What is Analysis: Definition and 999 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. BillTre

    Art O Brother, Where Art Thou? film analysis

    O Brother, Where Art Thou is one of my favorite movies. I always knew it was loosely based on the Odessey because it says so at the beginning, but the guy in this reaction video makes way more connections between the two then I would have ever thought of. I was amazed. Some comments on...
  2. Wrichik Basu

    Engineering Sinusoidal steady state analysis using Laplace transform

    ##\require{physics}##The given circuit is this: The question is taken from this video. The Professor has solved it using Phasor analysis, the final solution being $$\begin{equation} i_x(t) = 7.59 \sin \qty( 4t + 108.4^\circ )~\mathrm{amps}. \end{equation}$$My aim, however, is to use Laplace...
  3. L

    I Why is this function not ##L^1(\mathbb{R} \times \mathbb{R})##?

    Hi everyone in the following expression ##f(t)=\frac{1}{2 \pi} \int\left(\int f(u) e^{-i \omega u} d u\right) e^{i \omega t} d \omega ## the book says I can't swap integrals bacause the function ##f(u) e^{i \omega(t-u)}## is not ## L^1(\mathbb{R} \times \mathbb{R})## why ? complex...
  4. bhobba

    Insights What Are Numbers? - Insights for Beginners

    Hi Everyone I have been doing further investigation into infinitesimals since I wrote my insight article. I had an issue with the original article; the link to the foundations of natural numbers, integers, and rational numbers was somewhat advanced. I did need to write an insights article at...
  5. P

    A Topology - Boundary of a ball without a point

    Let us say we have f analytic in ##Ball_1(0)##. which means, radius 1, starting at ##z_0 = 0## point. If I want to find the boundary of ##Ball_1(0)##. Will the boundary be ##{0}## or ##{\emptyset}##? Not homework, just an intuition to understand ##f(z)=\frac 1 z## function ( for example ) better.
  6. MidgetDwarf

    I Sequentially Compact Sets

    Hi. Someone showed me a problem today regarding sequentially compact sets in ℝ. Ie., is the set of the image of sin(x) and x is an integer greater than one, sequentially compact? Yes or no. What is obvious is that we know that this set is a subset of [-1,1], which is bounded. So therefore...
  7. Slimy0233

    I Need Help understanding Griffiths [Vector Analysis]

    I don't understand anything about how 1.18 came to be (Red Rectangle), any help would be greatly appreciated!
  8. ARoyC

    Engineering Load Line Analysis of a Diode

    I have drawn the I-V characteristic graph of the diode. I am facing problems with drawing the load line. For what value of E (Source Voltage), should I draw the Load Line? To get the I-V graph, I had to continuously change E.
  9. FactChecker

    What analysis was done on Mike Lindell's election data?

    Mike Lindell offered a 5 million $ challenge to anyone who could prove that his election data was false. A computer expert won an arbitration decision that he had proven it. He thought that it would take a long time, or be impossible, but he says that it did not take long at all. Does anyone...
  10. Natanis_Likens

    Exploring the Works of Physicist Norma G. Sanchez & Star Analysis

    There's a some things I've been wanting to look into for a while now I just don't know where to access that kind of information. 1. I'm interested in any information/journals concerning Physicist Norma G. Sanchez (I might be wrong on her title, please correct me if I am) work on a unifying...
  11. A

    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##...
  12. S

    Circuit analysis via polyhedron

    (NOTE: I should state that my circuit analysis knowledge is up to the level of a single comprehensive EE course, as per ME students.) I was thinking about the famous resistor cube problem, and I've come up with some observations: - Every branch (I think that's the term - i.e., a branch is any...
  13. F

    I Exploring Nonlinear Least Squares for Regression Analysis

    Hello, Regression analysis is about finding/estimating the coefficients for a particular function ##f## that would best fit the data. The function ##f## could be a straight line, an exponential, a power law, etc. The goal remains the same: finding the coefficients. If the data does not show a...
  14. S

    Engineering Understanding stress analysis of truss bridges

    Hi im trying to understand truss bridges. Where would max stress be in single span truss bridge if it was evenly loaded across the bridge? I thought single span truss bridge, max stress would be in the ends of the bridge. it would act like a simple supported beam with max bending motion being...
  15. S

    Engineering FEA question (stuck): Stress analysis on this Connecting Rod

    Hey Im doing FEA and got stuck on question. The diagram go me stuck, because looking through the workbook it says high stresses are in red and low in blue, whereas they included both different kind of stresses on the same diagram and no data. Note: High Compressive strains are noted by blue...
  16. vinicius_linhares

    Analysis of the Ground Function: f(x) with $$f''(\bar{x})=0$$

    If f(x) is the function of the "ground": My first assumption is that in a certain $$\bar{x}$$, $$f''(\bar{x})=0$$, and from that point I will analyse the situation. The object has initial energy $$E_0=\frac{mv^2}{2}+mgf(x),$$ then $$v=\sqrt{\frac{2}{m}}\sqrt{E_0-mgf(x)}.$$ In each point the...
  17. D

    Manometer Force analysis

    Note: I am self-studying Material and Energy balance courses and I haven't done fluid mechanics yet. Case 1) Consider the manometer in the figure below. Levels on both sides of the manometer which are open to the atmosphere are equal. If we analyse the forces acting on the left side of the...
  18. B

    Help with finding mesh currents and nodal analysis

    Hi there, I have managed to work out the kirchoff's law bit but for the other two bits I just cant get started as I can get my head around it, its due in on monday I dont know why I left it so late!
  19. Artemisa

    Error floors in this Bayesian analysis

    In this article((https://arxiv.org/pdf/2001.04581.pdf)), the authors use a Bayesian analysis based on the positions of astrophysical bodies and their errors in the medians. This statistical analysis uses the markov chain monte carlo chains. The uncertainties in the positions are large, so what...
  20. etile

    Solve Angular Velocity & Acceleration - Dimensional Analysis

    Hi guys, Please see attached image - it's the part highlighted yellow that I'm stuck on. Here is what I got for linear velocity and angular velocity. (requested by mod) Thanks! c) Angular to velocity Dia=0.8m .. rad=0.4m v (linear velocity) = r (radius of circle) * omega (angular velocity)...
  21. M

    My proof of the Geometry-Real Analysis theorem

    Consider a convex shape ##S## of positive area ##A## inside the unit square. Let ##a≤1## be the supremum of all subsets of the unit square that can be obtained as disjoint union of finitely many scaled and translated copies of ##S##. Partition the square into ##n×n## smaller squares (see...
  22. H

    Dimensional Analysis: Forster Energy Transfer equation

    I'm trying to calculate Forster's Resonance Energy Transfer rate, but I just can't seem to get the units right. I'm trying to teach my students how to calculate them. Here is the (relatively) original technical note of FRET equation, made by the original author...
  23. simplemind

    B Question about a physical entity (dimensional analysis)

    i try to understand what physical entity mass or massdensity divided by speedquadrat could be? Any ideas? thanks
  24. M

    I Several Questions About Smooth Infinitesimal Analysis

    Hello. I read about smooth infinitesimal analysis and I have several questions: 1.What does "ε.1" and "ε.0" mean in this proof? (photo1) (https://publish.uwo.ca/~jbell/basic.pdf , page 5-6) 2. For what purpose do we use Kock-Lawvere axiom when we deal with law of excluded middle? (photo2)...
  25. M

    I Proving Continuous Functions in Smooth Infinitesimal Analysis

    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.
  26. M

    I Energy levels of atoms and spectroscopic analysis

    Hello ! My question is how the energy levels and sublevels that atoms are considered to have were obtained, and if these energy levels and sublevels were obtained as a result of the different spectroscopic analyzes of energy emission and absorption of atoms, or as it was concluded that atoms...
  27. ermia

    Finding Constants: Potential and Field Analysis

    I have wrote all feilds and potentials and I want to find the constants. My first question is " when we say in the a<x<2a the potential is V(x)" then the potential in the a is V(a) or V(0) ( cause it is 0 in our new area) ? Second one is " when I want to write the gausses law for the point x=a I...
  28. M

    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...
  29. J

    Proof for Real Analysis

    Proof: Suppose f is a function and x is in the domain of f s.t. there is a derivative at the point x and sppse. there are two tangent lines at the point (x,f(x)). Let t1 represent one of the tangent lines at (x,f(x)) and let t2 represent the other tangent line at (x,f(x)) s.t. the slopes of t1...
  30. A

    I Analysis of data from previous experiments

    Has it ever happened that after a discovery, data from previous experiments were analyzed and it was noticed that there was already some evidence of the phenomenon in question?
  31. B

    Two Particles Connected by Massless Rod: Dynamics Analysis

    Two point-like particles of mass m. The particles are rigidly connected to each other with a mass-less rod of length L. The particles are initially at rest in such a way that one particle is at the origin and the other is at the point (0, L). A point-like particle of mass M collides with a...
  32. H

    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...
  33. F

    Engineering Use voltage nodal analysis to solve for the current flowing in this AC circuit

    Hi first time using this forum, I have some homework that I am entirely stuck on, can anyone help me with this question? I understand the placement of nodes, and that I need to convert complex numbers, but apart from that I have no idea what direction that I'm supposed to take in this. I don't...
  34. G

    Capacitor circuit analysis

    Figure: My attempt at a solution: Once the capacitors have been applied, we see directly that ##\boxed{v_c=v_0=0\, \textrm{V}}## Wouldn't this one be done like this? I know the following: in a circuit with capacitors and coils, if we have Capacitor: we change for an open circuit. Coil: we...
  35. mcastillo356

    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}##...
  36. G

    Analysis of systems in equilibrium: articulated structures

    Figure: My attempt at a Solution: We calculate REACTIONS: $$\sum F_x=0\rightarrow \boxed{Bx=0}\qquad \sum F_y=0\rightarrow By+A=50$$ $$\sum M_B=0\rightarrow \boxed{A=25\, \textrm{kN}}\quad \boxed{B=25\, \textrm{kN}}$$ KNOT B: $$\alpha \rightarrow \alpha =90-40=50\, \textrm{º}$$ $$\beta...
  37. G

    Analysis of a Figure to Calculate Force and Moment

    Figure: My attempt at a Solution: $$\overrightarrow{TFD}=TFD\dfrac{(-0,16\widehat{i}+0,11\widehat{j}-0,08\widehat{k})}{0,21}$$ View from above: We calculate ##D##: $$\sigma =90-\arctan \left( \dfrac{0,07}{0,240}\right)=73,74\, \textrm{º}$$ $$d=0,16\cdot \sin (\sigma)=0,1536\, \textrm{m}$$...
  38. H

    Engineering Struggling with a Structural Analysis Problem?

    Diagram of the problem below. I have 45 minutes to finish all parts of this question without reference material. The Mechanics of Materials section of these tests are largely Structural Analysis, which I was not taught to this level in my undergrad degree so I am teaching myself via Youtube...
  39. C

    AI Computation (self study analysis, pointers welcome)

    In this thread I attempt to find a closed form solution to the gradient descent problem for a single sigmoid neuron using basic calculus. If you would like to give pointers feel free, if you see me make a mistake please let me know! Thank you!
  40. C

    Problems with Real Analysis class

    Hi. I'm nearing retirement so I thought I would take some math classes. This fall I took a Real Analysis class at a good school and dropped it because I did so bad on the first exam. I did great on the homework and quizzes. I also took Real Analysis about 47 years ago at a very good school...
  41. A

    Modal mass and kinetic energy in FEM modal analysis

    So, I use Ansys (well known FEM software) and get the next output for a modal analysis toy problem (If you happen to know Ansys that's a pre, but I promise it shouldn't matter). The problem is a simple beam, clamped at one end. I used 160 20-node brick elements to solve it (so no Timoshenko...
  42. JH_1870

    A A question about current density in finite element analysis

    I know that if there is only one conductor providing the current density, then the current density can be used. But if you apply Maxwell's equation when there are multiple current sources, I don't know which value to use. This is not an analysis using a tool, but a problem when I develop the...
  43. H

    I Analysis of converting a DE into complex DE

    In Lecture 7, Prof. Arthur Mattuck (MIT OCW 18.03) taught that the following equation $$ y’ +ky = k \cos(\omega t)$$ can be solved by replacing cos⁡(ωt) by ##e^{\omega t}## and, rewriting thus, $$ \tilde{y’} + k\tilde{y}= ke^{i \omega t} $$ Where ##\tilde{y} = y_1 + i y_2##. And the solution of...
  44. MexChemE

    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...
  45. Iqish

    Nodal analysis of this circuit with 2 sources and 4 resistors

    Here is the question a. Plot nodes in the problem and indicate how many equations will be obtained. answer: two unknown variables b. Use node analysis. Derive the system of equations for the voltages in the circuit. question: Did I calculate right V_a and V_b? C. Calculate the...
  46. M

    Free Decay analysis on SIMPACK

    I have been conducting free decay analysis on the NREL 5-MW-wind turbine model provided by SIMPACK . I wanted to compare the free decay results with OPENFAST simulation. Hence, I applied a initial displacement of 1m on the tower top in the Y direction. While comparing the results with OPENFAST...
  47. jim mcnamara

    Apparent death from supplement overdose: news analysis

    A non-technical opinion piece that claims there are many fatal and near-fatal events, ~23000 total, in the US every year -- with regard to supplement consumption/abuse. Additionally, it explains US labeling requirements and available remediation for problems related to supplement usage. Which...
  48. B

    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...