Abstractly, 1 / 8-schisma tuning may be considered the analog, among schismatic tunings, of 1 / 4-comma meantone among meantone tunings, as it also has pure interval ratios of 2: 1 and 5: 4, though with much more accurate interval ratios of 3: 2 and 6: 5 ( less than a quarter of a cent off from just intonation ) than its meantone counterpart.

Abstractly, this circle is a cyclic group of order twelve, and may be identified with the residue classes modulo twelve.

Abstractly, if one starts with a complex vector space W and takes the complexification of the underlying real space, one obtains a space isomorphic to the direct sum of W and its conjugate:

Abstractly, a hierarchy can be modelled mathematically as a rooted tree: the root of the tree forms the top level, and the children of a given vertex are at the same level, below their common parent.

Abstractly, an array reference is simply a procedure of two arguments: an array and a subscript vector, which could be expressed as ; but many languages provide special syntax like ; similarly an array element update is abstractly something like, but many languages provide syntax like.

Abstractly, we can say that D is a linear transformation from some vector space V to another one, W. We know that D ( c ) = 0 for any constant function c. We can by general theory ( mean value theorem ) identify the subspace C of V, consisting of all constant functions as the whole kernel of D. Then by linear algebra we can establish that D < sup >− 1 </ sup > is a well-defined linear transformation that is bijective on Im D and takes values in V / C.

Abstractly, we can identify the chain rule as a 1-cocycle.

Abstractly, Weyl groups are finite Coxeter groups, and are important examples of these.

Abstractly, a parametric equation defines a relation as a set of equations.

Abstractly, the matrix exponential gives the connection between a matrix Lie algebra and the corresponding Lie group.

Abstractly, in trying to prove a proposition P, one assumes that it is false, and that therefore there is at least one counterexample.

Abstractly, these are the left adjoint and right adjoint, respectively, to the inclusion functors of in D. In addition, the truncation functors fit into a triangle, and this is in fact the unique triangle satisfying the third axiom above:

Abstractly it is assumed that the economy consists only of capitalist production and that the capitalist economy is equal to capitalist society.

The convergence of programmable mechanical control with machine tool toolpath control was delayed many decades, in part because the programmable control methods of musical boxes and looms lacked the rigidity for machine tool toolpaths.

Starting with the programmable pixel shaders, people recognized the power of parallel operations on multiple pixels and began to perform general-purpose computing on ( the ) graphics processing units ( GPGPU ).

Dr. Eli Harari, an Israeli engineer, began making early contributions to EEPROM or electrically erasable programmable read-only memory, a precursor to flash memory.

The original Hand Held Products company, established in 1981 in Charlotte, North Carolina, began life as a manufacturer of memory modules for Hewlett-Packard and Texas Instruments programmable calculators, but quickly came to the industry spotlight in 1983 thanks to an RFP hand-off by Hewett-Packard, as a package delivery company Federal Express selected their Micro-Wand portable data collection device to begin its first major proof of delivery system for package tracking.

Based on a newer, slightly more flexible, version of the Asters original Motorola MC6845 video chip, the Rockwell 6545, it worked by adding a new video mode, one with the ability to reprogram an extended, ( 2048 characters instead of 256 characters ) version of the character set, supported by an extended character memory of the video card that did not use one ( 8 bit ) byte per character, but an 11 bit " word ", so it could address each one of the available 2048 unique programmable characters.

The distinction is not clear-cut: some devices classed as programmable calculators have programming functionality, sometimes with support for programming languages such as RPL or TI-BASIC.

The scientific programmable models were particularly poor in this respect, with the programmability comings at a heavy price in Transcendental function accuracy.

The first programmable pocket calculator was the HP-65, in 1974 ; it had a capacity of 100 instructions, and could store and retrieve programs with a built-in magnetic card reader.

The companies that survived making calculators tended to be those with high outputs of higher quality calculators, or producing high-specification scientific and programmable calculators.

Another approach to hardware support for database management was ICL's CAFS accelerator, a hardware disk controller with programmable search capabilities.

The need to break German codes in World War II led to advances in cryptography and theoretical computer science, with the first programmable digital electronic computer being developed at England's Bletchley Park.

One-time programmable but with window, can be erased with ultraviolet ( UV ) light.

Desktop sized programmable calculators by Hewlett Packard had evolved into the HP 9830 BASIC language computer by 1972, with IBM's releasing its own IBM 5100 in 1975.

It was a complete computer system programmable in BASIC or APL, with a small built-in CRT monitor, keyboard, and tape drive for data storage.

In 1972, Intel launched the 8008, the first 8-bit microprocessor .< ref > using enhancement load PMOS logic ( demanding 14V, achieving TTL-compatibility by having V < sub > CC </ sub > at + 5V and V < sub > DD </ sub > at-9V )</ ref > It implemented an instruction set designed by Datapoint corporation with programmable CRT terminals in mind, that also proved to be fairly general purpose.

Audio capabilities were CPU controlled with programmable frequency and duration.

These became essential with the appearance of complex synthesizers such as the Yamaha FS1R, which contained several thousand programmable parameters, but had an interface that consisted of fifteen tiny buttons, four knobs and a small LCD.

The copper is a programmable finite state machine that executes a programmed instruction stream, synchronized with the video hardware.

* 1984 – Roland Juno-106: Very popular programmable ( 128 patch memory locations ), digitally controlled 6-voice analog synthesizer, with MIDI and the ability to transmit button and slider information through SysEx.

To overcome these shortcomings, the 16550 series UARTs incorporated a 16-byte FIFO with a programmable interrupt trigger of 1, 4, 8, or 14 bytes.

The boot process was revolutionized by the introduction of integrated circuit read-only memory ( ROM ), with its many variants, including mask-programmed ROMs, programmable ROM ( PROM ), erasable ROM ( EPROM ), and flash memory.

For programmable devices like EPROMs and GALs, DIPs remained popular for many years due to their easy handling with external programming circuitry ( i. e., the DIP devices could be simply plugged into a socket on the programming device.

In modern Z-buffer systems, similar benefits can be gained through rough front-to-back sorting ( approaching the ' reverse painters algorithm '), early Z-reject ( in conjunction with hierarchical Z ), and less common deferred rendering techniques possible on programmable GPUs.

It was developed from EEPROM ( electrically erasable programmable read-only memory ) and must be erased in fairly large blocks before these can be rewritten with new data.

A sophisticated programmable calculator, the HP 9800 series desktop computers | HP 9830A was actually an early desktop computer with printer.

The very first large " programmable calculators / computers " ( machines lacking keyboards for text input ) were marketed in the second half of the 1960s, starting with Programma 101 ( 1965 ) and HP 9100 ( 1968 ).

This is done in order to solve boundary problems with neighborhoods, but another advantage of this system is that it is easily programmable using modular arithmetic functions.

A programmable automation controller ( PAC ) is a compact controller that combines the features and capabilities of a PC-based control system with that of a typical PLC.