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. J

    Mathematical induction inequality

    Homework Statement Prove; n^2 > n+1 for n = 2,3,4 by Induction Homework Equations The Attempt at a Solution p(n)= P(2) 2^2> 2+1 --> 4>3 Induction step: P(n+1): (n+1)^2 > (n+1) +1 (n+1)^2> n+2 n^2 + 2n + 1 > n+2 | -n n^2 +n + 1> 2 | -1 n^2 +n > 1 Is this correct, and how...
  2. K

    Amplituhedron: newly discovered mathematical object

    Is this anything to get excited about? I have no idea what it's about. https://www.simonsfoundation.org/quanta/20130917-a-jewel-at-the-heart-of-quantum-physics/ https://www.simonsfoundation.org/wp-content/uploads/2013/09/amplutihedron_span.jpg
  3. M

    Other great books like Mathematical Transition by Chartrand

    Hello, I was an electrical engineering student and knew how to utilize the necessary mathematical tools to solve anything that came my way. I always had a deep latent predilection for mathematics but due to some constraints I was unable to spare much time on improving my knowledge of the subject...
  4. B

    Describing A Mathematical Result

    Homework Statement Let u be an arbitrary fixed unit vector and show that an vector b satisfies b^2 = (\vec{u} \cdot \vec{b}) + (\vec{u} \times \vec{b})^2 Explain this result in words, with the help of a picture. Homework Equations The Attempt at a Solution I understand that the equations says...
  5. alyafey22

    Computer science and mathematical science

    How would you relate computer science to analysis ( real and complex ) ?
  6. S

    Topology of a mathematical plane

    Assuming a mathematical plane, does it have a top and a bottom or does defining them make the plane three dimensional? Example: Given a flat, transparent plastic sheet. One draws a picture on it with a marker. If one turns the sheet over, in other words looking at the bottom of the sheet...
  7. F

    What Is The Mathematical Relationship Between AC Power And DC Power?

    Is there a formula that shows the relationship between AC power and DC power? Something on the lines of, for example: Power of AC ... = Power of DC ... ? Thank you.
  8. J

    Need Some Mathematical Guidance Regarding Random Variables

    This is not a homework question but I project I am working on and need someone with more mathematical prowess than myself. I am using a computer program to draw random numbers from two independent distributions, x1 and x2, for two different cases and I want to establish a theoretical...
  9. T

    Holographic principle, mathematical universe & dimensional reduction

    If the holographic principle says that the the observable universe could be represented as a two dimensional surface, and we accept the idea that the universe is itself a mathematical object, and cantor showed that we could in some cases represent higher dimensional sets within lower dimensional...
  10. G

    Mathematical Induction where the base case starts above 1

    Homework Statement Find all natural numbers such that 2n ≥ (1+n)2, and prove your answer. 2. The attempt at a solution I can see this is true for n=0 and n>5. I try to prove this using induction as follows 20 =1≥ 1=(1+0)2 base case: 26 =64≥ 49=(1+6)2 so it is true for n=6 and suppose 2n...
  11. N

    How do I accurately model and calculate costs for different data plans?

    Re: Are my answers correct? Quadratic and logarithmic modeling I actually have a few more things that i'd like you to check for me please. Answers: Appetizer: {0<=x<=10 y=7 x>10 y=2(x-10)+7 or y=2x-13 Quest: {0<=x<=40 y=40 x>40 y=1(x-40)+40 or y=x) Voyager: {0<=x<=60 y=60 x>60...
  12. B

    Please help with an exercise in mathematical methods for physicists

    Homework Statement Derive the equation of motion of a particle of mass m and charge q moving in three dimensional space under a Coulombic attraction toward a fixed center. I have the answer for a plane but a i need it for the space Homework Equations Mathematical Methods for...
  13. W

    Mathematical Physics for Undergrad

    Hello; I wonder which one of the following textbooks is suitable for mathematical physics undergrad course: 1:) Mathematical Methods: For Students of Physics and Related Fields by "Sadri Hassani". 2:) Mathematical Methods in the Physical Sciences by "Mary L. Boas". Thank you in Advance
  14. C

    Improving Math Rigour - Tips & Advice

    I was told I lack mathematical rigour. But how do I go on improving on it? Is it only a matter of being very careful? Do I have to always support everything with a clear Euclidean succession of logical steps? Is it only a matter of 'believing' in the validity of the supporting tools? Then it's...
  15. C

    Is this a solid way to mark an exam on a Mathematical subject?

    In an Engineering course I was used to being rewarded for proving a good understanding of theory. For example, if I explained a basis of the following solution, it would still give some rewards, even if the solution itself had major computing errors or even if it was partially mathematically...
  16. W

    About Mathematical Analysis by Zorich

    I'm reading this book now. I like it because it's rigorous and interesting but I have too much difficulty solving the exercises. What's worse, Zorich didn't give the answers. Where can I find the answers? I mean is there a book or something that gives the answers or even just some hints to the...
  17. I

    Mathematical method in electrical potential ?

    Mathematical method in electrical potential ?? Hi All, I need mathematical help from the topic electrical potential for lectures on physics by Richard Feynman. Need some help to understand mathematical method used here. question : 1) From electrical potential.png ( how this partial...
  18. MarkFL

    JustCurious' question at Yahoo Answers regarding a mathematical model

    Here is the question: I have posted a link there to this topic so the OP can see my work.
  19. Seydlitz

    How to do Mathematical proof for beginner

    Hello all, I know that perhaps this question has cropped frequently in a lot of places, but I would like to be quite specific in my case. I've just finished High School with UK A level syllabus. The syllabus inside Math A level is mostly straight solving equation type of questions, such as for...
  20. D

    Best book on mathematical statistics

    Hi Please advice me a good book on mathematical statistics.I want it to be oriented in math students or engineering level. Best Regards
  21. T

    Mathematical map - diagram illustrating what you need to know to learn

    I was curious if anyone is aware of anything resembling a mathematical map, or learning tree. By this I mean a diagram that illustrates the sequence in which one must learn particular topics. I ask because I find myself encountering topics (today it was finite element method) and clicking the...
  22. M

    Relation between actual measurement and Mathematical observables

    I'm having a gap in understanding the relation between them, and resolving my confusion is really appreciated. For example, the Hamiltonian operator, why do we call its eigenvalues energies? how do we actually measure it in the laboratory, quantum mechanically? And maybe I need a better...
  23. matqkks

    Exploring Mathematical Induction: Impactful Examples and Real Life Applications

    Some students are not convinced that a proof by mathematical induction is a proof. I have given the analogy of dominoes toppling but still some remain unconvinced. Is there very convincing way of introducing mathematical induction? I need something which will have an impact. Are there any real...
  24. matqkks

    Is Mathematical Induction the Key to Understanding Finite and Infinite Concepts?

    Some students are not convinced that a proof by mathematical induction is a proof. I have given the analogy of dominoes toppling but still some remain unconvinced. Is there very convincing way of introducing mathematical induction? I need something which will have an impact. Are there any real...
  25. M

    Mathematical Integral in Neuroscience Paper

    Hi everyone, I'm having trouble getting a mathematical result in the following paper: http://link.springer.com/chapter/10.1007%2F978-3-540-88853-6_4#page-1 In particular, I can't figure out how they reach equation 7. They use equations (3) and (5), and I think they discard a lot of the...
  26. K

    Mathematical Link between Kinetic Force & Electro-Magnet's charge.

    Greetings! This is my first post and I am hopefully doing it in the right section of the site. Onto the topic: I am searching for the mathematical link between the kinetic force applied on two electro-magnets being forced against each other (assume repelling charges) and the effect that it...
  27. fluidistic

    Ladder operator for harmonic oscillator, I don't get a mathematical

    If the ladder operator ##a=\sqrt {\frac{m\omega}{2\hbar}}x+\frac{ip}{\sqrt{2m\hbar \omega}}## and ##a^\dagger=\sqrt {\frac{m\omega}{2\hbar}}x-\frac{ip}{\sqrt{2m\hbar \omega}}## then I get that the number operator N, defined as ##a^\dagger a## is worth ##\frac{m \omega...
  28. S

    A mathematical formula for p mod q

    What would be an expression for the modulo operation based only on basic arithmetic i.e. addition, subtraction, multiplication and division? For example, if I would like to define 7 mod 3 using only the basic arithmetic operations, how would I write a function for it in a programming language...
  29. Psinter

    Mathematical model for interaction force between magnetic fields?

    I wanted to know if there is a mathematical model to know the force with which a source of a magnetic field attracts or repels another source of magnetic field. For example, the mathematical model for the electrostatic force a charged particle exerts over another charged particle is: F =...
  30. J

    Mathematical Tripos, archived question about squares

    Homework Statement a, b, x, y are rational numbers (ay - bx)² + 4(a - x)(b - y) = 0 implies that x = a and y = b or 1 - ab and 1 - xy are squares of rational numbers 2. The attempt at a solution My attempts are hard to transcribe, is this mandatory for getting help? This is my first post, I...
  31. fluidistic

    Stability, Helmholtz free energy mathematical relation proof

    Stability, Helmholtz free energy mathematical relation "proof" Homework Statement I must show that \left ( \frac{\partial ^2 F}{\partial V ^2} \right ) _T=\frac{\frac{\partial ^2 U}{\partial S^2} \frac{\partial ^2 U}{\partial V ^2} - \left ( \frac{\partial ^2 U}{\partial S \partial V} \right...
  32. S

    Mathematical Induction on two Matrices

    Homework Statement (1 1)^n = (1 n) (0 1) (0 1) Prove this through mathematical induction. Homework EquationsThe Attempt at a Solution I've replaced n with 1, so I've done that far. Then I said k = n. Then replaced all n with (k+1). I'm really stuck...
  33. Y

    Mathematical tools for analysis of solutions of optim. control problem

    Hi, i’m looking for some mathematical tools for analysis or classification of solutions of optimal control problems/variational. Equations of motions and optimized function depend on some parameters eg. dX1/dt = f(X1,X2, u1, p1) dX2/dt = f(X1,X2, u2, p2) where X1,X2 are state variables, u1,u2...
  34. R

    Mathematical Logic and Structures

    URGENT ! Mathematical Logic and Structures Hey guys, i need a HUGE favor, i need the resolution for this 5 questions, its a question of end this year my degree or stay one year just with a subject. Please I am begging, who knows the resolution please say me something. Part 1...
  35. R

    Mathematical Logic and Structures

    URGENT ! Mathematical Logic and Structures Hey guys, i need a HUGE favor, i need the resolution for this 5 questions, its a question of end this year my degree or stay one year just with a subject. Please I am begging, who knows the resolution please say me something. Part 1...
  36. S

    Is there a field of Mathematical Astrophysics ?

    Is there a field of "Mathematical Astrophysics"? Just was wondering (for graduate school purposes), is there a field that does what mathematical physics does but for astrophysics? (e.g., perhaps simplifying or re-expressing the mathematics used in some subdiscipline of astrophysics?) Or perhaps...
  37. N

    Your biggest mathematical embarassment

    Have you ever struggle to do a simple math/physics problem correctly for a long time JUST because of a ridiculous oversight you repeatedly made?
  38. V

    Where can I find resources for learning mathematical proofs from scratch?

    Hey guys, I just recently transferred to the mathematics department at University of Illinois as a junior undergrad and one of the classes I'm required to take this fall is called Mathematical Reasoning and I'm pretty sure it has to do with proofs... Which I know absolutely nothing about! What...
  39. K

    Lattices, mathematical economics

    Homework Statement Let ≥ be a binary relation, defined on the set X as follows: for any x,y\inX, x≥y if x-y≥0 and x-y is even. Determine if the following are lattices: X={2,4,8} X={1,4,8,9} Homework Equations Note that ≥ in this case is just the symbol for relation as it relates...
  40. P

    Energy a mathematical expression?

    So, energy is simply a mathematical expression of how much work is done in the system? Since energy is not a "physical" object that has mass. It's not matter at all. If so, the laws of conservation makes sense, it can't be created nor destroyed. But I can't imagine how a mathematical...
  41. G

    Verify using Mathematical Induction

    Verify that 1(1!)+2(2!)+...+n(n!) = (n+1)! - 1 is true using induction This problem has me stumped...
  42. Barioth

    Correlation and mathematical expectation question

    Hi, I have these 2 problem, that I'm not so sure how to handle. 1-Let X_1,X_2,...,X_n independant Random variable that all follow a continuous uniform distribution in (0,1) a) Find E[Max(X_1,X_2,...,X_n)] b) Find E[Min(X_1,X_2,...,X_n)] where E is for the mathematical...
  43. F

    Mathematical physics line/surface integral problem

    Homework Statement i have to show that the length of line integral squared on surface (ds^2) , which has vectorequation r=r(u,v) (r here is a vector), can be given as ds^2=Edu^2+2Fdudv+Gdv^2 (Find functions E,G,F) Homework Equations I should use diferential dr=du(t)e1+dv(t)e2 (dr, e1 and...
  44. J

    Help me make a very mathematical encryption algorithm

    Suppose I make an application with a password of max 20 characters -- no special characters and not case-sensitive. So that means there is a 1-to-1 correspondence between the set of all passwords P and the set S = {1, 2, ..., 3720 - 1, 3720}. A simple bijective function f:P-->S could be...
  45. Y

    Prove the theorems using mathematical induction.

    Homework Statement Prove the theorems using mathematical induction. \forall n \in N, n\geq 4 \rightarrown2\leq n! Thanks in advance! Homework Equations The Attempt at a Solution First, check the base case which is n=4. \Rightarrown=4\geq4-True \Rightarrow42\leq4*3*2*1...
  46. phoenixthoth

    Definition of mathematical object

    definition of "mathematical object" ...And "mathematical existence." Do these phrases have accepted definitions? Back when Dedekind was rejecting Cantor's transfinite ideas, could there have been a definition Cantor would refer to and say definitively "my (infinite) sets have mathematical...
  47. G

    Mathematical Issues with Reif Thermodynamics Problem (BYO Math Chops)

    Hi there, I am a long time reader of the Physics Forums, but a first time poster. The problem I am requesting assistance with stems from a problem in Reif's Fundamentals of Statistical and Thermal Physics. 7.16 An aqueous solution at room temperature T contains a small concentration of...
  48. S

    Mathematical Induction: Find P(sub2)(A(subn)) & Prove (n*(n-1))/2

    Let A(subn) = {1,2,3,...,n} For any set B, let P(subk)B=the set of all subsets of B with exactly k elements. For example, P(sub2)({1,2,3})={{1,2},{1,3},{2,3}}. A) Find P(sub2)(A(sub1)), P(sub2)(A(sub2)), P(sub2)(A(sub4)), and P(sub2)(A(sub5)) B) Use mathematical induction to prove that the...
  49. M

    Nonlinear programming problem, mathematical programming, Summation Function.

    I was wondering if you might have some insight into a problem, where we consider an optimization problem: max ∑ from j=1 to n of fj(xj) such that ∑ to n of xj <=B xj>=0, integers where B is a positive integer and fj is real to real I am trying to formulate a solution using dynamic...
  50. S

    It makes logical sense but can't mathematical prove it

    Homework Statement Find the minimum and maximum of the given equation, f(x,y,z)=x+y+z, with the constraint 1/x+1/y+1/z=1. Homework Equations f(x,y,z)=x+y+z, with the constraint 1/x+1/y+1/z=1. The Attempt at a Solution I know the solution, which is minimum {(1,1,-1), (1,-1,1), (-1,1,1)} and...
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