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.
The following is the wave equation from Electrodynamics: $$\frac{\partial^2 \Psi}{\partial t^2} = c^2\frac{\partial^2 \Psi}{\partial x^2}$$ Where ##\Psi## is the wave function. But because of Heisenberg's Uncertainty, physicists had to come up with another equation (the Schrodinger equation)...
This is the code that i wrote
Clear["Global`*"]
Z = 500;
W = 100000;
G = 250;
H = 100;
K = 0.5;
T = 30;
L = 4000;
P = 5;
S = 2.5;
Y = 1;
A = 0.1;
V = 2.5;
J = 8000;
f[x_] := 1/
x {(J*Z*x*(2*Y - x))/(
2*Y) - ((W + T*G) + ((L + T*P)*2*Z*Y*(1 - ((Y - x)/Y)^1.5))/
3 + (H + T*S +...
The Korean textbook standard defines the convexity of the function as an open section. Many textbooks and university calculus textbooks define the convexity of the curve as an open section. However, some textbooks define convexity as closed sections.
Do you think it is right to define the...
For a nonconservative force,
What would be the dissipative function for a force f=-bvⁿ in Lagrangian
(Where v is the velocity)
[#qoute for a nonconservative force f=-bv
The dissipative function is D=-(1/2)bv² ]
Since ##f=\frac{\partial D}{\partial \dot x}## so the dissipative function should...
During a thermodynamic cycle, an ideal thermal machine absorbs heat Q2 > 0 from a hot source and uses it to perform Work W > 0, giving a cold source a heat Q1 < 0 with an efficiency of 20% . How much is the work done as a function of Q1 ?I have 2 question regarding this problem: 1) Why is Q1 the...
I believe it is the case that any linear second order ode with at most 3 regular singular points can be transformed into a hypergeometric function. I am trying to solve the following equation for a(x):
where E, m, v, k_{y} are all constants and I believe turning it into hypergeometric form will...
Hi, I am trying to get the transfer function from a wall between rooms. From one side I have the force of a hammer as an input ,and in the other side of the wall (next room) I have an accelerometer. Is it possible to get the TF without know the damping, stiffness and mass of the wall partition...
program main
! use ! some library that defines the function to calculate the determinant of a given matrix
implicit none
real,dimension(2,2)::A
real::det_val
A(1,1)=1
A(2,2)=1
A(2,1)=0
A(1,2)=0
! det_val=det(A)
print *,det_val ! Should print 1.
end program main
Hello!
Consider this filter,and that I have to find the transfer function U2/U1 with the norm Ω= ωRC (also double fractions are not allowed)
Now I can see that that the resistor and capacitor left as well as right are parallel to each other. So simplifying that
## Z_1 = \frac{R}{jωRC +1}...
Now, I don't understand how did author compute $F_{X_1}(x) = \displaystyle\sum_{j=1}^n \binom{n}{1} F^1(x) (1-F(x))^{n-1} = 1-(1-F(x))^n ?$ (I know L.H.S = R.H.S)
Would any member of Math help board explain me that? Any math help will be accepted.
Hi everyone
This is the solution for the problem.
I don't understand how they got from
To
This was my attempt at a solution
I can't seem to get rid of one of the y terms and am left with one on each side.
Could someone explain the solution to me please?
Thanks
I have a table of values (from my own analysis, not from a textbook) that represents a portion of a periodic function:
x
y
0
30
1
20
2
10
3
10
4
20
5
30
What function satisfies the table? What I know is that the function is periodic. I was thinking I could use cosine because its...
In the ADM decomposition, like in the construction of the FRW metric, the coordinates are defined to be co-moving, so we know $$d\tau = dt$$ (i.e. the lapse function is normalized away)
Starting from a five-dimensional embedded hyperboloid (as in carroll pg. 324) ## -u^2 + x^2 + y^2 + z^2 + w^2...
Hello everyone! I was doing some dimensional analysis to find an equation that gives a acceleration as a function of time, using constant power. I came up with the equation $$a = k\sqrt {\frac P {mt}}$$ I differentiated velocity with respect to time in order to check my work and also checked out...
In hamiltonian formalism we have the generalized coordinates ##q_i## and the conjugates moments ##p_i##.
For a dipole in a give magnetic field ##B## the Hamiltonian is ##H=-\mu B cos \theta## where ##\theta## is the angle between ##\vec \mu## and ##\vec B##.
Can i consider ##\theta## or ##cos...
Hello! I have this filter here
a)
Calculate the transfer function T(Ω) = Ua/Ue using voltage dividers.For this, use the normalized angular frequency Ω = ωRC and bring the result into the form ##T(Ω) = \frac{A+jB}{C+jD} ## . The result must not contain any double fractions.
I was able to that...
Hello,
I would like to understand a relation of this article by Volkov (eq. 4).
Let's define the Green function $$ G^{ij}_{ab} (1,2) = -i \langle T_c \Psi_a (1_i) \Psi_b (2_j) \rangle $$ where ##a,b = (1,2)## are the spin indices and ##i,j = (1,2) ## are the indices for the Keldysh contour ...
Hey! 😊
I am trying to write a code for a server in Python and I got stuck.
I gave as an input a csv file and using pandas we get a dictionary where the titles are the keys and the inputs are the values.
From that we get the below :
I have written the below endpoint to get all the...
Hey, I have a question about proving Saha's equation for ionizing hydrogen atoms.
The formula is
\frac{P_{p}}{P_{H}} = \frac{k_{B} T}{P_{e}} \left(\frac{2\pi m_{e} k_{B}T}{h^2} \right)^{\frac{3}{2}}e^{\frac{-I}{k_{B} T}}
with
P_{p} pressure proton's,
P_{H} pressure hydrogen atoms,
m_{e}...
I determined the partition function of the particle A, B and C.
C should be the same as B.
I then considered the situation, where all particles are in the system at the same time, and drew a diagram of all possible arrangements:
The grey boxes are the different partitions, given that we...
I'm not too sure how to account for both the mass and the rope at once.
I think the following are true for the two individually:
For the mass at the end, ## T = m ω^2 L ##, following from ##a = v^2/r##and ##v=ωr##.
For the rope, ##dT = ω^2 r dM##, where ##dM = λ dr## and λ is the mass per unit...
Prove directly that the transformation $$Q_{1} = q_{1}, P_{1} = p_{1} − 2p_{2}$$ $$Q_{2} = p_{2}, P_{2} = −2q_{1} − q_{2}$$ is canonical and find a generating functionSo the first part is easy and can be skipped here. I have some difficults regarding the second part, namely, the one that ask for...
Hi PF!
I have a function ##f(s,\theta) = r(s,\theta)\hat r + t(s,\theta)\hat \theta + z(s,\theta)\hat z##. How can I plot such a thing in Mathematica? Surely there's an easier way than decomposing ##\hat r, \hat \theta## into their ##\hat x,\hat y## components and then using ParametricPlot3D?
So what am I doing wrong here? I can clearly observe it, I'm nearly sure I can tell which particles are going throw each slit if I used another laser too. My suspicion is that the electrical current of the photon detector that uses germanium or silicon to detect the particles are influencing the...
Hello! So, I was beginning to skim Kleppner and Kolenkow for an upcoming course I’m taking over the summer. I saw this on pg. 17 and was wondering if I’m making a silly mistake in understanding what the book is saying. When they take the derivative of the position function, isn’t the velocity...
So the Langevin equation of Brownian motion is a stochastic differential equation defined as
$$m {d \textbf{v} \over{dt} } = - \lambda \textbf{v} + \eta(t)$$
where the noise function eta has correlation function $$\langle \eta_i(t) \eta_j(t') \rangle=2 \lambda k_B T \delta_{ij} \delta(t -...
Homework Statement:: Find the interference function ##I(\delta)## where The emission is analyze by a Michelson interferometer.
Relevant Equations:: ##I(\delta) = \frac{1}{2} \int_{-\infty}^{\infty} G(k) r^{ik \delta} dk## ##I(\vec{r}) = I_1 + I_i + 2 \sqrt(I_1 I_i) cos (k\delta)##
I have 5...
I came across this question on chegg for practice as I'm self learning complex analysis, but became stumped on it and without access to the solution am unable to check.
Let $$ f(z)=\frac{cos(z)} {(z-π/2)^7} $$. Then the singularity is at π/2. And on first appearance, it looks like a pole of...
Hello,
I try to solve a time dependent problem described by a Hamiltonian of the type $$ \mathcal{H}(t) = H_0 + V \delta(t) .$$
I started by trying to solve the Schrödinger equation with ##H_0 = p^2 / 2m##, but I'm getting a bit stuck.
I would like to know if you know of any books that deal...
Hi,
I was working through the following problem and I am getting confused with the solution's definition of the dual.
Problem:
Given the optimization problem:
minimize ## x^2 + 1 ##
s.t. ## (x - 2) (x - 4) \leq 0 ##
Attempt:
I can define the Lagrangian as:
L(x, \lambda) = (x^2 + 1) + \lambda...
Hi, PF, want to know how can I go from a certain error formula for linearization I understand, to another I do not
Error formula for linearization I understand:
If ##f''(t)## exists for all ##t## in an interval containing ##a## and ##x##, then there exists some point ##s## between ##a## and...
Hi,
I have some question about evaluating a cosine function from ##-\infty## to ##\infty##.
I saw for a cosine function evaluate from ##-\infty## to ##\infty## I can change the limits from 0 to ##\infty##. I have a idea why, but I can't convince myself, furthermore, is it always the case no...
Suppose i measure the phase and amplitude of some radiation, this might be a coherent state of an entangled state, how would i construct the Wigner function from these measurements?
At first, I inputted h(t), of which I solved for, into the mass flow rate formula.
So it looked something like this, m-dot(t) = -(density)*[sqrt(g*(H-(g*d^4*t^2/D^4)]*(pi/4)*d^2
But I don't think that's right? Any thoughts?
Hi,
I found Laplace transform of this Dirac delta function which is ##F(s) = e^{-st}## since ##\int_{\infty}^{-\infty} f(t) \delta (t-a) dt = f(a)##
and that ##\delta(x) = 0## if ##x \neq 0##
Then the Mellin transform
##f(t) = \frac{1}{2 \pi i} \int_{\gamma - i \omega}^{\gamma +i \omega}...
I would like a program to read in a file entered by the user via the command line, which is then used in the main body of the code.
See example code below:
#include <iostream>
#include <fstream>
struct run_t {
std::string file;
};
run_t run;
const POINTER* ptr = toy(run.file,0);
int...
Is there a general expression for the wave function $\psi$, which describes the electronic properties of an arbitrary covalent bond? For example is it equal to some sort of trigonometric expression?
Hi PF!
I followed someone's help on here and have the following code in python that performs monte carlo integration
from math import *
from random import *
def integrate(alpha):
# MONTE CARLO INTEGRATION OVER NON-RECTANGULAR DOMAINS
def f(pt):
# RETURN INTEGRAND AS FUNCTION...
So I've just started learning about Greens functions and I think there is some confusion. We start with the Stokes equations in Cartesian coords for a point force.
$$-\nabla \textbf{P} + \nu \nabla^2 \textbf{u} + \textbf{F}\delta(\textbf{x})=0$$
$$\nabla \cdot \textbf{u}=0$$
We can apply the...
Hi,
I am having to refresh my oscilloscope knowledge and am confused about one last function generator setting... Burst period.
If I have 1 cycle at say 700 kHz it is 1.43us. If I set number of cycles to 10 then that is 10 * 1.43us = 14.3us time. This is my ON burst.
So what is the burst...
Hi PF!
Given the following ODE $$(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,$$ where ##a = \sqrt{1-\cos^2\alpha} : \alpha \in (0,\pi)## and ##b = 1##, what is the Green's function?
This is the...
Hi PF!
I'm numerically integrating over a Green's function along with a few very odd functions. What I have looks like this
NIntegrate[-(1/((-1.` + x)^2 (1.` + x)^2 (1.` + y)^2))
3.9787262092516675`*^14 (3.9999999999999907` +
x (-14.99999999999903` +
x (20.00000000000097` -...
I have a question about statistical physics. Suppose we have a closed container with two compartments, each with volume V , in thermal contact with a heat bath at temperature T, and we discuss the problem from the perspective of a canonic ensemble. At a certain moment the separating wall is...
hi guys
I am recently taking a Nuclear structure course, and have a lot of questions regarding the nuclear rotor model.
in most nuclear physics books the I have, the wave function associated with the rotor model of the nucleus is written in terms of the Wigner D functions , like the expression...
I want to determine the normal flow depth in a perfectly horizontal circular conduit. The system characteristics are known (Internal pipe diameter, Mannings roughness, Discharge). However, I am not sure how to calculate the normal flow depth. When using Manning's equation one can find the normal...
From my reading of several quantum optics textbooks and spectroscopy texbooks, the emission and absorption spectrum of an atom or molecule are always given in terms of the time-correlation function, for example the emission spectrum of a two level atom is given by:
$$...