Introduction of Keto Pure Ireland

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Introduction of Keto Pure Ireland

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Reference request: Introduction to Finsler manifolds from the metric geometry point of view (possibly from the Busemann’s approach)

This question is a cross post from Math.SE. I have requested the migration of the question, but unfortunately it is not possible after two months of posting. I also have found this related question, but in my opinion it is not a duplicate from mine.

I was reading about geometry in metric spaces from different books, two of them are: (1) A course in metric geometry by Y. Burago, D. Burago and S. Ivanov; and (2) Metric spaces of non-positive curvature by M. Bridson and A. Häfliger. Both develop the Alexandrov’s approach to curvature, which uses comparison triangles with the constant curvature model spaces.

For a normed space $ X$ , the following statements are equivalent:

  1. $ X$ has curvature $ \leq\kappa$ in Alexandrov’s sense, for some real number $ \kappa$ .
  2. $ X$ has curvature $ \leq 0$ in Alexandrov’s sense.
  3. The norm on $ X$ is induced by an inner product.

So it seems to me that Alexandrov’s approach is not very informative in the normed case. On the other hand, a geodesic space has non-positive curvature in the Busemann’s sense if its metric is convex, in general this is a weaker notion than Alexandrov’s, and in the normed case the following statements are equivalent:

  1. $ X$ has non-positive curvature in the Busemann’s sense.
  2. $ X$ is uniquely geodesic, that is, every pair of points is joined by a unique geodesic (the linear segment between them).
  3. $ X$ is strinctly convex, that is, the ball in $ X$ is strictly convex which means that for every pair of different vectors $ v$ and $ w$ of norm equal to $ 1$ we have that $ tv+(1-t)w$ has norm strictly less than $ 1$ for every $ t$ in $ (0,1)$ .

So it seems to me that this weaker notion is the appropriate notion for non-positive curvature in the normed case and I think also for finsler manifolds. I have never studied finsler geometry, but I am very interested in studying metric geometry from this approach. And I do not know where I should start.

My question is: What is a good introductory book about finsler manifolds from the metric geometry point of view? What is a good introductory book for the Busemann’s approach? If there was not an introductory book available, a reference to an advanced one along with references that cover the necessary background would be very welcome.

In Math.SE, user @HK Lee has suggested the paper On intrinsic geometry of surfaces in normed spaces by D. Burago and S. Ivanov. And I have found the following references, although I need the advice of the experts:

  1. An introductory textbook by A. Papadopoulos about the Busemann’s approach: Metric Spaces, convexity and non-positive curvature.

  2. A textbook by H. Busemann: The geometry of geodesics

  3. Two interesting papers by H. Busemann: The geometry of finsler spaces and Spaces with non-positive curvature.

Thanks in advance!


Hi. My name is Daisy Simpson. I am new to this forum. I am writing for school, college or university students. It will be my pleasure to help any of students in their thesis writing

Introduction To Algorithms 3rd Edition MIT Press: Red Black Tree insertion error in pseudo-code?

I’m implementing the algorithm on page 316 of the book Introduction to Algorithms. When I look at the pseudo-code, I feel that there is something wrong between line 10 to 14. I feel it’s missing a check. There is a YouTube video explaining this whole function (and it includes the pseudo-code plus line numbers):

The thing is, I think that //case 2 needs its own check. The else if z == z.p.right is both meant for //case 2 and //case 3. However, the code from //case 2 shouldn’t always fire. It should only fire when there is a triangle formation according to the YouTube video. In my implementation it always fires, even when it’s a line.

So I feel the pseudo-code is wrong, it’s also weird that it has an indentation, but I see no extra check.

Am I missing something?

Maybe superfluous, but I also typed the pseudo code given from the book here:

RB-INSERT-FIXUP(T, z)   while  z.p.color == RED     if  z.p == z.p.p.left       y = z.p.p.right        if y.color == RED         z.p.color = BLACK   // case 1         y.color = BLACK     // case 1         z.p.p.color = RED   // case 1         z = z.p.p           // case 1        else if z == z.p.right           z = z.p           // case 2           LEFT-ROTATE(T, z) // case 2         z.p.color = BLACK   // case 3         z.p.p.color = RED   // case 3         RIGHT-ROTATE(T, z.p.p) // case 3      else (same as then clause with "right and "left" exchanged)    T.root.color = BLACK 

Book recommendations beyond an introduction

So I’ve scraped the surface of many topics, but I would like to go further. Can anyone recommend some continuations to the following introductory books? It’s okay if necessarily it needs to be a little more specialised.

  • Dynamical Systems – Brin & Stuck

  • Ergodic Theory – Ward

  • Stochastic Analysis – Le Gall

  • 3-Manifolds – Schultens

  • Geometric Measure Theory – Evans & Gariepy

  • Geometric Group Theory – Loh

Gameplay-to-cutscene ratio: how interactive should I make the introduction to the campaign?

I’m planning a campaign with a specific backdrop-the characters are employees of a certain organization and this is their first day on the job. The organization and the job will no longer actually exist by the end of that round or the next. However on that first day, I was planning on their taking orders and receiving orientation from a program coordinator (i.e. their manager). What are some ways I can make the orientation/first day engaging enough that it doesn’t just come across like a scripted one-man cutscene?

What are the k-collections described in ch. 8 of “An Introduction to the Analysis of Algorithms” by Sedgewick

In chapter eight of “An Introduction to the Analysis of Algorithms” by Sedgewick (1996 edition) the coupon collector problem is introduced on page 425.

My confusion is how to identify the k-collections. A k-collection is defined as: “to be a word that consists of k different letters with the last letter in the word being the only time that letter occurs

Exercise 8.6 of the book asks to find all the 2-collections and 3-collections in Table 8.1, where that table shows the configurations of 4 balls in 3 urns

If I give it a try, I’d say a 3-collection from Table 8.1 is 2213, where the last letter (number 3) occurring just at the end, but I’m pretty sure I”m wrong.

Can anybody help providing an example of a k-collection (2 or 3-collection) from Table 8.1? Thanks