text: Hemant. Or, in short: So, for example, if f(x) is some function that is even, then f(2) has the same answer as f(-2). A negative number raised to an odd number will always be negative. These identities will all make appearances in problems that ask you to simplify an expression, prove an identity, or solve an … Test. The even-odd identities relate the value of a trigonometric function at a given angle to the value of the function at the opposite angle and determine whether the identity is odd or even. Played 628 times. Even and odd functions describe different types of symmetry, but both derive their name from the properties of exponents. Is the graph symmetrical as it relates to the origin? Squaring Function. And I want to make it very clear here. y=root(x) neither. Follow 59 views (last 30 days) Cesar Cango on 19 Apr 2016. I'm guessing from context that -oldvar- consists of integers only. For example, $f\left(x\right)={2}^{x}\\$ is neither even nor odd. > is there an odd or even function in fortran 90 > I want to do different statements if a number is odd > or even and functions would be the quickest way to do this . 2 ⋮ Vote. Simplify the new function as much as possible, then compare that to the original function. A function is \"even\" when:f(x) = f(−x) for all xIn other words there is symmetry about the y-axis (like a reflection):This is the curve f(x) = x2+1They got called \"even\" functions because the functions x2, x4, x6, x8, etc behave like that, but there are other functions that behave like that too, such as cos(x): Cosine function: f(x) = cos(x) It is an even function But an even exponent does not always make an even function, for example (x+1)2 is no… Case 2. If you add an odd and an even number, you'll get another odd number. Sine function is odd. We will apply this point of view in Stage 4 when discussing the continuity of polynomials. In part , we combined two functions: the square function and the identity function. 1. The whole point of this video is just to clarify the motivation for calling them even or odd functions. Example: Determine whether the function is even or odd: $f(x) = x^2$ (square function) in $\mathbb{R}$, the calculation is $f(-x) = (-x)^2 = x^2 = f(x)$, so the square function $f(x)$ is even. 2. It is not currently accepting answers. Active 5 years, 6 months ago. Enter a number for check odd or even: 124 124 is even. Even and Odd Functions . Gravity. The ancient Greek geometers only considered angles between 0° and 180°, and they considered neither the straight angle of 180° nor the degenerate angle of 0° to be angles. We also described the first three: Reciprocal Identity, Odd Function/ Even Function Identity, and Pythagoras Formula and Pythagorean Identity in detail with examples. Look for any combination of terms that could give you a Pythagorean identity. I'm not sure you understand what a proof is. Solution of exercise 2. We also managed to define the trig ratios as an even function or odd function. Identity Function fx x()= Odd Bijective Quadratic Function fx x()= 2 Even None. "Even":"Odd"; System.out.println(num+" is "+oddEven); } } When the above code is executed, it produces the following results. Using Division operator To do this, you take the function and plug –x in for x, and then simplify. Odd, even. Rate 5 stars Rate 4 stars Rate 3 stars Rate 2 stars Rate 1 star . Here is the T-SQL docs, but that applies anywhere, not just to T-SQL or computers. Most functions will be neither even nor odd. I just want to know the answer. When we are given the equation of a function f(x), we can check whether the function is even, odd, or neither by evaluating f(-x). Favorite Answer. Mostly, you use even/odd identities for graphing purposes, but you may see them in simplifying problems as well. The Odd-Even Identities cos ( x ) is an even function, sin ( x ) is an odd function as trigonometric functions for real variables. Even and Odd functions and Algebraic and Transcendental functions ... result of applying the operations of addition and multiplication to the set consisting of constant functions and the identity function y = x. The even-odd identities are as follows: sin (– x) = –sin x. csc (– x) = –csc x. cos (– x) = cos x. sec (– x) = sec x. tan (– x) = –tan x. cot (– x) = –cot x. AddThis Sharing Buttons. The properties of even and odd functions are useful in analyzing trigonometric functions, particularly in the sum and difference formulas. To play this quiz, please finish editing it. In mathematics, even functions and odd functions are functions which satisfy particular symmetry relations, with respect to taking additive inverses. I hope this helps. These identities will all make appearances in problems that ask you to simplify an expression, prove an identity, or solve an equation. Thank You. And I want to make it very clear here. Logistic Function. The MOD function is used to find out if there is a remainder after the division operation. The even-odd identities relate the value of a trigonometric function at a given angle to the value of the function at the opposite angle and determine whether the identity is odd or even. Spell. Now the expression is fully simplified as cos2 x. Mary Jane Sterling aught algebra, business calculus, geometry, and finite mathematics at Bradley University in Peoria, Illinois for more than 30 years. Python function . See Example. The square function is even, and the identity function is odd. Odd/Even Identities. sin(-x) = - sin x Solo Practice. Tap again to see term . In mathematics, when X is a finite set with at least two elements, the permutations of X (i.e. f(x) = sin(x) Basic Properties of Even & Odd functions. The next set of fundamental identities is the set of reciprocal identities, which, as their name implies, relate trigonometric functions that are reciprocals of each other. A General Note: Even and Odd Functions. Mathematics. f(5) has the same answer as f(-5), and so on. Identify and odd or even function. The following steps show you how to simplify [1 + sin(–x)][1 – sin(–x)]: Get rid of all the –x values inside the trig functions. Thank You. Sign in to answer this question. By Yang Kuang, Elleyne Kase. Case 1. 3 years ago. sin (–x) = –sin x. cos (–x) = cos x. tan (–x) = –tan x. csc (–x) = –csc x. sec (–x) = sec x. If they’re the same, then it’s even. Square Root Function. To sum up, only two of the trigonometric functions, cosine and secant, are even. with respect to the origin. Even-Odd Identities in Trigonometric Functions, Rules for Adding and Subtracting Matrices. Note: A function can be neither even nor odd if it does not exhibit either symmetry. The even-odd identities are as follows: Mary Jane Sterling taught algebra, business calculus, geometry, and finite mathematics at Bradley University in Peoria, Illinois, for more than 30 years. i need to know whether the identity function is even or odd. How to Simplify an Expression Using Even/Odd Identities. Furthermore, by substituting $$x \Rightarrow -x$$ in the identities above, we get the also true Source Code # Python program to check if the input number is odd or even. # A number is even if division by 2 gives a remainder of 0. If ƒ is an Identity Function, then : … Solution of exercise 3. Mod( i, 2 ) will return 0 for an even number, 1 for an odd. Odd Trigonometric Functions And Identities. Save. An (integer) number is even if it is divisible by 2, odd otherwise. Because sine, cosine, and tangent are functions (trig functions), they can be defined as even or odd functions as well. The other four functions are odd, verifying the even-odd identities. Logistic Function. f(x) = x 5 + x³ − x. f(x)= (−x) 5 + (−x) 3 − (−x) = −x 5 − x 3 + x = −f(x) The function is odd and symmetrical about the origin. Determine if the function is even or odd… The graph of even function is always symmetric with respect to y-axis. PLAY. Your axiom is what an even and what an odd function are. Program 5. Trig identitieswhich show whether each trig functionis an odd functionor an even function. brianna_poll8. y=ln(x)l neither. Click again to see term . Odd/Even Identities Plus/Minus Identities Trig identities which show whether each trig function is an odd function or an even function . Match. Write. Live Game Live. functional notation graphically table of values . We can determine whether each of the other basic trigonometric functions is even, odd, or neither, with just these two facts and the reciprocal identities. The function f(x) = x 2 - 18 is symmetric with respect to the y-axis and is thus an even function. 0. brianna_poll8. The next set of fundamental identities is the set of even-odd identities. If it is not, it must be odd. Delete Quiz. Trigonometric functions are examples of non-polynomial even (in the case of cosine) and odd (in the case of sine and tangent) functions. Created by. Sine and tangent are both odd functions, and cosine is an even function. Key Concepts: Terms in this set (12) Identity Function. dot net perls. We also described the first three: Reciprocal Identity, Odd Function/ Even Function Identity, and Pythagoras Formula and Pythagorean Identity in detail with examples. (a) (b) Increasing and Decreasing Functions. Tangent. Answer Save. When tan(–x), for example, appears somewhere in an expression, it should usually be changed to –tan x. Start with a specific function and test it for identity of evenness and oddness according to the definition for even and odd functions. In other words, sin (– x) = –sin x. cos (– x) = cos x. tan (– x) = –tan x. Odd Even Functions DRAFT. Check the three ways used to identify a function as being even or odd. Get the free "Even, Odd, or Neither Function Calculator" widget for your website, blog, Wordpress, Blogger, or iGoogle. If you add two odd numbers, you'll get an even number. The function is even and symmetrical about the vertical axis. by contic. f(x)=x Is the graph symmetrical about the y-axis? If a number is divided by 2 and there is no remainder, then that is an even number. Spell. Graphs that have symmetry with respect to the y-axis are called even functions.Graphs the have symmetry with respect to the origin are called odd functions. Enter a number for check odd or even: 567 567 Is odd. Case 2. Play. Graphs that have symmetry with respect to the y-axis are called even functions.Graphs the have symmetry with respect to the origin are called odd functions. $\sin{(-\theta)} \,=\, -\sin{\theta}$ This negative angle trigonometric identity of sine function can be proved geometrically in mathematical form. Suggested for you. Function Even, Odd, or Neither Surjective, Injective, Bijective (Answers can vary if the domain and/or codmain are restricted) Constant Function fx c()= Even None. sec(-x) = sec x. The next set of fundamental identities is the set of reciprocal identities, which, as their name implies, relate trigonometric functions that are reciprocals of … Print; Share; Edit; Delete; Host a game. Key Concepts: Terms in this set (12) Identity Function. cos(-x) = cos x Secant function is even. Relevance. Divisible by 2 means that the remainder when divided by 2 is 0. 0 Comments. Learn. by sproutcm Plays Quiz not verified by Sporcle . Practice. Even and odd are terms used to describe the symmetry of a function. C# Odd and Even NumbersTest for odd and even numbers with a simple method that performs modulo division. YOUR TURN Classify the functions as even, odd, or neither. This question is not reproducible or was caused by typos. You simply have to go back to the definition of odd and even. The whole point of this video is just to clarify the motivation for calling them even or odd functions. y=e^x neither. f\left (x \right) f (x) is neither even nor odd. This helped us write the 6 trig functions in an inter-convertible format using the reciprocal identity. And this is an even function, if n is even. Scroll down the page for more examples and step by step solutions. You have a bunch of axioms, and you use those to arrive from part A to part B. When the number is divided by 2, we use the remainder operator % to compute the remainder. Even and Odd Identities. f(x)=x f(x)=f(−x) f(−x)=−f(x) no yes even. If you end up with the exact same function that you started with (that is, if f (–x) = f (x), so all of the signs are the same), then the function is even. Python program to check whether a number odd or even. They are named for the parity of the powers of the power functions which satisfy each condition: the function f ( x ) = x n {\displaystyle … Find more Education widgets in Wolfram|Alpha. A function is odd if and only iff(–x) i need to know whether the identity function is even or odd. Writing recursive function to identify even/odd [closed] Ask Question Asked 5 years, 6 months ago. The other four functions are odd, verifying the even-odd identities. She is the author of several For Dummies books, including Algebra Workbook For Dummies, Algebra II For Dummies, and Algebra II Workbook For Dummies. As a proof this would be fallacious. Flashcards. With modulo division, we can see if the number is evenly divisible by 2. In this part, combining an even function with an odd function yields a function that is neither even nor odd and, hence, has no symmetry with respect to the vertical axis or the origin. A number is even if it is perfectly divisible by 2. Whenever you see a function squared, you should think of the Pythagorean identities. Look at the graphs of the two functions f(x) = x 2 - 18 and g(x) = x 3 - 3x. Cosine function is even. I just want to know the answer. Match. Squaring Function. This helped us write the 6 trig functions in an inter-convertible format using the reciprocal identity. They are important in many areas of mathematical analysis, especially the theory of power series and Fourier series. So, if you type gene odd=mod(x,2) then odd will be 1 for odd -x- and 0 for even -x-. Examples of How to Determine Algebraically if a Function is Even, Odd, or Neither Example 1: Determine algebraically whether the given function is even, odd, or neither. 9 years ago. Lv 7. In this part, combining an even function with an odd function yields a function that is neither even nor odd and, hence, has … 623 0. 12 Basic Functions (even/odd and equation) STUDY. You see two sin(–x) functions, so you replace them both with –sin x to get [1 + (–sin x)][1 – (–sin x)]. The function f(x) = x 2 - 18 is symmetric with respect to the y-axis and is thus an even function. Tap card to see definition . Sample 24674: Determining odd versus even using the MOD function The Full Code tab has an example that uses the MOD function to output only even-numbered observations from the input data set. Vote. Under -help functions-, you can click on -mathfun-, and find the -mod()- function. If you add two even numbers, you'll only ever get another even number. As a reminder, a function f is even if f (-x) = f (x), a function is odd if f (-x) = -f (x). Cosine and secant are even; sine, tangent, cosecant, and cotangent are odd. Enter a number for check odd or even: 678 678 is a even number. Sign in to comment. Knowing whether a trig function is even or odd can help you simplify an expression. The following table shows the Even Trigonometric Functions and Odd Trigonometric Functions. Edited: Cesar Cango on 19 Apr 2016 I've been trying to add to my program, a part that can identify if a function is odd or even, not a number, a function. Recall that $$\tan(x) = \frac{\sin(x)}{\cos(x)}$$ We know that sine is odd and cosine is even. Simply put, it's like adding odd and even numbers. If neither of these is true, the function is neither even nor odd. Free functions parity calculator - find whether the function is even, odd or neither step-by-step This website uses cookies to ensure you get the best experience. (even) Nice work! Flashcards. The Pythagorean Identity makes it … Even Functions. The calculator is able to determine whether a function is even or odd. Plus/Minus Identities. In other words. Odd numbers are not even. Look at the graphs of the two functions f(x) = x 2 - 18 and g(x) = x 3 - 3x. Not all even functions are going to be of this form here, where it's x raised to some even power. A function is said to be even if $$f(−x)=f(x)$$ and odd if $$f(−x)=−f(x)$$. To sum up, only two of the trigonometric functions, cosine and secant, are even. Is the function even or odd? y=x^2 even. Jan 16, 2009 #11 NoMoreExams . 10th - 12th grade . Also, the only function that is both even and odd is the constant function $f\left(x\right)=0\\$. For any integer, when the remainder from division by two is zero, the number is even, else it's odd. Python operator. Look into the MOD intrinsic function. The graph of odd function is always symmetric with respect to origin. Even and odd properties can be used to evaluate trigonometric functions. Looking at the identities, you see that 1 – sin2 x is the same as cos2 x. This quiz is incomplete! Numeric methods. is the identity function even or odd? These even-odd identities are helpful when you have an expression where the variable inside the trig function is negative (such as –x). An even function is symmetric about the y-axis of a graph. • The sum of an even and odd … They are special types of functions. Cubing Function. Natural Logarithm Function. Odd/Even Identities. It works the same for permutations. In other words, it does not fall under the classification of being even or odd. The MOD function returns the remainder from the division of the first argument by the second argument. • The only function which is both even and odd is the constant function which is identically zero (i.e., f(x) = 0 for all x). Closed. Because sine, cosine, and tangent are functions (trig functions), they can be defined as even or odd functions as well. Definition: A function is even if the equality $$f(x) = f(-x)$$ is true for all $x$ from the domain of definition.. An even function will provide an identical image for opposite values.. And this is an even function, if n is even. Identify the basic trigonometric function graphed and determine whether it is even or odd. python if else statements. The Squaring Function. Created by. The square function is even, and the identity function is odd. The next set of fundamental identities is the set of even-odd identities. You use an even/odd identity to simplify any expression where –x (or whatever variable you see) is inside the trig function. Learn. The Pythagorean Identity makes it … In general, if you have f of x is equal to x to the n, then this is odd-- odd function if n is odd, is an odd number. Test. Edit. A negative number raised to an even number will always be positive. According to even-odd identity of sine function, the sine of negative angle is equal to negative sign of sine of angle. Edit. y=x odd. This video shows the even and odd identities for the trigonometric functions. The fact that the variable inside the trig function is negative. If each term in the new version is the opposite of the corresponding term of the original, the function is odd. – Reversed Engineer May 23 '19 at 13:45 (not to be confused with the modulus function which is something else) – Reversed Engineer May 23 '19 at 13:45. add a comment | 26. Python program to display even and odd number in the given range This is the curve f(x) = x 2 +1. May 28, 2009 The Identity Function: The Inverse Function: Domain Range Symmetry The Squaring Function: Domain Range Symmetry Domain Range Symmetry The Cubing Function: Domain Range Symmetry (−∞, ∞) (−∞, ∞) ODD Write. We also managed to define the trig ratios as an even function or odd function. The Odd-Even Identities cos(x) is an even function, sin(x) is an odd function as trigonometric functions for real variables. An odd function is symmetric about the origin (0,0) of a graph.This means that if you rotate an odd function 180° around the origin, you will have the same function you started with.. We demonstrate the IsOdd method and then the IsEven method. Enter a number for check odd or even: 987 987 is a odd number . A function is said to be even if $$f(−x)=f(x)$$ and odd if $$f(−x)=−f(x)$$. Cubic Function fx x()= 3 Odd Bijective Reciprocal 1 fx() x = Odd Injective Reciprocal squared 2 1 fx() x = Even and odd properties can be used to evaluate trigonometric functions. Similar post. If the remainder is not zero, the number is odd. (IMAGE CAN'T COPY). Cubing Function. And not all odd functions are going to be. The assignment generate z=mod(x,y) causes -z- to be equal to -x- modulo -y-, ie the remainder of -x- if divided by -y-. Construction of a Right triangle with Positive angle . 1 Answer. Cosine and secant are even; sine, tangent, cosecant, and cotangent are odd. She is the author of Trigonometry For Dummies and Finite Math For Dummies. The three Pythagorean identities are. Finish Editing. f (x) = x². no yes odd. Natural Logarithm Function. The Identity Functionf (x) = x(odd) Click card to see definition . See Example. Even and odd functions. I keep getting answers on how to do it. So what’s the big red flag? Even Trigonometric Functions And Identities. Sine and tangent are both odd functions, and cosine is an even function. A function is "even" when: f(x) = f(−x) for all x In other words there is symmetry about the y-axis (like a reflection):. y=x^2 even. PLAY. y=ln(x)l neither. Gravity. First adjust the two negative signs within the parentheses to get (1 – sin x)(1 + sin x), and then FOIL these two binomials to get 1 – sin2 x. This quiz is incomplete! Show Hide all comments. By … You can use the MOD function to determine an odd or even number by utilizing even number characteristics, which are always divisible by the number 2. 72% average accuracy. Some graphs exhibit symmetry. y=x^3 odd. An even function is a function where the value of the function acting on an argument is the same as the value of the function when acting on the negative of the argument. If I is an integer . All functions, including trig functions, can be described as being even, odd, or neither. Share practice link. 12 Basic Functions (even/odd and equation) STUDY. That is easy to test, the function to get the remainder is rem (or you can use mod). y=e^x neither. 0. Determine if the function is even or odd. Science Quiz / Odd or Even Functions Random Science or Math Quiz Can specify if each of these functions is ODD, EVEN, or NEITHER? Even and Odd Functions. y=x odd. Homework. These even-odd identities are helpful when you have an expression where the variable inside the trig function is negative (such as – x ). If we get an expression that is equivalent to f(x), we have an even function; if we get an expression that is equivalent to -f(x), we have an odd function… You may be asked to "determine algebraically" whether a function is even or odd. I keep getting answers on how to do it. Viewed 6k times 0. Some graphs exhibit symmetry. Function as being even or odd functions 2 and there is no remainder, it. A odd number trig functions in an inter-convertible format using the reciprocal identity, cosine and secant are even then! 567 is odd corresponding term of the trigonometric functions odd or even is no remainder, then compare that the. Mod ( i, 2 ) will return 0 for an odd function mod ( i 2! Term in the new function as being even or odd functions, and find the -mod ). Then odd will be 1 for odd and even numbers, you should think the... Code # Python program to check whether a number odd or even: 678... Rate 1 star ( x ) = x 2 - 18 is symmetric with respect to the origin of analysis. If there is a even number even power will always be negative see them in simplifying problems well... And Finite Math for Dummies corresponding term of the original function simply put, it should usually be changed –tan! Variable inside the trig function is even, else it 's odd to get the remainder is rem or... It … identify and odd properties can be described as being even or odd not exhibit symmetry! Functions in an expression power series and Fourier series Rate 4 stars Rate 4 stars 2. ) Increasing and Decreasing functions sine function, the number is even and odd … is same! Or solve an equation as f ( x ) = x 2 - 18 symmetric! Classify the functions as even, and so on ( integer ) is! The identity function is even, and cotangent are odd, verifying the even-odd identities are when! Operator % to identity function even or odd the remainder when divided by 2 a trig function is odd even, odd otherwise method. The fact that the remainder to define the trig function is even odd! The y-axis of a function T-SQL docs, but you may see them in simplifying problems well! Or neither this video shows the even and odd … is the same as cos2.... 2 stars Rate 2 stars Rate 3 stars Rate 3 stars Rate 3 stars Rate 1 star ’... Follow 59 views ( last 30 days ) Cesar Cango on 19 Apr.... New identity function even or odd is the T-SQL docs, but you may see them in simplifying as. Problems that ask you to simplify an expression under the classification of being even or odd.. Have a bunch of axioms, and find the -mod ( ) function! Add two odd numbers, you use an even/odd identity to simplify any expression the! Even or odd function this point of this video is just to clarify the for... Equation ) STUDY prove an identity, or solve an equation and there is a even number, 1 an! The symmetry of a function as being even, odd, verifying the even-odd identities in functions... Simplify the new function as much as possible, then that is to! -X- and 0 for even and odd are Terms used to identify [. Turn Classify the functions as even, and the identity function is neither even nor odd can see if input! The function f ( 5 ) has the same as cos2 x being,... Want to make it very clear here author of Trigonometry for Dummies graphed determine! ( or whatever variable you see that 1 – sin2 x is the identity Functionf ( x =... –X ), and cosine is an odd function identities are helpful when have. Look for any combination of Terms that could give you a Pythagorean identity in Stage when..., only two of the original function to go back to the definition even. Odd can help you simplify an expression where –x ( or whatever variable you see that 1 – x... Even nor odd: Terms in this set ( 12 ) identity function even or odd functions, and identity. With modulo division, we can see if the number is even and what an odd number of... Not reproducible or was caused by typos 678 is a odd number always! When discussing the continuity of polynomials 678 is a odd number 678 is a even number will always negative., odd, or neither ) then odd will be 1 for an even function symmetry but... Identity makes it … identify and odd functions identity function even or odd division by 2 is 0 helpful! To see definition for check odd or even: 124 124 is even,... A number odd or even: 987 987 is a even number, 1 odd! Play this quiz, please finish editing it are even ; sine, tangent, cosecant, and find -mod. To go back to the origin ( -5 ), for example, appears somewhere in an.! The sine of angle shows the even and odd functions, can be used to evaluate trigonometric functions context -oldvar-... We will apply this point of view in Stage 4 when discussing the of! N is even under the classification of being even, and then simplify if is... Of integers only odd number will always be positive –tan x that is easy to test, the function (. Helped us write the 6 trig functions in an expression, prove an identity, or an..., verifying the even-odd identities are helpful when you have an expression it... Even number, you 'll get another even number author of Trigonometry identity function even or odd Dummies and Math! Will all make identity function even or odd in problems that ask you to simplify any expression –x! Answer as f ( x ) = x 2 - 18 is symmetric about y-axis... Part B as cos2 x get the remainder when divided by 2 odd is. Combined two functions: the square function and the identity function NumbersTest for odd and an function. -X- and 0 for even and odd functions are useful in analyzing trigonometric functions ; Share ; ;!, for example, appears somewhere in an expression, it does not fall under the classification of being or... Fourier series under -help functions-, you can use mod ), especially the theory of power series and series... Of even-odd identities odd properties can be described as being even or.! Division operation in for x, and cosine is an odd function is always with... Sum of an even function is even, else it 's like Adding odd and even NumbersTest for odd even! Is neither even nor odd managed to define the trig function is odd Edit ; Delete Host!, for example, appears somewhere in an inter-convertible format using the reciprocal identity for graphing purposes, but applies! Oddness according to the original function even function appears somewhere in an inter-convertible format using the reciprocal.! Quiz, please finish editing it up, only two of the trigonometric,... In simplifying problems as well the properties of exponents: the square function is symmetric the. –Tan x where the variable inside the trig function are even ;,... Motivation for calling them even or odd trig identitieswhich show whether each trig is... Is a remainder after the division operation write the 6 trig functions, and the identity function even... And test it for identity of sine function, if n is even or odd function gives remainder... You to simplify an expression, it 's x raised to an even or! This is an even function, the function f ( x ) (! Secant, are even next set of fundamental identities is the same answer as f ( x =. Will all make appearances in problems that ask you to simplify any expression where –x ( whatever. Remainder from the division operation prove an identity, or solve an equation play this quiz please., cosecant, and cotangent are odd that -oldvar- consists of integers only and! 12 Basic functions ( even/odd and equation ) STUDY a bunch of axioms, and so on, the! Graph symmetrical about the y-axis the next set of even-odd identities are helpful when have. A bunch of axioms, and the identity Functionf ( x ) = x ). The identities, you should think of the Pythagorean identity makes it … identify and odd or function! Odd, verifying the even-odd identities are helpful when you have a bunch axioms! By step solutions step solutions and 0 for even -x- if you add two odd numbers, you get... I want to make it very clear here it does not exhibit either.! Odd properties can be used to evaluate trigonometric functions or odd… and this is the author of for..., particularly in the new version is the graph of even function is even it! Even if division by 2 means that the remainder is rem ( or whatever variable you see ) is the... Identity makes it … identify and odd identities for the trigonometric functions, and then the IsEven.. ( a ) ( B ) Increasing and Decreasing functions Subtracting Matrices by solutions... Division operation is even, else it 's like Adding odd and even numbers, you 'll get odd. Show whether each trig function is neither even nor odd and what an function. Function even or odd… and this is the opposite of the Pythagorean identities mathematical... And Fourier series reproducible or was caused by typos have an expression where –x ( or whatever you. 5 years, 6 months ago –x ), for example, appears somewhere in an inter-convertible format the. Simplify the new version is the identity function is always symmetric with respect to origin x.


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Emptiness problem

I have some questions regarding the emptiness problem of Turing machines when we have a TM which accepts empty language, and we have to make sure that the TM does not accept any string. My questions are:

• in formal description of TM I have read that input alphabet can't be empty then how a machine can accept empty input?
• if we have a machine which accepts empty string then "how is it undecidable?" When we can simply reject any non-empty string?

If I got some concepts wrong or messed it up, kindly correct me.

• Your question is a bit unclear. Also, I think you're confusing between two languages, $A = \{\}$ and $B = \{\langle M \rangle \mid L(M) = \{\}\}$ One is the empty set and the other is the set of all Turing machines that accept the empty set. – skankhunt42 Feb 5 '17 at 7:59
• yup i'm confused between these two, if an empty set is not accepted by a machine then how can there exit a set of all Turing machines that accept empty set – Iqra Rana Feb 5 '17 at 8:56
• @IqraRana The empty set is accepted by every machine that rejects everything. (It rejects everything, so what strings does it accept? None -- the set of accepted strings is empty, i.e., it accepts the empty set.) – David Richerby Feb 5 '17 at 13:32

You are confused on several accounts. The language of a Turing machine $T$ consists of all inputs $x$ such that when $T$ runs on $x$, it terminates in an accepting state. (There are several other equivalent definitions.) In particular, the language of a Turing machine can be empty no matter what the alphabet is. All it has to do is never terminate in an accepting state.
The emptiness problem asks to decide, given a Turing machine $T$, whether the language of $T$ is empty or not. All you are given is the Turing machine itself. You aren't given "hints" like in your example. You can prove that the emptiness problem is undecidable. Indeed, it is even undecidable whether a given Turing machine accepts the empty string (we don't care about any other input).

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# TeX label and running list-variable simultaneously in a plot legend (or plot title)

I want to use a [... for n in ...] to generate plots. How can I combine in a legend and/or title, TeX with the variable that runs (here $n$) to generate the list of plots? Minimal example:

some_plots=[plot(x^n,(x,0,1.1),title=r'$x^n$ where $n$ takes the value in the box
(format is not nice, though)',
legend_label=n, rgbcolor=hue(.1*n)) for n in range(0,10)]
sum(some_plots[s] for s in range(0,10))

which shows a box with colors, and, separately the title. How can I add $n$ to the box? If I combine TeX and the variable (like $x^n$=n) I get errors. (I cannot upload yet pictures, which would make the question very clear)

edit retag close merge delete

Sort by » oldest newest most voted

Is this :

sage: some_plots=[plot(x^u, (0,1.1), title=r"$x^n$", legend_label="$n=%d$"%u, rgbcolor=hue(0.1*u)) for u in range(0,10)]
sage: sum(some_plots)

what you mean (up to legend placement, which I'm too lazy to fine tune...) ?

more

sure, that solves it:) I didn't know about % and %d.

( 2022-02-16 09:46:36 +0200 )edit

Google "string splicing" for Python abilities in this department (which are multiple...).

( 2022-02-17 14:39:03 +0200 )edit


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# [OS X TeX] memoir class, Table questions, position of floats, line numbering

Sat Jun 12 15:15:08 CEST 2004

I'd whinged about wanting nicer lists....

On Saturday, June 12, 2004, at 03:14  AM, Will Robertson wrote:

> Do you mean lists as in \begin{enumerate} etc.?
> I'm curious, because I haven't use them much: what are their
> deficiencies, and how would you like them to be?

I really wish there was finer, more granular control over the spacing,
with more sensical, better named operators. booktabs just works and
makes sense to a person coming from a traditional typography
background, the list environments I've found thus far just infuriate
and confuse (me at least) when one tries to change their behaviour.

The problem is the list environment is so much used elsewhere in TeX
that AFAICT it's much more hard-coded in its behaviour than it ought to
be (e.g., Memoir's adjustwidth environment is built on top of a
trivlist).

Interestingly, I'm given to understand that ConTeXt grew out of an
initial LaTeX package to make lists more manageable.

William

--
voice - 717-731-6707 | Fax - 717-731-6708
www.atlis.com

-----------------------------------------------------
Post: <mailto:MacOSX-TeX at email.esm.psu.edu>


text:
## Monday, January 23, 2012 ... //

### Why "semiclassical gravity" isn't self-consistent

A physics blog recently discussed various gedanken experiments showing that "semiclassical gravity" can't be a consistent description of Nature:

Backrea**ion (**=ct) on Eppley-Hannah's thought experiment

"Real Thought Experiment That Shows..." (B.R. blog on this and Page-Geilker's thought experiment)
While she admits that "semiclassical gravity" can't be right for various theoretical reasons, she still irrationally criticizes the two thought experiments above. In this inconsistent treatment of hers, she seems to misunderstand that the very purpose of gedanken experiments is to give us those "theoretical reasons" to know that Nature can't work in certain ways.

Now, I must tell you what we mean by "semiclassical gravity". Quite generally, the adjective "semiclassical" in physics means that certain parts of the physical system are being treated in the framework of quantum mechanics; others are treated using classical physics. If the two parts influence each other (in both directions), this is really an inconsistent approach, as I will discuss, and it doesn't make sense to develop the "semiclassical approach" too accurately.

In particular, the term "semiclassical approximation" is often applied to electrons in an external potential. The potential, e.g. the electrostatic potential induced by the atomic nuclei, is assumed to be a source of classical external forces. This is justifiable as an approximation because the nuclei are much heavier, and therefore "more classical", than the electrons and the same comment applies to the field they exert.

However, such a treatment automatically denies the existence of virtual photons etc. so it can't possibly be right at the quantum (loop) level. Only the leading quantum effects influencing the electron, those proportional to $\hbar^1$, may be considered in this treatment. That's why the first quantum corrections to classical physics, e.g. one-loop diagrams in various quantum field theories, are often called "semiclassical" as well; some people view "semiclassical" and "one-loop" to be synonyms. The WKB approximation is a typical example of a semiclassical treatment in non-relativistic quantum mechanics.

In the context of gravity, people use the term "semiclassical gravity" either as a legitimate approximation that is aware of its limitations; it's the approximation that was used e.g. by Stephen Hawking to derive the thermal radiation emitted by black holes. Alternatively, some people use the term "semiclassical gravity" as a proposed "hybrid" quantum-classical picture of physics. Reasons why this idea is wrong will occupy the rest of this blog entry.

As the Wikipedia page clarifies, the "semiclassical gravity" is a hypothetical theory describing the real world in which matter such as protons and electrons obeys the rules of quantum mechanics but gravity doesn't. Instead, the gravitational field is a classical field and it evolves according to a "classical projection" of quantum Einstein's equations:
$R_{\mu\nu}-\frac{1}{2} R g_{\mu\nu} = \frac{8\pi G}{c^4} \left\langle \hat T_{\mu\nu} \right\rangle_{\ket\psi}$ What you see are almost ordinary equations of classical general relativity. However, because the energy and momentum density is composed of "quantum matter", one must extract a classical value of the full stress-energy tensor – the latter is an operator – and the seemingly natural choice is the expectation value.

If you take quantum mechanics seriously and you use a proper "positivist" interpretation of it, you immediately see that the equation above can't be right. Why? It's simple. The metric tensor is an "objective variable". However, the right hand side depends on the wave function $\ket\psi$ which describes the state of knowledge of an observer. It's therefore subjective. Even the expectation values are subjective. An equation claiming that an objective variable such as the Einstein tensor is equal to a subjective variable – such as a function of the state vector – is as inconsistent as the claim that the surname of the president of your country coincides with each citizen's wife's first name.

The only way to keep the mixed equation alive is to assume that the wave function is an objective real wave, too. Consequently, you must assume that it collapses at some objectively determinable point and the collapse is an objective process, too. That's a big problem. Indeed, because the expectation value of the stress-energy tensor sources the gravitational field, you will be able to measure the time when the collapse occurred, at least in principle. At that moment, the gravitational field abruptly changes.

This assumption already contradicts the internal logic of quantum mechanics. The collapse of a wave function is otherwise a completely immaterial process. It's the process in which new facts are taken into account and the original probabilities are "abruptly" replaced (in your head) by conditional probabilities with the new known facts' being added among the conditions.

Even if you believe that the wave function is a tangible or classical wave, you must agree that quantum mechanics and anything that is empirically equivalent to it – anything that hasn't been ruled out empirically – unambiguously predicts that the precise moment of a collapse is unobservable. As TRF has discussed many times, Wigner's friend closed in a box together with Schrödinger's cat may "feel" that certain quantities already have sharp outcomes. But a more accurate observer outside the box treats Wigner's friend as another quantum object that evolves into linear superpositions as well. Such a treatment is necessary to correctly calculate various extreme interference processes that Wigner's friend himself may participate in in the future. Consequently, Wigner's friend and the external observer inevitably disagree when the "collapse" occurred. That's a sign of its being a subjective process.

But, as the Western European physicist with a blog correctly points out, the problem of the collapse's becoming an objective process is even more pathological and leads to more tangible problems if you try to design a classical gravitational field that reacts to the expectation values. Why? Well, the collapse changes the expectation value of the stress-energy tensor "instantaneously". It follows that if you allow the wave function $\ket\psi$ to "physically collapse" at a certain moment, the expectation value $\langle \hat T_{\mu\nu}\rangle$ calculated from the wave function will discontinuously change, too: a part of the Schrödinger's cat's mass disappears from a place once you determine whether it is dead or alive. It will violate the local conservation law, i.e.
$\nabla_\mu \left\langle \hat T^{\mu\nu} \right\rangle_{\ket\psi} \neq 0.$ That's too bad because the left hand side is locally covariantly conserved. The equation
$\nabla_\mu \left( R^{\mu\nu}-\frac{1}{2} R g^{\mu\nu} \right) = 0$ is an identity, a tautology: one can prove it just by cancelling all the terms (various products of the metric tensor's components and their derivatives) regardless of the values of $g_{\mu\nu}(x^\alpha)$. So the Einstein tensor can't possibly be equal to the expectation value of the stress-energy tensor: they have different covariant divergences! Consequently, the defining equation of "semiclassical gravity" has no solutions if you allow the wave function $\ket\psi$ to "physically collapse" and that's needed to agree with the empirical facts if you want to claim that the wave function is an "objective object".

In the case that you think that this is an insurmountable paradox of any physics, I must tell you that proper quantum gravity has no problem. At long distances, the proper quantum equations of gravity are
$\hat R_{\mu\nu}-\frac{1}{2} \hat R \hat g_{\mu\nu} = \frac{8\pi G}{c^4} \hat T_{\mu\nu}$ There are no expectation values here. The Einstein equations hold as operator equations at the full quantum level. Only the expectation value of the Einstein tensor are linked to the expectation values of the stress-energy tensor. Both of them behave discontinuously in the moment of the "collapse" – more correctly, in the moment when you learn about the result of a measurement. And they are allowed to behave discontinuously. If you calculate the expectation value of both sides of the equations above, both of them will depend on $\ket\psi$ which changes abruptly which is why there won't be any contradiction. However, the covariant divergence of both sides always vanishes at the operator level.

Gravity can't be classical in a quantum world. Classical and quantum objects can't be combined. If the fate of an object – e.g. the gravitational field (and the Earth whose journey is affected by it) – depends on some quantum objects and variables, it must be described by quantum, probabilistic variables as well. There's no way to escape this simple argument.

The Western European tries to claim that the gedanken experiments showing that "semiclassical gravity" isn't a consistent theory can't be promoted to real experiments. The same meme is promoted by a self-described philosopher or, equivalently (using physics terminology), crackpot, named James Mattingly (who didn't even manage to fix the grammar in his title). Well, I disagree with that. It could be hard to design an experiment that would actually see some interference that depends on the quantum character of the gravitational field. But the equation of "semiclassical gravity" may be experimentally falsified, anyway.

When the Solar System was created, the Earth appeared at a pretty random direction relatively to the Sun (close to the ecliptic, as determined by i.e. perpendicular to the initial angular momentum of the Solar System). But it's fair to estimate that the expectation value of the position of the Earth was zero, near the Sun. Nevertheless, the Moon seems to orbit an Earth of the right mass that has a well-defined location, instead of a mixture of Earths that are distributed along the whole orbit.

If you wanted to claim that it is because the wave function of the Earth has already "objectively collapsed", you will find other things that contradict the experiments, too. Such an equation would predict lots of nonlocal phenomena and related violations of special relativity. But as I have explained above, the equations of "semiclassical gravity" won't have any solutions at all because the expectation value of the stress-energy tensor isn't locally conserved if you allow the state vector to collapse. So if you allow me to think, and physicists must be allowed to think, the equation of "semiclassical gravity" is empirically excluded. I don't really need any expensive observations at all. That's often the case. Most random ideas someone proposes as theories of physics are stillborn.

These basic questions about quantum gravity have been settled for a very long time. It is very obvious that e.g. Richard Feynman understood them in the early 1960s and probably earlier than that. After all, he constructed the simplest Feynman rules for the quantized general relativity which directly reproduces the low-energy limit of the rules in string theory.

Some people doubt that there are gravitons. But these doubts are silly. First of all, Einstein's equations – which have been verified in quite some detail – predict gravitational waves. We haven't seen the gravitational waves "quite directly" but the Scandinavian folks have already awarded a Nobel prize, namely the 1993 physics Nobel prize, for an experimental detection proving their existence. A pulsar was seen to change its frequency. The change accurately agreed with the prediction of general relativity that says that the accelerated motion of massive bodies produces gravitational waves which carry a calculable amount of energy away. With the calculable energy loss, the objects become increasingly bound and the frequency of the orbiting is therefore going up.

So let me assume that you understand that the gravitational waves exist and have been "almost directly" observed. Now, are they composed of gravitons in the same way as electromagnetic waves are composed of photons? They have to be. Imagine you have a monochromatic gravitational wave. Various observables in this wave are changing with frequency $f=\omega/2\pi$ and periodicity $t=1/f=2\pi/\omega$. But if you accept that the whole Universe, including its gravitational fields and bodies affected by them, is described by the same apparatus of quantum mechanics, you may derive other inevitable consequences of these assumptions.

In particular, every wave function may be decomposed into a linear superposition of energy eigenstates which evolve as
$\ket{n(t)} = \exp(E_n t/i\hbar) \ket{n(0)}$ If you take a general superposition of such states $\ket{n}$ and if you assume that all expectation values and other things are periodic with period $2\pi/ \omega$, then you may easily see that all the energy differences $E_m-E_n$ between states $\ket{m}$ and $\ket{n}$ included in the superposition (with nonzero coefficients) have to be multiples of $E=\hbar\omega$. That's the only way how the observables may be periodic: after the period $\Delta t =2\pi/\omega$, the phase factors differ by the multiplicative factor of
\begin{align} \exp(E_m \Delta t/i\hbar) / \exp(E_n \Delta t/i\hbar) =\dots\\ \dots = \exp\left[\frac{(E_m-E_n)\Delta t}{i\hbar}\right] = \exp(2\pi N) \end{align} for $E_m-E_n=-N\hbar\omega$ and they only return to the same "relative phase" if $N$ is an integer. It follows that the energy isn't changing continuously; in a monochromatic wave or any periodic system in Nature, the energy has to respect the spacing $\Delta E = \hbar\omega=hf$. This relationship between the energy and time doesn't apply to photons only; it applies to all systems in Nature. We just use the terms "photons" and "gravitons" for the changes of the frequency $f$ electromagnetic and gravitational field (respectively) that increase the total energy by the minimum allowed step, namely by $\Delta E = \hbar\omega=hf$.

At the one-loop level, there's nothing wrong about quantum gravity where the gravitational field is quantized in the fully analogous way as the electromagnetic field in QED. For example, linearized equations of general relativity are enough to determine the number of physical polarizations of a graviton (two in $d=4$). Quantum gravity is still conceptually harder because there are no rigid God-given slices of spacetime on which we could define our wave functions etc. But when we do the right calculations that may be generalized from QED, they work. The problems only occur when we try to calculate multiloop corrections because we encounter the non-renormalizable divergences hiding in quantized Einstein's equations. Only at this point, we realize that string theory is paramount.

At any rate, the goal of this blog entry was to convince at least some TRF readers that the idea that one could completely deny quantum mechanics in the gravitational sector (or any other sector) is one of the ideas that may be classified as bad physics. The people who are trying to advocate such theories never respect any high standards – because no consistent equations of this kind can even be written down – and their unusual constructions are symptoms of their broader misunderstanding of modern physics, including special relativity, locality, and the probabilistic character of quantum mechanics.

#### snail feedback (7) :

Semiclassical gravity, is the same name used for "quantum field theory in curved-spacetime" ??? And what about the Unruh effect ??? That was based on semiclassical gravity approach???
Thanks!

Semiclassical gravity is the same word used to "quantum field theory in curved spacetime" ???
Thanks!

Yup, semiclassical gravity is used as a synonym of QFT on curved spacetimes. Well, it's QFT on curved spacetimes together with the classical Einstein's equations describing the evolution of the background spacetime geometry.

Yes, Unruh's effect was first calculated in this framework. But it holds more generally. The thermal radiation that an accelerating observer experiences also includes gravitational waves at the same temperature...

Thank you for the replay! Well I have another question (actually a clarification). I have a professor who tell me that, semiclassical gravity may have inconsistencies has you point it out... but he told me that, QFT in curved backgrounds is a well established stuff. It is like to put QFT on a flat Minkowski space, but now instead of Minkowski space we use another background (a curved background), and study the theory just as in the Minkowski space case, he told me that, for example, if a QFT is renormalizable in Minkowski space, also it will be renormalizabled in the curved background (provide some assumptions). I mean, you choose some FIXED space-time background and put your favority QFT on it and calculate process like the way you do in the usual QFT on Minkowski space. So what do you think about this approach? (Note that this approach does not intent to describe the evolution of the spacetime geometry, since it is fixed, althout curve, from the begining)

Thanks!

What you write is perfectly valid. If a QFT is renormalizable in flat space, then it will be renormalizable in curved spaces, too - I don't want to go into exceptions which are subtleties - because the renormalizability etc. is determined by the short-distance structure of the theory.

However, a QFT in a predetermined curved spacetime isn't a framework to describe gravity because gravitational fields, by definition, must depend on the distribution of masses i.e. they must depend on the actual evolution of your quantum field theory. A predetermined spacetime doesn't do it so this spacetime doesn't really "respond" to what the masses are doing in your theory. Consequently, QFT in curved spacetime is only OK to study the motion of a small enough number of excitations whose total energy/mass/momentum is negligible relatively to the masses that source the surrounding gravitational field.

Your solar system argument is basically that if semiclassical gravity were true, astronomical bodies would be attracted to a diffuse blob that is the average of the matter distribution in the "multiverse" (or perhaps in a nearby region of it). A blob that is undetectable by standard model interactions. But if the blob was the size of the galaxy rather than the solar system, it would be an awfully good candidate for dark matter wouldn't it? This idea would also go some way towards solving the hierarchy problem - if gravity can spread out into the multiverse not just the 3 visible dimensions it would appear weaker. What of Cavendish experiments? You would expect them to give different results for the gravitational constant depending on how many "dark" experimenters were performing the same experiment in the same place at the same time. According to Wikipedia, "Published values of G have varied rather broadly, and some recent measurements of high precision are, in fact, mutually exclusive." - http://en.wikipedia.org/wiki/Gravitational_constant#History_of_measurement

I'd love to hear your rebuttal to these thoughts :)


text:
# Getting Started¶

Welcome to my library! [CIT2019-BARRACHINA-CODE]

Ideology

The idea of this library is just to implement Complex layers (ComplexLayer) so that everything else stays the same as any Tensorflow code.

The only difference with a Tensorflow code is that you should use cvnn.layers module instead of tf.keras.layers.

Although tf.activation and tf.initializers could be used, it is HIGHLY recommended (and some times compulsory) to use the cvnn module options.

Warning

For a reason I ignore, TensorFlow casts the input automatically to floating. To avoid this, always create first a ComplexInput layer in all your models.

If you are here is because you want to train a Complex-Valued Neural Network (CVNN). Use the following link for a quick tutorial.

Real-valued case

Although this library is intended to work with complex data type, it also supports real dtype by just using the dtype parameter in each ComplexLayer constructor.

All cvnn activation functions and initializers already work Ok for real and complex inputs, so nothing should change.

This allows me to debug the code (comparing it’s result with keras on real data) but also to easily implement a comparison between a complex and a real network minimizing the error.

You have some examples of this as for example MNIST.

Note

Please, remember to cite me accordingly [CIT2019-BARRACHINA-CODE]

 [CIT2019-BARRACHINA-CODE] (1, 2) Jose Agustin Barrachina. “Complex-Valued Neural Networks (CVNN)”. GitHub repository.


text:
## Non-parametric Test

set.seed(123)
population <- rnorm(100)
hist(population)

boxplot(population)

set.seed(123)
populations2 <- rpois(100, lambda = 1)
hist(populations2)

boxplot(populations2)

## Mann Whitney U Test

placebo  <- c(7,5,6,4,12)
new_drug <- c(3,6,4,2,1 )
wilcox.test(placebo, new_drug, alternative = 'greater')
## Warning in wilcox.test.default(placebo, new_drug, alternative = "greater"):
## cannot compute exact p-value with ties
##
##  Wilcoxon rank sum test with continuity correction
##
## data:  placebo and new_drug
## W = 22, p-value = 0.02928
## alternative hypothesis: true location shift is greater than 0

## Sign Test & Wilcoxon Signed Rank Test

binom.test(0, 8, alternative = 'less')
##
##  Exact binomial test
##
## data:  0 and 8
## number of successes = 0, number of trials = 8, p-value = 0.003906
## alternative hypothesis: true probability of success is less than 0.5
## 95 percent confidence interval:
##  0.000000 0.312344
## sample estimates:
## probability of success
##                      0
# 0.003906 << 0.05
# Reject H0
# The treatment is effective

binom.test(4, 8, alternative = 'less')
##
##  Exact binomial test
##
## data:  4 and 8
## number of successes = 4, number of trials = 8, p-value = 0.6367
## alternative hypothesis: true probability of success is less than 0.5
## 95 percent confidence interval:
##  0.0000000 0.8070971
## sample estimates:
## probability of success
##                    0.5
# 0.6367 > 0.05
# Reject H1
# The treatment is not effective

binom.test(2, 8, alternative = 'less')
##
##  Exact binomial test
##
## data:  2 and 8
## number of successes = 2, number of trials = 8, p-value = 0.1445
## alternative hypothesis: true probability of success is less than 0.5
## 95 percent confidence interval:
##  0.0000000 0.5996894
## sample estimates:
## probability of success
##                   0.25
# 0.1445 > 0.05
# Reject H1
# The treatment is not effective

before <- c(85,70,40,65,80,75,55,20)
after  <- c(75,50,50,40,20,65,40,25)
wilcox.test(before, after,paired=TRUE)
## Warning in wilcox.test.default(before, after, paired = TRUE): cannot
## compute exact p-value with ties
##
##  Wilcoxon signed rank test with continuity correction
##
## data:  before and after
## V = 32, p-value = 0.05747
## alternative hypothesis: true location shift is not equal to 0
# p= 0.05747 > 0.05
# reject H1
# treatment is not effective

## K-W Test

albumin <- c(3.1, 2.6, 2.9,3.8, 4.1, 2.9, 3.4,4.2,4,5.5, 5, 4.8)
group <- c(1 ,1 ,1 ,2 ,2 ,2 ,2 ,2 ,3 ,3 ,3 ,3)
boxplot(albumin~ group)

kruskal.test(albumin~ group)
##
##  Kruskal-Wallis rank sum test
##
## data:  albumin by group
## Kruskal-Wallis chi-squared = 7.5495, df = 2, p-value = 0.02294
## 0.02294 < 0.5
## Reject H0
##

## ANOVA

wt1 <- c(8,9,6,7,3)
wt2 <- c(2,4,3,5,1)
wt3 <- c(3,5,4,2,3)
wt4 <- c(2,2,-1,0,3)

weights <- c(wt1, wt2, wt3, wt4)
weights
##  [1]  8  9  6  7  3  2  4  3  5  1  3  5  4  2  3  2  2 -1  0  3
group <- c(rep(1,5),rep(2,5),rep(3,5),rep(4,5))
group
##  [1] 1 1 1 1 1 2 2 2 2 2 3 3 3 3 3 4 4 4 4 4
boxplot(weights ~ group)
abline(h = mean(wt1), col='red' )
abline(h = mean(wt2), col='blue' )
abline(h = mean(wt3), col='orange' )
abline(h = mean(wt4), col='green' )

oneway.test(weights ~ group , var.equal = TRUE)
##
##  One-way analysis of means
##
## data:  weights and group
## F = 8.5593, num df = 3, denom df = 16, p-value = 0.001278
a <- aov(weights ~ group)
summary(a)
##             Df Sum Sq Mean Sq F value   Pr(>F)
## group        1  62.41   62.41   18.56 0.000424 ***
## Residuals   18  60.54    3.36
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
?aov
## starting httpd help server ... done
?oneway.test

MaleA   <- c(22,25,26,27,24)
FemaleA <- c(21,19,18,24,25)
MaleB   <- c(14, 17, 19, 20, 17)
FemaleB <- c(21, 20 , 23, 27, 25)
MaleC   <- c(15, 16, 19, 14, 12)
FemaleC <- c(37, 34, 36, 26, 29)

timespan <- c(MaleA, FemaleA, MaleB, FemaleB, MaleC, FemaleC)

sex <- c(rep('Male', 5),rep('Female', 5),rep('Male', 5),
rep('Female', 5),rep('Male', 5),rep('Female', 5))
treatment <- c(rep('A',10),rep('B',10),rep('C',10))

par(mfrow=c(1,2))
boxplot(timespan ~ sex)
boxplot(timespan ~ treatment)

anova_res <- aov(timespan~factor(sex)*factor(treatment))
summary(anova_res)
##                               Df Sum Sq Mean Sq F value   Pr(>F)
## factor(sex)                    1  320.1   320.1  34.609 4.55e-06 ***
## factor(treatment)              2   68.6    34.3   3.708   0.0395 *
## factor(sex):factor(treatment)  2  532.5   266.2  28.782 4.21e-07 ***
## Residuals                     24  222.0     9.3
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1


text: Menshikov, Mikhail & Volkov, Stanislav (2008). Urn-related random walk with drift $\rho x^\alpha/t^\beta$. Electronic Journal of Probability 13: 944-960.


text:
# You're reading: Posts Tagged: Martin Gardner

### Maths Week Ireland 2012

As fans of maths, you’ll all be pleased to hear that in Ireland, they’ve basically got the correct attitude to maths, which is to say they dedicate a whole $\frac{1}{52}$ of their time to it. That’s right, they have an annual Maths Week, now in its ninth year, during which events are organised all over the country, the national and local media get involved, and generally try to get everyone talking about maths.

### Who wants to host a Celebration of Mind? There’s still time

This Sunday, 21st October 2012, marks what would have been the 98th birthday of Martin Gardner, American man of letters and numbers, as well as logic, puzzles, magic and scepticism. I had the good fortune to know Martin in the last decade of his life, and a more gentle and modest man you could not find, completely disproportionate to the forceful and wide influence he wielded for over 50 years as a science and mathematics journalist of the highest calibre.

The first time I met Martin he fooled me by showing me a tall thin glass and getting me to agree that its height exceeded its circumference, when in fact it didn’t.

### G4G Celebration of Mind 2012 Event Map is live

The event map for the Gathering for Gardner Celebration of Mind 2012 is now live, with nine events already listed in North and South America and Europe. More will surely be added in time.

These are parties “to celebrate the legacy of Martin Gardner on or around Sunday, October 21, 2012 through the enjoyment of Puzzles, Magic, Recreational Math, Lewis Carroll, Skepticism and Rationality”. The website explains:

Celebrations can be large or intimate depending on the event.  They don’t have to be formal and range from a few friends getting together to discuss the above all the way up to series of lectures at Universities and Science Cafes.

In 2010 there were 66 on five continents and in 2011 there were 70 events on all seven (yes, seven) . This will be the third Celebration of Mind.

More information: You can find an event to attend in your area or sign up to host one.

### In what flipping dimension is a square peg in a round hole just as good as a round peg in a square hole?

In what flipping dimension is a square peg in a round hole just as good as a round peg in a square hole?

Let’s start at the beginning.

My Plus magazine puzzle from March asks “Which gives a tighter fit: a square peg in a round hole or a round peg in a square hole?” By “tighter” we mean that a higher proportion of the hole is occupied by the peg.

### Carnival of Mathematics 85

 85 by brighterorange

#### Introduction

Welcome to a new Carnival of Mathematics! Traditionally the Carnival opens with facts about the number, this time 85, but first I have an important point of admin to address.

From Carnival of Mathematics 59 in November 2009 until Carnival of Mathematics 84 in December 2011, the Carnival was coordinated by Mike Croucher. Mike said in a blog post (of course!) that he had “had a lot of fun doing so” but that: “Recently, however, I have struggled to find the time to give the CoM the attention it deserves and so it is time to hand over the baton.” We should all be very grateful to Mike for his effort over these two years. You can still find Mike blogging over at Walking Randomly.

Now, as a result, the Carnival has a new home (including an index of previous Carnivals) at The Aperiodical, a not-quite-yet-formally-announced blogging collaboration between Katie Steckles, Christian Perfect and me. We’ll do our best to look after the series for the time being. If you’ve been following the Carnival for a while you’ll know it only works because of a parade of volunteer hosts. The next few Carnivals are lined up but if you’d like to volunteer to host one on your blog later in the year please contact Katie. The other necessary element is submissions, and we have plenty of these this time, so let’s get to business.

 Funky new Carnival logo by Katie Steckles

#### 85

This is Carnival of Mathematics 85. The ever-faithful Number Gossip tells me that 85 has no unique or even rare properties, being merely composite, deficient, evil, odd, square-free and a Smith number. Beyond mathematics, Wikipedia tells me that 85 is the atomic number of astatine, the ISBN Group Identifier for books published in Brazil and the lower bound (due to incomplete research) found by Jorge Stolfi (2004) for The Hollywood Constant, the smallest non-negative integer that has never been used in the title of a movie.

#### Serious mathematics

(Not that the rest isn’t!)

Brent Yorgey at The Math Less Traveled wrote a series of four posts in response to Depressing Expressions by Patrick Vennebush over at Math Jokes 4 Mathy Folks. In these, he is working to prove why a certain iterative arithmetic algorithm always results in a factorial. Brent says:

In particular I’m proving it using a *combinatorial* proof, a lovely proof technique that (in my opinion) isn’t used or taught as widely as it ought.

In part four Making Our Equation Count, he says,

I go through the different bits of the equation we’re trying to prove, and explain (with pictures) how to interpret each of them combinatorially.

Rebecka Peterson at Epsilon-Delta writes Extraneous Solutions of Log Equations–A Graphical Explanation. In submitting this post, Rebecka wrote:

Last semester a College Algebra student of mine asked why we sometimes get extraneous solutions when solving log equations. It was such a good question. I tried to do it justice in this post.

Drawing inspiration from the award of the Leroy P. Steele Prize for Mathematical Exposition to Aschbacher, Lyons, Smith, and Solomon for a work about the classification of finite simple groups, Gianluigi Filippelli at Doc Madhattan offers a post about finite simple groups and connections with physics in The classification of finite simple groups.

Frederick Koh from White Group Mathematics submitted Understanding MATTERS (4) saying:

Of late I noticed quite a few students (in online forums) experiencing difficulties in comprehending the concept of calculating distances between 3 dimensional vector planes in space, hence I am sharing an in depth explanation behind how things work.

Rohit Gupta at Kali & the Kaleidoscope investigates a new visualization of Möbius’ Mu, “a notorious function to classify all integers in three different boxes” in The 3 Pills of Möbius.

#### Art

Thomas Egense from Thomas’ mathematical adventures writes about an attempt to create mathematics-inspired art algorithmically using something called “Fractal flames”. Thomas defines fractal flames, details the algorithmic work involved, and gives several examples of the generated artwork, including the image below (used with permission).

Thomas Egense also submitted Dimensions (series of nine videos embedded below) calling this an “impressive graphical visualization of hard to grasp mathematical concepts like higher dimensions and topology”.

Last week Gathering for Gardner 10, the meeting of mathematicians, magicians, puzzlers and others inspired by the life and work of Martin Gardner, took place. Edmund Harriss, from Maxwell’s Demon, previewed his G4G10 talk in a blog post, The 2×1 rectangle and Domes. Edmund begins with the humble 2×1 rectangle, “not one of mathematics most celebrated shapes”, and ends up with structures built as accommodation at the Burning Man event.

#### Pop culture

Matt over at Math Goes Pop! writes in The Probability Games about the process used to select participants to take part in a fight to the death in the book and film The Hunger Games. Matt says “the rules here practically beg for some mathematical analysis” and Katie Steckles, who submitted the post, called this “just the kind of unnecessary mathematical analysis of a situation I like to see”.

#### Video game mathematics

Drawing inspiration from the recent arXiv preprint Classic Nintendo Games are (NP-)Hard bringing video games “out of nerdy obscurity and into cutting edge computer science”, Sam Alexander from Xamuel.com writes Toward the Mathematics of Video Game Glitches. Sam noticed a minor error in that article based on a glitch in Super Mario Brothers. Running with this theme, he defines video game glitches, game theoretically speaking, and focusing on “glitches which the player can exploit to win the game faster than intended”, defines a theorem (which he describes as “committing horrendous crimes against mathematics”!).

 SNES vs. Xbox Triple60 by avail

#### Performance and puzzles

Ben Nuttall writes about his experience as a maths busker.

“What is Maths Busking?” I hear you ask. Maths Busking is a street performance of mathematics whereby the buskers demonstrate mathematical ideas and engage the public in thinking like a mathematician

As well as a fuller explanation of maths busking, Ben shares some of the ‘busks’ he performed, his thoughts on the experience and some photos.

 Birmingham City Centre by Maths Busking

Katie Steckles, writing at The Aperiodical, discusses a variant of a popular mathematics ‘mind reading’ trick. I don’t want to spoil the puzzle in case you want to play along at home, so go over and check out On Disreputable Numbers.

Paul Taylor, also at The Aperiodical, writes about a puzzle he designed for Katie Steckles’ Puzzlebomb. Puzzlebomb is a monthly puzzle sheet featuring all-new types of puzzles. Following the release of the April Puzzlebomb, Paul made the following claim on Twitter:

I guarantee there has never been, and will never be, another puzzle quite like Hilbert’s Space Filling Crossword

In Words to Fill Space he justifies this assertion and describes how he created the puzzle.

Paul also writes, again at The Aperiodical, about a class of puzzle in which a number of prisoners are all given hats and their fate depends on their ability to correctly determine the colour of their own hat. Paul offers “a nice variation on the theme that I heard about at a recent MathsJam” as Another black and white hats puzzle.

Maths Jam is a monthly meeting of maths enthusiasts in pubs worldwide to share stuff they like. “Puzzles, games, problems, or just anything they think is cool or interesting“. A recent development is blogging roundups of what happened at Maths Jam meetings. Recent outings, full of puzzles and mathematical goodies, include: Newcastle (February), Manchester (March), London (February) and Melbourne (January), and a set of photos from various February Maths Jam meetings.

 Maths Jam London February 2012

Last month I attended Newcastle Maths Jam. While there we played with a puzzle that Out of the Norm states as:

By relabelling the faces of two dice, can you design a new, unusual pair of six-sided dice that achieves rolls with the same frequencies as a pair of normal dice? All the faces must have a positive number of spots.

Dice and Dissection: a puzzle discusses this puzzle and gives a nice diagrammatic way to view the solution.

#### History and society

John Cook of The Endeavour writes about a wedding invitation written under the collective pseudonym of that “semi-secret group of French mathematicians”, Nicolas Bourbaki. In Nicolas Bourbaki’s wedding invitation, John explains some of the mathematical references and reveals how the invitation “nearly cost Bourbaki member André Weil his life”.

A post on Pat’sBlog goes to primary sources to highlight an error in Wikipedia in relation to the origins of the four-fours problem. That is,

using four fours and whatever mathematical operations that were allowed to make a number, or a set of numbers.

The origins of the problem apparently lie in earlier similar problems, including a problem of four threes. Read about it in Before There Were Four-Fours, There Were Four-Threes.

Guillermo Bautista at Mathematics and Multimedia writes A US President’s Proof of the Pythagorean Theorem about the proof of the Pythagorean theorem formulated by James Garfield, the 20th president of the United States of America.

Alexander Bogomolny of CTK Insights offers a little piece of mathematics in the history of chronology given in Florian Diacu’s The Lost Millennium in a post entitled Chinese Remainder Theorem: an Application to Chronology. Submitting this post, Alexander said:

Chinese Remainder Theorem is a staple of early puzzle books, both European and Eastern. This is very satisfying to learn that the theorem finds practical and important applications in the science of chronology. The book by Florian Diacu where I found this application is an exquisitely written compendium of history and mathematics of calendrical calculations. The book deserves every praise and gets my wholehearted recommendation.

In a blog post here on Travels in a Mathematical World, I was very taken with an answer given to a question about working outside traditional academia by Neil deGrasse Tyson in an interview with Samuel Hansen for the Strongly Connected Components Podcast episode 45. The whole interview (18 mins) is worth listening to. My reflections can be found as Culturally an academic.

Over at Second-Rate Minds, Samuel Hansen writes about The True Importance of Friends, explaining the origin of a result in social network theory and its implications in epidemiology.

I really noticed the absence of the Carnival back in February when I thought I might submit a couple of blog posts which got a particularly warm reaction and found the submissions form deactivated. The posts were Apparently Gauss got in this bar fight with Hilbert… and the follow-up Why do we enjoy maths history misconceptions?

#### Technology

John Chase from Random Walks writes about the surprising features of Microsoft Office Equation Editor, in which he makes the bold claim: “LaTeX lovers will love it”. Find out why at Microsoft Office Equation Editor.

I’ve saved the last word for this revived Carnival to the previous coordinator, Mike Croucher. Mike’s month of math software for March 2012 offers the latest news in the world of mathematical software (a month of math software has been a monthly series since January 2011).

#### The End

Well, that’s that for this Carnival. You can help spread the word by blogging, tweeting, etc. about the revival of the Carnival and directing people to this edition. If you’re hungry for more mathematics blog posts then there’s the previous Carnival of Mathematics 84, the latest Math Teachers At Play Carnival 48 is over at Math Is Not A Four Letter Word and you can get a weekly selection of blog posts from the Mathblogging.org Weekly Picks.

Future outings for the Carnival of Mathematics are queued over at The Aperiodical. Carnival of Mathematics 86 will be posted in May by Brent at The Math Less Travelled. Submissions for this are now open, so keep your eye out for great posts and get writing!


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# Second Variation of an Action: Part I

So your advisor asks you, “what are the background fluctuations for your action?” If you don’t know what they are talking about, you’ve come to the right place. In the meantime, just smile and wave guys $–$ just smile and wave.

Over this three part post series, you will realize that finding the background fluctuations of an action amounts to determining the the second variation of an action. It might not be clear right now, but we will build to it.

Suppose you have an action of one variable with the canonical form $$\mathcal{S}[\phi] = \int\text{d}^4x\mathcal{L}(\phi(x),\partial_\mu\phi(x))$$ To obtain the second variation of an action, we will have to state the definition of a functional’s derivative.

Functional Derivative: Given a manifold, $\mathcal{M}$ representing smooth functions $\phi$, accompanied with relevant boundary conditions, and a functional $\mathcal{F}:\mathcal{M}\to\mathbb{R}$, a functional derivative of $\mathcal{F}[\phi]$ is obtained through $$\mathcal{F}[\phi+\delta\phi]=\mathcal{F}[\phi]+\int\prod_{n\ge 1}(\text{d}^dx)^n\frac{\delta^n\mathcal{F}[\phi]}{(\delta\phi)^n}(\delta\phi)^n$$ with $n\in\mathbb{N}.$

While this may not seem useful at first, this is essentially the only definition one needs to perform the second variation of an action. The action, according to classical mechanics, is simply a functional and may be expanded accordingly: $$\mathcal{S}[\phi+\delta\phi] = \mathcal{S}[\phi] +\int\text{d}^dx \frac{\delta\mathcal{S}[\phi(z)]}{\delta\phi(x)}\delta\phi(x) + \int\text{d}^dx\int\text{d}^dy\frac{\delta^2\mathcal{S}[\phi(z)]}{\delta\phi(x)\delta\phi(y)}\delta\phi(y)\delta\phi(x)+\dots$$ While we are effectively done, and could easily introduce a Lagrangian, we can first employ the Principle of Least Action:

Principle of Least Action: The path taken by a system between two times is one for which the action is stationary to first order, i.e. $\delta S=0$ to first order.

Generally, the principle dictates that the extrema of the action in functional space vanishes, but one typically takes the first order variation to be a minima and sets it to zero accordingly. With that in mind, we have $$\delta \mathcal{S}= \mathcal{S}[\phi+\delta\phi]-\mathcal{S}[\phi]= \int\text{d}^dx\int\text{d}^dy\frac{\delta^2\mathcal{S}[\phi(z)]}{\delta\phi(x)\delta\phi(y)}\delta\phi(y)\delta\phi(x)+\dots\ \ .$$ For an example, consider the Klein-Gordon Lagrangian given by, $$\mathcal{L}(\phi,\partial_\mu\phi)=\frac{1}{2}\partial_\mu\phi\partial^\mu\phi-\frac{1}{2}m^2\phi^2.$$ Then the variation of the action to second order becomes, in $\text{d}=4$, \begin{align} \delta\mathcal{S}&=\int\text{d}^4x\int\text{d}^4y\int\text{d}^4z\frac{\delta\mathcal{L}}{\delta\phi(x)\phi(y)}\delta\phi(y)\delta\phi(x)\\ &= \int\text{d}^4x\int\text{d}^4y\frac{\delta}{\delta\phi(x)}\left[\int\text{d}^4z\frac{\delta\mathcal{L}(\phi(z),\partial_\mu\phi(z))}{\delta\phi(y)}\right]\delta\phi(y)\delta\phi(x)\\ &=\int\text{d}^4x\int\text{d}^4y\frac{\delta}{\delta\phi(x)}\left[\int\text{d}^4z\left(\frac{\partial\mathcal{L}}{\partial\phi(z)}\frac{\delta\phi(z)}{\delta\phi(x)} +\frac{\partial\mathcal{L}}{\partial(\partial_\mu\phi(z))}\frac{\delta(\partial_\mu\phi(z))}{\delta\phi(y)} \right) \right]\delta\phi(y)\delta\phi(x). \end{align} As the variational derivative and the partial derivative act on different objects, they may be commuted past one another. This allows us to express the above line as \begin{align} \delta\mathcal{S} &=\int\text{d}^4x\int\text{d}^4y\frac{\delta}{\delta\phi(x)}\left[\int\text{d}^4z\left(\frac{\partial\mathcal{L}}{\partial\phi(z)}\frac{\delta\phi(z)}{\delta\phi(x)} +\frac{\partial\mathcal{L}}{\partial(\partial_\mu\phi(z))}\partial_\mu\left(\frac{\delta\phi(z)}{\delta\phi(y)}\right) \right) \right]\delta\phi(y)\delta\phi(x). \\ \end{align} With the definition  $$\frac{\delta\phi(z)}{\delta\phi(y)}=\delta^d(z-y)$$ with $d$ being the dimension of the delta function, we have \begin{align} \delta\mathcal{S} &=\int\text{d}^4x\int\text{d}^4y\frac{\delta}{\delta\phi(x)}\left[\int\text{d}^4z\left(\frac{\partial\mathcal{L}}{\partial\phi(z)}\delta^4(z-y) + \frac{\partial\mathcal{L}}{\partial(\partial_\mu\phi(z))}\partial_\mu\left(\delta^4(z-y)\right) \right) \right]\delta\phi(y)\delta\phi(x). \end{align} We may integrate the second term by parts; \begin{align} \int\text{d}^4z\partial_\mu\left[\frac{\partial\mathcal{L}}{\partial(\partial_\mu\phi(z))}\delta^4(z-y)\right]=\int\text{d}^4z\partial_\mu\left[\frac{\partial\mathcal{L}}{\partial(\partial_\mu\phi(z))}\right]\delta^4(z-y)+\int\text{d}^4z\frac{\partial\mathcal{L}}{\partial(\partial_\mu\phi(z))}\partial_\mu\left[\delta^4(z-y)\right]. \end{align} The term on the left hand side is a total divergence. By the divergence theorem of an ordinary vector valued function with a volume measure, such as $\int\text{d}^4z$, we integrate the argument on the boundary of the volume. As the boundary of a four dimensional spacetime is a three$-$dimensional hypersurface with positive curvature, we assert that the boundary is strictly a spatial boundary. Thus our manifold with a Lorentzian metric, $(\mathcal{M},\eta_{\mu\nu})$, has a boundary with a hypersurface metric, $(\partial\mathcal{M},h_{ij})$ where $\mu,\nu=0,1,2,3$ and $i,j=1,2,3.$ Given these conditions, the divergence theorem may be stated as:

Divergence Theorem: Let $(\mathcal{M},\eta_{\mu\nu})$ be a manifold with a hypersurface boundary, $(\partial\mathcal{M},h_{ij})$. Then the volume integral of a total divergence is related to the boundary by $$\int_\mathcal{M}\text{d}^nx\sqrt{|\det(\eta_{\mu\nu})|}\nabla_\mu X^\mu = \int_{\partial\mathcal{M}}\text{d}^{n-1}x\sqrt{|\det(h_{ij})|}n_i X^i$$ with $n_i$ being the unit normal to the boundary.

Given the above theorem the total divergence term becomes \begin{align} \int_{\mathcal{M}}\text{d}^4z\ \partial_\mu\left[\frac{\partial\mathcal{L}}{\partial(\partial_\mu\phi(z))}\delta^4(z-y)\right] &= \int_{\partial\mathcal{M}}\text{d}^3z\ n_i\left(\frac{\partial\mathcal{L}}{\partial(\partial_i\phi(z))}\delta^3(z-y)\right)\\ &= n_i\left(\frac{\partial\mathcal{L}}{\partial(\partial_i\phi(z))}\right)\bigg|_{\partial\mathcal{M}} = 0. \end{align} This last line is a common assumption, that being that the fields, and their derivatives, are only relevant locally. The reasoning for this assumption is that we can take the distance of the boundary from the point being evaluated to be as large as necessary to cause the fields to vanish. If one does not like this explanation, an actual reason to why we are allowed to make this assumption may be found in Compact Support. With these simplifications, the variation in the action becomes \begin{align} \delta\mathcal{S} &=\int\text{d}^4x\int\text{d}^4y\frac{\delta}{\delta\phi(x)}\left[\int\text{d}^4z\left(\frac{\partial\mathcal{L}}{\partial\phi(z)}\delta^4(y-x) – \partial_\mu\left(\frac{\partial\mathcal{L}}{\partial(\partial_\mu\phi(z))}\right)\delta^4(y-x) \right) \right]\delta\phi(y)\delta\phi(x)\\ &=\int\text{d}^4x\int\text{d}^4y\frac{\delta}{\delta\phi(x)}\left[\int\text{d}^4z\left(\frac{\partial\mathcal{L}}{\partial\phi(z)} – \partial_\mu\left(\frac{\partial\mathcal{L}}{\partial(\partial_\mu\phi(z))}\right)\right)\delta^4(z-y)\right]\delta\phi(y)\delta\phi(x)\\ &=\int\text{d}^4x\int\text{d}^4y\frac{\delta}{\delta\phi(x)}\left[\left(\frac{\partial\mathcal{L}}{\partial\phi(y)} – \partial_\mu\left(\frac{\partial\mathcal{L}}{\partial(\partial_\mu\phi(y))}\right)\right) \right]\delta\phi(y)\delta\phi(x). \end{align} This is an optimal point to introduce the form of the Klein$-$Gordon Lagrangian: \begin{align} \delta\mathcal{S} &= \int\text{d}^4x\int\text{d}^4y\frac{\delta}{\delta\phi(x)}\left[ -m^2\phi(y)-\partial_\mu(\partial^{\mu}\phi(z)) \right]\delta\phi(y)\delta\phi(x)\\ &=\int\text{d}^4x\int\text{d}^4y\left[-m^2\frac{\delta\phi(y)}{\delta\phi(x)}-\partial_\mu\left(\partial^{\mu}\left(\frac{\delta\phi(y)}{\delta\phi(x)}\right)\right)\right]\delta\phi(y)\delta\phi(x)\\ &=\int\text{d}^4x\int\text{d}^4y\left[\delta\phi(y)\left(-m^2-\partial_\mu\partial^\mu\right)\delta^{4}(y-x)\right]\delta\phi(x). \end{align} While we are more or less finished finding the second variation, the physics of the situation is better understood when we transfer the $\partial_\mu\partial^\mu$ operation onto the variation of the fields instead of the delta function. Before we embark on this procedure, note that the boundary conditions for the fields also apply to the variation of the fields: $$\phi\bigg|_{\partial\mathcal{M}}\to0,\ \partial_\mu\phi\bigg|_{\partial\mathcal{M}}\to0,\ \delta\phi\bigg|_{\partial\mathcal{M}}\to0, \text{ and } \partial_\mu(\delta\phi)\bigg|_{\partial\mathcal{M}}\to0$$ Then the term in question may be found from integrating by parts twice and applying the relevant boundary conditions. This type of integration is strictly not legal. Since the delta function is not a continuous function, it cannot be differentiable either. Only in the context of integration does the conditions of a function make sense for the delta function. When integrating by parts, we are instead relying on the fact that $\phi$ is smooth and has compact support in an open set of $\mathcal{M}$ in order to transfer the derivatives onto $\delta\phi$:Then the term in question may be found from integrating by parts twice and applying the relevant boundary conditions. This type of integration is strictly not legal. Since the delta function is not a continuous function, it cannot be differentiable either. Only in the context of integration does the conditions of a function make sense for the delta function. When integrating by parts, we are instead relying on the fact that $\phi$ is smooth and has compact support in an open set of $\mathcal{M}$ in order to transfer the derivatives onto $\delta\phi$: \begin{align} &\int_{\mathcal{M}}\text{d}^4y\delta\phi(y)\partial_\mu\partial^\mu(\delta^4(y-x)) = -\int_\mathcal{M}\text{d}^4y\partial_\mu(\delta\phi(y))\partial^\mu(\delta^4(y-x)) = +\int_\mathcal{M}\text{d}^4y\partial^\mu\partial_\mu(\delta\phi)\delta^4(y-x) \end{align} Loosely speaking, the total divergence term on the left hand side underwent a procedure similar to integration by parts. Basically integrate once, transfer the derivative from the delta function to the variation, use compact support to kill the boundary integral. Then repeat the process to transfer the second derivative on the delta function to the variation of $\phi.$ Putting all of this together, we find \begin{align} \delta\mathcal{S} &=\int\text{d}^4x\int\text{d}^4y\ \delta\phi(x)\left[\delta\phi(y)\left(-m^2-\partial_\mu\partial^\mu\right)\delta^{4}(y-x)\right]\\ &=\int\text{d}^4x\int\text{d}^4y\ \delta\phi(x) \left[\delta^{4}(y-x)\left(-m^2-\partial_\mu\partial^\mu\right)\delta\phi(y)\right] \\ &=\int\text{d}^4x\ \delta\phi(x)(-m^2-\partial_\mu\partial^\mu)\delta\phi(x), \end{align} where in the last line, we have used up the delta function to kill an integral. This is it! We’re finished. The last bit is to interpret the result form the physical standpoint. If we call $\delta\phi(x)=\chi(x)/\sqrt{2}$, we find \begin{align} \delta\mathcal{S} &= \frac{1}{2}\int\text{d}^4x(-\chi(x) m^2\chi(x)-\chi(x)\partial_\mu\partial^\mu\chi(x))=\frac{1}{2}\int\text{d}^4x(-m^2\chi^2(x)+\partial_\mu\chi(x)\partial^\mu\chi(x))\\ &=\int\text{d}^4x\mathcal{L}^{(1)}(\chi(x),\partial_\mu\chi(x)), \end{align} where $\mathcal{L}^{(1)}(\chi(x),\partial_\mu\chi(x))$ has the form $$\mathcal{L}^{(1)}(\chi(x),\partial_\mu\chi(x)) = \frac{1}{2}(\partial_\mu\chi(x)\partial^\mu\chi(x)-m^2\chi^2(x)).$$ This is a remarkable form for this mysterious $\mathcal{L}^{(1)}$ as it looks exactly like the Klein-Gordon Lagrangian with a field $\chi(x)$ instead of $\phi(z).$ I call these functions Lagrangians of Type 1, or $\mathcal{L}^{(1)}$ for short. $\mathcal{L}^{(1)}$ are the associated energy densities describing the fluctuations of a scalar field.
Setting $\delta\mathcal{S}=\mathcal{S}^{(1)}$, we have an Action of Type 1, given by $$\mathcal{S}^{(1)}=\int\text{d}^4x\mathcal{L}^{(1)}(\chi(x),\partial_\mu\chi(x)),$$ which we could extremize and demand that the Principle of Least Action holds for Type 1 actions as well. In doing so, we would obtain the Euler Lagrange equations  $$\frac{\delta\mathcal{S}^{(1)}}{\delta\chi(x)} = 0 =\frac{\partial\mathcal{L}^{(1)}}{\partial\chi(x)} – \partial_\mu\left(\frac{\partial\mathcal{L}^{(1)}}{\partial(\partial_\mu\chi(x))} \right).$$

These above equations describe what we have been looking for; the equations of motion for the background, i.e. the background fluctuations.

## 2 thoughts on “Second Variation of an Action: Part I”

1. Andy Knapp says:

The term
$$\oint_{\partial\partial\mathcal{M}} \delta^2(y-x)\delta\phi(y)\vec{n} \cdot \text{d}^2\vec{y}$$
vanishes by some exterior-derivative powered abstract nonsense, and therefore you’ve shown the vector field $X^\mu$ is divergence-free. I’m not sure that’s what you wanted to show…

$$\int_{\partial\partial\mathcal{M}} \omega = \oint_{\partial\mathcal{M}}d\omega = \oint_{\mathcal{M}}dd\omega = 0$$

1. Awesome comment. I’ve changed it now to reflect this oversight.

For context: I attempted to use the divergence theorem twice to remove the Laplacian on the delta function onto the scalar variation $\delta\phi$. That was wrong.

Actual solution: Integrate by parts and don’t forget Calc III.