Property Definition and 635 Threads

Homework Statement Hello fellow Mathematics enthusiasts. I was hoping someone could help me with the following problem from Terry Tao's Introduction to Measure Theory: Let ##[a,b]## be an interval, and let ##f,g:[a,b] \to \mathbb{R}## be Riemann integrable. Establish the following statement...

I need to prove that $$std(x+c) = std(x)$$ I have been trying to use the properties of the mean such as $$mean(x+c) = mean(x) + c$$ I am confused on the following property of the mean, is this statement correct? $$\sum\limits_{i=1}^{i=N}(x_i - mean(\{x\}))^2 = mean(\{x\}) \\$$ If that is...

The wave property of particles (like electrons) is due to : 1) The wave function 2) The underlying fermionic field 3) Just because of the existence of de broglie waves? Or maybe someshow the above three cases are unified ?

I'm building a linear actuator and I don't have much experience with magnetic circuits... A rough sketch of what I'm building is attached. I'm trying to determine if the materials I'm using for the shaft and shaft collar will have a detrimental effect on overall force output. Right now, my...

I am reading Chapter 2: Vector Spaces over \mathbb{Q}, \mathbb{R} \text{ and } \mathbb{C} of Anthony W. Knapp's book, Basic Algebra. I need some help with some issues regarding the Universal Mapping Property of direct sums of vector spaces as dealt with by Knapp of pages 60-61. I am not...

With groups, one often seeks to create larger groups out of smaller groups, or the reverse: break down large groups into easier-to-understand pieces. One construction often employed in this regard is the direct product. The normal way this is done is like so: The direct product of two groups...

Hi I am using relap to study a blowdown transient. If I let it run it with actual heat fluxes, the program terminates with error message saying thermodynamic property error with maximum time step. Reducing the time step delays the onset of the error by a fraction of a second. Runs much longer...

The usual "proof" entropy is a state property is like that: "Consider a system which undergoes a reversible process from state 1 to state 2 along path A, and let cycle be completed along path B, which is also reversible. Since the cycle is reversible we can write: ∫1-2 δQ / T + ∫2-1 δQ / T...

Homework Statement suppose f~:~A \rightarrow B be a surjective map of sets. Prove that the relation a Rb \iff f(a)=f(b) is a equivalence relation whose equivalence classes are the fibers of f. Homework Equations The Attempt at a Solution I was able to easily prove that the...

Hi there! I have the following property: If x(t) is a solution of \left\{ \begin{array}{l} \dot{x} = f(x) \\ x(t_0) = x_0 \end{array} \right. then the function y(t) = x(t+t_0) is a solution of the equation with initial data y(0) = x_0 . How could it be interpreted geometrically? Thanks!

The Schrodinger equation is of the form \frac{d^2 \psi}{dx^2}+[\varepsilon-v(x)] \psi=0. In a lecture, the lecturer said that if we have in a point x_0 , \psi(x_0)=\psi'(x_0)=0 , then \psi(x)=0 .(for a smooth v(x)!) Can anyone give a proof of this? Is it only for a equation of the form...

Hey! (Smile) Is it known that if $p,q$ primes with $p \neq q$ : $$a^p \equiv a \pmod q \Rightarrow a^{p \cdot q} \equiv a^q \pmod q$$ ? If so,why is it like that? Which property is used?

I have heard of various metals and other materials turning from solid to liquid or plasma when temperature and pressure goes up. Is there a metal or other material that goes otherwise , like turns from liquid to solid when temperature and pressure is increased? Is there a material that can...

Homework Statement I am asked to prove that if λ is an eigenvalue of A then λ + k is an eigenvalue of A + kI. The Attempt at a Solution ## A\vec{v}=\lambda\vec{v} ## ## (A+kI)\vec{v}=\lambda\vec{v} ## ## A\vec{v}+k\vec{v} = \lambda\vec{v} ## → ## A\vec{v} = \lambda\vec{v} -...

Difference between physical property and physical quantity?? Hi I'm confused after reading the definitions of physical Quantity and physical property: Definition from wikipedia: physical quantity= is a physical property of a phenomenon, body or substance that can be quantified by...

Ok, we have all read that space is expanding, especially in low mass areas between galaxies, as described by GR. And that mass bends space, like seen in gravitational lensing. My question is whether on larger scales and masses, like on the size of galaxies, space itself is warped? The properties...

Does the log property a*log(x)=log(x^a) still hold if a is even and x I imagine that ln(-1)+ln(-1) can't equal zero, even by some mysterious magic involving complex numbers.

Hi, In equation (7) of http://arxiv.org/pdf/gr-qc/9604038.pdf they consider a Sturm-Liouville problem of the form -(Rr')'-(8/R5)r = ω²r. with R(x) a positive function on (0, +∞) and r(x) the eigenfunction with eigenvalue ω². The goal is to show that there are negative eigenmodes...

Hey! :o Could you give me a hint how to prove the following property of the Fourier transform, when $F[f(x)]=\widetilde{f}(x)$, where $F[f(x)]$ is the Fourier transform of $f(x)$? $$F[ \widetilde{f}(x) ]= \frac{f(-k)}{2 \pi}$$ We know that: $ \widetilde{f}(k)=\int_{- \infty}^{+ \infty}{...

If I have a relation which is not only antisymmetric (##aRb\rightarrow{}\neg{}bRa##) but it also has a property that ##aRb\land{}bRc\rightarrow{}\neg{}cRa##. How can I be sure that this property holds for any string like that? So that ##aRb\land{}bRc\land{}cRd\rightarrow{}\neg{}dRa## without...

Hello, Does anyone have a reference to a proof of the reflective property of a hyperbola? I need a proof that uses the geometric definition of a hyperbola as the locus of points $X$ such that $|XF_1-XF_2|=2a$ for some fixed points $F_1$ and $F_2$ and a positive constant $a$. The proof may also...

Homework Statement Prove that for every two real numbers x and y ##|x+y| \leq |x| + |y| ##Homework Equations The Attempt at a Solution There are three cases. The easiest ones is when they are both positive and negative. The third one I have problems with. The numbers have different sign. Say...

It is known that:\mathcal{F}\{f\ast g\}=\mathcal{F}\{f\}\mathcal{F}\{g\}\mathcal{F}\{f g\}=\mathcal{F}\{f\}\mathcal\ast{F}\{g\} But this property is valid for inverse tranform too?\mathcal{F}^{-1}\{F\ast G\}=\mathcal{F}^{-1}\{F\} \mathcal{F}^{-1}\{G\}\mathcal{F}^{-1}\{F...

Suppose we have a methereological model which describes the weather change as a Markov chain: we assume that the weather on each day depends only on the weather of the previous day. Suppose further that we extended this model so as to consider season change. defining a different transition table...

Hello, The product of a 2x5 matrix P and a 5x3 matrix B shall be a 2x3 zero matrix. P and B are all matrices of integers. P = [6 2 -5 -6 1;3 6 1 -6 -5] One possible B is [0 -4 0;3 0 0;-1 -1 3;2 -3 -2;1 1 3] This solution B has a property: det(PPt) = det(BtB) = 7778 The question is: What...

Photons demonstrate superposition without having to be isolated from the enviorment. My question is does this only apply to light because I know you can't observe superposition in electrons unless theyre in a vaccum. This is my question in a nutshell. Are there any other propertys that...

Or is it a statistical calculation of where the particle could possibly be. If its a property of nature then it must have a limit, distance at which it seaces to exist. Also a second question does space time exist in QM and if it doesent then why does it exist in the classical world which it...

Use algebraic manipulation to prove that x+yz=(x+y)(x+z) Note that this is the distributive rule, So I have: (x + y)(x + z) = xx + xz + xy + zy = x + xz + xy + zy = x(1 + z + y) zy =x * 1 + (z + y)(zy) = x + ((zzy +...

Is it possible to prove the fact that any function of diagonal matrix is just a function of its element? I don't know how I could express the proof. I can prove that a multiplication of diagonal matrix will just be the multiplication of its element using summation notation, or diagonal matrix...

Hello friends, Most people that I heard put uncertainty as an intrinsic property of the universe which is evident when we make a measurement. But to me it seems that intrinsic property and making a measurement are two entirely different things. If uncertainty were to be just(purely)...

Say for example I discovered how to unify General Relativity with Quantum Mechanics and then someone else uses this improved understanding of physics to engineer a new product which would have been impossible without it. Am I entitled to a portion of the revenue generated by the new product...

Hi All, I'm really curious to know how we can predict the shapes of constant property lines (isenthalpic/constant internal energy/isotherms/isentropes/isobars) on any given plot, such as T-s and P-v or even u-s diagrams. Is there a rule to do so? Usually in practice we deal with p-v and T-s...

Homework Statement How to prove in the easiest way that Klein 4-group is associative. Homework Equations Four elements ##a^2=b^2=c^2=e^2=e##. The Attempt at a Solution If that is group with four elements, how many types of ##a*(b*c)=(a*b)*c## I need to have? 1) ##e*(e*e)=e*e=e##...

Homework Statement We have a finite group ##G## and a homomorphism ##\rho: G \rightarrow \mathbb{GL}_n(\mathbb{R})## where ##n## is a positve integer. I need to show that there's an ##n\times n## positive definite symmetric matrix that satisfies ##\rho(g)^tA\rho(g)=A## for all ##g \in G##...

Homework Statement x''(t)+r*x't+kx=0 Suppose that for some initial conditions the solution is given by x=e^(-2t)*(3cos(t)+4sin(t)) What are are and k? Homework Equations See aboveThe Attempt at a Solution I've tried to "brute force" the solution simply by sticking the expression for x...

How do you show that a linear transformation is idempotent? T:R^3 to R^3 T (x y z)^T = (0.5 (x-z) , y, 0.5 (z-x)) I have no idea where to begin. I know a few facts about idempotent properties e.g such as their eigenvalues are...

Prove that a set $A\subset\mathbb{R}^n$ is (Lebesgue) measurable $\iff$ there exist a set $B$ which is an $F_{\sigma}$ and a set $C$ which is a $G_{\delta}$ such that $B\subset A\subset C$ and $C$~$B$ (C without B) is a null set. $F_{\sigma}$ is a countable union of closed sets, and...

Homework Statement The statement that is purported to be true is \frac{a/b}{c/d} = \frac{ad}{bc}Homework Equations The Attempt at a Solution So, I am going along with my proof, and I believe it to be going nicely. However, there is one step that I am unsure of: \frac{\frac{a}{b} d}{c} =...

I'm not sure how they got the RHS of equation 349: where did the |y'(xj)| in the denominator come from? According to (343) the RHS is only f(x0) which in this case is the jth term of the sum that gives y(xj) = 0..

This problem has been taunting me for days now, and I still have no idea of where to start with it... Let \(m\) and \(n\) be non-zero integers. We say that \(k\) is a common divisor of \(m\) and \(n\) if \(k|m\) and \(k|n\). The greatest common divisor of \(m\) and \(n\), denoted as...

Hi everyone, :) I encountered this question and thought about it several hours. I am writing down my answer. I would greatly appreciate if somebody could find a fault in my answer or else confirm it is correct. :) Problem: Let \(V_1,\,\cdots,\,V_k\) be subspaces in a vector space \(V\)...

Are there anybody doing some research on quasicrystals ,especially its mechanical property? I am just wondering why so few people are working on it,because intuitively I think quasicrystals will have a bright future. So what do you think of quasicrystals? Is it worthy of research? I want...

Homework Statement Hello, I know the direct substitution property in calculus. But, the definition of a rational function still confuses me. For example, are these rational functions: y=(x^2+2x+1)/(x+1) y=((x^2+2)^(1/2))/(x+1) The denominator of the first one could cancel. So...

Homework Statement Prove this theorem regarding a property of the Dirac Delta Function: $$\int_{-\infty}^{\infty}f(x)\delta'(x-a)dx=-f'(a)$$ (by using integration by parts) Homework Equations We know that δ(x) can be defined as...

I ran into some formula: ^{a}_{0}∫J_{o}(kr) rdr= a/k J_{1}(ka) How can this be true? What property was used?

Homework Statement Consider the tangent surface of some regular differentiable curve given as X(t,v) = \alpha(t) + v \alpha'(t) . Show that the tangent planes along X(t,constant) are equal. Homework Equations N = \frac{X_{t} \wedge X_{v}}{|X_{t} \wedge X_{v}|} The general tangent...

Hi everyone, :) Trying hard to do a problem recently, I encountered the following question. Hope you can shed some light on it. :) Suppose we have a continuous mapping between two metric spaces; \(f:\, X\rightarrow Y\). Let \(A\) be a subspace of \(X\). Is it true that, \[f(A')=[f(A)]'\]...

So I have been asked to prove a result that is supposedly valid for any norm on any vector space. The statement to prove is: | ||x|| - ||y|| | <= ||x - y|| The problem is, I have no idea where to start with this proof. Maybe I'm missing some fundamental property of norms, but it seems that...

Homework Statement Air has a temperature of 227 Celsius and a pressure of 1000 kPa. Determine the internal energy, quality, enthalpy, and specific volume if it exists. Homework Equations A state is defined by two properties. All other properties can be derived from two independent...

Has anyone seen this before? Is this true? $$ \begin{vmatrix} a & b+c & 1\\ b & a+c & 1\\ c & a+b & 1 \end{vmatrix} = \begin{vmatrix} a & b & 1\\ b & a & 1\\ c & a & 1 \end{vmatrix} + \begin{vmatrix} a & c & 1\\ b & c & 1\\ c & b & 1 \end{vmatrix} $$ In this example this works but I don't know...