Isolated factorizations and their applications in simplicial affine semigroups

We introduce the concept of isolated factorizations of an element of a commutative monoid and study its properties. We give several bounds for the number of isolated factorizations of simplicial affine semigroups and numerical semigroups. We also generalize $\alpha$-rectangular numerical semigroups to the context of simplicial affine semigroups and study their isolated factorizations. As a consequence of our results, we characterize those complete intersection simplicial affine semigroups with only one Betti minimal element in several ways. Moreover, we define Betti sorted and Betti divisible simplicial affine semigroups and characterize them in terms of gluings and their minimal presentations. Finally, we determine all the Betti divisible numerical semigroups, which turn out to be those numerical semigroups that are free for any arrangement of their minimal generators

A snapshot of image pre-processing for convolutional neural networks: case study of MNIST

In the last five years, deep learning methods and particularly Convolutional Neural Networks (CNNs) have exhibited excellent accuracies in many pattern classification problems. Most of the state-of-the-art models apply data-augmentation techniques at the training stage. This paper provides a brief tutorial on data preprocessing and shows its benefits by using the competitive MNIST handwritten digits classification problem. We show and analyze the impact of different preprocessing techniques on the performance of three CNNs, LeNet, Network3 and DropConnect, together with their ensembles. The analyzed transformations are, centering, elastic deformation, translation, rotation and different combinations of them. Our analysis demonstrates that data-preprocessing techniques, such as the combination of elastic deformation and rotation, together with ensembles have a high potential to further improve the state-of-the-art accuracy in MNIST classification.Spanish Government TIN2014-524 57251-PAndalusian Research Plans P11-TIC-7765Spanish Government RYC-2015-1813

Fast sampling of satisfying assignments from random kk-SAT

We give a nearly linear-time algorithm to approximately sample satisfying assignments in the random $k$-SAT model when the density of the formula scales exponentially with $k$. The best previously known sampling algorithm for the random $k$-SAT model applies when the density $\alpha=m/n$ of the formula is less than $2^{k/300}$ and runs in time $n^{\exp(\Theta(k))}$ (Galanis, Goldberg, Guo and Yang, SIAM J. Comput., 2021). Here $n$ is the number of variables and $m$ is the number of clauses. Our algorithm achieves a significantly faster running time of $n^{1 + o_k(1)}$ and samples satisfying assignments up to density $\alpha\leq 2^{0.039 k}$. The main challenge in our setting is the presence of many variables with unbounded degree, which causes significant correlations within the formula and impedes the application of relevant Markov chain methods from the bounded-degree setting (Feng, Guo, Yin and Zhang, J. ACM, 2021; Jain, Pham and Vuong, 2021). Our main technical contribution is a $o_k(\log n )$ bound of the sum of influences in the $k$-SAT model which turns out to be robust against the presence of high-degree variables. This allows us to apply the spectral independence framework and obtain fast mixing results of a uniform-block Glauber dynamics on a carefully selected subset of the variables. The final key ingredient in our method is to take advantage of the sparsity of logarithmic-sized connected sets and the expansion properties of the random formula, and establish relevant properties of the set of satisfying assignments that enable the fast simulation of this Glauber dynamics.Comment: 47 page

Cyclotomic numerical semigroup polynomials with few irreducible factors

A numerical semigroup $S$ is cyclotomic if its semigroup polynomial $P_S$ is a product of cyclotomic polynomials. The number of irreducible factors of $P_S$ (with multiplicity) is the polynomial length $\ell(S)$ of $S.$ We show that a cyclotomic numerical semigroup is complete intersection if $\ell(S)\le 2$. This establishes a particular case of a conjecture of Ciolan, Garc\'{i}a-S\'{a}nchez and Moree (2016) claiming that every cyclotomic numerical semigroup is complete intersection. In addition, we investigate the relation between $\ell(S)$ and the embedding dimension of $S.$Comment: 14 page

Cyclotomic exponent sequences of numerical semigroups

We study the cyclotomic exponent sequence of a numerical semigroup $S,$ and we compute its values at the gaps of $S,$ the elements of $S$ with unique representations in terms of minimal generators, and the Betti elements $b\in S$ for which the set $\{a \in \operatorname{Betti}(S) : a \le_{S}b\}$ is totally ordered with respect to $\le_S$ (we write $a \le_S b$ whenever $a - b \in S,$ with $a,b\in S$). This allows us to characterize certain semigroup families, such as Betti-sorted or Betti-divisible numerical semigroups, as well as numerical semigroups with a unique Betti element, in terms of their cyclotomic exponent sequences. Our results also apply to cyclotomic numerical semigroups, which are numerical semigroups with a finitely supported cyclotomic exponent sequence. We show that cyclotomic numerical semigroups with certain cyclotomic exponent sequences are complete intersections, thereby making progress towards proving the conjecture of Ciolan, Garc\'ia-S\'anchez and Moree (2016) stating that $S$ is cyclotomic if and only if it is a complete intersection.Comment: 25 page

Cyclotomic exponent sequences of numerical semigroups

Part of the work on this paper was done during an internship in the Fall of 2016 carried out by the third author at the Max Planck Institute for Mathematics in Bonn and during a one-week visit in April 2019. He would like to thank the fourth author for the invitation and the institute staff for their hospitality and support. The project was completed during a stay of the first author at the same institute. Substantial progress on this paper was made in February 2017, when the first and the fourth author were invited by the second and third author for one week to the University of Granada. They are grateful for the hospitality, for the inspiring and cheerful atmosphere and, last but not least, for the excellent tapas and wine!We study the cyclotomic exponent sequence of a numerical semigroup S, and we compute its values at the gaps of S, the elements of S with unique representations in terms of minimal generators, and the Betti elements b∈S for which the set {a∈Betti(S):a≤Sb} is totally ordered with respect to ≤S (we write a≤Sb whenever a−b∈S, with a,b∈S). This allows us to characterize certain semigroup families, such as Betti-sorted or Betti-divisible numerical semigroups, as well as numerical semigroups with a unique Betti element, in terms of their cyclotomic exponent sequences. Our results also apply to cyclotomic numerical semigroups, which are numerical semigroups with a finitely supported cyclotomic exponent sequence. We show that cyclotomic numerical semigroups with certain cyclotomic exponent sequences are complete intersections, thereby making progress towards proving the conjecture of Ciolan, García-Sánchez and Moree (2016) stating that S is cyclotomic if and only if it is a complete intersection

Subcutaneous anti-COVID-19 hyperimmune immunoglobulin for prevention of disease in asymptomatic individuals with SARS-CoV-2 infection: a double-blind, placebo-controlled, randomised clinical trialResearch in context

Summary: Background: Anti-COVID-19 hyperimmune immunoglobulin (hIG) can provide standardized and controlled antibody content. Data from controlled clinical trials using hIG for the prevention or treatment of COVID-19 outpatients have not been reported. We assessed the safety and efficacy of subcutaneous anti-COVID-19 hyperimmune immunoglobulin 20% (C19-IG20%) compared to placebo in preventing development of symptomatic COVID-19 in asymptomatic individuals with SARS-CoV-2 infection. Methods: We did a multicentre, randomized, double-blind, placebo-controlled trial, in asymptomatic unvaccinated adults (≥18 years of age) with confirmed SARS-CoV-2 infection within 5 days between April 28 and December 27, 2021. Participants were randomly assigned (1:1:1) to receive a blinded subcutaneous infusion of 10 mL with 1 g or 2 g of C19-IG20%, or an equivalent volume of saline as placebo. The primary endpoint was the proportion of participants who remained asymptomatic through day 14 after infusion. Secondary endpoints included the proportion of individuals who required oxygen supplementation, any medically attended visit, hospitalisation, or ICU, and viral load reduction and viral clearance in nasopharyngeal swabs. Safety was assessed as the proportion of patients with adverse events. The trial was terminated early due to a lack of potential benefit in the target population in a planned interim analysis conducted in December 2021. ClinicalTrials.gov registry: NCT04847141. Findings: 461 individuals (mean age 39.6 years [SD 12.8]) were randomized and received the intervention within a mean of 3.1 (SD 1.27) days from a positive SARS-CoV-2 test. In the prespecified modified intention-to-treat analysis that included only participants who received a subcutaneous infusion, the primary outcome occurred in 59.9% (91/152) of participants receiving 1 g C19-IG20%, 64.7% (99/153) receiving 2 g, and 63.5% (99/156) receiving placebo (difference in proportions 1 g C19-IG20% vs. placebo, −3.6%; 95% CI -14.6% to 7.3%, p = 0.53; 2 g C19-IG20% vs placebo, 1.1%; −9.6% to 11.9%, p = 0.85). None of the secondary clinical efficacy endpoints or virological endpoints were significantly different between study groups. Adverse event rate was similar between groups, and no severe or life-threatening adverse events related to investigational product infusion were reported. Interpretation: Our findings suggested that administration of subcutaneous human hyperimmune immunoglobulin C19-IG20% to asymptomatic individuals with SARS-CoV-2 infection was safe but did not prevent development of symptomatic COVID-19. Funding: Grifols