What is Existence: Definition and 569 Discussions

I have attatched my Paradox for the existence of 4,5 and 7 using Brocard's problem . I don't know where i have gone wrong as 4,5,7 exist, surely.

As far as I can understand it, Picard's Existence and Uniqueness of ODEs theorem relies on the fact that a the given function f(x,t) in the initial value problem dx/dt = f(x,t) x(t0) = x0 is Lipschitz continuous and bounded on a rectangular region of the plane that it's defined on. And the...

Let A = [a b; c d] a 2x2 matrix with complex entries. Suppose that A is row-reduced and also that a+b+c+d =0 . Prove that there are exactly three such matrices... so i realize that there are seven possible 2x2 matrices that are row-reduced. [1 0; 0 1], [0 1; 1 0], [0 0; 1 0], [0 0;0 1]...

We've done a little bit on existence/uniqueness of solutions, and there's one thing that's a little confusing to me. We have a theorem which (paraphrased) says that if you have a linear ODE with an initial value problem, then a solution exists on the largest open interval containing t0 on which...

I have been thinking out of the box and I have come to the conclusion that something cannot exist without consuming something else. I know this sounds really wacky (for want of a better phrase :)) which is why I posted this in the Quantum forum, lol. We know plants consume nutrients and use...

How do you prove that if \textrm{card}(X)\leq\textrm{card}(Y) is not true, then \textrm{card}(X)\geq\textrm{card}(Y) must be true? In other words, if we know that no injection X\to Y exists, how do we prove that an injection Y\to X must exist? This is not the same thing as what...

Homework Statement Consider the IVP compromising the ODE. dy/dx = sin(y) subject to the initial condition y(X) = Y Without solving the problem, decide if this initial value problem is guaranteed to have a unique solution. If it does, determine whether the existence of that solution is...

why do minority carriers exist? in a p-type material, why don't the minority carriers recombine (and get annihlated) with excess holes?

Hi all, I was reading the book by Herbert Federer on Geometric Measure Theory and it seems he proves the existence of the Tensor Product quite differently from the rest. However it is not clear to me how to prove the existence of the linear map "g" in his construction. He defines F as the...

Is pure quantum gravity known to exist? I had thought it exists in 3D, but Strominger writes http://arxiv.org/abs/0906.1313 "Determining Z for pure 3D quantum Einstein gravity - if it exists - is an important open problem" Eg. Does the Turaev-Viro model not describe 3D QG?

Homework Statement for the differential equation t^2y''-2ty'+2y=0 with the general solutions y=C(t) + D(t^2) where C and D are constants. given the inital solution y(0)=1 and y'(0)=1 there are no solutions that exist. Why does this not contradict the Existence and Uniqueness Theorem...

I know this probably sounds weird, but I have a research problem that requires "random" analog circuits. Basically what this means is that I create Spice netlists by randomly adding linear and/or nonlinear components of random types with random node and parameter values. This works fine and I...

In certain philosophy discussions the concept of a square circle sometimes comes up as an example of something that can be proven not to exist. It occurred to me that the impossibility of its existence depends on: 1. the definitions one uses for square and circle; and 2. the geometry in...

dear friends, i've noticed that the main feature of my inner experience is the sense that something exists, and that the main feature of my outer experience is the sense that something is solid, and I've wondered if these two experiences might be two views of the same thing: solid existence...

Hello, I'm looking for a book on the connection between projective geometry and perspective. Many books vaguely mention perspective drawing as the historical reason for projective geometry, but they don't go into a deep connection between the two: they mostly just use it to argue the...

the big bang theory suggests that all the matter around us was once infinitely concentrated at some particular high density region...this matter then spread out across the universe following the big bang... but how did this energy/matter come into existence in the first place??..

I'm wondering if there is a monotonically increasing function with a jump discontinuity at every rational (or any other dense, countable subset of the reals). Here's a specific candidate that I've come up with: Let g:\mathbb{Q} \cap [0,1] \rightarrow \mathbb{R} take the rational p/q (p and q...

Homework Statement a) Does f(z)=1/z have an antiderivative over C/(0,0)? b) Does f(z)=(1/z)^n have an antiderivative over C/(0,0), n integer and not equal to 1. Homework Equations The Attempt at a Solution a) No. Integrating over C= the unit circle gives us 2*pi*i. So for at least one...

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

Hi everyone, Does someone knows where I can the statement about the existence theorem of caratheorory solutions? Thank you

I have a non-linear differential equation and I wonder whether it has a soliton solution or not. How can I approach to the problem? So far I have never dealt with non-linear differential equations, hence, any suggestion is appreciated.

Hi together ... I wonder how one can deduce the existence of antiparticles from the Klein-Gordon equation. Starting from (\frac{\partial^2}{\partial t^2} - \nabla^2 + m^2) \Psi(t,\vec{x})=0 one gets solutions \Psi(t,\vec{x})=\exp(\pm i (- E t + \vec{p} \cdot \vec{x})) leading to E^2=p^2 +...

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...

Imagine that you are an astronaut standing very far from a black hole.Now you throw a luminous body (a bulb may be) directly towards it.Now as it gets nearer the black hole,the light from the bulb as you observe it becomes more red-shifted.Eventually from your frame(consider it is an inertial...

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...

Hello all, I've recently used a property that seems perfectly valid, yet upon further scrutiny I could not come up with a way to prove it. Here is what I would like some help on. Given two sets X and Y and functions f and g mapping X into Y, with the property that f is injective and g is...

Homework Statement Is there a group of order 12 which contains one involution and ten elements of order 3? Give an example or otherwise prove that such a group cannot exist. 2. The attempt at a solution Let G be a group of order 12 = (p^k)*m where p is a prime number, k is greater than or...

Let f be a continuous on the closed and bounded interval [a,b] and x_1, x_2, …, x_n ∈ [a,b]. Show that there necessarily exists ξ ∈ [a,b] such that: f (ξ= [f(x_1) + f(x_2) + …f(x_n)] / n How can I start this problem i am really confused! please help !

Homework Statement Let's take function given by a condition: f(x) = \begin{cases} \frac{1}{q^2} \ iff \ x = \frac{p}{q} \ $nieskracalny$,\\ 0 \ iff \ x \notin \mathbb{Q} \end{cases} Prove the existence of the derivative of f in all points x \notin \mathbb{Q}. The Attempt at a...

I wonder if anyone knows or can figure out an answer to this question I've been thinking about: In a smooth pseudo-riemannian manifold like those in GR, and given some arbitrarily long spacelike geodesic, is it always ( or almost always, e.g. except for passing through a singularity) possible...

How we can prove the existence of black holes? And how they are located in in the space as they can absorb light too?

Homework Statement Just a clarification: the two last equations must hold in an open neighborhood of the point (2, 1, -1, -2), not just at that point. Homework Equations The Attempt at a Solution I have to do an existence proof. The shortest way of accomplishing this would...

Homework Statement Let k be any positive integer. Prove that there exists a positive integer multiple n of k such that the only digits in n are 0s and 1s. (Use the pigeonhole principle.) Homework Equations The General Pigeonhole Principle If more than mk things are distributed into k...

Unintelligent Design theory Take a look at the Sun (with proper darkened glasses of course). Have you ever wondered why is it there? In fact there are so many other places it could be (other galaxies, etc.) that it is quite improbable it is there. Therefore someone, let's call him the...

Here is a potentially neat problem. Let x(t),y(t) (for all t\in \mathbb{R}) be polynomials in t. Prove that for any x(t),y(t) there exists a non-zero polynomial f(x,y) in 2 variables such that f(x(t),y(t))=0 for all t. The strategy is to show that for n sufficiently large, the polynomials...

Do electric and magnetic fields occur simultaneously in the same spot anywhere around the globe? (other than during solar flares) If the field is named "electromagnetic" wouldn't that means exactly this simultaneity? Thank you.

Let V be a finite-dimensional vector space over the field F and let T be a linear operator on V. Let c be a scalar and suppose there is a non-zero vector \alpha in V such that t \alpha = c \alpha. Prove that there is a non-zero linear functional f on V such that T^{t}f=cf, where T^{t}f=f\circ T...

Given the equation y'= xg(x,y) , suppose that g and (partial) dg/dy are defined and continuous for all (x,y). Show the following: 1) y(x)=0 is a solution 2)if y=y(x), x in (a,b) is a solution and if y(x0)>0, x0 in (a,b), then y(x)>0 for all x in (a,b) Please i need your help.

Homework Statement Let T be a family of finite subsets of the natural numbers N = {1, 2, 3,...} such that if A and B are any members of T, then the intersection of A and B is nonempty. (a) Must N contain a finite subset F such that the intersection of A, B and F is nonempty for any sets A...

Example of a "pure existence metaproof" http://en.wikipedia.org/wiki/Existence_theorem A pure existence theorem is a theorem which states the existence of something, but the proof of the theorem does not indicate a construction of the thing in question. As the article mentions, this is...

Homework Statement Prove or disprove a) Let f:X---->Y. If f possesses more than 1 left inverse yet has no right inverse, then f has strictly more than 1 left inverse. b) If f and g are maps from a set X to X and fog is one to one, then f an g are both injective one to one. Homework Equations...

Prove that there exist (a) 5 points in the plane so that among all the triangles with vertices among these points there are 8 right-angled ones; (b) 64 points in the plane so that among all the triangles with vertices among these points there are at least 2005 right-angled ones.

Homework Statement suppose that h \inC(R) and f\inC(R^2) satisfies |f(t,x)|\leqh(t) |x| for all (t,x) \inR^2. Show that for any point , the IVP : x' = f( t, x) x( \tau)=\varsigma has a solution which exists for all t in R Homework Equations The Attempt at a Solution

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...

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...

Imagine viewing the world from state X at 1 pm, and viewing the world from state Y at 2 pm. State Y is required to be in existence in principle at the viewing of state Y, and at state X equally. This is because neither is more valid of a state to view the world from. Neither can claim it is more...

A torch (flashlight US) is positioned on a train going at 70kms/hour. Relative to someone standing beside the train track the torch will have a velocity of 70kms/hour since it is on the train. When you switch on the torch, a light beam emanates. However, the light was not actually inside the...

It is easy to see, from bianchi identities, that if energy-momentum tensor is not conserved, then Einstein's equation does not have a solution. But is there a proof that if energy momentum tensor IS conserved then Einstein's equation ALWAYS have a solution?

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...

In the another thread, I queried some posters comments which were along the lines that zero is a metaphysical concept / doesn't exist. Baywax responded, as below, and my additional comments are in blue. Zero exists as much as the number 1 in that these are both language equivalents that...