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

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, it can be defined as the integer triples ( a, b, c ), associated to 3 < sup > a </ sup > 5 < sup > b </ sup > 7 < sup > c </ sup >, where the distance measure is not the usual Euclidean distance but rather the Euclidean distance deriving from the vector space norm

Abstractly programmable toolpath guidance began with mechanical solutions, such as in musical box cams and Jacquard looms.

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, 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.

Most commercial uses of urethane foams require densities between 2 and 30 lb. / cu. ft. for rigid foams, between 1 and 3 lb. / cu. ft. for flexible foams.

The 1 / 4 abacus, which is suited to decimal calculation, appeared circa 1930, and became widespread as the Japanese abandoned hexadecimal weight calculation which was still common in China.

The New Mexico US Extension Service recommends a mix of not more than 1 / 3 clay, not less than 1 / 2 sand, and never more than 1 / 3 silt.

* by type: cargo 1, passenger / cargo 2, petroleum tanker 2, roll on / roll off 1

For example, the open interval ( 0, 1 ) does not have a least element: if x is in ( 0, 1 ), then so is x / 2, and x / 2 is always strictly smaller than x.

The yellow form has a density of 1. 97 g / cm < sup > 3 </ sup >.

Other examples include various prescription drugs ( e. g. most antiepileptic drugs have cerebellar ataxia as a possible adverse effect ), Lithium level over 1. 5mEq / L, cannabis ingestion and various other recreational drugs ( e. g. ketamine, PCP or dextromethorphan, all of which are NMDA receptor antagonists that produce a dissociative state at high doses ).

* Martin Gabel, General Electric Theater ( 4 / 1 / 1962 ; adaptation of " The Disappearance of Mr. Davenheim ")

: 16. 59 births / 1, 000 population ( 2009 est.

Historically significant is the 1 / 8-schisma tuning of Hermann von Helmholtz and Norwegian composer Eivind Groven.

The " Viennese tuning " ( A < sup > 1 </ sup >- D-F-A ) was popular, and in some cases a fifth string or even sixth string was added ( F < sup > 1 </ sup >- A < sup > 1 </ sup >- D-F-A ).

( Ordinarily, 5 mm " zither pins " are used, similar to, but smaller in diameter than piano tuning pins, which come in various sizes ranging upwards from " 1 / 0 " or 7 mm.

In modern tuning a half step has a frequency ratio of, approximately 1. 059.

The tuning of the laser wavelength of 0. 840 micrometres ( 1. 48 eV per photon ) and the PV panel bandgap of 1. 43 eV to each other produces an estimated conversion efficiency of 59 % and a predicted power density of up to 540 kW / m < sup > 2 </ sup >.

In general, a meantone is constructed the same way as Pythagorean tuning, as a stack of perfect fifths, but in meantone, each fifth is narrow compared to the ratio 2 < sup > 7 / 12 </ sup >: 1 used in 12 equal temperament.

Figure 1: The syntonic tuning continuum ( Milne 2007 ).

" The distinguishing feature of each unique syntonic tuning is the width of its generator in cents, as shown in the central column of Figure 1.

In Figure 1, the valid tuning ranges of 5-limit, 7-limit, and 11-limit syntonic tunings are shown, and can be seen to include many notable meantone tunings.

It is useful because it gives us an idea of the melodic qualities of the tuning, and because if R is a rational number N / D, so is ( 3R + 1 )/( 5R + 2 ) or ( 3N + D )/( 5N + 2D ), which is the size of fifth in terms of logarithms base 2, and which immediately tells us what division of the octave we will have.

Featuring a 1071 cc engine with a 70. 61 mm bore and nitrided steel crankshaft and strengthened bottom end to allow further tuning ; and larger servo-assisted disc brakes, 4, 030 Cooper S cars were produced and sold until the model was updated in August 1964. Cooper also produced two S models specifically for circuit racing in the under 1, 000 cc and under 1, 300 cc classes respectively, rated at and a, both had a bore and both of which were also offered to the public.

**** 311. 122. 1 Without tuning noose.

and includes upgraded brake pads, a 3. 16: 1 final drive ratio ( compared to the standard 2. 77: 1 ), recalibrated electronic power steering, further suspension tuning, a recalibrated ESC ( Electronic Stability Control ) with sport mode and summer compound Goodyear Eagle F1 245 / 45ZR20 tires.

** Schaller BM series open tuning heads M / C heads with a gear ratio of 1: 20.

1, 0 Channel Fine tuning

The No. 11 Set, a radio tranceiver featuring a single tuning unit, was designed in 1938 to replace the 1933 No. 1 Wireless Set.

This idea goes back to Salinas in the sixteenth century, and is interesting in part because it gives a system of meantone tuning, being close to 1 / 3 comma meantone.

MSSM predicts that the mass of the lightest Higgs boson should not be much higher than the mass of the Z boson, and, in the absence of fine tuning ( with the supersymmetry breaking scale on the order of 1 TeV ), should not exceed 130 GeV.

There are violin-making tools and Baroque fittings, early harpsichord and fortepiano tuning hammers, and 1, 000 brass instrument mouthpieces from virtually every turn-of-the-century manufacturer.

:( 5, 000, 000 persons in Chicago ) / ( 2 persons / household ) × ( 1 piano / 20 households ) × ( 1 piano tuning per piano per year )