Functions Definition and 1000 Threads

Assume s is a set such that Fs denotes the set of functions from S-->F where F is a field such as R, C or [0,1] etc. One requirement for F to be a vector space of these functions is closure- e.g. that sums of these functions are in the space: For f,g in Fs the sum f+g must be in Fs hence...

Hi all, I hope someone can help me with this: I am working with wien2wannier to obtain Wannier functions from wien2k calculations, but I am struggling to define the energy window needed for the init_w2w. The outputfind file shows emin = -999.00 eV, emax = 999.00 eV, and efermi = 0.6124 Ry...

Discuss uniform continuity of the following functions: ##\tan x## in ##[0,\frac{\pi}{2})## ##\frac{1}{x}\sin^2 x## in ##(0,\pi]## ##\frac{1}{x-3}## in ##(0,3),(4,\infty),(3,\infty)## I am completely new to this uniform continuity and couldn't find a lot of examples to learn the solving pattern...

Took calculus of a single variable almost a decade ago where every theorem had to be accepted without proof. Can I fill these gaps by studying a rigorous multivariable/vector analysis book? My justification for this is that R^1 is just a special case of R^n. Or am I looking at this the wrong way...

assalamualaikum (peace be upon you), brothers and sisters. i m new member. i m actually having trouble with understanding the functions and options of this site. i m unable to find my profile. hope to get help. [Translation by the Mentors via Google Translate] :smile:

*My attempt:*** We know that $$ (f-g)(x) =\begin{cases} 2x+3&, \ x\in \mathbb{Q}\\ 3x - \sqrt{5} &, \ x\in \mathbb{R} \setminus \mathbb{Q} \end{cases} $$ Now the problem is I don't understand how to navigate it further.. The two equations we obtain would have been straight line equations if...

TL;DR Summary: Writing functions for Bisection and Newtons Approximation in Mathematica Hello! I need to write 2 functions in mathematica, to find the roots of functions. The functions are the Bisection methods and Newtons Approximation. (b1) Write your own function ApproxBisect[a0_,b0_,n_]...

I resolved the numerator to ## cosx-sinx## We get $$mod\frac{cosx-sinx} {3(cosx+sinx)} $$ If we divide the numerator and denominator by cosx we get $$mod\frac{1-tanx} {3(1+tanx)}$$(eq1) We know that tan(π/4-x) is same as ##\frac{1-tanx} {1+tanx}## So re writing eq1 we get $$mod\frac{tan(π/4-x}...

For this problem, I am confused how they get all their negative definite, positive definite, and negative semidefinite domains. I agree that ##V_2## is positive definite by definition. However, for ##V_3## I think they made a mistake since by definition, ##V_3## is negative simi-definite. I'm...

My attempt: Say f(x)=y=##\frac{e^{2x}-e^x+1} {e^{2x}+e^x+1}## So I cant rearrange to write in terns of x ##e^{2x}(y-1)+e^x(y+1)+y-1,=0## Now we know that e^x will not be defined for y=1 Therefore when y≠1 ##e^x = \frac{ -y-1+-√((y+1)^2-4(y-1)^2)} {2(y-1)} ## So we can put an inequality on the...

Hi, The task is as follows For the proof I wanted to use the boundedness, in the script of my professor the following is given, since both ##(X,d)## and ##\mathbb{R}## are normalized vector spaces I have now proceeded as follows ##|d(x,p)| \le C |x|## according to Archimedes' principle, a...

Here are two examples: The first one is from Taylor's Mechanics, section 6.2: Taylor takes the function ##f(y+\alpha\eta, y' + \alpha\eta', x)## and differentiates it with respect to ##\alpha##. In the expression for the function, ##y## and ##\eta## depend on ##x##, ##\alpha## is an...

I decided to go through @psie's measure theory notes to refresh myself since it's been a while. I got to the theorem on page 60 which I will attempt to summarize my confusion as just this statement If X and Y are measure spaces and ##f:X\times Y\to \mathbb{C}## is measurable then the function...

For this problem, I am confused how they get $$| x - 4 | > \frac{1}{2}$$ from. I have tried deriving that expression from two different methods. Here is the first method: $$-1\frac{1}{2} < x - 4 < -\frac{1}{2}$$ $$1\frac{1}{2} > -(x - 4) > \frac{1}{2}$$ $$|1\frac{1}{2}| > |-(x - 4)| >...

In which sense(s) do square integrable functions go to zero at infinity? Of course, they cannot go to zero at infinity in the sense of point evaluation, because point evaluation is not the appropriate concept for square integrable functions. There was a recent discussion in the Quantum Physics...

EDIT: My Latex is not showing... Sorry. I attached a file with my "solution". I though this would be quite easy, but I can't seem to solve this system of equations. Should I solve for each mode, or both of them together? I tried to solve them together, here's how far I get: $\text{General...

By rewriting, for example, f(x)=2x+3, as y=2x+3, are we simply stating that something = 2x+3; and in the first case we’re calling that something f(x), and in the second case we’re calling it y? Does the y have anything to do with the y axis as in x,y coordinates axes? Or is just a randomly...

(Disclaimer: I don't know whether this type of post encouraging discussion on a function is allowed, if not please close this) Hello PF, If you're a fan of integrating, you'll hit a ton of special functions on the way. Things like the Harmonic Numbers, Digamma function, Exponential Integral...

This is a bit confusing...conflicting report from attached wolfram and symbolab. Which approach is correct?

A bit confusing here; what i did, Using gradient, we have, ##m=\dfrac{7-1}{0-3}=-2## ##y=-2x+7## Since there is a Vertex, we have the other ##m_2=2## thus, ##y=mx+c## ##1=2×3 +c## ##... y=2x-5## ##a=2, b=-5, c=12## or##a=-2, b=5, c=2##

The definition of a diffeomorphism involves the differentiable inverse of a function, so must the original function be one-to-one to make its inverse a function, or can the inverse be a relation and not a function? Sorry if it's a silly question, I am just a second semester calc student who...

Hi. I'm self-studying functions which relate to calculus. Let me post what I feel I know and what I'm not grasping yet. Please correct any mistakes I'm making. I'm just talking real numbers: A function is a rule that takes an input number and sends it to another number. We can describe it...

For part(a), The solution is, However, why do they not take the derivative of the inner function (if it exists) of f(x) or g(x) using the chain rule? For example if ##f(x) = \sin(x^2)## Many thanks!

For part(a), The solution is, However, I am having trouble understanding their finial formula. Does anybody please know what the floating ellipses mean? I have only seen ellipses that near the bottom like this ##...## I am also confused where they got the ##2 \cdot 1## from. When solving...

Does Precalculus by James Stewart teaches function from scratch ?

Good Morning As I continue to study the gyroscope with Tait-Bryan angles or Euler angles, and work out relationships to develop steady precession, I notice that the trig functions cancel. I stumble on terms like: 1. sin(theta)cos(theta) - cos(theta)sin(theta) 2. Cos_squared +...

Hello, My name is Josip Jakovac, i am a student of the theoretical solid state physics phd studies. First I want to apologize if my question has already been answered somewhere here, I googled around a lot, and found nothing similar. My problem is that I need to apply TBA to Graphene. I went...

Suppose ##f## is analytic in an open set ##\Omega \subset \mathbb{C}##. Let ##z_0\in \mathbb{C}## and ##r > 0## such that the closed disk ##\mathbb{D}_r(z_0) \subset \Omega##. If ##f## has a zero of order ##k## at ##z = z_0## and no other zeros inside ##\mathbb{D}_r(z_0)##, show that there an...

Hi everyone! Do you have an idea which organs/parts of the body are ONLY functional on glucose? I would say the brain, pancreas, liver and kidney, but I have to take into account only those organs that are ONLY functional on glucose

Rewrite the given equation, attempt 1: ##2\sin(x)\cos(x) + 2\sin(x) + 2\cos(x) = 0## ##\sin(x)\cos(x) + \sin(x) + \cos(x) = 0## ##\sin(x)(\cos(x) + 1) + \cos(x) = 0##, naaah, can't get any relevant out from here. Attempt 2: ##2\sin(x)\cos(x) + 2\sqrt{2}*\sin(x + \pi/4) = 0## ##\sin(x)\cos(x) +...

vector<OP> negate (vector<OP> a) { a.insert(a.begin(), neg); return a; } vector<vector<OP>> negate (vector<vector<OP>> a) { for (int i=0; i<a.size(); i++) a[i] = negate(a[i]); // reference to 'negate' is ambiguous? return a; } OP is an enum here. Why can't C++...

Trying to model friction of a linear motor in the process of creating a state space model of my system. I've found it easy to model friction solely as viscous friction in the form b * x_dot, where b is the coefficient of viscous friction (N/m/s) and x_dot represents the motor linear velocity...

I have a likelihood function that has one global minima, but a lot of local ones too. I attach a figure with the likelihood function in 2D (it has two parameters). I have added a 3D view and a surface view of the likelihood function. I know there are many global optimizers that can be used to...

Hello. Does anybody know a proof of this formula? $$J_{2}(e)\equiv\frac{1}{e}\sum_{i=1}^{\infty}\frac{J_{i}(i\cdot e)}{i}\cdot\frac{J_{i+1}((i+1)\cdot e)}{i+1}$$with$$0<e<1$$ We ran into this formula in a project, and think that it is correct. It can be checked successfully with numeric...

Hello. How to prove that in smooth infinitesimal analysis every function on R is continuous? (Every function whose domain is R, the real numbers, is continuous and infinitely differentiable.) Thanks.

If we have two functions ##f(x)## such that ##\lim_{x \to \infty}f(x)=0## and ##g(x)=\sin x## for which ##\lim_{x \to \infty}g(x)## does not exist. Can you send me the Theorem and book where it is clearly written that \lim_{x \to \infty}f(x)g(x)=0 I found that only for sequences, but it should...

Let ##f : \mathbb{R} \to \mathbb{R}## be a uniformly continuous function. Show that, for some positive constants ##A## and ##B##, we have ##|f(x)| \le A + B|x|## for all ##x\in \mathbb{R}##.

An example of physical applications for the gamma (or beta) function(s) is http://sces.phys.utk.edu/~moreo/mm08/Riddi.pdf (I refer to the beta function related to the gamma function, not the other functions with this name) The applications in Wikipedia...

Wiki defines orthogonal functions here https://en.wikipedia.org/wiki/Orthogonal_functions Here's one example, but it's an example that is only true for a specific interval https://www.wolframalpha.com/input?i=integral+sin(x)cos(x)+from+0+to+pi So are these functions orthogonal because there...

As an example, consider a vector-valued function of the form ##f(x,y) = (g_1(x,y),g_2(x,y))##. I typed up one example on wolfram to see if this could be visualized https://www.wolframalpha.com/input?i=plot+f(x,y)+=+(x+y,xy) which was inspired by this question...

If ##d(x,y)## is a metric, then it is said ##\frac{d}{1+d}## is also a metric. I don't know the proof of this, I'd appreciate a reference, but it got me wondering: If ##N(x)## is a norm on a Banach space ##x \in X##, then are there functions in a single real (or complex) variable ##f## (besides...

The problem reads: ##f:M \rightarrow N##, and ##L \subseteq M## and ##P \subseteq N##. Then prove that ##L \subseteq f^{-1}(f(L))## and ##f(f^{-1}(P)) \subseteq P##. My co-students and I can't find a way to prove this. I hope, someone here will be able to help us out. It would be very...

I am to consider a basis function ##\phi_i(x)##, where ##\phi_0 = 1 ,\phi_1 = cosx , \phi_2 = cos(2x) ## and where the scalar product in this vector space is defined by ##\braket{f|g} = \int_{0}^{\pi}f^*(x)g(x)dx## The functions are defined by ##f(x) = sin^2(x)+cos(x)+1## and ##g(x) =...

I am refreshing on this; of course i may need your insight where necessary...I intend to attempt the highlighted...this is a relatively new area to me... For part (a), We shall let ##f(x)=\dfrac{1}{x(2-x)}##, let ##g(x)## be the even function and ##h(x)## be the odd function. It follows...

I edited this to remove some details/attempts that I no longer think are correct or helpful. But my core issue is I have never seen this approach to approximating integrals that is used in the attached textbook image. Any more details on what is happening here, or advice on where to learn more...

(If this is in the wrong forum, please move it) Here is the potential energy of a spring Here is the strain energy function in elasticity The look alike -- I like that. If we want the force in the spring, we take the derivative of V with respect to the displacement and make the result...

In Wikipedia article on Bessel functions there is an integral definition of “non-integer order” a (“alpha”). For imaginary order ia I get that Jia* = J-ia, where * is complex conjugate and ia and -ia are subscripts. Then in same article there is a definition of Neumann function, again for...

The trig identities for adding trig functions can be seen: But here the amplitudes are identical (i.e. A = 1). However, what do I do if I have two arbitrary, real amplitudes for each term? How would the identity change? Analysis: If the amplitudes do show up on the RHS, we would expect them...

Hi PF Given the following def f1(var1, var2): var3 = var1 + var2 return var3 def f2(var1, var2, var4) var4 = 10 var5 = f1(var1, var2)*var4 return var5 it is obvious function f2 does not explicitly need variables var1 and var2. However, it needs the result of f1, which...