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.
Homework Statement
A high diver(m = 62kg) walks off a platform 15 meters above the water below (assume velocity inital = 0). The diver reaches a depth of 2.2 metres in the pool before coming to a stop.
1. What is the diver's change in kinetic energy (Answer: -9114J)
2. What is the average force...
I am reading Bruce Cooperstein's book: Advanced Linear Algebra ... ...
I am focused on Section 1.6 Bases and Finite-Dimensional Vector Spaces ...
I need help with the proof of Theorem 1.16 ...
Theorem 1.16 and its proof reads as follows:
Question 1
In the second paragraph of above proof...
Homework Statement
Here is a link to the paper I am working through: http://www.ams.org/journals/proc/1970-025-01/S0002-9939-1970-0262849-9/S0002-9939-1970-0262849-9.pdf
Homework EquationsThe Attempt at a Solution
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I am working on the first line of the proof. This is what I thus far...
Homework Statement
Hi everyone,
I have a problem that has me stumped and would appreciate some pointers as to where I am going wrong and maybe point me in the right direction for solving the problem.
The problem is in essence to use the "Work-Energy Theorem" to find the co-efficient of kinetic...
From the Ehrenfest theorem, we know that the equation below is correct for any state ## \psi ##.
##m\frac{d^2}{dt^2}\langle x \rangle_{\psi} =-\langle \frac{\partial V(x)}{\partial x} \rangle_{\psi} ##
But then one of the definitions of coherent states is states for which the expected value of...
Hi.
I've only ever seen Noether's theorem formulated ond proven in the framework of Lagrangian mechanics. Is it possible to do the same in Newtonian mechanics, essentially only using F=dp/dt ?
The "symmetries" in the usual formulation of the theorem are symmetries of the action with respect to...
Can you suggest any book for learning multinomial theorem and its application in permutation and combinations problems? I am also looking for a book for learning Permutations and Combinations. (Right now, I am using a problem oriented book by Marcus. But I want a book for learning the basics as...
Homework Statement
We have a system of two coupled Langevin equations
dr/dt=kr-yrr+nr(t)
dp/dt=kpr-ypp+np(t)
where the ki,yi are constants and ni(t) are noise terms satisfying <ni(t)>=0 and <ni(t')ni(t'')>=qiδ(t'-t'') (this is zero if the two indices differ).
The physical background of these...
Hello all,
Guys in my textbook they state that on a point mass at point outside spherical shell of uniform density, the gravitational force is just as if the entire mass of the shell is concentrated at the Centre of shell.
The text also states, the force of attraction due to a hollow...
Why are the solutions satisfying ##\psi(x+l)=\lambda\,\psi(x)## (4.191) the only physically admissible solutions? (##l## is the period of the periodic potential.)
We may argue that the probability of finding an electron at ##x##, ##|\psi(x)|^2##, must be the same at any indistinguishable...
I don't understand how the author get to these point. Please help me as i have been spending so much time trying to figure this out but to no avail. Thanks for your help
Source: http://phys.columbia.edu/~nicolis/NewFiles/Noether_theorem.pdf
Homework Statement
Prove that \sum\limits_{k=0}^l{n \choose k}{m \choose l-k} = {n+m \choose k}Homework Equations
Binomial theorem
The Attempt at a Solution
[/B]
We know that (1+x)^n(1+x)^m = (1+x)^{n+m}
which, by the binomial theorem, is equivalent to:
{\sum\limits_{k=0}^n{n \choose...
Usually its said that the violation of Bell's inequality means that any theory that contains the assumptions of locality and realism doesn't agree with QM and observations. But sometimes I hear people talk about counter-factual definiteness instead of realism(or maybe the presence of both!) as...
There's a theorem in Euclidean Geometry that says: "Let $\Delta$ and $\Delta'$ be two right triangles. If the hypotenuse and a leg of $\Delta$ has the same measure as the hypotenuse and a leg of $\Delta'$, then $\Delta\cong\Delta'$." Wikipedia says this is only a sufficient condition, by I don't...
Homework Statement
Using Parseval's theorem,
$$\int^\infty_{-\infty} h(\tau) r(\tau) d\tau = \int^\infty_{-\infty} H(s)R(-s) ds$$
and the properties of the Fourier transform, show that the Fourier transform of ##f(t)g(t)## is
$$\int^\infty_{-\infty} F(s)G(\nu-s)ds$$
Homework Equations...
(Edited to make an answer more likely)
So first let's quickly summarize what this is. If you have some closed curve c(t) around a set of fluid elements, Kelvin's circulation theorem says that the circulation around this curve is constant as the curve and its corresponding fluid elements move...
Assuming you are lifting a block up 1 meter from rest to rest with constant work. You know that the work is -deltaU or 10. However, you also know W=deltaKE which is 0. You finally know that W=Fx=10*F. How do you explain why the numbers are different? Thanks!
Hey!
I want to discretize a fluctuation dissipation theorem for the white noise ζ of a stochastic differential equation on a 2D domain (sphere). For that I integrate over "Finite Volume" elements with area A and A' (see below).
\begin{eqnarray*} \int_{A} d A \int_{A'} d A'...
I tried my best but I wasn't able to solve this can someone please provide me with a detailed solution.
Here 's the question : Establish the expression of Vs/Ve (complex) using Thevnin's theorem
Here is the circuit :
I spent 4 hours trying to solve this but I had no clue how. I'am having...
I am a little confused over part 1 of the fundamental theorem of calculus. Part 2 makes perfect sense to me. I guess my confusion is if we have an integral g(x) defined from [a, b], and we are looking at point x, how do we know that g'(x) = f(x)? It makes sense in the idea that they are...
Homework Statement
Calculate the moment of inertia of a uniform rigid rod of length L and mass M, about an axis perpendicular to the rod through one end.
Homework Equations
Parallel axis theorem: I = Icm + MD2
Long thin rod with rotation axis through centre: Icm = 1/12 ML2
Long thin rod with...
There's something I don't quite get about most illustrations about Bell's inequality theorem. I will explain what:
Consider a pair of entangled photons fired at two arbitrarily oriented polarizers. I most explications, it seems the author suggests that the hidden variable represents the binary...
Homework Statement
A uniform solid ball of mass m and radius R rolls without slipping down a plane inclined at an angle f above the horizontal. Find the frictional force and the acceleration of the center of mass.[/B]Homework Equations
τ=I*α
so:
fs*r=I*a
Mg-Fs=ma
Moment of inertia for...
Homework Statement
x(t) = 6cos(t)−cos(6t) y(t) = 6sin(t)−sin(6t) 0 <= t <= 2*pi
I need to find the area cm2 with Th Green.
I need to find the radius and the center coordinate
Homework EquationsThe Attempt at a Solution
$ = integral
1/2* ( 2*pi$0 ((x)dy - (y)dx) dt )
1/2 (2*pi$0...
Homework Statement
An engineer is measuring a quantity q. It is assumed that there is a random error in each measurement, so the engineer will take n measurements and reports the average of the measurements as the estimated value of q. Specifically, if Yi is the value that is obtained in the...
Using Abel's thrm, find the wronskian between 2 soltions of the second order, linear ODE:
x''+1/sqrt(t^3)x'+t^2x=0
t>0
I think I got the interal of 1/sqrt(t^3) to be 2t/sqrt(t^3) but this is very different to the other examples I've done where a ln is formed to cancel out the e in the formula...
I have been told on another forum I post to that there is a revolutionary theorem in physics which proves beyond doubt that the wavefunction (I presume meaning the one originally described by Schrodinger) is physically real. I have had various exchanges with the contributor who has told me this...
Given any integer A, and a positive integer B, there exist unique integers Q and R such that
$$A= B * Q + R$$ where $$ 0 ≤ R < B$$.
When is says that $$Q$$ and $$R$$ are unique, what does that mean? That they are different from each other?
For any int $$n $$ , prove that $$ 4 | n (n^2 - 1) (n + 2)$$.
I know I have to use the quotient remainder theorem, but I'm wondering how to go about this problem.
I'm not sure how to phrase this problem in English.
I've just watched half a dozen or so videos on shell theorem and I just can't get my head around something that none of the videos address directly, but seems so counter-intuitive I am assuming my understanding is incorrect. Can anyone help me out here?
With all the usual simplifying conditions...
Hi,
I am having some problems conceptualizing the Euler's Theorem. Any help will be greatly appreciated.
In Goldstein's book the Euler's theorem is stated as 'Any displacement of a rigid body, whose one point remains fixed throughout, is a rotation about some axis', then he has proven that the...
Noether's theorem said that because of homogeneity in time the law of conservation of energy exists. I am bit of confused and I am not sure is also time inversion some consequence of this. For example in the case of free fall we have symmetry ## t \rightarrow -t##. I am sometimes confused of...
This just showed up from a team led by Zeilinger, for those interested in loophole-free Bell tests:
http://arxiv.org/abs/1511.03190
A significant-loophole-free test of Bell's theorem with entangled photons
Marissa Giustina, Marijn A. M. Versteegh, Soeren Wengerowsky, Johannes Handsteiner...
Homework Statement
[/B]
Find the current that flows through the ##8 \Omega##
Homework Equations
Thévenin's theorem
The Attempt at a Solution
the theorem says that I can replace all the circuit to a power source and a resistor connected in series.
So first I need to connect all the power...
Some textbooks like (Numerical recipes the art of scientific computing) derive the DFT as a Riemann sum of the CTFT. With this in mind it would be natural then to approximate the identity
##y(t)=x*h=\mathcal{F}^{-1}\big\{XH\big\}##
with the mathlab code y=ifft(fft(x).*fft(h)) which roughly...
I'm having trouble understanding an example supposed to motivate Bayes' theorem.
Assume that 40% of all interstate highway accidents involve excessive speed on part of at least one of the drivers (event E) and that 30% involve alcohol use by at least one drives (event A). If alcohol is involved...
I am reading Joseph J. Rotman's book: Advanced Modern Algebra and I am currently focused on Section 6.1 Modules ...
I need some help with the proof of Corollary 6.25 ... Corollary to Theorem 6.22 (Correspondence Theorem) ... ...
Corollary 6.25 and its proof read as follows: Can someone explain...
Homework Statement
why the y bar is 0 ? according to the diagram , y ' has certain value , it's not 0 ! can someone help to explain ?
Homework EquationsThe Attempt at a Solution
I am reading Joseph J. Rotman's book: Advanced Modern Algebra and I am currently focused on Section 6.1 Modules ...
I need some help with the proof of Corollary 6.25 ... Corollary to Theorem 6.22 (Correspondence Theorem) ... ...
Corollary 6.25 and its proof read as follows:Can someone explain...
I am reading Joseph J. Rotman's book: Advanced Modern Algebra and I am currently focused on Section 6.1 Modules ...
I need some help with the proof of Theorem 6.22 (Correspondence Theorem) ... ...
Theorem 6.22 and its proof read as...
Homework Statement
A student experimenting with model rockets measures the speed of a vertically-launched rocket to be 18.0 m/s when it is 75.0 m above the ground on the way up. The rocket engine fires from when the rocket is at ground level to when it is 8.75 m above the ground. If the rocket...
I have made two posts recently concerning the composition series of groups and have received considerable help from Euge and Deveno regarding this topic ... in particular, Euge and Deveno have pointed out the role of the Correspondence Theorem for Groups (Lattice Isomorphism Theorem for Groups)...
I am reading Paolo Aluffi's book, Algebra: Chapter 0 ... I am currently focused on Chapter 4, Section 3: Composition Series and Solvability ...
I need help with an aspect of Aluffi's proof of the Jordan-Holder Theorem (Theorem 3.2, page 206) which reads as follows:
Theorem 3.2 and the early...
Homework Statement
Use Green''s Theorem in the plane to check:
\oint_C (xy+y^2) \> dx + x^2 \> dy
Where C is the closed curveof the region bound between the curve of y=x^2 and the line y=x
Homework Equations
\oint_C u \> dx + v \> dy = \int \int_A (\partial_x v - \partial_y u) \> dx \> dy...
So the theorem says:
Suppose that ##U## and ##V## are finite dimensional vector spaces, and that ##T:U\to V##, ##S: V \to W##. Then
##\text{dim Ker }ST \le \text{dim Ker }S + \text{dim Ker }T##.
Proof:
Set ##U_0 = \text{Ker }ST## and ##V_0 = \text{Ker }S##. ##U_0## and ##V_0## are subspaces of...
I'm trying to follow this mathematical explanation of Bell's theorem.
The problem I find is with the assumption of a probability density for the hidden variable. That implies - and my question is: am I wrong? why? - that you can expect the same distribution of such a variable for any repetition...
Find the set of solutions $x=x(r,s,t)$ such that $(r+2\mathbb{N})\cap (s+3\mathbb{N})\cap (t+5\mathbb{N})=x+n\mathbb{N}.$
Hello MHB :). Any hints for the problem?
Show that if $\text{gcd}(b,c)=1$, then $\forall r,s\in \mathbb{N}, \exists x\in \{1,...,bc\}$ such that $x\in (r+b\mathbb{N})\cap (s+c\mathbb{N}).$
Hello :). Can define an function $\varphi: \{1,...,bc\}\to \mathbb{Z}/b\mathbb{Z}\times \mathbb{Z}/c\mathbb{Z}$ at follow $x\mapsto ([x]_b,[x]_c)$...