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. E

    Anyone know of good Mathematical Modeling classes/books?

    I've recently gotten into Mathematics more and more, and I wanted to do some mathematical modeling just to see where I stand on it and maybe try to get better at it. I've looked around as well as I could, and I didn't really find much, so I was wondering if anyone here knew of any good...
  2. C

    Which mathematical subjects must I learn to understand basic QFT?

    So I did QFT at university and didn't feel that I really understood what was being done. We just did some calculations, heuristic guesswork and dwelled on phenomenology. I want to do it the way I personally understand things best: by learning the mathematics in detail, almost at the level of a...
  3. S

    Mathematical tools for quantum mechanics

    I am currently reading the book on quantum mechanics by N.Zettili. It's really a good book but has a lot of mathematical stuff which I am not able to absorb. There's a whole chapter on mathematical tools but it is difficult to.understand. Can anyone suggest me book or text on mathematics as...
  4. S

    What are some resources for understanding mathematical moments?

    Hey guys, I am trying to understand the concept of mathematical moments (as defined in this wikipedia page: http://en.wikipedia.org/wiki/Moment_%28mathematics%29), as some work I am doing relates to them. I am looking for more information to read up but when I search maths moments I get...
  5. T

    Mathematical induction - inequality

    Homework Statement Prove that \frac{1}{n}\sum_{i=1}^n x_i\geq {(\prod_{i=1}^n x_i)}^{1/n} for positive integers n and positive real numbers x_i Homework Equations There is also a hint. It states that it does not seem to be possible to prove it directly so you should prove it for n=2^m...
  6. S

    Is f(n) = 4^n + 6n - 1 Divisible by 9 for All Positive Integers n?

    Prove that f(n) = 4^n + 6n - 1 is divisible by 9 for all positive integers of n. I've proved this by considering that f(k) is divisible by 9, i.e f(k) = 4^k + 6k -1 = 9m where m is some integer. Rearranging to give 4^k = 9m - 4k + 1 and then considering f(k+1) = 4^(k+1) + 6(k+1) - 1 then...
  7. P

    Suppose traffic laws allowed mathematical reasoning

    Some time ago I had a thought regarding turning rules in the US, back when right turn on (a) red (light - unless otherwise marked) was becoming universal. If U-turn was legal on the cross street, then by smoothing out a sequence of legal maneuvers: - left turn on red should be legal (right...
  8. J

    Would Mathematical Physics benefit me?

    Hello everyone, i am just wondering if a mathematical physics honours degree would benefit me more if i am wanting to get into theoretical astrophysics or theoretical high energy physics afterwards for my graduate studies? Or should i just stick with a physics honours degree? i'm...
  9. N

    Proper mathematical notation in regarding integrals

    Hi I have a question regarding notation. I have a function f(x, y), which I would like to integrate as \int_{x>0,y<x\frac{1}{\sqrt{\pi}}+1}{f(x, y) dxdy} My question is very simple, and probably very silly: What there a notation which enables me to write the condition for the integral (x>0, y...
  10. O

    Understanding Measure Theory with Rudin's Principles of Mathematical Analysis

    Hello everyone, I needed to know more about measure theory so I'm reading in Rudin's Principle's of Mathematical Analysis, somewhere in the chapter, he says: We let E denote the family of all elementary subsets of $R^p$... E is a ring, but not a $\sigma$-ring. According to my understanding of...
  11. M

    What kind of mathematical order is this?

    What kind of mathematical "order" is this? Disclaimer: Sorry if this question is off-topic, I really don't yet know where it "belongs". I wrote a program that counts the number of "1" bits in a 16-bit value. My program is brute force right now (loop 0 to 65535, shift each number and count...
  12. 2

    Mathematical appraisal of Pascal's Wager

    For those who aren't familiar with it, http://en.wikipedia.org/wiki/Pascal's_Wager"]Pascal's[/PLAIN] Wager, in it's bluntest manifestation, is the idea that being religious (Christian or Muslim) is worth any finite cost, because the possible consequence (eternal damnation) of not being...
  13. B

    Books to Buy for Mathematical Maturity in Theory and Proofs

    I hope this is in the right section, otherwise please move it. This might be a long post and I hope someone will give it a shot. It is about books but not really a comparison between them. I will give some background first, so you might skip to the end (- Books * -) if you are only interested in...
  14. I

    Mathematical induction with a summation in the problem.

    Hi, I'm currently taking Discrete Mathematics and I'm working on a mathematical induction problem that's a little different than usual because it has a summation in it. What I basically want to know is did I do parts A and C correctly? Homework Statement Homework Equations The Attempt at a...
  15. A

    Mathematical Iteration for Alignment of 4 Points

    What is the mathematical equation or algorithm for alignment of four points in cartesian coordinates (X & Y). Say, we have 4 nominal points (Xn & Yn), and another 4 measured points (Xm & Ym)?. After alignment, what are the new locations of the measured points (Xmn & Ymn)?.
  16. M

    Revisiting an old problem with startling new mathematical results

    I've got the attached paper... It's interesting, because it deal with an old complex problem is a seemingly simple form, and just using high school algebra. Check it out! Here is the attackment:
  17. T

    How Many Ways to Place Divisors Around a Circle in Mathematical Induction?

    I want to now the answer of this question and I think it relates to mathematical induction. The question is: -Suppose is a natural number. In how many ways can we place numbers around a circle such that each number is a divisor of the sum of it's two adjacent numbers? Who can answer this...
  18. K

    Combining Equations with Different Domains: A Mathematical Challenge

    Please I need to combine these three equations to be one, though they have different domains, what can I do? RxLevel = -0.87RxQual - 72 RxLevel = -0.013Dis - 58 RxLevel - -0.2MOP - 81 Where RxQual, Dis and MOP are variables
  19. AnTiFreeze3

    Mathematical Proof That The Media is Sexist and Bad at Math

    "Mathematical Proof That The Media is Sexist and Bad at Math" Here's is a link to my favorite comedy website: Cracked.com (The website itself may be NSFW, but this particular article doesn't get vulgar). I thought that this was an interesting article that points out what is too obvious for the...
  20. S

    Element Selection for Voltage Doubler Design with Mathematical proof

    I need to design a voltage doubler that takes in 115V-50Hz a.c. and produces a doubled voltage output of 230V dc. I am using this circuit: I need help in the mathematics part: i) How do I find the optimum value of capacitance? (Using high capacitance values reduces ripples; but can...
  21. D

    Mathematical induction problem

    Hello! First of all I have like 5 exercises I don't quite understand so will it be a problem if I create 5 new topics in the next 24h? Homework Statement Prove, by using mathematical induction that if x+1 \geq 0 then (1+x)^n \geq 1+nx. Homework Equations The Attempt at a Solution Basic step...
  22. S

    Walter Rudin's Principles of Mathematical Analysis

    Is there any place on the Internet where I can pose questions on the text and get timely and reliable answers by those who are qualified enough? Besides the Harvey Mudd College, is there any other video lecture series on this text? Which is / are the most comprehensive and the most...
  23. N

    Combinatorics - Mathematical Induction?

    Hello, I am having trouble solving this problem. Maybe I'm just overreacting to it. In my two semesters in discrete math/combinatorics, I've never seen a problem like this (with two summations) and been asked to prove it. Can some one help? \sum^{n}_{i=1} i^3 = \frac{n^2(n+1)^2}{4} =...
  24. M

    Can you help me solve my mathematical problem?

    I'm definitely messing up somewhere :P The Larmor Equation is \Delta H = \frac{2\mu}{\hbar Mc^2 e} \frac{\partial V(r_{ij})}{\partial r_{ij}} ( L \cdot S) I derived an equation stated as \nabla \times \vec{F}_{ij} \cdot \hat{n} = \nabla \times \frac{\partial V(r_{ij})}{\partial r_{ij}} How...
  25. M

    Part III of the Mathematical Tripos in Cambridge

    Im studying my bachelors degree in physics and math outside the UK, but inside the EU. I am just wondering how hard is it to get into this program? I probably could get funding from my home country so let's assume that money isn't a problem. If I get my bachelors with 1st class honors and...
  26. S

    Lack of Mathematical Ability or Maturity?

    Lack of Mathematical Ability I'm currently in Grade Nine (by the Canadian Education System) and enrolled in Tenth Grade mathematics, scoring top-marks. I've self-studied Algebra I-II, Trigonometry and Calculus I (not including integration). However, I've noticed when I participate in...
  27. J

    Books recommendation for Mathematical proofs

    Hi, I was wondering if you can suggest good books on the techniques for mathematical proofs. I know this is quite a simple matter but I would be glad if you could suggest me with greats books for teaching proofs. More often than not I used the direct method and induction to prove problems in...
  28. D

    Mathematical induction problem

    Homework Statement Hello! This is I want to prove using Mathematical Induction: 1+3+5+...+(2n-1)=n^2. The problem is: I don`t understand very much about Mathematical Induction :(Homework Equations The Attempt at a Solution Suppose n=1. Then 1=1. Now suppose 1+3+5+...+(2n)=(n+1)^2. Then...
  29. A

    Bound charges - are they real or mathematical?

    In my chapter about electric fields in matter my book derives and expression for the potential due to the polarization of a dielectric material. For that you find that the polarization is equal to the potential of a collection of negative charges on the surface and positive charges inside the...
  30. A

    Mathematical Foundations of the Relativistic Theory of Quantum Gravity

    Hello. I would like to bring to discussion this paper called "Mathematical Foundations of the Relativistic Theory of Quantum Gravity", by Professor Fran de Aquino, from the Maranhao State University, in Brazil. The paper was published in the Pacific Journal of Science and Technology...
  31. H

    Where Can I Find a Good Introductory Book for Writing Mathematical Proofs?

    Hello Could anyone recommend a good introductory book for learning how to write mathematical proofs. Thank you.
  32. J

    How Does Rudin Prove the Existence of h in the Uniqueness of n-roots?

    Hi! I need some help here, please. In Principles of Mathematical Analysis, Rudin prove the uniqueness of the n-root on the real numbers. For doing so, he prove that both $y^n < 0$ and that $y^n > 0$ lead to a contradiction. In the first part of the proof, he chooses a value $h$ such...
  33. J

    Punctuation in mathematical writing

    Recently I went through a bunch of my old solution sets and realized that I've been quite sloppy with my use of punctuation; in particular, examining the quality of my writing in solution sets over a period time, it appears that the lack of proper punctuation is getting more and more pronounced...
  34. B

    Reading Mathematical Theorems And Their Proofs

    Lately, I have been experiencing a sort of anxiety over not understanding some of these proofs in my calculus textbook. I just finished a calculus I course, and we did not spend any time learning them; so, I thought, since it is summer time now, I would go back over and try to learn those...
  35. S

    Obtaining mathematical model for the kinetic system

    Homework Statement See attachment. f(t) is the input force and b1 and b2 are kinetic friction constants. There is no static friction. Homework Equations ƩF = m \ddot{x} F_s=kx F_f=b\dot{x} Ff is the force from friction and Fs is the force from spring. The Attempt at a Solution...
  36. J

    Mathematical physics, density of probability

    Homework Statement So I have to calculate the transmission of light, which goes through random black and "transparent" blocks (p=0.5), but the thickness (x) of each block is distributed exponentially dP/dx=1/t*exp(-x/t). When light goes through black block, it diminishes (I stands for...
  37. A

    Modeling mathematical representation for music signal

    Hello there I am sorry if this sounds silly or out dated i want to ask anyone, how can i model the mathematical representation from music signal(time domain). lets say i recorded the signal from guitar through my music production software and print screen that signal as shown here...
  38. A

    How to study mathematical analysis?

    I'm not quite sure if my question belongs to this section or not, so I hope I would be posting it in the right place. Anyway, I have troubles understanding Rudin's principals of mathematical analysis. To me, everything he says, is absurd. I can't understand what he says, neither do I want to...
  39. K

    Mathematical model, work done on a gas

    In a mathematical model, a gas is under a pressure of the form P=e-v2 (v is volume). Find the work (in Joules) done on the gas as its volume decreases from infinity to zero.
  40. F

    Good Mathematical Logic Textbook?

    I've been trying to decide on a mathematical logic textbook to teach myself a bit. I'm taking a course on it next semester, but I have never had a logic course before (I've had some CS courses though and proof-y math courses). I'm also taking a modal logic course the semester after math logic...
  41. S

    Frustated by the lack of mathematical rigour in physics topics

    I have become so annoyed by the lack of mathematical detail in the physics textbooks. For example, to study Foundations of Quantum Mechanics, I had to spend hours and hours doing background reading from books, the net and Wikipedia until I got my way aroung vectors spaces and its nooks and...
  42. T

    Super position and mathematical limitations

    I've been curious for some time about the concept of super position. my introduction to the idea was through the schrodingers cat paradox, which i assume most know, so i will stick with this example. is it not correct to assume super position is a concept to resolve the perceptual limitations of...
  43. D

    Proving inequality by mathematical induction

    Homework Statement I am asked to prove: 2n < (n+1)! , where n≥2 The Attempt at a Solution Base step: set n=2, then test 22 < (2+1)! 22 = 4 (2+1)!= 3! = 3(2)(1) = 6 so 4 < 6 , which is true. Induction hypothesis is 2k < (k+1)! Using this, prove 2(k+1) < [(k+1)+1]! = (k+2)! Attempt to...
  44. G

    Man gets mugged, transforms into mathematical savant?

    http://abcnews.go.com/blogs/health/2012/04/27/real-beautiful-mind-accidental-genius-draws-complex-math-formulas-photos/Don't know if he really is a genius at math now, but at least he can draw some cool pictures. So who wants to hit me on the head?
  45. P

    Can Mathematical Induction Prove the Existence of a Fourth Dimension?

    I am currently exploring if whether or not a fourth dimension exists or can be drawn. According to my professor, I have to use mathematical induction. I know that 2^n is the equation and "n" equals the dimension. Therefore 2^1 is 2. The first dimension is a line with 2 terminal points...
  46. P

    Mathematical Description of a Wave

    Here is the question: Transverse waves on a string have a wave speed of 8.00 m/s, amplitude of 0.0700m, and a wavelength of 0.320m. The waves travel in the -x direction, and at t=0, the end of the string (x=0) has its maximum upward displacement. a)What is the frequency, period, and wave...
  47. E

    Can you help me to solve this little mathematical game

    First of all sorry for the way I write, but I do not speak English very well. In the excel file attached there is a small puzzle that I can not fix, anyone can help me? Considering the initial value of 100,You can share this value so that instead of question marks in the cells H9, I9, J9...
  48. P

    Where to find mathematical problems for mathematics relating to electronics

    Does anyone know where i can score a site with lots of problems pertaining to electronics equipment. Trying to understand the basics (resistors now) and when i google it i just get tons of crap but no problems to solve. anyone? Thx :)
  49. C

    Mathematical Induction problem.

    Homework Statement (1/2!)+(2/3!)+(3/4!)+...+(n/(n+1)!) a) calculate for a few small values of n. b) Make a conjecture about a formula for this expression c)Prove your conjecture by mathematical induction. Homework Equations The Attempt at a Solution So for the first part I just used values...
  50. C

    Math Blogging: Interesting Problems & Representation Theory

    Hey. So I have decided that I want to try to produce a math blog. The blog would essencially just consist of interesting problems and proofs, as well as expository notes on stuff that I'm reading (the symmetric group and it's reprepresentation theory). However, I have found no useful information...
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