A study of a family of monomial ideals

In this paper, we study a family of rational monomial parametrizations. We investigate a few structural properties related to the corresponding monomial ideal [Formula: see text] generated by the parametrization. We first find the implicit equation of the closure of the image of the parametrization. Then we provide a minimal graded free resolution of the monomial ideal [Formula: see text], and describe the minimal graded free resolution of the symmetric algebra of [Formula: see text]. Finally, we provide a method to compute the defining equations of the Rees algebra of [Formula: see text] using three moving planes that follow the parametrization.

Parameter-free rendering of single-molecule localization microscopy data for parameter-free resolution estimation

Localization microscopy is a super-resolution imaging technique that relies on the spatial and temporal separation of blinking fluorescent emitters. These blinking events can be individually localized with a precision significantly smaller than the classical diffraction limit. This sub-diffraction localization precision is theoretically bounded by the number of photons emitted per molecule and by the sensor noise. These parameters can be estimated from the raw images. Alternatively, the resolution can be estimated from a rendered image of the localizations. Here, we show how the rendering of localization datasets can influence the resolution estimation based on decorrelation analysis. We demonstrate that a modified histogram rendering, termed bilinear histogram, circumvents the biases introduced by Gaussian or standard histogram rendering. We propose a parameter-free processing pipeline and show that the resolution estimation becomes a function of the localization density and the localization precision, on both simulated and state-of-the-art experimental datasets.

Resolution-enhanced OCT and expanded framework of information capacity and resolution in coherent imaging

Spatial resolution in optical microscopy has traditionally been treated as a fixed parameter of the optical system. Here, we present an approach to enhance transverse resolution in beam-scanned optical coherence tomography (OCT) beyond its aberration-free resolution limit, without any modification to the optical system. Based on the theorem of invariance of information capacity, resolution-enhanced (RE)-OCT navigates the exchange of information between resolution and signal-to-noise ratio (SNR) by exploiting efficient noise suppression via coherent averaging and a simple computational bandwidth expansion procedure. We demonstrate a resolution enhancement of 1.5× relative to the aberration-free limit while maintaining comparable SNR in silicone phantom. We show that RE-OCT can significantly enhance the visualization of fine microstructural features in collagen gel and ex vivo mouse brain. Beyond RE-OCT, our analysis in the spatial-frequency domain leads to an expanded framework of information capacity and resolution in coherent imaging that contributes new implications to the theory of coherent imaging. RE-OCT can be readily implemented on most OCT systems worldwide, immediately unlocking information that is beyond their current imaging capabilities, and so has the potential for widespread impact in the numerous areas in which OCT is utilized, including the basic sciences and translational medicine.

On the Betti numbers and Rees algebras of ideals with linear powers

An ideal $$I \subset \mathbb {k}[x_1, \ldots , x_n]$$ I ⊂ k [ x 1 , … , x n ] is said to have linear powers if $$I^k$$ I k has a linear minimal free resolution, for all integers $$k>0$$ k > 0 . In this paper, we study the Betti numbers of $$I^k$$ I k , for ideals I with linear powers. We provide linear relations on the Betti numbers, which holds for all ideals with linear powers. This is especially useful for ideals of low dimension. The Betti numbers are computed explicitly, as polynomials in k, for the ideal generated by all square-free monomials of degree d, for $$d=2, 3$$ d = 2 , 3 or $$n-1$$ n - 1 , and the product of all ideals generated by s variables, for $$s=n-1$$ s = n - 1 or $$n-2$$ n - 2 . We also study the generators of the Rees ideal, for ideals with linear powers. Particularly, we are interested in ideals for which the Rees ideal is generated by quadratic elements. This problem is related to a conjecture on matroids by White.

𝐿-dimension for modules over a local ring

We introduce a new homological dimension for finitely generated modules over a commutative local ring R R , which is based on a complex derived from a free resolution L L of the residue field of R R , and called L L -dimension. We prove several properties of L L -dimension, give some applications, and compare L L -dimension to complete intersection dimension.

An upper bound for the first Hilbert coefficient of Gorenstein algebras and modules

Let R R be a polynomial ring over a field and M = ⨁ n M n M= \bigoplus _n M_n be a finitely generated graded R R -module, minimally generated by homogeneous elements of degree zero with a graded R R -minimal free resolution F \mathbf {F} . A Cohen-Macaulay module M M is Gorenstein when the graded resolution is symmetric. We give an upper bound for the first Hilbert coefficient, e 1 e_1 in terms of the shifts in the graded resolution of M M . When M = R / I M = R/I , a Gorenstein algebra, this bound agrees with the bound obtained in [ES09] in Gorenstein algebras with quasi-pure resolution. We conjecture a similar bound for the higher coefficients.

From grids to pseudo-grids of lines: Resolution and seminormality

Over an infinite field [Formula: see text], we investigate the minimal free resolution of some configurations of lines. We explicitly describe the minimal free resolution of complete grids of lines and obtain an analogous result about the so-called complete pseudo-grids. Moreover, we characterize the total Betti numbers of configurations that are obtained posing a multiplicity condition on the lines of either a complete grid or a complete pseudo-grid. Finally, we analyze when a complete pseudo-grid is seminormal, differently from a complete grid. The main tools that have been involved in our study are the mapping cone procedure and properties of liftings, of pseudo-liftings and of weighted ideals. Although complete grids and pseudo-grids are hypersurface configurations and many results about such type of configurations have already been stated in literature, we give new contributions, in particular about the maps of the resolution.

On the configurations of codimension two linear subspaces of ℙN with the Waldschmidt constant less than 2

The purpose of this note is to generalize a result of [M. Dumnicki, T. Szemberg and H. Tutaj-Gasińska, Symbolic powers of planar point configurations II, J. Pure Appl. Alg. 220 (2016) 2001–2016] to higher-dimensional projective spaces and classify all configurations of [Formula: see text]-planes [Formula: see text] in [Formula: see text] with the Waldschmidt constants less than two. We also determine some numerical and algebraic invariants of the defining ideals [Formula: see text] of these classes of configurations, i.e. the resurgence, the minimal free resolution and the regularity of [Formula: see text], as well as the Hilbert function of [Formula: see text].