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 am reading Bruce N. Coopersteins book: Advanced Linear Algebra (Second Edition) ... ...
I am focused on Section 10.1 Introduction to Tensor Products ... ...
I need help in order to get a basic understanding of Theorem 10.2 regarding the basis of a tensor product ... ...
My apologies if my...
I uploaded a picture of what I am stuck on. I understand the equation of motion 3.4.5a for a damped oscillator but I don't understand how to use binomial theorem to get the expanded equation 3.4.5b. I am no where near clever enough to figure this one out. I know how to use binomial theorem to...
Hi everyone,
I am currently working on a Coanda UAV and I am aware that there's no mathematical model to express the lifting effects of Coanda. It is more of a physical description of airflow movement. Correct me if I am wrong!
Thus, I am using the generic expression of lift to describe the...
The Cayley-Hamilton Theorem can be used to express the third invariant of the characteristic polynomial obtained from the non-trivial solution of the Eigenvector/Eigenvalue problem. I follow the proof (in Chaves – Notes on Continuum Mechanics) down to the following equation, then get stuck at...
Homework Statement
why we can't form the pi group by using repeating variable of (μ, ρ , v) or (D, v , μ ) ?
http://www-mdp.eng.cam.ac.uk/web/library/enginfo/aerothermal_dvd_only/aero/fprops/dimension/node9.html
http://www.efm.leeds.ac.uk/CIVE/CIVE1400/Section5/dimensional_analysis.htm...
Hey all, the schema theorem shows that in all probability a genetic algorithm will converge to a solution. much like the second law of thermodynamics for optimization. Although, it is taught with the genes being $$ \in (0,1, *), * \in (0,1) $$ is there a proof for non binary genes? example...
Very simple question. So I am on a homework problem, and I want to make sure that I am using this theorem correctly. My book states that the Work-Kinetic Energy Theorem is:
W=ΔK
Now the solution to this problem involved multiple forces and thus each force is doing work. So my question is, is...
Homework Statement
Determine the voltage across ##R_3## in the following figure assuming an input voltage ##v_0## of 10V is applied across the open terminals.
Let ##R_1=3\Omega##, ##R_2=15\Omega##, ##R_3=10\Omega##, ##R_4=5\Omega## and ##R_5=2\Omega## and Homework Equations
The Attempt at a...
While studying energy conservation on Morin I found this explanation about the work-energy theorem for a system.
Using Koenig theorem $$\Delta K_\textrm{system}=\Delta K +\Delta K_\textrm{internal}$$ so we have
I've got two main question on that:
Why are only external forces considered for...
I am reading Bruce N. Coopersteins book: Advanced Linear Algebra (Second Edition) ... ...
I am focused on Section 10.2 Properties of Tensor Products ... ...
I need help with an aspect of the proof of Theorem 10.3 ... ... basically I do not know what is going on in the second part of the proof...
I am reading Bruce N. Coopersteins book: Advanced Linear Algebra (Second Edition) ... ...
I am focused on Section 10.2 Properties of Tensor Products ... ...
I need help with an aspect of the proof of Theorem 10.3 ... ... basically I do not know what is going on in the second part of the...
Homework Statement
Sorry- I've figured it out, but I am afraid I don't know how to delete the thread.
Thank you though :)
Homework Equations
Below
The Attempt at a Solution
Photo below- I promise its coming! I've started by using cylindrical coordinates, but I wasn't sure if spherical...
I am reading Bruce N. Coopersteins book: Advanced Linear Algebra (Second Edition) ... ...
I am focused on Section 10.2 Properties of Tensor Products ... ...
I need help with an aspect of the proof of Theorem 10.3 regarding a property of tensor products ... ...The relevant part of Theorem 10.3...
I am reading Bruce N. Coopersteins book: Advanced Linear Algebra (Second Edition) ... ...
I am focused on Section 10.2 Properties of Tensor Products ... ...
I need help with an aspect of the proof of Theorem 10.3 regarding a property of tensor products ... ... The relevant part of Theorem...
Hi,
I am reading "mathematical analysis" by Apostol right now for a course in analysis. Since I am trying to understand the author's proof of the above theorem(3.25 in the book), but I have something that I can't understand.
He assumes that each of the nested sets contains infinitely many...
Divergence theorem states that
$\int \int\vec{E}\cdot\vec{ds}=\int\int\int div(\vec{E})dV$
And Gauss law states that
$\int \int\vec{E}\cdot\vec{ds}=\int\int\int \rho(x,y,z)dV$
If $\vec{E}$ to be electric field vector then i could say that
$div(\vec{E})=\rho(x,y,z)$
However i can't see any...
Homework Statement
Homework Equations
Work energy theorem
The Attempt at a Solution
.5mv2 - .5mv2 = (kx - μmg)d
final velocity is 0
½mv2 =(kx - μmg)d
solve for v2
((kx - μmg)d2)m
When this was wrong I tried integrating Fx but it was still wrong
Homework Statement
Can anybody give me hint how to find K if F(x)= 3x+2
The integral lower part is not the same, , how to deal with his?
Homework EquationsThe Attempt at a Solution
Please ,I need hint to start
I am reading Bruce N. Coopersteins book: Advanced Linear Algebra (Second Edition) ... ...
I am focused on Section 10.1 Introduction to Tensor Products ... ...
I need help with another aspect of the proof of Theorem 10.2 regarding the basis of a tensor product ... ...Theorem 10.2 reads as...
I am reading Bruce N. Coopersteins book: Advanced Linear Algebra (Second Edition) ... ...
I am focused on Section 10.1 Introduction to Tensor Products ... ...
I need help with an aspect of Theorem 10.2 regarding the basis of a tensor product ... ...Theorem 10.2 reads as follows:
I do not...
I am reading Bruce N. Coopersteins book: Advanced Linear Algebra (Second Edition) ... ...
I am focused on Section 10.1 Introduction to Tensor Products ... ...
I need help with another aspect of the proof of Theorem 10.2 regarding the basis of a tensor product ... ... Theorem 10.2 reads as...
I am reading Bruce N. Coopersteins book: Advanced Linear Algebra (Second Edition) ... ...
I am focused on Section 10.1 Introduction to Tensor Products ... ...
I need help with an aspect of Theorem 10.2 regarding the basis of a tensor product ... ... Theorem 10.2 reads as follows:I do not...
How do I show that $\mathbb{Z}^{\times} _{20} ≅ \mathbb{Z}_{2} \times \mathbb{Z}_{4}$?
I read that the chinese remainder theorem is the way to go but there are many versions and I can't find the right one. Most versions that I have found are statements between multiplicative groups, not from...
I am reading Bruce N. Coopersteins book: Advanced Linear Algebra (Second Edition) ... ...
I am focused on Section 10.1 Introduction to Tensor Products ... ...
I need help with another aspect of the proof of Theorem 10.1 regarding the existence of a tensor product ... ... The relevant part of...
I am reading Anthony W. Knapp's book: Basic Algebra in order to understand tensor products ... ...
I need some help with a further aspect of the proof of Theorem 6.10 in Section 6 of Chapter VI: Multilinear Algebra ...
The text of Theorem 6.10 reads as follows:
The above proof mentions Figure...
I am reading Anthony W. Knapp's book: Basic Algebra in order to understand tensor products ... ...
I need some help with an aspect of Theorem 6.10 in Section 6 of Chapter VI: Multilinear Algebra ...
The text of Theorem 6.10 reads as follows:
The above proof mentions Figure 6.1 which is...
Torricelli's theorem relates the speed of a fluid exiting an opening in a reservoir to the height of the opening relative to the top of the reservoir... V=√2gh https://en.wikipedia.org/wiki/Torricelli%27s_law
As seen in the wiki-link provided, the equation is essentially a Bernoulli's...
Hi,
I was studying about the statistical ensemble theory and facing some problem to understand these concepts ,
I have understood that the ensemble is a collection of systems which are macroscopically identical but microscopically different . In some books they are called as systems with...
A waiter believes the distribution of his tips has a model that is slightly skewed to the left, with a mean of $\$8.20$ and a standard deviation of $\$5.60$. He usually waits on about 60 parties over a weekend of work. a) Estimate the probability that he will earn at least $\$600$. b) How...
What is the difference between quantum mechanics and realism ? quantum mechanics states on statistics while the hardy assumption of EPR is that hidden variables may describe exactly the outcomes of each individual test. Bell refutes the last idea. But, he didn't need to refute the case where...
Homework Statement
Take a standard deck of cards, and deal them out into 13 piles of 4 cards each. Then, using the marriage theorem, we can show that it is always possible to select exactly 1 card from each pile, such that the 13 selected cards contain exactly one card of each rank (Ace, 2, 3...
The dark matter crisis: falsification of the current standard model of cosmology
Pavel Kroupa (AIfA, Bonn)
(Submitted on 11 Apr 2012 (v1), last revised 20 Jun 2012 (this version, v2))
The current standard model of cosmology (SMoC) requires The Dual Dwarf Galaxy Theorem to be true according to...
Homework Statement
In the figure, a block of mass m is moving along the horizontal frictionless surface with a speed of 5.70 m/s. If the slope is 11.0° and the coefficient of kinetic friction between the block and the incline is 0.260, how far does the block travel up the incline?
Homework...
My homework problem is as follows:
Consider the Laplace transform shown below.
(4s3+15s2+s+30)/(s2+5s+6)
a. What is the value of f(t=0) and f(t=∞)? Use the initial and final value theorems.
b. Find the inverse transform f(t). Use this expression to find f(t=0) and f(t=∞) and compare with the...
Hello! (Wave)We have $G(x,y)=-\frac{1}{4 \pi} \frac{1}{||\overline{x}-\overline{y}||}$ for $x, y \in \mathbb{R}^3$.I want to show that $\int_{\mathbb{R}^3} \Delta{G(x,y)} dx= 1$.
It suffices to show that $\int_{\mathbb{R}^3} \Delta{G(x,0)} dx= 1$, since setting $\overline{x}=x-y$ we have...
I won't post the whole rigorous statement of the theorem, but basically the theorem states that
If ##F(x,y) = 0## on a neighborhood of the form ##[x-\delta ,x+\delta ]\times [y- \epsilon ,y+\epsilon ]## and if ##\frac{\partial F(x,y)}{\partial y} \neq 0##, then there exists a function ##y=\phi...
Incorrect Theorem:
Suppose x and y are real numbers and x + y = 10, then x != 3 and y != 8.
(a) What’s wrong with the following proof of the theorem?
Proof. Suppose the conclusion of the theorem is false. Then x = 3 and y = 8. But then x + y = 11, which contradicts the given information...
I've read the postulate that there could be a huge spherical dark matter halo extending far beyond the edges of the Milky Way.
However, according to Newton's shell theorem, there is no net gravitational pull within a shell.
How do they arrive at the conclusion of a halo so huge?
Simplify (find the sum) of {30 \choose 0} + \frac{1}{2}{30 \choose 1}+ \frac{1}{3}{30 \choose 2} + ... + \frac{1}{31}{30 \choose 30}.
Do this is two ways:
1. Write \frac{1}{i+1}{30 \choose i} in a different way then add
2. Integrate the binomial thorem (don't forget the constant of integration)...
Homework Statement
Verify that
a) ##\frac{1}{2\pi} \int_0^{2\pi} f(e^{it})\frac{e^{it}\bar z}{1-\bar z e^{it}}dt = 0##, if ##f(w)## is analytic for ##|w|<1+\epsilon##, and that
b) ##\frac{1}{2\pi} \int_0^{2\pi} f(e^{it})\frac{e^{it}}{e^{it}-z}dt = f(z).##
for ##z = re^{i\theta}## with ##r <...
At the bottom of the picture, I couldn't understand why differentiating with respect to x gives the first integral at the right-hand side 0. Thanks for reading.
I would like to post a comment for offtopic conversation in another thread.
This is the point of no-communication theorem that measuring one particle does not change anything measurable about the other particle.
But conclusions of no-communication theorem are limited by it's assumptions (as for...
Homework Statement
We have an ordinary LRC circuit with inductance L, capacitance C and resistance R with an oscillating voltage with low frequency (U^e). Using the energy conservation law and Poynting's theorem, find the differential equation:
$$L \frac{\partial ^2}{\partial t^2}I + R...
According to Castigliano's theorem, the local compliance of an elastic structure, e.g. a cantilever, can be determined by integrating the products of stress intensity factor weight functions over the length of said structure. However, if I do that for a double cantilever beam using simple beam...
Homework Statement
2. Homework Equations [/B]
De Moivres Theorem/ Eulers formula
The Attempt at a Solution
Honestly don't know where to go with this now. I already applied De Moivres theorem at the very end. It feels like I have to do something more with either De Moivres theorem or...
Homework Statement
We have the Lagrangian $$L=\frac{1}{2}\dot q^2-\lambda q^n$$
Determine the values for n so that the Lagrangian transform into a total derivative
$$\delta q = \epsilon (t\dot q - \frac{q}{2})$$
Homework Equations
The theorem says that if the variation of action
$$\delta S =...
Homework Statement
I and J are open subsets of the real line. The function f takes I to J, and the function g take J to R. The functions are in C1. Use the mean value theorem to prove the chain rule.
Homework Equations
(g o f)' (x) = g'(f (x)) f'(x)
MVT
The Attempt at a Solution
[/B]
I know...
Just got confused that while applying the Work - Energy Theorem in a vertical Spring-Block system performing SHM (considering no other external forces other than gravity), when I apply the theorem from equilibrium position, do I consider the work done by gravity?
Homework Statement
for the alternative form of second shift property (4.8) , why he integral of (e^-sp) g(p+a) dp isn't equal to integral of (e^-sp) g(t) dp ? why it will become L{ g(t+a) } ?
Homework EquationsThe Attempt at a Solution