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. K

    A mathematical issue raised from perturbation theory

    Take the usual time-independent perturbation theory in QM for example,H'=H_0+V, a basic assumption is we can expand the new states of H' in terms of the old ones of H_0, most of the textbooks justify this assumption by reasoning that the set of eigenfunctions of Hamiltonian is complete...
  2. D

    Wave Function mathematical vs real

    Hi, In Copenhagen Interpretation, the wave function is just a mathematical tool that has no physical reality. In Bohmian Mechanics, the wave function has objective reality as some kind of field. Why can't we have both where the wave function has objective reality as some kind of field...
  3. JeremyEbert

    HELP Mathematical notation needed for a prime mod 12 pattern

    OK, I need help putting this into mathematical notation. 2 and 3 being the first two prime numbers make up the basic pattern in primes of 6(n)+-1 which accounts for 2/3 of all factorable numbers giving way to highly composite numbers. This factorability is the reason a base 12 system lends...
  4. ╔(σ_σ)╝

    Mathematical Finance Program at UofT

    I just got of the phone with departement and I have heard good things about the program. Pros I may not need to retake math courses I already took in engineering. The program is only one year. 4 months school, 4 months internship and 4 months school. Department has 100% sucess rate of finding...
  5. L

    Mathematical definition of inertia

    I realize that inertia is the tendency of an object to maintain its present state of motion. Now I am attempting to attach a mathematical definition to this concept. In searching, however, I am finding instances of it being defined variously as kg, kg/m^2, m/s^2, as well as other creative...
  6. jal

    Is N=4 ONLY a mathematical construct?

    What are your thoughts. Some questions ... If we have a 4th space dimension: What does it look like? (A brane is 2d) What is in it? (Gravity, dark energy) Can a 2d or 3d “particle” be residing or coming from a one dimension? Do parts of a “particle” exist in ALL of the 4 dimension? Do...
  7. H

    Are variables mathematical objects?

    Just of out of curiosity, is it possible to rigorously define the notions of variable and constant? It seems to me that if these notions don't have rigorous definitions, then our way of thinking about mathematics pretty much falls apart.
  8. E

    What exactly is mathematical research?

    I am a high school senior now and I would love to pursue a career in math. I have been doing some research on the track from undergrad to grad school to tenure, and there's one thing I don't quite understand: research. I understand what research might be for life sciences or social...
  9. P

    A good book on Mathematical modelling

    i want to learn how to use experimental data to produce effective mathematical model for good prediction of any phenomenon. my mathematical background is : i know about single and multivariable calculus,college algebra,introductory probability theory,trigonometry,vector algebra and calculus,and...
  10. SkyWatcher

    Mathematical uses of the preposition of

    Mathematical uses of the preposition "of" Dear Professors, I am looking for any articles on the uses of the preposition "of" in mathematics. I once read in a teacher's edition of a pre-algebra textbook that suggested students should be taught, as early as possible, the many uses of the...
  11. P

    Mathematical Statistics question

    Homework Statement let X be a random variable with uniform p.d.f over the interval (0, theta) Determine whether 2X-bar (sample mean) is an unbiased and consistent estimator for theta Homework Equations bx(theta)=EX-theta The Attempt at a Solution By using normal...
  12. D

    What makes one type of mathematical modeling more difficult than the other?

    I'm guessing it depends on how many factors/how unpredictable a subject is, what is it really? What makes mathematical finance so rigorous- requiring a phd in math/finance- while something like systems biology or pharmacokinetics only requires knowledge up to linear algebra?
  13. N

    Need an opinion on this solution involving mathematical induction

    Hey guys, this is a problem given to us by our professor in one of the worksheets. I would like an opinion to see if this proof is valid Homework Statement Use mathematical induction to prove that 3^{n}+7^{n}-2 is divisible by 8 for \foralln\inZ^{+}Homework Equations Base step: n=0...
  14. J

    Help with text book mathematical notation

    Okay so let's say (hypothetically of course) I am on my third attempt at the final Calc course in the series at my college. I feel like my foundation of even simple mathematics and arithmetic is unstable to say the least for me. I enjoy the concepts of math and what it could be used for but...
  15. A

    Math Physics: Exploring Areas of Math & Physics Linked Together

    Some people call it physics that simply is more difficult or reliant on mathematics. I was wondering if anyone could enlighten me a bit more as to what types of maths and physics areas are generally equated with mathematical physics.
  16. B

    Developing my mathematical background

    Hi all, I have a question. I'm hoping to be a theoretical physicist, but I'm finding I really don't know the first thing about mathematics. I can plug and chug with the best of them, and I have a feel for various integration techniques, and I know enough tricks to get by (Fourier series, Fourier...
  17. J

    Lingusitics Are Natural and Mathematical Languages the Only Primary Forms of Communication?

    I think there are only two primary languages, i.e. natural language and mathematical one, which we use to communicate . If we go one step further, then I would say even mathematics is a subset of natural language where we use symbols and other signs to represent our concepts, ideas into...
  18. S

    Can you Prove that {2}_{}an+1 = _{}an _{}an+2 =(-1)n for the Fibonacci series?

    Mathematical induction...please help me! _{}a1 =1, _{}a2 =1,_{}a3 =2,_{}a4 =3..._{}an = _{}an-1 + _{}an-2 is a Fibonacci series...Prove That ^{2}_{}an+1 = _{}an _{}an+2 =(-1)n
  19. S

    Mathematical explanation of maximum speed limit c'

    mathematical explanation of maximum speed limit "c' i have tried many texts and searched but can't find particular mathematical proof or explanation of how sp.theory led to conclusion that maximum velocity limit is "c"
  20. H

    The mathematics required for mathematical physics (LQG, ST, FT, etc)?

    I realize this might be a little bit of academic advisement, but I think it'd be more useful to have some people who really know a lot about the subject post about this. I like math a lot, and specifically I was wondering about number theory or set theory and logic, which is interesting but I'm...
  21. M

    Unraveling the Universe: The Relationship Between Matter and Mathematics

    Are we made of pure mathematics? Is everything in universe mathematics? I mean, everything in physics is "written" in math. Looks like that the definitive theory of everything will be pure mathematical. Matter & Energy = Pure Math?
  22. K

    Mathematical definition of critical point？(thermodynamics)

    Homework Statement If a gas is described by Van der Waals eqn (p+a/v^2)(v-b)=kT, find the critical pint p_c, T_c,and v_c Homework Equations The Attempt at a Solution Actually this is just one intermediate step I got stuck for my stat mech homework, I don't even know the mathematical...
  23. Ed Aboud

    Mathematica Prompt user to enter mathematical function

    Hi all, I am trying to prompt the user to enter a mathematical function in matlab. Any suggestions on how to do this? I have looked at the inline functions but I am not sure how to code it. Any help is greatly appreciated. Ed
  24. S

    Self-Studying Mathematical Physics: Books & Approach

    hi, please suggest me a good book/s for selfstudy on mathematical physics, which gives a detailed explanation, and how should be the study approach for selfstudying, i feel I'm good at algebra :) Thanks :)
  25. Shackleford

    Can I Take Probability and Mathematical Intro to Options Concurrently?

    What is this class like? Is it very hard? I took an intermediate macroeconomic theory course a couple of semesters ago. One of the prerequisites for this class is Probability. I will be taking that next semester. I wonder if I could take these two classes concurrently.
  26. R

    Partial fraction with the expoential mathematical constant in the numerator

    (3*e^(-2s))/(s(s+5)) I was hoping someone could tell me how to find the partial fraction for the above; the answer i ended up with was: (3/5)/s - (3*e^(10))/(s+5) and i went about it by: (3*e^(-2s))/(s(s+5))=A/s+B/(s+5) (3*e^(-2s))=A(s+5) + B(s) set s=-5 and found B=-3*e^(10)/5 set s=0...
  27. M

    Mathematical Induction problem

    Im practicing inductions on my own and I am getting stuck on one in particular... S_n = \frac{3(3^n-1)}{2} for a_n = 3^n I know that S_1 = 3 when you plug 1 into the equation, because it is the first term in the sequence Therefore, S_1 = a_1 = 3 I need to prove then S_{n+1} =...
  28. T

    Proving Cardinal Number of R = Cardinal Number of {x|0<x<1}

    Hey guys, How can I proove the cardinal number of R is equal to the cardinal number of {x|0<x<1}?? Thanks~~~
  29. P

    Math Chemistry: Can an Undergrad Research?

    I'm not a chemist, but I was wondering there are any promising fields in mathematical chemistry. Not necessarily chemical engineering/physics-wise, more so to do with molecular structure, topology, etc. Is there any chance an undergrad would be able to participate in this area of research? Or is...
  30. B

    How Does the Inequality 2*3^k >= 3 Arise in Mathematical Induction?

    Hi, I'm trying to learn mathematical induction for proving inequalities, but there is just one step I cannot get past: finding another inequality that is added to the inductive hypothesis. For example, in this problem: Prove for all positive integers (n >= 1), prove 3^n + 2 >= 3n. I...
  31. 1

    What Are the Two Common Forms of the Mathematical Equation for Slope?

    Homework Statement Give the two common forms of the mathematical equation for slope (involving X & Y)? Homework Equations ? The Attempt at a Solution Can you guys please give me the two common forms of the mathematical equation for slope? I don't know what it is but I am guessing...
  32. K

    Exploring Infinity: Is it a Math Concept or Real?

    Is "Infinity" only a mathematical concept or is there anything infinite in reality? I mean ∞ is indeterminate in a way such that any of the following expressions can be constructed: ∞ - ∞ = 1, ∞ - ∞ = 0 ∞ - ∞ = ∞ Is there anything in reality that can actually behave like that?
  33. B

    Proving f(n) = 5 x 3^n - 4 with Mathematical Induction

    Homework Statement Let f(n + 1) = 3f(n) + 8, with f(1) = 11. Prove by induction that f(n) = 5 x 3^n - 4.Homework Equations The Attempt at a Solution I don't even know where to start! Any help would be appreciated. Thanks. :-)
  34. A

    MATLAB Mathematical engine modeling (FHM) on Matlab

    I am trying to replicate a JAVA based applet on MATLAB. Applet can be seen http://www.engr.colostate.edu/~allan/thermo/page6/EngineParm1/engine.html" . The theory behind the functions of the applet is http://www.engr.colostate.edu/~allan/thermo/page6/page6.html" . It is actually a finite...
  35. M

    Mathematical notation for elementwise multiplication

    Hi, I wonder if anyone knows of a mathematically established way of writing elementwise multiplication between two vectors? In MATLAB you can write A .* B to indicate that you want to multiply the vectors A and B elementwise. In my case, I have two column vectors, A and B, and I want to...
  36. C

    Exploring Factor Space: A Course in Modern Mathematical Physics

    FACTOR SPACE is defined in my book as the set whose elements are the equivalence classes defined by the equivalence relation R .denoted by S/R i cannot understand where this is different from equivalent class? page 8 - a coursein modern mathematical physics - szekeres any help will be...
  37. K

    Do you see anything mathematical in this figure?

    Is this mathematical art? What kind of mathematical concepts can you see or illustrate with this figure or its elements? I see patterns, geometry, sets & algorithms. what do you see? [PLAIN]http://www.galasdeguatemala.com/data/media/159/DSCF4310.jpg Mayan weavings from Guatemala (image...
  38. N

    Use mathematical induction to prove:

    Please help! Use mathematical induction to prove.
  39. M

    Kepler's First Law Mathematical Query

    Hello all! It's been a long summer, and I thought I'd warm up things by going over Kepler's Laws. I've been following the mathematical derivation in Introduction to Modern Astrophysics, and to be honest I am little stumped on a part of it: We have the derived definition of angular momentum...
  40. R

    Difference between applied statistics and mathematical statistics.

    What is the difference between applied and mathematical statistics ? Does applied statistics skip a great deal of necessary knowledge ? Is applied statistics enough to construct statistical models?
  41. S

    Proving the Intersection of Functions: A Mathematical Study

    Are there any kinds of functions which satisfy f(A1 ∩ A2) =f(A1) ∩ f(A2)? Prove your claim.?
  42. Z

    Are mathematical manipulations admisible if integral are divergents ?

    are mathematical manipulations admisible if integral are divergents ?? are formal manipulations of divergent integrals admisible whenever the integral are divergent \infty i mean if i have a 4-dimensional integral \int_{R^{4}}dxdydzdt F(x,y,z,t) why can we make a change of variable to...
  43. R

    Uncovering the Math Behind a Mysterious Paradox

    I discovered a mathamatical truth that seems to have no clear logical foundation. Could anyone here explain the math behind the following? A.Given the following recursive series: S(1) = 3 S(2) = 2 S(n) = 6*S(n-1) - S(n-2) - 4 Define a new series as follows N(i) = (S(i) + S(i+1) +...
  44. I

    Undergraduate mathematical logic questions

    Hello I'm reading Y. Manin's http://books.google.co.il/books?id=8NTWRFD5lZ8C&printsec=frontcover&dq=yuri+manin+introduction+to+mathematical+logic&hl=en&ei=cfp_TJ2vJ8KSjAeB-6xl&sa=X&oi=book_result&ct=result&resnum=1&ved=0CCkQ6AEwAA#v=onepage&q&f=false" and I've been having some difficulties. So...
  45. Q

    Is there a mathematical process for calculating an objects resonance?

    ^^^ Any way to get to it by using size / weight / etc.?
  46. P

    REAL ANALYSIS, Mathematical Induction

    Homework Statement What is wrong with my solution?... I don't quite understand where do I go from there... Homework Equations The Attempt at a Solution
  47. P

    Prime Numbers: A Mathematical Mystery

    Is there any mathematical formula to predict / generate / or test the prime number??
  48. P

    Difference between Strong Induction and Mathematical Induction?

    Homework Statement Explain the difference between The principle of Mathematical Induction and The principle of Strong Induction. One is easier than the other. Why? Homework Equations The Attempt at a Solution My book says that they are in fact identical. Proofs are almost the...
  49. F

    Mathematical Induction problem

    I'm trying to solve this problem from CH2 of spivak's calculus of which I am self-studying. Homework Statement Prove the following by mathematical induction: 1^3+...+n^3=(1+...+n)^2 Homework Equations To prove by mathematical induction, you test whether P(1) is true and if P(k) is...
  50. M

    Proof that (x^t)Ax>0 if Eigenvalues of Matrix A > 0

    I am looking for a rather simple proof that if the matrix A has eigenvalues>0, then (x^t)Ax>0 for any vector x not 0. My first tought was if the eigenvalues are bigger than 0, then (x^t)Ax=(x^t)"eigenvaulue"x="eigenvalue"(x^t)x>0, if x is nonzero and eigenvalues is bigger than 0. Is this...
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