Uniqueness Definition and 229 Threads

Homework Statement Is the ordering of a ordered field unique? That is, is it possible to have different ordering set(the order), we call P1 and P2, both able to make a field F into a ordered field? Homework Equations no. The Attempt at a Solution First I tried to assume now...

Hi, for awhile I was agonizing over part b) of this http://books.google.com/books?id=WZX4GEpxPRgC&lpg=PP1&dq=lang%20complex%20analysis&pg=PA62#v=onepage&q&f=false" of Theorem 3.2 in Lang's Complex Analysis. But I think part of the reason was that I kept concentrating on the second sentence...

Homework Statement I have set up this problem for myself. Let P be a system of the form x' = Ax + Bu y = Cx + Du The definition of a "state" is: "x(t) is a state for a system P if knowledge of x at some initial time t_{0} and the input u(t), t \geq t_{0} is sufficient to uniquely determine...

Are there cases where there's more than one binary operation to choose from by which to define a Lie algebra for a given vector space?

How does one prove that the smooth structure on 2-manifolds is unique? Source? Thx!

Hello All, I am trying to define a uniqueness of a member of a set, please bear with me as my notation is not as refined as it ought to be: For a set X: { x(i) } union { f(x(j)) = true, where j is not equal to i } = { x(i) } what I am trying to say is, for this set X there exists...

Homework Statement You have point charge a distance "d" above infinite conducting plane held at V = 0. What is the potential when you remove charge to infinity? Homework Equations The Attempt at a Solution I think I incorrectly used Coulomb's law between the charge (+q...

Hi.. The wikipedia article on euler angles claims that the Euler angles in zxz convention are unique if we constrain the range they are allowed to take (except in the case of the gimbal lock). This seems reasonable. But can someone give me a reference... a book or a paper where this is...

I had a few most questions which should be trivial for the group theorists out there, but since I'm still relatively new to this, they have me stumped: 1. Given a presentation, how can one verify it is unique? 2. Given a presentation, how can one verify it is minimal aside from the obvious...

Homework Statement Coeffcient Data and Existence and Uniqueness of Solutions. Assuming that a (not equal to) 0, and an equation that restricts a; b; c; d so that the following system has only the trivial solution. (1) ax1 + bx2 = 0 (2) cx1 + dx2 = 0 Hint: Find the echelon form of the...

There is one thing I don't understand about this and is that besides the Dirichlet and Neumann conditions there seems to be a third one which is important when the method of images is used and is never mentioned. The problem is that Newmann condition requires especification of \frac{\partial\phi...

I read a lot of books on the uniqueness theorem of Poisson equation,but all of them are confined to a bounded domain \Omega ,i.e. "Dirichlet boundary condition: \varphi is well defined at all of the boundary surfaces. Neumann boundary condition: \nabla\varphiis well defined at all of the...

Homework Statement Demonstrate that if u_1 and u_2 are solutions of the wave equation \frac{\partial ^2 u}{\partial t^2} - \triangle u=0 such that u_1 (0,x)=u_2(0,x), \partial _t u_1 (0,x)=\partial _t u_2(0,x) and such that the difference "tends to 0 at infinity" sufficiently quickly, then...

Investigate uniqueness of the differential equation the question is in the image attached

Function "uniqueness".. Ok, pardon the complete lack of terminology here. I can define a function with one parameter such that no two different inputs give the same output. Example: f(x) = x + 1 No value of x gives the same result as another value of x. I believe that it is impossible to...

Homework Statement Let the function: f : I→ I be continuous on I and differentiable on the open set I for I := [0,1] Now I need to use Rolle’s Theorem to show that if f'(x) is not equal to 1 in (0, 1), then there is exactly one such point t Homework Equations I know...

I've only learned differential equations for use in physics, and never took a rigorous math course on all their amazing features. So I'm hoping someone can teach me a bit here, in the context of this question: Consider Maxwell's equations in vacuum, units don't matter here so I'll get rid of...

Homework Statement Its number four on this link: http://www.math.pitt.edu/~dwang/math0280/math0280-r1.pdf" The Attempt at a Solution Well I reduced it to echelon form, and that's not really what I have the question on. But I have three equations now, but I am not sure what values of k would...

Homework Statement Can anyone help me with proving the uniqueness of a limit? The one that stated that a limit, L, only exists if the left and right hand limits at that point are the same? Homework Equations The Attempt at a Solution I started by saying that let us say a function f(x) has two...

Find integers s and t such that 1 = 7*s + 11*t. Show that s and t are not unique. I can find numbers that satisfy this question, t=2, s=-3 and t=-5, s=8, that show s and t are not unique, but this doesn't seem to be rigorous and I'm not sure where to start with proving this.

Hi, I have a question, which seems deceptively simple to me, but when I thought about it, I couldn't really come up with a rigorous proof. Here goes, Are the roots of a polynomial equation unique? Suppose we have a general monic polynomial equation: z^{n} + c_{1}z^{n-1} + c_{2}z^{n-2} +...

[for all of the following, "lim" means the limit as n->∞] Let an be a sequence of real numbers. Theorem: if lim an = L and lim an = M, then L=M. (Incorrect) "Proof": lim an = L and lim an = M Thus, L = lim an = lim an = M (transitive property) Therefore, L=M. To me, every step in...

If I have a PDE like Ux-Uy=0 and U(x,0)=f(x) when x in [0,1]. Then is there an uniqueness solution exist at point (5,1)? How can I explain it using characteristics lines? Thanks

We know that there are several different infinities, and there appears to be some kind of duality between infinity and zero. So how do we know that zero is unique? There as several distinct concepts of "nothing" in the english language that are often confused, as exemplified in the statement...

Homework Statement Show that the trace functional on n X n matrices is unique in the following sense. If W is the space of n X n matrices over the field F and if f is a linear functional on W such that f(AB) = f(BA) for each A and B in W, then f is a scalar multiple of the trace function. If...

Homework Statement Solve the IVP. Is your solution unique? Explain. ty' + (t-2)y = (t^4)*(e^t) y(0)=0 Homework Equations Theorem: If p(t) and g(t) are continuous functions on an open interval a< t < b and the interval contains t0, then there is a unique solution to the IVP on...

Homework Statement Show that the infinite cyclic group Z is the unique group that is isomorphic to all its non-trivial proper subgroups Homework Equations The Attempt at a Solution Due to the fact that Z is cyclic and that every subgroup is a cyclic group, every subgroup of Z is a...

Homework Statement Show that this problem has a unique solution: \frac {dy}{dx}=\frac{4x+2e^{y}}{2+2x^2} given that y(0) = 0. Homework Equations Test for exactness: If (when rewritten into (2+2x^2)y' - 4x+2e^y = 0 ; which i hope is correct) My = Nx then there is an exact...

Hi, This topic has been masterfully avoided in my classes, but several proofs of theorems in multivariate calculus use the existence of a parametrization like this: Let f:\mathbb{R}^2\to\mathbb{R}. Then we can write: f(x,y)=g(t)=f(x(t),y(t)) And from this, we can get some interesting...

Hello, I was trying to prove that the Laplace transform is unique and was wondering if anyone could tell me if I've made any errors in my attempt. Here it is: Suppose L(f) = L(g), where L() denotes the Laplace transform. We want to show that f = g. By linearity of the transform, L(f - g) = 0...

Hi All, I am dealing with the heat equation these days and in an attack of originality I thought I would find a new solution to it, namely (dT/dt)=d^2T/dx^2 has a solution of the type T(x,t) = ax^2+2t Now, I do not know much about the existence and uniqueness of PDE solutions, but...

Existence and Uniqueness of solutions (pretty urgent) Homework Statement I need to solve some problems and I've given one as an example. The question is if there is existence and uniqueness of solutions to the DE Homework Equations u'(x) = sin(u(x)) The Attempt at a Solution I...

Homework Statement I have a situation with a charge distribution for a system of static charges in a vacuum. It then asks to state the uniqueness theorem for such a system. Homework Equations The Attempt at a Solution I know that the uniquessness theorem means that once you have...

Homework Statement The differential equation that models the volume of a raindrop is \frac{dv}{dt} = kv^{2/3} where k = 3^{2/3}(4 \pi)^{1/3} A) Why doesn't this equation satisfy the hypothesis of the Uniqueness Theroem? B) Give a physical interpertation of the fact that solution to this...

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...

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.