Definition Definition and 1000 Threads

Questions; For the three definitions quoted in the below Background section, I would like to know what the subtle differences are. I know they all have to do with extracting diagonal sequence from a bunch of sequences and its convergence. Especially with Definition 2., why it seems to be...

The following are taken from Lectures on Algebra by Shreeram Shankar Abhyankar Commutative Algebra Volume 2 By Oscar Zariski, Pierre Samuel Abstract Algebra An Introduction by Thomas Hungerford Concepts in Abstract Algebra by Charles Lanski Background Questions The passages quoted...

The following are taken from Locally Compact Groups by Markus Stroppel Commutative Algebra Volume 1 By Oscar Zariski, Pierre Samuel Introduction to the Algebraic Geometry and Algebraic Groups Volume 39 by Michel Demazure Basic Algebra by Anthony Knapp General Topology by J. Dixmier...

I have found the following proof of remainder term for midpoint rule of integration: and I'm trying to understand the part of it where author is applying MVTI to composition of functions ##f''(\xi_i(x))## and move it out of integral sign as ##f''(\xi_i)##. If we solve Taylor's series for this...

Pretty much everywhere secondary electron has energy of 50 eV or less. I would like to know bit detail definition of secondary electron in the SEM. Why there is no SE with higher energy even beam energy is quite high like 10KeV and more. What is the difference between secondary electron and...

Please provide solution

Cobbling together a definition of the infinitesimal from bits and pieces of info gathered from books and the internet: The infinitesimal ##d## is the positive real number greater than ##0## but less than any other positive real number. My problem is how to express the above in logical...

I am learning analysis from Rudin's famous book (baby rudin). I am confused about how ##\mathbb{R}## is defined in this book. In the appendix of chapter 1, he says that members of ##\mathbb{R}## will be certain subsets of ##\mathbb{Q}##, called cuts. Is this definition different from the way we...

Here's the inverse function theorem as stated in Spivak's book: Then there's a paragraph in Folland's book: I have read the inverse function theorem and its proof in Spivak's Calculus on Manifolds and I have a hard time reconciling it with what Folland states in his book on the chapter on...

Why do we need second countable and Hausdorff conditions for manifold definition?

This is something that's bugged me for a long time and since I don't have any particular expertise in the subject, maybe I'm missing something obvious. But here goes. I was taught that the term particle in QM is synonymous with quanta. That is, that a particle is the lowest excitation level of a...

Here's my definition I've been working on. Comments? Suggestions for improvements? EDIT: The reason I'm looking for a sequential characterization of right continuous is because the way you check that ##F## is right continuous is through...

Is it correct to define Riemann integrability as follows: 'For any ϵ>0, there exists a δ>0 such that if the maximum interval length of the partition is less than δ, then the difference between the upper and lower Riemann sums is less than or equal to ϵ'? I wanted to define Riemann integrability...

Case one: ns is half-filled and (n-1)d is less than full. Is valence 1+number of electrons in (n-1)d or number of electrons in (n-1)d or 1? Case two: ns is fully filled and (n-1)d is less than full Is valence 2 or 2+ number of electrons in (n-1)d or number of electrons in (n-1)d? Case three...

Can someone please explain me the rationale for the terms circled in red on the attached copy of page 400 of "Riley, Hobson, Bence - Mathematical Methods for Physics and Engineering, 3rd edition"? Thank you. Mentor Note: approved - it is only a single book page, so no copyright issue.

Ha. Half-assed is usually an adjective or adverb. From now on I'm skipping past the half-assed efforts of Oxford Languages. They must be paying someone to uprate their stuff.

I thought vile meant "very unpalatable," as in a vile concoction. But the dictionary says the meaning is "wicked." Oh well. Wrong again. I guess it is close to villain, maybe that should have been a clue.

Hello! I would like to be sure about my understanding of the definition provided in screenshot below 1. What is this ##\mathcal{E}(G,N)##? I know that not all extension are isomorphic so I wonder What are the elements of ##\mathcal{E}## groups? Or maybe all Es diffeomorphic to each other...

For this problem, The solution is, However, does someone please know why we allowed to assume that the derivative exists for f i.e ##f'(a) = \lim_{x \to a} \frac{f(x) - f(a)}{x - a}##? Thanks!

The definition is, I rewrite it as $$(L[y_1] = L[y_2] = 0) \rightarrow (L[c_1y_1 + c_2y_2] = 0)$$. However, I also wonder, whether it could also be rewritten as, $$(L[c_1y_1 + c_2y_2] = 0) \rightarrow (L[y_1] = L[y_2] = 0) $$ And thus, combining, the two cases, Principle of superposition...

For this problem, My solution: Using definition of Supremum, (a) ##M ≥ s## for all s (b) ## K ≥ s## for all s implying ##K ≥ M## ##M ≥ s## ##M + \epsilon ≥ s + \epsilon## ##K ≥ s + \epsilon## (Defintion of upper bound) ##K ≥ M ≥ s + \epsilon## (b) in definition of Supremum ##M ≥ s +...

On page 3 of the lecture notes for Stochastic Analysis, it says '##B(s,t)## is the covariance function ##\mathbb{E}[X_sX_t]-\mathbb{E}[X_s]\mathbb{E}[X_t]##. Then On page 5, it says the notes also say that 'the covariance function ##B(s,t)## of a strongly stationary stochastic process is...

Just looking around for now, but would love to see links to attempts to define information in physics

I have read lots but still, there're some really unproductive explanations of dirac delta function. So hopefully, you can explain it by following my arguments and not formal definition because I've read it all. It's shown to be as ##\delta (x) = 0## when ##x \neq 0## and ##\delta (x) =...

My doubt arises over the definition of L'(v^2). If we are using ##x= v'^2##, shouldn't the derivative be made with respect to that very term? In essence, shouldn't it be: L'(v^2) = \frac{\partial L(v^2)}{\partial (v'^2)}? In the article I read, L'(v^2) = \frac{\partial L(v^2)}{\partial (v^2)} is...

I look on theh definition of "look on" (somebody) and I get: to consider or think of someone or something in a stated way In the link: https://dictionary.cambridge.org/dictionary/english/look-on-upon What is stated way? i have 2 definition: 1)fixed, settled and 2) said, expressed verbally or in...

I look in the Dictionary and his definitions is: Avoke = To call from or back again I don't understand nothing. Can someone explain to me the definition (i.e. in other words). Thanks in Advance.

Math qyuestion for AI (Skype) include an expression that x is resting on y (both straight line segments). AI insisted that x coincides with y, while my intent was only placing x on y. Does 'rest' have such a narrow definition?

I have been trying to understand this proof from the book 'Introduction to classical mechanics' by David Morin. This proof comes up in the first chapter of statics and is a proof for the definition of torque. I don't understand why the assumption taken in the beginning of the proof is...

I think definition (a) is not correct since the center of charge distribution rather than mass distribution is important here. The correct definition is the one given in (b). I am thinking that a distribution of charge will have a center of charge ##(x_c,y_c,z_c)## for -ve charges according to...

I use macro definition to model the results can be viewed in vised but vised does not display all the big guy know? Is there something wrong with my modeling?

C 1 1 -19.35 -7 1 -2 2 1 -19.35 -8 3 -4 3 1 -19.35 -5:-6 4 0 5 6 #1 #2 C 1 RCC 0 -10 0 0 -10 10 2 2 RCC 0 -10 0 0 -10 10 5 3 RCC 0 10 0 0 10 10 2 4 RCC 0 10 0 0 10 10 5 5 RPP 2 5 -10 10 0 10 6 RPP -3 0 -15 15 0 10 7 RPP 0 5 -15 -10 0 10 8 RPP 0 5 10 15 0 10 C M1 074184 1

So far, I have got the equations, ##u \cdot (\vec u \times \vec v) = 0## ##u_1a + u_2b + u_3c = 0## ##v_1a + v_2b + v_3c = 0## Could some please give me some guidance? Many thanks!

Hi all I am a little bit confused about the definition of angular frequency in the context of nuclear rotation, some times its defined in the regular way as $$ E=\hbar \omega $$ and other time from the rigid rotor formula $$ E=\frac{\hbar^{2}}{2I} J(J+1) $$ where ##I## is the moment of inertia...

I am seeing conflicting definitions of degree of freedom in my textbook. If I look at the definition given as per screenshot below then it is the number of independent terms/variables/coordinates used to define the energy of a molecule. But, if I look at the statement of Equipartition of energy...

Hi everyone, In MCNP manual there are often examples of Listing containing examples of tallies which have, in the definition of the cells/surfaces of the tally itself, the "<" symbol. I could not find in the document any reference to the use of logical expression in the definition of tallies...

Question: In defining adjacent transpositions in a permutation as swaps between neighbors, is one referring to the original set or to the last result before the transposition is applied? I clarify with an example. Suppose one assumes a beginning ordered set of <1,2,3> It is clear that (1,2)...

I am searching for a definition of “information” in the concept of physics and the theory that information is conserved. Rather than a general question, here are a couple of specific questions about information. Not limiting, just two possible examples of information. The magnetic state of...

Given a function ##f##, interval ##[a,b]##, and its tagged partition ##\dot P##. The Riemann Sum is defined over ##\dot P## is as follows: $$ S (f, \dot P) = \sum f(t_i) (x_k - x_{k-1})$$ A function is integrable on ##[a,b]##, if for every ##\varepsilon \gt 0##, there exists a...

The definition of the Wilson action relating to discrete Yang-Mills model is: $$ S_{plaq} (\sigma) := \frac{1}{2}\sum_{plaq}\|I_N - \sigma_p\|^2 $$ (from [here] at 5:55) It is mentioned that ##\sigma_p## is some kind of a matrix. Could anyone give an explicit example of what a ##\sigma_p##...

hello, I took an introductory course about statistics, we viewed the naive definition of probability which says "it requires equally likely outcomes and can't handle an infinite sample space ", I understood that it requires finite sample space but I didn't understand "equally likely outcomes "...

Hello everyone, Concerning the separation axioms in topology. Our topology professor introduced the equivalent definition for a topological space to be a ##T_{o}-space## as: $$ (X,\tau)\ is\ a\ T_{o}-space\ iff\ \forall\ x\ \in X,\ \{x\}^{\prime}\ is\ a\ union\ of\ closed\ sets. $$ The direction...

Question: There is a function ##f##, it is given that for every monotonic sequence ##(x_n) \to x_0##, where ##x_n, x_0 \in dom(f)##, implies ##f(x_n) \to f(x_0)##. Prove that ##f## is continuous at ##x_0## Proof: Assume that ##f## is discontinuous at ##x_0##. That means for any sequence...

I don't want to post this in a math forum because it's very basic and I just want a straightforward answer, not something math heavy . What's the definition of angle in a cuved space embedded in a higher eucledian space? Like when I have a spherical surface in 3d eucledian space and want to work...

I'm reading Tipler's 1976 paper, "Causality Violation in Asymptotically Flat Spacetimes" and he keeps using a symbol which seems to resemble the symbol for Future Null Infinity in a strange font, but it's usage doesn't make sense with what I would expect if that's what the symbol meant. He...

Wikipedia article on proper time "Given this differential expression for ##\tau##, the proper time interval is defined as ## \Delta \tau=\int_P d \tau=\int \frac{d s}{c} . ## Here ##P## is the worldline from some initial event to some final event with the ordering of the events fixed by the...

How do we define tangent line to curve accurately ? I cannot say it is a straight line who intersect the curve in one point because if we draw y = x^2 & make any vertical line, it will intersect the curve and still not the tangent we know. Moreover, tangent line may intersect the curve at other...

In texts on General Relativity, the proper time ##d\tau^2 = -ds^2## (with an appropriate choice of metric signature) is commonly said that the time measured by a timelike observer traveling along a path is given by the integral of ##d\tau## along this path. Of course it's possible to construct a...

I have always had trouble with formulas. Now the trouble is about a definiton. I would like to ask you how accurate the definition of heat is. It does not seem to me completely accurate. I think it is partly accurate. From the Fundamental of Engineering Thermodynamics by Sonntag\Borgnakke...

Hi Pfs i found a paper: https://arxiv.org/abs/math-ph/0306059 in which the author gives a definition of spin networks it is at the bottom of page 5 the words node or vertex does not appear. what do you think of it? Does the author think that all the information is in the hilbert spaces on which...