text: Trigonometry questions with answers. 20449 + 27225 = c2. com on November 14, 2020 by guest This quiz tests students' understanding of the three basic trigonometric ratios of sine, cosine, and tangent. 5, that is. MCR 3UI - Trigonometric Ratios of Special Angles 1. Identifying Trig Ratios. This means that in all right-angled triangles with an angle of 30 , the ratio of the side opposite the 30 to the hypotenuse is always 0. 11) cos Z 12 9 Z 15 Y X 0. This is an 8 question quiz. tan 42 ° Use the tangent ratio to find the variable. Round to the nearest tenth. Triangle sides a,b,c and angles A,B,C. 7070 θ = 45°. Practice: Relate ratios in right triangles. Example 1: Write each trigonometric ratio. Can be used in conjunction with the. What equation can be used to solve for x? Q. The cards are organized as follows:Cards 1-3: Find sine, cosine, and tangent ratios. View Worksheet Finding Trig Ratios (Answer Key). The matching angles of both triangles. 11) cos Z 12 9 Z 15 Y X 0. 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Given that 3cotθ = 4, then (5sinθ-3cosθ)/ (5sinθ+3cosθ) is equal to. About This Quiz & Worksheet. TRIGONOMETRIC FUNCTIONS Definitions of trig ratios and functions In Trigonometry there are six trigonometric ratios that relate the angle measures of a right triangle to the length of its sides. 3243 18) sin Z 30 40 50 Y X 0. a) sin e b) FRV e c) WDQ e d) FVF e e) VHF e f) FRW e g) How do the quadrant signs of the reciprocal trigonometric ratios (csc e, sec e, and cot e) compare. Susan is flying a kite, which gets caught in the top of a tree. Change answer; Math. Answer (round to 1 decimal): x =. It also includes a few application problems. Since 130 - 40 = 90, these two bearings will give me a right triangle. What is the trigonometric ratio for cos D? Enter your answer, as a simplified fraction, in the boxes. Our online ratio trivia quizzes can be adapted to suit your requirements for taking some of the top ratio quizzes. Trigonometric ratios are rounded to the nearest thousandth. 8000 12) cos C 36 27 45 C B A 0. Free student math practice. zero infinite. 2 Use your answers from Items 5 and 6 to write each trigonometric ratio as a fraction and as a decimal rounded to the nearest hundredth. Quiz by MRW412T. Questions and Answers. primary trigonometric ratios. Trigonometric Ratios QUIZ DRAFT. 1 CONDENSED Discovering Geometry Condensed Lessons CHAPTER 12 155 ©2003 Key Curriculum Press (continued) In this lesson, you Learn about the trigonometric ratios sine, cosine, and tangent Use trigonometric ratios to find unknown side lengthsin right triangles Use inverse trigonometric functionsto find unknown angle measuresin. 2 Worksheet M 1-18 Quiz Quiz 8. 06 25 6dG 17) cosc=. ANS: B PTS: 1 REF: 14-3 Right Triangles and Trigonometric Ratios. Trigonometry Task CardsStudents will practice finding sine, cosine, and tangent ratios, and find missing side and angle measures by working through these 20 task cards. one negative. A comprehensive database of more than 29 trigonometry quizzes online, test your knowledge with trigonometry quiz questions. Find x round to the nearest tenth. SIMPLIFY THE FRACTION! Q. 4) - Page 224 5c including work step by step written by community members like you. 7500 17) sin Z 35 12 37 ZY X 0. This is a quiz on writing trigonometric ratios (as fractions and decimals) and calculating trig functions and their inverses in the calculator. 1) x 10 27° 2) x 10 25° 3) x 7 39° 4) 8 x 46° 5) x 6 23° 6) 7 x 46° 7) x 20 72° 8) x 12 54°-1-. Simplify all fractions, rationalize and simplify all radicals, and convert all decimal answers to fractions. Mathematics. trig-ratios-multiple-choice-questions-and-answers 2/3 Downloaded from calendar. 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This clue was last seen on July 15 2019 New York Times Crossword Answers in the New York Times crossword puzzle. There are two other trigonometric ratios that compare the lengths of the sides of a right triangle. Here's a quiz on some of the basics of "Trig". , Nov 14th QUIZ Sec 4. ANS: B PTS: 1 REF: 14-3 Right Triangles and Trigonometric Ratios OBJ: 14-3. ) Find the sin, cos, and tan buttons on your calculator. 5: Read and copy examples from p. Trig Ratios HW Qs 1 - 4ace, 7 - 10 Answers to Trig Ratio HW Trig Ratios HW Qs #4, 6 - 10 Exact Trig Ratios HW All Questions Rotation Angles HO Rotation angles (filled in) Trig Ratios and special triangles note July 27 Solving Trig Equations AND Reciprocal Ratios The Ambiguous Case for Sine Law Trig Identities Solving Trig Equations HW. Find the value of each trigonometric ratio to the nearest ten-thousandth. Trigonometry Task CardsStudents will practice finding sine, cosine, and tangent ratios, and find missing side and angle measures by working through these 20 task cards. \displaystyle Sin A = \frac {12} {16} S inA = 1612. 1) x 10 27° 2) x 10 25° 3) x 7 39° 4) 8 x 46° 5) x 6 23° 6) 7 x 46° 7) x 20 72° 8) x 12 54°-1-. Created by. e, length of the the line along the axis which it is present. one negative. Copy and Edit. Your students will be given the value of sides and/or angles of a triangle, and they will. Solution: We first find the missing length of side RS. SIMPLIFY THE FRACTION! Q. State the reference angle and the standard position angle. Example 1: Write each trigonometric ratio. Right Triangle Trigonometry - SohCahToa. About this Quiz. We’ve not covered many of the topics from a trig class in this section, but we did cover some of the more important ones from a calculus standpoint. 8-2 Trigonometric Ratios Lesson Quiz: Part II Find each length. Terms in this set (5) Angle A is a right angle in ABC ABC. text should be lower case. Inverse trigonometric functions are simply defined as the inverse functions of the basic trigonometric functions which are sine, cosine, tangent, cotangent, secant, and cosecant functions. Topics you will need to know in order to pass the quiz include. Math Giraffe. 1a Worksheet R 1-14 8. If the lengths of each side of a rhombus is 8cm and its one angle is 60 degrees,then fing the length of diagonals. Algebra II Trig Right Triangle Trig Quiz Review Name:_____ 1. Chapter 13 - Trigonometric Ratios Answer Key CK-‐12 Algebra II with Trigonometry Concepts 4 13. Identifying Trig Ratios: Quick Formative Assessment. Sketch a graph of the angle. Questions on Angles in Standard Position. We know the side 6. Angles of Elevation and Depression. M Worksheet by Kuta Software LLC Kuta Software - Infinite Geometry Name_____ Trigonometric Ratios Date_____ Period____ Find the value of each trigonometric ratio. 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Below is the solution for Certain trig ratios crossword clue. \begin {align} \tan \beta & = \dfrac {PR} {QR} \\ \end {align} Step 2: Determine the value of for the given triangle. 4) - Page 224 5c including work step by step written by community members like you. With video solution. Cards 4-9: Find missing sides. 11) cos Z 12 9 Z 15 Y X 0. Which one is the easy way to remember trigonometric ratios? Set up Trig Ratios DRAFT. I can solve problems using trigonometric equations. Which trigonometric ratio is correct?. By expressing sin 170° in terms of trigonometrical ratios, answer will be _____. the sine θ is the ratio between _____and Hypotenuse. 5 Practice Exam Flashcards | Quizlet Extra Questions for Class 10 Maths Chapter 8 - Trigonometry. What equation can be used to solve for x? Q. Solving the equation ( ) 12 13RS 22 2+=, we obtain RS =5. Sample problems are solved and practice problems are provided. 13 miles 11. y 𝑥𝑥𝑧𝑥𝑐Type equation here. If the lengths of each side of a rhombus is 8cm and its one angle is 60 degrees,then fing the length of diagonals. In this section we see examples of how to use radians instead of degrees when finding the values of sin, cos, tan, csc, sec and cot of angles. 2 Exercises - Skill Practice - Page 863 16 including work step by step written by community members like you. Checking accuracy of answers [3] 2020/11/16 03:21 Under 20 years old / High-school/ University/ Grad student / Useful / To improve this 'Trigonometric functions. Sep 04, 2021 · Take the Quiz: Trigonometry. Some of the questions have students find angles and some require students to find missing sides. This quiz tests your knowledge of the trig rations of the angles 0°, 30°, 45°, 60°, 90°, and the equivalent angles in radians. So when the American student calculated $$3/5$$ as the value of $$\sin A$$ in Example 1. Fill in the missing information. For example: "1/2" and "r2/2". SIMPLIFY THE FRACTION! Q. Susan is flying a kite, which gets caught in the top of a tree. of or relating to or according to the principles of trigonometry. Complete the sentence: In a right triangle having an acute angle, Opposite _____. Find the value of the six trigonometric ratios from the given triangle. Download Free Practice 10 5 Trigonometry And Area Answers 10. 1) sec ° 2) cos Find the exact value of each trigonometric function. Solution: First use Pythagoras' Theorem to find the length of the hypotenuse c: 5 25 9 16 3 4 2. May 07, 2020 · Try this amazing Trigonometric Ratios Quiz: Math Trivia Test! quiz which has been attempted 1513 times by avid quiz takers. Key Concepts: Terms in this set (12) Right Triangle. Use the diagram to estimate the height of the tree. Questions and Answers. Finding the Trigonometric Ratios in a Right Triangle Example 1: In ∆ABC, ∠C = 90°, a = 3 and b = 4. Find x round to the nearest tenth. The matching angles of both triangles. A yacht is anchored 90 feet offshore from the base of a lighthouse. 6000 13) tan C 40 30 50 C B A 1. SIMPLIFY THE FRACTION! Q. These short objective type questions with answers are very important for Board exams as well as competitive exams. 1) x 10 27° 2) x 10 25° 3) x 7 39° 4) 8 x 46° 5) x 6 23° 6) 7 x 46° 7) x 20 72° 8) x 12 54°-1-. Radians and the Trigonometric Ratios. Use SOHCAHTOA to help you remember the tan ratio: TOA. 5 Practice Exam Flashcards | Quizlet Extra Questions for Class 10 Maths Chapter 8 - Trigonometry. sin A, cos A, tan A, sec A, csc A and cot A. 2 Use your answers from Items 5 and 6 to write each trigonometric ratio as a fraction and as a decimal rounded to the nearest hundredth. (3, 4, 5) and (6, 8, 10) are Pythagorean triples since 3 2 + 4 2 = 5 2 and 6 2 + 8 2 = 10 2. For a right triangle, there are six trigonometric ratios: sine, cosine (cos), tangent, cosecant, secant, and cotangent. 1 Finding the Lengths of Sides in a Right Triangle 2. Solution: We first find the missing length of side RS. I know the relationships between sine, cosine, and tangent. sin 400 = Ex 6. Find the value of the indicated trigonometric function of the angle in the figure. Practice: Reciprocal trig ratios. Key Concepts: Terms in this set (12) Right Triangle. Please check the answer provided below and if its not what you are looking for then head over to the main. Contents 1. Law of Sines and Cosines Worksheets Law of Sines and Cosines Worksheet (This sheet is a summative worksheet that focuses on deciding when to use the law of sines or cosines as well as on using both formulas to solve for a single triangle's side or angle) ; Law of Sines; Ambiguous Case of the Law of Sines; Law of Cosines. 31 738' 24 620 a7. If you are given the two sides that are not the hypotenuse, which trig function should you use? Q. Assess your students knowledge of the three trig ratios: sin, cosine and tangent. 8: Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. About This Quiz & Worksheet. The ratio opposite over adjacent, the tangent is; Tan angle = a/b. answer choices. What do we use to remember our trig ratios? Q. 9th - 11th grade. Use your calculator to evaluate each of the following. Here are some of my other products you might be interested in:Trig Ratio: Around. 15 Important Questions. Answers are included. Solving problems involving two right triangles in two dimensions. (cos f, sin f)-10-10 10-10 10-10 10 10 b) A circle centered at the origin with a radius of 1 has the. 0500 15) tan C 35 12 37 B C A 0. 17 plus review. Angles of Elevation and Depression. 8000 θ = 53. SWBAT: 1) Explore and use Trigonometric Ratios to find missing lengths of triangles, and 2) Use trigonometric ratios and inverse trigonometric relations to find missing angles. College Trigonometry Version bˇc Corrected Edition by Carl Stitz, Ph. one negative. Trig Ratios HW Qs 1 - 4ace, 7 - 10 Answers to Trig Ratio HW Trig Ratios HW Qs #4, 6 - 10 Exact Trig Ratios HW All Questions Rotation Angles HO Rotation angles (filled in) Trig Ratios and special triangles note July 27 Solving Trig Equations AND Reciprocal Ratios The Ambiguous Case for Sine Law Trig Identities Solving Trig Equations HW. ca Trigonometry LESSON TWO - The Unit Circle Lesson Notes a) A circle centered at the origin can be represented by the relation x2 + y2 = r2, where r is the radius of the circle. Since we defined the trigonometric functions in terms of ratios of sides, you can think of the units of measurement for those sides as canceling out in those ratios. Right Triangle Problems in Trigonometry. Find the following ratios (round to 4 decimal places. Give your answer correct to 2 decimal places. Simplify all fractions, rationalize and simplify all radicals, and convert all decimal answers to fractions. View Answer Answer: trigonometric ratios of the angles 16 If 0° θ 90°, then sec 0 is A >1. 06 25 6dG 17) cosc=. 6000 13) tan C 40 30 50 C B A 1. Sine & cosine of complementary angles. In this section we see examples of how to use radians instead of degrees when finding the values of sin, cos, tan, csc, sec and cot of angles. If you are given the two sides that are not the hypotenuse, which trig function should you use? Q. Below is the solution for Certain trig ratios crossword clue. cos 400 = 8. 02 Trigonometric Ratios Short Answer Quiz. Practice: Reciprocal trig ratios. Lakeland Community College Lorain County Community College. Identify and. Trigonometry, 10th Grade. Right Triangle Trigonometry - SohCahToa. `sin 35^"o" = a/c = a/15. 2: The Trigonometric Ratios MULTIPLE CHOICE. Topics you will need to know in order to pass the quiz include. 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Trigonometry questions with answers. B trigonometry. Trigonometry I can write the trigonometric ratios. About This Quiz & Worksheet. Day 2: Trig Review and Co-Functions SWBAT: 1) Solve problems involving angle of elevation/depression, and 2) Express sine and cosine. cos, and tan ratios (in decimal form) 1. Answer Section 1. sin 10° = 0. About this Quiz. Hypotenuse. Find the height of the building. SIMPLIFY THE FRACTION! Q. 02 Trigonometric Ratios Short Answer Quiz. ASSIGNMENT: Introduction to Trig Worksheet Grade: Monday, 1/22 Trigonometry I can find another trig function, given one. e, length of the the line along the axis which it is present. Create the worksheets you need with Infinite Calculus. Je Zeager, Ph. If the angle is in degrees, convert it to radians. We'll review your answers and create a Test Prep Plan for you based on your results. In the accompanying diagram of right. Sequential Easy First Hard First. Quiz Flashcard. 3 × 110 = 143. We know the side 6. 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While searching our database we found 1 possible solution matching the query “Certain trig ratios”. Using Trigonometry To Find Lengths Date_____ Period____ Find the missing side. Quiz Flashcard. Sine & cosine of complementary angles. ANS: B PTS: 1 REF: 14-3 Right Triangles and Trigonometric Ratios OBJ: 14-3. The trigonometric ratios are written in an equation form. Difficulty: Difficult. Questions and Answers. Start studying Trigonometric Ratios. The sine (sin) of an angle is the ratio of the length of the leg opposite the angle to the length of the hypotenuse. Do 4 problems. Round to the nearest tenth. 1) sec ° 2) cos Find the exact value of each trigonometric function. While searching our database we found 1 possible solution matching the query Trig ratio. This quiz tests students' understanding of the three basic trigonometric ratios of sine, cosine, and tangent. If tan A= 3/4 , then the value of sec A is 1v. Trigonometric Quiz; Trigonometric Quiz. Please check the answer provided below and if its not what you are looking for then head over to the main. Terms in this set (5) Angle A is a right angle in ABC ABC. Preview this quiz on Quizizz. Download Free Trigonometric Ratios Worksheet Answers Trigonometric Ratios Worksheet Answers When people should go to the book stores, search launch by shop, shelf by shelf, it is in reality problematic. 1b Worksheet F 1-15 8. 3 × 110 = 143. Use SOHCAHTOA to help you remember the tan ratio: TOA. 20 Questions Show answers. The law of cosines is for calculating one side of a triangle when the angle opposite and the other two sides are known. Round to the nearest tenth. Step 1: Find the ratio. ANS: B PTS: 1 REF: 14-3 Right Triangles and Trigonometric Ratios. Use the triangle at the right to determine the following ratios. 300 seconds. Source: s1. If you are given the hypotenuse and an adjacent side, which trig function should you use? Q. A comprehensive database of more than 29 trigonometry quizzes online, test your knowledge with trigonometry quiz questions. (cos f, sin f)-10-10 10-10 10-10 10 10 b) A circle centered at the origin with a radius of 1 has the. Explaination. Right Triangle Trigonometry Trigonometric Ratios Example Find the sine, cosine, and tangent ratios for each of the acute angles in the following triangle. 20449 + 27225 = c2. Use the given trigonometric ratio to determine which angle of the triangle is L A. 7 14 A B C θ 6) 5 B 4 A C θ 7) 11 4. 5 Practice Exam Flashcards | Quizlet Extra Questions for Class 10 Maths Chapter 8 - Trigonometry. pdf from MATH 6116 at Australian National University. Triangle sides a,b,c and angles A,B,C. About this Quiz. Do not reduce. Here are some Exercises on the evaluation of trigonometric functions. Tangent Quiz on _____ Lesson 2. Answers are included. If the lengths of each side of a rhombus is 8cm and its one angle is 60 degrees,then fing the length of diagonals. ; Daniels, Callie, ISBN-10: 0321671775, ISBN-13: 978--32167-177-6, Publisher: Pearson. Use the QR or test code that is at the top of the topic quiz. This unit covers sections 7. 20 days ago. As of Sep 04 21. Trig Ratios Practice Find the value of each trigonometric ratio using a calculator. About This Quiz & Worksheet. Please see below Trigonometry Class 10 Mathematics MCQ Questions, solve the questions and compare your answers with the solutions provided below. Jun 07, 2021 · Angles quiz review with answer key. Questions will ask about the lengths of a hypotenuse. Answers: 1) 2. 02 Trigonometric Ratios. Some of the questions have students find angles and some require students to find missing sides. Explore the surplus collection of trigonometry worksheets that cover key skills in quadrants and angles, measuring angles in degrees and radians, conversion between degrees, minutes and radians, understanding the six trigonometric ratios, unit circles, frequently used trigonometric identities, evaluating, proving and verifying trigonometric expressions and the list go on. For example: "1/2" and "r2/2". These ratios are given by the following trigonometric functions of the known angle A, where a, b and c refer to the lengths of the sides in the accompanying figure: * The sine function (sin), defined as the ratio of the side opposite the angle to the hypotenuse. 1) sin C 14 50 48 C B A 2) sin A 24 32 A 40 B C 3) cos Z 24 32 40 Y Z X 4) cos C 30 40 C 50 B A 5) tan X 30 40 50 X Z Y 6) tan Z 40 9 41 Z Y X Find the value of each trigonometric. Round answer to the nearest WHOLE NUMBER. 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Our online trigonometry trivia quizzes can be adapted to suit your requirements for taking some of the top trigonometry quizzes. Topics you will need to know in order to pass the quiz include. 2 9 13 10 1) Sin A = 2) cos G = 3) TanM= 4) Sin S = 5) CosD- 6) TanW= 7) SinJ= 8) cosv= 10 51 45 9) TanC= 2. 4 Application Problems Answers 1. You can also answer "u" for undefined. Using the table and your answers from Question 11, solve for sin θ and the angle θ for the right triangle problems that would be described by the table below. Oct 17, 2020 · mathematics trigonometry ratios & identities quiz-2 Dear Readers, As per analysis for previous years, it has been observed that students preparing for JEE MAINS find Mathematics out of all the sections to be complex to handle and the majority of them are not able to comprehend the reason behind it. Find the following ratios (round to 4 decimal places. In the accompanying diagram of right. 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Unit 6 - Geometry B - Trig Ratios and Solving Right Triangles. e, length of the the line along the axis which it is present. 5: Read and copy examples from p.


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Proving $\int_{1}^{\infty}\frac{\sin x}{\left(\log x\right)^{\frac{1}{2}}}dx$ converges

Prove that the following improper integral converges $$\int_{1}^{\infty}\frac{\sin x}{\left(\log x\right)^{\frac{1}{2}}}dx.$$

I see that you can show this using Dirichlet's Convergence Test, but how would you show it not using this test?

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It's somewhat nasty, but you can integrate by parts twice. Let $du$ be the numerator each time and $v$ be the denominator. –  Potato Jul 23 '12 at 0:13

You have two issues to address. First, is $\int_1^{1.1}\frac{\sin x}{\sqrt{\log x}}\,dx$ convergent? Second, is $\int_{1.1}^{\infty}\frac{\sin x}{\sqrt{\log x}}\,dx$ convergent?
The first integral blows up at its left end point. But we can show that for the some constant $C$, $\frac{\sin x}{\sqrt{\log x}}<\frac{C}{\sqrt{x-1}}$ for $x$ in $(1,1.1)$. Then by direct comparison, this piece is convergent. The numerator $\sin(x)$ is clearly positive and less than $1$. So we'd need to show that $\log x>k(x-1)$ for some constant $k$, for all $x\in(1,1.1)$. You can do this in the standard way by noting both are equal at $x=0$ and comparing derivatives.
The second integral breaks up into a sequence of alternating integrals $$\int_{1.1}^{\pi}\frac{\sin x}{\sqrt{\log x}}\,dx+\sum_{n=1}^{\infty}\int_{n\pi}^{n\pi+\pi}\frac{\sin x}{\sqrt{\log x}}\,dx$$ By the alternating test for integrals, we need to check that $\lim_{n\to\infty}\left|\int_{n\pi}^{n\pi+\pi}\frac{\sin x}{\sqrt{\log x}}\,dx\right|=0$. This is true, since \begin{align} \left|\int_{n\pi}^{n\pi+\pi}\frac{\sin x}{\sqrt{\log x}}\,dx\right|&<\frac{1}{\sqrt{\log(n\pi)}}\left|\int_{n\pi}^{n\pi+\pi}\sin(x)\,dx\right|\\ &=\frac{2}{\sqrt{\log(n\pi)}}\end{align} Joriki reminded me that the alternating series test also requires that the absolute values of the integrals be decreasing. (Otherwise you could have say, the positive integrals going down like $1/n$ with the negative ones going down like $1/2^n$, and the whole thing diverges.) So we should check that \begin{align} \left|\int_{n\pi}^{n\pi+\pi}\frac{\sin x}{\sqrt{\log x}}\,dx\right|&>\left|\int_{n\pi+\pi}^{n\pi+2\pi}\frac{\sin x}{\sqrt{\log x}}\,dx\right|\end{align} which is true since the numerator function just repeats another period with the same values, while the denominator function becomes larger.

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# Migrating from robot_pose_ekf¶

Migration from robot_pose_ekf is fairly straightforward. This page is meant to highlight relevant differences between the packages to facilitate rapid transitions.

## Covariances in Source Messages¶

For robot_pose_ekf, a common means of getting the filter to ignore measurements is to give it a massively inflated covariance, often on the order of 10^3. However, the state estimation nodes in robot_localization allow users to specify which variables from the measurement should be fused with the current state. If your sensor reports zero for a given variable and you don’t want to fuse that value with your filter, or if the sensor is known to produce poor data for that field, then simply set its xxxx_config parameter value to false for the variable in question (see the main page for a description of this parameter).

However, users should take care: sometimes platform constraints create implicit $$0$$ measurements of variables. For example, a differential drive robot that cannot move instantaneously in the $$Y$$ direction can safely fuse a $$0$$ measurement for $$\dot{Y}$$ with a small covariance value.

## The ‘’differential’’ Parameter¶

By default, robot_pose_ekf will take a pose measurement at time $$t$$, determine the difference between it and the measurement at time $$t-1$$, transform that difference into the current frame, and then integrate that measurement. This cleverly aids in cases where two sensors are measuring the same pose variable: as time progresses, the values reported by each sensor will start to diverge. If the covariances on at least one of these measurements do not grow appopriately, the filter will eventually start to oscillate between the measured values. By carrying out differential integration, this situation is avoided and measurements are always consistent with the current state.

This situation can be avoided using three different methods for the robot_localization state estimation nodes:

1. If fusing two different sources for the same pose data (e.g., two different sensors measuring $$Z$$ position), ensure that those sources accurately report their covariances. If the two sources begin to diverge, then their covariances should refect the growing error that must be occurring in at least one of them.
2. If available, fuse velocity data instead of pose data. If you have two separate data sources measuring the same variable, fuse the most accurate one as pose data and the other as velocity.
3. As an alternative to (2), if velocity data is unavailable for a given pose measurement, enable the _differential parameter for one of the sensors. This will cause it to be differentiated and fused as a velocity.


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mersenneforum.org > YAFU trivial snfs bug; advice ~9 days into job
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 2019-11-07, 06:30 #1 Belteshazzar   Feb 2011 25 Posts trivial snfs bug; advice ~9 days into job As I mentioned in the other subforum, I decided to try snfs for the first time, on the 203 digit cofactor of the 235-digit Pell(613). At first I didn't put the difficulty in my job file, and so I found that while factor/nfs/nfs.c tells you Code: nfs: detected snfs job but no snfs difficulty; assuming size of number is the snfs difficulty it actually does no such thing, instead assuming the snfs difficulty is 0, which gets translated to a gnfs difficulty of 30 (the .56*snfs_d +30 formula is nonsense for snfs_d< 68) and accordingly bizarre parameter choices. (BTW why not make some attempt to calculate snfs difficulty using the polynomials (as factmsieve does) rather than make such an assumption?) After I put size:235 in and ran a short test, I wrote that post asking for advice re params, but I didn't get much. I tried perturbing a couple parameterss away from yafu's defaults and doing -nt test sieving, but the defaults appeared best so I stuck with them. Maybe this isn't optimal, e.g. I found this thread suggesting lp31. A bit late to change now. I now have half the required relations. The sieving slowed down quite a bit, with the reported ETA staying almost constant (~330h) for around 7 days and only then beginning to drop, and I thought perhaps I'd try speeding things up a bit by doing some work on another computer and copying that to rels.add. But it looks like the only mechanism for telling one instance of yafu to do a different range is -ns start,end, which ends up with each siever sieving the entire (end-start)/#threads at once rather than biting off manageable chunks and checkpointing progress as nfs() normally does. Any recommendations on doing a rudimentary job of distributing some work?
 2019-11-07, 13:45 #2 swellman     Jun 2012 52·7·17 Posts Can you post your nfs.job and Yafu.ini files? What command did you use to invoke Yafu? Yafu is a great tool, but it’s parameters often need tweaking at higher difficulties. The above files should help us understand what is happening with your factorization.
 2019-11-07, 16:58 #3 Belteshazzar   Feb 2011 25 Posts The poly and skew are the same as in the other thread, so the job file just adds size:235 and the parameters yafu filled in by default: Code: n: 41992954986653668251498160346091865848259963960853060020750860632150880435788410917184988498688690850646615751221247309950434587076089709352088108916792779996577130820982289749315443146420724344354422429 Y1: 390458526450928779826062879981346977190 Y0: -942650270102046130733437746596776286089 skew: 1.75 type: snfs size: 235 c6: 1 c4: 15 c3: -40 c2: 75 c1: -72 c0: 29 rlim: 41800000 alim: 41800000 lpbr: 30 lpba: 30 mfbr: 60 mfba: 60 rlambda: 2.6 alambda: 2.6 I've done nothing fancy with yafu.ini or invoking yafu: Code: B1pm1=100000 B1pp1=20000 B1ecm=11000 rhomax=1000 threads=8 pretest_ratio=0.25 v=1 ggnfs_dir=C:\Users\Dan\Downloads\factor\GGNFS\ ecm_path=C:\Users\Dan\Downloads\factor\gmpecm-svn3052-znver1\ecm.exe tune_info=AMD Ryzen 5 2400G with Radeon Vega Graphics ,WIN64,9.0141e-006,0.205255,0.280054,0.10029,98.5466,3594.43 Code: .\yafu "nfs(41992954986653668251498160346091865848259963960853060020750860632150880435788410917184988498688690850646615751221247309950434587076089709352088108916792779996577130820982289749315443146420724344354422429)" -job snfs.job -R The most recent progress indicator is Code: nfs: found 49858664 relations, need at least 91912198 (filtering ETA: 242h 44m), continuing with sieving ... nfs.dat is 5.1GB and it's currently working on q around 67,500,000.
 2019-11-07, 17:14 #4 VBCurtis     "Curtis" Feb 2005 Riverside, CA 22·7·132 Posts What q did yafu start at? You might profit by using another machine to sieve low q values, while leaving this one to proceed normally. You could sieve q as low as 5M, though the duplicate rate may jump (which means you would need more than the forecast number of relations to build a matrix- by how much, we don't know).
 2019-11-07, 17:45 #5 Belteshazzar   Feb 2011 2016 Posts I don't have any record of that, but I can simply see what it does on another machine without any savefile: Code: nfs: continuing with sieving - could not determine last special q; using default startq nfs: commencing rational side lattice sieving over range: 21020000 - 21060000 nfs: commencing rational side lattice sieving over range: 20900000 - 20940000 etc, with the lowest q in the second one. so 20,900,000 seems to be the default start. also, in case it's informative, here's some typical output from its current range of q: Code: Total yield: 17840 0/0 mpqs failures, 14796/11935 vain mpqs milliseconds total: Sieve 1038043 Sched 0 medsched 403247 TD 172674 (Init 1976, MPQS 11586) Sieve-Change 1242786 TD side 0: init/small/medium/large/search: 22399 9895 10894 32381 8687 sieve: init/small/medium/large/search: 9896 144202 13329 320269 20051 TD side 1: init/small/medium/large/search: 24662 4282 9705 31614 4451 sieve: init/small/medium/large/search: 13712 176266 13310 324282 2726 total yield: 17977, q=67452023 (0.15930 sec/rel) ETA 0h00m) 1122 Special q, 7839 reduction iterations reports: 322740438->12683945->11614796->7695020->4282943->2623772 Number of relations with k rational and l algebraic primes for (k,l)=:
 2019-11-07, 18:12 #6 VBCurtis     "Curtis" Feb 2005 Riverside, CA 22·7·132 Posts That leaves you Q from 10M-20.9M (or lower, if you wish) that you can run manually on another machine, without worrying about trying to tell the main yafu process to skip any q values. So, you can use other machines like you wanted, while also reducing the number of q you'll have to run above 2*lim (that's the rough bound where sieving starts to get quite inefficient, and your job will slow even more than it has already). I'd strongly prefer to run Q from 8-80M than 20-110M, for example. Q from 10M-20.9M should be quite productive compared to your current progress; Q's under 10M may have the extra duplicates I mentioned earlier. That is, the progress will look fast, but you may need more relations (maybe 5% more than the expected total number, it's hard to predict). The lower you go, the more duplicates you should expect. 8-10M will be almost normal, but 1-2M may not help the overall job much. Edit: if you really get desperate, you could manually run some of the low Q's with 15e instead of 14e. It'll run slower, but (1) you'll get about twice as many relations per range, and (2) 15e on small Q is likely still faster than 14e up at Q=90M. Don't run the same range with 14e and 15e; 15e will repeat all the 14e relations. So, maybe run 15-20.9M on one machine with 14e, and 13-15M on another with 15e, and then see how close the job is to completing? Last fiddled with by VBCurtis on 2019-11-07 at 18:17
 2019-11-07, 18:44 #7 swellman     Jun 2012 52×7×17 Posts For a 30 bit job, use higher values for rlim and alim. Many use the rule of thumb rlim=2^(lpbr-4), same with alim. Yafu usually underestimates rlim and alim. The key metric is yield, i.e. relations per Q. Should be 1.5 to < 3. I suspect 30-bit is a tad low for a job of this difficulty, but like you said you’re committed now. Grunt it out is your best bet if you want to guarantee eventual success. In the past, I’ve stopped Yafu then tried to resume it only to have yafu fail to find the last stopping point and subsequently overwrite my data file with a new sieving effort. Good times... You can break up a job on different machines with Yafu using the -ns flag. Test sieve a bit to estimate the total yields (yield tends to drop as Q increases). Then specify a Q-range on each machine. After sieving, combine all the data files into one data file and start filtering. CADO is much better at managing a job over many machines. Yafu was never intended to run jobs of such size. Personally I love Yafu. Just takes some manual intervention. Last fiddled with by swellman on 2019-11-07 at 18:46
 2019-11-07, 19:44 #8 Belteshazzar   Feb 2011 25 Posts Thanks to both of you for your advice. For a second machine, as far as siever - or perhaps a -J flag - does the difference between the rational and algebraic norms affect what I should choose there?
 2019-11-07, 20:09 #9 VBCurtis     "Curtis" Feb 2005 Riverside, CA 22·7·132 Posts I think you'll be okay with the 14e siever just like the regular job is using, but you can use -J 14 with 14e to gain a little extra yield. That's, in effect, the 14.5 siever. A square sieve region (2^14 by 2^14 using -J 14 rather than default 2^14 by 2^13) isn't optimal, but neither are your parameters so we have to make do. I don't consider the norms when making such a choice; just the yield and the estimate of the number of relations I'll need. 14e will be fastest. 14e -J 14 (or 15e -J 13, which is 2^15 by 2^13 sieve region, again in effect a 14.5 siever) will be next fastest, and will find lots of relations on small Q. 15e will be slower still, but will find tons of relations. A little experimenting can show you the best path; I'd use the smallest siever I could such that Q from 5-20.9M produced enough relations to save me from Q>80M. I expect that's something like 10-20.9M on 14e, and 5-10M (or maybe 6-10M) on 15e. I changed my GGNFS script to use starting q of lim/5 for SNFS jobs like this; that is, I would have started my job at 8M. You have the chance to make that change now, but you can't profitably change alim, rlim, lpbr or lpba without starting over (or making a mess). Knowing what you know now about the size of the job, choosing alim = rlim = 67M, with Q from 12M till it finished and lpb of 30 or 31 would be about right for this job. I always choose a bigger lpb when it's close, but that's not really the mainstream opinion, so lpb=30 is likely standard. lim's just under powers of 2 are faster than just above; 67M is a bit below 2^26. That effect doesn't matter on small jobs, but becomes substantial at 134M or 268M (to the point that I don't ever select a lim between 134M and 200M- it either fits in the faster 134M selection, or I skip up 50%). Last fiddled with by VBCurtis on 2019-11-07 at 20:10
2019-11-07, 22:55   #10
swellman

Jun 2012

52×7×17 Posts

Quote:
 Originally Posted by Belteshazzar Thanks to both of you for your advice. For a second machine, as far as siever - or perhaps a -J flag - does the difference between the rational and algebraic norms affect what I should choose there?
FWIW, one trick is to sieve the same region on both the -r and -a sides simultaneously (one on each machine). This usually isn’t worth doing, as the “B-side” often has terrible yield. But it can save a job if yield fades out at high Q on the primary side.

But usually it’s best to pick the -r or -a side and sieve on all machines on that side.

And remember VBCurtis is the master siever. He’s forgotten more than I know!

 2019-11-08, 00:13 #11 bsquared     "Ben" Feb 2007 41×83 Posts Thanks swellman and VBCurtis for the help. Someday YAFU's NFS will get an overhaul to address some of these usability issues. It does fairly well for fire&forget type jobs (up to 130/140 digits, say) and SNFS poly generation is pretty capable. But for large job there are no doubt better tools. Backup data files whenever restarting if possible! Thanks for all of the feedback too, I can't fix things I don't know about. (I don't do NFS work much anymore; focusing on QS and ECM when I get time.)

 Similar Threads Thread Thread Starter Forum Replies Last Post tmp YAFU 11 2018-05-30 22:12 ryanp Factoring 69 2013-04-30 00:28 davieddy Information & Answers 6 2011-12-08 08:07 Christenson Forum Feedback 0 2011-03-21 03:49 fetofs Factoring 39 2006-07-26 12:32

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text:
Ex.10.1 Q1 Practical Geometry Soltion - NCERT Maths Class 7

Go back to  'Ex.10.1'

Question

Draw a line, say $$AB,$$ take a point $$C$$ outside it. Through $$C,$$draw a line parallel to $$AB$$ using ruler and compasses only.

Video Solution
Practical Geometry
Ex 10.1 | Question 1

Text Solution

What is known?

A line $$AB$$ and a point $$C$$ outside it.

To construct:

A line through $$C$$ parallel to $$AB$$ using ruler and compasses.

Reasoning:

Draw a line $$AB$$ and take a point $$C$$ outside it. Draw line $$AB$$ by using ruler and compasses, follow the steps given below.

Steps:

Steps of construction

1. Draw a line,  $$AB,$$ take a point $$C$$ outside this line. Take any point $$P$$ on $$AB$$. Join $$C$$ to $$P$$
2. Taking $$P$$ as centre and a convenient radius draw an arc intersecting line $$AB$$ at $$D$$ and $$PC$$ at $$E.$$
3. Taking $$C$$ as the centre and the same radius in previous step, draw an arc $$FG$$ intersecting $$PC$$ at $$H.$$
4. Adjust the compasses up to the length of $$DE$$. Without changing the opening of compasses and taking $$H$$ as the centre, draw an arc to intersect arc $$HG$$ at point $$I.$$
5. Join the point $$C$$ and $$I$$ to draw a line $$L.$$

This is the required line $$L$$ which is parallel to $$AB.$$

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# Time required to cool an object using a fan

#### samiers92

Hello,

A flat object with a known surface area is at a temperature of 285F. The ambient temperature in the room is 80F. The object needs to be cooled down to 100F within a time limit of 5 minutes. To accelerate the heat transfer process, there is a fan in the setup.

The objective is to calculate the air flow rate required in the fan (in CFM) to achieve the above mentioned cooling process.

#### benit13

Whenever you have a heat transfer problem, consider the following:

1. Whether the system is going to be in thermal equilibrium or not
2. What kind of heat transfer processes there (i.e. radiation, conduction and convection)
3. Whether the system is isolated or not

In your case, I would recommend a system that is not in thermal equilibrium, is not isolated by the surroundings and then consider convection heat transfer from a substance to the surroundings.

Then, you can consider a control volume around your system of interest (draw a diagram!), draw the heat transfer processes occurring in the your volume and determine a conservation of energy equation describing your volume.

In your case, you probably have something like a lump of mass, $$\displaystyle M$$, with initial temperature, $$\displaystyle T_0$$, and heat capacity, $$\displaystyle c_p$$, surrounded by air with a constant ambient temperature, $$\displaystyle T_a$$. Then:

$$\displaystyle Q_{system} = Q_{gen} + Q_{in} - Q_{out}$$

Where
$$\displaystyle Q_{gen} = 0$$ (no internal heating),
$$\displaystyle Q_{in} = 0$$ (no heat gain by any sources), and
$$\displaystyle Q_{out} = h_cA(T_a - T)$$ (heat loss due to convection)

where $$\displaystyle h_c$$ is the convective heat transfer coefficient and $$\displaystyle A$$ is the surface area.

The heat transfer to the control volume can be sensible (not latent), so

$$\displaystyle Q_{system} = Mc_p \frac{\partial T}{\partial t}$$

Substituting into the formula above gives

$$\displaystyle Mc_p \frac{\partial T}{\partial t} = - h_cA(T_a - T)$$

This gives a first order separable ODE that can be solved for temperature as a function of time with the boundary condition $$\displaystyle T = T_0$$ when $$\displaystyle t = 0$$.

Note that the convective heat transfer coefficient is given as a function of air velocity, so you can solve the equation above to get the heat transfer coefficient with the boundary condition $$\displaystyle T = T_{target}$$ when t = $$\displaystyle t_{target}$$. You can then relate it to the air velocity required and finally the fan volume flow rate.


text:
# Italics in definition and proof environment not working simultaneously

I am writing my thesis and am using the amsmath package for the theorem/lemma/definition/proof environments. I am having trouble figuring out how to simultaneously have the "definition" environment in plain text and then I manually italicize the definition and having the proof environment work correct. I have been able to get both to work independently following the information on overleaf's site:

https://www.overleaf.com/learn/latex/Theorems_and_proofs

To get the proof environment to work, my preamble looks like the below code:

\usepackage{amsmath}
\newtheorem{thm}{Theorem}[section]
\newtheorem{prop}[thm]{Proposition}
\newtheorem{lemma}[thm]{Lemma}
\newtheorem{ex}[thm]{Example}
\theoremstyle{definition}
\newtheorem{dfn}[thm]{Definition}

As an example, consider this snippet of code:

\begin{dfn}
A \textit{group} is a set blah blah blah
\end{dfn}
\begin{proof}
does it work?
\end{proof}

This is what the output looks like using the above preamble:

As you can see, the defintion is fully italicized. This doesn't look the way I want it to until I add \usepackage{amsthm}, like below:

\usepackage{amsmath}
\usepackage{amsthm}
\newtheorem{thm}{Theorem}[section]
\newtheorem{prop}[thm]{Proposition}
\newtheorem{lemma}[thm]{Lemma}
\newtheorem{ex}[thm]{Example}
\theoremstyle{definition}
\newtheorem{dfn}[thm]{Definition}

When I compile my document using the second example for the preamble, my definitions look how I want them to, but the proof environment no longer works correctly. The word "proof" is no longer in bold and the standard square QED icon does not show up. I get the error "Something's wrong--maybe missing a \item" all over my document.

Does anyone know how to resolve both of these issues simultaneously? This is not my entire preamble, I can add the rest if needed. Also, I am not typing my text directly in the file that is loading the document. I am using a separate file introduction.tex file and then using the command "\include{introduction}. I am not sure if this matters but I thought it was somewhat relevant.

edit: My documentlcass is unique to my university because all of the formatting and whatnot is regulated. A template can be found on overleaf:

https://www.overleaf.com/latex/templates/purdue-university-thesis-template/rkjkbcgbcdck

• Why defining theorems before loading amsthm? – egreg Jan 17 at 17:54
• If I'm not mistaken, amsmath contains amsthm, so they should still be correctly loading when I load amsmath. In any event, if I load amsthm immediately after amsmath, the problem still persists – Jake Jan 17 at 17:54
• No, amsmath and amsthm are independent of each other. You don't have a proof environment without amsthm. – egreg Jan 17 at 17:59
• In any event, like I mentioned earlier, the problem persists if I load amsthm immediately after amsmath. I will edit my question to reflect this. Do you have any other suggestions? – Jake Jan 20 at 19:00
• Could you add a minimal working example showing the problem? It should start with \documentclass{...} and end with \end{document}. – Vincent Jan 20 at 19:04

The puthesis class already defines a proof environment in a way that I find no polite word to describe, namely

\newsavebox{\proofbox}
\sbox{\proofbox}{\rule{7pt}{7pt}}
\newtheorem{Proof}{Proof}
\renewcommand{\theProof}{}
\newenvironment{proof}{\begin{Proof}\rm}{\hfill \usebox{\proofbox} \end{Proof}}

What's the problem here? First: \rm has been deprecated for more than 25 years at the time of writing. Second: the end-of-proof marker is not guaranteed to be typeset at the end of a line. And more.

Of course, if you load amsthm, you will get errors.

You can do like this:

\documentclass[ece,dissertation]{puthesis}
\usepackage{amsmath}

\let\proof\relax\let\endproof\relax % use proof from amsthm
\usepackage{amsthm,xpatch}

\newtheorem{thm}{Theorem}[section]
\newtheorem{prop}[thm]{Proposition}
\newtheorem{lemma}[thm]{Lemma}
\newtheorem{ex}[thm]{Example}
\theoremstyle{definition}
\newtheorem{dfn}[thm]{Definition}

% tombstone like in puthesis
\renewcommand{\qedsymbol}{\rule{7pt}{7pt}}
% Proof in boldface like in puthesis
\xpatchcmd{\proof}{\itshape}{\bfseries}{}{}

\begin{document}

\begin{thm}
A theorem statement.
\end{thm}

\begin{proof}
Its proof.
\end{proof}

\begin{dfn}
A definition.
\end{dfn}

\end{document}


text:
0
RESEARCH PAPERS

# Thermal Phenomena in Nanoscale Transistors

[+] Author and Article Information
Eric Pop

Department of Electrical Engineering, Stanford University, Stanford, CA 94305

Kenneth E. Goodson

Department of Mechanical Engineering, Stanford University, Stanford, CA 94305goodson@stanford.edu

J. Electron. Packag 128(2), 102-108 (Dec 14, 2006) (7 pages) doi:10.1115/1.2188950 History: Received January 21, 2005; Revised December 14, 2006

## Abstract

As CMOS transistor gate lengths are scaled below $45nm$, thermal device design is becoming an important part of microprocessor engineering. Decreasing dimensions lead to nanometer-scale hot spots in the drain region of the device, which may increase the drain series and source injection electrical resistances. Such trends are accelerated with the introduction of novel materials and nontraditional transistor geometries, like ultrathin body, surround-gate, or nanowire devices, which impede heat conduction. Thermal analysis is complicated by subcontinuum phenomenan including ballistic electron transport, which reshapes the hot spot region compared with classical diffusion theory predictions. Ballistic phonon transport from the hot spot and between material boundaries impedes conduction cooling. The increased surface to volume ratio of novel transistor designs also leads to a larger contribution from material boundary thermal resistance. In this paper we survey trends in transistor geometries and materials, from bulk silicon to carbon nanotubes, along with their implications for the thermal design of electronic systems.

<>

## Figures

Figure 9

Current-voltage characteristics of a 3μm long metallic single-wall carbon nanotube up to breakdown by oxidation (burning) in air, when the middle of the SWNT reaches about 600°C at 15V bias. The data (symbols) and inset (AFM image showing place of breakdown) are from Ref. 43. The model (solid line) is described in Ref. 47.

Figure 1

Trends of on-chip power density for the past 10+ years. Note the vertical axis is logarithmic and the horizontal one (years) is linear. By comparison, typical power densities for a hot plate and a rocket nozzle are listed, while the surface of the sun puts out approximately 7000W∕cm2. The solid line marks an exponential trend. (Data compiled by F. Labonte, Stanford.)

Figure 2

False-color Transmission Electron Microscopy (TEM) cross-section of a future-generation strained silicon-on-insulator (SSOI) transistor. Image courtesy IBM.

Figure 5

Evolution of transistor designs from traditional bulk silicon (a) to (strained) silicon or germanium on insulator (b), to multiple-gate or FinFET devices (c). Non-traditional device designs like (b) and (c) introduce more complicated geometries and lower thermal conductivity materials, making heat removal more difficult. The gate length (currently near 50nm) is expected to be scaled to the 10nm range, while the semiconductor film and “fin” thickness of (b) and (c), respectively, are approximately one third to one half the gate length. Figure (c) is from the technology roadmap (ITRS) (1).

Figure 4

Snapshot of electron transport in a future-generation thin-body silicon-on-insulator (SOI) device simulated with the Monte Carlo code MONET (13,15). The color scale is the electron energy in eV and the device dimensions are in nm.

Figure 6

Estimated reduction in silicon and germanium thin film thermal conductivity due to phonon boundary scattering alone (i.e., no confinement). Thin silicon films suffer from a stronger reduction in thermal conductivity (vs germanium) due to the longer bulk silicon phonon mean free path (30).

Figure 7

Thin film transistor thermal resistance model (also see Fig. 5). The dark gray areas represent the metalized gate and contacts, the light gray is the surrounding oxide insulator (21,30).

Figure 8

Suspended silicon nanowire field effect device fabricated with electron beam lithography and a buffered HF underetch (40). The cross section of the wire is 23×80nm. The confined dimensions significantly alter both current and heat transport through the wire.

Figure 3

Phonon dispersion in the (100) direction of silicon (a) and computed net (emission minus absorption) phonons generated by Joule heating (b). The symbols in (a) are from neutron scattering data and the lines are a quadratic fit (14). Solid lines are for longitudinal, and dashed lines are for transverse phonons. Optical modes are above and acoustic modes have energies below approximately 50meV, respectively. Note the vertical axes are matched (E=ℏω) to facilitate comparison of the dispersion and generated modes.

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text:
# MILLIMETER WAVE SPECTRA AND CENTRIFUGAL STRETCHING CONSTANTS OF THE METHYL $HALIDES^{*}$

Please use this identifier to cite or link to this item: http://hdl.handle.net/1811/7268

Files Size Format View
1954-H-6.jpg 111.4Kb JPEG image

 Title: MILLIMETER WAVE SPECTRA AND CENTRIFUGAL STRETCHING CONSTANTS OF THE METHYL $HALIDES^{*}$ Creators: Orville-Thomas, W. J.; Cox, J. T.; Gordy, Walter Issue Date: 1954 Publisher: Ohio State University Abstract: The extension of the workable millimeter wave region1 has made possible measurements of high J transitions in the methyl halides, where centrifugal distortion displacements of rotational lines amount to many megacycles. High J transitions of these substances have been measured for a more accurate evaluation of the distortion constants. The new $D_{J}$ value for $CH_{3}Cl^{35}, 18.1 \pm 0.5$ Kc, is in good agreement with the value 18.4 Mc, theoretically predicted by Chang and $Dennison.^{2}$ Earlier $measurements^{3}$ on lower J transitions gave $D_{JJ} = 26.4$ Kc, while recent infrared measurements gave an estimated $D_{JJ} = 24 Kc.^{4}$ Description: $^{*}$Supported by a contract with the United States Air Force. $^{**}$Visiting Fulbright Scholar, 1953-54. On Sabbatical leave from Edward Davies Chemical Laboratories, University College of Wales, ABERYSTWYTH. $^{1}$ W. C. King and W. Gordy, Phys. Rev. 90:319 (1953) $^{2}$ Tsu-Shen Chang and D. M. Dennison, J. Chem. Phys. 21:1293 (1953) $^{3}$ J. W. Simmons and W. E. Anderson, Phys. Rev. 80:338 (1950) $^{4}$J. Pickworth and H. W. Thompson, Trans. Faraday Soc. 50:218, (1954) Author Institution: Department of Physics, Duke University URI: http://hdl.handle.net/1811/7268 Other Identifiers: 1954-H-6


text:
Browse Questions

# If $f: R \to R$ is defined by $f(x)=[2x]-2[x]$ for $x \in R,$ where $[x]$ is the greatest integer not exceeding x, then the range of f is :

$\begin {array} {1 1} (a)\;\{x \in R:0 \leq x \leq 1\} & \quad (b)\;\{0,1 \} \\ (c)\;\{x \in R : x > 0\} & \quad (d)\;\{x \in R: x \leq 0\} \end {array}$

$(b)\;\{0,1 \}$


text:
# Construct the Identity Matrix

The challenge is very simple. Given an integer input n, output the n x n identity matrix. The identity matrix is one that has 1s spanning from the top left down to the bottom right. You will write a program or a function that will return or output the identity matrix you constructed. Your output may be a 2D array, or numbers separated by spaces/tabs and newlines.

Example input and output

1: [[1]]
2: [[1, 0], [0, 1]]
3: [[1, 0, 0], [0, 1, 0], [0, 0, 1]]
4: [[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]]
5: [[1, 0, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0], [0, 0, 0, 0, 1]]

1
===
1

2
===
1 0
0 1

3
===
1 0 0
0 1 0
0 0 1

etc.

This is , so the shortest code in bytes wins.

• Given an integer input n ... -- I assume you mean a natural number? – Jonathan Frech Aug 28 '18 at 17:00

# PHP, 64 bytes

for(;$i<$argn**2;$m?:$k++)echo$k==$m?:0,($m=++$i%$argn)?" ":" ";  Try it online! # tinylisp repl, 107 bytes (d =(q((A R C S)(i C(=(c(e R C)A)R(s C 1)S)A (d #(q((A R S)(i R(#(c(=()R S S)A)(s R 1)S)A (d f(q((S)(#()S S  Using the repl saves 6 bytes in implied closing parentheses at the ends of lines. Defines a function f which takes a positive integer and returns the identity matrix of that size as a nested list. Try it online! ### Explanation The function = constructs a row in the matrix. Its arguments are (in order) an Accumulator, the Row number, the current Column number, and the Size of the matrix. If the column is not zero (i C ...), then cons a number to the accumulator (1 if the row and column are equal, 0 otherwise) (c (e R C) A), decrease the column number (s C 1), and recurse (= ...). Otherwise, return the accumulator A. The function # constructs the matrix as a list of rows. Its arguments are an Accumulator, the current Row number, and the Size of the matrix. If the row number is not zero (i R ...), then generate a row (=()R S S) and cons it to the accumulator (c ... A), decrease the row number (s R 1), and recurse (# ...). Otherwise, return the accumulator A. Finally, the function f is simply a single-argument wrapper function that calls #. ## Clojure, 45 bytes #(for[i(range %)](assoc(vec(repeat % 0))i 1))  # PHP, 49 bytes function($n){for(;$n--;)$k[$n][$n]=1;return $k;};  There is a bit of a caveat with this answer, and I will delete it or mark it as non-competing if it is deemed invalid. The array returned by this function technically only has n elements, those being a 1 at every intersection. ie: 0:0 1:1, 2:2, etc However, not only does this satisfy the definition of the identity matrix as given by the question: The identity matrix is one that has 1s spanning from the top left down to the bottom right It also works if used as an identity matrix, as an empty value in an array in PHP is treated as 0 if used arithmetically. # Axiom, 68 bytes f(n:PI):Matrix INT==matrix([[(i=j=>1;0)for i in 1..n]for j in 1..n])  test and results (30) -> for i in 1..3 repeat output f(i) [1] +1 0+ | | +0 1+ +1 0 0+ | | |0 1 0| | | +0 0 1+ Type: Void  # Mathematica ,26 bytes without using IdentityMatrix Boole[#==#2]&~Array~{#,#}&  # Python 2, 52 bytes y=x=2**input() while~-x:x/=2;print str(bin(y+x))[3:]  Try it online! Using binary representation of powers of 2. ## PowerShell, 45 bytes {param($n)0..--$n|%{"$(,0*$_+1+,0*($n-$_))"}}  Try It Online! # Bash (w/ coreutils and awk), 51 50 bytes yes 0|head -$((N*N))|xargs -n$N echo|awk {$NR=1}1  ## Usage Set the N environment variable to the matrix size  N=5 yes 0|head -((N*N))|xargs -nN echo|awk {$NR=1}1 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1  ## Explanation • yes 0 generates an endless stream of 0s to stdout; • head -$((N*N)) takes the first N2 lines;
• xargs -n$N echo groups every N lines and outputs them together; • awk {\$NR=1}1 changes the value of the field whose index matches the line number to a 1 (otherwise it remains 0) and then outputs the whole record.

There may be a shorter way to generate N2 lines of 0s, but I haven't found one yet...

# Vim, 47 keystrokes

"zy$␘a@q␛"xYoa␛"zPy$qqo␛@"0␛q@x3Gqqr1jlq@x1G2dd

␘ is CtrlX and ␛ is Esc

## Explanation

"zy$␘ Save n for use later a@q␛"xY Create the macro x that repeats the macro q n-1 times oa␛"zPy$        Create an anonymous macro that inserts a character n times
qqo␛@"0␛q@x    Create the macro q that adds a new line with n zeroes and run it n times
3G               Go to the top-left corner of the matrix
qqr1jlq@x        Redefine q as a macro that replaces the character at the cursor with
a one and moves the cursor one step down-right. Run q n times
1G2dd            Delete the two lines used in macro defenitions

# Excel VBA, 55 52 43 Bytes

Anonymous VBE immediate window function that takes input as number from cell [A1] and outputs the identity matrix to the range of [A1].Resize(n,n)

[A1].Resize([A1],[A1])="=1*(Row()=Column())

-2 bytes thanks to Engineer Toast

### Previous Version

n=[A1]:[A1].Resize(n,n)=0:For i=1To n:Cells(i,i)=1:Next

• Save 1 byte: "=1*(Row()=Column())" – Engineer Toast Apr 25 '18 at 15:18

# Husk, 4 bytes

´Ṫ=ḣ

Try it online!

### Explanation

´Ṫ=ḣ  -- example input: 2
ḣ  -- range: [1,2]
´     -- duplicate argument: [1,2] [1,2]
Ṫ    -- outer product by
=   -- | equality: [[1==1,1==2],[2==1,2==2]]
-- : [[1,0],[0,1]]

# Tcl, 92 bytes

proc I n {time {incr j;set a "";set i 0;time {lappend a [expr [incr i]==$j]}$n;puts $a}$n}

Try it online!

# Gol><>, 10 bytes

IR0PlF}D|;

Try it online!

The output for n = 5 looks like this:

[1 0 0 0 0]
[0 1 0 0 0]
[0 0 1 0 0]
[0 0 0 1 0]
[0 0 0 0 1]

### How it works

IR0PlF}D|;

IR0P       Take input n, then fill the stack with that many 0s
and then make the top 1; stack = [0 ... 0 1]
lF..|  Do the following n times...
}D   Rotate the stack once to the right, and print the whole stack
; Terminate

# 05AB1E, 6 bytes

Lã€Ësô

Try it online.

Explanation:

L         # List in range [1,n]
#  i.e. 3 → [1,2,3]
ã        # Cartesian power of this list
#  i.e. [1,2,3] → [[1,1],[1,2],[1,3],[2,1],[2,2],[2,3],[3,1],[3,2],[3,3]]
€Ë      # Check for every pair if they are equal or not
#  i.e. [[1,1],[1,2],[1,3],[2,1],[2,2],[2,3],[3,1],[3,2],[3,3]]
#   → [1,0,0,0,1,0,0,0,1]
sô    # Split it into parts equal to the input
#  i.e. [1,0,0,0,1,0,0,0,1] and 3 → [[1,0,0],[0,1,0],[0,0,1]]

• Alternative 6-byter °¨IF=Á – Kaldo Aug 28 '18 at 15:01
• @Kaldo Ah, that's quite a bit different than my approach, but also pretty cool. And I just saw your 4-byte answer, +1 from me. :) – Kevin Cruijssen Aug 28 '18 at 15:22

## Seriously, 24 bytes

,;╗r╙k╜#"{:0%sb}"%f#MR

Try it online!

Explanation:

,;╗r╙k╜#"{:0%sb}"%f#MR
,;╗                       get input (n), save a copy in register 0
r                      push [0..n-1]
                M   map the function:
╙k                     push [2**a]
╜#"{:0%sb}"%         push "{:0nb}"
f#       format and explode string
(push binary representation, padded with zeroes to n digits)
R  reverse list

## Sage, 15 bytes

identity_matrix

Exactly what it says on the tin

Try it online

Thanks to Lynn for pointing out that I was being a doofus.

• Why not eta-reduce this to identity_matrix? – Lynn Aug 21 '16 at 18:21

# V, 16, 11 bytes (non-competing?)

Ài0 ÀÄòjl

Try it online!

# PHP, 55 bytes

while($p**.5<$z=+$argn)echo$p%-~$z?0:1," "[++$p%$z>0];  requires PHP 5.6 or later for the power operator. Note that the second line begins with a space! Run as pipe with -nR or try it online. # Charcoal, 7 bytes ＥθＥθ⁼κμ  Try it online! This returns an array, which in Charcoal is not readable in its default format. A version using Wolfram Lists is included below for readability. ### Explanation Ｅθ Map for each number from 0 to first input Ｅθ Map for each number from 0 to first input ⁼κμ Return whether both loop indices (yes, indices) are the same. The variables here are: i = first loop variable, k = first loop index, l = second loop variable, m = second loop index.  # Common Lisp, 96 94 92 bytes (lambda(n)(set'a(make-array(list n n):initial-element 0))(dotimes(i n)(setf(aref a i i)1))a)  Try it online! ## PHP 88 Bytes Try it online Code <?php for($i=0;$i++<$argv;){$a=array_fill(1,$argv,0);$a[$i]=1;echo implode(",",$a)." ";}  Explanation <?php for($i=0;$i++<$argv;){          // loop up to the argument
$a=array_fill(1,$argv,0);   // create an array fill with 0 with length = $argv value$a[$i]=1; // use the current iteration to place the "1" echo implode(",",$a)."      // implode to output the matrix
";}

Output

  \$argv=5

1,0,0,0,0
0,1,0,0,0
0,0,1,0,0
0,0,0,1,0
0,0,0,0,1

# Forth (gforth), 48 bytes

: f dup 0 do dup 0 do 0 i j = - . loop cr loop ;

Try it online!

### Explanation

uses nested loops to output the matrix. For each element it outputs 1 if the indexes are equal, and 0 otherwise.

### Code Explanation

:f           \ start a new word definition
dup        \ duplicate input
0 do       \ outer counted loop from 0 to input - 1
dup      \ duplicate input again
0 do     \ inner counted loop from 0 to input - 1
0      \ put a 0 on the stack (used to convert -1 to 1)
i j =  \ -1 if i equals j, 0 otherwise
- .    \ subtract from 0 to convert -1 to 1, and output
loop     \ end inner loop
cr       \ output a newline
loop       \ end outer loop
;            \ end word definition

# D, 114 bytes

void f(int n){import std.stdio;int N=n*n;while(N--){if(N%(n+1))write("0 ");else write("1 ");if(N%n==0)writeln();}}

Try it online!


text:
# American Institute of Mathematical Sciences

• Previous Article
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## Stabilization of a discrete-time system via nonlinear impulsive control

 1 School of Software Engineering, Huazhong University of Science and Technology, Wuhan, Hubei 430074, China 2 School of Information Science and Engineering, Fujian University of Technology, Fuzhou, Fujian 350118, China 3 Business School, Hunan Normal University, Changsha 410081, China

* Corresponding author: Jing Huang

Received  March 2018 Revised  August 2018 Published  September 2019

An impulsive control is one of the important stabilizing control strategies and exhibits many strong system performances such as shorten action time, low power consumption, effective resistance to uncertainty. This paper develops a nonlinear impulsive control approach to stabilize discrete-time dynamical systems. Sufficient conditions for asymptotical stability of discrete-time impulsively controlled systems are derived. Furthermore, an Ishi chaotic neural network is effectively stabilized by a designed nonlinear impulsive control.

Citation: Shaohong Fang, Jing Huang, Jinying Ma. Stabilization of a discrete-time system via nonlinear impulsive control. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020106
##### References:
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##### References:
 [1] S. Ishi, K. Fukumizu and S. Watanabe, A network of chaotic elements for information processing, Neural Networks, 1 (1996), 25-40.   Google Scholar [2] A. Khadra, X. Z. Liu and X. Shen, Synchronizing chaotic systems with delay and applications to secure communication, Automatica, 41 (2005), 1491-1502.  doi: 10.1016/j.automatica.2005.04.012.  Google Scholar [3] V. Lakshmikantham, D. D. Bainoov and P. S. Simeonov, Theory of Impulsive Differential Equations, Singapore: World Scientific, 1989. doi: 10.1142/0906.  Google Scholar [4] B. Liu and X. Liu, Robust stability of uncertain discrete impulsive systemss, IEEE Trans. Circuit Syst. II, Exp. Brief, 54 (2007), 455-459.   Google Scholar [5] X. Liu and K. L. Teo, Impulsive control of chaotic system, International Journal of Bifurcation and Chaos, 12 (2002), 1181-1190.  doi: 10.1142/S0218127402005029.  Google Scholar [6] T. Ushio, Chaotic synchronization and controlling chaos based on constraction mapping, Physics Letters A, 198 (1995), 14-22.  doi: 10.1016/0375-9601(94)01015-M.  Google Scholar [7] X. Xie, H. Xu, X. Cheng and Y. Yu, Improved results on exponential stability of discrete-time switched delay systems, Discrete & Continuous Dynamical Systems-Series B, 22 (2017), 199-208.  doi: 10.3934/dcdsb.2017010.  Google Scholar [8] X. Xie, H. Xu and R. Zhang, Exponential stabilization of impulsive switched systems with time delays using guaranteed cost control, Abstract and Applied Analysi, 2014 (2014), Art. ID 126836, 8 pp. doi: 10.1155/2014/126836.  Google Scholar [9] H. Xu and K. L. Teo, Stabilizability of discrete chaotic systems via unified impulsive control, Physics Letters A, 374 (2009), 235-240.  doi: 10.1016/j.physleta.2009.10.065.  Google Scholar [10] T. Yang, Impulsive Systems and Control: Theory and Applications, Huntington, New York: Nova Science Publishers, Inc., 2001. Google Scholar [11] T. Yang and L. O. Chua, Impulsive stabilization for control and synchronization of chaotic systems: Theory and application to secure communication, IEEE Trans. Circuit Syst. I, Fundam. Theory Appl., 44 (1997), 976-988.  doi: 10.1109/81.633887.  Google Scholar [12] T. Yang, L. Yang and C. Yang, Impulsive control of Lorenz system, Physica D, 110 (1997), 18-24.  doi: 10.1016/S0167-2789(97)00116-4.  Google Scholar
State trajectory of $x_1(m)$ without nonlinear impulsive control
State trajectory of $x_2(m)$ without nonlinear impulsive control
State trajectory of $x_1(m)$ under nonlinear impulsive control
State trajectory of $x_2(m)$ under nonlinear impulsive control
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