Uniqueness Definition and 246 Threads

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...

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...

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...

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...

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...

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...

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 =...

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...

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...

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...

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...

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...

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

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...

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...

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...

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.

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...

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...

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...

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...

[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...

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!

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.

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?

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 ..

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 +...

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...

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...

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

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...

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)...

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...

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...

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}...

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...

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...

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

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...

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...

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

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...

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...

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...

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...

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...