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. S

    How to find uniqueness in first order pde

    Hi guys, I have a general problem that I'm not quite sure how to solve. Suppose you have a first order pde, like Ut=Ux together with some boundary conditions. You'd do the appropriate transformations that lead to a solution plus an arbitrary function defined implicitly. How would you know...
  2. W

    Finding uniqueness of PDE via. energy method

    Homework Statement consider a solution such that: -\triangle u + b\triangledown u + cu = f in domain Ω and \delta u/\delta n=g in domain δΩ where b is a constant vector and c is a constant scalar. Show that if c is large enough compared to |b|, there is uniqueness Homework Equations Energy...
  3. JJBladester

    Interval of existence / uniqueness

    Homework Statement Problem 1 of 2: Why is it that the continuity of a function in a region R and the continuity of the first partial derivative on R enables us to say that not only does a solution exist on some interval I0, but it is the only solution satisfying y(x0) = y0? Problem 2 of...
  4. I

    Prove uniqueness of solution to a simple equation

    Homework Statement Prove that the equation e^x = 1+x admits the unique solution x_0 = 0. 2. The attempt at a solution I think there should be a very simple proof based on monotonicity or the absence of inflection points, etc. But I have no idea how to do it, and what theorems are to...
  5. C

    Exploring the Uniqueness of Quark Triplets

    during the big bang there was said to be at one time a giant soup of quarks for a split of a second before atoms were formed. But why is it quarks always link up in triplets to form protons and neutrons with up and down quarks. Why didnt five or six quarks join up together with the strong...
  6. S

    Establishing uniqueness of an isomorphism

    Homework Statement Let G=[a] and G'= be cyclic groups of the same order. Show, that among the isomorphisms \theta from G to G', there is exactly one with \theta(a)=c if and only if c is a generator of G. Homework Equations The Attempt at a Solution I have managed to show the...
  7. M

    Solving Uniqueness Questions for Laplace Problem

    I am struggling with proving uniqueness and have multiple HW questions, which basically just have different boundary conditions. Something just isn't clicking for me when doing uniqueness analyses. The approach we were taught is the following: del^2(phi) = 0 in a domain del(phi) * n =...
  8. S

    Proof of uniqueness of square root

    Homework Statement Let G be a finite group in which every element has a square root. That is, for each x\epsilon G, there exists y \epsilon G such that \(y^2=x.\)Prove that every element in G has a unique square root. The Attempt at a Solution Proof: Assume not. Let k be the order of G...
  9. S

    Differential Existence and Uniqueness

    Homework Statement a) Verify that both y1(t)= 1-t and y2(t)= (-t^2)/4 are solutions of the initial value problem y-prime = (-t + (t^2 + 4y)^(1/2)) / 2 , for y(2) = -1 Where are these solutions valid? b) Explain why the existence of two solutions of the given problem does not...
  10. H

    Uniqueness of solutions to EFE?

    To what extent in general relativity do we get unique solutions to the Einstein field equations given the topology of space-time and a boundary condition? What if we're given only the boundary condition, but not the topology of space-time? I know that symmetry under diffeomorphisms means...
  11. nicksauce

    Uniqueness of Analytic Functions on a Disc

    Homework Statement Find all functions f(z) satisfying a) f(z) is analytic in the disc |z-1| < 1, and b) f(n/(n+1)) = 1 - 1 / (2n^2 + 2n+1).Homework Equations The Attempt at a Solution One can deduce by algebraic re-arrangement that one solution is f(z) = 2z / (1+z^2). But how can I show that...
  12. S

    Existence and Uniqueness of Solutions for ODE with Initial Conditions y(1)=0

    Homework Statement given this ODE with initial conditions y(1)=0 \[ (x + y^2 )dx - 2xydy = 0 \] Homework Equations solving this ODE gives us \[y = \sqrt {x\ln (x)} \] as we can see this equation is true only for x>=1 in order to use the theorem on existence and uniqueness we isulate...
  13. M

    Uniqueness Theorem: Complex Analysis Explained

    Will anybody please tell me what is the statement of the "Uniqueness theorem" in Complex analysis??
  14. quasar987

    Something strange about uniqueness of the derivative in higher dimensions

    Recall that for a function f:A\subset \mathbb{R}^n\rightarrow \mathbb{R}^m, the derivative of f at x is defined as the linear map L:R^n-->R^m such that ||f(x+h)-f(x)-L(h)||=o(||h||) if such a linear map exists. We can show that for certain geometries of the set A, when the derivative exists...
  15. C

    Proving Existence and Uniqueness of Y(x) for 0<Y(x)<1

    Hi! Thanks for reading! :) Homework Statement Y(x) is the solution of the next DFQ problem: y' = [(y-1)*sin(xy)]/(1+x^2+y^2), y(0) = 1/2. I need to prove that for all x (in Y(x)'s definition zone), 0<Y(x)<1. Homework Equations I just know that this excercise is under the title of "The...
  16. marcus

    Uniqueness of deSitter spacetime and the Standard Model (hints from two papers)

    Loll will deliver three one-hour talks at Oporto in mid July Here's the abstract Renate Loll, Quantum Gravity from Causal Dynamical Triangulations Abstract: I discuss motivation, implementation and results of the nonperturbative approach to quantum gravity based on Causal Dynamical...
  17. E

    Uniqueness of solutions to maxwell eqns

    I am wondering if anyone knows of any conditions for uniqueness of solutions to maxwells equations. For electrostatics, I have seen uniqueness formulated in terms of the potential. I am asking here how this result generalizes to the non-electrostatic case.
  18. P

    Uniqueness of quantization of Dirac field

    Let's have a theory involving Dirac field \psi. This theory is decribed by some Lagrangian density \mathcal{L}(\psi,\partial_\mu\psi). Taking \psi as the canonical dynamical variable, its conjugate momentum is defined as \pi=\frac{\partial\mathcal{L}}{\partial(\partial_0\psi)} Than the...
  19. P

    How Does the Second Uniqueness Theorem Determine the Electric Field in a Volume?

    it states that in a given volume V surrounded by conductors or for that matter infinity if the charge density \rho and the charge on each conductor is fixed then the electric field is uniquely determined in that volume V Can someone use this find the field in certain situations. For Example...
  20. F

    Uniqueness of Vector x: Exploring Scalars and Free Variables in Linear Algebra

    Homework Statement Determine if the vector x is unique. x = (-15, -3, 0) + x_3 (10, 0, 1) Note: The scalars should be vertically placed instead of horizontal. The Attempt at a Solution Seeing that there is a free variable, I said that x is not unique, but my teacher marked it wrong. Why...
  21. L

    What Are the Conditions for Uniqueness in Nonlinear Differential Systems?

    I am familiar with the existence and uniqueness of solutions to the system \dot{x} = f(x) requiring f(x) to be Lipschitz continuous, but I am wondering what the conditions are for the system \dot{q}(x) = f(x) . It seems like I could make the same argument for there existing a...
  22. E

    Griffith's Second Uniqueness Theorem

    [SOLVED] Griffith's Second Uniqueness Theorem Homework Statement I am having trouble understanding the Second uniqueness theorem in Griffith's Electrodynamics book which states that "In a volume V surrounded by conductors and containing a specified charge density rho, the electric field is...
  23. M

    Pψ=aψ and wave function uniqueness

    I want to know whether the wave function of particle is unique? If not, could we find a ψ to rationalize the equation Pψ=Aψ, in which P is the momentum operator and A is a constant. Thank you!
  24. O

    Theorem of the uniqueness and existence of a solution of ODE

    Could you please explain the theory intuitively and provide a proof to it. I understand how to apply it but i want to understand the logic behind it.
  25. O

    Existence and Uniqueness of a solution for ordinary DE

    I just don't understand the idea behind it. I hate it when they throw these theories at us without proofs or elaborate explanations and just ask us to accept and applym mthem. Anyone care to enlighten me?
  26. P

    Is there a converse of uniqueness theorem

    is there a converse of uniqueness theorem for circuits have for charged conductors. or atleast is there such a thing in case of circuit analysis ..
  27. V

    Proving the Uniqueness of a System of Equations

    Here's a problem from one of the last (or previous of last) year, which bothers me ssssssoooooooo much. I've been working on this like a day or so, and haven't progressed very far. So, I'd be very glad if someone can give me a push on this. \left\{ \begin{array}{ccc} e ^ x - e ^ y & = & \ln(1 +...
  28. S

    Uniqueness of Laplace's equation

    Homework Statement Prove the uniqueness of Laplace's equation Note that V(x,y,z) = X(x) Y(y) Z(z)) Homework Equations \frac{d^2 V}{dx^2} + \frac{d^2 V}{dy^2}+ \frac{d^2 V}{dz^2} = 0 The Attempt at a Solution Suppose V is a solution of Lapalce's equation then let V1 also be a...
  29. M

    Urgent: Existence and Uniqueness theorem

    Greating my friends, I have just returned home today from heart surgery. I still feeling the effects of the operation, because I'm affried the hospital send me home a bit to early. But I have to have these questions finished before tomorrow. So therefore I would very much appreciate...
  30. M

    Proving the Uniqueness of the Sum of 3 Primes

    if u have 3 primes: x,y,z then prove its sum m=x+y+z is unique ? Thank you
  31. S

    Can Two Distinct Functions Share Magnitude and Fourier Magnitude?

    I'm trying to uniquely determine a complex function given pairs of real valued functions derived from it. For example, if you have its real and imaginary parts, or phase and the magnitude, the function is uniquely determined from them. But what if you have the magnitude of the function and...
  32. U

    Need help with ODE and Existence and Uniqueness Thm

    Need help with ODE and "Existence and Uniqueness Thm" I'm currently helping my neice study for her exams and going though last years test there was this one question that I wasn't sure about. ------------- Considering the initial value problem, \frac{dy}{dx} = f(x,y) Where f(x,y) = (1+x)...
  33. G

    Proving the uniqueness of a polynomial

    If a polynomial p(x)=a_0+a_1x+a_2x^2+ \ldots +a_{n-1}x^{n-1} is zero for more than n-1 x-values, then a_0=a_1= \ldots =0. Use this result to prove that there is at most one polynomial of degree n-1 or less whose graph passes through n points in the plane with distinct x-coordinates. Let p(x) be...
  34. S

    General and specific existence and uniqueness proofs

    Short question: Can anyone provide me with a nice synopsis of how to go about proving the "existence" of some object as often requested in math questions such as, "prove that X really exists and is unique"? In other owrds, in general, when presented with an "existence" question, is there a nice...
  35. dextercioby

    Exploring the Unitarity Problem in Non-Renormalizable Theories

    Using this convention A \overleftrightarrow{\partial }_{\mu} B =:A \overrightarrow{\partial}_{\mu} B - A \overleftarrow{\partial}_{\mu} B one can write the QED Lagrangian density simply as \mathcal{L}_{QED} =\frac{i}{2} \bar{\Psi}_{\alpha} \left(\gamma^{\mu}\right)^{\alpha}{}_{\beta}...
  36. J

    Negate Uniqueness: Check Correctness

    I think I've got this one, I'd just like someone to check my work Negate the statement (\exists! x \in S) P(x) Since (\exists ! x \in S) P(x) \Longleftrightarrow \{(\exists x \in S) (P(x) \} \wedge \{(\forall x,y \in S) [P(x) \wedge P(y) \longrightarrow x = y \} The negation would be...
  37. K

    Lipschitz Continuity and Uniqueness

    Dear all, If a differential equation is Lipschitz continuous, then the solution is unique. But what about the implication in the other direction? I know that uniqueness does not imply Lipschitz continuity. But is there a counterexample? A differential equation that is not L-continuous, still...
  38. H

    Existence and Uniqueness Theorem

    Can anybody help me with the proof of Existence and Uniqueness Theorems.
  39. matthyaouw

    Forensic Science and Fingerprint Uniqueness: Investigating the Truth

    I often hear that fingerprints are unique, and you will share yours with no one. How much truth is there to this? I'll accept that there is very little chance you will find someone with identical finger prints, but I'm not entirely convinced that no two people will ever have identical ones. On...
  40. S

    A necessary condition for uniqueness

    Hello guys. I've been looking at uniqueness requirements for the differential equation: \frac{dy}{dx}=f(x,y);\qquad y(a)=b And the extension of this to higher-ordered equations. I'd like to understand the sufficient and necessary conditions for uniqueness. Most proofs require that...
  41. A

    Uniqueness/ Non-uniquenss of Cartesian & Polar Coordinates

    What is the difference in the "uniqueness" of the representations of Cartesian coordinates and in polar coordinates? :confused: Also, what is the non-uniqueness?
  42. M

    Uniqueness of cubic interpolating polynomial

    This is a numerical analysis question, and I am trying to prove that the p(0), p'(0), p(1), p'(1) define a unique cubic polynomial, p. More precisely, given four real numbers, p00, p01, p10, p11, there is one and only one polynomial, p, of degree at most 3 such that p(0) = p00, p'(0) = p01...
  43. C

    Uniqueness Theorem for homogenous linear ODEs

    Consider the system of linear differential equations: X' = AX where X is a column vector (of functions) and A is coefficient matrix. We could just as well consider a first order specific case: y'(x) = C(x)y We know that the soltuion will be a subset of the vector space of continuous...
  44. E

    How Do Splitting Fields Relate to Each Other?

    In my class notes, I have two theorems which don't quite seem to fit together. Maybe you can help me out. Thm 1 If p(x) in F[x] splits in K, then E=F(a1,...,an) is the splitting field of p(x) in K (the a_i's are the roots of p(x)). Thm 2 If p(x) in F[x], then the splitting field of p(x) is...
  45. K

    Help with proof of the uniqueness of limits.

    Good evening, I am a first year engineer here and a first time poster also. I had a problem that has been bugging me for the last few days; after much head-scratching and tree-killing, I may have solved it. I am, however, not sure at all if all my assumptions along the way are correct. So...
  46. P

    Uniqueness Theorem's for Vector Fields

    I recall an prof of mine (from a grad EM course) telling me that once the divergence and curl are of a vector field is specified then the vector field is determined up to an additive constant. Helmhotz's Theorem states that any vector field whose divergence and curl vanish at infinity can be...
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