What is Theorem: Definition and 999 Discussions

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.

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1. Quantum Virial Theorem Derivation

Using the time derivative of an operator, and expanding out, I got to this: $$\frac{d}{dt}\langle\hat{x}\hat{p}\rangle=\frac{i}{\hbar}\left\langle\left[\hat{H},\hat{x}\hat{p}\right]\right\rangle+\left\langle\frac{\partial}{\partial t}\left(\hat{x}\hat{p}\right)\right\rangle$$ Expanding using...
2. I Bloch Analysis proof of Theorem 2.5.5 (Definition by recursion)

Want to understand how set C contains ##N## x H. H is only defined to be a set with element e and as the domain/range of function k. Is this enough information to conclude that the second set in the cartesian product W is H and not a subset of H? My thinking is to show that ##N## and H satisfy...
3. Help me with Taylor's theorem please

I am trying to grasp how the last equation is derived. I understand everything, but the only thing problematic is why in the end, it's ##+O(\epsilon)## and not ##-O(\epsilon)##. It will be easier to directly attach the image, so please, see image attached.
4. A Extended version of Cochran's Theorem

Hi, Anyone know if Cochran's Theorem can be extended to many-factor Anova, to determine the distribution of statistics used therein? Maybe similar other results can be used for determining relevant stats in use in multifactor Anova?
5. To prove the "##m^{\text{th}}## Powers Theorem"

Statement : Let me copy and paste the statement as it appears in the text on the right. Attempt : I could attempt nothing to prove the identity. The best I could do was to verify it for a given value of the ##a's, m, n##. I am not even sure what this identity is called but I will take the...
6. B Why is the growth of exps & logs so fast & slow, respectively?

Hi, PF, there goes the theorem, questions, and attempt THEOREM 4 If ##x>0##, then ##\ln x\leq{x-1}## PROOF Let ##g(x)=\ln x-(x-1)## for ##x>0##. Then ##g(1)=0## and ##g'(x)=\displaystyle\frac{1}{x}-1\quad\begin{cases}>0&\mbox{if}\,0<x<1\\ <0&\mbox{if}\,x>1\end{cases}## (...). Thus...
7. I Question from a proof in Axler 2nd Ed, 'Linear Algebra Done Right'

My question is motivated by the proof of TH 5.13 on p 84 in the 2nd edition of Linear Algebra Done Right. (This proof differs from that in the 4th ed - online at: https://linear.axler.net/index.html chapter 5 ) In the proof we arrive at the following situation: ##T## is a linear operator on a...
8. B Is everything in math either an axiom or a theorem?

Considering math as a collection of true/logically consistent statements, I see only two possibilities: either the statement is true and can be proven, which means it's a theorem. Or it's true but cannot be proven, which means it's an axiom. Is there a third possibility? Or maybe more? I feel...
9. Help please with De Moivre's Theorem (raising a complex number to a power)

Hi all any help on this would be great I cant seem to progress with the theorem, z= -2 + j > R sqrt (-2)'2 + (-1)'2 r = 2.24 0= Arctan(-1) = 26.57 Polar form = 2.24(cos(26.58)+jsin(26.58) -2 Demoivre - (cos0+jsin0)'n = cosn0 +jsinno Could some one...
10. A About Proof of Engel's Theorem

in the Proof of Engel's Theorem. (3.3), p. 13: please, how we get this step: ##L / Z(L)## evidently consists of ad-nilpotent elements and has smaller dimension than ##L##. Using induction on ##\operatorname{dim} L##, we find that ##L / Z(L)## is nilpotent. Thanks in advance,
11. Work energy KE theorem for a book being lifted up in a gravitional field

For this, From the work kinetic energy theorem, if we assume that the book and the earth is the system, and that the finial and inital speed of the system is zero, then is the work KE theorem there is no net work done on the system. However, clearly there is work done on the system is shown by...
12. Engineering Solve Superposition Theorem for R1, R2 Circuit

i have this question and i just want to make sure i am on the right track as i know there are quite a few steps to this to get to the final soultions i need to find Current at r1 voltage at r1 current at r2 voltage at r2 i have so far split this into 2 drawings so i am dealing with one power...
13. Why is continuity necessary before applying the Extreme Value Theorem?

For this problem, Why cannot we say that ##f(2.999999999) ≥ f(x)## and therefore absolute max at f(2.99999999999999) (without reasoning from the extreme value theorem)? Many thanks!
14. I Proof of Column Extraction Theorem for Finding a Basis for Col(A)

Theorem: The columns of A which correspond to leading ones in the reduced row echelon form of A form a basis for Col(A). Moreover, dimCol(A)=rank(A).
15. A Virial theorem and translational invariance

According to the virial theorem, $$\left\langle T\right\rangle =-{\frac {1}{2}}\,\sum _{k=1}^{N}{\bigl \langle }\mathbf {F} _{k}\cdot \mathbf {r} _{k}{\bigr \rangle }$$ where ##N## is the number of particles in the system and ##T## is the total kinetic energy. It is often claimed that this...
16. Work-Energy Theorem and Friction

My final answer is different from the official one in the back of the book, and I can't figure out what I did wrong. This is my attempt: Let block 1 be the vertically moving block and let block 2 be the horizontally moving one. Also, let ##m_1 = 6.00 ~\rm{kg}##, ##m_2 = 8.00 ~\rm{kg}##, ##v_0...
17. A Main Theorem of Renormalization, but in physicist-speak

I have been reading a few things about the mathematical formulation of perturbative QFT, specifically in terms of the Stuckelberg-Petermann RG, the Gell-Mann-Low RG, and their difference. Unfortunately I lack the mathematical background to understand these things in depth, and I'm having a...
18. I Using the orbit-stabilizer theorem to identify groups

I want to identify: ##S^n## with the quotient of ##O(n + 1,R)## by ##O(n,R)##. ##S^{2n+1}## with the quotient of ##U(n + 1)## by ##U(n)##. The orbit-stabilizer theorem would give us the result, but my problem is to apply it. My problem is how to find the stabilizer. In 1 how to define the...
19. Work-Energy Theorem Exercise

My solution is different from the official solution and I don't understand what I did wrong. Here is my solution: The magnitude of the initial velocity is ##|v_0| = 12.0~\rm{m/s}##, so the vertical component of the initial velocity is ##v_{0-y} = (12.0 \sin{25^{\circ}})~\rm{m/s}##. Then I use...
20. A Von Neumann's uniqueness theorem (CCR representations)

Hi Pfs, Please read this paper (equation 4): https://ncatlab.org/nla b/files/RedeiCCRRepUniqueness.pdf It is written: Surprise! P is a projector (has to be proved)... where can we read the proof?
21. I Martingale, Optional sampling theorem

In this exercise, we consider simple, nonsymmetric random walk. Suppose 1/2 < q < 1 and ##X_1, X_2, \dots## are independent random variables with ##\mathbb{P}\{X_j = 1\} = 1 − \mathbb{P}\{X_j = −1\} = q.## Let ##S_0 = 0## and ##S_n = X_1 +\dots +X_n.## Let ##F_n## denote the information...
22. Residue Theorem applied to a keyhole contour

I'm really struggling with this one. A newbie to using the residue theorem. I'm trying to solve this by factorising the denominator to find values for z0 and I have: ##z=\frac{-\sqrt{2}+i\sqrt{2}}{2}## and ##z=\frac{-\sqrt{2}-i\sqrt{2}}{2}## I also know that sin(3π/8)=...
23. Logical Proof: Theorem (Truths of Logic) A iff ~~A

My thought was to break up the sentence into its equivalent form: (A ->~~A) & (~~A -> A) From there I assumed the premise of both sides to use indirect proofs, so: 1. ~(A -> ~~A) AP 2. ~(~A or ~~A) 1 Implication 3. ~~A & ~~~A 2...
24. I Some reasoning about Alfvén’s frozen-in flux theorem

Alfvén’s theorem is very famous in plasma physics. It is also often used in astrophysics. The link in Wiki: https://en.wikipedia.org/wiki/Alfv%C3%A9n%27s_theorem However, after a series of continuous reasoning, it seems that this theorem has problem. What errors can be hidden in the reasoning...
25. I Disproving Carnot's Theorem: Is There an Error in My Thinking?

But if I would assume that the efficiency of the carnot's engine is greater than the other engine and the carnot engine is driving the other engine backward as a refrigerator ,that would lead to the same contradiction hence disproving carbot's theorem! Is there something wrong I have done...
26. Solving a Motion Problem with Work-Energy Theorem

I have been trying to solve the following problem: Point-like object at (0,0) starts moving from rest along the path y = 2x2-4x until point A(3,6). This formula gives the total force applied on the object: F = 10xy i + 15 j. a) Find the work done by F along the path, b) Find the speed of the...
27. I Differential operator in multivariable fundamental theorem

I'm referring to this result: But I'm not sure what happens if I apply a linear differential operator to both sides (like a derivation ##D##) - more specifically I'm not sure at what point should each term be evaluated. Acting ##D## on both sides I'll get...
28. Understanding the Equipartition Theorem for Ideal Gases

Hi, I am unfortunately stuck with the following task I started once with the hint that at very low temperatures the diatomic ideal gas behaves like monatomic gas and has only three degrees of freedom of translation ##f=3##. If you then excite the gas by increasing the temperature, you add two...
29. A Hidden Assumptions in Bell's Theorem?

There has been a lot of discussion on Bell's theorem here lately. Superdeterminism as a Bell's theorem loophole has been discussed extensively. But I have not seen discussion about Karl Hess, Hans De Raedt, and Kristel Michielsen's ideas, which essentially suggest that there are several hidden...
30. I The far reaching ramifications of the work-energy theorem

The work-energy theorem is the connection between expressing mechanics taking place in terms of force-and-acceleration, ##F=ma## and representing mechanics taking place in terms of interconversion of kinetic energy and potential energy. The following statements are for the case that there is a...
31. I Lars Olsen proof of Darboux's Intermediate Value Theorem for Derivatives

Here is Lars Olsen's proof. I'm having difficulty in understanding why ##y## will lie between ##f_a (a)## and ##f_a(b)##. Initially, we assumed that ##f'(a) \lt y \lt f'(b)##, but ##f_a(b)## doesn't equal to ##f'(b)##.
32. Use the Virial theorem to show the following...

The expression ##\langle \cal H \rangle_k## is the expected value of the canonical ensemble. The Hamiltonian is defined as follows, with the scaling ##\lambda## ##\lambda \cal H ## : ##\lambda H(x_1, ...,x_N)=H(\lambda^{a_1}x_1,....,\lambda^{a_N}x_N)## As a hint, I should differentiate the...
33. Applying Stoke's Theorem: A Hint

I considered the vector field ##a \times F##, and applied Stoke's theorem. I obtained that $$\int_C (a \times F) \cdot dr = \int_C (F \times dr ) \cdot a.$$ Now, $$\nabla \times (a \times F) = a (\nabla \cdot F) - (a \cdot \nabla) F.$$ Using Stoke's theorem for the vector field ##a \times F##...
34. My proof of the Geometry-Real Analysis theorem

Consider a convex shape ##S## of positive area ##A## inside the unit square. Let ##a≤1## be the supremum of all subsets of the unit square that can be obtained as disjoint union of finitely many scaled and translated copies of ##S##. Partition the square into ##n×n## smaller squares (see...

50. How and why does it mention Theorem 13?

Solution Write the Taylor formula for ##e^x## at ##x=0##, with ##n## replaced by ##2n+1##, and then rewrite that with ##x## replaced with ##-x##. We get: $$e^x=1+x+\dfrac{x^2}{2!}+\dfrac{x^3}{3!}+\cdots+\dfrac{x^{2n}}{(2n)!}+\dfrac{x^{2n+1}}{(2n+1)!}+O(x^{2n+2})$$...