In the notation of the proof of Theorem 12, let us take a look at the special case in which the minimal polynomial for T is a product of first-degree polynomials, i.e., the case in which each Af is of the form Af.

Using the Dirac notation, let the system initially be in the state where

To describe the construction let us fix some notation.

The modern convention regarding the choice between bottom-up or top-down interval notation is to specify the main note ( either the bass line or the melody line ) and let the intervals go up or down from there, as appropriate.

Using the abstract index notation, let a, b, c, ... refer to and μ, ν, ... refer to the tangent bundle.

Before showing how to do this, let us simplify the notation.

: Cement chemist notation: C < sub > 3 </ sub > S + H → C-S-H + CH

Many Americans attribute the dish to France and offer as evidence a notation by U. S. President Thomas Jefferson.

In Light path notation global lighting the paths of the type L ( D | S ) corresponds * E.

Hand-written musical notation by Johann Sebastian Bach | J. S. Bach: beginning of the Prelude from the Suite for Lute in G minor BWV 995 ( transcription of Cello Suite No. 5, BWV 1011 ) BR Bruxelles II.

In an alternate notation for block designs, an S ( t, k, n ) would be a t -( n, k, 1 ) design.

In 1909, Charles S. Peirce proposed a graphical notation of nodes and edges called " existential graphs " that he called " the logic of the future ".

S was also used for the knight in the early days of algebraic notation, from the German Springer, and is still used in chess problems ( where N stands for the nightrider, a popular fairy chess piece ).

John Florian Sowa is the computer scientist who invented conceptual graphs, a graphic notation for logic and natural language, based on the structures in semantic networks and on the existential graphs of Charles S. Peirce.

An element of order 6 in the group S < sub > 5 </ sub > can be written in cycle notation as ( 1 2 ) ( 3 4 5 ).

In notation, where S < sub > i </ sub > is the ith state of the shift register, a << x is a shifted up x bits, head ( a, x ) is the x highest bits of a and n is number of bits of IV:

An example of such a function notation would be S ( 1, 3 ) in which the function S denotes addition: S ( 1, 3 )

In notation of first-order logic, this type of fallacy can be expressed as (∃ x ∈ S: φ ( x )) → (∀ x ∈ S: φ ( x )), meaning " if there exists any x in the set S so that a property φ is true for x, then for all x in S the property φ must be true.

The notation S < sub > n </ sub > ( S for Spiegel, German for mirror ) denotes the symmetry group generated by an n-fold improper rotation ( not to be confused with the same notation for symmetric groups ).

The multi-line format was designed to permit writing of HAL / S code that is similar to mathematical notation.

The notation below describes the relationship under the Galilean transformation between the coordinates and of a single arbitrary event, as measured in two coordinate systems S and S ', in uniform relative motion ( velocity v ) in their common directions, with their spatial origins coinciding at time t = t '= 0:

In 1960, Łukasiewicz notation concepts and stacks were used as the basis of the Burroughs B5000 computer designed by Robert S. Barton and his team at Burroughs Corporation in Pasadena, California.

In this system, the number ten may be written as " A ", " T " or " X ", and the number eleven as " B " or " E " ( another common notation, introduced by Sir Isaac Pitman, is to use a rotated " 2 " for ten and a reversed " 3 " for eleven ).

Temperatures are specified by the notation T < sub > s </ sub >/ T < sub > r </ sub >) with T < sub > s </ sub > representing the temperature at which the sample's density was determined and T < sub > r </ sub > the temperature at which the reference ( water ) density is specified.

), functional notation ( e. g. sin x or sin ( x )), and superscripts ( e. g. transpose A < sup > T </ sup >).

The notation A < sup >− T </ sup > is often used to represent either of these equivalent expressions.

No notation is used to describe the set of all λ, which satisfy this condition, but for a subset: If λ-T does not have dense range but is injective, λ is said to be in the residual spectrum of T, denoted by σ < sub > r </ sub >( T ).

In this notation he perhaps followed some older contemporaries, such as Petrus Ramus, who designated the points in geometrical figures by vowels, making use of consonants, R, S, T, etc., only when these were exhausted.

The indices for the fiber of T will follow a similar notation, except that they will be hatted like this:.

where T is the total kinetic energy of the system, and the notation

where T ( V ) is the tensor algebra on V, and the notation means " the ideal generated by ".

This is a guide for a " softer " knot ( in Fink and Mao's notation Li Ro Li Ro Ci Ro Li Co T, knot 26 ):

The notation T for the circle group stems from the fact that T < sup > n </ sup > ( the direct product of T with itself n times ) is geometrically an n-torus.

In a simple notation, these three sounds are represented by three letters: D, T, and K. When capitalized, the beat is emphasized, and when lower-case, it is played less emphatically.

The notation X ≤< sub > T </ sub > Y indicates that X is Turing reducible to Y.

The notation X ≡< sub > T </ sub > Y indicates that X and Y are Turing equivalent.

in the Dirac notation, where each | i > is a normalized eigenvector of T. The set

The notation here says that X is a configuration of n points in D, and T is a triangle with three points of X as vertices.

Because this defines T, F, NOT ( as a postfix operator ), OR ( as an infix operator ), and AND ( as a postfix operator ) in terms of SKI notation, this proves that the SKI system can fully express Boolean logic.

In the coordinate notation, with the coordinates Σ < sup > a </ sup >, a = 1 ,..., n where n is the dimension of T,

The same notation is used with sets to denote cardinality ; the meaning depends on context.

We will use the notation to denote the multiplicative inverse of, it is defined exactly when and are coprime ; the following construction explains why the coprimality condition is needed.

The alternative notation can be used to denote the equivalence class of the element specifically with respect to the equivalence relation.

He also introduced the modern notation for the trigonometric functions, the letter for the base of the natural logarithm ( now also known as Euler's number ), the Greek letter Σ for summations and the letter to denote the imaginary unit.

The j-j coupling method, which is historically linked to X-ray notation, is almost always used to denote Auger transitions.

We usually denote this set using set-builder notation as

In elementary mathematics the notation or is used to denote a divided by b. This notation was first introduced by Michael Stifel in Arithmetica integra, published in 1544.

The notation |:| may also denote the index of a subgroup.

* In chess notation, the dagger may be suffixed to a move to signify the move resulted in a check, and a double dagger is used to denote checkmate.

This earliest question mark was a decoration of one of these dots, with the " lightning flash " perhaps meant to denote intonation ( or a tilde or titlo, named after the Latin word titulus, as in “ ·~ ”, like those wavy and more or less slanted marks used in lots of medieval texts for denoting various things such as abbreviations, and that would become later various diacritics or ligatures or modified letters used in the Latin script ), and perhaps associated with early musical notation like neumes.

The Rigveda has elements of present Indian music, with a musical notation to denote the metre and the mode of chanting.

We use the shorthand notation to denote the joint probability of by.

For instance, a special metal alloy might expand 1. 2 micrometers per meter of length for every degree Celsius and this would be expressed as “ α = 1. 2 ppm /° C .” Parts-per notation is also employed to denote the change, stability, or uncertainty in measurements.

Newton < nowiki >' s </ nowiki > notation for differentiation, or dot notation, uses a dot placed over a function name to denote the time derivative of that function.

The differential operator is linear — if we use the Heaviside D notation to denote this, we may extend D < sup >− 1 </ sup > to mean the first integral.

For the sake of clarity and simplicity in the present introduction, it is convenient to use the two-sign notation, but allow the sign "→" to denote the boolean function that is associated with the truth table of the material conditional.

Note that the asterisk is placed above the function symbol to denote the lower adjoint, in contrast to the notation from Erné et al.

The notions of key and absolute pitch are deeply rooted in Western music, whereas the Carnatic notation does not specify the key and prefers to use scale degrees ( relative pitch ) to denote notes.

A grace note is a kind of music notation used to denote several kinds of musical ornaments.

This type of notation is reminiscent of the notation used in chemical formulas to denote the composition of chemical compounds.

The notation is commonly used to denote with n arrows.

It is often written in the same notation used to denote IP addresses.