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.
I just wanted to point out a resource useful for dealing with claims of violating Bell's theorems. You can point the claimant at https://algassert.com/quantum/2015/10/11/Bell-Tests-vs-No-Communication.html and say "I won't believe you unless you can make the 'Write Your Own Classical CHSH...
The articles:
https://lmcs.episciences.org/5328/pdf
http://argo.matf.bg.ac.rs/publications/2013/2013-icga-krk-sat.pdf
http://archive.ceciis.foi.hr/app/public/conferences/1/papers2012/dkb3.pdf
KRK endgame is a win for white regardless of starting position, with the trivial drawing exception in...
Hi,
In Chapter 5 Munkres proves the Tychonoff Theorem and after proving the theorem the first exercise is: Let ##X## be a space. Let ##\mathcal{D}## be a collection of subsets of ##X## that is maximal with respect to finite intersection property
(a) Show that ##x\in\overline{D}## for every...
I don't know how to solve this proof
Prove that a set $M \subset C([a, b])$ for which there exist $m. L> 0$ and $x_0 \in [a; b]$ such that $|f(x_0)| \leq{} m$ for all $f \in M$ and $|f(x)-f(y)| \leq{} L |x-y|$ for all $f\in M$ and for all $x,y \in [a,b]$ is relatively compact in $C([a, b])$
My...
I’m looking over a recent paper mentioned in another thread. It claims to refute Bell’s theorem. At first glance, the model presented in the paper doesn’t appear consistent with QM. Here’s a simple example.
Suppose we set both polarizers to the same angle ##\alpha = \pi /4##. In the model...
I hate to create a thread for a step in a calculation, by I don't know what else to do. I'm having a lot of trouble reproducing E. Weinberg's index calculation (found here https://inspirehep.net/literature/7539) that gives the dimension of the moduli space generated by BPS solutions in the...
The Paper “On a contextual model refuting Bell’s theorem” has now been published by the journal EPL (Europhysics Letters) and is available under
https://iopscience.iop.org/article/10.1209/0295-5075/134/10004
In this paper a contextual realistic model is presented which correctly predicts...
Hi,
I have a question, or am looking for clarification, about the no-cloning theorem and state tomography. My understanding is that the theorem states one cannot make an exact copy of a quantum state. I was also reading about state state tomography where it was said*
'On the other hand, the...
Guys, I have a problem that really needs you guys to help, I know it is a stupid question but please bear with me:
Context:
You have a block on a slope(has friction) you use a string to pull the block up with constant speed.
Problem:
So according to the network theorem, the work net is equal...
If ##f## is a constant function, then choose any point ##x_0##. For any ##x\in K##, ##f(x_0)\geq f(x)## and there is a point ##x_0\in K## s.t. ##f(x_0)=\sup f(K)=\sup\{f(x_0)\}=f(x_0)##.
Now assume that ##f## is not a constant function.
Construct a sequence of points ##x_n\in K## as follows...
The Earnshaw’s theorem comes directly from Maxwell equation so it should be unavoidable in any classical situation. The theorem usually disallows magnetic levitation. However, there are loopholes. Quoting wikipedia "Earnshaw's theorem has no exceptions for non-moving permanent ferromagnets...
I'm trying to prove Plancherel's theorem for functions $$f\in L^1\cap L^2(\mathbb{R})$$. I've included below my attempt and I would really appreciate it if someone could check this for me please, and give me any feedback they might have.
**Note:** I am working with a slightly different...
Loosely speaking, the Bell theorem says that any theory making the same measurable predictions as QM must necessarily be "nonlocal" in the Bell sense. (Here Bell locality is different from other notions of locality such as signal locality or locality of the Lagrangian. By the Bell theorem, I...
(1) We want to find the voltage across ##R_L##
(2) We remove the load and label the terminals ##V_T##
(3) The equivalent network of (2)
So basically the voltage across the load is ##V_{th}## but when we find the equivalent network we put that same voltage behind a Thevenin...
I attached a screenshot of the book (sorry no pdf available for this book). Right above the somewhat central line they give the theorem that if there are m currents and n nodes, then there will be n - 1 independent equations from the current law and m - n - 1 from the voltage law.
I count 4...
The Fundamental Theorem of Arithmetic essentially states that any positive whole number n can be written as:
##n = p_1^{a_1} \cdot p_2^{a_2} \cdot p_3^{a_3} \cdot \dots##
where ##p_1##, ##p_2##, ##p_3##, etc. are all the primes, and ##a_1##, ##a_2##, ##a_3##, etc. are non-negative integers...
Use the Squeeze Theorem to find the limit.
lim [x^2 • (1 - cos(1/x)]
x--> 0
Let me see.
-1 ≤ cos (1/x) ≤ 1
-x^2 ≤ x^2 • [1 - cos(1/x)] ≤ x^2
-|x^2| ≤ x^2 • [1 - cos(1/x)] ≤ |x^2|
lim -|x^2| as x tends to 0 = 0.
lim |x^2| as x tends to 0 = 0.
.
By the Squeeze Theorem, [x^2 • (1 -...
Use the Squeeze Theorem to find the limit.
lim (x^2 • sin(1/x))
x--> 0
Let me see.
-1 ≤ sin (1/x) ≤ 1
-x^2 ≤ x^2 • sin(1/x) ≤ x^2
-|x^2| ≤ x^2 • sin(1/x) ≤ |x^2|
lim -|x^2| as x tends to 0 = 0.
lim |x^2| as x tends to 0 = 0.
.
By the Squeeze Theorem, x^2 • sin(1/x) was squeezed between...
If 0 ≤ f(x) ≤ 1 for every x, show that
lim [x^2 • f(x)] = 0.
x--> 0
Let me see.
0 ≤ f(x) ≤ 1
Multiply all terms by x^2.
0 • x^2 ≤ x^2• f(x) ≤ 1 • x^2
0 ≤ x^2 • f(x) ≤ x^2
Is this right so far? If correct, what's next?
Proof goes like this:
(1) Prove the existence of open intervals centered around the end-points of the domain such that the image of the points in these intervals through ##f## has the same sign as the image of the end-point through ##f##. In other words, prove that there is a ##\delta>0## such...
Explain why the Intermediate Value Theorem gives no information about the zeros of the function
f(x) = ln(x^2 + 2) on the interval [−2, 2].
Let me see.
Let x = -2.
f(-2) = ln((-2)^2 + 2)
f(-2) = ln(4 + 2)
f(-2) = ln (6). This is a positive value.
When I let x be 2, I get the same answer...
So I've just learned Norton's Theorem and I have this problem on my homework assignment that is wrong. I've checked the answer with a circuit simulator(PSPICE) and the simulation said that V0 should be a drop of 2V. However, my simplified circuit shows a voltage drop of 4V. I have been staring...
I wanted to ask about a step I couldn't understand in Tong's notes$$\int_M d^n x \partial_{\mu}(\sqrt{g} X^{\mu}) = \int_{\partial M} d^{n-1}x \sqrt{\gamma N^2} X^n = \int_{\partial M} d^{n-1}x \sqrt{\gamma} n_{\mu} X^{\mu}$$we're told that in these coordinates ##\partial M## is a surface of...
Hello at all!
I have to solve this exercise:
A tampon diagnostic test provides 1% positive results. The positive predictive values (probabilities of positive test disease) and negative (absence disease given negative test) are respectively 0.95 and 0.98.
What is the prevalence of the disease...
I have tried many times to solve this network, but can't understand how to get current in each resistors by superposition theorem. Please help me to solve and find currents in each 3 resistors with solution.
Note:- The figure is attached below.
The 4-colour theorem states that the maximum number of colours required to paint a map is 4.
The proof requires exhaustive computation with a help of a computer.
But I thought that one can visually prove the theorem in the following way;
If one replaces the map with a graph where each region...
Poynting's Theorem (https://en.wikipedia.org/wiki/Poynting's_theorem) says:
The rate of energy transfer (per unit volume) from a region of space equals the rate of work done on a charge distribution plus the energy flux leaving that region.
$$-\frac{\partial u}{\partial...
Hi All, being ##p## a prime number, is there a way to solve the congruence ##(p^k-1)! \equiv X \mbox{ (mod p)}## for ##X## using Wilson's theorem: $$ (p-1)! \equiv -1 \mbox{(mod p)} $$?
[Moderator's note: Spun off from another thread due to topic and subforum change.]
I think Ballentine's interpretation is ruled out by the PBR theorem. Maybe we could discuss that?
I have a question related to the following passage in the quantum mechanical scattering textbook by Taylor,
Here Taylor makes the choice to use a basis of total angular momentum eigenvectors instead of using the simple tensor product given in the first equation above (6.47). I understand that...
In the book: SET THEORY AND LOGIC By ROBERT S.STOLL in page 19 the following theorem ,No 5.2 in the book ,is given:
If,for all A, AUB=A ,then B=0
IS that true or false
If false give a counter example
If true give a proof
So I have often heard it argued that "super-determinism" is a loophole to Bell's theorem, that allows a local hidden variable theory. Bell himself alluded to it in a 1980s BBC interview.
But why is this the case? And how is super-determinism different to regular determinism. And the many-world's...
Hello, I am trying to solve this question:
Assume that the Sun's energy production doesn't happen by fusion processes, but is caused by a slow compression and that the radiated energy can be described by the Virial Theorem: $$L_G = - \frac{1}{2} \frac{GM^2}{R^2} \frac{dR}{dt} $$
How much must...
In Hamiltonian statement the Noether theorem is read as follows. Consider a system with the Hamiltonian function $$H=H(z),\quad z=(p,x),\quad p=(p_1,\ldots,p_m),\quad x=(x^1,\ldots,x^m)$$ and the phase space ##M,\quad z\in M.## Assume that this system has a one parametric group of symmetry...
So i have this question. If a projectile is fired from a spring loaded system and when it goes pass a chronograph, reads 300FPS and has a mass of 0.12grams. Is there any way to use the proportionality theorem (1/3=x/6 example) to approximate how fast a mass of 0.25grams is when fired from same...
I get a nonsensical result. I am unable to understand where I go wrong.
Let's consider a material with a temperature independent Seebeck coefficient, thermal conductivity and electrochemical potential to keep things simple. Let's assume that this material is sandwiched between 2 other materials...
Summary:: Be the set X of vectors {x1,...,xn} belong to the vector space E. If this set X is convex, prove that all the convex combination of X yet belong to X. Where convex combination are the expression t1*x1 + t2*x2 + ... + tn*xn where t1,...,tn >= 0 and t1 + ... + tn = 1
I tried to suppose...
Hi everyone hope you are well, I would like to express what I have done for this question:
Proving and employing caratheodory theorem we can say that any point in polyhedron can be expressed as a convex combination of at most n+1 points (where n is the space dimension) in same polyhedron that...
Is there a theorem that states that a set of binary swaps can result in any permutation?
For example, the original set (1,2,3,4,5) can have the swap (24) and result in (1,4,3,2,5). is there a set of specific swaps for each net result permutation?
The integral that I have to solve is as follows:
\oint_{s} \frac{1}{|r-r'|}da', \quad\text{ integrating with respect to r '}, integrating with respect to r'
Then I apply the divergence theorem, resulting in:
\iiint \limits _{v} \nabla \cdot \frac{1}{|r-r'|}dv' =...
According to the book I am using, one can decompose a finite abelian group uniquely as a direct sum of cyclic groups with prime power orders.
Uniquely meaning that the structures in the group somehow force you to one particular decomposition for any given group.
Unfortunately, the book gives no...
The potential inside the crystal is periodic ##U(\vec{r} + \vec{R}) = U(\vec{r})## for lattice vectors ##\vec{R} = n_i \vec{a}_i##, ##n_i \in \mathbb{Z}## (where the ##\vec{a}_i## are the crystal basis), and Hamiltonian for an electron in the crystal is ##\hat{H} = \left( -\frac{\hbar^2}{2m}...
So i derived the moment of inertia about the axis of symmetry (with height h) and I am confused about the perpendicular axis theorem.
The problem ask to find the moment of inertia perpendicular to axis of symmetry
So the axis about h, i labelled as z, the two axis that are perpendicular to z, i...
1. The factor theorem states that (x-a) is a factor of f(x) if f(a)=0
Therefore, suppose (x+1) is a factor:
f(-1)=3(-1)^3-4(-1)^2-5(-1)+2
f(-1)=0
So, (x+1) is a factor.
3x^3-4x^2-5x+2=(x+1)(3x^2+...)
Expand the RHS = 3x^3+3x^2
Leaving a remainder of -7x^2-5x+2
3x^3-4x^2-5x+2=(x+1)(3x^2-7x+...)...