Existence Definition and 543 Threads

I'm currently working my way through the existence theorem of strong solutions for the stochastic differential equation ## X_t = X_0 + \int_0^t b(s,X_s)ds + \int_0^t \sigma(s,X_s)Bs ##, Where ## \int_0^t \sigma(s,X_s)Bs ## is the Ito integral. The assumptions are: 1: ## b,\sigma ## are jointly...

Hello! (Wave) Sentence: If $A,B$ are sets, there is the (unique) set, of which the elements are exactly the following: $\langle a,b\rangle: a \in A \wedge b \in B$. Proof: Remark: $\langle a,b\rangle:=\{ \{a\},\{a,b\}\}$ If $a \in A$, then $\{ a \} \subset A \rightarrow \{ a \} \in...

Hi guys, I stumbled upon this lovely quote from the philosopher of science Wesley Salmon: "The fool hath said in his heart that there is no null set. But if that were so, then the set of all such sets would be empty, and hence, it would be the null set. Q.E.D." (in Martin Gardner, Mathematical...

Homework Statement For every real x>0 and every n>0 there is one and only one positive real y s.t. yn=x Homework Equations 0<y1<y2 ⇒ y1n<y2n E is the set consisting of all positive real numbers t s.t. tn<x t=[x/(x+1)]⇒ 0≤t<1. Therefore tn≤t<x. Thus t∈E and E is non-empty. t>1+x ⇒ tn≥t>x, s.t...

We have a general spacetime interval ##ds^2 = g_{\mu \nu} dx^\mu dx^\nu##. One way to define an affine parameter is to define it to be any parameter ##u## which is related to the path length ##s## by ##u = as + b## for two constants ##a,b##. One can show that for the tangent vector ##u^\alpha =...

Dear friends, I read in Kolmogorov-Fomin's that the following property of measurable real or complex valued functions ##\varphi,f## defined on measure space ##X##, proven in the text for ##\mu(X)<\infty## only, is also valid if ##X=\bigcup_n X_n## is not of finite measure, but it is the union of...

Given the differential equation y'=4x^3y^3 with initial condition y(1)=0determine what the existence and uniqueness theorem can conclude about the IVP. I know the Existence and Uniquness theorem has two parts 1)check to see if the function is differentiable and 2)check to see if...

Dear friends, I read that, if ##A## is a bounded linear operator transforming -I think that such a terminology implies that ##A## is surjective because if ##B=A## and ##A## weren't surjective, that would be a counterexample to the theorem; please correct me if I'm wrong- a Banach space ##E##...

Suppose all second partial derivatives of F = F (x, y) are continuous and F_{xx} + F_{yy} = 0 on an open rectangle R. Show that F_ydx - F_xdy = 0 is exact on R, and therefore there’s a function G such that G_x = −F_y and Gy = F_x in R. ≈≈≈≈≈≈≈≈To prove that F_ydx + F_xdy = 0 is exact on R...

How to understand Uniqueness and existence theorem for first order and second order ODE's intuitively?

I am reading Beachy and Blair's book: Abstract Algebra (3rd Edition) and am currently studying Theorem 6.5.7. I need help with the proof of the Theorem. Theorem 6.5.7 and its proof read as follows:In the above proof, Beachy and Blair write: By Lemma 6.5.4, the set of all roots of $$f(x)$$ is...

Homework Statement Given: 1. a_{1} < b_{1} 2. a_{n} = \sqrt{a_{n-1}b_{n-1}} 3. b_{n} = \frac{a_{n-1} + B_{n-1}}{2} 4. The sequences a_{n} and b_{n} are convergent. Prove: The sequences a_{n} and b_{n} have the same limit. The Attempt at a Solution Assume by contradiction that...

Does the existence of the "Axis of Evil" observed by WMAP and Planck mean that there is a special frame of reference in the universe?

How does the blackbody radiation prove the existence of photons or quanta as Planck described it, I've understood how the photoelectric effect proves the existence of photons, but the blackbody radiation seems quite vague to me. I would like a basic explanation for this, thanks in advance.

Homework Statement For what values of a, from the reals, does the limit exist? lim_{x\rightarrow2} (\frac{1}{2-x}-\frac{a}{4-x^{2}}) Homework Equations I chose a so that the denominator would be one. By putting the fractions together. The Attempt at a Solution When a = 4 the...

Hi everyone, I've been studying the so-called XYZ spectroscopy and the existence of possible 4-quark states. The LHCb collaboration recently confirmed the existence of a particle called Z(4430)^-. This particle is the unambiguous evidence for the existence of 4-quark states. From what I...

Homework Statement a.) Prove ## \exists x, x \in \mathbb{R} | x^3 -x^2 = 5## I know that x = 2.1163, but how do I find this without a calculator? b.) Prove that ## \not \exists x, x \in \mathbb{R} | x^4 - 2x^2 +2 =0##The Attempt at a Solution x^2 ( x-1 )=5 for part a)? =====================...

What is 'Quantum Domain'? I read something about it being a level of existence... If it is a level of existence, what sort of classification of existence are we talking about?

If counting/positive numbers exist, do they imply the existence of negative numbers? I'd say yes, because there's always a bijection that maps the lowest counting number of the set to the highest, then the second lowest to the second highest, etc. This reversal of order/mirroring is possible...

Hi to everyone! I'm searching information about evidences of photons existence. It seems like the photoelectric effect isn't for itself a proof of photons existence. Some people tried a semi-classical discussion of this effect (Lamb - "The Photoelectric effect without photons"). I'm...

Homework Statement Given that the divergence of a vector C = 0, show that there exists a vector A such that C = curl A. Homework Equations See above. The Attempt at a Solution No clue. Can this be proved with introductory vector calculus? That's all I know, including many of the...

I am given that a DE with the form x' = f(x) is defined on the interval (c,b) where f has continuous derivative on its domain How do i show that if f(p) = f(q) = 0 and x(t) is between p and q then the maximal interval of existence of x is (-∞, ∞)

i was given that f is a real alued function defined on an open interval I with IVP x'(t) = f(x(t)) where x(s) = b how would I go to prove that if I is continuous on I and b is in I then there exists a postive number say k and a solution x for the initial value problem defined on (s-k,s+k)

Hey! :o I am looking at the following that is related to the existence of the optimal approximation. $H$ is an euclidean space $\widetilde{H}$ is a subspace of $H$ We suppose that $dim \widetilde{H}=n$ and $\{x_1,x_2,...,x_n\}$ is the basis of $\widetilde{H}$. Let $y \in \widetilde{H}$ be...

Homework Statement Prove Existence Unique Real Solution to ## x^{3} + x^{2} -1 =0 ## between ## x= \frac{2}{3} \text{and} x=1## The Attempt at a Solution ## x^{2} ( x+1) =1 ## I know that the solution is x =0.75488, but this came from some website. How do I find this number without a calculator?

Hello, My question is this. Is it possible to prove that there exist an eigenvectors for a symmetric matrix without discussing about what eigenvalues are and going into details with characteristic equations, determinants, and so on? This my short proof for that: (The only assumption is ##A##...

I don't know very much about differential geometry but from the things I know I think that the metric is somehow the quantity which specifies what kind of a geometry we're talking about(Though not sure about this because different coordinate systems on the same manifold can lead to different...

Does Casimir plates prevent photon existence only perpendicular to them? I mean, Casimir attraction arises from the fact that the plates prevent some wavelenghts of photons to exist in between them, so an imbalance arises and pushes the plates together, right? But what about photons in other...

Some sources I have checked define the Hodge dual of a form \omega \in \Omega^p as the object such that \forall \eta \in \Omega^p: \eta \wedge \omega^\star = g(\eta,\omega) \textrm{ Vol} (where "Vol" is a chosen volume form). I can see that there can be only one form with such a solution...

I am reading and trying to follow the notes of Keith Conrad on Tensor products, specifically his notes: Tensor Products I (see attachment ... for the full set of notes see Expository papers by K. Conrad ). I would appreciate some help with Theorem 3.2 which reads as follows: (see attachment...

Hello Everyone. I have a question. Suppose I have a differential equation for which I want to find the values at which \displaystyle f(x,y) and \displaystyle \frac{\partial f}{\partial y} are discontinuous, that I might know the points at which more than one solution exists. Suppose that...

Prove the following Suppose that $f$ is piecewise continuous on $$[0,\infty) $$ and of exponential order $c$ then $$\int^\infty_0 e^{-st} f(t)\, dt $$ is analytic in the right half-plane for $$\mathrm{Re}(s)>c$$

Hi everyone, :) Here's a question I am stuck on. Hope you can provide some hints. :) Problem: Let \(U\) be a 4-dimensional subspace in the space of \(3\times 3\) matrices. Show that \(U\) contains a symmetric matrix.

Homework Statement . Let ##(M,d)## be a metric space and let ##f:M \to M## be a continuous function such that ##d(f(x),f(y))>d(x,y)## for every ##x, y \in M## with ##x≠y##. Prove that ##f## has a unique fixed point The attempt at a solution. The easy part is always to prove unicity...

Let n be a positive integer, and for each $j = 1,..., n$ define the polynomial $f_j(x)$ by f_j(x) = $\prod_{i=1,i \ne j}^n(x-a_i)$ The factor $x−a_j$ is omitted, so $f_j$ has degree n-1 a) Prove that the set $f_1(x),...,f_n(x)$ is a basis of the vector space of all polynomials of degree ≤ n -...

Homework Statement Prove that if ##S## is a nonempty closed subset of ##E^n## and ##p_0\in E^n## then ##\min\{d(p_0,p):p\in S\}## exists. 2. The attempt at a solution If ##p_0## was in ##S## why would ##\min\{d(p_0,p):p\in S\} = 0?## Is it just because it is the minimum? How about if ##p_0...

Homework Statement Let ##f## be a continuous map from ##[0,1]## to ##[0,1].## Show that there exists ##x## with ##f(x)=x.## 2. The attempt at a solution I have ##f## being a continuous map from ##[0,1]## to ##[0,1]## thus ##f: [0,1]\to [0,1]##. Then I know from the intermediate value...

A common definition of an inertial frame is that it is a reference frame in which space and time are homogeneous and isotropic; see, for instance, Landau and Lifshitz's Classical Mechanics. L&L also use homogeneity and isotropy to justify the functional form of the Lagrangian. But intuitively...

Iam wondering whether 'infinity' has real physical existence or just a mathematical paradox? If it does have a physical existence why don't we come across any quantity which is physically eternal? Someone please help..

I would like to know how this article applies to the possible existence of FTL particles. Does it point to a possible violation of c as the ultimate speed limit of a particle? In layman's terms what is this paper saying? http://arxiv.org/abs/1309.3713 Thanks

My book has a theorem of the uniqueness of the Lebesgue measure. But my question is: Is it necessarily a good thing that something in mathematics is unique and seems to indicate that this is very important. But my question is? Would the theory of measures fail if there existed another measure...

Hi all, I have a quick question about limits. This is something I should know but shame on me I forgot. If a function is bounded both above and below but isn't monotonic and is not necessarily continuous at all points, how do I go about proving its limit exists? In particular I am thinking...

Hi all, I have my exam in differential equations in one week so I will probably post a lot of question. I hope you won't get tired of me! Homework Statement This is Legendres differential equation of order n. Determine an interval [0 t_0] such that the basic existence theorem guarantees...

Question : suppose space comes into existence (expansion of space) where ever there is a local lack of mass or energy and also suppose things do come in and out of existence (for example virtual particles). If so, could the singularity of the big bang have undergone a cascade of points of...

Prove or disprove for every prime P there is a K such that $$10^k=1\text{mod}P$$. I arrived at this statement while proving something and can't find progress here is the problem which may doesn't matter but if you wan't to find the origin [here]

It seems to me that the question as to whether the universe is infinite or not carries the same validity as the question as to electron, quarks, etc. being infinitesimal or otherwise stated being modeled as point particles. It seems to me that these two quandaries are linked and perhaps can...

in waves and oscillations i read that any bounded medium oscillates in a particular freuency...why is it so?i need a proper reason for this

The "Dark Flow" & the Existence of Other Universes --New Claims of Har I just saw this big news story, "The "Dark Flow" & the Existence of Other Universes --New Claims of Hard Evidence" and thought that others would be interested in hearing this here. Dark Flow isn't new is it? Is this bold...

Hi! I was wondering: is it possible to have a non-orientable surface in 3D which is parametrized by u and v, with u and v periodic (i.e. is it possible to map the torus continuously into a non-orientable surface in 3D?) If so, does anyone have any explicit examples?

P(x) = "x is Provable" axiom 1 : P(x)→x "Statement x can be proven true." 1. (x∧¬x) consider a contradiction 2. x simplification(1) 3. ¬x simplification (1) 4. x∨∀sP(s) addition (2) 5. ∀sP(s) disjunctive syllogism (3,4) 6. (x∧¬x)→∀sP(s) conditional proof (1,5) "Anything is provable if it follows...