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Lattice Extensions And Zeros Of Multilinear Polynomials, Maxwell Forst
Lattice Extensions And Zeros Of Multilinear Polynomials, Maxwell Forst
CGU Theses & Dissertations
We treat several problems related to the existence of lattice extensions preserving certain geometric properties and small-height zeros of various multilinear polynomials. An extension of a Euclidean lattice $L_1$ is a lattice $L_2$ of higher rank containing $L_1$ so that the intersection of $L_2$ with the subspace spanned by $L_1$ is equal to $L_1$. Our first result provides a counting estimate on the number of ways a primitive collection of vectors in a lattice can be extended to a basis for this lattice. Next, we discuss the existence of lattice extensions with controlled determinant, successive minima and covering radius. In …
On Symmetric Operator Ideals And S-Numbers, Daniel Akech Thiong
On Symmetric Operator Ideals And S-Numbers, Daniel Akech Thiong
CGU Theses & Dissertations
Motivated by the well-known theorem of Schauder, we study the relationship between various s-numbers of an operator T and its adjoint T∗ between Banach spaces. For non-compact operator T ∈ L(X, Y ), we do not have a lot of information about the relationship between n-th s-number, sn(T), with sn(T∗ ), however, in chapter 2, by considering X and Y , with lifting and extension properties, respectively, we were able to obtain a relationship between sn(T) with sn(T∗ ) for certain …