Definitions Definition and 263 Threads

Below in the quoted passage is expressing the definition of group action in terms of the definition of a group and vice versa. Can someone check if it there any mistakes please. I had helped with one of the LLMs. I always want to know for the two definitions, if one can be written in terms of...

Hello, I've been doing Calculus for a few days now and I'm beginning to wonder whether or not I should be memorizing definitions. By memorizing definitions I mean memorizing them by heart (word for word). This type of memorization hasn't worked very well for me in the past. At this very moment...

Hi, I was reading the interesting lecture of Feynman about Characteristics of Force -- https://www.feynmanlectures.caltech.edu/I_12.html He basically says that nominal definitions like mathematical definitions of "abstract" objects have actually no physical meaning. For instance take the...

Is there a word for when when the purpose of some activity is forgotten, and the activity is confused as an end in itself. The means becomes the end? Examples: You need a ladder, so you cut down a tree to make one. Then remember that the reason you wanted the ladder was to climb the tree...

Obviously, we know intuitively what they mean, but it seems that physicists have developed an objective definition for all of these. If I were to guess, I'd say that: - a gas is vastly less compressible than a liquid or solid (i.e., which are considered in thermodynamics as basically...

https://www.feynmanlectures.caltech.edu/I_10.html (From the paragraphs after equation 10.5) 'It is not just a definition to say the masses are equal when the velocities are equal, because to say the masses are equal is to imply the mathematical laws of equality, which in turn makes a prediction...

In the definition of entropy, there are two. One is about the degree of randomness and one is about energy that is not available to do work. What is the relationship between them?

I have read a bunch of articles online regarding my question, and none have helped. Newton's Laws: 1. In an inertial reference frame, an object's momentum doesn't change unless acted upon by a force. 2. In an inertial reference frame, the force on an object equals the time derivative of its...

I have read three definitions of electric flux in textbook which is confusing me.. 1. Electric flux is the number of electric lines passing through any area of a surface. 2. Electric flux is the number of electric lines passing through unit area per second held perpendicularly. 3. Electric flux...

Hi All, I would like to know how can one connect the two definitions of entropy ##\Delta S = \int_{T_i}^{T_f} \frac{dQ}{T} ## and ##\Delta S = k_B \ln (\frac{W_f}{W_i})##, particularly I am interested in how the logarithm emerges. Does it have to do with some linear dependence of the heat with...

Hi, I have seen two versions of definitions of material dissipation factor Df: The first one: Dk (dielectric constant) = K = relative permittivity = ɛ -jɛ’ , ɛ = energy stored and ɛ’ = energy lost. Df (dissipation factor/loss tangent) = ratio of ɛ’ and ɛ The second one: On slide 8 of webinar...

Hello. Questions: how to give new definitions of things in abstract spaces?What are the criteria? Is it just acceptable to define things that do not contradict with other things on the abstract space?What are the motivations to give definitions of things?Also, what are the motivations for...

My probability class has me wondering about pure math questions now. We started with the axioms and are slowly building up the theory. Everything was fine but then a definition of Conditional Probability P[A|B] = \frac{P[AB]}{P} appeared and it's just not sitting right with me. I know that...

The definitions of currently popular computer languages assume only the technology of printing. For example, they provide syntax for printed expressions in one location that refer to printed expressions in other locations (e.g. "include" type syntax ). However, they don't assume that such...

I've been looking into definitions for SI units (pre 2019) I'm fine with length based on a certain transition of a Krypton atom and the wavelength associated with it, a spectrometer can be used to find the wavelength of the spectral line and then multiply up... but with the ampere, the idea of...

Good Morning (I see this has been discussed here, but I am more interested in two specific definitions and whether the conflict.) I had always thought that the definition of an inertial frame was: "A frame in which Newton's Laws are valid." A person has been arguing with me that the definition...

Fred H. Croom (Principles of Topology) and Tej Bahadur Singh (Elements of Topology) define local basis (apparently) slightly differently ... Croom's definition reads as follows:... and Singh's definition reads as follows: The two definitions appear different ... ... Croom requires that each...

I am reading Wilson A. Sutherland's book: "Introduction to Metric & Topological Spaces" (Second Edition) ... I am currently focused on Chapter 8: Continuity in Topological Spaces; bases ... I need some help in order to prove Definition 8.1 is essentially equivalent to Definition 8.2 ... ...

Formal definition (epsilon-delta definition) of limit is symbolically as follows: $$\lim_{x \to c}f(x) = L \iff [\forall \epsilon > 0,\ \exists \delta > 0,\ \forall x \in I,\ (0 < |x - c| < \delta \implies |f(x) - L| < \epsilon)]$$ Now I want to create alternative definitions out of this by...

I am reviewing and comparing a wide range of mathematical models that are being applied to a specific realm of wildlife biology. For the comparison of these models, and to weigh advantages/disadvantages of different aspects with regard to application, I need to characterize each model. As I do...

What is opposite in Mathematics? I argued with my engineering brother and his pal that "inverse" is indeed "opposite" if you use the general concept of "opposite." My brother claims that the idea of opposite in mathematics only means returning to null. Opposite: 1.) Having a position on the...

The Definition of a Neighborhood and the Definition of an Open Set ... Carothers, Chapters 3 & 4 ... I am reading N. L. Carothers' book: "Real Analysis". ... ... I am focused on Chapter 3: Metrics and Norms and Chapter 4: Open Sets and Closed Sets ... ... I need help with an aspect of...

So, I was looking into Einstein's 1907 paper where he derived the specific heat of solids using quantum mechanics and I found that Einstein just took the derivative of Planck's equation from 1900 for the average energy, U, as a function of time (and multiplied it by 3N for the three dimensions)...

I'm trying to learn about Abstract Wiener Spaces and Gaussian Measures in a general context. For that I'm reading the paper Abstract Wiener Spaces by Leonard Gross, which seems to be where these things were first presented. Now, I'm having a hard time to grasp the idea/motivation behind the...

I'm troubled by what I think the 'community' considers them to be, but I'm not sure if I'm correct. It appears as though finite is thought to have both an end and a beginning, but is it true that infinite (infinity) is thought to only have no end? Is this accurate? If so, then it would seem like...

Hi Lets start off with the definition of diffeomorphism from Wolfram MathWorld: The issue is that I am learning about smooth manifolds, and in the books I've read, the map has to be smooth and have a smooth inverse. Also, the definition above doesn't say that it has to be bijective. However...

Homework Statement (a) State what is meant by a ‘distortionless’ and a ‘lossless’ transmission line.(b) A transmission line has the primary coefficients as given below. Determine the line’s secondary coefficients Zo, α and β at a frequency of 1 GHz. R = 2 Ω/m L = 8 nH/m G=0.5 mS/m C=0.23...

while I study in cont. mech. I found this equation and in the index he mention that "the difference between the volume supply and the production phi can be recognized only in relation to the constitutive definition of the material" what does it mean?

Are the following definitions correct? Work done as energy transferred The energy transferred when the forces between two objects interact Work done by a force Work done = force x distance moved in the direction of force applied (W=F*Δs) Work done by a gas Work done = pressure x change...

I was studying Group Theory on my own from a mathematics journal and got confused at some point where it defines Cartesian products, from binary one, say (A × B), to n-tuples one, say (A_1 × A_2 × ... × A_n). What confuses me when I tried to read it is that the definition made for infinite...

Math definition: integral of function within limits divided by difference of limits. QM definition: integral of complex conjugate of wave equation times function times wave equation within limits of minus to plus infinity.

So I'm studying electrostatics and I came across to two different definitions of potential difference/voltage (because we're in stationary regimes) and I'm having trouble understanding how the expressions are equivalent. They are for a voltage between point A and point B $$U=V_a - V_b...

The general definition for a sequence to diverge is the negation of what it means for a sequence to converge: ##\forall L\in\mathbb{R}~\exists\epsilon>0~\forall N\in\mathbb{N}~\exists n\ge N##, ##|a_n - L| \ge \epsilon##. How does this general definition of divergence relate to the definition of...

Hi I have always had an issue with understanding the definitions used in mathematics. I need examples before I can start using and reasoning with them. However, with tensor products, I have been completely stuck. Stillwell's Elements of Algebra was that made abstract algebra "click" for me...

Many books sometimes for example define energy as quantity and sometimes as property. Also the definition of energy is the ability to do work or the meter of the ability to do work ? we define for example force as a quantity or as some quality and then we quantify this ?

Homework Statement Define the interior A◦ and the closure A¯ of a subset of X. Show that x ∈ A◦ if and only if there exists ε > 0 such that B(x,ε) ⊂ A.The Attempt at a Solution [/B]

Hello all, I am an undergraduate student who is studying real analysis from Rudin's POMA and I am trying to prove that these two definitions that I have for dense sets are equivalent: 1) Given a metric space X and E ⊂ X ; E is dense in X iff every point of X is a limit point of E or E = X or...

Hi, from the books I have, it appears that some rules for operators, boundedness, positivity and possibly the definition of the spectrum regard real operators, and not complex operators. From the complex operator ##i\hbar d^3/dx^3 ## it appears that it can be defined as not bounded (unbounded)...

My question is regarding a few descriptions of Entropy. I'm actually unsure if my understanding of each version of entropy is correct, so I'm looking for a two birds in one stone answer of fixing my misunderstanding of each and then hopefully linking them together. 1) A measure of the tendency...

Feel free to test out the code. I use these nearly anytime I load up a new file in LaTex. They are very simple and easy to use versus typing out the full latex command. \usepackage{amsmath} \usepackage{amssymb} \usepackage{amsthm} \newtheorem{definition}{Definition}...

We have the 3 equivalent definition for solvable groups: There exists a chain of subgroups 1 < G1 ...< Gi + < G i+1 < Gr = G such that Gi is normal in Gi+1 and Gi+1/Gi is abelian. Another definition is there exists 1 < H1 ...< Hi + < H i+1 < Hs = H such that Hi is normal in Hi+1, and...

Hello! I want to make sure I understand these definitions (mainly the difference between them), so please let me know if what I am saying is correct. So a ##\textbf{homeomorphism}## between 2 topological spaces, means that the 2 can be continuously deformed from one to another, while keeping a...

Hi I am a person who always have had a hard time picking up new definitions. Once I do, the rest kinda falls into place. In the case of abstract algebra, Stillwell's Elements of Algebra saved me. However, in the case of Spivak's Calculus on Manifolds, I get demotivated when I get to concepts...

Hello everyone, I am taking my third term of physics right now, and we are talking about flux at the moment. This terms is supposed to be a lot about E&M. Though I find that our textbook (Physics for scientists and engineers a strategic approach 4/e) is very textbooky. I was wondering if there...

Hello, According to the Wikipedia article on random variables: If the above statement is true, then, instead of defining a (real) random variable as a function from a sample space of some probability space to the reals, could we equivalently define it as a subset of ℝ associated with a CDF?

I am reading D. J. H. Garling: "A Course in Mathematical Analysis: Volume I Foundations and Elementary Real Analysis ... ... and I am also referencing concepts in Derek Goldrei's book, "Classic Set Theory for Guided Independent Study" ... I am currently focused on Garling's Section 1.3...

I am reading D. J. H. Garling: "A Course in Mathematical Analysis: Volume I Foundations and Elementary Real Analysis ... ... and I am also referencing concepts in Derek Goldrei's book, "Classic Set Theory for Guided Independent Study" ... I am currently focused on Garling's Section 1.3...

In some elementary introductions to integration I have seen the Riemann integral defined in terms of the limit of the following sum $$\int_{a}^{b}f(x)dx:=\lim_{n\rightarrow\infty}\sum_{i=1}^{n}f(x^{\ast}_{i})\Delta x$$ where the interval ##[a,b]## has been partitioned such that...

What is electrical charge? is it a measure of the abundance of electrons within a system? Current is defined as the amount of charge flowing between any two points in a system over a given period of time, correct? but what is elctrical charge if possible to define it better/further. I am...

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...