Property Definition and 635 Threads

Hello, I am embarking to read Spivak's book on Calculus, and have come across some difficulty with something that is perhaps rather trivial. In the third edition, there is a section entitled Basic Properties of Numbers. Near the end of page 7, the author begins discussing how he will use...

Prove the following logic property using a Truth Table (perfect induction). What is this property called? x + y * z = (x + y)(x + z) My Answer: Distributive Property? Truth table. x y z f [0]0 0 0 0 [1]0 0 1 0 [2]0 1 0 0 [3]0 1 1 1 [4]1 0 0 0 [5]1 0 1 1 [6]1 1 0 0 [7]1 1 1 1 By truth...

Hi, I'm following the proof of the "Scaling Property of the Fourier Transform" from here: http://www.thefouriertransform.com/transform/properties.php ...but don't understand how they went from the integral to the right hand term here: The definition of the Fourier Trasform they...

Attached is a copy of p181 of Strauss Partial Differential Equation. This is part of proof of Mean Value Property using Green's 1st identity: \int_Dv\nabla u+\nabla v\cdot\nabla u \;dV=\oint_A v\frac {\partial u}{\partial r}dA Let ##v=1## and let ##u(r,\theta,\phi)## be a harmonic function ie...

Ok, I don't think I'm on the right track here. I ASSUMED that the set of all countable collections \{I_k\}_{k = 1}^\infty of nonempty open, bounded intervals such that E \subseteq \bigcup_{k = 1}^\infty I_k is a countable set itself, which it probably isn't. I'm not even sure where to start...

Can you explain what this image is saying. I'm confused.

So there is a proof that the sum of any two even numbers is an even number. 2k + 2l = 2(k +l) We have written the sum as 2 times an integer. Therefore the sum of any two even numbers is an even number. An essential part of this proof is that k + l is an integer. How do we know this? Is it an...

Hi I am not sure where to post this question but I am trying to simplify this expression: r*c'w+c (As in R AND NOT C AND W OR C) to c+wr (As in C OR W AND R) and I know that it simplifies to this and they are both equivalent; however my question is which boolean simplification property is...

hello! I'm trying to understand the following property: Let X and Y be independent random variables z: = X + Y. Then http://imageshack.us/a/img268/9228/71pe.png where fZ (z) is the probability mass function for a discrete random variable defined as follows...

Recently I review some old text and browse around the internet to read about definition of strain and stress. I come across the following document http://web.mit.edu/emech/dontindex-build/full-text/emechbk_4.pdf Previously I have been facing difficulty trying to come out with an...

Suppose you have the following definition of Dirac-delta function, or the so called sifting property: \int^{d}_{c}f(t)δ(t-a)dt =\left\{\begin{array}{cc}f(a),&\mbox{ if } c\leq x \leq d\\0, & \mbox{ if } x>d \mbox{ or } x<c \\ \mbox{undefined}, & \mbox {if } x = d \mbox{ or } x = c...

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 φ(r):=exp(-2μr). Show that M_{t}:=φ(X_{t}) for t>=0 is...

Hello all, I came across this rather mysterious, for me of course, statement. It is stated that, starting from the solution of the Laplacian under definite boundary conditions and for a given geometry, if a portion of the domain is altered in such a way that the maximum distance between...

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...

The lines of flux of the magnet possesses the following properties: 1)Forms closed loop 2)Starts from N-pole and closes at S-pole 3)Do not intersect each other 4)Parallel line and same direction repels each other 5)Parallel line and opposite direction attract each other So...

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

Homework Statement S is a ball of radius 1 in R^2; Δu=0 in S u=g in ∂S, g(x1,x2)>1 for any (x1,x2) in ∂S. Show that for any r satisfying 0<r<1 there is a point (x1,x2) in S such that u(x1, x2) >=1. Homework Equations using mean value formula: ∫u(y)dy=1/Vr^n(∫u(y)dy) The...

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...

Hi, 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...