What is Uniqueness: Definition and 244 Discussions
Uniqueness is a state or condition wherein someone or something is unlike anything else in comparison. When used in relation to humans, it is often in relation to a person's personality, or some specific characteristics of it, signalling that it is unlike the personality traits that are prevalent in that individual's culture. When the term uniqueness is used in relation to an object, it is often within the realm of product, with the term being a factor used to publicize or market the product in order to make it stand out from other products within the same category.The notion of American exceptionalism is premised on the uniqueness of the West, particularly its well-defined secularism.
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?
I have tried to follow "Symmetry, Uniqueness, and the Coulomb Law of Force" by Shaw (1965) in both asking and solving this question, but to no avail. Some of the mathematical arguments there are a bit too quick for me but, it suffices to say, the paper tries to make the "by symmetry" arguments...
Much of the theory of ordinary differential equations is based around continuous derivatives. A lot of nice theories came together with semi-group theory of linear systems and the Banach contraction theorem, but these are limited to continuous functions. Then you get into partial differential...
I would appreciate help walking through this. I put solid effort into it, but there's these road blocks and questions that I can't seem to get past. This is homework I've assigned myself because these are nagging questions that are bothering me that I can't figure out. I'm studying purely on my...
Existence: Ax = b has at least 1 solution x for every b if and only if the columns span Rm. I don't understand why then A has a right inverse C such that AC = I, and why this is only possible if m≤n.
Uniqueness: Ax = b has at most 1 solution x for every b if and only if the columns are...
Does the second uniqueness theorem just say that if there is an electric field that satisfies Gauss's law for a surface surrounding each conductor + a surface enclosing all the conductors, it is indeed the true electric field, and no other electric field will satisfy those conditions?
in this example in Griffiths' electrodynamics, he says the following :(Figure 3.7 shows
a simple electrostatic configuration, consisting of four conductors with charges
±Q, situated so that the plusses are near the minuses. It all looks very comfort-
able. Now, what happens if we join them in...
There is a nice uniqueness theorem of electrostatics, which I have found only after googling hours, and deep inside some academic site, in the lecture notes of Dr Vadim Kaplunovsky:
Notice that the important thing here is that only the NET charges on the conductors are specified, not their...
Let me first list the four axioms that a determinant function follows:
1. ## d (A_1, \cdots, t_kA_k, \cdots, A_n)=t_kd(A_1, \cdots A_k, \cdots, A_n)## for any ##A_k## and ##t_k##
2. ##d(A_1, \cdots A_k + C , \cdots A_n)= d(A_1, \cdots A_k, \cdots A_n) + d(A_1, \cdots C, \cdots A_n)## for any...
I'm studying ODEs and have understood most of the results of the first chapter of my ODE book, this is still bothers me. Suppose
$$\begin{cases}
f \in \mathcal{C}(\mathbb{R}) \\
\dot{x} = f(x) \\
x(0) = 0 \\
f(0) = 0 \\
\end{cases}.
$$
Then,
$$
\lim_{\varepsilon \searrow...
Let ##x\in\ker(T_\lambda^2)\cap\ker(T_\mu^2)##. Then the following must hold:
\begin{eqnarray}
(A^2-2\lambda\cdot A+\lambda^2I)x=0\\
(A^2-2\mu\cdot A+\mu^2I)x=0
\end{eqnarray}
Subtracting the latter equation from the former gives us:
\begin{eqnarray}
0-0&=&0\\
&=&(-2\lambda\cdot...
1. For regions that contain charge density, does the 1st uniqueness theorem still apply?
2. For regions that contain charge density, does the 'no local extrema' implication of Laplace's equation still apply? I think not, since the relevant equation now is Poisson's equation. Furthermore...
I am dealing with restricted Boltzmann machines to model distributuins in my final degree project and some question has come to my mind.
A restricted Boltzmann machine with v visible binary neurons and h hidden neurons models a distribution in the following manner:
## f_i= e^{ \sum_k b[k]...
My guess is that since there are no rows in a form of [0000b], the system is consistent (the system has a solution).
As the first column is all 0s, x1 would be a free variable.
Because the system with free variable have infinite solution, the solution is not unique.
In this way, the matrix is...
I'm new to learning about ODE's and I just want to make sure I am on the right track and understanding everything properly.
We have our ODE which is y' = 6x3(y-1)1/6 with y(x0)=y0.
I know that existence means that if f is continuous on an open rectangle that contains (x0, y0) then the IVP has...
Show that the equation $x(x+1)(x+2)\cdots(x+2020)-1=0$ has exactly one positive solution $x_0$ and prove that this solution $x_0$ satisfies $\dfrac{1}{2020!+10}<x_0<\dfrac{1}{2020!+6}$.
Hey! 😊
Show that the interpolation exercise for cubic splines with $s(x_0), s(x_1), , \ldots , s(x_m)$ at the points $x_0<x_1<\ldots <x_m$, together with one of $s'(x_0)$ or $s''(x_0)$ and $s'(x_m)$ or $s''(x_m)$ has exactly one solution.
Could you give me a hint how we could show that? Do...
We work with Maxwell's equations in the frequency domain.
Let's consider a bounded open domain ## V ## with boundary ## \partial V ##.
1. The equivalence theorem tells me that if the field sources in ## V ## are assigned and if the fields in the points of ## \partial V ## are assigned, then I...
Hi, I am writing a report on uniqueness theorems and I am at the section for non asymptotically flat spacetime. I know that if we request certain restrictions, there are the existence of certain uniqueness theorems, however for the most part there are (so far) not many and they are hard to find...
I don't understand proof of uniqueness theorem for polynomial factorization, as described in Stewart's "Galois Theory", Theorem 3.16, p. 38.
"For any subfield K of C, factorization of polynomials over K into irreducible polynomials in unique up to constant factors and the order in which the...
Following my instructor's notes the statement of the Uniqueness Theorem(s) are as follows
"If ##\rho_{inside}## and ##\phi_{boundary}## (OR ##\frac{d \phi_{boundary}}{dn}## ) are known then ##\phi_{inside}## is uniquely determined"
A few paragraphs later the notes state
"For the field inside...
I am under the impression, there is no unique solutions to Einstein's field equations for a cosmological constant, or for higher dimensional spacetimes. Has anybody got a counter example for a solution including the cosmo constant to show there are multiple solutions, for example, i know of the...
Homework Statement
how do we establish failure of uniqueness on this first order differential equation
## y(x)= x y'+(y')^2##Homework EquationsThe Attempt at a Solution
[/B]
general solutions are ## y= cx^2+c^2## where c = constant and
## y= -0.25x^2##
## -0.25x^2+cx+4c^2=0##
##x= -2c ⇒...
Hey! :o
We have the initital value problem $$\begin{cases}y'(t)=1/f(t, y(t)) \\ y(t_0)=y_0\end{cases} \ \ \ \ \ (1)$$ where the function $f:\mathbb{R}^2\rightarrow (0,\infty)$ is continuous in $\mathbb{R}^2$ and continuously differentiable as for $y$ in a domain that contains the point $(t_0...
Hello! (Wave)
Let $\mathbb{R}[x]_{ \leq n}$ be the vector space of the real polynomials of degree $\leq n$, where $n$ a natural number. I want to show that there is a unique $q(x) \in \mathbb{R}[x]_{\leq n}$, with the property that $\int_{-1}^1 p(x) e^x dx=\int_0^1 p(x) q(x) dx$, for each $p(x)...
How do you show that there can be only one tangent space at a given point of a manifold? Geometrically it's pretty obvious in 3 dimensions, as one notices that there can be only one tangent plane at a point. But how could we show that using equations?
Homework Statement
I am looking at the wikipedia proof of uniqueness of laurent series:
https://en.wikipedia.org/wiki/Laurent_seriesHomework Equations
look above or belowThe Attempt at a Solution
I just don't know what the indentity used before the bottom line is, I've never seen it before...
I'm trying to really get a grasp on proofs of uniqueness.
Here is a model problem: Prove that ##x=-b/a## is the unique solution to ##ax+b=0##.
First method:
First we show existence of a solution: If ##x = -b/a##, then ##a(-b/a)+b = -b+b = 0##.
Now, we show uniqueness: If ##ax+b=0##, then...
Homework Statement
[/B]
The proposition that I intend to prove is the following. (From Terence Tao "Analysis I" 3rd ed., Proposition 6.1.7, p. 128).
##Proposition##. Let ##(a_n)^\infty_{n=m}## be a real sequence starting at some integer index m, and let ##l\neq l'## be two distinct real...
I am trying to negate ##\exists ! x P(x)##, which expanded means ##\exists x (P(x) \wedge \forall y (P(y) \rightarrow y=x))##. The negation of this is ##\forall x (\neg P(x) \lor \exists y (P(y) \wedge y \ne x))##. How can this be interpreted in natural language? Is it logically equivalent to...
Homework Statement
Show that for every α ∈ ℂ with α ≠ 0, there exists a unique β ∈ ℂ such that αβ = 1
Homework Equations
Definition[/B]: ## \mathbb {F^n} ##
## \mathbb {F^n} ## is the set of all lists of length n of elements of ## \mathbb {F} ## :
## \mathbb {F} ## = {## (x_1,...,x_n) : x_j...
Dear Everyone,
Directions: Decide whether the statement is a theorem. If it is a theorem, prove it. if not, give a counterexample.
There exists a unique integer n such that $$n^2+2=3$$.
Proof:
Let n be the integer.
$$n^2+2=3$$
$$n^2=1$$
$$n=\pm1$$
How show this is unique or not? Please...
Hello, I am learning about smooth analytic functions and smooth nonanalytic functions, and I am wondering the following:
Is there a theorem that states that for any real analytic functions f and g and a point a, that if at a f=g and all of their derivatives are equal, that then f=g?
Consider y' = 1/sqrt(y)
I seem to be able to find a unique solution given the initial condition of the form y(c) = 0, but the theorem says I won't be able to do so, so I am kind of confused.
I just want some clarifications. Does the uniqueness and existence theorem say anything about the...
I am working on a 2-D planar problem in the x-y direction, dealing with stresses, strain, displacements. Under the linear elastic relation and after substitution I can write the following:
##
\begin{bmatrix}
\sigma_{xx} & \sigma_{xy} \\
\sigma_{xy} & \sigma_{yy}
\end{bmatrix} = \mu...
So if E and E' are both extensions of K so that both E and E' are splitting fields of different families of polynomials in K[x], then E and E' are not isomorphic, correct? They need to be splitting fields for the same family of polynomials in K[x], correct?
If you have a density matrix \rho, there is a basis |\psi_j\rangle such that
\rho is diagonal in that basis. What are the conditions on \rho such that the basis that diagonalizes it is unique?
It's easy enough to work out the answer in the simplest case, of a two-dimensional basis: Then \rho...
That the metric tensor is not uniquely determined by the EFE and what this might entail has been a source of debate for about a century.
A way to view the problem is to decide what the manifold that has the property of diffeomorphism invariance and background independence exactly is in the...
Hi,
I am learning ODE and I have some problems that confuse me.
In the textbook I am reading, it explains that if we have a separable ODE: ##x'=h(t)g(x(t))##
then ##x=k## is the only constant solution iff ##x## is a root of ##g##.
Moreover, it says "all other non-constant solutions are separated...
Hello! (Wave)
Let $(\star)\left\{\begin{matrix}
\Delta u=0 & \text{ in } B_R \\
u|_{\partial{B_R}}=\phi &
\end{matrix}\right.$.
Theorem: If $\phi \in C^0(\partial{B_R})$ then there is a unique solution of the problem $(\star)$ and $u(x)=\frac{R^2-|x|^2}{w_n R} \int_{\partial{B_R}}...
Problem:
y'=((x-1)/(x^2))*(y^2) , y(1)=1 . Find solutions satisfying the initial condition, and determine the intervals where they exist and where they are unique.
Attempt at solution:
Let f(x,y)=((x-1)/(x^2))*(y^2), which is continuous near any (x0,y0) provided x0≠0 so a solution with y(x0)=y0...
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...
Homework Statement
Homework Equations
Leibniz notation: dy/dx = f(x) g(y)
integral 1/g(y) dy = integral f(x) dx
The Attempt at a Solution
integral 1/y dy = integral sqrt (abs x) dx
ln (y) = ? because sqrt (abs x) is not integrable at x =0
Then my thought is that y=0 is not unique
When I was taking a look at this page, I noticed that she is "known for proving the local existence and uniqueness of solutions to the vacuum Einstein Equations". But this doesn't make sense to me(the uniqueness part). Just consider the Minkowski and Schwarzschild solutions. They're both vacuum...
Homework Statement
Determine whether existence of at least one solution of the given initial value problem is guaranteed and, if so, whether uniqueness of the solution is guaranteed.
dy/dx=y^(1/3); y(0)=0
Homework Equations
Existence and Uniqueness of Solutions Theorem:
Suppose that both...
Homework Statement
Here, V is a vector space.
a) Show that identity element of addition is unique.
b) If v, w and 0 belong to V and v + w = 0, then w = -v
Homework EquationsThe Attempt at a Solution
a)
If u, 0', 0* belong to V, then
u + 0' = u
u + 0* = u
Adding the additive inverse on both...
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 the proof of Lemma 10.1 on the uniqueness of a tensor product ... ... Before proving the uniqueness (up to an...
Spivak proves that limit of function f (x) as x approaches a is always unique.
ie...If lim f (x) =l
x-> a
and lim f (x) =m
x-> a
Then l=m.
This definition means that limit of function can't approach two different values.
He takes definition of both the limits.
He...
My question is best illustrated by an example from a Griffiths book on E&M:
"A point charge q is situated a distance ##a## from the center of a grounded conducting sphere of radius R (##a>R##). Find the potential outside the sphere... With the addition of a second charge you can simulate any...