text:
11-23-2011 , 06:01 AM
Quote:
Originally Posted by imolin
you dont think tiffany lip sync that well?
.
11-23-2011 , 06:30 AM
Browsing TL, found a Lex Veldhuis blog
http://www.teamliquid.net/blogs/view...opic_id=271590

Says he watches 2-3 hours of sc2 a day... sc2 is preventing him from losing money!
... on the other hand, maybe he's grinding nosebleeds while watching sc2 at the same time, I do it at micros wouldn't be surprised if a sicko like Lex does it high stakes
11-23-2011 , 06:46 AM
11-23-2011 , 09:13 AM
Quote:
Originally Posted by Karak
Yo hppe you ask dir especially end breed 77
Don't delete this.
11-23-2011 , 12:58 PM
Someone has to make a Dreamhack thread. Probably not going to be around for the next 48 hours and I want to post about it come Friday. Going to be an awesome tourney.
11-23-2011 , 01:01 PM
Quote:
Originally Posted by skater3598
Man, is there anything ElKy can't do?
11-23-2011 , 01:12 PM
11-23-2011 , 02:29 PM
fight results:

Spoiler:
raszi kicks elky in the head twice and KO-s him in the second LOL
11-23-2011 , 02:42 PM
Quote:
Originally Posted by Karak
Yo hppe you ask dir especially end breed 77
i was trying to post from my phone and the above happened due to swype being unable to correctly interpret my drunk finger strokes (herp)

i actually was able to divine what im trying to say under a careful analysis this morning:

"Yo hope you all die, especially needle77."
11-23-2011 , 03:04 PM
Quote:
Originally Posted by Karak
i was trying to post from my phone and the above happened due to swype being unable to correctly interpret my drunk finger strokes (herp)

i actually was able to divine what im trying to say under a careful analysis this morning:

"Yo hope you all die, especially needle77."
something something girls, something something drunk fingers strokes.

That's all the wit I have atm...
11-23-2011 , 03:13 PM
Quote:
Originally Posted by Firefly
Man, is there anything ElKy can't do?
Spoiler:
By the looks of things, fight.
11-23-2011 , 06:08 PM
11-23-2011 , 10:27 PM
the queen!!

11-23-2011 , 11:45 PM
1-(1 - p)^k

someone isolate p...i don't feel like it.
11-23-2011 , 11:47 PM
no u
11-24-2011 , 12:52 AM
p = -(sq rt of k) - 1

can you have a negative sq rt? i dont remember, i havent done algebra in like 7 years.
11-24-2011 , 01:06 AM

WHAT THE **** IS THIS ****

alright, what the ****'s the error fucntion
wait it's the ****in inverse error function?
GOD DAMMIT PIECE OF **** I HATE IT SO MUCH
11-24-2011 , 01:14 AM
Quote:
Originally Posted by pwncakery

WHAT THE **** IS THIS ****

alright, what the ****'s the error fucntion
wait it's the ****in inverse error function?
GOD DAMMIT PIECE OF **** I HATE IT SO MUCH
Substitute y for x in your equation then solve for y =. For example, f(x) = 2x. y=2x. Inverse f^-1 is x=2y. y =x/2.

For the other, use log or ln to undo the exponent. Then use properties of logs.

In before cuey.
11-24-2011 , 01:45 AM
The error function is an integral.

$img=http://latex.codecogs.com/gif.latex?\text{Erf}(x)=\frac{2}{\sqrt{\pi}}\int_0^x+e^{-t^2}\+dt$

11-24-2011 , 02:01 AM
No analysis pwn?
11-24-2011 , 02:03 AM
I just answered your question in SMP. Is this a different one?

PM me if it is.
11-24-2011 , 02:05 AM
Looking for masters terran player to practice w/. PM me.
11-24-2011 , 02:08 AM
Out of everyone in this forum, I don't want any of you as students.
11-24-2011 , 02:16 AM
Different questions. The SMP one really really helped me out. It translates really easily into code. The one in this thread was for the standard normal distribution, but translating that into code looks a little daunting. The hint given with the question was "sin-cos technique" but I'm unfamiliar with that, because **** trig. I'm supposed to be creating a program that will generate various distributions. The programming is fairly straight forward, but this math **** is a big bucket of what the ****.
11-24-2011 , 02:27 AM
as long as you dont use hotkeys, you'll be fine. and watch out for cueball trying to kill you if you dont get what hes saying

m


text:
## Stream: metaprogramming / tactics

### Topic: Trace to stdout

#### Logan Murphy (Sep 22 2020 at 15:43):

Sorry if this is a noobish question. Let's say that i have something like the following in the file even.lean

import system.io
open  io

inductive even : ℕ → Prop
| zero : even 0
| add_two : ∀ n, even n → even (n+2)

meta def tactic_even : tactic unit :=
do
tactic.repeat
((tactic.trace "Inductive step"; tactic.applyc even.add_two)
<|>
(tactic.trace "Base Case"; tactic.applyc even.zero))

def six_is_even : even 6 := by tactic_even

The trace of six_is_even is

Inductive step
Inductive step
Inductive step
Inductive step
Base Case

Is there a way I can write a function def main : io unit such that invoking lean --run even.lean would print the above trace to std output?

#### Simon Hudon (Sep 22 2020 at 16:09):

Yes you can call it using io.run_tactic in your main function

Oh great!

#### Logan Murphy (Sep 22 2020 at 19:14):

@Simon Hudon can you give any example of how to use it? I can't seem to figure how it works from the code itself

#### Simon Hudon (Sep 22 2020 at 19:16):

Have you tried def main : io unit := io.run_tactic tactic_even

#### Logan Murphy (Sep 22 2020 at 19:33):

Yes, but invoking

meta def main : io unit :=
do
io.run_tactic tactic_even

prints

Inductive step
Base Case

which I'm assuming just means it traces out everything in the tactic. I'm wondering how I would use it in conjunction with a particular goal, so if I use tactic_even to show that, say, 4 is even, it would print 2 isntances of Inductive step and one instance of Base Case

(although the above version of tactic_even I provided actually prints an extra line)

#### Logan Murphy (Sep 22 2020 at 19:35):

oh, maybe I can just provide an argument to tactic_even

#### Simon Hudon (Sep 22 2020 at 19:48):

If you call lean even.lean you'll already see the printout with regards to your specific proof. Why do you need it to be called from main?

Regardless, you can do that with:

meta def main : io unit :=
do
io.run_tactic \$ do [have : even 6], tactic_even

#### Logan Murphy (Sep 22 2020 at 19:50):

What I actually want to do is get some information from stdin in main` and then show the trace of a proof about that information.

Thanks for the help!

Last updated: May 09 2021 at 21:10 UTC

format: plain


text:
## Calculus (3rd Edition)

$$\sqrt{2}(b-a)$$
Since $0 \leq f^{\prime}(x) \leq 1,$ then $$\sqrt{1+\left(f^{\prime}(x)\right)^{2}} \leq \sqrt{1+1}=\sqrt{2}$$ By comparison, we have $$\int_{a}^{b} \sqrt{1+\left(f^{\prime}(x)\right)^{2}} d x \leq \int_{a}^{b} \sqrt{2} d x=\sqrt{2}(b-a)$$ For $f(x)=x,$ we have $$f^{\prime}(x)=1$$ Therefore, we get $$\int_{a}^{b} \sqrt{1+\left(f^{\prime}(x)\right)^{2}} d x=\int_{a}^{b} \sqrt{2} d x=\sqrt{2}(b-a)$$


text:
Next: Multiphase mixtures Up: Special geometries Previous: Hedgehogs, Spirals, etc.

## Self-repeating structures

Here we describe an elegant yet unexpected approach to the calculation of polycrystal structures suggested in [#!Nesi:1991:PCM!#].

Consider an anisotropic material . Let us laminate this material with some anisotropic material in a laminate with a normal and a volume fraction of . Denote the resulting composite by . Suppose that it is possible to choose the parameters , , and the unknown material so that

 (23)

where is the tensor of rotation through an angle. The formula 23 states that lamination of the given anisotropic material with a material with properties leads to the composite with the properties equal to rotated on some angle.

If the relation 23 is satisfied, then is a polycrystal assembled of variously oriented fragments of .

To show this, we again use the process with an infinite number of steps. At the first step, we obtain the composite 23. At the second step, we laminate the composite with the rotated material in the same way as in the first step. Obviously, the result of this lamination is , i.e., the material rotated on the double angle. This step can be repeated infinitely many times; still, the resulting material remains equal to the rotated material .

Let us calculate the volume fractions of materials in a composite obtained at the th step. The material is used only on the first step of the procedure. Its volume fraction is equal to in the composite obtained after the first step. After the second step this fraction becomes , because the material has been added to the mixture in the volume fraction . Similarly, after the th iteration the fraction of the initial seed'' is equal to . When increases, the volume fraction of the seed'' becomes arbitrarily small, and almost all the volume is occupied with differently oriented fragments of material . In the limit , the described composite becomes a polycrystal of .

To find this polycrystal, we solve equation 23 for the unknown tensor . The parameter and the rotation tensor can be arbitrarily assigned. Any solution represents a polycrystal of . The set of solutions represents a class of polycrystals of that can be obtained by the described method.

Here, an animated graph will show the process of formation of a repeated structure

Next: Multiphase mixtures Up: Special geometries Previous: Hedgehogs, Spirals, etc.
Andre Cherkaev
2001-07-31


text:
1

### AIPMT 2012 Prelims

A metal crystallises with a face-centred cubic lattice. The edge of the unit cell is 408 pm. The diameter of the metal atom is
A
288 pm
B
408 pm
C
144 pm
D
204 pm

## Explanation

Given $a$ = 408 pm

For the face centred cubic structure

r = ${a \over {2\sqrt 2 }}$ = ${{408} \over {2\sqrt 2 }}$ = 144 pm

$\therefore$ Diameter = 2r = 2$\times$144 = 288 pm
2

### AIPMT 2011 Mains

A solid compound XY has NaCl structure. If the radius of the cation is 100 pm, the radius of the anion (Y$-$) will be
A
275.1 pm
B
322.5 pm
C
241.5 pm
D
165.7 pm

## Explanation

For NaCl crystal, ${{{r_ + }} \over {{r_ - }}} = 0.414$

Given, r+ = 100 pm

$\therefore$ ${{100} \over {{r_ - }}} = 0.414$

$\Rightarrow$ ${r_ - } = {{100} \over {0.414}}$ = 241.5 pm
3

### AIPMT 2010 Prelims

AB crystallizes in a body centered cubic lattice with edge length '$a$' equal to 387 pm. The distance between two oppositely charged ions in the lattice is
A
335 pm
B
250 pm
C
200 pm
D
300 pm

## Explanation

For a bcc lattice, 2(r+ + r) = $\sqrt 3 a$

$\therefore$ r+ + r = ${{\sqrt 3 \times 387} \over 2}$ = 335 pm
4

### AIPMT 2009

Copper crystallises in a face-centred cubic lattice with a unit cell length of 361 pm. What is the radius of copper atom in pm?
A
157
B
181
C
108
D
128

## Explanation

Since Cu crystallises in a face-centred cubic lattice,

Atomic radius, r = ${a \over {2\sqrt 2 }}$

$\therefore$ r = ${{361} \over {2\sqrt 2 }}$ = 128 pm


text:
## Want to keep learning?

This content is taken from the UNSW Sydney's online course, Maths for Humans: Linear, Quadratic & Inverse Relations. Join the course to learn more.
2.2

## UNSW Sydney

Skip to 0 minutes and 12 seconds So our story really beings more than 2,000 years ago with the ancient Greeks, who were very interested in geometry. And in Euclid’s geometry, the line and the circle are considered the most perfect figures. Apollonius, who was a successor of Euclid and considered himself a rival to Archimedes, was the greatest geometer of the ancient world. And he took this idea further and said, well, why don’t we look in three dimensions at what we get when we combine a line with a circle? We can form a cone. So here’s an example of a cone. There’s the apex there. But the ancient Greeks actually considered the cone as actually extending also below, so there was a top part and also a bottom part.

Skip to 0 minutes and 59 seconds So here we have one of these cones that Apollonius studied. So this is a three-dimensional figure obtained by starting with a point and a circle, which is above that point, and connecting each point of the circle with the given point here with a straight line. This is supposed to be an infinite straight line, so this cone actually extends up and below as well. So it has these two parts of it– the top part and the bottom part. And Apollonius realised it was interesting to think about what happens when you slice this cone with a plane. That’s called a conic section.

Skip to 1 minute and 37 seconds So for example, if we slice it with a plane, perhaps, in a direction like that, then we would get a curve of intersection that looks like this. That’s an ellipse.

Skip to 2 minutes and 0 seconds On the other hand, if we take a plane and slice the cone in such a way that we’re touching both the top and the bottom parts of the cone, then we get something that’s quite different. For example, if we take a plane like this and slice the cone, then we get a curve that has a branch in this direction and also another branch in this direction. And that’s a hyperbola.

Skip to 2 minutes and 34 seconds And then there’s a third case, which is intermediate between these two. When we take a plane which is just touching one of the branches but almost touching the other one but not quite, the way we do that is we take a plane– say this one here– which is parallel to one of the sides or one of the lines in the cone. So if we take this line in the cone and take a plane which is parallel to that, then it will slice this bottom part but not touch the top part. And that will give us a curve that looks something like that, and that’s a parabola.

Skip to 3 minutes and 19 seconds These are the names given to these curves by Apollonius of Perga. He proved many things about these curves. He knew hundreds of interesting facts about them. And it was really only 2,000 years later the Europeans were able to go beyond Apollonius’s understanding. And that happened with the discoveries of Descartes and his introduction of the Cartesian coordinates that we talked about last week. So in Descartes’s framework, it turned out that you could also describe ellipses or hyperbolas or parabolas in a very precise algebraic way. So these conic sections as they’re called, because they’re sections of a cone, to Descartes were second degree curves.

Skip to 4 minutes and 21 seconds That means they were curves that had an equation that maybe was like this– x squared minus 2y squared plus 3x minus 5y equals 7. That’s an equation of degree 2 in x and y. And it turns out that that’s exactly one of these conic sections. So this remarkable correspondence between this classical subject from ancient Greece and this purely algebraic formulation of things discovered by Descartes. And it was a very, very powerful tool, and that’s really going to be a central tool for us in understanding the geometry of the parabola. So it will turn out that quadratic relations that come up in practise are often mostly conveniently described by this parabola.

Skip to 5 minutes and 13 seconds So we have to turn to understanding this parabola next, which is poised very delicately between the cases of the ellipse and the hyperbola. That’s actually very interesting both geometrically and practically.

# Conic sections

Conic sections go back to the ancient Greek geometer Apollonius of Perga around 200 B.C. He viewed these curves as slices of a cone and discovered many important properties of ellipses, parabolas and hyperbolas.

Apollonius considered the cone to be a two-sided one, and this is quite important. It means that if we slice the cone with a plane, then we can get not just a circle or ellipse, but also more subtle curves with two separate branches called hyperbolas, and balanced between these two, a very special kind of curve called a parabola.

The parabola will turn out to be particularly important for us, as this is exactly the shapes assumed by the graphs of quadratic functions of the form

$\Large{y=ax^2+bx+c}.$

So in this video we span several thousand years, from the ancient world’s greatest geometer, to one of the most innovative and influential geometers of the modern era. If you are interested in the historical aspects of the subject (and all students of mathematics really ought to be!) then check out Norman’s YouTube lecture on Greek geometry.


text:
# Laplace equation with integral source terms

I am not specifically asking for a solution, but any reference on any method i could read about would be a big help. This I clarify as I am aware of the fact that MathOverflow only deals with research level questions. A starting point is all I aim to get from posting this question here

I have the following coupled PDEs: $$\begin{eqnarray} \frac{\partial \theta_h}{\partial x} + \beta_h (\theta_h - \theta_w) &=& 0,\\ \frac{\partial \theta_c}{\partial y} + \beta_c (\theta_c - \theta_w) &=& 0,\\ \lambda_h \frac{\partial^2 \theta_w}{\partial x^2} + \lambda_c V \frac{\partial^2 \theta_w}{\partial y^2} - \frac{\partial \theta_h}{\partial x} - V\frac{\partial \theta_c}{\partial y} &=& 0 \end{eqnarray}$$ The boundary conditions are :

The PDE needs to be solved on a rectangular region where $$x$$ varies between $$0$$ to $$1$$ and $$y$$ varies between $$0$$ to $$1$$.

$$\frac{\partial \theta_w(0,y)}{\partial x}=\frac{\partial \theta_w(1,y)}{\partial x}=0$$

$$\frac{\partial \theta_w(x,0)}{\partial y}=\frac{\partial \theta_w(x,1)}{\partial y}=0$$

$$\theta_h(0,y)=1$$$$\theta_c(x,0)=0$$

$$\beta_h,\beta_c,\lambda_h,\lambda_c,V$$ are all constants $$>0$$

Attempt

The third equation can be written as $$\lambda_h \frac{\partial^2 \theta_w}{\partial x^2} + \lambda_c V \frac{\partial^2 \theta_w}{\partial y^2} = \frac{\partial \theta_h}{\partial x} + V\frac{\partial \theta_c}{\partial y}$$

Then, from the first two equations the following can be written :-

$$\lambda_h \frac{\partial^2 \theta_w}{\partial x^2} + \lambda_c V \frac{\partial^2 \theta_w}{\partial y^2} =\beta_h e^{-\beta_h x} \int e^{\beta_h x} \theta_w(x,y) \, \mathrm{d}x + \beta_c e^{-\beta_c y} \int e^{\beta_c y} \theta_w(x,y) \, \mathrm{d}y$$

This resulting equation looks a lot like Laplace equation with integral source terms (if the LHS is taken into canonical form).

What kind of procedure should I take up to solve this problem? Is there any standard problem type that corresponds to this situation.

Attempt taken forward

It can be seen that the resulting equation can be variable separated if we consider the following ansatz:

$$\theta_w(x,y) = e^{-\beta_h x} f(x) e^{-\beta_c y} g(y)$$

with $$F(x) := \int f(x) \, \mathrm{d}x$$ and $$G(y) := \int g(y) \, \mathrm{d}y$$

to get:

$$\begin{eqnarray} \lambda_h F''' - 2 \lambda_h \beta_h F'' + \left( (\lambda_h \beta_h - 1) \beta_h - \mu \right) F' + \beta_h^2 F &=& 0,\\ V \lambda_c G''' - 2 V \lambda_c \beta_c G'' + \left( (\lambda_c \beta_c - 1) V \beta_c + \mu \right) G' + V \beta_c^2 G &=& 0, \end{eqnarray}$$

for some separation constant $$\mu$$

## 1 Answer

More a long comment than an answer, but perhaps it could be useful to you: one reference I know that deals extensively with non-local equation (read "integrodifferential" and "with integral type boundary conditions") is the monograph of Chia-Ven Pao [1]. For example, elliptic integrodifferential equations are analyzed in chapter 3, §3.7, while boundary conditions of integral type (though for parabolic equations, and of Volterra type) are analyzed in chapter 4, §4.3 (this may be relevant for your later question here). The whole monograph is however devoted to non-classical boundary value problems so, even if it don't seem to suit your precise needs, I advice you to have a look at it. Note that

• Pao develops the theory in Ḧolder spaces, i.e. $$C^{k+\alpha}\equiv C^{(k,\alpha)}$$, $$k\in\Bbb N$$ and $$\alpha\in (0,1)$$ and uses $$\min$$-$$\max$$-solutions constructed by iteration (convergence of the processes is proved by using maximum principles for the problem at hand): this implies that no complicated functional analysis is used.
• Though been mathematically sound and rigorous, the book deals extensively with the analysis of models derived from biology, engineering and physics, which seems the stuff you (and perhaps the author of this answer...) are interested to.

[1] Chia-Ven Pao (1992), Nonlinear Parabolic and Elliptic Equations, Plenum Press, xv+777, MR1212084 Zbl 0777.35001.

• Thanks. I would now try to get a hand on this text. Will revert back, with what I find. This is much appreciated. – Indrasis Mitra Mar 10 '19 at 12:59
• For the sake of completion, I have updated my question here with further modifications of the BC that I carried out, since I posted it. – Indrasis Mitra Mar 11 '19 at 3:00


text:
### SectionVSVector Spaces

In this section we present a formal definition of a vector space, which will lead to an extra increment of abstraction. Once defined, we study its most basic properties.

#### SubsectionVSVector Spaces

Here is one of the two most important definitions in the entire course.

##### Definition VS Vector Space

Suppose that $V$ is a set upon which we have defined two operations: (1) vector addition, which combines two elements of $V$ and is denoted by “+”, and (2) scalar multiplication, which combines a complex number with an element of $V$ and is denoted by juxtaposition. Then $V$, along with the two operations, is a vector space over $\complexes$ if the following ten properties hold.

If $\vect{u},\,\vect{v}\in V$, then $\vect{u}+\vect{v}\in V$.

• SC Scalar Closure
If $\alpha\in\complex{\null}$ and $\vect{u}\in V$, then $\alpha\vect{u}\in V$.

• C Commutativity
If $\vect{u},\,\vect{v}\in V$, then $\vect{u}+\vect{v}=\vect{v}+\vect{u}$.

If $\vect{u},\,\vect{v},\,\vect{w}\in V$, then $\vect{u}+\left(\vect{v}+\vect{w}\right)=\left(\vect{u}+\vect{v}\right)+\vect{w}$.

• Z Zero Vector
There is a vector, $\zerovector$, called the zero vector, such that $\vect{u}+\zerovector=\vect{u}$ for all $\vect{u}\in V$.

If $\vect{u}\in V$, then there exists a vector $\vect{-u}\in V$ so that $\vect{u}+ (\vect{-u})=\zerovector$.

• SMA Scalar Multiplication Associativity
If $\alpha,\,\beta\in\complex{\null}$ and $\vect{u}\in V$, then $\alpha(\beta\vect{u})=(\alpha\beta)\vect{u}$.

• DVA Distributivity across Vector Addition
If $\alpha\in\complex{\null}$ and $\vect{u},\,\vect{v}\in V$, then $\alpha(\vect{u}+\vect{v})=\alpha\vect{u}+\alpha\vect{v}$.

• DSA Distributivity across Scalar Addition
If $\alpha,\,\beta\in\complex{\null}$ and $\vect{u}\in V$, then $(\alpha+\beta)\vect{u}=\alpha\vect{u}+\beta\vect{u}$.

• O One
If $\vect{u}\in V$, then $1\vect{u}=\vect{u}$.

The objects in $V$ are called vectors, no matter what else they might really be, simply by virtue of being elements of a vector space.

Now, there are several important observations to make. Many of these will be easier to understand on a second or third reading, and especially after carefully studying the examples in Subsection VS.EVS.

An axiom is often a “self-evident” truth. Something so fundamental that we all agree it is true and accept it without proof. Typically, it would be the logical underpinning that we would begin to build theorems upon. Some might refer to the ten properties of Definition VS as axioms, implying that a vector space is a very natural object and the ten properties are the essence of a vector space. We will instead emphasize that we will begin with a definition of a vector space. After studying the remainder of this chapter, you might return here and remind yourself how all our forthcoming theorems and definitions rest on this foundation.

As we will see shortly, the objects in $V$ can be anything, even though we will call them vectors. We have been working with vectors frequently, but we should stress here that these have so far just been column vectors — scalars arranged in a columnar list of fixed length. In a similar vein, you have used the symbol “+” for many years to represent the addition of numbers (scalars). We have extended its use to the addition of column vectors and to the addition of matrices, and now we are going to recycle it even further and let it denote vector addition in any possible vector space. So when describing a new vector space, we will have to define exactly what “+” is. Similar comments apply to scalar multiplication. Conversely, we can define our operations any way we like, so long as the ten properties are fulfilled (see Example CVS).

In Definition VS, the scalars do not have to be complex numbers. They can come from what are called in more advanced mathematics, “fields”. Examples of fields are the set of complex numbers, the set of real numbers, the set of rational numbers, and even the finite set of “binary numbers”, $\set{0,\,1}$. There are many, many others. In this case we would call $V$ a vector space over (the field) $F$.

A vector space is composed of three objects, a set and two operations. Some would explicitly state in the definition that $V$ must be a nonempty set, but we can infer this from Property Z, since the set cannot be empty and contain a vector that behaves as the zero vector. Also, we usually use the same symbol for both the set and the vector space itself. Do not let this convenience fool you into thinking the operations are secondary!

This discussion has either convinced you that we are really embarking on a new level of abstraction, or it has seemed cryptic, mysterious or nonsensical. You might want to return to this section in a few days and give it another read then. In any case, let us look at some concrete examples now.

#### SubsectionEVSExamples of Vector Spaces

Our aim in this subsection is to give you a storehouse of examples to work with, to become comfortable with the ten vector space properties and to convince you that the multitude of examples justifies (at least initially) making such a broad definition as Definition VS. Some of our claims will be justified by reference to previous theorems, we will prove some facts from scratch, and we will do one nontrivial example completely. In other places, our usual thoroughness will be neglected, so grab paper and pencil and play along.

##### Example VSMThe vector space of matrices, $M_{mn}$

So, the set of all matrices of a fixed size forms a vector space. That entitles us to call a matrix a vector, since a matrix is an element of a vector space. For example, if $A,\,B\in M_{34}$ then we call $A$ and $B$ “vectors,” and we even use our previous notation for column vectors to refer to $A$ and $B$. So we could legitimately write expressions like \begin{equation*} \vect{u}+\vect{v}=A+B=B+A=\vect{v}+\vect{u} \end{equation*} This could lead to some confusion, but it is not too great a danger. But it is worth comment.

The previous two examples may be less than satisfying. We made all the relevant definitions long ago. And the required verifications were all handled by quoting old theorems. However, it is important to consider these two examples first. We have been studying vectors and matrices carefully (Chapter V, Chapter M), and both objects, along with their operations, have certain properties in common, as you may have noticed in comparing Theorem VSPCV with Theorem VSPM. Indeed, it is these two theorems that motivate us to formulate the abstract definition of a vector space, Definition VS. Now, if we prove some general theorems about vector spaces (as we will shortly in Subsection VS.VSP), we can then instantly apply the conclusions to both $\complex{m}$ and $M_{mn}$. Notice too, how we have taken six definitions and two theorems and reduced them down to two examples. With greater generalization and abstraction our old ideas get downgraded in stature.

Let us look at some more examples, now considering some new vector spaces.

##### Example VSFThe vector space of functions

Here is a unique example.

##### Example VSSThe singleton vector space

Perhaps some of the above definitions and verifications seem obvious or like splitting hairs, but the next example should convince you that they are necessary. We will study this one carefully. Ready? Check your preconceptions at the door.

#### SubsectionVSPVector Space Properties

Subsection VS.EVS has provided us with an abundance of examples of vector spaces, most of them containing useful and interesting mathematical objects along with natural operations. In this subsection we will prove some general properties of vector spaces. Some of these results will again seem obvious, but it is important to understand why it is necessary to state and prove them. A typical hypothesis will be “Let $V$ be a vector space.” From this we may assume the ten properties of Definition VS, and nothing more. It is like starting over, as we learn about what can happen in this new algebra we are learning. But the power of this careful approach is that we can apply these theorems to any vector space we encounter — those in the previous examples, or new ones we have not yet contemplated. Or perhaps new ones that nobody has ever contemplated. We will illustrate some of these results with examples from the crazy vector space (Example CVS), but mostly we are stating theorems and doing proofs. These proofs do not get too involved, but are not trivial either, so these are good theorems to try proving yourself before you study the proof given here. (See Proof Technique P.)

First we show that there is just one zero vector. Notice that the properties only require there to be at least one, and say nothing about there possibly being more. That is because we can use the ten properties of a vector space (Definition VS) to learn that there can never be more than one. To require that this extra condition be stated as an eleventh property would make the definition of a vector space more complicated than it needs to be.

##### Theorem ZVU Zero Vector is Unique

Suppose that $V$ is a vector space. The zero vector, $\zerovector$, is unique.

##### Theorem AIU Additive Inverses are Unique

Suppose that $V$ is a vector space. For each $\vect{u}\in V$, the additive inverse, $\vect{-u}$, is unique.

As obvious as the next three theorems appear, nowhere have we guaranteed that the zero scalar, scalar multiplication and the zero vector all interact this way. Until we have proved it, anyway.

##### Theorem ZSSM Zero Scalar in Scalar Multiplication

Suppose that $V$ is a vector space and $\vect{u}\in V$. Then $0\vect{u}=\zerovector$.

Here is another theorem that looks like it should be obvious, but is still in need of a proof.

##### Theorem ZVSM Zero Vector in Scalar Multiplication

Suppose that $V$ is a vector space and $\alpha\in\complex{\null}$. Then $\alpha\zerovector=\zerovector$.

Here is another one that sure looks obvious. But understand that we have chosen to use certain notation because it makes the theorem's conclusion look so nice. The theorem is not true because the notation looks so good; it still needs a proof. If we had really wanted to make this point, we might have used notation like $\vect{u}^\sharp$ for the additive inverse of $\vect{u}$. Then we would have written the defining property, Property AI, as $\vect{u}+\vect{u}^\sharp=\zerovector$. This theorem would become $\vect{u}^\sharp=(-1)\vect{u}$. Not really quite as pretty, is it?

##### Theorem AISM Additive Inverses from Scalar Multiplication

Suppose that $V$ is a vector space and $\vect{u}\in V$. Then $\vect{-u}=(-1)\vect{u}$.

Because of this theorem, we can now write linear combinations like $6\vect{u}_1+(-4)\vect{u}_2$ as $6\vect{u}_1-4\vect{u}_2$, even though we have not formally defined an operation called vector subtraction.

Our next theorem is a bit different from several of the others in the list. Rather than making a declaration (“the zero vector is unique”) it is an implication (“if…, then…”) and so can be used in proofs to convert a vector equality into two possibilities, one a scalar equality and the other a vector equality. It should remind you of the situation for complex numbers. If $\alpha,\,\beta\in\complexes$ and $\alpha\beta=0$, then $\alpha=0$ or $\beta=0$. This critical property is the driving force behind using a factorization to solve a polynomial equation.

##### Theorem SMEZV Scalar Multiplication Equals the Zero Vector

Suppose that $V$ is a vector space and $\alpha\in\complex{\null}$. If $\alpha\vect{u}=\zerovector$, then either $\alpha=0$ or $\vect{u}=\zerovector$.

#### SubsectionRDRecycling Definitions

When we say that $V$ is a vector space, we then know we have a set of objects (the “vectors”), but we also know we have been provided with two operations (“vector addition” and “scalar multiplication”) and these operations behave with these objects according to the ten properties of Definition VS. One combines two vectors and produces a vector, the other takes a scalar and a vector, producing a vector as the result. So if $\vect{u}_1,\,\vect{u}_2,\,\vect{u}_3\in V$ then an expression like \begin{equation*} 5\vect{u}_1+7\vect{u}_2-13\vect{u}_3 \end{equation*} would be unambiguous in any of the vector spaces we have discussed in this section. And the resulting object would be another vector in the vector space. If you were tempted to call the above expression a linear combination, you would be right. Four of the definitions that were central to our discussions in Chapter V were stated in the context of vectors being column vectors, but were purposely kept broad enough that they could be applied in the context of any vector space. They only rely on the presence of scalars, vectors, vector addition and scalar multiplication to make sense. We will restate them shortly, unchanged, except that their titles and acronyms no longer refer to column vectors, and the hypothesis of being in a vector space has been added. Take the time now to look forward and review each one, and begin to form some connections to what we have done earlier and what we will be doing in subsequent sections and chapters. Specifically, compare the following pairs of definitions:


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# ALMA SURVEY of LUPUS PROTOPLANETARY DISKS. I. DUST and GAS MASSES

21 Nov 2017

We present the first high-resolution sub-mm survey of both dust and gas for a large population of protoplanetary disks. Characterizing fundamental properties of protoplanetary disks on a statistical level is critical to understanding how disks evolve into the diverse exoplanet population. We use ALMA to survey 89 protoplanetary disks around stars with $M_{\ast}>0.1~M_{\odot}$ in the young (1--3~Myr), nearby (150--200~pc) Lupus complex. Our observations cover the 890~$\mu$m continuum and the $^{13}$CO and C$^{18}$O 3--2 lines. We use the sub-mm continuum to constrain $M_{\rm dust}$ to a few Martian masses (0.2--0.4~$M_{\oplus}$) and the CO isotopologue lines to constrain $M_{\rm gas}$ to roughly a Jupiter mass (assuming ISM-like $\rm {[CO]/[H_2]}$ abundance). Of 89 sources, we detect 62 in continuum, 36 in $^{13}$CO, and 11 in C$^{18}$O at $>3\sigma$ significance. Stacking individually undetected sources limits their average dust mass to $\lesssim6$ Lunar masses (0.03~$M_{\oplus}$), indicating rapid evolution once disk clearing begins. We find a positive correlation between $M_{\rm dust}$ and $M_{\ast}$, and present the first evidence for a positive correlation between $M_{\rm gas}$ and $M_{\ast}$, which may explain the dependence of giant planet frequency on host star mass. The mean dust mass in Lupus is 3$\times$ higher than in Upper Sco, while the dust mass distributions in Lupus and Taurus are statistically indistinguishable. Most detected disks have $M_{\rm gas}\lesssim1~M_{\rm Jup}$ and gas-to-dust ratios $<100$, assuming ISM-like $\rm {[CO]/[H_2]}$ abundance; unless CO is very depleted, the inferred gas depletion indicates that planet formation is well underway by a few Myr and may explain the unexpected prevalence of super-Earths in the exoplanet population.


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# The Greta Effect

Bloomberg is reporting that Germany and Sweden are seeing a trend of reduced domestic air travel and greater rail usage. In Germany, intercontinental air traffic is up 2% year-over-year and international European traffic is down 2%, but domestic traffic has crashed in the last few months and is down 12% now. In Sweden, domestic air traffic is down 11%.

The Greta effect

Greta Thunberg famously crossed the Atlantic by sailboat to avoid personally contributing to greenhouse gas emissions. But she’s fairly practical about alternatives and said right out that she travels in such conditions to highlight how difficult complete decarbonization is. She is also very insistent on the fact that while changes in behavior are nice, collective political action is still needed.

Moreover, the young (as in, younger than me) Greens I meet in Germany are themselves practical as well. The more committed might take a train to France or Italy, but there’s not much interest in back-to-the-land 1960s communes, degrowth, or political revolution in the sense of the socialists and anarchists. Nor have I seen anti-nuclear sentiments recently – the one anti-nuclear sign I saw at the September 20th climate march, which had 100,000 people in attendance, was held by a pensioner and someone who looked 40, whereas the median age at the rally looked like 20.

It’s relatively easy to change travel behavior to avoid domestic flying in Germany as well as Sweden. Domestic rail travel pain in Germany means hourly Hamburg-Munich and Berlin-Stuttgart trains take 5:40 each. International rail travel pain means Berlin-Paris trains take 8:11 with a short transfer that I don’t trust DB or SNCF to meet. Domestic trains only get this long if many transfers are needed, in which case the main competition to the train is the car rather than the airplane, or if one needs to travel between Umeå (population 123,000) and central or southern Sweden. It’s thus likely that the shift in travel pattern reflects a change in consumer desires to avoid polluting – other explanations, such as the grounding of the 737 MAX, would equally affect domestic and European air travel.

Upcoming carbon taxation

Germany has been planning climate legislation for years, but the September 20th protest created a lot of pressure on the government to enact an aggressive package. A carbon tax will begin at €25/t-CO2 in 2021 and rise to €55/t by 2025, where the original plan was to only go up to €35/t. Sweden has had a carbon tax going back to 1991; starting in 2014, the Löfven cabinet has hiked the tax on industry to match the tax on transportation, both currently at €114/t. The effects on the German economy are to be seen, but in Sweden, economic growth has been healthy throughout this period, ahead of any not-newly-industrialized developed country save Australia (although the differences near the top are small).

In addition to the German carbon tax package, the EU is planning to levy a carbon tax on jet fuel for internal flights; so far, international emissions, including international aviation and shipping, are not subject to carbon tax. A leaked report suggests the EU is considering a tax of €330 per 1,000 liters of jet fuel, which corresponds to a hefty €130/t-CO2, the high figure coming from the fact that a ton of CO2 emitted at high altitude causes more global warming than one emitted at ground level. A very fuel-efficient plane like the A320neo consumes 2.25 liters per 100 seat-km on a 1,200 km flight, raising fares on a full flight by €9.06, which is not a game changer but is noticeable at low-cost carrier rates.

Planning for busier trains

The upshot is that demand for flights in Europe is likely to go down, shifting toward rail. The article linked above about the Greta effect says that DB expects its intercity rail traffic to double to 260 million passengers a year by 2040. The article makes no mention of which further investments in intercity rail DB is assuming, but a virtuous cycle is likely: higher ridership justifies more investment, and faster and more convenient trains attract higher ridership.

Of note, the weakness of international rail in Europe points to international connections as an investment priority. In Sweden, trains from Stockholm are fast toward Gothenburg and Malmö, averaging almost 140 km/h, and there are unfunded plans for high-speed rail connecting the three largest cities. However, Stockholm-Oslo trains are quite slow (about 6 hours for what looks like 500 km), even though Oslo is bigger than Gothenburg and Malmö and there are extensive economic and cultural connections between the Nordic countries. The Greens have called for Stockholm-Oslo high-speed rail, and the government should work with Norway on establishing such a line.

In Germany, the situation is different. London and Paris are vast cities, and Paris is within reasonable high-speed rail distance of most of Germany, with good connections on the French side and poor ones on the German one. Trains between Paris and Frankfurt take about 3:48, of which 1:47 is between Paris and Saarbrücken on the German side of the border, a distance of 380 km, and then 2:00 is between Saarbrücken and Frankfurt, a distance of about 200 km by rail and 160 by air. In Belgium, the existing high-speed line east of Brussels is compromised to the point of being slower between Brussels and Liège or Aachen than legacy lines like Stockholm-Gothenburg or London-Manchester.

The reason the map of the high-speed rail I think Germany should build is heavy on international connections is mostly that Europe is gradually building thicker international economic and social connections. However, a future with more expensive air travel and a consumer taste for greener ways of travel does not change the basic picture, and makes it more urgent.

(Map legend: blue is existing or under-construction lines, red is lines that are either in planning or not even in planning but should be built.)

Speed and capacity

DB’s forecast for 260 million annual rail travelers argues in favor of building more capacity. However, in no way does this conflict with building a dedicated high-speed rail network for Germany. On the contrary, the bypasses providing relief to congested lines are already planned to be high-speed: this was the case for the Tokaido Shinkansen and LGV Sud-Est decades ago, and this is now the case for HS2 and the planned Frankfurt-Mannheim express connection.

A largely dedicated network for high-speed passenger rail, with freight using the legacy lines, improves intercity rail reliability, allowing average speeds to rise to be closer to their theoretical technical maximums. Average speeds of 250 km/h on a few lines are plausible, as on Paris-Strasbourg or Madrid-Barcelona. Moreover, through-tunnels enabling intercity trains to run through Frankfurt and possibly Munich without reversing direction facilitate planning high-speed rail as a separate system. Timed connections with regional trains remain important, but critical trunks like Frankfurt-Cologne and Berlin-Hanover can run very frequently.

The schedule I tried writing for the above map in which domestic city pairs mostly run every half hour all day, interlining on a few trunks, assumes ridership of about 250 million. This is not the same as DB’s forecast of 260 million: this counts only high-speed rail riders, and assumes the average trip is 350 km long. To get from DB’s forecast to 87.5 billion p-km per year requires the virtuous cycle of higher ridership and more investment to work over time, but this is plausible given high levels of investment.

Green convenience

When Greta talks about systemic solutions, she understands that it’s important to make it easier to live a comfortable life without greenhouse gas emissions and harder to live one with high emissions. There are many aspects to green convenience: carbon-free electricity (largely achieved in Sweden but not in Germany), pedestrian- and bike-friendly streets, urban and periurban public transport, intercity and freight rail, passive solar design, urban density, carbon-free industrial power generation.

In every case, it’s important to seize upon any social, economic, or political trend that facilitates the green option. If people want to live in big central cities, then governments should make it easy to build housing there so that more people can enjoy the low-carbon wealth of Munich or Stockholm rather than live in cheap declining rural areas and drive. If people support solar power, then governments should leverage its political popularity and subsidize it to decarbonize electricity.

In the case of intercity transportation, a shift in taste toward intercity rail is a cause for celebration. Europe is full of intercity trunk lines ranging from ones that scream “build me now” no matter what (HS2, completing Berlin-Munich, etc.) to speculative ones. Any positive shift toward rail justifies adding ever more marginal intercity rail lines to the network. Perhaps if the network I mapped was justified before the Greta effect, after the Greta effect the most marginal parts of the network (like Stuttgart-Würzburg) are on more solid footing, while unmapped marginal lines like Munich-Prague or even Bremen-Oldenburg-Groningen become plausible.

But celebration does not mean idleness. Climate change is a systemic issue. The state must plan ahead, using the shift toward rail to plan further investments now so that they open in the 2020s and early 30s. This way, the rail network will meet near- and medium-term growth in demand, while stimulating long-term growth, to be satisfied through future investment, paid by taxes on the richer Germany of the 2030s. Good transit activists should take a page from Greta’s refusal to treat good news as grounds for letting up, and demand intensive investment in Europe’s rail network to ensure that green travel will be more convenient, featuring higher speeds rather than more sitting on luggage in the corridors of full trains.

# The Brooklyn Bus Redesign is Out!

Marron just published my and Eric Goldwyn’s Brooklyn bus redesign proposal (with many thanks to Juliet Eldred for doing the graphics and design). The substance isn’t really changed from what we discussed last year. The delay in publication has had a few causes, of which I believe the biggest is that I completely missed that the links to many of the references in the lit review were dead and thus could not be typeset.

Instead of retyping an old blog post, I want to emphasize a few things that have come up in the last year. Some are specific to New York, others more general within the US. The idea of a bus redesign, introduced to the American discourse by Jarrett at the beginning of this decade, has gotten steadily more popular, and New York is beginning its own process, starting with the Bronx; in that context, it’s worthwhile pointing out specifics that Eric and I have learned from the Brooklyn process.

The redesign is a process, not a one-and-done program

Cities change. The point of a bus redesign is to let the bus network reflect the city of today and not that of when bus routes were set, typically when the streetcars were removed in the postwar era. The upshot is that the city can expect to change in the future, which means further bus redesigns may be necessary.

Instead of letting bus networks drift away from serving the city as is and doing a big redesign once in a generation, cities should change buses on an ongoing basis. American transit agencies are learning the principles of bus redesign this decade. They can and should use these principles for forward planning, tweaking bus routes as needed. Any of the following changes can trigger small changes in bus service:

• New development
• Shifts in commuting patterns even without new development
• Changes in traffic patterns
• Changes in the urban rail network
• Long-term changes in driver labor, maintenance, etc.
• Changes in bus technology, such as ride quality, dispatching, or pollution levels

In New York, the biggest ongoing change is probably the urban rail network. There are no subway extensions planned for Brooklyn, but there is expansion of subway accessibility, which changes the optimal bus network since some buses, like the B25 and B63, have no reason to exist if the subway lines they parallel are made accessible. There has been extensive activism about priorities here. To its credit, the MTA is accelerating accessibility retrofits, even though construction costs are extremely high.

New York’s current redesign process is flawed

Eric and I have heard negative feedback from various people involved in the process. Some are planners. One is a community activist, enough of a railfan and busfan not to NIMBY changes for the sake of NIMBYism, but nonetheless disaffected with how the Bronx redesign went.

As far as I can tell, the problem with the current process is that it’s too timid. In the Bronx, this timidity is understandable. The borough’s bus network is mostly good enough. The most important change in the Bronx is to speed up the buses through off-board fare collection, stop consolidation, bus lanes on main streets, and conditional signal priority, and plug the extra speed into higher frequency.

The MTA treats it as part of a separate process – select bus service (“SBS”) – and even though planning these two aspects separately is workable, the MTA does not understand that they are related and that speedups provide crucial resources for higher frequency. The problem here is with operating cost estimation. Like the other American agencies where I’ve asked, the MTA assumes bus costs scale with service-km, and thus higher speeds don’t change frequency. In reality, bus costs, dominated by driver wages, scale with service-hours. Higher speeds can be plugged one-to-one into higher frequency. In Brooklyn, only 30% of the benefits we estimate come from changing the network, and the other 70% come from speeding up the buses.

But Brooklyn is not the Bronx. The Bronx is largely good enough, in ways Brooklyn isn’t. Brooklyn is not terrible, but the bus network has too many circuitous or duplicative routes. Eric and I have consolidated about 530 km of bus route down to 350, without any of the coverage vs. ridership tradeoffs common to areas with less isotropic population density than Brooklyn. The MTA needs to be bolder in Brooklyn, and even bolder than that in Queens, if the redesign is to succeed.

The 14th Street bus lane

Eric and I encountered some political resistance to the idea of mass installation of bus lanes. Local interests listen to people with local connections, who are usually drivers. Transit riders are disproportionately riding to city center jobs, and have citywide rather than local political identities. When I went to an Open New York meeting, people began with a round of introductions in which people say their names and where they live, and the about 20 attendees represented maybe 15 different city neighborhoods. The upshot is that like Open New York’s mission of building more housing, the mission of diverting scarce street space from drivers to bus riders is best done on a citywide rather than street-by-street basis.

There is some hope of such a transformation happening. The bus lane on 14th Street survived a nuisance lawsuit, and ridership rose 17% almost immediately after it opened. The success is stark enough that a citywide increase in installation is plausible. City council speaker Corey Johnson promised to install 48 km of bus lane per year were he to be elected mayor, which is too passive but could do some good on the busiest routes.

# Quick Note: What is Culture, Anyway?

Six weeks ago, I talked about the Anglosphere in context of its high construction costs, especially recently. In comes Bella Wang, and in a much greater generality, asserts,

In the context of transportation, there are some empirical observations from construction cost and mode share data:

• American transit usage underperforms any other first-world standard
• Anglosphere construction costs are very high
• Ex-colonies in the third world have very high construction costs

We can take all three observations to be matters of culture, but really culture is a measure of ignorance. It’s easy to list so many US-rest-of-world cultural differences, and still possible to list Anglosphere-rest-of-world differences that cover Singapore. But the question, which of them are relevant and which aren’t?, is still critical.

Separately, there’s the question, how deep is a specific cultural attribute? The example I want to zoom in on is the issue of hyperlocalism and too many stakeholders. In Brooks-Liscow, it’s identified as a key contributing factor to rising highway construction costs in the US since the 1960s (“citizen voice”) alongside rising incomes. In addition, one expert Eric and I talked to mentioned the multiplicity of stakeholders, as well as many other issues, not all of which I think are relevant.

From one angle, hyperlocalism goes very deep in American culture. Some of it is relatively recent, coming from the white middle class’s desire to maintain local control as the only way to legally prevent integration. Some of it is older – New England had a lot of local empowerment in the 18th and 19th centuries, and unlike in Europe, local elites were viewed as leaders who brought freedom rather than as the main obstacles to freedom.

But from another angle, the specific mechanism through which hyperlocalism acts is not that deep. The local gadfly who launches nuisance lawsuits against everything is a figure of derision; the politician who cuts through the red tape and knocks some heads together and gets things done is a figure of worship and a prime candidate for higher office. If anything, the reason things do not get done in the United States is that politicians prefer to play it safe and knock heads together on low-risk, low-reward projects, hence for example Andrew Cuomo’s proposal for a LaGuardia air train that goes the wrong way but avoids a NIMBY fight from 20 years ago.

The example of Cuomo’s air train, in turn, introduces another attribute: do-nothing politicians. That’s a fairly American problem – other high-cost countries, like Britain and Canada, have politicians that build extravagant projects at high cost, but those projects (HS2, Ontario Line, etc.) are actually useful. Is it a result of an American legal regime that favors the state against the individual and therefore cannot guarantee security of property unless the government credibly pledges to be slow and stupid? Or is it a contingent effect of a handful of governors being slow and stupid in 2019, which may change if someone more competent is elected in the future?

The ultimate question is “can anything get better?”. There’s a lot of evidence in both directions when it comes to American construction costs; when it comes to transit usage in the vast majority of the United States where there is no public transit, the same is true but right now I believe the evidence is stronger on the “no” side.

# What I Mean When I Say Cities Have no Transit

On social media and various forums, I have an expression for a variety of cities: “it has no public transportation.” This concerns just about the entire United States excluding a handful of cities like New York, San Francisco, and Chicago; Los Angeles notably is not among these handful, but has no public transportation, and neither do any cities in the South or the Midwest except Chicago. I want to talk a bit more about what I mean by this. I obviously don’t mean that literally there is no scheduled public transportation in these cities; I’ve taken these non-existent transit systems, in Los Angeles on a visit and in Providence when I lived there. But I mean that there’s something about such places distinguishing them from the bad-but-existing transit category of Boston, Chicago, Nice, etc.

Whatever you’re doing isn’t working

Let’s use an 8% cutoff for trips to work. This number is fully motivated reasoning: the metro area (MSA, not CSA) of Philadelphia is just above this cutoff, and I would not say it has no public transportation, at least not in the current state of the system. Bad, yes, but it exists. I may be missing some areas, but I don’t think I am: the list of American metro areas that meet that cutoff is New York, San Francisco, Boston, Washington, Chicago, Fairfield County, Seattle, Kitsap County, Philadelphia, Honolulu. 70% of American transit commuters live in one of these MSAs. Go down to 6% and you also get Portland and Baltimore, adding about 2.5% of US transit commuters.

Nor are things improving. Some parts of the US are seeing rising mode shares. The most notable is Seattle, which is serious about permitting urban housing, and has tunneling construction costs that would only get Europeans fired rather than simply not existing in democratic Continental Europe. But other cities that occasionally win accolades from American urbanists for investing in public transportation just aren’t cutting it. In the 2006-17 period, chosen because that’s what the ACS makes available, Denver went from 4.6% to 4.4%, Los Angeles from 6.1% to 4.8%, and Portland from 6.4% to 6.3%; in the praised-by-urbanists set, only Minneapolis went up, from 4% to 4.8%.

Let’s unpack what this means: whatever Los Angeles has been doing in the last 10+ years has gotten its mode share down – and that’s without counting the fact that the Inland Empire, officially a separate metro area, is growing much faster and has an even lower mode share, as people drive further and further from jobs to qualify for a mortgage. Portland and Denver have done a lot of supposedly good things with their light rail networks, but are standing still. Portland’s stagnation goes back at least to 1980, while Vancouver has built SkyTrain, a high-rise downtown, and Metrotown, and at 20% has a higher (and rising) mode share than any American metro area save New York.

Tabula rasa

When a metro area has 2-3% mode share, it’s best to treat it as tabula rasa. Yes, there are people who ride the buses and trains today, but so few that the advantages of from-scratch design are usually greater than the disadvantage coming from the risk to current ridership. The 2-3% figure really depends on the situation – I don’t want to give it as an ironclad figure.

Suburbs of very large cities (read: New York) approaching 10% may still be best treated the same way: commuter rail systems like the LIRR are really shuttles that extend auto-oriented suburbia into the city rather than the reverse. Sadly, where I say such suburbs have no transit as a positive statement, an MBTA general manager said “commuter rail is not public transit” as a normative statement.

The situations of extremely low-mode share metro areas and low-mode share suburbs are not exactly the same. For one, existing ridership is higher on Long Island than in Cleveland or St. Louis so there’s more risk if (for example) supernumerary workers go on strike to fight efficiency improvements, but the reward is much greater. We know how to squeeze high ridership out of regional rail in the suburbs, even low-density ones, since the city has so many jobs in the center. Moreover, we know which ready sources of ridership are suppressed by current operating patterns: working-class reverse-commuters, people who work non-traditional hours regardless of class, and peak-direction commuters getting off short of city center.

The tabula rasa concept notably does not mean the infrastructure doesn’t exist. Los Angeles has the physical infrastructure of a rail network. Long Island and Westchester have many rail lines pointing toward Manhattan. However, the operating patterns and development are deficient and little to no accommodation should be made for them. In the suburbs of New York and a handful of other American cities this concerns premium fares, low off-peak frequency, and lack of integration with local buses. In American metro areas with low overall ridership this concerns weak city centers, lack of TOD even when it could succeed (for example in Los Angeles and San Diego), local political systems that view transit as an excuse to get federal funds for other things such as road repaving, and, as in suburbia, low off-peak frequency. The problems vary, but the fact that there are severe problems remains.

The other element of tabula rasa is social. There is almost never any knowledge base in those areas about how good transit works, because people who’ve only lived there have by definition not regularly used even bad-but-existing public transport. Whatever local activists of all stripes have been doing in Los Angeles is not working. Understanding why from them can be valuable, for the same reason I talk to planners at poorly-run agencies like the MTA and the MBTA to understand what’s wrong, but all local practices should be considered suspect unless corroborated in an area with at least decent public transportation.

On giving offense

The people who complain about my use of “no transit” to refer to the vast majority of the United States are not making a semantic nitpick or asking for clarification. They specifically complain I give offense by erasing 2-3% of the population of Cleveland and St. Louis, or 1% of the population of Kansas City. (I name these cities and not 6% Portland because that’s what people have complained about to me.)

So let’s unpack what this means. I point out that in the vast majority of the United States, excepting a handful of regions all of which are politically stereotyped as Not Real America partly because they have public transit, has buses and trains that are so useless they might as well not exist. I point out that this remains the case despite extensive construction in many cities – Dallas has 150 km of light rail, which is respectable for a city of its size, Denver keeps expanding its network and has something resembling frequent regional rail, and so on. The problem is that I do not conveniently blame this on a political faction of others, be it Republicans, unions, moderates, drivers, or whoever. I genuinely think it’s the fault of everyone who’s had any amount of power, and this includes community organizations that keep identifying as always losing even when they repeatedly succeed in blocking changes they dislike.

This is American culture. Even the denigration of New York and other cities where there is public transportation is part of that culture; there are certain aspects of San Francisco, Boston, and Philadelphia that are useful for other parts of the US to emulate. But accepting that requires understanding that there is to a good approximation no contribution coming from no-transit cities (and this again includes Portland and Los Angeles, it’s not just Cleveland or Dallas).

Part of the problem is that the US defines itself so much around cars and car culture that the presence of public transportation is enough to make something feel not really American. The result is that any exhortation to learn from places with trains with decent ridership is bound to offend; I might as well tell Americans to move to Tokyo and learn Japanese and never come back to the West. But sadly for Americans, reality can be offensive. The culture of Real America has to change, at least when it comes to how to treat transportation and cities.

# I’m Giving a Talk About Construction Costs Tomorrow

By popular demand, I’m giving the talk I gave 2 weeks ago at NYU, again. The database will be revised slightly to include more examples (like Ukraine, which I added between when I gave the talk and when I blogged about it), and I may switch around a few things, but it should be similar to what I already said.

Where? Halyards in Brooklyn at 3rd Avenue and 6th Street, near the 4th Avenue/9th Street subway stop where the F/G and R intersect.

When? Monday December 2nd at 9 pm, for an hour.

Do I need to RSVP? No.

Will there be food? To some extent – the bar has minimal selection, although what it does have on the menu seems better for the price than most American bar food (which, to be fair, is like saying “better public transportation than Los Angeles”).

# Little Things That Matter: Jerk

When you ride a subway train, and the train decelerates to its station, you feel your body pulled forward, and your muscles tense to adjust, but then when the train reaches a sudden stop, you are suddenly flung backward, since you are no longer decelerating, but your muscles take time to relax and stop fighting a braking that no longer exist. This effect is called jerk, and is defined to be change in acceleration, just as acceleration is change in speed and speed is change in position. Controlling jerk is crucial for a smooth railway ride. Unfortunately, American mainline rail is not good at this, leading to noticeable jolts by passengers even though speed limits on curves and acceleration rates are very conservative.

This is particularly important for speeding up mainline trains around New York and other legacy cities in the US, like Boston. Speeding up the slowest segments is more important than speeding up the fastest ones; my schedules for New York-New Haven trains, cutting trip times from 2:09 to 1:24, save 4 minutes between Grand Central and 59th Street just through avoiding slowdowns in the interlocking. The interlocking is slow because the switches have very conservative speed limits relative to curve radius (that is, lateral acceleration), which in turn is because they are not designed with good lateral jerk control. The good news is that replacing the necessary infrastructure is not so onerous, provided the railroads know what they need to do and avoid running heavy diesel locomotives on delicate infrastructure.

Spirals and jerk

In practice, the worst jerk is usually not forward or backward, except in the last fraction of a second at the end of acceleration. This is because it takes about a second for train motors to rev up, which controls jerk during acceleration. Rather, the worst is sideways, because it is possible to design curves that transition abruptly from straight track, on which there is no lateral acceleration, to curved track, on which there is, in the form of centrifugal force centripetal force.

To reduce jerk, the transition from straight track to a circular arc is done gradually. There are a number of usable transition curve (see Romain Bosquet’s thesis, PDF-p. 36), but the most common by far is called the clothoid, which has the property of having constant change in curvature per unit of arc length – that is, constant jerk. Different countries have different standards for how long the clothoid should be, that is what the maximum lateral jerk is. Per Martin Lindahl’s thesis, the limit in Sweden is 55 mm/s (PDF-p. 30) and that in Germany is 69.44 mm/s (PDF-p. 38), both measured in units of cant deficiency; in SI units, this is 0.367 m/s^3 and 0.463 m/s^3 respectively. In France, the regular limit is 50 mm/s (Bosquet’s thesis, PDF-p. 35), that is 0.333 m/s^2, but it is specifically waived in turnouts.

Track switches are somehow accepted as sites of very high jerk. A presentation about various technical limits in France notes on p. 106 that in switches (“appareils de voie” or “aiguilles” or “aiguillages,” depending on source, just like “switch” vs. “turnout” in English), the jerk can be increased to 100 and even 125 mm/s. On p. 107 it even asserts that in exceptional circumstances, abrupt change in cant deficiency of up to 50 mm on main track and 100 on the diverging direction on a switch is allowed; see also PDF-pp. 13-15 of a pan-European presentation. Abrupt changes are not good for passengers, but will not derail a train.

Second derivative control, that is acceleration and cant deficiency, can be done using calculus and trigonometry tools. Third derivative control, that is clothoids and jerk, requires numerical calculations, but fortunately they are approximated well by pretending the clothoid is half straight line, half circular arc, with the length determined by the maximum jerk. Working from first principles, it’s possible to figure out that at typical turnout needs – e.g. move a train from one track to a parallel track 4 meters away – the clothoid is far longer than the curve itself, and at 50 mm/s jerk and 150 mm cant deficiency it’s not even possible to hit a curve radius of 250 meters.

Turnouts are inherently compromises. The question is just where to compromise. Here, for example, is a French turnout design, in two forms: 0.11 and 0.085. The numbers denoting the tangent of the angle at the frog, and the radius is proportional to the inverse square of the number, thus the speed is proportional to the inverse of the number. The sharper turnout, the 0.11, has a radius of 281 meters, a maximum speed of 50 km/h, and a total length of 26 meters from point to frog (“lead” in US usage), of which the clothoid curve (“point”) takes up 11, to limit jerk to 125 mm/s at a cant deficiency of 100 mm. The 0.085 turnout has a radius of 485 meters, a maximum speed of 65 km/h, a lead of about 38 meters, and a point of about 14.5 meters.

In Germany, turnouts have somewhat independent numbers and radii – some have shorter leads than others. The numbers are the inverse of those of France, so what France calls 0.11, Germany calls 1:9, but at the end of the day, the curve radius is the important part, with a cant deficiency of 100 mm. A higher cant deficiency may be desirable, but lengthening the point requires almost as much space as just increasing the curve radius, so might as well stick with the more comfortable limits.

American turnouts look similar to French or German ones, at first glance. I’ve seen a number of different designs; here’s one by CSX, on PDF-pp. 22 (#8) and 24 (#10), the numbers being very roughly comparable to German ones and inverses of French ones. CSX’s #10 has a curve radius of 779.39′, or 238 meters, and a lead of 24 meters, both numbers slightly tighter than the French 0.11. The radius is proportional to the square of the number, and so speed is proportional to the number.

However, the cant deficiency is just 50 mm. The point is not always curved; Amtrak’s low-number switches are not, so the change in cant deficiency is abrupt. Judging by what I experience every time I take a train between New York and New Haven, Metro-North’s switches have abrupt change in cant deficiency even on the mainline. The recommended standards by AREMA involve a curved point, but the point is still much shorter than in France (19.5′, or just under 6 meters, on a #12), so a 125 mm/s jerk only gets one up to about 62 mm cant deficiency.

The reason for this is that European turnouts are curved through the frog, whereas American ones are always straight at the frog. Extremely heavy American freight trains do not interact well with curved frogs and long points.

One might ask, why bother with such turnout design on rail segments that never see a heavy freight locomotive or 130-ton freight car? And on segments that do see the odd freight locomotive, like the approaches to Grand Central and Penn Station with the rare dual-mode locomotive, why not kick out anything that doesn’t interact well with advanced track design? Making a handful of passengers transfer would save around 4 minutes of trip time on the last mile into Grand Central alone for everyone else, not to mention time savings farther up the line.

# Fare Regulations

Public transportation companies may have the ability to raise fares arbitrarily based on market demands, for examples British buses outside London and American freight railroads. Or they may be subject to regulations capping the fare, for example Japanese railroads. Mixed systems exist as well, such as British rail fares. In Britain, the privatized, mostly deregulated approach is so commonly accepted that a Conservative recently called Labour dangerous socialists for proposing municipalizing bus systems, as in such socialist states as the US, Japan, Germany, etc. In reality, in the case of rail specifically (and perhaps buses as well), there’s a theoretical case with some empirical backing for why reasonable fare caps as in Japan can lead to more investment and more capacity, whereas wholly unregulated fares lead to hoarding and capacity cuts to create shortages.

The model

I’m stealing the economic model for this post from Paul Krugman, who used it to explain the California blackouts of 2000-1. The demand curve is inelastic: the demand is 1,000 units at $20/unit, decreasing to 900 units at$1,000/unit, at which point the curve goes flat. The supply curve is a constant $20/unit, but the market is oligopolistic (say, there are very high barriers to entry because building your own power plant is hard), and there are 5 producers, each with 200 units. If the price is regulated at$20/unit, each producer will supply 200 units. If the price is unregulated, then each producer alone gets an incentive to hold back production, since 100*1000 > 200*20, and then production will be curtailed to 900 units.

The model is simplified in a number of ways: real supply curves slope up; the part about demand going flat at 900 units is unrealistic and exists purely to avoid dealing with optimizing where at 800-something units each producer has an incentive to go back to producing more; capacity constraints involve escalating production costs rather than a God-given restriction on the number of suppliers and their capacity. But with all these caveats, it fits markets that have the following characteristics:

• There are steep barriers to entry, for example if large amounts of capital are required to enter (to build a power plant, set up a rail operating company, etc.).
• Demand is highly inelastic.
• Adding new capacity is expensive.

The issue of capacity

In rail, we can start plugging real numbers for both demand elasticity and the cost of new capacity.

In the above model the price elasticity is -0.0244 in the 900-1,000 units range, which is ridiculously inelastic, on purpose so as to highlight how the model works. TCRP Report 95 says the elasticity in a number of large cities studied is about -0.18, and a VTPI review in a mixture of cities and circumstances (peak vs. off-peak, bus vs. rail, etc.) asserts a short-term average of about -0.3. Unregulated fares will lead to supply reductions if the elasticity times the number of producers is more than -1 (or less than 1 if you flip signs); if no producer has <18% of the market, there will be supply restrictions under unregulated fares, just as a monopolist will hold back supply and raise fares if demand is inelastic.

The cost of new capacity of course depends on the line and the characteristics of competition between different railroads. It’s higher in Japan, where separate railroads run their own lines and trains, than in Britain, where different companies franchise to run trains on the same tracks. But even in Britain, getting a franchise requires a commitment to running service for many years. The significance of this is that the long-run public transport ridership elasticity with respect to fare is more elastic (VTPI recommends a range of -0.6 to -0.9), with a few estimates even going below -1.

For the purposes of this section, we do not distinguish capital from operating costs. Thus, the cost of new capacity is not given in units of capital costs, but in units of operating costs: if increasing service by 1% raises operating expenses by 2% counting the extra investment required, then we say the supply elasticity is 2. Note that supply curves slope up so the elasticity is always positive, but the elasticity can be below 1, for example if economies of scale are more important than the need to invest in new capacity.

Set the following variables: u is quantity of service, r is total revenue (thus, fare is r/u), c is total costs. The railroad is assumed profitable, so r > c. We are interested in the change in profit based on quantity of service, i.e.

$\frac{dr}{du} - \frac{dc}{du}.$

The important thing to note is that price controls keep dr/du higher in an oligopoly (but not in a competitive environment, like housing – a single landlord can’t meaningfully create a housing shortage). With price controls, we get

$\frac{dr}{du} = \frac{r}{u} = \mbox{constant fare}$

whereas without price controls, with elasticity $e_{d} < 0$, we get

$\frac{dr}{du} = \frac{r}{u} + \frac{r}{ue_{d}} = \frac{r}{u}(1 + 1/e_{d}).$

And likewise, with supply elasticity $e_{s} > 0$, we get

$\frac{dc}{du} = \frac{c}{ue_{s}}.$

Note, moreover, that price controls as construed in Japan let operating companies recover profits, letting them raise prices if they invest in more capacity, so that dr/du is actually higher than r/u.

The real world

I do not know to what extent the lack of fare regulation on many British trains contributes to capacity shortages. However, there is some evidence that the same situation is holding back investment in the United States, on Amtrak. Amtrak is a monopolist facing some fare regulations, for example congressional rules limiting the spread between the lowest and highest fares on a given train, but within its ability to set its own capacity in the medium run, it has relatively free hand, and in fact a strong incentive to maximize fares, in order to subsidize money-losing trains outside the Northeast Corridor.

Amtrak generally runs the trains it has on the Northeast Corridor, without explicitly holding back on capacity. However, this is in an environment with very low utilization rates. There are 20 Acela trainsets, but only 16 run in service at a given time, giving them the moniker “hangar queens.” There is no real interest within Amtrak at raising speed just enough to be able to run consistent service intervals, for example hourly with two trainsets coupled to form a 16-car train south of New York. Nor is there any interest in making small investments to permit such long trainsets to run – most Acela stops from New York to the south have platforms long or almost long enough for such trains, but the rest need to be lengthened, within right-of-way so that the cost is positive but low.

In the future, capacity cliffs may prove serious enough to stymie American passenger rail development. Right now the main obstacle are Amtrak itself and obstructive commuter railroads such as Metro-North, but assuming competent, profit-maximizing investment plans, it is not so expensive to invest in capacity and speed so as to permit around 4 long high-speed trains per hour north of New York (or even New Haven) and 6 south of it. But then the next few trains per hour require further bypasses, for example four-tracking most of the Providence Line. High supply elasticity – let’s say around 2 – is plausible. Then eventually a dedicated pathway to intercity trains through New York becomes necessary, raising supply elasticity even higher. In an environment with uncapped, profit-maximizing fares, a rational Amtrak management may well just keep what it has and jack up prices rather than build more capacity.

# I Gave a Talk About Transportation in Connecticut

I gave a second talk this week about transportation, this time at Hartford Station, concerning the plans for Connecticut transportation. The starting point is Governor Lamont’s $21 billion plan for investment, including both expansion and repairs (read: the State of Good Repair black hole), of which$14 billion is highways, $6.2 billion is rail, and$450 million is buses. But most of the talk concerns what Connecticut should be doing, rather than the specifics of Lamont’s plan.

Here are my slides. The talk itself took around 40-45 minutes out of a nearly 2-hour meeting, so it was designed around taking many questions, and around further explanations. Something I didn’t put in the slides but explained verbally is how easy the modern track renewal process is. Nowadays, there are machines that use no infrastructure except the tracks themselves, running on the tracks at very low speed (slower than walking) and systematically replacing the rails, ties, and ballast. They can also regrade the tracks’ superelevation angle independently of the drainage angle, changing the tracks’ cant as they go. The upshot is that increasing the cant on tracks is almost cost-free, and would enable large increases in train speed on both regional and intercity trains.

Other technology that has negative cost in the future is getting higher-performance EMUs than the current equipment. The current trains are obsolete technology, built around superseded federal regulations. There’s no point in getting more of the same. They’re okay to run until end of life, but new purchases should involve electrification and modern European EMUs. Whereas infrastructure costs are rising (see here and here), technology costs are falling in real terms. The fall in train costs is not so quick as that of computer costs, but still the rolling stock factories are designed around making products for the 2020s, not the 1990s, and retooling them for older technology costs extra.

Hence my slogan from the talk: better things are possible, on a budget.

One question I was asked at the talk that I didn’t have an answer to was, why is construction in Connecticut so expensive? Plans for infill stations are budgeted extravagantly, ranging between $50 million and$100 million without any special construction difficulties. Boston builds infill stations (counting high-platform upgrades as infill since the preexisting stations have no facilities) for \$20-30 million counting various hidden costs (e.g. regular MBTA employees, like project managers, count as operating and not capital costs even if they only work on capital costs); Berlin does for €10-20 million.

After the talk, Roger Senserrich explained to me (and a planner at the MBTA confirmed to me) that in Connecticut there’s no in-house design at all. Massachusetts has a mix of in-house design review, with the team stymied by uncompetitive wages making hiring and retention difficult, and outsourcing work to consultants. CDOT exclusively outsources to consultants, and has no in-house expertise to evaluate whether the contracts are fair or whether it’s being overcharged.

# I Gave a Talk About Construction Costs

Two years ago, I gave a talk at NYU about regional rail, and as promised, uploaded slides the next day for discussion. Yesterday I gave another such talk, about construction costs.

But here there are two things to upload: the slides, and the data table. I’ve been intermittently adding cities to a spreadsheet of various urban rapid transit lines and their construction costs, and by now there is a total of 207 distinct items, ranging from 1 km extensions to 3-figure packages like 200-km GPX and 160-km Delhi Metro phases. The total length of the lines in this database right now is 3610 km, of which 2090 are underground. These are almost exclusively new lines – most of them aren’t even open, and most of the rest opened this decade, so be cautious since much of the cost estimation is ex ante and a number of the soon-to-open line on the list have had serious cost overruns.

I hope people make use of this dataset and the preliminary analysis contained in the slides, and I ask that people look at both, since the slides do have some interpretive notes about confounding variables. One note that I did not include in the slides and explained verbally is what source means in the table: media means I’m drawing costs from popular media, trade means trade media like Railway Gazette, plan means official plans (either ex ante or ex post), wiki means Wikipedia (as always, a reliable source for line length and station count, never cost), measured means I measured line length on Google Earth lacking any alternative. One item, Crossrail, has its tunnel cost coming from a freedom of information request submitted by an alert reader who I will credit upon request; the headline budget is somewhat higher as it includes surface improvements, a common confounder for regional rail projects (the RER E extension, for example, splits its budget about 50/50 between the tunnel and above-ground works).

More detailed analysis is forthcoming, either here or in print.

# Urban Freeways and Rapid Transit

A ride-hailing trip today reminded me of something about freeway travel in cities – namely, it is untethered from the surface street network. Oddly enough, for a different reason this is equally true of rapid transit. The commonality to these two ways of travel is that they change the geography of the city, rather than just extending the range of walking along the usual paths as surface arterial streets and surface transit do.

Rapid transit compression

Rapid transit networks compress distances along the lines, and by the same token magnify distances in orthogonal directions. Manhattan is a good example of how this works: north of Midtown the subway only runs north-south, not east-west, so there are separate East Side and West Side cultures. Moreover, as middle-class gentrifiers are displaced by rising rents coming from even richer gentrifiers, they tend to move along subway lines, and thus people from the Upper West Side and Columbia end up in Washington Heights and Inwood.

The contrast here is with surface transit. Bus networks are far too dense to have the same effect. A citywide bus grid would offer 15 km/h transit in all directions in New York, and a tramway grid like what parts of Berlin have (and what big Eastern European cities like Prague and Budapest have) offers 15-20 km/h transit in all directions. It extends walking, in the sense that the most important throughfares probably get their own routes, or if they don’t they are closely parallel with roads with surface transit.

This is not how rapid transit works. A handful of very strong orthogonal routes can and should get rapid transit, hence the Ringbahn, M2/M6 in Paris, and the under-construction M15 – and by the same token, 125th Street in New York should get a subway extension off of Second Avenue. But that still leaves the city with a wealth of major routes that have no reason to get rapid transit, ever. Most of these are crosstown routes, for example the east-west streets of Manhattan, but in less gridded cities they can just be major streets that don’t quite fit into a regionwide radial metro network.

Rapid transit spikiness

I get a lot of pushback when I talk about this, but rapid transit encourages spiky density. This does not mean that every transit city is spiky and every spiky city is a transit city. Density in Paris within the city is fairly uniform, aided by zoning rules that prohibit high-rises even though many could succeed commercially on top of Métro transfer points or RER stations. In the other direction, some American auto-oriented cities have spiky density near transit, like San Diego’s Mission Valley or Atlanta’s Buckhead, but it’s not big enough a development to permit people to comfortably walk and take transit to all destinations.

Nonetheless, for the most part, rapid transit tends to be associated with spiky development forms, especially if it’s been built more recently and if the interstation is long (as in Vancouver, Singapore, Hong Kong, or Stockholm). This isn’t really how a pedestrian city works: pedestrians have no need for spikiness because they don’t have particular distinguished stations – at most, the corner nodes are distinguished, but that includes all corners, which are placed at far shorter intervals than subway stops.

Freeways as street bypasses

Surface transit promotes urban forms that look like an extended pedestrian city. This is equally true of surface roads designed around car access. The car was originally not supposed to take over the entire city, but merely provide convenient intra-urban transportation at a faster speed than walking. It was originally just a faster, more private, more segregated streetcar. The effect on urbanism was to reduce overall density (as did the streetcars and rapid transit in New York, which used to have inhuman overcrowding levels on the Lower East Side), but not to change the urban form beyond that.

Freeways, like rapid transit, are completely different. This does not mean that they change the city in the same way as rapid transit, just that both operate independently of the usual street grid. Freeways, like rapid transit, compress travel distances along the freeway, and simultaneously lengthen them in all other directions because of the effect of traffic congestion.

Moreover, freeways are different from rapid transit in typical alignments. They are far more land-intensive, which is why they tend to be placed in formerly marginal parts of the city. This can include the waterfront if it is originally industrial and low-value, as it was in midcentury America, rather than a place of high-end residential consumption because of the views.

Interface with the street

How does a surface street transit network interface with either rapid transit or freeways?

With rapid transit, the answer is that surface transit is slow, so it should feed rapid transit using transfers, which may be timed if the trains are not so frequent (say, 15 minutes or worse, as is common on suburban rail branches). Rapid transit should then be constructed to connect with surface transit this way, that is the stations should be at intersections with arterial corridors for bus connections.

With freeways, the answer is that often interface is impossible. San Diego provides a convenient example: there is an arterial road that’s great for buses running northward from city center toward the beachfront neighborhood of Pacific Beach. But there’s also a parallel freeway inland, so drivers mostly use the freeway, and commerce on the north-south arterial is neglected. In contrast, the main east-west arterials feeding the freeway are bustling, and one of them has of the city’s strongest buses. Buses can make stops on these arterials and then express to city center on the freeway, but on the freeway itself the buses are not very efficient since there’s minimal turnover, and chaining a few neighborhoods together on one frequent route is usually not possible.


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# Complexity of finding the pseudoinverse matrix

How many arithmetic operations are required to find a Moore–Penrose pseudoinverse matrix of a arbitrary field?

If the matrix is invertible and complex valued, then it's just the inverse. Finding the inverse takes $O(n^\omega)$ time, where $\omega$ is the matrix multiplication constant. It is Theorem 28.2 in Introduction to Algorithms 3rd Edition.

If the matrix $A$ has linearly independent rows or columns and complex valued, then the pseudoinverse matrix can be computed with $A^*(A A^*)^{-1}$ or $(A A^*)^{-1}A^*$ respectively, where $A^*$ is the conjugate transpose of $A$. In particular, this implies an $O(n^\omega)$ time for finding the pseudoinverse of $A$.

For general matrix, the algorithms I have seen uses QR decomposition or SVD, which seems to take $O(n^3)$ arithmetic operations in the worst case. Is there algorithms that uses fewer operations?

-

First of all, people tend to forget that $\omega$ is an infimum. Whenever we write $O(n^\omega)$, we actually mean for all $\gamma > \omega$, there is an algorithm running in time $O_\gamma(n^\gamma)$.
Keller-Gehrig showed (among else) how to present a matrix $A$ in rank normal form in time $O(n^\omega)$. If $A$ has rank $r$, then a rank normal form of $A$ is $$S \begin{pmatrix} I_r & 0 \\ 0 & 0 \end{pmatrix} T$$ for some invertible $S,T$ of the appropriate dimensions; see also Algebraic Complexity Theory, Proposition 16.13 on page 435.
Rank normal form is similar to the rank decomposition mentioned in the Wikipedia article, $A = XY$ where $X$ has $r$ columns and $Y$ has $r$ rows. Indeed, we can take $X$ to be the first $r$ columns of $S$, and $Y$ to be the first $r$ rows of $T$. Given this decomposition, Wikipedia gives a formula for the pseudoinverse using only Hermitian adjoint, matrix multiplication and matrix inverse. Therefore the pseudoinverse can be computed in time $O(n^\omega)$.
So $O(n^ω)$ holds in general? Can you give me a hint what the "Matrix multiplication constant" $w$ is? – ben Apr 27 at 20:18
We don't know the value of $\omega$. The best upper bound, due to Le Gall, is $\omega < 2.3728639$. It is conjectured that $\omega = 2$. – Yuval Filmus Apr 27 at 21:35


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2d turn-based tile movement

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Now, im definately sure that this is probably a simple equation, but ive never been very good with nested loops. What I want to do is this: think back to games like Shining Force or Vandal Hearts, on the players turn they select a unit and up pops a highlighted portion of the map that the player can move within. So far im going for a simple highlight, ive got an array constructs itself and returns to the main form and writes into a label. But so far it creates a box, which I dont want. Ill show you what I want below. 0,0,0,0,0,0,0,0,0,0 0,0,0,0,3,0,0,0,0,0 0,0,0,3,2,3,0,0,0,0 0,0,3,2,1,2,3,0,0,0 0,3,2,1,X,1,2,3,0,0 0,0,3,2,1,2,3,0,0,0 0,0,0,3,2,3,0,0,0,0 0,0,0,0,3,0,0,0,0,0 0,0,0,0,0,0,0,0,0,0 0,0,0,0,0,0,0,0,0,0 Say I have a 10 by 10 array, X is the current X,Y position of the selected Character. If his movement is 1 then he can move as far as the positions marked "1", if his movement is 2 then he can move as far as the positions marked "2" etc. Any help or points in the right direction here would be greatly appreciated :)

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Since your character can't move diagonally, you probably want what is sometimes called the 'Manhattan distance', that is, the sum of the displacements along the Cardinal axes of the space in question.

Here's some pseudocode that applies this to your situation. I'll assume that the grid is indexed as [x][y], and I'm going to ignore range-checking (you can add that yourself easily enough):
void HighlightCells(int x, int y, int distance){    for (int i = x - distance; i <= x + distance; ++i) {        for (int j = y - distance; j <= y + distance; ++j) {            if (abs(x-i) + abs(y-j) <= distance) {                cells[j].hightlighted = true;            }        }    }}
That should do what you want (assuming I didn't mess anything up).

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if (abs(x-i) + abs(y-j) <= distance)

So ^THATS^ what I was mising , i knew it was simple.

It isnt the entire solution but it is a step in the right direction, ill muck around with it and see if I can straighten it out, that general code returns this (X = 3, Y = 3, Distance = 3) (where "2" is where the user can move to and "X" is where the player started):

0,0,0,2,2,2,2,0,0,0
0,0,2,2,2,2,2,0,0,0
0,2,2,2,2,2,2,0,0,0
2,2,2,X,2,2,2,0,0,0
2,2,2,2,2,2,2,0,0,0
2,2,2,2,2,2,2,0,0,0
2,2,2,2,2,2,2,0,0,0
0,0,0,0,0,0,0,0,0,0
0,0,0,0,0,0,0,0,0,0
0,0,0,0,0,0,0,0,0,0

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Okay ive got most of it working, its just the bottom-right corner thats beeing a nuisance.

This is the code that im using (posX = X Position, posY = Y Position, Mo = Movement, all integers)

(posX = 3, posY = 3, Mo = 3)

0,0,0,2,0,0,0,0,0,0
0,0,2,2,2,0,0,0,0,0
0,2,2,2,2,2,0,0,0,0
2,2,2,X,2,2,2,0,0,0
0,2,2,2,2,2,2,0,0,0
0,0,2,2,2,2,2,0,0,0
0,0,0,2,2,2,2,0,0,0
0,0,0,0,0,0,0,0,0,0
0,0,0,0,0,0,0,0,0,0
0,0,0,0,0,0,0,0,0,0

This code returns that array above, what am I missing?

int[,] arrMove = new int[11,11];

for(int i = (posX - Mo); i <= (posX + Mo); i++)
{
for(int j = (posY - Mo); j <= (posY + Mo); j++)
{
if((posX - i) + (posY - j) <= Mo & (posX + i) - (posY + j) <= Mo)
{
if((posX + j) - (posY + i) <= Mo & (posX - j) + (posY - i) <= Mo)
{
arrMove[i,j] = 2;
}
}
}
}
arrMove[XLoc, YLoc] = 7;//Show Starting Point
return arrMove;

*Edit* Sorry for the lack of tabbing but for some reason both normal spaces and tab-spacers seem to not work...

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You should try it with absolute value, like in my earlier example. I didn't try to figure out what your code would do, but it seems unlikely that it would produce the correct results.

For proper formatting of code you can use [ source ][ /source ], or < pre >< /pre > for small snippets (remove the spaces from the tags).

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Your method is ok if you don't have any obstacles or walls in your map. Once you add those you can't use it anymore, since you suddenly might be able to walk through a wall.

The method i usually use is kind of recursive. You create a second array the same size of the map. Init this distance array to zero. Set the current location of your player to 1.

Now iterate max_walkable_distance times over the array. If an array spot equals the current iteration value check all four connecting tiles if they're walkable for the player. If they are, set the connecting tile value one higher than the current spot. Only do this if the connecting tile has not already a value set.

This method basically walks all possible tiles starting from the current player and will nicely walk around walls.

Some clarification, imagine the main map:

000000000000000000000000000010000000000010A0000000001000000000B011111100000000000000000000000000

Let A be the current player, B be his wanted target position and 1 be unwalkable tiles.

Now lets look at the distance array which will build step by step here:

Start out with the start location:
000000000000000000000000000000000000000000100000000000000000000000000000000000000000000000000000

Next step, all tiles around the player are accessable:
000000000000000000000000000000200000000002120000000000200000000000000000000000000000000000000000

Next step, we hit the first wall to the left and down:
000000000000000000300000000003230000000002123000000003230000000000000000000000000000000000000000

Next step:
000000400000000004340000000003234000000002123400000003234000000000000000000000000000000000000000

Next step, we get around the wall on top:
000005450000000054345000000003234500000002123450000003234500000000000000000000000000000000000000

Of course you stop the iteration once you reach the players max walk count.

And yes, this method is fast enough, unless you use a 1024x1024 map. And even then probably.

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Well I do have a terrain array (Walls, switches, etc), a player array, and this was the selection/movement/attacking array, depending on what action was selected would determine which tiles needed to be highlighted, if you get what im saying.

Now that I think of it, that method will be easier than drawing the array, THEN having to code which tiles are unreachable/unwalkable. Ill work on it and if ive got any problems (I most probably will lol) ill post back here.

Thanks for the nudge in the right direction here Endurion, its been fun and frustrating getting my feet wet in these kinds of calculations :)

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