Acta Mathematica Vietnamica最新文献

Let ((R,mathfrak m,Bbbk )) be a regular local ring of dimension 3. Let I be a Gorenstein ideal of R of grade 3. Buchsbaum and Eisenbud proved that there is a skew-symmetric matrix of odd size such that I is generated by the sub-maximal pfaffians of this matrix. Let J be the ideal obtained by multiplying some of the pfaffian generators of I by (mathfrak m); we say that J is a trimming of I. Building on a recent paper of Vandebogert, we construct an explicit free resolution of R/J and compute a partial DG algebra structure on this resolution. We provide the full DG algebra structure in the appendix. We use the products on this resolution to study the Tor algebra of such trimmed ideals and we use the information obtained to prove that recent conjectures of Christensen, Veliche and Weyman on ideals of class (textbf{G}) hold true in our context. Furthermore, we address the realizability question for ideals of class (textbf{G}).

The purpose of this paper is to characterize Noetherian local rings ((R, mathfrak {m})) such that the Chern numbers of certain (mathfrak {m})-primary ideals in R are bounded above or range among only finitely many values. Consequently, we characterize the Gorensteinness, Cohen-Macaulayness, and generalized Cohen-Macaulayness of local rings in terms of the behavior of their Chern numbers.

Let R be a commutative Noetherian ring of prime characteristic p. The main goal of this paper is to study in some detail when

is an open set in the Zariski topology, where (mathcal {F}^{E_{mathfrak {p}}}) denotes the Frobenius algebra attached to the injective hull of the residue field of (R_{mathfrak {p}}.) We show that this is true when R is a Stanley–Reisner ring; moreover, in this case, we explicitly compute its closed complement, providing an algorithmic method for doing so.

In this paper, we use composite ring extensions to construct a new class of Noetherian rings. Composite ring extensions are examples of pullback constructions, and they are useful in constructing of (counter)-examples.

This paper is concerned with the inverse medium scattering problem of determining the location and shape of penetrable scattering objects from measurements of the scattered field. We study a sampling indicator function for recovering the scattering object in a fast and robust way. A flexibility of this indicator function is that it is applicable to data measured in near-field regime or far-field regime. The implementation of the function is simple and does not involve solving any ill-posed problems. The resolution analysis and stability estimate of the indicator function are investigated using the factorization analysis of the far-field operator along with the Funk-Hecke formula. The performance of the method is verified on both simulated and experimental data.

A Taylor variety consists of all fixed order Taylor polynomials of rational functions, where the number of variables and degrees of numerators and denominators are fixed. In one variable, Taylor varieties are given by rank constraints on Hankel matrices. Inversion of the natural parametrization is known as Padé approximation. We study the dimension and defining ideals of Taylor varieties. Taylor hypersurfaces are interesting for projective geometry, since their Hessians tend to vanish. In three and more variables, there exist defective Taylor varieties whose dimension is smaller than the number of parameters. We explain this with Fröberg’s Conjecture in commutative algebra.

In this paper, we provide the structure of Hopf graphs associated to pairs ((G, mathfrak {r})) consisting of groups G together with ramification datas (mathfrak {r}) and their Leavitt path algebras. Consequently, we characterize the Gelfand-Kirillov dimension, the stable rank, the purely infinite simplicity and the existence of a nonzero finite dimensional representation of the Leavitt path algebra of a Hopf graph via properties of ramification data (mathfrak {r}) and G.

We study the exploration of an Erdös-Rényi random graph by a respondent-driven sampling method, where discovered vertices reveal their neighbors. Some of them receive coupons to reveal in their turn their own neighborhood. This leads to the study of a Markov chain on the random graph that we study. For sparse Erdös-Rényi graphs of large sizes, this process correctly renormalized converges to the solution of a deterministic curve, solution of a system of ODEs absorbed on the abscissa axis. The associated fluctuation process is also studied, providing a functional central limit theorem, with a Gaussian limiting process. Simulations and numerical computation illustrate the study.

In this note, we give some results on maximal subextensions of plurisubharmonic functions on hyperconvex domains in (mathbb C^n) and introduce the notion about cone of maximal subextensions of plurisubharmonic functions. Furthermore, we establish the invariant of this cone through proper holomorphic surjections.

Motivated by the Jacobian problem, this article is concerned with the density of the image set (F( mathbb {Z}^n)) of polynomial maps (Fin mathbb {Z}[X_1,dots ,X_n]^n) with (det DFequiv 1). It is shown that if such a map F is not invertible, its image set (F( mathbb {Z}^n)) must be very thin in the lattice ( mathbb {Z}^n): (1) for almost all lines l in ( mathbb {Z}^n) the numbers (texttt {#}(F^{-1}(l) cap mathbb {Z}^n)) are uniformly bounded; (2) (texttt {#}{ zin F( mathbb {Z}^n): vert z_ivert le B} ll B^{n-1}) as (Brightarrow +infty ), where the implicit constants depend on F.