Let-M-be-a-structure-of-signature-σ-and-N-a-substr

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Page "Elementary equivalence" ¶ 16

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Let and M

Let ( m, n ) be a pair of amicable numbers with m < n, and write m = gM and n = gN where g is the greatest common divisor of m and n. If M and N are both coprime to g and square free then the pair ( m, n ) is said to be regular, otherwise it is called irregular or exotic.

Let M be a smooth manifold and let x be a point in M. Let T < sub > x </ sub > M be the tangent space at x.

Let M be a smooth manifold and let x be a point in M. Let I < sub > x </ sub > be the ideal of all functions in C ∞( M ) vanishing at x, and let I < sub > x </ sub > 2 be the set of functions of the form, where f < sub > i </ sub >, g < sub > i </ sub > ∈ I < sub > x </ sub >.

Let M be a smooth manifold and let f ∈ C ∞( M ) be a smooth function.

Let r be a non zero real number and let the r th power mean ( M r ) of a series of real variables ( a < sub > 1 </ sub >, a < sub > 2 </ sub >, a < sub > 3 </ sub >, ... ) be defined as

Let M be a ( pseudo -) Riemannian manifold, which may be taken as the spacetime of general relativity.

Let M and N be ( left or right ) modules over the same ring, and let f: M → N be a module homomorphism.

Let M be an n × n Hermitian matrix.

Let P − 1 DP be an eigendecomposition of M, where P is a unitary complex matrix whose rows comprise an orthonormal basis of eigenvectors of M, and D is a real diagonal matrix whose main diagonal contains the corresponding eigenvalues.

Let ( M, g ) be a Riemannian manifold and ƒ: M m → R n a short C ∞- embedding ( or immersion ) into Euclidean space R n , where n ≥ m + 1.

Let M and N be smooth manifolds and be a smooth map.

Let and be

Let the open enemy to it be regarded as a Pandora with her box opened ; ;

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 him be now ''!!

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 T be a linear operator on the finite-dimensional vector space V over the field F.

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 Af be the null space of Af.

Let N be a linear operator on the vector space V.

Let T be a linear operator on the finite-dimensional vector space V over the field F.

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 this be denoted by Af.

Let it be granted then that the theological differences in this area between Protestants and Roman Catholics appear to be irreconcilable.

Let not your heart be troubled, neither let it be afraid ''.

The same God who called this world into being when He said: `` Let there be light ''!!

For those who put their trust in Him He still says every day again: `` Let there be light ''!!

Let us therefore put first things first, and make sure of preserving the human race at whatever the temporary price may be ''.

Let her out, let her out -- that would be the solution, wouldn't it??

Let and structure

Ancient resources as well as recent archaeological evidence suggest that, at one point, Caligula had the palace extended to annex this structure. When several kings came to Rome to pay their respects to him and argued about their nobility of descent, he cried out " Let there be one Lord, one King ".

Let G denote the set of bijective functions over A that preserve the partition structure of A: ∀ x ∈ A ∀ g ∈ G ( g ( x ) ∈ ).

Let us briefly recall the structure theory in the case of a finitely generated module over a PID.

Let be a ( positive ) integer and consider the elliptic curve ( a set of points with some structure on it ).

Let X be locally ringed space with structure sheaf O < sub > X </ sub >; we want to define the tangent space T < sub > x </ sub > at the point x ∈ X.

Let P → M be a principal bundle over a manifold M with structure Lie group G and a principal connection ω.

Let be any set, let be an algebraic structure of type generated by.

Let the underlying set of this algebraic structure, sometimes called universe, be, and let be a function.

Let be a spin structure on a Riemannian manifold that is, an equivariant lift of the oriented orthonormal frame bundle with respect to the double covering

The frame bundle F ( E ) can be given a natural topology and bundle structure determined by that of E. Let ( U < sub > i </ sub >, φ < sub > i </ sub >) be a local trivialization of E. Then for each x ∈ U < sub > i </ sub > one has a linear isomorphism φ < sub > i, x </ sub >: E < sub > x </ sub > → R k .

Let π: E → X be a fibre bundle over a topological space X with structure group G and typical fibre F. By definition, there is a left action of G ( as a transformation group ) on the fibre F. Suppose furthermore that this action is effective.

Let us extend to compact Lie group and consider " integrable " orbits for which the symplectic structure comes from a line bundle then quantization leads to the irreducible representations of.

Let ρ: TP / G → M be the projection onto M. The fibres of the bundle TP / G under the projection ρ carry an additive structure.

Let L be the language of first-order arithmetic, and let N be the standard structure for L. Thus ( L, N ) is the " interpreted first-order language of arithmetic.

Let us note, that these three axioms of a curvature structure give rise to a well-developed structure theory, formulated in terms of projectors ( a Weyl projector, giving rise to Weyl curvature and an Einstein projector, needed for the setup of the Einsteinian gravitational equations ).

Let V i be the subspace of V on which L < sub > 0 </ sub > has eigenvalue i. Assume that V is acted on by a group G which preserves all of its structure.

The idea that all things were created “ by mine Only Begotten ” ( i. e., Jesus Christ, in his premortal state ) is made clear, as is the Son ’ s identity as the co-creator at the time when God said “ Let us make man .” Otherwise, the structure and basic premises of the Genesis account of the Creation were left intact.

Let H be a topological group and let G be a covering space of H. If G and H are both path-connected and locally path-connected, then for any choice of element e * in the fiber over e ∈ H, there exists a unique topological group structure on G, with e * as the identity, for which the covering map p: G → H is a homomorphism.

The minimal separators also form an algebraic structure: For two fixed vertices a and b of a given graph G, an ( a, b )- separator S can be regarded as a predecessor of another ( a, b )- separator T, if every path from a to b meets S before it meets T. More rigorously, the predecessor relation is defined as follows: Let S and T be two ( a, b )- separators in ' G '.

Let V be a real vector space with a complex structure J.

Let be an almost complex manifold with almost complex structure.

Let be a smooth Riemann surface ( also called a complex curve ) with complex structure.

Let: the Deligne – Mumford moduli space of curves of genus g with n marked points and denote the moduli space of stable maps into X of class A, for some chosen almost complex structure J on X compatible with its symplectic form.

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