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. I

    Problems with proofs of Robert Geroch mathematical physics

    Hello guys, I'm new in this forum, this is my first Thread. I've started reading Robert Geroch's Mathematical Physics recently and I've been having problems with some of the proofs that involve monomorphism. He defines monomorphism the following way (pg 4): let ψ be a morphism between A...
  2. M

    How can i show my mathematical theory?

    I have a math theory about finding the root of negetive one however now i have no clue how to get it out. How can i trademark it or patent it... I am a 10th grader in South Africa. I can't post it on some site as people may steal or something.
  3. A

    Assumptions in Mathematical Word Problems

    Hi there, Consider a related rates problem, Gas is being added into a balloon at a rate of dE/dt = x ft^3/sec Find the rate at which the radius is changing at time h. This is a completely false problem; I am just giving an example. My question is, How would you know if there is a...
  4. I

    Is Mathematical Induction Proven Correctly for $S_{k+1}:2^{k+1}>(k+1)^2$?

    $S_k>k^2$ $S_{k+1}:2^{k+1}>(k+1)^2$ $2*2^{k+1}>2(k+1)^2$ $2^{k+2}>2(k+1)^2$ Assume $x=k+1$ $\frac{2^{x+1}}{2}>x^2$ $2^{x+1}*2^{-1}>x^2$ $2^x>x^2$ right? $2^{k+1}>(k+1)^2$
  5. W

    Why is the mathematical model favored over the mechanical model?

    In theoretical physics, why is the mathematical model favored over the mechanical model? Awhile back, I posted a thread asking about what each theory posits as real. For example, quantum field theory might limit the set of real things to fields, field quanta, the universe, and causality. It...
  6. I

    Using Mathematical Induction to Prove a Summation Formula

    $S_k:5\cdot 6 +5\cdot 6^2+5\cdot 6^3+ ...+5\cdot 6^k=6(6^k-1)$$S_k:5\cdot 6 +5\cdot 6^2+5\cdot 6^3+ ...+5\cdot 6^k+ 5\cdot 6^{k+1}=6(6^k-1)+5\cdot 6^{k+1}$ what do i do now? to prove $S_{k+1}$
  7. Greg Bernhardt

    What is mathematical induction

    Definition/Summary Mathematical Induction is a method of proving a series of mathematical statement labelled by natural numbers. This method usually involves two steps. First one proves the base case, then one shows that if the statement holds for some natural number, it holds for the...
  8. I

    Is $k \geq 3$ enough to conclude $(k - 1)^2 > 2$ in mathematical induction?

    Did i do this right? $2^n>n^2$, $n \ge 5$ $S_{k+1}: 2^{k+1}>k^2+2k+1$ $2^k>k^2$ $2(2^k)>2k^2$ $2^{k+1}>k^2+k^2$ And $k^2+k^2>k^2+2k+1$ RHS: $k^2+2k+1$ So $2^{k+1}>(k+1)^2$
  9. I

    How Does Mathematical Induction Work?

    I don't know what forum to post this under. PLEASE HELP ME THOUGH! Principle of Mathematical induction: Let $S_n$ be a statement concerning the positive integer n. Suppose that, $S_n$ is true. For any positive integer k, k $\le$ n, if $S_k$ is true, then $S_{k+1}$ is also true. Then $S_n$ is...
  10. H

    How Do I Simplify the Right Side of the Mathematical Induction Equation?

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  11. D

    Mathematical proofs and an understanding of them for scientists

    To what extent, if any, is an understanding of mathematical proofs required for a scientist? I can empathize with a need for an understanding of the general machinery of the tools you are using (understanding, for example, how it is the chain rule came about, ie, how it was derived) but, using...
  12. gfd43tg

    Mathematical Induction: Proving Inequalities with k and k+1

    Homework Statement Homework Equations The Attempt at a Solution Hello, I am working on the problem in the attached image regarding induction based on the inequality 2^n \geq n + 1 I am confused how to do this by proving that if it is assumed to be true for ## n = k ##, then how it is true...
  13. C

    Mathematical Physics as a graduate program

    Hi guys, I was wondering whether atypical preparation is needed for admission to a mathematical physics program? i.e. Do you need to take Real Analysis, Topology, and other "rigorous" mathematics even if you are only a physics undergrad, or will such mathematics be taught from the ground up...
  14. I

    What mathematical backgrounds is needed for cosmology?

    Hi there! I would like to know, what kind of mathematical background is needed for cosmology? I'm about to reach 50% of undergraduate physics and I appreciate if you could point it out some interesting courses I can take to build a strong basis for a «future» cosmology masters or phd degree...
  15. H

    Mathematical falacy. ln(1+x) series exapansion:proving 2=1

    series expansion: ln(1+x)=1-x^2/2+x^3/3-x^4/4+x^5/5+......∞ ln(1+1)=1-1/2+1/3-1/4+1/5...∞ ln(2)=(1+1/3+1/5+1/7...)-(1/2+1/4+1/6+1/8...) ln(2)=(1+1/3+...)-2(1/2+1/4+1/6+1/8...)+(1/2+1/4+1/6+1/8...) ln(2)=(1+1/2+1/3+1/4+1/5+1/6...)-2(1/2+1/4+1/6...)...
  16. W

    Mathematical and physical concepts in nanoscience

    Dear all, I am going to attend masters program in micro and nano technology. I would like to know what concepts of maths and physics I should be clear at and some good textbooks for the same. My study module includes nanoscience, nanomaterials and nano electronic design.
  17. A

    Mathematical Model Homework: Lengthen Time until Shock Stabilised

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  18. P

    Where should I go after Boas' Mathematical Methods in Physics?

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  19. N

    Understanding ∏ and other mathematical constants

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  20. P

    "Horizon" A Mathematical Mystery Tour

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  21. O

    Mathematical Induction questions

    Are these satisfactory answers? Question 1: http://i.imgur.com/EElOqwM.jpg Question 2: http://i.imgur.com/SlE2jza.jpg?1 Thank you :)
  22. W

    Please help me refine a dissertation topic in mathematical logic

    I'm looking to write a dissertation in the field of logic (for a philosophy degree). I'm deeply interested in logic, but whenever I consider the material beyond my courses it becomes pretty daunting. I'm reasonably familiar with: *First Order Logic *Set Theory and ZFC *Cantor's Diagonal...
  23. P

    Ant on a rubber string, mathematical series

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  24. R

    Mathematical Difference Between Mean Free Path vs RMS Free Path?

    I was wondering if there is a mathematical difference between the RMS free path and the mean free path of molecules in an ideal gas. For example, When I calculate the mean free path, I use use the average velocity and the scattering rate which is a function of the average velocity. I then...
  25. J

    What is the proposed scale for rating mathematical acumen?

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  26. U

    Surface charge and volume charge density mathematical confusion

    If you have a charged solid sphere with uniform volume charge density ρ, then the total charge on the sphere is Q = ρ*4/3*∏*R^3 , where R is the radius of the sphere. Now...
  27. R

    Mathematical Biology (Reaction-Diffusion equation)

    A spatially varying competition model between red and grey squirrels is given by \begin{equation} \frac{\partial R}{\partial t}= D_R \frac{\partial^2 R}{ \partial x^2} + aR(1-R-b_2G),\tag 1 \end{equation} \begin{equation} \frac{\partial G}{\partial t}= D_G \frac{\partial^2 G}{ \partial x^2} +...
  28. R

    Mathematical Biology (infectious disease)

    Consider the infectious disease model defined by \begin{equation} \frac{dS_3}{dt}= -\rho I_3S_3+\gamma I_3+\mu-\mu S_3\tag 1 \end{equation} \begin{equation} \frac{dI_3}{dt}=\rho I_3S_3-\gamma I_3-\mu S_3 \tag 2 \end{equation} with initial conditions $S_3(0)=S_{30}$ and $I_3(0)=I_{30}$ at $t=0$...
  29. R

    Mathematical Biology (non-dimensionalise)

    Consider the predator-prey system defined by \begin{equation} \frac{dP}{dt}= aP(1-\frac{N}{K})-bPN= f(N,P)----------(1) \end{equation} \begin{equation} \frac{dN}{dt}=cNP-dP= g(N,P)--------------(2) \end{equation} with initial conditions $P=P_{0}$ and $N=N_{0}$, Where $N=N(t)$ is the prey density...
  30. T

    Which graduate math courses for mathematical physics

    Hello, I've started a MS in mathematics at my university and I'm interested in mathematical physics or going to to physics for a PhD. The classes required for the MS are 2 semesters of real analysis, linear algebra, and one semester of algebra, everything else is up to the student to pick and...
  31. S

    How Do You Calculate the Distance ACD with Six Variable Geometric Parameters?

    See attachment, please. Six variables that can change individually and hence altering all connected. The goal is always to find the distance between A & C, "ACD" if you will. ACD = ... Given 6 flexible variables: (Abbreviated to perhaps make things easier: r, abd, wt, abl, pll, pld. Use or not...
  32. T

    Master degree - Mathematical modelling or Astro?

    I'm an udergrad, 2nd year physics. I recently got interested in solar physics - one reason being, that I was offered to cooperate on a project, where I'll be doing some sunspot simulations. I really like mathematics and I kinda like the mathematical modelling curriculum. However the...
  33. BadgerBadger92

    Is Natural Mathematical Ability Required To Be A Physicist?

    Hello, all. My whole life, I've had a passion for physics and understanding the universe in general. For years since I was younger I've read layman's books on the subject, reading articles and journals, etc. I have deeply thought about physics (theoretical physics in particular) as being my...
  34. Demystifier

    Math is Beautiful: Evidence for Mathematical Aesthetics

    When a mathematician says that mathematics is "beautiful", is it so in the same sense as e.g. art is beautiful? A recent research presents evidence that it is: http://journal.frontiersin.org/Journal/10.3389/fnhum.2014.00068/full
  35. R

    Mathematical Biology (steady states)

    non-dimensionalisation equation: \begin{equation} \frac {du}{d\tau}=\frac{\overline{\lambda}_{1} u}{u+1} -\overline{r}_{ab}uv -\overline{d}u \end{equation} where $\overline{\lambda}_{1}= \frac {\lambda_{1}}{\lambda_{2} K_{1}}$ Another non-dimensionalisation equations \begin{equation} \frac...
  36. R

    Mathematical Biology and Modelling

    Consider the two species competition model given by $$ \frac{da}{dt }= \frac {λ_1 a} {a+K_1} - r_{ab}\cdot ab - da, \ \ \ \ \ \ \ \ \ \ (1)$$ $$\frac{db}{dt }= λ_2 b (1-\frac{b}{K_2}) - r_{ba}\cdot ab , \ \ \ \ \ \ \ t>0,\ \ \ \ \ \ \ \ (2)$$ for two interacting species denoted a=a(t) and...
  37. R

    How Do Interactions Between Two Species Model Their Population Dynamics?

    Consider the two species competition model given by da/dt = [λ1 a /(a+K1)] - r_(ab) ab - da, (1) db/dt = [λ2 b *(1-b/K2)] - r_(ba) ab , t>0, (2) for two interacting species denoted a=a(t) and b=b(t), with...
  38. Giant

    Confused between mathematical physics and theoretical physics

    Hello! This is my second thread in here. I'm just finished my high school and waiting for the results to get admission in an undergrad physics program. I was confused about career guidance and academic guidance feel free to move my thread if I'm in the wrong place I was wondering about major...
  39. D

    Does point really exist?a mathematical concep

    does point really exist in reality or it is mere a mathematical concep
  40. D

    How Can You Prove That PV Equals E in Thermodynamics?

    Homework Statement Show that PV = E Homework Equations E= \int^\infty_0 D(\epsilon)n_{FD}(\epsilon) \epsilon \cdot d\epsilon n_{FD}=\frac{1}{(1+ e^{-(\alpha +\beta \epsilon_k)})} \psi(\alpha ,\beta, V) =\beta PV =\sum_\vec{k} \ln{(1+e^{-(\alpha +\beta \epsilon_k)}) } and in an...
  41. MexChemE

    Mathematical interpretation of work done by a gas

    Hello everybody! First of all, I would like to say that this is my first post in this forum, even though I've occasionally read some posts before. I'm a ChemE major from Mexico! I am currently taking Thermodynamics I, and I have trouble figuring out the expression: W = \int_{V_1}^{V_2} PdV I...
  42. R

    What Are the Dynamics of a Single Species Population Model?

    Consider the single species population model defined by dR/dt = [gR/(R+R_m)] - dR, t>0, where g,R_m and d are all positive parameters and R(0) =R_0 (a) Describe the biological meaning of each term in the equation. (b) Determine the steady-states of the system and discuss any...
  43. K

    Computational, theoretical or mathematical physics for finance (Quan)?

    Hi. My purpose is to take a phd in physics, and after few years as researcher to become a Quant. What phd is better for this purpose: computational, theoretical or mathematical physics? http://en.wikipedia.org/wiki/Computational_physics
  44. V

    Redundancy in mathematical properties

    I've been thinking about some common properties of mathematical objects and I've been wondering if they are redundant. Like: Aren't all associative operations also closed under a set? Doesn't the existence of inverses imply the existence of an identity element? So that stating associativity...
  45. E

    Good foundational mathematical basics.

    I'm trying to get get a good base for learning further mathematics. Would an intro abstract algebra, analysis, and category theory text be good places to start? There are so many different places you could start and I'd like to first learn about those areas of math which help most when learning...
  46. kq6up

    Boas: Mathematical Methods for Phys Sci Pr.1.13.25

    Mclaurin Series with Division by Zero? Boas: Mathematical Methods for Phys Sci Pr.1.13.25 Homework Statement Using the methods of this section: (a) Find the first few terms of the Maclaurin series for each of the following functions. (b) Find the general term and write the series in...
  47. kq6up

    Mary Boas: Mathematical Methods Problem 1.13.8

    Homework Statement Using the methods of this section: (a) Find the first few terms of the Maclaurin series for each of the following functions. (b) Find the general term and write the series in summation form. (c) Check your results in (a) by computer. (d) Use a computer to plot the...
  48. kq6up

    Not understanding textbook solution: Mary Boas mathematical methods

    Homework Statement 4. Write the Maclaurin series for 1/√(1 + x) in ∑ form using the binomial coefficient notation. Then find a formula for the binomial coefficients in terms of n as we did in Example 2 above.Homework Equations { \left( 1+x \right) }^{ P }=\sum _{ n=0 }^{ \infty }{ \left(...
  49. Avatrin

    Memorizing Mathematical Definitions

    Hi Usually when learning math, understanding the theorems and ideas helps tremendously to remember math. I get that... I got through calculus, linear algebra and complex analysis easily. The problem for me started with three branches in mathematics: Real analysis, measure theory and abstract...
  50. V

    Mathematical connection in the cartesian product

    mathematical "connection" in the cartesian product What is the mathematical connection between elements of a cartesian product ##A\times{}B## and the elements of the sets ##A## and ##B##? In other words, what is the difference between the set ##A\times{}B## and just any set ##Z## with...
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