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.
"Write out a series of three or more different whole numbers in geometric progression, starting from one, so that the numbers should add up to a square. So like, 1 + 2 + 4 + 8 + 16 + 32 = 63 (one short of a square)"(can't find an actual real life example)
I can't seem to find an answer for this?
Homework Statement
This theorem comes from the book "The Real Numbers and Real Analysis" by Bloch. I am having a hard time understanding a particular part of the proof given in the book.
Prove the following theorem:
There is a unique binary operation +:ℕ×ℕ→ℕ that satisfies the following two...
Hello, It's been puzzling for me to try to understand this issue. To begin with it is clear that there are basically two principles, the Position-Momentum uncertainty and the Time-Energy uncertainty. It is also clear that there are at least two different interpretations attached to both. One is...
Okay, I am studying Baby Rudin and I am in a lot of trouble.
I want to show that a closed interval [a,b] is compact in R. The book gives a proof for R^n but I am trying a different proof like thing.
Since a is in some open set of an infinite open cover, the interval [a,a+r_1) is in that open set...
Homework Statement
I was given the problem of determining if the Converse of the Intermediate Value Theorem in my book was true. Below is my theorem from the book.
Homework EquationsThe Attempt at a Solution
I had looked at the converse and tried to draw some examples, and I am thinking it...
Hi... New to this forum. Be kind!
I did not study physics at university, and consider myself an armchair physicist. I am a computer programmer by trade. I first came across Bells inequalities a few years ago, while working with a fello programmer who did have a PHD in physics. Its pretty...
So I have this question that goes like this, for a classical 1D system we are given an Hamiltonian of the form of an Harmonic Oscilator. However the term for the potential is infite when ##x\leq0## and the usual harmonical oscillator potential otherwise. The question is: is the equipartition...
Homework Statement
in this problem , the author make π1 = D(dp/ dx) / ρ( V^2) , and make π3 as μ/ ρVD , how if i want to make μ/ ρVD (reciprocal of reynold number ) as π1 and make D(dp/ dx) / ρ( V^2) as π3 ?
Homework EquationsThe Attempt at a Solution
since we know that π1 is function of (...
Hi i am fatih from turkey.i am high school student.question is "how many squares are in an rectangle subdivided into unit squares?"(a<=b)
My theorem about this question.Please write your comments.Thanks For your time, thanks all mathematicians !:)
Homework Statement
A polynomial P(x) is divided by (x-1), and gives a remainder of 1. When P(x) is divided by (x+1), it gives a remainder of 3. Find the remainder when P(x) is divided by (x^2 - 1)
Homework Equations
Remainder theorem
The Attempt at a Solution
I know that
P(x) = (x-1)A(x) +...
Homework Statement
Suppose you're at a college campus. 3/4 of the people on the campus are students or professors from that college, and the rest 1/4 aren't. When asked a question, students and professors from that college will give you a correct answer every time, and those that aren't from...
Let's say there is a 5 sided cube that is missing the bottom face.
Obviously, there is a boundary curve at the middle of this cube that goes around the 4 sides, front, right, back, and left.
This boundary curve forms the boundary of the top half of the cube with the 5 faces and the bottom...
the first step of the Plancherel's Theorem proof found in: http://mathworld.wolfram.com/PlancherelsTheorem.html, says:
let be a function that is sufficiently smooth and that decays sufficiently quickly near infinity so that its integrals exist. Further, let and be FT pairs so that...
Homework Statement
why there is a need to change π3 to π3 ' (which is inverse of reynold number)? (in 2nd picture)
Homework EquationsThe Attempt at a Solution
why can we do so ? i was told that π1 = f( π2 , π3 , ...)
if we use π3' , which is this will change the original meaning of π1 = f(...
I often read (for example, in Wikipedia on "Rosser's Trick") that in order for a proof of Gödel's First Incompleteness Theorem, one assumes an efficient consistent theory of numbers which includes a "sufficient fragment of elementary arithmetic". What minimum would qualify? Is Robinson's Q a...
Today I heard the claim that its wrong to use Stokes(more specificly divergence/Gauss) theorem when trying to get the Einstein equations from the Einstein-Hilbert action and the correct way is using the non-Abelian stokes theorem. I can't give any reference because it was in a talk. It was the...
Homework Statement
http://www-mdp.eng.cam.ac.uk/web/library/enginfo/aerothermal_dvd_only/aero/fprops/dimension/node9.html
in the link above , the author stated that F / (rho)(D^2)(v^2) = f( (rho)(v)(D) / (μ) ) ,
Homework EquationsThe Attempt at a Solution
can i rewrite in in anotgher way ...
Is this an abuse of Rolle's theorem?
Rolle's theorem
If a real-valued function f is continuous on a proper closed interval [a, b], differentiable on the open interval (a, b), and f(a) = f(b), then there exists at least one c in the open interval (a, b) such that f'(c) = 0.
##[x_1, x_1]##...
Hi All.
I have a doubt concerning the limit:
$$ \lim_{n \to \infty} \frac{\pi (n)}{Li(n)} = 1 $$.
This mathematical statement does not imply that both functions converge to the same value. The main reason is that both tend to infinity as n tend to infinity. I would like to ask you if it is...
Hi,
I have a question about identifying closed and open surfaces.
Usually, when I see some exercises in the subject of the divergence theorem/flux integrals, I am not sure when the surface is open and needed to be closed or if it is already closed.
I mean for example a cylinder that is...
Homework Statement
Prove the following statement:
Let f be a polynomial, which can be written in the form
fix) = a(n)X^(n) + a(n-1)X^(n-1) + • • • + a0
and also in the form
fix) = b(n)X^(n) + b(n-1)X^(n-1) + • • • + b0
Prove that a(i)=b(i) for all i=0,1,2,...,n-1,n
Homework Equations
3. The...
Homework Statement
https://projects.exeter.ac.uk/fluidflow/Courses/FluidDynamics3211-2/DimensionalAnalysis/dimensionalLecturese4.htmlaccording to this link , when we form the pi group , we need to put an exponent for the non-repeating variable ,( in this case , delta P is non-repeating variable...
Homework Statement
[/B]
Polynomial f(x) is divisible by ##x^2-1##. If f(x) is divided by ##x^3-x##, then the remainder is...
A. ##(x^2-x)f(-1)##
B. ##(x-x^2)f(-1)##
C. ##(x^2-1)f(0)##
D. ##(1-x^2)f(0)##
E. ##(x^2+x)f(1)##
Homework Equations
Remainder theorem
The Attempt at a Solution
[/B]...
Homework Statement
i was told by my lecturer that when we choose the repeating variables in pi buckingham theorem , we can choose based on 3 property , which is geometry property which consists of (length , width and area) , then followed by flow property ( velocity , acceleartion, discharge)...
Hi,
Etherington't reciprocity theorem states that distances measured by angular separation and by luminosity differ. My question is which one (if any of them) is the actual distance. I can understand they might differ in an expanding universe, but there's still a physical distance in such one...
I am reading Bruce N. Coopersteins book: Advanced Linear Algebra (Second Edition) ... ...
I am focused on Section 10.3 The Tensor Algebra ... ...
I need help in order to get an understanding of an aspect of Example 10.11 and Definition 10.7 in Section 10.3 ...
The relevant text in...
Bloch's theorem states that the wave functions for electrons in a periodic potential have the form:
ψn,k(r) = un(r)exp(ik⋅r)
, where un has the same periodicity as the potential.
Bloch's theorem is used to calculate energy bands, and my question is:
Does the n in un label the different bands...
Homework Statement
Hi everybody! I have a math problem to solve, I'd like to check if I understand well the Banach fixed-point theorem in the case of Euclidean norm and how to deal with maximum norm.
Check if the following functions ƒ: ℝ2 → ℝ2 are strictly contractive in relation to the given...
Homework Statement
Homework Equations
one dimensional Reynold's transport theorem
The Attempt at a Solution
[/B]
I started with this equation, and tried to expand it using the equation given in #2.
This is the farthest I have gotten so far. I got stuck from here. I do not know how to...
Hi,
Does anyone know why k has to be real in an infinite system for bloch's theorem. I understand that the wavefunction becomes unphysical in an infinite system as it diverges. Why does that mean k has to be real?
f(x)=u(x)exp(ikx)
Homework Statement
For the network of constant current shown in Figure 4 it is known that R1 = 50 Ω and , R = 10 Ω. When the switch P is
in the 1-position , current I = 50 mA and Ip = 70 mA known i . When the switch P is in
the 2-position , current I' = 40 mA and Ip' = 90 mA are known ...
Or basically anything that isn't a positive integer.
So I can prove quite easily by induction that for any integer n>0, De Moivre's Theorem (below) holds.
If ##\DeclareMathOperator\cis{cis} z = r\cis\theta, z^n= r^n\cis(n\theta)##
My proof below:
However I struggle to do this with...
Homework Statement
Homework Equations
The Attempt at a Solution
I only know that they gave the parameterization of the circle: r(t) = <cost, sint, 2>.
My problem is, did they already give the curl of F in the line integral? I don't understand why dx, dy, and dz are separated like that.
In de Broglie's original proof of the theorem of phase harmony, the frequency of the moving wave of energy mc^2 (not the internal periodic phenomenon wave) is multiplied by the following term
##freq * ( t - \frac{\beta * x}{c} ) ##
Does anyone have an idea where the fraction comes from? All...
I am reading Bruce N. Coopersteins book: Advanced Linear Algebra (Second Edition) ... ...
I am focused on Section 10.3 The Tensor Algebra ... ...
I need help in order to get a basic understanding of Theorem 10.8 which is a Theorem concerning the direct sum of a family of subspaces as a...
I am reading Bruce N. Coopersteins book: Advanced Linear Algebra (Second Edition) ... ...
I am focused on Section 10.3 The Tensor Algebra ... ...
I need help in order to get a basic understanding of Theorem 10.8 which is a Theorem concerning the direct sum of a family of subspaces as a...
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 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...