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Let L be an ordered set, called a concrete set and let L ′ be another ordered set, called an abstract set.
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Let and L
Let P be the root of the unbalanced subtree, with R and L denoting the right and left children of P respectively.
Suppose that in a mathematical language L, it is possible to enumerate all of the defined numbers in L. Let this enumeration be defined by the function G: W → R, where G ( n ) is the real number described by the nth description in the sequence.
Let L < sub > 1 </ sub >, L < sub > 2 </ sub >, L ′< sub > 1 </ sub > and L ′< sub > 2 </ sub > be ordered sets.
Let us suppose that L is a complete lattice and let f be a monotonic function from L into L. Then, any x ′ such that f ′( x ′) ≤ x ′ is an abstraction of the least fixed-point of f, which exists, according to the Knaster – Tarski theorem.
Theorem: Let R be a Dedekind domain with fraction field K. Let L be a finite degree field extension of K and denote by S the integral closure of R in L. Then S is itself a Dedekind domain.
Let us now consider a three-dimensional cubical box that has a side length L ( see infinite square well ).
Let and be
Let every policeman and park guard keep his eye on John and Jane Doe, lest one piece of bread be placed undetected and one bird survive.
Let us assume that it would be possible for an enemy to create an aerosol of the causative agent of epidemic typhus ( Rickettsia prowazwki ) over City A and that a large number of cases of typhus fever resulted therefrom.
Let p be the minimal polynomial for T, Af, where the Af, are distinct irreducible monic polynomials over F and the Af are positive integers.
Let V be a finite-dimensional vector space over an algebraically closed field F, e.g., the field of complex numbers.
Let N be a positive integer and let V be the space of all N times continuously differentiable functions F on the real line which satisfy the differential equation Af where Af are some fixed constants.
Let Q be a nonsingular quadric surface bearing reguli Af and Af, and let **zg be a Af curve of order K on Q.
Let us take a set of circumstances in which I happen to be interested on the legislative side and in which I think every one of us might naturally make such a statement.
Let the state of the stream leaving stage R be denoted by a vector Af and the operating variables of stage R by Af.
Let it be granted then that the theological differences in this area between Protestants and Roman Catholics appear to be irreconcilable.
Let us therefore put first things first, and make sure of preserving the human race at whatever the temporary price may be ''.
Let and ordered
* Let be the set of ordered pairs of integers with not zero, and define an equivalence relation on according to which if and only if.
Suppose G is an ordered abelian group, meaning an abelian group with a total ordering "<" respecting the group's addition, so that a < b if and only if a + c < b + c for all c. Let I be a well-ordered subset of G, meaning I contains no infinite descending chain.
Let be a probability space with a filtration, for some ( totally ordered ) index set ; and let be a measurable space.
: Let L be a partially ordered set with the smallest element ( bottom ) and let f: L → L be an order-preserving function.
Let be a probability space with a filtration, for some ( totally ordered ) index set ; and let be a measurable space.
Let E be a vector bundle of fibre dimension k over a differentiable manifold M. A local frame for E is an ordered basis of local sections of E.
Let L be a Lie algebra over K and X a totally ordered basis of L. A canonical monomial over X is a finite sequence ( x < sub > 1 </ sub >, x < sub > 2 </ sub > ..., x < sub > n </ sub >) of elements of X which is non-decreasing in the order ≤, that is, x < sub > 1 </ sub > ≤ x < sub > 2 </ sub > ≤ ... ≤ x < sub > n </ sub >.
Another, more general definition may be given in terms of a filtration: Let be an ordered index set ( often or a compact subset thereof ), and let be a filtered probability space, i. e. a probability space equipped with a filtration.
Let V be a finite-dimensional real vector space and let b < sub > 1 </ sub > and b < sub > 2 </ sub > be two ordered bases for V. It is a standard result in linear algebra that there exists a unique linear transformation A: V → V that takes b < sub > 1 </ sub > to b < sub > 2 </ sub >.
Let us consider the disordered phase ( > 0 ), ordered phase ( < 0 ) and critical temperature ( = 0 ) phases separately.
0.536 seconds.