Acta Mathematica Vietnamica最新文献

We overview a web of conjectures about torsors under reductive groups over regular rings and survey some techniques that have been used for making progress on such problems.

We give a proof of the combinatorial Brill-Noether conjecture for cactus graphs. This conjecture was formulated by Baker in 2008 when studying the interaction between algebraic curves theory and graph theory. By analyzing the treelike structure of cactus graphs, we produce a construction proof that is based on the Chip Firing Game theory.

We introduce the notion of (formal) partial derivative and develop an application of it to get a new proof for the commutativity of the classical squaring and the Kameko squaring.

Consider the ideal ((x_{1} , dotsc , x_{n})^{d} subseteq k[x_{1} , dotsc , x_{n}]), where k is any field. This ideal can be resolved by both the L-complexes of Buchsbaum and Eisenbud, and the Eliahou-Kervaire resolution. Both of these complexes admit the structure of an associative DG algebra, and it is a question of Peeva as to whether these DG structures coincide in general. In this paper, we construct an isomorphism of complexes between the aforementioned complexes that is also an isomorphism of algebras with their respective products, thus giving an affirmative answer to Peeva’s question.

In this paper, we explore the almost Cohen-Macaulayness of the associated graded ring of stretched ({mathfrak m})-primary ideals with small first Hilbert coefficient in a Cohen-Macaulay local ring ((A,{mathfrak m})). In particular, we explore the structure of stretched ({mathfrak m})-primary ideals satisfying the equality e₁(I) = e₀(I) − ℓ_A(A/I) + 4, where e₀(I) and e₁(I) denote the multiplicity and the first Hilbert coefficient, respectively.

The conformal module of conjugacy classes of braids is an invariant that appeared earlier than the entropy of conjugacy classes of braids, and is inversely proportional to the entropy. Using the relation between the two invariants, we give a short conceptional proof of an earlier result on the conformal module. Mainly, we consider situations, when the conformal module of conjugacy classes of braids serves as obstruction for the existence of homotopies (or isotopies) of smooth objects involving braids to the respective holomorphic objects, and present theorems on the restricted validity of Gromov’s Oka principle in these situations.

In this paper, basing on the forward-backward method and inertial techniques, we introduce a new algorithm for solving a variational inequality problem over the fixed point set of a nonexpansive mapping. The strong convergence of the algorithm is established under strongly monotone and Lipschitz continuous assumptions imposed on the cost mapping. As an application, we also apply and analyze our algorithm to solve a convex minimization problem of the sum of two convex functions.

Schreier bases are introduced and used to show that skew polynomial rings are free ideal rings, i.e., rings whose one-sided ideals are free of unique rank, as well as to compute a rank of one-sided ideals together with a description of corresponding bases. The latter fact, a so-called Schreier-Lewin formula (Lewin Trans. Am. Math. Soc. 145, 455–465 1969), is a basic tool determining a module type of perfect localizations which reveal a close connection between classical Leavitt algebras, skew polynomial rings, and free associative algebras.

In this article, we give explicit minimal generators of the first syzygy of the Hibi ring for a planar distributive lattice in terms of sublattices. We also give a characterization when it is linearly related and derive an exact formula for the first Betti number of a planar distributive lattice.