Uniqueness Definition and 246 Threads

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. arivero

    I The uniqueness of D=4

    I have found an interesting rabbit hole, because I thought the question of why we live in 3+1 was mainly a matter of footnotes and off-press debates. But it seems if was touched early by Weyl, Ehrenfest and Whitrow https://einsteinpapers.press.princeton.edu/vol13-doc/764 And then elaborated...
  2. P

    I A non-empty intersection of closures of level sets implies discontinuity

    Let X and Y be topological spaces, and suppose f: X \to Y is such that there exist distinct points c and c' of Y such that S = \overline{f^{-1}(\{c\})} \cap \overline{f^{-1}(\{c'\})} \neq \varnothing. What conditions must be placed on X and Y so that it follows that f is discontinuous at each...
  3. H

    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?
  4. E

    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...
  5. A

    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...
  6. A

    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...
  7. J

    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...
  8. Ahmed1029

    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?
  9. Ahmed1029

    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?
  10. Ahmed1029

    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...
  11. C

    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...
  12. H

    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...
  13. M

    A Help with understanding why limit implies uniqueness

    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...
  14. Eclair_de_XII

    B Proving the uniqueness of eigenspaces

    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...
  15. P

    I Confusion over applying the 1st uniqueness theorem to charged regions

    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...
  16. J

    I Restricted Boltzmann machine uniqueness

    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]...
  17. S

    Linear Algebra uniqueness of solution

    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...
  18. V

    Existence and Uniqueness For ODE

    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...
  19. anemone

    MHB Proving the Uniqueness of Positive Solution of $x(x+1)(x+2)\cdots(x+2020)-1=0$

    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 $\dfrac{1}{2020!+10}<x_0<\dfrac{1}{2020!+6}$.
  20. M

    MHB Uniqueness of Cubic Spline Interpolation: How Can We Prove It?

    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...
  21. U

    I Equivalence principle and the Uniqueness theorem

    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...
  22. M

    I Uniqueness Theorems in Non-Flat Spacetime

    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...
  23. S

    I Don't understand proof of uniqueness theorem for polynom factorization

    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...
  24. P

    Corollary of the Uniqueness Theorem in Electrostatics

    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...
  25. M

    I Uniqueness theorems for black holes

    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...
  26. chwala

    Failure of uniqueness for this first-order differential equation

    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 ⇒...
  27. M

    MHB Existence and uniqueness of solution

    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...
  28. evinda

    MHB Can a Unique Polynomial Satisfy Specific Integral Equations?

    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)...
  29. K

    I Uniqueness of tangent space at a point

    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?
  30. binbagsss

    Quick question on Laurent series proof uniqueness

    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...
  31. Mr Davis 97

    I Understanding Proof of Uniqueness

    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...
  32. L

    Proof of uniqueness of limits for a sequence of real numbers

    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...
  33. Mr Davis 97

    I Negating the uniqueness quantifier

    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...
  34. Bishamonten

    Understanding Existence and Uniqueness

    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...
  35. C

    MHB How to show uniqueness in this statement for integers

    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...
  36. J

    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?
  37. A

    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...
  38. P

    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...
  39. PsychonautQQ

    I Are Splitting Fields Unique for Different Polynomial Families?

    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?
  40. stevendaryl

    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...
  41. R

    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...
  42. mr.tea

    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...
  43. evinda

    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}}...
  44. A

    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...
  45. 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...
  46. nysnacc

    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
  47. ShayanJ

    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...
  48. Ian Baughman

    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...
  49. S

    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...
  50. Math Amateur

    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...
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