Surjectivity of near-square random matrices

We show that a nearly square independent and identically distributed random integral matrix is surjective over the integral lattice with very high probability. This answers a question by Koplewitz [6]. Our result extends to sparse matrices as well as to matrices of dependent entries.

INVESTIGATION OF THE SPECTRUM OF A GENERALIZED FRIEDRICHS MODEL: NON-INTEGRAL LATTICE CASE

The article investigates the essential and discrete spectrum of the self-adjoint generalized Friedrichs model. This model corresponds to a system consisting of no more than two particles on a non-integral lattice, and operates in a truncated subspace of Fock space. The number and location of eigenvalues is determined according to the "interaction parameter". Anobvious form of the eigenvectors is found

The Spread Philosophy in the Study of Algebraic Cycles

This chapter discusses the spread philosophy in the study of algebraic cycles, in order to make use of a geometry by considering a variation of Hodge structure where D is the Hodge domain (or the appropriate Mumford–Tate domain) and Γ‎ is the group of automorphisms of the integral lattice preserving the intersection pairing. If we have an algebraic cycle Z on X, taking spreads yields a cycle Ƶ on X. Applying Hodge theory to Ƶ on X gives invariants of the cycle. Another related situation is algebraic K-theory. For example, to study Kₚsuperscript Milnor(k), the geometry of S can be used to construct invariants.

ENUMERATING PRIME LINKS BY A CANONICAL ORDER

The first author defined a well-order in the set of links by embedding it into a canonical well-ordered set of (integral) lattice points. He also gave elementary transformations among lattice points to enumerate the prime links in terms of lattice points under this order. In this paper, we add some new elementary transformations and explain how to enumerate the prime links. We show a table of the first 443 prime links arising from the lattice points of lengths up to 10 under this order. Our argument enables us to find 7 omissions and 1 overlap in Conway's table of prime links of 10 crossings.