# metrics

1. ### Integrals of Different Metrics (Power & Volumetric Flow Rates)

Hi all, I am trying to understand the implications of taking the integral of different metrics/measurements, specifically energy and volumes. Here's my current understanding as it relates to energy: Energy = power * time. For example, kWh = kW * h. If you take the integral of power over a...

Let p,q be two prime numbers. Define d:\mathbb{Q} \times \mathbb{Q} \to [0,\infty) by d(x,y) = \max\{|x-y|_p,|x-y|_q\}. Is there an algebraic analogue to the completion of the rationals by this metric similar to the inverse limit of \Bbb{Z}/p^n\Bbb{Z} for the completion of the rationals by a...
3. ### A Question Involving Metrics and Topologies

So, my professor asked me to prove a theorem, and I only have a question about PART of it. Theorem: Let X be a set and let \mu be a metric on X. The set B = \{\emptyset\} \cup \{B_r(x) : x \in X, r \in \mathbb{R}\} forms a basis for a topology on X. My question is this: Is it necessary to...
4. ### Metrics close implies volume forms close

Let g be a metric on the S^2 which is close to the standard metric \gamma, i.e. \sup_{\theta \in S^2}\vert g - \gamma \vert \leq \varepsilon for some \varepsilon small, where \vert \cdot \vert is the norm with respect to \gamma (say). Is there an easy way of showing that the volume forms are...

The question: What permutation(s) \sigma \in S_n† give the maximum values for D_n? Definition: Let D_n: S_n \rightarrow \mathbb{Z}_{\geq 0} be a map such that D(\sigma) = \sum_{i=1}^{n}{ \sum_{j=i}^{n}{ [[i - j] - [\sigma(i) - \sigma(j)]] } } ‡ Explanation: I want to maximize D_n (when given a...
6. ### Relationship metrics

Hi all, I'm not a maths guru by any means, so maybe this is something that is easy to you guys. What I'm trying to do is work out how to build a relationship between two metrics. The first metric is something that we will be adjusting up or down to great extent with the second getting no action...
7. ### inequality with two metrics

I want to prove that sup \ x \in C[0,1] |f_n(x)-f(x) | \geq ({\displaystyle\int^1_0 |f_n(x)-f(x)|^2 \ dx})^{1/2} ({\displaystyle\int^1_0 |f_n(x)-f(x)|^2 \ dx})^{1/2} \leq (by c-s inequality) ({\displaystyle\int^1_0 |f_n(x)|^2 \ dx})^{1/2} + ({\displaystyle\int^1_0 |f(x)|^2 \ dx})^{1/2}...
8. ### Searching and understanding lorentzian metrics w/ timelike closed curves on cylinder

I defined a covering map ℝ^2 → S^1 x ℝ in order to work with the manifold. 1) How can I find lorentzian metrics (=metric tensors) on S^1 x ℝ (cylinder that is a 2-dimensional manifold)? I know that the diagonal matrix (2x2 matrix) of such a lorentzian metric must have signature 1. and...
9. ### Fun with metrics

I'm a first year grad student. After listening to my algebra professor's explanation that for any finite set A, P(A) is a vector space under the operation of symmetric difference over the field \mathbf{Z}_2. It got me to wondering what a metrization of the power set might look like. So, I came...
10. ### Topology and metrics

find two metric functions (distance) d1 , d2 on the space V=(0,1) (the interval 0,1). d1 , d2 must support: a. V is complete with the metric d1 and incomplete with d2. b. d1 , d2 induce the same topology on V (same topological space). I apologize for any spelling mistakes and appreciate...
11. ### Combining Performance Metrics

Okay I need some help. Im trying to figure out my performance metrics, I havent been in college in a couple years & not for sure if Im doing this correct. Say an employee has a benchmark of 90% or better in a metric he exceeds his goal at 91.39%, next his benchmark is 15% or below, he also...
12. ### Equivalent Metrics

Define \rho on X \times X by \rho(x,y) = min(1,d(x,y)), \ \ \ \ \ \ \ \ \ \ x,y \in X Show that \rho is a metric that is equivalent to d My solution: Two metrics are equivalent if and only if the convergent sequences in (X,d) are the same as the convergent sequences in (X,\rho) Let...
13. ### Are equivalent metrics comparable on compact sets?

Suppose I have a set X equipped with two equivalent metrics, d_1 and d_2, meaning that the metrics induce the same topology on X. I know that d_1 and d_2 need not be comparable on all of X, however, is it true that they will be comparable on compact subsets of X?
14. ### Equivalent Metrics

Hi, I'd be greatful for any help on the following problem; I have to show that two metrics \rho and \sigma are equivalent if and only if every open ball B_s^{\sigma}(x) contains an open ball B_r^{\rho}(x) and every open ball B_r^{\rho}(x) contains an open ball B_s^{\sigma}(x). Here is my proof...
15. ### Discrete metrics (basic topology)

Let X be a set donated by the discrete metrics d(x,y) = 0 if x=y, 1 if x̸=y. Show that a subset Y of X is compact iff this set is closed
16. ### metrics

Supoose that f:D -> X where D is any set and x is any metric space with metric d. We can define a function d*:DxD->R given by: d*(x,y) = d(f(x),f(y)) for any x,y in D. If f is injective prove that d* is a metric on D! Any help would be great!
17. ### Discrete metrics, open and closed sets

Let X be set donoted by the discrete metrics d(x; y) =(0 if x = y; 1 if x not equal y: (a) Show that any sub set Y of X is open in X (b) Show that any sub set Y of X is closed in X
18. ### Just starting metrics

Consider the set S of all real numbers, but with a new distance function defined: d(x,y) = |\frac{x}{1 + x} - \frac{y}{1 + y}| Add two new points, +\infty and -\infty to the set S. call the resulting set \bar{S} = S \cup \{+\infty,-\infty\}. Now extend d to \bar{S} by setting d(x,+\infty) =...
19. ### Metrics Spaces: Ball

Hello, I have a problem with the following exercise: For this question, I wrote, but I am not sure about this. B(f,1)={g\inC[0,1]: d1(f,g)<1} B(f,1)={g\inC[0,1]: \int|e^(-x)-g(x)|dx<1} For the second question, I m don't know the way how can I prove that? Can someone help me? Thank...
20. ### Challenging Problem!!! (Equivalent Metrics)

I need to show that if (X,p) is a non-compact metric space, then there exists a metric p* equivalent to p such that (X,p*) is not complete. I greatly appreciate your help!