closure Definition and 171 Threads

Assume s is a set such that Fs denotes the set of functions from S-->F where F is a field such as R, C or [0,1] etc. One requirement for F to be a vector space of these functions is closure- e.g. that sums of these functions are in the space: For f,g in Fs the sum f+g must be in Fs hence...

The Dirichlet problem asks for the solution of Poisson or Laplace equation in an open region ##S## of ##\mathbb R^n## with a condition on the boundary ##\partial_S##. In particular the solution function ##f()## is required to be two-times differentiable in the interior region ##S## and...

Hi together! Say we have ## \Lambda_q{(A)} = \{\mathbf{x} \in \mathbb{Z}^m: \mathbf{x} = A^T\mathbf{s} \text{ mod }q \text{ for some } \mathbf{s} \in \mathbb{Z}^n_q\} ##. How can we proof that this is a subgroup of ##\mathbb{Z}^m## ? For a sufficient proof we need to check, closure...

For every instance of addition or multiplication there is an inverse, closed on the naturals. Not every instance of subtraction and division is defined, so not closed on the naturals. This looks like two kinds of inverse. Instance inverse - the inverse of instances of addition and...

Problem: Let ## (X,d) ## be a metric space, denote as ## B(c,r) = \{ x \in X : d(c,x) < r \} ## the open ball at radius ## r>0 ## around ## c \in X ##, denote as ## \bar{B}(c, r) = \{ x \in X : d(c,x) \leq r \} ## the closed ball and for all ## A \subset X ## we'll denote as ## cl(A) ## the...

Summary:: Can someone point me to an example solution? Hello The attached figure is a four bar link. Each of the four bars has geometry, mass, moment of inertia, etc. A torque motor drives the first link. I am looking for an example (a simple solution so I can ground my self before...

Here is this week's POTW: ----- Let $X$ and $Y$ be topological spaces. If $Y$ is compact, show that the projection map $p_X : X \times Y \to X$ is closed. -----

Water is flowing in the pipe with velocity v0. Upon sudden closure of the valve at T, a pressure wave travels in the -ve x direction with speed c. The task is to find ##\alpha##, where ##\Delta P = \rho_0 c (\Delta v) \alpha##. The 1st step is to set up an equation using conservation of mass...

This is the exact definition and I've summarized it, as I understand it above. Why is it, that for elements in the third subspace, closure will be lost? Wouldn't you still get another vector (when you add two vectors in that subspace), that's still a linear combination of the vectors in the...

I am reading Stephen Willard: General Topology ... ... and am studying Chapter 2: Topological Spaces and am currently focused on Section 3: Fundamental Concepts ... ... I need help in order to prove Theorem 3.11 Part 1-a using the duality relations between closure and interior ... ..The...

I am reading Stephen Willard: General Topology ... ... and am currently focused on Chapter 2: Topological Spaces and am currently focused on Section 3: Fundamental Concepts ... ... I need help in order to fully understand a result or formula given by Willard concerning a link between...

I am reading Stephen Willard: General Topology ... ... and am currently reading Chapter 2: Topological Spaces and am currently focused on Section 1: Fundamental Concepts ... ... I need help in order to fully understand an aspect of the proof of Theorem 3.7 ... ..Theorem 3.7 and its proof...

I am reading Tej Bahadur Singh: Elements of Topology, CRC Press, 2013 ... ... and am currently focused on Chapter 1, Section 1.2: Topological Spaces ... I need help in order to fully understand Singh's proof of Theorem 1.3.7 ... (using only the definitions and results Singh has established to...

I'm watching this video to which discusses how to find the domain of the self-adjoint operator for momentum on a closed interval. At moment 46:46 minutes above we consider the constant function 1 $$f:[0,2\pi] \to \mathbb{C}$$ $$f(x)=1$$ The question is that: How can we show that the...

I am reading Andrew Browder's book: "Mathematical Analysis: An Introduction" ... ... I am reading Chapter 6: Topology ... ... and am currently focused on Section 6.1 Topological Spaces ... I need some help in order to fully understand a statement by Browder in Section 6.1 ... ... The...

Homework Statement Let ##M## be a metric space. Prove that ##\overline{\overline{S}} = \overline{S}## for ##S\subseteq M##. Homework EquationsThe Attempt at a Solution First we know that ##\overline{S} \subseteq \overline{\overline{S}}## is true (just take this for granted, since I know how to...

I'm trying to prove that the set of all square integrable functions f(x) for which ∫ab |f(x)|^2 dx is finite is a vector space. Everything but the proof of closure is trivial. To prove closure, obviously we should expand out |f(x)+g(x)|^2, which turns our integral into one of |f(x)|^2 (finite)...

Homework Statement Identify ##\bar{c}##, ##\bar{c_0}## and ##\bar{c_{00}}## in the metric spaces ##(\ell^\infty,d_\infty)##. Homework Equations The ##\ell^\infty## sequence space is $$ \ell^\infty:=\left\{\{x_n\}_{n=1}^\infty\in\mathbb{R}^\mathbb{N}\,:\...

Homework Statement Define the interior A◦ and the closure A¯ of a subset of X. Show that x ∈ A◦ if and only if there exists ε > 0 such that B(x,ε) ⊂ A.The Attempt at a Solution [/B]

I am reading "Multidimensional Real Analysis I: Differentiation by J. J. Duistermaat and J. A. C. Kolk ... I am focused on Chapter 1: Continuity ... ... I need help with an aspect of the proof of Lemma 1.2.10 ... Duistermaat and Kolk"s proof of Lemma 1.2.10 (including D&K's definition of a...

Let me give some context. Let X be a compact metric space and ##C(X)## be the set of all continuous functions ##X \to \mathbb{R}##, equipped with the uniform norm, i.e. the norm defined by ##\Vert f \Vert = \sup_{x \in X} |f(x)|## Note that this is well defined by compactness. Then, for a...

Homework Statement If ##C## is a connected space in some topological space ##X##, then the closure ##\overline{C}## is connected. Homework EquationsThe Attempt at a Solution Suppose that ##\overline{C} = A \cup B## is separation; hence, ##A## and ##B## are disjoint and do not share limit...

Homework Statement I am trying to determine whether the closure of a path-connected set is path-connected. Homework EquationsThe Attempt at a Solution Let ##S = \{(x, \sin(1/x) ~|~ x \in (0,1] \}##. Then the the closure of ##S## is the Topologist's Sine Curve, which is known not to be...

Hello everyone, I was wondering if someone could assist me with the following problem: Let T be the topology on R generated by the topological basis B: B = {{0}, (a,b], [c,d)} a < b </ 0 0 </ c < d Compute the interior and closure of the set A: A = (−3, −2] ∪ (−1, 0) ∪ (0, 1) ∪ (2, 3) I...

Hey! :o Let $E/F$ be an algebraic extension and $C$ the algebraic closure of $E$. I want to show that the field $C$ is the algebraic closuree also for $F$. We have that $C=\{c\in E\mid c \text{ algebraic over } E\}$, i.e., every polynomial $f(x)\in E[x]$ splits completely in $C$. Since...

Problem. Let $K$ be the field obtained from $\mathbf F_p$ by adjoining all primitive $\ell$-th roots of unity for primes $\ell\neq p$. Then $K$ is algebraically closed. It suffices to show that the polynomial $x^{p^n}-x$ splits in $K$ for all $n$. In order to show this, it in turn suffices to...

Homework Statement Let E be a bounded set in R^n Show that E and the closure of E have the same Jordan outer measure Homework Equations Jordan outer measure is defined as m^* J(E)=inf(m(b)) where B \supset E B is elementary. 3. The Attempt at a Solution [/B] If E and the closure...

Question. Is it true that a finite extension $K:F$ is simple iff the purely inseprable closure is simple over $F$? I think have an argument to support the above. First we show the following: Lemma. Let $K:F$ be a finite extension and $S$ and $I$ be the separable and purely inseparable...

Hi everyone, didn't know where to post question on sigma algebra so here it is:- What I've tried till now: Let C\in G 1) For C=X, f^{-1}(B)=X which will be true for B=Y (by definition) 2) For closure under complementation, to show C^{c}\in G. So, C^{c}=X\setminus C=X\setminus...

does anyone know what the boundary condition is for a closing valve using the wave equation pde?

Hello, I'm trying to calculate a recurrence relation of the phases of 3 telescopes in a closure phase. Usually in a stellar interferometer we have 3 telescopes, located in a triangle, measuring intensity of light in 3 points on a far field plane. I found an article, describing how the phase is...

Homework Statement Let ##E'## be the set of all limit points of a set ##E##. Prove that ##E'## is closed. Prove that ##E## and ##\bar E = E \cup E'## have the same limit points. Do ##E## and ##E'## always have the same limit points? Homework Equations Theorem: (i) ##\bar E## is closed (ii)...

Hey guys, I really need help in revising my Axiom 6 for my Linear Algebra course. My professor said, "You need to refine your statement. You want to show rx1 and rx2 are real numbers. You should not state they are real numbers." Here is my work: Proof of Axiom 6: rX is in R2 for X in R2...

Homework Statement Show that ##\overline{B(a,r)} = \{ x \in \mathbf R^n ; |a-x| \le r \}## in ##\mathbf R^n## for all points ##a \in \mathbf R^n## and ##r>0##. Is it possible to generalize the statement to any normed vector space? Give a example of a metric space where the statement is not...

The closure relation in infinite dimension is : ∫|x><x|dx =I (identity operator),but if we apply the limit definition of the integral the result is not logic or intuitive. The limit definition of the integral is a∫b f(x)dx=lim(n-->∞) [i=1]∑[i=∞]f(ci)Δxi, where Δxi=(b-a)/n (n--.>∞) and...

I am building a walk in size faraday cage. In order to make it re-positionable, I am creating 6 wooden frames (top, bottom and sides) covered in 2 or 3 layers of aluminum screen from a hardware store. I am using at least 2 layers of screen, because that is how many layers it takes to cause my...

I'm confused on why exactly the following two statements are equivalent for a finite field K: -If K has no proper finite extensions, then K is algebraically closed. -If every irreducible polynomial p with coefficients in K is linear then K is closed. Can somebody help shed some light on this?

Hello, This is not a homework problem, nor a textbook question. Please do not remove. Is there a concrete example of the following setup : R is an integrally closed domain, a is an integral element over R, S is the integral closure of R[a] in its fraction field, S is not of the form R{[}b{]} for...

This is probably a stupid question. Let R be a domain, K its field of fractions, L a finite (say) extension of K, and S the integral closure of R in L. Is the quotient field of S equal to L ? I believe that not, but I have no counter-example.

Homework Statement . Let ##X## be a nonempty set and let ##x_0 \in X##. (a) ##\{U \in \mathcal P(X) : x_0 \in U\} \cup \{\emptyset\}## is a topology on ##X##. (b) ##\{U \in \mathcal P(X) : x_0 \not \in U\} \cup \{X\}## is a topology on ##X##. Describe the interior, the closure and the...

In showing diam(cl(A)) ≤ diam(A), (cl(A)=closure of A) one method of proof* involves letting x,y be points in cl(A) and saying that for any radius r>0, balls B(x,r) and B(y,r) exist such that the balls intersect with A. But if x,y is in cl(A), isn't there the possibility that x,y are...

Homework Statement Determine the interior, the boundary and the closure of the set {z ε: Re(z2>1} Is the interior of the set path-connected? Homework Equations Re(z)=(z+z*)/2 The Attempt at a Solution Alright so z2=(x+iy)(x+iy)=x2+2ixy-y2 so Re(x2+2ixy-y2)= x2-y2 >1 So would...

Homework Statement -∏<arg(z)<∏ (z≠0) Homework Equations arg(z) is the angle from y=0The Attempt at a Solution Arg(z) spans the entire graph since -pi to pi is the full 360 degrees so I put: -∏<arg(z)<∏ --> 0<arg(z)<2∏+k∏, (k ε Z) --> arg(z) \subset R --> arg(z) = R: all real numbers but I...

If the tangent space at p is a true vector space, then it must be that the sum of two vectors is itself a directional derivative operator along some path passing through p. I've been trying to prove that this is true without any luck. My textbook "proves" that vector addition is closed by...

Homework Statement Show that if a set has 3 elements, then we can find 8 relations on A that all have the same symmetric closure. Homework Equations Symmetric closure ##R^* = R \cup R^{-1} ## The Attempt at a Solution If the symmetric closures of n relations are the same then...

My first analysis/topology text defined the boundary of a set S as the set of all points whose neighborhoods had some point in the set S and some point outside the set S. It also defined the closure of a set S the union of S and its boundary. Using this, we can prove that the closure of S is...

Homework Statement Homework Equations The Attempt at a Solution Well thankfully I just have to present closure under mult. inverses and closure under addition. But I seem to be going in circles...if a is in G, then we need to show that a-1 is also in G. So a*a-1 = 1F, but is...

I intend to show, for a set ##X## containing ##A_i## for all ##i##, $$\overline{\bigcup A_i}\supseteq \bigcup \overline{A_i}.$$ //Proof: We proceed to prove that ##\forall x\in X,~x\in\bigcup\overline{A_i}\implies x\in\overline{\bigcup A_i}##. Equivalently, ##\forall x\in...

Let G be a group and my book defines closure as: For all a,bε G the element a*b is a well defined element of G. Then G is called a group. When they say well defined element does that mean I have to show a*b is well defined and it is a element of the group? Or do I just show a*b is closed under...