In mathematics, a function is a binary relation between two sets that associates to each element of the first set exactly one element of the second set. Typical examples are functions from integers to integers, or from the real numbers to real numbers.
Functions were originally the idealization of how a varying quantity depends on another quantity. For example, the position of a planet is a function of time. Historically, the concept was elaborated with the infinitesimal calculus at the end of the 17th century, and, until the 19th century, the functions that were considered were differentiable (that is, they had a high degree of regularity). The concept of a function was formalized at the end of the 19th century in terms of set theory, and this greatly enlarged the domains of application of the concept.
A function is a process or a relation that associates each element x of a set X, the domain of the function, to a single element y of another set Y (possibly the same set), the codomain of the function. It is customarily denoted by letters such as f, g and h.If the function is called f, this relation is denoted by y = f (x) (which reads "f of x"), where the element x is the argument or input of the function, and y is the value of the function, the output, or the image of x by f. The symbol that is used for representing the input is the variable of the function (e.g., f is a function of the variable x).A function is uniquely represented by the set of all pairs (x, f (x)), called the graph of the function. When the domain and the codomain are sets of real numbers, each such pair may be thought of as the Cartesian coordinates of a point in the plane. The set of these points is called the graph of the function; it is a popular means of illustrating the function.
Functions are widely used in science, and in most fields of mathematics. It has been said that functions are "the central objects of investigation" in most fields of mathematics.
Hi,
I am having to refresh my oscilloscope knowledge and am confused about one last function generator setting... Burst period.
If I have 1 cycle at say 700 kHz it is 1.43us. If I set number of cycles to 10 then that is 10 * 1.43us = 14.3us time. This is my ON burst.
So what is the burst...
Hi PF!
Given the following ODE $$(p(x)y')' + q(x)y = 0$$ where ##p(x) = 1-x^2## and ##q(x) = 2-1/(1-x^2)## subject to $$y'(a) + \sec(a)\tan(a)y(a) = 0$$ and $$|y(b)| < \infty,$$ where ##a = \sqrt{1-\cos^2\alpha} : \alpha \in (0,\pi)## and ##b = 1##, what is the Green's function?
This is the...
Hi PF!
I'm numerically integrating over a Green's function along with a few very odd functions. What I have looks like this
NIntegrate[-(1/((-1.` + x)^2 (1.` + x)^2 (1.` + y)^2))
3.9787262092516675`*^14 (3.9999999999999907` +
x (-14.99999999999903` +
x (20.00000000000097` -...
I have a question about statistical physics. Suppose we have a closed container with two compartments, each with volume V , in thermal contact with a heat bath at temperature T, and we discuss the problem from the perspective of a canonic ensemble. At a certain moment the separating wall is...
hi guys
I am recently taking a Nuclear structure course, and have a lot of questions regarding the nuclear rotor model.
in most nuclear physics books the I have, the wave function associated with the rotor model of the nucleus is written in terms of the Wigner D functions , like the expression...
I want to determine the normal flow depth in a perfectly horizontal circular conduit. The system characteristics are known (Internal pipe diameter, Mannings roughness, Discharge). However, I am not sure how to calculate the normal flow depth. When using Manning's equation one can find the normal...
From my reading of several quantum optics textbooks and spectroscopy texbooks, the emission and absorption spectrum of an atom or molecule are always given in terms of the time-correlation function, for example the emission spectrum of a two level atom is given by:
$$...
Write and test the fib function in two linked files (Fib.asm, fib_main.asm). Your solution must be made up of a function called fib(N, &array) to store the first N elements of the Fibonacci sequence into an array in memory. The value N is passed in $a0, and the address of the array is passed in...
Given a wavefunction ψ(x, 0) of a free particle at initial time t=0, I need to write the general expression of the function at time t. I used a Fourier transform of ψ(x, t) in terms of ψ(p, t), but, i don't understand how to use green's functions and the time dependent schrodinger equation to...
Hi,
I was reading through some slides about graph clustering. In the slides was a very short discussion about 'eigenvectors and segmentation'. I don't quite understand where one of the formulae comes from.
Context: We have some undirected graph with an affinity matrix (i.e. weighted adjacency...
Hello all, see below for a snippet of a header file, class1.h, and source code file, class1.cpp, adjusted to reproduce the issue I am having. I have declared a series of functions in the .h file and their corresponding definitions in the .cpp file but when I compile I get the error...
Hi,
I was looking at the following example, and I cannot really understand the justification for why this function is positive semi-definite.
Example Problem: We are looking at the function: ## f(x, y) = \frac{x^2}{y}, x \in \mathbb{R} , y > 0 ##. Is this function positive semi-definite...
I tried to code spinoperators who act like $S_x^iS_x^j$ (y and z too) and to apply them to the states, which works fine. I am not sure about how to code the expectation value in the product Space. Has anyone pseudo Code to demonstrate that?
Im having trouble understanding the wording to this problem. When it says "from r=0 to r=infinity". My Qenc would zero out. I guess it makes sense that from infinitely far away you wouldn't "feel' the electric field but considering this question leads to 4 other questions I don't think I am...
What I do is set the two equations equal to one another and solve for z.
This gives:
$$z = \sqrt{x^2+2y^2-4x}$$
which is a surface and not a curve.
What am I doing wrong?
It is asking to derive the time-independent wave function and has managed to get the answer of
and i am very confused as where (ix/a) and (-x^2/2a) came from ?
Thanks.
Greetings!
My question is: is it possible to use the green theorem to compute the circulation while in presence of a scalar function ? I know how to solve by parametrising each part but just in case we can go faster? thank you!
I have to find 2 solutions of this Bessel's function using a power series.
##x^2 d^2y/dx^2 + x dy/dx+ (x^2 -9/4)y = 0##
I'm using Frobenius method.
What I did so far
I put the function in the standard form and we have a singularity at x=0. Then using ##y(x) = (x-x_0)^p \sum(a_n)(x-x_0)^n##...
Greetings!
Here is the solution that I understand very well I reach a point I think the Professor has mad a mistake , which I need to confirm
after putting x-1=t
we found:
But in this line I think there is error of factorization because we still need and (-1)^(n+1) over 3^n
Thank you...
Before starting, I will leave the link to the article I am talking about here: http://www.msc.univ-paris-diderot.fr/~phyexp/uploads/LaimantParesseux/aimant2.pdf
I am conducting a similar experiment to the one discussed in the paper above. Basically, I am rolling a neodium supermagnet down a...
In an article written by Richard Rollleigh, published in 2010 entitled The Double Slit Experiment and Quantum Mechanics, he argues as follows:
"For something to be predictable, it must be a consistent measurement result. The positions at which individual particles land on the screen are not...
I am desperate. I've scoured the web for the formula for the probability density function for the interference pattern obtained in the double slit experiment with both slits open. So I want to know the probability density function and not the intensity function. I prefer not to have references...
Hey! :giggle:
For $n \in \mathbb{N}$ we consider the discrete statistical product model $(X ,(P_{\theta})_{\theta \in\Theta})$ with $X = \mathbb{N}^n$, $\Theta = (0, 1)$ and $p_{\theta}(x_i) = \theta(1 -\theta)^{x_i−1}$
for all $x_i \in \mathbb{N}, \theta \in \Theta$. So $n$ independent...
Hello
I have a script a13_roman2num.py that reads
def value(r):
if (r == 'I'):
return 1
if (r == 'V'):
return 5
if (r == 'X'):
return 10
if (r == 'L'):
return 50
if (r == 'C'):
return 100
if (r == 'D'):
return 500
if...
I tried using substitution ##u=\sqrt{16x-x^8}##, didn't work
Tried factorize ##x## from the denominator and then used ##u=\sqrt{16-x^7}##, didn't work
Tried using ##u=x^4## also didn't work
How to approach this question? Thanks
I think the answer is an even function as the function ##x^2## is an even function and thus, is symmetrical w.r.t. Y axis. The question I have is how to do this problem algebraically. I tried to graph some functions on GeoGebra to verify my answer.
a) ##y = ln(x^2)##
b) ##y = sin(x^2)##...
There is a typo. It should say ##h=\frac{f}{g}##.
Attempt: ##f## and ##g## are holomorphic on ##\Omega##. Homomorphic functions form a ##\mathcal{C}^*## algebra, so ##h## is holomorphic on ##\Omega## where ##g\neq 0##.
If ##z_0## is a removal singularity of ##h##, then ##Res(h,z_0)=0## by...
Hello to all.
I am trying to design a ripple free (read as ripple free as possible) power supply (PS) for my DIY DDS function generator.
I am (was) in the possession of the hyland 5v to 12v PS which wrecked due to a stupid action on my side, my bad.
so i was going to repair it, but i found that...
First of all I am not sure which type of singularity is ##z=0##?
\ln\frac{\sqrt{z^2+1}}{z}=\ln (1+\frac{1}{z^2})^{\frac{1}{2}}=\frac{1}{2}\ln (1+\frac{1}{z^2})=\frac{1}{2}\sum^{\infty}_{n=0}(-1)^{n}\frac{(\frac{1}{z^2})^{n+1}}{n+1}
It looks like that ##Res[f(z),z=0]=0##
Why do we want to always deal with single valued functions?
In the classical treatment a function is a rule which assigned to one number another number. In the modern sense, it is a rule which assigns to each element in a set called the domain an element (one element) in a set called the range...
False
The reasoning for answer:
The absolute value function is is not analytic wherever its argument equals zero. ##f## is not analytic at ##z=0## so it is not entire.
I've never actually done this, so I was wondering if someone could show me how this is done. One way I tried was by simply using ##cos^{-1}## in order to cancel the cosine, but that gave me a different value, so I assume this is not how you are supposed to do this.
--> I know I am supposed to...
I need to know how to predict particle size of a water driplet produced by a given ultrasonic frequency? For example, an ultrasonic fogger will create ~5 micron water driplets at a frequency of 1.75 MHz. I do know that the higher the frequency the smaller the driplet diameter. How is this...
Here is what the problem looks like. The thing is I don't remember what π1is exactly and I don't really know much group theory or know what equivalence classes are. I remember learning some group theory fact that f*(n) = n*f*(1). So, I think (a) was just equal to m since f(1) = 1 and (b) was...
Hi,
I have a class master_t which is composed by two other classes, dev_a, dev_b. I would like that a member function from the dev_b object (within master_t) could use a member function of dev_a object (within master_t). This is a minimal working code, where line 26 implements this feature...
I'm learning about Fourier theory from my lecture notes and I have a few questions that I wasn't able to concretely find answers to:
1. What's the definition of periodic extension? I think the definition is as follows ( Correct me if I'm wrong please ):
for ## f: [ a,b) \to \mathbb{R} ## its...
Consider item ##vii##, which specifies the function ##f(x)=\sqrt{|x|}## with ##a=0##
Case 1: ##\forall \epsilon: 0<\epsilon<1##
$$\implies \epsilon^2<\epsilon<1$$
$$|x|<\epsilon^2\implies \sqrt{|x|}<\epsilon$$
Case 2: ##\forall \epsilon: 1\leq \epsilon < \infty##
$$\epsilon\leq\epsilon^2...
For a Prandtl stress function to be valid, it must be zero on the boundary. For a circular bar, both of these work:
$$\phi_1 = C\left(\frac{x^2}{r^2}+ \frac{y^2}{r^2} - 1\right)$$
$$\phi_2 = C \left(x^2+ y^2- r^2\right)$$
But performing the integration for the internal torque M gives...
(a)
i tried to decompose the fracion as a sum of fractions of form ##\frac{1}{1-g}##
$$f=\frac{-z}{(1+z)(2-z)}=\frac{a}{1+z}+\frac{b}{2-z}$$
$$a=\frac{1}{3}, b=-\frac{2}{3}$$
$$f=\frac{1}{6}\frac{1}{1+z}-\frac{1}{3}\frac{1}{1-\frac{z}{2}}$$
$$f=\frac{1}{6}\sum_{n=0}^\infty...
I learned that ##f## has another singular point at ##z=1.715##, but i don't think this would be related to the pole at ##z=0##
I tried substitutine ##u=2\cos z-2+z^2##
and $$f(u)=\frac{1}{u^2}$$ has a pole of order 2 at ##u=0## which happens i.f.f. ##z=0## or ##z=1.715##.
so ##f## has a pole...
Dear Everybody,
I need some help understanding how to use pade approximations with a given data points (See the attachment for the data).
Here is the basic derivation of pade approximation read the Derivation of Pade Approximate.
I am confused on how to find a f(x) to the data or is there a...
Denote wheel turning angle as theta--> Induced EFM (Em)=Kb*Theta_dot.
Voltages on the wheel: R*i= V - Em
Moments on engine's axis: Kt*i-C*a + a (m+M)*x_double_dot=0 (As Jm negligible).
From here I would find another equation to have x and theta expressing each other, but i think I'm...