hobyrne's New Writeups

Coherence (thing)

2004-06-22T18:51:44Z

Coherence is a property of information-carrying signals. There are two types of coherence: Fully-coherent and differentially-coherent.

Fully-coherent signals are signals where the information mapped directly onto a property of the carrier signal. For instance, amplitude modulation is almost always fully-coherent; a high amplitude corresponds to the bit '1', a low amplitude corresponds to a bit '0' (or vice-versa).

Differentially-coherent signals are signals where the information is encoded onto changes in the carrier signal. Phase modulation more often uses differential coherence, because two systems with different clocks may not be able to easily agree (long-term) on what a reference phase is, but either or both should be able to recognise a change in phase when it happens.

For example, imagine there are two states for a transmitter to be in: A and B. Fully-coherent transmission of data might map the value '0' to state A, and…

curvature (idea)

2004-03-15T16:59:42Z

The curvature of a function of two variables, f(x₁, x₂) is the product of all partial second derivatives with respect to like variables (e.g. ∂²f/∂x₁²), minus the product of all second partial derivatives with respect to different variables (e.g. ∂²f/∂x₁∂x₂).

For two variables, this is (∂²f/∂x₁²)(∂²f/∂x₂²) - (∂²f/∂x₁∂x₂)² (because ∂²f/∂x₁∂x₂ = ∂²f/∂x₂∂x₁).

In Eindhoven notation, this is (* : 0 ≤ …

point of inflection (thing)

2004-02-17T23:45:48Z

A point of inflection is, as Ryouga describes, a point on a graph where concavity changes (making it the second derivative equivalent of a local extremum). He's also correct that most points of inflection are zeroes of the second derivative. However, not every zero of the second derivative is an inflection point.

Take, for example, the function f(x)=x⁴. The first derivative is f'(x)=4x³, the second derivative is f''(x)=12x². Finding the zeroes of the second derivative, 12x²=0, gives x=0. However, x=0 is not the location of an inflection point.

The second derivative, f''(x), is 12x². The graph is concave up where the second derivative is positive - 12x² is positive everywhere except at zero, so the graph is…

less vs. fewer (idea)

2004-01-27T20:53:31Z

The opposite of 'fewer' is 'more', and the opposite of 'less' is also 'more'. This is part of the reason behind the confusion between the two words.

One way to test which word is appropriate is that something you can have 'fewer' of is something you can have 'a few' of (something quantum), something you can have 'less' of is something you can have 'a little' of (something continuous). For example, 'a little oranges' just sounds wrong - you say, 'a few oranges', so the opposite of 'more oranges' is 'fewer oranges'. Likewise, 'a few orange juice' is clearly wrong, it's 'a little orange juice', so the opposite of 'more orange juice' is 'less orange juice'.

mob (thing)

2003-08-12T20:27:36Z

One very important aspect of the mob is this: No one person in the mob can control what the mob is going to do, yet every person who participates shares responsibility for what the mob does. This is why I find flash mobs a scary idea: to volunteer to be part of a mob?

So why go to a flash mob? There is an appeal there - no doubt about it. The mob mentality. Participants are freed from making decisions, and thus feel like they're absolved from individual responsibility. There's a feeling of solidarity, so strong, very rarely achieved in many people's lives. Those are powerful incentives; after getting criticized by your boss, nagged at by your spouse, teased by your friends, finding a few hundred people to do something with, together, even for a few minutes, in a common (if inane) task, is invigorating.

The problem is this. A flash mob is still, fundamentally, a mob. Any research into the history…

x-simple (idea)

2003-08-02T06:34:54Z

A region is x-simple if every line parallel to the x-axis satisfies one of:
(1) The line does not touch the region.
(2) The line touches the border of the region, either at a single point or along an unbroken line segment.
(3) The line passes through the interior of the region, once.
Another way of saying this is that for any two points on the line that are in the region, all the points in between are in the region too.

The terms y-simple and z-simple have similar definitions, for the y-axis and z-axis respectively.

Regions which are x-simple are relatively easy to integrate over with x as the dependent variable. In a 2-D system, then, the interval of integration (on x) will be from f(y) to g(y), two functions of y, and y would be the independent variable (meaning the interval of integration will be from some…