Uniqueness Definition and 246 Threads

Hi all. I need more brain power than I can muster. This is a two part question for a software function I need to write. I'm trying to 1) pre-determine how many n% unique ways there are to intermix items from various lists and 2) produce output of only combinations that are n%...

Hi, Everyone: A question about finding the inverse of a matrix M using elementary row operations (ERO's) E_k (where E_k is either a row-exchange, a scaling of a row by k, or adding the multiple of one row to another row ) to do row-reduction in reduced-row-echelon format, to end...

What's so unique about sodium chloride dissolving in H20? I mean, does the electronic, rotational, and vibrational degree of freedom differ from other compound dissolving in water? Any special characteristics in*its wave function or other properties?

uniqueness of limits: please check which answer i should use :) Prove that in a metric space (X, d) limits are unique. [xn] -> x and xn ->y then x = y By contradiction: Assume x =y. let |x - y|/3 (Can I just make a random assumption like this?) |x - y| = |f(x) - x|+|f(x) - y| defn of a...

In your own words, what exactly is the purpose of the Existence Uniqueness Theorem and why is it useful

Let p be a continuous function such that for all r1, r2 in [0,R], ∫r2r1p(r)dr=(r22-r12)/R2. I'm trying to prove that p(r)=(2/R2)r. Question: Must p be unique? I'm not sure how to prove/disprove this.

I'm studying for my numerical analysis final on tuesday, and I know this is going to be one of the problems, so any help is greatly appreciated. Homework Statement State and prove existence and uniqueness for the solution of the linear least squares problem. Homework Equations y \approx...

Using the existence and uniqueness criteria, give the region (call it D) in the x-y plane consisting of all points (xo, yo) such that there is a unique solution. Choose a point in D as your initial condition, show that the equation is exact, then use the fact to solve the associated initial...

Homework Statement The Attempt at a Solution I can show that the complementary solution y_c solves L[y]=0 and any initial conditions for a unique choice of the c_i's, using the standard "Wronskian and invertible matrix proof". I'm stuck on this part though, how can I prove it for y(x) = y_c...

Hi I'm trying to learn more about the Unruh effect, and was wondering if someone could comment on how exactly the lack of Poincare symmetry in a general curved space leads ambiguity in the notion of "particles". Why exactly do we associate particles in QFT with positive frequency modes...

Homework Statement Show that the DE y'=\sqrt{y} has more than one solution when y(0) = 0 by finding two of them. Why does Picard's uniqueness theorem not apply? The Attempt at a Solution By standard ODE techniques, y=\frac{1}{4}[2c+c^2+t^2] but if y(0)=0 then y=\frac{t^2}{4} only...

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?

Homework Statement Let p(t) and q(t) be continuous on \mathbb{R}. Is it possible for the function y=e^t-(t^2/2)-t-1 to be a solution of the equation y''+p(t)y'+q(t)y=0 ? Why or why not? Homework Equations Existence/uniqueness theorem. The Attempt at a Solution Supposedly I...

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...

sorry about my English Homework Statement In Purcell 3.7 (Problem) and Griffith there is a question,look at fig. (we have 4 conductors with charges +Q,-Q,+Q,-Q (b),what will happen if we connect them with tiny wires in pairs ) Griffith say that c distribution of charge ,can't be a...

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

Homework Statement Suppose that f is an integrable function (and suppose it's real valued) on the circle with c_n=0 for all n, where c_n stands for the coefficient of Fourier series. Then f(p)=0 whenever f is continuous at the point p. Homework Equations The Attempt at a Solution...

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 Solve the IVP and determine if the solution is unique and explain why. Homework Equations t*y'+(t-2)y=t^4*e^t, y(0)=0 The Attempt at a Solution t*y'+(t-2)y=t^4*e^t y'+ ((t-2)/t)y=t^3*e^t integration factor=e^(integral((t-2)/t dt)=e^t/(t^2)...

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...

Hey all, I was working a little on parabolic pde, and came across this (comes up in regularity theory). Consider a Hilbert triple V\subset H\subset V^* (continuous embeddings) and a linear operator A(t) from V to V*, where t ranges in some interval [0,T]. Now let w\in H^1(0,T;V^*)\cap L^2(0,T;V)...

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...