Abstract Definition and 506 Threads

Homework Statement Hello guys So I have the following problem, given the mapping above I have to check weather it's ring homomorphism, and maybe monomorphism or epimorphism. The Attempt at a Solution So the mapping is obviously well defined, and I have proven it's homomorphism, and it's...

I am well aware of an abstract definition of a general tensor as a map: \mathbf{T}:\overbrace{V\times\cdots\times V}^{n}\times\underbrace{V^{\star}\times \cdots\times V^{\star}}_{m}\longrightarrow\mathbb{R} I am happy with this definition, it makes a lot of sense to me. However, the physics...

Dear Physics Forum advisers, My name is Phoenix, a sophomore with major in mathematics and an aspiring applied mathematician in the theoretical computing. I wrote this email to seek your recommendation on the textbooks for abstract algebra. I want to self-study the abstract algebra during...

Hello, I'm having a little trouble figuring out the following problem: Consider the set of number a, 2a, 3a, ..., na where a and n are positive integers. (i) Show that the expression for the mean of this set is \frac{a(n+1)}{2}. So far the only work I've been able to muster up is: Mean =...

Dear Physics Forum advisers, I am a college sophomore in US with a major in mathematics, and an aspiring algebraic number theorist and cryptographer. I wrote this email to seek your advice about taking the Analysis I (Real Analysis I), Abstract Algebra I, and Linear Algebra with Proofs. At...

Hi, Are Calculus I, II, III courses a prerequisite requirement for studying Abstract Algebra? I have read that Proofs and a willingness to work hard is. I am studying Logic and Set Theory and want to study Abstract Algebra in the distant future. I am focused on Foundational and Pure...

The overall question is on non steady fluid mechanics however the part on stuck on boils down to the two equations below, which I am unable to solve. X = 122.3 (2 - Y) Y= 0.18 * SQRT( 100 + X ) the text states the equations are satisfied by Y = 1.903 and X = 11.82. To prove this isn't a...

Hello, A couple of years ago I studied abstract algebra from Dummit and Foote. However, I was never able to gain the intuition on the subject that I would like from that book. I want to study the subject again, and I want to use a different book this time around - one that covers a lot of...

Hi I recently read a book called "The fundamental theorem of algebra" by Fine and Rosenberger. It focused specifically on polynomials, and proved the theorem using several fields of mathematics; Two of the proofs were algebraic. Abstract algebra has been very difficult for me; Mostly because...

Homework Statement Suppose that ## u = s_1i + s_2j ## and ## v = t_1i + t_2j ##, where s1, s2, t1 and t2 are real numbers. Find a necessary and sufficient condition on these real numbers such that every vector in the plane of i and j can be expressed as a linear combination of the vectors u and...

I found out I can pick up a second major in math should I elect to take a two semester sequence in abstract algebra. My first major is in chemical engineering. Right now, I plan on taking a two semester sequence in either: 1) probability with measure theory, 2) abstract algebra (Dummit and...

Let N be a normal subgroup of a group G and let f:G→H be a homomorphism of groups such that the restriction of f to N is an isomorphism N≅H. Prove that G≅N×K, where K is the kernel of f. I'm having trouble defining a function to prove this. Could anyone give me a start on this?

Homework Statement How can I return a new class object in an abstract class?Homework Equations None The Attempt at a Solution Suppose I have an abstract class named Animal (there are many other animal types that are made as classes). In this abstract class it has two methods...

Homework Statement If ≡(mod), ≡(mod),and gcd(,)=1,provethat ≡ (mod ). Homework Equations If ≡(mod)→n|ab-cd ≡(mod)→n|b-d gcd(,)=1→ relatively prime. So bx+ny=1 Need to show n|a-c→a-c=nw The Attempt at a Solution If n|ab-cd, then nk=ab-cd If n|b-d, then nl=b-d If n|ab-cd AND n|b-d, then...

Dear all, Can anyone please explain how the linear combination of non-coplanar and non-orthogonal coordinate axes representing a point x as shown below is derived. Please use the reference text attached in this post to explain to me as i will find it a bit relevant. I want to...

Entering my third year of my bachelor of science majoring in maths/physics and having some trouble deciding what courses to do this semester. I know for sure I will be taking complex analysis and 3rd year quantum however am having trouble picking between 3 in particular for my final two courses...

Hello everyone, I have been trying to teach myself GR for sometime now, and while I started to learn through books like Schutz and Hartle, I reached a point where the motivation behind the mathematics they used didn't satisfy me nor did it appeal to me. I think the worst I remember is when...

Homework Statement Let V be a finite dimensional vector space over ℂ . Show that any linear transformation T:V→V has at least one eigenvalue λ and an associated eigenvector v. Homework EquationsThe Attempt at a Solution Hey everyone I've been doing sample questions in the build up to an exam...

Homework Statement Let V be a finite-dimensional real vector space with inner product <⋅,⋅> and L: V → R a linear transformation. Show that there exists a unique vector a ∈ V such that L(x) = <a,x>. Homework Equations Hey everyone, so I'm a physics student who had to choose a few electives in...

Not quite sure, could someone kindly explain the concept of the topic to me? Thanks, it really means a lot. :)

I'm good at math like stats calc and others that r more process based. But I suck at things like abstract math, pure math , proofs etc. Am I an idiot? I'm OK with being stupid, I know it does not define a persons worth..

Hello, I'm debating between taking either abstract algebra, theory of numbers, or intermediate symbolic logic as a math elective. Does anyone have any idea which would make my life easier?

In GR an important, purely geometric equation is called Raychoudhuri's equation governing the behaviour of geodesic congruences which states that $$\frac{d\theta}{d\tau} = - \frac{1}3 \theta^2 - \sigma^{ab}\sigma_{ab} + \omega^{ab}\omega_{ab} -R_{ab} u^a u^a$$ where ##R_{ab}## is the ricci...

Homework Statement Let ab=a and ba=b, show that a^2 = a and that b^2 = b Homework Equations none The Attempt at a Solution Not sure if I did this correct.. but here is what I did. Given: ab = a. Multiply both by left hand multiplication by a^-1 a^-1*a*b = 1. where a^-1*a is obviously...

Assuming the axiom of choice, show that a group $G$ has trivial automorphism group if and only if $G$ has less than three elements.

A truck is moving with constant acceleration "a" up a hill that makes an angle phi with the horizontal. A small sphere of mass "m" is suspended from the ceiling of the truck by a light cord. If the pendulum makes a constant angle theta with the perpendicular to the ceiling, what is a? What...

Hi! I couldn't find good enough logic here. What's the correct answer (A,B,C,D, or E), what I am missing? Zoki

Has anyone else found that this is not the best way of describing higher math? I feel like abstract is not a good categorization of higher maths. I've found the hard parts, what I spend most my time on and what's the hardest to keep track of are technical things and subtle details. Random...

I've read Spivak 's calculus so I tried Sternberg 's book,IT IS kind of hard,because the abstraction is put upfront,I know of his other book with Bamberg, but a review on amazon deterred me from buying it,was I right in doing so? Any adv.Calculus book that have some physics in it like Sternberg's?

I was wondering if there is a field of mathematics which lies beyond and higher than abstract algebra? If it exists could someone tell me the name of that field? Thanks.

Hi, There is a theorem in Abstract which said if g.c.d(x,y)= d (g.c.d the greatest common divisor between x and y) then there exist an integers a,b such that ax + by = d It is a corollary from Euclidean algorithm. Does it has a name ? Thanks in advance.

Which course do you think is more important or interesting to take for someone interested in theoretical computer science or theoretical mathematics, number theory or abstract algebra? I am mainly interested acquiring skills and knowledge that will enable me to prove something significant...

I have a question about abstract algebra so if someone could help me answering this question please ... Suppose P,P' are 3-Sylow subgroup, and let Q be their intersection and N the normalizer of Q. Problem: Explain why is the order of N divisible by 9 ? Thanks for your help. Regards,

Homework Statement Let F be a field with p\inN, a prime natural number. Show that either X^{p}-\alpha is irreducible in F[X] or \alpha has a pth root in F Homework Equations The Attempt at a Solution I'm trying to do this without making reference to the field norm, so far I've...

Homework Statement Let G be an abelian group and let x, y be elements in G. Suppose that x and y are of finite order. Show that xy is of finite order and that, in fact, o(xy) divides o(x)o(y). Assume in addition that (o(x),(o(y)) = 1. Prove that o(xy) = o(x)o(y). The Attempt at a...

Homework Statement define a function f:H--> gHg^{-1} Homework Equations prove if f is 1-1 and onto.The Attempt at a Solution 1-1: f(h1)=f(h2) gh1g^{-1}=gh2g^{-1} h1=h2 (left and right cancellations) onto: f(g^{-1}hg)=gg^{-1}hgg^{-1}=h so every h belonging to H has an image of g^{-1}hg...

Homework Statement Theorem 8.1 of Dan Saracino: Let f ε S_{n}. Then there exist disjoint cycles f_{1},f_{2} .. in S such that f= f_{1}°f_{2}... In proving this theorem, it considers a finite group S_n={1,2,..,n} and chooses x_1 ε S_n. Then it defines x_2= f(x_1), x_3=f(x_2) and so on. The...

I'm trying to round out my math skills in order to apply to graduate school for physics and I've already taken all of the calculus offered along with linear algebra, power series etc... I'm wondering which would be better should I choose to take a math course this term: abstract algebra or set...

1. Show that S42 contains multiple subgroups that are isomorphic to S41. Choose one such subgroup H and find σ1,...,σ42 such that How can you solve this?? I am confused if anyone can help me to solve this!

Homework Statement Let A=C_{p^k} where p is a prime and k>0. Let _{p^m} A consist of all element a of A such that a^{p^m}=e. Prove that _{p^m} A/_{p^m-1} A\cong C_p if m\leq k, \frac{_{p^m} A}{_{p^m-1} A}=e if m>kThe Attempt at a Solution Please could someone explain how to get started with...

In high school I absolutely hated Physics. The class was just all about memorizing formulas and stuff, and I thought I wasn't learning anything useful for real life. Yet now that I'm older I have re-discovered my passion for science and I'd like to give the subject another try. The Internet...

Hey guys, I'm in the middle of writing a lab report and I'm having trouble with my abstract. I know as a 3rd year uni student I should know how to write one by now. I feel it's a bit long, what do you think? How could I improve? I'm not asking you to write it for me just some hints. The...

Hi, I'm trying to justify to myself the abstract notion of a vector space and I would really appreciate if people wouldn't mind taking a look at my description and letting me know if it's correct, and if not, what is the correct explanation? : "Vectors are most often introduced as ordered...

Homework Statement Prove that (\mathbb{Z}/2\mathbb{Z})[x,y]/(x^2 + xy + x^2, x^2 + x + 1, y^2 + y + 1) is isomorphic to F_4[z]/(z^2 + z + 1) by showing that the kernel of \phi : (\mathbb{Z}/2\mathbb{Z})[x,y] \to (\mathbb{Z}/2\mathbb{Z})[x,y]/(x^2 + xy + x^2, x^2 + x + 1, y^2 + y + 1) is the...

Homework Statement Show via induction that the nth root of (a1 * a2 * a3 * ... an) ≤ 1/ (n) * ∑ ai, where i ranges from 1 to n. Homework Equations Induction The Attempt at a Solution Let Pn be the statement above. It is clear that P1 holds since a1 ≤ a1. Now let us assume that Pn...

Homework Statement Suppose H is a subgroup of G. For g in G, define fg : G/H > G/H by fg (aH) = gaH for a in G, where G/H is the set of left cosets of H in G. I know that fg is a well-defined permutation. However, we have not established (yet) that G/H is a group. 2 parts to the...

Greetings, For a homomorphism \varphi, I'm trying to show that elements of a fiber, say the fiber above a, X_a, are writable as a given element of X_a times an element of the kernel K. So, if a\in X_a and b\in X_a, then \exists k\in K such that b=ak. I want to do this without using the...

Homework Statement The Attempt at a Solution I'm very new to this kind of maths, so don't quite know how to get started. If I understood the question at all we have g_i \mapsto \phi_i and so I have a homomorphism if I can show that \pi(g \cdot g_i) = \pi(g) \circ \pi(g_i) I'm thinking...

Prove that if G is a group and aεG, then o(a-1)=o(a) This is all I have so far: Assume G is a group and aεG. Because G is a group a has an inverse in the group, a-1 s.t. aa-1=e, which is also in G. <a>={an|nεZ}. |<a>| is the number of elements in <a> before it cycles back. Basically all I've...

I have two more loosely based questions about PDEs and the separation of variables technique: In the intro of this chapter the author imposed that we "assume" the the solution to a set of special PDEs is: U(x,t) = X(x)T(t) where X and T are the eigenfunctions. My question is how did...