Analog Signal

Sampled Signal

In mathematics, a partition of an interval [a,b] is a finite sequence

Each [x_i,x_{i + 1}] is called a subinterval of the partition. The mesh of a partition is defined to be the length of the longest subinterval [x_i,x_{i + 1}], that is, it is max(x_{i + 1} − x_i) where . It is also called the norm of the partition.

A tagged partition of an interval is a partition of an interval together with a finite sequence of numbers subject to the conditions that for each i, . In other words, it is a partition together with a distinguished point of every subinterval. The mesh of a tagged partition is the same as that of an ordinary partition.

Suppose that together with are a tagged partition of [a,b], and that together with are another tagged partition of [a,b]. We say that and together are a refinement of together with if for each integer i with , there is an integer r(i) such that x_i = y_r(i) and such that t_i = s_j for some j with . (It is not correct to allow j to equal r(i + 1) because s_{r(i + 1)} is greater than or equal to x_{i + 1}.) Said more simply, a refinement of a tagged partition takes the starting partition and adds more tags, but does not take any away.

We can define a partial order on the set of all tagged partitions by saying that one tagged partition is bigger than another if the bigger one is a refinement of the smaller one.

There is a pretty obvious similarity between those two notions : the refinement of a tagged partition represents an increase of the sampling rate.

Step 1: Installation

It seems like Microsoft has made a huge progress in making the installation procedure as compact and rapid as possible. All a user has to declare is the language and the regional settings and, after just a few restarts, the operating system is set up within 20 minutes! Nevertheless, the advanced users will feel very unhappy with the lack of option to select parts of the software to be installed. Perhaps an ‘advanced setup’ feature would be more convenient.

Step 2: Booting up for the first time

For those who upgrade from Windows Vista, the new interface experience will not be as dramatic as for those who replace Windows XP. ‘Aero Peek’ will surely get most of the users excited, while the new taskbar will provide an aroma of different platforms – Mac OS X and KDE. The notification system and the user account control (UAC) seem to be a lot more quiet than the ones of Windows Vista. The brand new ‘action centre’, which replaces the ’security centre’ of Windows XP, will be in charge for announcing critical news about the operating system.

Step 3: Setting up the hardware

In most cases, Windows itself will recognize and install the appropriate drivers for each part of the hardware – no need for frustrating searches over the Internet or for CDs that came with your hardware. Most motherboard parts are properly installed with the included drivers database. For anything else, the ‘windows update’ will get the job done for you.  You only have to separately install drivers for devices like external audio interfaces. In the case of Firewire audio interfaces, be extra cautious with the driver installed for the IEEE 1394 chipset. In my case, it had to be replaced with the generic ‘1394 OHCI Compliant Host Controller (Legacy)’ driver to get my TC Konnekt 8 up and running, smoothly. A blue screen might appear when changing the driver – nothing to worry about. Everything will be normal, afterwards. Needless to say, the ASIO4ALL driver is, still, a neat solution for seamless workflow on audio and MIDI operations.

Step 4: Recommended Software

It would be useful to add some utilities to your software arsenal:

• CCleaner (http://www.ccleaner.com), to clean the cache memory appropriately.
• Defraggler (http://www.defraggler.com), to defrag your hard disk drives efficiently.
• RevoUninstaller (http://www.revouninstaller.com), to uninstall applications radically.

If you run Defraggler, you ‘ll be happy to see that just after the installation there’s no urge for a defrag, as in previous versions of Windows.

Terence Tao – Solving Mathematical Problems : A Personal Perspective

Donald M. Davis – The Nature and Power of Mathematics

Rudy Rucker – Infinity and the Mind : The Science and Philosophy of the Infinite