Proportional and Relative Changes

There is an excellent description of the difference between logarithmic and linear scales in the book "Mathematical Analysis for Economists" by R Allen which I describe here.

Consider the following two sequences of numbers:
[latexpage]

100,  150,  200,  250,    300,     $\ldots$
100,  150,  225,  337.5, 506.25, $\ldots$

the first sequence shows a regular increase of 50 units and the second a regular increase of 50 per cent from one number to the next. On a linear scale, the points representing the first sequence appear as equal distances from each other and those representing the second sequence at increasing distances. If we now take the logarithm to the base 10 of each number we will generate the following two new sequences:

2,  2.176,  2.301,  2.398,  2.477,  $\ldots$
2,  2.176,  2.352,  2.528,  2.704,  $\ldots$

What is striking here is that it is the second sequence that now shows points at equal distances (1.176 apart). It would seem, therefore, that equal distances between points on a linear scale indicate equal absolute changes in the variable and equal distances between points on a logarithmic scale indicate equal proportional changes in the variable. Before taking the logarithm, the second sequence increased by 50% each time, in log form however, it increased by a constant absolute amount of 1.176.

We can show this property quite easily as follows. If $y_1$, $y_2$, $y_3$ and $y_4$ are values shown by points at equal distances on a linear scale, then it must be true that

$$ y_3 - y_2 = y_2 - y_1 $$

The same property on a logarithmic scale implies that

$$ \log y_3 - \log y_2 = \log y_2 - \log y_1. $$

Using the rule that the logarithm of a quotient is the difference of the separate logarithms:

$$ \log \frac{y_3}{y_2} = \log \frac{y_2}{y_1} $$

and taking anti-logarithms on both sides yields:

$$ \frac{y_3}{y_2} = \frac{y_2}{y_1}. $$

that is, in the linear scale,we have equal proportional changes in the variable. In practical terms, if we plot an exponential growth curve on a semi-logarithmic scale, we will observe a linear relationship because exponential growth means a fixed percentage change in the variable. This example why the following relationship is true:

$$ \frac{d\ln y}{dx} = \frac{1}{u} \frac{du}{dx}. $$

that is the change of a variable, $y$, in logarithmic space is equal to a proportional change of the same variable in linear space.

The Shrinking Transistor

 Originally posted: April 2011

How big are modern transistors? Surely they must be much bigger than an enzyme? Well, not quite. The picture below shows two transistors, both about 22 nm wide. Superimposed on top of the right-hand transistor is a single molecule of phosphofructokinase.

Transistor image from Tech-On
Image of phosphofructokinase from Molecule of the Month PDB

Installing MathTime Professional II in MiKTeX

September 20, 2010 11:25 am

How to install MathTime Professional II Fonts in MiKTeX.

The following instructions apply to MikTex 2.6 and above (including 2.9)

1. If you haven't already, create your own root directory. See http://docs.miktex.org/manual/localadditions.html for instructions on how to do this.
2. Unzip the mtp2fonts.zip file into a temporary directory
3. In the zip file you'll find the folder called texmf
4. Move the contents of texmf to the "Local tex files" directory you created in 1.
5. From a command line run

initexmf --edit-config-file updmap

It doesn't actually matter where you do this, for example c:\Users\MyUserName will do. This will open the config file into an editor.

5. Add the following two lines to the file

# MathTimePro II
Map mtpro2.map

6. Save and exit
7. From a command line run (don't change directory here, call the following from wherever you called initexmf):

updmap

You might also have to do a" Refresh FNDB" and an "Update Formats" from the MiKTeX Settings (Admin) dialog windows (See MiKTeX under start menu) if you have any issues.

8. Instructions on using MathTimePro 2 can also be found in your root directory and also in

C:\Local TeX Files\doc\fonts\mtpro2

To use, the following examples illustrate one way to use the fonts, add the following to the start of your LaTeX document:

\usepackage[T1]{fontenc}
\usepackage{textcomp}
\renewcommand{\rmdefault}{ptm}
\usepackage[scaled=0.94]{helvet}
\usepackage[subscriptcorrection,slantedGreek,nofontinfo,mtpcal]{mtpro2}

Never say never….

Someone posted these quotes in response to a skeptic’s comment that solar, wind, and other renewable energy sources will never economically match petroleum, coal, or nuclear: 

 “Man will never reach the moon regardless of all future scientific advances. –Dr. Lee DeForest, “Father of Radio & Grandfather of Television.” 

 “The bomb will never go off. I speak as an expert in explosives.” –Admiral William Leahy, US Atomic Bomb Project 

 “There is no likelihood man can ever tap the power of the atom.” –Robert Millikan, Nobel Prize in Physics, 1923 

 “I think there is a world market for maybe five computers.” –Thomas Watson, chairman of IBM, 1943 

Meanwhile: Startup Aims to Cut the Cost of Solar Cells in Half

 Although I was brought up on digital computers I have quite a soft spot for analog computers. I used to have an old web page that had some notes and pictures plus some rare analog construction notes that I copied from a 1978 article I had. I thought I’d rejuvenate the page and notes here.

The digital computer is a sequential device, in general, operating on data one step at a time, in addition, the digital computer represents data internally using a verbose but very robust form of representation called binary. Thus a single transistor in a digital computer can only store two states, on and off. Obviously, to store a number to any sensible degree of precision, many transistors are required.

An analog computer operates in a quite different way. For a start, all operations in an analog computer are performed in parallel. Secondly, data are represented in an analog computer as voltages, a very compact but not necessarily robust form of storage (prone to noise corruption). A single capacitor (equivalent to the digital’s computer use of a transistor) in an analog computer can represent one entire continuous variable.

EC-1 Educational Analog Computer, introduced in 1960 by Heathkit.

The Heathkit Educational Analog Computer is completely self-contained and contains nine DC operational amplifiers with provision for balancing without removing problem setup. It also features three initial condition power supplies, five coefficient potentiometers, four sets of relay contacts, an electronically regulated power supply and a built-in repetitive oscillator for automatic operation. The complete EC-1 kit also contains an assortment of precision resistors, capacitors,

special silicon diodes and patch cords for setting up scores of complex computer problems easily and accurately.

Vannevar Bush’s differential analyzer

(Image from http://archive.computerhistory.org/)

The first mechanical/electrical analog computers were developed in the early part of the 20th century. One of the most well known of these was the Vannevar Bush’s differential analyzer that filled a room at the Massachusetts Institute of Technology. Vannevar Bush’s differential analyzer crunched through calculus in seconds, although technicians often spent hours setting it up to solve an equation. A multitude of wheels, discs, shafts and gears handled the computations with precision unmatched by any contemporaneous machine. With the advent of the digital computer after World War II, the development of the analog computer slowed and by the early 1970s it was being rapidly supplanted by digital machines. One wonders whether with today’s electronic technology the analog computer might reemerge as a general-purpose computer.

Designs for Analog Computers

Practical Electronics (UK), 1978

Part 1
Part 2
Part 3
Part 4

Practical Electronics (UK), 1968

http://www.archive.org/details/anacomp