol. 2. Step 2: Press “Alt” and “0176” from the numeric keypad. This will add the degree symbol to the cell. 3. Step 3: For a bulk addition of degree symbols, select the data range and press “Ctrl+1.” 4. Step 4: From the format cells dialog box, select “Number” and then “Custom.” In the “Type” option, add the degree sign by the “%°” format code, click “Ok,” and you’re done. It’s worth noting that after entering the degree symbol in a cell, it becomes a text value, and you can use it for further calculations. It’s also important to double-check the data type format before proceeding to use any calculations for accuracy. Recently, a customer had an issue inputting degree symbols in their Excel sheet while preparing for an annual report. After trying various methods unsuccessfully, they got in touch with Excel support, who provided a similar guide and resolved the issue immediately. Some Facts About A Guide to Using the Degree Symbol Shortcut in Excel: • ✅ You can insert the degree symbol (°) in Excel using the shortcut key combination “Alt + 0176”. (Source: Excel Easy) • ✅ The degree symbol is commonly used to represent temperature, angles, and geographic coordinates. (Source: Techwalla) • ✅ You can also use the “Insert Symbol” feature in Excel to insert the degree symbol or other special characters. (Source: dummies) • ✅ The degree symbol shortcut works in both Windows and Mac versions of Excel. (Source: Business Insider) • ✅ Knowing how to insert the degree symbol can save time and improve the visual appeal of your Excel spreadsheets. (Source: Ablebits) FAQs about A Guide To Using The Degree Symbol Shortcut In Excel What is the degree symbol shortcut in Excel? The degree symbol is a symbol (°) used to represent degrees, which is often used when measuring temperature, angles, and other values. In Excel, there is a keyboard shortcut to insert the degree symbol into a cell. What is the keyboard shortcut to insert the degree symbol in Excel? To insert the degree symbol in Excel, you can use the keyboard shortcut “Alt + 0176” (without the quotes). This works on both Windows and Mac versions of Excel. Can I assign a different keyboard shortcut to insert the degree symbol in Excel? Yes, you can assign a different keyboard shortcut to insert the degree symbol in Excel. To do this, go to the “File” tab, click “Options”, click “Customize Ribbon”, then click “Customize” next to “Keyboard Shortcuts”. In the “Categories” list, select “Symbols”, then select “Degree”. Choose a new keyboard shortcut in the “Press new shortcut key” field, then click “Assign”. Can I insert the degree symbol using the Symbol dialog box in Excel? Yes, you can insert the degree symbol using the Symbol dialog box in Excel. To do this, click on the cell where you want to insert the degree symbol, go to the “Insert” tab, click on “Symbols” in the “Symbols” group, choose “More Symbols”, select the “Symbol” tab, choose “Latin-1 Supplement” from the “Subset” list, then select the degree symbol and click “Insert”. What are some common uses for the degree symbol in Excel? The degree symbol is often used in Excel to represent temperature values, such as the temperature of a room or of a cup of coffee. It can also be used to represent angles, such as the angles of a triangle or the rotation angle of an object. Other common uses include representing geographic coordinates, latitude and longitude, and wind direction. Can I use the degree symbol in Excel formulas? Yes, you can use the degree symbol in Excel formulas. Simply type the degree symbol into a cell or formula as you would any other character, and Excel will recognize it as the degree symbol. For example, you could enter “=C2+45°” to add 45 degrees to the value in cell C2.	{"url":"https://exceladept.com/a-guide-to-using-the-degree-symbol-shortcut-in-excel/","timestamp":"2024-11-07T06:26:16Z","content_type":"text/html","content_length":"65175","record_id":"<urn:uuid:8089f4cb-dd4f-4ea5-8b4c-76a2a428edb2>","cc-path":"CC-MAIN-2024-46/segments/1730477027957.23/warc/CC-MAIN-20241107052447-20241107082447-00602.warc.gz"}
Efficient quantum measurement of Pauli operators Title Efficient quantum measurement of Pauli operators Publication Journal Article Year of 2021 Authors Crawford, O, van Straaten, B, Wang, D, Parks, T, Campbell, E, Brierley, S Journal Quantum Volume 5 Start Page 385 Date 01/19/2021 Estimating the expectation value of an observable is a fundamental task in quantum computation. Unfortunately, it is often impossible to obtain such estimates directly, as the computer is restricted to measuring in a fixed computational basis. One common solution splits the observable into a weighted sum of Pauli operators and measures each separately, at the cost of many measurements. An improved version first groups mutually commuting Pauli operators together and then measures all operators within each group simultaneously. The effectiveness of this depends on two factors. First, to enable simultaneous measurement, circuits are required to rotate each group to the computational basis. In our work, we present two efficient circuit Abstract constructions that suitably rotate any group of k commuting n-qubit Pauli operators using at most kn−k(k+1)/2 and O(kn/logk) two-qubit gates respectively. Second, metrics that justifiably measure the effectiveness of a grouping are required. In our work, we propose two natural metrics that operate under the assumption that measurements are distributed optimally among groups. Motivated by our new metrics, we introduce SORTED INSERTION, a grouping strategy that is explicitly aware of the weighting of each Pauli operator in the observable. Our methods are numerically illustrated in the context of the Variational Quantum Eigensolver, where the observables in question are molecular Hamiltonians. As measured by our metrics, SORTED INSERTION outperforms four conventional greedy colouring algorithms that seek the minimum number of groups. URL https://arxiv.org/abs/1908.06942 DOI 10.22331/q-2021-01-20-385	{"url":"https://quics.umd.edu/publications/efficient-quantum-measurement-pauli-operators","timestamp":"2024-11-11T10:05:48Z","content_type":"text/html","content_length":"22444","record_id":"<urn:uuid:4e0c3e0e-c768-4690-9c6c-f35ea4f2d1e5>","cc-path":"CC-MAIN-2024-46/segments/1730477028228.41/warc/CC-MAIN-20241111091854-20241111121854-00864.warc.gz"}
: Magneti What is the direction of the magnetic force on a positive charge that moves as shown in each of the six cases shown in Figure 22.56? What is the direction of the magnetic force on a negative charge that moves as shown in each of the six cases shown in Figure 22.50? (a) At what speed will a proton move in a circular path of the same radius as the electron in Exercise 22.12? (b) What would the radius of the path be if the proton had the same speed as the electron? (c) What would the radius be if the proton had the same kinetic energy as the electron? (d) The same momentum? What Hall voltage is produced by a 0.200-T field applied across a 2.60-cm-diameter aorta when blood velocity is 60.0 cm/s? A nonmechanical water meter could utilize the Hall effect by applying a magnetic field across a metal pipe and measuring the Hall voltage produced. What is the average fluid velocity in a 3.00-cm-diameter pipe, if a 0.500-T field across it creates a 60.0-mV Hall voltage? Calculate the Hall voltage induced on a patient’s heart while being scanned by an MRI unit. Approximate the conducting path on the heart wall by a wire 7.50 cm long that moves at 10.0 cm/s perpendicular to a 1.50-T magnetic field. Show that the Hall voltage across wires made of the same material, carrying identical currents, and subjected to the same magnetic field is inversely proportional to their diameters. (Hint: Consider how drift velocity depends on wire diameter.) A patient with a pacemaker is mistakenly being scanned for an MRI image. A 10.0-cm-long section of pacemaker wire moves at a speed of 10.0 cm/s perpendicular to the MRI unit’s magnetic field and a 20.0-mV Hall voltage is induced. What is the magnetic field strength? What is the direction of the magnetic force on the current in each of the six cases in Figure 22.59? What force is exerted on the water in an MHD drive utilizing a 25.0-cm-diameter tube, if 100-A current is passed across the tube that is perpendicular to a 2.00-T magnetic field? (The relatively small size of this force indicates the need for very large currents and magnetic fields to make practical MHD drives.) A wire carrying a 30.0-A current passes between the poles of a strong magnet that is perpendicular to its field and experiences a 2.16-N force on the 4.00 cm of wire in the field. What is the average field strength? The force on the rectangular loop of wire in the magnetic field in Figure 22.56 can be used to measure field strength. The field is uniform, and the plane of the loop is perpendicular to the field. (a) What is the direction of the magnetic force on the loop? Justify the claim that the forces on the sides of the loop are equal and opposite, independent of how much of the loop is in the field and do not affect the net force on the loop. (b) If a current of 5.00 A is used, what is the force per tesla on the 20.0-cm-wide loop? (a) By how many percent is the torque of a motor decreased if its permanent magnets lose 5.0% of their strength? (b) How many percent would the current need to be increased to return the torque to original values? (a) The hot and neutral wires supplying DC power to a light-rail commuter train carry 800 A and are separated by 75.0 cm. What is the magnitude and direction of the force between 50.0 m of these wires? (b) Discuss the practical consequences of this force, if any. The force per meter between the two wires of a jumper cable being used to start a stalled car is 0.225 N/m. (a) What is the current in the wires, given they are separated by 2.00 cm? (b) Is the force attractive or repulsive? A 2.50-m segment of wire supplying current to the motor of a submerged submarine carries 1000 A and feels a 4.00-N repulsive force from a parallel wire 5.00 cm away. What is the direction and magnitude of the current in the other wire? An AC appliance cord has its hot and neutral wires separated by 3.00 mm and carries a 5.00-A current. (a) What is the average force per meter between the wires in the cord? (b) What is the maximum force per meter between the wires? (c) Are the forces attractive or repulsive? (d) Do appliance cords need any special design features to compensate for these forces? Figure 22.63 shows a long straight wire near a rectangular current loop. What is the direction and magnitude of the total force on the loop? Find the direction and magnitude of the force that each wire experiences in Figure 22.58(a) by, using vector addition. Find the direction and magnitude of the force that each wire experiences in Figure 22.64(b), using vector addition. Indicate whether the magnetic field created in each of the three situations shown in Figure 22.59 is into or out of the page on the left and right of the current. What are the directions of the fields in the center of the loop and coils shown in Figure 22.66? What are the directions of the currents in the loop and coils shown in Figure 22.61? Inside a motor, 30.0 A passes through a 250-turn circular loop that is 10.0 cm in radius. What is the magnetic field strength created at its center? Nonnuclear submarines use batteries for power when submerged. (a) Find the magnetic field 50.0 cm from a straight wire carrying 1200 A from the batteries to the drive mechanism of a submarine. (b) What is the field if the wires to and from the drive mechanism are side by side? (c) Discuss the effects this could have for a compass on the submarine that is not shielded. How strong is the magnetic field inside a solenoid with 10,000 turns per meter that carries 20.0 A? Measurements affect the system being measured, such as the current loop in Figure 22.62. (a) Estimate the field the loop creates by calculating the field at the center of a circular loop 20.0 cm in diameter carrying 5.00 A. (b) What is the smallest field strength this loop can be used to measure, if its field must alter the measured field by less than 0.0100%? Find the magnitude and direction of the magnetic field at the point equidistant from the wires in Figure 22.64(a), using the rules of vector addition to sum the contributions from each wire. Find the magnitude and direction of the magnetic field at the point equidistant from the wires in Figure 22.58(b), using the rules of vector addition to sum the contributions from each wire. What current is needed in the top wire in Figure 22.64(a) to produce a field of zero at the point equidistant from the wires, if the currents in the bottom two wires are both 10.0 A into the page? Calculate the size of the magnetic field 20 m below a high voltage power line. The line carries 450 MW at a voltage of 300,000 V. Find the radius of curvature of the path of a 25.0-MeV proton moving perpendicularly to the 1.20-T field of a cyclotron. To construct a nonmechanical water meter, a 0.500-T magnetic field is placed across the supply water pipe to a home and the Hall voltage is recorded. (a) Find the flow rate in liters per second through a 3.00-cm-diameter pipe if the Hall voltage is 60.0 mV. (b) What would the Hall voltage be for the same flow rate through a 10.0-cm-diameter pipe with the same field applied? A current balance used to define the ampere is designed so that the current through it is constant, as is the distance between wires. Even so, if the wires change length with temperature, the force between them will change. What percent change in force per degree will occur if the wires are copper? A cyclotron accelerates charged particles as shown in Figure 22.70. Using the results of the previous problem, calculate the frequency of the accelerating voltage needed for a proton in a 1.20-T Frustrated by the small Hall voltage obtained in blood flow measurements, a medical physicist decides to increase the applied magnetic field strength to get a 0.500-V output for blood moving at 30.0 cm/s in a 1.50-cm-diameter vessel. (a) What magnetic field strength is needed? (b) What is unreasonable about this result? (c) Which premise is responsible? Assume for simplicity that the Earth’s magnetic north pole is at the same location as its geographic north pole. If you are in an airplane flying due west along the equator, as you cross the prime meridian (0° longitude) facing west and look down at a compass you are carrying, you see that the compass needle is perpendicular to your direction of motion, and the north pole of the needle dipole points to your right. As you continue flying due west, describe how and why the orientation of the needle will (or will not) change. Describe what steps must be undertaken in order to convert an unmagnetized iron rod into a permanently magnetized state. As part of your answer, explain what a magnetic domain is and how it responds to the steps described. Iron is ferromagnetic and lead is diamagnetic, which means its magnetic domains respond in the opposite direction of ferromagnets but many orders of magnitude more weakly. The two blocks are placed in a magnetic field that points to the right. Which of the following best represents the orientations of the dipoles when the field is present? A weather vane is some sort of directional arrow parallel to the ground that may rotate freely in a horizontal plane. A typical weather vane has a large cross-sectional area perpendicular to the direction the arrow is pointing, like a “One Way” street sign. The purpose of the weather vane is to indicate the direction of the wind. As wind blows past the weather vane, it tends to orient the arrow in the same direction as the wind. Consider a weather vane’s response to a strong wind. Explain how this is both similar to and different from a magnetic domain’s response to an external magnetic field. How does each affect its surroundings? Electrons starting from rest are accelerated through a potential difference of 240 V and fired into a region of uniform 3.5-mT magnetic field generated by a large solenoid. The electrons are initially moving in the +x-direction upon entering the field, and the field is directed into the page. Determine (a) the radius of the circle in which the electrons will move in this uniform magnetic field and (b) the initial direction of the magnetic force the electrons feel upon entering the uniform field of the solenoid. Imagine the xy coordinate plane is the plane of the page. A wire along the z-axis carries current in the +z-direction (out of the page, or ⊙ ). Draw a diagram of the magnetic field in the vicinity of this wire indicating the direction of the field. Also, describe how the strength of the magnetic field varies according to the distance from the z-axis.	{"url":"https://collegephysicsanswers.com/chapter-22-magnetism?textbook=ap","timestamp":"2024-11-04T00:54:02Z","content_type":"text/html","content_length":"402378","record_id":"<urn:uuid:21f19287-42c1-40fc-9ff1-d7ac71b52b8b>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00446.warc.gz"}
How To Decrypt Magnetic Card Data With DUKPT Recently I found myself in a position where I needed to decrypt card data coming off of a magnetic stripe scanner. I originally thought this was going to be straight forward. Get a key and pass it into some predefined decryption algorithm. Not quite. It turns out these types of scanners often use a schema known as DUKPT (Derived Unique Key Per Transaction). The idea behind this schema is that for every transaction (or in this case for every card swipe) the data is encrypted using a key specific to that card swipe. In order to decrypt data that was encrypted using this schema you have to be able to generate the key for that specific card swipe. The process to generate this key (session key) is far from straight The process is described in ANSI X9.24 part 1. However, this document costs about $140. Finding free, easily accessible documentation describing this process is difficult to come by. The best resource I was able to find was this; a pretty good explanation of how to generate the IPEK (Initial Pin Encryption Key). Unfortunately this is only part of the full solution. In this post I am going to attempt to comprehensively explain the DUKPT schema. Some Terms to Know BDK: This is an acronym for Base Derivation Key. This key is known only to the manufacturer and the software developer interfacing with the magstripe scanner. IPEK: This is an acronym for Initial Pin Encryption Key. This key is derived from the BDK. This key is injected onto the device by the manufacturer and is used to derive future keys. Compromising the IPEK does not compromise the BDK. KSN: This is an acronym for Key Serial Number. The KSN is a combo of the serial number of the magstripe scanner and a counter representing the number of swipes that have taken place on the device. How It Works The BDK is used by the manufacturer to generate the IPEK which is injected onto the device during the manufacturing process. The device uses the IPEK and the KSN to generate a session key that is used to encrypt the data coming off the card. The BDK is required by the software developer so they can also generate the IPEK. Once they have the IPEK they can query the device for the KSN. The IPEK and KSN are then used to derive the key for that specific transaction/swipe. Once the developer has this key they can easily decrypt the card data. Generating the IPEK In order to generate the Initial Pin Encryption Key (IPEK) we need the Base Derivation Key (BDK) and the Key Serial Number (KSN). The IPEK is derived using TripleDES encryption. TripleDES is just DES encryption daisy chained together three times. The algorithm takes a 24 byte key. It runs a DES algorithm three times using bytes 1-8, then again using bytes 9-16 and then a final time with bytes 17-24 as the key for each round respectively. If the TripleDES algorithm is given a 16 byte key it should use what is known as the EDE3 method. What this means is it will use bytes 1-8 , then bytes 9-16, then bytes 1-8 again to fake a 24 byte Some implementations of TripleDES will not do this automatically. I learned this the hard way and spent many hours trying to solve the problem. I was assuming that the particular implementation of TripleDES I was using would fake a 24 bytes key from a 16 byte key. After a lot of frustration it occurred to me to try taking the first 8 bytes and append it to the end of my key before I fed it into the TripleDES algorithm. This fixed my issue. The Details The IPEK is going to consist of two 8 byte registers that are produced by two separate TripleDES algorithms. Both will encrypt the 8 most significant bytes of the KSN with the counter zeroed out. The difference between the left and right register is that the right register is going to encrypt the KSN with a slightly modified version of the BDK. I will describe the process below. Let's assume we have a 16 byte BDK represented as the hexadecimal string "0123456789ABCDEFFEDCBA9876543210". We also have a 10 byte KSN with an 8 count represented as the hexadecimal string We are going to form the BDK used to encrypt the left register of our IPEK by appending the first 8 bytes to the end of the BDK to give us the following 24 byte key. Now if the KSN is not already 10 bytes in length then pad it to 10 bytes with hexadecimal "F" (1111). It is important to note that the IPEK represents the very first key on the device. This means that we want to generate it with the counter portion of the KSN set to 0. In order to get our KSN with a 0 counter we want to bitmask it with the hexadecimal represented by the string and FFFFFFFFFFFFFFE00000 = FFFF9876543210E00000 Great. Now we have our 0 counter KSN. But we only want the most significant 8 bytes of this KSN. We get this by bit shifting this KSN to the right 16 bits. FFFF9876543210E00000 >> 16 = FFFF9876543210E0 Perfect. The next step is to TripleDES encrypt "FFFF9876543210E0" with the 24 byte "0123456789ABCDEFFEDCBA98765432100123456789ABCDEF" BDK. The result of this encryption should give us the left register of our IPEK. If you remember, I mentioned that the right register would use a slightly modified version of the BDK to encrypt our KSN. In order to do this we want to start with our original 16 byte BDK "0123456789ABCDEFFEDCBA9876543210" and XOR it with the following mask "C0C0C0C000000000C0C0C0C000000000". This seems to be a totally arbitrary mask, but alas, it's required to get the right IPEK. xor C0C0C0C000000000C0C0C0C000000000 = C1E385A789ABCDEF3E1C7A5876543210 We are going to do the same thing we did with the key for the left register and take the most significant 8 bytes and append it to the end to get the following 24 byte key. Go ahead and take the most significant 8 bytes of the KSN with the counter zeroed (as we computed earlier) and TripleDES encrypt it with this new key we just generated. This will give you the right register of your IPEK, producing the following 16 byte IPEK (I've separated the left and right registers for clarity). 6AC292FAA1315B4D 858AB3A3D7D5933A Generating Future Keys We now have the IPEK. We need to get from the IPEK to the unique key for a specific card swipe (the session key). In order to get to this point I am going to define the existence of a black box subroutine that has the soul purpose of returning a single future key. What happens in this black box we will not concern ourselves with at the moment. Right now we are only concerned with the preparation of this subroutines inputs. This black box subroutine takes two inputs. One is going to be a key and the other is a message to encrypt. This message is a modification of the KSN. If you remember, the least significant 21 bits of the KSN holds a counter representing how many card swipes have occurred on the device. We are going to pass a modified KSN into this subroutine as many times as there are 1's in the binary representation of that counter. The key that is passed in along with this modified KSN will be the IPEK on the first iteration and on subsequent iterations it will be the last key produced by the black box subroutine. Let's begin to modify the KSN. To start we are only concerned with the lease significant 8 bytes of the KSN. We also want to zero out the counter portion of the KSN. This can be done by bitmasking the KSN using the mask below. and 0000FFFFFFFFFFE00000 = 00009876543210E00000 This resulting number is what we are going to use to generate each message we pass into the black box. In our example with the 8 counter we only have to pass our inputs through the black box once due to the nature of the binary representation of the number 8 (1000). So for demonstration let's assume our counter is actually something more complex like 10 (1010). We are going to derive a set of binary numbers from the counter. In the case of a number like 10 (1010), there exists two binary numbers in this set: 1000, and 0010. Do you see the pattern? We construct a binary number to represent each 1 in the binary form of 10 so that if you added this set together it would equal 11. We take the first of these numbers and OR it with our 8 LSB zeroed counter KSN we prepared previously as follows (note that this is in hex so our first number is represented as 0008 in hex). OR 0000000000000008 = 9876543210E00008 We now pass the IPEK as the key and this newly generated KSN variation into the blackbox. The blackbox is going to return a new key. This key is the first future key (represented here in hex): Now we are going to repeat the process for the next number in the previously constructed binary set, 2 (0010). This time, however, we are going to use the future key we just produced as the key and we are going to generate a new variation of the KSN. To generate this new variation of the KSN we are going to perform the OR operation again using the last variation we generated: 9876543210E00008. OR 0000000000000002 = 9876543210E0000A Now we pass in our new key, "27f66d5244ff62e1aa6f6120edeb4280", and our new KSN variation "9876543210E0000A" into our blackbox and get out another future key, "6cf2500a22507c7cc776ceadc1e33014". This is our session key for this device with a counter of 10. However, our actual counter in this blog post was originally 8, so our real session key is actually the "27F66D5244FF62E1AA6F6120EDEB4280" we computed on the first round. There is one last operation we have to perform on this value before we've generated our final permutation of the key that is going to allow us to decrypt our data. We must XOR it with XOR 00000000000000FF00000000000000FF = 27F66D5244FF621EAA6F6120EDEB427F This is the final key we need to decrypt our data. The Black Box This black box I've been referring to is the algorithm that generates our future keys. This black box takes the current session key, which I will refer to as current_sk, and a modification of the KSN, which I will refer to as ksn_mod. If we begin with the assumption that the IPEK we generated above was passed in as the current_sk and that our ksn_mod is "9876543210E00008" that we also generated above. To begin with we want to take the current_sk and get the most significant 8 bytes and bit shift it 64 bits to the right to get "6AC292FAA1315B4". This can be done by performing the following AND FFFFFFFFFFFFFFFF0000000000000000 = 6AC292FAA1315B4D0000000000000000 At this point we just need to bit shift it 64 bits to the right to get "6AC292FAA1315B4D". We will call this value left_key (seeing as how it's the left side of the current_sk). Next we want to get the 8 least significant bytes of the current_sk which can be done by the following bitmask. AND 0000000000000000FFFFFFFFFFFFFFFF = 0000000000000000858AB3A3D7D5933A Let's call this value (you guessed it) right_key. Now we are going to take the right_key and XOR it with the ksn_mod value "9876543210E00008". AND 9876543210E00008 = 1DFCE791C7359332 This value we will call the message. Next we are going to take this message value DES encrypt it (that's single DES). We are going to pass the message into the DES algo as the content to be encrypted and we are going to pass in left_key as the key to encrypt it with. This should give us "2FE5D2833A3ED1BA". We now need to XOR this value with right_key. XOR 858AB3A3D7D5933A = AA6F6120EDEB4280 This value is the least significant 8 bytes of our session key! We now just need to repeat the above operation once more with different inputs. This time we are going to take current_sk and XOR it with "C0C0C0C000000000C0C0C0C000000000". As far as I can tell this value is arbitrary but is part of the ANSI standard so you will just have to take my word for it. XOR C0C0C0C000000000C0C0C0C000000000 = AA02523AA1315B4D454A7363D7D5933A If we take this new value "AA02523AA1315B4D454A7363D7D5933A" and use it in place of the current_sk in the operation I described above we should get "27F66D5244FF62E1". This is the most significant 8 bytes of our session key. Combined it should be "27F66D5244FF62E1AA6F6120EDEB4280". I hope I have explained the DUKPT schema as it pertains to magnetic stripe scanners effectively. I encourage any corrections or questions in the comments section.	{"url":"https://www.parthenonsoftware.com/blog/how-to-decrypt-magnetic-stripe-scanner-data-with-dukpt/","timestamp":"2024-11-07T18:25:08Z","content_type":"text/html","content_length":"33180","record_id":"<urn:uuid:6120965e-9546-4a64-a580-d4b5389e87ee>","cc-path":"CC-MAIN-2024-46/segments/1730477028009.81/warc/CC-MAIN-20241107181317-20241107211317-00750.warc.gz"}
Understanding Mathematical Functions: How To Find Range Of Multivariab Understanding mathematical functions is crucial for solving a wide range of problems in various fields such as engineering, physics, economics, and computer science. One important aspect of understanding functions is being able to find the range of a multivariable function. In this blog post, we will provide a brief overview of the importance of understanding mathematical functions and delve into the process of finding the range of a multivariable function. Key Takeaways • Understanding mathematical functions is essential for problem-solving in various fields. • Finding the range of a multivariable function is a crucial aspect of understanding functions. • Methods for finding the range of multivariable functions include visualization, applying constraints, and using technology. • Constraints and conditions can significantly impact the range of multivariable functions. • Awareness of common pitfalls and challenges can help in overcoming obstacles when finding the range of multivariable functions. Defining Multivariable Functions A multivariable function can be defined as a function that takes in multiple input variables and produces a single output variable. In other words, it is a function of two or more independent variables. These functions are often used in fields such as physics, engineering, and economics to model complex relationships between multiple variables. Explanation of multivariable functions When dealing with multivariable functions, the input consists of multiple independent variables, often denoted as x, y, z, and so on. The output, typically denoted as f(x, y, z), is a single dependent variable that is determined by the values of the input variables. For example, a multivariable function could represent the temperature at different points in a room, where the input variables are the coordinates (x, y, z) and the output variable is the temperature at that point. Examples of multivariable functions One common example of a multivariable function is the distance formula, which calculates the distance between two points in a two-dimensional or three-dimensional space. Another example is the production function in economics, which describes the relationship between inputs (such as labor and capital) and output (such as goods or services). These examples illustrate how multivariable functions can be used to model relationships between multiple variables in various contexts. Finding the Range of Multivariable Functions Understanding the range of a multivariable function is essential in mathematical analysis. It helps us to comprehend the possible outputs or values that a function can produce based on its input Explanation of what the range of a function represents The range of a function represents the set of all possible output values that the function can produce when the input variables are varied. In other words, it is the collection of all the attainable values of the function. Methods for finding the range of multivariable functions • Graphical Analysis: One method to find the range of multivariable functions is by graphing the function and observing the highest and lowest points on the graph. • Algebraic Manipulation: Another method involves algebraic manipulation of the function equation to determine the possible range of values for the output variables. • Partial Differentiation: For functions with more than one input variable, partial differentiation can be used to find the maximum and minimum values of the function, hence determining its range. Examples of finding the range of multivariable functions Let's consider a multivariable function, f(x, y) = x^2 + y^2. To find its range, we can use the method of graphical analysis by plotting the function and observing the range of possible output values. Another example could involve algebraic manipulation of the function equation to determine the range of values that the function can produce. Constraints and Conditions When dealing with multivariable functions, constraints and conditions play a crucial role in determining the range of the function. Let's discuss how constraints and conditions affect the range of multivariable functions and explore some examples of applying these concepts to find the range. A. Discussion of how constraints and conditions affect the range of multivariable functions Constraints and conditions impose limitations on the input variables of a multivariable function, which in turn affects the possible outputs or the range of the function. These limitations can arise from physical or mathematical considerations, and they often restrict the domain of the function. For example, a multivariable function representing the temperature distribution in a room may be subject to the constraint that the temperature cannot exceed a certain limit. This constraint will impact the range of the function, as it restricts the possible values that the function can output. B. Examples of applying constraints and conditions to find the range Let's consider a simple example of a multivariable function f(x, y) = x^2 + y^2, with the constraint x + y = 1. This constraint limits the possible values of x and y, and thus restricts the domain of the function. To find the range of the function subject to this constraint, we can use techniques such as Lagrange multipliers to optimize the function within the given constraint. Another example involves a multivariable function representing the profit of a company in terms of the quantities of two products sold, subject to the constraint that the total production capacity cannot exceed a certain limit. By applying this constraint, we can determine the range of possible profits for the company under the given conditions. Visualizing the Range Understanding the range of a multivariable function is crucial in mathematical analysis. Visualizing the range of a function can provide valuable insights into its behavior and help in solving a wide range of problems. A. Explanation of how to visualize the range of a multivariable function When dealing with a multivariable function, it is important to understand that the range is the set of all possible output values that the function can produce for a given input. Visualizing the range involves considering all possible combinations of input values and observing the corresponding output values. B. Using graphs to illustrate the range Graphs are powerful tools for visualizing the range of a multivariable function. By plotting the function in a coordinate system with multiple dimensions, one can observe how the output values vary as the input values change. This can provide a clear picture of the range of the function and how it behaves across different input ranges. C. Using technology to visualize the range Advancements in technology have made it easier to visualize the range of multivariable functions. Utilizing software such as graphing calculators, computer software, and programming languages, one can generate visual representations of the range with greater precision and detail. This allows for a more comprehensive understanding of the function's behavior. Common Pitfalls and Challenges When it comes to finding the range of multivariable functions, there are several common mistakes that students and even experienced mathematicians often encounter. These pitfalls can make the process challenging and sometimes frustrating. Understanding these common mistakes and learning strategies to overcome them is essential for successfully finding the range of multivariable functions. Identification of common mistakes when finding the range of multivariable functions • Not considering all variables: One of the most common mistakes when finding the range of multivariable functions is not considering all the variables involved. It's crucial to take into account all the variables in the function. Failure to do so can result in an incomplete or incorrect range. • Ignoring constraints: Another common mistake is overlooking the constraints or domain of the function. Constraints can significantly impact the range of the function, and ignoring them can lead to inaccurate results. • Incorrectly applying techniques: Applying the wrong techniques or methods for finding the range of multivariable functions can also lead to mistakes. It's important to have a clear understanding of the appropriate techniques and how to apply them correctly. • Overlooking critical points: Critical points play a crucial role in determining the range of multivariable functions. Failing to identify and consider critical points can result in an incomplete or inaccurate range. Strategies for overcoming challenges in finding the range Overcoming the challenges of finding the range of multivariable functions requires a systematic approach and attention to detail. Here are some strategies to help navigate through these challenges: • Thoroughly analyze all variables: Take the time to thoroughly analyze and consider all the variables involved in the function. This includes understanding their relationships and dependencies on each other. • Pay attention to constraints: Ensure that you carefully consider any constraints or domain restrictions on the function. Incorporating these constraints into your analysis is essential for accurately determining the range. • Master the appropriate techniques: Develop a strong understanding of the techniques and methods for finding the range of multivariable functions. Practice applying these techniques to different functions to build proficiency and confidence. • Identify and evaluate critical points: Be diligent in identifying and evaluating critical points within the function. Critical points often provide valuable insights into the behavior and range of the function. Understanding multivariable functions is crucial for solving complex mathematical problems and real-world applications. The ability to find the range of a multivariable function is an important skill that allows us to understand the possible output values of the function. In summary, we discussed the importance of understanding multivariable functions and the method for finding the range of a multivariable function through analyzing the critical points and boundaries. We encourage further exploration of multivariable functions as they play a significant role in various fields such as physics, engineering, and economics. The more we understand and master these concepts, the better equipped we will be to tackle the challenges of the modern world. ONLY $99 Immediate Download MAC & PC Compatible Free Email Support	{"url":"https://dashboardsexcel.com/blogs/blog/understanding-mathematical-functions-how-to-find-range-of-multivariable-function","timestamp":"2024-11-15T01:27:18Z","content_type":"text/html","content_length":"214947","record_id":"<urn:uuid:5c3a7eda-256f-445e-a38c-8e42541bc132>","cc-path":"CC-MAIN-2024-46/segments/1730477397531.96/warc/CC-MAIN-20241114225955-20241115015955-00260.warc.gz"}
June 2022 - Howard Fensterman Minerals Quantum computing. Close-up of optical CPU process light signal. Photo: iStock Quantum Introduction The term ‘Quantum Computing’ hasn’t gotten the much-needed traction in the tech world as yet and those that have traversed through this subject might find it a bit confusing, to say the least. Some experts believe that this is not just the future, but the future of humanity. Quantum theory moves ahead of the binary computer and ventures into the world of computing that resides at the subatomic level. If you don’t have a clue what we are talking about, you are not alone. Stay with us through this article where we will discuss quantum computing in great detail—what it is—how it will change the tech world and its practical implications (both for better or worse). Before we usher in the discussion of this potential life-changing advancement, it is necessary to discuss the platform on which quantum computing is based – Quantum theory. What is Quantum? Particles of the atom: protons, electrons, and neutrons. Nucleus. Photo: iStock Also known as quanta, in simple terms, it represents the minimum amount of energy that can be used within any physical interaction. Using examples of particle interaction within the atom, a quantum of light would be a photon, and a quantum of electricity would be an electron. There can be no activity smaller than when these particles have an interaction. In the Beginning The industrial revolution of the 20th century was one of the greatest milestones of modern history. From the invention of the automobile to industrial steel, elevators, and aircraft, it gave birth to a plethora of things that now define our civilization and will continue to shape the history of our future. Enter the 21st century and we are watching a transition from the tangible to the intangible (virtual) world; notably, computer technology, its hardware, software, and the world wide web. Among the many incredible things that are ensuing during this technological revolution is the colossal development in physics, specifically quantum theory. We will try to keep the explanation of quantum theory as simple as possible to make this as interesting and informative as possible. Modern Physics The field of physics is divided into two definite branches: classical and modern. The former branch was established during the period of the Renaissance and continued to progress after that. Classical physics is based on the ideas by Galileo and Newton. Their concepts are focused on the macroscopic (visible to the naked eye) of the world around us. Conversely, modern physics is about analyzing matter and energy at microscopic levels. While we are at it, it is important to clarify that quantum theory doesn’t just refer to one idea or hypothesis. It is a set of several principles. We will discuss them simply and remain focused on the items that are relevant to quantum computing. □ The work of physicists Max Plank and Albert Einstein in the 20th century theorized that energy can exist in discrete units called ‘quanta’. This hypothesis contradicts the principle of classical physics which states that energy can only exist in a continuous wave spectrum. □ In the following years, Louis de Broglie extended the theory by suggesting that at microscopic (atomic and subatomic) levels, there is not much difference between matter particles and energy and both of them can act as either particles or waves as per the given condition. □ Lastly, Heisenberg proposed the theory of uncertainty, which entails that the complementary values of a subatomic particle can’t be simultaneously measured to give accurate values. Neil Bohr’s Interpretation of Quantum Theory: The Primal Basis of Quantum Computing Image by Pete Linforth from Pixabay During the period when quantum theory was extensively discussed among top physicists, Neil Bohr came up with an important interpretation of this theory. He suggested that light cannot be determined if it is composed of particles or waves, called wave-particle duality until they are particularly found out. The infamous Schrodinger’s Cat experiment is an easy way to understand this concept. The experiment entails that a cat enclosed in a box with poison could be considered both dead and alive until the box is opened and the cat is observed. Computer Algorithms Now, this is the point where the theory demonstrates its potential, but first, a definition of an algorithm – a set of instructions that a computer reads to carry out a function. E..G. You tell the computer to print a document. The computer will read the instructions (algorithm) and perform the printing function. To understand the quantum-based algorithm, it is essential to understand how contemporary/conventional computing systems work. Whether it’s a handheld gadget or a supercomputer working in the server room of Google, every current computing device employs the binary language, where every bit of information can exist in either one of two states: 0 or 1 (hence ‘binary’), but not both states at once. When we discuss quantum algorithms, they follow the idea that any particle-wave system can exist in multiple states at any given time. This means when data is stored in a quantum system, it can be stored in more than two states. This makes quantum bits (also referred to as ‘qubits’) more powerful than the conventional method of binary computing. Standard Binary Computing Vs. Quantum Computing 4 rows of 8 bits = 4 rows of bytes. Photo: iStock The fact that a quantum bit can exist in multiple states gives quantum computing an edge over conventional binary computing. With the help of a simple example, we will try to demonstrate how quantum theory is superior to its classical counterpart. Picture a cylindrical rod, and each end of the rod is a bit, labeled 1 or 0. When one side is a 1, then the other side must be a 0. On the other hand, the quantum bit exists in every possible state simultaneously, between the 1 and 0 together. The above explanation exhibits that quantum bits can hold an unprecedented amount of information and hence the computing governed by this type of algorithm can exceed the processing of any classical computing machine. A quantum computer can compute every instance between 0 and 1 simultaneously, called parallel computing. Quantum Entanglement Apart from storing more information than classical computers, quantum computing can also implement the principle of entanglement. In simple words, this principle will enable every quantum bit to be processed separately. Beneficial Uses of Quantum Computing The processing capabilities of quantum computing make it an ideal machine to carry out many tasks where conventional computers fall short. Science and Life Sciences The study of complex atomic and molecular structures and reactions is no easy task. A lot of computing capacity is required to simulate such processes. For instance, the complete simulation of a molecule as simple as hydrogen is not possible with the available conventional computing technology. So, quantum computing can play a significant role in understanding many of the concealed facts of nature and more particularly, of life. Many chemicals and physical and biological research work previously stalled for years can take off after quantum computers become a reality. Artificial Intelligence and Machine Learning Even though scientists have made significant inroads in the area of machine learning and AI with the existing computing resources, quantum computing can help take the giant leap to make a machine as intelligent as human cognition. Machine learning feeds on big data, which is the processing of humongous databases; in other words, big data contains a colossal amount of information, above and beyond what conventional databases contain. And the more information you have, the more intelligent you become! With the fast processing of quantum computing, even conventional AI will become obsolete, revamping it into a new and more powerful artificial intelligence. Improvement of General Optimization Procedures With the addition of big data, the processing that takes place involves more than just reading information. It also involves the ability to make more decisions. It is called if/then conditions, meaning if something exists and something else acts on it, what could be the outcome? The conditions use variables to calculate each condition. So, the more data, the more variables to calculate. Putting it another way, the number of permutations and combinations increases, and thus, the amount of processing power increases. When this happens, the amount of data to be processed can increase exponentially. Some examples would be the optimization of a financial plan might need the processing of several petabytes, equivalent to about 500 billion pages of printed text. Implementation of such extensive computing can only be achieved with the processing power of quantum machines. Other Side of the Coin: The Dangers Involved with Quantum Computing One should not be surprised by this heading. We have seen through the course of history how the advent of new technology, intended for the benefit of humankind, is followed by its misuse. One example is Einstein’s famous equation E = mc2, which gave scientists the idea of building an atomic bomb. Although Einstein was a man of peace and his theory was never indented to be used for destructive purposes, it became so anyway; hence, with quantum computing, this unrestrained processing power can be harnessed for nefarious purposes. Quantum Computing Puts Data Encryption Practices in a Great Peril Photo by Shopify Photos from Burst And as we know every precious commodity is vulnerable to vandalism, breaches, and thefts. So, to address this vulnerability, computer scientists have developed encryption modules that are used to lock the data, and only those that have the encryption key can access it, with such a password. Unauthorized parties can’t get around this encryption without a technique called brute force cracking. But it is important to mention that brute force attacks might only work to crack simple passwords that consist of only a few bytes. Let’s try to better understand this with the help of numbers With today’s computers, It could take more than a billion, billion years to crack data that is protected by what is called a 128-bit encryption key, widely used by financial resources on the A standard 128-bit key can’t get cracked by the brute force algorithm using the conventional binary coding system, but when we replace this two-state concept with a quantum bit of unlimited existing states, the tables surely get turned. The result is that a 128-bit Key that is so formidable against the brute force of classical binary supercomputers will fall flat when quantum computing is used to carry out the brute-force algorithm. No operating quantum machine exists today, but experts have estimated that a quantum supercomputer would be able to crack 128-bit encryption keys within 100 seconds. Compare that to the billon-billion years it would take a binary computer to crack the same code! If data encryption becomes ineffective, it will expose everything to criminal elements. To understand just a fraction of this devastation, imagine that every person on earth linked to the banking system loses access to their account. The mere idea of such a situation can send chills down your spine. Apart from that, the neutralization of data encryption can lead to cyber warfare between nation-states. Here also, rogue elements will easily be able to capitalize on the situation. A global outbreak of war in a world with the existing eight nuclear powers can end up with a dreadful outcome. All things considered, the manifestation of quantum computing can bring along many irretrievable repercussions. Preparation to Protect Against the Nefarious Use of Quantum Computing Google and IBM have successfully carried out quantum computing in a controlled environment. So, to think that quantum computers are a distant reality won’t be deemed an insightful judgment. For that matter, businesses should start preparing against this abuse. There is no point in waiting for formal rules and protocols to be issued. Experts working in the area of digital security and cryptography recommend some measures to protect business data in the future from any exploitation of the quantum era. How technology has progressed in the last few decades is indicative of the fact that quantum computing is the reality of the future. So, the arrival of quantum computers is not the question of ‘if’ – it’s the question of ‘when’. Quantum theory, with all its benefits for the development of life sciences, the financial sector, and AI poses a great threat to the existing encryption system, which is central for the protection of any type of confidential data. The proper approach for any nation and business is to accept this unwanted aspect of quantum mechanics as a technological hazard and start preparing against it with the help of experts. With that said, it will also be a blessing when used proactively for the benefit of humankind and we look forward to a better lifestyle for each of us when quantum computing becomes a reality. Units of Power and How They are Related to Electricity Before we learn about kilowatts and kilowatt-hours, let’s get a jump start (pun intended ????) on what these terms mean. The Units of Electrical Power Note: If you are not a physics enthusiast and want to skip the calculations, you can jump to this section. Let’s travel into our way back machine and go back to high school physics 101. These terms and measurements are for background purposes only. We will not be using them later on, but understanding these concepts can help you better comprehend how power (energy) is referenced in units of watts (w) and how they are calculated. Let’s do it! The rate of time at which an object is moving along a path. Units: Length, Time Example: The car traveled 1 mile in 60 seconds or 1 mile/minute. Further Reading: What is speed in physics? The rate of time at which an object is moving along a path in a particular direction. Units: Length, Time, Direction. More precisely, length/time (speed) in a particular direction. Example: The car traveled 1 mile/minute going west. Further Reading: What is the difference between speed and velocity? When we speak about acceleration, it is the rate at which the velocity changes. In other words, velocity doesn’t stay constant. Units: Feet per second per second or feet/second squared. Example: A plane traveling south accelerates from 550 m/h (mph) to 600 m/h over a time period of 40 seconds. It has a change in velocity from 550 m/h to 600 m/h and the time period that this occurs in 40 seconds. Further Reading: Speed, velocity, and acceleration. Here we add a new component – Force. When we talk about the measurements of Newtons, we are talking about an acceleration (remember, acceleration means just a change in velocity) of an object. By Mhermsenwhite – Own work, CC BY-SA 4.0, https://commons.wikimedia.org/w/index.php?curid=70624309 One newton is the force needed to accelerate one kilogram of mass at the rate of one meter per second squared in the direction of the applied force. Simply put, this is the amount of push (force) of one kilogram of an object that weighs one kilogram at a changing velocity (acceleration) of one meter per second per second. Units: 1 kg⋅m/s^2 Example: Joe is pushing a box weighing one kg down the road at 1 m/s[2 ] Joules refer to the amount of work done. A joule is equal to the work done by a force of one newton moving one meter, so Joe has pushed the box weighing one kg down the road at 1 m/s squared for a distance of 1 meter. A joule is also referred to as energy. Say Watt? The number of Joules that an electrical device (e.g a lightbulb) is burning per second. Joules and watts both refer to work and equate to power, but both are interchangeable. Here is the connection: 1 Watt = 1 Joule per second (1W = 1 J/s), so a watt is the amount of energy (in Joules) that an electrical device (such as a light) is running per second. So if a device is burning 500 watts for 60 seconds, then a Joule would equate to 500 * 60 = 30,000 J. Moving ahead, if an air conditioner is burning 1000 watts for 1 hour (60 sec * 60 min = 3600 seconds), then that equates to 1000 watts * 3600 seconds = 3,600,000 Joules (of energy that was used for that hour). A kilowatt is equal to 1000 watts, so 1 kWh represents the amount of energy transfer that occurs over one hour from a power output of 1000 watts (i.e., joules per second). Thus 1 kWh is equal to 3,600,000 joules of energy transfer (work). What Does This Mean? It means that the work of one newton is being performed in the form of electrons that are being pushed through the wire per meter. Saying it in a simpler form, one watt is one joule of energy running a device per second. Just Tell Me in Plain English What a Watt is! Consider this to be a one-watt light bulb. If it was a two-watt light bulb, it would be about twice as bright. If it was a 500-watt bulb, more power is needed to provide that additional wattage; hence, more power or we can say more current or voltage is needed, and up goes your electric bill! See how it works? Photo by LED Supermarket Glad you asked. 1 watt is equal to voltage times current: W=EI (don’t worry, you don’t have to memorize this formula). Also known as power, a watt is a unit of power. The more the voltage and/or current that flows through the wire, the more power (watts) is used to run the device. Let’s Talk About Time Devices run for a period of time, right? So we have to add this value to our watt calculations. That way, we will know how many watts are used for a certain period of time, and as we will see later, this will help us determine what it costs to run electrical devices, or more specifically, what the electric company charges us and why. Examples: Joe turned on a one-watt lightbulb for 60 seconds, so that is equal to 60 watts. Now Joe turned on a 250-watt lightbulb for 2 minutes, so that is equal to (250 * 0.333 hours) = 83.25 watts. (Remember, for you physics guys, 83.25 watts is the same as saying that 4995 joules of power have been generated). We’ll be going into this in another article, but just to enlighten you, if your electric company charges you 14.34 cents per 1000 watts used per hour (that’s what they generally charge in New York), then, using the example above, you have paid the company 14.34 cents * (per 1000 watts) * 0.25 watts * 0.0333 / hour (2 minutes) = .036 cents per hour. If Joe ran the 250-watt bulb for 1 hour, then he would be paying 3.6 cents per hour, but if Joe ran a 1000-watt device for 1 hour, he would be paying 14.34 cents. OK, but if Joe ran the 1000 watt bulb for 10 hours, then he would owe the energy company $143 cents or $1.43. OK forget about Joe. What if your electric company charges you 14.34 cents per hour for a 2000-watt air conditioner? You would be paying 29 cents per hour, so if you run the air conditioner for 10 hours each day, you would be paying $2.90 every day. That’s $29.00 every 10 days or close to $100 per month. Say 1000 Watts! Are you getting tired of hearing of thousands of watts? This author is also, so let’s call 1000 watts – 1 kilowatt. There you go. Kilo means 1000 so 1kw is 1000 watts. If you run a 1000-watt device for 1 hour, then the designation is 1Kwh (1 kilowatt-hour or you can say a 1-kilowatt device is running for one hour), denoted as kWh. So, 1 kilowatt is equal to 1,000 watts. If a unit consumes 60 watts hourly and runs for 60 hours, then the energy consumption rate will be 60 watts x 60 hours to equal 3,600 watts per hour, which is equal to 3.6 kWh of electricity. Ok we know, you want to know what it cost to run your electrical devices in your home and you probably want to know about your air conditioner for starters. Let’s just say that a typical air conditioner runs about 3 kWh per day. To calculate how much that costs you, just call your local energy company to get the correct number. For our area, Nassau County, the cost is 7 cents per kWh. If you want to know more about your air conditioner costs, check it out here. Gas Cars Vs. EV Cars Costs Comparisions 2023 Austin, Texas, 2-1-2021: Tesla Model 3 charging at home in front of a house on an L2 charger. Photo: iStock Note: If you want to bypass the calculations below and go directly to the actual costs of charging an EV against today’s gas prices, go to our Costs of Charging an EV in this article. 2023 Update PSEG is now providing the monthly costs for home charging. Below is an example of a real Long Island homeowner’s EV statement from PSEG. Photo: SMS © Why Electric? There are a number of benefits of driving an electric vehicle (EV). One is the cost savings on gas. The other is the environment. We will concentrate on the former now and will talk about the environment in a separate article. Before we start discussing how EV costs are calculated, make sure you have read our articles on the atom, electric current and Units of Power and How They are Related to Electricity so that you will be able to keep up with our cost calculations that involve knowledge about watts and kilowatts, but if you haven’t, no worries. You can skip to the bottom to get our estimate of EV electrical costs when charging from the home, or just read the review below. Here’s a brief overview for those who didn’t read the articles mentioned above. □ Electrons are subatomic particles (one of the entities within an atom) that travel through the wire when power is applied (the wire is attached to an electrical socket). This is known as electrical current and is referred to in units of amps. More on this here. □ Voltage is the force that pushes the electrons through the wire. Similar to turning on the pressure of a water faucet. □ Current usually flows through a copper wire which is the conductor and the wire is covered by an insulator (rubber packaging around the wire so that the copper is not bear). □ Resistance is the opposition to the current (electrons) that is flowing in an electrical circuit. Think of it as the friction that brushes along the side of the current. □ A watt is the energy (power) that runs the electric device. It is a product of how much electrical current is running and how much voltage (push) is occurring. It is determined by multiplying the voltage times the current. The formula is E=IR (E=voltage, I=current, and R=resistance). □ A kilowatt is 1000 watts (kW). □ A kilowatt-hour (kWh) equates to 1kw that runs a device for 1 hour. Example: If you run an air conditioner for one hour and that air conditioner uses 70 kilowatts of electricity per hour, then you have used 70 kilowatts of electrical energy for that hour. If you run the air conditioner for two hours, you would have used up 140 kilowatts of energy. Most EVs, with the exception of the high-end luxury ones, have batteries that consist of a 60-65kWh capacity. Sparing you the formula, a battery of this size will equate to about 260 miles after a full (100%) charge. Note: Most EVs are set to charge to 80% only. Constant charging to 100% diminishes the battery’s lifetime. 80% of a 65kWh battery equates to about 230 miles. How Do Kilowatts Relate to Electrical Costs? High voltage transmission towers with red glowing wires against blue sky – Energy concept. iStock Conventional Gas Cars We will use a 2021, 4-cylinder Nissan Altima as our example. Gas tank size: 16.2 gals and MPG: 31 average. If we multiply 31 miles/gals * 16.2 gals, we can determine the total mileage that this car can run on a full tank of gas, which is 502 miles. As of this writing, the price for a gallon of gas is $5.00 on average across the United States. So $5.00 * 16.2 gallons (a full tank) equals $81 to fill up. Electrical Vehicles Photo by Michael Fousert on Unsplash For EVs, we calculate units per mile instead of MPG. For this example, we will use a 2020 Kia Niro EV, which is a fully electric vehicle and contains a 65kWh battery. As mentioned, the industry standard for charging a 65kWh EV to 80% is about 230 miles. Note: If you have an EV, never let it go below 30%, as you may run into trouble if you are on the road and can’t find a charging station, especially in the winter time. Let’s review what we know so far: □ Filling up a gas tank of a 2021 Nissan Altima will take you about 502 miles without having to fill up again. □ The cost to fill up this car as of this writing is $81.00. □ To charge a 2020 Kia Niro’s battery to 80%, the car can go about 230 miles without having to recharge. Local Averages Using Electric Utility Calculations Photo by LED Supermarket, Pexels We called PSEGLI directly to find out the average cost of electrical consumption for a typical home in Nassau County. Keeping it simple, an average home uses about $.33 per kWh (this includes delivery and service charges). According to one source, 7.2 kWh is used each hour to charge the battery and if it takes approximately 4 hours to charge, the total kWh is 28.8 kWh. 28.8 kWh x $.33 = $9.5. Rounded off, it costs about $10.00 to charge a 65kWh battery, which equates to 230 miles, but if you’d like to be a little more cautious if you think that might be too low (since there are so many variables involved that might not meet your particular driving habits or lifestyle, we can say the approximate cost for charging a 65kWh battery from a 220/240-volt level 2 charger is $15.00. How’s We will now compare filling a gas tank of a conventional car which equates to the same mileage (230 miles). Here are the steps: □ Divide the total mileage to charge the battery to 80% by the total mileage to fill a gas tank to get the percentage between the two: 230 mi / 502 mi = 45% □ Multiply this percentage by the total cost to gas up a car: To get the cost for a conventional car to go 230 miles, we multiply the cost to fill up the gas tank ($81.00) by 45% to match the 230 miles, and that cost would be 0.45 *$85 = $38.7. Using an average of today’s gas prices ($5.00 as of today), it would cost a gas car $38.7 to go 230 miles of highway driving and an EV car would cost $15 to go the same distance (230 miles) in Nassau County, New York. Note: As of October 2022, the price of gas fell to $3.5 / gallon, so proportioning this price, we get the cost to fill a gas tank to go 230 miles is – ($3.5 x 16.2) x 0.45 = $25.5, which is about 7 gallons of gas. That’s still a savings of $15.5 for every 230 the gas car drives. Cost of Charging an EV Update: As of January 2023, PSEG and other utilities are now using disaggregation. A technique that breaks down energy utilization by appliance via AI computer algorithms. Below is an example of disaggregation of a common household’s individual energy usage by appliance. Notice that $91 was spent on EV charging for the 30 days of November 11, 2022, to December 12, 2022. That’s $22.75 per week using standard electrical charges (not Time of Use as described below). In comparison, one SUV that averages 25 MPG and traveled 1,100 miles for that same time period would have cost $149.60 at today’s price of $3.40 per gallon. Similarly, a typical mid-sized sedan traveling 1,100 miles would run $124.66. You can calculate your specific mileage costs here. Selective Electric Utility Plans Overview Most electric utility companies provide more than one plan that you can select for your household. Besides the default plan which provides the same price for electric consumption 24×7, there is a plan that can allow you to select lower rates based on different times of the day. This plan, called Time of Use (TOU) is available at PSEGLI and NYC’s Con Edison, as well as many other utility companies nationwide. Refer to their brochure as to exactly how this works. If you have not already done so, change your plan to TOU and schedule your EV charging for after midnight on weekdays. You can also apply the same schedule for your dishwasher, washer and dryer and any other appliance that uses electricity. PSEGLI TOU Chart Take a look at the electric bill above from PSEG of Long Island (PSEGLI) above, which powers Nassau County and where the offices of Howard Fensterman are located. Electrical power companies charge per kWh and we did some preliminary calculations starting with the delivery charges in the bill, and that doesn’t include the actual electrical costs after that. Note: It can take up to four hours to charge an EV using a level 2 charger. Image capture: PSEGLI If you are looking to save money on gas, EV cars are the way to go. Yes, these vehicles are more expensive than conventional gas cars, but at $3,50 per gallon, you will be pleasantly surprised how much your savings can accumulate. Finally, we leave you with this. Below is a copy of the estimated charges that accrued for the month of July 2022, from a 1,100-square-foot home that has an EV in its garage in Nassau County, NY. The family charges the car to its 80% capacity about three-four times per month. Notice that the cost in the Electronics category is only 10% of the total usage in the house. Something to think about! Photo: SS What is Voltage and Electrical Current? (A Brief Guide) High voltage transmission towers with red glowing wires against blue sky – Energy concept. IStock. Electrical current is the measure of electrical flow. It’s measured in amperes, or amps for short. The current refers to the number of electrons that pass by a point in an electrical conductor in one second, and it’s usually given in units as milliamps (mA) or microamps (μA). This article explains what electrical current is and how it works. Keep reading to learn more about this topic! How Does Electrical Current Work? Electrical current travels through a wire (conductor) to reach a device (eg. light bulb) which causes the device to enable. This traveling of electrons through the wire to the device is called a circuit. It is the pathway for an electrical current to flow from the source to the load. Copper cables are surrounded by rubber insulation. The copper wire is the pathway from the source to the load. iStock There are three basic parts to a circuit: □ The “source,” or “sourcing device,” is where the electrons come from. This can be a battery, a generator, or the flow of electricity from a wall outlet. □ The “load,” or “dumping device,” is where the electrons go after completing the circuit. This could be a light bulb, an appliance, or some other device. □ The “pathway,” or “wiring,” is the middle part that brings the electrons from the sourcing device to the dumping device. The wiring is almost always made of copper, iron, or in electronic devices, a semiconductor. The current can only flow when the circuit is complete. When the circuit is broken, the current stops. What Is Electrical Conductivity? Electrical conductivity is the ability of a material to allow an electrical current to flow through it. The term conductivity is used to describe the extent to which a material will allow the flow of an electrical current. If a material has high conductivity, such as copper, it means that it is very good for allowing electrons to flow rather freely through the wire, while low conductivity, such as rubber will inhibit the electron flow to a greater extent, known as resistance. The harder it is for the electrons to flow, the more resistance the material has. That’s why the rubber is used to insulate the copper wire in almost all manufacturing that will transmit electric current. Rubber has a high resistance rating. Wood and glass are two types of materials that have very low conductivity ratings. Have you ever used wood to connect to an electrical circuit or battery? On the other end, copper is one of the most conductive materials around and that is why you see so many wires and/or cables that have copper wiring. Besides the type of material that is used, electrical conductivity can be affected by several factors. For example, temperature, and the presence of contaminants like dust and water. What is Voltage? Turn on your water faucet about a quarter of the way and place a cup under it. Notice how fast (or slow) the water is running to fill the cup. How long did it take? Now turn the faucet to make the water run faster. When you do this, the water fills up the cup sooner. This is your voltage (actually an equivalent of voltage). The faster the water comes out, the more the force or pressure of water will be used. In electricity, this means that the more the pressure, the faster the electric current will come out to power an electrical device. The bulb will light up quicker, which you won’t notice, since it happens so quickly, but that is what will happen. Ohm’s Law A law that states the relationship between voltage, current, and resistance in a conductor (or insulator). It states that voltage is equal to current times resistance or E=IR. So the voltage equates to the amount of current that flows through the wire but includes the amount of resistance the current is subjected to. Types of Electrical Current There are two basic types of electrical current: Direct Current (DC) and Alternating Current (AC). A direct current is a constant flow of electrons that always flows in the same direction. It can flow in one direction or it can flow in both directions. It is provided by batteries, solar cells, and hydroelectric plants. Electrical current can be changed from DC to AC by using a device called a transformer. Transformers are used to change the voltage of the electricity. Electrical current is the flow of electrons through a conductor. A complete circuit is where electrons flow from the source to the load through a pathway or wiring. Electrical current works when a circuit is complete. A circuit is a pathway for an electrical current to flow from the source to the load. There are 3 basic parts in a circuit. The source is where the electrons come from. The load is where the electrons go after completing the circuit. The pathway is the middle part that brings the electrons from the sourcing device to the dumping device. There are two basic types of electrical current: Direct Current (DC) and Alternating Current (AC). A direct current is a constant flow of electrons that always flows in the same direction. AC can change from DC to AC by using a device called a transformer.	{"url":"https://howardfenstermanminerals.com/2022/06/","timestamp":"2024-11-14T21:29:32Z","content_type":"text/html","content_length":"168724","record_id":"<urn:uuid:369b36c2-143d-466e-a270-47c99109d847>","cc-path":"CC-MAIN-2024-46/segments/1730477395538.95/warc/CC-MAIN-20241114194152-20241114224152-00375.warc.gz"}
Hugo Duminil-Copin: The Maths, the Man Laureates of mathematics and computer science meet the next generation Hugo Duminil-Copin will be a name familiar to mathematicians worldwide. He was awarded the 2022 Fields Medal for his work in statistical physics, making him just about as famous as a mathematician could get. Meeting him, though, you would never know it. He is warm and down-to-earth, nothing like the stereotypical picture of a genius that one might expect him to project. This year was Duminil-Copin’s first time at the Heidelberg Laureate Forum, and he was very generous with his time, appearing as part of a panel discussion on the Thursday, giving a lecture on the Friday, and still finding time to talk to the young researchers and give a press conference. Throughout the week, I got to learn a lot more about Duminil-Copin, and was able to discover a bit more about the maths behind the man and the man behind the maths. [Hugo Duminil-Copin at a press conference at the 10th Heidelberg Laureate Forum, 2023, in Heidelberg, Germany. Image credits: HLFF] The Maths Duminil-Copin works in statistical physics, looking at the probabilistic theory of phase transitions. This is a bit of a mouthful but what it essentially means is that he uses statistical models to investigate changes in state. The work which won him the Fields Medal focussed on a specific mathematical model: the Ising Model. The Ising Model, as Duminil-Copin explained in his lecture, looks at alloys composed of precisely two elements (binary alloys) and attempts to explain their magnetism. Magnets are made of dipoles, which have a magnetic moment known as “spin”. Spins can point in one of two opposing directions. When arranged in a lattice, each spin will interact with its neighbours. When neighbouring atoms spin in the same direction, they have a lower energy and hence the system would naturally tend to this state. The introduction of heat, however, can disrupt this, which can lead to a variety of structural phases and hence there could be a phase transition. The Ising model is named after Ernst Ising, who was set the problem by his supervisor Wilhelm Lenz. Ising proved in 1925 that there was no transition between phases in the one-dimensional case. It took until 1944 for Lars Onsager, an American physical chemist and recipient of the 1968 Nobel Prize in Chemistry, to provide a solution for the two-dimensional case. The amount of time that it took between Ising and Onsager’s solutions gives a useful insight into the comparative difficulties of the two cases. It is not surprising that third and fourth dimensions proved to be harder still. In fact, the solutions in three and four dimensions eluded mathematicians, until finally Duminil-Copin was able to compute the continuity and sharpness of the phase transition in three dimensions. He has made great strides in the fourth dimension as well, which is also mentioned in his Fields Medal citation. The Models Much of Duminil-Copin’s work deals with probability theory, and he pointed out in his press conference that probability is something people use in their daily lives, even without realising it. For example, you may assess the probability that it might rain, when deciding whether to wear a coat in the morning. Or you might work out the chance that your bus will get caught at a red light when telling a friend how long your journey will take. Thinking probabilistically is second nature to people. Duminil-Copin elaborated that in order to work with complex information, our brains need to be able to assess uncertainty. “The irony is that we’re catastrophically bad machines of consciously understanding probability,” he laughed. When it comes to independence, conditionality, and assessing small probability events, humans slip up frequently. One only needs to look at the birthday paradox to see this. The birthday paradox is the counterintuitive fact that in a room of 23 or more people, there is a greater than even chance that at least two share a birthday. Probabilistically we can show that this is true, but to many people it just feels wrong. The way the human brain deals with probabilities and decision making closely resembles a technique mathematicians regularly use – it simplifies. When determining the trustworthiness of a new friend, there is a wealth of information that could be drawn upon – the thousands of clues in their body language, whether you have mutual friends, the veracity of what they are saying. It is humanly impossible to think through every factor, and even if it were possible, it would take a long time. So, our brains filter the data, sacrificing some accuracy for speed. We do the same in mathematics, and this is called creating a mathematical model. Mathematical models attempt to describe a situation in the language of maths. This will normally involve some sort of equation, but it does not have to. Models are almost always a simplification of reality but can still give a very useful insight into what is going on. An example of a model which shot up in popularity in 2020 is the SIR model. This model explains the dynamics of diseases – such as COVID – with the S representing the susceptible population, I representing the Infected population, and R representing the Removed population, i.e. those who are no longer able to catch COVID, hopefully through immunity but sometimes through death. Duminil-Copin explained in his lecture the key ingredients needed to create a mathematical model, such as the Ising Model: First, you need to start with an experience, something you notice. Then you want to define your parameters. Duminil-Copin had a catchy slogan for this stage: “Be wise, discretise.” Finally, you introduce a bit of randomness into your model, take limits (see what happens when you get closer to the extreme cases of the model) , and try to break the symmetry. The Methods In the Ising model, the spins are arranged in a lattice and can each take one of two values, $+$ or $-$. Therefore, if there are $L$ spin sights, there are $2^L$ possible states that the lattice can be in. This lends the problem to a simulation using Monte Carlo methods. Monte Carlo methods are computer algorithms that repeatedly run randomised simulations to get a sense if what results can be expected. Often an average will be taken over the many tests that are run. By running the experiments as a computer simulation, time and money are saved. This allows orders of magnitude more results to be obtained than if the experiments were done physically. The law of large numbers gives that the distribution of the repeated events should tend to the true distribution of the random variable. In the case of the Ising model, the simulation runs as follows. The experiment starts in a (possibly randomly chosen) state. Then a random spin site $v$ is selected according to some probabilities $g(\mu,v) = \mathbb{P}(v \text{ is chosen }\|\text{ in state } \mu)$. Normally it is assumed that every spin site has the same chance of being chosen: $g(\mu,v) = 1/L$ for all $v$. Now this is where it gets a bit fiddly. Let us call the state with spin site $v$ flipped (from $+$ to $–$ or from $–$ to $+$) $\nu$. We define some acceptance probabilities $A(\mu,\nu) = \mathbb{P}(\text{ state }\nu\text{ is accepted given the previous state was state }\mu)$. Then we randomly decide whether to move to state $\nu$ or stay in state $\mu$ with probability $A(\mu,\ nu)$ that we swap to $\nu$. We repeat the process until a defined end point. Normally in the case of the Ising model, this end point is when all the spins are aligned, i.e. the alloy has become ferrous (magnetic). The meat of this method is in determining the probabilities $A(\mu,\nu)$. These probabilities rely on a parameter $\beta$, given by $\beta = \frac{1}{k_B T}$ for $T$ the temperature and $k_B $ the Boltzmann constant. Therefore, $\beta$ is the reciprocal of the “thermodynamic temperature”. The other parameters which $A(\mu,\nu)$ depend upon are the energies at states $\mu$ and $\ nu$, denoted $H_\mu$ and $H_\nu$ respectively. Through a principle known as “detailed balance” we get that \[A(\mu,\nu) = \begin{cases} e^{-\beta(H_\nu – H_\mu)} & \text{ if } H_\nu – H_\mu >0, \\ 1 & \text{otherwise.} \end{cases} \] I will not go into the intricacies of what detailed balance means, but we can see that the greater the increase in energy, the less likely it is that we move to the new state. On the flipside, if the new state has less energy than the initial state, we always swap to it. This fits with our intuition, as nature tends to move towards configurations with minimal energy. Another fact that can be seen from the expression for the transition probabilities A is that the as the temperature increases, it becomes more likely that we keep the new state, even when the new state is of higher energy. Again, this makes sense because a higher temperature means that there is more energy in the system. There is a particular name in mathematics for a series of states with transition probabilities between them. What I have described above is known as a Markov Chain. In a Markov Chain, as seen above, the probability of transitioning to a future state can depend only on the current state. Knowing this means that the known mathematical properties of Markov Chains can be used to analyse the Ising model, giving more tools in a mathematician’s arsenal. Unfortunately, as with any model, this is a massive oversimplification, and it is not sufficient in order to determine phase transitions in three or four dimensions. To do this, Hugo Duminil-Copin had to employ tools from quantum field theory among other disciplines, in the process solving conjectures that had been open since the seventies – years before he was born in 1985. The Man Duminil-Copin comes from a family of teachers and athletes. His mother was a dancer who then became a primary school teacher, and his father was a sports teacher. He has inherited his parents’ love for sport, and famously his childhood dream was to become a handball player. In fact, it was only with one year remaining at high school that he decided instead to become a mathematician. According to Duminil-Copin, one learns sport in the same ways as one learns maths – by repeating the same movements again and again. Duminil-Copin also explained how sport helped him become adaptable and gain resilience. He swapped sports every two years or so, which helped him become very good at starting from scratch and learning something new. Duminil-Copin derives the same enjoyment from maths and sport. “You get pride in maths because you sweat,” he explained. He also is a fan of the collaborative aspects both have the capacity for: “I mostly like team sports. I think that math is a team science.” Duminil-Copin stresses the importance of his collaborators, noting that none of the papers for which he earned his Fields Medal were written alone. Hugo Duminil-Copin is not afraid to show his fallibility and his humanity. When asked by a journalist about how it felt to become a professor at such a young age, he was keen to point out that up until that point, he had done everything at the standard age. He explained how he thinks it is incredibly unhelpful to only talk about those who are successful from a young age and have illustrious careers from the moment they begin. He strongly believes we need to break the fantasy that mathematicians understand everything immediately; he tells without a trace of embarrassment of how he came second to last in a maths test in his penultimate year of high school. Hugo Duminil-Copin breaks many of the stereotypes of mathematicians. He is sporty, he is down-to-earth, he is young and charismatic. So where does that leave his view of himself as a mathematician? In this respect, Duminil-Copin breaks one more misconception about maths, namely that it is not a creative subject. “I like to think of myself as an artist more than as a scientist discovering things,” he says. So, there it is from the lion’s mouth. Hugo Duminil-Copin: the very model of a modern mathematician. 2 comments 1. Es geht darum, wissenschaftlich und der Idee der Falsifikation folgend, eine Hypothese, eine sog. Nullhypothese, die Hypothese meint die Unterstellung, die Möglichkeit, sie darf bis muss rational (vs. irrational) gehalten sein, so-o mit den Mitteln der Empirie, mit Experimenten, die Daten, Gemessenes und das Weltliche meinen, naturwissenschaftlich soz. so-o zu beschießen, dass die Komplementärhypothese wahrscheinlicher wird, wichtig ist hier die Gegensätzlichkeit zur zu testenden Hypothese .Zu : Much of Duminil-Copin’s work deals with probability theory, and he pointed out in his press conference that probability is something people use in their daily lives, even without realising i your journey will take. Thinking probabilistically is second nature to people. Ganz genau. Mit freundlichen Grüßen Dr. Webbaer 2. Bonuskommentar, sich diesen gerne übersetzen lassen, danke, hierzu : The way the human brain deals with probabilities and decision making closely resembles a technique mathematicians regularly use – it simplifies. When determining the trustworthiness of a new friend, there is a wealth of information that could be drawn upon – the thousands of clues in their body language, whether you have mutual friends, the veracity of what they are saying. It is humanly impossible to think through every factor, and even if it were possible, it would take a long time. So, our brains filter the data, sacrificing some accuracy for speed. Einfachheit ‘opfert’ sich sozusagen der ‘Einfachheit’, nicht der ‘Geschwindigkeit’. [1] [2] Einfachheit ist ein im wissenschaftlichen Sinne anzustrebendes Konstrukt. Sicherlich ist zum Wesen der menschlichen Verständigkeit auch so anzumerken : -> Das Volumen eines menschlichen Gehirns liegt bei einem Mann bei durchschnittlich etwa 1,27 Litern, bei einer Frau bei etwa 1,13 Litern. [Quelle]. MFG – WB -> https://de.wikipedia.org/wiki/Sparsamkeitsprinzip Die Moderne Naturwissenschaftlichkeit nähert sich der Welt, im oben genannten skeptizistischen Sinne, an Beleg-Lagen gebunden, auch näherungsweise, ausschnittsartig und an Interessen (!) Wissenschaftlichkeit ist eine Veranstaltung.	{"url":"https://scilogs.spektrum.de/hlf/hugo-duminil-copin-the-maths-the-man/","timestamp":"2024-11-02T18:00:59Z","content_type":"text/html","content_length":"90082","record_id":"<urn:uuid:60764264-f13b-4307-9e1e-df0460e470aa>","cc-path":"CC-MAIN-2024-46/segments/1730477027729.26/warc/CC-MAIN-20241102165015-20241102195015-00128.warc.gz"}
Pierre Alliez, Clément Jamin, Laurent Rineau, Stéphane Tayeb, Jane Tournois, Mariette Yvinec This package is devoted to the generation of isotropic simplicial meshes discretizing 3D domains. The domain to be meshed is a subset of 3D space, required to be bounded. The domain may be connected or composed of multiple components and/or subdivided in several subdomains. Boundary and subdivision surfaces are either smooth or piecewise smooth surfaces, formed with planar or curved surface patches. Surfaces may exhibit 1-dimensional features (e.g. crease edges) and 0-dimensional features (e.g. singular points as corners tips, cusps or darts), that have to be fairly approximated in the mesh. The output mesh is a 3-dimensional triangulation, including subcomplexes that approximate each input domain feature: subdomain, boundary surface patch or input domain feature with dimension 0 or 1. Thus, the output mesh includes a 3D submesh covering each subdomain, a surface mesh approximating each boundary or subdividing surface patch, a polyline approximation for each 1-dimensional feature and of course a vertex on each corner. The main entry points of the package are two global functions that respectively generate and refine such meshes. The mesh generator is customized to output a mesh that fits as much as possible the user needs, for instance in terms of sizing field or with respect to some user customized quality criteria. The meshing engine used in this mesh generator is based on Delaunay refinement [9], [15], [16]. It uses the notion of restricted Delaunay triangulation to approximate 1-dimensional curves and surface patches [2]. Before the refinement, a mechanism of protecting balls is set up on 1-dimensional features, if any, to ensure a fair representation of those features in the mesh, and also to guarantee the termination of the refinement process, whatever may be the input geometry, in particular whatever small angles the boundary and subdivision surface patches may form [7], [8]. The Delaunay refinement is followed by a mesh optimization phase to remove slivers and provide a good quality mesh. Optionally, the meshing and optimization algorithms support multi-core shared-memory architectures to take advantage of available parallelism. Input Domain The domain to be meshed is assumed to be bounded and representable as a pure 3D complex. A 3D complex is a set of faces with dimension 0, 1, 2, and 3 such that all faces are pairwise interior disjoint and the boundary of each face of the complex is the union of lower-dimensional faces of the complex. The 3D complex is pure, meaning that each face is included in a face of dimension 3, so that the complex is entirely described by the set of its 3D faces and their subfaces. However the 3D complex needs not be connected. The set of faces with dimension lower or equal than 2 forms a 2D subcomplex which needs not be manifold, neither pure, nor connected: some 3D faces may have dangling 2D or 1D faces in their boundary faces. In the rest of the documentation, we will refer to the input 3D complex as the input domain. The faces of the input domain with dimension 0, 1, 2, and 3 are called respectively corners, curves, surface patches, and subdomains to clearly distinguish them from the faces of the mesh that are called vertices, edges, facets, and cells. Note that the input complex faces are not required to be linear nor smooth. Surface patches, for instance, may be smooth surface patches or portions of surface meshes with boundaries. Curves may be for instance straight segments, parameterized curves or polylines. Each of those features will be accurately represented in the final mesh. The 0 and 1-dimensional features of the input domain are usually singular points of the subdomain boundaries, however this is not required. Furthermore, those features are not required to cover all the subdomains boundaries singularities but only those that need to be accurately represented in the final mesh. In the following, we say that a domain has features when it has 0 and 1-dimensional features that need to be accurately represented in the mesh, and we call those features exposed features. Therefore, a domain may be without features either because all boundary surface patches are smooth closed surfaces, or simply because the curves joining different surface patches and the singularities of those patches need not be accurately approximated in the final mesh. Note also that input complex faces are not required to be connected. Faces of the input domain are identified by indices. If a subdomain is not connected, its different components receive the same index. Likewise different surface patches, curves, or corners may share the same index. Each connected component of a feature will be accurately represented in the final mesh. Note however that the occurrence of multiply connected faces in the input complex may affect the relevance of internal topological checks performed by the mesh generator. The domain is passed to the mesh generation function as a domain class, often called the oracle, that provides predicates and constructors related to the domain, the subdomains, the boundary surface patches, and also the 0 and 1-dimensional exposed features, if any. Mainly, the oracle provides a predicate to test if a given query point belongs to the domain or not and to find in which subdomain it lies in, in the affirmative case. The domain class also provides predicates and constructors to test the intersection of a query line segment with the boundary surface patches and to construct intersection points, if any. Lastly, if the input domain includes 1-dimensional exposed features, the domain class provides a way to construct sample points on these features. The current implementation provides classes to represent domains bounded by isosurfaces of implicit functions, polyhedral domains, and domains defined through 3D labeled images. Currently, 1-dimensional features may be defined as segments and polyline segments. Output Mesh The resulting mesh is output as a subcomplex of a weighted Delaunay 3D triangulation, in a class providing various iterators on mesh elements. The 3D triangulation provides approximations of the subdomains, surface patches, curves, and corners according to the restricted Delaunay triangulation paradigm. This means that each subdomain is approximated by the union of the tetrahedral cells whose circumcenters are located inside the domain (or subdomain). Each surface patch is approximated by the union of the Delaunay mesh facets whose dual Voronoi edges intersect the surface patch. Such mesh facets are called surface facets in the following. The 1-dimensional exposed features are approximated by sequences of mesh edges and the 0-dimensional exposed features are represented by mesh vertices. It is possible to extract the facets of the complex as a FaceGraph, using the function facets_in_complex_3_to_triangle_mesh(). Delaunay Refinement The mesh generation algorithm is mainly a Delaunay refinement process. The Delaunay refinement is preceded by a protecting phase to ensure an accurate representation of 1-dimensional features, if any, and followed by an optimization phase to achieve a good quality mesh. The Delaunay refinement process is driven by criteria concerning either the size and shape of mesh cells and surface facets. The refinement process terminates when there are no more mesh cells or surface facets violating the criteria. The criteria are designed to achieve a nice spread of the mesh vertices while ensuring the termination of the refinement process. Those criteria may be somehow tuned to the user needs to achieve for instance the respect of a sizing field by mesh elements, some topological conditions on the representation of boundary surfaces in the mesh, and/or some error bound for the approximation of boundary surfaces. To some extent, the user may tune the Delaunay refinement to a prescribed trade-off between mesh quality and mesh density. The mesh density refers to the number of mesh vertices and cells, i.e. to the complexity of the mesh. The mesh quality referred to here is measured by the radius edge ratio of surface facets end mesh cells, where the radius edge ratio of a simplex (triangle or tetrahedron) is the ratio between its circumradius and its shortest edge length. Protection of 0 and 1-dimensional Exposed Features If the domain description includes 0 dimensional features, the corresponding points are inserted into the Delaunay triangulation from the start. If the domain has 1-dimensional exposed features, the method of protecting balls [7], [8] is used to achieve an accurate representation of those features in the mesh and to guarantee that the refinement process terminates whatever may be the dihedral angles formed by input surface patches incident to a given 1-feature or the angles formed by two 1-features incident to a 0-feature. According to this method, the 1-dimensional features are sampled with points and covered by protecting balls centered on the sample points, in such a way that: • no three balls intersect; • no pair of balls centered on different 1-features intersect. The triangulation embedding the mesh is in fact a weighted Delaunay triangulation, and the triangulation is initialized by the insertion of all the protecting balls, regarded as weighted points. The Delaunay refinement process is then launched as before except that refinement points are no longer circumcenters but are weighted circumcenters. All Steiner vertices inserted by the refinement process are given a zero weight. The method guarantees: • that each segment joining two successive centers on a 1-dimensional feature will stay in the triangulation, thus ensuring an accurate approximation of the 1-dimensional features; • that the refinement process will never try to insert a refinement point in the union of the protecting balls, which ensures the termination of the refinement process. Optimization Phase Any tetrahedron that is quasi degenerate has a big radius edge ratio, except those belonging to the family of slivers. A sliver is easily obtained as the convex hull of 4 points close to the equatorial circle of a 3D ball and roughly equally spread along this circle. The Delaunay refinement tracks tetrahedra with big radius edge ratio and therefore eliminates all kinds of badly shaped tetrahedra except slivers. Therefore, some sliver-shaped tetrahedra may still be present in the mesh at the end of the refinement process. The optimization phase aims to eliminate these slivers. The optimization phase is a sequence of optimization processes, amongst the following available optimizers: an ODT-smoother, a Lloyd-smoother, a sliver perturber, and a sliver exuder. The Lloyd and ODT-smoother are global optimizers moving the mesh vertices to minimize a mesh energy. Those optimizers are described respectively in [11], [10] and in [5], [1]. In both cases the mesh energy is the L1 error resulting from the interpolation of the function $ f(x) =x^2$ by a piecewise linear function. In the case of the Lloyd smoother, the interpolation is linear in each Voronoi cell of the set of mesh vertices. In the case of the ODT-smoother, the interpolation is linear in each cell of the Delaunay triangulation of the mesh vertices, hence the name ODT which is an abbreviation for Optimal Delaunay Triangulation. The Lloyd optimizer is known to be blind to the occurrence of slivers in the mesh while the ODT-smoother tends to chase them out. Both of them are global optimizers, meaning that they try to improve the whole mesh rather than focusing on the worst elements. However, both are empirically known to be very efficient as a preliminary step of optimization as they tend to enhance the efficiency of the perturber and/or the exuder applied afterwards, see Figure 60.3 The perturber and the exuder focus on improving the worst mesh elements. The perturber [18] improves the meshes by local changes in the vertices positions aiming to make slivers disappear. The exuder [6] chases the remaining slivers by re-weighting mesh vertices with optimal weights. Each optimization process can be activated or not, and tuned according to the user requirements and the available time. By default, only the perturber and the exuder are activated. Optimization processes are designed to improve mesh quality. However, beware that such an improvement is obtained by perturbing mesh vertices and modifying the mesh connectivity which has an impact on the strict compliance to the refinement criteria. Though a strict compliance to mesh criteria is granted at the end of the Delaunay refinement, this may no longer be true after some optimization processes. Also beware that the default behavior does involve some optimization processes. The tetrahedral mesh generation algorithm by Delaunay refinement provided by this package does not guarantee that all the vertices of the output mesh are actually present in the final mesh. In most cases, all points are used, but if the geometry of the object has small features, compared to the size of the simplices (triangles and tetrahedra), it might be that the Delaunay facets that are selected in the restricted Delaunay triangulation miss some vertices of the triangulation. The concurrent version of the tetrahedral mesh generation algorithm also inserts a small set of auxiliary vertices that belong to the triangulation but are isolated from the complex at the end of the meshing process. These so-called isolated vertices belong to the triangulation but not to any cell of the C3T3. They can be removed using the function remove_isolated_vertices() of As of CGAL 5.6, this package uses Named Parameters to set parameters. More details are provided in Upgrading Code using Boost Parameters to CGAL Named Function Parameters. The Global Functions A 3D mesh generation process is launched through a call to one of the two following functions: template <class C3T3, class MeshDomain, class MeshCriteria, class NamedParameters> const MeshCriteria& criteria, const NamedParameters& np); template <class C3T3, class MeshDomain, class MeshCriteria, class NamedParameters> const MeshDomain& domain, const MeshCriteria& criteria, const NamedParameters& np); void refine_mesh_3(C3T3 &c3t3, const MeshDomain &domain, const MeshCriteria &criteria, const NamedParameters &np=parameters::default_values()) The function refine_mesh_3() is a 3D mesh generator. Definition: refine_mesh_3.h:264 C3T3 make_mesh_3(const MeshDomain &domain, const MeshCriteria &criteria, const NamedParameters &np=parameters::default_values()) The function make_mesh_3() is a 3D mesh generator. Definition: make_mesh_3.h:468 The function make_mesh_3() generates from scratch a mesh of the input domain, while the function refine_mesh_3() refines an existing mesh of the input domain. Note that as the protection of 0 and 1-dimensional features does not rely on Delaunay refinement, the function refine_mesh_3() has no parameter to preserve features. The following sections describe the different template parameters (and their requirements) of these two global functions. The Data Structure The template parameter C3T3 is required to be a model of the concept MeshComplex_3InTriangulation_3, a data structure devised to represent a three dimensional complex embedded in a 3D triangulation. In both functions, an instance of type C3T3 is used to maintain the current approximating simplicial mesh and to represent the final 3D mesh at the end of the procedure. The embedding 3D triangulation is required to be the nested type CGAL::Mesh_triangulation_3::type, provided by the class template CGAL::Mesh_triangulation_3. The type for this triangulation is a wrapper around the class CGAL::Regular_triangulation_3 whose vertex and cell base classes are respectively models of the concepts MeshVertexBase_3 and MeshCellBase_3. The Domain Oracle and the Features Parameter The template parameter MeshDomain is required to be a model of the concept MeshDomain_3. The argument domain of type MeshDomain is the sole link through which the domain to be discretized is known by the mesh generation algorithm. This concept provides, among others, member functions to test whether or not a query segment intersects boundary surfaces, and to compute an intersection point in the affirmative. The MeshDomain_3 concept adds member functions which given a query point tell whether the point lies inside or outside the domain and in which subdomain the point lies, if inside. If the domain description includes 0 and 1-dimensional features that have to be accurately represented in the final mesh, the template parameter MeshDomain is required to be of a model of the concept MeshDomainWithFeatures_3. The concept MeshDomainWithFeatures_3 mainly provides the incidence graph of 0, 1 and 2-dimensional features, and a member function to construct sample points on curves. Users whose domain is a model of MeshDomainWithFeatures_3 can choose to have the corners and curves of the domain represented in the mesh or not, using the following parameters: • parameters::features(domain) sets features according to the domain, i.e. 0 and 1-dimensional features are taken into account if domain is a MeshDomainWithFeatures_3. • parameters::no_features() prevents the representation of 0 and 1-dimensional features in the mesh. This is useful to get a smooth and rough approximation of a domain with features. The Meshing Criteria The template parameter MeshCriteria must be a model of the concept MeshCriteria_3, or a model of the refined concept MeshCriteriaWithFeatures_3 if the domain has exposed features. The argument of type MeshCriteria passed to the mesh generator specifies the size and shape requirements for the tetrahedra in the mesh and for the triangles in the boundary surface mesh. These criteria condition the rules that drive the refinement process. At the end of the refinement process, mesh elements satisfy the criteria. Note that this may not be strictly true anymore after the optimization phase, but this last phase is devised to only improve the mesh quality. The criteria for surface facets are governed by the four following parameters: • facet_angle. This parameter controls the shape of surface facets. Specifically, it is a lower bound for the angle (in degrees) of surface facets. When boundary surfaces are smooth, the termination of the meshing process is guaranteed if this angular bound is at most 30 degrees [9]. • facet_size. This parameter controls the size of surface facets. Each surface facet has a surface Delaunay ball which is a ball circumscribing the surface facet and centered on the surface patch. The parameter facet_size is either a constant or a spatially variable scalar field, providing an upper bound for the radii of surface Delaunay balls. • facet_distance. This parameter controls the approximation error of boundary and subdivision surfaces. Specifically, it is either a constant or a spatially variable scalar field. It provides an upper bound for the distance between the circumcenter of a surface facet and the center of a surface Delaunay ball of this facet. • facet_topology. This parameter controls the set of topological constraints which have to be verified by each surface facet. By default, each vertex of a surface facet has to be located on a surface patch, on a curve, or on a corner. It can also be set to check whether the three vertices of a surface facet belongs to the same surface patch. This has to be done cautiously, as such a criterion needs that each intersection of input surface patches is an input 1-dimensional feature. The criteria for mesh cells are governed by two parameters: • cell_radius_edge_ratio. This parameter controls the shape of mesh cells (but can't filter slivers, as we discussed earlier). It is an upper bound for the ratio between the circumradius of a mesh tetrahedron and its shortest edge. There is a theoretical bound for this parameter: the Delaunay refinement process is guaranteed to terminate for values of cell_radius_edge_ratio bigger than 2. • cell_size. This parameter controls the size of mesh tetrahedra. It is either a scalar or a spatially variable scalar field. It provides an upper bound on the circumradii of the mesh tetrahedra. Figure 60.4 shows how the mesh generation process behaves with respect to these parameters. If the domain has 1-dimensional exposed features, the criteria includes a sizing field to guide the sampling of 1-dimensional features with protecting balls centers. • edge_size. This constant or variable scalar field is used as an upper bound for the distance between two protecting ball centers that are consecutive on a 1-feature. This parameter has to be set to a positive value when 1-dimensional features protection is used. The Optimization Parameters The four additional parameters are optimization parameters. They control which optimization processes are performed and enable the user to tune the parameters of the activated optimization processes. These parameters have internal types which are not described but the library provides global functions to generate appropriate values of these types: These parameters are optional and can be passed in any order. If one parameter is not passed the default value is used. By default, only the perturber and the exuder are activated. Note that whatever may be the optimization processes activated by make_mesh_3() or refine_mesh_3(), they are always launched in the order that is a suborder of the following: ODT-smoother, Lloyd-smoother, perturber, and exuder. The package also provides four global functions to launch each optimization process independently. These functions are useful for advanced experimentation on the efficiency of each optimization method. Note however that the exuder adds on mesh vertices weights that are conditioned by vertices positions. Therefore an exudation process should never be run before a smoother or a perturber. For a maximum efficiency, whatever may be the optimization processes activated, they should be launched in the order that is a suborder of the following: ODT-smoother, Lloyd-smoother, perturber, and template< class C3T3, class MeshDomain > template< class C3T3, class MeshDomain > template< class C3T3, class MeshDomain > template< class C3T3 > Mesh_optimization_return_code perturb_mesh_3(C3T3 &c3t3, const MeshDomain &domain, const NamedParameters &np=parameters::default_values()) The function perturb_mesh_3() is a mesh optimizer that improves the quality of a Delaunay mesh by cha... Definition: perturb_mesh_3.h:105 Mesh_optimization_return_code odt_optimize_mesh_3(C3T3 &c3t3, const MeshDomain &domain, const NamedParameters &np=parameters::default_values()) The function odt_optimize_mesh_3() is a mesh optimization process based on the minimization of a glob... Definition: odt_optimize_mesh_3.h:133 Mesh_optimization_return_code lloyd_optimize_mesh_3(C3T3 &c3t3, const MeshDomain &domain, const NamedParameters &np=parameters::default_values()) The function lloyd_optimize_mesh_3() is a mesh optimization process based on the minimization of a gl... Definition: lloyd_optimize_mesh_3.h:138 Mesh_optimization_return_code exude_mesh_3(C3T3 &c3t3, const NamedParameters &np=parameters::default_values()) The function exude_mesh_3() performs a sliver exudation process on a Delaunay mesh. Definition: exude_mesh_3.h:95 Note that the global functions activating the optimization processes or launching those processes have themselves parameters (see details in reference pages) to tune the optimization process. Parallel Algorithms Enabling parallel meshing and optimization algorithms is achieved through setting the third template parameter of the Mesh_triangulation_3 class to Parallel_tag, when defining the triangulation type. Note that when the user provides his/her own vertex and cell base classes, the MeshVertexBase_3 and MeshCellBase_3 concepts impose additional requirements. Parallel algorithms require the executable to be linked against the Intel TBB library. To control the number of threads used, the user may use the tbb::task_scheduler_init class. See the TBB documentation for more details. Several file formats are supported for writing a mesh: 3D Domains Bounded by Isosurfaces 3D Domains Bounded by Implicit Isosurfaces The following code produces a 3D mesh for a domain whose boundary surface is an isosurface defined by an implicit function. Figure 60.5 shows a cut view of the resulting mesh. Note the use of named parameters in the constructor of the Mesh_criteria instance. File Mesh_3/mesh_implicit_sphere.cpp Mesh_domain domain = Mesh_domain::create_implicit_mesh_domain( sphere_function, Mesh_criteria criteria(params::facet_angle(30).facet_size(0.1).facet_distance(0.025). C3t3 c3t3 = CGAL::make_mesh_3<C3t3>(domain, criteria); Meshing Multiple Domains Construction from a Vector of Implicit Functions The following code produces a 3D mesh for a domain consisting of several subdomains. It uses Implicit_multi_domain_to_labeling_function_wrapper as model of ImplicitFunction which is required by Figure 60.6 shows a view and a cut view of the resulting mesh. File Mesh_3/mesh_implicit_domains.cpp #include <CGAL/Exact_predicates_inexact_constructions_kernel.h> #include <CGAL/Mesh_triangulation_3.h> #include <CGAL/Mesh_complex_3_in_triangulation_3.h> #include <CGAL/Mesh_criteria_3.h> #include <CGAL/Implicit_to_labeling_function_wrapper.h> #include <CGAL/Labeled_mesh_domain_3.h> #include <CGAL/make_mesh_3.h> #include "implicit_functions.h" // IO #include <CGAL/IO/File_medit.h> #ifdef CGAL_CONCURRENT_MESH_3 // Domain typedef FT_to_point_function_wrapper<double, K::Point_3> Function; typedef Function_wrapper::Function_vector Function_vector; // Triangulation // Mesh Criteria typedef Mesh_criteria::Facet_criteria Facet_criteria; typedef Mesh_criteria::Cell_criteria Cell_criteria; int main() // Define functions Function f1(&torus_function); Function f2(&sphere_function<3>); Function_vector v; // Domain (Warning: Sphere_3 constructor uses square radius !) // Set mesh criteria Facet_criteria facet_criteria(30, 0.2, 0.02); // angle, size, approximation Cell_criteria cell_criteria(2., 0.4); // radius-edge ratio, size Mesh_criteria criteria(facet_criteria, cell_criteria); // Mesh generation C3t3 c3t3 = CGAL::make_mesh_3<C3t3>(domain, criteria, params::no_exude().no_perturb()); // Perturbation (maximum cpu time: 10s, targeted dihedral angle: default) // Exudation // Output std::ofstream medit_file("out.mesh"); return 0; The class Implicit_multi_domain_to_labeling_function_wrapper is a helping class to get a function wit... Definition: Implicit_to_labeling_function_wrapper.h:169 Construction from a Vector of Implicit Functions and a Vector of Strings The following code produces a 3D mesh for a domain consisting of several subdomains too. Here, the set of subdomains is given by a vector of vector of signs, whereas the set was built automatically in the previous example. Each subdomain corresponds to a sign vector [s1, s2, ..., sn] where si is the sign of the function fi(p) at a point p of the subdomain. Figure 60.7 shows a view of the resulting mesh. File Mesh_3/mesh_implicit_domains_2.cpp #include <CGAL/Exact_predicates_inexact_constructions_kernel.h> #include <CGAL/Mesh_triangulation_3.h> #include <CGAL/Mesh_complex_3_in_triangulation_3.h> #include <CGAL/Mesh_criteria_3.h> #include <CGAL/Implicit_to_labeling_function_wrapper.h> #include <CGAL/Labeled_mesh_domain_3.h> #include <CGAL/make_mesh_3.h> #include "implicit_functions.h" // IO #include <CGAL/IO/File_medit.h> // Domain typedef FT_to_point_function_wrapper<double, K::Point_3> Function; typedef Function_wrapper::Function_vector Function_vector; // Triangulation // Mesh Criteria typedef Mesh_criteria::Facet_criteria Facet_criteria; typedef Mesh_criteria::Cell_criteria Cell_criteria; int main() // Define functions Function f1(&torus_function); Function f2(&sphere_function<3>); Function_vector v; std::vector<std::string> vps; Mesh_domain domain(Function_wrapper(v, vps), // Set mesh criteria Facet_criteria facet_criteria(30, 0.2, 0.02); // angle, size, approximation Cell_criteria cell_criteria(2., 0.4); // radius-edge ratio, size Mesh_criteria criteria(facet_criteria, cell_criteria); // Mesh generation C3t3 c3t3 = CGAL::make_mesh_3<C3t3>(domain, criteria, params::no_exude().no_perturb()); // Perturbation (maximum cpu time: 10s, targeted dihedral angle: default) // Exudation // Output std::ofstream medit_file("out.mesh"); return 0; Construction of a Hybrid Domain : From an Implicit and a Polyhedral Domain The example code of Mesh_3/mesh_hybrid_mesh_domain.cpp produces a 3D mesh for a domain consisting of several subdomains too. Here, the set of subdomains is given by two subdomains of different types : a domain defined by an implicit sphere, and a domain defined by the triangulated surface of a cube. Figure 60.8 shows the results with or without the protection of the 1D-features. Polyhedral Domains 3D Polyhedral Domains The following code produces a 3D mesh for a domain defined by a polyhedral surface. Figure 60.9 shows the resulting mesh. File Mesh_3/mesh_polyhedral_domain.cpp Polyhedron polyhedron; std::ifstream input(fname); input >> polyhedron; Mesh_domain domain(polyhedron); Mesh_criteria criteria(params::facet_angle(25).facet_size(0.15).facet_distance(0.008). Mesh_criteria new_criteria(params::cell_radius_edge_ratio(3).cell_size(0.03)); Remeshing a Polyhedral Domain with Surfaces In the following example we have a "bounding polyhedron" which defines the meshing domain and two surfaces inside this domain. The surfaces inside the domain may be closed surfaces as well as surfaces with boundaries. In the case of a closed surface the volume delimited by this surface is also considered as inside the domain. previous subsection. File Mesh_3/mesh_polyhedral_domain_with_surface_inside.cpp #include <CGAL/Exact_predicates_inexact_constructions_kernel.h> #include <CGAL/Mesh_triangulation_3.h> #include <CGAL/Mesh_complex_3_in_triangulation_3.h> #include <CGAL/Mesh_criteria_3.h> #include <CGAL/Polyhedron_3.h> #include <CGAL/Polyhedral_mesh_domain_with_features_3.h> #include <CGAL/make_mesh_3.h> #include <CGAL/Timer.h> // Domain // Triangulation Tr,Mesh_domain::Corner_index,Mesh_domain::Curve_index> C3t3; // Criteria int main(int argc, charargv[]) const std::string fname = (argc>1)?argv[1]:CGAL::data_file_path("meshes/horizons.off"); std::ifstream input(fname); const std::string fname2 = (argc>2)?argv[2]:CGAL::data_file_path("meshes/horizons-domain.off"); std::ifstream input2(fname2); Polyhedron sm, smbounding; input >> sm; input2 >> smbounding; std::cerr << "Error: Cannot read file " << fname << std::endl; return EXIT_FAILURE; // Create domain Mesh_domain domain(sm, smbounding); // Get sharp features // Mesh criteria Mesh_criteria criteria(params::edge_size(0.025). // Mesh generation C3t3 c3t3 = CGAL::make_mesh_3<C3t3>(domain, criteria, std::cerr << t. () << " sec." << std::endl; // Output dump_c3t3(c3t3, "out"); The class Polyhedral_mesh_domain_with_features_3 implements a domain whose boundary is a simplicial p... Definition: Polyhedral_mesh_domain_with_features_3.h:110 Remeshing a Polyhedral Surface The following code creates a polyhedral domain, with only one polyhedron, and no "bounding polyhedron", so the volumetric part of the domain will be empty. This enables to remesh a surface, and is equivalent to the function make_surface_mesh(). File Mesh_3/remesh_polyhedral_surface.cpp #include <CGAL/Exact_predicates_inexact_constructions_kernel.h> #include <CGAL/Mesh_triangulation_3.h> #include <CGAL/Mesh_complex_3_in_triangulation_3.h> #include <CGAL/Mesh_criteria_3.h> #include <CGAL/Polyhedral_mesh_domain_with_features_3.h> #include <CGAL/make_mesh_3.h> // Domain // Polyhedron type // Triangulation Tr,Mesh_domain::Corner_index,Mesh_domain::Curve_index> C3t3; // Criteria int main() // Load a polyhedron Polyhedron poly; std::ifstream input(CGAL::data_file_path("meshes/lion-head.off")); input >> poly; std::cerr << "Input geometry is not triangulated." << std::endl; return EXIT_FAILURE; // Create a vector with only one element: the pointer to the polyhedron. std::vector<Polyhedron> poly_ptrs_vector(1, &poly); // Create a polyhedral domain, with only one polyhedron, // and no "bounding polyhedron", so the volumetric part of the domain will be // empty. Mesh_domain domain(poly_ptrs_vector.begin(), poly_ptrs_vector.end()); // Get sharp features domain.detect_features(); //includes detection of borders // Mesh criteria Mesh_criteria criteria(params::edge_size(0.025). // Mesh generation C3t3 c3t3 = CGAL::make_mesh_3<C3t3>(domain, criteria, params::no_perturb().no_exude()); // Output the facets of the c3t3 to an OFF file. The facets will not be // oriented. std::ofstream off_file("out.off"); return off_file.fail() ? EXIT_FAILURE : EXIT_SUCCESS; Domains From 3D Images 3D Domains Bounded by Isosurfaces in 3D Gray-Level Images The following example produces a 3D mesh for a domain whose boundary surface is the isosurface associated to an isovalue inside the input gray-level 3D image. In the distribution you can also find the example Mesh_3/mesh_3D_gray_vtk_image.cpp which can deal with .nii as well as DICOM files as input. File Mesh_3/mesh_3D_gray_image.cpp #include <CGAL/Exact_predicates_inexact_constructions_kernel.h> #include <CGAL/Mesh_triangulation_3.h> #include <CGAL/Mesh_complex_3_in_triangulation_3.h> #include <CGAL/Mesh_criteria_3.h> #include <CGAL/Labeled_mesh_domain_3.h> #include <CGAL/make_mesh_3.h> #include <CGAL/Image_3.h> #include <functional> typedef float Image_word_type; // Domain // Triangulation // Criteria int main(int argc, charargv[]) const std::string fname = (argc>1)?argv[1]:CGAL::data_file_path("images/skull_2.9.inr"); // Load image std::cerr << "Error: Cannot read file " << fname << std::endl; return EXIT_FAILURE; Mesh_domain domain = Mesh_domain::create_gray_image_mesh_domain(image, params::iso_value(2.9f).value_outside(0.f)); // Mesh criteria Mesh_criteria criteria(params::facet_angle(30).facet_size(6).facet_distance(2). // Meshing C3t3 c3t3 = CGAL::make_mesh_3<C3t3>(domain, criteria); // Output std::ofstream medit_file("out.mesh"); return 0; The class Image_3 is a C++ wrapper around the InrImage library. Definition: Image_3.h:11 bool read(const char file) Open a 3D image file. Domains From Segmented 3D Images The following code produces a 3D mesh from a 3D image. The image is a segmented medical image in which each voxel is associated a label in accordance with the tissue the voxel belongs to. The domain is therefore a multi-domain where each subdomain corresponds to a specific tissue. In the following example, the image is read from the file liver.inr.gz which is encoded in the format of the library Inrimage https://www-pequan.lip6.fr/~bereziat/inrimage/. The resulting mesh is shown in Figure 60.12. File Mesh_3/mesh_3D_image.cpp #include <CGAL/Exact_predicates_inexact_constructions_kernel.h> #include <CGAL/Mesh_triangulation_3.h> #include <CGAL/Mesh_complex_3_in_triangulation_3.h> #include <CGAL/Mesh_criteria_3.h> #include <CGAL/Labeled_mesh_domain_3.h> #include <CGAL/make_mesh_3.h> #include <CGAL/Image_3.h> // Domain #ifdef CGAL_CONCURRENT_MESH_3 // Triangulation // Criteria int main(int argc, char argv[]) const std::string fname = (argc>1)?argv[1]:CGAL::data_file_path("images/liver.inr.gz"); std::cerr << "Error: Cannot read file " << fname << std::endl; return EXIT_FAILURE; Mesh_domain domain = Mesh_domain::create_labeled_image_mesh_domain(image); // Mesh criteria Mesh_criteria criteria(params::facet_angle(30).facet_size(6).facet_distance(4). C3t3 c3t3 = CGAL::make_mesh_3<C3t3>(domain, criteria); // Output std::ofstream medit_file("out.mesh"); return 0; Domains From Segmented 3D Images, with Weights When a segmented image is given as input, the generated mesh surface sometimes sticks too closely to the voxels surface, causing an aliasing effect. A solution to generate a smooth and accurate output surface was described by Stalling et al in [17]. It consists in generating a second input image, made of integer coefficients called weights, and use those weights to define smoother domain boundaries. The 3D image of weights can be generated using CGAL::Mesh_3::generate_label_weights() as shown in the following example. File Mesh_3/mesh_3D_weighted_image.cpp Mesh_domain domain = Mesh_domain::create_labeled_image_mesh_domain(image, Mesh_criteria criteria(params::facet_angle(30).facet_size(6).facet_distance(0.5). C3t3 c3t3 = CGAL::make_mesh_3<C3t3>(domain, criteria, params::no_exude(), params::no_perturb()); Domains From 3D Images, with a Custom Initialization The example Mesh_3/mesh_3D_image_with_custom_initialization.cpp is a modification of Mesh_3/mesh_3D_image.cpp. The goal of that example is to show how the default initialization of the triangulation, using random rays, can be replaced by a new implementation. In this case, the initialization detects all connected components in the 3D segmented image, and inserts points in the triangulation for each connected component. For the meshing, in the previous example (Mesh_3/mesh_3D_image.cpp), we called make_mesh_3() as follows. C3t3 c3t3 = CGAL::make_mesh_3<C3t3>(domain, criteria); In the example Mesh_3/mesh_3D_image_with_custom_initialization.cpp, that call is replaced by: 1. the creation of an empty c3t3 object, 2. a call to a non-documented function initialize_triangulation_from_labeled_image() that inserts points in the triangulation, 3. then the call to refine_mesh_3(). C3t3 c3t3; static_cast<unsigned char>(0)); The code of the function initialize_triangulation_from_labeled_image() is in the non-documented header CGAL/Mesh_3/initialize_triangulation_from_labeled_image.h. As it is undocumented and may be removed or modified at any time, if you wish to use it then you should copy-paste it to your user code. The code of that function is rather complicated. The following lines show how to insert new points in the c3t3 object, with the calls to MeshVertexBase_3::set_dimension() and MeshVertexBase_3::set_index(). Vertex_handle v = tr.insert(pi); // `v` could be null if `pi` is hidden by other vertices of `tr`. CGAL_assertion(v != Vertex_handle()); c3t3.set_dimension(v, 2); // by construction, points are on surface c3t3.set_index(v, index); The value of index must be consistent with the possible values of Mesh_domain::Index. In CGAL/Mesh_3/initialize_triangulation_from_labeled_image.h, it is constructed using the API of the mesh domain, as follows. First the functor construct_intersect is created typename Mesh_domain::Construct_intersection construct_intersection = then the Mesh_domain::Intersection object (a tuple with three elements) is constructed using a call to the functor construct_intersection const typename Mesh_domain::Intersection intersect = construct_intersection(Segment_3(seed_point, test)); and eventually index is the element #1 of intersect. const typename Mesh_domain::Index index = std::get<1>(intersect); The result of the custom initialization can be seen in Figure 60.14. The generated 3D image contains a big sphere at the center, and 50 smaller spheres, generated randomly. Without the custom initialization, only the biggest component (the sphere at the center) was initialized and meshed. With the custom initialization, the initial c3t3 object contains points on all connected components, and all spheres are meshed. Note that the example Mesh_3/mesh_3D_image_with_custom_initialization.cpp also shows how to create a 3D image using the undocumented API of CGAL_ImageIO. The code of the function random_labeled_image() is in the header file Mesh_3/random_labeled_image.h. The example Mesh_3/mesh_3D_gray_image_with_custom_initialization.cpp is another custom initialization example, for meshing of 3D gray-level images. Similarly to the segmented image example above, the code consists in: 1. the creation of an empty c3t3 object, 2. a call to a non-documented function initialize_triangulation_from_gray_image() that inserts points in the triangulation, 3. then the call to refine_mesh_3(). C3t3 c3t3; The code of the function initialize_triangulation_from_gray_image() is in the non-documented header CGAL/Mesh_3/initialize_triangulation_from_gray_image.h. As it is undocumented and may be removed or modified at any time, if you wish to use it then you should copy-paste it to your user code. Using Variable Sizing Field Sizing Field as an Analytical Function The following example shows how to use an analytical function as sizing field. File Mesh_3/mesh_implicit_sphere_variable_size.cpp #include <CGAL/Exact_predicates_inexact_constructions_kernel.h> #include <CGAL/Mesh_triangulation_3.h> #include <CGAL/Mesh_complex_3_in_triangulation_3.h> #include <CGAL/Mesh_criteria_3.h> #include <CGAL/Labeled_mesh_domain_3.h> #include <CGAL/make_mesh_3.h> // Domain typedef K::FT FT; typedef K::Point_3 Point; typedef FT (Function)(const Point&); #ifdef CGAL_CONCURRENT_MESH_3 // Triangulation // Criteria // Sizing field struct Spherical_sizing_field typedef ::FT FT; typedef Point Point_3; FT operator()( Point_3& p, const int const Index // Function FT sphere_function (const Point& p) int main() Mesh_domain domain = Mesh_domain::create_implicit_mesh_domain // Mesh criteria Spherical_sizing_field size; Mesh_criteria criteria(params::facet_angle(30).facet_size(0.1).facet_distance(0.025). // Mesh generation C3t3 c3t3 = CGAL::make_mesh_3<C3t3>(domain, criteria, params::no_exude(). // Output std::ofstream medit_file("out.mesh"); return 0; unspecified_type no_perturb() The function parameters::no_perturb() enables the user to tell mesh generation global functions make_... Different Sizing Field for Different Subdomains The following example shows how to use different size for different organs in a 3D medical image. File Mesh_3/mesh_3D_image_variable_size.cpp #include <CGAL/Exact_predicates_inexact_constructions_kernel.h> #include <CGAL/Mesh_triangulation_3.h> #include <CGAL/Mesh_complex_3_in_triangulation_3.h> #include <CGAL/Mesh_criteria_3.h> #include <CGAL/Mesh_constant_domain_field_3.h> #include <CGAL/Labeled_mesh_domain_3.h> #include <CGAL/make_mesh_3.h> #include <CGAL/Image_3.h> // Domain #ifdef CGAL_CONCURRENT_MESH_3 // Triangulation // Criteria Mesh_domain::Index> Sizing_field; int main(int argc, char* argv[]) const std::string fname = (argc>1)?argv[1]:CGAL::data_file_path("images/liver.inr.gz"); // Loads image std::cerr << "Error: Cannot read file " << fname << std::endl; return EXIT_FAILURE; // Domain Mesh_domain domain = Mesh_domain::create_labeled_image_mesh_domain(image); // Sizing field: set global size to 8 and kidney size (label 127) to 3 double kidney_size = 3.; int volume_dimension = 3; Sizing_field size(8); size.set_size(kidney_size, volume_dimension, // Mesh criteria Mesh_criteria criteria(params::facet_angle(30).facet_size(6).facet_distance(2). // Meshing C3t3 c3t3 = CGAL::make_mesh_3<C3t3>(domain, criteria); // Output std::ofstream medit_file("out.mesh"); return 0; The class Mesh_constant_domain_field_3 is a model of concept MeshDomainField_3. Definition: Mesh_constant_domain_field_3.h:51 Lipschitz Sizing Field The following example shows how to use another custom sizing function, that is k-Lipschitz. For each subdomain, the user provides the parameter k, a minimal size and maximal size for cells. File Mesh_3/mesh_polyhedral_domain_with_lipschitz_sizing.cpp #include <CGAL/Exact_predicates_inexact_constructions_kernel.h> #include <CGAL/Mesh_triangulation_3.h> #include <CGAL/Mesh_complex_3_in_triangulation_3.h> #include <CGAL/Mesh_criteria_3.h> #include <CGAL/Polyhedral_mesh_domain_with_features_3.h> #include <CGAL/make_mesh_3.h> #include <CGAL/Mesh_3/experimental/Lipschitz_sizing_polyhedron.h> typedef K::FT FT; // Domain #ifdef CGAL_CONCURRENT_MESH_3 // Triangulation Tr,Mesh_domain::Corner_index,Mesh_domain::Curve_index> C3t3; // Criteria // Sizing field typedef CGAL::Mesh_3::Lipschitz_sizing<K, Mesh_domain, Mesh_domain::AABB_tree> Lip_sizing; int main(int argc, charargv[]) const std::string fname = (argc>1) ? argv[1] : CGAL::data_file_path("meshes/fandisk.off"); std::ifstream input(fname); Polyhedron polyhedron; input >> polyhedron; if (input.fail()){ std::cerr << "Error: Cannot read file " << fname << std::endl; return EXIT_FAILURE; std::cerr << "Input geometry is not triangulated." << std::endl; return EXIT_FAILURE; // Create domain Mesh_domain domain(polyhedron); // Get sharp features // Create Lipschitz sizing field Lip_sizing lip_sizing(domain, &domain.aabb_tree()); FT min_size = 0.02; lip_sizing.add_parameters_for_subdomain(1, //subdomain id 0.3, //k 0.5); //max_size // Mesh criteria Mesh_criteria criteria(params::edge_size(min_size). // Mesh generation C3t3 c3t3 = CGAL::make_mesh_3<C3t3>(domain, criteria); // Output std::ofstream medit_file("out.mesh"); return EXIT_SUCCESS; Meshing Domains with Sharp Features 3D Polyhedral Domain with Edges The following example shows how to generate a mesh from a polyhedral surface. The output mesh conforms to the sharp features of the input surface. File Mesh_3/mesh_polyhedral_domain_with_features.cpp #include <CGAL/Exact_predicates_inexact_constructions_kernel.h> #include <CGAL/Mesh_triangulation_3.h> #include <CGAL/Mesh_complex_3_in_triangulation_3.h> #include <CGAL/Mesh_criteria_3.h> #include <CGAL/Polyhedral_mesh_domain_with_features_3.h> #include <CGAL/make_mesh_3.h> #include <CGAL/IO/output_to_vtu.h> // Domain #ifdef CGAL_CONCURRENT_MESH_3 // Triangulation Tr,Mesh_domain::Corner_index,Mesh_domain::Curve_index> C3t3; // Criteria int main(int argc, charargv[]) const std::string fname = (argc>1)?argv[1]:CGAL::data_file_path("meshes/fandisk.off"); std::ifstream input(fname); Polyhedron polyhedron; input >> polyhedron; std::cerr << "Error: Cannot read file " << fname << std::endl; return EXIT_FAILURE; std::cerr << "Input geometry is not triangulated." << std::endl; return EXIT_FAILURE; // Create domain Mesh_domain domain(polyhedron); // Get sharp features // Mesh criteria Mesh_criteria criteria(params::edge_size(0.025). // Mesh generation C3t3 c3t3 = CGAL::make_mesh_3<C3t3>(domain, criteria); // Output std::ofstream file("out.vtu"); // Could be replaced by: // CGAL::IO::write_MEDIT(file, c3t3); return EXIT_SUCCESS; void output_to_vtu(std::ostream &os, const C3T3 &c3t3, Mode mode=BINARY) Implicit Domain With 1D Features The following example shows how to generate a mesh from an implicit domain. We add by hand the intersection of the spheres as a sharp feature. File Mesh_3/mesh_two_implicit_spheres_with_balls.cpp #include <CGAL/Exact_predicates_inexact_constructions_kernel.h> #include <CGAL/Mesh_triangulation_3.h> #include <CGAL/Mesh_complex_3_in_triangulation_3.h> #include <CGAL/Mesh_criteria_3.h> #include <CGAL/Labeled_mesh_domain_3.h> #include <CGAL/Mesh_domain_with_polyline_features_3.h> #include <CGAL/make_mesh_3.h> // Kernel // Domain typedef K::FT FT; typedef K::Point_3 Point; typedef FT (Function)(const Point&); // Polyline typedef std::vector<Point> Polyline_3; typedef std::list<Polyline_3> Polylines; #ifdef CGAL_CONCURRENT_MESH_3 // Triangulation Tr,Mesh_domain::Corner_index,Mesh_domain::Curve_index> C3t3; // Criteria // Function FT sphere_function1 (const Point& p) FT sphere_function2 (const Point& p) FT sphere_function (const Point& p) if(sphere_function1(p) < 0 \|\| sphere_function2(p) < 0) return -1; return 1; #include <cmath> int main() // Domain (Warning: Sphere_3 constructor uses squared radius !) Mesh_domain domain = K::Sphere_3(Point(1, 0, 0), 6.)); // Mesh criteria Mesh_criteria criteria(params::edge_size(0.15). // Create edge that we want to preserve Polylines polylines (1); Polyline_3& polyline = polylines.front(); for(int i = 0; i < 360; ++i) Point p (1, std::cos(iCGAL_PI/180), std::sin(iCGAL_PI/180)); polyline.push_back(polyline.front()); // close the line // Insert edge in domain domain.add_features(polylines.begin(), polylines.end()); // Mesh generation without feature preservation C3t3 c3t3 = CGAL::make_mesh_3<C3t3>(domain, criteria, std::ofstream medit_file("out-no-protection.mesh"); // Mesh generation with feature preservation c3t3 = CGAL::make_mesh_3<C3t3>(domain, criteria); // Output return 0; The class Mesh_domain_with_polyline_features_3 enables the user to add some 0- and 1-dimensional feat... Definition: Mesh_domain_with_polyline_features_3.h:535 Domains from Segmented 3D Images, with 1D Features The example Mesh_3/mesh_3D_image_with_features.cpp is a modification of Mesh_3/mesh_3D_image.cpp. That example shows how to generate a mesh from a 3D labeled image (also known as "a segmented image"), that has 2D surfaces that intersect the box corresponding to the boundary of the image. The intersection of the 2D surface with the bounding box of the image is composed of 1D curves, and must be defined as 1D-features in the domain. The first modification is the type of the mesh domain. Instead of being Labeled_mesh_domain_3, it is a Mesh_domain_with_polyline_features_3 templated by a Labeled_mesh_domain_3. #include <CGAL/Exact_predicates_inexact_constructions_kernel.h> #include <CGAL/Mesh_domain_with_polyline_features_3.h> #include <CGAL/Labeled_mesh_domain_3.h> In the main() function, the domain is created with an additional argument - a dedicated functor that computes the one-dimensional features, which are then added to the domain. Mesh_domain domain = Mesh_domain::create_labeled_image_mesh_domain(image, Functor for feature detection in labeled images. Definition: Detect_features_in_image.h:263 The CGAL::Mesh_3::Detect_features_in_image functor is defined in its own header file. It computes the one-dimensional features that correspond to the intersections of the bounding box of the image with the surfaces defined by the image, as well as polylines that lie at the intersection of three or more subdomains (including the outside). It then constructs a graph of these polyline features. The named constructor adds this feature graph to the domain for later feature protection. The original feature detection algorithm was described in [12], which provides a list of possible voxel configurations. The feature detection implemented in CGAL generalizes this description. The example Mesh_3/mesh_3D_image_with_features.cpp shows how user-specified input polylines can further be added as 1D features to the mesh domain. Mesh_domain domain = Mesh_domain::create_labeled_image_mesh_domain(image, params::input_features = std::cref(features_inside));//use std::cref to avoid a copy Functor for feature detection in labeled images. Definition: Detect_features_on_image_bbox.h:76 In the meshing criteria, if 1D features are added to the domain, the user can define the parameter edge_size of the criteria class Mesh_criteria_3, as follows, to set up an upper bound on the length of the edges of the 1D-mesh corresponding to the 1D-features. Mesh_criteria criteria(params::edge_size = 6., params::facet_angle = 30, params::facet_size = 6, params::facet_distance = 4, params::cell_radius_edge_ratio = 3, params::cell_size = 8); The rest of the example is similar to Mesh_3/mesh_3D_image.cpp. Figure 60.20 shows the results without or with the protection of the 1D-features. Polyhedral Complex The following example shows how to generate a mesh from a polyhedral complex that forms a bounded domain. The domain is defined by a group of polyhedral surfaces that are conformal. For each facet of the input polyhedral surfaces, the ids of incident subdomains are known. The input surfaces intersect along polylines that are considered as sharp features, and protected as for all domain types in this Section. File Mesh_3/mesh_polyhedral_complex.cpp #include <CGAL/Exact_predicates_inexact_constructions_kernel.h> #include <CGAL/Mesh_triangulation_3.h> #include <CGAL/Mesh_complex_3_in_triangulation_3.h> #include <CGAL/Mesh_criteria_3.h> #include <CGAL/Polyhedral_complex_mesh_domain_3.h> #include <CGAL/make_mesh_3.h> #include <cstdlib> #include <cassert> // Domain #ifdef CGAL_CONCURRENT_MESH_3 // Triangulation Tr,Mesh_domain::Corner_index,Mesh_domain::Curve_index> C3t3; // Criteria const char* const filenames[] = { const std::pair<int, int> incident_subdomains[] = { std::make_pair(0, 1), std::make_pair(1, 3), std::make_pair(2, 0), std::make_pair(2, 1), std::make_pair(2, 3), std::make_pair(3, 0), int main() const std::size_t nb_patches = sizeof(filenames) / sizeof(const char); assert(sizeof(incident_subdomains) == nb_patches sizeof(std::pair<int, int>)); std::vector<Polyhedron> patches(nb_patches); for(std::size_t i = 0; i < nb_patches; ++i) { std::ifstream input(CGAL::data_file_path(filenames[i])); if(!(input >> patches[i])) { std::cerr << "Error reading " << CGAL::data_file_path(filenames[i]) << " as a polyhedron!\n"; return EXIT_FAILURE; // Create domain Mesh_domain domain(patches.begin(), patches.end(), incident_subdomains, incident_subdomains+nb_patches); domain.detect_features(); //includes detection of borders // Mesh criteria Mesh_criteria criteria(params::edge_size(8). // Mesh generation C3t3 c3t3 = CGAL::make_mesh_3<C3t3>(domain, criteria); // Output std::ofstream medit_file("out.mesh"); return EXIT_SUCCESS; The class Polyhedral_complex_mesh_domain_3 implements a domain defined by a collection of polyhedral ... Definition: Polyhedral_complex_mesh_domain_3.h:120 Figure 60.21 shows the results without or with the protection of the 1D-features. Sizing Field for Feature Edges The following example shows how to generate a mesh from a polyhedral complex or a polyhedral surface with polyline features. Polyline features are covered by a set of so-called "protecting balls" which sizes are highly related to the edge length, and are driven by the size component of Mesh_edge_criteria_3 (see Section Protection of 0 and 1-dimensional Exposed Features). The ideal size can be computed using Sizing_field_with_aabb_tree that helps start the feature protection and one-dimensional meshing process with a good initial guess. To fit the protecting balls requirements, no protecting ball can have its radius larger than half of the distance from the corresponding vertex (its center), to surface patches the current polyline feature does not belong to. File Mesh_3/mesh_polyhedral_domain_with_features_sizing.cpp #include <CGAL/Exact_predicates_inexact_constructions_kernel.h> #include <CGAL/Mesh_triangulation_3.h> #include <CGAL/Mesh_complex_3_in_triangulation_3.h> #include <CGAL/Mesh_criteria_3.h> #include <CGAL/Surface_mesh.h> #include <CGAL/Polyhedral_mesh_domain_with_features_3.h> #include <CGAL/make_mesh_3.h> #include <CGAL/Sizing_field_with_aabb_tree.h> // Domain using Polyhedron = CGAL::Surface_mesh<K::Point_3>; #ifdef CGAL_CONCURRENT_MESH_3 // Triangulation Tr,Mesh_domain::Corner_index,Mesh_domain::Curve_index> C3t3; // Criteria int main(int argc, charargv[]) const std::string fname = (argc>1)?argv[1]:CGAL::data_file_path("meshes/fandisk.off"); std::ifstream input(fname); Polyhedron polyhedron; input >> polyhedron; std::cerr << "Error: Cannot read file " << fname << std::endl; return EXIT_FAILURE; // Create domain Mesh_domain domain(polyhedron); // Get sharp features // Mesh criteria Features_sizing_field edges_sizing_field(0.07, domain); Mesh_criteria criteria(params::edge_size(edges_sizing_field). // Mesh generation C3t3 c3t3 = CGAL::make_mesh_3<C3t3>(domain, criteria, params::no_exude().no_perturb()); // Output CGAL::dump_c3t3(c3t3, "out_sizing_field_with_aabb_tree"); return EXIT_SUCCESS; The class Sizing_field_with_aabb_tree is a model of concept MeshDomainField_3. Definition: Sizing_field_with_aabb_tree.h:73 Approximation Criterion for Feature Edges The following example shows how to generate a mesh with an approximation error criterion. Polyline features are covered by a set of so-called "protecting balls" which sizes are highly related to the edge_size and edge_distance criteria. This edge distance is driven by the distance component of Mesh_edge_criteria_3 (see Section Protection of 0 and 1-dimensional Exposed Features). The parameter edge_distance enables the subdivision of edges that are too far away from the corresponding input sharp features. It may result in an output mesh with shorter edges along the input polyline features. File Mesh_3/mesh_polyhedral_domain_with_edge_distance.cpp #include <CGAL/Exact_predicates_inexact_constructions_kernel.h> #include <CGAL/Mesh_triangulation_3.h> #include <CGAL/Mesh_complex_3_in_triangulation_3.h> #include <CGAL/Mesh_criteria_3.h> #include <CGAL/Polyhedral_mesh_domain_with_features_3.h> #include <CGAL/make_mesh_3.h> // Domain #ifdef CGAL_CONCURRENT_MESH_3 // Triangulation Tr,Mesh_domain::Corner_index,Mesh_domain::Curve_index> C3t3; // Criteria int main(int argc, charargv[]) const std::string fname = (argc>1)?argv[1]:CGAL::data_file_path("meshes/u_arch.off"); std::ifstream input(fname); Polyhedron polyhedron; input >> polyhedron; std::cerr << "Error: Cannot read file " << fname << std::endl; return EXIT_FAILURE; std::cerr << "Input geometry is not triangulated." << std::endl; return EXIT_FAILURE; // Create domain Mesh_domain domain(polyhedron); // Get sharp features // Mesh criteria Mesh_criteria criteria(params::edge_size(4).edge_distance(0.01). // Mesh generation C3t3 c3t3 = CGAL::make_mesh_3<C3t3>(domain, criteria); // Output //CGAL::dump_c3t3(c3t3, "out"); return EXIT_SUCCESS; Tuning Mesh Optimization In the previous examples, the mesh generation is launched through a call make_mesh_3() with a minimal number of parameters. In such cases, the default optimization strategy is applied: after the Delaunay refinement process two optimization steps are performed, a perturbation and a sliver exudation. The following examples show how to disable default optimization steps and how to tune the parameters of optimization steps. Disabling Exudation and Tuning Perturbation In this first example, we show how to disable the exudation step. The optimization phase after the refinement includes only a perturbation phase which is launched with no time bound and an objective of 10 degrees for the minimum dihedral angle of the mesh. The example shows two ways of achieving the same result. The first way issues a single call to make_mesh_3() with the required optimization process activated and tuned. In the second way, make_mesh_3() is first called without any optimization process and the resulting mesh is next optimized through a call to perturb_mesh_3() with tuned File Mesh_3/mesh_optimization_example.cpp #include <CGAL/Exact_predicates_inexact_constructions_kernel.h> #include <CGAL/Mesh_triangulation_3.h> #include <CGAL/Mesh_complex_3_in_triangulation_3.h> #include <CGAL/Mesh_criteria_3.h> #include <CGAL/Labeled_mesh_domain_3.h> #include <CGAL/make_mesh_3.h> #include <CGAL/Image_3.h> // Domain #ifdef CGAL_CONCURRENT_MESH_3 // Triangulation // Mesh Criteria int main(int argc, char* argv[]) const std::string fname = (argc>1)?argv[1]:CGAL::data_file_path("images/liver.inr.gz"); // Domain std::cerr << "Error: Cannot read file " << fname << std::endl; return EXIT_FAILURE; Mesh_domain domain = Mesh_domain::create_labeled_image_mesh_domain(image); // Mesh criteria Mesh_criteria criteria(params::facet_angle(30).facet_size(5).facet_distance(1.5). // Mesh generation and optimization in one call (sliver_bound is the // targeted dihedral angle in degrees) C3t3 c3t3 = CGAL::make_mesh_3<C3t3>(domain, criteria, // Mesh generation and optimization in several call C3t3 c3t3_bis = CGAL::make_mesh_3<C3t3>(domain, criteria, // Output std::ofstream medit_file("out.mesh"); std::ofstream medit_file_bis("out_bis.mesh"); return 0; Using Lloyd Global Optimization In this second example, we show how to call the Lloyd optimization on the mesh, followed by a call to exudation. We set a time bound of 30s for the Lloyd optimization. We set a time bound of 10s and a sliver bound of 10 degrees for the exuder. File Mesh_3/mesh_optimization_lloyd_example.cpp #include <CGAL/Exact_predicates_inexact_constructions_kernel.h> #include <CGAL/Mesh_triangulation_3.h> #include <CGAL/Mesh_complex_3_in_triangulation_3.h> #include <CGAL/Mesh_criteria_3.h> #include <CGAL/Labeled_mesh_domain_3.h> #include <CGAL/make_mesh_3.h> #include <CGAL/Image_3.h> // Domain #ifdef CGAL_CONCURRENT_MESH_3 // Triangulation // Mesh Criteria int main(int argc, char*argv[]) const std::string fname = (argc>1)?argv[1]:CGAL::data_file_path("images/liver.inr.gz"); // Domain std::cerr << "Error: Cannot read file " << fname << std::endl; return EXIT_FAILURE; Mesh_domain domain = Mesh_domain::create_labeled_image_mesh_domain(image); // Mesh criteria Mesh_criteria criteria(params::facet_angle(30).facet_distance(1.2). // Mesh generation and optimization in one call C3t3 c3t3 = CGAL::make_mesh_3<C3t3>(domain, criteria, // Mesh generation and optimization in several call C3t3 c3t3_bis = CGAL::make_mesh_3<C3t3>(domain, criteria, // Output std::ofstream medit_file("out.mesh"); std::ofstream medit_file_bis("out_bis.mesh"); return 0; We provide here some benchmarks of the performance of the mesh generation algorithms. Delaunay Refinement The computer used for benchmarking is a PC running Linux64 with two Intel Xeon CPU X5450 clocked at 3.00 GHz with 32GB of RAM. The program has been compiled with g++ v4.3.2 with the -O3 option. These benchmarks have been done using CGAL v3.8. Note that these benchmarks were obtained with the sequential version of the algorithm, which does not take advantage of multi-core architectures. See the next section for performance of parallel algorithms. We study the refinement part of the mesh generation engine in this section. We give the CPU time (measured by Timer) using the 3 provided oracles. In all experiments, we produce well shaped elements: we set the facet angle bound and the radius edge bound to their theoretical limit (resp. 30 degrees and 2). We also use the same uniform sizing field for facets and cells. Implicit Function We mesh an analytical sphere of radius 1. Size bound vertices nb facets nb tetrahedra nb CPU Time (s) vertices/second 0.2 499 488 2,299 0.0240 20,800 0.1 3,480 2,046 18,756 0.146 23,800 0.05 25,556 8,274 149,703 1.50 17,000 0.025 195,506 33,212 1,194,727 17.4 11,200 0.0125 1,528,636 134,810 9,547,772 179 8,530 Polyhedral Domain We mesh a volume bounded by a closed triangulated surface made of about 50,000 vertices and 100,000 triangles. Figure 60.23 shows the mesh obtained when size is set to 0.005. Size bound vertices nb facets nb tetrahedra nb CPU Time (s) vertices/second 0.04 423 717 1,332 0.488 866 0.02 2,638 3,414 10,957 2.64 998 0.01 18,168 15,576 90,338 13.9 1,310 0.005 129,442 64,645 722,018 66.7 1,940 0.0025 967,402 263,720 5,756,491 348 2,780 3D Image We mesh image number 2 from the 3D-IRCADb-01 (available at https://www.ircad.fr/research/data-sets/liver-segmentation-3d-ircadb-01/) public database. The size of this image is 512x512x172 voxels (about 45M voxels). The size of the voxels is 0.78mm x 0.78mm x 1.6mm. Figure 60.24 shows the mesh obtained for size set to 4. Size bound (mm) vertices nb facets nb tetrahedra nb CPU Time (s) vertices/second 16 3,898 4,099 20,692 0.344 11,300 8 34,117 27,792 199,864 3.09 11,000 4 206,566 86,180 1,253,694 22.4 9,230 2 1,546,196 329,758 9,617,278 199 7,780 Parallel Performance We provide below speed-up charts generated using the parallel version of the meshing algorithms of CGAL 4.5. The machine used is a PC running Windows 7 64-bits with two 6-core Intel Xeon CPU X5660 clocked at 2.80 GHz with 32GB of RAM. The program has been compiled with Microsoft Visual C++ 2012 in Release mode. Figure Figure 60.25 shows mesh refinement speed-up, and figure Figure 60.26 shows Lloyd optimization speed-up. ODT optimization exhibits similar speed-up. Design and Implementation History Theoretical Foundations The CGAL mesh generation package implements a meshing engine based on the method of Delaunay refinement introduced by Chew [9] and Ruppert [15] and pioneered in 3D by Shewchuk [16]. It uses the notion of restricted Delaunay triangulation to approximate 1-dimensional curved features and curved surface patches and rely on the work of Boissonnat and Oudot [2] and Oudot et al. [13] to achieve accurate representation of boundary and subdividing surfaces in the mesh. The mechanism of protecting balls, used to ensure a fair representation of 1-dimensional features, if any, and the termination of the refinement process whatever may be the input geometry, in particular whatever small dihedral angles may form the boundary and subdivision surface patches, was pioneered by Cheng et al. [8] and further experimented by Dey, Levine et al. [7]. The optimization phase involves global optimization processes, a perturbation process and a sliver exudation process. The global optimizers are based on Lloyd smoothing [11], [10] and odt smoothing [5], [1], where odt means optimal Delaunay triangulation. The perturbation process is mainly based on the work of Tournois [20] and Tournois et al. [19], while the exudation process is, the now famous, optimization by weighting described in Edelsbrunner et al. [6]. Implementation History Work on the package Mesh_3 started during the PhD thesis of Laurent Rineau advised by Mariette Yvinec. A code prototype, together with a first version of design and specifications [14] came out of their collaboration. From the beginning of 2009, most of the work has been performed by Stéphane Tayeb, in collaboration with Mariette Yvinec, Laurent Rineau, Pierre Alliez and Jane Tournois. First, Stéphane released the first public version of the package, implementing the specifications written by Laurent and Mariette. The optimization processes are heavily based on the work of Jane Tournois and Pierre Alliez during the PhD of Jane advised by Pierre. The optimization phase was imported in the mesh generation package by Stéphane Tayeb and appeared first in release 3.6 of CGAL. In collaboration with Laurent Rineau, Stéphane also added demos and examples. After some experiments on medical imaging data performed by Dobrina Boltcheva et al. [4], [3], the handling of 1-dimensional features was worked out by Laurent Rineau, Stéphane Tayeb and Mariette Yvinec. It appeared first in the release 3.8 of CGAL. In 2013, Clément Jamin made the meshing and optimization algorithms parallel on multi-core shared-memory architectures.	{"url":"https://cgal.geometryfactory.com/CGAL/doc/master/Mesh_3/index.html","timestamp":"2024-11-09T00:12:12Z","content_type":"application/xhtml+xml","content_length":"239729","record_id":"<urn:uuid:054b72f0-9e9f-4599-ab85-47e123ba76b6>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00672.warc.gz"}
3029 - Minimum Time to Revert Word to Initial State I3029 - Minimum Time to Revert Word to Initial State I Welcome to Subscribe On Youtube You are given a 0-indexed string word and an integer k. At every second, you must perform the following operations: • Remove the first k characters of word. • Add any k characters to the end of word. Note that you do not necessarily need to add the same characters that you removed. However, you must perform both operations at every second. Return the minimum time greater than zero required for word to revert to its initial state. Example 1: Input: word = "abacaba", k = 3 Output: 2 Explanation: At the 1st second, we remove characters "aba" from the prefix of word, and add characters "bac" to the end of word. Thus, word becomes equal to "cababac". At the 2nd second, we remove characters "cab" from the prefix of word, and add "aba" to the end of word. Thus, word becomes equal to "abacaba" and reverts to its initial state. It can be shown that 2 seconds is the minimum time greater than zero required for word to revert to its initial state. Example 2: Input: word = "abacaba", k = 4 Output: 1 Explanation: At the 1st second, we remove characters "abac" from the prefix of word, and add characters "caba" to the end of word. Thus, word becomes equal to "abacaba" and reverts to its initial state. It can be shown that 1 second is the minimum time greater than zero required for word to revert to its initial state. Example 3: Input: word = "abcbabcd", k = 2 Output: 4 Explanation: At every second, we will remove the first 2 characters of word, and add the same characters to the end of word. After 4 seconds, word becomes equal to "abcbabcd" and reverts to its initial state. It can be shown that 4 seconds is the minimum time greater than zero required for word to revert to its initial state. • 1 <= word.length <= 50 • 1 <= k <= word.length • word consists only of lowercase English letters. Solution 1: Enumeration Let’s assume that if we can restore word to its initial state with only one operation, it means that word[k:] is a prefix of word, i.e., word[k:] == word[:n-k]. If there are multiple operations, let’s assume $i$ is the number of operations, then it means that word[ki:] is a prefix of word, i.e., word[ki:] == word[:n-ki]. Therefore, we can enumerate the number of operations and check whether word[ki:] is a prefix of word. If it is, then return $i$. The time complexity is $O(n^2)$, and the space complexity is $O(n)$. Here, $n$ is the length of word. • class Solution { public int minimumTimeToInitialState(String word, int k) { int n = word.length(); for (int i = k; i < n; i += k) { if (word.substring(i).equals(word.substring(0, n - i))) { return i / k; return (n + k - 1) / k; • class Solution { int minimumTimeToInitialState(string word, int k) { int n = word.size(); for (int i = k; i < n; i += k) { if (word.substr(i) == word.substr(0, n - i)) { return i / k; return (n + k - 1) / k; • class Solution: def minimumTimeToInitialState(self, word: str, k: int) -> int: n = len(word) for i in range(k, n, k): if word[i:] == word[:-i]: return i // k return (n + k - 1) // k • func minimumTimeToInitialState(word string, k int) int { n := len(word) for i := k; i < n; i += k { if word[i:] == word[:n-i] { return i / k return (n + k - 1) / k • function minimumTimeToInitialState(word: string, k: number): number { const n = word.length; for (let i = k; i < n; i += k) { if (word.slice(i) === word.slice(0, -i)) { return Math.floor(i / k); return Math.floor((n + k - 1) / k);	{"url":"https://leetcode.ca/2024-02-09-3029-Minimum-Time-to-Revert-Word-to-Initial-State-I/","timestamp":"2024-11-11T11:50:40Z","content_type":"text/html","content_length":"34349","record_id":"<urn:uuid:ab2bf46a-dc25-4455-8efd-1d7490580d0b>","cc-path":"CC-MAIN-2024-46/segments/1730477028228.41/warc/CC-MAIN-20241111091854-20241111121854-00483.warc.gz"}
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Adding Fractions Simplifying Complex Fractions Registered: Solutions to Linear 10.07.2002 Equations in Two From: NW AR, USA Quadratic Expressions Completing Squares Dividing Radical Rise and Run Graphing Exponential Multiplying by a The Cartesian Coordinate Writing the Terms of a Polynomial in Descending Quadratic Expressions Solving Inequalities Solving Rational Inequalities with a Sign Solving Linear Equations Solving an Equation with Two Radical Terms Simplifying Rational Intercepts of a Line Completing the Square Order of Operations Factoring Trinomials Solving Linear Equations Solving Multi-Step Solving Quadratic Equations Graphically and Algebraically Collecting Like Terms Solving Equations with Radicals and Exponents Percent of Change Powers of ten (Scientific Notation) Comparing Integers on a Number Line Solving Systems of Equations Using Factoring Out the Greatest Common Factor Families of Functions Monomial Factors Multiplying and Dividing Complex Numbers Properties of Exponents Multiplying Square Roots Adding or Subtracting Rational Expressions with Different Expressions with Variables as Exponents The Quadratic Formula Writing a Quadratic with Given Solutions Simplifying Square Roots Adding and Subtracting Square Roots Adding and Subtracting Rational Expressions Combining Like Radical Solving Systems of Equations Using Dividing Polynomials Graphing Functions Product of a Sum and a Solving First Degree Solving Equations with Radicals and Exponents Roots and Powers Multiplying Numbers decimals yr 7 common test Related topics: Home math taks hints \| algerbra calculator \| good calculator for college algebra \| ti-84 plus calculator program downloads \| algebra 3 radicals homework \| printable Rotating a Parabola math transformation quiz \| cube root on calculator Multiplying Fractions Finding Factors Miscellaneous Equations Author Message Mixed Numbers and Improper Fractions Acecaon Posted: Wednesday 27th of Dec 19:13 Systems of Equations in Hello dude , can anyone help me out with my assignment in Intermediate algebra. It would be good if you could just give me heads up about Two Variables the urls from where I can acquire help on hyperbolas. Literal Numbers Adding and Subtracting Polynomials Registered: Subtracting Integers 13.07.2002 Simplifying Complex From: Behind my Fractions computer Decimals and Fractions Multiplying Integers Logarithmic Functions Multiplying Monomials ameich Posted: Thursday 28th of Dec 16:27 Mixed Well of course there is. If you are determined about learning decimals yr 7 common test, then Algebrator can be of great benefit to you. It The Square of a Binomial is designed in such a manner that almost anyone can use it. You don’t need to be a computer expert in order to use the program. Factoring Trinomials The Pythagorean Theorem Solving Radical Registered: Equations in One 21.03.2005 Variable From: Prague, Czech Multiplying Binomials Republic Using the FOIL Method Imaginary Numbers Solving Quadratic Equations Using the DVH Posted: Saturday 30th of Dec 12:52 Quadratic Formula I second that. Algebrator has already helped me solving problems on decimals yr 7 common test in the past, and I’m sure that you would like Solving Quadratic it. I have never been to a high ranking school, but thanks to this software my math problem solving skills are as good as than students Equations studying in one of those fancy schools. Order of Operations Registered: Dividing Complex Numbers 20.12.2001 Polynomials From: The Appearance of a Polynomial Equation Standard Form of a Line Positive Integral patnihstal Posted: Sunday 31st of Dec 09:41 Divisors Algebrator is one of the best tools that would provide you all the fundamentals of decimals yr 7 common test. The detailed training offered Dividing Fractions by the Algebrator on graphing equations, factoring expressions, roots and linear inequalities is second to none. I have tried 4-5 home Solving Linear Systems tutoring algebra software and I found this to be remarkable . The Algebrator not only provides you the primary principles but also aids you of Equations by in figuring out any tough Remedial Algebra question with ease. The quick formula list that comes with Algebrator is very descriptive and Elimination Registered: has almost every formula relating to Algebra 1. Factoring 12.02.2006 Multiplying and Dividing From: Las Vegas Square Roots moving to Fulshear, Functions and Graphs TX Dividing Polynomials Solving Rational Numbers TihBoasten Posted: Monday 01st of Jan 10:10 Use of Parentheses or I remember having problems with proportions, geometry and point-slope. Algebrator is a really great piece of math software. I have used it Brackets (The through several math classes - Intermediate algebra, Algebra 2 and Pre Algebra. I would simply type in the problem and by clicking on Distributive Law) Solve, step by step solution would appear. The program is highly recommended. Multiplying and Dividing by Monomials Registered: Solving Quadratic 14.10.2002 Equations by Graphing From: Multiplying Decimals Use of Parentheses or Brackets (The Distributive Law) Admilal`Leker Posted: Tuesday 02nd of Jan 08:31 Simplifying Complex Well, you don’t have to wait any longer. Go to https://gre-test-prep.com/multiplying-and-dividing-by-monomials.html and get yourself a copy Fractions 1 for a very small price. Good luck and happy learning! Adding Fractions Simplifying Complex Fractions Registered: Solutions to Linear 10.07.2002 Equations in Two From: NW AR, USA Quadratic Expressions Completing Squares Dividing Radical Rise and Run Graphing Exponential Multiplying by a The Cartesian Coordinate Writing the Terms of a Polynomial in Descending Quadratic Expressions Solving Inequalities Solving Rational Inequalities with a Sign Solving Linear Equations Solving an Equation with Two Radical Terms Simplifying Rational Intercepts of a Line Completing the Square Order of Operations Factoring Trinomials Solving Linear Equations Solving Multi-Step Solving Quadratic Equations Graphically and Algebraically Collecting Like Terms Solving Equations with Radicals and Exponents Percent of Change Powers of ten (Scientific Notation) Comparing Integers on a Number Line Solving Systems of Equations Using Factoring Out the Greatest Common Factor Families of Functions Monomial Factors Multiplying and Dividing Complex Numbers Properties of Exponents Multiplying Square Roots Adding or Subtracting Rational Expressions with Different Expressions with Variables as Exponents The Quadratic Formula Writing a Quadratic with Given Solutions Simplifying Square Roots Adding and Subtracting Square Roots Adding and Subtracting Rational Expressions Combining Like Radical Solving Systems of Equations Using Dividing Polynomials Graphing Functions Product of a Sum and a Solving First Degree Solving Equations with Radicals and Exponents Roots and Powers Multiplying Numbers Rotating a Parabola Multiplying Fractions Finding Factors Miscellaneous Equations Mixed Numbers and Improper Fractions Systems of Equations in Two Variables Literal Numbers Adding and Subtracting Subtracting Integers Simplifying Complex Decimals and Fractions Multiplying Integers Logarithmic Functions Multiplying Monomials The Square of a Binomial Factoring Trinomials The Pythagorean Theorem Solving Radical Equations in One Multiplying Binomials Using the FOIL Method Imaginary Numbers Solving Quadratic Equations Using the Quadratic Formula Solving Quadratic Order of Operations Dividing Complex Numbers The Appearance of a Polynomial Equation Standard Form of a Line Positive Integral Dividing Fractions Solving Linear Systems of Equations by Multiplying and Dividing Square Roots Functions and Graphs Dividing Polynomials Solving Rational Use of Parentheses or Brackets (The Distributive Law) Multiplying and Dividing by Monomials Solving Quadratic Equations by Graphing Multiplying Decimals Use of Parentheses or Brackets (The Distributive Law) Simplifying Complex Fractions 1 Adding Fractions Simplifying Complex Solutions to Linear Equations in Two Quadratic Expressions Completing Squares Dividing Radical Rise and Run Graphing Exponential Multiplying by a The Cartesian Coordinate Writing the Terms of a Polynomial in Descending Quadratic Expressions Solving Inequalities Solving Rational Inequalities with a Sign Solving Linear Equations Solving an Equation with Two Radical Terms Simplifying Rational Intercepts of a Line Completing the Square Order of Operations Factoring Trinomials Solving Linear Equations Solving Multi-Step Solving Quadratic Equations Graphically and Algebraically Collecting Like Terms Solving Equations with Radicals and Exponents Percent of Change Powers of ten (Scientific Notation) Comparing Integers on a Number Line Solving Systems of Equations Using Factoring Out the Greatest Common Factor Families of Functions Monomial Factors Multiplying and Dividing Complex Numbers Properties of Exponents Multiplying Square Roots Adding or Subtracting Rational Expressions with Different Expressions with Variables as Exponents The Quadratic Formula Writing a Quadratic with Given Solutions Simplifying Square Roots Adding and Subtracting Square Roots Adding and Subtracting Rational Expressions Combining Like Radical Solving Systems of Equations Using Dividing Polynomials Graphing Functions Product of a Sum and a Solving First Degree Solving Equations with Radicals and Exponents Roots and Powers Multiplying Numbers decimals yr 7 common test Related topics: math taks hints \| algerbra calculator \| good calculator for college algebra \| ti-84 plus calculator program downloads \| algebra 3 radicals homework \| printable math transformation quiz \| cube root on calculator Author Message Acecaon Posted: Wednesday 27th of Dec 19:13 Hello dude , can anyone help me out with my assignment in Intermediate algebra. It would be good if you could just give me heads up about the urls from where I can acquire help on hyperbolas. From: Behind my ameich Posted: Thursday 28th of Dec 16:27 Well of course there is. If you are determined about learning decimals yr 7 common test, then Algebrator can be of great benefit to you. It is designed in such a manner that almost anyone can use it. You don’t need to be a computer expert in order to use the program. From: Prague, Czech DVH Posted: Saturday 30th of Dec 12:52 I second that. Algebrator has already helped me solving problems on decimals yr 7 common test in the past, and I’m sure that you would like it. I have never been to a high ranking school, but thanks to this software my math problem solving skills are as good as than students studying in one of those fancy schools. patnihstal Posted: Sunday 31st of Dec 09:41 Algebrator is one of the best tools that would provide you all the fundamentals of decimals yr 7 common test. The detailed training offered by the Algebrator on graphing equations, factoring expressions, roots and linear inequalities is second to none. I have tried 4-5 home tutoring algebra software and I found this to be remarkable . The Algebrator not only provides you the primary principles but also aids you in figuring out any tough Remedial Algebra question with ease. The quick formula list that comes with Algebrator is very descriptive and has almost every formula relating to Algebra 1. From: Las Vegas moving to Fulshear, TihBoasten Posted: Monday 01st of Jan 10:10 I remember having problems with proportions, geometry and point-slope. Algebrator is a really great piece of math software. I have used it through several math classes - Intermediate algebra, Algebra 2 and Pre Algebra. I would simply type in the problem and by clicking on Solve, step by step solution would appear. The program is highly recommended. Admilal`Leker Posted: Tuesday 02nd of Jan 08:31 Well, you don’t have to wait any longer. Go to https://gre-test-prep.com/multiplying-and-dividing-by-monomials.html and get yourself a copy for a very small price. Good luck and happy learning! From: NW AR, USA Author Message Acecaon Posted: Wednesday 27th of Dec 19:13 Hello dude , can anyone help me out with my assignment in Intermediate algebra. It would be good if you could just give me heads up about the urls from where I can acquire help on hyperbolas. From: Behind my ameich Posted: Thursday 28th of Dec 16:27 Well of course there is. If you are determined about learning decimals yr 7 common test, then Algebrator can be of great benefit to you. It is designed in such a manner that almost anyone can use it. You don’t need to be a computer expert in order to use the program. From: Prague, Czech DVH Posted: Saturday 30th of Dec 12:52 I second that. Algebrator has already helped me solving problems on decimals yr 7 common test in the past, and I’m sure that you would like it. I have never been to a high ranking school, but thanks to this software my math problem solving skills are as good as than students studying in one of those fancy schools. patnihstal Posted: Sunday 31st of Dec 09:41 Algebrator is one of the best tools that would provide you all the fundamentals of decimals yr 7 common test. The detailed training offered by the Algebrator on graphing equations, factoring expressions, roots and linear inequalities is second to none. I have tried 4-5 home tutoring algebra software and I found this to be remarkable . The Algebrator not only provides you the primary principles but also aids you in figuring out any tough Remedial Algebra question with ease. The quick formula list that comes with Algebrator is very descriptive and has almost every formula relating to Algebra 1. From: Las Vegas moving to Fulshear, TihBoasten Posted: Monday 01st of Jan 10:10 I remember having problems with proportions, geometry and point-slope. Algebrator is a really great piece of math software. I have used it through several math classes - Intermediate algebra, Algebra 2 and Pre Algebra. I would simply type in the problem and by clicking on Solve, step by step solution would appear. The program is highly Admilal`Leker Posted: Tuesday 02nd of Jan 08:31 Well, you don’t have to wait any longer. Go to https://gre-test-prep.com/multiplying-and-dividing-by-monomials.html and get yourself a copy for a very small price. Good luck and happy learning! From: NW AR, USA Posted: Wednesday 27th of Dec 19:13 Hello dude , can anyone help me out with my assignment in Intermediate algebra. It would be good if you could just give me heads up about the urls from where I can acquire help on hyperbolas. Posted: Thursday 28th of Dec 16:27 Well of course there is. If you are determined about learning decimals yr 7 common test, then Algebrator can be of great benefit to you. It is designed in such a manner that almost anyone can use it. You don’t need to be a computer expert in order to use the program. Posted: Saturday 30th of Dec 12:52 I second that. Algebrator has already helped me solving problems on decimals yr 7 common test in the past, and I’m sure that you would like it. I have never been to a high ranking school, but thanks to this software my math problem solving skills are as good as than students studying in one of those fancy schools. Posted: Sunday 31st of Dec 09:41 Algebrator is one of the best tools that would provide you all the fundamentals of decimals yr 7 common test. The detailed training offered by the Algebrator on graphing equations, factoring expressions, roots and linear inequalities is second to none. I have tried 4-5 home tutoring algebra software and I found this to be remarkable . The Algebrator not only provides you the primary principles but also aids you in figuring out any tough Remedial Algebra question with ease. The quick formula list that comes with Algebrator is very descriptive and has almost every formula relating to Algebra 1. Posted: Monday 01st of Jan 10:10 I remember having problems with proportions, geometry and point-slope. Algebrator is a really great piece of math software. I have used it through several math classes - Intermediate algebra, Algebra 2 and Pre Algebra. I would simply type in the problem and by clicking on Solve, step by step solution would appear. The program is highly recommended. Posted: Tuesday 02nd of Jan 08:31 Well, you don’t have to wait any longer. Go to https://gre-test-prep.com/multiplying-and-dividing-by-monomials.html and get yourself a copy for a very small price. Good luck and happy learning!	{"url":"https://gre-test-prep.com/algebra-1-practice-test/exponent-rules/decimals-yr-7-common-test.html","timestamp":"2024-11-02T21:29:55Z","content_type":"text/html","content_length":"117269","record_id":"<urn:uuid:d9cb6ff8-bf1d-4cf7-9e15-3973d8d23cc5>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00039.warc.gz"}
Alan's baseball aerodynamics model I look at it and dig through it for cross-references all the time but never write it down. So here's how Alan Nathan does the physics of a baseball, for my reference, and possibly for yours as well! First, the drag model is given by: C_d= C_{d, 0} + C_{d, \omega} \|\boldsymbol\omega(t)\| where, in Nathan's spreadsheet, $\omega$ is the spin rate in $\textrm{rev min}^{-1}$ per-thou (i.e. thousandths of an RPM). The spin rate decays as: \boldsymbol\omega(t)= \boldsymbol\omega_0 \exp(-\frac{t}{\tau} \frac{\|\mathbf{V}\|}{V_\mathrm{set}}) where $\mathbf{V}$ is the velocity in feet per second, and the constants are given by $\tau= 30 \textrm{ s}$ and $V_\mathrm{set}= 100 \textrm{ mph}$. The exponential term is spin decay term with a decay parameter and scaled with velocity. The lift model is given by: C_L= \frac{C_{L, 2} S}{C_{L, 0} + C_{L, 1} S} with parameters $C_{L, 2}= 1.120$, $C_{L, 1}= 2.333$, and $C_{L, 0}= 0.583$. Last but not least, there's an approximator for the density that he uses, which corrects for altitude, barometric pressure, altitude, and relative humidity: \rho= \rho_0 \left( \frac{T_0}{T + T_0} \frac{p}{p_0} \exp(-\beta h_0) - 0.3783 x_{RH} \frac{p_{SVP}}{p_0} \right) where $\rho_0= 1.2929 \mathrm{~kg/m^3}$ is a reference density, $T_0= 273 \mathrm{~K}$ and $p_0= 101,325 \mathrm{~Pa}$ are the standard sea level temperature and pressures, $h_0$ is the elevation of the stadium above sea level in meters, $\beta= 121.7 \times 10^{-6} \mathrm{~m^{-1}}$ is a pressure decay rate. Last but not least, $x_{RH}$ is the relative humidity fraction, and $p_{SVP}$ is the saturation vapor pressure of air in Pascals. The saturation vapor pressure is a function of the temperature by: p_{SVP}= 4.5841 \exp(\frac{18.687 \mathrm{~K} - T}{234.5 \mathrm{~K}} \frac{T}{257.14 \mathrm{~K} + T})	{"url":"https://coryfront.in/sports/alans-physical-model/","timestamp":"2024-11-07T15:09:56Z","content_type":"text/html","content_length":"14202","record_id":"<urn:uuid:3c391a37-9590-4f9d-ade7-06c695df90d2>","cc-path":"CC-MAIN-2024-46/segments/1730477028000.52/warc/CC-MAIN-20241107150153-20241107180153-00528.warc.gz"}
This class was created by Brainscape user John Georgiades. Visit their profile to learn more about the creator. Learn faster with Brainscape on your web, iPhone, or Android device. Study John Georgiades's SAT Math II flashcards for their Central High School class now! Brainscape's adaptive web mobile flashcards system will drill you on your weaknesses, using a pattern guaranteed to help you learn more in less time. Either request "Edit" access from the author, or make a copy of the class to edit as your own. And you can always create a totally new class of your own too! Brainscape is a digital flashcards platform where you can find, create, share, and study any subject on the planet. We use an adaptive study algorithm that is proven to help you learn faster and remember longer....	{"url":"https://www.brainscape.com/packs/sat-math-ii-13130798","timestamp":"2024-11-07T01:13:14Z","content_type":"text/html","content_length":"41645","record_id":"<urn:uuid:b6ddffb7-fb8d-4cdd-8f40-bcb0d71aa6b1>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00255.warc.gz"}
Statamperes to Milliamperes Conversion (statA to mA) Statamperes to Milliamperes Converter Enter the electric current in statamperes below to convert it to milliamperes. Do you want to convert milliamperes to statamperes? How to Convert Statamperes to Milliamperes To convert a measurement in statamperes to a measurement in milliamperes, multiply the electric current by the following conversion ratio: 3.3356E-6 milliamperes/statampere. Since one statampere is equal to 3.3356E-6 milliamperes, you can use this simple formula to convert: milliamperes = statamperes × 3.3356E-6 The electric current in milliamperes is equal to the electric current in statamperes multiplied by 3.3356E-6. For example, here's how to convert 500,000 statamperes to milliamperes using the formula above. milliamperes = (500,000 statA × 3.3356E-6) = 1.667821 mA Statamperes and milliamperes are both units used to measure electric current. Keep reading to learn more about each unit of measure. What Is a Statampere? The statmpere is the electrical current constant equal to the flow of one statcoulomb per second, or 0.33356 nanoamperes in the International System of Units. The statampere is a centimeter-gram-second (CGS) electrostatic unit of electric current. Statamperes can be abbreviated as statA, and are also sometimes abbreviated as A-esu. For example, 1 statampere can be written as 1 statA or 1 A-esu. Learn more about statamperes. What Is a Milliampere? One milliampere is equal to 1/1,000 of an ampere, which is the electrical current equal to the flow of one coulomb per second. The milliampere is a multiple of the ampere, which is the SI base unit for electric current. In the metric system, "milli" is the prefix for thousandths, or 10^-3. A milliampere is sometimes also referred to as a milliamp. Milliamperes can be abbreviated as mA; for example, 1 milliampere can be written as 1 mA. Learn more about milliamperes. Statampere to Milliampere Conversion Table Table showing various statampere measurements converted to Statamperes Milliamperes 1 statA 0.0000033356 mA 2 statA 0.0000066713 mA 3 statA 0.000010007 mA 4 statA 0.000013343 mA 5 statA 0.000016678 mA 6 statA 0.000020014 mA 7 statA 0.000023349 mA 8 statA 0.000026685 mA 9 statA 0.000030021 mA 10 statA 0.000033356 mA 100 statA 0.000334 mA 1,000 statA 0.003336 mA 10,000 statA 0.033356 mA 100,000 statA 0.333564 mA 1,000,000 statA 3.3356 mA More Statampere & Milliampere Conversions	{"url":"https://www.inchcalculator.com/convert/statampere-to-milliampere/","timestamp":"2024-11-07T02:58:32Z","content_type":"text/html","content_length":"65195","record_id":"<urn:uuid:c728fbeb-000e-4986-ae05-a2819a4185e2>","cc-path":"CC-MAIN-2024-46/segments/1730477027951.86/warc/CC-MAIN-20241107021136-20241107051136-00120.warc.gz"}
Federico L. What do you want to work on? About Federico L. Algebra, Elementary (3-6) Math, Chemistry, Midlevel (7-8) Math, MS Excel Bachelors in Chemical Engineering from Universidad de Buenos Aires Career Experience I am a chemical engineering student, and I am a web developer. I helped many students from my university and some other places. I Love Tutoring Because I have a vocation to it, and I have always pictured myself as a teacher. It is so satisfying when my students resolve their problems, and they leave with all the tools I provided to them. Tutoring is an entertaining way to boost our communication skills. Other Interests Cooking, Driving, Hiking, Music, Programming, Skiing, Volleyball Technology - MS Excel The tutor was patient and answered my question. Science - Chemistry Frederico was spectacular!!!!! THank you for your knowedge in both chemistry and Excel! Brilliant!!!! Technology - MS Excel Awesome, great showed me lots i didn't know. Technology - Computer Science - Webdesign	{"url":"https://origin-www.princetonreview.com/academic-tutoring/tutor/federico%20l--9538434","timestamp":"2024-11-11T09:56:59Z","content_type":"application/xhtml+xml","content_length":"222296","record_id":"<urn:uuid:5ffb5627-9158-4ca8-beea-5b3bdfe7bf5b>","cc-path":"CC-MAIN-2024-46/segments/1730477028228.41/warc/CC-MAIN-20241111091854-20241111121854-00598.warc.gz"}
The Laws of Physics for the Atmosphere – The Computer Model The second requirement to produce a forecast is an understanding of how one state of the atmosphere evolves to another. This evolution is encapsulated within the laws of physics as developed for the Earth's atmosphere. The following are the five required laws. • Ideal gas law • Conservation of mass (air) • Conservation of mass (water) • How wind changes • How temperature changes Ideal Gas Law The ideal gas law, also known as the equation of state, describes the relationship between the pressure ($p$), density ($\rho$), and temperature ($T$) of the air in the atmosphere. The letter $R$ is called the ideal gas constant. The ideal gas law has several different forms, and the form below is commonly used. $$p=\rho RT$$ FACT BOX What is the gas constant? The gas constant is a quantity that is universal and does not change with time. Used in the ideal gas law, it equals 8.314 J K^−1 mol^−1. Conservation of Mass (Air) Matter can neither be created nor destroyed. As such, a rather simple equation can be written for the conservation of mass in the atmosphere. It is called the continuity equation. It describes the change in air density ($\rho$) with time as the result of the three-dimensional divergence of air. In other words, if more air flows out of a region than enters the region (mass divergence), then the density will decrease. Alternatively, if more air flows into a region than exits the region (mass convergence), then the density will increase. Conservation of Mass (Water) Water is an important chemical constituent in the Earth's atmosphere. Without the water in the atmosphere, we'd have no rain. Expressing the conservation of water mass in the atmosphere is simple. In the equation above, the amount of water in a small volume of air is called the mixing ratio $q$. The mixing ratio will increase ($\frac{dq}{dt} > 0$) if water vapor is evaporated into the air ($Q_ {E}$). The mixing ratio will decrease ($\frac{dq}{dt} < 0$) if water vapor is condensed out of the air ($Q_{C}$). Equation of Motion Another equation essential for prediction is the momentum equation, also known as the equation of motion. The equation of motion is based on Newton's second law, which states that an acceleration (or a change in the momentum or velocity) is caused by the application of a force. $F$ = force $m$ = mass $a$ = acceleration If all the forces acting in the atmosphere are included, we end up with the following for the equation of motion for the atmosphere. $$\frac{d\mathbf{V}}{dt}+2\Omega\times\mathbf{V}=\frac{1}{\rho}\nabla p-g\mathbf{k}+v\nabla^{2}\mathbf{V}$$ The equation above says the following. The change in the three-dimensional velocity (or an acceleration, $\frac{d\mathbf{V}}{dt}$) plus the Coriolis force ($2\Omega \times \mathbf{V}$) is equal to the gradient of pressure ($\nabla p$) minus gravity ($g$) plus the friction within the fluid due to viscosity ($v\nabla^{2}\mathbf{V}$). Gravity only works in the vertical direction and is largely offset by the vertical gradient of pressure (i.e., pressure decreases with height, creating a vertical pressure gradient force). A similar expression can be derived for the forces acting in the horizontal direction. In that direction, the dominant forces are the horizontal pressure gradient force and the Coriolis force. FACT BOX The Coriolis Effect The Coriolis effect was discovered in the 19th Century by French mathematician Gustave Coriolis. The Coriolis effect is the apparent deflection of air in the atmosphere because the Earth rotates underneath it. An observer in space would not see the deflection, which is why the Coriolis force is referred to as an apparent force, because it needs to be accounted for within a rotating frame of reference. On Earth, the Coriolis effect affects large-scale motions, on the scale of several hours and longer because of the Earth's low rate of rotation (one revolution per day). Thermodynamic Equation The last equation describes how temperature changes in the atmosphere. The first term on the left-hand side describes the rate of change of temperature with time ($\frac{dT}{dt}$). The other term on the left-hand side says that the horizontal divergence of air ($\nabla\ cdot\mathbf{V}$) leads to vertical motions that warm or cool the air. The two $Q$ terms on the right-hand side express heat fluxes from the Earth's surface to the atmosphere ($Q_{H}$; e.g., cold air moving over water water is warmed by sensible heat fluxes from the water to the air) and diabatic heating ($Q_{D}$) caused by condensation (warming the atmosphere) and evaporation (cooling the atmosphere). Calculating these terms will yield how the temperature changes with time in the atmosphere. Programming the model These are the laws of physics written in a mathematical form. This mathematical form, however, is not how a computer would recognize these equations. So, the equations need to be rewritten from their mathematical form to what is called finite-difference form in order to be programmed into an algorithm (or computer code) which is used in the forecast model. With the initial conditions derived from the global observations and the computer model derived from the laws of physics, we are ready to run the model and get a forecast. These pages are written for ManUniCast by David Schultz, Fiona Lomas, and Katy Mulqueen, University of Manchester. Photos and graphics are credited individually.	{"url":"https://manunicast.seaes.manchester.ac.uk/how/physics.html","timestamp":"2024-11-07T19:55:49Z","content_type":"text/html","content_length":"18955","record_id":"<urn:uuid:996f335e-638b-4148-adc4-cd3994dea6b3>","cc-path":"CC-MAIN-2024-46/segments/1730477028009.81/warc/CC-MAIN-20241107181317-20241107211317-00790.warc.gz"}
[Solved] Bond P is a premium bond with a coupon of \| SolutionInn Answered step by step Verified Expert Solution Bond P is a premium bond with a coupon of 8.6 percent , a YTM of 7.35 percent, and 15 years to maturity. Bond D Bond P is a premium bond with a coupon of 8.6 percent , a YTM of 7.35 percent, and 15 years to maturity. Bond D is a discount bond with a coupon of 8.6 percent, a YTM of 10.35 percent, and also 15 years to maturity. If interest rates remain unchanged, what do you expect the price of these bonds to be 1 year from now? In 5 years? In 10 years? In 14 years? In 15 years? (Input all amounts as positive values. Do not round intermediate calculations. Round your answers to 2 decimal places.) There are 3 Steps involved in it Step: 1 Get Instant Access to Expert-Tailored Solutions See step-by-step solutions with expert insights and AI powered tools for academic success Ace Your Homework with AI Get the answers you need in no time with our AI-driven, step-by-step assistance Get Started	{"url":"https://www.solutioninn.com/study-help/questions/bond-p-is-a-premium-bond-with-a-coupon-of-6563791","timestamp":"2024-11-06T09:18:43Z","content_type":"text/html","content_length":"93993","record_id":"<urn:uuid:e0a1fb93-337a-4304-abb8-64728c80bc8d>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00696.warc.gz"}
Hook and simulate global keyboard events on Windows and Linux. \| PythonRepo Take full control of your keyboard with this small Python library. Hook global events, register hotkeys, simulate key presses and much more. • Global event hook on all keyboards (captures keys regardless of focus). • Listen and send keyboard events. • Works with Windows and Linux (requires sudo), with experimental OS X support (thanks @glitchassassin!). • Pure Python, no C modules to be compiled. • Zero dependencies. Trivial to install and deploy, just copy the files. • Python 2 and 3. • Complex hotkey support (e.g. ctrl+shift+m, ctrl+space) with controllable timeout. • Includes high level API (e.g. record and play, add_abbreviation). • Maps keys as they actually are in your layout, with full internationalization support (e.g. Ctrl+ç). • Events automatically captured in separate thread, doesn't block main program. • Tested and documented. • Doesn't break accented dead keys (I'm looking at you, pyHook). • Mouse support available via project mouse (pip install mouse). Install the PyPI package: pip install keyboard or clone the repository (no installation required, source files are sufficient): git clone https://github.com/boppreh/keyboard or download and extract the zip into your project folder. Then check the API docs below to see what features are available. import keyboard keyboard.press_and_release('shift+s, space') keyboard.write('The quick brown fox jumps over the lazy dog.') keyboard.add_hotkey('ctrl+shift+a', print, args=('triggered', 'hotkey')) # Press PAGE UP then PAGE DOWN to type "foobar". keyboard.add_hotkey('page up, page down', lambda: keyboard.write('foobar')) # Blocks until you press esc. # Record events until 'esc' is pressed. recorded = keyboard.record(until='esc') # Then replay back at three times the speed. keyboard.play(recorded, speed_factor=3) # Type @@ then press space to replace with abbreviation. keyboard.add_abbreviation('@@', '[email protected]') # Block forever, like `while True`. Known limitations: • Events generated under Windows don't report device id (event.device == None). #21 • Media keys on Linux may appear nameless (scan-code only) or not at all. #20 • Key suppression/blocking only available on Windows. #22 • To avoid depending on X, the Linux parts reads raw device files (/dev/input/input*) but this requires root. • Other applications, such as some games, may register hooks that swallow all key events. In this case keyboard will be unable to report events. • This program makes no attempt to hide itself, so don't use it for keyloggers or online gaming bots. Be responsible. Table of Contents class keyboard.KeyboardEvent KeyboardEvent.to_json(self, ensure_ascii=False) = {'alt', 'alt gr', 'ctrl', 'left alt', 'left ctrl', 'left shift', 'left windows', 'right alt', 'right ctrl', 'right shift', 'right windows', 'shift', 'windows'} = {'alt', 'ctrl', 'shift', 'windows'} Returns True if key is a scan code or name of a modifier key. keyboard.key_to_scan_codes(key, error_if_missing=True) Returns a list of scan codes associated with this key (name or scan code). Parses a user-provided hotkey into nested tuples representing the parsed structure, with the bottom values being lists of scan codes. Also accepts raw scan codes, which are then wrapped in the required number of nestings. parse_hotkey("alt+shift+a, alt+b, c") # Keys: ^~^ ^~~~^ ^ ^~^ ^ ^ # Steps: ^~~~~~~~~~^ ^~~~^ ^ # ((alt_codes, shift_codes, a_codes), (alt_codes, b_codes), (c_codes,)) keyboard.send(hotkey, do_press=True, do_release=True) Sends OS events that perform the given hotkey hotkey. • hotkey can be either a scan code (e.g. 57 for space), single key (e.g. 'space') or multi-key, multi-step hotkey (e.g. 'alt+F4, enter'). • do_press if true then press events are sent. Defaults to True. • do_release if true then release events are sent. Defaults to True. send('alt+F4, enter') Note: keys are released in the opposite order they were pressed. Presses and holds down a hotkey (see send). Releases a hotkey (see send). Returns True if the key is pressed. is_pressed(57) #-> True is_pressed('space') #-> True is_pressed('ctrl+space') #-> True keyboard.call_later(fn, args=(), delay=0.001) Calls the provided function in a new thread after waiting some time. Useful for giving the system some time to process an event, without blocking the current execution flow. keyboard.hook(callback, suppress=False, on_remove=<lambda>) Installs a global listener on all available keyboards, invoking callback each time a key is pressed or released. The event passed to the callback is of type keyboard.KeyboardEvent, with the following attributes: • name: an Unicode representation of the character (e.g. "&") or description (e.g. "space"). The name is always lower-case. • scan_code: number representing the physical key, e.g. 55. • time: timestamp of the time the event occurred, with as much precision as given by the OS. Returns the given callback for easier development. keyboard.on_press(callback, suppress=False) Invokes callback for every KEY_DOWN event. For details see hook. keyboard.on_release(callback, suppress=False) Invokes callback for every KEY_UP event. For details see hook. keyboard.hook_key(key, callback, suppress=False) Hooks key up and key down events for a single key. Returns the event handler created. To remove a hooked key use unhook_key(key) or unhook_key(handler). Note: this function shares state with hotkeys, so clear_all_hotkeys affects it as well. keyboard.on_press_key(key, callback, suppress=False) Invokes callback for KEY_DOWN event related to the given key. For details see hook. keyboard.on_release_key(key, callback, suppress=False) Invokes callback for KEY_UP event related to the given key. For details see hook. Removes a previously added hook, either by callback or by the return value of hook. Removes all keyboard hooks in use, including hotkeys, abbreviations, word listeners, recorders and waits. Suppresses all key events of the given key, regardless of modifiers. keyboard.remap_key(src, dst) Whenever the key src is pressed or released, regardless of modifiers, press or release the hotkey dst instead. Parses a user-provided hotkey. Differently from parse_hotkey, instead of each step being a list of the different scan codes for each key, each step is a list of all possible combinations of those scan codes. keyboard.add_hotkey(hotkey, callback, args=(), suppress=False, timeout=1, trigger_on_release=False) Invokes a callback every time a hotkey is pressed. The hotkey must be in the format ctrl+shift+a, s. This would trigger when the user holds ctrl, shift and "a" at once, releases, and then presses "s". To represent literal commas, pluses, and spaces, use their names ('comma', 'plus', 'space'). • args is an optional list of arguments to passed to the callback during each invocation. • suppress defines if successful triggers should block the keys from being sent to other programs. • timeout is the amount of seconds allowed to pass between key presses. • trigger_on_release if true, the callback is invoked on key release instead of key press. The event handler function is returned. To remove a hotkey call remove_hotkey(hotkey) or remove_hotkey(handler). before the hotkey state is reset. Note: hotkeys are activated when the last key is pressed, not released. Note: the callback is executed in a separate thread, asynchronously. For an example of how to use a callback synchronously, see # Different but equivalent ways to listen for a spacebar key press. add_hotkey(' ', print, args=['space was pressed']) add_hotkey('space', print, args=['space was pressed']) add_hotkey('Space', print, args=['space was pressed']) # Here 57 represents the keyboard code for spacebar; so you will be # pressing 'spacebar', not '57' to activate the print function. add_hotkey(57, print, args=['space was pressed']) add_hotkey('ctrl+q', quit) add_hotkey('ctrl+alt+enter, space', some_callback) Removes a previously hooked hotkey. Must be called with the value returned by add_hotkey. Removes all keyboard hotkeys in use, including abbreviations, word listeners, recorders and waits. keyboard.remap_hotkey(src, dst, suppress=True, trigger_on_release=False) Whenever the hotkey src is pressed, suppress it and send dst instead. remap_hotkey('alt+w', 'ctrl+up') Builds a list of all currently pressed scan codes, releases them and returns the list. Pairs well with restore_state and restore_modifiers. Given a list of scan_codes ensures these keys, and only these keys, are pressed. Pairs well with stash_state, alternative to restore_modifiers. Like restore_state, but only restores modifier keys. keyboard.write(text, delay=0, restore_state_after=True, exact=None) Sends artificial keyboard events to the OS, simulating the typing of a given text. Characters not available on the keyboard are typed as explicit unicode characters using OS-specific functionality, such as alt+codepoint. To ensure text integrity, all currently pressed keys are released before the text is typed, and modifiers are restored afterwards. • delay is the number of seconds to wait between keypresses, defaults to no delay. • restore_state_after can be used to restore the state of pressed keys after the text is typed, i.e. presses the keys that were released at the beginning. Defaults to True. • exact forces typing all characters as explicit unicode (e.g. alt+codepoint or special events). If None, uses platform-specific suggested value. keyboard.wait(hotkey=None, suppress=False, trigger_on_release=False) Blocks the program execution until the given hotkey is pressed or, if given no parameters, blocks forever. Returns a string representation of hotkey from the given key names, or the currently pressed keys if not given. This function: • normalizes names; • removes "left" and "right" prefixes; • replaces the "+" key name with "plus" to avoid ambiguity; • puts modifier keys first, in a standardized order; • sort remaining keys; • finally, joins everything with "+". get_hotkey_name(['+', 'left ctrl', 'shift']) # "ctrl+shift+plus" Blocks until a keyboard event happens, then returns that event. Blocks until a keyboard event happens, then returns that event's name or, if missing, its scan code. Similar to read_key(), but blocks until the user presses and releases a hotkey (or single key), then returns a string representing the hotkey pressed. # "ctrl+shift+p" keyboard.get_typed_strings(events, allow_backspace=True) Given a sequence of events, tries to deduce what strings were typed. Strings are separated when a non-textual key is pressed (such as tab or enter). Characters are converted to uppercase according to shift and capslock status. If allow_backspace is True, backspaces remove the last character typed. This function is a generator, so you can pass an infinite stream of events and convert them to strings in real time. Note this functions is merely an heuristic. Windows for example keeps per- process keyboard state such as keyboard layout, and this information is not available for our hooks. get_type_strings(record()) #-> ['This is what', 'I recorded', ''] Starts recording all keyboard events into a global variable, or the given queue if any. Returns the queue of events and the hooked function. Use stop_recording() or unhook(hooked_function) to stop. Stops the global recording of events and returns a list of the events captured. keyboard.record(until='escape', suppress=False, trigger_on_release=False) Records all keyboard events from all keyboards until the user presses the given hotkey. Then returns the list of events recorded, of type keyboard.KeyboardEvent. Pairs well with play(events). Note: this is a blocking function. Note: for more details on the keyboard hook and events see hook. keyboard.play(events, speed_factor=1.0) Plays a sequence of recorded events, maintaining the relative time intervals. If speed_factor is <= 0 then the actions are replayed as fast as the OS allows. Pairs well with record(). Note: the current keyboard state is cleared at the beginning and restored at the end of the function. keyboard.add_word_listener(word, callback, triggers=['space'], match_suffix=False, timeout=2) Invokes a callback every time a sequence of characters is typed (e.g. 'pet') and followed by a trigger key (e.g. space). Modifiers (e.g. alt, ctrl, shift) are ignored. • word the typed text to be matched. E.g. 'pet'. • callback is an argument-less function to be invoked each time the word is typed. • triggers is the list of keys that will cause a match to be checked. If the user presses some key that is not a character (len>1) and not in triggers, the characters so far will be discarded. By default the trigger is only space. • match_suffix defines if endings of words should also be checked instead of only whole words. E.g. if true, typing 'carpet'+space will trigger the listener for 'pet'. Defaults to false, only whole words are checked. • timeout is the maximum number of seconds between typed characters before the current word is discarded. Defaults to 2 seconds. Returns the event handler created. To remove a word listener use remove_word_listener(word) or remove_word_listener(handler). Note: all actions are performed on key down. Key up events are ignored. Note: word matches are case sensitive. Removes a previously registered word listener. Accepts either the word used during registration (exact string) or the event handler returned by the add_word_listener or add_abbreviation functions. keyboard.add_abbreviation(source_text, replacement_text, match_suffix=False, timeout=2) Registers a hotkey that replaces one typed text with another. For example add_abbreviation('tm', u'™') Replaces every "tm" followed by a space with a ™ symbol (and no space). The replacement is done by sending backspace events. • match_suffix defines if endings of words should also be checked instead of only whole words. E.g. if true, typing 'carpet'+space will trigger the listener for 'pet'. Defaults to false, only whole words are checked. • timeout is the maximum number of seconds between typed characters before the current word is discarded. Defaults to 2 seconds. For more details see add_word_listener. Given a key name (e.g. "LEFT CONTROL"), clean up the string and convert to the canonical representation (e.g. "left ctrl") if one is known.	{"url":"https://pythonrepo.com/repo/boppreh-keyboard-python-programming-with-hardware","timestamp":"2024-11-09T21:06:22Z","content_type":"text/html","content_length":"210247","record_id":"<urn:uuid:500a6c3c-3aba-4651-8a3c-972400c5842a>","cc-path":"CC-MAIN-2024-46/segments/1730477028142.18/warc/CC-MAIN-20241109182954-20241109212954-00028.warc.gz"}
Estimating how disease severity varies over the course of an outbreak The severity of a disease (commonly understood as the case fatality risk, CFR) might change during the course of an outbreak for multiple biological, epidemiological and behavioural reasons. cfr_time_varying() offers a convenient way to understand changes in disease severity over time using an approach with a finer time resolution that calculates severity over a moving time window, using methods from Nishiura et al. (2009). Use case There are substantial changes to the characteristics of an outbreak over time — such as the introduction of therapeutics or a changing case definition. We want to estimate how disease severity in the form of the case fatality risk (CFR) changes over time while correcting for the delay in reporting the outcomes of cases. What we have • A time-series of cases and deaths, (cases may be substituted by another indicator of infections over time); • Data on the distribution of delays, describing the probability an individual will die $t$ days after they were initially infected. Potential reasons for changing disease severity • Change in the probability of infection being reported as a case, • Transmission dynamics within specific subgroups of differing risk of severe outcomes, • Introduction of vaccines or therapeutics reducing the relative risk of death, • Emergence of pathogen variants which may alter the mortality risk associated with infection. Changing severity of the Covid-19 pandemic in the U.K. This example shows time-varying severity estimation using cfr and data from the Covid-19 pandemic in the United Kingdom. Preparing the raw data We load example Covid-19 daily case and death data provided with the cfr package as covid_data, and subset for the first year of U.K. data. We would expect the estimated CFR to change over this period due to changes in pandemic response policy, such as changes in case definitions, implementation and relaxation of lockdowns, and new variants emerging. # get Covid data loaded with the package # filter for the U.K df_covid_uk <- filter( country == "United Kingdom", date <= "2020-12-31" # View the first few rows and recall necessary columns: date, cases, deaths #> date country cases deaths #> 1 2020-01-03 United Kingdom 0 0 #> 2 2020-01-04 United Kingdom 0 0 #> 3 2020-01-05 United Kingdom 0 0 #> 4 2020-01-06 United Kingdom 0 0 #> 5 2020-01-07 United Kingdom 0 0 #> 6 2020-01-08 United Kingdom 0 0 Onset-to-death distribution for Covid-19 We retrieve the appropriate distribution of the duration between symptom onset and deaths reported in Linton et al. (2020); this is a lognormal distribution with $\mu$ = 2.577 and $\sigma$ = Linton et al. (2020) fitted a discrete lognormal distribution — but we use a continuous distribution here. See the vignette on delay distributions for more on when using a continuous instead of discrete distribution is acceptable, and on using discrete distributions with cfr. Note also that we use the central estimates for each distribution parameter, and by ignoring uncertainty in these parameters the uncertainty in the resulting CFR is likely to be underestimated. Estimating the naive and corrected CFR We use the cfr_time_varying() function within the cfr package to calculate the time-varying CFR for the Covid-19 pandemic in the U.K., and plot the results. The burn_in time is used to determine how many days at the start of the outbreak are excluded from the CFR calculation, potentially due to poor data quality at the beginning of an outbreak. The default value is 7, which ignores the first week of data. The smoothing_window is used to smooth the case and death data using a rolling median with a window of the corresponding size (in days) using stats::runmed() internally — this is disabled by default. Users should apply smoothing if there are reporting artefacts such as lower reporting on weekends. # calculating the naive time-varying CFR df_covid_cfr_uk_naive <- cfr_time_varying( burn_in = 7, smoothing_window = 7 # calculating the corrected time-varying CFR df_covid_cfr_uk_corrected <- cfr_time_varying( delay_density = function(x) dlnorm(x, meanlog = 2.577, sdlog = 0.440), burn_in = 7, smoothing_window = 7 # assign method tag and plot df_covid_cfr_uk_naive$method <- "naive" df_covid_cfr_uk_corrected$method <- "corrected" df_covid_cfr_uk <- bind_rows(df_covid_cfr_uk_naive, df_covid_cfr_uk_corrected) ggplot(df_covid_cfr_uk) + x = date, ymin = severity_low, ymax = severity_high, fill = method alpha = 0.5 ) + x = date, y = severity_estimate, colour = method ) + date_labels = "%b-%Y" ) + labels = percent ) + palette = "Dark2", name = NULL, labels = c("Naive CFR", "Corrected CFR") ) + palette = "Dark2", name = NULL, labels = c("Naive CFR", "Corrected CFR") ) + x = "Date", y = "CFR (%)" ) + expand = FALSE ) + theme_classic() + theme(legend.position = "top") #> Warning: Removed 93 rows containing missing values (`geom_line()`). Note that the severity estimates and confidence intervals in cfr_time_varying() are obtained from a Binomial test on deaths (treated as ‘successes’) and estimated outcomes or cases (depending on whether delay correction is applied; treated as ‘trials’). Severity of Covid-19 in multiple countries cfr_time_varying() and other cfr functions can be conveniently applied over nested data to estimate the time-varying severity of Covid-19. We use the example Covid-19 cases and deaths data provided with the package as covid_data, while excluding four countries which only provide weekly data (with zeros for dates in between). # countries with weekly reporting weekly_reporting <- c("France", "Germany", "Spain", "Ukraine") covid_data <- filter(covid_data, !country %in% weekly_reporting) # for each country, get the time-varying severity estimate, # correcting for delays and smoothing the case and death data # first nest the data; nest() from {tidyr} df_covid_cfr <- nest( .by = country # define delay density function delay_density <- function(x) dlnorm(x, meanlog = 2.577, sdlog = 0.440) # to each nested data frame, apply the function `cfr_time_varying` # overwrite the `data` column, as all data will be preserved df_covid_cfr <- mutate( # using map() from {purrr} data = map( .x = data, .f = cfr_time_varying, # arguments to the function delay_density = delay_density, smoothing_window = 7, burn_in = 7 # unnest the cfr data; unnest() from {tidyr} df_covid_cfr <- unnest(df_covid_cfr, cols = data) For simplicity, we use the same delay distribution between onset and death for all countries — users should note that this likely introduces biases given inter-country differences in testing or reporting policies. Finally we plot the time-varying CFR for a selection of three countries with large outbreaks of Covid-19: Brazil, India, and the United States. filter(df_covid_cfr, country %in% c("Brazil", "India", "United States")) %>% ggplot() + x = date, ymin = severity_low, ymax = severity_high, group = country fill = "grey" ) + x = date, y = severity_estimate, colour = country ) + date_labels = "%b-%Y" ) + labels = percent ) + palette = "Dark2" ) + x = "Date", y = "CFR (%)" ) + ylim = c(0, 0.15), expand = FALSE ) + theme_classic() + theme(legend.position = "top") #> Warning: Removed 215 rows containing missing values (`geom_line()`). Details: Adjusting for delays between two time series cfr_time_varying() estimates the number of cases which have a known outcome over time following Nishiura et al. (2009), by calculating a quantity $k_t$ for each day within the input data, which represents the number of cases with a known adverse outcome (usually death), on day $t$. \[ k_t = \sum_{j = 0}^t c_t f_{j - t}. \] We then assume that the severity measure (usually CFR) is binomially distributed in the following way \[ d_t \sim {\sf Binomial}(k_t, \theta_t). \] We use maximum likelihood estimation to determine the value of $\theta_t$ for each $t$, where $\theta$ represents the severity measure of interest. The precise severity measure — case fatality risk (CFR), infection fatality risk (IFR), hospitalisation fatality risk (HFR), etc. — that $\theta$ represents depends upon the input data given by the Note that the function arguments burn_in and smoothing_window are not explicitly used in this calculation. burn_in controls how many estimates at the beginning of the outbreak are replaced with NAs — the calculation above is not applied to the first burn_in data points. The calculation is applied to the smoothed data, if a smoothing_window is specified. Linton, Natalie M., Tetsuro Kobayashi, Yichi Yang, Katsuma Hayashi, Andrei R. Akhmetzhanov, Sung-mok Jung, Baoyin Yuan, Ryo Kinoshita, and Hiroshi Nishiura. 2020. “Incubation Period and Other Epidemiological Characteristics of 2019 Novel Coronavirus Infections with Right Truncation: A Statistical Analysis of Publicly Available Case Data.” Journal of Clinical 9 (2): 538. Nishiura, Hiroshi, Don Klinkenberg, Mick Roberts, and Johan A. P. Heesterbeek. 2009. “Early Epidemiological Assessment of the Virulence of Emerging Infectious Diseases: A Case Study of an Influenza Pandemic.” PLOS ONE 4 (8): e6852.	{"url":"https://cran.ma.imperial.ac.uk/web/packages/cfr/vignettes/estimate_time_varying_severity.html","timestamp":"2024-11-05T06:52:06Z","content_type":"text/html","content_length":"571574","record_id":"<urn:uuid:db68b45e-2bc5-41c0-9459-8008d3317a83>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00493.warc.gz"}
An Easy Way to Solve Complex Optimization Problems in Machine Learning - DataScienceCentral.com Source: here There are numerous examples in machine learning, statistics, mathematics and deep learning, requiring an algorithm to solve some complicated equations: for instance, maximum likelihood estimation (think about logistic regression or the EM algorithm) or gradient methods (think about stochastic or swarm optimization). Here we are dealing with even more difficult problems, where the solution is not a set of optimal parameters (a finite dimensional object), but a function (an infinite dimensional object). The context is discrete, chaotic dynamical systems, with applications to weather forecasting, population growth models, complex econometric systems, image encryption, chemistry (mixtures), physics (how matter reaches an equilibrium temperature), astronomy (how celestial man-made or natural bodies end up having stable or unstable orbits), or stock market prices, to name a few. These are referred to as complex systems. The solutions to the problems discussed here requires numerical methods, as usually no exact solution is known. The type of equation to be solved is called functional equation or stochastic integral. We explore a few cases where the exact solution is actually known: this helps assess the efficiency, accuracy and speed of convergence of the numerical methods discussed in this article. These methods are based on the fixed-point algorithm applied to infinite dimensional problems. 1. The general problem We are dealing with a discrete dynamical system defined by xn+1 = T(xn), where T is a real-valued function, and x0 is the initial condition. For the sake of simplicity, we restrict ourselves to the case where xn is in [0, 1]. Generalizations, for instance with xn being a vector, are described here. The most well known example is the logistic map, with T(x) = λx(1-x), exhibiting a chaotic behavior or not, depending on the value of the parameter λ. In our case, the function T(x) takes the following form: T(x) = p(x) – INT(p(x)), where INT denote the integer part function, p(x) is positive, monotonic, continuous and decreasing (thus bijective) with p(1) = 1 and p(0) infinite. For instance p(x) = 1 / x corresponds to the Gauss map associated with continued fractions; it is the most fundamental and basic example, and I discuss it here as well as below in this article. Another example is the Hurwitz-Riemann map, discussed here. 1.1. Invariant distribution and ergodicity The invariant distribution of the system is the one followed by the successive xn‘s, or in other words, the limit of the empirical distribution attached to the xn‘s, given an initial condition x0. A lot of interesting properties can be derived if the invariant density f(x) (the derivative of the invariant distribution) is known, assuming it exists. This only works with ergodic systems. All systems under consideration here are ergodic. The invariant distribution applies to almost all initial conditions x0, though some x0‘s called exceptions, violate the law. This is a typical feature of all these systems. For some systems (the Bernoulli map for instance), the x0‘s that are not exceptions are called normal numbers. By ergodic, I mean that for almost any initial condition x0, the sequence (xn) eventually visits all parts of [0, 1], in a uniform and random sense. This implies that the average behavior of the system can be deduced from the trajectory of a “typical” sequence (xn) attached to an initial condition x0. Equivalently, a sufficiently large collection of random instances of the process (also called orbits) can represent the average statistical properties of the entire process. Invariant distributions are also called equilibrium or attractor distributions in probability theory. 1.2. The functional equation to be solved Let us assume that the invariant distribution F(x) can be written as F(x) = r(x+1) − r(1) for some function r. The support domain for F(x) is [0, 1], thus F(0) = 0, F(1) = 1, F(x) = 0 if x < 0, and F(x) = 1 if x > 1. Define R(x) = r(x+1) − r(x). Then we can retrieve p(x) (under some conditions) using the formula Thus r(x) must be increasing on [1,2] and r(2) = 1 + r(1). Not any function can be an invariant distribution. In practice, you know p(x) and you try to find the invariant distribution F(x). So the above formula is not useful, except that it helps you create a table of dynamical systems, defined by their function p(x), with known invariant distribution. Such a table is available here, see Appendix 1 in that article, in particular example 5 featuring a Riemann zeta system. It is useful to test the fixed point algorithm described in section 2, when the exact solution is known. If you only know p(x), to retrieve F(x) or its derivative f(x), you need to solve the following functional equation, whose unknown is the function f. where q is the inverse of the function p. Note that R(x) = F(q(x)) or alternatively, R(p(x)) = F(x), with p(q(x)) = q(p(x)) = x. Also, here x is in [0, 1]. In practice, you get a good approximation if you use the first 1,000 terms in the sum. Typically, the invariant density f is bounded, and the weights \|q‘(x+k)\| are decaying relatively fast as k increases. The theory behind this is beyond the scope of this article. It is based on the transfer operator, and also briefly discussed in one of my previous articles, here: see section “Functional equation for f“. The invariant density is the eigenfunction of the transfer operator, corresponding to the eigenvalue 1. Also, if x is replaced by a vector (for instance, if working with bivariate dynamical systems), the above formula can be generalized, involving two variables x, y, and the derivative of the (joint) distribution is replaced by a Jacobian. 2. Numerical solution via the fixed point algorithm The last formula in section 1.2. suggests a simple iterative algorithm to solve this type of equation. You need to start with an initial function f0, and in this case, the uniform distribution on [0, 1] is usually a good starting point. That is, f0(x) = 1 if x is in [0, 1], and 0 elsewhere. The iterative step is as follows: with x in [0, 1]. Each iteration n generates a whole new function fn on [0, 1], and the hope is that the algorithm converges as n tends to infinity. If convergence occurs, the limiting function must be the invariant density of the system. This is an example of the fixed point algorithm, in infinite dimension. In practice, you compute f(x) for only (say) 10,000 values of x evenly spaced between 0 and 1. If for instance, fn+1(0.5) requires the computation of (say) fn(0.879237…) and the closest value in your array is fn(0.8792), you replace fn(0.879237…) by fn(0.8792) or you use interpolation techniques. This is more efficient than using a function defined recursively in a programming language. Surprisingly the convergence is very fast and in the examples tested, the error between the true solution and the one obtained after 3 iterations, is very small, see picture below. In the above picture, p(x) = q(x) = 1 / x, and the invariant distribution is known: f(x) = 1 / ((1+x)(log 2)). It is pictured in red, and it is related to the Gauss-Kuzmin distribution. Note that we started with the uniform distribution f0 pictured in black (the flat line). The first iterate f1 is in green, the second one f2 is in grey, and the third one f3 is in orange, and almost undistinguishable from the exact solution in red (I need magnifying glasses to see it). Source code for these computations is available here. In the source code, there are two extra parameters α, λ. When α = λ = 1, it corresponds to the classic case p(x) = 1 / x. 3. Applications One interesting concept associated with these dynamical systems is that of digit. The n-th digit dn is defined as INT(p(xn)) where INT is the integer part function. I call it “digit” because all these systems have a numeration system attached to them, generalizing standard numeration systems which are just a particular case. If you know the digits attached to an initial condition x0, you can retrieve x0 with a simple algorithm. Start with n = N large enough and xn+1 = 0 (you will get about N digits of accuracy for x0), and compute iteratively xn backward from n = N to n = 0 using the recursion xn = q(xn+1 + dn) – INT(q(xn+1 + dn)). These digits can be used in encryption systems. This will be described in detail in my upcoming book Gentle Introduction to Discrete Dynamical Systems. However, the interesting part discussed here is related to statistical modeling. As a starter, let’s look at the digits of x0 = π – 3 in two different dynamical systems: • Continued fractions. Here p(x) = 1 / x. The first 20 digits are 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, 14, 3, 3, 23, 1, 1, 7, 4, 35, see here. • A less chaotic dynamical system. Here p(x) = (-1 + SQRT(5 +4/x)) / 2. The first 20 digits are 2, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 26, 1, 3, 1, 10, 1, 1. We also have F(x) = 2x / (x+1). The distribution of the digits is known in both cases. For continued fractions, it is the Gauss-Kuzmin distribution. For the second system, the probability that a digit is equal to k, is 4 / (k(k+1)( k+2)), see Example 1 in this article. In general, the probability in question is equal to F(q(k)) – F(q(k+1)) for k = 1, 2, and so on. Clearly, the distribution of these digits can be used to quantify the level of chaos in the system. For continued fractions, the expected value of an arbitrary digit is infinite (though it is finite and well known for the logarithm of a digit, see here), while it is finite (equal to 2) for the second system. Yet each system, given enough time, will shoot arbitrarily large digits. Another way to quantify chaos in a dynamical system is to look at the auto-correlation structure of the sequence (xn). Auto-correlations very close to zero, decaying very fast, are associated with highly chaotic systems. In the case of continued fraction, the lag-1 auto-correlation, defined as the limit of the empirical auto-correlation on a sequence starting with (say) x0 = π – 3, is where γ is the Euler–Mascheroni constant, see Appendix 2 in this article. This is probably a new result, never published before. Below is a picture featuring the successive values of p(xn) for the smoother dynamical system mentioned above. These values are close to the digits dn. the initial condition is x0 = π – 3. In my next article, I will further discuss a new way to define and measure chaos in these various systems. The first 5,500 values of p(xn), for n = 0, 1, 2 and so on, are featured in the above picture. Think about what business, natural or industrial process could be modeled by such kinds of time series! The possibilities are endless. For instance, it could represent meteorite hits over a large time period, with a few large values representing massive impacts. Clearly, it can be used in outlier, extreme events, and risk modeling. Finally, here is another example, this time based on an unrelated different bivariate dynamical system on the grid (the cat map), used for image encryption. This is a mapping on a picture of a pair of cherries. The image is 74 pixels wide, and takes 114 iterations to be restored, although it appears upside-down at the halfway point (the 57th iteration). Source: here. To receive a weekly digest of our new articles, subscribe to our newsletter, here. About the author: Vincent Granville is a data science pioneer, mathematician, book author (Wiley), patent owner, former post-doc at Cambridge University, former VC-funded executive, with 20+ years of corporate experience including CNET, NBC, Visa, Wells Fargo, Microsoft, eBay. Vincent is also self-publisher at DataShaping.com, and founded and co-founded a few start-ups, including one with a successful exit (Data Science Central acquired by Tech Target). You can access Vincent’s articles and books, here.	{"url":"https://www.datasciencecentral.com/an-easy-way-to-solve-complex-optimization-problems/","timestamp":"2024-11-04T13:47:56Z","content_type":"text/html","content_length":"173276","record_id":"<urn:uuid:b9273314-b3ec-47f9-ac50-7b1d8ba11c63>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00349.warc.gz"}
Fitting a Sine Function to Scatter Plot Data What will you learn? In this comprehensive tutorial, you will master the art of fitting a sine function to scatter plot data using Python. This skill is invaluable for various applications such as signal processing and analyzing periodic phenomena. Introduction to the Problem and Solution Encountering datasets with periodic tendencies, like seasonal temperature variations or sound waves, is common. By modeling such data with a sine function, we can unveil underlying patterns. The challenge lies in determining the optimal parameters (amplitude, frequency, phase shift) for the sine wave that best fits our data. To tackle this challenge, we will leverage curve fitting techniques available in Python libraries like NumPy and SciPy. These tools empower us to define a generic sine function and iteratively adjust its parameters until we achieve the ideal fit. This process not only ensures accurate modeling of our dataset but also reveals insightful characteristics within the data. import numpy as np from scipy.optimize import curve_fit import matplotlib.pyplot as plt # Define our sine function: f(x) = A * sin(Bx + C) def sine_function(x, A, B, C): return A * np.sin(B * x + C) # Example data (replace these with your actual data points) x_data = np.linspace(0, 4np.pi, 100) y_data = 3 np.sin(2 * x_data + 1.5) + np.random.normal(size=len(x_data)) # Fit our sine function to the data params, params_covariance = curve_fit(sine_function, x_data, y_data) # Plotting original data vs fitted curve plt.figure(figsize=(10, 6)) plt.plot(x_data,sine_function(x_data,params[0],params[1],params[2]),color='red',label='Fitted function') # Copyright PHD The solution commences by defining sine_function, representing our fitting model that computes (A \cdot \sin(Bx + C)), where (A), (B), and (C) denote amplitude, frequency (angular), and phase shift Subsequently, synthetic sample data (x_data and y_data) mimicking real-world scenarios is generated; noise is introduced into y_data through np.random.normal. In practical scenarios, this step would involve loading your dataset. The crux of our approach utilizes curve_fit from SciPy’s optimization module. This function takes our defined sine_function, along with initial parameter estimates (if needed), refining these values iteratively until they minimize the disparity between model predictions and observed values – effectively tailoring our model to fit observations accurately. Finally plotting both original scattered datapoints alongside our fitted curve visually confirms how well our sine model approximates real behavior within given dataset. 1. How do I choose initial parameter estimates for better convergence? One way is by visually inspecting your dataset�s pattern or utilizing domain knowledge about expected frequency/amplitude ranges which may inform better starting guesses. 2. Can I fit functions other than a sine wave? Absolutely! The approach remains similar; just define your target function with unknown parameters you aim to optimize. 3. What if my fitting doesn’t converge? Try different initial parameter estimates or increase max iterations allowed (maxfev parameter in curve_fit). Sometimes preprocessing steps like normalization might help too. 4. How do I evaluate my fit’s quality? Common metrics include sum of squared errors (SSE) available via covariance matrix returned by curve_fit or R-squared statistic comparing fitted values against mean of observed ones. 5. Can I apply weights to my datapoints during fitting? Yes! Use ‘sigma’ argument in curve_fit if some datapoints are more reliable than others so they have more influence on final fitted Mastering the technique of fitting a sine wave onto scatter plot data unlocks powerful analysis capabilities across various domains where cyclic behaviors prevail. Python’s scientific stack simplifies intricate numerical computations ensuring accessible yet robust solutions suitable for beginners seeking fundamental comprehension and experts aiming at precision-engineered outputs alike. Leave a Comment	{"url":"https://pythonhelpdesk.com/2024/02/25/fitting-a-sine-function-to-scatter-plot-data/","timestamp":"2024-11-10T01:59:08Z","content_type":"text/html","content_length":"43660","record_id":"<urn:uuid:ae69ae56-5579-45d6-9704-4300dffaa7e8>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00763.warc.gz"}
pi^(2) ms^(-2) and direction along the radius towards the centre A stone tied to the end of a string 1m long is whirled in a horizontal circle with a constant speed. If the stone makes 22 revolution in 44 seconds, what is the magnitude and direction of acceleration of the stone π2/4ms−2 and direction along the radius towards the centre π2ms−2 and direction along the radius away from the centre π2ms−2 and direction along the radius towards the centre π2ms−2 and direction along the tangent to the circle The correct Answer is:C r = 1m, frequency(f) =2244=12 sec acceleration (a) =v2r=ω2r=(2πf)2×r directed toward centre.	{"url":"https://www.doubtnut.com/qna/649422811","timestamp":"2024-11-10T18:28:42Z","content_type":"text/html","content_length":"213523","record_id":"<urn:uuid:81608e27-d906-4ff5-be81-478c68c6d870>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.61/warc/CC-MAIN-20241110170046-20241110200046-00408.warc.gz"}
Problem E Languages en is The University of Iceland is estimating how long it takes to walk between classrooms to make sure students have enough time between classes to get around. In a prior contest we asked you to find the worst case, this will not be required again. The question now is what the best cases are. You should find the two buildings that are the closest and the two buildings that are the second closest. The first line of the input is a single integer $3 \leq n \leq 10^5$, the number of buildings. The follow $n$ lines, each with two floating point numbers $-10^9 \leq x, y \leq 10^9$ denoting the entrance to a building at location $(x, y)$. We will assume that students move between these locations along straight lines with no regard to any obstacles that may be in their way. These numbers are given with at most $6$ digits after the decimal. Print two lines. The first containing the distance between the closest buildings. The second containing the distance between the second closest buildings. Your answers should be correct within absolute or relative error less than $10^{-6}$. Sample Input 1 Sample Output 1 0.0 0.0 1.0 2.0 1.0000000000 2.0 2.0 1.4142135624 3.0 0.0 3.0 3.0 Sample Input 2 Sample Output 2 1.0 0.0 0.0 1.0 1.0000000000 1.0 1.0 1.0000000000 2.0 1.0 1.0 2.0	{"url":"https://ru.kattis.com/courses/T-414-AFLV/aflv22/assignments/r3xzyr/problems/fkhi21.naestbest","timestamp":"2024-11-07T10:14:14Z","content_type":"text/html","content_length":"29546","record_id":"<urn:uuid:cc8006c9-18fb-4c14-9c95-2bb3d431ebb2>","cc-path":"CC-MAIN-2024-46/segments/1730477027987.79/warc/CC-MAIN-20241107083707-20241107113707-00366.warc.gz"}
Intrinsic dynamics of randomly clustered networks generate place fields and preplay of novel environments This study presents an important finding on the spontaneous emergence of structured activity in artificial neural networks endowed with specific connectivity profiles. The evidence supporting the claims of the authors is convincing, providing direct comparison between the properties of the model and neural data although investigating more naturalistic inputs to the network would have strengthened the main claims. The work will be of interest to systems and computational neuroscientists studying the hippocampus and memory processes. Significance of the findings: Important: Findings that have theoretical or practical implications beyond a single subfield • Landmark • Fundamental • Important • Valuable • Useful Strength of evidence: Convincing: Appropriate and validated methodology in line with current state-of-the-art • Exceptional • Compelling • Convincing • Solid • Incomplete • Inadequate During the peer-review process the editor and reviewers write an eLife Assessment that summarises the significance of the findings reported in the article (on a scale ranging from landmark to useful) and the strength of the evidence (on a scale ranging from exceptional to inadequate). Learn more about eLife Assessments During both sleep and awake immobility, hippocampal place cells reactivate time-compressed versions of sequences representing recently experienced trajectories in a phenomenon known as replay. Intriguingly, spontaneous sequences can also correspond to forthcoming trajectories in novel environments experienced later, in a phenomenon known as preplay. Here, we present a model showing that sequences of spikes correlated with the place fields underlying spatial trajectories in both previously experienced and future novel environments can arise spontaneously in neural circuits with random, clustered connectivity rather than pre-configured spatial maps. Moreover, the realistic place fields themselves arise in the circuit from minimal, landmark-based inputs. We find that preplay quality depends on the network’s balance of cluster isolation and overlap, with optimal preplay occurring in small-world regimes of high clustering yet short path lengths. We validate the results of our model by applying the same place field and preplay analyses to previously published rat hippocampal place cell data. Our results show that clustered recurrent connectivity can generate spontaneous preplay and immediate replay of novel environments. These findings support a framework whereby novel sensory experiences become associated with preexisting “pluripotent” internal neural activity patterns. The hippocampus plays a critical role in spatial and episodic memory in mammals (Morris et al., 1982; Squire et al., 2004). Place cells in the hippocampus exhibit spatial tuning, firing selectively in specific locations of a spatial environment (Moser et al., 2008; O’Keefe and Nadel, 1978). During sleep and quiet wakefulness, place cells show a time-compressed reactivation of spike sequences corresponding to recent experiences (Wilson and McNaughton, 1994; Foster and Wilson, 2006), known as replay. These replay events are thought to be important for memory consolidation, often referred to as memory replay (Carr et al., 2011). The CA3 region of the hippocampus is a highly recurrently connected region that is the primary site of replay generation in the hippocampus. Input from CA3 supports replay in CA1 (Csicsvari et al., 2000; Yamamoto and Tonegawa, 2017; Nakashiba et al., 2008; Nakashiba et al., 2009), and peri-ripple spiking in CA3 precedes that of CA1 (Nitzan et al., 2022). The recurrent connections support intrinsically generated bursts of activity that propagate through the network. Most replay models rely on a recurrent network structure in which a map of the environment is encoded in the recurrent connections of CA3 cells, such that cells with nearby place fields are more strongly connected. Some models assume this structure is pre-existing (Haga and Fukai, 2018; Pang and Fairhall, 2019), and some show how it could develop over time through synaptic plasticity ( Theodoni et al., 2018; Jahnke et al., 2015). Related to replay models based on place-field distance-dependent connectivity is the broader class of synfire-chain-like models. In these models, neurons (or clusters of neurons) are connected in a one-dimensional feed-forward manner (Diesmann et al., 1999; Chenkov et al., 2017). The classic idea of a synfire-chain has been extended to included recurrent connections, such as by Chenkov et al., 2017; however, such models still rely on an underlying one-dimensional sequence of activity propagation. A problem with these models is that in novel environments place cells remap immediately in a seemingly random fashion (Leutgeb et al., 2005; Muller et al., 1987). The CA3 region, in particular, undergoes pronounced remapping (Leutgeb et al., 2004; Leutgeb et al., 2005; Alme et al., 2014). A random remapping of place fields in such models that rely on environment-specific recurrent connectivity between place cells would lead to recurrent connections that are random with respect to the novel environment, and thus would not support replay of the novel environment. Rather, these models require a pre-existing structure of recurrent connections to be created for each environment. A proposed solution to account for remapping in hippocampal models is to assume the existence of multiple independent and uncorrelated spatial maps stored within the connections between cells. In this framework, the maximum number of maps is reached when the noise induced via connections needed for alternative maps becomes too great for a faithful rendering of the current map (Samsonovich and McNaughton, 1997; Battaglia and Treves, 1998; Azizi et al., 2013). However, experiments have found that hippocampal representations remain uncorrelated, with no signs of representation re-use, after testing as many as 11 different environments in rats (Alme et al., 2014). Rather than re-using a previously stored map, another possibility is that a novel map for a novel environment is generated de novo through experience-dependent plasticity while in the environment. Given the timescales of synaptic and structural plasticity, one might expect that significant experience within each environment is needed to produce each new map. However, replay can occur after just 1–2 laps on novel tracks (Foster and Wilson, 2006; Berners-Lee et al., 2022), which means that the synaptic connections that allow the generation of the replayed sequences must already be present. Consistent with this expectation, it has been found that decoded sequences during sleep show significant correlations when decoded by place fields from future, novel environments. This phenomenon is known as preplay and has been observed in both rodents (Dragoi and Tonegawa, 2011; Dragoi and Tonegawa, 2013; Grosmark and Buzsáki, 2016; Liu et al., 2019) and humans (Vaz et al., 2023 The existence of both preplay and immediate replay in novel environments suggests that the preexisting recurrent connections in the hippocampus that generate replay are somehow correlated with the pattern of future place fields that arise in novel environments. To reconcile these experimental results, we propose a model of intrinsic sequence generation based on randomly clustered recurrent connectivity, wherein place cells are connected within multiple overlapping clusters that are random with respect to any future, novel environment. Such clustering is a common motif across the brain, including the CA3 region of the hippocampus (Guzman et al., 2016) as well as cortex (Song et al., 2005; Perin et al., 2011), naturally arises from a combination of Hebbian and homeostatic plasticity in recurrent networks (Bourjaily and Miller, 2011; Litwin-Kumar and Doiron, 2014; Lynn et al., 2022), and spontaneously develops in networks of cultured hippocampal neurons (Antonello et al., 2022). As an animal gains experience in an environment, the pattern of recurrent connections of CA3 would be shaped by Hebbian plasticity (Debanne et al., 1998; Mishra et al., 2016). Relative to CA1, which has little recurrent connectivity, CA3 has been found to have both more stable spatial tuning and a stronger functional assembly organization, consistent with the hypothesis that spatial coding in CA3 is influenced by its recurrent connections (Sheintuch et al., 2023). Gaining experience in different environments would then be expected to lead to individual place cells participating in multiple formed clusters. Such overlapping clustered connectivity may be a general feature of any hippocampal and cortical region that has typical Hebbian plasticity rules. Sadovsky and MacLean, 2014 , found such structure in the spontaneous activity of excitatory neurons in primary visual cortex, where cells formed overlapping but distinct functional clusters. Further, such preexisting clusters may help explain the correlations that have been found in otherwise seemingly random remapping (Kinsky et al., 2018; Whittington et al., 2020) and support the rapid hippocampal representations of novel environments that are initially generic and become refined with experience (Liu et al., 2021). Such clustered connectivity likely underlies the functional assemblies that have been observed in hippocampus, wherein groups of recorded cells have correlated activity that can be identified through independent component analysis (Peyrache et al., 2010; Farooq et al., 2019). Since our model relies on its random recurrent connections for propagation of activity through the network during spontaneous activity, we also sought to assess the extent to which the internal activity within the network can generate place cells with firing rate peaks at a location where they do not receive a peak in their external input. While the total input to the network is constant as a function of position, each cell only receives a peak in its spatially linearly varying feedforward input at one end of the track. Our reasoning is that landmarks in the environment, such as boundaries or corners, provide location-specific visual input to an animal, but locations between such features are primarily indicated by their distance from them, which in our model is represented by reduction in the landmark-specific input. One can therefore equate our model’s inputs as corresponding to boundary cells (Savelli et al., 2008; Solstad et al., 2008; Bush et al., 2014), and the place fields between boundaries are generated by random internal structure within the network. Further, variations in spatial input forms do not affect the consistency and robustness of the model. In our implementation of this model, we find that spontaneous sequences of spikes generated by a randomly clustered network can be decoded as spatial trajectories without relying on pre-configured, environment-specific maps. Because the network contains neither a preexisting map of the environment nor an experience-dependent plasticity, we refer to the spike-sequences it generates as preplay. However, the model can also be thought of as a preexisting network in which immediate replay in a novel environment can be expressed and then reinforced through experience-dependent plasticity. We find that preplay in this model occurs most strongly when the network parameters are tuned to generate networks that have a small-world structure (Watts and Strogatz, 1998; Haga and Fukai, 2018; Humphries and Gurney, 2008). Our results support the idea that preplay and immediate replay could be a natural consequence of the preexisting recurrent structure of the hippocampus. We propose a model of preplay and immediate replay based on randomly clustered recurrent connections (Figure 1). In prior models of preplay and replay, a preexisting map of the environment is typically assumed to be contained within the recurrent connections of CA3 cells, such that cells with nearby place fields are more strongly connected (Figure 1a). While this type of model successfully produces replay (Haga and Fukai, 2018; Pang and Fairhall, 2019), such a map would only be expected to exist in a familiar environment, after experience-dependent synaptic plasticity has had time to shape the network (Theodoni et al., 2018). It remains unclear how, in the absence of such a preexisting map of the environment, the hippocampus can generate both preplay and immediate replay of a novel environment. Illustration of the randomly clustered model. Our proposed alternative model is based on a randomly clustered recurrent network with random feed-forward inputs (Figure 1b). In our model, all excitatory neurons are randomly assigned to overlapping clusters that constrain the recurrent connectivity, and they all receive the same linear spatial and contextual input cues which are scaled by randomly drawn, cluster-dependent connection weights (see Methods). This bias causes cells that share cluster memberships to have more similar place fields during the simulated run period, but, crucially, this bias is not present during sleep simulations so that there is no environment-specific information present when the network generates preplay. An example network with 8 clusters and cluster participation of 1.5 (the mean number of clusters to which an excitatory neuron belongs) is depicted in Figure 1c. Excitatory neurons are recurrently connected to each other and to inhibitory neurons. Inhibitory cells have cluster-independent connectivity, such that all E-to-I and I-to-E connections exist with a probability of 0.25. Feed-forward inputs are independent Poisson spikes with random connection strength for each neuron (Figure 1d). Excitatory cells are randomly, independently assigned membership to each of the clusters in the network. All neurons are first assigned to one cluster, and then randomly assigned additional clusters to reach the target cluster participation (Figure 1e). Given the number of clusters and the cluster participation, the within-cluster connection probability is calculated such that the global connection probability matches the parameter $pc=0.08$ (Figure 1f). The left peak in the distribution shown in Figure 1f is from cells in a single cluster and the right peak is from cells in two clusters, with the long tail corresponding to cells in more than two clusters. For a given $pc$, excitatory connectivity is parameterized by the number of clusters in the network and the mean cluster participation. The small-world index (SWI; Neal, 2015; Neal, 2017) systematically varies across this 2-D parameterization (Figure 1g). A high SWI indicates a network with both clustered connectivity and short path lengths (Watts and Strogatz, 1998). A ring lattice network (Figure 1—figure supplement 1a) exhibits high clustering but long path lengths between nodes on opposite sides of the ring. In contrast, a randomly connected network (Figure 1—figure supplement 1c) has short path lengths but lacks local clustered structure. A network with small world structure, such as a Watts-Strogatz network (Watts and Strogatz, 1998) or our randomly clustered model (Figure 1—figure supplement 1b), combines both clustered connectivity and short path lengths. In our clustered networks, for a fixed connection probability, SWI increases with more clusters and lower cluster participation, so long as cluster participation is greater than one to ensure sparse overlap of (and hence connections between) clusters. Networks in the top left corner of Figure 1g are not possible, since in that region all within-cluster connections are not sufficient to match the target global connectivity probability, $pc$. Networks in the bottom right are not possible because otherwise mean cluster participation would exceed the number of clusters. The dashed red line shows an example contour line where $SWI=0.4$. Our randomly clustered model produces both place fields and preplay with no environment-specific plasticity or preexisting map of the environment (Figure 2). Example place cell activity shows spatial specificity during linear track traversal (Figure 2a–c). Although the spatial tuning is noisy, this is consistent with the experimental finding that the place fields that are immediately expressed in a novel environment require experience in the environment to stabilize and improve decoding accuracy (Tang and Jadhav, 2022; Shin et al., 2019; Hwaun and Colgin, 2019). Raster plots of network spiking activity (Figure 2a) and example cell membrane potential traces (Figure 2b) demonstrate selective firing in specific track locations. Place fields from multiple networks generated from the same parameters, but with different input and recurrent connections, show spatial tuning across the track (Figure 2c). Spatially correlated reactivations in networks without environment-specific connectivity or plasticity. To test the ability of the model to produce preplay, we simulated sleep sessions in the same networks. Sleep sessions were simulated in a similar manner to the running sessions but with no location cue inputs active and a different, unique set of context cue inputs active to represent the sleep context. The strength of the context cue inputs to the excitatory and inhibitory cells were scaled in order to generate an appropriate level of network activity, to account for the absence of excitatory drive from the location inputs (see Methods). During simulated sleep, sparse, stochastic spiking spontaneously generates sufficient excitement within the recurrent network to produce population burst events resembling preplay (Figure 2d–f). Example raster and population rate plots demonstrate spontaneous transient increases in spiking that exceed 1 standard deviation above the mean population rate denoting population burst events (PBEs; Figure 2d). We considered PBEs that lasted at least 50 ms and contained at least five participating cells candidates for Bayesian decoding (Shin et al., 2019). Bayesian decoding of an example PBE using the simulated place fields reveals a spatial trajectory (Figure 2e). We use the same two statistics as Farooq et al., 2019 to quantify the quality of the decoded trajectory: the absolute weighted correlation (r) and the maximum jump distance (jd; Figure 2f). The absolute weighted correlation of a decoded event is the absolute value of the linear Pearson’s correlation of space-time weighted by the event’s derived posteriors. Since sequences can correspond to either direction along the track, the sign of the correlation simply indicates direction while the absolute value indicates the quality of preplay. The maximum jump distance of a decoded event is the maximum jump in the location of peak probability of decoded position across any two adjacent 10 ms time bins of the event’s derived posteriors. A high-quality event will have a high absolute weighted correlation and a low maximum jump distance. Together, these results demonstrate that the model can reproduce key dynamics of hippocampal place cells, including spatial tuning and preplay, without relying on environment-specific recurrent To compare the place fields generated by the model to those from hippocampal place cells of rats, we calculated several place-field statistics for both simulated and experimentally recorded place fields (Figure 3). Because our model assumes no previous environment-specific plasticity, we analyzed data from place cells in rats on their first exposure to a W-track (Shin et al., 2019). Equivalent statistics of place-field peak rate, sparsity, and spatial information are shown for experimental data (Figure 3a) and simulations (Figure 3b). We found that the model produces qualitatively similar (but not quantitatively identical) distributions for the fiducial parameter set. The model produces place fields with similar properties to hippocampal place fields. These place-field properties depend on the network parameters (Figure 3c). With fewer clusters and lower cluster overlap (lower cluster participation), place fields have higher peak rates, sparsity, and spatial information (Figure 3c, top row and bottom left). However, lower overlap reduces the uniformity of place-field locations, measured by KL-divergence (Figure 3c bottom middle) and the fraction of place fields in the central third of the track (Figure 3c bottom right). To verify that our simulated place cells were more strongly coding for spatial location than for elapsed time, we performed simulations with additional track traversals at different speeds and compared the resulting place fields and time fields in the same cells. We find that there is significantly greater place information than time information (Figure 3—figure supplement 1). Having found that the model produces realistic place-field representations with neither place-field like inputs nor environment-specific spatial representation in the internal network connectivity ( Figure 3), we next examined whether the same networks could generate spontaneous preplay of novel environments. To test this, for the same set of networks characterized by place-field properties in Figure 3, we simulated sleep activity by removing any location-dependent input cues and analyzed the resulting spike patterns for significant sequential structure resembling preplay trajectories ( Figure 4). We find significant preplay in both our reference experimental data set (Shin et al., 2019; Figure 4a and b; see Figure 4—figure supplement 1 for example events) and our model (Figure 4c and d) when analyzed by the same methods as Farooq et al., 2019, wherein the significance of preplay is determined relative to time-bin shuffled events (see Methods). The distribution of absolute weighted correlations of actual events was significantly greater than the distribution of absolute weighted correlations of shuffled events for both the experimental data (Figure 4a, KS-test, p=2 × 10^–12, KS-statistic=0.078) and the simulated data (Figure 4c, KS-test, p=3 × 10^–16, KS-statistic=0.29). Additionally, we found that this result is robust to random subsampling of cells in our simulated data (Figure 4—figure supplement 2). Our analyses of the hippocampal data produce similar results when analyzing each trajectory independently (Figure 4—figure supplement 3). Figure 4 with 4 supplements see all Preplay depends on modest cluster overlap. For each event, we also calculated the maximum spatial jump of the peak probability of decoded position between any two adjacent time bins as a measure of the continuity of the decoded trajectory. The absolute weighted correlation (high is better) and maximum jump (low is better) were then two different measures of the quality of a decoded trajectory. We performed a bootstrap test that took both of these measures into account by setting thresholds for a minimum absolute weighted correlation and a maximum jump distance and then calculating the fraction of events meeting both criteria of quality. The significance of the fraction of events meeting both criteria was then determined by comparing it against a distribution of such fractions generated by sets of the time-bin shuffled events. We systematically varied both thresholds and found that the actual events are of significantly higher quality than chance for a wide range of thresholds in both the hippocampal (Figure 4b) and simulated (Figure 4d) data. The upper right corner of these grids cannot be significant since 100% of all possible events would be included in any shuffle or actual set. Points in the left-most column are not all significant because the strictness of the maximum jump distance means that very few events in either the actual or shuffled data sets meet the criterion, and therefore the analysis is underpowered. This pattern is similar to that seen in Farooq et al., 2019 (as shown in their Figure 1e). Both PBEs and preplay are significantly affected by the two network parameters (Figure 4e and f). The number of clusters and the extent of cluster overlap (indicated via mean cluster participation) affects PBE participation (Figure 4e, top left), firing rates (Figure 4e, top right), event durations (Figure 4e, bottom left), and event frequency (Figure 4e, bottom right). We find that significant preplay occurs only at moderate cluster overlap (Figure 4f, top left), where we also find the greatest increase from chance in the linearity of decoded trajectories (Figure 4f, top right). The fraction of events that are individually significant (determined by comparing the absolute weighted correlation of each decoded event against the set of absolute weighted correlations of its own shuffles) is similarly highest for modest cluster overlap (Figure 4f, bottom left). The mean entropy of position probability of each time bin of decoded trajectories is also highest for modest cluster overlap (Figure 4f, bottom right), meaning that high cluster overlap leads to more diffuse, less precise spatial decoding. To test the robustness of our results to variations in input types, we simulated alternative forms of spatially modulated feedforward inputs. We found that with no parameter tuning or further modifications to the network, the model generates robust preplay with variations on the spatial inputs, including inputs of three linearly varying cues (Figure 4—figure supplement 4a) and two stepped cues (Figure 4—figure supplement 4b–c). The network is impaired in its ability to produce preplay with binary step location cues (Figure 4—figure supplement 4d), when there is no cluster bias (Figure 4—figure supplement 4e), and at greater values of cluster participation (Figure 4—figure supplement 4f). Preplay is due to successive activations of individual clusters Figure 4f indicates that PBEs are best decoded as preplay when cluster participation is only slightly above one, indicating a small, but non-zero, degree of cluster overlap. We hypothesized that this can be explained as balancing two counteracting requirements: (1) Sufficient cluster overlap is necessary for a transient increase in activity in one cluster to induce activity in another cluster, so as to extend any initiated trajectory; and (2) Sufficient cluster isolation is necessary so that, early in a transient, spikes from an excited cluster preferentially add excitement to the same cluster. A network with too much cluster overlap will fail to coherently excite individual clusters—rendering decoded positions to be spread randomly throughout the track—while a network with too little cluster overlap will fail to excite secondary clusters—rendering decoded positions to remain relatively localized. We find that the dependence of preplay on cluster overlap can indeed be explained by the manner in which clusters participate in PBEs (Figure 5). An example PBE (Figure 5a) shows transient recruitment of distinct clusters, with only one cluster prominently active at a time. We define a cluster as ‘active’ if its firing rate exceeds twice the rate of any other cluster. We calculated the number of active clusters per event (Figure 5b) and the duration of each active cluster period (Figure 5d). We find that these statistics vary systematically with the network parameters (Figure 5c and e), in a manner consistent with the dependence of preplay on cluster overlap (Figure 4f). When there is modest overlap of an intermediate number of clusters, events involve sequential activation of multiple clusters that are each active sufficiently long to correspond to at least one of the time bins used for decoding (10 ms). Figures 4 and 5 together indicate that high-quality preplay arises via a succession of individually active clusters. Such succession requires a moderate degree of cluster overlap, but this must be combined with sufficient cluster isolation to promote independent activation of just one cell assembly for the duration of each time-bin used for decoding. Coherent spiking within clusters supports preplay. The results of Figure 5 suggest that cluster-wise activation may be crucial to preplay. One possibility is that the random overlap of clusters in the network spontaneously produces biases in sequences of cluster activation which can be mapped onto any given environment. To test this, we looked at the pattern of cluster activations within events. We found that sequences of three active clusters were not more likely to match the track sequence than chance (Figure 5—figure supplement 1a). This suggests that preplay is not dependent on a particular biased pattern in the sequence of cluster activation. We then asked if the number of clusters that were active influenced preplay quality. We split the preplay events by the number of clusters that were active during each event and found that the median preplay shift relative to shuffled events with the same number of active clusters decreased with the number of active clusters (Spearman’s rank correlation, p=0.0019, ρ=−0.13; Figure 5—figure supplement 1b). Cluster identity is sufficient for preplay The pattern of preplay significance across the parameter grid in Figure 4f shows that preplay only occurs with modest cluster overlap, and the results of Figure 5 show that this corresponds to the parameter region that supports transient, isolated cluster-activation. This raises the question of whether cluster-identity is sufficient to explain preplay. To test this, we took the sleep simulation population burst events from the fiducial parameter set and performed decoding after shuffling cell identity in three different ways. We found that when the identity of all cells within a network are randomly permuted the resulting median preplay correlation shift is centered about zero (t-test 95% confidence interval, –0.2018–0.0012) and preplay is not significant (distribution of p-values is consistent with a uniform distribution over 0–1, chi-square goodness-of-fit test p=0.4436, chi-square statistic = 2.68; Figure 6a). However, performing decoding after randomly shuffling cell identity between cells that share membership in a cluster does result in statistically significant preplay for all shuffle replicates, although the magnitude of the median correlation shift is reduced for all shuffle replicates (Figure 6b). The shuffle in Figure 6b does not fully preserve cell’s cluster identity because a cell that is in multiple clusters may be shuffled with a cell in either a single cluster or with a cell in multiple clusters that are not identical. Performing decoding after doing within-cluster shuffling of only cells that are in a single cluster results in preplay statistics that are not statistically different from the unshuffled statistics (t-test relative to median shift of un-shuffled decoding, p=0.1724, 95% confidence interval of –0.0028–0.0150 relative to the reference value; Figure 6c). Together these results demonstrate that cluster-identity is sufficient to produce preplay. Preplay is abolished when events are decoded with shuffled cell identities but is preserved if cell identities are shuffled only within clusters. Mean relative spike rank correlates with place field location While cluster-identity is sufficient to produce preplay (Figure 6b), the shuffle of Figure 6c is incomplete in that cells belonging to more than one cluster are not shuffled. Together, these two shuffles leave room for the possibility that individual cell-identity may contribute to the production of preplay. It might be the case that some cells fire earlier than others, both on the track and within events. To test the contribution of individual cells to preplay, we calculated for all cells in all networks of the fiducial parameter point their mean relative spike rank and tested if this is correlated with the location of their mean place field density on the track (Figure 7). We find that there is no relationship between a cell’s mean relative within-event spike rank and its mean place field density on the track (Figure 7a). This is the case when the relative rank is calculated over the entire network (Figure 7, ‘Within-network’) and when the relative rank is calculated only with respect to cells with the same cluster membership (Figure 7, ‘Within-cluster’). However, because preplay events can proceed in either track direction, averaging over all events would average out the sequence order of these two opposite directions. We performed the same correlation but after reversing the spike order for events with a negative slope in the decoded trajectory (Figure 7b). To test the significance of this correlation, we performed a bootstrap significance test by comparing the slope of the linear regression to the slope that results when performing the same analysis after shuffling cell identities in the same manner as in Figure 6. We found that the linear regression slope is greater than expected relative to all three shuffling methods for both the within-network mean relative rank correlation (Figure 6c) and the within-cluster mean relative rank correlation (Figure 6d). Place cells’ mean event rank are correlated with their place field location when accounting for decode direction. Small-world index correlates with preplay We noticed that that the highest quality of decoded trajectories (Figure 4f) seemed to arise in networks with the highest small-world index (SWI; Figure 1g). In order to test this, we simulated different sets of networks with both increased and decreased global E-to-E connection probability, $pc$. Changing $pc$, in addition to varying the number of clusters and the mean cluster participation, impacted the SWI of the networks (Figure 8, left column). The Small-World Index of networks correlates with preplay quality. We hypothesized that independent of $pc$, a higher SWI would correlate with improved preplay quality. To test this, we simulated networks across a range of parameters for three $pc$ values: a decrease of $pc$ by 50% – 0.04, the fiducial value of 0.08, and an increase by 50% – 0.12 (Figure 8a–c, respectively). For the decreased and increased $pc$ cases, the E-to-E connection strength was respectively doubled or reduced to 2/3 of the fiducial strength to keep total E-to-E input constant. For each parameter combination, we quantified preplay quality as the rightward shift in median absolute weighted correlation of decoded preplay events versus shuffled events (as in Figure 4f, top right). We then asked if there was a correlation between that quantification of preplay quality and SWI. Across all three $pc$ values, SWI significantly correlated with improved preplay both across parameter sets (Figure 8, center column) and across individual networks (Figure 8, right column). These results support our prediction that higher small-world characteristics correspond to higher-quality preplay dynamics regardless of average connectivity. Preplay significantly decodes to linear trajectories in arbitrary environments Information about each environment enters the network via the feed-forward input connection strengths, which contain cluster-dependent biases. A new environment is simulated by re-ordering those input biases. We first wished to test that a new environment simulated in such a manner produced a distinct set of place fields. We therefore simulated place maps for leftward and rightward trajectories on linear tracks in two distinct environments (Figure 9a). The two maps with different directions of motion showed very high correlations when in the same environment (Figure 9b, blue) while the comparisons of trajectories across environments show very low correlations (Figure 9b, red). Cells that share membership in a cluster will have some amount of correlation in their remapping due to the cluster-dependent cue bias, which is consistent with experimental results (Hampson et al., 1996; Pavlides et al., 2019), but the combinatorial nature of cluster membership renders the overall place field map correlations low (Figure 9b). We also performed simulations with extra laps of running and calculated the correlations between paired sets of place fields produced by random, independent splits of trials of the same trajectory. The distribution of these correlations was similar to the distribution of within-environment correlations (comparing opposite trajectories with the same spatial input), showing no significant de novo place-field directionality. This is consistent with hippocampal data in which place-field directionality is initially low in novel environments and increases with experience (Frank et al., 2004; Navratilova et al., 2012; Shin et al., 2019). Trajectories decoded from population-burst events are significantly correlated with linear trajectories in arbitrary environments. Because we simulated preplay without any location-specific inputs, we expected that the set of spiking events that significantly decode to linear trajectories in one environment (Figure 4) should decode with a similar fidelity in another environment. Therefore, we decoded each PBE four times, once with the place fields of each trajectory (Figure 9c–e). Since the place field map correlations are high for trajectories on the same track and near zero for trajectories on different tracks, any individual event would be expected to have similar decoded trajectories when decoding based on the place fields from different trajectories in the same environment and dissimilar decoded trajectories when decoding based on place fields from different environments. A given event with a strong decoded trajectory based on the place fields of one environment would then be expected to have a weaker decoded trajectory when decoded with place fields from an alternative environment (Figure 9c). The distributions of absolute weighted correlations arising from decoding of PBEs according to each of the four sets of place fields was consistent across environments (Figure 9d, colored lines) and all were significantly rightward shifted (indicating greater absolute weighted correlation) when compared to those absolute weighted correlations arising from the corresponding shuffled events ( Figure 9d, overlapping black lines). If we consider both absolute weighted correlation and jump-distance thresholds as in Figure 4d, we find that the matrices of p-values are consistent across environments (Figure 9e). In summary, without environment-specific or place-field dependent pre-assigned internal wiring, the model produces population-burst events, which, as an ensemble, show significant preplay with respect to any selected environment. Our work shows that spontaneous population bursts of spikes that can be decoded as spatial trajectories can arise in networks with clustered random connectivity without pre-configured maps representing the environment. In our proposed model, excitatory neurons were randomly clustered with varied overlap and received feed-forward inputs with random strengths that decayed monotonically from the boundaries of a track (Figure 1). Even though the model neural circuit lacked place-field like input and lacked environment-specific internal wiring, the network exhibited both realistic place fields (Figures 2 and 3) and spontaneous preplay of novel, future environments (Figures 2 and 4). We validated our modeling results by applying the same analyses to a previously collected experimental data set (Shin et al., 2019). Indeed, we replicated the general finding of hippocampal preplay found previously in Farooq et al., 2019, although the p-value matrix for our experimental data (Figure 4b) is significant across a smaller range of threshold values than found in their prior work. This is likely due to differences in statistical power. The pre-experience sleep sessions of Shin et al., 2019 were not longer than half an hour for each animal, while the pre-experience sleep sessions of Farooq et al., 2019 lasted 2–4 hr. However, finding statistically significant hippocampal preplay in an experiment not designed for studying preplay shows that the general result is robust to a number of methodological choices, including shorter recording sessions, use of a W-track rather than linear track, and variations in candidate event detection criterion. Although our model is a model of the recurrently connected CA3 region and the data set we analyze (Shin et al., 2019) comes from CA1 cells, the qualitative comparisons we make here are nevertheless useful. Despite some statistically significant quantitative differences, the general properties of place fields that we consider are qualitatively similar across CA1 and CA3 (Sheintuch et al., 2023; Harvey et al., 2020), and CA3 and CA1 generally reactivate in a coordinated manner (O’Neill et al., 2008; Karlsson and Frank, 2009). The model parameters that controlled the clustering of the recurrent connections strongly influenced preplay and place-field quality. Moderate overlap of clusters balanced the competing needs for both (a) sufficiently isolated clusters to enable cluster-wise activation and (b) sufficiently overlapping clusters to enable propagation of activity across clusters (Figure 5). In our clustered network structure, such a balance in cluster overlap produces networks with small-world characteristics (Watts and Strogatz, 1998) as quantified by a small-world index (SWI; Neal, 2015; Neal, 2017). Networks with a high SWI, indicating high clustering (if two neurons are connected to the same third neuron, they are more likely than chance to be connected to each other) yet short paths (the mean number of connections needed to traverse from one neuron to any other), showed optimal preplay dynamics (Figure 8). The same networks could flexibly represent distinct remapped environments (Leutgeb et al., 2004; Leutgeb et al., 2005; Alme et al., 2014) solely through differences in scaling of feed-forward spatially linear input (Figure 9). Across many species, small-world properties can be found at both the local neuronal network scale and the gross scale of the network of brain regions. At the neuronal connection scale, small-world properties have been reported in a number of networks, such as the C. elegans connectome (Watts and Strogatz, 1998; Humphries and Gurney, 2008), the brainstem reticular formation (Haga and Fukai, 2018), mouse visual cortex (Sadovsky and MacLean, 2014), cultured rat hippocampal neurons (Antonello et al., 2022), mouse prefrontal cortex (Luongo et al., 2016), and connectivity within the entorhinal-hippocampal region in rats (She et al., 2016). At the level of connected brain regions, small-world properties have been reported across the network of brain regions activated by fear memories in mice (Vetere et al., 2017), in the hippocampal-amygdala network in humans (Zhang et al., 2022), and across the entire human brain (Liao et al., 2011). Our results suggest that the preexisting hippocampal dynamics supporting preplay may reflect general properties arising from randomly clustered connectivity, where the randomness is with respect to any future, novel experience. The model predicts that preplay quality will depend on the network’s balance of cluster isolation and overlap, as quantified by small-world properties. Synaptic plasticity in the recurrent connections of CA3 may primarily serve to reinforce and stabilize intrinsic dynamics, which could be established through a combination of developmental programming (Perin et al., 2011; Druckmann et al., 2014; Huszár et al., 2022) and past experiences (Bourjaily and Miller, 2011), rather than creating spatial maps de novo. The particular neural activity associated with a given experience would then selectively reinforce the relevant intrinsic dynamics, while leaving the rest of the network dynamics unchanged. Our model provides a general framework for understanding the origin of pre-configured hippocampal dynamics. Hebbian plasticity on independent, previously experienced place maps would produce effectively random clustered connectivity. The spontaneous dynamics of such networks would influence expression of place fields in future, novel environments. Together with intrinsic sequence generation, this could enable preplay and immediate replay generated by the preexisting recurrent connections. Future modeling work should explore how experience-dependent plasticity may leverage and reinforce the dynamics initially expressed through preexisting clustered recurrent connections to produce higher-quality place fields and decoded trajectories during replay (Shin et al., 2019; Farooq et al., 2019). Plasticity may strengthen connectivity along frequently reactivated spatiotemporal patterns. Clarifying interactions between intrinsic dynamics and experience-dependent plasticity will provide key insights into hippocampal neural activity. Additionally, the in vivo microcircuitry of CA3 is complex and includes aspects such as nonlinear dendritic computations and a variety of inhibitory cell types (Rebola et al., 2017). This microcircuitry is crucial for explaining certain aspects of hippocampal function, such as ripple and gamma oscillogenesis (Ramirez-Villegas et al., 2018), but here we have focused on a minimal model that is sufficient to produce place cell spiking activity that is consistent with experimentally measured place field and preplay statistics. To investigate what network properties could support preplay, we simulated recurrently connected networks of spiking neurons and analyzed their dynamics using standard hippocampal place cell We simulate networks of Leaky Integrate-and-Fire (LIF) neurons, which have leak conductance, $gL$, excitatory synaptic conductance, $gE$, inhibitory synaptic conductance, $gI$, spike-rate adaptation (SRA) conductance, $gSRA$, and external feed-forward input synaptic conductance, $gext$. The membrane potential, $V$, follows the dynamics ${\tau }_{m}\frac{dV}{dt}=-{g}_{L}\left(V-{E}_{L}\right)-{g}_{E}\left(V-{E}_{E}\right)-{g}_{I}\left(V-{E}_{I}\right)-{g}_{SRA}\left(V-{E}_{SRA}\right)-{g}_{ext}\left(V-{E}_{E}\right)$ where $τm$ is the membrane time constant, $EL$ is the leak reversal potential, $EE$ is the excitatory synapse reversal potential, $EI$ is the inhibitory synapse reversal potential, $ESRA$ is the SRA reversal potential, and $Eext$ is the external input reversal potential. When the membrane potential reaches the threshold $Vth$, a spike is emitted and the membrane potential is reset to $Vreset$. The changes in SRA conductance and all synaptic conductances follow ${\tau }_{i}\frac{d{g}_{i}}{dt}=-{g}_{i}$ to produce exponential decay between spikes for any conductance $i$. A step increase in conductance occurs at the time of each spike by an amount corresponding to the connection strength for each synapse ($WE-E$ for E-to-E connections, $WE-I$ for E-to-I connections, and $WI-E$ for I-to-E connections), or by $δSRA$ for $gSRA$. Initial feed-forward input conductances were set to values approximating their steady-state values by randomly selecting values from a Gaussian with a mean of $WinrGτE$ and a standard deviation of $Win2rGτE$. Initial values of the recurrent conductances and the SRA conductance were set to zero. Parameter Fiducial value Description $τm$ 40 ms Membrane time constant $Cm$ 0.4 nF Membrane capacitance $dt$ 0.1 ms Simulation time step $gL$ 10 nS Leak conductance $EL$ -70 mV Leak reversal potential $EE$ 0 mV Excitatory synaptic reversal potential $EI$ -70 mV Inhibitory synaptic reversal potential $ESRA$ -80 mV SRA reversal potential $Vth$ -50 mV Spike threshold $Vreset$ -70 mV Reset potential $τE$ 10 ms Excitatory time constant $τI$ 3 ms Inhibitory time constant $τSRA$ 30 ms Spike-rate adaptation time constant $δSRA$ 3 pS Spike-rate adaptation strength We simulated networks of $n=500$ neurons, of which 75% were excitatory. Excitatory neurons were randomly, independently assigned membership to each of $nc$ clusters in the network. First, each neuron was randomly assigned membership to one of the clusters. Then, each cluster was assigned a number—$nE(μc-1)/nc$ rounded to the nearest integer—of additional randomly selected neurons such that each cluster had identical numbers of neurons, $nE,clust=nE(μc/nc)$,and mean cluster participation, $μc$, reached its goal value. E-to-E recurrent connections were randomly assigned on a cluster-wise basis, where only neurons that shared membership in a cluster could be connected. The within-cluster connection probability was configured such that the network exhibited a desired global E-to-E connection probability $pc$. Given the total number of possible connections between excitatory neurons is $Ctot=nE(nE-1)$ and the total number of possible connections between excitatory neurons within all clusters is $Cclust=nE,clust(nE,clust−1)nc$, we calculated the within-cluster connection probability as $pc(Ctot/Cclust)$. That is, given the absence of connections between clusters (clusters were coupled by the overlap of cells) the within-cluster connection probability was greater than $pc$ so as to generate the desired total number of connections equal to $pcCtot$. All E-to-I and I-to-E connections were independent of cluster membership and existed with a probability $pcI$. There were no I-to-I connections. $pc$, $nc$, and $μc$ were varied for some simulations. Except where specified otherwise, all parameters took the fiducial value shown in the table below. The network visualization in Figure 1c was plotted based on the first two dimensions of a t-distributed stochastic neighbor embedding of the connectivity between excitatory cells using the MATLAB function tsne. The feature vector for each excitatory cell was the binary vector indicating the presence of both input and output connections. Parameter Fiducial value Description $n$ 500 Number of neurons $nE$ 375 Number of excitatory neurons n[c] or ‘cluster’ 15 Number of clusters μ[c] or ‘cluster participation’ 1.25 Mean cluster membership per neuron $pc$ 0.08 E-to-E connection probability $pcI$ 0.25 E-to-I and I-to-E connection probability $WE-E$ 220 pS E-to-E synaptic conductance step increase $WE-I$ 400 pS E-to-I synaptic conductance step increase $WI-E$ 400 pS I-to-E synaptic conductance step increase All excitatory neurons in the network received three different feed-forward inputs (Figure 1b). Two inputs were spatially modulated, with rates that peaked at either end of the track and linearly varied across the track to reach zero at the opposite end. One input was a context cue that was position independent. All excitatory cells received unique Poisson spike trains from each of the three inputs at their position-dependent rates. Inhibitory cells received only the context input. The connection strength of each feed-forward input to each neuron was determined by an independent and a cluster-specific factor. First, strengths were randomly drawn from a log-normal distribution $eμ+σN$, where $N$ is a zero-mean, unit variance Normal distribution, $μ=lnWin2σin+Win2$ and $σ=lnσinWin2+1$ for mean strength $Win$ and standard deviation $σin$ for the location cues, with $σin$ replaced by $σcontext$ for the context cue. Each environment and the sleep session had unique context cue input weights. For model simplicity, the mean input strength $Win$ for all inputs was kept the same for both E and I cells in both the awake and sleep conditions, but the strength of the resulting context input was then scaled by some factor $fx$ for each of the four cases to accommodate for the presence, or lack thereof, of the additional current input from the location cues. These scaling factors were set at a level that generated appropriate levels of population activity. During simulation of linear track traversal, the context cue to excitatory cells was scaled down by $fE-awake$ to compensate for the added excitatory drive of the location cue inputs, and the context cue input to I cells was not changed ($fI-awake=1$). During sleep simulation, the context cue input to E cells was not scaled ( $fE-awake=1$) but the context cue input to I cells was scaled down by $fI-sleep$. Second, to incorporate cluster-dependent correlations in place fields, a small ($≤4%$) location cue bias was added to the randomly drawn feed-forward weights based on each neuron’s cluster membership. For each environment, the clusters were randomly shuffled and assigned a normalized rank bias value, such that the first cluster had a bias of –1 (corresponding to a rightward cue preference) and the last cluster had a bias of +1 (leftward cue preference). A neuron’s individual bias was calculated as the mean bias of all clusters it belonged to, multiplied by the scaling factor $σbias$. The left cue weight for each neuron was then scaled by 1 plus its bias, and the right cue weight was scaled by 1 minus its bias. In this way, the feed-forward input tuning was biased based on the mean rank of a neuron’s cluster affiliations for each environment. The addition of this bias produced correlations in cells’ spatial tunings based on cluster membership, but, importantly, this bias was not present during the sleep simulations, and it did not lead to high correlations of place-field maps between environments (Figure 9b). Parameter Value Description $rG$ 5000 Hz Peak Poisson input rate $Win$ 72 pS Mean strength of the input synapses $σin$ 5 pS Standard deviation of the location cue input synapses $σcontext$ 1.25 pS Standard deviation of the context cue input synapses $σbias$ 0.04 Location bias scale $fE-awake$ 0.1 E-cell context cue input scaling during awake simulation $fE-sleep$ 1 E-cell context cue input scaling during sleep simulation $fI-awake$ 1 I-cell context cue input scaling during awake simulation $fI-sleep$ 0.75 I-cell context cue input scaling during sleep simulation For a given parameter set, we generated 10 random networks. We simulated each network for one sleep session of 120 s and for five 2 s long traversals of each of the two linear trajectories on each track. For the parameter grids in Figures 3 and 4, we simulated 20 networks with 300 s long sleep sessions in order to get more precise empirical estimates of the simulation statistics. For analysis comparing place-field reliability, we simulated 10 traversals of each trajectory. To compare coding for place vs time, we performed repeated simulations for the same networks at the fiducial parameter point with 1.0 x and 2.0 x of the original track traversal speed. We then combined all trials for both speed conditions to calculate both place fields and time fields for each cell from the same linear track traversal simulations. The place fields were calculated as described below (average firing rate within each of the fifty 2 cm long spatial bins across the track) and the time fields were similarly calculated but for fifty 40 ms time bins across the initial two seconds of all track traversals. We followed the methods of Shin et al., 2019 to generate place fields from the spike trains. We calculated for each excitatory cell its trial-averaged occupancy-discounted firing rate in each 2 cm spatial bin of the 1 m long linear track. Note that the occupancy-discounting term is uniform across bins, so it has no impact in our model, because we simulated uniform movement speed. We then smoothed this with a Gaussian kernel with a 4 cm standard deviation. For statistics quantifying place-field properties and for Bayesian decoding, we considered only excitatory cells with place-field peaks exceeding 3 Hz as in Shin et al., 2019. Place-field specificity was defined as 1 minus the fraction of the spatial bins in which the place field’s rate exceeded 25% of its maximum rate (Shin et al., 2019). Place-field spatial information The spatial information of each cells’ place field was calculated as $\text{Spatial Information}=\sum _{i}{p}_{i}\left(\frac{{r}_{i}}{\overline{r}}\right)lo{g}_{2}\left(\frac{{r}_{i}}{\overline{r}}\right)$ where $pi$ is the probability of being in spatial bin $i$, $ri$ is the place field’s rate in spatial bin $i$, and $r¯$ is the mean rate of the place field (Sheintuch et al., 2023). Given the division of the track into 50 spatial bins, spatial information could vary between 0 for equal firing in all bins and $log250≅5.6$ for firing in only a single bin. Spatial information of 1 is equivalent, for example, to equal firing in exactly one half of the bins and no firing elsewhere. We used two measures to quantify the extent to which place-field peaks were uniformly distributed across the track. In our first measure, we calculated the Kullback-Leibler divergence of the distribution of peaks from a uniform distribution, as ${D}_{KL}=-\sum _{i}{p}_{i}^{\text{data}}lo{g}_{2}\left(\frac{{p}_{i}^{\text{uniform}}}{{p}_{i}^{\text{data}}}\right)$ where $pidata$ is the fraction of cells with peak firing rates in the $ith$ spatial bin and $piuniform$ is 1/50, that is the fraction expected from a uniform distribution (Sheintuch et al., 2023). Similarly, the range for spatial information, $DKL$ is bounded between zero for a perfectly uniform distribution of peaks and $log250≅5.6$ if all peaks were in a single bin. $DKL$ of 1 is equivalent, for example, to all peaks being uniformly spread over one half of the bins in the track. For our second measure, we calculated the fraction of place cells whose peak firing rate was in the central third of the track. Since inputs providing spatial information only peaked at the boundaries of the track, the central third was ubiquitously the most depleted of high firing rates. Place-field map correlations To compare the similarity of place fields across different trajectories, we calculated the correlation between the place-field rate maps of each pair of trajectories. For each spatial bin, we calculated the Pearson correlation coefficient between the vector of the population place-field rates of the two trajectories. We then averaged the correlation coefficients across all spatial bins to get the correlation between the two trajectories. We detected candidate preplay events in the simulated data by identifying population-burst events (PBEs). During the simulated sleep period, we calculated the mean rate of the population of excitatory cells, which defines the population rate, smoothed with a Gaussian kernel (15 ms standard deviation). We then detected PBEs as periods of time when the population rate exceeded 1 standard deviation above the mean population rate for at least 30 ms. We also required the peak population rate to exceed 0.5 Hz (corresponding to 5–6 spikes per 30 ms among excitatory cells) in order for the rate fluctuation to qualify as a PBE. We then combined PBEs into a single event if their start and end times were separated by less than 10 ms. Sharp-wave ripple detection Because of the reduced number of recorded cells relative to the simulated data, we detected candidate events in the Shin et al., 2019 data with a method that incorporated the ripple band oscillation power in the local field potential (LFP) in addition to the population spiking activity. We first calculated the smoothed firing rate for each excitatory neuron by convolving its spikes with a Gaussian kernel (100 ms standard deviation) and capping at 1 to prevent bursting dominance. We then computed the z-scored population firing rate from the capped, smoothed single-neuron rates. Additionally, we calculated the z-scored, ripple-filtered envelope of the tetrode-averaged LFP. We then summed these two z-scores and detected peaks that exceeded 6 for at least 10 ms and exceeded the neighboring regions by at least 6 (MinPeakHeight, MinPeakWidth, and MinPeakProminence of the MATLAB function findpeaks, respectively). Candidate events were defined as periods around detected peaks, spanning from when the z-score sum first dipped below 0 for at least 5 ms before the peak to after the peak when it again dipped below 0 for at least 5 ms. We additionally required that the animal be immobile during the event. We performed Bayesian decoding of candidate preplay events following the methods of Shin et al., 2019. We performed decoding on all candidate events that had at least 5 active cells and exceeded at least 50 ms in duration. Spikes in the event were binned into 10 ms time bins. We decoded using the place fields for each trajectory independently. The description provided below is for the decoding using the place fields of one particular trajectory. For each time bin of each event, we calculated the location on the track represented by the neural spikes based on the place fields of the active cells using a memoryless Bayesian decoder where $P(x\|s)$ is the probability of the animal being in spatial bin $x$ given the set of spikes $s$ that occurred in the time bin, $P(s\|x)$ is the probability of the spikes $s$ given the animal is in spatial bin $x$ (as given by the place fields), $P(x)$ is the prior probability of the animal being in spatial bin $x$, and $P(s)$ is the probability of the spikes $s$. We assumed a uniform prior probability of position, $P(x)$. We assumed that the $N$ cells firing during the event acted as independent Poisson processes in order to calculate $P\left(s\|x\right)=\prod _{i}^{N}\frac{\left(\tau {r}_{i}\left(x\right){\right)}^{{s}_{i}}{e}^{-\tau {r}_{i}\left(x\right)}}{{s}_{i}!}$ where $τ$ is the time bin window duration (10 ms), $ri(x)$ is the place-field rate of cell $i$ in spatial bin $x$ and $si$ is the number of spikes from cell $i$ in the time bin. This allows us to calculate the posterior probability of position for each time bin as $P\left(x\|s\right)=C\left(\prod _{i}^{N}{r}_{i}\left(x{\right)}^{{s}_{i}}\right){e}^{-\tau \sum _{i}^{N}{r}_{i}\left(x\right)}$ where $C$ is a normalization constant, which accounts for the position-independent term, $Ps$. Bayesian decoding statistical analyses We analyzed the significance of preplay using the methods of Farooq et al., 2019 (see also Silva et al., 2015). We computed two measures of the sequence quality of each decoded event: the event’s absolute weighted correlation and its jump distance. The absolute weighted correlation is the absolute weighted Pearson’s correlation of decoded position across the event’s time bins. For each decoded event, we calculate the weighted correlation between space and time with MATLAB’s fitlm function using the decoded probability in each space-time bin (10 ms by 2 cm) as the weight for the corresponding location in the correlation. The absolute value of the weighted correlation is used in order to account for both forward and reverse preplay. The jump distance is the maximum of the distance between the positions of peak probability for any two adjacent 10 ms time bins in the event, quantified as fraction of the track length. For each event, we generated 100 shuffled events by randomly permuting the order of the 10 ms time bins. We then calculated the weighted correlation and jump distance for each shuffled event in the same manner as for the actual events. For each simulated parameter set, we combined all events from the 10 simulated networks. Following the methods of Farooq et al., 2019, we calculated the statistical significance of the population of preplay events using two different methods. First, we used the Kolmogorov-Smirnov (KS) test to compare the distributions of absolute weighted correlations obtained from the actual events and the shuffled events (Figure 4a and c). Second, we used a bootstrap test to compare the fraction of high-quality events—defined as having both high absolute weighted correlations and low maximum jump distance—relative to shuffles (Figure 4b and d). To perform the bootstrap test, we created a grid of thresholds for minimum absolute weighted correlation and maximum jump distance, and for each combination of thresholds we calculated the fraction of actual events that exceeded the minimum absolute weighted correlation threshold and did not exceed the maximum jump distance threshold. Then, we generated 100 data sets of shuffled events by randomly permuting the order of the 10 ms time bins for each actual event and calculated the fraction of events meeting the same pairs of thresholds for each shuffled data set. The p-value of the fraction of high-quality events was then calculated as the fraction of shuffled data sets with a higher fraction of high-quality events. To test the significance of each event’s absolute weighted correlation individually, we calculated the event’s p-value as the fraction of the event’s own shuffles that had a higher absolute weighted correlation than the un-shuffled event (Figure 4f, bottom left). The spatial entropy $H$ of a decoded event was calculated as the mean over its time bins of the entropy of the decoded position probability in each time bin, using the equation $H=-\sum _{i}{p}_{i}\phantom{\rule{thinmathspace}{0ex}}lo{g}_{2}\left({p}_{i}\right)$ for each time bin, where $pi$ is the decoded position probability for spatial bin $i$. Cell identity shuffled decoding We performed Bayesian decoding on the fiducial parameter set after shuffling cell identities in three different manners (Figures 6 and 7). To shuffle cells in a cluster-independent manner (‘Across-network shuffle’), we randomly shuffled the identity of cells during the sleep simulations. To shuffle cells within clusters (‘Within-cluster shuffle’), we randomly shuffled cell identity only between cells that shared membership in at least one cluster. To shuffle cells within only single clusters (‘Within-single-cluster shuffle’), we shuffled cells in the same manner as the within-cluster shuffle but excluded any cells from the shuffle that were in multiple clusters. To test for a correlation between spike rank during sleep PBEs and the order of place fields on the track (Figure 7), we calculated for each excitatory cell in each network of the fiducial parameter set its mean relative spike rank and correlated that with the location of its mean place field density on the track (Figure 7a). To account for event directionality, we calculated the mean relative rank after inverting the rank within events that had a negatively sloped decoded trajectory (Figure 7b). We calculated mean relative rank for each cell relative to all cells in the network (‘Within-network mean relative rank’) and relative to only cells that shared cluster membership with the cell (‘Within-cluster mean relative rank’). We then compared the slope of the linear regression between mean relative rank and place field location against the slope that results when applying the same analysis to each of the three methods of cell identify shuffles for both the within-network regression (Figure 7c) and the within-cluster regression (Figure 7d). The small-world index (SWI) was calculated following the method of Neal, 2015 (see also Neal, 2017). It was defined as where $L$ is the mean path distance and $C$ is the clustering coefficient of the network. We calculate $L$ as the mean over all ordered pairs of excitatory cells of the shortest directed path length from the first to the second cell. We calculate $C$ as the ratio of the number of all triplets of excitatory cells that are connected in either direction over the number of all triplets that could form, following the methods of Fagiolo, 2007 for directed graphs. $Ll$ and $Cl$ are the expected values for a one-dimensional ring lattice network with the same size and connection probability (in which connections are local such that there are no connections between cells with a greater separation on the ring than that of any pairs without a connection). And $Lr$ and $Cr$ are the expected values for a random network of the same size and connection probability. A network with a high SWI index is therefore a network with both a high clustering coefficient, similar to a ring lattice network, and small mean path length, similar to a random network. For directed graphs of size $n$, average degree $k$, and global connection probability $p$: $Cr=p$ (Fagiolo, 2007), $Lr=ln(n)-γln(k)+0.5$ (Fronczak et al., 2004), $Cl=3(k−2)4(k−1)$ (Neal, 2015) $Ll=n2k+0.5$ (Neal, 2015; Fronczak et al., 2004) where $γ$ is the Euler-Mascheroni constant. To quantify cluster activation (Figure 5), we calculated the population rate for each cluster individually as the mean firing rate of all excitatory cells belonging to the cluster smoothed with a Gaussian kernel (15 ms standard deviation). A cluster was defined as ‘active’ if at any point its population rate exceeded twice that of any other cluster during a PBE. The active clusters’ duration of activation was defined as the duration for which it was the most active cluster. To test whether the sequence of activation in events with three active clusters matched the sequence of place fields on the track, we performed a bootstrap significance test (Figure 5—figure supplement 1). For all events from the fiducial parameter set that had three active clusters, we calculated the fraction in which the sequence of the active clusters matched the sequence of the clusters’ left vs right bias on the track in either direction. We then compared this fraction to the distribution expected from randomly sampling sequences of three clusters without replacement. To determine if there was a relationship between the number of active clusters within an event and it’s preplay quality, we performed a Spearman’s rank correlation between the number of active clusters and the normalized absolute weighted correlation across all events at the fiducial parameter set. The absolute weighted correlations were z-scored based on the absolute weighted correlations of the time-bin shuffled events that had the same number of active clusters. Electrophysiological data was reanalyzed from the hippocampal CA1 recordings first published in Shin et al., 2019. All place-field data (Figure 3a) came from the six rats’ first experience on the W-track spatial alternation task. All preplay data (Figure 4a and b) came from the six rats’ first sleep-box session, which lasted 20–30 min and occurred immediately before their first experience on the W-track. Simulations and analysis were performed in MATLAB with custom code. Code available at https://github.com/primon23/Preplay_paper, copy archived at Miller, 2024. All computer codes, which can reproduce all simulated data and carry out analyses can be found at GitHub, copy archived at Miller, 2024. The experimental data has been deposited at DANDI Archive. 51. Book The Hippocampus as a Cognitive Map Oxford : New York: Clarendon Press ; Oxford University Press. Article and author information Author details National Institutes of Health (R01NS104818) • Jordan Breffle • Hannah Germaine • Paul Miller National Institutes of Health (R01MH112661) • Justin D Shin • Shantanu P Jadhav National Institutes of Health (R01MH120228) • Shantanu P Jadhav • Justin D Shin Brandeis University (Neuroscience Graduate Program) • Jordan Breffle • Hannah Germaine The funders had no role in study design, data collection and interpretation, or the decision to submit the work for publication. NIH/NINDS R01NS104818, NIH/NIMH R01MH112661, NIH/NIMH R01MH120228, and Brandeis University Neuroscience Graduate Program. This study was performed in strict accordance with the recommendations in the Guide for the Care and Use of Laboratory Animals of the National Institutes of Health. All of the animals were handled according to approved institutional animal care and use committee (IACUC) protocol #24001-A of Brandeis University. All surgery was performed under ketamine, xylazine, and isoflurane anesthesia, and every effort was made to minimize suffering. You can cite all versions using the DOI https://doi.org/10.7554/eLife.93981. This DOI represents all versions, and will always resolve to the latest one. © 2024, Breffle et al. This article is distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use and redistribution provided that the original author and source are credited. Views, downloads and citations are aggregated across all versions of this paper published by eLife. A two-part list of links to download the article, or parts of the article, in various formats. Downloads (link to download the article as PDF) Open citations (links to open the citations from this article in various online reference manager services) Cite this article (links to download the citations from this article in formats compatible with various reference manager tools) 1. Jordan Breffle 2. Hannah Germaine 3. Justin D Shin 4. Shantanu P Jadhav 5. Paul Miller Intrinsic dynamics of randomly clustered networks generate place fields and preplay of novel environments eLife 13:RP93981. Further reading 1. Brain water homeostasis not only provides a physical protection, but also determines the diffusion of chemical molecules key for information processing and metabolic stability. As a major type of glia in brain parenchyma, astrocytes are the dominant cell type expressing aquaporin water channel. How astrocyte aquaporin contributes to brain water homeostasis in basal physiology remains to be understood. We report that astrocyte aquaporin 4 (AQP4) mediates a tonic water efflux in basal conditions. Acute inhibition of astrocyte AQP4 leads to intracellular water accumulation as optically resolved by fluorescence-translated imaging in acute brain slices, and in vivo by fiber photometry in mobile mice. We then show that aquaporin-mediated constant water efflux maintains astrocyte volume and osmotic equilibrium, astrocyte and neuron Ca^2+ signaling, and extracellular space remodeling during optogenetically induced cortical spreading depression. Using diffusion-weighted magnetic resonance imaging (DW-MRI), we observed that in vivo inhibition of AQP4 water efflux heterogeneously disturbs brain water homeostasis in a region-dependent manner. Our data suggest that astrocyte aquaporin, though bidirectional in nature, mediates a tonic water outflow to sustain cellular and environmental equilibrium in brain parenchyma. 2. Neural implants have the potential to restore lost sensory function by electrically evoking the complex naturalistic activity patterns of neural populations. However, it can be difficult to predict and control evoked neural responses to simultaneous multi-electrode stimulation due to nonlinearity of the responses. We present a solution to this problem and demonstrate its utility in the context of a bidirectional retinal implant for restoring vision. A dynamically optimized stimulation approach encodes incoming visual stimuli into a rapid, greedily chosen, temporally dithered and spatially multiplexed sequence of simple stimulation patterns. Stimuli are selected to optimize the reconstruction of the visual stimulus from the evoked responses. Temporal dithering exploits the slow time scales of downstream neural processing, and spatial multiplexing exploits the independence of responses generated by distant electrodes. The approach was evaluated using an experimental laboratory prototype of a retinal implant: large-scale, high-resolution multi-electrode stimulation and recording of macaque and rat retinal ganglion cells ex vivo. The dynamically optimized stimulation approach substantially enhanced performance compared to existing approaches based on static mapping between visual stimulus intensity and current amplitude. The modular framework enabled parallel extensions to naturalistic viewing conditions, incorporation of perceptual similarity measures, and efficient implementation for an implantable device. A direct closed-loop test of the approach supported its potential use in vision restoration. 3. Neuromodulatory inputs to the hippocampus play pivotal roles in modulating synaptic plasticity, shaping neuronal activity, and influencing learning and memory. Recently, it has been shown that the main sources of catecholamines to the hippocampus, ventral tegmental area (VTA) and locus coeruleus (LC), may have overlapping release of neurotransmitters and effects on the hippocampus. Therefore, to dissect the impacts of both VTA and LC circuits on hippocampal function, a thorough examination of how these pathways might differentially operate during behavior and learning is necessary. We therefore utilized two-photon microscopy to functionally image the activity of VTA and LC axons within the CA1 region of the dorsal hippocampus in head-fixed male mice navigating linear paths within virtual reality (VR) environments. We found that within familiar environments some VTA axons and the vast majority of LC axons showed a correlation with the animals’ running speed. However, as mice approached previously learned rewarded locations, a large majority of VTA axons exhibited a gradual ramping-up of activity, peaking at the reward location. In contrast, LC axons displayed a pre-movement signal predictive of the animal’s transition from immobility to movement. Interestingly, a marked divergence emerged following a switch from the familiar to novel VR environments. Many LC axons showed large increases in activity that remained elevated for over a minute, while the previously observed VTA axon ramping-to-reward dynamics disappeared during the same period. In conclusion, these findings highlight distinct roles of VTA and LC catecholaminergic inputs in the dorsal CA1 hippocampal region. These inputs encode unique information, with reward information in VTA inputs and novelty and kinematic information in LC inputs, likely contributing to differential modulation of hippocampal activity during behavior and learning.	{"url":"https://elifesciences.org/articles/93981","timestamp":"2024-11-08T14:37:12Z","content_type":"text/html","content_length":"523052","record_id":"<urn:uuid:68575fe5-8c23-4a3f-a046-04880e83099a>","cc-path":"CC-MAIN-2024-46/segments/1730477028067.32/warc/CC-MAIN-20241108133114-20241108163114-00219.warc.gz"}
Unlocking the Lock: Solving the “Open the Lock” Problem with BFS Today, we’re diving into a classic interview problem, Open the Lock. This problem requires us to use Breadth-First Search (BFS) to find the shortest path by simulating a lock mechanism with four wheels. Let’s walk through the problem step-by-step, analyze the solution, and cover key details in the code. The problem describes a lock with four rotating wheels, each numbered from 0 to 9. The lock starts at “0000”, and each wheel can be rotated up or down to the next number. For instance, “0” can rotate to “1” or “9”. We have a list of deadend combinations. If we reach one of these, the lock stops, and we can’t proceed further from that position. Given these constraints, we need to find the minimum number of moves to turn the lock from “0000” to a target combination target. If it’s impossible, we return -1. Since we’re trying to find the shortest path to the target, this problem is a perfect candidate for Breadth-First Search (BFS). BFS explores all paths level by level, ensuring we reach the target in the fewest moves possible. Step-by-Step Solution Outline: 1. Initialization: Store all deadends in a Set for quick lookup. 2. BFS: Begin from “0000” and explore every possible combination by turning each wheel up or down. 3. Track Moves: Use a queue to store each combination and the current move count, increasing it by one each level. Code Implementation with Comments * @param {string[]} deadends - List of dead-end combinations * @param {string} target - Target combination * @return {number} - Minimum moves to reach target from "0000", or -1 if impossible var openLock = function(deadends, target) { // Initialize a Set for deadends for quick lookup const seen = new Set(deadends); // Handle edge cases: if "0000" or target is in deadends, we can't proceed if (seen.has(target) \|\| seen.has('0000')) return -1; // If the start is already the target, no moves are needed if (target === '0000') return 0; // Initialize BFS queue and moves counter const queue = ["0000"]; seen.add("0000"); // Mark "0000" as visited let steps = 0; // Main BFS loop while (queue.length > 0) { // Number of nodes in the current level let size = queue.length; // Process each node in the current level while (size > 0) { const current = queue.shift(); // Remove the first element in queue // If we've reached the target, return the steps taken if (current === target) return steps; // Generate all possible next moves and add to queue const comb = generateComb(current); for (const next of comb) { if (!seen.has(next)) { // Only add unvisited nodes queue.push(next); // Add new combination to queue seen.add(next); // Mark new combination as visited size--; // Decrement level size steps++; // Move to the next level (increment moves) // If BFS completes without finding target, return -1 return -1; * generateComb - Generates all possible next moves for a given lock state * @param {string} input - Current lock state * @return {string[]} - Returns all possible next moves var generateComb = function(input) { const combinations = []; // Loop through each of the four wheels for (let i = 0; i < 4; i++) { // Get the current digit of the wheel const digit = parseInt(input[i]); // Roll the digit up by one (wrap around from 9 to 0) const up = input.slice(0, i) + ((digit + 1) % 10) + input.slice(i + 1); combinations.push(up); // Add the new combination to the result array // Roll the digit down by one (wrap around from 0 to 9) const down = input.slice(0, i) + ((digit - 1 + 10) % 10) + input.slice(i + 1); combinations.push(down); // Add the new combination to the result array return combinations; Detailed Code Explanation 1. Main Function openLock: □ The seen set stores all visited states to prevent revisiting deadends and previous combinations. □ We initialize a queue to store each combination and mark “0000” as visited to prevent reprocessing. □ The BFS loop processes nodes level by level, allowing us to find the shortest path to target. 2. Combination Generator generateComb: □ For each digit in the 4-digit lock, we generate two new states: one by rolling the digit up and one by rolling it down. □ Each new combination is added to the result array, which is then returned to the BFS loop for further processing. 3. Core Logic: □ BFS systematically expands from the start state, checking all possible next moves. □ The algorithm uses steps to count levels, ensuring that when we reach target, it’s in the minimum moves. □ If BFS finishes without finding target, we return -1, as it’s impossible to reach. Example Walkthrough Consider the case where deadends = ["0201","0101","0102","1212","2002"] and target = "0202": • Initial State: queue = ["0000"], steps = 0 • First Level (step = 1): From “0000”, we expand to “1000”, “9000”, “0100”, “0900”… adding all valid moves to the queue. • Subsequent Levels: We keep expanding outward, avoiding deadends and marking visited states. • Final Steps: Eventually, we reach “0202” in the minimum moves and return the step count. This problem is an excellent example of BFS’s utility in finding shortest paths. Key takeaways include: • Using BFS: For shortest path problems where each move has equal weight, BFS provides an efficient approach. • Tracking Visited States: By storing visited states, we avoid deadends and reduce redundant processing. • Generating Moves: By iterating through each digit and generating moves, we simulate all possible paths from a given state. By practicing with problems like this, you’ll improve your understanding of BFS and how to apply it to real-world scenarios! This site uses Akismet to reduce spam. Learn how your comment data is processed.	{"url":"https://maxinrui.com/unlocking-the-lock-solving-the-open-the-lock-problem-with-bfs/","timestamp":"2024-11-05T10:17:54Z","content_type":"text/html","content_length":"95823","record_id":"<urn:uuid:b03fe6b6-13bd-4290-a228-0be285d02b91>","cc-path":"CC-MAIN-2024-46/segments/1730477027878.78/warc/CC-MAIN-20241105083140-20241105113140-00313.warc.gz"}
Distance oracles for vertex-labeled graphs Given a graph G = (V,E) with non-negative edge lengths whose vertices are assigned a label from L = {λ[1],...,λ[ℓ]}, we construct a compact distance oracle that answers queries of the form: "What is δ(ν,λ)?", where ν ∈ V is a vertex in the graph, λ ∈ L a vertex label, and δ(ν,λ) is the distance (length of a shortest path) between ν and the closest vertex labeled λ in G. We formalize this natural problem and provide a hierarchy of approximate distance oracles that require subquadratic space and return a distance of constant stretch. We also extend our solution to dynamic oracles that handle label changes in sublinear time. Original language English Title of host publication Automata, Languages and Programming - 38th International Colloquium, ICALP 2011, Proceedings Pages 490-501 Number of pages 12 Edition PART 2 State Published - 11 Jul 2011 Externally published Yes Event 38th International Colloquium on Automata, Languages and Programming, ICALP 2011 - Zurich, Switzerland Duration: 4 Jul 2011 → 8 Jul 2011 Publication series Name Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) Number PART 2 Volume 6756 LNCS ISSN (Print) 0302-9743 ISSN (Electronic) 1611-3349 Conference 38th International Colloquium on Automata, Languages and Programming, ICALP 2011 Country/Territory Switzerland City Zurich Period 4/07/11 → 8/07/11 ASJC Scopus subject areas • Theoretical Computer Science • General Computer Science Dive into the research topics of 'Distance oracles for vertex-labeled graphs'. Together they form a unique fingerprint.	{"url":"https://cris.bgu.ac.il/en/publications/distance-oracles-for-vertex-labeled-graphs-2","timestamp":"2024-11-12T04:01:30Z","content_type":"text/html","content_length":"57066","record_id":"<urn:uuid:51f2baf4-fd55-448a-8e39-1149743d43ed>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00598.warc.gz"}
Numbers Divisible By 12 \| Learn and Solve Questions Imagine 3 friends trying to share 10 biscuits. Each of them gets 3 biscuits, and there’s one biscuit left over. They are unsure what to do with it; would one person get an extra biscuit? That did not seem fair to everyone, who loved sharing everything equally. If there were 9 biscuits, they would have divided the biscuits equally, and there would have been no confusion. Because 9 is divisible by 3. This means that 9 biscuits could have been divided into three equal parts without any extra biscuits left. What is Divisibility Rule? Divisibility rules are a set of general rules often used to determine whether or not a number is divisible by a particular number. Divisibility rules can help kids determine whether a number will be divisible by another. Divisibility Rule of 12 The divisibility chart or rule of 12 is - If the number is divisible by both 3 and 4, then the number is divisible by 12 exactly. Example: 5844 Sum of the digits = 5 + 8 + 4 + 4 = 21 (multiple of 3) Last two digits = 44 (divisible by 4) The number 5844 is divisible by 4 and 3; hence, it is divisible by 12. Divisibility by 12 Solved Worksheet Following are the divisibility rule of 12 with examples - Example 1: Check whether 840 is divisible by 12 or not. Solution: We know that - If the given number is divisible by both 3 and 4, then it is also divisible by 12. 1. First, check whether the given number is divisible by 3. Sum of the digits :8 + 4 + 0 = 12 (multiple of 3) 2. Now, check whether the given number is divisible by 4. In the given number 840, the number formed by the last two digits is 40, divisible by 4. So, the number 840 is divisible by 4. Now, it is clear that the given number 840 is divisible by both 3 and 4. Therefore, the number 840 is divisible by 12 Example 2: Check whether 9140 is divisible by 12 or not. Solution: We know that - If the given number is divisible by both 3 and 4, then it is also divisible by 12. 1. Firstly, check whether the given number is divisible by 3. Sum of the digits : 9 + 1 + 4 + 0 = 14 (not a multiple of 3) Because the number 9140 is not divisible by 3, the number 9140 is not divisible by 12. Example 3: Check whether 2370 is divisible by 12 or not. Solution: We know that - If the given number is divisible by both 3 & 4, then it is also divisible by 12. 1. divisible by 12. 1. Firstly, check whether the given number is divisible by 3. Sum of the digits: 2 + 3 + 7 + 0 = 12 (multiple of 3) 2. Now, check whether the given number is divisible by 4. In the given number 2370, the number formed by the last two digits is 70, which is not divisible by 4. So, the number 2370 is not divisible by 4. Therefore, the number 2370 is not divisible by 12. In this article, we have learned about the divisibility rule for 12. For instance, a number divisible by 12 will also be divisible by 3 and 4. Many mathematical rules and properties are required or beneficial to understand when solving a Maths problem. Learning and comprehending these rules provides students with a foundation to solve problems and tackle more advanced mathematical concepts. FAQs on Numbers Divisible By 12 1. What does the term "divisibility test" mean? A divisibility test determines whether a given integer is divisible by a fixed divisor without performing the division, typically by examining its digits. A number is said to be divisible by another number if the result of the division is a whole number. 2. How many divisibility rules exist? A divisibility rule is a heuristic for determining whether a positive integer can be divided evenly by another positive integer (i.e. there is no remainder left over). For example, determining whether a number is even is as simple as looking at its last digit: 2, 4, 6, 8, or 0. 3. What is the definition of a perfectly divisible number? This method generally employs digits to determine whether a given number is divided by a divisor. If one number is perfectly divisible by another, the remainder should be zero, and the quotient should be a whole number. We have rules for divisibility for 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, and so on.	{"url":"https://www.vedantu.com/maths/numbers-divisible-by-12","timestamp":"2024-11-11T09:49:37Z","content_type":"text/html","content_length":"181866","record_id":"<urn:uuid:0c035c52-1baf-4820-905e-10964096b3ec>","cc-path":"CC-MAIN-2024-46/segments/1730477028228.41/warc/CC-MAIN-20241111091854-20241111121854-00732.warc.gz"}
Axial map The axial map is constructed by taking an accurate map and drawing a set of intersecting lines through all the spaces of the urban grid so that the grid is covered and all rings of circulation are Hillier, B. & Hanson, J. (1984), The Social Logic of Space, Cambridge University Press: Cambridge. pp.17, 91; Turner, A., Penn, A., & Hillier, B. (2005), An algorithmic definition of the axial map. Environment and Planning B: Planning and Design 32(3):425-444 Vaughan L., Geddes I. (2009), “Urban form and deprivation: a contemporary proxy for Charles Booth’s analysis of poverty” Radical Statistics 99 46-73	{"url":"https://www.spacesyntax.online/term/axial-map/","timestamp":"2024-11-10T10:45:50Z","content_type":"text/html","content_length":"58174","record_id":"<urn:uuid:d29d912c-4f92-4441-9dcf-41ffc9069dd9>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00694.warc.gz"}
Q3 Vector Distances Q3 Vector Distances 25 Points Purpose: Write array functions to solve a mathematical problem. I'm sure you have all heard of Euclidean distance. In two dimensions, the Euclidean distance between (21,2₂) and (3.3/2) is For example, the distance between (3, 1) and (1, 1) is √(21-31)² + (22-2². and the Euclidean distance between (1, 1) and (0,0) is ((3-1)²+(1-1)²)¹/² 2 The Euclidean distance bewteen (0, 0) and (1,1) is approximately 1.4142. This notion of Euclidean distance can easily be extended to vectors of arbitrary dimension. Indeed, if is a vector of dimension n (versus dimension 2), then its Euclidean norm, which we will denote by \|\|*\|\|2, is given by ((1-0)²+(1-0)²)¹/2 1.4142 There, is a shorthand for "sum," as we saw in lecture and z; is just the absolute value of the real number zi. By the way, this is quite common notation and such distances are used in machine learning applications like image and speech recognition (you will see an intro later on the \|\|176 — 36\|²2) ¹1/²2 — (153 — 301)² + \|2₂-M₂² ++ \|³n-M³₁ (²) ¹/2 It turns out that the Euclidean distance is just one of many possible vector norms. Another common one is the 1-norm, aka the "Manhattan distance," so called since that is the distance in city blocks a taxicab must travel between two points on the streets and avenues of Manhattan. The formula for the 1-norm is: \|\|z-y\|\|1=(x₁ - y₁\|) = (x₁ − Y₁\| + x2 - y2 + ... + In - Yn), See also, the figure below, which is reproduced from https://en.wikipedia.org/wiki/Taxicab geometry There, the Euclidean or "straight-line" distance is shown in green and the 1-norm or "Manhattan distance" are shown by the equally shortest "taxicab" paths in red, blue, and yellow./n(a: 7 points) Use a function to compute the usual, Euclidean distance between the two filled circles in the example above: i. Point your browser to https://colab.research.google.com/drive/1kCbywGZ20mmkkncYzMUWJXB zgVKaT? File -> Save a copy in Drive and rename it pynb ii. Read through the function. iii. Add comments to the function. iv. Underneath the function, add a single line that both calls it to compute the Euclidean distance (length of green line) between the point (0,0) and (6,6) in the figure above and also prints the v. Copy and paste the output in a comment at the end of the block. vi. Turn in a cut-and-paste of your entire block in the box below: (b; 3 points) Don't do it, but explain in words how you would alter the function to work for and y of arbitrary dimension (that is, not 2)? You may assume that and have the same length./n(b; 3 points) Don't do it, but explain in words how you would alter the function to work for x and y of arbitrary dimension that is, not 2)? You may assume that and have the same length. Save Answer (b; 3 points) Don't do it, but explain in words how you would alter the function to work for and of arbitrary dimension (that is, not 2)? You may assume that and have the same length. Enter your answer in the box below. (c; 15 points) Next, you will a function that computes the vector norm 21. It should work for arbitrary vectors z and y of the same length. That is, the function header might look like the following, where and are lists det nori, y If your function is called as below normi, 1. (1. 1]) then it should return 2. Here, the or dimension of 1,115 11, 13 which is 1 1. In the same notebook as used in (a) above, make a copy of the first block. Edit the program to compute the Manhattan distance for arbitrary vectors x and y of the same length (again, arbitary) Also, edit the comments as necessary. iii. Test your method by computing (and printing) the Manhattan distance between (0,0) and (6, 6), as seen in red/blue/yellow in the picture above. Again, place the answer as a comment at the end of your block. vi. Turn in a cut-and-paste of the entire block in the box below: Fig: 1 Fig: 2 Fig: 3	{"url":"https://tutorbin.com/questions-and-answers/q3-vector-distances-25-points-purpose-write-array-functions-to-solve-a-mathematical-problem-i-m-sure-you-have-all-heard","timestamp":"2024-11-12T15:44:52Z","content_type":"text/html","content_length":"81780","record_id":"<urn:uuid:629c37d1-cbbb-4951-87e2-150c313ca4a4>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.63/warc/CC-MAIN-20241112145015-20241112175015-00104.warc.gz"}
Julia sets with Ahlfors-regular conformal dimension one by InSung Park Geometry Topology Seminar Monday, February 22, 2021 - 2:00pm for 1 hour (actually 50 minutes) InSung Park – Indiana University Bloomington – park433@iu.edu – https://sites.google.com/view/insung-park/ Please Note: Office hours will be held 3-4pm EST. Complex dynamics is the study of dynamical systems defined by iterating rational maps on the Riemann sphere. For a rational map f, the Julia set Jf is a beautiful fractal defined as the repeller of the dynamics of f. Fractal invariants of Julia sets, such as Hausdorff dimensions, have information about the complexity of the dynamics of rational maps. Ahlfors-regular conformal dimension, abbreviated by ARconfdim, is the infimum of the Hausdorff dimension in a quasi-symmetric class of Ahlfors-regular metric spaces. The ARconfdim is an important quantity especially in geometric group theory because a natural metric, called a visual metric, on the boundary of any Gromov hyperbolic group is determined up to quasi-symmetry. In the spirit of Sullivan's dictionary, we can use ARconfdim to understand the dynamics of rational maps as well. In this talk, we show that the Julia set of a post-critically finite hyperbolic rational map f has ARconfdim 1 if and only if there is an f-invariant graph G containing the post-critical set such that the dynamics restricted to G has topological entropy zero.	{"url":"https://math.gatech.edu/seminars-colloquia/series/geometry-topology-seminar/insung-park-20210222","timestamp":"2024-11-14T08:07:56Z","content_type":"text/html","content_length":"32464","record_id":"<urn:uuid:5d38801b-0895-42e2-a5f7-fd52f9db33f3>","cc-path":"CC-MAIN-2024-46/segments/1730477028545.2/warc/CC-MAIN-20241114062951-20241114092951-00500.warc.gz"}
Suffix Automaton detailed A Free Trial That Lets You Build Big! Start building with 50+ products and up to 12 months usage for Elastic Compute Service • Sales Support 1 on 1 presale consultation • After-Sales Support 24/7 Technical Support 6 Free Tickets per Quarter Faster Response • Alibaba Cloud offers highly flexible support services tailored to meet your exact needs.	{"url":"https://topic.alibabacloud.com/a/suffix-automaton-detailed_8_8_10259404.html","timestamp":"2024-11-11T07:30:15Z","content_type":"text/html","content_length":"89173","record_id":"<urn:uuid:315acd2b-6b8b-4a73-8858-4a537ac7dcda>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00314.warc.gz"}