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 kernel of h to be the set of elements in G which are mapped to the identity in H

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 say that a strict chain of inclusions of prime ideals of the form:

< BLOCKQUOTE > We ... see events not as isolated incidents, but as part of a long chain of conspiracies, a war of annihilation ... In southern Sudan hundreds of thousand were killed .</ BLOCKQUOTE >

We also see cooperativity in large chain molecules made of many identical ( or nearly identical ) subunits ( such as DNA, proteins, and phospholipids ), when such molecules undergo phase transitions such as melting, unfolding or unwinding.

We apologize for the late notice, but a seemingly endless chain of unexpected events blocked us previously and made us wary of predicting the Great Transition.

" We ’ ve learnt an incredibly painful lesson through this and we will be much, much more suspicious worldwide on ensuring the safety and integrity of our supply chain everywhere in the world ," said Ferrier.

We have developed a general purpose graph-based ranking algorithm like Page / Lex / TextRank that handles both " centrality " and " diversity " in a unified mathematical framework based on absorbing Markov chain random walks.

We believe all people have a right to express their unique joy and beauty and we use humor and irreverent wit to expose the forces of bigotry, complacency and guilt that chain the human spirit.

We will number the terms of the cochain complex by n. Later, we will also assume that the filtration is Hausdorff or separated, that is, the intersection of the set of all F < sup > p </ sup > C < sup >•</ sup > is zero, and that the filtration is exhaustive, that is, the union of the set of all F < sup > p </ sup > C < sup >•</ sup > is the entire chain complex C < sup >•</ sup >.

We have the chain

Known for his somewhat bombastic speaking style, Smith once said: We breathe the free air, we have the best looking men and handsomest women, and if they ( Non-Mormons ) envy us our position, well they may, for they are a poor, narrow-minded, pinch-backed race of men, who chain themselves down to the law of monogamy, and live all their days under the dominion of one wife.

We first have to show that the ascending Kleene chain of f exists in L. To show that, we prove the following lemma:

In a 1957 interview with the German weekly Der Spiegel, Weizsäcker frankly admitted to the scientific ambitions of those years " We wanted to know if chain reactions were possible.

Two years after the group's first break-up, the video for the song We are Sudamerican rockers was chosen to be the first aired by the then-nascent chain MTV Latin America, October 1, 1993.

On October 1, 1993 was opened for the first time MTV chain and the first video broadcast was We are sudamerican rockers.

We adopt the convention the countable chain condition means the downwards countable chain condition, in other words no two elements have a common lower bound.

We peeled open the source, examined the chain of acquisition.

We know the line of this frontier which ran from the Main across the upland Odenwald to the upper waters of the Neckar and was defended by a chain of forts.

We know that Śāntarakṣita focused his early teachings in Tibet directed to the ' seven that were tested ' upon the ' ten virtues ' (; Sanskrit: daśakuśala ; Pali: dasa sikkhapadani or dasa sila ) and ' the chain of casual relation ' ( Sanskrit: pratītyasamutpāda ).