# Analysis Definition and 149 Discussions

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).

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1. ### A A claim about smooth maps between smooth manifolds

Given the definition of a smooth map as follows: A continuous map ##f : X → Y## is smooth if for any pair of charts ##\phi : U →R^m, \psi:V →R^n## with ##U ⊂ X, V ⊂Y##, the map ##\phi(U ∩f^{-1}(V)) → R^n## given by the composition $$\psi ◦ f ◦ \phi^{-1}$$ is smooth. Claim: A map ##f : X → Y##...
2. ### I Method for experimental results analysis

Hello guys, I have conducted an experiment and got some results. I have 3 variables to vary, for example, five x1, five x2, and two x3 and 2 observation results, like y1, y2 I already make y1 y2 and x1 x2 x3 dimensionless since plot is 2D, what I am doing now is just plot when x3=1, x2=1, plot...
3. ### Good introductory book on statistical/data analysis?

TL;DR Summary: I'm looking for a book on statistical/data analysis. Hey all. I've been doing statistical analysis in my research (such as using PCA and LDA), but I have never received a formal education on statistical analysis or data mining, and what I know about analysis is quite scattered...
4. ### I Understanding Theorem 13 from Calculus 7th ed, R. Adams, C. Essex, 4.10

The following properties of big-O notation follow from the definition: (i) if ##f(x)=O(u(x))## as ##x\rightarrow{a}##, then ##Cf(x)=O(u(x))## as ##x\rightarrow{a}## for any value of the constant ##C##. (ii) If ##f(x)=O(u(x))## as ##x\rightarrow{a}## and ##g(x)=O(u(x))## as ##x\rightarrow{a}##...
5. ### Analysis Study plan for Functional Analysis - Recommendations and critique

Hello, PF! It’s been a while since I last posted. I am looking for a critique and recommendations regarding my study plan towards Functional Analysis and applications (convex optimization, optimal control), but first, some background: - This plan is in preparation for my master’s thesis, I...
6. ### I Filling in Missing Values in a string of data

hello, I’m trying to figure out if I’m doing this correctly or if there’s a different way that I should be finding a missing value. I’m trending data for an automatic transformer. Every month I collect the operations counter value and at the end of the year sum the number of tap changes...
7. ### A Anti-dual numbers and what are their properties?

In [this post] user William Ryman asked what would happen if we try to build "complex numbers" with shapes other than circle or hyperbola in the role of a "unit circle". [Here] I proposed three shapes that could work. The common principle behind them being that if the unit curve is...
8. ### I Looking for an expert on integrals

I’ve written an insight article on what I think is original material (at least I’ve not seen it in my reading nor google): A Novel Technique of Calculating Unit Hypercube Integrals I am looking first for someone that can follow my work, I’ve had some mathematicians look over it but none whose...
9. ### I Demonstration of inequality between 2 variance expressions

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)...
10. ### Finding the maximum of a function

Why does f attain its local maximum at r' in (p,q). Is it because we have f(x)<= f(r') for all x in (p,p+delta)?
11. ### Forces on a slider crank

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...
12. ### Analysis 1 Homework Help with Complex Numbers

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)) =...
13. ### Foundations I would love to see this book translated to English (Klein's Encyclopedia of Mathematical Sciences)

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...
14. ### Inequalities Since ε is arbitrarily small, do the inequalities hold?

#### 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##...
15. ### Prove the lower bound for a sequence (Buck, Advanced Calculus)

Clearly, ##x_{n+1}>x_n \because x_n + \sqrt{x_n} > x_n## \begin{align*} x_{n+1} &= x_n+ \sqrt{x_n} \\ &= x_1 + \sqrt{x_1} + \sqrt{x_2} + \cdots \sqrt{x_n} \\ &>n+1 \end{align*} ##\because \sqrt{x_n}>\sqrt{x_1}=1## In fact, $$x_{n+1} > 1+ \sqrt{1} + \sqrt{2}+ \sqrt{3} + \cdots \sqrt{n}$$...

17. ### A Applications of analysis in signal processing/machine learning?

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...
18. ### Showing that a function is surjective onto a set

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.
19. ### A Help with the Derrick scaling argument and topological solitons

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...
20. ### What approach should be used when solving a circuit?

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 I am solving for...
21. ### Analysis Books for learning Fourier series expansion

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...
22. ### I Find the only periodic solution of an ODE

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...
23. ### Instrument name for water sampling and analysis

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...
24. ### Resistor Network analysis

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...
25. ### In AC analysis do I alternate the current and voltage?

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.
26. ### I Understanding why ##(y_n)_n## is a bounded sequence

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##...
27. ### Analysis of an absolutely convergence of series

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...
28. ### Showing a sequence of functions is Cauchy/not Cauchy in L1

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} & ...
29. ### Calculus Multivariable calculus without forms or manifolds

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...
30. ### Engineering Derive expressions for the voltage gain of this opamp circuit

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...
31. ### B Spivak's Calculus as a Prerequisite for General Topology

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...
32. ### Show that the integral converges

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...
33. ### Convex Set in R^n Problem

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...
34. ### Extracting data from a spectrometer to Excel

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...
35. ### Proof involving convex function and concave function

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...
36. ### Mathematical Analysis Proof: |x-y|<= |x|+|y|

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...
37. ### A Convergence of a subsequence of a sum of iid r.v.s

##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...
38. ### I Limit of an extension

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...
39. ### Prove that this function is holomorphic

Homework Statement Prove that the function ## f(z)= 1/\sqrt{2}(\sqrt{\sqrt{x^{2}+y^{2}}+x}+i*sgn(y)\sqrt{\sqrt{x^{2}+y^{2}}-x})## is holomorphic on the domain ## \Omega = \left \{ z: z \neq 0, \left | \arg{z} \right | <\pi\right \} ## and further that in this domain ##f(z)^{2} = z. ##...
40. ### Prove a statement using Peano's Axioms

Homework Statement Let, m, n be natural numbers and S(n) the succesor of n. If S(n)*m = nm + m Prove that m*S(n) = nm + m Homework Equations The Attempt at a Solution
41. ### Finding the min value using the derivative

Homework Statement Hi I'm having a trouble with finding min value of given function: f(x) = sqrt((1+x)/(1-x)) using derivative. First derivative has no solutions and it is < 0 for {-1 < x < 1} when f(x) is given for {-1 < x <= 1}. For x = - 1 there is a vertical asymptote and f(x) goes to +...
42. ### Proof of uniqueness of limits for a sequence of real numbers

Homework Statement [/B] The proposition that I intend to prove is the following. (From Terence Tao "Analysis I" 3rd ed., Proposition 6.1.7, p. 128). ##Proposition##. Let ##(a_n)^\infty_{n=m}## be a real sequence starting at some integer index m, and let ##l\neq l'## be two distinct real...
43. ### I Why do systematic uncertainties disappear using ratios?

Hello, I often hear the phrase "Well, since you are taking a ratio bin-by-bin, you don't have to care about the luminosity syst. uncertainty and the trigger efficiency syst. uncertainty". I think I understand qualitatively why this is the case (It cancels out in the ratio, since both...
44. ### Physics Over what field I'm going to choose

Hello dear members, I couldn't figure out whether I should post this in the academic guidance or career guidance section, yet the latter sounded more convenient for my situation. I have also seen many similar threads but I believe everybody has different thoughts& aspirations as we all know...
45. ### Proof of sequence convergence via the "ε-N" definition

Homework Statement Prove that \lim \frac{n+100}{n^{2}+1} = 0 Homework Equations (x_{n}) converges to L if \forall \hspace{0.2cm} \epsilon > 0 \hspace{0.2cm} \exists \hspace{0.2cm} N\in \mathbb{N} \hspace{0.2cm} \text{such that} \hspace{0.2cm} \forall n\geq N \hspace{0.2cm} , |x_{n}-L|<...
46. ### Big Oh for a Fraction of 'n'

Let me start by saying that this is from a 30 question assessment on Big Oh, Big Theta, and Big Omega. I understood every other question, however, even after being given the correct answer, I do not understand why my answer was wrong for this one. If you could point me in the direction of any...
47. ### Show that this function is differentiable

Homework Statement [/B] 2. The attempt at a solution I'm not really sure where to start. We just want to show that ##\lim_{x \to c} \frac{f(x) - f(c)}{x - c} = 0##. I see that ##\lim_{x \to c} (x - c)^2 = 0##. I feel that this may be a simple trick of inequalities, but I am having a complete...
48. L

### I Circular Functions

Does a circular function with complex variable represent a three-dimensional graph? For example cosiz
49. ### I Must functions really have interval domains for derivatives?

Nearly every analysis reference I come across defines the derivative for functions on an open interval ##f:(a, b) \rightarrow \mathbb{R}##. I understand that, in constructing the definition of ##f## being differentiable on a point ##c##, we of course want it to first be a point it's domain, so...
50. ### Continuity implies bounded.

1. The problem statement: Let ##f:[a, b] \rightarrow \mathbb{R}##. Prove that if ##f## is continuous, then ##f## is bounded. 2. Relevant Information This is the previous exercise. I have already proved this result, and the book states to use it to prove the next exercise. It also hints to use...