22 | September | 2009 | A kind of library

502 – Tilings

September 22, 2009

Since we covered some material on tilings, I figured you may find interesting the following nice expository paper by Richard Stanley and my friend and colleague Federico Ardila.

Leave a Comment » | 502: Logic and set theory | Tagged: tilings | Permalink
Posted by andrescaicedo

Here is a link to the Wikipedia page on Dandelin spheres, giving us the “modern” proof of the equivalence between the definition of conic sections as regions of intersection of planes and cones, with the standard definition in terms of distance to foci. The links on the Wikipedia page provide further explanations and nice graphics illustrating the argument.

Leave a Comment » | 175: Calculus II | Tagged: conic sections, Dandelin spheres, Germinal Pierre Dandelin | Permalink
Posted by andrescaicedo

You are currently browsing the A kind of library blog archives for the day Tuesday, September 22nd, 2009.
Categories

Categories
Search

Mathematical links
My pages
Others
Mathoverflow
- Answer by Andrés E. Caicedo for Where is the Erdős–Rado theorem stated in Erdős and Rado's Bull AMS paper? December 23, 2014
  This is Theorem 39 in the paper (see Theorem 4.(i) for a user-friendly preview). But the fact that $(2^\kappa)^+\to(\kappa^+)^2_\kappa$ is older (1946) and due to Erdős, see here: Paul Erdős. Some set-theoretical properties of graphs, Univ. Nac. Tucumán. Revista A. 3 (1942), 363-367 MR0009444 (5,151d). (Anyway, it is probably easier to read a more modern pre […]
  Andrés E. Caicedo
- Answer by Andrés E. Caicedo for The Erdős-Turán conjecture or the Erdős' conjecture? June 3, 2013
  One of the best places to track these things down is The mathematical coloring book, by Alexander Soifer, Springer 2009. Chapter 35 is on "Monochromatic arithmetic progressions", and section 35.4, "Paul Erdős’s Favorite Conjecture", is on the problem you ask about. As far as I can tell, the question is sometimes called the Erdős-Turán con […]
  Andrés E. Caicedo
- Relative densities September 12, 2013
  Throughout the question, we only consider primes of the form $3k+1$. A reference for cubic reciprocity is Ireland & Rosen's A Classical Introduction to Modern Number Theory. How can I count the relative density of those $p$ (of the form $3k+1$) such that the equation $2=3x^3$ has no solutions modulo $p$? Really, even pointers on how to say anything […]
  Andrés E. Caicedo
- On the division paradox December 22, 2022
  This question is partly motivated by Timothy Chow's recent question on the division paradox. Say that a set $X$ admits a paradoxical partition if and only if there is an equivalence relation $\sim$ on $X$ such that $|X|
  Andrés E. Caicedo
- Answer by Andrés E. Caicedo for Relative densities December 22, 2022
  A solution can be obtained as suggested by Keith Conrad in the comments, via Chebotarëv's theorem. Details can be found in $\S3.4$ of Coloring the $n$-Smooth Numbers with $n$ Colors Andrés Eduardo Caicedo, Thomas A. C. Chartier, Péter Pál Pach The Electronic Journal of Combinatorics 28 (1) (2021), #P1.34, 79 pp. DOI: https://doi.org/10.37236/8492 Many t […]
  Andrés E. Caicedo
Math.StackExchange
- Answer by Andrés E. Caicedo for cardinals such that $\alpha < \alpha ^ \beta$ January 31, 2023
  Recall that the beth ($\beth$) numbers are defined by transfinite recursion as $\beth_0=0$, $\beth_{\alpha+1}=2^{\beth_\alpha}$ and $\beth_\lambda=\sup_{\alpha
  Andrés E. Caicedo
- Answer by Andrés E. Caicedo for Is it possible for a function to be smooth everywhere, analytic nowhere, yet Taylor series at any point converges in a nonzero radius? January 7, 2014
  No, this is not possible. Dave L. Renfro wrote an excellent historical Essay on nowhere analytic $C^\infty$ functions in two parts (with numerous references). See here: 1 (dated May 9, 2002 6:18 PM), and 2 (dated May 19, 2002 8:29 PM). As indicated in part 1, in Zygmunt Zahorski. Sur l'ensemble des points singuliers d'une fonction d'une variab […]
  Andrés E. Caicedo
- Answer by Andrés E. Caicedo for How to check the real analyticity of a function? December 3, 2013
  This is a difficult question in general. Ideally, to show that $f$ is analytic at the origin, you show that in a suitable neighborhood of $0$, the error of the $n$-th Taylor polynomial approaches $0$ as $n\to\infty$. For example, for $f(x)=\sin(x)$, any derivative of $f(x)$ is one of $\sin(x)$, $\cos(x)$, $-\sin(x)$, or $-\cos(x)$, and the error given by the […]
  Andrés E. Caicedo
- Answer by Andrés E. Caicedo for Borel hierarchy doesn't "collapse" before $\omega_1$ October 1, 2013
  To complement Yann's answer: This is a nice problem, the possible length of Borel hierarchies in different spaces or without assuming the axiom of choice. It has been studied in detail by Arnie Miller. See Arnold W. Miller. On the length of Borel hierarchies, Ann. Math. Logic, 16 (3), (1979), 233–267. MR0548475 (80m:04003), Arnold W. Miller. Long Borel […]
  Andrés E. Caicedo
- Answer by Andrés E. Caicedo for consistency strength of PA September 3, 2013
  This is a good question, because a priori $\mathsf{PA}$ lacks the flexibility of $\mathsf{ZFC}$ that allows us to deal with consistency problems semantically (by building models) and, anyway, the obvious model of most subtheories of $\mathsf{PA}$ is just the standard model. The way this is done in the context of $\mathsf{ZFC}$ is using the reflection theorem […]
  Andrés E. Caicedo
September 2009

M T W T F S S

1 2 3 4 5 6

7 8 9 10 11 12 13

14 15 16 17 18 19 20

21 22 23 24 25 26 27

28 29 30

« Aug Oct »
Archives
- June 2019 (2)
- February 2019 (7)
- December 2018 (1)
- October 2018 (2)
- August 2018 (1)
- February 2018 (1)
- October 2017 (1)
- July 2017 (1)
- September 2016 (2)
- August 2016 (3)
- July 2016 (2)
- April 2016 (1)
- October 2015 (1)
- June 2015 (1)
- May 2015 (1)
- April 2015 (7)
- March 2015 (5)
- February 2015 (7)
- January 2015 (11)
- December 2014 (2)
- November 2014 (5)
- October 2014 (6)
- September 2014 (3)
- August 2014 (3)
- April 2014 (3)
- March 2014 (5)
- February 2014 (2)
- January 2014 (2)
- December 2013 (2)
- November 2013 (10)
- October 2013 (6)
- September 2013 (7)
- August 2013 (5)
- June 2013 (1)
- May 2013 (8)
- April 2013 (7)
- March 2013 (5)
- February 2013 (9)
- January 2013 (2)
- November 2012 (1)
- September 2012 (1)
- August 2012 (3)
- May 2012 (3)
- April 2012 (10)
- March 2012 (4)
- February 2012 (13)
- January 2012 (15)
- November 2011 (8)
- October 2011 (14)
- September 2011 (7)
- August 2011 (1)
- April 2011 (1)
- March 2011 (3)
- February 2011 (4)
- January 2011 (7)
- December 2010 (1)
- November 2010 (7)
- October 2010 (12)
- September 2010 (13)
- August 2010 (3)
- May 2010 (4)
- April 2010 (11)
- March 2010 (6)
- February 2010 (12)
- December 2009 (9)
- November 2009 (11)
- October 2009 (13)
- September 2009 (19)
- August 2009 (10)
- June 2009 (1)
- May 2009 (2)
- April 2009 (14)
- March 2009 (17)
- February 2009 (19)
- January 2009 (15)
- December 2008 (5)
- November 2008 (12)
- October 2008 (17)
- September 2008 (10)
- August 2008 (3)
- June 2008 (3)
- May 2008 (15)
- April 2008 (21)
- March 2008 (10)
- February 2008 (13)
- January 2008 (11)
- September 2007 (1)
- July 2007 (1)
- March 2007 (4)
- February 2007 (20)
- January 2007 (23)
- December 2006 (4)
Alan Turing Alexander S. Kechris Alfred Tarski algebraic numbers Andras Hajnal Andreas Blass Andrew Michael Bruckner arithmetic mean-geometric mean inequality Arnoud C. M. van Rooij Augustin Cauchy Axel Thue Bernhard Riemann BEST bpfa compactness completeness covering property determinacy Donald A. Martin elementary embedding Emile Borel Euclidean algorithm extension field field field extension Francisco Frank Plumpton Ramsey Fred Galvin Galvin-Hajnal theorem GCH Georg Cantor Godfrey Harold Hardy Grigor Sargsyan Harvey Friedman ideal Isaac Newton James Baumgartner Jerome Keisler Johan Thim John R. Steel Karl Theodor Wilhelm Weierstrass Koenig's lemma Kurt Goedel Lech Maligranda Marion Scheepers measurable cardinal Menachem Magidor midterm mm Nowhere differentiable function Paul Erdos Paul Larson pfa propositional logic quartic equation Ramsey theorem Rene Baire Richard Rado Robert Solovay Saharon Shelah Samuel Coskey sch Silver's theorem Stefan Banach Stevo Todorcevic supercompact cardinals Tarski's conjecture Turing degrees ultrafilter ultrapower Undecidability and incompleteness Unsolvable problems W. Hugh Woodin Wacław Franciszek Sierpiński Wilhelmus Hendricus Schikhof

	Richard Kimberly Hec… on Woodin’s proof of the se…
	Is there a Borel-mea… on 414/514 Examples of Baire clas…
	Simple bijectiveness… on 580 -Some choiceless results…
	Schur’s theore… on 580 -III. Partition calcu…
	Uncountable linearly… on 403/503 – Homework …
	What is a non-constr… on On strong measure zero se…
	function from Nmathb… on Hindsight
	Cauchy Schwarz Inequ… on 275 -Positive polynomials
	What is a Cantor-sty… on 580 -Some choiceless results…
	Question about Gener… on 580 -Some choiceless results…
	Set has Measure Zero… on On strong measure zero se…
	Non-measurable subse… on 580 -Cardinal arithmetic …
	hmeng1 on 414/514 The theorems of Rieman…
	580 -Cardinal arithm… on 580 -Cardinal arithmetic …
	580 -Cardinal arithm… on 580 -Cardinal arithmetic …

A kind of library

502 – Tilings

175 – Dandelin spheres

Categories

Recent Comments

Mathematical links

My pages

Others

Mathoverflow

Math.StackExchange

Archives