text:
# Polynomial partitioning: the basics (part 2)

This is the second in my series of posts about the basics of polynomial partitioning. In the first post of the series we presented the basics of polynomial partitioning and showed how the technique can be used to prove the Szemerédi-Trotter theorem. In this post we present Guth and Katz’s proof of the polynomial partitioning theorem. We first repeat the statement of the theorem.

Theorem 1 (Polynomial partitioning). Let $\cal P$ be a set of $m$ points in ${\mathbb R}^d$. Then for every $1< r \le m$, there exists an $r$-partitioning polynomial $f \in {\mathbb R}[x_1,\ldots, x_d]$ of degree $O(r^{1/d})$.

To prove Theorem 1, we recall the discrete version of the ham sandwich theorem (e.g., see this paper). A hyperplane $h$ in ${\mathbb R}^d$ bisects a finite point set $S\subset {\mathbb R}^d$ if each of the two open halfspaces bounded by $h$ contains at most $|S|/2$ points of $S$. The bisecting hyperplane may contain any number of points of $S$.

Theorem 2. Every $d$ finite point sets $S_1,\ldots,S_d \subset {\mathbb R}^d$ can be simultaneously bisected by a hyperplane.

A planar example of Theorem 2 is depicted in the following figure, where both the blue set and the orange set are bisected.

We use Theorem 2 to prove the following theorem, which is the discrete version of the polynomial ham sandwich theorem. A polynomial $f$ in ${\mathbb R}[x_1,\ldots,x_d]$ bisects a finite point set $S\subset {\mathbb R}^d$ if $f(p) > 0$ for at most $|S|/2$ points $p \in S$ and $f(p) < 0$ for at most $|S|/2$ points $p\in S$. The zero set $Z(f)$ may contain any number of points of $S$.

Theorem 3 (Stone and Tukey). Let $S_1,\ldots, S_t \subset {\mathbb R}^d$ be $t$ finite point sets, and let $D$ be the smallest integer such that $\binom{D+d}{d}-1\ge t$. Then there exists a nonzero polynomial $f\in {\mathbb R}[x_1,\ldots,x_d]$ of degree at most $D$ that simultaneously bisects all of the sets $S_i$.

Proof.     The number of monomials that a polynomial of degree at most $D$ in ${\mathbb R}[x_1,\ldots,x_d]$ can have is $\binom{D+d}{d}$; this is illustrated in the following figure.

Set $m=\binom{D+d}{d}-1$, the number of nonconstant monomials, and let

$U = \{ (i_1,\ldots,i_d) \mid 1 \le i_1 + \cdots + i_d \le D \}$;

clearly $|U|=m.$ The Veronese map $\nu:{\mathbb R}^d \to {\mathbb R}^m$ is defined as

$\nu(x_1,\ldots,x_d) := (x_1^{u_1}x_2^{u_2}\cdots x_d^{u_d})_{u\in U}\ .$

Every coordinate in ${\mathbb R}^m$ corresponds to a nonconstant monomial of degree at most $D$ in ${\mathbb R}[x_1,\ldots,x_d]$, and $\nu(\cdot)$ maps a point $p$ in ${\mathbb R}^d$ to the tuple of the values of these monomials at $p$. For $1 \le i \le t$, we set $S'_i = \nu(S_i)$. That is, every $S_i'$ is a finite point set in ${\mathbb R}^m$. By Theorem 2, since $m\ge t$, there exists a hyperplane $h\subset {\mathbb R}^m$ that simultaneously bisects all of the sets $S'_i$. If we denote the coordinates of ${\mathbb R}^m$ as $y_u$, for $u\in U$, then $h$ can be defined as the zero set of a linear equation of the form $h_0 + \sum_{u\in U}y_uh_u$, for a suitable set of constants $h_u$.

Returning to ${\mathbb R}^d$, we obtain the polynomial $f(x_1,\ldots,x_d) = h_0 + \sum_{u\in U} h_ux_1^{u_1}x_2^{u_2}\cdots x_d^{u_d}$. Consider a point $a\in {\mathbb R}^d$ and let $a' = \nu(a)$. Notice that $h_0+(h_u)_{u\in U}\cdot a' = f(a)$. That is, for every point $a \in {\mathbb R}^d$, $f(a)>0$ (resp., $f(a)<0$) if and only if $\nu(a)$ is in the positive side of $h$ (resp., in the negative side of $h$). Since $h$ bisects every $S'_i$, then $Z(f)$ bisects every $A_i$.     $\Box$

In other words, Theorem 3 implies the existence of a bisecting polynomial of degree at most

$D\le\lceil(d!\cdot(t+1))^{1/d}\rceil \le 2(d!\cdot t)^{1/d}.$

Recent uses of the polynomial ham sandwich theorem already appeared before Guth and Katz’s distinct distances paper (e.g., see Gromov and Guth). However, Guth and Katz were the first to apply the polynomial ham sandwich theorem to obtain a polynomial partitioning. We now present a variant of their proof.

Proof of Theorem 1.     Let $c_d = 2(d!)^{1/d}$. To prove the theorem, we show that there exists a sequence of polynomials $f_1,f_2,\ldots$ such that the degree of $f_i$ is smaller than $c_d 2^{(i+1)/d}/(2^{1/d}-1)$ and every connected component of ${\mathbb R}^d\setminus Z(f_i)$ contains at most $m/2^i$ points of $\cal P$. An example is depicted in the following figure.

This would complete the proof since we can then choose $f=f_t$, where $t$ is the minimum integer satisfying $2^t>r$.

We prove the existence of $f_1,f_2,\ldots$ by induction. The existence of $f_1$ is immediately implied by Theorem 2. For the induction step, assume that there exists a polynomial $f_i$ of degree smaller than $c_d 2^{(i+1)/d}/(2^{1/d}-1)$ such that every connected component of ${\mathbb R}^d\setminus Z(f_i)$ contains at most $m/2^i$ points of $\cal P$. Since $|{\cal P}|=m$, the number of these connected components that contain more than $m/2^{i+1}$ points of $\cal P$ is smaller than $2^{i+1}$. Let $S_1,\ldots,S_n \subset {\cal P}$ be the subsets of $\cal P$ that are contained in each of these connected components (that is, $|S_i|>m/2^{i+1}$ for each $i$, and $n<2^{i+1}$). By Theorem 3 (and the remark following it), there is a polynomial $g_i$ of degree smaller than $c_d2^{(i+1)/d}$ that simultaneously bisects every $S_i$. We can set $f_{i+1} = f_i \cdot g_i$, since every connected component of ${\mathbb R}^d\setminus Z(f_i \cdot g_i)$ contains at most $m/2^{i+1}$ points of $\cal P$ and $f_i \cdot g_i$ is a polynomial of degree smaller than

$\frac{c_d 2^{(i+1)/d}}{2^{1/d}-1} + c_d2^{(i+1)/d} = c_d2^{(i+1)/d}\cdot \left(\frac{1}{2^{1/d}-1}+1\right) = \frac{c_d2^{(i+2)/d}}{2^{1/d}-1}.$

This completes the induction step, and thus also the proof.     $\Box$


text:
# Chapter 10 - Exponents and Radicals - 10.2 Rational Numbers as Exponents - 10.2 Exercise Set - Page 641: 55

$\dfrac{1}{\sqrt[4]{8r^3s^3}}$

#### Work Step by Step

Using the laws of exponents, the given expression, $(2rs)^{-3/4} ,$ is equivalent to \begin{array}{l}\require{cancel} \dfrac{1}{(2rs)^{3/4}} .\end{array} Using $a^{m/n}=\sqrt[n]{a^m}=(\sqrt[n]{a})^m$, the expression, $\dfrac{1}{(2rs)^{3/4}} ,$ simplifies to \begin{array}{l}\require{cancel} \dfrac{1}{\sqrt[4]{(2rs)^{3}}} \\\\= \dfrac{1}{\sqrt[4]{2^3r^3s^3}} \\\\= \dfrac{1}{\sqrt[4]{8r^3s^3}} .\end{array}

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.

format: plain


text:
location:  Publications → journals
Search results

Search: All articles in the CMB digital archive with keyword measure

 Expand all        Collapse all Results 1 - 25 of 26

1. CMB 2015 (vol 58 pp. 415)

Willson, Benjamin
 A Fixed Point Theorem and the Existence of a Haar Measure for Hypergroups Satisfying Conditions Related to Amenability In this paper we present a fixed point property for amenable hypergroups which is analogous to Rickert's fixed point theorem for semigroups. It equates the existence of a left invariant mean on the space of weakly right uniformly continuous functions to the existence of a fixed point for any action of the hypergroup. Using this fixed point property, a certain class of hypergroups are shown to have a left Haar measure. Keywords:invariant measure, Haar measure, hypergroup, amenability, function translationsCategories:43A62, 43A05, 43A07

2. CMB 2014 (vol 57 pp. 780)

Erzakova, Nina A.
 Measures of Noncompactness in Regular Spaces Previous results by the author on the connection between three of measures of non-compactness obtained for $L_p$, are extended to regular spaces of measurable functions. An example of advantage in some cases one of them in comparison with another is given. Geometric characteristics of regular spaces are determined. New theorems for $(k,\beta)$-boundedness of partially additive operators are proved. Keywords:measure of non-compactness, condensing map, partially additive operator, regular space, ideal spaceCategories:47H08, 46E30, 47H99, 47G10

3. CMB 2013 (vol 57 pp. 598)

Lu, Yufeng; Yang, Dachun; Yuan, Wen
 Interpolation of Morrey Spaces on Metric Measure Spaces In this article, via the classical complex interpolation method and some interpolation methods traced to Gagliardo, the authors obtain an interpolation theorem for Morrey spaces on quasi-metric measure spaces, which generalizes some known results on ${\mathbb R}^n$. Keywords:complex interpolation, Morrey space, Gagliardo interpolation, CalderÃ³n product, quasi-metric measure spaceCategories:46B70, 46E30

4. CMB 2012 (vol 57 pp. 240)

Bernardes, Nilson C.
 Addendum to Limit Sets of Typical Homeomorphisms'' Given an integer $n \geq 3$, a metrizable compact topological $n$-manifold $X$ with boundary, and a finite positive Borel measure $\mu$ on $X$, we prove that for the typical homeomorphism $f : X \to X$, it is true that for $\mu$-almost every point $x$ in $X$ the restriction of $f$ (respectively of $f^{-1}$) to the omega limit set $\omega(f,x)$ (respectively to the alpha limit set $\alpha(f,x)$) is topologically conjugate to the universal odometer. Keywords:topological manifolds, homeomorphisms, measures, Baire category, limit setsCategories:37B20, 54H20, 28C15, 54C35, 54E52

5. CMB 2012 (vol 56 pp. 759)

Issa, Zahraa; Lalín, Matilde
 A Generalization of a Theorem of Boyd and Lawton The Mahler measure of a nonzero $n$-variable polynomial $P$ is the integral of $\log|P|$ on the unit $n$-torus. A result of Boyd and Lawton says that the Mahler measure of a multivariate polynomial is the limit of Mahler measures of univariate polynomials. We prove the analogous result for different extensions of Mahler measure such as generalized Mahler measure (integrating the maximum of $\log|P|$ for possibly different $P$'s), multiple Mahler measure (involving products of $\log|P|$ for possibly different $P$'s), and higher Mahler measure (involving $\log^k|P|$). Keywords:Mahler measure, polynomialCategories:11R06, 11R09

6. CMB 2011 (vol 56 pp. 388)

Mursaleen, M.
 Application of Measure of Noncompactness to Infinite Systems of Differential Equations In this paper we determine the Hausdorff measure of noncompactness on the sequence space $n(\phi)$ of W. L. C. Sargent. Further we apply the technique of measures of noncompactness to the theory of infinite systems of differential equations in the Banach sequence spaces $n(\phi)$ and $m(\phi)$. Our aim is to present some existence results for infinite systems of differential equations formulated with the help of measures of noncompactness. Keywords:sequence spaces, BK spaces, measure of noncompactness, infinite system of differential equationsCategories:46B15, 46B45, 46B50, 34A34, 34G20

7. CMB 2011 (vol 56 pp. 326)

Erdoğan, M. Burak; Oberlin, Daniel M.
 Restricting Fourier Transforms of Measures to Curves in $\mathbb R^2$ We establish estimates for restrictions to certain curves in $\mathbb R^2$ of the Fourier transforms of some fractal measures. Keywords:Fourier transforms of fractal measures, Fourier restrictionCategories:42B10, 28A12

8. CMB 2011 (vol 55 pp. 646)

Zhou, Jiang; Ma, Bolin
 Marcinkiewicz Commutators with Lipschitz Functions in Non-homogeneous Spaces Under the assumption that $\mu$ is a nondoubling measure, we study certain commutators generated by the Lipschitz function and the Marcinkiewicz integral whose kernel satisfies a HÃ¶rmander-type condition. We establish the boundedness of these commutators on the Lebesgue spaces, Lipschitz spaces, and Hardy spaces. Our results are extensions of known theorems in the doubling case. Keywords:non doubling measure, Marcinkiewicz integral, commutator, ${\rm Lip}_{\beta}(\mu)$, $H^1(\mu)$Categories:42B25, 47B47, 42B20, 47A30

9. CMB 2011 (vol 55 pp. 297)

Glasner, Eli
 The Group $\operatorname{Aut}(\mu)$ is Roelcke Precompact Following a similar result of Uspenskij on the unitary group of a separable Hilbert space, we show that, with respect to the lower (or Roelcke) uniform structure, the Polish group $G= \operatorname{Aut}(\mu)$ of automorphisms of an atomless standard Borel probability space $(X,\mu)$ is precompact. We identify the corresponding compactification as the space of Markov operators on $L_2(\mu)$ and deduce that the algebra of right and left uniformly continuous functions, the algebra of weakly almost periodic functions, and the algebra of Hilbert functions on $G$, i.e., functions on $G$ arising from unitary representations, all coincide. Again following Uspenskij, we also conclude that $G$ is totally minimal. Keywords:Roelcke precompact, unitary group, measure preserving transformations, Markov operators, weakly almost periodic functionsCategories:54H11, 22A05, 37B05, 54H20

10. CMB 2011 (vol 55 pp. 225)

Bernardes, Nilson C.
 Limit Sets of Typical Homeomorphisms Given an integer $n \geq 3$, a metrizable compact topological $n$-manifold $X$ with boundary, and a finite positive Borel measure $\mu$ on $X$, we prove that for the typical homeomorphism $f \colon X \to X$, it is true that for $\mu$-almost every point $x$ in $X$ the limit set $\omega(f,x)$ is a Cantor set of Hausdorff dimension zero, each point of $\omega(f,x)$ has a dense orbit in $\omega(f,x)$, $f$ is non-sensitive at each point of $\omega(f,x)$, and the function $a \to \omega(f,a)$ is continuous at $x$. Keywords:topological manifolds, homeomorphisms, measures, Baire category, limit setsCategories:37B20, 54H20, 28C15, 54C35, 54E52

11. CMB 2011 (vol 55 pp. 26)

Bertin, Marie José
 A Mahler Measure of a $K3$ Surface Expressed as a Dirichlet $L$-Series We present another example of a $3$-variable polynomial defining a $K3$-hypersurface and having a logarithmic Mahler measure expressed in terms of a Dirichlet $L$-series. Keywords:modular Mahler measure, Eisenstein-Kronecker series, $L$-series of $K3$-surfaces, $l$-adic representations, LivnÃ© criterion, Rankin-Cohen bracketsCategories:11, 14D, 14J

12. CMB 2011 (vol 54 pp. 544)

Strungaru, Nicolae
 Positive Definite Measures with Discrete Fourier Transform and Pure Point Diffraction In this paper we characterize the positive definite measures with discrete Fourier transform. As an application we provide a characterization of pure point diffraction in locally compact Abelian groups. Keywords:pure point diffraction, positive definite measure, Fourier transform of measuresCategory:43A25

13. CMB 2011 (vol 55 pp. 146)

Li, Songxiao; Wulan, Hasi; Zhu, Kehe
 A Characterization of Bergman Spaces on the Unit Ball of ${\mathbb C}^n$. II It has been shown that a holomorphic function $f$ in the unit ball $\mathbb{B}_n$ of ${\mathbb C}_n$ belongs to the weighted Bergman space $A^p_\alpha$, $p>n+1+\alpha$, if and only if the function $|f(z)-f(w)|/|1-\langle z,w\rangle|$ is in $L^p(\mathbb{B}_n\times\mathbb{B}_n,dv_\beta \times dv_\beta)$, where $\beta=(p+\alpha-n-1)/2$ and $dv_\beta(z)= (1-|z|^2)^\beta\,dv(z)$. In this paper we consider the range $0n+1+\alpha$ is particularly interesting. Keywords:Bergman spaces, unit ball, volume measureCategory:32A36

14. CMB 2011 (vol 54 pp. 739)

Samuels, Charles L.
 The Infimum in the Metric Mahler Measure Dubickas and Smyth defined the metric Mahler measure on the multiplicative group of non-zero algebraic numbers. The definition involves taking an infimum over representations of an algebraic number $\alpha$ by other algebraic numbers. We verify their conjecture that the infimum in its definition is always achieved, and we establish its analog for the ultrametric Mahler measure. Keywords:Weil height, Mahler measure, metric Mahler measure, Lehmer's problemCategories:11R04, 11R09

15. CMB 2010 (vol 54 pp. 193)

Bennett, Harold; Lutzer, David
 Measurements and $G_\delta$-Subsets of Domains In this paper we study domains, Scott domains, and the existence of measurements. We use a space created by D.~K. Burke to show that there is a Scott domain $P$ for which $\max(P)$ is a $G_\delta$-subset of $P$ and yet no measurement $\mu$ on $P$ has $\ker(\mu) = \max(P)$. We also correct a mistake in the literature asserting that $[0, \omega_1)$ is a space of this type. We show that if $P$ is a Scott domain and $X \subseteq \max(P)$ is a $G_\delta$-subset of $P$, then $X$ has a $G_\delta$-diagonal and is weakly developable. We show that if $X \subseteq \max(P)$ is a $G_\delta$-subset of $P$, where $P$ is a domain but perhaps not a Scott domain, then $X$ is domain-representable, first-countable, and is the union of dense, completely metrizable subspaces. We also show that there is a domain $P$ such that $\max(P)$ is the usual space of countable ordinals and is a $G_\delta$-subset of $P$ in the Scott topology. Finally we show that the kernel of a measurement on a Scott domain can consistently be a normal, separable, non-metrizable Moore space. Keywords:domain-representable, Scott-domain-representable, measurement, Burke's space, developable spaces, weakly developable spaces, $G_\delta$-diagonal, Äech-complete space, Moore space, $\omega_1$, weakly developable space, sharp base, AF-completeCategories:54D35, 54E30, 54E52, 54E99, 06B35, 06F99

16. CMB 2010 (vol 54 pp. 12)

Bingham, N. H.; Ostaszewski, A. J.
 Homotopy and the Kestelman-Borwein-Ditor Theorem The Kestelman--Borwein--Ditor Theorem, on embedding a null sequence by translation in (measure/category) large'' sets has two generalizations. Miller replaces the translated sequence by a sequence homotopic to the identity''. The authors, in a previous paper, replace points by functions: a uniform functional null sequence replaces the null sequence, and translation receives a functional form. We give a unified approach to results of this kind. In particular, we show that (i) Miller's homotopy version follows from the functional version, and (ii) the pointwise instance of the functional version follows from Miller's homotopy version. Keywords:measure, category, measure-category duality, differentiable homotopyCategory:26A03

17. CMB 2008 (vol 51 pp. 57)

Dobrowolski, Edward
 A Note on Integer Symmetric Matrices and Mahler's Measure We find a lower bound on the absolute value of the discriminant of the minimal polynomial of an integral symmetric matrix and apply this result to find a lower bound on Mahler's measure of related polynomials and to disprove a conjecture of D. Estes and R. Guralnick. Keywords:integer matrices, Lehmer's problem, Mahler's measureCategories:11C20, 11R06

18. CMB 2007 (vol 50 pp. 191)

Drungilas, Paulius; Dubickas, Artūras
 Every Real Algebraic Integer Is a Difference of Two Mahler Measures We prove that every real algebraic integer $\alpha$ is expressible by a difference of two Mahler measures of integer polynomials. Moreover, these polynomials can be chosen in such a way that they both have the same degree as that of $\alpha$, say $d$, one of these two polynomials is irreducible and another has an irreducible factor of degree $d$, so that $\alpha=M(P)-bM(Q)$ with irreducible polynomials $P, Q\in \mathbb Z[X]$ of degree $d$ and a positive integer $b$. Finally, if $d \leqslant 3$, then one can take $b=1$. Keywords:Mahler measures, Pisot numbers, Pell equation, $abc$-conjectureCategories:11R04, 11R06, 11R09, 11R33, 11D09

19. CMB 2006 (vol 49 pp. 536)

Dostál, Petr; Lukeš, Jaroslav; Spurný, Jiří
 Measure Convex and Measure Extremal Sets We prove that convex sets are measure convex and extremal sets are measure extremal provided they are of low Borel complexity. We also present examples showing that the positive results cannot be strengthened. Keywords:measure convex set, measure extremal set, faceCategories:46A55, 52A07

20. CMB 2005 (vol 48 pp. 523)

Düvelmeyer, Nico
 Angle Measures and Bisectors in Minkowski Planes \begin{abstract} We prove that a Minkowski plane is Euclidean if and only if Busemann's or Glogovskij's definitions of angular bisectors coincide with a bisector defined by an angular measure in the sense of Brass. In addition, bisectors defined by the area measure coincide with bisectors defined by the circumference (arc length) measure if and only if the unit circle is an equiframed curve. Keywords:Radon curves, Minkowski geometry, Minkowski planes,, angular bisector, angular measure, equiframed curvesCategories:52A10, 52A21

21. CMB 2005 (vol 48 pp. 147)

 Baker-Type Estimates for Linear Forms in the Values of $q$-Series We obtain lower estimates for the absolute values of linear forms of the values of generalized Heine series at non-zero points of an imaginary quadratic field~$\II$, in particular of the values of $q$-exponential function. These estimates depend on the individual coefficients, not only on the maximum of their absolute values. The proof uses a variant of classical Siegel's method applied to a system of functional Poincar\'e-type equations and the connection between the solutions of these functional equations and the generalized Heine series. Keywords:measure of linear independence, $q$-seriesCategories:11J82, 33D15

22. CMB 2003 (vol 46 pp. 373)

Laugesen, Richard S.; Pritsker, Igor E.
 Potential Theory of the Farthest-Point Distance Function We study the farthest-point distance function, which measures the distance from $z \in \mathbb{C}$ to the farthest point or points of a given compact set $E$ in the plane. The logarithm of this distance is subharmonic as a function of $z$, and equals the logarithmic potential of a unique probability measure with unbounded support. This measure $\sigma_E$ has many interesting properties that reflect the topology and geometry of the compact set $E$. We prove $\sigma_E(E) \leq \frac12$ for polygons inscribed in a circle, with equality if and only if $E$ is a regular $n$-gon for some odd $n$. Also we show $\sigma_E(E) = \frac12$ for smooth convex sets of constant width. We conjecture $\sigma_E(E) \leq \frac12$ for all~$E$. Keywords:distance function, farthest points, subharmonic function, representing measure, convex bodies of constant widthCategories:31A05, 52A10, 52A40

23. CMB 2002 (vol 45 pp. 231)

Hironaka, Eriko
 Erratum:~~The Lehmer Polynomial and Pretzel Links Erratum to {\it The Lehmer Polynomial and Pretzel Links}, Canad. J. Math. {\bf 44}(2001), 440--451. Keywords:Alexander polynomial, pretzel knot, Mahler measure, Salem number, Coxeter groupsCategories:57M05, 57M25, 11R04, 11R27

24. CMB 2002 (vol 45 pp. 196)

Dubickas, Artūras
 Mahler Measures Close to an Integer We prove that the Mahler measure of an algebraic number cannot be too close to an integer, unless we have equality. The examples of certain Pisot numbers show that the respective inequality is sharp up to a constant. All cases when the measure is equal to the integer are described in terms of the minimal polynomials. Keywords:Mahler measure, PV numbers, Salem numbersCategories:11R04, 11R06, 11R09, 11J68

25. CMB 2002 (vol 45 pp. 97)

Haas, Andrew
 Invariant Measures and Natural Extensions We study ergodic properties of a family of interval maps that are given as the fractional parts of certain real M\"obius transformations. Included are the maps that are exactly $n$-to-$1$, the classical Gauss map and the Renyi or backward continued fraction map. A new approach is presented for deriving explicit realizations of natural automorphic extensions and their invariant measures. Keywords:Continued fractions, interval maps, invariant measuresCategories:11J70, 58F11, 58F03
 Page 1 2 Next
 top of page | contact us | privacy | site map |


text:
# volume of ellipsoid

## Primary tabs

Synonym:
ellipsoid volume
Type of Math Object:
Result
Major Section:
Reference
Groups audience:

## Mathematics Subject Classification

### Ellipsoid Volume

Why do some people (particularly from the medical field) use the following equation to calculate ellipsoid volume?

volume = length Ã— width Ã— thickness Ã— Ï€/6

### Re: Ellipsoid Volume

Volume = Length X width X thickness X pi/6
rather than using Length X width X thickness X 4/3 X pi

### Re: Ellipsoid Volume

> Volume = Length X width X thickness X pi/6
> rather than using Length X width X thickness X 4/3 X pi

That's easy. The second formula would be valid if you were using the half-length, half-width and half-thickness (like the radius of the sphere). If you use the full lengths (like the diameter) then you must divide the answer by 2x2x2=8, so 4/3 becomes 1/6.

### Re: Ellipsoid Volume

Hi ansonkoehler,
See the last Property in http://planetmath.org/encyclopedia/Similar2.html
Regards,
Jussi


text:
# Abstract Nonsense

## Review of Group Theory: Alternate Proof to the Sylow Theorems

Point of post: This is a continuation of this post.

We lastly prove that all Sylow $p$-subgroups of a group $G$ are conjugate. This is what was initially the second Sylow theorem.

Theorem: Let $G$ be a group and $p$ a prime such that $p\mid |G|$. Then, for any $H\in\text{Syl}_p\left(G\right)$ one has that $C(H)=\left\{gHg^{-1}:g\in G\right\}=\text{Syl}_p\left(G\right)$.

Proof: Clearly we have that $G$ acts on $\text{Syl}_p\left(G\right)$ by conjugation. Our claim is equivalent to $\mathcal{O}_{G,H}=\text{Syl}_p\left(G\right)$ where $H$ is the orbit representative and $G$ is to keep track that this is the orbit under the $G$-action (there will be more actions to come). So, choose $H_0\in\text{Syl}_p\left(G\right)$ and suppose that $\text{Syl}_p\left(G\right)$ admits more orbits than $\mathcal{O}_{G,H_0}$, suppose they are $\mathcal{O}_{G,H_1},\cdots,$. We know that $H_0$ acts on $\text{Syl}_p\left(G\right)$ by conjugation though and that this action induces its own orbits; denote such orbits in general as $\mathcal{O}_{H_0,L}$ (where $L$ is the orbit representative and $H_0$ is to remind us that these are orbits under the $H_0$-action). We claim that for each $\mathcal{O}_{G,H_k}$ if we define

$\Omega_{H_k}=\left\{\mathcal{O}_{H_0,L}:\mathcal{O}_{H_0,L}\cap\mathcal{O}_{G,H_k}\ne\varnothing\right\}$

then

$\displaystyle \mathcal{O}_{G,H_k}=\bigcup_{\mathcal{O}_{H_0,L}\in\Omega_{H_k}}\mathcal{O}_{H_0,L}$

Indeed, it’s clear that

$\displaystyle \mathcal{O}_{G,H_k}\subseteq\bigcup_{\mathcal{O}_{H_0,L}\in\Omega_{H_k}}\mathcal{O}_{H_0,L}$

Conversely though, if $h_0Lh_0^{-1}\in\mathcal{O}_{G,H_k}\cap\mathcal{O}_{H_0,L}$ then we have in particular that $h_0Lh_0^{-1}\in\mathcal{O}_{G,H_k}$. But, since the orbits are equivalence classes it follows that

$\mathcal{O}_{G,H_k}=\mathcal{O}_{G,h_0Lh_0^{-1}}=\mathcal{O}_{G,L}\supseteq \mathcal{O}_{H_0,L}$

It clearly follows then using the same methodology as in our last proof and noticing that $\mathcal{O}_{H_0,H_0}\subseteq\mathcal{O}_{G,H_0}$ (it contains the only trivial orbit, and $\mathcal{O}_{G,H_1}$ being distinct from $\mathcal{O}_{G,H_0}$ doesn’t) that

$\#\left(\mathcal{O}_{G,H_0}\right)\equiv 1\text{ mod }p\;\;\text{ and }\;\;\#\left(\mathcal{O}_{G,H_2}\right)\equiv 0\text{ mod }\cdots$

But, notice that there was nothing special about using the $H_0$ action of conjugation on $\text{Syl}_p\left(G\right)$. We could easily repeat the above argument for $\mathcal{O}_{G,H_2}$ to arrive at

$\#\left(\mathcal{O}_{G,H_0}\right)\equiv 0\text{ mod }p\;\;\text{ and }\;\;\#\left(\mathcal{O}_{G,H_1}\right)\equiv 1\cdots$

which is evidently a contradiction. It follows that there can be at most one distinct orbit, from where by prior discussion the conclusion follows. $\blacksquare$

References:

1. Simon, Barry. Representations of Finite and Compact Groups. Providence, RI: American Mathematical Society, 1996. Print.

2. Dummit, David Steven., and Richard M. Foote. Abstract Algebra. Hoboken, NJ: Wiley, 200

3.Grillet, Pierre A. Abstract Algebra. New York: Springer, 2007. Print.


text:
# -40 (number)

 Cardinal -40 Binary -101000 or 11011000in two's complementin a signed byte Hexadecimal -28, or D8in two's complementin a signed byte

Minus forty is:

• the temperature (in degrees) at which Celsius and Fahrenheit scales read the same; often referred to as 40 below.

This can be shown easily. To convert from Fahrenheit to Celsius, one subtracts 32, then multiples by 5/9. This is expressed mathematically as C = (5/9)(F − 32), where F = Fahrenheit and C = Celsius. Alternatively F = C(9/5) + 32 .

Therefore, to find this special temperature, choose a formula and set the variables equal, denoting a temperature that is the same in both scales:

      F = F(9/5) + 32
(−4/5)F = 32
F = −40

or

     C = (5/9)(C − 32)
(9/5)C = C − 32
(4/5)C = −32
C = −40

These are mathematically equivalent demonstrations.

Also, the Fahrenheit and Celsius temperature conversion formulae can be represented in the particularly nice forms (using -40)

[itex] F = \frac{9(C - (-40))}{5} + (-40)\,[itex]

and

[itex] C = \frac{5(F - (-40))}{9} + (-40)\,[itex]

which may be easier for mental calculation.

sl:-40 (število)

• Art and Cultures
• Musical Instruments (http://academickids.com/encyclopedia/index.php/List_of_musical_instruments)
• Countries of the World (http://www.academickids.com/encyclopedia/index.php/Countries)
• Ancient Civilizations (http://www.academickids.com/encyclopedia/index.php/Ancient_Civilizations)
• Industrial Revolution (http://www.academickids.com/encyclopedia/index.php/Industrial_Revolution)
• Middle Ages (http://www.academickids.com/encyclopedia/index.php/Middle_Ages)
• United States (http://www.academickids.com/encyclopedia/index.php/United_States)
• World History (http://www.academickids.com/encyclopedia/index.php/History_of_the_world)
• Human Body (http://www.academickids.com/encyclopedia/index.php/Human_Body)
• Physical Science (http://www.academickids.com/encyclopedia/index.php/Physical_Science)
• Social Studies (http://www.academickids.com/encyclopedia/index.php/Social_Studies)
• Space and Astronomy
• Solar System (http://www.academickids.com/encyclopedia/index.php/Solar_System)


text:
Cabal-syntax-3.9.0.0: A library for working with .cabal files
Backwards compatibility reexport of most things you need to know about .cabal files.


text:
# Core Features¶

This section discusses the core features of the ASDF data format, and provides examples and use cases that are specific to the Python implementation.

## Data Model¶

The fundamental data object in ASDF is the tree, which is a nested combination of basic data structures: dictionaries, lists, strings and numbers. In Python, these types correspond to dict, list, str, and int, float, and complex, respectively. The top-level tree object behaves like a Python dictionary and supports arbitrary nesting of data structures. For simple examples of creating and reading trees, see Overview.

One of the key features of ASDF is its ability to serialize numpy arrays. This is discussed in detail in Array Data.

While the core ASDF package supports serialization of basic data types and Numpy arrays, its true power comes from its ability to be extended to support serialization of a wide range of custom data types. Details on using ASDF extensions can be found in Using extensions. Details on creating custom ASDF extensions to support custom data types can be found in Writing ASDF extensions.

## Array Data¶

Much of ASDF’s power and convenience comes from its ability to represent multidimensional array data. The asdf Python package provides native support for numpy arrays.

## Using extensions¶

According to Wikipedia, serialization “is the process of translating data structures or object state into a format that can be stored…and reconstructed later” [1].

The power of ASDF is that it provides the ability to store, or serialize, the state of Python objects into a human-readable data format. The state of those objects can later be restored by another program in a process called deserialization.

While ASDF is capable of serializing basic Python types and Numpy arrays out of the box, it can also be extended to serialize arbitrary custom data types. This section discusses the extension mechanism from a user’s perspective. For documentation on creating extensions, see Writing ASDF extensions.

Even though this particular implementation of ASDF necessarily serializes Python data types, in theory an ASDF implementation in another language could read the resulting file and reconstruct an analogous type in that language. Conversely, this implementation can read ASDF files that were written by other implementations of ASDF as long as the proper extensions are available.

## Schema validation¶

Schema validation is used to determine whether an ASDF file is well formed. All ASDF files must conform to the schemas defined by the ASDF Standard. Schema validation occurs when reading ASDF files (using asdf.open), and also when writing them out (using AsdfFile.write_to or AsdfFile.update).

Schema validation also plays a role when using custom extensions (see Using extensions and Writing ASDF extensions). Extensions must provide schemas for the types that they serialize. When writing a file with custom types, the output is validated against the schemas corresponding to those types. If the appropriate extension is installed when reading a file with custom types, then the types will be validated against the schemas provided by the corresponding extension.

### Custom schemas¶

Every ASDF file is validated against the ASDF Standard, and also against any schemas provided by custom extensions. However, it is sometimes useful for particular applications to impose additional restrictions when deciding whether a given file is valid or not.

For example, consider an application that processes digital image data. The application expects the file to contain an image, and also some metadata about how the image was created. The following example schema reflects these expectations:

%YAML 1.1
---
$schema: "http://stsci.edu/schemas/yaml-schema/draft-01" id: "http://stsci.edu/schemas/asdf/core/asdf-1.1.0" tag: "tag:stsci.edu:asdf/core/asdf-1.1.0" type: object properties: image: description: An ndarray containing image data.$ref: "ndarray-1.0.0"

type: object
properties:
time:
description: |
A timestamp for when the image was created, in UTC.
type: string
format: date-time
resolution:
description: |
A 2D array representing the resolution of the image (N x M).
type: array
items:
type: integer
number: 2

This schema restricts the kinds of files that will be accepted as valid to those that contain a top-level image property that is an ndarray, and a top-level metadata property that contains information about the time the image was taken and the resolution of the image. Note that the schema uses the same id and tag as the top-level core schema from the ASDF Standard. This is because it is validating the file at the top level, but is imposing restrictions beyond what is normally required for an ASDF file.

In order to use this schema for a secondary validation pass, we pass the custom_schema argument to either asdf.open or the AsdfFile constructor. Assume that the schema file lives in image_schema.yaml, and we wish to open a file called image.asdf. We would open the file with the following code:

import asdf
af = asdf.open('image.asdf', custom_schema='image_schema.yaml')

Similarly, if we wished to use this schema when creating new files:

new_af = asdf.AsdfFile(custom_schema='image_schema.yaml')
...

## Versioning and Compatibility¶

There are several different versions to keep in mind when discussing ASDF:

• The software package version
• The ASDF Standard version
• The ASDF file format version
• Individual tag and schema versions

Each ASDF file contains information about the various versions that were used to create the file. The most important of these are the ASDF Standard version and the ASDF file format version. A particular version of the ASDF software package will explicitly provide support for a specific combination of these versions.

Tag and schema versions are also important for serializing and deserializing data types that are stored in ASDF files. A detailed discussion of tag and schema versions from a user perspective can be found in Custom types, extensions, and versioning.

Since ASDF is designed to serve as an archival format, the software attempts to provide backwards compatibility when reading older versions of the ASDF Standard and ASDF file format. However, since deserializing ASDF types sometimes requires other software packages, backwards compatibility is often contingent on the available versions of such software packages.

In general, forward compatibility with newer versions of the ASDF Standard and ASDF file format is not supported by the software. However, if newer tag and schema versions are detected, the software will attempt to process them.

When creating new ASDF files, it is possible to control the version of the file format that is used. This can be specified by passing the version argument to either the AsdfFile constructor when the file object is created, or to the AsdfFile.write_to method when it is written. By default, the latest version of the file format will be used. Note that this option has no effect on the versions of tag types from custom extensions.

## External References¶

### Tree References¶

ASDF files may reference items in the tree in other ASDF files. The syntax used in the file for this is called “JSON Pointer”, but users of asdf can largely ignore that.

First, we’ll create a ASDF file with a couple of arrays in it:

from asdf import AsdfFile
import numpy as np

tree = {
'a': np.arange(0, 10),
'b': np.arange(10, 20)
}

target = AsdfFile(tree)
target.write_to('target.asdf')

target.asdfb'#ASDF' 1.2.0
%YAML 1.1
%TAG ! tag:stsci.edu:asdf/
--- !core/asdf-1.1.0
asdf_library: !core/software-1.0.0 {author: Space Telescope Science Institute, homepage: 'http://github.com/spacetelescope/asdf',
name: asdf, version: 2.1.0.dev1406}
history:
extensions:
extension_class: asdf.extension.BuiltinExtension
software: {name: asdf, version: 2.1.0.dev1406}
a: !core/ndarray-1.0.0
source: 0
datatype: int64
byteorder: little
shape: [10]
b: !core/ndarray-1.0.0
source: 1
datatype: int64
byteorder: little
shape: [10]
...BLOCK 0:
allocated_size: 80
used_size: 80
data_size: 80
data: b'0000000000000000010000000000000002000000...'BLOCK 1:
allocated_size: 80
used_size: 80
data_size: 80
data: b'0a000000000000000b000000000000000c000000...'#ASDF BLOCK INDEX
%YAML 1.1
--- [599, 733]
...

Then we will reference those arrays in a couple of different ways. First, we’ll load the source file in Python and use the make_reference method to generate a reference to array a. Second, we’ll work at the lower level by manually writing a JSON Pointer to array b, which doesn’t require loading or having access to the target file.

ff = AsdfFile()

with AsdfFile.open('target.asdf') as target:
ff.tree['my_ref_a'] = target.make_reference(['a'])

ff.tree['my_ref_b'] = {'$ref': 'target.asdf#b'} ff.write_to('source.asdf')  source.asdfb'#ASDF' 1.2.0 %YAML 1.1 %TAG ! tag:stsci.edu:asdf/ --- !core/asdf-1.1.0 asdf_library: !core/software-1.0.0 {author: Space Telescope Science Institute, homepage: 'http://github.com/spacetelescope/asdf', name: asdf, version: 2.1.0.dev1406} history: extensions: - !core/extension_metadata-1.0.0 extension_class: asdf.extension.BuiltinExtension software: {name: asdf, version: 2.1.0.dev1406} my_ref_a: {$ref: target.asdf#a}
my_ref_b: {\$ref: target.asdf#b}
...

Calling find_references will look up all of the references so they can be used as if they were local to the tree. It doesn’t actually move any of the data, and keeps the references as references.

with AsdfFile.open('source.asdf') as ff:
ff.find_references()
assert ff.tree['my_ref_b'].shape == (10,)

On the other hand, calling resolve_references places all of the referenced content directly in the tree, so when we write it out again, all of the external references are gone, with the literal content in its place.

with AsdfFile.open('source.asdf') as ff:
ff.resolve_references()
ff.write_to('resolved.asdf')

resolved.asdfb'#ASDF' 1.2.0
%YAML 1.1
%TAG ! tag:stsci.edu:asdf/
--- !core/asdf-1.1.0
asdf_library: !core/software-1.0.0 {author: Space Telescope Science Institute, homepage: 'http://github.com/spacetelescope/asdf',
name: asdf, version: 2.1.0.dev1406}
history:
extensions:
extension_class: asdf.extension.BuiltinExtension
software: {name: asdf, version: 2.1.0.dev1406}
my_ref_a: !core/ndarray-1.0.0
source: 0
datatype: int64
byteorder: little
shape: [10]
my_ref_b: !core/ndarray-1.0.0
source: 1
datatype: int64
byteorder: little
shape: [10]
...BLOCK 0:
allocated_size: 80
used_size: 80
data_size: 80
data: b'0000000000000000010000000000000002000000...'BLOCK 1:
allocated_size: 80
used_size: 80
data_size: 80
data: b'0a000000000000000b000000000000000c000000...'#ASDF BLOCK INDEX
%YAML 1.1
--- [613, 747]
...

A similar feature provided by YAML, anchors and aliases, also provides a way to support references within the same file. These are supported by asdf, however the JSON Pointer approach is generally favored because:

• It is possible to reference elements in another file
• Elements are referenced by location in the tree, not an identifier, therefore, everything can be referenced.

Anchors and aliases are handled automatically by asdf when the data structure is recursive. For example here is a dictionary that is included twice in the same tree:

d = {'foo': 'bar'}
d['baz'] = d
tree = {'d': d}

ff = AsdfFile(tree)
ff.write_to('anchors.asdf')

anchors.asdfb'#ASDF' 1.2.0
%YAML 1.1
%TAG ! tag:stsci.edu:asdf/
--- !core/asdf-1.1.0
asdf_library: !core/software-1.0.0 {author: Space Telescope Science Institute, homepage: 'http://github.com/spacetelescope/asdf',
name: asdf, version: 2.1.0.dev1406}
history:
extensions:
extension_class: asdf.extension.BuiltinExtension
software: {name: asdf, version: 2.1.0.dev1406}
d:
baz:
baz: &id001
baz: *id001
foo: bar
foo: bar
foo: bar
...

### Array References¶

ASDF files can refer to array data that is stored in other files using the ExternalArrayReference type.

External files need not be ASDF files: ASDF is completely agnostic as to the format of the external file. The ASDF external array reference does not define how the external data file will be resolved; in fact it does not even check for the existence of the external file. It simply provides a way for ASDF files to refer to arrays that exist in external files.

Creating an external array reference is simple. Only four pieces of information are required:

• The name of the external file. Since ASDF does not itself resolve the file or check for its existence, the format of the name is not important. In most cases the name will be a path relative to the ASDF file itself, or a URI for a network resource.
• The data type of the array data. This is a string representing any valid numpy.dtype.
• The shape of the data array. This is a tuple representing the dimensions of the array data.
• The array data target. This is either an integer or a string that indicates to the user something about how the data array should be accessed in the external file. For example, if there are multiple data arrays in the external file, the target might be an integer index. Or if the external file is an ASDF file, the target might be a string indicating the key to use in the external file’s tree. The value and format of the target field is completely arbitrary since ASDF will not use it itself.

As an example, we will create a reference to an external CSV file. We will assume that one of the rows of the CSV file contains the array data we care about:

import asdf

csv_data_row = 10 # The row of the CSV file containing the data we want
csv_row_size = 100 # The size of the array
extref = asdf.ExternalArrayReference('data.csv', csv_data_row, "int64", (csv_row_size,))

tree = {'csv_data': extref}
af = asdf.AsdfFile(tree)
af.write_to('external_array.asdf')

external_array.asdfb'#ASDF' 1.2.0
%YAML 1.1
%TAG ! tag:stsci.edu:asdf/
--- !core/asdf-1.1.0
asdf_library: !core/software-1.0.0 {author: Space Telescope Science Institute, homepage: 'http://github.com/spacetelescope/asdf',
name: asdf, version: 2.1.0.dev1406}
history:
extensions:
extension_class: asdf.extension.BuiltinExtension
software: {name: asdf, version: 2.1.0.dev1406}
csv_data: !core/externalarray-1.0.0
datatype: int64
fileuri: data.csv
shape: [100]
target: 10
...

When reading a file containing external references, the user is responsible for using the information in the ExternalArrayReference type to open the external file and retrieve the associated array data.

## Saving history entries¶

asdf has a convenience method for notating the history of transformations that have been performed on a file.

Given a AsdfFile object, call add_history_entry, given a description of the change and optionally a description of the software (i.e. your software, not asdf) that performed the operation.

from asdf import AsdfFile
import numpy as np

tree = {
'a': np.random.rand(32, 32)
}

ff = AsdfFile(tree)
"Initial random numbers",
{'name': 'asdf examples',
'author': 'John Q. Public',
'homepage': 'http://github.com/spacetelescope/asdf',
'version': '0.1'})
ff.write_to('example.asdf')

example.asdfb'#ASDF' 1.2.0
%YAML 1.1
%TAG ! tag:stsci.edu:asdf/
--- !core/asdf-1.1.0
asdf_library: !core/software-1.0.0 {author: Space Telescope Science Institute, homepage: 'http://github.com/spacetelescope/asdf',
name: asdf, version: 2.1.0.dev1406}
history:
entries:
- !core/history_entry-1.0.0
description: Initial random numbers
software: !core/software-1.0.0 {author: John Q. Public, homepage: 'http://github.com/spacetelescope/asdf',
name: asdf examples, version: '0.1'}
time: 2018-05-25 20:42:28
extensions:
extension_class: asdf.extension.BuiltinExtension
software: {name: asdf, version: 2.1.0.dev1406}
a: !core/ndarray-1.0.0
source: 0
datatype: float64
byteorder: little
shape: [32, 32]
...BLOCK 0:
allocated_size: 8192
used_size: 8192
data_size: 8192
data: b'26b1c1139d2fe03fa9959f66d26eec3fd896f963...'#ASDF BLOCK INDEX
%YAML 1.1
--- [783]
...

ASDF automatically saves history metadata about the extensions that were used to create the file. This information is used when opening files to determine if the proper extensions are installed (see Extension checking for more details).

## Saving ASDF in FITS¶

Note

This section is about packaging entire ASDF files inside of FITS data format files. This is probably only of interest to astronomers. Making use of this feature requires the astropy package to be installed.

Sometimes you may need to store the structured data supported by ASDF inside of a FITS file in order to be compatible with legacy tools that support only FITS.

First, create an HDUList object using astropy.io.fits. Here, we are building an HDUList from scratch, but it could also have been loaded from an existing file.

We will create a FITS file that has two image extensions, SCI and DQ respectively.

from astropy.io import fits

hdulist = fits.HDUList()
hdulist.append(fits.ImageHDU(np.arange(512, dtype=np.float), name='SCI'))
hdulist.append(fits.ImageHDU(np.arange(512, dtype=np.float), name='DQ'))

Next we make a tree structure out of the data in the FITS file. Importantly, we use the same array references in the FITS HDUList and store them in the tree. By doing this, ASDF will automatically refer to the data in the regular FITS extensions.

tree = {
'model': {
'sci': {
'data': hdulist['SCI'].data,
},
'dq': {
'data': hdulist['DQ'].data,
}
}
}

Now we take both the FITS HDUList and the ASDF tree and create an AsdfInFits object.

from asdf import fits_embed

ff = fits_embed.AsdfInFits(hdulist, tree)
ff.write_to('embedded_asdf.fits')

The special ASDF extension in the resulting FITS file contains the following data. Note that the data source of the arrays uses the fits: prefix to indicate that the data comes from a FITS extension:

content.asdfb'#ASDF' 1.2.0
%YAML 1.1
%TAG ! tag:stsci.edu:asdf/
--- !core/asdf-1.1.0
asdf_library: !core/software-1.0.0 {author: Space Telescope Science Institute, homepage: 'http://github.com/spacetelescope/asdf',
name: asdf, version: 2.1.0.dev1406}
history:
extensions:
extension_class: asdf.extension.BuiltinExtension
software: {name: asdf, version: 2.1.0.dev1406}
model:
dq:
data: !core/ndarray-1.0.0
source: fits:DQ,1
datatype: float64
byteorder: big
shape: [512]
sci:
data: !core/ndarray-1.0.0
source: fits:SCI,1
datatype: float64
byteorder: big
shape: [512]
...

To load an ASDF-in-FITS file, simply open it using asdf.open. The returned value will be an AsdfInFits object, which can be used in the same way as any other AsdfFile object.

with asdf.open('embedded_asdf.fits') as asdf_in_fits:
science = asdf_in_fits.tree['model']['sci']

Footnotes


text:
# Hyperbolic entire functions and the Eremenko-Lyubich class: Class B or not class B?

Math. Z. (2016) 1-18. Also available on the arXiv. This is a joint work with Lasse Rempe-Gillen.  Hyperbolicity plays an important role in the study of dynamical systems, and is a key concept in the iteration of rational functions of one complex variable. Hyperbolic systems have also been considered in the study of transcendental entire functions. There does not appear to be an agreed definition of the concept in this context, due to complications arising from the non-compactness of the phase space.

In this article, we consider a natural definition of hyperbolicity that requires expanding properties on the preimage of a punctured neighbourhood of the isolated singularity. We show that this definition is equivalent to another commonly used one: a transcendental entire functions is hyperbolic if and only if its postsingular set is a compact subset of the Fatou set. This leads us to propose that this notion should be used as the general definition of hyperbolicity in the context of entire functions, and, in particular, that speaking about hyperbolicity makes sense only within the Eremenko-Lyubich class $\mathcal{B}$ of transcendental entire functions with a bounded set of singular values.

We also considerably strengthen a recent characterisation of the class $\mathcal{B}$, by showing that functions outside of this class cannot be expanding with respect to a metric whose density decays at most polynomially. In particular, this implies that no transcendental entire function can be expanding with respect to the spherical metric. Finally we give a characterisation of an analogous class of functions analytic in a hyperbolic domain.


text:
## Robust facility location

• Let A be a nonempty finite subset of R^2 representing the geographical coordinates of a set of demand points (towns, ...), to be served by a facility, whose location within a given region S is sought. Assuming that the unit cost for a in A if the facility is located at x in S is proportional to dist(x,a) - the distance from x to a - and that demand of point a is given by w_a, minimizing the total trnsportation cost TC(w,x) amounts to solving the Weber problem. In practice, it may be the case, however, that the demand vector w is not known, and only an estimator {hat w} can be provided. Moreover the errors in sich estimation process may be non-negligible. We propose a new model for this situation: select a threshold valus B 0 representing the highest admissible transportation cost. Define the robustness p of a location x as the minimum increase in demand needed to become inadmissible, i.e. p(x) = min{||w^*-{hat w}|| : TC(w^*,x) B, w^* = 0} and solve then the optimization problem max_{x in S} p(x) to get the most robust location.

$Rev: 13581$