Mathematical Definition and 1000 Threads

Mathematics (from Greek: μάθημα, máthēma, 'knowledge, study, learning') includes the study of such topics as quantity (number theory), structure (algebra), space (geometry), and change (analysis). It has no generally accepted definition.Mathematicians seek and use patterns to formulate new conjectures; they resolve the truth or falsity of such by mathematical proof. When mathematical structures are good models of real phenomena, mathematical reasoning can be used to provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation, measurement, and the systematic study of the shapes and motions of physical objects. Practical mathematics has been a human activity from as far back as written records exist. The research required to solve mathematical problems can take years or even centuries of sustained inquiry.
Rigorous arguments first appeared in Greek mathematics, most notably in Euclid's Elements. Since the pioneering work of Giuseppe Peano (1858–1932), David Hilbert (1862–1943), and others on axiomatic systems in the late 19th century, it has become customary to view mathematical research as establishing truth by rigorous deduction from appropriately chosen axioms and definitions. Mathematics developed at a relatively slow pace until the Renaissance, when mathematical innovations interacting with new scientific discoveries led to a rapid increase in the rate of mathematical discovery that has continued to the present day.Mathematics is essential in many fields, including natural science, engineering, medicine, finance, and the social sciences. Applied mathematics has led to entirely new mathematical disciplines, such as statistics and game theory. Mathematicians engage in pure mathematics (mathematics for its own sake) without having any application in mind, but practical applications for what began as pure mathematics are often discovered later.

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  1. T

    Regarding mathematical operators

    One basic operator is addition. In order to add the any number x number of times, multiplication was invented. In order to multiply any number x number of times, exponentiation was invented. What if we want to raise a number to a power x number of times? How come we didn't invent that? Also...
  2. M

    Mathematical misconception in scattering: switching from cartesian to spherical

    If we were to consider a nucleon-nucleon interaction: We know that the incident wave (plane wave) is ψ= Ae^{ikz}, propagating in z direction But for some mathematical facilities, we tend to use spherical coordinates, the wave becomes = \frac{A}{2ik}[e^{ikr}/r - e^{-ikr}/r] How come...
  3. L

    Is There a Mathematical Physics Degree?

    Hi, I am wondering if there is a mathematical physics degree. I was thinking about doing consultant/contract work for applied or research physicists, insofar that I would be paid to help establish/confirm the mathematical foundations of their work or research. Thank-you.
  4. W

    Mathematical model of continuous and batch (discrete) system combined

    I'm having difficulties trying to establish the best approach to create a mathematical model of a process that has a combined continuous and discrete (batch) element to it. I explain as follows: The system is a hopper (vessel), open to atmosphere, with dry granular material being fed in by...
  5. S

    How to create a simple mathematical model

    Hello, I am really not very good with math... over the years, I totally lost it. But today, it caught up with me... LOL! I hope someone can help me. In my spare time I am fiddling around with proximity sensors. I have a proximity sensor that accepts 135 steps. Therefore, when I am at step 135...
  6. T

    How useful is mathematical proof as a mechanical engineer?

    Specifically I plan on specializing as a Mechatronics engineer. I recently bought the book "Mathematical Proofs: A Transition to Advanced Mathematics" and I plan to study it on my own due to curiosity and interest towards mathematics. I would like to listen on one's opinion on how useful it...
  7. E

    How Can I Solve This Mathematical Induction Problem?

    Hello folks, I have an induction problem that I have been working on, and I am at my wit's end. If anybody could provide me with a nudge in the correct direction, I would greatly appreciate it. I have attempted to factor out a number of terms to find something remotely intuitive here to present...
  8. A

    What are some recommended books for improving mathematical physics skills?

    So I really want to get good at mathematical physics. Below is a list of books I could get from various libraries, but I don't know which one's I should start with and which to do next and so on. I currently understand intro CM, EM, and QM. My math skills are very basic (ODES as of now). I thank...
  9. J

    Leibniz formula using mathematical induction

    Homework Statement Here is this problem: I have the solution http://www.proofwiki.org/wiki/Leibniz%27s_Rule/One_Variable This is where I get stuck.. Where it says: 'For the first summation, we separate the case k=n and then shift the indices up by 1.' Why does this lead to the...
  10. M

    Exploring the Mathematical Conversion between ∆ and dX

    hi, Reading a book on thermodynamics and the guy often uses something like this : ∫1/T dQ = ΔS and then he says "this in differential form" : dQ/T = dS I kind of get the idea visually that one slice of "integral" will be dQ and you can think of it this way. But my question is how...
  11. F

    Splitting light into colors, mathematical expression (fourier transforms)

    I am trying to solve a problem that includes a function of the light hitting a certain area. My question is, how would I change a function G(x) of photons hitting a certain area to include just photons of a certain wavelength, say red light. I feel like this could be accomplished using a Fourier...
  12. E

    MATLAB Writing mathematical equations in MATLAB

    Hello, I have the attached paper, and I want to write equations 4, 5 and the equation in the first line after equation 5 in MATLAB. I did it as following: Np=24; L=19; TapsL=[1 19 20]; S=length(TapsL); Index=zeros(S,1);%Tap Locations c=zeros(L+1,1); cHatMP=zeros(L+1,1)...
  13. B

    Good programs for mathematical physics in canada

    im looking to become a theoretical physicist and i think i might want to do a program in mathematical physics for an undergrad. does anyone know which are best in Canada? i know the obvious ones, (i think) U of T, Waterloo, UBC, McGill, but can anyone be more specific please?
  14. H

    Math Structures: Proving Existence w/ Accurate Solutions?

    Hey guys, let's say I were to define a new mathematical object, a novel type of number for example, and I am trying to determine its various properties (arithmetic, exponential, logarithmic, etc). Now, let's say I am able to use these numbers to produce solutions that agree with expected...
  15. B

    Developing mathematical maturity?

    One of the problems that I come across is my bad habit of glancing at the solutions. I usually look at them after I struggle on a problem for 20 minutes. Almost always its a proof problem like in linear algebra such as "Prove that there is one and only one way to write A as the sum of a...
  16. W

    Mathematical treatment of signals and systems

    Hi there, I am looking for books that treat signals and systems from a mathematical perspective. When I say "signals and systems", I am referring to the typical second/third year material taught in EE programs: system characteristics (causality, memory, stability), LTI systems, linearization...
  17. S

    Show the mathematical relationship between Relative Risk and Difference in

    Smoker Non-smoker Disease A B No disease C D Show the mathematical relationship between Relative Risk and Difference in Conditional Proportions. (Work the formulas algebraically so RR is on one side of the equation and the difference (plus some other...
  18. C

    Mathematical logic and statistics

    Let's say that a treatment A has been proven to have an impact on the levels of B with a given confidence interval. Let's also say that we know that the treatment C causes the treatment A to be imposed on our sample. Before the testing on the effects of C has been done, which statistical models...
  19. T

    Curvatures in space-time: actual reality or mathematical concept?

    According to GR is space actually supposed to be bending around mass, or is the concept of "curvature" just abstract and purely mathematical idea?
  20. W

    Help with a Mathematical Induction Problem

    Suppose we want to prove that: 1/2 * 3/4 ... 2n-1/2n < 1/sqrt(3n) for all positive integers. (a) Show that if we try to prove this inequality using mathematical induction, the basis step works, but the inductive step fails.(b) Show that mathematical induction can be used to prove the stronger...
  21. M

    Which Mathematical Physics Text Should I Choose?

    I found myself having a hard time choosing what mathematical physics text should I stick with. I'd like to think of myself as mathematically inclined, and I would really prefer a mathematical physics book that emphasizes 'the maths' and the proofs and not just the methods while maintaining its...
  22. M

    Proficiency in Mathematical Proofs

    Where does one attain the skill/ability (skill vs. ability is a good question in itself) to do proofs? By "do proofs" I mean prove existing theorems. For instance I am really no good with ε-δ proofs. The concept is so easy, but when I have to prove something using ε-δ I either have no idea what...
  23. K

    Otto cycle mathematical model - Heat added by combustion?

    I am attempting to improve a spreadsheet I created back in uni some 5-6 years ago, which models all the velocities and accelerations of the pistons and rods, as well calculating pressure, temperature and resulting torque. It is only a simple model so far, originally it started simply by...
  24. L

    A mathematical explanation of hysteresis?

    I wish to know why and how does hysteresis occur from a purely mathematical perspective, using analytical functions and multivariable calculus. Thank you!
  25. V

    Pairwise disjoint mathematical proof

    Set 1 {{1, 2}, {2, 3}, {1, 2, 3}, {2, 3, 4}, {2, 3, 5}, . . . , {2, 3, n}}; Set 2 {{3, 5}, {5, 4}, {3, 5, 4}, {4, 5, 1}, {4, 5, 2},{3, 5, 6}, {3, 5, 7}, . . . , {3, 5, n}}; Set 3 {{3, 1}, {1, 4}, {3, 1, 4}, {1, 4, 2}, {1, 3, 5}, {1, 3, 6}, . . . , {1, 3, n}}; Set 4 {{1, 5}, {5, 2}, {1...
  26. K

    Using induction technique in mathematical logic

    Homework Statement Prove the following LEMMA: For every proposition A[P_{1}, \dots, P_{n}] and any two interpretations v and v', if v(P_{i})=v'(P_{i}) for all i=1, \dots,n, then v^{*}(A)=v'^{*}(A). Homework Equations The Attempt at a Solution Sure this is obviously an...
  27. P

    Mathematical description of Refractive index in crystal/glass

    Hi, I have read on a website that the refractive index of glass is related to the number of electrons it has per unit volume. I don't recall which website it was, but is it true? & if so what is the quantitative description behind it? or is it a lot more complicated :) Many thanks for...
  28. C

    Can anyone solve this mathematical induction problem?

    Homework Statement Prove that for all n>1, P(n) :1 + 1/2 + 1/3 +...+1/n = k/m where k is an odd number an m is an even number.Homework Equations The Attempt at a Solution 1)Base Case n =2 P(2) = k/m 3/2 = k/m which is true. 2) Inductive Step Assume P(n) is true for some arbitrary n. 3)...
  29. Crazymechanic

    How can I calculate gas pressure in a cylinder using mathematical formulas?

    Hi could you please help me out with a formula? So imagine this abstract cylinder and a piston moving in it , I have let's say an x amount of gas in it and all the other parameters like stroke diameter and so on , by what formula could I calculate the pressure that the gas will have when the...
  30. C

    Mathematical logic & its branches

    What branches of mathematical logic are there? I've taken formal logic (that is, the logic where one has various operators, like conjunction, disjunction, etc). What other fields of logic should I take to become better at mathematical logic? Which fields of logic are the most useful and essential?
  31. M

    Filling holes or gaps in mathematical knowledge.

    I graduated high school nine years ago. The highest math I took then was Algebra II, as I had little interest then. After a seven year career, I was able to go back to school and have made it to Calculus I this semester. I struggled some in college algebra, finding out that there were...
  32. B

    My Mathematical Methods in Physics is written to weed out students, help me

    what is a mathematical methods in physics book that you all would recommend? I need One that shows enough steps, explains clearly, etc. At ASU, my mathematical methods in phyiscs book covers the following topics with excruciatingly little detail and cryptic wording/notation, skipping so many...
  33. L

    Proving n3 + n < 3n for n >= 4 using Mathematical Induction

    Homework Statement show n3 + n < 3n for all n >= 4 Homework Equations The Attempt at a Solution I.H : n3 + n < n for all n >= 4 3(n3 + n) < 3(3n) then (3n+1) = 3 x 3 n > 3((n3) + n ) by I.H...
  34. H

    Best place to study mathematical logic uk

    I was wondering where the best place to study logic at postgraduate level in the UK. As far as I know it's Manchester but I could be wrong? Thanks for any help
  35. M

    Visualising mathematical concepts in QM - comments needed

    Hi, I'm unsure if I'm posting this in the correct place, so apologies in advance if not. I looked at the study board, but the post template doesn't apply to my question. I'm a layperson who had previously only read popular science books on QM, but has recently been watching some more...
  36. T

    Standard Model Mathematical Representation

    Hello, everyone. I understand the Standard Model and all of its quarks, leptons and bosons, but could someone show me the equation and simply explain it? I haven't been able to find it or an explanation of how it relates to the graphical representation of the Standard Model. Thanks in advance!
  37. M

    Mathematical Induction - Algebra Manipulation

    Homework Statement I am working on a mathematical induction problem, where I need to prove: (1 - (-7) ^ (k + 2)) / 4Homework Equations (1 - (-7) ^ (k + 1)) / 4 + 2(-7) ^ (k + 1) The Attempt at a Solution So I just need to add the two items in section two above. Now I know I need a common...
  38. G

    Exploring the Pauli Vector: Mathematical Definition

    Hello I'm reading my old notes of QM, I found the definition of Pauli vector, as follow \vec{\sigma}=\sigma_1 e_x+\sigma_2e_y + \sigma_3 e_z Where e_x. e_y and e_z are unit vectors. So, here is my question. \sigma_i and e_i are elements of different nature. How can we define the product...
  39. D

    Use mathematical induction to prove the following statements are true

    Homework Statement Use mathematical induction to prove the following statements are true for n≥1 a) 1^2+3^2+5^2+...+(2n-1)^2= [n(4n-1)]/3 Homework Equations The Attempt at a Solution Attempt at showing for n+1 is true: n[4(n+1-1)]/3+ 2[k+1-1)^2
  40. A

    Step-by-step mathematical physics from classical to quantum physics

    Dear folks, Under the aalaniz blog, I just posted step-by-step notes starting for ODEs and PDEs both linear and nonlinear, of the type you will see in grad physics programs. The notes connect Lagrangian, Hamiltonian and Poisson Bracket methods in classical physics to the various approaches...
  41. M

    Can I Visualize Mathematical Terms with Wolfram Alpha?

    hello! I think it would help a lot to visualize mathematical terms I would like to see the graphs of various f(x), then their f'(x), their f''(x), in order to understand what derivatives mean example: http://en.wikipedia.org/wiki/File:Graph_of_sliding_derivative_line.gif same with...
  42. R

    Using second Principle of Mathematical Induction Proof

    Homework Statement Let F: N->N be defined by f(1)=1 and f(2)=2 and f(n+2)=1/2(f(n+1)+f(n)) Use Theorem 1.3 to prove that 1<=f(n)<=2 for all n in N Homework Equations For each n in N let P(n) be a statement about the positive integer n. If a) P(1) is true, and b) for k>1, P(k) is...
  43. M

    Mathematical Physics vs. Honours Physics vs. Physics and Astronomy

    Hi, I'm a Grade 12 student and I'm very interested in physics. I would like to apply to Waterloo University becasue of their renowned Co-op program, but they have the three types on Physics Undergraduate Degrees; Mathematical Physics...
  44. H

    Momentum, impulse and force (mathematical)

    Homework Statement See the attachment Once you read the question (attached) you'll notice that there are two different parts of the question, the first one asks to find the combined velocity of the system after collision while the 2nd one uses the result of the first part and states to find...
  45. A

    Understanding Mathematical Induction: Why Do We Use n Instead of k?

    what is mathematical induction? explain with an example.
  46. M

    Complete list of mathematical notations

    hello! is there a complete list of mathematical notations? thanks!
  47. G

    What Are the Best Resources for Learning Mathematical Finance?

    Can anybody provide some resources (books, websites etc. - preferably websites or course notes) for a (somewhat rigorous) introduction to Mathematical Finance? Basically, I want to have the background to understand stuff like this http://www3.stat.sinica.edu.tw/library/c_tec_rep/c-2002-10.pdf...
  48. E

    Explaining the Mathematical Reasoning of Finding Image in Convex Mirrors

    I am having a problem understanding the mathematical reasoning of finding the description of image in convex mirror... We know the relation, 1/v + 1/u =1/f is true for all mirrors... we can prove this for both concave and convex mirrors... In case of convex mirrors we use the conventional...
  49. TrickyDicky

    Some doubts concerning the mathematical bases of GR

    I'm a little confused about certain assumptions usually made in GR as to how rigorous they are mathematically speaking. For instance the assumption generally presented without proof that GR spacetime manifold is a Hausdorff space seems not to be warranted given the fact that pseudometric spaces...
  50. I

    Mathematical Quantum Statistics: Why is A*rho of trace class?

    Hi, as we know a density operator \rho is defined to be a non-negative definite operator of trace class (with trace 1). We also know that for a given observable A, which is a (possibly unbounded) self-adjoint operator, the expectation value can be calculated as \operatorname{tr}(A \cdot...
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