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Then and any
Then in 2 we show that any line involution with the properties that ( A ) It has no complex of invariant lines, and ( B ) Its singular lines form a complex consisting exclusively of the lines which meet a twisted curve, is necessarily of the type discussed in 1.
I said quietly, gaining nerve, ready to ask any question at all, no matter how intimate, ready to be rebuffed, `` Then why did she leave Israel??
Then, one must locate any relevant statutes and cases.
Then, for any given sequence of integers a < sub > 1 </ sub >, a < sub > 2 </ sub >, …, a < sub > k </ sub >, there exists an integer x solving the following system of simultaneous congruences.
Then along with Archbishop Wilhelm von Brandenburg of The Archbishopric of Riga and his Coadjutor Christoph von Mecklenburg, Kettler gave to Magnus the portions of The Kingdom of Livonia, which he had taken possession of, but they refused to give him any more land.
Then to define multiplication, it suffices by the distributive law to describe the product of any two such terms, which is given by the rule
Then Fischer and Rabin ( 1974 ) proved that any decision algorithm for Presburger arithmetic has a worst-case runtime of at least, for some constant c > 0.
Then in 726, Leo issued an iconoclast edit, condemning possession of any icon of the saints.
For f a real polynomial in x, and for any a in such an algebra define f ( a ) to be the element of the algebra resulting from the obvious substitution of a into f. Then for any two such polynomials f and g, we have that ( fg ) ( a )
Then the expected number of i type organisms produced by any j type parent is,
Then, in 1964 the new Chrysler 426 Hemi engine so dominated the series in a Plymouth Belvedere " Sport Fury ", the homologation rules were changed so that 1, 000 of any engine and car had to be sold to the public to qualify as a stock part, instead of just 500.
Then, for any non-zero element x of M, the cyclic submodule xR must equal M. Fix such an x.
Then, these philosophers say, it is rational to act on those beliefs that have best withstood criticism, whether or not they meet any specific criterion of truth.
Then for any
Then the formula Cold ( r )→ High ( t ) is true for any t and therefore any t gives a correct control given r. A rigorous logical justification of fuzzy control is given in Hájek's book ( see Chapter 7 ) where fuzzy control is represented as a theory of Hájek's basic logic.
Then in 1991, President George H. W. Bush made an attempt to abolish affirmative action altogether claiming that “ any regulation, rule, enforcement practice or other aspect of these programs that mandates, encourages, or otherwise involves the use of quotas, preferences, set-asides or other devices on the basis of race, sex, religion or national origin are to be terminated as soon as is legally feasible.
Then, given any point x and neighbourhood G of x, there is a closed neighbourhood E of x that is a subset of G.
Then write a review without using any of the following words: " Progressive rock ", " Genesis ", " Fish ", " heavy metal ", " dinosaurs ", " predictable ", " concept album ".
", Yossarian replies, " Then I'd certainly be a damned fool to feel any other way.
Then S is not a base for any topology on R. To show this, suppose it were.
Then is an open cover of S, but any finite subcollection of has the form of C discussed previously, and thus cannot be an open subcover of S. This contradicts the compactness of S. Hence, every accumulation point of S is in S, so S is closed.
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.

Then and vector
Then I < sub > x </ sub > and I < sub > x </ sub >< sup > 2 </ sup > are real vector spaces and the cotangent space is defined as the quotient space T < sub > x </ sub >< sup >*</ sup > M = I < sub > x </ sub > / I < sub > x </ sub >< sup > 2 </ sup >.
Then I and I < sup > 2 </ sup > are real vector spaces, and T < sub > x </ sub > M may be defined as the dual space of the quotient space I / I < sup > 2 </ sup >.
Then consider a vector tangent to:
Let X be a normed topological vector space over F, compatible with the absolute value in F. Then in X *, the topological dual space X of continuous F-valued linear functionals on X, all norm-closed balls are compact in the weak -* topology.
Then the joint distribution of is multivariate normal with mean vector and covariance matrix
Then the coordinates of the vector V in the new coordinates are required to satisfy the transformation law
In particular, let p define the coordinates of points in a reference frame M coincident with a fixed frame F. Then, when the origin of M is displaced by the translation vector d relative to the origin of F and rotated by the angle φ relative to the x-axis of F, the new coordinates in F of points in M are given by
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.
Then the two equations still allow the normal to rotate around the view vector, thus additional constraints are needed from prior geometric information.
Then the contravariant coordinates of any vector v can be obtained by the dot product of v with the contravariant basis vectors:
Then the zero vector of this space can be expressed as a linear combination of no elements, which again is an empty sum.
Then every Hodge class on X is a linear combination with rational coefficients of Chern classes of vector bundles on X.
Let X be a g-dimensional torus given as X = V / L where V is a complex vector space of dimension g and L is a lattice in V. Then X is an abelian variety if and only if there exists a positive definite hermitian form on V whose imaginary part takes integral values on L × L.
Then k < sub > x </ sub > := R < sub > x </ sub >/ m < sub > x </ sub > is a field and m < sub > x </ sub >/ m < sub > x </ sub >< sup > 2 </ sup > is a vector space over that field ( the cotangent space ).
Let v ∈ T < sub > p </ sub > M be a tangent vector to the manifold at p. Then there is a unique geodesic γ < sub > v </ sub > satisfying γ < sub > v </ sub >( 0 )
Then, where is the vector ( 1, 0 ,..., 0 )< sup > T </ sup >, ||·|| is the Euclidean norm and is an m-by-m identity matrix, set
Then applying the Gram – Schmidt process to the three vectors ( A < sub > 2 </ sub >− A < sub > 1 </ sub >, A < sub > 3 </ sub >− A < sub > 1 </ sub >, V ) produces an orthonormal basis of space, the third vector of which will be normal to plane A.
Then a representation of Q is just a covariant functor from this category to the category of finite dimensional vector spaces.
Then and are equivalent: The functor which maps the object of to the vector space and the matrices in to the corresponding linear maps is full, faithful and essentially surjective.
Then the propositions of incidence are derived from the following basic result on vector spaces: given subspaces U and V of a vector space W, the dimension of their intersection is at least dim U + dim V − dim W. Bearing in mind that the dimension of the projective space P ( W ) associated to W is dim W − 1, but that we require an intersection of subspaces of dimension at least 1 to register in projective space ( the subspace
Let E → M be a vector bundle of rank k and let F ( E ) be the principal frame bundle of E. Then a ( principal ) connection on F ( E ) induces a connection on E. First note that sections of E are in one-to-one correspondence with right-equivariant maps F ( E ) → R < sup > k </ sup >.
Let V a representation of G, and form the vector bundle V = Q ×< sub > G </ sub > V over M. Then the principal G-connection α on Q induces a covariant derivative on V, which is a first order linear differential operator
Then, A is a vector potential for v, that is,

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