Definition Definition and 1000 Threads

I'm particularly interested in the foundation of mathematics. I've read various ways of defining the integers and addition. But I never encountered a formal definition of equality. It's seems (at least for what i read) that the equality is treated as something fundamental that does not need to...

We physicists must be careful to insure that theories begin with correct principles. One basic principle is that all quantities must be capable of being observed or measured. If a theory uses a quantity that cannot be observed, then it is not a physics theory, but a hypothesis or a...

Hello everyone. I am slightly confused by these ideas so i would like your help. How is additive inverse defined? Is unary negation an operation in its own right just like those more familiar, like addition, multiplication? Or something else?

Hi all, I have a general question about relative error. Suppose that we have a vector of measurements \hat{b}=\left(\hat{b_{1}},\hat{b_{2}},...,\hat{b_{n}}\right). Furthermore, suppose that these measurements are accurate to 10%. My natural interpretation of this statement is that there is...

Hello I'm reading my old notes of QM, I found the definition of Pauli vector, as follow \vec{\sigma}=\sigma_1 e_x+\sigma_2e_y + \sigma_3 e_z Where e_x. e_y and e_z are unit vectors. So, here is my question. \sigma_i and e_i are elements of different nature. How can we define the product...

Homework Statement For all x\in R , f(x)>0 . Using precise definition of limits and infinite limits, prove that \lim_{x\to a}f(x)=\infty if and only if \lim_{x\to a}\frac{1}{f(x)}=0 Homework Equations The Attempt at a Solution I know the precise definition of limits and infinite limits...

First off I want to apologize for bombarding this subforum with my gazillion questions. If my continuous barrage of questions poses a problem just let me know and I'll stop. Homework Statement For each value of ε, find a positive value of δ such that the graph of the function leaves the window...

According to this news article , genome analysis has shown that the understanding of what constitutes a gene has to reveiwed and redefined in light of new evidence. Btw there is no junk DNA in our genome. http://medicalxpress.com/news/2012-09-encode-massive-genome-analysis-gene.html

Homework Statement Use the definition of the derivative to show that if G(x)=\int^{u(x)}_{a}f(z)dz, then \frac{dG}{dx}=f(u(x))\frac{du}{dx}. This is called Leibniz's rule. Also, by thinking of the value of an integral as the area under the curve of the integrand (and drawing a picture of...

I am a bit confused about this matter. While I was studying Calculus I saw an excercise like this: The domain of f(x) [0,2] and the range is [0,1], it also shows its graphic, though it is not important it is something like a parabola, its maximum point is (1,1) and its intersection points are...

Homework Statement Using the epsilon and delta definition, prove that: \mathop {\lim }\limits_{x \to - 3} \sqrt {{x^2} + 16} = 5 Homework Equations The Attempt at a Solution Given epsilon > 0. Choose \delta {\rm{ = min}}\left\{ {{\rm{1,}}\frac{{\left( {5 + \sqrt {20} } \right)\varepsilon...

Hello, I just began my Discrete Mathematics class. It is rather interesting, but I have a few questions regarding the definitions of logical connectives. For instance, my book states that the conditional statement,p \rightarrow q serving as an example, is false when p is true and and q is...

What does it mean to give a group structure? I'm working on a problem and part of it asks for the structure of the group. The law of composition and generators seem to be given already (and an expression that says that a^2 = 1 for any elt a of the group). Is there anything to do other than...

In the definition, 1) why must you find a n_0 \in N such that \forall N \geq n_0? You might as well say find a n_0 \in R such that \forall N > n_0. Just a matter of simplicity? 2) Why must |x_n - a| < \epsilon hold? I think |x_n - a| \leq \epsilon is fine as well, given that it must hold...

Hiah, I've got a question concerning a t-pipe configuration and the corresponding friction coefficient values because there are two different friction coefficients stated in literature. Let's assume we have a simple t-pipe where the main passage is larger than the side branch. The friction...

Hello, I have a doubt on the definition of Lie groups that I would like to clarify. Let's have the set of functions G=\{ f:R^2 \rightarrow R^2 \; | \; < f(x),f(y)>=<x,y> \: \forall x,y \in R^2 \}, that is the set of all linear functions ℝ2→ℝ2 that preserve the inner product. Let's associate the...

Warning: Semantics battle may ensue, tread lightly So I was wondering the other day, the repeated-addition definition of multiplication only works for integers, for example you cannot use this to calculate the square of e or pi. So is there a rigorous definition for multiplication that is...

Hi, Could someone explain how the following two definitions of the displacement operator are equal? The first is the standard one 1) e^{\alpha a^{\dagger}-\alpha*a} But what about this one? This is from a Fock state decomposition of a coherent state. 2) e^{\frac{-|\alpha^{2}|}{2}}e^{\alpha...

A dihedral group of an n-gon denoted by Dn, whose corresponding group is called the Dihedral group of order 2n? What I gather from that is a square has 8 symmetries, an octagon has 16, a hexagon 12, etc?

Dirac's "Quantum Mechanics" - the definition of the time evolution operator I'm reading Dirac's "Principles Of Quantum Mechanics" to learn more about the formal side of the subject. I have a question about the way he defines the time evolution operator in the book. Either there's a mistake or...

One concept in physics that has never set well with me is the way work and energy are defined. According to all the physics sources I've looked at, work is defined as: W = \vec{F} \cdot \vec{d} (for a constant force over a distance) However, intuitively the notion of taking the dot...

Could someone explain what a gauge theory is, both in general and how it applies to physics? Please try to keep definitions relatively simple, even though the topic is exceedingly complicated. Examples are also greatly appreciated. Thanks!

I am still a physics novice and am learning new things everyday. I've been looking at tensors recently and I'm finding that I can't really understand what they are. Could someone explain in relatively simple words what the definition of a tensor is and why they are so significant? Also, what is...

Hi, been reading about GR and am quite confused about the new definition of vectors. My main problem with this is that the text uses partial derivatives as the vector basis, I understand this is related to directional derivatives but cannot see the direct mathematical relation. Secondly, how...

In studying SR, I've been subscribing to a particular definition of a Frame of Reference that makes sense to me. Recently, I've been made aware by another PF member that there may be other, broader, definitions that are valid and that people use. I would like to know more about these broader...

Let $\mathbb{G}$ be a set with a map $(\xi, ~ \eta) \mapsto f(\xi, ~\eta)$ from $\mathbb{G}\times\mathbb{G}$ into $\mathbb{G}$. For every pair $(\xi, ~ \eta)$ in $\mathbb{G}$ let $f(\xi, ~\eta) = f(\eta, ~ \xi)$. Suppose there are elements $\omega$ and $\xi'$ in $\mathbb{G}$ such that for every...

physics textbook, replace sine with its definition (?) What on Earth do they mean? "That will introduce either sin(theta) or cos(theta). Reduce the resulting two variables, x and theta, to one, x, by replacing the trigonometric function with an expression (its definition) involving x and y."...

I just started studying set theory, and I've seen this definition for an ordered pair (a,b) = {{a}, {a,b}} However, I don't understand how this definition makes sense. Could someone explain this definition to me? Maybe use a concrete example too?

Hello all, For a few months, I've been (off and on) trying to come up with a more intuitive definition for Electric Potential (or Voltage, if you prefer), as all I can seem to find are mathematical equations. I believe I have finally come up with a satisfactory result, and I merely wanted to...

I am totally confused about the Lorentz Group at the moment. According to wikipedia, the Lorentz group can be defined as the General Orthogonal Lie Group##O(1,3)##. However, the definition of the GO Lie Group that I know only works when there is a single number inside the bracket, not 2, e.g...

hi i have a few questions about entropy: why does the definition of entropy stress the fact that the heat exchange by the system is reversible(dS=dQ_rev/T)? am i right, that processes are only reversible iff ΔS=0 and therefore e.g. isothermic, isobaric and isochoric processes even of...

Homework Statement Find the convolution of g(x) = e^{-πx^{2}} with itself from -∞ to ∞ using the definition of convolution, not the Fourier Transform. The Attempt at a Solution See my attachment. My professor said that you have to use integration by parts, but I keep getting stuck...

Hello, The definition is ln(x) = \int_1^x\frac{1}{t}dt I have read several sources regarding this, but what I can't seem to find is why it was defined this way. What is the justification for defining it this way, and how was ln (x) found to be the same as the that particular integral?

I understand the definition to be the amount of work done in moving a charge from one point to another divided by the charge. If you have a standard 1.5 volt battery, the charge should move easily from one end of the circuit to the other because the positive terminal attracts the electrons...

for derivative sinx = cosx, by setting up into formal definition formula limΔx->0 \frac{f(x+Δx)-f(x)}{Δx} this formal definition of derivative is formulated from the cartesian coordinate system where the horizontal is x and verticle is y. But sinx is a trig function and trig functions...

Hi comrades. According to spivak, the defition of limit goes as follows: " For every ε > 0, there is some δ > 0, such that, for every x, if 0 < |x-a| < δ, then |f(x) - l |< ε. " After some exercices, I came across with a doubt. Say that I could prove that | f(x) - l |< 5ε, for some...

Hi all, I'm quite confused concerning the definition of tangent vectors and tangent spaces as presented in Munkres's Analysis on Manifolds. Here is the book's definition: Given ##\textbf{x} \in \mathbb{R}^n##, we define a tangent vector to ##\mathbb{R}^n## at ##\textbf{x}## to be a pair...

It is kinda strange. There is no agreement on the definition of a relation. Some books says it is a set of ordered pairs. Other books says it is a subset of a cartesian product. How nice if everything can be agreed down to a few axioms like Euclid's elements. What is your favourite...

Homework Statement Using the definition of the derivative, find the derivative of g(t) = 1 / sqrt(t).Homework Equations I was told I could solve it by rationalizing it. I asked a question on Yahoo! Answers and saw someone work it out step by step, but I don't understand any of why they did what...

What is the definition of a "coefficient" How would you define what a coefficient is in the context of differential equations? How do they influence the graph of a DE (variable and constant)? Thank you in advance

I'm reading through a multivariable calculus book and it starts off with some linear algebra. It defines vector addition as V \times V \rightarrow V. My text describes V as a set and describes the above process as a mapping. I believe the \times may represent a Cartesian product. Could someone...

I understood that Newton has introduced a concept called "Force" which is basically a cause for an effect i.e. if an object is in a state of rest and if applied a "force" then the object moves (change in velocity, ∴ accelerates) also if an object moves with a constant velocity and is disturbed...

Hi there, I have a question regarding this statement: My question is whether we can say so... Thank you very much!

Hi, I am scratching the surface of information regarding particle physics. I have a basic understanding of standard model. What I am not quite understanding is what 'spin' is. I know that all fermions have a spin of 1/2, but what exactly is spin? Thanks

I'm reading a paper and have came across the term 'Cn-close' in the sense of a curve being C1-close to a circle for example, but can't find a definition of this term anywhere, and would be grateful if anyone could help.

Hello I read the following sentence when reading about ion traps: "By changing the trapping voltage we are changing the depth of the potential trapping well, therefore at the same axial position there is a corresponding increase in the potential well, which means that the ion will have to...

Just a quick question of something I found in my textbook but can't get how they produced it. C_p =(∂Q/∂T)_p that is the definition of heat capacity at a constant pressure p. Q is heat and T is temperature. This equation is fine and I know how to derive it. Now it is the next line which...

I searched for "definition" and "planet" but found no thread which matched this purpose; if one already exists then it is significantly old, but I will apologize anyways. It used to be that our solar system had Nine planets. Then some trans-Neptunian object (Eris) forced some astrological...

I've been doing a lot of readings on NLO calculations for high energy physics, and several papers I have read mention "Born final state" particles, "Born level" processes/trees/diagrams etc. None of them seem to define them, however, and my searches on the Google and in textbooks have been...

So far, all I understand is that the definition proves that there's a value of f(x,y) as f(x,y) approaches (x0,y0) which is sufficiently close to but not exactly the value at f(x0,y0). I am probably completely off... but I just don't understand the purpose of proving this. I also don't...