Functions Definition and 1000 Threads

Hello! (Wave) In $C^1[0,1]$ calculate the distances between the functions $y_1(x)=0$ and $y_2(x)=\frac{1}{100} \sin(1000x)$ in respect to the weak and strong norm.That's what I have tried:Weak norm: $||y_1-y_2||_w=\max_{0 \leq x \leq 1} |y_1(x)-y_2(x)|+ \max_{0 \leq x \leq 1} |y_1'(x)-y_2'(x)...

Homework Statement (a) Show that the set of all square-integrable functions is a vector space. Is the set of all normalised functions a vector space? (b) Show that the integral ##\int^{a}_{b} f(x)^{*} g(x) dx## satisfies the conditions for an inner product. Homework Equations The main...

Can you show me as explicitly as possible through equations?

More than just a few problems that happen to pop up in the textbook, I mean.

I am trying to calculate a pole of f(z)=http://www4b.wolframalpha.com/Calculate/MSP/MSP86721gicihdh283d613000033ch4ae4eh37cbd4?MSPStoreType=image/gif&s=35&w=44.&h=40. . The answer in the textbook is: Simple pole at...

As part of a personal musicology project I found myself with the mathematical model of a geometry which utilizes the equation a*(a/b)sin(pi*x) The only problem with this is that I need to take the integral from -1/2 <= x <= 1/2, and according to Wolfram Alpha no such integral exists. I can...

In the GRW spontaneous collapse model (for example) the wave-function evolves by linear Schrödinger equation, except, at random times, wave-function experiences a jump of the form: \psi_t(x_1, x_2, ..., x_n) \rightarrow \frac{L_n(x)\psi_t(x_1, x_2, ..., x_n)}{||\psi_t(x_1, x_2, ..., x_n)||}...

In other words, why does a function have to be onto (or surjective, i.e. Range=Codomain) for its inverse to exist?

Hey! :o Can we enumerate the primitive recursive functions?

Where can I find and how can I derive the orthogonality relations for Hankel's functions defined as follows: H^{(1)}_{m}(z) \equiv J_{n}(z) +i Y_{n}(z) H^{(2)}_{m}(z) \equiv J_{n}(z) - i Y_{n}(z) Any help is greatly appreciated. Thanks

Suppose we have a complex function f(z) with simple poles on the complex plane, and we know exactly where these poles are located (but we don't know how the function depends on z) Is there any way to build up the exact form of f(z) just from its poles?

I am feeling stupid today...so: Is the following statement true? \int_{-\infty}^{+\infty} f(x) g(x) \ge 0 if f(x) \ge 0 and 0< \int_{-\infty}^{+\infty} g(x) < \infty (integral converges)? If yes, then how could I show that? (It's not a homework) I am trying to understand how the...

Let me post this question again in a slightly modified form. On the attached picture the path integral for the partion function: Z = Tr(exp(-βH)) Now according to what it says on the picture it should be easy from this to get the Green's function in the path integral formalism. The Green's...

Hello! I'm currently taking a course on partial differential equations, and we're using Asmar's textbook. We've reached Green's functions, and even though Asmar is a great book, I feel like I need a deeper study of the subject. Which book would you recommend to help me better grasp the theory...

Homework Statement (4a/a+4)+(a+2/2a) Homework Equations Just combine and then factor out The Attempt at a Solution It's actually fairly simple, but I'm having difficulty at the end. /multiply each term by opposite denominator 4a(2a)/a+4(2a) + a+2(a+4)/2a(a+4) /combine 4a(2a)+(a+2)(a+4) /...

I work a good deal better when the equation is in x and y form, is it possible to set up a trig expression like 5Cos(x)/(Sin(x)-1)and substitute the proper x or y equivalent so long as I remember to replace the trig identities later when the problem is finished? Or can you just not solve these...

I'm trying to create a vector field plot of an equation in x and y. Basically, I would like to create a function F(x, y) = p(x, y)i + q(x, y)j that defines a force field, and have the field direction and magnitude plotted at points in the x-y plane, and both components of the vector are...

I am trying to solve a Duffing's equation ##\ddot{x}(t)+\alpha x(t)+\beta x^3(t)=0## where ##\alpha## is a complex number with ##Re \alpha<0## and ##\beta>0##. The solution can be written as Jacobi elliptic function ##cn(\omega t,k)##. Then both ##\omega## and ##k## are complex. The solution to...

Hi, I am trying to solve a model where Non-interacting Green functions take part it. It has happened something that is spinning my head and I hope someone could help. The non interacting Green function for a chanel of electrons is...

Homework Statement The problem ask for points of intersection of two functions Homework Equations 1: 2x+y-4=0 2: (y^2)-4x=0 The Attempt at a Solution My attempt of solution its in a picture attached below... I get stuck in this two equations 1: ((y^2)/4)+(y/2)-2=0 2: square...

Hey! :o According to the book that I'm reading, we can define the $\mu-$recursive functions inductively, as follows: The constant, projection, and successor functions are all $\mu-$recursive. If $g_1, \dots , g_m$ are $n-$variable $\mu-$recursive functions and $h$ is an $m-$variable...

1) Problem: given that x is an obtuse angle for which cos^2x/(1 + 5sin^2x) = 8/35, find the value of cosx/(1 - 5 sin x) without evaluating x. 2) Relevent equations: sin(-x) = - sin x cos(-x) = cos x sin(180° - x) = sin x cos(180° - x) = - cos x sin^2x + cos^2x = 1 3) Attempt: cos^2x/(1 +...

I was wondering what the physical insight is of integrating a product of two functions. When we do that for a Fourier transform, we decompose a function into its constituent frequencies, and that's because the exponential with an imaginary x in the transform can be seen as a weighting function...

Homework Statement I am trying to prove without using the mean value theorem that two different functions with the same derivative differ only by a constant. Is it possible to do this without the mean value theorem? If so, would someone help guide me towards the right solution. Homework...

Homework Statement Show that the number of injections from ##E \rightarrow F ##, ##E,F## finite sets, ## p = \#E ,\ n = \#F,\ p\le n ##, is ##\frac{n!}{(n-p)!} ## Then, find the number of functions from ##E \rightarrow F ##. Homework Equations ## A = \{ \text{injections from }E \rightarrow F...

I am confused about the ''4 basic types'' of generating functions. I have searched for this a bit on google but haven't found anything that truly made the click for me on this concept so I'll try here: What I do understand and need no elaboration on: 1) When considering the Hamiltonian and...

Homework Statement Homework Equations [/B] ---------------------- [FONT=Courier New]function [C]=mymatmult(A,B) [L1 C1]=size(A); [L2 C2]=size(B); if C1 ~= L2 error('dimension mismatch'); end %if ERROR C=zeros(L1,C2); for i=1:L1 for j=1:C2 C(i,j)=A(i,:)*B(:,j); end %in for end %out...

I've run across a system of recursive functions (call them f and g). The system looks like this: f(x) = a f(x-1) + b g(x-1) g(x) = a g(x-1) + c f(x-1) I also know that f(0) = 0 and g(0)>0. Finally, I know for other reasons that are too complicated to go into here that the system is somehow...

New to composite functions here. Lesson has been vague and unhelpful.. again. Here is what I've worked on so far but not sure on the last equation in particular, or that I have done my multiplication properly when working with a squared set of brackets, multiplied by an number.. (b and c) Any...

Homework Statement A rectangular region of 125,000 sq ft is fenced off. A type of fencing costing $20 per foot was used along the back and front of the region. A fence costing $10 per foot was used for the other sides. What were the dimensions of the region that minimized the cost of the...

what is the relationship between special functions and integration ? why integral of some function like (sqrt(ln(x)) and (cos(1/x) and more) are entering us to special functions?? PLEASE HELP ME TO UNDERSTAND.

Hi guys, just having some confusions on the Delta-Epsilon proofs for multivariable limit functions. here is my question: Apply Delta-Epsilon proof for the Lim (x,y) --> (0,0) of (y^3 + 5x^2y)/(y^2 + 3y^2) to show the limit exists. The part that has me confused is the y to the power of 3, where...

Hi. Here is one example from my book. Calculate Fourier transform of signal: Here is solution: We can write x(n) as: , where x1(n) is u(n+N)-u(n-N-1). We can write: (we used that cos(n)=(1/2)*(exp(j*n)+exp(-j*n)). Using properties of Fourier transform of discrete signal: , Fourier...

Hi PF! Can any of you help me determine a good measure for how "different" two functions are from each other? I've thought of using something like ##\int_\Omega (f-g)^2 \, dx##. Can anyone recommend a good technique and direct me to the theory so I can understand it well? Thanks so much! Josh

Homework Statement Determine whether or not the following functions are harmonic: u = z + \bar{z} u = 2z\bar{z} Homework Equations z = u(x,y) + v(x,y)i \bar{z} = u(x,y) - v(x,y)i A function is harmonic if Δu = 0. The Attempt at a Solution Δu = Δz +Δ \bar{z} = u_{xx} + v_{xx} +...

Hello everybody, For my thermodynamics test I have to tell whether or not a quantity is a state function, which is obviously not all too difficult when regarding entropy, enthalpy etc. on their own. However there are a lot of questions where it is about "H-S" or "G-H". Are these not always...

Write the piecewise function \[ f(t) = \begin{cases} 2t, & 0\leq t < 3 \\ 6, & 3 \le t < 5 \\ 2t, & t \ge 5 \\ \end{cases} \] in terms of unit step functions. So here is what i;ve got just guessing , I don't think I'm correct. I really need some help. But I got...

Homework Statement Homework EquationsThe Attempt at a Solution For x>b, Ψ(x) = Ae-ikx + Beikx , where k = (√2mE)/hbar a<x<b Ψ(x) = Ce-ik'x + Deik'x , where k = (√2m(U2 - E)/hbar This is the problem part 0<x<a Ψ(x) = Fsink''x...

Determine the value of $\dfrac{2k^2}{k-1}$ given $\dfrac{k^2}{k-1}=k^2-8$.

hey! How do you work out the domain and range of fg(x), do you work out what range of g(x) will fit the domain of f(x)? I have no clue. Thanks

People say that if you could break a function down into these three functions (constant, successor, projection or sometimes called initial/basic functions) using some operators, then it is primitive recursive. What makes these three functions so special?

y=CektA) First find k. [Hint:Use the given information of y=100 when t=2, and y=300 when t=4 to compute k.] B) Finally, find the value for C. [Hint use ine of the two pieces of information given in the problem to solve for C. in other words, use either y=100 when t=2 or use y=300 when 4=4 to...

Homework Statement ∫δ(x3 - 4x2- 7x +10)dx. Between ±∞. Homework EquationsThe Attempt at a Solution Well I don't really know how to attempt this. In the case where inside the delta function there is simply 2x, or 5x, I know the answer would be 1/2 or 1/5. Or for say δ(x^2-5), the answer would...

Homework Statement If f:(2,4)-->(1,3) where f(x)=x-[x/2] (where[.] denotes the greatest integer function), then find the inverse function of f(x). Homework Equations (None I believe.) The Attempt at a Solution I know that for a function to be invertible, it must be both one-one and onto...

Homework Statement 1. Range of the function ## \sqrt {x^2+x+1} ## is equal to? 2.ƒ:R---->R is defined as ƒ(x) = x2 -3x +4, then f -1 (2) is equal to?Homework Equations NA The Attempt at a Solution For the first one tried squaring on both the sides but that does not give linear x in terms of...

Homework Statement Recall that we have defined the Gaussian ##f_s## by ##f_s (t) = \sqrt{s}e^{-st^2}## and shown that ##\hat{f_s}(\lambda) = \frac{1}{\sqrt{2}}e^{\frac{-\lambda^2}{4s}}##. Show that ##f_3 \ast f_6 (t) = \sqrt{\pi}f_{1/2}(t) = \sqrt{\pi/2}e^{-t^{2}/2}## The Attempt at a...

1. Homework Statement A graph of y=f(x) is shown. Find the following function values and justify your answers. f(30)= f(-14)= Homework EquationsThe Attempt at a Solution I know the graph is periodic, I know it's max and min, and I know it's amplitude because of that. But I don't know what...

Hi All, Having a tough time with this one and I'm not sure why. Need to state amplitude, period and phase shift of f(x)=3cos2[x-(π/4)]+1. Amplitude being 3, period being 2π/2=π and phase shifted (π/4) to the right. Midline would also be at y=1 Good so far? Right, so I know that 1/4 phase...

Homework Statement 1 kg air at the pressure ##10^6##Pa and the temperature ##125^\circ C = 398K## expand until the volume is 5 times larger. The expansion is done with change in heat at every moment being ##1/4## of the work done by the gas. Calculate the end pressure.Homework Equations ##dU...

Homework Statement Specify the domain and range of f(x, y) = arccos(y − x2). Indicate whether the domain is (i) open or closed, and (ii) bounded or unbounded. Give a clear reason in each case.Homework EquationsThe Attempt at a Solution y-x2 >= -1 y >= x2 -1 y-x2 <= 1 y <= x2 +1 I sketched it...