We first define a function b{t} as follows: given the set of squares such that each has three corners on C and vertex at t, b{t} is the corresponding set of positive parametric differences between T and the backward corner points.

We define these values as Af, and define g{t} in the same way for each T.

Articles of faith are sets of beliefs usually found in creeds, sometimes numbered, and often beginning with " We believe ...", which attempt to more or less define the fundamental theology of a given religion, and especially in the Christian Church.

We then define f ( x, y ) to be this z.

We can then define the differential map d: C < sup >∞</ sup >( M ) → T < sub > x </ sub >< sup >*</ sup > M at a point x as the map which sends f to df < sub > x </ sub >.

We can now define the value

We should not define ' wisdom ' as the absence of folly, or a healthy thing as whatever is not sick.

We cannot define a point except as ' something with no parts ', nor blindness except as ' the absence of sight in a creature that is normally sighted '.

We define the periodic Bernoulli functions P < sub > n </ sub > by

We then define a contravariant functor F from C to D as a mapping that

We need a " mark " to define where we are and which direction we are heading to see if we ever get back to exactly the same pixel.

Verner wrote, " We can conclude that although the ancient Egyptians could not precisely define the value of π, in practice they used it ".

We define the degree of to be the number of universal quantifier blocks, separated by existential quantifier blocks as shown above, in the prefix of.

We define the inverse limit of the inverse system (( A < sub > i </ sub >)< sub > i ∈ I </ sub >, ( f < sub > ij </ sub >)< sub > i ≤ j ∈ I </ sub >) as a particular subgroup of the direct product of the A < sub > i </ sub >' s:

We wish to maximize total value subject to the constraint that total weight is less than or equal to W. Then for each w ≤ W, define m to be the maximum value that can be attained with total weight less than or equal to w. m then is the solution to the problem.

We could also define a Lie algebra structure on T < sub > e </ sub > using right invariant vector fields instead of left invariant vector fields.

* We then define truth-functional operators, beginning with negation.

We can define addition, subtraction, and multiplication on by the following rules:

We can define methods to manipulate these new complex types.

We assume that A is an m-by-n matrix over either the real numbers or the complex numbers, and we define the linear map f by f ( x ) = Ax as above.

We can define a multiplication on the set S using the Steiner triple system by setting aa

We may define a possibly different topology on X using the continuous ( or topological ) dual space X < sup >*</ sup >.

We can also define it as a second component of business which includes all activities, functions and institutions involved in transferring goods from producers to consumer.

We thus obtain the inequality in terms of dimensions of kernel, which can then be converted to the inequality in terms of ranks by the rank-nullity theorem.

We are looking for a kernel vector a = such that the matrix product of M on a yields the zero vector.

We simply construct the equaliser of two morphisms f and g as the kernel of their difference g − f ; similarly, their coequaliser is the cokernel of their difference.

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.

We say P is a projection along V onto U ( kernel / range ) and Q is a projection along U onto V.

We say that H is a reproducing kernel Hilbert space if the linear map

We call the orthogonal complement of the kernel of W the initial subspace of W, and the range of W is called the final subspace of W.

We use the stronger statement that every odd ( antipode-preserving ) mapping h: S < sup > n-1 </ sup > → S < sup > n-1 </ sup > has odd degree.

# We apply this construction to the case when the manifold M is the underlying space of a Lie group G, with G acting on G = M by left translations L < sub > g </ sub >( h ) = gh.

We define headers as finite subsets of C. A relational database schema is defined as a tuple S = ( D, R, h ) where D is the domain of atomic values ( see relational model for more on the notions of domain and atomic value ), R is a finite set of relation names, and

We note that h is equal to the area, ΔA, swept out by the radius divided by the time, Δt, and also related to the parameter, p

We now assume that g is semisimple, with a chosen Cartan subalgebra h and corresponding root system.

We say that if the function θ oscillates, it represents a new type of quantum-mechanical wave, and this new wave has its own momentum p = h / λ, which turns out to patch up the discrepancies that otherwise would have broken conservation of momentum.

We have explicitly extracted the exponential phase factors exp (- iE < sub > n </ sub > t /< strike > h </ strike >) on the right hand side.

We can summarize this lifting property as follows: a module P is projective if and only if for every surjective module homomorphism f: N ↠ M and every module homomorphism g: P → M, there exists a homomorphism h: P → N such that fh = g. ( We don't require the lifting homomorphism h to be unique ; this is not a universal property.

We think of this as a family of coordinate systems on M, parametrized by the points of M. Two such parametrized coordinate systems φ and φ ′ are H-related if there is an element h < sub > p </ sub > ∈ H, parametrized by p, such that

In the year 2007 a team called We Ain't Stupid used modified 1972 Cadillac Eldorado called T-47 to break the Art Car World Land Speed Record setting a speed of 125 m. p. h ..

We also form the harmonic mean of x and y and call it h < sub > 1 </ sub >, i. e. h < sub > 1 </ sub > is the reciprocal of the arithmetic mean of the reciprocals of x and y.

We should be uttering: thoo t ( h ) aakur thum pehi aradhaas || You are our Lord and Master ; to You, I offer this prayer-slowly, with meaning, word-by-word in our mind and talking along with the Sangat.

We will abbreviate this data by ( A, C, f, g, h ).

We can iterate this procedure to get exact couples ( A < sup >( n )</ sup >, C < sup >( n )</ sup >, f < sup >( n )</ sup >, g < sup >( n )</ sup >, h < sup >( n )</ sup >).

We let E < sub > n </ sub > be C < sup >( n )</ sup > and d < sub > n </ sub > be g < sup >( n )</ sup > h < sup >( n )</ sup >.

We express this relation by means of the notation ∠( h, k ) ≅ ( h ′, k ′) Every angle is congruent to itself ; that is, ∠( h, k ) ≅ ( h, k ) or ∠( h, k ) ≅ ( k, h )