Proof Definition and 999 Threads

I want to know whether the following the counts as a proof that infimum of the set $S = \left\{2(-1)^n+\frac{5}{n^2+2}: n \in \mathbb{N}^{+} \right\}$ is $\text{inf}(S) = -2$. Let $A \subseteq X$, where $X$ is some ordered field. Then $\text{inf}(A)$ is $m \in X$ such that for any $x \in A$...

any particular solution plus the general solution to the homogeneous equation. I'm having difficuilty understanding this proof from my lecture notes Theorem : Let T : V → W be a linear transformation. Let w ∈ W and suppose T(u0) = w T(v) = 0. where v ∈ V (the kernel ) to prove: T(u) = w...

I am reading Bruce N. Coopersteins book: Advanced Linear Algebra (Second Edition) ... ... I am focused on Section 10.1 Introduction to Tensor Products ... ... I need help with another aspect of the proof of Theorem 10.1 regarding the existence of a tensor product ... ...The relevant part of...

I am reading Bruce N. Coopersteins book: Advanced Linear Algebra (Second Edition) ... ... I am focused on Section 10.1 Introduction to Tensor Products ... ... I need help with another aspect of the proof of Theorem 10.1 regarding the existence of a tensor product ... ... The relevant part of...

Homework Statement Prove by contradiction that if b is an integer such that b does not divide k for every natural number k, then b=0. Homework EquationsThe Attempt at a Solution I know that proof by contradiction begins by assuming the false statement: If b is an integer such that b does not...

Homework Statement Find the limit \lim_{(x,y)\to(2,2)}\frac{x^3-y^3}{x-y} Homework Equations \epsilon - \delta, baby: If the limit L exists, \forall \: \epsilon \: \exists \: \delta: 0 < \sqrt{(x-a)^2+(y-b)^2} < \delta \rightarrow |f(x,y)-L| < \epsilon The Attempt at a Solution By...

I am reading Bruce N. Coopersteins book: Advanced Linear Algebra (Second Edition) ... ... I am focused on Section 10.1 Introduction to Tensor Products ... ... I need help with the proof of Theorem 10.1 on the existence of a tensor product ... ...Theorem 10.1 reads as follows: In the above text...

I am reading Bruce N. Coopersteins book: Advanced Linear Algebra (Second Edition) ... ... I am focused on Section 10.1 Introduction to Tensor Products ... ... I need help with the proof of Theorem 10.1 on the existence of a tensor product ... ... Theorem 10.1 reads as follows:In the above...

To prove that n log n is big oh of log(n!), I did: n log n <= C log(n!) n log n/ log(n!) <= C Let k = 1 n > k, so for n = 2 2 log 2 / log 2 <= C 2 <= C C is an element of [2, infinity) Taking C = 2 and k = 1 can we say, n log n <= 2 log(n!) and hence n log n is big oh of log(n!) ?

Homework Statement Let S \subset \mathbb{R} be bounded above. Prove that s \in \mathbb{R} is the supremum of S iff. s is an upper bound of S and for all \epsilon > 0 , there exists x \in S such that |s - x| < \epsilon . Homework Equations **Assume I have only the basic proof...

Homework Statement The BCH formula states that the product of two exponentials of non commuting operators can be combined into a single exponential involving commutators of these operators. One may write that ##\ln(e^A e^B) = \sum_{n \geq 1} c_n(A,B)## where $$c_{n+1} = \frac{1}{n+1} \left(...

Hi, I don't know how to prove ((Ǝx) F(x) →(Ǝx) (G(x)) with conditional proof from: ((Ǝx) F(x) → (∀z) H(z)) H(a) →G(b) Thanks

Homework Statement The problem is attached. I don't get this part. Let G = Sn be the group of all permutations of S. S is a set, so how can we permute something in a set?. Neither I know if the 4 power in the S is a typo. Homework EquationsThe Attempt at a Solution

Homework Statement The problem is attached and it's part A. There is no need to put problem 4 hence the problem is fully explained in the file attached Homework Equations Zk is mod k basically. The Attempt at a Solution I know that we have to prove that the transformation is onto,one to one...

Every math major eventually learns logic and standard proof techniques. For example, to show that a rigorous statement P implies statement Q, we suppose the statement P is true and use that to show Q is true. This, along with the other general proof techniques are very broad. A math major would...

During lecture, the professor gave us a theorem he wants us to prove on our own before he goes over the theorem in lecture. Theorem: Let ##V_1, V_2, ... V_n## be subspaces of a vector space ##V##. Then the following statements are equivalent. ##W=\sum V_i## is a direct sum. Decomposition of...

Well, let's look at how this works. Quadratic equations can have either 1, 2, or no zeroes. If it has no real zeroes, the zeroes it DOES have are complex, so that's obviously not it. Let's imagine $$ax^2 + bx + c = 0$$ has one zero, call it $$ \alpha$$ (Cuz it looks pretty). Then that means...

Hey all, since I was programming a polynomial interpolater i found it easier to use the expanded divided difference $$ f[x_0 ,...,x_n] = \sum_{j=0}^{n} \frac{f(x_j)}{\Pi_{k}^{n,j \neq k} (x_j - x_k)} $$ , it works, but I can find no proof, any help/ references appreciated. Second question: how...

Homework Statement Take the expression 21.11 (pictured below, specifically the bottom one) for the electric field above the center of a uniformly charged disk with radius R and surface charge density σ, and show that when one is very far from the disk, the field decreases with the same square...

1. The question is. Show that if |nx| <1, the following is exact up to (and including) the x^2 order. The hint giving says to use the Taylor Expansion for both sides of the equation2. (1+x)^n = e^n(x-(1/2)x^2) ; the n(x-(1/2)x^2) is all an exponent3. My first attempt was to take the taylor...

Hi. In the attached proof for Lorentz transformation for momentum http://www.colorado.edu/physics/phys2170/phys2170_sp07/downloads/lorentz_transformation_E_p.pdf, there is this step that I don't understand: 1/√1-u'2/c2 = γ(1-vux/c2)/√1-u2/c2 Can someone explain how they derived this? Thanks! :)

Homework Statement If gcd(f(x),g(x)) = 1 and m,n ∈ ℕ, show that gcd(f(x)^m, g(x)^n) = 1. Homework EquationsThe Attempt at a Solution So I had previously proved this for non-polynomials: gcd(a,b)=1 then gcd(a^n,b^n)=1 Proof: a = p1*p2*...*pn b = p1*p2*...*pm then a^n = p1^n*p2^n*...*pn^n...

I'm going through Bishop and Goldberg's "Tensor Analysis on Manifolds" right now and I'm stuck in Chapter 0. :H They give a proof of the statement "A compact subset of a Hausdorff space is closed" that I can't seem to wrap my head around. I'm reprinting the proof here: "Suppose that A is a...

Homework Statement Attached is the problem. I didn't find anyway to apply the hamming distance to this problem, but I hope that at least this is close to show something. Homework EquationsThe Attempt at a Solution Lets consider Rn over Z 2 n, so the basis of R n under Z 2 is (0,0,………0 )all the...

Homework Statement Show that all square matrix (A whatever) can be written as the sum of a symmetric matrix and a anti symmetric matrix. Homework Equations I think this relation might be relevant : $$ A=\frac{1}{2}*(A+A^{T})+\frac{1}{2}*(A-A^{T}) $$ The Attempt at a Solution I know that we...

Using these equations I am about to prove that photons have a rest mass of zero (mathematically) ________________________________________________________________________________________ E=hc/λ Photon Energy Equation E2=(pc+mc2)2 Mass-Energy Equivalence with Momentum Equation p=h/λ Momentum...

I'd like to discuss the question in the title, following up on my remark quoted below. Note that I don't want to repeat the discussion in https://www.physicsforums.com/threads/tracks-in-particle-detectors-and-quantum-paths.758778 so maybe reread that one first! The traditional analysis is...

Homework Statement Let r be an element of an integral domain R such that r^2 = r. Show that either r = 0_R or 1_R Homework Equations integral domain means no zero divisors. The Attempt at a Solution This is fundamental as 0 and 1 solve r^2 = r and are the only solutions. However, I'm not...

Homework Statement Let T:R-> S be a homomorphism of rings. Show that T(0_r) = 0_s. Homework EquationsThe Attempt at a Solution First off, the terminology used is kinda confusing. I take 0_r to be the zero in R. Is this correct? For some reason I recall my teacher quickly saying that it was...

From Baby Rudin "Thm: Let P be a non-empty, perfect subset of R^k. Then P is uncountable. Pf: Since P has limit points, P must be infinite. Suppose P is countable, list the point of P {x1 ...xn }. Construct a sequence of nbhds. as follows. Let V1 be any nbhd of x1 . Suppose Vn has been...

Homework Statement Let ##\sigma_4## denote the group of permutations of ##\{1,2,3,4\}## and consider the following elements in ##\sigma_4##: ##x=\bigg(\begin{matrix}1&&2&&3&&4\\2&&1&&4&&3\end{matrix}\bigg);~~~~~~~~~y=\bigg(\begin{matrix}1&&2&&3&&4\\3&&4&&1&&2\end{matrix}\bigg)##...

I've just encountered this somewhere and I need some sort of formal proof for why a continuous function ##f(x)## can equal zero because its integral is zero. Are there any out there? I've seen similar forum posts on places like Stack Exchange and one here, but I can't exactly follow the logic...

In Andrew McInerney's book: First Steps in Differential Geometry, Theorem 2.4.3 reads as follows:https://www.physicsforums.com/attachments/5252McInerney leaves the proofs for the Theorem to the reader ... I am having trouble formulating a proof for Part (3) of the theorem ... Can someone help...

Could anyone recommend some books or exercises to practice proofs? Or even post some theorems to prove? Thanks in advance, Ajay

Homework Statement Homework Equations Prof. Note's. The Attempt at a Solution I'm on the 3 line where my Prof. combines both equations, I'm confused on what my equation should look. Her's was (n+1)(n+1)+1)/2

Homework Statement Let R be a ring and suppose r ∈R such that r^2 = 0. Show that (1+r) has a multiplicative inverse in R. Homework Equations A multiplicative inverse if (1+r)*x = 1 where x is some element in R. The Attempt at a Solution We know we have to use two facts. 1. Multiplicative...

Homework Statement Suppose R is a commutative ring with only a finite number of elements and no zero divisors. Show that R is a field. Homework Equations Unit is an element in R which has a multiplicative inverse. If s∈R with r*s = 1. A zero divisor is an element r∈R such that there exists...

I am trying to understand the following basic proposition about invertibility: a linear map is invertible if and only if it is injective and surjective. Now suppose ##T## is a linear map ##T:V\rightarrow W##. The book I read goes the following way in proving the proposition in the direction when...

Prof. S Lakshmi Bala from Department of Physics, Madras, India writes a blackboard of equations which show how beamsplitters used alone affect the wavefunctions of input photons. It seems that it depends on the number of photons you use and in which input port to get you a different entangled...

Hello, I'm trying to self teach myself Fundamental Mathematics. I looked around, but I wasn't sure what to look for exactly. I read the part on Set Theory in "Book of Proof" by Richard Hammack. I enjoyed it, but I wasn't sure if it is rigorous enough to stand against a college level course...

Homework Statement Show that every partition of X naturally determines an equivalence relation whose equivalence classes match the subsets from the partition. Homework Equations ( 1 ) we know that equivalence sets on X can either be disjoint or equal The Attempt at a Solution Let Ai be a...

Let G be a subgroup of Sym(X) and ρ ∈ G. Prove that Gρ(x) = ρGxρ−1, where ρGxρ−1 = {ρgρ−1|g ∈ Gx} What I Know: I need to somehow prove the left is contained in the right and the right is contained in the left. What I Have Done: Well based on the definition of a stabilizer Gx I assumed that...

I don't know how to Google appropriately for this, since the kind of keywords I use present me with search results that try to define the exponential function using limits instead of what I am trying to ask: What does the proof look like for the following (assuming f(x) is "nice"). Any sites...

m and n are integers. log2(i) = m/n 2^(m/n) = i 2^m = i^n 2^0 = i^4 = 1 so that means that log2(i) is rational because there are integers n and m so that log2(i) = m/n , they are m=0 and n=4. But what I do get about this proof is that it seems to imply that log2(i) = 0/4 = 0 while google says...

Hello Physicsforum Homework Statement I have a problem proving this: Given C(x)=[0, 3/x] for all x\in\chi, with \chi=\Omega being the sample space and P_q=Geom(q) being the geometric distribution. I have to show that C(x) is a confidence Interval for q but I don't know how to get started...

I'm reading the book by Zee, I came across a paragraph saying that the world is not flat. "Given an airline table of distances, you can deduce that the world is curved without ever going outside. If I tell you the three distances between Paris, Berlin, and Barcelona, you can draw a triangle on...

Which books in QFT give representations about general proof of renormalization?I know that the book of QFT of Peskin&Schroder does not give the full demontration.

I am trying to prove ##||A||_{\infty} = max_i \sum_{j} |a_{ij}|## which reads as the ##\infty## norm is the max row sum of matrix A. ##i## is the row index and ##j## is the column index. Here is what I thought of: ##||A||_{\infty} = sup_{x\neq 0} \frac{||Ax||_{\infty}}{||x||_{\infty}}## The...

The thought experiment used to prove Lorentz transform uses a light signal as an assumption. What if there was something other than the light signal then Lorentz transformation would have totally different term in place of 'c'(speed of light).

Homework Statement I am trying to craft a hypothesis regarding factors that affect the coefficient of friction. I know that it is determined by the triboforces and asperity interactions at the interface between the materials (among other factors, but right now I'm just going to focus on this)...