We compute the Gorenstein weak dimension of a coherent power series rings over a commutative rings and we show that, in general, Gwdim (R) ≤ 1 does not imply that R is an arithmetical ring.

Let Vn be the SL2-module of binary forms of degree n and let V = Vn1 ⊕ . . . ⊕ Vnp . We consider the algebra R = O(V )2 of polynomial functions on V invariant under the action of SL2. The measure of the intricacy of these algebras is the length of their chains of syzygies, called homological dimension hdR. Popov gave in 1983 a classi cation of the cases in which hdR ≤ 10 for a single binary for...

We define homological matrices, construct examples of one-dimension restricted homo-logical quantum field theories, and show a relationship between the two theories.

In this paper, we explore hidden persistent homological structures in sparse regressions. Sparse regressions are usually parameterized by tuning parameters that determine the sparsity of solutions. By treating the tuning parameters as additional dimension, we can have multi-scale representations. We can show there exist hidden persistent homological structures in these dimensions. By exploiting...

Let A be a connected cochain DG algebra such that its underlying graded A# is the skew polynomial k⟨x1,x2,x3⟩/x1x2+x2x1x2x3+x3x2x3x1+x1x3,|x1|=|x2|=|x3|=1. Then differential ∂A determined by ∂A(x1)∂A(x2)∂A(x3)=Mx12x22x32 for some M∈M3(k). When rank r(M) of M belongs to {1,2,3}, we compute H(A) case case. The computational results in this paper give substantial support research various homologic...

Pinched Veronese rings are formed by removing an algebra generator from a subring of polynomial ring. We study the homological properties such rings, including Cohen-Macaulay, Gorenstein, and complete intersection properties. Greco Martino classified Cohen-Macaulayness pinched maximum entry exponent vector monomial; we re-prove their results with semigroup methods correct omission small class e...