Acta Mathematica Vietnamica最新文献

We discuss certain results related to the linear independence of alternating multiple zeta values introduced by Harada in 2021.

Given a base field (Bbbk ) of characteristic zero, for each graph G, we associate the artinian algebra A(G) defined by the edge ideal of G and the squares of the variables. We study the weak Lefschetz property of A(G). We classify some classes of graphs with relatively few edges, including paths and cycles, such that its associated artinian ring has the weak Lefschetz property.

We introduce a new approach by using unconstrained optimization to find a solution to the system of the split equality problems in real Hilbert spaces. Our new algorithms do not depend on the norm of the transfer mappings. We also give the relaxed iterative algorithms corresponding to the proposed algorithms. Finally, we present some numerical experiments to demonstrate the performance of the main results.

In this paper we introduce minimal balanced neighborly polynomials and show some methods to construct such polynomials. In particular, using this notion, we prove the existence of balanced neighborly polynomials of the following types: (i) type ((p,dots ,p)) for most prime numbers p, (ii) types ((d-1,d,d,d)), ((d-1,d-1,d,d)) and ((d-1,d-1,d-1,d)) when d is odd or is divisible by 4. We also construct balanced neighborly simplicial spheres of type ((2,4k-1,4k-1,4k-1)).

Let ((R,mathfrak {m})) be a Noetherian local ring such that (widehat{R}) is reduced. We prove that, when (widehat{R}) is (S_2), if there exists a parameter ideal (Qsubseteq R) such that (bar{e}_1(Q)=0), then R is regular and (nu (mathfrak {m}/Q)le 1). This leads to an affirmative answer to a problem raised by Goto-Hong-Mandal [Goto, S., Hong, J., Mandal, M.: The positivity of the first coefficients of normal Hilbert polynomials. Proc. Amer. Math. Soc. 139(7), 2399–2406 (2011)]. We also give an alternative proof (in fact a strengthening) of their main result. In particular, we show that if (widehat{R}) is equidimensional, then (bar{e}_1(Q)ge 0) for all parameter ideals (Qsubseteq R), and in characteristic (p>0), we actually have (e_1^*(Q)ge 0). Our proofs rely on the existence of big Cohen-Macaulay algebras.

In this paper, we establish a series of new results on strong duality and solution existence for conic linear programs in locally convex Hausdorff topological vector spaces and finite-dimensional Euclidean spaces. Namely, under certain regularity conditions based on quasi-relative interiors of convex sets, we prove that if one problem in the dual pair consisting of a primal program and its dual has a solution, then the other problem also has a solution, and the optimal values of the problems are equal. In addition, we show that if the cones are generalized polyhedral convex, then the regularity conditions can be omitted. Moreover, if the spaces are finite-dimensional and the ordering cones are closed convex, then instead of the solution existence condition, it suffices to require the finiteness of the optimal value. The present paper complements our recent research work [Luan, N.N., Yen, N.D.: Strong duality and solution existence under minimal assumptions in conic linear programming. J. Optim. Theory Appl. (https://doi.org/10.1007/s10957-023-02318-w)].

Let (H = langle n_1, n_2,n_3rangle ) be a numerical semigroup. Let (widetilde{H}) be the interval completion of H, namely the semigroup generated by the interval (langle n_1,n_1+1, ldots , n_3rangle ). Let K be a field and K[H] the semigroup ring generated by H. Let (I_H^{*}) be the defining ideal of the tangent cone of K[H]. In this paper, we describe the defining equations of (I_H^{*}). From that, we prove the Herzog-Stamate conjecture for monomial space curves stating that (beta _i(I_H^{*}) le beta _i(I_{widetilde{H}}^{*})) for all i, where (beta _i(I_H^{*})) and (beta _i(I_{widetilde{H}}^{*})) are the ith Betti numbers of (I_H^{*}) and (I_{widetilde{H}}^{*}) respectively.

Let K be a division ring with center Z(K), and n a positive integer. Let (textrm{SL}(n,K)) be the special linear group of degree n over K and (textrm{SD}(n,K)) its subgroup consisting of all diagonal matrices whose Dieudonne’s determinant is (overline{1}). We prove that (textrm{SD}(n,K)) is weakly pronormal, but not pronormal in (textrm{SL}(n,K)) provided either Z(K) is an infinite field in case (nge 3) or Z(K) is a finite field containing at least seven elements in case (nge 5).

Let G be a graph with n vertices and let (S=mathbb {K}[x_1,dots ,x_n]) be the polynomial ring in n variables over a field (mathbb {K}). Assume that I(G) and J(G) denote the edge ideal and the cover ideal of G, respectively. We provide a combinatorial upper bound for the index of depth stability of symbolic powers of J(G). As a consequence, we compute the depth of symbolic powers of cover ideals of fully clique-whiskered graphs. Meanwhile, we determine a class of graphs G with the property that the Castelnuovo–Mumford regularity of S/I(G) is equal to the induced matching number of G.

We prove Bertini type theorems and give some applications of them. The applications are in the context of Lefschetz theorem for Nori fundamental group for normal varieties as well as for geometric formal orbifolds. In another application, it is shown that a certain class of a smooth quasi-projective variety contains a smooth curve such that irreducible lisse (ell )–adic sheaves on the variety with “ramification bounded by a branch data” remains irreducible when restricted to the curve.