# 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.

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1. ### A Von Neumann's uniqueness theorem (CCR representations)

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?
2. ### I The implications of symmetry + uniqueness in electromagnetism

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...
3. ### I Discontinuous systems? And why do we need uniqueness anyway?

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...
4. ### Prove the identity matrix is unique

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...
5. ### I Existence and Uniqueness of Inverses

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...
6. ### I When does the second uniqueness theorem apply?

For the second uniqueness theorem of electrostatics to apply, does the outer boundary enclosing all the conductors have to be at a constant potential?
7. ### I The second uniqueness theorem in electrostatics

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?
8. ### I A question about the Second Uniqueness Theorem in electrostatics

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...
9. ### I Where to find this uniqueness theorem of electrostatics?

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...
10. ### I How is uniqueness about the determinant proved by this theorem?

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...

34. ### B Uniqueness of Analytic Functions

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?
35. ### I Uniqueness and Existence Theorm

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...
36. ### A Stress-strain; strain-displacement in 2-D; uniqueness

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...
37. ### I Uniqueness of splitting fields

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?
38. ### I On uniqueness of density matrix description as mixed state

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...
39. ### A Lack of uniqueness of the metric in GR

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...
40. ### I Constant solution and uniqueness of separable differential eq

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...
41. ### MHB Uniqueness of Solution for $\Delta u=0$ in a Ball with Boundary Condition $\phi$

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}}...
42. ### I Interval of existence and uniqueness of a separable 1st ODE

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...
43. M

### I Recursion theorem: application in proof

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...
44. ### Differential equation uniqueness

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
45. ### A Local Existence & Uniqueness of Vacuum EFE Solutions

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...
46. ### Determining Existence and Uniqueness

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...
47. ### Uniqueness of identity element of addition

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...
48. ### MHB The Uniqueness of a Tensor Product

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...
49. ### Uniqueness of limit proof

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...
50. ### A question related to the method of images and uniqueness theorems

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...