text:
# Count the reviews!

## Introduction and Credit

Imagine you're Amazon. Imagine you have a large number of products for which a lot of people have written reviews scoring 1-5 stars. Now you need to give your customers a summary for each product based on the simple product-scoring association. Because you're Amazon, time is money and as such you need to fit this into the smallest code possible because that's the code you can write the fastest.
Credit: This task appeared in 2014 in a programming mid-term exam in the functional programming course at my university and since then an example solution has been provided. I have the lecturer's and author's (same person) permission to post this question publicly.

## Input

Your input will be a list of pairs of natural numbers, where the second value of each pair is always either 1, 2, 3, 4 or 5. Take the input using your preferred, generally accepted way.

## Output

Your output will be a list of pairs of a number and a list of 5 numbers. Give the output using your preferred, generally accepted way.

## What to do?

Given the (unsorted) list of product-id and scoring pairs, count for each product how often each score appears and return the amounts in the second list of the output pair at the appropriate position.
So for product #5 with 10x 1-star, 15x 2-star, 25x 3-star, 50x 4-star and 150x 5-star would result in the entry [5 [10 15 25 50 150]].

## Potential Corner Cases

The input list may be empty in which case your output should be as well.
You don't need to consider the case where the input violates the above constraints.
You can safely assume all numbers to be representable by a signed 32-bit integer.
If you have no review(s) for a product (ie it doesn't appear in the output) it must not appear in the output either, i.e. 5x zeroes doesn't happen and can always be dropped.

## Who wins?

This is so the shortest answer in bytes wins. Built-ins are allowed and standard rules apply.

## Examples

Example input:

[(1,4),(2,5),(1,5),(1,1),(1,5),(1,4)]

Example corresponding output:

[
(1,[1, 0, 0, 2, 2]),
(2,[0, 0, 0, 0, 1])
]

Other test vectors:

[] -> []
[(1,1),(2,2),(3,3),(4,4),(5,5)] ->
[(1, [1,0,0,0,0]),(2, [0,1,0,0,0]),(3, [0,0,1,0,0]),(4, [0,0,0,1,0]),(5, [0,0,0,0,1])]

• Can / must we output a list of 5 zeros for product 2 if only products 1 are given in the input? Or can we assume that will never happen? – Luis Mendo Dec 5 '16 at 22:53
• @LuisMendo if you have no reviews for a product "it doesn't exist". Or in another way: If you have no review then there's no way for you to tell product 2 or 3 or 4 is a thing and thus you can obviously leave it out. Also see the edit. – SEJPM Dec 5 '16 at 22:57
• Would a JS object such as {"1":[1,0,0,2,2],"2":[0,0,0,0,1]} be a valid output? – Arnauld Dec 6 '16 at 0:15
• @Arnauld well, you pair the item number with a list so it's OK. – SEJPM Dec 6 '16 at 8:52

## R, 64 bytes

function(m)if(length(m)>1,by(m[,2],m[,1],tabulate,n=5),list())

This solution works if the I/O is somewhat flexible. It's an unnamed function that works by taking an R-matrix as input and tabulates the frequency of each type of review. The only problem is dealing with the empty list. We need to introduce a control whether this is actually the case and because an empty matrix matrix() is not actually empty, we return an empty list() instead.

Try it on R-fiddle

Example output for the last test case:

m[, 1]: 1
[1] 1 0 0 0 0
---------------------------------------------------------------------------------------------
m[, 1]: 2
[1] 0 1 0 0 0
---------------------------------------------------------------------------------------------
m[, 1]: 3
[1] 0 0 1 0 0
---------------------------------------------------------------------------------------------
m[, 1]: 4
[1] 0 0 0 1 0
---------------------------------------------------------------------------------------------
m[, 1]: 5
[1] 0 0 0 0 1

• Nice, didn't know you could use 'if' that way. – JAD Dec 6 '16 at 9:26
• @JarkoDubbeldam I was surprised too when I found out. It also appears to be completely undocumented. But it basically works as a non-vectorized ifelse() – Billywob Dec 6 '16 at 9:29
• I knew that you could use it on binary operators: 1+1 -> '+'(1,1), but this is more useful. – JAD Dec 6 '16 at 9:34

# MATL, 8 bytes

5B#uilZ?

Input format is a column array of products followed by a column array of scores:

[1; 2; 1; 1; 1; 1]
[4; 5; 5; 1; 5; 4]

Output format is a vertical array of products followed by a matrix of score occurrences:

1
2
1 0 0 2 2
0 0 0 0 1

Try it online!

### Explanation

5B#u   % Implicitly input array of products. Push unique values and integer labels
% of the input entries. So input [1; 3; 1; 1; 1] would give outputs [1; 3]
% and [1; 2; 1; 1; 1]. The latter is lik the input but without "holes", i.e.
% uses consecutive integers. This is required to avoid rows of zeros in the
% result for non-existing intermediate products
i      % Input array of scores
l      % Push 1
Z?     % Build sparse matrix with those row indices, column indices, and value 1.
% Implicitly display

# Python 2 - 88727169 67 bytes

def s(l,d={}):
for p,s in l:d.setdefault(p,[0]*5)[s-1]+=1
print d

Input to the function:
[(1,4),(2,5),(1,5),(1,1),(1,5),(1,4)]

Output:
{1: [1, 0, 0, 2, 2], 2: [0, 0, 0, 0, 1]}

Edits:

• Thanks to nedla2004 for saving 16 bytes!
• Changed to Python 2 to save 1 byte :D
• Saved 3 bytes by taking newline byte count as 1 byte.
• Thanks flp-tkc for saving 2 bytes. (There was an unnecessary extra space too :D)

Golfing suggestions are welcome :)

• You could put the forth line on the same line as the for loop. – nedla2004 Dec 6 '16 at 13:51
• @nedla2004 Thanks.I did not know that we do that with for loops. – Gurupad Mamadapur Dec 6 '16 at 14:23
• Not sure, but you might be able to print d instead of returning it. – nedla2004 Dec 6 '16 at 16:14
• Does this work in Python 2? I don't see why not. – nedla2004 Dec 6 '16 at 16:25
• Could you save bytes using def s(l,d={})? – FlipTack Dec 17 '16 at 8:47

# Mathematica, 68 bytes

{#[[1,1]],Last/@Tally@Join[Range@5,#2&@@@#]-1}&/@GatherBy[#,#&@@#&]&

Unnamed function taking as input a list of ordered pairs of integers such as {{1, 4}, {2, 5}, {1, 5}, {1, 1}, {1, 5}, {1, 4}}. To start, GatherBy[#,#&@@#&] groups together all the pairs with the same first element (which can be any identifier, actually, not just an integer).

Each of those groups is then operated on by the function {#[[1,1]],Last/@Tally@Join[Range@5,#2&@@@#]-1}, which produces a two-element list as output, the first element of which is just the common identifier. Join[Range@5,#2&@@@#] generates a list of integers starting with 1,2,3,4,5 and ending with all of the scores corresponding to the given identifier. Last/@Tally then computes how many time each score was found, returning the appropriate 5-tuple; then -1 makes up for the fact that we introduced each score one extra time, leaving zeros for any missing score (which otherwise would be skipped completely).

# Jelly, 18 17 bytes

Fm2QṢµ5,@þċ@€€³ż@

TryItOnline!

### How?

Fm2QṢµ5,@þċ@€€³ż@ - Main link: List of (id, score) pairs
F                 - flatten
m2               - mod 2 of list (gets the IDs)
Q              - unique items by first appearance (one of each ID)
Ṣ             - sorted (IDs in order)
µ            - monadic chain separation (call the above z)
þ        - table the result of
,@         -    pair z (with reversed @rguments) with
5           -    5 (implicit range, so [1,2,3,4,5]) ...we now have:
-        [[id,1],[id,2],[id,3],[id,4],[id,5]] for every id, in order.
ċ@€€    - count occurrences of (with reversed @rguments) for €ach for €ach in
³   - the first input (the list of (id, score) pairs)
ż@ - zip (with reversed @rguments) with z

## JavaScript (ES6), 58 bytes

Returns an object whose keys are the product IDs, associated to a list of review scores.

a=>a.map(([i,n])=>(o[i]=o[i]||[0,0,0,0,0])[n-1]++,o={})&&o

### Test cases

let f =

a=>a.map(([i,n])=>(o[i]=o[i]||[0,0,0,0,0])[n-1]++,o={})&&o

console.log(JSON.stringify(f([])));
console.log(JSON.stringify(f([[1,4],[2,5],[1,5],[1,1],[1,5],[1,4]])));
console.log(JSON.stringify(f([[1,1],[2,2],[3,3],[4,4],[5,5]])));

Clojure, 79 bytes

(fn[c](reduce(fn[r[i v]](update r i #(update(or %[0 0 0 0 0])(dec v)inc))){}c))

Must called with a sequence of vectors so that item id and ratinv value are correctly mapped to function arguments. Returns a hash-map of vectors.

(def f (fn[c]( ... )))
(f [[1 4] [2 5] [1 5] [1 1] [1 5] [1 4]])
{1 [1 0 0 2 2], 2 [0 0 0 0 1]}


text:
# factoring polynomials

Polynomials can be factored with factor. Factorization works in polynomial rings over prime finite fields, ZZ, or QQ.
 i1 : R = ZZ/10007[a,b]; i2 : f = (2*a+3)^4 + 5 4 3 2 o2 = 16a + 96a + 216a + 216a + 86 o2 : R i3 : g = (2*a+b+1)^3 3 2 2 3 2 2 o3 = 8a + 12a b + 6a*b + b + 12a + 12a*b + 3b + 6a + 3b + 1 o3 : R i4 : S = factor f 2 o4 = (a - 402)(a + 405)(a + 3a - 2301)(16) o4 : Expression of class Product i5 : T = factor g 3 o5 = (a - 5003b - 5003) (8) o5 : Expression of class Product

The results have been packaged for easy viewing. The number of factors is obtained using

 i6 : #T o6 = 2
Each factor is represented as a power (exponents equal to 1 don't appear in the display.) The parts can be extracted with #.
 i7 : T#0 3 o7 = (a - 5003b - 5003) o7 : Expression of class Power i8 : T#0#0 o8 = a - 5003b - 5003 o8 : R i9 : T#0#1 o9 = 3

format: plain


text:
# Calculating the jacobian of norm and square root terms in the Finite Element Method

In the code that my group is writing (Lethe) we use a stabilized approach to solve the Navier-Stokes equation. The GLS stabilized method we use has a stabilisation term which contains a stabilization parameter such as: $$\tau =\left[\left( \frac{2\rho |\mathbf{u}|}{h} \right)^2+\left( \frac{4 \mu}{\gamma h^2} \right) \right]^{0.5}$$

where $$\gamma$$ is a constant that depends on the type of element, $$\rho$$ is the density, $$h$$ is the element size and $$\mu$$ is the dynamic viscosity.

This stabilization parameter does not depend on the flow variable except for the first term which contains the absolute value of the velocity. I find myself unable to calculate the jacobian of that term, so right now we assemble an analytical jacobian for everything but this term.

Is there anyway to proceed to establish the jacobian of this term analytically? I understand that the $$u^2$$ term can be treated straightforwardly, but I am unsure on how to calculate the Frechet derivative of the square root.

• The term is squared. You could just think of it as having a $u^2$ term instead, which is amenable to analytical derivatives. – Paul Nov 16 '19 at 23:27
• Yeah that make sense. I am more confused about the square root part. I am under the impression that it would lead to an atrocious number of terms :( – BlaB Nov 16 '19 at 23:33
• @Paul do you mean $\tau^2$? or I totally misunderstood both the question and your comment? – Anton Menshov Nov 17 '19 at 19:46
• @AntonMenshov what I am actually wondering is how to calculate the jacobian of $\tau$ with respect to $\mathbf{u}$. So what you are suggesting is that I derive $\tau^2$ with respect to $\mathbf{u}$, which is actually very doable, and use that to establish my jacobian by dividing the result by $\tau$? – BlaB Nov 17 '19 at 22:01
• A good sanity check for any solution you do come up with is that the quantity you describe is convex with respect to $u$. – Daniel Shapero Nov 21 '19 at 18:44

You can arrive at the Jacobian analytically it just takes a few steps

So assuming we have our typical FE field values:

$$u_i = \sum_j \phi^j u_i^j$$

Where $$i$$ represents coordinate direction, $$\phi$$ our shape function, and $$j$$ index to DOF.

$$\tau = \sqrt{\alpha \lVert u \rVert + \beta}$$

Define some functions for ease of writing derivatives

$$g(\mathbf{u}) = \alpha f(\mathbf{u}) + \beta$$ $$f(\mathbf{u}) = \lVert u \rVert = \sqrt{u_x^2 + u_y^2 + u_z^2}$$

So now stabilization parameter can be written as

$$\tau = \sqrt{g(\mathbf{u})}$$

And our derivatives will be in the form

$$\frac{\partial\tau}{\partial u_i} = \frac{\partial g / \partial u_i}{2 \tau}$$

And so we find derivative of $$g$$

$$\frac{\partial g}{ \partial u_i} = \alpha \frac{\partial f}{\partial u_i}$$

and finally derivative of $$f$$

$$\frac{\partial f}{ \partial u_i} = \frac{u_i \phi^j}{f(\mathbf{u})}$$

Putting it all together:

$$\frac{\partial\tau}{\partial u_i} = \frac{\alpha u_i \phi^j}{2 \tau \lVert u \rVert}$$

You may want to double-check my math but it should give you the idea of how to do it.

• Explained this way it is perfectly clear. Thanks :)! – BlaB Nov 22 '19 at 7:44


text:
# Onset of Interlayer Phase Coherence in a Bilayer Two-Dimensional Electron System: Effect of Layer Density Imbalance

@article{Spielman2004OnsetOI,
title={Onset of Interlayer Phase Coherence in a Bilayer Two-Dimensional Electron System: Effect of Layer Density Imbalance},
author={I. B. Spielman and Melinda Kellogg and James Philip Eisenstein and Loren N. Pfeiffer and Ken W. West},
journal={Physical Review B},
year={2004},
volume={70},
pages={081303}
}
• Published 2 June 2004
• Physics
• Physical Review B
Tunneling and Coulomb drag are sensitive probes of spontaneous interlayer phase coherence in bilayer two-dimensional electron systems at total Landau level filling factor nu(T)=1. We find that the phase boundary between the interlayer phase coherent state and the weakly coupled compressible phase moves to larger layer separations as the electron density distribution in the bilayer is imbalanced. The critical layer separation increases quadratically with layer density difference.
33 Citations

## Figures from this paper

### Evolution of the bilayer ν = 1 quantum Hall state under charge imbalance

• Physics
• 2005
We use high-mobility bilayer hole systems with negligible tunneling to examine how the bilayer nu=1 quantum Hall state evolves as charge is transferred from one layer to the other at constant total

### Precursors to Exciton Condensation in Quantum Hall Bilayers.

• Physics
Physical review letters
• 2019
Tunneling spectroscopy reveals evidence for interlayer electron-hole correlations in quantum Hall bilayer two-dimensional electron systems at layer separations near, but above, the transition to the

### CRITICAL CURRENTS AND NEW EXPERIMENTS ON EXCITONIC ELECTRON BILAYER SYSTEMS

• Physics
• 2009
Pure dc interlayer tunneling experiments have found critical currents Icritical in electron double layers at a total filling factor of one. Below Icritical, the four-terminal interlayer resistance is

### Pairing transition in a double layer with interlayer Coulomb repulsion

• Physics
• 2019
We study the effect of interlayer Coulomb interaction in an electronic double layer. Assuming that each of the layers consists of a bipartite lattice, a sufficiently strong interlayer interaction

### Clausius-Clapeyron relations for first-order phase transitions in bilayer quantum Hall systems

• Physics
• 2010
A bilayer system of two-dimensional electron gases in a perpendicular magnetic field exhibits rich phenomena. At total filling factor ν_(tot)=1, as one increases the layer separation, the bilayer

### Critical tunneling currents in the regime of bilayer excitons

• Physics
• 2008
We have investigated the tunneling properties of an electron double quantum well system where the lowest Landau level of each quantum well is half filled. This system is expected to be a Bose

### Critical tunneling currents in quantum Hall superfluids: Pseudospin-transfer torque theory

• Physics
• 2010
At total filling factor $\nu=1$ quantum Hall bilayers can have an ordered ground state with spontaneous interlayer phase coherence. The ordered state is signaled experimentally by dramatically

### Theory of activated transport in bilayer quantum Hall systems.

• Physics
Physical review letters
• 2008
It is demonstrated using a bosonic Chern-Simons theory that in drag geometries current in a single layer directly leads to forces on only two of the four types of merons, inducing dissipation only in the drive layer.

### Quantum phase transitions in bilayer quantum hall systems at a total filling factor nuT=1.

• Physics
Physical review letters
• 2007
A quantum Ginsburg-Landau theory is constructed to study the quantum phase transition from the excitonic superfluid to a possible pseudospin density wave (PSDW) and it is suggested that the PSDW takes a square lattice structure.

### Exciton condensate at a total filling factor of one in Corbino two-dimensional electron bilayers

• Physics
• 2008
Magnetotransport and drag measurements on a quasi-Corbino two-dimensional electron bilayer at a total filling factor of 1 (νT=1) reveal a drag voltage that is equal in magnitude to the drive voltage


text:
# Math Help - Simplifying matrices

1. ## Simplifying matrices

The question:
Given that A, B and C are invertible n x n matrices, simplify

$A^2(B^{-1}CA)^{-1}B^{-1}C^T$

My attempt:
First I expanded $A^2$ to $AA$:
$AA(B^{-1}CA)^{-1}B^{-1}C^T$

Then I found the inverse of the matricies within the parenthesis:

$AAA^{-1}C^{-1}BB^{-1}C^T$

$AA^{-1}$and $BB^{-1}$ are just I (the identity matrix) and can be removed:

$AC^{-1}C^T$

Does this look correct? My text has no solution for this question. I'm trying to think if I can do something with the C matrices.

Thanks!

2. Looks great! I don't think you can do anything more, unless you happen to know before-hand that C is unitary.

3. Thanks for the quick reply.

4. You're welcome. Have a good one!


text:
## Stellantis to invest $229M in Kokomo operations to produce 4th-gen 8-speed transmission; ICE, MHEV and PHEV ##### 08 October 2021 Stellantis will invest a total of$229 million in three of its Kokomo, Indiana, plants to help the company reach its goal of achieving 40% low-emission vehicle sales in the US by 2030. This announcement, which will retain 662 jobs, comes three months after the company laid out its long-term electrification strategy, where it committed to investing $35 billion through 2025 in electrification and software. The investment will support retooling of the Kokomo Transmission (KTP), Kokomo Casting and Indiana Transmission plants to produce the fourth-generation eight-speed transmission. This new transmission will have the flexibility to be paired with internal combustion engines (ICE), as well as mild hybrid (MHEV) and plug-in hybrid (PHEV) propulsion systems, for vehicles across the Jeep, Ram, Chrysler and Dodge brands. It will deliver improved fuel economy, provide pure-electric range and help vehicles meet future emission regulations. This investment and this transmission represent an important stepping stone in getting us to 2030 and further expanding our battery electric offerings. This new eight-speed is the ‘multi-tool’ of transmissions. It is a versatile transmission that can be adapted for a variety of applications, allowing us the flexibility to adjust our production as consumer demand for electrified vehicles grows over the next decade. It also offers consumers the opportunity to get familiar with this technology and explore all of the benefits that can come from electrification. —Stellantis Head of Powertrain Operations Brad Clark The new, fourth-generation, eight-speed transmission will capitalize on a number of common parts within the different versions—transmission gear train/clutch, mechatronics, mechanical pump and E-pump motor. It also will have the flexibility for unique parts to adapt to the various electrification options. These parts include the eMotor, torque converter and power electronics. Adding electrified transmission assembly also diversifies the product portfolio of the Kokomo operations. In March 2020, Stellantis announced it would invest$400 million to convert Indiana Transmission Plant II to the Kokomo Engine Plant to produce the GMET4 engine, the company’s 2.0-liter Global Medium Engine inline four-cylinder turbo. Production is expected to launch later this year.

This investment also leverages the core manufacturing competencies of the Kokomo operations and the local workforce in the areas of casting, machining and assembly, all of which will be needed as the market transitions to an electrified future.

The second-generation eight-speed transmission has been a workhorse of the company’s powertrains since production began in 2012. KTP celebrated the 5 millionth eight-speed transmission built on 30 September. It is the current transmission option on the Jeep Wrangler, Gladiator and Grand Cherokee; Wagoneer/Grand Wagoneer; Ram 1500; Dodge Charger, Challenger, Durango; and Chrysler 300 models.


text:
# Chapter 6 Inference for the SLR Model

In Chapter 4 we saw how to conduct inference about the parameters $$\beta_0$$ and $$\beta_1$$. That allowed to answer quesitons about the nature of the relationship between $$x_i$$ and $$y_i$$.

A related question we might ask is: Is the model useful overall? Rather than testing a specific parameter, this questions asks about the overall utility of the entire regression model. In this chapter, we explore how to address this question using F-tests.

## 6.1 Sum of Squares Decomposition

Decomposing the variability in the data provides the pieces necessary to assess overall model performance. The total variability in outcome for a specific dataset is given by the sample variance of the $$y_i$$’s. In regression this quantity is commonly called the total sum of squares:

$SS_{tot} = \displaystyle\sum_{i=1}^n(y_i - \overline{y})^2$

This is the sum of the squared distances between each observation and the overall mean, which can be visually represented by the sum of the squared lengths of the green lines in Figure 6.1.

The total sum of squares can be decomposed into two pieces:

\begin{align} \sum_{i=1}^n (y_i - \overline{y})^2 &= \sum_{i=1}^n \left(y_i - \hat y_i + \hat y_i - \overline{y}\right)^2 \notag\\ &= \sum_{i=1}^n (y_i - \hat y_i)^2 + 2 \sum_{i=1}^n(y_i - \hat y_i)(\hat y_i - \overline{y}) + \sum_{i=1}^n (\hat y_i - \overline y)^2 \notag\\ &= \sum_{i=1}^n (y_i - \hat y_i)^2 + 2 \sum_{i=1}^n(y_i - \hat y_i)\hat y_i \notag \\ & \qquad \qquad \qquad - 2 \sum_{i=1}^n(y_i - \hat y_i)\overline{y}+ \sum_{i=1}^n (\hat y_i - \overline y)^2 \notag\\ &= \sum_{i=1}^n (y_i - \hat y_i)^2 + \sum_{i=1}^n (\hat y_i - \overline y)^2 \tag{6.1} \end{align} The first term in (6.1) is called the residual sum of squares:

$SS_{res} = \sum_{i=1}^n (y_i - \hat y_i)^2$ This should look familiar–it is the same things as the sum of squared residuals ($$\sum_{i=1}^n e_i^2 = (n-2)\hat\sigma^2$$). This is variability “left over” from fitted regression model, and can be visualized as the sum of the squared distances in the following figure:

The other term in (6.1) is the regression sum of squares: $SS_{reg} = \sum_{i=1}^n (\hat y_i - \overline y)^2$. This is the variability that is explained by the regression model:

## 6.2 Coefficient of Determination ($$R^2$$)

One way to determine how “good” a model fit is, is to compute the proportion of variability in the outcome that is accounted for by the regression model. This can be represented by the following ratio:

$\frac{SS_{reg}}{SS_{tot}} = 1 - \frac{SS_{res}}{SS_{tot}} = R^2$ The quantity $$R^2$$ is commonly called the coefficient of determination. The denominator $$SS_{tot}$$ is a fixed quantity and the numerator $$SS_{reg}$$ can vary by model (i.e., by the chosen predictor variable). Larger values of $$R^2$$ mean that greater amounts of variability in the outcome is explained by the model. When there are multiple predictor variables in a model, we will need to use an alternative form of $$R^2$$ (see Section 11.3).

Since the amount of variation explained by the model cannot be greater than the total variation in the outcome, $$R^2$$ must lie between 0 and 1. The range of $$R^2$$ values corresponding to a “good” or “bad” model fit varies by context. In most settings, values above 0.8 represent a “good” model. But in some contexts, $$R^2$$ as low as 0.3 can be “good”. What is “good” for prediction may differ from what is “good” for learning about the underlying science, and vice versa. Making a qualitative judgment about $$R^2$$ requires familiarity with what is typical for the type of data being analyzed.

R will compute the value of $$R^2$$ for you in the output from summary():

penguin_lm <- lm(body_mass_g~flipper_length_mm,
data=penguins)
summary(penguin_lm)
##
## Call:
## lm(formula = body_mass_g ~ flipper_length_mm, data = penguins)
##
## Residuals:
##      Min       1Q   Median       3Q      Max
## -1058.80  -259.27   -26.88   247.33  1288.69
##
## Coefficients:
##                    Estimate Std. Error t value Pr(>|t|)
## (Intercept)       -5780.831    305.815  -18.90   <2e-16 ***
## flipper_length_mm    49.686      1.518   32.72   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 394.3 on 340 degrees of freedom
##   (2 observations deleted due to missingness)
## Multiple R-squared:  0.759,	Adjusted R-squared:  0.7583
## F-statistic:  1071 on 1 and 340 DF,  p-value: < 2.2e-16

The value of $$R^2$$ is listed as Multiple R-squared: 0.759 in the second line from the bottom. Adjusted $$R^2$$ will be discussed in Section 11.3.

## 6.3 F-Test for Regression

In regression, we can ask whether there is an overall linear relationship between the predictor variable and the average values of the outcome variable. This is commonly called a test for significance of regression or a global F-test, and the corresponding null and alternative hypotheses are:

• $$H_0 =$$ There is no linear relationship between the $$x$$’s and the average value of $$y$$
• $$H_A =$$ There is a linear relationship between the $$x$$’s and the average value of $$y$$

In SLR there is only one $$x$$ variable, so this is exactly equivalent to testing $$H_0: \beta_1 = 0$$ vs. $$H_A: \beta_1 \ne 0$$. However, when there are multiple predictor variables in the model, the global F-test and a test for a single $$\beta$$ parameter are no longer the same (see Section 11).

The test statistic for the test for significance of regression is:

$$$\label{eq:slrfstat} f = \frac{MS_{reg}}{MS_{res}} = \frac{SS_{reg}/df_{reg}}{SS_{res}/df_{res}} = \frac{SS_{reg}/1}{SS_{res}/(n-2)}$$$

In equation (??), the statistic can be heuristically thought of as a signal-to-noise ratio. The numerator $$SS_{reg}/1$$ is the average amount of variabiliy explained by each predictor variable, which is a measure of “signal”. The denominator $$SS_{res}/(n-2)$$ is the average amount of variability left unexplained, which is a measure of “noise”.

When the value of $$f$$ is large enough, we conclude that the variability explained by the model is sufficiently larger than the variability left unexplained. Thus, we use a one-sided test and reject $$H_0$$ if $$f$$ is large enough. If the null hypothesis is true and the $$\epsilon_i$$ are normally distributed,8 then $$f$$ follows an $$F$$ distribution with 1 and $$n-2$$ degrees of freedom, i.e., $$f \sim F_{1, n-2}$$.

### 6.3.1 F-test in R “by hand”

Example 6.1 In the penguin data from Example 4.7, is there a linear relationship between flipper length and body mass?

To answer this using a global F-test, we set up the null and alternative hypotheses as:

$$H_0 =$$ There is no linear relationship between penguin flipper length and average body mass. $$H_A =$$ There is a linear relationship between penguin flipper length and average body mass.
We now need to compute $$SS_{res}$$:

SSres <- sum(residuals(penguin_lm)^2)
SSres
## [1] 52854796

and $$SS_{reg}$$:

SSreg <- sum((fitted(penguin_lm) - mean(penguin_lm$model$body_mass_g))^2)
SSreg
## [1] 166452902

Note that when computing $$SS_{reg}$$ in this way, we are using penguin_lm$model$body_mass_g rather than penguins\$body_mass_g. This is because the version of body mass stored within the penguin_lm object will contain exactly the observations used to fit the model. This can be important if there is missing data that lm() drops automatically.

With both $$SS_{res}$$ and $$SS_{reg}$$ calculated, we can compute the F-statistic:

f <- (SSreg/1)/(SSres/(nobs(penguin_lm)- 2))
f
## [1] 1070.745

and $$p$$-value (being careful to set lower=FALSE to do a one-sided test):

pf(f, df1=1, df2=nobs(penguin_lm)- 2, lower=FALSE)
## [1] 4.370681e-107

### 6.3.2 F-test in R

Of course, it is faster and simpler to let R compute the F-statistic and $$p$$-value for you. This information is available at the bottom of the output from summary():

summary(penguin_lm)
##
## Call:
## lm(formula = body_mass_g ~ flipper_length_mm, data = penguins)
##
## Residuals:
##      Min       1Q   Median       3Q      Max
## -1058.80  -259.27   -26.88   247.33  1288.69
##
## Coefficients:
##                    Estimate Std. Error t value Pr(>|t|)
## (Intercept)       -5780.831    305.815  -18.90   <2e-16 ***
## flipper_length_mm    49.686      1.518   32.72   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 394.3 on 340 degrees of freedom
##   (2 observations deleted due to missingness)
## Multiple R-squared:  0.759,	Adjusted R-squared:  0.7583
## F-statistic:  1071 on 1 and 340 DF,  p-value: < 2.2e-16

Regardless of which approach to calculating the test statistic that we use, we can summarize our conclusions by saying:

We reject the null hypothesis that there is no linear relationship between penguin flipper length and average body mass ($$p < 0.0001$$).

## 6.4 ANOVA Table

$$F$$-tests are widely used in the Analysis of Variance (ANOVA) approach to analyzing data. Information can be summarized in an “ANOVA Table”:

Source of Variation Sum of Squares Degrees of Freedom MS F
Regression $$SS_{reg}$$ $$1$$ $$MS_{reg}$$ $$MS_{reg}/MS_{res}$$
Residual $$SS_{res}$$ $$n-2$$ $$MS_{res}$$
Total $$SS_{tot}$$ $$n-1$$

This format arises most frequently for designed experiments, where there is a rich history of decomposing the variability of data into distinct sources.

1. In many cases, it is sufficient for sample sizes to be large so that the central limit theorem applies. But care should be taken in small samples.↩


text:
Press "Enter" to skip to content

Problem:

There is a garden with N slots. In each slot, there is a flower. The N flowers will bloom one by one in N days. In each day, there will be exactly one flower blooming and it will be in the status of blooming since then.

Given an array flowers consists of number from 1 to N. Each number in the array represents the place where the flower will open in that day.

For example, flowers[i] = x means that the unique flower that blooms at day i will be at position x, where i and x will be in the range from 1 to N.

Also given an integer k, you need to output in which day there exists two flowers in the status of blooming, and also the number of flowers between them is k and these flowers are not blooming.

If there isn’t such day, output -1.

Example 1:

Input:
flowers: [1,3,2]
k: 1
Output: 2
Explanation: In the second day, the first and the third flower have become blooming.

Example 2:

Input:
flowers: [1,2,3]
k: 1
Output: -1

Note:

1. The given array will be in the range [1, 20000].

Idea:

BST/Buckets

# Solution 1: TLE with latest test cases

## Python

If you like my articles / videos, donations are welcome.

Buy anything from Amazon to support our website

Paypal
Venmo
huahualeetcode


text:
Accelerating the pace of engineering and science

# lassoglm

Lasso or elastic net regularization for generalized linear model regression

## Syntax

B = lassoglm(X,Y)
[B,FitInfo] = lassoglm(X,Y)
[B,FitInfo] = lassoglm(X,Y,distr)
[B,FitInfo] = lassoglm(X,Y,distr,Name,Value)

## Description

B = lassoglm(X,Y) returns penalized maximum-likelihood fitted coefficients for a generalized linear model of the response Y to the data matrix X. Y are assumed to have a Gaussian probability distribution.

[B,FitInfo] = lassoglm(X,Y) returns a structure containing information about the fits.

[B,FitInfo] = lassoglm(X,Y,distr) fits the model using the probability distribution type for Y as specified in distr.

[B,FitInfo] = lassoglm(X,Y,distr,Name,Value) fits regularized generalized linear regressions with additional options specified by one or more Name,Value pair arguments.

## Input Arguments

 X Numeric matrix with n rows and p columns. Each row represents one observation, and each column represents one predictor (variable). Y When distr is not 'binomial', Y is a numeric vector or categorical array of length n, where n is the number of rows of X. Y(i) is the response to row i of X. When distr is 'binomial', Y is either a: Numeric vector of length n, where each entry represents success (1) or failure (0)Logical vector of length n, where each entry represents success or failureCategorical array of length n, where each entry represents success or failureTwo column numeric matrix, where the first column contains the number of successes for each observation, and the second column contains the total number of trials distr Distributional family for the nonsystematic variation in the responses, a string. Choices: 'normal''binomial''poisson''gamma''inverse gaussian' By default, lassoglm uses the canonical link function corresponding to distr. Specify another link function using the 'link' name-value pair.

### Name-Value Pair Arguments

Specify optional comma-separated pairs of Name,Value arguments. Name is the argument name and Value is the corresponding value. Name must appear inside single quotes (' '). You can specify several name and value pair arguments in any order as Name1,Value1,...,NameN,ValueN.

'Alpha'

Scalar value from 0 to 1 (excluding 0) representing the weight of lasso (L1) versus ridge (L2) optimization. Alpha = 1 represents lasso regression, and other values represent elastic net optimization. Alpha close to 0 approaches ridge regression. See Definitions.

Default: 1

'CV'

Method lassoglm uses to estimate deviance:

• K, a positive integer — lassoglm uses K-fold cross validation.

• cvp, a cvpartition object — lassoglm uses the cross-validation method expressed in cvp. You cannot use a 'leaveout' partition with lassoglm.

• 'resubstitution'lassoglm uses X and Y to fit the model and to estimate the deviance, without cross validation.

Default: 'resubstitution'

'DFmax'

Maximum number of nonzero coefficients in the model. lassoglm returns results for Lambda values that satisfy this criterion.

Default: Inf

'Lambda'

Vector of nonnegative Lambda values. See Lasso.

• If you do not supply Lambda, lassoglm estimates the largest value of Lambda that gives a nonnull model. In this case, LambdaRatio gives the ratio of the smallest to the largest value of the sequence, and NumLambda gives the length of the vector.

• If you supply Lambda, lassoglm ignores LambdaRatio and NumLambda.

Default: Geometric sequence of NumLambda values, the largest just sufficient to produce B = 0

'LambdaRatio'

Positive scalar, the ratio of the smallest to the largest Lambda value when you do not explicitly set Lambda.

If you set LambdaRatio = 0, lassoglm generates a default sequence of Lambda values, and replaces the smallest one with 0.

Default: 1e-4

Specify the mapping between the mean µ of the response and the linear predictor Xb.

ValueDescription
'comploglog'

log( –log((1–µ))) = Xb

'identity', default for the distribution 'normal'

µ = Xb

'log', default for the distribution 'poisson'

log(µ) = Xb

'logit', default for the distribution 'binomial'

log(µ/(1 – µ)) = Xb

'loglog'

log( –log(µ)) = Xb

'probit'

Φ–1(µ) = Xb, where Φ is the normal (Gaussian) CDF function

'reciprocal', default for the distribution 'gamma'

µ–1 = Xb

p (a number), default for the distribution 'inverse gaussian' (with p = –2)

µp = Xb

Cell array of the form {FL FD FI}, containing three function handles, created using @, that define the link (FL), the derivative of the link (FD), and the inverse link (FI). Equivalently, can be a structure of function handles with field Link containing FL, field Derivative containing FD, and field Inverse containing FI.

User-specified link function (see Custom Link Function)

'MCReps'

Positive integer, the number of Monte Carlo repetitions for cross validation.

• If CV is 'resubstitution' or a cvpartition of type 'resubstitution', MCReps must be 1.

• If CV is a cvpartition of type 'holdout', MCReps must be greater than 1.

Default: 1

'NumLambda'

Positive integer, the number of Lambda values lassoglm uses when you do not set Lambda. lassoglm can return fewer than NumLambda fits if the deviance of the fits drops below a threshold fraction of the null deviance (deviance of the fit without any predictors X).

Default: 100

'Offset'

Numeric vector with the same number of rows as X. lassoglm uses Offset as an additional predictor variable, but keeps its coefficient value fixed at 1.0.

'Options'

Structure that specifies whether to cross validate in parallel, and specifies the random stream or streams. Create the Options structure with statset. Option fields:

• UseParallel — Set to true to compute in parallel. Default is false.

• UseSubstreams — Set to true to compute in parallel in a reproducible fashion. To compute reproducibly, set Streams to a type allowing substreams: 'mlfg6331_64' or 'mrg32k3a'. Default is false.

• StreamsRandStream object or cell array consisting of one such object. If you do not specify Streams, lassoglm uses the default stream.

'PredictorNames'

Cell array of strings representing names of the predictor variables, in the order in which they appear in X.

Default: {}

'RelTol'

Convergence threshold for the coordinate descent algorithm (see Friedman, Tibshirani, and Hastie [3]). The algorithm terminates when successive estimates of the coefficient vector differ in the L2 norm by a relative amount less than RelTol.

Default: 1e-4

'Standardize'

Boolean value specifying whether lassoglm scales X before fitting the models.

Default: true

'Weights'

Observation weights, a nonnegative vector of length n, where n is the number of rows of X. At least two values must be positive.

Default: 1/n * ones(n,1)

## Output Arguments

B

Fitted coefficients, a p-by-L matrix, where p is the number of predictors (columns) in X, and L is the number of Lambda values.

FitInfo

Structure containing information about the model fits.

Field in FitInfoDescription
AlphaValue of Alpha parameter, a scalar.
DevianceDeviance of the fitted model for each value of Lambda, a 1-by-L vector.
If cross validation was performed, the values for Deviance represent the estimated expected deviance of the model applied to new data, as calculated by cross validation. Otherwise, Deviance is the deviance of the fitted model applied to the data used to perform the fit.
DFNumber of nonzero coefficients in B for each Lambda value, a 1-by-L vector.
InterceptIntercept term β0 for each linear model, a 1-by-L vector.
LambdaLambda parameters in ascending order, a 1-by-L vector.

If you set the CV name-value pair to cross validate, the FitInfo structure contains additional fields.

Field in FitInfoDescription
IndexMinDevianceIndex of Lambda with value LambdaMinDeviance, a scalar.
Index1SEIndex of Lambda with value Lambda1SE, a scalar.
LambdaMinDevianceLambda value with minimum expected deviance, as calculated by cross validation, a scalar.
Lambda1SELargest Lambda such that Deviance is within one standard error of the minimum, a scalar.
SEStandard error of Deviance for each Lambda, as calculated during cross validation, a 1-by-L vector.

## Examples

expand all

### Lasso Regularization of a Generalized Linear Model

Construct data from a Poisson model, and identify the important predictors using lassoglm.

Create data with 20 predictors, and Poisson responses using just three of the predictors, plus a constant.

rng('default') % for reproducibility
X = randn(100,20);
mu = exp(X(:,[5 10 15])*[.4;.2;.3] + 1);
y = poissrnd(mu);

Construct a cross-validated lasso regularization of a Poisson regression model of the data.

[B FitInfo] = lassoglm(X,y,'poisson','CV',10);

Examine the cross-validation plot to see the effect of the Lambda regularization parameter.

lassoPlot(B,FitInfo,'plottype','CV');

The green circle and dashed line locate the Lambda with minimal cross-validation error. The blue circle and dashed line locate the point with minimal cross-validation error plus one standard deviation.

Find the nonzero model coefficients corresponding to the two identified points.

minpts = find(B(:,FitInfo.IndexMinDeviance))
minpts =

3
5
6
10
11
15
16
min1pts = find(B(:,FitInfo.Index1SE))
min1pts =

5
10
15

The coefficients from the minimal plus one standard error point are exactly those coefficients used to create the data.

expand all

A link function f(μ) maps a distribution with mean μ to a linear model with data X and coefficient vector b using the formula

f(μ) = Xb.

Find the formulas for the link functions in the Link name-value pair description. Here, "typical" means a link function that is typically used for the listed distribution.

Distributional FamilyLink Function (typical, {default})
'normal'{'identity'}
'binomial''comploglog', 'loglog', 'probit', {'logit'}
'poisson'{'log'}
'gamma'{'reciprocal'}
'inverse gaussian'{-2}

### Lasso

For a nonnegative value of λ, lasso solves the problem

$\underset{{\beta }_{0},\beta }{\mathrm{min}}\left(\frac{1}{N}\text{Deviance}\left({\beta }_{0},\beta \right)+\lambda \sum _{j=1}^{p}|{\beta }_{j}|\right),$

where

• Deviance is the deviance of the model fit to the responses using intercept β0 and predictor coefficients β. The formula for Deviance depends on the distr parameter you supply to lassoglm. Minimizing the λ-penalized deviance is equivalent to maximizing the λ-penalized log likelihood.

• N is the number of observations.

• λ is a nonnegative regularization parameter corresponding to one value of Lambda.

• Parameters β0 and β are scalar and p-vector respectively.

As λ increases, the number of nonzero components of β decreases.

The lasso problem involves the L1 norm of β, as contrasted with the elastic net algorithm.

### Elastic Net

For an α strictly between 0 and 1, and a nonnegative λ, elastic net solves the problem

$\underset{{\beta }_{0},\beta }{\mathrm{min}}\left(\frac{1}{N}\text{Deviance}\left({\beta }_{0},\beta \right)+\lambda {P}_{\alpha }\left(\beta \right)\right),$

where

${P}_{\alpha }\left(\beta \right)=\frac{\left(1-\alpha \right)}{2}{‖\beta ‖}_{2}^{2}+\alpha {‖\beta ‖}_{1}=\sum _{j=1}^{p}\left(\frac{\left(1-\alpha \right)}{2}{\beta }_{j}^{2}+\alpha |{\beta }_{j}|\right).$

Elastic net is the same as lasso when α = 1. For other values of α, the penalty term Pα(β) interpolates between the L1 norm of β and the squared L2 norm of β. As α shrinks toward 0, elastic net approaches ridge regression.

## References

[1] Tibshirani, R. Regression Shrinkage and Selection via the Lasso. Journal of the Royal Statistical Society, Series B, Vol. 58, No. 1, pp. 267–288, 1996.

[2] Zou, H. and T. Hastie. Regularization and Variable Selection via the Elastic Net. Journal of the Royal Statistical Society, Series B, Vol. 67, No. 2, pp. 301–320, 2005.

[3] Friedman, J., R. Tibshirani, and T. Hastie. Regularization Paths for Generalized Linear Models via Coordinate Descent. Journal of Statistical Software, Vol. 33, No. 1, 2010. http://www.jstatsoft.org/v33/i01

[4] Hastie, T., R. Tibshirani, and J. Friedman. The Elements of Statistical Learning, 2nd edition. Springer, New York, 2008.

[5] Dobson, A. J. An Introduction to Generalized Linear Models, 2nd edition. Chapman & Hall/CRC Press, New York, 2002.

[6] McCullagh, P., and J. A. Nelder. Generalized Linear Models, 2nd edition. Chapman & Hall/CRC Press, New York, 1989.

[7] Collett, D. Modelling Binary Data, 2nd edition. Chapman & Hall/CRC Press, New York, 2003.


text:
# logarithmic question

• November 22nd 2010, 11:20 PM
patci
logarithmic question
I need help with some examples :

a) 100 ^1+log5
b) 8^log2(5 - 1/3)

what kinds of formulas should I use?

In the first example I'm thinking about changing 1 to log5 (5), but I don't know if it has any sense
• November 22nd 2010, 11:31 PM
emakarov
Quote:

a) 100 ^1+log5
Do you mean $100^1+\log 5$ or $100^{1+\log 5}$, and log to what base?

Quote:

b) 8^log2(5 - 1/3)
Rewrite 8 as 2^3 and use $(a^b)^c=a^{bc}$ and $2^{\log_2x}=x$.
• November 23rd 2010, 06:59 AM
patci
Quote:

Originally Posted by emakarov
or $100^{1+\log 5}$

I meant this one. Exactly, index of the power amount to 1 + log5
• November 23rd 2010, 07:05 AM
Plato
Assuming you mean $\log_{10}(5)$ then $\displaystyle 10^{\log(5)}=5$.


text:
# Could someone explain how to do this? For the next three problems, consider 1.0 L of a solution...

## Question:

Could someone explain how to do this? For the next three problems, consider 1.0 L of a solution which is{eq}0.35 M HC_2H_3O_2 and 0.2 M NaC_2H_3CO_2 (K_a for HC_2H_3O_2 = 1.8 times 10^-5) {/eq}. Assume 2 significant figures in all of the given concentrations so that you should calculate all of the following pH values to two decimal places. Calculate the pH of this solution. Calculate the pH after 0.10 mol of HCl has been added to the original solution. Assume no volume change on addition of HCl. Calculate the pH after 0.20 mol of NaOH has been added to the original buffer solution. Assume no volume change on addition of NaOH.

## Henderson-Hasselbalch Equation

One can use the Henderson-Hasselbalch equation to predict the pH of a buffer solution. Similarly, the equation can be used to calculate the recipe needed to make a buffer of the desired solution. The Henderson-Hasselbalch equation is

$$pH = pK_a + log \frac {[A^-]}{[HA]}$$

• a typo in the sodium salt (extra carbon) is ignored in the following answer.

The pH of the original solution is a buffer because it contains both a weak acid and its conjugate base. Its pH can be calculated from the Henderson-Hasselbalch equation. Sodium ion is ignored because it is pH neutral.

$$pH = pK_a + log \frac { [A^-] }{ [HA] }\\ pH = pK_a + log \frac { [C_2H_3O_2^-] }{ [HC_2H_3CO_2] } = -\log (1.8 \times 10^{-5} ) + log \frac {0.2 \ M}{0.35 \ M} =\boxed {4.50}$$

After addition of 0.10 mole of HCl, some of the conjugate base will react with it and form more of the weak acid.

$$C_2H_3O_2^- + H^+ \rightarrow HC_2H_3O_2$$

The new equilibrium concentrations are

$$[C_2H_3O_2^-] = (0.2 - 0.10) \ M \\ [HC_2H_3O_2] = (0.35 + 0.10) \ M$$

The new pH is calculated

$$pH = pK_a + log \frac { [C_2H_3O_2^-] }{ [HC_2H_3CO_2] } = -\log (1.8 \times 10^{-5} ) + log \frac {0.1 \ M}{0.45 \ M} =\boxed {4.09}$$

As expected, adding HCl lowers the pH.

After addition of 0.20 mole of NaOH, some of the acid will react with it and form more of the conjugate base

$$HC_2H_3O_2 + NaOH \rightarrow C_2H_3O_2^-$$

The new equilibrium concentrations are

$$[C_2H_3O_2^-] = (0.2 + 0.20) \ M \\ [HC_2H_3O_2] = (0.35 - 0.20) \ M$$

The new pH is calculated

$$pH = pK_a + log \frac { [C_2H_3O_2^-] }{ [HC_2H_3CO_2] } = -\log (1.8 \times 10^{-5} ) + log \frac {0.40 \ M}{0.15 \ M} =\boxed {5.17}$$

As expected, adding NaOH increases the pH.


text:
# How to change bibliography to webography in lyx 2.0.2?

I'm writing a report with lyx 2.0.2 and I want to insert in the end a webography which is the same as bibliography but its called webography instead. So what I want basically is either to change the bibliography to webography, or if there's another way of inserting a webography in my report.

## update:

Well it's not exactly like a bibliography, the references in the webography are just urls.

## update 2:

Actually, I don't just want to change the heading of the bibliography to 'Webography' I want to change the behavior too, 'Webography' will contain references for Urls, just Urls not like bibliography author, and other stuff.
And I'm using the report class.

## update 3:

I have this in the head of the lyx document:

\makeatletter
\let\ps@plain=\ps@empty
\makeatother
\widowpenalty=10000
\clubpenalty=10000
\raggedbottom
\pagenumbering{arabic}

And now after trying @Torbjørn's answer, I keep having this error:

! Package natbib Error: Bibliography not compatible with author-year citations.

And putting the cursor at this line:

\makeatother

And that happen even when I try to see the DVI version and when I remove the bibliography everything works fine.
By the way this is my first time writing something with lyx or latex.

You can create a 'webography' with the thebibliography environment and the \url command. To redefine the name, you can use the techniques described in How to change the name of document elements like "Figure", "Contents", "Bibliography", "Appendix", etc.?, but it seems a small change must be made due to how LyX loads packages (specifically, babel).

Where you want to have this webography, choose the Bibliography-element from the drop-down menu in the top left. This will add a thebibliography environment to your document, and it will look like this:

Place the cursor right after the key-1 [1] inset, and choose Insert --> URL. Type a URL in the box that appears. To add a new URL to the webography, move the cursor outside the URL-inset, and hit enter. Finally, you can change the label (key-N) by clicking it.

LyX loads babel by default as far as I know, so in How to change the name of document elements like "Figure", "Contents", "Bibliography", "Appendix", etc.? it is the 'With babel' section that is relevant, but with a small change. Add the following to Document --> Settings --> LaTeX Preamble:

\AtBeginDocument{%
\renewcommand{\bibname}{Webography}
}}

If English is not your chosen language, you have to change that.

The reason for the \AtBeginDocument is that when I tried this, LyX added babel to the very end of the preamble, the stuff you add to the preamble yourself is placed before the loading of babel. Therefore, \addto\captionsenglish is not available at the time it is used, but with AtBeginDocument the use is postponed until after the entire preamble.

Note that the heading will not change within LyX, but it will change in the PDF.

## Don't use natbib

LyX seems to behave a little silly (or maybe it's just me), so in Document --> Settings --> Bibliography, make sure that Default (Numerical) is selected, and not Natbib.

• Hi, thanks for answering but I got some errors, check update 3. – Yahya KACEM May 4 '13 at 21:45
• @YahyaKACEM See my update. – Torbjørn T. May 4 '13 at 21:54
• That did it but now I get this (author?) [1]: in the place where I want to put the citation, how to remove this. – Yahya KACEM May 4 '13 at 22:36
• @YahyaKACEM Oh, damn, sorry about that. Try with Default (numerical) instead of Natbib. – Torbjørn T. May 4 '13 at 22:40


text:
Dynamics of Generalized Nevanlinna Functions

# Dynamics of Generalized Nevanlinna Functions

Tao Chen and Linda Keen
###### Abstract.

In the early 1980’s, computers made it possible to observe that in complex dynamics, one often sees dynamical behavior reflected in parameter space and vice versa. This duality was first exploited by Douady, Hubbard and their students in early work on rational maps. See [DH, BH] for example.

Here, we continue to study these ideas in the realm of transcendental functions.

In [KK1], it was shown that for the tangent family, , the way the hyperbolic components meet at a point where the asymptotic value eventually lands on infinity reflects the dynamic behavior of the functions at infinity. In the first part of this paper we show that this duality extends to a much more general class of transcendental meromorphic functions that we call generalized Nevanlinna functions with the additional property that infinity is not an asymptotic value. In particular, we show that in “dynamically natural” one dimensional slices of parameter space, there are “hyperbolic-like” components with a unique distinguished boundary point whose dynamics reflect the behavior inside an asymptotic tract at infinity. Our main result is that every parameter point in such a slice for which the asymptotic value eventually lands on a pole is such a distinguished boundary point.

In the second part of the paper, we apply this result to the families , , to prove that all hyperbolic components of period greater than are bounded.

###### 2010 Mathematics Subject Classification:
37F10, 30D05, 37F30, 30D30

### Introduction

In the early 1980’s, computers made it possible to observe that in complex dynamics, one often sees dynamical behavior reflected in parameter space and vice versa. This duality was first exploited by Douady, Hubbard and their students in early work on rational maps. See [DH, BH] for example. Here, we continue to study these ideas in the realm of transcendental functions.

In [KK1], it was shown that for the tangent family, , the way the hyperbolic components meet at a point where the asymptotic value eventually lands on infinity, reflects the dynamic behavior of the functions at infinity. This is very different from what one sees for the exponential family (see [DFJ, RG, Sch]) where all the hyperbolic components are unbounded. A crucial difference between these families is that for the exponential family infinity is an asymptotic value whereas for the tangent family it is not.

In the first part of this paper we define a much more general class of transcendental meromorphic functions that we call generalized Nevanlinna functions. We show that families of this class with the additional property that infinity is NOT an asymptotic value exhibit this duality; that is, in a one dimensional slice of the parameter space, the way the hyperbolic components meet at a point where the asymptotic value eventually lands on infinity reflects the dynamical behavior of the functions at infinity.

To describe these functions and state our theorems, we need some background. The dynamical plane of a meromorphic map is divided into two sets: the Fatou or stable set on which the iterates are well defined and form a normal family, and the Julia set, its complement. The Julia set can be characterized as the closure of the set of repelling periodic points, or equivalently, the closure of the set of pre-poles, points that map to infinity after finite iteration. A good introduction to meromorphic dynamics can be found in [Ber].

The points over which a meromorphic function is not a regular covering map are called singular values. There are two types of singular values: critical values (images of zeroes of , the critical points) and asymptotic values (points where as ). If an asymptotic value is isolated, the local inverse there is the logarithm. We denote the class of meromorphic functions with finitely many singular values by . This class is particularly tractable: families in have parameter spaces with natural embeddings into , where is a simple function of the number of singular values; all asymptotic values are isolated; their Fatou domains have a simple classification because there are no wandering domains (see Section 1 for definitions).

The focus in this paper is on the subclass of functions in for which infinity is not an asymptotic value. These necessarily have infinitely many poles and this behavior at infinity has consequences for the dynamical and parameter spaces (see e.g. [BK]). In particular, it is very different from that for entire functions or transcendental functions with finitely many poles.

A general principle in dynamics is that each stable dynamical phenomenon is “controlled” by a singular point. For example, each attracting or parabolic periodic cycle always attracts a singular value. Using this principle, one can define one dimensional slices of the parameter space of a family in that are “compatible with the dynamics” by keeping all but one of the dynamical phenomena fixed and letting the remaining one, controlled by the “free singular value ”, vary. Looking for regions in the slice where the free singular value is attracted to an attracting cycle gives a picture of how the “hyperbolic-like” components of the slice fit together around the bifurcation locus.

Dynamically natural slices were defined in [FK] for families of functions in . It was shown there that for those slices, as for slices of parameter spaces of rational maps, the bifurcation locus contains parameters distinguished by functional relations. There are Misiurewicz points where lands on a repelling periodic cycle and centers where lands on a super-attracting cycle. Another type of distinguished parameter, not seen for rational maps, is a virtual cycle parameter, where lands on a pole.

Dynamically natural slices contain two different kinds of “hyperbolic-like” domains in which the functions are all quasiconformally conjugate on their Julia sets: capture components, where the free asymptotic value is attracted to one of the fixed phenomena — an attracting or super-attracting cycle, and shell components, where the free asymptotic value is the only singular value attracted to an attracting cycle. In a shell component, the period of the attracting cycle is constant and the multiplier map is a well-defined universal covering map onto the punctured disk. Properties of shell components for general families in were studied in detail in [FK]. In particular, it was proved that the boundary of every shell component contains a special point, the virtual center where the limit of the multiplier map is zero. One of the main results proved there is

###### Theorem A.

For families in , a virtual center on the boundary of a shell component in a dynamically natural slice is a virtual cycle parameter and any virtual cycle parameter on the boundary is a virtual center.

In this paper, we concern ourselves with a special subclass in . We start with the Nevanlinna functions, the family of functions with asymptotic values and no critical values. We next form the family by pre- and post-composing by polynomials of degrees and respectively. We then look at , the subset of functions in all of whose asymptotic values are finite. We first define the embedding of into and then study shell components of dynamically natural slices for this embedding. We include the proof, given originally in [FK], that

###### Proposition A.

If is topologically conjugate to a meromorphic function in , and if is meromorphic, then it is also in ; that is there is a function and polynomials of degrees respectively such that .

A corollary is

###### Corollary A.

There is a natural embedding of into and hence an embedding of into .

Theorem A does not preclude the existence of virtual cycle parameters that are not on the boundary of a shell component. For example, such a parameter might be buried in the bifurcation locus. Our first new theorem says this cannot happen.

###### Theorem B.

In a dynamically natural slice in , every virtual cycle parameter lies on the boundary of a shell component.

Corollary B, which follows directly from Theorems A and B, says that in the notions of virtual center and virtual cycle parameter are equivalent.

###### Corollary B.

In a dynamically natural slice of , every virtual cycle parameter is a virtual center and vice versa.

In [FK] it was proved that for families in , shell components of period in dynamically natural slices are always unbounded and it was conjectured that, in contrast to families of entire functions, those of period greater than are always bounded. The conjecture was proved true in the tangent family in [KK1]. In this paper we prove it for the generalization of the tangent family, . This is a subfamily of .

###### Theorem C.

Every shell component of period greater than in the plane of the family is bounded.

The paper is organized as follows. Part I is a discussion of the general family and dynamically natural slices of its parameter space. In Section 1 we give a brief overview of the basic theory, set our notation and discuss the theorem of Nevanlinna, Theorem 1.1, which we use to define the class of Nevanlinna functions. Next, in Section 2, we define the classes and , and prove Proposition A and Corollary A. In Section 2.1 we recall the properties of Fatou components that we need and define dynamically natural slices of parameter spaces and the shell components in these slices. In Section 3, we define virtual cycle parameters and virtual centers, state Theorem A and prove Theorem B. Part II is a discussion of the special symmetric subfamily . In Section 4 we classify the shell components by period and discuss the special properties of the components of periods and . Finally, in Section 5 we prove Theorem C.

## Part I The family Mp,q,r

### 1. Basics and Tools

#### 1.1. Meromorphic functions.

In this paper, unless we specifically say otherwise, we always assume that an entire or meromorphic map is transcendental and so has infinite degree. If we mean a map of finite degree we call it polynomial or rational. We also always assume infinity is an essential singularity. We need the following definitions:

###### Definition 1.

A point is called a singular value111In the literature these are sometimes called singularites of . of if, for some small neighborhood of , some branch of is not well defined. If is a zero of , it is a critical point and its image is a critical value. The branch with is not well defined so a critical value is singular. A point is also singular if there is a path such that and . The limit is singular and called an asymptotic value of . Singular values may be critical, asymptotic or accumulations of such points. We denote the set of singular points of by .

If an asymptotic value is isolated, and is an asymptotic path for , we can find a nested sequence of neighborhoods of , with , and a particular branch of such that is a nested sequence of neighborhoods containing and . Then can be chosen small enough so that the map is a universal covering map. In this case is called an asymptotic tract for the asymptotic value and is called a logarithmic singular value. The number of distinct asymptotic tracts of a given asymptotic value is called its multiplicity222Note this notion is not the same as multiplicity of a critical point..

###### Definition 2.

A ray approaching infinity is called a Julia ray or Julia direction for the meromorphic function if assumes all (but at most two) values infinitely often in any sector containing .

An example to keep in mind is with asymptotic values at and . The asymptotic tracts are the left and right half planes and the Julia directions are parallel to the positive and negative imaginary axes. Another example is with asymptotic values , asymptotic tracts the upper and lower half planes and Julia directions parallel to the positive and negative real axes.

In this paper, whenever we talk about the number of critical points and/or asymptotic values, we tacitly assume that we count with multiplicity.

An entire function always has an asymptotic value at infinity.

We will only be interested in meromorphic functions with . These are called functions of finite type. Note that all the asymptotic values of a finite type function are isolated and so are logarithmic.

##### 1.1.1. Nevanlinna Functions

Nevanlinna, in [Nev, Nev1], Chap X1, characterized families of meromorphic functions with finitely many asymptotic values, finitely many critical points and a single essential singularity at infinity. (See [DK, KK1, EreGab] for further discussion.)

Recall that the Schwarzian derivative of a function is defined by

 (1) S(g)=(g′′/g′)′−12(g′′/g′)2.

Because Schwarzian derivatives satisfy the cocycle relation

 S(f∘g)(z)=S(f)(g′(z))2+S(g)(z)

and the Schwarzian derivative of a Möbius transformation is zero, solutions to equation (1) are determined only up to post-composition by a Möbius transformation.

Nevanlinna’s theorem says

###### Theorem 1.1 (Nevanlinna).

Every meromorphic function with asymptotic values and critical points has the property that its Schwarzian derivative is a rational function of degree . If , the Schwarzian derivative is a polynomial . In the opposite direction, for every polynomial function of degree , the solution to the Schwarzian differential equation is a meromorphic function with exactly asymptotic values and no critical points.

###### Remark 1.1.

The proof of the first part of this theorem involves the construction of the function as a limit of holomorphic functions whose Schwarzians are rational of bounded degree. The proof of the second part of the theorem involves understanding the asymptotic properties of solutions to the equation . In particular, there are exactly “truncated solutions” that, for any , have asymptotic developments of the form

 loggk(z)∼(−1)k+1zp/2

defined in the sector . Each is entire and tends to zero as tends to infinity along each ray of the sector and tends to infinity in the adjacent sectors. The rays separating the sectors are the Julia rays for . See [Hille, Nev] for details.

###### Definition 3.

We denote the family of meromorphic functions with asymptotic values and no critical values by and call the functions Nevanlinna functions.

One immediate corollary is that Nevanlinna functions cannot have exactly one asymptotic value. Moreover, since polynomials of degree depend on their coefficients, and the solutions are determined up to post-compostion by Möbius transformations which depend on three coefficents, another immediate corollary is

###### Corollary 1.2.

Nevanlinna functions with asymptotic values have a natural embedding into .

We are interested in the dynamical systems generated by these Nevanlinna functions and some generalizations of them. Infinity plays a special role so we separate the cases where infinity is an asymptotic value from those where it is not. For these functions, the dynamics are unchanged by conjugation by an affine transformation. Thus suitably normalized solutions have a natural embedding into

In his proof of the above theorem, (see also [DK], Section 1), Nevanlinna shows that for , a neighborhood of infinity is divided into exactly disjoint sectors, , each with angle . Each of these is an asymptotic tract for one asymptotic value. The sectors are separated by the rays that bound them, and these are Julia rays. Although two tracts may map to the same asymptotic value, tracts in adjacent sectors must map to different asymptotic values.

Another immediate corollary of Nevanlinna’s theorem is that the family is topologically closed. That is,

###### Corollary 1.3.

If is topologically conjugate to a meromorphic function in , and if is meromorphic, then it is also in .

### 2. The family Mp,q,r

A family that is more general than is the family of functions

 ~Mp,q,r={f=P∘g∘Q|g∈Nr,P,Q polynomials of degrees p,q}.

This family is also topologically closed. We have333This result is proved in [FK]. Since the proof is short and the ideas used are relevant we include it here.

###### Proposition A.

If is topologically conjugate to a meromorphic function in , and if is meromorphic, (with essential singularity at infinity), then it is also in ; that is there is a function and polynomials of degrees respectively such that .

The proof uses quasiconformal mappings and the Measurable Riemann mapping theorem. We refer the reader to the standard literature ([A, LV] and for a more dynamical discussion, [BF].

###### Proof.

We prove the theorem for quasiconformally conjugate to . It then follows from Theorem 3.3 of [KK1] that it is true for topologically conjugate to .

Let be a quasiconformal homeomorphism with Beltrami coefficient such that

 h=ϕμ∘f∘(ϕμ)−1=ϕμ∘P∘g∘Q∘(ϕμ)−1

is meromorphic. We can use to pull back the complex structure defined by to obtain a complex structure such that the map

 ~P=ϕμ∘P∘(ϕν)−1

is holomorphic. Note that this is not a conjugacy since it involves two different homeomorphisms. Similarly, we can use to pull back the complex structure defined by to obtain a complex structure such that the map

 ~g=ϕν∘g∘(ϕη)−1

is meromorphic. Again this is not a conjugacy. The function

 ~P∘~g=ϕμ∘P∘(ϕν)−1∘ϕν∘g∘(ϕη)−1

is a composition of meromorphic functions and is meromorphic. Now set

 ~Q=ϕη∘Q0∘(ϕμ)−1.

Although is not conjugate to , we were given that

 h=ϕμ∘P∘(ϕν)−1∘ϕν∘g∘(ϕη)−1∘ϕη∘Q0∘(ϕμ)−1=~P∘~g∘~Q

is meromorphic so that is also meromorphic.

The main point here is that although is not a conjugate of , since the quasiconformal maps and are homeomorphisms, the map is a meromorphic map with the same topology as ; that is, it has asymptotic values and no critical values so that by Corollary 1.3 belongs to . Similarly, although and are not respective conjugates of and , because the quasiconformal maps , and are homeomorphisms, the maps and are holomorphic maps with the same topology as and ; that is, they are respectively a degree and a degree branched covering of the Riemann sphere with the same number of critical points and the same branching as and . The maps , and are defined up to normalization. If we want to keep the essential singularity of at infinity, we normalize so that and are polynomials of degrees and respectively, (that is, infinity is a fixed critical point with respective multiplicities and ). ∎

A corollary to this theorem is

###### Corollary A.

The space of functions has a natural embedding into and the subspace has dimension .

###### Proof.

A polynomial of degree is determined by its coefficients. If , set . It is determined by its coefficients. It is straightforward to check that if are linearly independent solutions of the equation

 w′′+12Sgw=0

then . Since the space of solutions to this linear differential equation has dimension , and the solution of the Schwarzian equation is the quotient of two such solutions, is determined by parameters. Requiring that infinity cannot be an asymptotic value restricts one parameter. Therefore has a natural embedding into and the subspace has dimension . ∎

#### 2.1. Dynamics of meromorphic functions

Let be a transcendental meromorphic function and let denote the iterate of , that is for . Then is well-defined, except at the poles of , which form a countable set. These points have finite orbits that end at infinity.

The basic objects studied are the Fatou set and Julia set of the function . The Fatou set of is defined by

 F(f)={z∈C | fn is defined and normal in a neighborhood % of z},

and the Julia set by

 J(f)=ˆC∖F(f).

Note that the point at infinity is always in the Julia set. If is a meromorphic function with more than one pole, then the set of prepoles, is infinite. By the Picard theorem, is normal on . Since it is not normal on , , (see also [BKL1]).

A point is called a periodic point of , if and for any . The multiplier of the cycle is defined to be The periodic point is attracting if , super-attracting if , parabolic if , where is rational number, and neutral if is not rational. It is repelling if .

If is a component of the Fatou set, then is either a component of the Fatou set or a component missing one point. For the orbit of under , there are only two cases:

• there exist integers such that , and is called eventually periodic;

• for all , , and is called a wandering domain.

Suppose that is a periodic cycle of Fatou components, then either:

1. The cycle is (super)attracting: each contains a point of a periodic cycle with multiplier and all points in each domain are attracted to this cycle. If , the critical point itself belongs to the periodic cycle and the domain is called super-attracting.

2. The cycle is parabolic: the boundary of each contains a point of a periodic cycle with multiplier , , a divisor of , and all points in each domain are attracted to this cycle.

3. The components of the cycle are Siegel disks: that is, each contains a point of a periodic cycle with multiplier , where is irrational and there is a holomorphic homeomorphism mapping each to the unit disk , and conjugating the first return map on to an irrational rotation of . The preimages under this conjugacy of the circles , foliate the disks with forward invariant leaves on which is injective.

4. The components of the cycle are Herman rings: each is holomorphically homeomorphic to a standard annulus and the first return map is conjugate to an irrational rotation of the annulus by a holomorphic homeomorphism. The preimages under this conjugacy of the circles , foliate the disks with forward invariant leaves on which is injective.

5. is an essentially parabolic (Baker) domain: the boundary of each contains a point (possibly ) and for all , but is not holomorphic at . If , then .

###### Definition 4.

Define the post-singular set of as the closure of the orbits of the singular values; that is,

 PSf=¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯∪∞n=0fn(Sf).

For notational simplicity, if a pre-pole is a singular value, is a finite set and includes infinity.

Each non-repelling periodic cycle is associated to some singular point. In particular we have, see e.g. [M2], chap 8-11 or [Ber], Sect.4.3,

###### Theorem 2.1.

If is an attracting, superatttracting, parabolic or Baker periodic cycle of Fatou components, then for some , contains a singular value. If is a cycle of rotation domains (Siegel disks or Herman rings) the boundary of each contains the accumulation set of some singular value.

We use the following definition of a hyperbolic function in 444This definition is equivalent to standard definitions of hyperbolic functions because for these function is finite.

###### Definition 5.

An is called hyperbolic if

 PSf∩J(f)=∅.

Note that if all the singular values of are attracted to attracting or super-attracting cycles then is hyperbolic.

For a discussion of hyperbolicity for more general meromorphic functions, see the discussion in [Z] and the references therein.

Singularly finite maps may have Baker domains but

###### Theorem 2.2.

[BKL4] If is finite, then there are no wandering domains in the Fatou set.

#### 2.2. Holomorphic families

In this paper, we are interested in the family of meromorphic functions in for which infinity is not an asymptotic value. For each choice of triples this is an example of a holomorphic family. Below we state the general definitions and results we need.

###### Definition 6 (Holomorphic family).

A holomorphic family of meromorphic maps over a complex manifold is a map , such that is meromorphic for all and is holomorphic for all .

###### Definition 7 (Holomorphic motion).

A holomorphic motion of a set over a connected complex manifold with basepoint is a map given by such that

1. for each , is holomorphic in ,

2. for each , is an injective function of , and,

3. at , .

A holomorphic motion of a set respects the dynamics of the holomorphic family if whenever both and belong to .

The following equivalencies are proved for rational maps in [McM] and extended to the transcendental setting in [KK1].

###### Theorem 2.3.

Let be a holomorphic family of meromorphic maps with finitely many singularities, over a complex manifold , with base point . Then the following are equivalent.

1. The number of attracting cycles of is locally constant in a neighborhood of .

2. There is a holomorphic motion of the Julia set of over a neighborhood of which respects the dynamics of .

3. If in addition, for , is are holomorphic maps parameterizing the singular values of , then the functions form a normal family on a neighborhood of .

###### Definition 8 (J−stability).

A parameter is a -stable parameter for the family if it satisfies any of the above conditions.

The set of non -stable parameters is precisely the set where bifurcations occur, and it is often called the bifurcation locus of the family , and denoted by . In families of maps with more than one singular value, however, it makes sense to consider subsets of the bifurcation locus where only some of the singular values are bifurcating, in the sense that the families are normal in a neighborhood of for some values of , but not for all. We define

 BX(si)={x0∈X∣{gin(x)}\ is not normal% in any neighborhood of x0.}

In this paper we investigate dynamically natural one dimensional slices of the holomorphic family . In these slices, roughly speaking, all the dynamic phenomena but one are fixed and the last is determined by a “free asymptotic value”. We will study the components of the complement of the bifurcation locus in these slices. The parameters in them are -stable.

Precisely, (see [FK])

###### Definition 9.

A one dimensional subset is a dynamically natural slice with respect to if the following conditions are satisfied.

1. is holomorphically isomorphic to the complex plane punctured at points where the function is not in the family; for example, points where the number of singular values is reduced. The removed points are called parameter singularities. By abuse of notation we denote the image of in by again, and denote the variable in by .

2. The singular values are given by distinct holomorphic functions , , and an asymptotic value that is an affine function of .555This is only for convenience of exposition and can be arranged by a holomorphic change of coordinates in . We call the free asymptotic value; we require that .

3. The poles (if any) are given by distinct holomorphic functions , , .

4. The critical values and some of the asymptotic values are attracted to an attracting or parabolic cycle666If the attracting cycle is parabolic, the functions in the slice are not hyperbolic but the results hold. See [FK] for further discussion whose period and multiplier are constant for all . We call these dynamically fixed singular values. The remaining singular values include ; if there is only one remaining value, it may have arbitrary behavior. If there are several, they satisfy a relation that persists throughout so that the remaining dynamical behavior is controlled by the behavior of . Examples of how this may work are discussed in Section 4.

5. Suppose is attracted to , the basin of attraction of an attracting cycle that does not attract any of the dynamically fixed singular values. Then the slice contains, up to affine conjugacy, all meromorphic maps that are quasiconformally conjugate to in and conformally conjugate to on .

6. is maximal in the sense that if where is a parameter singularity, then does not satisfy at least one of the conditions above.

In the components of the complement of the bifurcation locus in these slices is attracted to an attracting cycle of fixed period; this is the period of the component. We distinguish two cases:

1. does not attract one of the dynamically fixed singular values. We call these Shell components and denote the individual components by and the collection by .

2. attracts one of the dynamically fixed singular values in addition to attracting . We call these Capture components. Their properties are very different from those of the shell components and we leave a discussion of them to future work.

### 3. Shell components

The properties of shell components are described in detail in [FK]. We summarize them here. We assume is the restriction of a holomorphic family in to a dynamically natural slice and denote a function in by . We need the following definitions

###### Definition 10.

Suppose that for some , satisfies , where and ; that is, is a prepole of order . Then is called a virtual cycle parameter of order . This is justified by that fact that if is an asymptotic path for and if is the branch of the inverse of taking to the prepole then

 limt→∞fk(h(γ(t))=vλ=a1.

We can think of the points as forming a virtual cycle.

###### Definition 11.

Let be a shell component in and let

 aλ={a0,a1,…,ak−2,ak−1}

be the attracting cycle of period that attracts . Suppose that as , and the multiplier Then is called a virtual center of .

Since the attracting basin of the cycle must contain , we will assume throughout that the points in the cycle are labeled so that and are in the same component of the immediate basin.

The next theorem collects the main the results in [FK] about shell components in a dynamically natural slice for a holomorphic family of transcendental functions with finite singular set, none of whose asymptotic values is at infinity. Note that Theorem A of the introduction is part (c) of the theorem.

###### Theorem 3.1.

Let be a shell component in . Then if

1. The map is a universal covering map. It extends continuously to and is piecewise analytic; either is simply connected and is infinite to one or is isomorphic a punctured disk and the puncture is a parameter singularity.

2. There is a unique virtual center on . If the period of the component is , the virtual center is at infinity.

3. Every (finite) virtual center of a shell component is a virtual cycle parameter and any virtual cycle parameter on the boundary of a shell component is a virtual center.

Here we prove a stronger theorem for slices of the family ; recall that all their asymptotic values are finite. Note that because the functions are of the form , if is an asymptotic value of , then is an asymptotic value of . There are distinct asymptotic tracts corresponding to each asymptotic tract of so that there are distinct asymptotic tracts at infinity separated by the Julia directions. The asymptotic values, tracts and Julia directions depend holomorphically on the parameters. At each pole of of order there are pre-asymptotic tracts and the pull-backs of the rays in the Julia directions separate them.

###### Theorem B.

Let be a dynamically natural slice for the meromorphic family consisting of meromorphic functions of the form all of whose asymptotic values are finite, and let be a virtual center parameter of order . Then is on the boundary of a shell component of order in . That is, in any neighborhood of there exists such that has an attracting cycle of period .

As an immediate corollary to Theorem 3.1 and Theorem B is

###### Corollary B.

In a dynamically natural slice of , every virtual cycle parameter is a virtual center and vice versa.

###### Remark 3.1.

The essence of the theorem is that the dynamic picture at the poles is reflected in the parameter picture at the virtual center parameters.777 This is analogous to the situation for Misiurewicz points in the parameter plane for quadratic polynomials. If infinity is an asymptotic value, both the dynamics and parameter pictures are different.

###### Proof.

Since is a virtual center parameter of order , has a virtual cycle . For each , the cycle uniquely determines a branch of the inverse of , such that . By abuse of notation, for readability, we drop the and denote all of these branches by . Because we are in a dynamically natural slice of a holomorphic family, the analytic continuations of the , denoted by are well defined.

Note however, that in a neighborhood of the asymptotic value , the inverse branch of is not uniquely determined. Since is part of a virtual cycle, a neighborhood of has at least one pre-image that is in an asymptotic tract. If the multiplicity of is one, then by definition there is a unique asymptotic tract that is determined by the virtual cycle of ; we denote it by and take as the branch that maps to . Then taking the analytic continuation of this we obtain as the analytic continuation of . If the multiplicity of is greater than one there will be a choice among the tracts corresponding to (and hence the inverse branch ). Similarly, if there is more than one asymptotic value that varies with (and satisfies a functional relation with ), we choose the tract (or one of them if there is more than one) corresponding to the free asymptotic value888See the examples in Part 2.. In the argument below, we take , or choose one of the tracts as , along with the corresponding branch . Since, in general, the asymptotic value is not omitted, there may also be infinitely many inverse branches of the neighborhood that are bounded. We don’t need to concern ourselves with them here.

Because we are in a dynamically natural slice we have the following holomorphic functions:

1. is the free asymptotic value of .

2. If , then is the holomorphic function defining the pole of such that .

3. Note that for in a neighborhood of , the affine function determines a corresponding neighborhood of in the dynamic plane of and vice versa. Define the map by . Then if is small enough, both and are in , a small neighborhood of in the dynamic plane of .

4. In the dynamic plane of , set . Then is the preimage of the pole in a neighborhood of the asymptotic value .

5. Each has infinitely many inverses in the asymptotic tract ; denote these by , .

The main ideas for the proof are first to use the relation between and the free asymptotic value to carefully choose and fix a close to and then to construct a domain such that . It will then follow from Schwarz’s lemma that has an attracting fixed point in .

Choosing : (See figure 1.) The asymptotic tract lies between two Julia rays and these span an angle . We choose a small enough neighborhood of such that for each , the Julia rays of , lie within of respectively. Each has an asymptotic tract between these rays and we can find a set which lies inside the sector in bounded by the rays and such that is a sector in with vertex at infinity and angle .

As above, let be a neighborhood of and let be the corresponding neighborhood in parameter space. For each , and is defined. Define by . Since contains and is holomorphic, we can find such that Set and let be the “triangular” subset of with vertex at . Note that the number of such triangular sets is equal to the order of the pole. If it is one, the set is unique. If it is greater than one, we choose one arbitrarily.

Constructing : (See figure 2.) Now we work with a fixed and we set . Since we have . In we have and . Let be a circle with center and radius . Taking closer to if necessary, we may assume lies in a compact subset of ; that is the disk . Now and is an infinite to one cover so contains preimages , , of .

We want to approximate the distance from a point on to . Let and be a conformal homeomorphisms from the left half plane to and the punctured disk to respectively chosen such that .

Because is a homeomorphism, is bounded above and below for in a closed disk containing . Applying Koebe’s distortion theorem, there are positive constants such that for ,

 K1ηλ(1+ηλ)2<|ψ−1(z)|

We are assuming is small so we may assume it is less than . Thus we can find closed annuli in and containing and respectively.

We then have, for appropriate positive constants, ,

 (2) K2+|logηλ|<|Relogψ−1(z)|

This says that the lift of the circle lies in a vertical strip of bounded horizontal width in Let denote ; it is the lift of the annulus in to .

The covering group for the exponential map acting on pulls back to an infinite cyclic group generated by a map that is the covering group acting on under the map so that

 ϕ(γ(z))=w+2πi.

Thus

 ϕ′(z)dz=dw and ϕ′(γ(z))γ′(z)dz=dw.

This says that

 dwdz=ϕ′(z)=ϕ′(γ(z))γ′(z),

and because is bounded in the closure of a fundamental domain for intersected with , it is bounded for all in . Now because is a conformal homeomorphism it preserves the hyperbolic density so that

 ρAλ(γ(z))|dγ(z)|=|dw||Rew|
 ρAλ(γ(z))|γ′(z)dz|=|dw+2πi||Re(w+2πi|)=|dw||Rew|.

Therefore the hyperbolic density in is invariant under the action of . Equation (2), says that is comparable to which gives us the estimate we want; that is, there are positive constants such that

 minζ∈∂Aλ,y∈ˆC|y−ζ|∼K3+K′3|logηλ|.

Note that because we have a group action on , there will be a and a in each fundamental region. Now let be a triangular region in with one vertex at , two sides spanning an angle of , where is the order of the pole , joining to , and third side an arc of the circle ; the sides are chosen so that contains in its interior. Next set . First note that and is a doubly infinite curve in that stays a bounded distance from , where the bound depends on and . The inverse images of sides of the triangle form scallops “above” (further inside ).

Now look at . This is a triangular shaped region contained in with a boundary point ; it contains . Choose realizing the minimum above so that is contained in . Then, as is a pole of order , we have

 (3) |f−1λ(y)−pλ|∼K3|logηλ|1m.

Since the derivative of along the orbit of from to is bounded and varies holomorphically with , the derivative of along the orbits of from to and to are also bounded for . Because and is small, when we map by the images of and points on are comparably close to the pole . Specifically, for some positive constant , we have

 |fk−2λ(vλ)−pλ|∼K4ηλ<

Comparing this with the estimate (3) it follows that

 fk−1λ(C∩~T)⊂T and fk−1λ(vλ)∈fkλ(T).

Therefore because is inside ,

 fkλ(T)⊂T

so that by the Schwarz lemma has a fixed point in . ∎

## Part II The Extended Tangent Family λtanpzq.

In this part of the paper we use the results above to prove that for family , every shell component of period is bounded. A corollary is that every capture component is bounded as well. The proof will follow from by studying the period and period components.

### 4. The shell components of F

The family is a subfamily of with and and in the one dimensional slice of consisting of functions that fix and have symmetric asymptotic values. The functions in have one fixed critical point at and have either one asymptotic value with mulitplicity or two asymptotic values with multiplicity that are opposite in sign. Specifically, the map has two distinct asymptotic tracts and two asymptotic values, . Each of these tracts has pre-images under . If is even, all of the asymptotic tracts map onto punctured neighborhoods of the single point which is the free asymptotic value . If is odd, of the asymptotic tracts map onto punctured neighborhoods of and the other tracts map onto punctured neighborhoods of . In this case we choose as the free asymptotic value. The other asymptotic value satisfies the relation .

The punctured plane is thus a dynamically natural slice in and the dynamics are determined by the forward orbit of .

The full set of shell components in this slice is denoted by and we divide it into subsets depending on the period of the cycle as follows:

###### Definition 12.

If is even

 Sn={Ωn| fλ has an attracting cycle of % period n},

otherwise

 Sn={Ωn | fλ has one attracting cycle % of period 2n or
 Ω′n |fλ  has two attracting cycles of period % n}.

For readability we only include the subscript on if the period is not obvious from the context.

Figure 3 shows the parameter plane for the family . The period  shell components are yellow, the period  shell components are cyan blue. The capture components are green.

#### 4.1. Symmetries

Note that for any , . In addition, if , are the roots of unity,

 fωkλ(ωkz)=ωkλtanp(ωkz)q=ωkfλ(z).

It follows that if then


text:
## Chemistry: The Central Science (13th Edition)

$As_{4}$(g)
$As_{4}$ has a higher molar mass than $P_{4}$. Increasing molar mass generally increases standard entropy.