Definition Definition and 1000 Threads

Let ##S = \lbrace a, b \rbrace## and define ##F_S## to be the free group, i.e. the set of reduced words of ##\lbrace a, b \rbrace## with the operation concatenation. We then have the universal mapping property: Let ##\phi : S \rightarrow F_S## defined as ##s \mapsto s## and suppose ##\theta : S...

Summary: What is meant by port CARP ASIC functionality to a field programmable Gate Array FPGA. And what is Acronym CARP? What is meant by port CARP ASIC functionality to a field programmable Gate Array FPGA. And what is Acronym CARP?

--- Say we have the dictionary in Anaconda 2.7 Jupyter :--- B={'A1':'Jan1' , 'A2':'Feb2',...} of Birthdates. ---I can call a name , e.g., 'A1' , using Birth[name]. As in :--- print('What is your name '\? ') name=input() print('Yes, Birth[name] is in our list') ---How do I call a Date? I want...

If 'ground state' is the lowest energy state, and H atom takes -13.6eV to ionise while H- takes -14.35eV, why is H- not considered the 'ground state'? Is that definition wrong? What is the definition of 'ground state' that "fixes" this?

Is the wikipedia definition of Fermi coordinates accurate? https://en.wikipedia.org/wiki/Fermi_coordinates

I need some help defining a tally volume. I want a volume bounded by three surfaces, but when I do an initial plot the volume is in red dashed lines. I know that each cell needs to be uniquely defined, but I am not seeing how my volume is not unique. The cell in question is cell 200 in the code...

Hi, today I stumbled upon a 2016 article in Scientific American about the (then) possibility of re-defining the kilogram through Planck's constant. The article is really a very quick review of the topic. At some point the author states the following "So for years, physicists have chased an...

Consider limx→3x^2=9. Find a maximum value of δ such that: |x2 - 9|<0.009 if |x-3|<δ I just learned how to do this today and I am quite comfortable doing this if the function is linear, however now I am struggling with working with quadratic functions. So far this is what I have come up with...

The best-known classes of nuclear transmutations are fission and fusion: Nuclear Fusion Under normal condition, nuclei do not stick together; because they repel each other at large distances (due to the electrostatic repulsion 'barrier') and thus the strong nuclear force cannot act. But if these...

Dear all, I'm having a small issue with the notion of Lie-derivatives after rereading Carroll's notes https://arxiv.org/abs/gr-qc/9712019 page 135 onward. The Lie derivative of a tensor T w.r.t. a vector field V is defined in eqn.(5.18) via a diffeomorphism ##\phi##. In this definition, both...

This is going to sound like a silly question, but here we go anyway! I've always thought about a definite integral being used for modelling a change in some quantity whilst an indefinite integral is employed to find the defining function of that quantity. For example, consider the...

As far as I understand, when we want to differentiate a vector field along the direction of another vector field, we need to define either further structure affine connection, or Lie derivative through flow. However, I don't understand why they are needed. If we want to differentiate ##Y## in...

The definition Let a, b be any two real numbers Let c > 0 We define a function f to be funny iff For all x, y belonging to [a,b], |f (x) - f (y)| ≤ c |x - y| Question Let a < b (arbitrarily) Let c > 0 Assume function g is funny on [a, b] Let x, y ∈ [a, b] Therefore, |g (x) - g (y)| ≤ c |x - y|...

The solar constant is only on one side of the Earth, however it moves around and the Sun gives out it's energy/heat all over the Earth in 24 hours. And the seas and land have heated up over time. However I got told this, No its not. By definition the solar constant is referring to a 2...

The wording of the definition of the meter has apparently changed recently. I'm wondering about the motivation for the change. The current definition is: The older definition (1983, I think) was Does anyone know when and why the change was made?

Dictionary says it is the opposite of clockwise. But if a clock is transparent and you look at it from behind, the same direction that was anticlockwise before is now clockwise. So it seems to depend on the point of view. But there are some electromagnetic equations that depend on clockwise...

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...

Not satisfied with the following definition of calculus. What is a better definition? More detailed? 1a : a method of computation or calculation in a special notation (as of logic or symbolic logic) b : the mathematical methods comprising differential and integral calculus —often used with the...

Hi All, Considering a set of many many small hard balls which start colliding inside a box. The velocities of these balls being mostly greater than c/2. Is it possible, in this case, to speak of convergence to a thermal state in the same sense of ordinary thermodynamics (i.e., using...

Hi, I'm a bit confused about the locally euclidean request involved in the definition of manifold (e.g. manifold ): every point in ##X## has an open neighbourhood homeomorphic to the Euclidean space ##E^n##. As far as I know the definition of homeomorphism requires to specify a topology for...

I have attempted to sketch the timing and got the following graph From reading the VHDL code my understanding is io each rising edge clock tick the system will carry out one of the following (choosing based on priority): reset if Clear_L is high load in a value if Clear_D is high UP count if...

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Zemansky defines Heat as : When a closed system whose surroundings are at a different temperature and on which diathermic work may be done undergoes a process, then the energy transferred by non mechanical means, equal to the difference between the change in internal energy and the diathermic...

In https://en.wikipedia.org/wiki/Recursively_enumerable_set, in the introductory section one reads "...a set S of natural numbers is called recursively enumerable... if: ...There is an algorithm that enumerates the members of S." and in a later section it says "According to the...

By the Wigner theorem, symmetries transformations are implemented by operators ##\hat{U}## that are unitary or antiunitary. This is what is written in most books. But I have read somewhere that, to ##\hat{U}## represent a symmetrie, it's necessary that ##\hat{U}^{\dagger} \hat{H} \hat{U} =...

In quantum mechanics, I can write the hamiltonian as ##\hat{H} = \hat{p}^{2}/2m + \hat{V}##. I am confusing with the definition of the operator ##\hat{V}##, who represents the potential energy. If the potential energy depend only on the position, is it correct write ##\hat{V} = V(\hat{x})##...

My understanding is that to define the entropy of a system what you have to do is as follows: Define the boundaries of your system. Define a set of "microstates" of the system. Define a partition of microstates of the system where each element of the partition is measurable and known as a...

In general, the textbooks says that, if the set ##G## is a group, so to every element ##g \in G## there is other element ##g^{-1} \in G## such that ##g g^{-1} = g^{-1}g = e##, where ##e## is the identity of the group. But I am reading a book where this propriete is write only as ##g^{-1} g =...

I don't get it - the definition of a chemical element is: "Cannot be chemically interconverted or broken down into simpler substances" -- But isn't every atom of an element made up of further sub-atomic particles, and atoms can be then broken apart, etc...? Also, how did people tell the...

In light of the modern definition of what is a coordinate system, namely it's a pair (U, f) with U a region of a m-dimensional manifold, and f a bijection from U to ##\mathbb R^m##, can we say that the polar coordinates on ##\mathbb R^2## are a coordinate system? I was thinking about this and...

I am reading Andrew McInerney's book: First Steps in Differential Geometry: Riemannian, Contact, Symplectic ... and I am focused on Chapter 3: Advanced Calculus ... and in particular on Section 3.1: The Derivative and Linear Approximation ... I am trying to fully understand Definition 3.1.1 and...

I am reading Andrew McInerney's book: First Steps in Diofferential Geometry: Riemannian, Contact, Symplectic ... and I am focused on Chapter 3: Advanced Calculus ... and in particular on Section 3.1: The Derivative and Linear Approximation ... I am trying to fully understand Definition 3.1.1...

I'm reading a book from the authors Halliday and Resnick and it says that The meter is the length of the path traveled by light in a vacuum during a time interval of 1/299 792 458 of a second. This time interval was chosen so that the speed of light c is exactly c = 299 792 458 m/s. I...

## \newcommand{\ihat}{\hat{\boldsymbol{\imath}}} \newcommand{\jhat}{\hat{\boldsymbol{\jmath}}} \newcommand{\khat}{\hat{\boldsymbol{k}}} ## Several times now I've seen the following technique for deriving the component form of the dot product. It always felt clean and simple until last night when...

Dear experts, I'm currently working my way through the paper Masanes, Galley, Müller, https://arxiv.org/abs/1811.11060. On page 3, they define what they call a bi-local measurement: If we have two systems a and b and we define an outcome probability function for some measurement f on system a...

Hi i have seen in abook the dot product defined as follows: Dot(A,B)=(1/4)[Norm(A+B)^2-Norm(A-B)^2] how this definition connect with the common one: Dot(A,B)=Sum(ai*bi) Thanks!

Homework Statement Problem is part of a double integral. but my boundries are: 1<=x^2 + y^2 <=9 so between 2 circles with r1=1 and r2=3 and x<=y and y<=sqrt(3x) the first boundry is obviously pi/4 and/or 3pi/4 the answer is pi/3 and i have no idea how u get that. u obviously have to...

The current definition of Gravity is: The force of attraction between bodies as a result of their mass. Gravity affects both the space and time of the area surrounding a mass, diminishing with distance, so is the current accepted definition truly accurate? Do I have the correct definition and if...

I'm learning about solitons from a book called Solitons and Instantons by R. Rajaraman. He defines (page 14-15) a soliton as a solution to a (possibly non-linear) PDE where the energy density of the system is of the form ##\epsilon (x,t) = \sum_i \epsilon_0(x-a_i-u_i t-\delta_i)##, as ##t...

I will use 3 way power splitter and power detector. 1. This is power splitter data sheet. In this data sheet, there are different isolation values. What does it mean?? Also in that point, I wonder definition of isolation at power splitter. 2. This is power detector data sheet. In this data...

I was asking myself what is the definition of a Cartesian Coordinate System. Can we say that it's a coordinate system such that - the basis vectors are the same ##\forall x \in R^n## - the basis vectors are orthonormal at each ##x \in R^n## So for instance, normalized polar coordinates do not...

Hello, I've just been slightly unsure of something and would like to get secondary confirmation as I've just begun a book on tensor analysis. I would also preface this by saying my linear algebra is somewhat rusty. Suppose you have the inertia tensor in some unprimed coordinate system such that...

Hello, In the sources I have looked into (textbooks and articles on differential geometry), I have not found any abstract definition of the electromagnetic fields. It seems that at most the electric field is defined as $$\bf{E}(t,\bf{x}) = \frac{1}{4\pi \epsilon_0} \int \rho(t,\bf{x}')...

A total derivative dU = (dU/dx)dx + (dU/dy)dy + (dU/dz)dz. I am unsure of how to use latex in the text boxes; so the terms in parenthesis should describe partial differentiations. My question is, where does this equation comes from?

At the time ##t _0##, a ray of light goes out ##A##, reflected at ##B## at time ## t_1##, and arrives back at ##A## at time ##t_2##. So Einstein provides: ##t_2-t_0 = (t_1-t_0) + (t_2 - t_1) = \frac{l _{rod}}{c - v _{rod} } + \frac{l _{rod}}{c + v _{rod} }## Where: Rod with ends A and B ##v...

"Assuming that the system is in dynamical equilibrium – that the magnitude of the total potential energy is equal to twice the kinetic energy – a so-called “dynamical mass” can be derived..." https://astrobites.org/2012/03/16/what-defines-a-galaxy/ I am trying to understand what is meant by...

Hey. The series $a_n$ is defined by a recursive formula $a_n = a_{n-1} + a_{n-3}$ and its base case is $a_1 = 1 \ a_2 = 2 \ a_3 = 3$. Prove that every natural number can be written as a sum (of one or more) of different elements of the series $a_n$. Now, I know that is correct intuitively but...

I have read through the 2013 thread https://www.physicsforums.com/threads/definition-of-species-under-attack.688447/ . This was the only thread title I found when I searched for "species definition". Although the question about a definition for "species" was discussed, I found no clarity about...

Why are there (at least) two definitions of a tensor? For some people a tensor is a product of vectors and covectors, but for others it's a functional. While it's true that the two points of view are equivalent (there's an isomorphism) I find having to switch between them confusing, as a...