What is Definition: Definition and 1000 Discussions
A definition is a statement of the meaning of a term (a word, phrase, or other set of symbols). Definitions can be classified into two large categories, intensional definitions (which try to give the sense of a term) and extensional definitions (which try to list the objects that a term describes). Another important category of definitions is the class of ostensive definitions, which convey the meaning of a term by pointing out examples. A term may have many different senses and multiple meanings, and thus require multiple definitions.In mathematics, a definition is used to give a precise meaning to a new term, by describing a condition which unambiguously qualifies what a mathematical term is and is not. Definitions and axioms form the basis on which all of modern mathematics is to be constructed.
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...
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?
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...
I just started to study thermodynamics and very often I see formulas like this:
$$ \left( \frac {\partial V} {\partial T} \right)_P $$
explanation of this formula is something similar to:
partial derivative of ##V## with respect to ##T## while ##P## is constant.
But as far as I remember...
In the book Quantum Field Theory for the Gifted Amateur, they define the functional derivative as:
$$ \frac{\delta F}{\delta f(x))} = \lim_{\epsilon\to 0} \frac{F[f(x') + \delta(x'-x)) ] - F[ f(x') ]}{\epsilon} $$
Why do they use the delta function and not some other arbitrary function?
My question is about the precise definition of what is being referred to as “physical frame”, in particular in the context of cosmology. Is it simply the observational frame in which physical units are held constant? Is the FLRW frame physical? A good reference would also be helpful. Thanks for...
I have read that non-inertial frames are those, where time is not orthogonal on space. Does it just mean that the speed of light is not isotropic there or does it mean anything else? How can I picture more easily this concept (for space orthogonality I just imagine perpendicularity of one axis...
Hi,
reading this old thread Second postulate of SR quiz question I'd like to ask for a clarification on the following:
Here the Einstein definition of simultaneity to a given event on the Langevin observer's worldline locally means take the events on the 3D spacelike orthogonal complement to...
Homework Statement:: The SI definition of unit of time says the following.
"The second is the duration of 9192631770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium 133 atom."
Relevant Equations:: None
I know an...
I have a rather general question about the definition of entropy used in most textbooks:
S = k ln Ω, where Ω is the number of available microstates.
Boltzmann wrote W rather than Ω, and I believe this stood for probability (Wahrscheinlichkeit).
Obviously this is not a number between 0 and 1, so...
I know the logic of proving/disproving mathematical statements, I learned it by reading books, texts regarding to the matter, lots of exercises ( in the subject of how to do mathematical proofs and in the subject of proving/disproving statements in my math courses [ e.g. Linear Algebra, Real...
Hi,
I was working through the following problem and I am getting confused with the solution's definition of the dual.
Problem:
Given the optimization problem:
minimize ## x^2 + 1 ##
s.t. ## (x - 2) (x - 4) \leq 0 ##
Attempt:
I can define the Lagrangian as:
L(x, \lambda) = (x^2 + 1) + \lambda...
According to Wikipedia, the definition of the Riemann Tensor can be taken as ##R^{\rho}_{\sigma \mu \nu} = dx^{\rho}[\nabla_{\mu},\nabla_{\nu}]\partial_{\sigma}##. Note that I dropped the Lie Bracket term and used the commutator since I'm looking at calculating this w.r.t. the basis. I...
abstraction levels in programming define different approaches with a varying degree of detail for representing, accessing and manipulating data.
What I understand by abstraction is that we hide what is not necessary. Isn't that.
Hi,
a clarification about the following: consider a smooth curve ##γ:\mathbb R→\mathbb R^2##. It is a injective smooth map from ##\mathbb R## to ##\mathbb R^2##. The image of ##\gamma## (call it ##\Gamma##) is itself a smooth manifold with dimension 1 and a regular/embedded submanifold of...
Hello everyone, I'd like to share a doubt I am currently struggling with.
So we know that ΔU=−W, where ΔU is the difference of potential energy and Wthe work done by the force to move the body from point A to point B.
When analyzing this for the gravitational force, since we have U=−GmM/R, with...
So the part in italics "an operation performed on one quantity which when performed on unity produces the other." I do not understand. Can anyone help me understand what this means? I know how to multiply fractions, but this explanation is confounding to me.
Hello.
Considering this DE;
$$ x^7 x' = (x^8-300)t^6 $$ with inital value x(0) = -2
Now the solution for the initial value should be
C = -44;
And for x(t) I get ;
$$x(t) = (-44 e^{\frac{8}{7} t^7} + 300)^{\frac{1}{8}}$$
Now to get the biggest domain of definition I did this;
$$ -44...
Hi,
although there is a lot of discussion here in PF, I'd like to ask for a clarification about the definition of 'spatial x direction' in the context of flat or curved spacetime.
Consider a set of free-falling gyroscopes (zero proper acceleration) passing through an event A with different...
Hello!
I have some troubles with the definition of the so called super Lie module. In Alice Rogers' textbook "Supermanifolds theory and applications" definition goes as follows
Suppose that ##\mathbb{A}## is a super algebra and that #\mathfrak{u}# is a super Lie algebra which is also a super...
Mathematicians will use the term "elementary functions," often in the context of integration wherein some integrals cannot be expressed in elementary functions.
The elementary functions are usually listed as being arithmetic, rational, polynomial, exponential, logarithmic, trigonometric...
Hi all,
I would like to understand the definition of finite size correction, radiative correction and weak magnetism correction, with their impacts on the beta spectrum. I'm not a physics student, thus I would like to seek for a help about the simple explanation that can be understand by...