RULED SURFACES WITH NON-TRIVIAL SURJECTIVE
... C with π(C) = B is linearly equivalent to aC0 + π ∗E for some a > 0 and a divisor E of P1 . Since 0 ≤ C0 · C = aC02 + deg E, we have deg E > 0 and C 2 = a2C02 + 2a deg E > 0. Hence, there is no section C0 with C02 < 0. Therefore π is associated with a semi-stable vector bundle of rank two on B. By [ ...
... C with π(C) = B is linearly equivalent to aC0 + π ∗E for some a > 0 and a divisor E of P1 . Since 0 ≤ C0 · C = aC02 + deg E, we have deg E > 0 and C 2 = a2C02 + 2a deg E > 0. Hence, there is no section C0 with C02 < 0. Therefore π is associated with a semi-stable vector bundle of rank two on B. By [ ...
Vector bundles and torsion free sheaves on degenerations of elliptic
... curves. However, we provide non-classical proofs which, as we think, are simpler and fit well in our approach to coherent sheaves over singular curves. The behavior of the category of vector bundles on a smooth projective curve X is controlled by its genus g(X). If g(X) = 0 then X is a projective li ...
... curves. However, we provide non-classical proofs which, as we think, are simpler and fit well in our approach to coherent sheaves over singular curves. The behavior of the category of vector bundles on a smooth projective curve X is controlled by its genus g(X). If g(X) = 0 then X is a projective li ...
On Exact Controllability and Complete Stabilizability for Linear
... The operator Cλ (A, B) is surjective if and only if for some constant c and for all x ∈ X, we have kCλ∗ (A, B)xk2 ≥ ckxk2 . This equivalent to (K(λ)x, x) ≥ 2λckxk2 , which means that K(λ) is uniformly positive definite and therefore invertible. For the case of a bounded operator, the theorem gives t ...
... The operator Cλ (A, B) is surjective if and only if for some constant c and for all x ∈ X, we have kCλ∗ (A, B)xk2 ≥ ckxk2 . This equivalent to (K(λ)x, x) ≥ 2λckxk2 , which means that K(λ) is uniformly positive definite and therefore invertible. For the case of a bounded operator, the theorem gives t ...
Previous1-LinearAlgebra-S12.pdf
... [1] (5 pts) [2] (5 pts) [3] (5 pts) [4] (5 pts) [5] (5 pts) [6] (5 pts) ...
... [1] (5 pts) [2] (5 pts) [3] (5 pts) [4] (5 pts) [5] (5 pts) [6] (5 pts) ...
Algebra
... • When you have multiple powers, look for things to factor. If you have an expression that = 0 and something is factorable that you can assume is nonzero (e.g., a parameter, such as population growth rate, that has to be positive), then divide it out. If something is factorable that you cannot assu ...
... • When you have multiple powers, look for things to factor. If you have an expression that = 0 and something is factorable that you can assume is nonzero (e.g., a parameter, such as population growth rate, that has to be positive), then divide it out. If something is factorable that you cannot assu ...
Appendix 3 - UCLA Department of Mathematics
... Proof. We first observe that it is enough to construct a ccc, dense, linear ordering (X; R) that is not separable. If we have such an (X; R), then we can let X 0 be the set of all Dedekind cuts in (X; R), i.e., the set of all bounded initial segments of (X; R) without R-greatest elements, and we can ...
... Proof. We first observe that it is enough to construct a ccc, dense, linear ordering (X; R) that is not separable. If we have such an (X; R), then we can let X 0 be the set of all Dedekind cuts in (X; R), i.e., the set of all bounded initial segments of (X; R) without R-greatest elements, and we can ...
Isotriviality and the Space of Morphisms on Projective Varieties
... In general, ψ will not be unique. So we define the space of endomorphisms of X as the quotient: Homd (X, L) ∶= {ψ ∈ Homn,d ∶ ψ(X) = X}/{ψ ∼ ψ ′ ⇐⇒ ψ∣X = ψ ′ ∣X } as the analog of Homn,d for Pn . Remark 2.1. The construction of Homd (X, L) is similar to the construction of the space of polynomials in ...
... In general, ψ will not be unique. So we define the space of endomorphisms of X as the quotient: Homd (X, L) ∶= {ψ ∈ Homn,d ∶ ψ(X) = X}/{ψ ∼ ψ ′ ⇐⇒ ψ∣X = ψ ′ ∣X } as the analog of Homn,d for Pn . Remark 2.1. The construction of Homd (X, L) is similar to the construction of the space of polynomials in ...
Basic Concepts of Linear Algebra by Jim Carrell
... This textbook is meant to be an introduction to abstract linear algebra for first, second or third year university students who are specializing in mathematics or a closely related discipline. We hope that parts of this text will be relevant to students of computer science and the physical sciences. ...
... This textbook is meant to be an introduction to abstract linear algebra for first, second or third year university students who are specializing in mathematics or a closely related discipline. We hope that parts of this text will be relevant to students of computer science and the physical sciences. ...
THE PUK´ANSZKY INVARIANT FOR MASAS IN
... masa V N (H) ⊆ V N (G) arising from an abelian subgroup H ⊆ G satisfying the properties already discussed. In the previous section we introduced an equivalence relation on the nontrivial double cosets H \G/H in terms of the commensurability of the stabilizer subgroups Kc for elements c ∈ G\H. The fi ...
... masa V N (H) ⊆ V N (G) arising from an abelian subgroup H ⊆ G satisfying the properties already discussed. In the previous section we introduced an equivalence relation on the nontrivial double cosets H \G/H in terms of the commensurability of the stabilizer subgroups Kc for elements c ∈ G\H. The fi ...
1 Valuations of the field of rational numbers
... space. We define TrL/Q (α) as the trace of this transformation and NmL/Q (α) as its determinant, the subscript L/Q can be dropped if no confusion is possible. The Q-linear transformation induces by the multiplication by α in L has a characteristic polynomial ch(α) = x n − c 1 x n−1 + · · · + (−1)n c ...
... space. We define TrL/Q (α) as the trace of this transformation and NmL/Q (α) as its determinant, the subscript L/Q can be dropped if no confusion is possible. The Q-linear transformation induces by the multiplication by α in L has a characteristic polynomial ch(α) = x n − c 1 x n−1 + · · · + (−1)n c ...
Polynomial Factoring Algorithms and their Computational Complexity
... and so forth, and using similar manipulations to achieve a larger, set of smaller numbers being multiplied and added. Asymptotically, the computational time of such an modification of the algorithm would be faster, but since the number of shifts and additions would be greater as r increases, it is l ...
... and so forth, and using similar manipulations to achieve a larger, set of smaller numbers being multiplied and added. Asymptotically, the computational time of such an modification of the algorithm would be faster, but since the number of shifts and additions would be greater as r increases, it is l ...
Basis (linear algebra)
Basis vector redirects here. For basis vector in the context of crystals, see crystal structure. For a more general concept in physics, see frame of reference.A set of vectors in a vector space V is called a basis, or a set of basis vectors, if the vectors are linearly independent and every vector in the vector space is a linear combination of this set. In more general terms, a basis is a linearly independent spanning set.Given a basis of a vector space V, every element of V can be expressed uniquely as a linear combination of basis vectors, whose coefficients are referred to as vector coordinates or components. A vector space can have several distinct sets of basis vectors; however each such set has the same number of elements, with this number being the dimension of the vector space.