What is Mathematical: Definition and 1000 Discussions

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. P

    Mathematical induction problem

    Homework Statement Show that it is possible to arrange the numbers 1, 2, ... n in a row so that the average of any two of these numbers never appears between them. [Hint: Show that it suffices to prove this fact when n is a power of 2. Then use mathematical induction to prove the result when n...
  2. P

    Mathematical induction question

    Homework Statement Suppose a and b are real numbers with 0 < b < a. Show that, if n is a positive integer, then a^n - b^n \leq na^{n-1}(a-b) Homework Equations The Attempt at a Solution I'm trying to show this by induction. Let P(n) be the proposition that a^n - b^n \leq na^{n-1}(a-b)...
  3. L

    Why you can't prove a single mathematical thing beyond a shadow of a doubt

    Oyez! I'm an amateur philosopher and I'd like to start up an interesting debate. The topic will be mathematics (and, in general, everything) and how it is not logically possible to prove such a system. This is my informal argument: 1. Humans live in a "reality" governed by laws, axioms...
  4. N

    Knowing more about the discipline of mathematical neuroscience

    I've been looking for curriculum (e.g. classes needed) for mathematical/theoretical neuroscience but with no success. Math, physics, and neuroscience are some of my prime interests, but I have no idea on how to go about thinking if I should pursue theoretical neuroscience instead of theoretical...
  5. V

    Surface tension mathematical formula.

    consider a U shaped wire piece. a movable wire is fixed on it and their is liquid bubble in between them. If we move wire through distance l what force we need to apply surface tension basic formula is F=Sl. so it should Sl but it is 2Sl i cn't understand can you please help.
  6. L

    Mathematical Axioms of General Relativity

    What are the equations from which all of GR can be derived? Obviously one of the equations is Einstein's Field Equation: G^{\alpha\beta}=8\pi T^{\alpha\beta}. I would also guess that you would need the Euler-Lagrange Equations: -\frac{d}{d\sigma}(\frac{\partial L}{\partial...
  7. P

    Mathematical Induction (The inductive step)

    1. I don't understand how to prove this. for all n≥1, 10n - 1 is divisible by 9. 3. I've done the basis step. Now I'm on the inductive step. I'm using (10k+1-1)/9=1. I don't know where to go from there. Using algebra just gets me down to 10k+1= 10. And I really don't think that's...
  8. fluidistic

    Mathematical methods of Physics, ODE

    Homework Statement For what values of K does the DE xy''-2xy'+(K-3x)y=0 (1) has a bounded solution in (0, \infty)?Homework Equations Not so sure, Frobenius method maybe.The Attempt at a Solution First, I check what happens when x tends to infinity. I see that the DE behaves like \phi ''-2 \phi...
  9. L

    Is Mathematical Visualization a Discipline or Just a Tool?

    What fields / disciplines of that field does mathematical visualization fall under. I know numerical computational geometry is one , where you wright algorithms to develop programs to visualize mathematics.
  10. fluidistic

    Mathematical methods of Physics, ODE

    Homework Statement I must find the constant K such that y''-\left ( \frac{1}{4}+\frac{K}{x} \right )y=0 for x>0 has a non trivial solution that is worth 0 when x tends to 0 and when x tends to infinity.Homework Equations Frobenius method.The Attempt at a Solution I proposed a solution of the...
  11. E

    Exploring the MUSIC Algorithm to Understand its Mathematical Derivations

    I'm studying the MUSIC algorithm in order to implement it in some project of mine, but I have some difficulties understanding the mathematical derivations done in the original Schmidt paper. For those of you who have access to this paper, I'll appreciate your time and help. The author begins...
  12. M

    Some kind of mathematical formula that I obviously dont know about?

    Hello there. Can you please explain the following: Let us suppose that δ is very much less than r, and expand |r+δ|^-1 in powers of δ/r. Keeping the lower order terms gives: |r+δ|^-1 ≈ 1/r.(1-r.δ/r^2) + 1/2r [3(δ.r/r)^2-(δ/r)^2] How did they reach here? is this kind of a mathematical...
  13. N

    What do mathematical physicists work on?

    How do they differ from theoretical physicists? I was looking into arXiv papers in the mathematical physics portion and saw things like the physics of hoolahoops..I hope no offense is taken but I thought that was bit funny haha. On a more serious note, how are theoretical and mathematical...
  14. S

    Ways to answer mathematical posts

    Mathematics is supposed to abstract the essential features of a situation. I have yet to see this done for the common task of answering a question about mathematics posted to the forum. For questions of the form: "How do I do ... X ...?", I offer the following general general patterns...
  15. D

    How can I prove that n^3 + (n+1)^3 + (n+2)^3 is a multiple of 9?

    Hello everybody, I am doing my reading lately to prepare for some exams to join a mathematics department. And I would very much like, if anyone could help, the solution (or a hint) to the following induction proof. " Show that n^3 + (n+1)^3 + (n+2)^3 is a multiple of 9 " :smile: I...
  16. E

    Highest authority on Mathematical Physics?

    Anyone have any candidates, and yes i know its a broad term.
  17. Z

    Mathematica Hard mathematical induction question

    Hi everyone, I have been trying to do this induction question... but can't seem to understand how to do it. Prove by induction that, for all integers n is greater than or equal to 1, (n+1)(n+2)...(2n-1)2n = 2^n[1x3x...x(2n-1)] This is what someone else has given as the answer: For n...
  18. G

    Mathematical Basis for Learning Relativity

    Hi, I seem to remember there is a book by Steven Weinberg that gives the mathematical basis for tensor calculus for relativity, but the name escapes me. Anyone know what I'm talking about?
  19. K

    Fit in some Mathematical Finance in before end of degree?

    I'll be in my 4th year of a Pure Math degree next year, and I'm think of what courses to fill up my timetable with. I'll be taking an overload, so I'll be taking 6.0 credits worth of courses (3.0 per semester). I'm only required to take 3.0 more courses in Pure Math to get my degree, so I have...
  20. M

    Help with mathematical assertions for natural numbers

    Homework Statement Prove by Mathematical Induction that the assertion, n ∑ r^2 = n/6 (n+1)(2n+1) r=1 holds for every natural number n. Homework Equations Ok, so basically, how do you solve this question? I have got to the Induction step but I'm not sure how to do it...
  21. A

    Where can I find classic mathematical treatises?

    I'm looking for an English translation of Lagrange's treatise "Reflections on the algebric solutions of equations" (Reflexions sur la Resolution Algebrique des Equations). I've searched the internet for it but haven't found anything about it yet. Where can I find such truly classic treatises?
  22. H

    How indicative is olympiad mathematics on a person's mathematical strength?

    I have always loved math and have done independent theorem-proving and research. However, I have had absolutely no olympiad training but it seems that the math olympiad is quite important in applying to universities. I have started to do some of the questions myself and find that I progress...
  23. I

    Mathematical Notation/Symbol Reference Sheet?

    Hey all! For awhile, I have been looking for a .pdf version of a math notation sheet, similar to these sites: http://www.rapidtables.com/math/symbols/Basic_Math_Symbols.htm http://en.wikipedia.org/wiki/List_of_mathematical_symbols I like the way they have the notation in LaTeX, have a meaning...
  24. A

    Which mathematical topics should I attempt to self-study over the Summer?

    Just as a little background, by the time summer comes around, I will have taken mathematics classes up to basic differential equations, multivariable calculus, linear algebra, and some probability theory. In Physics, I will have taken up to classical mechanics and mathematical physics. I want to...
  25. C

    String theory and mathematical physics in Germany

    Hey guys! Which research departments are best? I was hoping to get the top 3-4 institutions names. Basically I am into mathematical string theory and geometry/topology-physics in general.
  26. T

    Are physics laws mathematical models?

    Are physics laws (even the laws) mathematical models of reality? Or they describe reality as it is? Are those laws just models, like for example a population's growth mathematical model (just an example that I remembered)? A maths teacher told me that in physics everything are...
  27. A

    Mathematical objects, structures and spaces

    What are they, how are they visualized? Is it like how basic equations when plotted form a shape (e.g. circle) but much more complex? When I think of the those words, I think of some sort of weird/cool looking shape.
  28. A

    Principles of Quantum Mechanics - mathematical introduction

    I'm going through the book: Principles of Quantum Mechanics 2nd edition by R. Shankar 1. In the mathematical introduction to projection operators (page 22), it writes: "Consider the expansion of an arbitrary ket |V\rangle in a basis: |V\rangle=\sum_{i}^{} |i\rangle\langle i|V\rangle"...
  29. H

    Mathematical Induction Problem: Finding a Formula for a Sequence of Numbers

    Homework Statement Okay, so I'm going to be completely honest, I am really bad at math, and I have been struggling the past couple of weeks in my Quantitative Reasoning class. I am so lost. I don't know if it's my teacher's teaching method or what, but nothing is clicking for me at the moment...
  30. A

    Is there a limit to mathematical progress?

    I'm not sure myself.
  31. B

    Mathematical Modelling question

    Though this question is about medicine, the actual question has little to do with medicine and more to do with modelling .. Homework Statement A patient is put on an intravenous drip at time t=0, the drip supplies a drug into the patients bloodstream at a constant rate λ. At the same time...
  32. B

    How do you evaluate mathematical discoveries?

    Hi all, Can someone help me understand how mathematical discoveries are evaluated? If a new finding is made, how do you know if it's significant or not? What differentiates the content of the prestigious journals from others? I can understand that solving problems that have been unsolved...
  33. S

    Looking for an interesting Mathematical problem family

    Looking for an interesting Mathematical problem "family" And what do I mean by that. All right, I think this was the most fitting category, I hope I'm not too off-topic. So I want to write a puzzle game, but, as everyone who loses his time playing puzzle flash games (like me), knows that you...
  34. R

    Principle of Relativity, from a mathematical perspective

    I didn't know where to post this question, but the general math forum seemed like the best idea. Newton's Laws are invariant with respect to the Galilean transformations. Maxwell's equations are invariant with respect to the Lorentz transformations. My question is, is it necessary...
  35. T

    Proving r^n > r^m through mathematical induction

    Homework Statement I need to prove that for any real number r, if 0 < r < 1, then for all positive integers n and m, if n < m, then r^n > r^m. Homework Equations No calculus techniques are permitted, only mathematical induction. The Attempt at a Solution I know that any...
  36. K

    What are some recommended books on mathematical logic and set theory?

    Hello, I would like to know about a good introductory book on mathematical logic. It should start from set theory , include ZFC axioms and also touch on Godel's theorems.
  37. S

    Mathematical Methods in the Physical Sciences (3rd edition)

    Mathematical Methods in the Physical Sciences (3rd edition) by Mary .L. Boas, Does this book has separate solution manual?
  38. M

    What sort of mathematics for mathematical physics?

    I am hoping to do mathematical or theoretical physics after my undergraduate science degree. I have been advised to do more maths subjects than physics subjects in order to achieve this. I am double majoring in maths and physics and I am just wondering what area of maths I should be focusing on...
  39. A

    Understanding Mathematical Induction: Proving Implications and When to Use It

    Hello, I have a question about inductive reasoning... Earlier this week my intro proofs class went over the logical structure of induction, and an example. The example was a proof of \Sigmai = n*(n + 1)/2 My main issue is the assumption that "p(k)" is true. What if it's not? I asked this in...
  40. S

    Is there anything in our universe that we can't model with a mathematical set?

    Wondering if this is mathematically feasible.
  41. V

    Mathematical modeling: physics degree or math degree

    I am currently a double major in biology and math, interested in mathematical modeling of ecosystems and physiological systems. The math portion involves modeling and pure math, but there isn't a lot of concentration on knowing physical laws. I'm not sure if I would miss all the knowledge...
  42. D

    Implementation of Schroedinger's equation - mathematical concepts.

    Hi all! I'm not sure whether this is the correct section to post my question. I'm learning the program OpenFOAM, implemented in C++ and I'd like to implement schroedinger's equation i\hbar\frac{\partial}{\partial t}\Psi(\mathbf{x},t)=-\frac{\hbar^2}{2m}\nabla^2 \Psi(\mathbf{x},t) +...
  43. matqkks

    How to Help Students Understand Mathematical Proofs

    I am having difficulty with students understanding mathematical proofs. Even if they do understand them in the class they cannot reproduce them in the final examination. Majority of the students avoid doing the proof questions and concentrate on the numerical or technique application of...
  44. R

    Please I want mathematical model of cpu,HDD

    Please I need Mathematical Model of Processor and Mathematical Model of Hard Disk because i want to study thermal behavior to discover strategy for cooling it
  45. Z

    Unconventional Mathematical Theorems Beyond Textbooks

    I'm looking for something like a books filled with theorems not typically covered in textbooks. For example Viete's formulas for roots and coefficients of polynomials, which, strangely enough, I find useful. They speed up my computational speed and give interesting relationships between objects...
  46. R

    Mathematical Methods in the Physical Sciences

    I have been self studying this book by Mary L.Boas for a while which is a treat to study for its great depth. However, the problem of mine is though I have been able to solve most of the problems of this book, there are some really tough Questions in this book which are very hard to solve. So...
  47. K

    Equilibrium solutions in mathematical modelling

    Hi, I don't want to list the exact problem I have to solve, as I would rather do it myself, all I would like is some guidance. The problem is a function like: Pk+1=a (Pk-1) with a few more constants and some brackets thrown about on the right hand side. I need to find all the equilibrium...
  48. T

    Intermediate methods of mathematical physics

    Hi my school has these two courses: Specific mathematical methods used in advanced courses in physics. 233. Differential equations, vector differential and integral calculus. 234. Laplace transforms, Fourier analysis, complex analysis. There is no indication that 233 is a prereq for...
  49. G

    Terms in a Mathematical Expression

    When we refer to terms in an equation, what EXACTLY are we referring to? Suppose for example we have: ab + bc + cd = A Suppose somebody refers to term "ab". Are they referring to the syntactical NAME "ab", the IMAGE of ab (i.e., it's value), or the ARGUMENT (a,b) which belongs to some...
  50. B

    How can I effectively learn mathematical modeling for social science research?

    Hi All, Just wanted to know, what is a good way to learn mathematical modeling? I'm a social scientist working with simulation and data, and would like approach the issue more formally. Before trying modeling textbooks, what should I know beyond calculus? Any suggestions (books, tutorials...
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