Analysis is the process of breaking a complex topic or substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics and logic since before Aristotle (384–322 B.C.), though analysis as a formal concept is a relatively recent development.The word comes from the Ancient Greek ἀνάλυσις (analysis, "a breaking-up" or "an untying;" from ana- "up, throughout" and lysis "a loosening"). From it also comes the word's plural, analyses.
As a formal concept, the method has variously been ascribed to Alhazen, René Descartes (Discourse on the Method), and Galileo Galilei. It has also been ascribed to Isaac Newton, in the form of a practical method of physical discovery (which he did not name).
Just to remind, ##C_\ell## is the variance of random variables ##a_{\ell m}## following a Gaussian PDF (in spherical harmonics of Legendre) :
##C_{\ell}=\left\langle a_{l m}^{2}\right\rangle=\frac{1}{2 \ell+1} \sum_{m=-\ell}^{\ell} a_{\ell m}^{2}=\operatorname{Var}\left(a_{l m}\right)##
1)...
I am in a mechanical design class that has been focusing on the slider crank mechanism. My professor tends to just provide derived equations without showing the analysis. I feel like I am missing out on some key understanding because of this. Specifically, I am trying to do what should be a...
I need help actually creating the proof. I've done the scratch needed for the problem, it's just forming the proof that I need help in.
Bar(a+bi/c+di)= (a-bi) / (c-di)
Bar ((a+bi/c+di)*(c-di/c-di)) = ((a-bi/c-di)*(c+di/c+di))
Bar((ac+bd/c^2 +d^2)+(i(bc-ad)/c^2+d^2)) =...
https://en.wikipedia.org/wiki/Klein%27s_Encyclopedia_of_Mathematical_Sciences
Originals are in German or French, the Japanese version cut all the historical content :(
Do you think that some day we will see this published in English?
Size is big, 20k pages, but it cannot be more interesting I...
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If ##b \leq x_n \leq c## for all but a finite number of n, show that ##b \leq \operatorname{lim inf}_{n \to \infty} x_n## and ##\operatorname{lim sup}_{n \to \infty} x_n \leq c_n##
(Buck, Advanced Calculus, Section 1.6, Exercise 24)
Let ##\beta =\operatorname{lim inf}_{n \to \infty} x_n##...
Hello everyone,
My question for this thread concerns the application of (mainly) mathematical analysis to fields such as signal processing and machine learning. More specifically, I was wondering if you happen to know of some interesting application of things like measure theory or functional...
I have to show that $\forall z\in B(0,0.4)$, there exists an $x\in B(0,1)$ such that $f(x)=z$ but I am not sure how to show this. From the reverse triangle inequality
$$-|f(x)-f(y)|+|x-y|\leq 0.1|x-y|\implies |f(x)-f(y)|\geq 0.9|x-y|$$
im not sure if this helps.
I have been reading Manton & Sutcliffe for some time now and can't quite wrap my head around something.
If you take the Hopf invariant N of a topological soliton ϕ then its Skyrme-Faddeev energy (which I hope I've gotten right up to some constants)
E=∫∂iϕ⋅∂iϕ+(∂iϕ×∂jϕ)⋅(∂iϕ×∂jϕ) d3x
satisfies...
I am close to graduating as an EE major but I have never been able to organize a step by step method on analyzing a circuit. It seems to me that every time I am trying to analize a circuit I end up with a bunch of equations and nothing more. I know that I should:
1. Know what im solving for...
Hello Everyone!
I want to learn about Fourier series (not Fourier transform), that is approximating a continuous periodic function with something like this ##a_0 \sum_{n=1}^{\infty} (a_n \cos nt + b_n \sin nt)##. I tried some videos and lecture notes that I could find with a google search but...
Find the only periodic solution for 𝑦′+𝑦=𝑏(𝑥) with 𝑏:ℝ→ℝ has a period of 2𝑇 and is 1 for 𝑥(0,𝑇) and −1 for 𝑥(−𝑇,0).
The ODE is easy to solve: 𝑦(𝑥)=exp(−𝑥)⋅𝑐+1 and 𝑦(𝑥)=exp(−𝑥)⋅𝑐−1. But how can I find the 𝑐 such that the solution is periodic with a period of 2𝑇?
The solution is...
Hello, thank you for your attention in my thread.
I'm the head of Youth Science Club on one of the many High School in Indonesia, I got many members and I want to guide them for doing water sampling and analysis. In the history of my high school, no one is able to do water sampling and...
Problem Statement: Finding the resistance when probed at point bc, cd and da
Relevant Equations: Series and Parallel resistance equation derived from kirchhoff's law with application of ohm's law
I genuinely don't know what to do on this one. The example our professor made isn't exactly clear...
Lets say I am analyzing a simple AC voltage source with a resistor. In the positive voltage peak then I will use V and I say current is flowing clockwise. When I am analyzing the -Vpk iteration then do I make the current counter clockwise too or do i keep it clockwise? Thanks.
Suppose ##(y_n)_n## is a sequence in ##\mathbb{C}## with the following property: for each sequence ##(x_n)_n## in ##\mathbb{C}## for which the series ##\sum_n x_n## converges absolutely, also the series ##\sum_n \left(x_ny_n\right)## converges absolutely. Can you then conclude that ##(y_n)_n##...
Homework Statement
- Given a bounded sequence ##(y_n)_n## in ##\mathbb{C}##. Show that for every sequence ##(x_n)_n## in ##\mathbb{C}## for which the series ##\sum_n x_n## converges absolutely, that also the series ##\sum_n \left(x_ny_n\right)## converges absolutely.
- Suppose ##(y_n)_n## is...
Homework Statement
Determine whether or not the following sequences of real valued functions are Cauchy in L^{1}[0,1]:
(a) f_{n}(x) = \begin{cases} \frac{1}{\sqrt{x}} & , \frac{1}{n+1}\leq x \leq 1 \\ 0 & , \text{ otherwise } \end{cases}
(b)
f_{n}(x) = \begin{cases} \frac{1}{x} & ...
Hi there all,
I'm currently taking a course in Multivariable Calculus at my University and would appreciate any recommendations for a textbook to supplement the lectures with. Thus far the relevant material we've covered in a Single Variable course at around the level of Spivak and some Linear...
Homework Statement
Derive the expressions for the voltage gain (Gv) of the following op amp:
Homework Equations
In = Ip = 0
Vp =Vn
The Attempt at a Solution
I can use KCL, and the fact that In and Ip are both 0, to derive the two equations, one from the top node and the other from...
High school student here...
Recently, I've found an interest in topology and am trying to figure out the correct path for self-studying. I am familiar with set theory and some concepts of abstract algebra but have not really studied any form of analysis, which from what I've read is a...
Homework Statement
(FYI It's from an Real Analysis class.)
Show that $$\int_{0}^{\infty} (sin^2(t) / t^2) dt $$ is convergent.
Homework Equations
I know that for an integral to be convergent, it means that :
$$\lim_{x\to\infty} \int_{0}^{x} (sin^2(t) / t^2) dt$$ is finite.
I can also use...
Homework Statement
Let ##C \subset \mathbb{R}^n## a convex set. If ##x \in \mathbb{R}^n## and ##\overline{x} \in C## are points that satisfy ##|x-\overline{x}|=d(x,C)##, proves that ##\langle x-\overline{x},y-\overline{x} \rangle \leq 0## for all ##y \in C##.
Homework Equations
By definition...
Good morning,
I used the Laser beam with HR4000 spectrometer with Ocean View software when saving the files it is saved by (.ocv) format. when trying to extract information to excel I get some unreadable data like (bkg thin sheet gel.png) attached. I used the same instrument and software with...
Homework Statement
[/B]
Let X be a vector space over ##\mathbb{R}## and ## f: X \rightarrow \mathbb{R} ## be a convex function and ##g: X \rightarrow \mathbb{R}## be a concave function. Show: The set {##x \in X: f(x) \leq g(x)##} is convex.
Homework Equations
[/B]
If f is convex...
Homework Statement
1. Show that for all real numbers x and y:
a) |x-y| ≤ |x| + |y|
Homework Equations
Possibly -|x| ≤ x ≤ |x|,
and -|y| ≤ y ≤ |y|?
The Attempt at a Solution
I tried using a direct proof here, but I keep getting stuck, especially since this is my first time ever coming...
##X_i## is an independent and identically distributed random variable drawn from a non-negative discrete distribution with known mean ##0 < \mu < 1## and finite variance. No probability is assigned to ##\infty##.
Now, given ##1<M##, a sequence ##\{X_i\}## for ##i\in1...n## is said to meet...
When we define a limit of a function at point c, we talk about an open interval. The question is, can it occur that function has a limit on a certain interval, but it's extension does not? To me it seems obvious that an extension will have the same limit at c, since there is already infinitely...