Definitions Definition and 274 Threads

Dear PF Forum, Perhaps it's not a question, just a light discussion. Time From this, we get the standard time https://en.wikipedia.org/wiki/Caesium_standard, it's 9,192,631,770 ticks per second. I think this number should be discreet. No need to tune it to some figures after decimal point. Is...

I am reading Dummit and Foote's book "Abstract Algebra" (3rd Edition) and am focused on Chapter 15: Commutative Rings and Algebraic Geometry ... ... On page 657 D&F give a definition of a k-algebra ... as follows: I have to say I do not find that this definition gives me a good intuitive idea...

I am reading Dummit and Foote's book "Abstract Algebra" (3rd Edition) and am focused on Chapter 15: Commutative Rings and Algebraic Geometry ... ... On page 657 D&F give a definition of a k-algebra ... as follows:I have to say I do not find that this definition gives me a good intuitive idea...

Hi everybody! I'm currently studying integrals, and I would like to clarify a few definitions, especially about the criterions of convergence/divergence of an integral. Basically if that's okay for you guys I'm going to list and number a few statements and I'd like to know if they are true or...

Hi. I am reading a paper about gaussian beams and the author says that gaussian beams have simultaneously minimal divergence and minimal transversal extension. In order to prove it, the author states that \mathrm{divergenece} \propto \int_{-\infty}^{+\infty} \frac{d\,k_{x}}{2\pi}...

I' m studing the hodge helmotz decomposition of a flow Field, and i have Found different definitions. I'm Not sure to have assigned the rigth meaning to the terms of the decomposition. Look At The picture( i don't write here cose there are several equations).

Definition 1: The expectation value of the observable related to the parity operator ##\hat{P}## is constant over time. That is, \frac{d}{dt}\langle P\rangle=0 \int\Psi^*(r)\ \hat{P}\ \Psi(r)\ dr=constant \begin{align}\int\Psi^*(r)\ \Psi(-r)\ dr=constant\end{align} Definition 2: If the...

When I read any physics articles online I always end up receiving the wrong message. especially in terms such as 'observe'. what does this term mean when its used to describe exeperiments such as the double slit

Just want to see if I actually understand what these all mean. Partition: is like the x-coordinate values, also gives the number of times the graph was chopped up. We need them in order to find the distance or length of each rectangle. The distance is found by taking the further point minus...

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I teach both physics and math at a community college, and I've volunteered to give a short talk for students at our weekly math colloquium that has to do with curvature and non-curvature singularities in relativity. This is a tall order, given that I can't even assume that all the students will...

I am trying to get a full understanding of the notion of an algebra ... I have thus consulted two books - Cohn: "Introduction to Ring Theory and Dummit and Foote: Abstract Algebra. Cohn defines a k-algebra (or linear algebra) as follows: If we want to amend the above definition to an R-Algebra...

I was going to ask a question about whether or not pi was constant or changed with curved space. I found the answer on here that it does indeed change. Then I started thinking about the ramifications of that. sine waves are dependent on pi, so they should change too. Does sin(theta) =...

Is the claim that all definitions are biconditionals, true? Is the converse statement true as well, that all biconditionals are defintions?

So I have a friend who wants to become an engineer who is overtly obsessed with his mathematical foundations at the moment. He has confessed recently that he didn't understand the definition of the derivative, and asked me to elaborate. And so I did. However, what he asked next kind of...

First off: I am not complaining about any of the members here or elsewhere. I do not post questions expecting help. When I post a question I would like an answer but I do not require it. Getting an answer depends on who is around and who is willing to help when I post the question. Still...

I know there are three definitions of humidity. Absolute , relative and specific. Can you tell me if on a typical commercial device (such as the one i have in a clock i have) that measures humidity as percentage is it the relative humidity that it measures? Can the percentage become greater...

Hello, I am self-studying an introductory mechanics textbook and while I feel I understand the material there pretty well, I came across across an online definition of position which seems at odds with the explanations and definitions in my book. The definition that confused me is: ""Position...

First, I am not sure that this is the right place for this post, because earlier Physics Forums broke up the "computing and technology" into several subheadings, such as information science as opposed to hardware, and so forth, that I don't see anymore. Anyway, so here is an elementary question...

1. With respect to any oscillating system, what is the difference between ΔPE, PE, and PEaverage?Homework Equations ---- The Attempt at a Solution Hi all. I want to preface this by saying that we have been discussing the Lennard-Jones potential and particle theory in class recently, after...

Dear friends,I know the definition, from A.N. Kolmogorov and S.V. Fomin's Элементы теории функций и функционального анализа, of Lebesgue integral of measurable function ##f:X\to \mathbb{C}## on ##X,\mu(X)<\infty## as the limit ##\int_X...

Hi everyone, I am in second year university and am taking linear algebra this semester. Never having been a strong maths student, I am certainly struggling with some basic concepts and especially notation. I have tried searching on the web but have had difficulty in finding something which...

I am reading "Introduction to Ring Theory" by P. M. Cohn (Springer Undergraduate Mathematics Series) In Cohn's book, in Chapter 2: Linear Algebras and Artinian Rings, we find the definition of an algebra ... ... but in Jonathan Golan's book ["The Linear Algebra a Beginning Graduate Student...

I am reading "Introduction to Ring Theory" by P. M. Cohn (Springer Undergraduate Mathematics Series) In Chapter 2: Linear Algebras and Artinian Rings we find the definition of an algebra ... ... but in the chapter on module theory on page 342 of Dummit and Foote we find a different definition...

During my studies I failed to understand thermonynamics and compared the whole subject to black magic. This frustrated me a lot. Years later I tried to restudy it by myself reading sources with less conventional approaches. I had finally come to believe I could make sense of it. But re-reading...

There are two "formal" definitions of vectors (and tensors in general) which I've learned. The first is what I consider the "better" definition, one I learned in linear algebra. We call a set X a vector space over a field F whenever that set has properly defined operations of scalar...

Hey guys, Here's another quick question this time from a problem set I'm having trouble with at the moment. Question: So, for a, I computed f(x) = x^5 This is because of the numberator's right side. If 2+h is raised to the 5, this must be the function. Moreover, a=2 because 2 is already on...

Hey guys, I have a couple more questions about this problem set I've been working on. I'm doubting some of my answers and I'd appreciate some help. Question: For 1a, I just took \lim_{{h}\to{0}} of the function using [ f(x+h)-f(x) / h ] and simplified. Ultimately, this gave me...

Homework Statement S(E,V) = kln(\Gamma(E) )\\ S(E,V) = kln(\omega(E) )\\ S(E,V) = kln(\Sigma(E) )\\ S entropy, k Boltzmann's constant. Prove these 3 are equivalent up to an additive constant. Homework Equations \Gamma(E) = \int_{E<H<E+\Delta}^{'}dpdq\\...

I think the first thing that is confusing me is the terminology. There are too many similar terms (e.g. probability measure, probability distribution, probability density function, probability mass function) What are the general concepts and what are the instances of those concepts? Like, are...

If work ##W = \Delta E = \int_{s} \vec{F} \cdot d\vec{s}##, so work can't be ##\frac{dW}{d\vec{s}} = \vec{F}## like is here: http://en.wikipedia.org/wiki/Work_%28physics%29#Path_dependence cause this implies that ##W = \int \vec{F} \cdot d\vec{s} = E##, but the work is the variation of...

Hi, this question seem to fall somewhere between Analysis and Algebra; I just choose this section; sorry if it is the wrong one. I would appreciate any suggestions, refs., etc. I'm basically trying to see if the different definitions of adjoint maps can be unified into a single...

I'm not sure if this is particularly important, but so far through my studies I've only encountered DE with two related variables (e.g. ## \frac {dy}{dx} = 3x##). Now, given another function with an additional variable that is UNRELATED to the two other variables, can this still be considered...

What's the original definition of Ohm, ampere, volt and coulomb, And which unit was defined according to the other ??!

In the wiki, I found this definition for the argument: http://en.wikipedia.org/wiki/List_of_trigonometric_identities#Exponential_definitions However, in other page of the wiki (http://en.wikipedia.org/wiki/Complex_conjugate#Use_as_a_variable), I found this definition for...

I want to know the difference between various kinds of series like Taylor, Laurent and Asymptotic. I have some understanding but I want some clarifications. Here is what I understand:- 1) Taylor series is just f(0) + x.f'(0) + x2.f''(0)/2! + ... 2) Laurent series is applied when taylor series...

Hi Usually when learning math, understanding the theorems and ideas helps tremendously to remember math. I get that... I got through calculus, linear algebra and complex analysis easily. The problem for me started with three branches in mathematics: Real analysis, measure theory and abstract...

I've seen open sets ##S## of a bigger set ##X## being defined as 1) for every ##x\in S## one can find an open disk ##D(x,\epsilon)## centered at ##x## of radius ##\epsilon## such that ##D## is entirely contained in ##S##. Where $$D(x,\epsilon)= \left\{y \in X: d(x,y) < \epsilon\right\}$$...

I'm learning algebra by myself and this concept is confusing me. Please excuse me if I define anything wrong... I've never expressed myself in this language before. Lets say we have a group G and a group G' and there exists a homomorphism R: G → G' and for any element g \in G, the...

What are considered fundamental definitions? For context, a question was posed where the prof. provided parametric equations for the motion of a satellite in orbit and said that we can use the provided equations, fundamental definitions, and no other equations to solve the problems. What does...

Can I check whether these are right? Here let's define the ADM mass as length of the ADM energy-momentum vector. In the Schwarzschild spacetime ADM energy = Schwarzschild mass parameter In a spacetime in which the ADM energy and the Komar mass are both defined ADM energy = Komar mass

I have an example given in the textbook that defines port A bits 5 as PA5 equ 0x40004080. Port A is 0x40004000, and the first 7 bits are for data. I do not see how bit 5 is 0x80. I figured that would be bit 7 and bits 5 would be 0x20. We use ARM assembly language for LM4F120H5QR microcontroller.

Hello, While riding home today, I started to wonder what was the definition of a second in the context of time. I looked it up and found out that the definition has changed somewhat over the years and it is now defined as (paraphrased) 9,192,631,770 periods of the cesium atom. A meter is...

I just need to know the basic ideas of topology, and the most important results, because I'll have differential geometry the next semester. Does anyone have a good material for this? Or you can just say what to search for and I'll search it. Thank you :)

Homework Statement \mathbf{D1:}\forall\varepsilon>0,\exists K\epsilon\mathbb{N},\forall n\epsilon\mathbb{N},n\geq K\Longrightarrow|x_{n}-x|<\varepsilon \mathbf{D2:}\forall\rho>0,\exists M\epsilon\mathbb{N},\forall n\epsilon\mathbb{N},n>M\Longrightarrow|x_{n}-x|\leq\rho Show these two...

Homework Statement e^{x}=\sum\limits_{k=0}^{\infty}\frac{x^{k}}{k!} and e^{x}=\lim\limits_{n\rightarrow\infty}\left(1+\frac{x}{n}\right)^{n}. I want to show that \sum\limits_{k=0}^{\infty}\frac{x^{k}}{k!}=\lim\limits_{n\rightarrow\infty}\left(1+\frac{x}{n}\right)^{n}. 2. The attempt at a...

Not sure if this is in the right section, but I'm not sure where else it would fit. I'm currently researching a variety of optics-based topics, and I'm a bit confused by what appear to be some conflicting definitions of transmittance. I've seen the following: 1) It's the ratio of monochromatic...

My high school physics textbook gives much emphasis on memorizing definitions.We have to memorize the definition of force, inertia, etc...We have to write the definitions as it is in the textbooks... For example, the definition of "force" in physics textbook is, "Force is defined as that...

Hi all! In a paper they say that a certain quantity is a rank-2 tensor because it transforms like a spin-2 object under rotations, that is: if the basis vectors undergo a rotation of angle \phi, then this quantity, say A, transforms like A\mapsto Ae^{i2\phi} As far as I knew, a rank-2...

Hello all, In the equation V= apd/Ln(pd)+b as described by Paschens law regarding breakdown voltage, I was wondering what the "L" and the "n" stood for. Example of what I'm looking for: p stands for pressure. Thanks.