A short proof of Kneser's addition theorem for abelian groups

Martin Kneser proved the following addition theorem for every abelian group $G$. If $A,B \subseteq G$ are finite and nonempty, then $|A+B| \ge |A+K| + |B+K| - |K|$ where $K = \{g \in G \mid g+A+B = A+B \}$. Here we give a short proof of this based on a simple intersection union argument.Comment: 3 page

Schrijver graphs and projective quadrangulations

In a recent paper [J. Combin. Theory Ser. B}, 113 (2015), pp. 1-17], the authors have extended the concept of quadrangulation of a surface to higher dimension, and showed that every quadrangulation of the $n$-dimensional projective space $P^n$ is at least $(n+2)$-chromatic, unless it is bipartite. They conjectured that for any integers $k\geq 1$ and $n\geq 2k+1$, the Schrijver graph $SG(n,k)$ contains a spanning subgraph which is a quadrangulation of $P^{n-2k}$. The purpose of this paper is to prove the conjecture

Averages of Fourier coefficients of Siegel modular forms and representation of binary quadratic forms by quadratic forms in four variables

Let $-d$ be a a negative discriminant and let $T$ vary over a set of representatives of the integral equivalence classes of integral binary quadratic forms of discriminant $-d$. We prove an asymptotic formula for $d \to \infty$ for the average over $T$ of the number of representations of $T$ by an integral positive definite quaternary quadratic form and obtain results on averages of Fourier coefficients of linear combinations of Siegel theta series. We also find an asymptotic bound from below on the number of binary forms of fixed discriminant $-d$ which are represented by a given quaternary form. In particular, we can show that for growing $d$ a positive proportion of the binary quadratic forms of discriminant $-d$ is represented by the given quaternary quadratic form.Comment: v5: Some typos correcte

Prodsimplicial-Neighborly Polytopes

Simultaneously generalizing both neighborly and neighborly cubical polytopes, we introduce PSN polytopes: their k-skeleton is combinatorially equivalent to that of a product of r simplices. We construct PSN polytopes by three different methods, the most versatile of which is an extension of Sanyal and Ziegler's "projecting deformed products" construction to products of arbitrary simple polytopes. For general r and k, the lowest dimension we achieve is 2k+r+1. Using topological obstructions similar to those introduced by Sanyal to bound the number of vertices of Minkowski sums, we show that this dimension is minimal if we additionally require that the PSN polytope is obtained as a projection of a polytope that is combinatorially equivalent to the product of r simplices, when the dimensions of these simplices are all large compared to k.Comment: 28 pages, 9 figures; minor correction

On the Volume of Boolean expressions of Large Congruent Balls

We consider the volume of a Boolean expression of some congruent balls about a given system of centers in the d-dimensional Euclidean space. When the radius r of the balls is large, this volume can be approximated by a polynomial of r, which will be computed up to an O(r^{d−3}) error term. We study how the top coefficients of this polynomial depend on the set of the centers. It is known that in the case of the union of the balls, the top coefficients are some constant multiples of the intrinsic volumes of the convex hull of the centers. Thus, the coefficients in the general case lead to generalizations of the intrinsic volumes, in particular, to a generalization of the mean width of a set. Some known results on the mean width, along with the theorem on its monotonicity under contractions are extended to the "Boolean analogues" of the mean width

The Kodaira dimension of the moduli of K3 surfaces

The moduli space of polarised K3 surfaces of degree 2d is a quasi-projective variety of dimension 19. For general d very little has been known about the Kodaira dimension of these varieties. In this paper we present an almost complete solution to this problem. Our main result says that this moduli space is of general type for d>61 and for d=46,50,54,58,60.Comment: 47 page

On the critical pair theory in abelian groups : Beyond Chowla's Theorem

We obtain critical pair theorems for subsets S and T of an abelian group such that |S+T| < |S|+|T|+1. We generalize some results of Chowla, Vosper, Kemperman and a more recent result due to Rodseth and one of the authors.Comment: Submitted to Combinatorica, 23 pages, revised versio

On complex surfaces diffeomorphic to rational surfaces

In this paper we prove that no complex surface of general type is diffeomorphic to a rational surface, thereby completing the smooth classification of rational surfaces and the proof of the Van de Ven conjecture on the smooth invariance of Kodaira dimension.Comment: 34 pages, AMS-Te

Association of age with perioperative morbidity among patients undergoing surgical management of minor burns

INTRODUCTION: Burn injuries are associated with significant morbidity, often necessitating surgical management. Older patients are more prone to burns and more vulnerable to complications following major burns. While the relationship between senescence and major burns has already been thoroughly investigated, the role of age in minor burns remains unclear. To better understand differences between elderly and younger patients with predominantly minor burns, we analyzed a multi-institutional database. METHODS: We reviewed the 2008-2020 ACS-NSQIP database to identify patients who had suffered burns according to ICD coding and underwent initial burn surgery. RESULTS: We found 460 patients, of which 283 (62%) were male and 177 (38%) were female. The mean age of the study cohort was 46 ± 17 years, with nearly one-fourth (n = 108; 23%) of all patients being aged ≥60 years. While the majority (n = 293; 64%) suffered from third-degree burns, 22% (n = 99) and 15% (n = 68) were diagnosed with second-degree burns and unspecified burns, respectively. An average operation time of 46 min, a low mortality rate of 0.2% (n = 1), a short mean length of hospital stay (1 day), and an equal distribution of in- and outpatient care (51%, n = 234 and 49%, n = 226, respectively) indicated that the vast majority of patients suffered from minor burns. Patients aged ≥60 years showed a significantly prolonged length of hospital stay (p0.0001), creatinine (p>0.0001), white blood cell count (p=0.02), partial thromboplastin time (p = 0.004), and lower levels of albumin (p = 0.0009) and hematocrit (p>0.0001) were identified as risk factors for the occurrence of any complication. Further, complications were more frequent among patients with lower body burns. DISCUSSION: In conclusion, patients ≥60 years undergoing surgery for predominantly minor burns experienced significantly more complications. Minor lower body burns correlated with worse outcomes and a higher incidence of adverse events. Decreased levels of serum albumin and hematocrit and elevated values of blood urea nitrogen, creatinine, white blood count, and partial thromboplastin time were identified as predictive risk factors for complications