What is Abstract: Definition and 532 Discussions

Abstract expressionism is a post–World War II art movement in American painting, developed in New York City in the 1940s. It was the first specifically American movement to achieve international influence and put New York at the center of the Western art world, a role formerly filled by Paris. Although the term "abstract expressionism" was first applied to American art in 1946 by the art critic Robert Coates, it had been first used in Germany in 1919 in the magazine Der Sturm, regarding German Expressionism. In the United States, Alfred Barr was the first to use this term in 1929 in relation to works by Wassily Kandinsky.

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  1. S

    Proving something is commutative in abstract algebra

    If \ast : (f \ast g)(n) = \sum\limits_{d|n}f(d)g(\frac{n}{d}), show that \ast is commutative. Note that d|n says d divides n. Now I was not sure how to do this from an abstract algebra point of view although when I stare at it my though process was to maybe rewrite it somehow, which will then be...
  2. alyafey22

    MHB Abstract algebra recommendation

    I am reading at the moment about abstract algebra. It is a very interesting field. I was amazed by the number of examples, applications and related concepts. Never seen something similar in any other mathematical field. I saw lots and lots of theorems and I was wondering whether I should...
  3. S

    Training yourself to compulsively/always reflect on abstract ideas

    I work retail, and spend hours each day folding clothes. I would like to ponder interesting facets of mathematics and logic, but am unable to (or plainly, do not). When I go to bed at night, I'd like to ponder abstract questions, yet do not. Basically, I see people (and read of people) who...
  4. stripes

    More intro abstract algebra problems

    Homework Statement Define the set Q[√2] to be the set {a + b√2 | a, b are rationals}, and define addition and multiplication as "usual" (so 2×4 = 8, 2 + 4 = 6, you know, the usual). Show that for any nonzero A in the set Q[√2], there exists an inverse element so that A×A-1 = 1Q[√2]. There...
  5. stripes

    Intro abstract algebra along with basic set theory

    Homework Statement An interesting example of a ring: Begin with a nonempty set X and form the power set of X, P(X), which is the set of all subsets of X. On P(X), define addition + and multiplication × as follows: For A, B in P(X): A × B = A ∩ B A + B = (A\B) ∪ (B\A), where as...
  6. C

    Why don't most textbooks help the readers to get abstract concepts?

    I was always good at maths, just because primary/high school math was simple enough to find concrete examples for the abstract concepts, and that helped me a lot on exams. Since then I tried to grasp more advanced concepts. But I always faced with pure overformalized, overgeneralized stuff...
  7. J

    Abstract Algebra or Topology: Which is the Better Choice for a Math Major?

    Hi there, Need one upper div math class to fill out my schedule. It looks like it's a choice between intro to abstract algebra or intro to topology. Which would benefit me more, as a student looking towards grad school?
  8. R

    Abstract Algebra: Relations; Find a symmetric and transitive relation in Z x Z

    Abstract Algebra: Relations; Find a relation that is symmetric, etc Homework Statement Find a relation that is symmetric and transitive but not reflexive. Homework Equations None, other than my chosen condition on the relation, namely: xy > |x + y|. The Attempt at a Solution...
  9. K

    Abstract Algebra Proof Concerns

    Hello. I started Gallians Contemporary Abstract Algebra today. Is it wise to go through each of the given proofs for all of the theorems. For example I just studied the proof for division algorithm. Took Quite some time. I don't know if I could have produced this proof without peeking at the...
  10. J

    Have You Seen This Abstract Art Inspired by Physics?

    Hey everyone.. Not sure why I made this but I'm going to post it anyways haha. If you like to graphic design or have seen abstract art similar please post here:) I created this art from scratch. I attached the image.. created by using GIMP on a linux distro:)
  11. A

    Which abstract algebra textbook is most cummulative

    If I were to use an abstract algebra book for quick and easy reference which one would it be? Dummit and Foote is very cumulative, is there anything better in the market? And how long would it take to work out all of D + F for an average student with basic background in Algebra?
  12. N

    Abstract Algebra: Unnecessary Information in D&F Problem Statement?

    Homework Statement Problem 35, Section 7.3 of Dummit and Foote: Let I, J, and K be ideals of R. (a) Prove that I(J+K) = IJ+IK and IJ+IK = I(J+K). (b) Prove that if J \subseteq I then I \cap (J + K) = J + (I \cap K). 2. Concern/Question Despite the problem statement specifically...
  13. D

    Help with an Abstract Binary Math Problem

    So I was trying to figure out a straightforward method to calculating the possible number of combinations on a beginner minesweeper game (81 squares, 9x9, 10 mines) I figure that i can attribute this to binary. Because the 9x9 part shouldn't really matter. It is essentially a 81 bit...
  14. T

    Scheduling: Abstract algebra, numerical analysis, Probability, or?

    I need to choose one more math class to reach a full-time status for next fall. So far I am already taking Classical Mech I from Physics Dept, Analysis I and PDE from Math Dept. I hear Analysis is already time-consuming hard class and I guess PDE isn't easy either, so I am considering to...
  15. N

    Abstract Algebra: Non-trivial Rings Containing Only Zero-Divisors

    Homework Statement Is there a finite non-trivial ring such that for some a, b in R, ac = bc for all c in R? Does there exist finite non-trivial rings all of whose elements are zero-divisors or zero? 2. The attempt at a solution Let a, b ≠ 0 in R such that ac=bc for all c in R...
  16. C

    The sounds of thought: Detectable or too abstract?

    Technological advances in the last several years (such as Japan's "dream machine") have given us crude glimpses into the visual component of the thought process. Is it possible to do the same with the auditory component of thoughts? Is there any scientific evidence suggesting that thought sounds...
  17. L

    Abstract in my bachelor's thesis

    well,I'm not quite sure is it appropiate to post this here. I'm just here for some help... I‘m a Chinese senior student in college. When I'm writing the abstract in my bachelor's thesis,I've got some trouble—— terminology and grammar. As you know,my English is not that good to write an english...
  18. J

    Proving this basic fact about the annihilator in abstract algebra

    Maybe I'm misinterpreting the question, I'm not sure how to prove that n_0 i = 0.
  19. C

    Abstract Algebra- Finding the Minimal Polynomial

    Homework Statement Given field extension C of Q, Find the minimal polynomial of a=sqrt( 5 + sqrt(23) ) (element of C).Homework Equations The Attempt at a Solution I may be complicating things, but let me know if you see something missing. Doing the appropriate algebra, I manipulated the above...
  20. C

    Abstract Algebra- Conjugate Problem

    Homework Statement Let G be a group of odd order, and a an element of G (not identity). Show that a and a^-1 are not conugate. Homework Equations The Attempt at a Solution The only hint I have is to consider action of G on itself by conjugation.
  21. D

    Where can I find helpful resources for abstract algebra?

    I was wondering if anyone has compiled a list of AA resources. Recently, I have found that I practically need to learn everything from the class outside of class all over again. I have been playing around with YouTube, but haven't really found anything worthwhile. So, what about you guys...
  22. Spinnor

    Can unquantized fields be considered smooth curved abstract manifolds?

    Can unquantized fields be considered smooth curved abstract manifolds? Say free particle solutions of the Dirac equation or the Klein Gordon equation? Can quantized fields also be considered curved abstract manifolds? Thanks for any help!
  23. N

    Tricky abstract algebra problem

    Homework Statement Prove that SL_{2}(ℝ) is generated by the set: [1 a], [1 0] [0 1], [b 1], a,b \in ℝ Homework Equations GCD (Greatest common divisor) The property of special linear group Some basic linear algebra, like determinant The Attempt at a Solution SL_{2}(ℝ) is the group...
  24. L

    Abstract Algebra during final year?

    Good morning everyone. So I've been thinking quite a bit about it and recently switched from applied math to pure math, and I wish to attend grad school, if not PhD then at least a master's with thesis. I'm in the middle of my 2nd year, so next Fall I plan on taking Analysis, and then the fall...
  25. K

    Is it normal to be so discouraged by abstract algebra?

    I'm currently in my first abstract algebra course, focused on sets, groups, arithmetic modulo, rings, fields etc. I've never taken an abstract course before. I've taken: Pre-calc Calc 1-2 Linear Algebra Advanced Applied Linear Algebra so the concept of abstraction is very new to me; I...
  26. K

    Abstract Algebra, Sets Proof

    Homework Statement Question 1. Let U be a universal set, A and B two subsets of U. (1) Show that B ⊆ A ∪ (B ∩ A^c). (2) A = B if and only if there exists a subset X of U such that A ∪ X = B ∪ X and X\A^c = X\B^c. The Attempt at a Solution My attempt at a solution is as follows...
  27. A

    Abstract Algebra - Natural Numbers Proof

    The question is which sets of natural numbers are closed under addition. I know that odd is not, and I know how to prove that sets of multiples are, but my professor said there is something more and that is has to do with greatest common divisor. He said to pick numbers like 3 and 5 or 5 and 8...
  28. micromass

    Algebra Abstract Algebra by Dummit and Foote

    Author: David Dummit, Richard Foote Title: Abstract Algebra Amazon link https://www.amazon.com/dp/0471433349/?tag=pfamazon01-20 Prerequisities: Being acquainted with proofs and rigorous mathematics. Level: Undergrad Table of Contents: Preface Preliminaries Basics Properties of the...
  29. micromass

    Algebra A book of Abstract Algebra by Pinter

    Author: Charles Pinter Title: A book of Abstract Algebra Amazon link https://www.amazon.com/dp/0486474178/?tag=pfamazon01-20 Prerequisities: High-school algebra Level: Undergrad Table of Contents: Preface Why Abstract Algebra? History of Algebra New Algebras Algebraic Structures...
  30. B

    Abstract Algebra Proof Using the First Isomorphism Theory

    Homework Statement See attatchment. I couldn't upload the picture. 2. The attempt at a solution I have the following: Define mapping f: ℝ2 -> ℝ as follows: f(x,y) = 3x - 4y Claim: f is a homomorphism Pick any (x,y) in ℝ2. Then f(x,y) = f(x)*f(y) = 3x - 4y = (x+x+x)-(y+y+y+y) =...
  31. T

    A mapping from an integral domain to non-negative integers, Abstract Algebra

    So just had this question as extra credit on a final: Let D be an integral domain, and suppose f is a non-constant map from D to the non-negative integers, with f(xy) = f(x)f(y). Show that if a has an inverse in D, f(a) = 1. Couldn't figure it out in time. I was thinking the way to go...
  32. S

    Abstract Algebra - Group of Order 12 with Conjugacy Class of Order 4

    Homework Statement A group G of order 12 contains a conjugacy class C(x) of order 4. Prove that the center of G is trivial.Homework Equations |G| = |Z(x)| * |C(x)| (Z(x) is the centralizer of an element x\inG, the center of a group will be denoted as Z(G)) The Attempt at a Solution Let G...
  33. S

    Abstract Algebra: Finite Field

    Show that every finite field with p+1 elements, where p is a prime number, is commutative. I know this has something to do with composite numbers, but I'm not quite sure how to show this.
  34. S

    Abstract Algebra: Rings, Unit Elements, Fields

    1) Show that (R,*,+) is a ring, where (x*y)=x+y+2 and (x+y)=2xy+4x+4y+6. Find the set of unit elements for the second operation. I understand that the Ring Axioms is 1. (R,+) is an albein group. 2. Multiplication is associative and 3. Multiplication distributes. I just don't understand how to...
  35. A

    Abstract Algebra homework Direct products

    Homework Statement We've shown if G_{1},G_{2},...,G_{n} are subgroups of G s.t. 1)G_{1},G_{2},...,G_{n} are all normal 2)Every element of G can be written as g_{1}g_{2}...g_{n} with g_{i}\inG 3)For 1\leqi\leqn, G_{i}\capG_{1},G_{2},...,G_{i-1}=e then G\congG_{1}xG_{2}x...xG_{n}...
  36. N

    Abstract Algebra HW: Show nk=kn for N,K ∈ G

    Homework Statement Suppose N \lhd G and K \vartriangleleft G and N \cap K = \{e\}. Show that if n \in N and k \in K, then nk = kn. Hint: nk = kn if and only if nkn^{-1}k^{-1} = e. Homework Equations These "relevant equations" were not provided with the problem I'm just putting them here to...
  37. F

    Abstract Algebra Order of Permutation

    Homework Statement See image. Homework Equations The Attempt at a Solution I am finding the orders of permutations. I know that you first find the orbits or cycles I don't know the difference (but I should). This is what my professor said: If you have (1345)(897)...
  38. H

    What Are the Proofs for Powers in Normal Subgroups and Orders in Homomorphisms?

    Homework Statement a) Let H be a normal subgroup of G. If the index of H in G is n, show that y^n \in H for all y \in G. b) Let \varphi : G \rightarrow G' be a homomorphism and suppose that x \in G has order n. Prove that the order of \varphi(x) (in the group G') divides n. (Suggestion: Use...
  39. O

    Proof of \sqrt{ab}>\frac{2ab}{a+b} for Positive and Unequal Integers

    Can anyone help me confirm if I've solved this correctly? Many thanks. Homework Statement Prove that \sqrt{ab}>\frac{2ab}{a+b} if a & b are positive & unequal. Homework Equations The Attempt at a Solution if (\sqrt{ab})^2>(\frac{2ab}{a+b})^2 if ab>\frac{4a^2b^2}{(a+b)^2} if...
  40. O

    Solving Abstract Inequalities: Proving (a+b)(a-b) = 0 for Positive a and b

    Homework Statement The final answer I have of (a+b)(a-b) does not appear to fit the textbook's required "results of inequalities which hold true for all real no.s", i.e. either: 1. (a)^2 or (a-b)^2 or 2. -(a+b)^2. Can anyone confirm if I have solved this correctly, in line with the conditions...
  41. J

    Quantum Mechanics Before Classical Mechanics: Implications and Speculations

    Just an abstract question here. How different do you think the world would be if we were taught quantum mechanics before classical mechanics, given the prerequisite that we already have a good knowledge of the mathematics? Of course, it's a highly unlikely scenario, but an interesting one none...
  42. Z

    Abstract Algebra, order of ab is equal to the order of a times the order of b?

    Abstract Algebra, order of ab is equal to the order of a times the order of b?? Hi! I am working on some problems in abstract algebra and I am stuck at the moment. I hope some of you guys could help me out a little. Homework Statement a and b are two elements in a group G. Assume that...
  43. R

    Abstract algebra, finite A-module

    Homework Statement Let A be an integral domain with field of fractions K, and suppose that f\in A is non zero and not a unit. Prove that A[\frac{1}{f}] is not a finite A-module. [Hint: if it has a finite set of generators then prove that 1,f^{-1},f^{-2},...,f^{-k} is a set of generators for...
  44. A

    Bridge to abstract math: what is wrong with following proof

    See attached picture. The question asks to prove that the statement which I have written on the first line is true. But I somehow proceeded to proving it is false. Basically what I did was simplify the given expression into the form (P or Q) => R and said this is equivalent to (P=>R) ^...
  45. U

    Unique Decomposition of Elements in an Abelian Group

    Homework Statement Let A be an abelian group, written additively, and let n be a positive integer such that nx=0 for all x \in A. Such an integer n is called an exponent for A. Assume that we can write n=rs, where r, s are positive relatively prime integers. Let A_{r} consist of all x \in A...
  46. C

    Abstract Linear Algebra, Linear Functional

    Homework Statement problem didn't state, but I assume let V be a vector space: V = C^3 and scalar is C Homework Equations Define a non-zero linear functional T on C^3 such that T ((1, 1, 1)) = T ((1, 1, −1)) = 0 The Attempt at a Solution So let X1 = (1, 1, 1); X2 = (1, 1, -1); It...
  47. G

    Testing Screwed up Abstract Algebra exam unsure if I have the ability to do math.

    After getting back a result in an Abstract Algebra exam (In which I only got 70%), a result just below the class average I am having extreme doubts about my ability to become a mathematician. The real shock was that I believed I understood the material well enough to get at least 90%. I am...
  48. H

    Abstract Algebra: repeating decimals and prime factors

    Homework Statement Prove if m/n has a repeating decimal expansion of period k, and n has no repeated prime factors, then some prime factor of n divides 10k-1 and no number of the form 10j-1 for 1 ≤ j < k Homework Equations The Attempt at a Solution I know that if a decimal...
  49. srfriggen

    Abstract Algebra: List elements of Subgroup

    Homework Statement List the elements of the subgroups <3> and <7> in U(20). Homework Equations The Attempt at a Solution U(20)= {1, 3, 7, 9, 11, 13, 17, 19} = <3> = <7>. So basically I have that the common elements of, <3> and <7> and U(20), under + modulo 20, are all...
  50. R

    Unique factorization domain, roots of a polynomial, abstract algebra

    Homework Statement let A be a UFD and K its field of fractions. and f\in A[x] where f(x)=x^{n}+a_{n-1}x^{n-1}+...+a_{1}x+a_{0} is a monic polynomial. Prove that if f has a root \alpha=\frac{c}{d}\in K,K=Frac(A) then in fact \alpha\in A I need some guidance with the proof. Proof...
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