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Full-Text Articles in Physical Sciences and Mathematics
On An Effective Variation Of Kronecker's Approximation Theorem, Lenny Fukshansky
On An Effective Variation Of Kronecker's Approximation Theorem, Lenny Fukshansky
CMC Faculty Publications and Research
Let Λ ⊂ Rn be an algebraic lattice, coming from a projective module over the ring of integers of a number field K. Let Z ⊂ Rn be the zero locus of a finite collection of polynomials such that Λ |⊂ Z or a finite union of proper full-rank sublattices of Λ. Let K1 be the number field generated over K by coordinates of vectors in Λ, and let L1, . . . , Lt be linear forms in n variables with algebraic coefficients satisfying an appropriate linear independence condition over K1. For each ε > 0 and a ∈ Rn, …
Bounds For Solid Angles Of Lattices Of Rank Three, Lenny Fukshansky, Sinai Robins
Bounds For Solid Angles Of Lattices Of Rank Three, Lenny Fukshansky, Sinai Robins
CMC Faculty Publications and Research
We find sharp absolute constants C1 and C2 with the following property: every well-rounded lattice of rank 3 in a Euclidean space has a minimal basis so that the solid angle spanned by these basis vectors lies in the interval [C1,C2]. In fact, we show that these absolute bounds hold for a larger class of lattices than just well-rounded, and the upper bound holds for all. We state a technical condition on the lattice that may prevent it from satisfying the absolute lower bound on the solid angle, in which case we …
Algebraic Points Of Small Height Missing A Union Of Varieties, Lenny Fukshansky
Algebraic Points Of Small Height Missing A Union Of Varieties, Lenny Fukshansky
CMC Faculty Publications and Research
Let K be a number field, Q, or the field of rational functions on a smooth projective curve over a perfect field, and let V be a subspace of KN where N≥ 2. Let ZK be a union of varieties defined over K such that V ⊈ ZK. We prove the existence of a point of small height in V \ ZK, providing an explicit upper bound on the height of such a point in terms of the height of V and the degree of hypersurface containing ZK, where dependence on …
On Similarity Classes Of Well-Rounded Sublattices Of Z², Lenny Fukshansky
On Similarity Classes Of Well-Rounded Sublattices Of Z², Lenny Fukshansky
CMC Faculty Publications and Research
A lattice is called well-rounded if its minimal vectors span the corresponding Euclidean space. In this paper we study the similarity classes of well-rounded sublattices of Z2. We relate the set of all such similarity classes to a subset of primitive Pythagorean triples, and prove that it has the structure of a non-commutative infinitely generated monoid. We discuss the structure of a given similarity class, and define a zeta function corresponding to each similarity class. We relate it to Dedekind zeta of Z[i], and investigate the growth of some related Dirichlet series, which reflect on …