In computational complexity, a complexity class is given by a set of problems or functions, and a basic challenge is to show separations of complexity classes A 6= B especially when A is known to be a subset of B. In this paper we introduce a homological theory of functions that can be used to establish complexity separations, while also providing other interesting consequences. We propose to a...

Several representations of the symmetric group, arising from different combinatorial, algebraic and geometric constructions, have lead to the same character, up to multiplication by the sign character: the homology of partition lattice (cf. [5, 7, 13]), the top component of a special quotient of the Stanley-Reisner ring of this same lattice [4], the top component of the cohomology algebra of th...

The Zykov ring of signed finite simple graphs with topological join as addition and compatible multiplication is an integral domain but not a unique factorization domain. We know that because by the graph complement operation it is isomorphic to the strong Sabidussi ring with disjoint union as addition. We prove that the Euler characteristic is a ring homomorphism from the strong ring to the in...

A result of G. Walker and R. Wood states that the space indecomposable elements in degree 2 n -1-n polynomial algebra

Abstract A local ring R is regular if and only every finitely generated -module has finite projective dimension. Moreover, the residue field k a test module: This characterization can be extended to bounded derived category $\mathsf {D}^{\mathsf f}(R)$ , which contains small objects regular. Recent results of Pollitz, completing work initiated by Dwyer–Greenlees–Iyengar, yield an analogous for ...

We study combinatorial aspects of the representation theory of the 0-Hecke algebra Hn(0), a deformation of the group algebra of the symmetric group Sn. We study the action of Hn(0) on the polynomial ring in n variables. We show that the coinvariant algebra of this action naturally carries the regular representation of Hn(0), giving an analogue of the well-known result for the symmetric group by...

In computational complexity, a complexity class is given by a set of problems or functions, and a basic challenge is to show separations of complexity classes A 6= B especially when A is known to be a subset of B. In this paper we introduce a homological theory of functions that can be used to establish complexity separations, while also providing other interesting consequences. We propose to a...

Adviser: Carina Curto Neurons in the brain represent external stimuli via neural codes. These codes often arise from stimulus-response maps, associating to each neuron a convex receptive field. An important problem confronted by the brain is to infer properties of a represented stimulus space without knowledge of the receptive fields, using only the intrinsic structure of the neural code. How d...

Let C be a clutter with a perfect matching e1, . . . , eg of König type and let ∆C be the Stanley-Reisner complex of the edge ideal of C. If all c-minors of C have a free vertex and C is unmixed, we show that ∆C is pure shellable. We are able to describe in combinatorial terms when ∆C is pure. If C has no cycles of length 3 or 4, then it is shown that ∆C is pure if and only if ∆C is pure shella...

For a graph G=(V,E) the edge ring k[G] is k[x1,…,xn]/I(G), where n=|V| and I(G) generated by {xixj;{i,j}∈E}. The conjecture we treat following.Conjecture 1. If has 2-linear resolution, then projective dimension of K[G], pd (k[G]), equals maximal degree vertex in G.As far as know, this first mentioned paper Gitler Valencia [7, Conjecture 4.13], there it called Eliahou-Villarreal conjecture. trea...