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
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##C_\rho## is a semicircle of radius ##\rho## in the upper-half plane.
What is
$$\lim_{\rho\rightarrow 0} \int_{C_{\rho}} \frac{e^{iaz}-e^{ibz}}{z^2} \,dz$$Homework Equations
If ##C## is a closed loop and ##z_1, z_2 ... z_n## are the singular points inside ##C##...
Consider the lagrangian of the real scalar field given by $$\mathcal L = \frac{1}{2} (\partial \phi)^2 - \frac{1}{2} m^2 \phi^2 - \frac{\lambda}{4!} \phi^4$$
Disregarding snail contributions, the only diagram contributing to ## \langle p_4 p_3 | T (\phi(y)^4 \phi(x)^4) | p_1 p_2 \rangle## at...
Homework Statement
Consider the beam shown in (Figure 1) . Suppose that a = 15 in. , b = 8 in. , c = 1 in., and d = 4 in.
Determine the moment of inertia for the beam's cross-sectional area about the x axis...
Homework Statement
Find the solution of the following integral
Homework Equations
The Attempt at a Solution
I applied the above relations getting that
Then I was able to factor the function inside the integral getting that
From here I should be able to get a solution by simply finding the...
1. Homework Statement
i attached the problem statement as an image file
Homework Equations
p(x) = (x-c)q(x) + r
The Attempt at a Solution
i've simplified it down to ((x-1)^114) / (2^114)(x+1). is there a practical way to approach this besides long division? wolfram alpha gave an extremely...
I just suppose the Bell's Ansatz for the result of measurement to be $$A (\theta,\lambda) $$
Now the parameter lambda could be anything :
-a physical quantity like the polarization angle of the incoming photon
-the coordinate of a 'world'
- the whole wavefunction.
...
In the case of the...
I'm having a little trouble starting on this problem. Can someone help? I was trying to solve it out, but I just ended up telling how to draw the center. Attached is the problem:
Hello again, everyone. Have a multivariate calculus question this time around. If anyone can point me in the right direction and help me see where WebAssign finds me wrong, it would be greatly appreciated.
1. Homework Statement
Homework Equations
∫∫ScurlF ⋅ dS = ∫CF ⋅ dr
The Attempt at a...
Homework Statement
I am trying to express ##T(\phi(x1)\Phi(x2)\phi(x3)\Phi(x4)\Phi(x5)\Phi(x6))## in terms of the Feynman propagators ##G_F^{\phi}(x-y)## and ##G_F^{\Phi}(x-y)##
where ##G_F^{\phi}(x-y) =\int \frac{d^{4}k}{(2\pi)^{4}}e^{ik(x-y)} \frac{ih}{-k.k - m^2 -i\epsilon} ##
and...
<Moderator's note: Moved from a technical forum, so homework template missing>
Sorry
i have one question to ask
how to check the v.dl part in this problem
i cannot do this problem as it is too hard to integrate the equation
How did griffith get this long-horrible equation(see the orange...
I am reading Matej Bresar's book, "Introduction to Noncommutative Algebra" and am currently focussed on Chapter 1: Finite Dimensional Division Algebras ... ...
I need help with some aspects of the proof of Theorem 1.4 ... ...
Theorem 1.4 reads as follows:
Questions 1(a) and 1(b)
In...
I am reading Matej Bresar's book, "Introduction to Noncommutative Algebra" and am currently focussed on Chapter 1: Finite Dimensional Division Algebras ... ...
I need help with some aspects of the proof of Theorem 1.4 ... ...
Theorem 1.4 reads as follows:
Questions 1(a) and 1(b)
In the above...
The thread I wanted to post my question on got closed. Recapitulating:
The best (simplest) account I have found to date for the Bell inequality (SPOT stands for Single Photon Orientation Tester):
Imagine that each random sequence that comes out of the SPOT detectors is a coded message. When...
1. The problem statement, all variables and given/
You must push a box up an incline plane (the angle being constant : a), to a person waiting to receive it, who is a distance of h(constant) vertically above you. Though the slope is slippery, there is a small amount of friction with kinetic...
I don't know how to start proving this theorem, so can someone please help? I need to prove that the circumcircles all intersect at a point M. Thank you!
Miquel's Theorem: If triangleABC is any triangle, and points D, E, F are chosen in the interiors of the sides BC, AC, and AB, respectively...
Homework Statement
The moment of inertia for a perpendicular axis through the center of a uniform, thin, rectangular metal sheet with sides a and b is (1/12)M(a2 + b2). What is the moment of inertia if the axis is through a corner?
The answer is given as this was a powerpoint lecture and it...
Homework Statement
Find the remainder of ##4^{87}## in the division by ##17##.
Homework Equations
Fermat's Little Theorem:
If ##p## is prime and ##a## is an integer not divisible by ##p##, then
##a^{p-1} \equiv 1 (\mod \space p)##
or equivalently,
##a^p \equiv a (\mod \space p)##
The...
Hello! I am a bit confused about the first Sylow theorem. So it says that if you have a group of order ##p^mn##, with gcd(n,p)=1, you must have a subgroup H of G of order ##p^m##. So, if I have a group G of order ##p^k##, there is only one subgroup of G of order ##p^k## which is G itself. Does...
Hello! (Wave)
We consider the following problem.
$$Lu=f(x) \text{ in } \Omega \\ u|_{\partial{\Omega}}=0$$
I want to show that if $c(x) \leq -c_0<0$ in $\overline{\Omega}$, then it holds that $\min\{ 0, \frac{\min_{\Omega}f(x)}{-c_0}\}\leq u(x) \leq \max_{\Omega} \{ 0...
Liouville's theorem states that the total time-derivative of the distribution function is zero along a system trajectory in phase-space. Where the system follows a trajectory that satisfies the Hamilton's equations of motion.
I have a hard time getting an inuitive understanding of this...
Homework Statement
Homework Equations
Green's theorem
The Attempt at a Solution
DO I first parametrize? For 1st part, I have 3 parametrizations, which I can then find the normal vector, and use in the integrals?
Hawking area theorem says that area of black hole generally never decrease. Penrose process says that energy can be extracted from black hole. Energy extraction will decrease mass? if yes then if mass is decreased then will area also decrease?
I am confusing things here :(
According to DeMorgan’s theorem (break the bar and change the sign), the complement of ܽa⋅b+c⋅d is a'+b'⋅c'+d' Yet both functions are 1 for ܾܽܿ abcd 1110. How can both a function and its complement be 1 for the same input combination? What’s wrong here?I honestly have no idea. I mean, shouldn't...
I am reading D. J. H. Garling: "A Course in Mathematical Analysis: Volume I Foundations and Elementary Real Analysis ... ...
I am currently focused on Garling's Section 1.7 The Foundation Axiom and the Axiom of Infinity ... ...
I need some help with Theorem 1.7.4 ... and in particular with...
Homework Statement
(See attachment: if it doesn't work, see below for poorer formatting)[/B]
Theorem 6.1 (Centered Formula of Order O(h2)). Assume that f ∈ C^3[a, b] and that x − h, x, x + h ∈ [a, b]. Then (3) f (x) ≈ f (x + h) − f (x − h) 2h . Furthermore, there exists a number c = c(x) ∈ [a...
I am reading D. J. H. Garling: "A Course in Mathematical Analysis: Volume I Foundations and Elementary Real Analysis ... ...At present I am focused on Chapter 1: The Axioms of Set Theory and need some help with Theorem 1.2.2 and its relationship to the Separation Axiom ... ...
The...
I am reading D. J. H. Garling: "A Course in Mathematical Analysis: Volume I Foundations and Elementary Real Analysis" ... ...
At present I am focused on Chapter 1: The Axioms of Set Theory and need some help with Theorem 1.2.2 and its relationship to the Separation Axiom ... ...
The...
Obviously ##\mathbb{R^2}## is convex, that is, any points ##a,b\in\mathbb{R^2}## can be connected with a line segment. In addition, ##f## is differentiable as a composition of two differentiable functions. Thus, the conditions of mean value theorem for vector functions are satisfied. By applying...
The population mean and standard deviation are given below. Find the required probability and determine whether the given sample mean would be considered unusual.
μx̄ = μ = 12,749
σ = 1.2
n = 35
For the given sample n = 35, the probability of a sample mean being less than 12,749 or greater...
I am wondering is someone could comment on a question I have recently answered. I have attached the question and my answer. Apologies for not following the standard procedure of Latex but there are drawings associated with this question. I answered section A and my results are written on the...
Homework Statement
Using paper, pencil and the Virial theorem, calculate the position uncertainty (an estimate of the vibration amplitude) of the H atom in its ground state C-H stretching mode. In more precise language, calculate the bond length uncertainty in a C-H bond due to the C-H...
Homework Statement
Using the maximum power transfer theorem, find the value of R which will result in the maximum power being delivered to R.
Homework Equations
P_max= (V_oc)^2/4(R_th))
R_L=R_th
P_RL=(V_th/(R_th+R_L))^2*R_L
The Attempt at a Solution
I have no clue where to even begin with...
I am having trouble doing this problem from my textbook... and have
no idea how to doit.
1. Homework Statement
I am having trouble doing this problem from my textbook...
Show that the equation x + y - z + cos(xyz) = 0 can be solved for z = g(x,y) near the origin. Find dg/dx and dg/dy
(dg/dx...
I know that for rigid bodies only the work-energy theorem states that the net work done on the body equals the change in kinetic energy of the body since a rigid body has no internal degrees of freedom and hence no other forms of energy such as potential energy. Is there a most generalized form...
Homework Statement
Homework Equations
Parallel axis theorem: Ip = Icm + Md^2
Icm = I = ML²/12 + 2 * mr²
3. The attempt
Ip = Icm + Md^2 ==> wrong
I = Md^2 ==> right
Why don't I need to add "Icm"?
Thanks.
Hello everyone. I need help on proofs. I have to proof the Angle-Angle-Side theorem. Can someone help me with this?
The AAS states : If triangles ABC and DEF are two triangles such that angle ABC is congruent to angle DEF, angle BCA is congruent to angle EFD, and segment AC is congruent to DF...
Homework Statement
I need to accommodate a dashpot in an intentionally simple work-kinetic energy analysis method. For example, for a box being dragged up a ramp via a rope while attached to a spring, I can deal with the work done by gravity, rope tension, spring force, and friction via the...
Hello everyone,
I am trying to find a proof for:
\[\vdash \left ( \sim \alpha \rightarrow \sim \left ( \sim \alpha \right ) \right )\rightarrow \alpha\]
I am using the L inference system, which includes the modus ponens inference rule, and the axioms and statements attached below. That's the...
Hi
I need a little help in my homework. It is not a direct problem to be solved. Rather I am supposed to find an application of Noether's theorem. All the article or papers I have found are very difficult for me to understand. In fact, I still don't understand any application of Noether's...
Homework Statement
Use the Intermediate Value Theorem to prove that any continuous function with domain [0,1] and range in [0,1] must have a fixed point.
Homework Equations
Intermediate Value Theorem (IVT) states that if a function ##f(x)## of domain [##a,b##] takes values ##f(a)## and...
Hi, I wonder why wronskian must be constant.
I know that p(x)W[u1(x),u2(x)]=constant, according to the Abel's theorem, but
wouldnt there a special case that W[u1(x),u2(x)]=c/p(x).
Then for this special case, W[u1(x),u2(x)]=/=c and satisfies Abel's theorem.
Is it ok to ignore this special case?
I'm curious if anyone has ever simulated the infinite monkeys on typewriters using a computer, and managed to generate short sentences or phrases that have appeared in books/print media before.
That would demonstrate the effectiveness of the infinite monkey theorem.
Hi All,
I would like to know if is there any problem to present and prove a theorem and a Lemma (in this order) and after that use this theorem and this lemma to prove a corollary (which is simpler to prove and not so important as the theorem).
I have looked up in some papers but with no...
I have been curious for some time, does the incompleteness theorem of mathematics have any consequences in physics? In order that I may understand your response you should know I'm was a senior math major at the university when last I was in school and my only physics background is the standard...
Hello,I had a discussion with my professor. He tried to convince me but I couldn't understand the idea.
The Stokes Theorem (Curl Theorem) is the following:
My professor says that the value of the equation should be zero whenever the area of integration is closed! (which will make a volume in...
Homework Statement
An employee goes to work from 9 am to 4 pm. He takes a nap for an average of 2 hours if he starts napping before 1 pm and naps for an average of 1 hours if he starts napping after 1 pm. His boss randomly checks up on him once during his shift. If his boss finds him napping...
I was looking at the proof of Lagrange's theorem (that the order of a group ##G## is a multiple of the order of any given subgroup ##H##) in Wikipedia:
I understand this proof fine, but I was wondering, instead of finding a bijection between cosets, is it enough to find a bijection between an...
I have read a proof but I have a question. To give some context, I first wrote down this proof as written in the book. First, I provide the recursion theorem though.
Recursion theorem:
Let H be a set. Let ##e \in H##. Let ##k: \mathbb{N} \rightarrow H## be a function. Then there exists a...
I wonder why projected area has been of much interest among physics communities, while the surface area could well be the solution unless any complex geometries are involved.
The question popped up in my head when the surface tension in a water jet was derived. Clearly the jet has a circular...