text:
upper cervical instability treatment

Mar 21, 2022 · 11. Determine the domain and range of the graph.Hint: Not looking for a set of ordered pairs!Make sure you label the Domain and Range when answering! 12. What is the value of f (x) when x = 0?Check all that apply. 13. What is the value of x when f (x) = 3?Check all that apply.. The graph of such a function will be symmetrical in the y-axis. Even functions which are polynomials have even degrees (e.g. y = x²). A function is odd if the sign of the function is changed when x is replaced by -x . The graph of the function will have rotational symmetry about. Graphs and charts - quiz L1 Level C 1. We collected data about ....

juniper ex3400 reset to factory default
cost to repair cracked septic tankbobwhite quail eggs for hatching for sale
webview2 wpf sample

rwby fanfiction aokiji

Quiz: What Is a Function? Take a quiz to assess your understanding of the material. Duration: 0 hrs 20 mins Scoring: 20 points LESSON 2: GRAPHING FUNCTIONS Study: Graphing Functions Learn the vertical line and horizontal line tests for evaluating a function. Evaluate a function for given values and explore special functions. Duration: 0 hrs 35 mins. Quiz 9 Rational Functions Level 2 1. For the following graph, identify the following: a. Vertical asymptotes b. Horizontal asymptotes c. Holes d. End behavior e. Domain f. ... functions to their corresponding graphs Determine asymptotes and holes from an equation Write the equation given a graph Graph the function given a set of. (3.3) Rational Functions and Their Graphs Quiz Part 1 - Quiz (1-12) 10. The amount of oil used by a ship traveling at a uniform speed varies jointly with the distance and the square of the speed. Review for Unit 6 Part II test - on Tues. 3/10 Unit 6 Test Part II - Thursday, March 12th & Monday, March 16th Weeks 22-24:January 28th-Feb. 13, 2015.

yamaki deluxe folk

blocking someone on muzmatch

given an array find two statistical indicators for the array and report the difference

Example: f (x)=√x and g (x)=√ (3−x) The domain for f (x)=√x is from 0 onwards: The domain for g (x)=√ (3−x) is up to and including 3: So the new domain (after adding or whatever) is from 0 to 3: If we choose any other value, then one or the other part of the new function won't work. In other words we want to find where the two. A function is even if it is unchanged when x is replaced by -x . The graph of such a function will be symmetrical in the y-axis. Even functions which are polynomials have even degrees (e.g. y = x²). A function is odd if the sign of the function is changed when x is replaced by -x . The graph of the function will have rotational symmetry about.Graphs and charts - quiz L1 Level C 1. The logarithm base e is called the natural logarithm and is denoted. Logarithmic functions with definitions of the form have a domain consisting of positive real numbers and a range consisting of all real numbers The y -axis, or , is a vertical asymptote and the x -intercept is. To graph logarithmic functions we can plot points or identify the. Mar 22, 2022 · Trivia Quiz. A function is a twofold relation between two sets that connects to each element of the first set, precisely one element of the second set. In relational database theory, a relation is between the x values and y values of... Questions: 21 | Attempts: 9283 | Last updated: Mar 22, 2022. Sample Question.. Teaching with Desmos. 1. Select an activity. Browse our collections of free, engaging, and customizable activities that’ll help you invite, celebrate, and develop your students’ thinking. Students will share their ideas using card sorts, sketches, images, multiple choice responses, and a large and growing list of other components.

jeremy hales daughters

the peaks resort and spa

midea 50pint energystar smart dehumidifier for wet rooms with pump black mad50ps1wbl

Quiz & Worksheet Goals. This quiz will test your ability to: Identify the number of points needed to graph a linear function. Describe the linear function graph. Identify linear function equations .... Preview this quiz on Quizizz. Any function that can be defined as a rational fraction. Rational Functions and Their Graphs - Quiz DRAFT. 9th - 12th grade. 0 times. Mathematics. 0% average accuracy. a day ago. jholloway7059. 0. Save. Edit. Edit. Rational Functions and Their Graphs - Quiz DRAFT. a day ago. by jholloway7059. Played 0 times. 0. 9th. If h < 0 , the graph would be shifted right. Consider the logarithmic function y = [ log 2 ( x + 1) − 3] . This can be obtained by translating the parent graph y = log 2 ( x) a couple of times. Consider the graph of the function y = log 2 ( x) . Since h = 1 , y = [ log 2 ( x + 1)] is the translation of y = log 2 ( x) by one unit to the left. Piecewise Linear Functions. PDF DOCUMENT. VIDEO. PDF ANSWER KEY. WORD DOCUMENT. WORD ANSWER KEY. Lesson 7. Systems of Linear Equations (Primarily 3 by 3) PDF DOCUMENT.

allstate pay my bill

the little cbt workbook pdf

Functions and their graphs, after studying this section, you will be able to: understand function notation; apply transformations to the graphs of various functions; Functions. y = f(x) stands for 'y is a function of x' When y = x 2 + 13 then f(x) = x 2 + 13. Therefore from the above f(x) + x = x 2 + 13 + x. Transforming graphs of functions. Bing: precalculus chapter 1 functions and their graphs test A General Note: Graphical Interpretation of a Linear Function . In the equation $f\left(x\right)=mx+b$ b is the y-intercept of the graph and indicates the point (0, b) at which the graph crosses the y-axis.; m is the slope of the line..

SOLUTION STEP 1 STEP 2 Plot the vertex ( h , k ) = ( - 2, 5) and draw the axis of symmetry x = - 2. 14 Identify the constants a = - , h = - 2, and k = 5. Because a < 0, the parabola opens down. 14. 20. EXAMPLE 1 Graph a quadratic function in vertex form STEP 3 Evaluate the function for two values of x . Plot the points (0, 4) and (2, 1.

hobbywing esc setup sheets

ayahuasca new york

Match the quadratic function to its graph. Beware! I am unable to use plus signs so I have had to rearrange. ... This is an online quiz called Quadratic Functions. ... Pick an audience - or yourself - and it'll end up in their play queue. Tags. math. function. maths. quadratic. graph. Games by same creator. Surface area of cubes 9p Image Quiz.

snake mystery box

1-9 Bell Work - Patterns, Equations, and Graphs. 1-9 Exit Quiz - Patterns, Equations, and Graphs. 1-9 Guide Notes SE - Patterns, Equations, and Graphs. 1-9 Guide Notes TE - Patterns, Equations, and Graphs. ... Rational Functions and Their Graphs - Group Activity Algebraic Expressions Worksheet and Activity - Mazing! Holiday Algebra 2 Activities.

dance lessons atlanta couples

real number line. Sometimes, this is related to a point on the graph of f. Example 1 (Evaluating the Limit of a Polynomial Function at a Point) Let fx()= 3x2 + x 1. Evaluate lim x 1 fx(). § Solution f is a polynomial function with implied domain Dom()f = . We substitute ("plug in") x =1 and evaluate f ()1 . WARNING 2: Sometimes, the limit. Identify a function graphically using the vertical line test [ Lesson 6.4, Lesson 4.6, Lesson 7.5] 2) Evaluating fu nctions. a. Given input and function, determine output.b. Given oup ut and function, determine input.3) Analyzing functions and their graphs [ Exploration of Functions in Everyday Situations] a. Transformations of functions are algebraic operations on the function that.

sql server srid

lady amherst pheasant eggs for sale

crash on 275 yesterday

home depot deep freezer

Quiz & Worksheet Goals. This quiz will test your ability to: Identify the number of points needed to graph a linear function. Describe the linear function graph. Identify linear function equations ....

Observe that when the function is positive, it is symmetric with respect to the equation $\mathbf{y = x}$.Meanwhile, when the function is negative (i.e., has a negative constant), it is symmetric with respect to the equation $\mathbf{y = -x}$. Summary of reciprocal function definition and properties Before we try out some more problems that involve reciprocal functions, let's summarize.

twisted wonderland x mother figure reader

2006 sprinter limp mode bypass

Precalculus (10th Edition) answers to Chapter 2 - Functions and Their Graphs - 2.1 Functions - 2.1 Assess Your Understanding - Page 56 1 including work step by step written by community members like you. Textbook Authors: Sullivan, Michael, ISBN-10: -32197-907-9, ISBN-13: 978--32197-907-0, Publisher: Pearson. A function is even if it is unchanged when x is replaced by -x . The graph of such a function will be symmetrical in the y-axis. Even functions which are polynomials have even degrees (e.g. y = x²). A function is odd if the sign of the function is changed when x is replaced by -x . The graph of the function will have rotational symmetry about.Graphs and charts - quiz L1 Level C 1.

Piecewise Linear Functions. PDF DOCUMENT. VIDEO. PDF ANSWER KEY. WORD DOCUMENT. WORD ANSWER KEY. Lesson 7. Systems of Linear Equations (Primarily 3 by 3) PDF DOCUMENT. Functions and Graphs 2.4 Polynomial and Rational Functions Consider g(x) = jxj 2. g(x) is continuous, but not smooth due to a sharp corner at (0; 2). 5 3 1 1 3 5 4 2 2 4 x y g(x) Consider h(x) = 2 x 1. h(x) has a discontinuous break at x = 1. 5 3 1 1 3 5 5 3 1 1 3 5 x y h(x) Graph of a Polynomial Function The graph of a polynomial function is.

In mathematics, some functions or groups of functions are important enough to deserve their own names.This is a listing of articles which explain some of these functions in more detail. There is a large theory of special functions which developed out of statistics and mathematical physics.A modern, abstract point of view contrasts large function spaces, which are infinite-dimensional and. 2.1 Functions and function notation Parent Graphs Polynomial Project DUE! HW 2.1 9/22, 9/23 Evaluating Functions Activity to discover 2.2 (do first) 2.2 The graph of a function HW 2.2 9/24*, 9/27 2.3 Modelling Functions and Word Applications HW 2.3 9/28, 9/29 QUIZ on 2.1-2.2 2.4 Key features of graphs Factoring, exponent rules, and.

over 400 companies have made a pledge

2007 honda crv blue book value

The Graph of a Quadratic Function. A quadratic function is a polynomial function of degree 2 which can be written in the general form, f(x) = ax2 + bx + c. Here a, b and c represent real numbers where a ≠ 0. The squaring function f(x) = x2 is a quadratic function whose graph follows. This general curved shape is called a parabola and is. Use the vertical line test to determine whether or not a graph represents a function. If a vertical line is moved across the graph and, at any time, touches the graph at only one point, then the graph is a function. If the vertical line touches the graph at more than one point, then the graph is. linux tc htb. note 20 ultra texting issues.

Functions and their graphs DRAFT. 9th - 12th grade. 2 times. Mathematics. 35% average accuracy. a year ago. mrshearne. 0. Save. Edit. Edit. Functions and their graphs DRAFT. a year ago. by mrshearne. ... This quiz is incomplete! To play this quiz, please finish editing it. Delete Quiz. This quiz is incomplete! To play this quiz, please finish.

pole setting foam price

hornady 300 win mag ballistics chart

8-3 Rational Functions and Their Graphs 9th - 12th grade 0 times Mathematics 3 hours ago papashell 0 Save Edit Start a multiplayer game Play Live Assign HW Solo Practice Practice Preview (15 questions) Show answers Question 1 300 seconds Q. What is the X-Intercept? answer choices x= 0 x= 4 x= 5 x= -5 Question 2 300 seconds Q.

bedford cf parts

19/11/21 23:57 W08 Reading Quiz: Functions and Their Graphs: Math for the Real World W08 Reading Quiz: Functions and Their Graphs Due Nov 4 at 11:59pm Points 10 Questions 8 Available until Nov 4 at 11:59pm Time Limit None Allowed Attempts 2 Instructions This quiz was locked Nov 4 at 11:59pm. Attempt History Attempt Time Score LATEST Attempt 1 38 minutes 8 out of 10 Correct answers are hidden.

The logarithm base e is called the natural logarithm and is denoted. Logarithmic functions with definitions of the form have a domain consisting of positive real numbers and a range consisting of all real numbers The y -axis, or , is a vertical asymptote and the x -intercept is. To graph logarithmic functions we can plot points or identify the. Evaluate functions from their graph - Promoting Literacy ... My Courses. thundermax touring. ain nadia wattpad; crime scene photos of layne staley; nerf gun party florida esp32 nimble tutorial; indian roommates morrisville nc arca rail width mitsubishi customer.

buffalo bullet company

SAT information-Includes Test Prep Sites and Apps. Unit 1: Graphs. unit 2: Functions. Unit 2: Functions and Their Graphs. Unit 3: Linear and Quadratic Functions. Unit 3: Linear and Quadratic Functions. Unit 4 notes. Unit 4: Polynomial and Rational Functions. Unit 5 Notes.

ngrok reverse shell

indefinite supervision virginia

Figure 1 compares relations that are functions and not functions. Figure 1 (a) This relationship is a function because each input is associated with a single output. Note that input q and r both give output n. (b) This relationship is also a function. In this case, each input is associated with a single output.

what happens to child support arrears when custodial parent dies in va

treasures or trophies chords

Mixed Transformations. Most of the problems you'll get will involve mixed transformations, or multiple transformations, and we do need to worry about the order in which we perform the transformations. It usually doesn't matter if we make the $$x$$ changes or the $$y$$ changes first, but within the $$x$$'s and $$y$$'s, we need to perform the transformations in the order below. A parent function is the simplest function that still satisfies the definition of a certain type of function. For example, when we think of the linear functions which make up a family of functions, the parent function would be y = x. This is the simplest linear function. Furthermore, all of the functions within a family of functions can be.

You are given 4 graphs, select the best possible answer. Question 1: Identify the graph of the rational function f. f (x) = 1 / x. Click here to see the 4 graphs. Question 2: Identify the graph of the rational function f. f (x) = 1 / (x - 1) Click here to see the 4 graphs. Question 3: Identify the graph of the rational function f.

jaguar xke restoration cost

v power casino review

Mar 19, 2022 · Create your own Quiz. Questions and Answers. 1. What family of functions has the parent function: 2. What family of functions has the parent function: 3. What family of functions has the parent function: 4..

1. When it comes to relations in math, what does each line represent in a mapping? 2. When it comes to relations in math, the range is the set of _____. 3. If we multiply a function, f (x) by a .... The graph of a function f is a drawing hat represents all the input-output pairs, (x, f(x)). In cases where the function is given by an equation, the graph of a function is the graph of the equation y = f(x). The vertical line test - a graph represents a function if it is impossible to draw a vertical line that intersects the graph more than once. The goal of these algebra lessons is help you become very familiar with the following topics: Learn to add, subtract, multiply, and divide integers. Learn the difference between rational numbers and irrational numbers. Learn to solve linear equations, absolute value equations, and system of linear equations. Learn to add, subtract, and multiply.

big bend national park maps

st martin parish arrests records

Mixed Transformations. Most of the problems you'll get will involve mixed transformations, or multiple transformations, and we do need to worry about the order in which we perform the transformations. It usually doesn't matter if we make the $$x$$ changes or the $$y$$ changes first, but within the $$x$$'s and $$y$$'s, we need to perform the transformations in the order below.

1960s bathroom vanity with sink

Quiz & Worksheet Goals. You will be asked to do the following when you use this assessment: Note a fact about the slope of a derivative based on the graph of a parabola. Identify the equation for .... When graphing certain polynomial functions, we can use the graphs of monomials we already know, and transform them using the techniques we learned earlier. Example 1: Sketch the graph of the function . Px x ( ) =−+ 3. 2. n. by transforming the graph of an appropriate function of the form . y = x. Indicate all . x - and . y-intercepts on the.

1. Hormones of Thyroid. Thyroid gland basically releases two hormones Triiodothyronine (T3) and Thyroxine (T4), which helps in controlling the metabolism of our body. Further, these hormones. Try It 10.20. Graph: g ( x) = 5 x. If we look at the graphs from the previous Example and Try Its, we can identify some of the properties of exponential functions. The graphs of f ( x) = 2 x and g ( x) = 3 x, as well as the graphs of f ( x) = 4 x and g ( x) = 5 x, all have the same basic shape. This is the shape we expect from an exponential.

Mar 22, 2022 · Table And Bar Graph Test! Trivia Questions Quiz. Study the following table and answer the questions. Number of Candidates Appeared and Qualified in a Competitive Examination from the Different States Over the Years. - The total number of candidates qualified from all the states together in 1997 is approximately what percentage of the total ....

hot wheels super ultimate garage replacement track

dark psychology 101 free

1.!Given the function 1(()=−(3+4(+21, use the following steps to graph 1(() on the coordinate plane on the ....

• why is my con edison delivery charges so high – The world’s largest educational and scientific computing society that delivers resources that advance computing as a science and a profession
• 2001 ford f350 replacement seats – The world’s largest nonprofit, professional association dedicated to advancing technological innovation and excellence for the benefit of humanity
• livery yard worthing – A worldwide organization of professionals committed to the improvement of science teaching and learning through research
• hydraulic accumulator troubleshooting –  A member-driven organization committed to promoting excellence and innovation in science teaching and learning for all
• can peripheral neuropathy be cured – A congressionally chartered independent membership organization which represents professionals at all degree levels and in all fields of chemistry and sciences that involve chemistry
• atf sims 4 – A nonprofit, membership corporation created for the purpose of promoting the advancement and diffusion of the knowledge of physics and its application to human welfare
• armor games swf – A nonprofit, educational organization whose purpose is the advancement, stimulation, extension, improvement, and coordination of Earth and Space Science education at all educational levels
• best age to transition ftm – A nonprofit, scientific association dedicated to advancing biological research and education for the welfare of society

prius c battery price

verizon store northpark mall

Bing: precalculus chapter 1 functions and their graphs test A General Note: Graphical Interpretation of a Linear Function . In the equation $f\left(x\right)=mx+b$ b is the y-intercept of the graph and indicates the point (0, b) at which the graph crosses the y-axis.; m is the slope of the line.. Try It 10.20. Graph: g ( x) = 5 x. If we look at the graphs from the previous Example and Try Its, we can identify some of the properties of exponential functions. The graphs of f ( x) = 2 x and g ( x) = 3 x, as well as the graphs of f ( x) = 4 x and g ( x) = 5 x, all have the same basic shape. This is the shape we expect from an exponential.

suddenly no likes on hinge

mfc cam recorder

SECTION 1.4 Operations on Functions MATH 1330 Precalculus 107 Section 1.4: Operations on Functions Combining Functions by Addition, Subtraction, Multiplication, ... Sketch the graph of f g. (Hint: For any x value, add the y values of f and g.) (g) What is the domain of f g? Explain how you obtained your answer. 8.

cummins def dosing unit relay

html number input limit decimal places

Math; Precalculus; Precalculus questions and answers = Quiz: Quiz on Chapter 2 Functions and their Graphs Question 10 This Quiz: 10 pts possible Use the graph of the function f shown to the right to answer parts (a)-(n). ..... Bing: precalculus chapter 1 functions and their graphs test A General Note: Graphical Interpretation of a Linear Function . In the equation $f\left(x\right)=mx+b$ b is the y-intercept of the graph and indicates the point (0, b) at which the graph crosses the y-axis.; m is the slope of the line.

Question 2 1 / 1 pts Apply the order of operations to simplify the following expression. 3 + 1 - (8 * 4 + 3) -31 Question 3 1 / 1 pts What type of function is this? f (x)=4 ( 34)x+3 f (x)=4 (34)x+3 Linear function Quadratic function Exponential function. 8-3 Rational Functions and Their Graphs 9th - 12th grade 0 times Mathematics 3 hours ago papashell 0 Save Edit Start a multiplayer game Play Live Assign HW Solo Practice Practice Preview (15 questions) Show answers Question 1 300 seconds Q. What is the X-Intercept? answer choices x= 0 x= 4 x= 5 x= -5 Question 2 300 seconds Q.

harley 3 wire to 2 wire turn signal

starseed soulmate

4 stages of trauma recovery
Overview. In this lesson students interpret key features of graphs for both linear and quadratic functions in the context of total and marginal production. The lesson begins with a short video about a young entrepreneur who designed his own line of bowties. Students then predict the relationship between number of workers and production of bowties. Graphing Rational Functions. Review: What Are Rational Functions? X and Y Intercepts. Vertical Asymptotes. Horizontal and Slant (Oblique) Asymptotes. Putting It All Together. Increasing and Decreasing Revisited.
fastapi log response time cabins for sale on mississippi river in illinois property for sale in giffnock what is capsaicin cream used for unsecured network macbook


text:
# Pandit S.N. Shukla University, Shahdol (M.P) M.Sc. MATHEMATICS SEMESTER-IV Paper II-Advanced Special Function-II Syllabus

## Pandit S.N. Shukla University, Shahdol (M.P) M.Sc. MATHEMATICS SEMESTER-IV Paper II-Advanced Special Function-II Syllabus

Theory Paper : Max. Marks- 42

Internal Assessment: Max. Marks- 08

Note: The question paper will consist of three sections A, B&C. Section A will consist of objeclive type questions each carrving 1 marks, section B will consist of 5 short answer type questions cach earrying 3 marks and section C will consist of 2 long answer type questions each carrying 10 marks. Section B & C will have internal choice.

Unit 1- Bessel Function and Legendre polynomials: Definition of J(Z). Bessel's differential equation, Generating function, Bessel's integral with index half and an odd integer.

Unit 2- Generating function for Legendre polynomials, Rodrigues formula, Bateman's generating function, Additional generating functions. Hypergeometric forms of P.(X).

Unit 3- Special properties of Pa(X). Some more generating functions, Laplace's first integral form, Orthogonality.

Unit 4- Definition of Hermite polynomials H-(X). Pure recurrence relations, Differential recurrence relations, Rodrigue's formula, Other generating functions, Orthogonality, Expansion of polynomials, more generating functions.

Unit 5- The Laguerre Polynomials L-(X). Generating functions, Pure recurrence relations, Differential recurrence relation, Rodrigue's formula, Orthogonality, Expansion of polynomials, special properties, Other generating functions.

Recommended Books:

[1]E.D. Rainville, Special Functions, The Macmillan Co., New York, 1971.

[2] H.M.Srivastava, K.C.Gupta and S.P.Goyal, The H-function of One and Two variables with applications, South Asian Publication, New Delhi,

[3] N . Saran, S.D.Sharma and Trivedi, Special Functions with applications, Pragati Prakashan, 1986.

[4] N.N.Lebdev, Special functions and their applications, Prentice Hall, Englewood Cliffs, New Jersey, USA, 1965.

format: plain


text:
NEW
New Website Launch
Experience the best way to solve previous year questions with mock tests (very detailed analysis), bookmark your favourite questions, practice etc...
1

### AIEEE 2005

If the roots of the equation $${x^2} - bx + c = 0$$ be two consecutive integers, then $${b^2} - 4c$$ equals
A
$$-2$$
B
$$3$$
C
$$2$$
D
$$1$$

## Explanation

Let n and (n + 1) be the roots of x2 $$-$$ bx + c = 0.

Then, n + (n + 1) = b and n(n + 1) = c

$$\therefore$$ b2 $$-$$ 4c = (2n + 1)2 $$-$$ 4n(n + 1)

= 4n2 + 4n + 1 $$-$$ 4n2 $$-$$ 4n = 1

2

### AIEEE 2005

The value of $$a$$ for which the sum of the squares of the roots of the equation
$${x^2} - \left( {a - 2} \right)x - a - 1 = 0$$ assume the least value is
A
$$1$$
B
$$0$$
C
$$3$$
D
$$2$$

## Explanation

$${x^2} - \left( {a - 2} \right)x - a - 1 = 0$$

Let $$\alpha$$ and $$\beta$$ are the roots of the equation.

$$\therefore$$ $$\alpha$$ + $$\beta$$ = $$a - 2$$

and $$\alpha$$$$\beta$$ = $$- a - 1$$

Now $${\alpha ^2} + {\beta ^2} = {\left( {\alpha + \beta } \right)^2} - 2\alpha \beta$$

$$\Rightarrow$$ $${\alpha ^2} + {\beta ^2} = {\left( {a - 2} \right)^2} + 2\left( {a + 1} \right)$$

$$\Rightarrow$$ $${\alpha ^2} + {\beta ^2} = {a^2} - 2a + 6$$

$$\Rightarrow$$ $${\alpha ^2} + {\beta ^2} = {\left( {a - 1} \right)^2} + 5$$

$$\Rightarrow$$ The value of $${\alpha ^2} + {\beta ^2}$$ will be minimum, when $${a - 1}$$ = 0

$$\Rightarrow$$ $${a = 1}$$
3

### AIEEE 2005

If both the roots of the quadratic equation $${x^2} - 2kx + {k^2} + k - 5 = 0$$ are less than 5, then $$k$$ lies in the interval
A
$$\left( {5,6} \right]$$
B
$$\left( {6,\,\infty } \right)$$
C
$$\left( { - \infty ,\,4} \right)$$
D
$$\left[ {4,\,5} \right]$$

## Explanation

both roots are less than $$5,$$

then $$(i)$$ Discriminant $$\ge 0$$

$$\left( {ii} \right)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,p\left( 5 \right) > 0$$

$$\left( {iii} \right)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{{Sum\,\,of\,\,roots} \over 2} < 5$$

Hence $$\left( i \right)\,\,\,\,\,\,4{k^2} - 4\left( {{k^2} + k - 5} \right) \ge 0$$

$$4{k^2} - 4{k^2} - 4k + 20 \ge 0$$

$$4k \le 20 \Rightarrow k \le 5$$

$$\left( {ii} \right)\,\,\,\,\,f\left( 5 \right) > 0;25 - 10k + {k^2} + k - 5 > 0$$

or $${k^2} - 9k + 20 > 0$$

or $$k\left( {k - 4} \right) - 5\left( {k - 4} \right) > 0$$

or $$\left( {k - 5} \right)\left( {k - 4} \right) > 0$$

$$\Rightarrow k \in \left( { - \infty ,4} \right) \cup \left( { - \infty ,5} \right)$$

$$\left( {iii} \right)\,\,\,\,\,\,{{Sum\,\,of\,\,roots} \over 2}$$

$$= - {b \over {2a}} = {{2k} \over 2} < 5$$

The intersection of $$(i)$$, $$(ii)$$ & $$(iii)$$ gives

$$k \in \left( { - \infty ,4} \right).$$
4

### AIEEE 2005

In a triangle $$PQR,\;\;\angle R = {\pi \over 2}.\,\,If\,\,\tan \,\left( {{P \over 2}} \right)$$ and $$\tan \left( {{Q \over 2}} \right)$$ are the roots of $$a{x^2} + bx + c = 0,\,\,a \ne 0$$ then
A
$$a = b + c$$
B
$$c = a + b$$
C
$$b = c$$
D
$$b = a + c$$

## Explanation

$$\angle$$R = 90o $$\therefore$$ $$\angle$$P + $$\angle$$Q = 90o

$$\Rightarrow$$ $${P \over 2} + {Q \over 2} = {{90} \over 2} = 45$$o

$$\tan \left( {{P \over 2}} \right),\tan \left( {{Q \over 2}} \right)$$ are the roots of $$a{x^2} + bx + c = 0$$

$$\therefore$$ $$\tan \left( {{P \over 2}} \right) + \tan \left( {{Q \over 2}} \right) = - {b \over a},\,\,$$

and $$\tan \left( {{P \over 2}} \right).\tan \left( {{Q \over 2}} \right) = {c \over a}$$

$${{\tan \left( {{P \over 2}} \right) + \tan \left( {{Q \over 2}} \right)} \over {1 - \tan \left( {{P \over 2}} \right)\tan \left( {{Q \over 2}} \right)}}$$

$$= \tan \left( {{P \over 2} + {Q \over 2}} \right)$$= tan 45o = 1

$$\Rightarrow {{ - {b \over a}} \over {1 - {c \over a}}} = 1$$

$$\Rightarrow - {b \over a} = {a \over a} - {c \over a}$$

$$\Rightarrow - b = a - c$$ or $$c = a + b.$$

### Joint Entrance Examination

JEE Main JEE Advanced WB JEE

### Graduate Aptitude Test in Engineering

GATE CSE GATE ECE GATE EE GATE ME GATE CE GATE PI GATE IN

NEET

Class 12


text:
# Map Operator

Base.mapFunction
map(::Type{R}, mappingFn::F) where { F <: Function }

Creates a map operator, which applies a given mappingFn to each value emmited by the source Observable, and emits the resulting values as an Observable. You have to specify output R type after mappingFn projection.

Arguments

• ::Type{R}: the type of data of transformed value, may be or may not be the same as source type
• mappingFn::Function: transformation function with (data::L) -> R signature, where L is type of data in input source

Producing

Stream of type <: Subscribable{R}

Examples

using Rocket

source = from([ 1, 2, 3 ])
subscribe!(source |> map(Int, (d) -> d ^ 2), logger())
;

# output

[LogActor] Data: 1
[LogActor] Data: 4
[LogActor] Data: 9
[LogActor] Completed
source

## Description

Like map(f, array), the map operator applies a function to each source value.


text:
# Time Evolution of Coronal Magnetic Helicity in the Flaring Active Region NOAA 10930

Research paper by Sung-Hong Park, Jongchul Chae, Ju Jing, Changyi Tan, Haimin Wang

Indexed on: 18 Aug '10Published on: 18 Aug '10Published in: arXiv - Astrophysics - Solar and Stellar Astrophysics

#### Abstract

To study the three-dimensional (3D) magnetic field topology and its long-term evolution associated with the X3.4 flare of 2006 December 13, we investigate the coronal relative magnetic helicity in the flaring active region (AR) NOAA 10930 during the time period of December 8-14. The coronal helicity is calculated based on the 3D nonlinear force-free magnetic fields reconstructed by the weighted optimization method of Wiegelmann, and is compared with the amount of helicity injected through the photospheric surface of the AR. The helicity injection is determined from the magnetic helicity flux density proposed by Pariat et al. using Solar and Heliospheric Observatory/Michelson Doppler Imager magnetograms. The major findings of this study are the following. (1) The time profile of the coronal helicity shows a good correlation with that of the helicity accumulation by injection through the surface. (2) The coronal helicity of the AR is estimated to be -4.3$\times$10$^{43}$ Mx$^{2}$ just before the X3.4 flare. (3) This flare is preceded not only by a large increase of negative helicity, -3.2$\times$10$^{43}$ Mx$^{2}$, in the corona over ~1.5 days but also by noticeable injections of positive helicity though the photospheric surface around the flaring magnetic polarity inversion line during the time period of the channel structure development. We conjecture that the occurrence of the X3.4 flare is involved with the positive helicity injection into an existing system of negative helicity.


text:
# American Institute of Mathematical Sciences

May  2017, 37(5): 2717-2743. doi: 10.3934/dcds.2017117

## Diagonal stationary points of the bethe functional

 1 Faculty of Physics, Warsaw University of Technology, Faculty of Mathematics and Information Science, Warsaw University of Technology, Koszykowa 75, PL-00-662 Warsaw, Poland 2 Faculty of Mathematics and Information Science, Warsaw University of Technology, Koszykowa 75, PL-00-662 Warsaw, Poland

* Corresponding author: g.swiatek@mini.pw.edu.pl

Received  March 2016 Revised  December 2016 Published  February 2017

Fund Project: Both authors supported in part by Narodowe Centrum Nauki - grant 2015/17/B/ST1/00091.

We investigate stationary points of the Bethe functional for the Ising model on a $2$-dimensional lattice. Such stationary points are also fixed points of message passing algorithms. In the absence of an external field, by symmetry reasons one expects the fixed points to have constant means at all sites. This is shown not to be the case. There is a critical value of the coupling parameter which is equal to the phase transition parameter on the computation tree, see [13], above which fixed points appear with means that are variable though constant on diagonals of the lattice and hence the term “diagonal stationary points”. A rigorous analytic proof of their existence is presented. Furthermore, computer-obtained examples of diagonal stationary points which are local maxima of the Bethe functional and hence stable equilibria for message passing are shown. The smallest such example was found on the $100× 100$ lattice.

Citation: Grzegorz Siudem, Grzegorz Świątek. Diagonal stationary points of the bethe functional. Discrete and Continuous Dynamical Systems, 2017, 37 (5) : 2717-2743. doi: 10.3934/dcds.2017117
##### References:
 [1] R. J. Baxter, Exactly solved models in statistical mechanics, Integrable Systems in Statistical Mechanics, 1 (1985), 5-63.  doi: 10.1142/9789814415255_0002. [2] H. A. Bethe, Statistical theory of superlattices, Selected Works of Hans A Bethe, 18 (1997), 245-270.  doi: 10.1142/9789812795755_0010. [3] S. Dorogovtsev, A. Goltsev and J. Mendes, Critical phenomena in complex networks, Rev. Mod. Phys., 80 (2008), 1275-1335.  doi: 10.1103/RevModPhys.80.1275. [4] C. Fortuin, P. Kasteleyn and J. Ginibre, Correlation inequalities on some partially ordered sets, Commun. Math. Phys., 22 (1971), 89-103.  doi: 10.1007/BF01651330. [5] T. Heskes, Stable fixed points of loopy belief propagation are local minima of the Bethe free energy, In S. Thrun S. Becker and K. Obermayer, editors, Advances in Neural Information Processing Systems 15, MIT Press, Cambridge, MA, (2003), 343-350. [6] J. M. Mooij and H. J. Kappen, On the properties of the Bethe approximation and loopy belief propagation on binary networks J. Stat. Mech. Theor. Exp. 11 (2005), P11012. doi: 10.1088/1742-5468/2005/11/P11012. [7] J. M. Mooij and H. J. Kappen, Sufficient conditions for convergence of the sum-product algorithm, IEEE Transactions on Information Theory, 53 (2007), 4422-4437.  doi: 10.1109/TIT.2007.909166. [8] S. Newhouse, Diffeomorphisms with infinitely many sinks, Topology, 13 (1974), 9-18.  doi: 10.1016/0040-9383(74)90034-2. [9] J. Pearl, Reverend Bayes on inference engines: A distributed hierarchical approach, Proceedings of the Second National Conference on Artificial Intelligence, (1982), 133-136. [10] T. G. Roosta and M. J. Wainwright snd S. S. Sastry, Convergence analysis of reweighted sum-product algorithms, IEEE Transactions on Signal Processing, 56 (2008), 4293-4305.  doi: 10.1109/ICASSP.2007.366292. [11] J. Shin, The complexity of approximating a bethe equilibrium, IEEE Transactions on Information Theory, 60 (2014), 3959-3969.  doi: 10.1109/TIT.2014.2317487. [12] G. Siudem and G. Świątek, Dynamics of the belief propagation for the ising model, Acta Physica Polonica A, 127 (2015), 3A145-3A149.  doi: 10.12693/APhysPolA.127.A-145. [13] S. Tatikonda and M. Jordan, Loopy belief propagation and Gibbs measures, in Proceedings of the 18th Conference on Uncertainty in Artificial Intelligence, Morgan Kaufmann, San Francisco, (2002), 493-500, [14] M. J. Wainwright and M. I. Jordan, Graphical models, exponential families, and variational inference, Foundations and Trends in Machine Learning, 1 (2008), 1-305.  doi: 10.1561/2200000001. [15] A. Weller and T. Jebara, Bethe bounds and approximating the global optimum, Journal of Machine Learning Research W& CP, 31 (2013), 618-631. [16] M. Welling and Y. -W. Teh, Belief Optimization for Binary Networks: A Stable Alternative to Loopy Belief Propagation in Proc. 17th Conference on Uncertainty in Artificial Intelligence (UAI), 2001. [17] J. Yedidia, W. Freeman and Y. Weiss, Constructing free-energy approximations and generalized belief propagation algorithms, IEEE Trans. on Information Theory, 51 (2005), 2282-2312.  doi: 10.1109/TIT.2005.850085.

show all references

##### References:
 [1] R. J. Baxter, Exactly solved models in statistical mechanics, Integrable Systems in Statistical Mechanics, 1 (1985), 5-63.  doi: 10.1142/9789814415255_0002. [2] H. A. Bethe, Statistical theory of superlattices, Selected Works of Hans A Bethe, 18 (1997), 245-270.  doi: 10.1142/9789812795755_0010. [3] S. Dorogovtsev, A. Goltsev and J. Mendes, Critical phenomena in complex networks, Rev. Mod. Phys., 80 (2008), 1275-1335.  doi: 10.1103/RevModPhys.80.1275. [4] C. Fortuin, P. Kasteleyn and J. Ginibre, Correlation inequalities on some partially ordered sets, Commun. Math. Phys., 22 (1971), 89-103.  doi: 10.1007/BF01651330. [5] T. Heskes, Stable fixed points of loopy belief propagation are local minima of the Bethe free energy, In S. Thrun S. Becker and K. Obermayer, editors, Advances in Neural Information Processing Systems 15, MIT Press, Cambridge, MA, (2003), 343-350. [6] J. M. Mooij and H. J. Kappen, On the properties of the Bethe approximation and loopy belief propagation on binary networks J. Stat. Mech. Theor. Exp. 11 (2005), P11012. doi: 10.1088/1742-5468/2005/11/P11012. [7] J. M. Mooij and H. J. Kappen, Sufficient conditions for convergence of the sum-product algorithm, IEEE Transactions on Information Theory, 53 (2007), 4422-4437.  doi: 10.1109/TIT.2007.909166. [8] S. Newhouse, Diffeomorphisms with infinitely many sinks, Topology, 13 (1974), 9-18.  doi: 10.1016/0040-9383(74)90034-2. [9] J. Pearl, Reverend Bayes on inference engines: A distributed hierarchical approach, Proceedings of the Second National Conference on Artificial Intelligence, (1982), 133-136. [10] T. G. Roosta and M. J. Wainwright snd S. S. Sastry, Convergence analysis of reweighted sum-product algorithms, IEEE Transactions on Signal Processing, 56 (2008), 4293-4305.  doi: 10.1109/ICASSP.2007.366292. [11] J. Shin, The complexity of approximating a bethe equilibrium, IEEE Transactions on Information Theory, 60 (2014), 3959-3969.  doi: 10.1109/TIT.2014.2317487. [12] G. Siudem and G. Świątek, Dynamics of the belief propagation for the ising model, Acta Physica Polonica A, 127 (2015), 3A145-3A149.  doi: 10.12693/APhysPolA.127.A-145. [13] S. Tatikonda and M. Jordan, Loopy belief propagation and Gibbs measures, in Proceedings of the 18th Conference on Uncertainty in Artificial Intelligence, Morgan Kaufmann, San Francisco, (2002), 493-500, [14] M. J. Wainwright and M. I. Jordan, Graphical models, exponential families, and variational inference, Foundations and Trends in Machine Learning, 1 (2008), 1-305.  doi: 10.1561/2200000001. [15] A. Weller and T. Jebara, Bethe bounds and approximating the global optimum, Journal of Machine Learning Research W& CP, 31 (2013), 618-631. [16] M. Welling and Y. -W. Teh, Belief Optimization for Binary Networks: A Stable Alternative to Loopy Belief Propagation in Proc. 17th Conference on Uncertainty in Artificial Intelligence (UAI), 2001. [17] J. Yedidia, W. Freeman and Y. Weiss, Constructing free-energy approximations and generalized belief propagation algorithms, IEEE Trans. on Information Theory, 51 (2005), 2282-2312.  doi: 10.1109/TIT.2005.850085.
Illustration of the diagonal matrix which can be obtained from means given by Eq. (11).
Numerical evidence that the means (from Eq. (11), visualized on Fig. 1) in fact define a stationary point of the Bethe functional. The dots on the graph show values of the negative Bethe functional computed for the means given by vector $B_{\eta}$ given by formula (12) with $\eta$ shown on the horizontal axis and various randomly chosen $(X_{\ell})$.
Values of the negative Bethe functional for the diagonal stationary point $\mathcal{B}_0$ perturbed in the direction of $P$ according to formula (12).
The stability test algorithm.
Stable fixed point given by Eq. (13).
 [1] Velimir M. Ilić, Miomir S. Stanković, Branimir T. Todorović. Computation of cross-moments using message passing over factor graphs. Advances in Mathematics of Communications, 2012, 6 (3) : 363-384. doi: 10.3934/amc.2012.6.363 [2] Gabriella Bretti, Roberto Natalini, Benedetto Piccoli. Fast algorithms for the approximation of a traffic flow model on networks. Discrete and Continuous Dynamical Systems - B, 2006, 6 (3) : 427-448. doi: 10.3934/dcdsb.2006.6.427 [3] Michele Gianfelice, Marco Isopi. On the location of the 1-particle branch of the spectrum of the disordered stochastic Ising model. Networks and Heterogeneous Media, 2011, 6 (1) : 127-144. doi: 10.3934/nhm.2011.6.127 [4] Robert Baier, Matthias Gerdts, Ilaria Xausa. Approximation of reachable sets using optimal control algorithms. Numerical Algebra, Control and Optimization, 2013, 3 (3) : 519-548. doi: 10.3934/naco.2013.3.519 [5] Manuel González-Navarrete. Type-dependent stochastic Ising model describing the dynamics of a non-symmetric feedback module. Mathematical Biosciences & Engineering, 2016, 13 (5) : 981-998. doi: 10.3934/mbe.2016026 [6] Lori Badea, Marius Cocou. Approximation results and subspace correction algorithms for implicit variational inequalities. Discrete and Continuous Dynamical Systems - S, 2013, 6 (6) : 1507-1524. doi: 10.3934/dcdss.2013.6.1507 [7] Fanghua Lin, Juncheng Wei. Superfluids passing an obstacle and vortex nucleation. Discrete and Continuous Dynamical Systems, 2019, 39 (12) : 6801-6824. doi: 10.3934/dcds.2019232 [8] Fengmin Wang, Dachuan Xu, Donglei Du, Chenchen Wu. Primal-dual approximation algorithms for submodular cost set cover problems with linear/submodular penalties. Numerical Algebra, Control and Optimization, 2015, 5 (2) : 91-100. doi: 10.3934/naco.2015.5.91 [9] Hua-Ping Wu, Min Huang, W. H. Ip, Qun-Lin Fan. Algorithms for single-machine scheduling problem with deterioration depending on a novel model. Journal of Industrial and Management Optimization, 2017, 13 (2) : 681-695. doi: 10.3934/jimo.2016040 [10] Farrukh Mukhamedov, Otabek Khakimov. Chaotic behavior of the P-adic Potts-Bethe mapping. Discrete and Continuous Dynamical Systems, 2018, 38 (1) : 231-245. doi: 10.3934/dcds.2018011 [11] Bernard Bonnard, Thierry Combot, Lionel Jassionnesse. Integrability methods in the time minimal coherence transfer for Ising chains of three spins. Discrete and Continuous Dynamical Systems, 2015, 35 (9) : 4095-4114. doi: 10.3934/dcds.2015.35.4095 [12] Gonzalo Galiano, Julián Velasco. Finite element approximation of a population spatial adaptation model. Mathematical Biosciences & Engineering, 2013, 10 (3) : 637-647. doi: 10.3934/mbe.2013.10.637 [13] Jishan Fan, Tohru Ozawa. An approximation model for the density-dependent magnetohydrodynamic equations. Conference Publications, 2013, 2013 (special) : 207-216. doi: 10.3934/proc.2013.2013.207 [14] Stephen Thompson, Thomas I. Seidman. Approximation of a semigroup model of anomalous diffusion in a bounded set. Evolution Equations and Control Theory, 2013, 2 (1) : 173-192. doi: 10.3934/eect.2013.2.173 [15] Tong Zhang, Jinyun Yuan. Two novel decoupling algorithms for the steady Stokes-Darcy model based on two-grid discretizations. Discrete and Continuous Dynamical Systems - B, 2014, 19 (3) : 849-865. doi: 10.3934/dcdsb.2014.19.849 [16] Luís Tiago Paiva, Fernando A. C. C. Fontes. Sampled–data model predictive control: Adaptive time–mesh refinement algorithms and guarantees of stability. Discrete and Continuous Dynamical Systems - B, 2019, 24 (5) : 2335-2364. doi: 10.3934/dcdsb.2019098 [17] Lin Lu, Qi Wang, Yongzhong Song, Yushun Wang. Local structure-preserving algorithms for the molecular beam epitaxy model with slope selection. Discrete and Continuous Dynamical Systems - B, 2021, 26 (9) : 4745-4765. doi: 10.3934/dcdsb.2020311 [18] Xin Du, M. Monir Uddin, A. Mostakim Fony, Md. Tanzim Hossain, Md. Nazmul Islam Shuzan. Iterative Rational Krylov Algorithms for model reduction of a class of constrained structural dynamic system with Engineering applications. Numerical Algebra, Control and Optimization, 2021  doi: 10.3934/naco.2021016 [19] Gilles Carbou, Bernard Hanouzet. Relaxation approximation of the Kerr model for the impedance initial-boundary value problem. Conference Publications, 2007, 2007 (Special) : 212-220. doi: 10.3934/proc.2007.2007.212 [20] Pierre Degond, Maximilian Engel. Numerical approximation of a coagulation-fragmentation model for animal group size statistics. Networks and Heterogeneous Media, 2017, 12 (2) : 217-243. doi: 10.3934/nhm.2017009

2020 Impact Factor: 1.392


text:
# Center of Gravity

1. Nov 20, 2007

### Superfluous

The center of gravity of an irregular object of mass 4.50 g is shown in the figure. You need to move the center of gravity 2.00 cm to the left by gluing on a tiny 1.60-g mass, which will then be considered as part of the object. Where should you attach this additional mass? Express your answer in cm to the left of the original center of gravity.

Well, to tell the truth, I don't know how to go about solving this at all. My physics instructor skipped over the section that included this, but still assigned the homework. The chapter that this problem is in includes a bunch of rigid-body equilibrium problems, which I have been solving by using Newton's laws and summing up torques, but this problem has really thrown me for a loop. Would someone please describe the process I use to solve this type of problem, or the relevant equations even?

Thanks.

2. Nov 20, 2007

### TVP45

Can you think of anything unique or special about the CG?

3. Nov 20, 2007

### Superfluous

No, unforunately I do not understand what you're trying to point me towards.

4. Nov 20, 2007

### Staff: Mentor

What is the relevant equation here -- the equation for the center of mass of a set of objects?

5. Nov 20, 2007

### Superfluous

I am happy to report that I figured it out. I guess taking a break for awhile and coming back to it can do wonders. I re-read the section and took a stab at it:

$$\frac{m_{1}(0)+m_{2}(x)}{m_{1}+m_{2}}=2$$

Solved for x:

$$x=\frac{2(m_{1}+m_{2})}{m_{2}}$$

Plugged in values... got a result of about 7.63 cm. Seemed reasonable, so I tried it. Turned out to be correct. Thanks to everyone who tried to help.


text:
# Number of 5x5 matrix permutations without repetitions in rows or columns

Context

In the boardgame Azul, your goal is to complete as much as possible of a $$5\times5$$ board by placing 25 tiles of 5 different colours (5 tiles of each colour) so that no colour appears twice in a row or column. For the normal mode, the tiles must be placed following a predefined pattern, which can be seen here and that I represent with the following matrix $$P$$, where each letter represents a different colour:

$$P = \begin{bmatrix}a & b & c & d & e\\ e & a & b & c & d\\ d & e & a & b & c \\ c & d & e & a & b \\ b & c & d & e & a\end{bmatrix}$$

The advanced playing mode has no predefined pattern, so you can come up with your own, while respecting the constraint that no colour appears twice in each row or column.

I've realised that I can build valid patterns by permutating the rows and columns of the predefined pattern, as these operation preserve the number of different colours in each row or column. The resulting pattern $$P'$$ can be represented by $$R \times P \times C$$, where $$R$$ and $$C$$ are two permutation matrices indicating the rows and columns to permutate, respectively. For example:

$$P' = \begin{bmatrix} 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 & 0\end{bmatrix} \times \begin{bmatrix} a & b & c & d & e\\ e & a & b & c & d\\ d & e & a & b & c \\ c & d & e & a & b \\ b & c & d & e & a\end{bmatrix} \times \begin{bmatrix} 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1\end{bmatrix} = \begin{bmatrix} a & c & b & d & e\\ d & a & e & b & c\\ b & d & c & e & a\\ c & e & d & a & b\\ e & b & a & c & d\\\end{bmatrix}$$

Which is a valid pattern.

Since there are $$5!$$ permutation matrix, I have managed to create $$(5!)^2 = 14400$$ valid patterns this way, although each pattern appears 5 times, so only 2880 of them are distinct.

Questions

1. Is there a valid pattern that can't be created by the permutation of rows and columns of $$P$$? Does the same answer hold true for matrices of higher order?

For patterns of order 3, I checked and all the valid patterns are permutations of rows and columns of $$P$$, but brute-forcing this doesn't scale particularly well.

2. Given $$P$$ and $$P'$$, how can I find the permutations matrices $$C$$ and $$R$$ that transform $$P$$ into $$P'$$?

• You might want to read up on Latin squares. – Richard Lyons Oct 16 '20 at 16:42

The answer to Question 1 is yes. What you have described is called a Latin square. Two Latin squares are isotopic if one can be obtained from the other by permuting rows, columns, and permuting the names of the symbols. Note that isotopy is stronger than what you describe since we are allowed to permute the names of the symbols. It is known (see OEIS A040082) that there are exactly two isotopy classes of $$5 \times 5$$ Latin squares. Thus, every $$P'$$ which is not in the isotopy class of your $$P$$ cannot be obtained from $$P$$ by permuting rows and columns. Of course, this also holds for larger Latin squares, since the number of isotopy classes increases with $$n$$.


text:
# What does the value of reactive power represent physically?

In my power engineering course, we were introduced to reactive power. I believe I understand what reactive power is in essence: there is energy transfer between the source and inductor/capacitor. This makes sense to me, and I can do many calculations just fine. However, I am failing to understand what the value of reactive power represents. From my notes:

\begin{align*} P=VI\cos\psi \\ Q=VI\sin\psi \end{align*} where P and Q are active and reactive power respectively. I understand the formula for P, and its derivation is presented in my notes. Physically, P represents the amount of energy dissipated through the resistor. What does Q represent?

Considering the graphs of power vs time in capacitor/inductor only circuits, the average power is 0. Shouldn't this mean that on average reactive power is just 0?

Apologies if the question is a bit unclear, I am having a difficult time just explaining my confusion.

• So Q is a quantification of the power transferred between source and load. Positive, power is absorbed by device, negative, returned due to collapse of magnetic or electric field, – StainlessSteelRat Apr 18 '17 at 16:15
• You can also look at this from an energy view point. Reactive power is the work that is done to establish and collapse magnetic and electric fields in components (specifically inductors and capacitors). This power is "not real" in the sense it does no useful work, but is required to be put into the system to make it operational to begin with. – Envidia Apr 18 '17 at 16:50
• Just to add another analogy. Think of walking across a long trampoline. If you go by leaps and bounds, you will be transferring energy between sprung weight on the trampoline (call that electrostatic potential stored in capacitance) and potential energy in the air when out of contact with the surface (call that magnetic potential stored in inductance.) Either way, the energy needed to get to the end is the same, whether you walk or bounce your way. But the headroom you need is different, depending. – jonk Apr 18 '17 at 16:57
• Reactive power makes for a great beer analogy – Mast Apr 18 '17 at 22:12
• 0 on average is very different from just 0. AC voltage and current are also 0 on average. – Dmitry Grigoryev Apr 19 '17 at 8:44

Let's ignore the power aspect for a second, and think about what reactance really is.

You know a math and theory, you can talk about things in abstract theoretical constructs using complex numbers, phasors, and all that. But abstract models are exactly that: abstract models. The math can model a thing, but it won't really help you understand the physical system it is modeling except in other abstract ways.

So let's ignore the math for a second, and talk about what reactance really is.

Actually, let's talk about resistance. The real component to impedance. Ultimately, resistance represents energy loss. Resistance consumes some of the kinetic energy of electrons moving through the circuit, and that manifests as the familiar ohmic voltage drop we see across any resistive load. Electrons smack into stuff, set them vibrating, and the resistive load heats up as joules are lost by the electrons and transferred into the load. The faster the rate of energy that moves through this load, the faster the rate of power lost, and the harder you have to push to make that happen.

But that's just one side of the coin. Besides simply dissipating energy to the environment, there is another option that can happen: energy can be stored. Capacitance and inductance are often talked about as being 'duals' of each other because they're both measures of energy storage. Capacitance is a measure of energy stored in an electric field, while inductance is a measure of energy stored in a magnetic field.

Energy being stored looks just like energy being dissipated, at least at first. In both cases, energy that was in the circuit is no longer present. The only difference between resistance and reactance here is that with resistance, that energy is gone for good, but reactance will eventually return that energy back to the circuit at a later time. Well, and of course as a measure of storage, they eventually reach a maximum amount of storage given a static circuit. A capacitor will need a higher voltage to store more energy, an inductor likewise will need higher current to store more energy. This is the 'reactance' aspect. As energy is stored, less power is seemingly dissipated by this reactance until it vanishes entirely. If the power starts going down, the stored energy is released back into the circuit.

So what is apparent power? It's simply the rate at which a circuit or part of a circuit (depending on what you're calculating for/looking at) is storing energy, or, if the magnitude is opposite, the rate at which it is releasing energy. That's all. It's not weird, or strange, and it is a real, physical, quantifiable thing. If you charge a massive capacitor bank, from a battery, it will consume joules from that battery, and it will do so at a certain rate, one that is highest at first, but will eventually fall to zero. This is, technically, reactive power. But it's still measured in watts, and watts are, well, always watts. You just are measuring the rate at which something stored joules, rather than the rate at which it simply dissipated them.

Your confusion I think is that you've actually sort of arrived at the answer already without realizing it. If you have a circuit with only capacitors and inductors, then there is no 'P' as there is no energy being dissipated at some number of joules per second. There is only energy being stored, and it will eventually get released, and so yes, it averages to 0. Reactive power always does. It's ultimately just storage, not consumption, so yes, the it will always always always average to 0. Those joules got loaned out, but inductors and capacitors have terrific credit ratings and always pay you back eventually, so you didn't actually lose any money/joules in the long run.

So, you don't need to talk about this in terms of math at all. In fact, if you understand that reactive power is simply the rate energy is being stored and then released, measured in joules per second or watts like anything else involving power, then the behavior and math should all just make logical sense, because that's ultimately what you are modeling with said math.

Now, one might wonder why reactive power even matters if it averages to zero.

Let's talk power factor real quick. Power factor is, of course, the ratio of real to apparent power. This might seem like a rather strange or pointless thing to have a ratio of. I mean, who cares? The apparent power isn't actually being lost, why even measure it?

The issue is that this energy storage is never (excepting in the case of perhaps superconductors) totally efficient. Electrons have to move onto the negative plate of a capacitor, while an equal number of electrons are pushed off the positive plate. Moving charge is current. Conductors (again, except in the case of superconduction) always have some resistance, so you have losses. In the context of alternating current, where storing energy will have a profound effect in this regard, you wind up having electrons flowing in and out over and over, uselessly storing energy for no reason. So even though the energy is getting returned to the circuit, you're still suffering losses in the form of current flowing but without it doing any work. Really, the idea of current and voltage phase is just a way of looking at how reactance is effectively dropping voltage but because it is storing energy, or maintaining voltage (or increasing it to maintain current instead), by releasing energy.

Never forget that important concept, that all of this is ultimately just different sort of abstract ways of looking at or modeling one true, physical process going on, which is actually very simple at its core. The storage of energy, and that there are two different fields with which it can be stored. From that concept, everything else can be derived.

• So does adding a thermo-electric generator next to a resister turn it into a reactor? – JDługosz Apr 19 '17 at 6:00
• This talk of energy storage, while correct, can be a bit misleading in this context. You really should point out that the energy is being stored and unstored twice per cycle. This is not what is meant by storage on the power grid. It's not like pumping water uphill during low demand times, then using it to run generators during peak times, for example. – Olin Lathrop Apr 19 '17 at 10:38
• "reactive power is simply the rate energy is being stored and then released, measured in joules per second or watts like anything else involving power" That's instantaneous reactive power, not reactive power. The absolute value of reactive power is the amplitude of that oscillation of energy. – Alejandro Nava Oct 18 '20 at 4:52

When you have a complex load, its associated power can be modeled with a complex number:

$$S = P+jQ$$

Where S is the complex power, usually measured in VA, P is the real power, measured in W, and Q is the reactive power, measured in VAR.

As op noted, the only "useful" power is the real power P, since it is the only power dissipated in the load that can do useful work.

Suppose you have a purely capacitive load, and a sinusoidal power source, and you need to size the conductors from the source to the capacitor. You calculate the real power, and it is zero! You use very thin wires to connec the cap and... BANG they vaporize. What happened?

The reactive power Q, even if it is not dissipated, must be carried to and from the load and must be taken in account when sizing all the components making a power transmission line. Since usually $P\neq0$, to size a power line power engineers use apparent power, i.e. the magnitude of S, or $\sqrt{P^2+Q^2}$.

To answer your question, Q physically represents the power that is continuously "bounced" between the source and the load, and it is fundamental to size conductors, transformers, switches and everything that constitutes a power line.

"Reactive power" is one of several possible ways of reconciling the fact that in AC systems, the voltage times the current isn't the average power.

The reason rms(V) x rms(I) is not ave(V x I) is phase shift. When V and I have the same phase, then rms(V) x rms(I) is ave(V x I), which is the power delivered. As the phase between V and I increases, the power delivered goes down while rms(V) and rms(I) stay the same.

This can be accounted for by tracking the phase angle, looking at V or I as a phaser, or treating the 0° and 90° parts separately.

Reactive power is basically a bookkeeping method that accounts for the 0° and 90° components of the current separately. Since the phase shift of voltage and current is only relative to each other, you can pick one as the reference and consider the other phase shifted relative to that reference. In the reactive power scheme, the voltage is the reference and the current is shifted in phase relative to the voltage.

Note that the phase shift of the current can be expressed as a angle, or equivalently, as the 0° and 90° components. Real power is the voltage times the 0° current component, and reactive power the voltage times the 90° current component.

You are right in that reactive power doesn't actually deliver any power. The confusion is that the word "power" is applied. Calling it the "reactive VA" might be more accurate, but right or wrong, the industry has converged on the term "reactive power".

This bookkeeping scheme has some useful properties. By thinking of and accounting for the real and reactive power separately, they can more easily be dealt with separately. For example, capacitors are reactive power generators. This may sound odd at first glance, but once you buy into the definition of reactive power, that's what capacitors are. Note that inductors are therefore reactive power consumers. The electric grid usually looks a little inductive, so "reactive power" has been defined to be positive to feed this inductance.

Capacitors really do offset the phase shift caused by inductors regardless of what bookkeeping convention is used. When dealing with the electric grid, the "reactive power" bookkeeping method has proven to be useful in quickly evaluating what is going on and what is required to offset reactive loads.

For example, you could describe a situation as 101.5 MVA with 10° lagging phase, or you could describe it as 100 MW real power and 17.4 MW reactive power. The two are equal mathematically. However, the latter is usually a more immediately useful expression when dealing with this situation in the power utility world.

• Thank you very much for the response. The mathematical equivalence you mentioned at the end makes a lot of sense now. One thing I don't see is is why the reactive power can be positive or negative, since it refers to a zero-sum transfer of energy between the load and the source? If it is going back and forth in equal amounts, what makes an inductive load 'positive' and a capacitive load 'negative'? I will try to understand this, thanks very much! – masiewpao Apr 18 '17 at 17:05
• @masi: Again, it's not really power, more like VA. Inductors and capacitors have 90 deg phase shift in current with respect to voltage, but each with opposite sign. Since reactive power is only concerned with the current component along the 90 deg axis, capacitors and inductors will "produce" opposite polarities of reactive power. By convention, capacitors "generate" positive reactive power and inductors negative. That's another way of saying inductors are reactive power loads. This polarity convention is used because that's the most common situation on real power lines. – Olin Lathrop Apr 18 '17 at 17:11
• I strongly disagree with this answer. Reactive power is a measure of a real, physical process and is not at all a 'book keeping method' and does not exist to 'reconcile' anything. Reactive power is a measure of the rate of energy storage (or energy release when it is negative). It takes time to store energy, and we measure that in watts, and they are watts like any other watts. Unless you're saying no energy is stored in electric fields or magnetic fields, then you cannot say that apparent power is just some sort of theoretical convenience. – metacollin Apr 18 '17 at 17:49
• @meta: But this energy storage you talk about gets filled and unfilled twice per power line cycle. The storage is so short term, it's really more of a sloshing of energy back and forth. I didn't say reactive power isn't real, but it is one of several possible ways to describing the same thing. Presenting it as a bookkeeping scheme I think is a useful way to help people understand it. Your bookkeeping scheme could be based on phase angles and be just as valid, but in this case, less immediately useful to the utility power industry. – Olin Lathrop Apr 18 '17 at 18:19
• @Olin: Thank you, I think I understand what you mean. Essentially we have chosen a convenient sign convention. – masiewpao Apr 18 '17 at 18:50

Reactive power $Q$ is a measure of the energy $E$ that is oscillating between the generator and the load without loss. At any instant energy might be moving from load to generator or the other way, and the energy stored in the load will vary over time. But the sum of the energy in both is constant, since by definition there's no loss.

Specifically:

$$E = {Q \over 2 \pi f}$$

or

$$Q = 2\pi fE$$

where $f$ is the frequency.

If you follow the convention of using a negative reactance for capacitors, you might end up with a "negative power". But this is just a mathematical convention to distinguish between a 90° or a -90° phase shift; either way the total energy oscillating between the load and generator is the same.

Derivation:

Let's say the load is just an ideal inductor.

The reactance $X$ will tell us the ratio of RMS voltage $V$ to current $I$:

$$X = {V/I} \tag 1$$

And reactive power $Q$ is just the product of voltage and current since our load is purely reactive:

$$Q = V I \tag 2$$

Reactance is related to frequency $f$ and inductance $L$:

$$L = {X \over 2 \pi f} \tag 3$$

Combine (1) and (2) to get:

$$I = \sqrt{Q/X} \tag 4$$

Energy $E$ stored in an inductor is a function of the inductance and current $I$:

$$E = 1/2\: LI^2 \tag 5$$

When current is at a peak, so too is the stored energy in the load inductor. We can convert RMS current to a peak instantaneous current like so:

$$I_\text{peak} = \sqrt{2} \cdot I_\text{RMS} \tag 6$$

Let's combine these equations, starting with (5):

$$E = {1\over 2}\: LI_\text{peak}^2$$

Sub in equations 3 and 6:

$$E = {1\over 2}\: {X \over 2 \pi f}\: (\sqrt{2}\: I_\text{RMS})^2$$

Simplify:

$$E = {1\over 2}\: {X \over 2 \pi f}\: 2I_\text{RMS}^2$$

Ohm's law or (4):

$$E = {1\over 2}\: {X \over 2 \pi f}\: 2(\sqrt{Q/X})^2$$

Simplify:

$$\require{cancel} E = {1\over 2}\: {X \over 2 \pi f}\: {2Q\over X } \ = {Q \over 2\pi f}$$

Again, that's the peak energy stored in our load inductor, a purely reactive load.

By laws of energy conservation it should be apparent that when the energy stored in the inductor is at a maximum, energy stored in the generator is zero, and energy can't go anywhere else, so this is also at any time the total energy in the system.

• I bet there's a relation between reactive power and the rate of change of stored energy in the inductor as well, though I don't have time to derive that. – Phil Frost Apr 18 '17 at 18:49
• Side note: Equation #5 could be misleading because you're using the same notation for instantaneous current that you used for RMS current in equations #1, #2, and #4. Also, in equation #5 you wrote instantaneous energy as E, but in the following equations you used peak energy yet still used the letter E (my suggestion is to change the latter to E_peak, although at the end of the derivation you clarified it was peak energy.) – Alejandro Nava Oct 18 '20 at 5:08

The simplest way to think about reactive power is to realize that some kinds of a load will spend a portion of each cycle taking energy from the line in excess of what they'll use, and then spend another portion of each cycle giving the excess energy back. If a load consists of a passive network of resistors, inductors, and capacitors, then any time the reactive component of instantaneous power is positive, the device will be absorbing energy that it is ultimately going to return, and whenever the reactive component is negative it will be returning power that it had previously stored up.

• That makes sense! In the RLC circuit, if the source has constant frequency, then won't the phase angle be constant? Then from equation Q=VI*(psi), Q will be a constant value, either positive or negative. I'm getting confused because the energy is meant to be returned (or absorbed), and this is associated with the sign of Q. But if Q is constant and not changing, how does it account for the energy 'sloshing' between the source and load! – masiewpao Apr 18 '17 at 18:56
• @masiewpao: Both P and Q describe long-term averages or, if preferred, an average over any integer number of cycles, or the energy used or transferred in any one of many equal cycles. – supercat Apr 18 '17 at 19:11
• I thought I understood this, but I've looked at it again and I don''t think I do. The long term average of P is just the integral of vi over 2pi. But in terms of Q, because the power curve is a sine wave, any average over its cycle comes out to zero. I tried to derive equation for Q but integrating from 0 to pi/2 instead (I thought this may be valid since Q represents the rate of energy absorption/release, so integrating over half a cycle made sense to me). The result however is still a cos term, and not a sin term like it should be. Any idea why? – masiewpao Apr 19 '17 at 12:38
• @masiewpao: I should have clarified; since P is in-phase voltage times in-phase current, the instantaneous value is always positive, and the long-term average will be half the peak value. For Q, what's of interest is the average absolute value, since as noted the positive portion and negative portion will precisely cancel. – supercat Apr 19 '17 at 14:08

Physically represent while still taking about electricity is a tough one. If you can accept an analogy, this is my favorite explination of reactive power it you look at it from the utility companies perspective:

• The beer analogy is wrong. It suggests apparent power is the sum of reactive power and active power, but actually is the square root of the sum of the squares of these two powers. – Alejandro Nava Oct 18 '20 at 4:43
• @AlejandroNava That’s correct. It’s not to scale but shows that real power is always equal or smaller than apparent power. – winny Oct 18 '20 at 7:32

What does the value of reactive power represent physically?

What does Q represent?

I already explained it in the physics Stack Exchange, here. But I can elaborate more if you want, just ask in the comments.

The absolute value of reactive power is the maximum/minimum rate of transfer of energy between source and load. The sign of reactive power indicates the instantaneous direction of flow of energy due to the reactive component of current.


text:
MAT331 homework problems
17.
(expires 18 March)     Consider the differential equation , where the vector and the field . Plot a few solutions. What happens to them when ? Give a Maple-proof'' that this is a general fact for every solution. [A Maple-proof'' is an argument that is rigorous once we accept Maple results as incontrovertibly true.]

18.
(expires 18 March)    (No Maple.) For the equation , , with the vector field

prove that the origin is an attractor in the future, i.e., every solution verifies

[You can ask around how to do this, but then you have to show clearly that you have understood it.]

19.
(expires 18 March)     We will study the Lotke-Volterra predator-prey equations: In a very simple ecosystem, at the time (which is expressed, say, in years), there is a population of foxes and rabbits. The evolution of these quantities obeys the system

where and are the growth rates for the foxes and the rabbits, respectively, in the absence of each other. is the probability of a fatal encounter between a fox and a rabbit (normalized per number of foxes and rabbits).

First, write some words to explain why these equations make sense. Then, fix , (it's notorius that rabbits have the tendency to reproduce quickly) and . For a few initial conditions of your choice, plot the trajectories in the -plane (say, with and ). For the same initial conditions, plot the actual solutions too (i.e, against , and against ). Write some comments interpreting how the behaviour of the solutions relates to what happens to the two species.

Finally, repeat the same procedure with . Things change substantially. Again, what is the physical'' interpretation of this?


text:
# Hermitian¶

Module: pennylane

class Hermitian(A, wires)[source]

An arbitrary Hermitian observable.

For a Hermitian matrix $$A$$, the expectation command returns the value

$\braket{A} = \braketT{\psi}{\cdots \otimes I\otimes A\otimes I\cdots}{\psi}$

where $$A$$ acts on the requested wires.

If acting on $$N$$ wires, then the matrix $$A$$ must be of size $$2^N\times 2^N$$.

Details:

• Number of wires: Any
• Number of parameters: 1


text:
# IEEE Transactions on Information Theory

## Filter Results

Displaying Results 1 - 25 of 40

Publication Year: 2009, Page(s):C1 - C4
| PDF (51 KB)
• ### IEEE Transactions on Information Theory publication information

Publication Year: 2009, Page(s): C2
| PDF (42 KB)
• ### Interference Alignment and the Degrees of Freedom of Wireless $X$ Networks

Publication Year: 2009, Page(s):3893 - 3908
Cited by:  Papers (237)  |  Patents (3)
| | PDF (988 KB) | HTML

We explore the degrees of freedom of M times N user wireless X networks, i.e., networks of M transmitters and N receivers where every transmitter has an independent message for every receiver. We derive a general outer bound on the degrees of freedom region of these networks. When all nodes have a single antenna and all channel coefficients vary in time or frequency, we show that the total ... View full abstract»

• ### Cooperative Multiplexing: Toward Higher Spectral Efficiency in Multiple-Antenna Relay Networks

Publication Year: 2009, Page(s):3909 - 3926
Cited by:  Papers (11)
| | PDF (523 KB) | HTML

Previous work on cooperative communications has concentrated primarily on the diversity benefits of such techniques. This paper, instead, considers the multiplexing benefits of cooperative communications. First, a new interpretation on the fundamental tradeoff between the transmission rate and outage probability in multiple-antenna relay networks is given. It follows that multiplexing gains can be... View full abstract»

• ### Source and Channel Coding for Correlated Sources Over Multiuser Channels

Publication Year: 2009, Page(s):3927 - 3944
Cited by:  Papers (40)
| | PDF (336 KB) | HTML

Source and channel coding over multiuser channels in which receivers have access to correlated source side information are considered. For several multiuser channel models necessary and sufficient conditions for optimal separation of the source and channel codes are obtained. In particular, the multiple-access channel, the compound multiple-access channel, the interference channel, and the two-way... View full abstract»

• ### Cognitive Radio: An Information-Theoretic Perspective

Publication Year: 2009, Page(s):3945 - 3958
Cited by:  Papers (274)  |  Patents (3)
| | PDF (345 KB) | HTML

In this paper, we consider a communication scenario in which the primary and the cognitive radios wish to communicate to different receivers, subject to mutual interference. In the model that we use, the cognitive radio has noncausal knowledge of the primary radio's codeword. We characterize the largest rate at which the cognitive radio can reliably communicate under the constraint that 1) no View full abstract»

• ### On Capacity Scaling in Arbitrary Wireless Networks

Publication Year: 2009, Page(s):3959 - 3982
Cited by:  Papers (56)
| | PDF (581 KB) | HTML

In recent work, Ozgur, Leveque, and Tse (2007) obtained a complete scaling characterization of throughput scaling for random extended wireless networks (i.e., n nodes are placed uniformly at random in a square region of area n). They showed that for small path-loss exponents alpha isin (2,3], cooperative communication is order optimal, and for large path-loss exp... View full abstract»

• ### Parallel Gaussian Interference Channels Are Not Always Separable

Publication Year: 2009, Page(s):3983 - 3990
Cited by:  Papers (51)
| | PDF (244 KB) | HTML

It is known that the capacity of parallel (multicarrier) Gaussian point-to-point, multiple access and broadcast channels can be achieved by separate encoding for each subchannel (carrier) subject to a power allocation across carriers. In this paper we show that such a separation does not apply to parallel Gaussian interference channels in general. A counterexample is provided in the form of a 3 us... View full abstract»

• ### Rate Region of Gaussian Multiple Description Coding With Individual and Central Distortion Constraints

Publication Year: 2009, Page(s):3991 - 4005
Cited by:  Papers (22)  |  Patents (1)
| | PDF (351 KB) | HTML

The rate region of Gaussian multiple description coding with individual and central distortion constraints is completely characterized. Specifically, a lower bound and an upper bound are derived for each supporting hyperplane of the rate region, where the lower bound is associated with a max-min game while the upper bound is associated with a min-max game; furthermore, it is shown that these two b... View full abstract»

• ### On the Duality Between Slepian–Wolf Coding and Channel Coding Under Mismatched Decoding

Publication Year: 2009, Page(s):4006 - 4018
Cited by:  Papers (6)
| | PDF (304 KB) | HTML

In this paper, Slepian-Wolf coding with a mismatched decoding metric is studied. Two different dualities between Slepian-Wolf coding and channel coding under mismatched decoding are established. These two dualities provide a systematic framework for comparing linear Slepian-Wolf codes, nonlinear Slepian-Wolf codes, and variable-rate Slepian-Wolf codes. In contrast with the fact that linear codes s... View full abstract»

• ### Precoding for the AWGN Channel With Discrete Interference

Publication Year: 2009, Page(s):4019 - 4032
Cited by:  Papers (2)
| | PDF (407 KB) | HTML

For a state-dependent discrete memoryless channel with input alphabet X, state alphabet S, and output alphabet Y where the independent and identically distributed (i.i.d.) state sequence is known causally at the transmitter, it is shown that by using at most min{|X||S|-|S| + 1,|Y|} out of |X||S| inputs of the Shannon's derived channel, the capacity is achievable. As... View full abstract»

• ### Towards the Secrecy Capacity of the Gaussian MIMO Wire-Tap Channel: The 2-2-1 Channel

Publication Year: 2009, Page(s):4033 - 4039
Cited by:  Papers (161)
| | PDF (164 KB) | HTML

We find the secrecy capacity of the 2-2-1 Gaussian MIMO wiretap channel, which consists of a transmitter and a receiver with two antennas each, and an eavesdropper with a single antenna. We determine the secrecy capacity of this channel by proposing an achievable scheme and then developing a tight upper bound that meets the proposed achievable secrecy rate. We show that, for this channel, Gaussian... View full abstract»

• ### Optimality of Myopic Sensing in Multichannel Opportunistic Access

Publication Year: 2009, Page(s):4040 - 4050
Cited by:  Papers (173)  |  Patents (1)
| | PDF (230 KB) | HTML

This paper considers opportunistic communication over multiple channels where the state (ldquogoodrdquo or ldquobadrdquo) of each channel evolves as independent and identically distributed (i.i.d.) Markov processes. A user, with limited channel sensing capability, chooses one channel to sense and decides whether to use the channel (based on the sensing result) in each time slot. A reward is obtain... View full abstract»

• ### End-to-End Energy–Bandwidth Tradeoff in Multihop Wireless Networks

Publication Year: 2009, Page(s):4051 - 4066
Cited by:  Papers (54)
| | PDF (1726 KB) | HTML

In this paper, energy-constrained wireless multihop networks with a single source-destination pair are considered. A network model that incorporates both the energy radiated by the transmitter and the energy consumed by the circuits that process the received signals is proposed. The rate of communication is the number of information bits transmitted (end-to-end) per coded symbol transmitted by any... View full abstract»

• ### Interference and Outage in Clustered Wireless Ad Hoc Networks

Publication Year: 2009, Page(s):4067 - 4086
Cited by:  Papers (152)
| | PDF (1271 KB) | HTML

In the analysis of large random wireless networks, the underlying node distribution is almost ubiquitously assumed to be the homogeneous Poisson point process. In this paper, the node locations are assumed to form a Poisson cluster process on the plane. We derive the distributional properties of the interference and provide upper and lower bounds for its distribution. We consi... View full abstract»

• ### Queue Back-Pressure Random Access in Multihop Wireless Networks: Optimality and Stability

Publication Year: 2009, Page(s):4087 - 4098
Cited by:  Papers (27)
| | PDF (697 KB) | HTML

A model for wireless networks with slotted-Aloha-type random access and with multihop flow routes is considered. The goal is to devise distributed algorithms for utility-optimal end-to-end throughput allocation and queueing stability. A class of queue back-pressure random access algorithms (QBRAs), in which actual queue lengths of the flows in each node's close neighborhood are used to determine t... View full abstract»

• ### Analysis of Connections Between Pseudocodewords

Publication Year: 2009, Page(s):4099 - 4107
Cited by:  Papers (12)
| | PDF (234 KB) | HTML

The role of pseudocodewords in causing non-codeword outputs in linear programming decoding, graph cover decoding, and iterative message-passing decoding is investigated. The three main types of pseudocodewords in the literature-linear programming pseudocodewords, graph cover pseudocodewords, and computation tree pseudocodewords-are reviewed and connections between them are explored. Some discrepan... View full abstract»

• ### Error Exponents for Two Soft-Decision Decoding Algorithms of Reed–Muller Codes

Publication Year: 2009, Page(s):4108 - 4118
Cited by:  Papers (1)
| | PDF (281 KB) | HTML

Error exponents are studied for recursive and majority decoding of general Reed-Muller (RM) codes RM(r, m) used on the additive white Gaussian noise (AWGN) channels. Both algorithms have low complexity and correct many error patterns whose weight exceeds half the code distance. Decoding consists of multiple consecutive steps, which repeatedly recalculate the input symbols and determi... View full abstract»

• ### Rateless Coding for Arbitrary Channel Mixtures With Decoder Channel State Information

Publication Year: 2009, Page(s):4119 - 4133
Cited by:  Papers (10)
| | PDF (356 KB) | HTML

Rateless coding has recently been the focus of much practical as well as theoretical research. In this paper, rateless codes are shown to find a natural application in channels where the channel law varies unpredictably. Such unpredictability means that to ensure reliable communication block codes are limited by worst case channel variations. However, the dynamic decoding nature of rateless codes ... View full abstract»

• ### Linear-Programming Decoding of Nonbinary Linear Codes

Publication Year: 2009, Page(s):4134 - 4154
Cited by:  Papers (38)  |  Patents (1)
| | PDF (715 KB) | HTML

A framework for linear-programming (LP) decoding of nonbinary linear codes over rings is developed. This framework facilitates LP-based reception for coded modulation systems which use direct modulation mapping of coded symbols. It is proved that the resulting LP decoder has the ldquomaximum-likelihood (ML) certificaterdquo property. It is also shown that the decoder output is the lowest cost pseu... View full abstract»

• ### Single-Exclusion Number and the Stopping Redundancy of MDS Codes

Publication Year: 2009, Page(s):4155 - 4166
Cited by:  Papers (4)
| | PDF (421 KB) | HTML

For a linear block code C, its stopping redundancy is defined as the smallest number of check nodes in a Tanner graph for C, such that there exist no stopping sets of size smaller than the minimum distance of C. Schwartz and Vardy conjectured that the stopping redundancy of a maximum-distance separable (MDS) code should only depend on its length and minimum distance. We define the (n, t)-single-ex... View full abstract»

• ### An Efficient Algorithm to Find All Small-Size Stopping Sets of Low-Density Parity-Check Matrices

Publication Year: 2009, Page(s):4167 - 4178
Cited by:  Papers (29)
| | PDF (358 KB) | HTML

In this work, we introduce an efficient algorithm to find all stopping sets, of size less than some threshold, of a fixed low-density parity-check (LDPC) matrix. The solution is inspired by the algorithm proposed by Rosnes and Ytrehus in 2005 to find an exhaustive list of all small-size turbo stopping sets in a turbo code. The efficiency of the proposed algorithm is demonstrated by several numeric... View full abstract»

• ### List Decoding of Burst Errors

Publication Year: 2009, Page(s):4179 - 4190
Cited by:  Papers (4)
| | PDF (265 KB) | HTML

A generalization of the Reiger bound is presented for the list decoding of burst errors. It is then shown that Reed-Solomon codes attain this bound. View full abstract»

• ### Maximum-Likelihood Priority-First Search Decodable Codes for Combined Channel Estimation and Error Correction

Publication Year: 2009, Page(s):4191 - 4203
Cited by:  Papers (3)
| | PDF (850 KB) | HTML

The coding technique that combines channel estimation and error correction has received attention recently, and has been regarded as a promising approach to counter the effects of multipath fading. It has been shown by simulation that a proper code design that jointly considers channel estimation can improve the system performance subject to a fixed code rate as compared to a conventional system w... View full abstract»

• ### Improvement of Gallager Upper Bound and its Variations for Discrete Channels

Publication Year: 2009, Page(s):4204 - 4210
| | PDF (222 KB) | HTML

A new tight upper bound on the maximum-likelihood (ML) word and bit-error decoding probabilities for specific codes over discrete channels is presented. It constitutes an enhanced version of the Gallager upper bound and its variations resulting from the Duman-Salehi second bounding technique. An efficient technique is developed that, in the case of symmetric channels, overcomes the difficulties as... View full abstract»

## Aims & Scope

IEEE Transactions on Information Theory publishes papers concerned with the transmission, processing, and utilization of information.

Full Aims & Scope

## Meet Our Editors

Editor-in-Chief
Prakash Narayan

Department of Electrical and Computer Engineering


text:
## Saturday, December 08, 2012 ... //

### Last man on the Moon: 40 years ago

Some technologies such as the communication technologies have experienced unbelievable progress which seems to be continuing.

But there are some technological challenges in which the mankind has apparently become weaker. We may argue hundreds of times that it's silly and unscientific to send the humans into the outer space. But aren't you worried that we may have grown unable – or unwilling so that it's effectively equivalent to unable – to send the people to the Moon? Isn't it obvious that such a fading potency affects more useful dreams as well?

In total, 12 Americans have walked on the Moon – and no one else. The last folks walking on the Moon went there with the Apollo 17 mission that was launched on December 7th, 1972, i.e. fourty years ago. It's a pretty long time ago, isn't it? I wasn't born yet. In twelve days, it landed back on Earth.

The Apollo 17 crew consisted of Commander Eugene Čerňan and pilots (of two different kinds) Ronald Evans and (global warming skeptic) Harrison Schmitt. Schmitt has been the last man who arrived to the Moon; Čerňan was the last man walking on the Moon in general.

As you may have noticed, I am writing Čerňan [translated as a "Blackie" of a sort] with the Czechoslovak diacritical signs. He brought the first Czechoslovak flag to the Moon. He didn't forget to bring the flag back to Earth so it's located in a Czech observatory these days. On the other hand, he did forget his camera on the Moon. ;-)

His father was originally Slovak and his mother was Czech. Interestingly enough, Vladimír Remek, the world's first non-Soviet non-American astronaut, also had a Slovak father and a Czech mother.

In a hybrid jargon, both of them were "hinnies". That's somewhat peculiar because the stereotypical couples – especially in TV News – consisted of Czech males and Slovak females so they could give rise to "mules" rather than "hinnies". But as you can see, "hinnies" are more genetically ready to become astronauts. :-)

I think it would be good if the Moon – and perhaps Mars and Mercury whose poles could actually be more life-friendly than Mars – had some base that would make those places more controllable, at least as tools to communicate with/via the planets and moons and, perhaps later, to use them for transfer flights or even to mine natural resources. Much of it is the music of a distant future but I do feel that we're getting further from such goals, not closer to them. Of course, Cernan is also annoyed by the faded prestige of NASA. Incidentally, he opened a Czech community center in Chicagoland a few weeks ago.

Golden Spike, a company in Colorado, plans to make routine flights to the Moon. You just pay a symbolic amount of \$1.4 billion and a round-trip ticket is yours. I have certain doubts whether they will manage to fulfill these commitments anytime soon – and whether they will find some passengers at all even if they're successful – but I would love to be proved wrong.

#### snail feedback (7) :

Perhaps NASA is looking at it from the long term historical point of view. 12 is a number with religious and astrological connotations. In 2000+ years in the future they could be seen as the "12 apostles of the Space Age" or something similar. Who knows what mumbo-jumbo is in the heads of the NASA management.

In view of this animalistic talk of hinnies and mules, allow me to note that Eugene comes from Greek and combines "εὐ" (well, good) with "γενής" (born). While we get made fun of and called geekish or nerdy a lot, we are found not only on the moon but in many languages including Russian, Spanish, and Polish.

(People named Vladimir do not need defending, they are scary and powerful.)

Whole hearted agreement with you to go with my old guy nostalgia. I well remember the first moon landing. It was the first live television broadcast in my home Alaska - provided by our military to us. Then thought bases and colonies were only a few years away.Thank you for reminding me and for teaching also of the Czech heritage and connection.
I would have preferred that my home country never had turned away from the big science. I favor much that politically is not accepted but advance of science, technology or humanity are not always linear, I accept. Humans do change political climates and politicians may prefer to follow the political winds regardless that science would lead elsewhere.
A belated happy birthday.

Sent this yesterday – it vanished!

Having been on Earth (just barely) when the Battle of Monte
Cassino began I am old enough to remember when Sputnik (some say this began the
Space Age) went up. The US programs had
spectacular successes and some serious failures, one being Apollo 1, with its
high pressure pure oxygen atmosphere. It
is disheartening to know that we can no longer do what we paid so dearly for.

I think, maybe, I did not click the correct place.

Apparently (I don't have the references) academic research maps from the just deceased (today) celebrated astronomy presenter of BBC's 50 year old The Sky At Night programmewere used by the Americans and Russians in their lunar space programmes.

Lubos might be interested in the comment war going on at the BBC's site after his death:

http://www.bbc.co.uk/news/uk-20657939

You see, Sir Patrick (in his later years) became very un-pc regarding immigration and "women".

Anyway, mostly the BBC moderators are suppressing the silly comments suggesting his slightly non-pc comments in later life might be more important than a lifetimeof service to science.


text:
# How do I calculate the landing weight in this situation?

Empty weight 538kg
Pilot and passengers are 150kg with full fuel tanks

If we use 2 US gallons for taxi and 10 US gallons for the trip, what will be our landing weight?

• Fuel Capacity
• Total Fuel Quantity: 24.5 US gal. (93.0 liters)
• Usable Fuel: 24.0 US gal. (91.0 liters)
• Unusable Fuel: 0.5 US gal. (2.0 liters)
• Its a bit unclear what you are asking. If 538 KG is your empty weight and 150 KG is your pax weight we would need to know how much fuel the plane holds to get the total weight. What ever the total is you will burn ~32.7 KG's of fuel and can use that for the subtraction. We would need a bit more info to help out. – Dave Dec 3 '18 at 23:53
• Fuel Capacity Total Fuel Quantity: : 24.5 US gal. (93.0 liters) Usable Fuel: : 24.0 US gal. (91.0 liters) Unusable Fuel: : 0.5 US gal. (2.0 liters) – Enki Sumerian Dec 4 '18 at 0:07
• There isn't enough information to calculate the balance, but the weight can be calculated (from your original title) – Ron Beyer Dec 4 '18 at 1:37

## 1 Answer

First calculate your weight on the ramp:

$$\mathrm{Empty~Weight} + \mathrm{Crew} + \mathrm{Fuel} = \mathrm{Ramp~Weight}\\ 538\, \mathrm{kg} + 150\, \mathrm{kg} + 93\, \mathrm{L} \cdot \Bigg[\frac{0.72\, \mathrm{kg}}{\mathrm{L}}\Bigg] = 754.96\, \mathrm{kg}$$

During Taxi, you said 2 gallons of fuel will be used: $$\mathrm{Ramp~Weight} - \mathrm{Taxi~Fuel} = \mathrm{Takeoff~Weight}\\ 754.96\, \mathrm{kg} - 2\, \mathrm{gal} \cdot \Bigg[\frac{2.72\, \mathrm{kg}}{\mathrm{gal}}\Bigg] = 749.52\, \mathrm{kg}$$

Then subtract the weight of the fuel that will be used on the trip: $$\mathrm{Takeoff~Weight} - \mathrm{Trip~Fuel} = \mathrm{Landing~Weight}\\ 749.52\, \mathrm{kg} - 10\, \mathrm{gal} \cdot \Bigg[\frac{2.72\, \mathrm{kg}}{\mathrm{gal}}\Bigg]= 722.32\, \mathrm{kg}$$


text:
# Getting MIDI SoundNote Pitches as Numeric Values

I'm trying to do some evaluation on notes (specifically the pitches) imported from a MIDI file. I've imported a list containing tracks of sound notes with Import["path/to/midi.mid", "SoundNotes"] and a track's sound notes look like the following:

{ SoundNote["G4", {0., 0.5}, "Piano", SoundVolume -> 0.501961],
SoundNote["C5", {0.5, 1.}, "Piano", SoundVolume -> 0.501961],
SoundNote["D5", {1., 1.5}, "Piano", SoundVolume -> 0.501961],
...
}

I'm attempting to get the pitch out of each SoundNote, treating each SoundNote as a list and grabbing the first value with aNote[[1]]. However, this returns a note string such as "C5". Is there a way to get a numeric representation of the pitch, in either MIDI scale (Middle C = 60) or Mathematica's scale (Middle C = 0)? I haven't seen any way to pull information from a SoundNote in the docs, unless I'm missing something.

-
Good question. If my memory serves me right, last time I did this I just created a replacement table "C4" -> 0, etc. and used that. There might be something better though. –  Szabolcs Feb 26 at 0:37

## 1 Answer

If someone has a native solution I'll mark it as the correct answer, but I ended up making a quick function to convert the string into a numeric pitch. I go off the assumption that the MIDI import only uses sharps and no flats as representation for SoundNote pitch.

(* Converts note string into pitch relative to middle C = 0 *)
stringToPitch[string_String] :=
Module[{noteValues, noteList, pitch},
noteValues = {"C", "C#", "D", "D#", "E", "F", "F#", "G", "G#", "A", "A#", "B"};
noteList = StringCases[string, {RegularExpression["[A-G]#?"], RegularExpression["\\d+"]}];
pitch = Position[noteValues, First[noteList]][[1, 1]] - 1;
((ToExpression[noteList[[2]]] - 4)*12) + pitch
]

Note that this is relative to middle C like Mathematica tones (where C4 = MiddleC = 0). The function can be called as

stringToPitch["C#4"]

(* ==> 1 *)

-
+1 for providing a ready to use function. I cleaned it up a bit, I hope you don't mind. It is recommended not to use Return when it's not actually necessary and pitch was not localized. –  Szabolcs Feb 26 at 2:48
Still getting used to the language, thanks for cleaning it up! –  pk-nb Feb 26 at 2:49
Noticed the + pitch was accidentally removed, added it back to make sure the function is still correct. Feel free to further edit if necessary. –  pk-nb Feb 26 at 2:59


text:
# Program to find length of longest chunked palindrome decomposition in Python

PythonServer Side ProgrammingProgramming

Suppose we have a text. We have to find the largest possible k such that there exists a[1], a[2], ..., a[k] such that: Each a[i] is a string which is not blank. And their concatenation a[1] + a[2] + ... + a[k] is equal to the given text; For all i in range 1 to k, a[i] = a[{k+1 - i}].

So, if the input is like text = "antaprezatepzapreanta", then the output will be 11 because we can split it like "a|nt|a|pre|za|tpe|za|pre|a|nt|a".

To solve this, we will follow these steps −

• counter := 0

• i := 1, j := size of text - 1

• ic := 0, jc := size of text

• while i <= j, do

• if substring of text[from index ic to i-1] is same as substring of text[from index j to jc-1], then

• counter := counter + 2

• ic := i

• jc := j

• i := i + 1

• j := j - 1

• if ic is not same as jc, then

• counter := counter + 1

• return counter

## Example

Let us see the following implementation to get better understanding

def solve(text):
counter = 0
i, j = 1, len(text) - 1
ic, jc = 0, len(text)

while i <= j:
if text[ic:i] == text[j:jc]:
counter += 2
ic = i
jc = j
i += 1
j -= 1

if ic != jc:
counter += 1

return counter

text = "antaprezatepzapreanta"
print(solve(text))

## Input

[3,4,5,2,1,7,3,4,7], 3

## Output

11
Published on 08-Oct-2021 07:37:58