Let R be a standard graded K-algebra, that is, an algebra of the form R = K[x1, . . . , xn]/I where K[x1, . . . , xn] is a polynomial ring over the field K and I is a homogeneous ideal with respect to the grading deg(xi) = 1. Let M be a finitely generated graded R-module. Consider the (essentially unique) minimal graded Rfree resolution of M · · · → Ri → Ri−1 → · · · → R1 → R0 → M → 0 The rank ...

Non-Boolean computations implementing operations on multi-valued variables beyond base 2 allow enhanced computational complexity. We introduce DNA as a functional material for ternary computing, and in particular demonstrate the use of three-valued oligonucleotide inputs to construct a 3 × 3 multiplication table. The system consists of two three-valued inputs of -1; 0; +1 and a fluorophore/quen...

We present novel algorithms to factor polynomials over a finite field Fq of odd characteristic using rank 2 Drinfeld modules with complex multiplication. The main idea is to compute a lift of the Hasse invariant (modulo the polynomial f(x) ∈ Fq[x] to be factored) with respect to a Drinfeld module φ with complex multiplication. Factors of f(x) supported on prime ideals with supersingular reducti...

We describe a natural structure of an abelian intertwining algebra (in the sense of Dong and Lepowsky) on the direct sum of the untwisted vertex operator algebra constructed from the Leech lattice and its (unique) irreducible twisted module. When restricting ourselves to the moonshine module, we obtain a new and conceptual proof that the moonshine module has a natural structure of a vertex oper...

We present an Avoidance Principle for certain graded rings. As an application we fill a gap in the proof of a result by Brodmann, Rohrer and Sazeedeh about the antipolynomiality of the Hilbert-Samuel multiplicity of the graded components of the local cohomology modules of a finitely generated module over a Noetherian homogeneous ring with two-dimensional local base ring. Let R = ⊕ n∈N0 Rn be a ...

The purpose of this paper is to explore some basic facts from radical of submodules in the free multiplication R-module M = Rn. Mathematics Subject Classification: 13C13, 13C99

Primary-like and weakly primary-like submodules are two new generalizations of primary ideals from rings to modules. In fact, the class of primary-like submodules of a module lie between primary submodules and weakly primary-like submodules properly. In this note, we show that these three classes coincide when their elements are submodules of a multiplication module and satisfy the primeful pro...