In mathematics and logic, a theorem is a non-self-evident statement that has been proven to be true, either on the basis of generally accepted statements such as axioms or on the basis of previously established statements such as other theorems. A theorem is hence a logical consequence of the axioms, with a proof of the theorem being a logical argument which establishes its truth through the inference rules of a deductive system. As a result, the proof of a theorem is often interpreted as justification of the truth of the theorem statement. In light of the requirement that theorems be proved, the concept of a theorem is fundamentally deductive, in contrast to the notion of a scientific law, which is experimental.Many mathematical theorems are conditional statements, whose proofs deduce conclusions from conditions known as hypotheses or premises. In light of the interpretation of proof as justification of truth, the conclusion is often viewed as a necessary consequence of the hypotheses. Namely, that the conclusion is true in case the hypotheses are true—without any further assumptions. However, the conditional could also be interpreted differently in certain deductive systems, depending on the meanings assigned to the derivation rules and the conditional symbol (e.g., non-classical logic).
Although theorems can be written in a completely symbolic form (e.g., as propositions in propositional calculus), they are often expressed informally in a natural language such as English for better readability. The same is true of proofs, which are often expressed as logically organized and clearly worded informal arguments, intended to convince readers of the truth of the statement of the theorem beyond any doubt, and from which a formal symbolic proof can in principle be constructed.
In addition to the better readability, informal arguments are typically easier to check than purely symbolic ones—indeed, many mathematicians would express a preference for a proof that not only demonstrates the validity of a theorem, but also explains in some way why it is obviously true. In some cases, one might even be able to substantiate a theorem by using a picture as its proof.
Because theorems lie at the core of mathematics, they are also central to its aesthetics. Theorems are often described as being "trivial", or "difficult", or "deep", or even "beautiful". These subjective judgments vary not only from person to person, but also with time and culture: for example, as a proof is obtained, simplified or better understood, a theorem that was once difficult may become trivial. On the other hand, a deep theorem may be stated simply, but its proof may involve surprising and subtle connections between disparate areas of mathematics. Fermat's Last Theorem is a particularly well-known example of such a theorem.
https://lh4.googleusercontent.com/FCqUErWAqlG8w0CskhcsLgpG91xyxzAkV_nD-bZAq8147-_RKesQDpglwqF5ylKZ0Q6VW88jX-KNuIpSXi9vhw5AiWmwiv_fMyyUo_WWZJG4uwWS0aB-3rGMA0h0PDo7ZpolexCe
this is the question
Here is a tutorial video but his steps are very confusing to me. I personally know bayes theorem and...
Homework Statement:: Can someone explain the finite number of equilibria outcome of the Poincaré-Bendixson Theorem?
Relevant Equations:: Poincaré-Bendixson Theorem
[Mentor Note -- General question moved from the schoolwork forums to the technical math forums]
Hi,
I was reading notes in...
“Given any class of mutually exclusive classes, of which none is null, there is at least one class which has exactly one term in common with each of the given classes…”
The reason this statement sounds like one of those theora is that I recall reading a Time-Life book on Mathematics, and there...
Hi.
I am working through some notes on the above 2 theorems.
Liouville's Theorem states that the volume of a region of phase space is constant along Hamiltonian flows so i assume this means dV/dt = 0
In the notes on the Poincare Recurrence Theorem it states that if V(t) is the volume of phase...
I attached a file which shows my attempt to resolve this problem with the possible two pair interaction which gives us the kinetic energy of the cluster in an expanding system, at least I think so. But to be honest I´m more or less completely stuck with this question and I would be glad if...
when the net force is constant then
Q1. rate of change of momentum (dp/dt) is zero or constant
Q2. assuming dp/dt is constant we replaced it with ----> p2-p1(total change in momentum ) ? how?
Let ##X## be a bounded subset of ##\mathbb{R}## with infinite cardinality. We consider a countably-infinite subset of ##X##. We write this set as a sequence to be denoted ##\{a_n\}_{n\in\mathbb{N}}##.
Now define ##A## to be the set of points in the sequence with the property that for each...
Let ##X## be a closed and bounded subset of the real numbers. Let ##\{x_i\}_{i\in I}##, for some index ##I##, represent the set of limit points of ##X##. Since ##X## is closed, it must follow that ##\{x_i\}_{i\in I}\subset X##. Hence, the set of limit points must be bounded.
Let ##\{U_j\}_{j\in...
Here is the link.
https://www.grc.nasa.gov/www/BGH/sincos.html
Sorry just a little rusty on Pythagoras theorem.
I mean the formula still holds but in order to find the opposite and the adjacent the opposite becomes the adjacent and vice versa .
Hello! Consider this circuit
Now I want to calculate the current Ik. The values are given as follows;
Uq1 = 12 V
Uq2 = 18V
R1 = R2 = 8 Ohm
R3 = R4 = 20 Ohm
My approach was using the Superposition theorem. First I deactived Uq2 and left Uq1 active.
Now if I am not mistaken the resistors R2 R3...
Hi,
I am missing the point about the application of Darboux theorem to symplectic manifold case as explained here Darboux Theorem.
We start from a symplectic manifold of even dimension ##n=2m## with a symplectic differential 2-form ##w## defined on it. Since by definition the symplectic 2-form...
Hello! I have a setup consisting of some charged particles each of which is produced at a different position, ##(x_i,y_i,z_i)##. I don't know the exact position, but I know that each of the 3 variables is normally distributed with mean zero and standard deviation of 3 mm. What I measure in the...
Could someone explain why ##[a][x_0]=[c]\iff ax_0\equiv c\, (mod\, m)##?
My instructor said it came from the definition of congruence class. But I am not convinced.
##lim_{|z|\rightarrow \infty}\frac{f}{g}=1\neq \frac{\infty}{\infty}##
so ##lim_{|z|\rightarrow \infty}f\neq \infty## and ##lim_{|z|\rightarrow \infty}g\neq \infty##.
Because f(z) and g(z) are bounded and entire, f(z) and g(z) are constant functions by Liouville's theorem
.
f(z) and g(z) are...
The solution is from ##fdt=d(mv)=mdv+vdm## and separate the variables and then integrate them.
But at first I tried this method. At time ##t##, suppose the mass of the cart is ##m##, and its velocity is ##v##. And suppose at time ##t+dt##, its mass will be ##m-\rho dt##, and its velocity...
The question arises the way Goldstein proves Euler theorem (3rd Ed pg 150-156 ) which says:
" In three-dimensional space, any displacement of a rigid body such that
a point on the rigid body remains fixed, is equivalent to a single rotation about some axis that runs through the fixed point"...
Darboux theorem says a derivative function must have an intermediate value theorem without requiring the derivative function to be continuous. Why is this property not true for any continuous function in its intermediate value theorem?
Hi,
Question(s):
1. Are there any good resources that explain, at a very simple level, how Mercer's theorem is related to valid covariance functions for gaussian processes? (or would anyone be willing to explain it?)
2. What is the intuition behind this condition for valid covariance...
Wikipedia while deriving Bertrands theorem writes after some steps :
However by a similar argument we can say that since ##β## is a constant then we can directly solve
##J^{\prime}(u)=1-\beta^{2}## and find that ##J=\left(1-β^{2}\right) u##) which is wrong.
What went wrong?
Bertrand's Theorem says : the only forces whose bounded orbits imply closed orbits are the Hooke's law and the attractive inverse square force.
I'm looking at the hookes law ##f=-k r## and try to see explicitly that the orbit is indeed closed.
I use the orbit equation ##\frac{d^{2} u}{d...
In learning about translational and rotational motion, I solved a problem involving a wheel rolling down an inclined plane without slipping.
There are multiple ways to solve this problem, but I want to focus on solutions using energy.
Now to my questions. The reference frame in the posted...
Hi,
So my goal is to compute the integral of the "curl" of the vector field ##A_i(x_i)## over a 2-dimensional surface. Following a physics book that I am reading, I introduce the antisymmetric 2-nd rank tensor ##\Omega_{ij}##, defined as:
$$\Omega_{ij} = \frac {\partial A_i}{\partial x_j} -...
So Noether's Theorem states that for any invarience that there is an associated conserved quantity being:
$$ \frac {\partial L}{\partial \dot{Q}} \frac {\partial Q}{\partial s}$$
Let $$ X \to sx $$
$$\frac {\partial Q}{\partial s} = \frac {\partial X}{\partial s} = \frac {\partial...
Goldstein 2nd ed.
In its Appendix is given the derivation of Bertrands Theorem.Here ##x=u-u_0## is the deviation from circularity and ##J(u)=-\frac{m}{l^{2}} \frac{d}{d u} V\left(\frac{1}{u}\right)=-\frac{m}{l^{2} u^{2}} f\left(\frac{1}{u}\right)##
If the R.H.S of A-10 was zero, the solution...
Wikipedia on Bertrands theorem, when discussing the deviations from a circular orbit says:
>..."The next step is to consider the equation for ##u## under small perturbations ##{\displaystyle \eta \equiv u-u_{0}}## from perfectly circular orbits"
(Here ##u## is related to the radial distance...
Using the transformation for ##t##, I obtained
$$\mathrm{d}t'=\left(1+\frac{\partial f}{\partial t}\right)\mathrm{d}t+\frac{\partial f}{\partial r}\mathrm{d}r$$.
Using this equation, I substituted it into the general line element to obtain
\begin{align*}...
Preceding Problem. Let ##y=f(x)## be a continuous function defined on a closed interval ##[0, b]## with the property that ##0 < f(x) < b## for all ##x## in ##[0, b]##. Show that there exist a point ##c## in ##[0, b]## with the property that ##f(c) = c##.
This problem can be solved by letting...
So I've seen other threads on here with the same problem from a few years ago, and I'm just not getting the same answers. However, I followed along with a similar problem in the textbook and used all the same methods, so can't understand where I've gone wrong, or if I even am wrong. Also not...
Mihăilescu's theorem proves that Catalan's conjecture is true. That is for x^a - y^b = 1, the only possible solution in naturual numbers for this equation is x=3, a=2, y=2, b=3. What is not clear to me is this. Does Mihăilescu's theorem prove that the difference between any other two...
Theorem: Let ## f(x), g(x) \in \mathbb{F}[ x] ## by polynomials, s.t. the degree of ## g(x) ## is at least ## 1 ##. Then: there are polynomials ## q(x), r(x) \in \mathbb{F}[ x] ## s.t.
1. ## f(x)=q(x) \cdot g(x)+r(x) ##
or
2. the degree of ## r(x) ## is less than the degree of ## g(x) ##
Proof...
Hi,
I was attempting the following question and just wanted to check whether my working was correct:
Question: A bag has three coins in it which are visually indistinguishable, but when flipped, one coin has a 10% chance of coming up heads, another as a 30% chance of coming up heads, and the...
If you go to "The Abel Prize Interview 2016 with Andrew Wiles" on YouTube, there is a statement made by Andrew Wiles beginning at about 4:10 and ending about 4:54 where he mentions there are some new number systems possible where the fundamental theorem of arithmetic does not hold. I don't...
A simulation/animation/explanation based on the inertial frame only:
The previous videos referenced there are here:
See also this post for context on the Veritasium video: https://mathoverflow.net/a/82020
Note to mods: The previous thread is not open anymore so I opened a new one. Feel free...
Given surface ##S## in ##\mathbb{R}^3##:
$$
z = 5-x^2-y^2, 1<z<4
$$
For a vector field ##\mathbf{A} = (3y, -xz, yz^2)##. I'm trying to calculate the surface flux of the curl of the vector field ##\int \nabla \times \mathbf{A} \cdot d\mathbf{S}##. By Stokes's theorem, this should be equal the...
Here I'm asking solely about the circle pictograms. Please eschew referring to, or using, numbers as much as possible. Please explain using solely the circle pictograms. Undeniably, I'm NOT asking about how to divide numbers.
I don't understand
1. How do I "visually" divide Circle 1...
I recently learned how to calculate the centroid of a semi-circular ring of radius ##r## using Pappus's centroid theorem as
##\begin{align}
&4 \pi r^2=(2 \pi d)(\pi r)\nonumber\\
&d=\frac {2r}{\pi}\nonumber
\end{align}##
Where ##d## is the distance of center of mass of the ring from its base...
1. For regions that contain charge density, does the 1st uniqueness theorem still apply?
2. For regions that contain charge density, does the 'no local extrema' implication of Laplace's equation still apply? I think not, since the relevant equation now is Poisson's equation. Furthermore...
The correct answer is ##\frac{\pi a^2 h} 2## by using the standard approach. However when I tried using the divergence theorem to solve this problem, I got a different answer. My work is as follows:
$$\iint_S \vec F\cdot\hat n\, dS = \iiint_D \nabla\cdot\vec F\,dV$$
$$= \iiint_D \frac{\partial...
It is more or less a generic problem of stokes theorem:
##\int_{\gamma} F dr##, where ##F = (-y/(x²+y²) + z,x/(x²+y²),ln(2+z^10))## and gamma is the intersection of ##z=y^2, x^2 + y^2 = 9## oriented in such way that its projection in xy is traveled clockwise.
So, i decided to apply stokes...
Sorry if there's latex errors. My internet connection is so bad I can't preview.
Here's the wikipedia proof I'm referring to. I'm fine with the steps up to $$W(x,0) = W_0 (x) [1 + \beta f_0 (x(0) - \langle x \rangle_0) ]$$ where ##W(x,t)## is the probability density of finding the system at...
Greetings!
here is the following exercice
I understand that when we follow the traditional approach, (prametrization of the surface) we got the answer which is 8/3
But why the divergence theorem can not be used in our case? (I know it's a trap here)
thank you!