Closed Definition and 1000 Threads

Homework Statement A set F\subseteqR is closed iff every Cauchy sequence contained in F has a limit that is also an element of F. Homework Equations The Attempt at a Solution Let F be closed. Then F contains its limit points. This means x=lima_{n} are elements of F.

Suppose I have N ideal particles in an enclosure, be it a ball or a cube or some other form. The particles shall bounce off the walls of the enclosure and against each other without losing speed. The velocity of each particle i shall be such that it fullfills |v_i|=\rho, where \rho is constant...

A field K is called algebraically closed field if any no-zero polynomial has at least one root in K. Given finite field F_q, q=p^m, p is a prime and m is non-negative integer. A famous property of finite field is any element in F_q satisfies: x^q=x. Then I have such an assumption...

Homework Statement Prove that for a linear set M a subset of Hilbert space, that the set perpendicular to the set perpendicular to M is equal to M iff M is closed. The Attempt at a Solution I already have my proof -- but what is an example of a linear subset of H that is not closed? I think...

Homework Statement As the title says Homework Equations Definitions of "open" and "closed" The Attempt at a Solution Suppose a finite set S is not closed. Then Sc is not open, and there exists an element x of Sc, so that for all µ > 0, either x + u, or x - u, is an element of S...

Homework Statement Prove that SU(n) is closed and bounded Homework Equations The Attempt at a Solution So in order to prove this, I first mapped SU(n) to be a subset of R^{{2n}^2}. To prove the closed portion, I tried mapping a sequence in SU(n) to a sequence in R^{{2n}^2}. However, I...

Homework Statement Show that: Let S be a subset of the real numbers such that S is bounded above and below and if some x and y are in S with x not equal to y, then all numbers between x and y are in S. then there exist unique numbers a and b in R with a<b such that S is one of the...

I'm sure there is an easy solution to my problem, but I very little electronics background and don't yet have a breadboard to even experiment on. In fact, it was only two days ago that I learned a 555 doesn't need to (or can) be programmed by a computer to use. I want to make a buzzer that...

Homework Statement L is the set of limit point of A in the real space, prove that L is closed. Homework Equations The Attempt at a Solution L may or may not have limit points. If L does not have limit points, then it's obviously closed. If L has limit points, the let l be a...

Homework Statement Give a nontrivial example of an infinite dimensional subspace in l2(R) that is closed. Also find an example of an infinite dimensional subspace of l2(R) that is not closed. Repeat the same two questions for L2(R). The Attempt at a Solution To my understanding, l2 is...

Homework Statement Prove that if S is open and Sc is open then boundary of S must be empty The Attempt at a Solution S is open means boundary of S is a subset of Sc Sc is open means boundary of Sc is a subset of S (By taking complement of both sides from the definition ?) This means that they...

Homework Statement I need to show that the subset of R^2 given with A = {(x, y) : xy = 1} is closed by using the "closed set formulation" of continuity. The Attempt at a Solution So, if a function f : X --> Y is continuous, then for every closed subset B of Y, its preimage f^-1(B) is...

If I am within that closed system, can I make the rate of entropy of the entire system any faster or slower overall? Does the existence of life within the system decrease or increase the overall system entropy rate?

How were they able to derive this?

I'm sure this has been asked before, but the proofs I've seen use the fact R is connected or continuous functions is some way. I'm trying to prove it with the things that have only been presented in the book so far (Mathematical Analysis by Apostol). So, let A be a subset of R which is both...

Homework Statement let |e-x-e-y| be a metric, x,y over R. let X=[0,infinity) be a metric space. prove that X is closed, bounded but not compact. Homework Equations The Attempt at a Solution there is no problem for me to show that X is closed and bounded. but how do I prove...

Homework Statement Homework Equations know dot product The Attempt at a Solution [SIZE="5"]PART A [SIZE="5"]PART B not sure what's it asking for help would be great

My professor proved this result in class, but I don't understand the "simple" direction. He said that the above result is in another words proving that Cl(a) = intersection of all closed sets that contain A. So he proved Cl(A) subset of intersection of all of the closed sets containing A...

Would the conservation of mass apply to the universe? Sorry I looked for awhile for the answer to this question but I couldn't really find anything.

Hello, Can a closed system change the position of its center of mass if no external force is exerted on it?

Is the interval I of an autonomous diff closed? Homework Statement Given this autonomous diff.eqn Where we have an open set E defined on R^n and f \in \mathcal{C}^1(E) x' = f(x) where x(t_a) = x(t_b) and t_a,t_b \in I and where t_a < t_b. Show for n = 1, that the solution x is...

Ignore the above, I was haveing problems with the symbol... Convert each to closed form: 1. Sum from i=1 to n of: \frac{n}{a^n} 2. Sum from i=1 to n of: \frac{1}{a^n} Thanks. P.S. I know how to do it if it was an infinite series, but not for this.

ln(-1)/i=pi this equation does not use limits or integrals, as you can see, but it does involve imaginary numbers. Does this make it an open expression, or does the fact that it uses i not matter?

So here are my steps, which for some reason I feel are very wrong: Well in closed form would be [n(n+1)]/2 so 2k would be 2*[n(n+1)]/2 For 1.7^k, I used a different form, which I don't have the formula for in front of me, but the final result for that part is [1.7^(n+1) - 1] /[1.7 - 1] So...

Definition: A semigroup is a pair (R,op) where R is a set an op is a binary operation that is closed and associative. A commutative semigroup is a semigroup where op satisfies for all a,b in R, op(a,b) = op(b,a). A monoid is a semigroup where with an identity,e, for op, satisfying for all r in...

Homework Statement Prove or disprove the following statement: The closure of a set S is closed. Homework Equations Definition of closure: set T is the closure of set S means that T is the union of S and the set of limit points of S. Definition of a closed set: set S is...

Is it correct to make the following statement? If a point x in E is not a limit point of E, then any neighborhood V of x will--at most--contain finitely many points of E. Thus, its possible for V to contain only one point, namely, x. Thanks, M

Homework Statement 5 A closed wire loop in the form of a square of side 4.0 cm is mounted with its plane horizontal. The loop has a resistance of 2.0 x 10-3Ω, and negligible self-inductance. The loop is situated in a magnetic field of 0.70 T, directed vertically downwards. When the field is...

Am I missing something and doing this all wrong? Experiment and data: The level of water in the glass tube was adjusted by raising the supply tank until the sound from the tuning fork was at its loudest. This level corresponds to first resonance position and it was recorded. The...

Homework Statement Let X be an ordered set where every closed interval is compact. Prove that X has the least upper bound property. Homework Equations X having the least upper bound property means that every nonempty subset that is bounded from above has a least upper bound, in other...

Has anyone ever seen the treatment of a closed classical string loop. Like if you had a loop of string on the space shuttle and subject it to accoustic driving or initial impulses. I post this here in beyond the standard model because no one in the classical physics section seems to have heard...

Has anyone seen a treatment of how to use the wave equation to describe a closed loop of string. I am talking ordinary strings here not the fancy string theory kind.

Hi All, So all closed interval [a,b] is compact (see Theorem 2.2.1 in Real Analysis and Probability by RM Dudley) Now, Let's say I have [0,10] as my closed interval. Let My Open Cover be (0, 5) (5, 7.5) (7.5, 8.75) (8.75, 9.375) ... Essentially, The length of each open...

Homework Statement proove is either true of false let A be a set of integer closed under subtraction. if x and y are element of A, then x-ny is in A for any n in Z. Homework Equations n/a The Attempt at a Solution is there any proof, without induction? i suspect its true because any...

I am working on an incubator/shaker for laboratory use. I am trying to work out a temperature failure and repair it, but that is besides the point here. I was looking through the user manual to try to get some clues about the failure and I came across this: "Depending on various conditions...

Dear all, Is there a method to find the maximum and minimum dimension of an irregular closed loop? This is a problem when we want to define the full-width - half maximum of a image. The level contour of this image at its half maximum can be an irregular closed loop. Any reference or...

Consider a vertical pipe partially filled with liquid. The pipe is open at the lower end and closed at the top. See the attached picture. Will the liquid fall out or not? In a small diameter pipe a stable meniscus will form due to surface tension and prevent the water from falling out. In a...

Me again. Problem. Let X be a topological space, and Y a T2-space (i.e. a Haussdorf topological space). Let f : X --> Y be a continuous function. One needs to show that the graph of , i.e. the set G = {(x, f(x)) : x is in X} is closed in X x Y. Attempt of proof. To show what we need to show...

"Open" and "closed" in the geometrical sense vs the thermodynamic sense Perhaps this is a silly question, but what is the relationship between the words "open" and "closed" in the geometrical sense (open, flat, closed universes) and in the thermodynamic sense (open and closed systems) in the...

In case you have never heard of the elephant toothpaste experiment, take a look at this: http://www.using-hydrogen-peroxide.com/elephant-toothpaste.html" I was just wondering, if you put the chemicals together in an air tight container, would the air pressure increase?

im trying top find the flow rate of water from a closed pipe. one thing that i think i can work with is a hose. there is a hose branching of the main flow pipe, which can be used to clean floor etc. im thinking that i can not open the hose and thus measure the stagnatiob pressure at the...

So, I'm going through a proposition, which states that if (X, d) is a metric space, then any set {x}, where x e X, is a closed subset of X. First of all, could we do this proof to assume the contrary? Since then obviously for the point x from {x} there doesn't exist any real number r > 0 such...

First of all I just want to rant why is the Latex preview feature such a complete failure in Firefox? Actually it is really bad and buggy in IE too... So I am reading into Foundations of geometry by Abraham and Marsden and there is a basic topology proof that's giving me some trouble. They...

Homework Statement Giving 2 closed curves in 3-dimension space C1 and C2, prove that:\oint _{C1} \oint _{C2}\frac{(\vec{dl_2}.\hat{r_{12}})\vec{dl_1}}{r^2_{12}}=0 Where: _ \vec{dl_1} and \vec{dl_2} are the vector elements of the curves C1 and C2 respectively. _ r_{12} is the distance between...

Why does work done by a conservative force = 0 in a closed path? I know this sounds foolish :rolleyes: but how can some forces have such a property? Can anybody give a satisfactory physical explanation?:confused:

Homework Statement A long bar magnet is bent into the form of a closed ring. If the intensity of magnetisation is M, and ignoring any end effects due to the join, find the magnetic field H and the induction B: (a) Inside the material of the magnet (b) just outside Homework Equations...

Homework Statement Find the closed form value for n SIGMA e^(i/n) i= 0 Homework Equations ? The Attempt at a Solution summation expands to 1 + e^(1/n) + e^(2/n) - - - - - e^1 To be honest i have no clue how to go about these kinds of problems so a general help would...

Homework Statement In Rosenlicht's Intro to Analysis, there is a proposition (p. 52). A Cauchy sequence of points in a metric space is bounded. Proof: For if the sequence is P1, P2, P3, ... and ε is any positive number and N an integer such tat d(Pn, Pm) < ε if n, m > N, then for any...

Homework Statement I am using Rosenlicht's Intro to Analysis to self-study. 1.) I learn that the complements of an open ball is a closed ball. And... 2.) Some subsets of metric space are neither open nor closed. Homework Equations Is something amiss here? I do not understand how...

First post, please excuse my ignorance. If the Universe is closed, then at the end of the expansion, micro gravity eventually pulls all objects together. Black holes absorb more and more stars and whole galaxies and eventually each other until there is just one black hole and no matter left...