Existence Definition and 570 Threads

Hey! :o We have the system \begin{align*}&x_1=\left (5+x_1^2+x_2^2\right )^{-1} \\ &x_2=\left (x_1+x_2\right )^{\frac{1}{4}}\end{align*} and the set $G=\{\vec{x}\in \mathbb{R}^2: \|\vec{x}-\vec{c}\|_{\infty}\leq 0.2\}$ where $\vec{c}=(0.2,1)^T$. I want to show with the Banach fixed-point...

I am still puzzled by the issue of existence of electrostatic field. According to the classical electromagnetic theory, electrostatic field can be created by an electrically charged particle. The electrostatic field surrounding the electrically charged particle does not stop close to the...

So, I am confused about the following.I learned in high school and in my first year of university that there is something called time dilation that observers observe that time is slower for objects that are moving faster. As in if there were two initially synchronized clocks that you could...

If we have solution of an equation as x=1, it may be expressing, depending on context, 1 apple, 1 excess certain thing, etc. And, if we have solution of an equation as x=-1, it may be expressing, depending on context, 1 deficient apple, 1 deficient certain thing, etc. Is there any experience...

Hello! (Wave) Let $(a_n)$ be a bounded sequence such that $\inf_{\ell} a_{\ell}<a_n< \sup_{\ell} a_{\ell}$ for each $n=1,2, \dots$ I want to show that there are subsequences $(a_{k_n})$ increasing and $(a_{m_n})$ decreasing such that $a_{k_n} \to \sup_{\ell} a_{\ell}$ and $a_{m_n} \to...

Given a finite-dimensional normed linear space ##(L,\lVert \cdot \rVert)##, is there anything that suggests that at every point ##x_0 \in L##, there exists a direction ##\delta \in L## such that that ##\lVert x_0 + t\delta \rVert \geqslant \lVert x_0 \rVert## for all ##t \in \mathbb{R}##?

I mean , it is said that universe started with a big bang and before it it was a point .so what can we say about where the point was , i mean like when we talk about ,where are galaxies , the answer is universe . So like that where did that point exist . Or there was just "Nothing"?

Problem: Let $E$ have finite outer measure. Show that $E$ is measurable if and only if there is a $F_\sigma$ set $F \subset E$ with $m^*\left(F\right)=m^*\left(E\right)$. Proof: "$\leftarrow$" To Show: $E=K\cup N$ where $K$ is $F_\sigma$ and $m^*(N)=m(N)=0$. By assumption, $\exists F$, and...

Hello! (Wave) Let $m$ be a natural number. I want to check the sequence $\left( \binom{n}{m} n^{-m}\right)$ as for the convergence and I want to show that there exist constants $C_1>0, C_2>0$ (independent of $n$) and a positive integer $n_0$ such that $C_1 n^m \leq \binom{n}{m} \leq C_2 n^m$...

Having trouble with something that's likely too trivial, but here goes.. In Optimization theory and nonlinear programming [Sun, Yuan] the following is discussed at section 8.2. Consider the optimisation of f:\mathbb R^n\to\mathbb R with constraints g_j:\mathbb R^n\to\mathbb R, g_j(x) = 0...

Hey! :o We have the initital value problem $$\begin{cases}y'(t)=1/f(t, y(t)) \\ y(t_0)=y_0\end{cases} \ \ \ \ \ (1)$$ where the function $f:\mathbb{R}^2\rightarrow (0,\infty)$ is continuous in $\mathbb{R}^2$ and continuously differentiable as for $y$ in a domain that contains the point $(t_0...

No physicist has ever seen an electron. Yet, all physicists believe in the existence of electrons. An intelligent but superstitious man advances this analogy to argue that 'ghosts' exist even though no one has seen one. How will you refute his argument?

Out of curiosity i was watching some physics documentaries. They threw out quantum theory and relativity and i have taken a modern physics course. I can't say i remember antimatter ever coming up in lecture. I am curious about the math.

A question arose over a simplification I wrote on another subject. My i information say's that matters antimatter pairs are generated in what is known to be very empty space such as the voids within the cosmic web. When I read about this it was considered anomalous but definitely verified. Now...

This might be a stupid question so please bear with me. I am having a debate with a friend of mine over the Internet. It is about science. His argument is that although science works (that is the technology part), science cannot actually tell anything about "reality". He sites for example...

Homework Statement Prove that for every n greater than or equal to two, there exists a number with exactly n divisors. Homework Equations I used induction: The Attempt at a Solution Base case: assume n=2 which implies there exists a number with 2 divisors, that is the case for all prime...

Existence of Partial Derivatives and Continuity ... Kantorovitz's Proposition pages 61-62 ... I am reading the book "Several Real Variables" by Shmuel Kantorovitz ... ... I am currently focused on Chapter 2: Derivation ... ... I need help with another element of the proof of Kantorovitz's...

I am reading the book "Several Real Variables" by Shmuel Kantorovitz ... ... I am currently focused on Chapter 2: Derivation ... ... I need help with another element of the proof of Kantorovitz's Proposition on pages 61-62 ... Kantorovitz's Proposition on pages 61-62 reads as follows: In the...

I'm trying to read this paper. At some point, it tries to find the minimums of an action. At first it deduces the existence of two independent solutions and says that one of them is a minimum and the other is a saddle point. Then goes on to mention that this means that actually there is another...

Hey! :o Let $f:[-1,1]\rightarrow \mathbb{R}$ be defined by \begin{equation*}f(x)=\begin{cases}\frac{1}{k} & \text{ for } x\in \left (\frac{1}{k+1}, \frac{1}{k}\right ], \ k \in \mathbb{N}\\ 0 & \text{ for } x=x_0=0 \\ \frac{1}{k} & \text{ for } x\in \left [\frac{1}{k}, \frac{1}{k-1}\right ), \...

By requiring the inner product in two points ##x## and ##x'## having metrics ##g## and ##g'## to be invariant, i.e. ##g'(x') = g(x)##, one is lead to the Killing equation. Does general relativity forbiddes spaces where the Killing equation cannot be satisfied? It seems obvious that we want...

Hi! I'm completely new to this website, and I have no background. I'm just a curious kid who googles stuff for the fun of it, and I've just had this burning question for a while now. I watched this interview with Michio Kaku on YouTube, and he discussed how we shouldn't exist. Anti-matter and...

As an alternative to posting on arXiv (due to the endorsement requirement, mainly) -- how sound are the following methods : 1. Use a blockchain based service like stamp.io ... then upload / share on blog, web page, whatever 2. Self-publish a small e-book on Amazon, for example.

Homework Statement Show that for every α ∈ ℂ with α ≠ 0, there exists a unique β ∈ ℂ such that αβ = 1 Homework Equations Definition[/B]: ## \mathbb {F^n} ## ## \mathbb {F^n} ## is the set of all lists of length n of elements of ## \mathbb {F} ## : ## \mathbb {F} ## = {## (x_1,...,x_n) : x_j...

Some will claim that RF energy being composed of photons can only be accepted on faith because there is no experimental evidence and there probably will be no experimental evidence due to the comparatively long wavelenghts of RF waves. But the technique of NMR (nuclear magnetic resonance)...

In the classical electromagnetic field theory, the field density of energy is given by: $$u = (\epsilon/2)E^2 + (\mu/2)H^2$$ One of the differences between the classical electromagnetic theory and the real world is that in classical EM all charge and current density, (ρ(r), J(r)), is...

I have a few questions about the Many Worlds Interpretation. I read the article https://plato.stanford.edu/entries/qm-manyworlds/ but was having trouble understanding what the "measure of existence" is supposed to represent in the theory, and why a believer in the idea should adhere to either...

Lets suppose we have an object, could be a quark or proton or a chair. Is there some physics description's that we can say, If these conditions satisfy, the particle will exist. This question may seems metaphysical but I am asking in physics perspective. It seems like to me that things are...

I am reading David S. Dummit and Richard M. Foote : Abstract Algebra ... I am trying to understand the example on Finite Fields in Section 13.5 Separable and Inseparable Extensions ...The example reads as follows: My questions are as follows: Question 1In the above text from D&F we read the...

I am reading David S. Dummit and Richard M. Foote : Abstract Algebra ... I am trying to understand the example on Finite Fields in Section 13.5 Separable and Inseparable Extensions ...The example reads as follows: My questions are as follows: Question 1In the above text from D&F we read the...

Consider y' = 1/sqrt(y) I seem to be able to find a unique solution given the initial condition of the form y(c) = 0, but the theorem says I won't be able to do so, so I am kind of confused. I just want some clarifications. Does the uniqueness and existence theorem say anything about the...

My question is very simple but it is something I have been thinking about for some time. Every time a person needs to have the expansion of the Universe explained or the question "What does the Universe expand into?", people who know a bit about the topic answer that it expands into and onto...

in a P-N junction diode, in an equilibrium state, a depletion layer forms in the surface of contact of N type and P type. But the problem is, this layer is formed by diffusion of electron to a lower electron density zone. Why does it always have to form right in the contactt surface? when...

Homework Statement I post here to check if I am in the right way to understand this point in the book. The wave function of free particle is ##Ae^{\frac{i}{\hbar}(px-Et)}##.This could be regarded as ##{\phi}(x,t)=Ae^{\frac{i}{\hbar}S(x,t)}##. ##S(x,t)## is the free particle's least action...

As I understand it, there is no solid definition of time other than the entropy is an arrow of time flow. I believe that physics equation would work regardless of the direction of time flow. What would be the implications if we assume that past, present, and future events are part of the...

Homework Statement Let γ : R → Rn be a regular (smooth) closed curve with period p. Show that there exist an orientation preserving diffeomorphism ϕ: R → R, a number p' ∈ R such that ϕ(s + p') = ϕ(s) + p and γ' = γ ◦ ϕ is an arclength parametrized closed curve with period p' Homework...

Hey guys, Does black holes exist? Is it even real?

I know that the span of any subset of vectors in a vector space is also a vector space (subspace), but is it true that every vector space has a generating set? That is, the moment that we define a vector space, does there necessarily exist a spanning set consisting of its vectors?

I did not understand of the non-existence of variance. What does it mean?

Hello Everyone, While deriving potential energy stored in space due to two stationary opposite charges we end up with negative value of energy which upon dividing by c square provides us with negative value of mass. What is the significance of this mass other than reducing the total mass of...

I cannot conceive of time as an actual physical existence. What are the current opinions on what time is?

I have the following question: Is there a basis for the vector space of polynomials of degree 2 or less consisting of three polynomial vectors ##\{ p_1, p_2, p_3 \}##, where none is a polynomial of degree 1? We know that the standard basis for the vector space is ##\{1, t, t^2\}##. However...

Problem: y'=((x-1)/(x^2))*(y^2) , y(1)=1 . Find solutions satisfying the initial condition, and determine the intervals where they exist and where they are unique. Attempt at solution: Let f(x,y)=((x-1)/(x^2))*(y^2), which is continuous near any (x0,y0) provided x0≠0 so a solution with y(x0)=y0...

If there was no mass/energy would space time still exist? In other words, does on space time only exist because of an interaction between two points of energy?

Hi, Just curious if someone knows of any Characteristic class used to determine if a manifold allows a Complex structure? It seems strange that Complex Space C^n is topologically Identical to R^{2n} yet I believe not all R^{2n}s ( if any) allow Complex structures. Thanks for any comments, refs...

Hi. Different interpretations of QM have different opinions about the ontology of the wavefunction, i.e. if it really, physically exists or if it is "just" a mathematical tool needed to calculate the outcome of measurements. The QM interpretations comparison table on Wikipedia summarises the...

When I was taking a look at this page, I noticed that she is "known for proving the local existence and uniqueness of solutions to the vacuum Einstein Equations". But this doesn't make sense to me(the uniqueness part). Just consider the Minkowski and Schwarzschild solutions. They're both vacuum...

Homework Statement Determine whether existence of at least one solution of the given initial value problem is guaranteed and, if so, whether uniqueness of the solution is guaranteed. dy/dx=y^(1/3); y(0)=0 Homework Equations Existence and Uniqueness of Solutions Theorem: Suppose that both...

Homework Statement I have the function: f(x,y)=x-y+2x^3/(x^2+y^2) when (x,y) is not equal to (0,0). Otherwise, f(x,y)=0. I need to find the partial derivatives at (0,0). With the use of the definition of the partial derivative as a limit, I get df/dx(0,0)=3 and df/dy(0,0)=-1. However, my...

Finding a limit entails understanding how a function behaves near a particular value of x. So what do we mean when we say that a limit doesn't exist (in context to the upper statement)? (From what i studied, i noticed that limit exists only for those functions which have a discontinuity in the...