Betti diagrams from graphs

The emergence of Boij-S\"oderberg theory has given rise to new connections between combinatorics and commutative algebra. Herzog, Sharifan, and Varbaro recently showed that every Betti diagram of an ideal with a k-linear minimal resolution arises from that of the Stanley-Reisner ideal of a simplicial complex. In this paper, we extend their result for the special case of 2-linear resolutions using purely combinatorial methods. Specifically, we show bijective correspondences between Betti diagrams of ideals with 2-linear resolutions, threshold graphs, and anti-lecture hall compositions. Moreover, we prove that any Betti diagram of a module with a 2-linear resolution is realized by a direct sum of Stanley-Reisner rings associated to threshold graphs. Our key observation is that these objects are the lattice points in a normal reflexive lattice polytope.Comment: To appear in Algebra and Number Theory, 15 pages, 7 figure

On properties of modified degree six chordal rings network

Chordal rings are circulant graphs and have deserved significant attention in the last decade. Chordal rings were used to build interconnection networks for distributed and parallel systems. There are many of publications analyzing their networks properties. In this paper, we present the main properties of the latest method of chordal rings called Modified Chordal Rings Degree Six (CHRm6). The properties are connectivity, Hamiltonian cycle and asymmetry. We prove some lemmas and theorems for every property. All of these properties are useful for further works especially in developing a routing scheme

Mitral Valve Surgery for Rheumatic Lesions in Young Patients

BACKGROUND: The appropriateness of rheumatic mitral valve repair remains controversial due to the risks of recurrent mitral dysfunction and need for reoperation. The aims of this study were to determine the overall short- and long-term outcomes of pediatric rheumatic mitral valve surgery in our center. METHODS: Single-center, observational, retrospective study that analyzed the results of rheumatic mitral valve surgery in young patients, consecutively operated by the same team, between 1999 and 2014. RESULTS: We included 116 patients (mean age = 12.6 ± 3.5 years), of which 66 (57%) were females. A total of 116 primary surgical interventions and 22 reoperations were performed. Primary valve repair was possible in 86 (74%) patients and valve replacement occurred in 30 (26%). Sixty percent of the patients were followed up beyond three months after surgery (median follow-up time = 9.2 months [minimum = 10 days; maximum = 15 years]). Long-term clinical outcomes were favorable, with most patients in New York Heart Association functional class I (89.6%) and in sinus rhythm (85%). Freedom from reoperation for primary valve repair at six months, five years, and ten years was 96.4% ± 0.25%, 72% ± 0.72%, and 44.7% ± 1.34%, respectively. Freedom from reoperation for primary valve replacement at six months, five years, and ten years was 100%, 91.7% ± 0.86%, and 91.7% ± 0.86%, respectively. Mitral stenosis as the primary lesion dictated early reintervention. CONCLUSIONS: Despite the greater rate of reoperation, especially when the primary lesion was mitral stenosis, rheumatic mitral valve repair provides similar clinical outcomes as compared with replacement, with the advantage of avoiding anticoagulation

Soundtracks and the moving image

This study is a survey and analysis of modern soundtrack pieces accompanying recent movies. I will analyze the form and chord progressions within the scores to better understand the role each musical element plays in the piece. Then, I will compare these details to find common themes to see if they correspond more to the Western classical tradition, or to modern pop music, and how each textural element enhances its visual counterpart. This paper is an attempt to define and understand the formal procedures are common across the audio that accompanies modern movies