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. G

    Mathematical induction question

    Hi all, I am revising on Proof by mathematical induction and I have came across a question I haven't found a way to work it out. \sum_{r=1}^n r(r!) = (n + 1)! -1 I understand the steps of proving by mathematical induction question. The ! is causing the confusion.
  2. G

    How Does Series Notation Equal the Highlighted Part in Induction Proof?

    Hi all, I am so sorry I don't know whether to post this. I am revising Proof by mathematical induction Further maths using an Edexcel FP1 book. The following is from chapter 6 example 3. I know how to work Proof by mathematical induction it is part of the example that I don't understand. I...
  3. M

    Nonlinear programming problem, mathematical programming

    Hello, I need to minimize {- f (x) | a <=x <= b} where f ( x) is a concave and twice differentiable function. In addition, a and b are positive constants such that a <b. Assume that -f (x) exists in the given interval [a, b] . Show that if the optimal solution is at x*= a , then delta f...
  4. C

    Math Induction: Where Does the >2xk Come From?

    Please refer to the image attached. Where does the > 2 x k come from? Based on the proposition, shouldn't it be > k+1?
  5. M

    Mathematical Reasoning and Writing:

    Homework Statement Let f : A → B be a function and let S, T \subseteq A and U, V \subseteq B: Prove that if U \subseteq V; then f-1(U) \subseteq f-1(V): Homework Equations Preimage: f-1(U) = {x \in f-1(U) \ni f(x) \subseteq U} The Attempt at a Solution Assume U \subseteq...
  6. M

    Mathematical Reasoning and Writing - Counterexamples with subsets.

    Homework Statement Let f: A --> B be a function and let S, T \subseteq A and U, V \subseteq B. Give a counterexample to the statement: If f (S) \subseteq f (T); then S \subseteq T: Homework Equations The Attempt at a Solution PF: Assume f(S) \subseteq f(T). Let x \in...
  7. S

    Mathematical Methods book for Undergraduate

    Greetings everyone, I am posting this question here since I cannot post it in the math and science learning materials section of the forum. My question is that sometimes in physics I get into a lot of involved math particularly linear algebra, Hilbert space, partial differential equations and...
  8. T

    Which Implications are True in Mathematical Logic?

    Hi This is a question from my self study of ch.2 of Alfred Tarski's Introduction to Logic Which of the following implications are true from the perspective of mathematical logic? a) If a number x (assuming x is an integer) is divisible by 2 or by 6, then it is divisible by 12 b) if 18 is...
  9. J

    Mastering Mathematical Induction: Solving for the nth Term

    Homework Statement 1-1/2+1/4-1/8+...nth term= [(2^n-(-1)^n)]/3(2^(n-1))] I just can't get the nth term. I realize that the denominator should be 2^(n-1) and that, because the sign constantly changes there needs to be something like -1^n so I came up with the equation -1^n/-2^(n-1) but this...
  10. djh101

    Physical Chemistry Electives: ODE, Mechanics, or Mathematical Methods?

    Hello, everyone. I am currently a junior [physical] chemistry major and am picking out my future upper division electives. I've narrowed them down to a handful of classes and what I'm looking for is just a little background information on them, which ones might be better than others, general...
  11. A

    Exploring the Logic of Mathematical Induction

    Hello I'm learning about proofs and in my book there's a sect. On mathematical induction. And I'm trying understand why this makes it true for all values. 1+3+5...2n-1=n^2 Suppose that the formula is known to be true for n=1, and suppose that as a result of assuming that it is true for n=k...
  12. J

    What is the mathematical intuition behind operator embedding?

    Can someone explain to me the mathematical intuition that motivates the embedding of quantum operators between the conjugate wave function and the (non-conjugated) wave function? That is, we write: \Psi^{*}\hat{H}\Psi, that is: \Psi^{*}(\hat{H}\Psi), so that \hat{H} operates on \Psi (not...
  13. G

    LQG, string theory and spinors - all achieve mathematical miracles

    LQG, strink theory and Penrose's spinor theory, or maybe it's twistor theory, I don't know, all I know is that all three theories achieve mathematical miracles in their attempts to go beyond the Standard Model - how can all three theories do this but be mutually exclusive at the same time. Or...
  14. M

    First Year Calculus Course Mathematical Induction Problem

    Prove by mathematical induction: (2n)! < (2^(2n))*(n!)^2 , for all n=2,3,4... I know that to start you must prove that it is true for n=2, (2*2)! = 24 < 64 = (2^4)(2!)^2 Then you assume that n=k and show tha n=k implies that n=(k+1) (2k)! < (2^(2k))*(k!)^2 ... At this point I...
  15. B

    Which Advanced Mathematical Physics Books Should I Read?

    Hey guys, I have seen many posts on the same topic but I was unable to come to a coherent conclusion. I have seen people reffering to Arfken or Mary Boas. I have sufficient knowledge in linear algebra, statistics and calculus in many variables. I am presently taking a course in complex analysis...
  16. Vahsek

    Mathematical physics or Math/physics double major?

    Hi, I am a high school student and my fields of interest are physics and mathematics. I've been advised to apply for the mathematical physics undergraduate program but I don't want to rush so that I won't have to regret my decision afterwards. I have a few questions about the paths I can...
  17. S

    Good introductory book on mathematical proofs?

    My 7th grader has enough mathematical background in algebra, geometry and trigonometry to start learning how to write out proofs. Are their good books that teach this step by step? I can certainly teach him myself with examples, but I figured there must be a systematic way to teach this...
  18. M

    Magnetic Fields - Just Convenient Mathematical Conceptualizations?

    With my term-end examinations fast-approaching, I sat down to revise my entire senior year physics syllabus. As I was reading through electromagnetic induction, I found myself in the same state of utter confusion that I found myself in when I first studied about Lenz's Law and Eddy Currents. But...
  19. A

    Need Help with Mathematical Proof? - Introducing Myself & Seeking Assistance

    Dear all, I am new to this forum and this is my first post. I hope this is the right place to open this topic. I have a homework for my mathematical methods for physics course and i am kind of stuck into a question: v=ψ∇β show that ∇xv is perpendicular to v. Here v is a vector x is cross...
  20. A

    Is the B.Sc. Physics offered Chennai Mathematical Institute reputed?

    Hi, I hail from Chennai, India. 1. I would like to know if the B.Sc Physics(Hons.) course offered by the Chennai Mathematical Institute (CMI) is a reputed one. 2. What university is CMI affiliated to? 3. I read that the course span is 3 years. Can I pursue my Masters abroad after...
  21. PhizKid

    Is there a mathematical way to balance equations (by hand)?

    As opposed to guess and checking by inspection. Is there a method mathematically like with an algorithm or something to balance equations by hand (without the aid of any computer programs or anything like that except a simple calculator)?
  22. D

    Advancing Mathematical Knowledge

    Repeatedly in the past few months any attempt at learning more advanced physics has ended in hitting a brick wall: my maths knowledge is really not up to par. To put this into perspective, I am an A Level student but feel as though the current syllabus for the sciences is really not enough. I...
  23. M

    Khinchin: Mathematical Foundations Of Statistical Mechanics

    Hello, I'm a first year graduate student, and one of the required texts for my class is the book alluded to in the title. However, it seems to me that this book is written in the language of a mathematician, and is a bit inaccessible to me. Particularly, I'm having trouble following the...
  24. J

    Proving the Solution to a Recurrence Relation Using Mathematical Induction

    Hi all, I've been struggling with a large piece of coursework. been quite stressed latley and now I am struggling while aproaching my deadline. i need help answering this question. Use mathematical induction to show that S(n) = 3 × 2 n-1 -2 is the solution for the recurrence relation: T(n)...
  25. E

    Book on the mathematical theory of continuum mechanics

    I was wondering if anyone knows of a good book on the mathematical theory of continuum mechanics. I have looked online, and the only ones I can seem to find are like your average physics or applied mathematics book. I want something with rigorous theoretical formulation of the subject. It...
  26. J

    How to prove by mathematical induction?

    How do I prove a formula/rule or something by mathematical induction? Please give me a few examples or resources and explain it as best you can because I think I'm messing up some how.
  27. micromass

    Problems in mathematical analysis by Demidovich

    Author: B.P. Demidovich Title: Problems in mathematical analysis Amazon link https://www.amazon.com/dp/B0007BNL1W/?tag=pfamazon01-20 Table of Contents: Preface Introduction to Analysis Functions Graphs of Elementary Functions Limits Infinitely Small and Large Quantities Continuity...
  28. R

    Is the magnetic field a mathematical abstraction?

    Is the magnetic field purely a mathematical abstraction or is there actually something there? In other words, if a proton floating in deep space, is there actually something that shoots out in all directions from the proton that will interact with another charge particle?
  29. N

    Mathematical physics textbooksuggestions please

    I am an high school student starting grade 11 this year . I have been interested in physics since class 6 and i have been studying advanced physics since then . IN grade 9 and 10 i have read and solved problems from Resnick ,Halliday and walker's book and completed it . I thought myself...
  30. X

    Can Mathematical Objects Be Structured Like Probability Models?

    I was wondering what the typical approach is for creating a "mathematical object" such as the probability model kolmogorov made (I've also heard it called a probability space...not really sure what the difference is)... <\Omega,\mathcal{F},P> , where \Omega is the sample space...
  31. X

    Mathematical physics textbook with solutions in it?

    Hello I'm taking a mathematical physics course which has heavy emphasis on algebraic structures. The course has no required textbook but I want one anyway for extra practice. All of the books I see at my library all have no solutions in them which makes it frustrating because then I don't know...
  32. micromass

    Methods of Mathematical Physics by Hilbert and Courant

    Author: Richard Courant, David Hilbert Title: Methods of Mathematical Physics Amazon Link: https://www.amazon.com/dp/0471504475/?tag=pfamazon01-20 https://www.amazon.com/dp/0471504394/?tag=pfamazon01-20 Table of Contents for Volume I: The Algebra of Linear Transformations and Quadratic...
  33. micromass

    Schaum's Outline of Mathematical Handbook of Formulas and Tables

    Author: Seymour Lipschutz, Murray Spiegel, John Liu Title: Schaum's Outline of Mathematical Handbook of Formulas and Tables Amazon Link: https://www.amazon.com/dp/0071795375/?tag=pfamazon01-20 Prerequisities: None Table of Contents: Formulas Elementary Constants, Products, Formulas...
  34. A

    Could this be a mathematical way to obtain longitude without clock? Please help

    archeoastronomist are bufled at polinesian finding out longitude without a clock here the concept i just found out how to prove: you have a dial with two geared counterotating needles you move by hand every rotation of the sun around you you rotate a needle ten times with which the...
  35. Greg Bernhardt

    Mathematical Methods for Physicists by Arfken and Weber

    Author: George B. Arfken (Author), Hans J. Weber (Author), Frank E. Harris (Author) Title: Mathematical Methods for Physicists Amazon Link: https://www.amazon.com/dp/0123846544/?tag=pfamazon01-20 Prerequisities: Calculus 1,2,3 Table of Contents: Preface Vector Analysis Definitions...
  36. M

    Next great mathematical invention

    Hi , I was thinking about this , as you know calculus was used many years before the explicit work of both Newton and Leibniz , it was used the hard way , unrecognized , scattered and buried among unrelated formulas and topics. The golden question is ; what if there is another revolutionary...
  37. T

    Proving 1 + 1/√2 +...+ 1/√n < 2√n Using Mathematical Induction

    Prove that: 1 + 1/√2 +...+ 1/√n < 2√n Work: So I've done the base case of n = 1, and I've set up the Indutive hypothesis as assuming n=k as 1 + 1/√2 +...+ 1/√k < 2√k and for the inductive step: 1 + 1/√2 +...+ 1/√k + 1/√(k+1) < 2√k + 1/√(k+1) now I'm having trouble trying...
  38. Greg Bernhardt

    Mathematical Methods of Classical Mechanics by V.I. Arnol'd

    Author: V.I. Arnol'd (Author), K. Vogtmann (Translator), A. Weinstein (Translator) Title: Mathematical Methods of Classical Mechanics Amazon Link: https://www.amazon.com/dp/1441930876/?tag=pfamazon01-20 Prerequisities: Contents:
  39. Greg Bernhardt

    Mathematical Methods in the Physical Sciences by Mary L. Boas

    Author: Mary L. Boas Title: Mathematical Methods in the Physical Sciences Amazon Link: https://www.amazon.com/dp/0471198269/?tag=pfamazon01-20 Prerequisities: Calculus Table of Contents: Infinite Series, Power Series The Geometric Series Definitions and notation Applications of...
  40. Greg Bernhardt

    Principles of Mathematical Analysis by Walter Rudin

    Author: Rudin Title: Principles of Mathematical Analysis Amazon Link: https://www.amazon.com/dp/007054235X/?tag=pfamazon01-20 Prerequisities: Rigorous calculus, including epsilon-delta proofs. Spivak's "Calculus" would be more than sufficient preparation. Table of Contents: Preface The Real...
  41. A

    Mathematical proof for positive feedback

    Is there a mathematical proof that the positive feedback makes the op-amp saturated? http://i.imgur.com/71PNh.png
  42. marellasunny

    Analytical proofs in mathematical papers

    I am writing my first paper in applied mathematics.I would like to know how important is it to write a analytical proof for the numerical solutions arrived?I always thought these sort of things were left to the 'pure mathematicians' and one didn't need to bother.As a engineer,I was always used...
  43. Astrum

    Having Trouble With Mathematical Proofs - How Important for Physics?

    I'm having a lot of difficulty with proofs. I can understand them when I read them, but when asked to prove things myself, I usually fall flat on my face. How important are proofs for physics? Do I really need an intimate understand of the mathematics, or would that be irrelevant? I was...
  44. H

    Depression, doubt in my mathematical abilities

    Hello all, I am currently an undergraduate student (just finished my first semester) at Stony Brook University. Basically, I'm fairly certain that I want to go into pure math or theoretical physics, it's just that I feel like I may not be up to par with everyone else who will be competing...
  45. Weissritter

    The Number Devil: A Mathematical Adventure

    HE'S A DEMON! Yep, a Demon, a Teufel if you like. Have you read this book there ↑ ? Plain statiscal question for statistical answers. Anything not covered in the poll shall be posted here. Go wild.
  46. R

    Mathematical Physics vs. Physics: Impact on Grad School/Jobs

    The importance of the title "Mathematical Physics" vs. "Physics" for grad school/jobs I've got another quick question for anyone that cares and knows how to help. As I've mentioned elsewhere, I'm a third year physics student, currently enrolled in the "regular", straight-up physics stream...
  47. D

    Mathematica Allotting more resources to mathematical software

    Is there a way to allot more resources for Mathematica to use? I have 16GB of RAM and the Intel i5 but Mathematica is still too slow at many things. Is there a way I can allot more of my computers resources to the program?
  48. A

    Graphical to Mathematical representation of changing the order of some elements

    I have a question that is a little hard to explain, since i don't know the name of this method, but I'll try my best, if anyone knows the name please do tell me. So let's say we have three numbers, 1 2 3 (in this order) and we have a container for this numbers: C123 and we have some...
  49. R

    Which Allocation Maximizes Factory Profits?

    The table below records the profits that are made when one unit of material is sold by factories A,B, C to dealers L,M,N, P. The output of each factory and demand from each dealer are given in brackets ( A,B,C ) -> L(30) M(30) N(30) P(45) A(100) -> 25 30 20 20 B(20) -> 30 25 15 10 C(15) -> 10 35...
  50. R

    How to Solve a Mixed Kuhn-Tucker/Lagrange Problem with an Inequality Constraint?

    Queston From Mathematical Programming:? Find the minimum value of P(x,y,z)=Z+1/2(x^2+ y^2+1/10 z^2) constrained by and where r is a positive constant. Solve the problem when r=1, r=5. Can you find the solution for general r? (Note this is a mixed Kuhn-Tucker/ Lagrange problem. The inequality...
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