Property Definition and 635 Threads

the basic property of light is that ,it travels in a straight line but what causes it to travel in a straight line .in refraction of light ,what causes the refracted ray to bend towards the normal ,i am aware that it is related to velocity n material of medium but how does it influence the angle...

Two particles cannot be entangled in respect to position and momentum, I've read. But can particles be entangled by all other properties, including either position or momentum? For example by energy, momentum, spin and polarization (+some other(s)?) between two photons. I've read in some other...

Homework Statement understanding the following equation: ##(n+1)^n(n+1)+1=n(n+1)^n+(n+1)^n+1##. I don't know from where the first ##n## on the RHS comes and how this is related to a single addition of ##(n+1)^n##. Homework Equations ##a(b+c)=ab+ac## The Attempt at a Solution My guess is...

This page (https://shiyuzhao.wordpress.com/2011/06/08/rotation-matrix-angle-axis-angular-velocity/), gives the following relation: \left[R\vec{\omega}\right]_{\times}=R\left[\vec{\omega}\right]_{\times}R^{T} Where: * ##R## is a DCM (Direction Cosine Matrix) * ##\vec{v}## is the angular...

Does the property that two bodies connected together and if one of them is accelerating with some acceleration the second also accelerates with the same acceleration, have a name? Like for example, if there is a person in an elevator and the elevator is accelerating with acceleration ##a##, then...

I am reading Paolo Aluffi's book, Algebra: Chapter 0. I am currently focused on Chapter II, Section 3: The Category Grp. I need some help in getting started on Problem 3.3 in this section. Problem 3.3 at the end of Chapter III, Section 3 reads as...

what is the difference between property and characteristic of a material?

Hello! (Wave) The following algorithm is given: Algorithm FractionalKnapsack(S,W): Input: Set S of items, such that each item i∈S has a positive benefit b_i and a positive weight w_i; positive maximum total weight W Output: Amount x_i of each item i ∈ S that maximizes the...

I am asked: Prove that each of the following is an ideal of $\mathcal{F}(\mathcal{R})$: a. The set of all f such that f(x)=0 for every rational x b. The set of all f such that f(0)=0 My question is how do I know what the multiplicative operation is within the ring? Is multiplication the...

Let $f$ be a solution of the following equation $y''+p(x)y=0$, $p$ is continuous on $\mathbb{R}$ such that $p(x)\leq 0$ for all $x\in\mathbb{R}$. Suppose that $f$ is defined on $[a,+\infty)$, $f(a)>0$, $f'(a)>0$, $a\in\mathbb{R}$ . Prove $f(x)>0$ for all $x\in[a,\infty)$. Any help would be...

The speed of light is a parameter that attaches itself to what exactly, an inertial frame of reference or a massless particle moving therein?IH

Hello everyone. At first, I appreciate your click this page. I have a book named 'A first Course in Abstract Algebra 7th' by Fraleigh. I have a question about 'relation algebraic property with conjugate' in automorhisms of fields. in page415, this book explains "Let E is algebraic extension...

I was watching a video on youtube with a theory of time, the video explains 'Time' as a physical process supported by mathematics) I want to know what you think about this? Pseudoscience, or have any validity? "Could the future be an emergent interactive property with 'time' formed by the...

hello guys, i just want to know if space had been there before the big bang, or is it a property of the big bang :)

Homework Statement -\left ( \frac{\partial U}{\partial V} \right )_{S, N} is a definition of an imporant thermodynamic property,where S denote the entropy and the subscript 0 denotes reference state, so they must be constant. show what is this property. In your analysis, use the equation...

Considering the interval [0,1], say for each number (binary) on the interval you form the sequence of numbers: number of zeros up to the nth place/number of ones up to the nth place. Then as n goes to infinity the sequence of numbers (for the given binary number) will go to somewhere in...

Point $H$ is the orthocenter of $\triangle ABC$ prove :$HA^2+BC^2=HB^2+AC^2=HC^2+AB^2$

In physics, it seems like everything is ultimately reduced to a property. I used to believe that everything reduces down to matter and energy. But it seems as though matter and energy are made up of properties. For example, pure energy such as a photon, seems to be its parts/properties. Its...

The integers $x$ and $y$ have the property that for every non-negative integer $n$, the number $2^nx+y$ is a perfect square. Show that $x=0$.

Homework Statement Let u, and v be vectors in Rn, and let c be a scalar. c(u+v)=cu+cv The Attempt at a Solution Proof: Let u, v ERn, that is u=(ui)ni=1, and v=(vi)ni=1. Therefore c(ui+vi)ni=1 At this point can I distribute the "c" into the parenthesis? For example: =(cui+cvi)ni=1...

Homework Statement I am given f(t) = e^-|t| and I found that F(w) = ##\sqrt{\frac{2}{\pi}}\frac{1}{w^2 + 1}## The question says to use the nth derivative property of the Fourier transform to find the Fourier transform of sgn(t)f(t), and gives a hint: "take the derivative of e^-|t|" I also...

First of all, apologies as I've asked this question before a while ago, but I never felt the issue got resolved on that thread. Is it valid to prove that \int_{a}^{c}f(x)dx=\int_{a}^{b}f(x)dx+\int_{b}^{c}f(x)dx using the fundamental theorem of calculus (FTC)?! That is, would it be valid to do...

Hello; I'm reading "principles of quantum mechanics" by R.Shankar. I reached a theorem talking about Hermitian operators. The theorem says: " To every Hermetian operator Ω,there exist( at least) a basis consisting of its orthonormal eigenvectors.Its diagonal in this eigenbasis and has its...

(All vector spaces are over a fixed field $F$). Universal Property of Tensor Product. Given two finite dimensional vector spaces $V$ and $W$, the tensor product of $V$ and $W$ is a vector space $V\otimes W$, along with a multilinear map $\pi:V\times W\to V\otimes W$ such that whenever there is...

I've been reading Wald's book on general relativity and in one of the questions at the end of chapter 2 he gives a hint which says to make use the following integral identity (for a smooth function in): F(x)=F(a)+\int_{0}^{1}F'(t(x-a)+a)dt Is this result true simply because...

A simple question: If we have $$z$$ is a complex function, and we have here $$\omega_\mu^{ij}$$ represents some spin connection where $$\mu$$ is spacetime corrdinate. And say we have $$z + \omega_\mu^{12}$$ no matter for now what the metric is, if I want to take the conjugate of this, is the...

Hi! (Mmm) I want to write a function that takes as argument a pointer A to the root of a binary tree that simulates a (not necessarily binary) ordered tree. We consider that each node of the tree saves apart from the necessary pointers LC and RS, an integer number. The function should...

In Dirac's book on GRT, top of page 17, he has this: (I'll use letters instead of Greeks) gcdgac(dva/ds) becomes (dvd/ds) I seems to me that that only works if the metric matrix is diagonal. (1) Is that correct? (2) If so, that doesn't seem to be a legitimate limitation on the property of...

Hi! (Nerd) I want to prove for any cardinal numbers $m,n,p$ it holds that: $$m \cdot (n+p)=m \cdot n+m \cdot p$$ Could we prove this using induction on m ? Or could we maybe show that $A \times (B \cup C)=(A \times B) \cup (A \times C)$ where $card(A)=m, card(B)=n, card(C)=p$ ? (Thinking)

1+3+5+...+(2n-1)=∑(2k-1) but (2k-1)=k2-(k-1)2 summing we use the telescoping property and deduce that ∑(2k-1)=n2-02=n2 This seems accurate to me. Now my question is this a proper use of the telescoping property. In the least it reveals the proper answer, which can then be proved by induction.

I am reading Rudin's proof of this property, but I find one assertion he makes quite disagreeable to my understanding; I am hoping that someone could expound on this assertion. Here is the statement and proof of the archimedean property: (a) If ##x \in R##, ##y \in R##, and ##x > 0##, then...

Is the fact that all manifolds are hausdorff spaces a part of the definition, or can this be proven from the fact that it is a set which is locally isomorphic to open subsets of a hausdorff space? P.S. if it can be proven I don't want to know the proof, I want to keep working on it, I just...

Homework Statement Let ##R## be an ordered relation on a set ##A## that is reflexive and anti-symmetric. If there is a chain of elements in ##R## that begins and ends with the same element, say the element ##x \in A## is it true that all the elements of ##R## sandwiched in between the ones...

Homework Statement I could prove a, trying b now. Homework Equations The definition of the cross prod.? The Attempt at a Solution https://www.dropbox.com/s/0sauaexkl4j2yko/proof_cross_prod.pdf?dl=0 I did not manage to get a scalar times v and a scalar times w. (No need to point this...

Researchers from the University of Manchester were surprised to find that positively charged hydrogen atoms - protons - can pass through it http://www.independent.co.uk/news/science/scientists-predict-green-energy-revolution-after-incredible-new-graphene-discoveries-9885425.html Does that mean...

Hi! (Smirk) It is a given binary tree $T$, for each node $ n$ of which , all the keys of the nodes of the left subtree of $n$ are greater than the key of $n$ and all the keys of the nodes of the right subtree of $n$ are smaller than the key of $n$. We suppose that $T$ contains the nodes...

Homework Statement Prove if an ordered set A has the least upper bound property, then it has the greatest lower bound property. Homework Equations Definition of the least upper bound property and greatest lower bound property, set theory. The Attempt at a Solution Ok, I think that my main...

Dear everyone, I have a question about a property of square root. $${\frac{1}{x}\sqrt{x^2}}$$$\implies$ $\sqrt{\frac{x^2}{x^2}}$=$\left| 1 \right|$ Is that property of a square root? Since $$\sqrt{x^2}$$= $\left| x \right|$.

Weird question, but does anyone have any feelings on whether parity can be classified as a kinematic property? It doesn't scale with energy and so in that sense doesn't seem to be classifiable as a dynamic property, nor do objects interact through it; but parity is of course violated by the...

Hi all. My task is to prove the property of covariance function: ##(r(n)-r(m))^2≤2r(0)(r(0-r(n-m)))## My solution: ##1) (r(n)-r(m))^2=r(n)^2-2r(n)r(m)+r(m)^2## ##2) 2r(0)(r(0)-r(n-m)))=2r(0)^2-2r(0)r(n-m)## From covariance function properties I know that ##2r(0)^2≥r(n)^2+r(m)^2## So now I...

Hello I don't quiet understand how the integration in the picture works... I must have forgotten something... Can anyone explain what is used?

I just want to confirm a statement the - Light travels in the form of electromagnetic waves in open space, not particles, but converts to a particle while encountering an obstacle deserting its wave form. So is the statement correct or not? And does it persists any anomaly or exception while...

Show that $(\forall x\in \mathbb{R})(\exists p\in \mathbb{Z}):\, p\le x\le p+1.$Hello :). The Hint is use the Axiom of Archimedes and the Principle of Well Order

Homework Statement A generalized helix is a space curve whose unit tangent makes ##T## makes a constant angle ##\theta## with the a fixed unit vector ##A## in Euclidean space, I.e ##T \cdot A = \cos \theta = \text{const}##. Prove that if the torsion ##\tau \neq 0## everywhere then the space...

Hi, I have some troubles understanding the definition of the Markov property in the general case since I'm struggling with conditional expectations. Let $(X(t), t \in T)$ be a stochastic process on a filtered probability space $(\Omega, \mathcal{F}, \mathcal{P})$ with adapted filtration...

Is this statement true: "If two meromorphic functions have the same poles(all simple) and the same zeros(all simple), than they are proportional."? If it is true, than why? Thanks for the help...

Hey again! (Nerd) According to my notes, the Kuratowski definition for the ordered pair is the following: Let $a,b$ sets. We define the ordered pair of $a,b$ like that: $$<a,b>=\{ \{a \}, \{ a, b \} \}$$ So, when $x \in <a,b>$, which property does it satisfy? (Thinking)

Sorry, I wasn't sure of the best way to phrase this. This is a common problem I keep having. Here's the definition of a norm: Let E be a vector space V defined over a field F. A norm on V is a function p: V \rightarrow \mathbb{R} such that: \forall a \in F and \forall u,b \in V: (i) p(av)...

Hello, While studying dual vectors in general relativity, it was written as we all know that dual vectors (under Lorentz Transformation) transform as follows: \tilde{u}_{a} = \Lambda^{b}_{a}μ_{b} where \Lambda^{b}_{a}= η_{ac}L^{c}_{d}η^{db} I was wondering if one can prove the latter...

I show that the assoc. property applies to addition and multiplication in my book: (a+b)+c = a+(b+c) (ab)c = a(bc) But what about subtraction and division?