Local moduli of holomorphic bundles

We study moduli of holomorphic vector bundles on non-compact varieties. We discuss filtrability and algebraicity of bundles and calculate dimensions of local moduli. As particularly interesting examples, we describe numerical invariants of bundles on some local Calabi-Yau threefolds.Comment: 18 pages. Revision history: v1: As submitted for publication. v2: minor corrections, as publishe

Numerically flat Higgs vector bundles

After providing a suitable definition of numerical effectiveness for Higgs bundles, and a related notion of numerical flatness, in this paper we prove, together with some side results, that all Chern classes of a Higgs-numerically flat Higgs bundle vanish, and that a Higgs bundle is Higgs-numerically flat if and only if it is has a filtration whose quotients are flat stable Higgs bundles. We also study the relation between these numerical properties of Higgs bundles and (semi)stability.Comment: 11 page

Computing Tropical Varieties

The tropical variety of a $d$-dimensional prime ideal in a polynomial ring with complex coefficients is a pure $d$-dimensional polyhedral fan. This fan is shown to be connected in codimension one. We present algorithmic tools for computing the tropical variety, and we discuss our implementation of these tools in the Gr\"obner fan software \texttt{Gfan}. Every ideal is shown to have a finite tropical basis, and a sharp lower bound is given for the size of a tropical basis for an ideal of linear forms.Comment: 22 pages, 2 figure

A clinical and haemodynamic investigation into the role of calf perforating vein surgery in patients with venous ulceration and deep venous incompetence

Objective:To determine the clinical efficacy and local haemodynamic effects of perforating vein surgery in ulcerated limbs with combined deep and perforating vein incompetence.Design:Prospective, interventional study.Materials and methods:Seven ulcerated limbs with combined primary deep and perforating vein incompetence were studied. Clinical efficacy was determined by ultimate ulcer healing and reduction in ulcer area, local haemodynamics were assessed at three sites with photoplethysmographic 90% venous refilling times (PPG RT90); both assessments were performed pre- and 1-month postoperatively.Results:None of the ulcers healed following perforating vein surgery, the median (range) ulcer areas pre- and postoperatively were 31 (7–685) cm2 and 35.5 (7–796) cm2 (Wilcoxon p = 0.07). Preoperative PPG RT90 demonstrated a global abnormality of venous function at all sites examined that persisted after perforating vein surgery.Conclusion:In the presence of deep venous incompetence perforating vein surgery had no influence on venous function or ulcer healing. We conclude that perforating vein surgery is not indicated for the treatment of venous ulceration in limbs with primary deep venous incompetence

On special quadratic birational transformations of a projective space into a hypersurface

We study transformations as in the title with emphasis on those having smooth connected base locus, called "special". In particular, we classify all special quadratic birational maps into a quadric hypersurface whose inverse is given by quadratic forms by showing that there are only four examples having general hyperplane sections of Severi varieties as base loci.Comment: Accepted for publication in Rendiconti del Circolo Matematico di Palerm

Affine T-varieties of complexity one and locally nilpotent derivations

Let X=spec A be a normal affine variety over an algebraically closed field k of characteristic 0 endowed with an effective action of a torus T of dimension n. Let also D be a homogeneous locally nilpotent derivation on the normal affine Z^n-graded domain A, so that D generates a k_+-action on X that is normalized by the T-action. We provide a complete classification of pairs (X,D) in two cases: for toric varieties (n=\dim X) and in the case where n=\dim X-1. This generalizes previously known results for surfaces due to Flenner and Zaidenberg. As an application we compute the homogeneous Makar-Limanov invariant of such varieties. In particular we exhibit a family of non-rational varieties with trivial Makar-Limanov invariant.Comment: 31 pages. Minor changes in the structure. Fixed some typo

A Product Formula for the Normalized Volume of Free Sums of Lattice Polytopes

The free sum is a basic geometric operation among convex polytopes. This note focuses on the relationship between the normalized volume of the free sum and that of the summands. In particular, we show that the normalized volume of the free sum of full dimensional polytopes is precisely the product of the normalized volumes of the summands.Comment: Published in the proceedings of 2017 Southern Regional Algebra Conferenc

Five-year study assessing the clinical utility of anti-Müllerian hormone measurements in reproductive-age women with cancer

An important discussion point before chemotherapy is ovarian toxicity, a side-effect that profoundly affects young women with cancer. Their quality of life after successful treatment, including the ability to conceive, is a major concern. We asked whether serum anti-Müllerian hormone (AMH) measurements before chemotherapy for two most common malignancies are predictive of long-term changes in ovarian reserve? A prospective cohort study measured serum AMH in 66 young women with lymphoma and breast cancer, before and at 1 year and 5 years after chemotherapy, compared with 124 healthy volunteers of the same age range (18-43 years). Contemporaneously, patients reported their menses and live births during 5-year follow-up. After adjustment for age, serum AMH was 1.4 times higher (95% CI 1.1 to 1.9; P < 0.02) in healthy volunteers than in cancer patients before chemotherapy. A strong correlation was observed between baseline and 5-year AMH in the breast cancer group (P < 0.001, regression coefficient = 0.58, 95% CI 0.29 to 0.89). No significant association was found between presence of menses at 5 years and serum AMH at baseline (likelihood ratio test from logistics regression analysis). Reproductive-age women with malignancy have lower serum AMH than healthy controls even before starting chemotherapy. Pre-chemotherapy AMH was significantly associated with long-term ovarian function in women with breast cancer. At key time points, AMH measurements could be used as a reproductive health advisory tool for young women with cancer. Our results highlight the unsuitability of return of menstruation as a clinical indicator of ovarian reserve after chemotherapy. [Abstract copyright: Crown Copyright © 2019. Published by Elsevier Ltd. All rights reserved.

Reprinted Article “Subintimal Angioplasty of Femoropopliteal Artery Occlusions: The Long-term Results”

AbstractThe technique of subintimal angioplasty has been attempted on 200 consecutive femoropopliteal artery occlusions of median (range) length 11 (2–37) cm. The principle of the technique is to traverse the occlusion in the subintimal plane and recanalise by inflating the angioplasty balloon within the subintimal space. The technical success rate was 159/200 (80%) and was not significantly different for occlusions <10 cm (81%, n = 73), 11–20 cm (83%, n = 63) or >20 cm (68%, n = 23), p = 0.20. There were no deaths nor limb loss resulting from the procedure. The median (range) ankle-brachial pressure index increased from 0.61 (0.21–1.0) preangioplasty to 0.90 (0.26–1.50) postangioplasty. The actuarial haemodynamic patencies of technically successful procedures at 12 and 36 months were 71% and 58% respectively, the symptomatic patencies were 73% and 61%. A multiple regression analysis showed that smoking multiplied the risk of reocclusion by 2.70 (p < 0.001), each additional run-off vessel reduced the risk by 0.54 (p < 0.001) and the risk increased by 1.73 (p = 0.020) for every 10 cm of occlusion length. In conclusion, the technical success rate (80%) of subintimal angioplasty for femoropopliteal occlusions is unrelated to occlusion length and for all procedures, including technical failures, cumulative symptomatic and haemodynamic patencies of 46 and 48% can be achieved at 3 years. The factors influencing long-term patency were smoking, the number of calf run-off vessels and occlusion length