Property Definition and 608 Threads

Homework Statement I am working through an introductory real analysis textbook and am having a little trouble with certain aspects of the proof of the well ordering property (I am new to proving). Theorem: Every nonempty subset of the natural numbers (N) has a smallest element. Proof: Let...

The book proves this limit and I am a bit confused how all the pieces fit together. So the book proves "If (s_n) converges to s and (t_n) converges to t, then (s_nt_n) converges to st . That is, lim(s_nt_n) = (lim s_n)(lim t_n). The proof goes like this Let \epsilon> 0 . By Theorem 9.1...

I am struggling to draw this point home: To prove that R has LUB property, we used the following reasoning: First we defined R to be set of cuts (having certain properties) where each cut corresponds to a real number and then we proved any subset A of R has LUB (least upper bound) property...

I was solving a question which is the following : Give examples of 3 sets W,X,Y such that W ε X and X ε Y but W doesn't ε Y . I solved the question by taking the following 3 sets: W = {1,2} X = { 7 , 8 , W} Y = { 3 , 4 , X} looking it from the theory point of you I find that W is not...

Homework Statement Let A and P be square matrices of the same size with P invertible, Prove detA=det(P-1AP) Homework Equations Suppose that A and B are square matrices of the same size. Then det(AB)=det(A)det(B) The Attempt at a Solution detA=det(P-1AP) detA=det(P-1PA) detA=det(IA)...

Hi! I need some help at the following exercise... Let $$B$$ be a typical brownian motion with $$μ>0$$ and $$x$$ ε $$R$$. $$ X_{t}:=x+B_{t}+μt$$, for each $$t>=0$$, a brownian motion with velocity $$μ$$ that starts at $$x$$. For $$r$$ ε $$R$$, $$T_{r}$$:=inf{$$s>=0:X_{s}=r$$} and...

In the back of my thermodynamics book it has large quantities of thermodynamics properties listed for water--ie temperature, pressure, specific volume, internal energy, enthalpy, and enthalpy. I would like to know how these tables are built and the methods used to ascertain the data in...

Let X be a metric separable metric and zero dimensional space.Then X is homeomorphic to a subset of Cantor set. How can it be proved? Thank's a lot, Hedi

Greetings, I was trying to prove a theorem regarding degeneracy, and I succeeded. However, I also proved the converse of the if-then part of the theorem (underlined below), which I know is wrong. I can't spot my mistake though. The theorem and my proof are written below - could someone...

Homework Statement Find the Fourier transform of (1/p)e^{[(-pi*x^2)/p^2]} for any p > 0 Homework Equations The Fourier transform of e^{-pi*x^2} is e^{-pi*u^2}. The scaling property is given to be f(px) ----> (1/p)f(u/p) The Attempt at a Solution Using the information above, I got...

Homework Statement Let f(z) = \sum_{n =-\infty}^{\infty} e^{2 \pi i n z} e^{- \pi n^2}. Show that f(z+i) = e^{\pi} e^{-2\pi i z}f(z). Homework Equations Nothing specific I can think of; general complex analysis/summation techniques. The Attempt at a Solution f(z+i) = \sum_{n...

$$ \zeta(s) = s \int^{\infty}_1 \,\frac{ [ t ] }{t^{s+1}} \, =\,\frac{s}{s-1} \, -s \int^{\infty}_1 \frac{ \{ t \} } {t^{s+1}}\,dt $$

Assume for some real number L and c \displaystyle\lim_{x\rightarrow c} f(x) = ∞ and \displaystyle\lim_{x\rightarrow c} g(x) = L We must prove \displaystyle\lim_{x\rightarrow c} [f(x) + g(x)] = ∞ Let M > 0. We know \displaystyle\lim_{x\rightarrow c} f(x) = ∞. Thus, there exists...

Prove that. \int_a^b f(x)g' (x)\, dx = -f(0) This is supposed to be a delta Dirac function property. But i can not prove it. I thought using integration by parts. \int_a^b f(x)g' (x)\, dx = f(x)g(x) - \int_a^b f(x)'g (x)\, dx But what now? Some properties: \delta...

Hi All, I am teaching my self algebra using khan academy and I've come across a problem I can't figure out. I am trying to solve composite functions and I can't figure out why a Plus sign is added to the equation, Could I be missing something that I am suppose to distrubute? The functions...

I am trying to follow examples solved by the publisher of my book in order to understand the problem. However, I can't understand why he is solving it like this. What is confusing me, is why v1=(12+8)*1/8 why is v1 not 12*(1/8). Why is he adding the 8ohm resistor in there? Any help would be...

Homework Statement Let f_k\rightarrow f in L^2(\Omega) where |\Omega| is finite. If \int_{\Omega}{f_k(x)}dx=0 for all k=1,2,3,\ldots, then \int_{\Omega}{f(x)}dx=0. Homework Equations The Attempt at a Solution I started by playing around with Holder's inequality and constructing...

[FONT=Verdana]Hi, [FONT=Verdana]For any odd composite 'N', let u = (N-1)/2, v = u+1, then u^2(mod p) = v^2(mod p) if and only if 'p' is a factor of 'N'.

At time 2:20, the woman says, "Using trigonometry, we know that this angle is the same as this angle." Which trigonometry or geometry property is she referring to that is needed in order to determine those two angles are equal? Is there a name for it? I don't have a geometry book, but I am...

If (E_{n})) is either an increasing or decreasing sequence of events, then lim n\rightarrow∞ P(E_{n}) = P(lim n\rightarrow∞ (E_{n})) This seems to be saying that the limit as n goes to infinity of the probability of an increasing or decreasing sequence of events is equal to the probability...

Hi everyone, I am working on simplifying a differential equation, and I am trying to figure out if a simplification is valid. Specifically, I'm trying to determine if: \frac{\del^2 p(x)}{\del p(x) \del x} = \frac{\del^2 p(x)}{\del x \del p(x)} where p(x) is a function of x. Both p(x)...

can particles be entangled on any property having more than two states? Photons can be entangled on spin. however spin has only two states: Up or down, plus or minus So the question is: is there any property (having more than two states) on which photons/electrons/bucky-ball can be...

Hello, I wanted to ask if this is a correct move, A and B are matrices, a is a scalar, thank you ! A^{2}\cdot B^{t}-aA=A(A\cdot B^{t}-aI)

Given that a random variable X follows an Exponential Distribution with paramater β, how would you prove the memoryless property? That is, that P(X ≤ a + b|X > a) = P(X ≤ b) The only step I can really think of doing is rewriting the left side as [P((X ≤ a + b) ^ (X > a))]/P(X > a). Where...

Hello all, while practicing set theory, I cam across this problem: If A and B are sets, prove that A x (B-C) = (AxB) - (BxC). This looks suspiciously like the distributive property but it's not. Is this simply a typo? Shouldn't the problem look like this: A x (B-C) = (AxB) - (AxC) Thanks...

Hello everyone! I have three questions: (1) If $x\in R$, is it true that $f ^{-1} (f(x)) = x$? (2) If $y\in R$, is it true that $f (f^{-1}(y)) = y$? (3) If $B\subset R$, is it true that $f(f ^{-1} (B)$? I think I have showed it for (3), but not sure of my proof. For (1) and (2), I considered...

Hi friends what are intrinsic properties of spacetime ? curvature & torsion ? or they are just properties of connections ? Since in teleparallel gravity we consider them as properties of connections. thank u

How to prove that 3 x 2 = 2 x 3?

Let S1*(S2*) be the polar cone of the set S1(S2) (http://en.wikipedia.org/wiki/Dual_cone_and_polar_cone). How can I show that if S1 is contained in S2 then S2* is contained in S1*. It looks obvious (especially if we think in R^2), but I do not find a way to prove it.

i am reading equivalence relation on wikipedia. and in this examply. i don't see the transitity property according to the definition of transitivity which is: For every three elements a, b, and c in X, if a ~ b and b ~ c, then a ~ c (transitivity). any explanation please?

Homework Statement Let A be an invertible matrix. Show that (A^n)^{-1} = (A^{-1})^n Homework Equations The Attempt at a Solution I want to begin on the left side of the equality sign; but I am having a little difficulty on expanding it. I started to--(A^n)^{-1} = AAA...A^{-1}--but...

I found an example like the problem asks, but I'm still trying to show the first part. You want the maximum number of subsets such that you can guarantee none are pairwise disjoint. I'm trying to apply my specific case to the whole problem. For a set with 3 elements, I chose all of the sets...

Hello everyone. I desperately need clarifications on the least upper bound property (as the title suggests). Here's the main question: Why doesn't the set of rational numbers ℚ satisfy the least upper bound property? Every textbook/website answer I have found uses this example: Let...

Hi, the attached picture shows a derivation of what I can only assume to be the property that the lagrange equations are invariant under a transformation of the coordinates. But I have some trouble understanding how you go from the term pointed out the rear of the arrow to the point pointed...

Hi everybody... I've been working a bit with models of chemical oscillators and I've run into something that isn't quite clear to me. Chemical reaction systems are typically regarded as having the Markov property -- they lack memory and their evolution depends only on their current state...

Homework Statement Determine all continuous functions g: R -> R such that g(x + y) = g(x) + g(y) for all x, y \in \mathbf{R} The Attempt at a Solution g(x) = g(x + 0) = g(x) + g(0). Hence G(0) = 0. G(0) = g(x + -x) = g(x) + g(-x) = 0. Therefore g(x) = -g(-x). It seems obvious that the only...

I would like to show that if a prime number P mod 8 is a) 1 or 7 or b) 3 or 5 then a) \frac{(P+1)}{2}(1-sqrt{2})(3+sqrt{8})^\frac{P-1}{2}+ \frac{(P+1)}{2}(1+sqrt{2})(3-sqrt{8})^\frac{P-1}{2} = (\frac{P-3}{2} + 2) mod P b) \frac{(P+1)}{2}(1-sqrt{2})(3+sqrt{8})^\frac{P-1}{2}+...

Hi Does anybody know if the irrational numbers have the least upper bound property?

Homework Statement I am using the time differentiation property to find the Fourier transform of the following function: Homework Equations f(t)=2r(t)-2r(t-1)-2u(t-2) The Attempt at a Solution f'(t)=2u(t)-2u(t-1)-2δ(t-2) f''(t)=2δ(t)-2δ(t-1)-?? Can somebody explain what the...

I'm guessing that if z\in \mathbb C, then we have \left| z^{-1/2} \right|^2 = |z^{-1}| = |z|^{-1} = \frac{1}{|z|}. Proving this seems to be a real headache though. Is there a quick/easy way to do this?

How can I calculate degrees of freedom of a rank (o,3) tensor, Aabc, that is mixed symmetry and antisymmetric in the first 2 indices? By mixed symmetry I mean this: Aabc+Acab+Abca=0.

OK - this one has been argued to death in several different threads, but the answers have been less than satisfactory... so someone provide a reason why I am wrong here: 48 / 2(9 + 3) = 2. Why? Because the distributive property of multiplication means that 2(9+3) = (2*9+2*3). For...

As a preface to a different question, it is valid to think of the property of spin of elementary and related particles as basically just tiny magnets, right?

Could you please give me a hint on how to show that a set of operators with a property P is closed under addition? In other words, how one could prove that a sum of any two operators from the set still possesses this property P. The set is assumed to be infinite. Any references, comments...

Dirac Matrix Property? Possible Book mistake? Derive KG from Dirac I got a copy of QFT demystified and on pg. 89 he says he can write \gamma_{\nu} \gamma^{\mu} = g_{\nu \sigma} \gamma^{\sigma} \gamma^{\mu} = g_{\nu \sigma} \frac{1}{2} (\gamma^{\sigma} \gamma^{\mu} + \gamma^{\mu}...

Homework Statement Given Pressure=P=700 KPa Specific Entropy = s = 7.6953 KJ / ( K Kg ) Find the Specific Enthalpy (h) Homework Equations No equation Am using the property tables at the end of the following book : Thermodynamics, an engineering approach, by Yunus A. Cengel and...

Hi, I'm working through a paper and I am quite stupid so some things that maybe obvious are not obvious to me. Say you have some have some complex analytic function that is defined on some simply closed curve, and the index of this function defined on this curve is zero, \int_C...

Hello, I am looking at the proof (Theorem 2.5 (b) Apostol) of $$ \phi (mn) = \phi(m) \phi(n) \frac{d}{\phi(d)} $$ where $$ d = (m, n) $$. Can someone explain how they go from $$ \prod_{p|mn} \left( 1 - \frac{1}{p} \right) = \frac{\prod_{p|m} \left( 1 - \frac{1}{p} \right) \prod_{p|n}...

I have already solved it, but I need confirmation: Are there other ways of proving this? Thanks in advance!

A power tower (x^^n) is a variable raised to the power of itself n amount of times. x^^4 = x^x^x^x x^^3 = x^x^x x^^2 = x^x x^^1 = x I was wondering if an associative property for power towers exists. Does x^(x^x) equal the same thing as (x^x)^x? Is x^(x^^n) equal to x^^(n + 1)? If anybody...