Hyperforests on the Complete Hypergraph by Grassmann Integral Representation

We study the generating function of rooted and unrooted hyperforests in a general complete hypergraph with n vertices by using a novel Grassmann representation of their generating functions. We show that this new approach encodes the known results about the exponential generating functions for the different number of vertices. We consider also some applications as counting hyperforests in the k-uniform complete hypergraph and the one complete in hyperedges of all dimensions. Some general feature of the asymptotic regimes for large number of connected components is discussed.Comment: 35 page

A fermionic field theory for spanning hyperforests

Il modello di Potts generalizza il modello di Ising del ferromagnetismo assumendo che le variabili di spin possano assumere q stati distinti. L'interazione tra i primi vicini è a due soli valori a seconda che questi si trovino nello stesso stato o in stati differenti. Nonostante questo modello abbia ricevuto inizialmente poca attenzione, fin dagli anni '70 è stato oggetto di grande interesse a seguito del suo ricco comportamento critico e dei suoi stretti legami con alcuni problemi di statistica su reticolo, di combinatoria e di teoria dei grafi. Nel 1972 Fortuin e Kasteleyn mostrarono che è possibile estendere la definizione del modello di Potts anche a valori di q non interi. Nel caso in cui l'interazione sia esclusivamente ferromagnetica, questa estensione definisce una misura di probabilità, nota con il nome di random-cluster model, che include come caso particolare (q=1) il già noto modello di percolazione. In questa tesi considereremo in particolar modo il limite q -> 0 del modello di Potts. Questo caso limite ha un particolare significato combinatorio, infatti la funzione di partizione del modello di Potts corrisponde per q -> 0 alla funzione generatrice delle foreste massimali sul grafo in cui il modello è definito. Il limite q -> 0 del modello di Potts acquista ulteriore interesse a seguito della recente scoperta per cui esso può essere descritto da una teoria fermionica contenente un termine Gaussiano e uno speciale accoppiamento a quattro fermioni. Questa teoria fermionica risulta essere equivalente, ad ogni ordine perturbativo, al modello O(N) prolungato analiticamente a N = -1 e ad un modello sigma non lineare con gruppo di (super) simmetria OSP(1|2). Questa corrispondenza, seppur perturbativa, ci segnala che, in due dimensioni, questa teoria è asintoticamente libera come gran parte dei modelli sigma non-lineari e le teorie di gauge non-abeliane in quattro dimensioni. In questo lavoro viene sviluppata un estensione della teoria fermionica sopracitata al caso in cui siano presenti interazioni a più corpi. Generalizzando opportunamente il modello di Potts per includere tali interazioni, si mostra come nel limite q -> 0 la funzione di partizione di questo modello si riconduca alla funzione generatrice delle iperforeste massimali sull'ipergrafo definito dall'interazione a più corpi. Viene quindi formulata in termini di variabili di Grassmann una teoria fermionica che descrive tali oggetti combinatori. Successivamente questa teoria viene studiata nell'ipotesi che le interazioni formino un ipergrafo completo. In questo caso, che fisicamente corrisponde ad una teoria di campo medio, il modello è esattamente risolubile e la funzione di partizione può essere calcolata esplicitamente. Ciò costituisce di per sé un risultato di interesse combinatorio in quanto fornisce il conteggio delle iperforeste massimali sull'ipergrafo completo. Si mostra infine questa teoria sia anch'essa in corrispondenza (sempre perturbativa) con un modello sigma non lineare con supersimmetria OSP(1|2). Viene osservato come la supersimmetria del modello \sigma non lineare induca nella teoria puramente fermionica una super-simmetria non manifesta e come questa sia in relazione con l'algebra dei prodotti scalari per N = -1

A tree-decomposed transfer matrix for computing exact Potts model partition functions for arbitrary graphs, with applications to planar graph colourings

Combining tree decomposition and transfer matrix techniques provides a very general algorithm for computing exact partition functions of statistical models defined on arbitrary graphs. The algorithm is particularly efficient in the case of planar graphs. We illustrate it by computing the Potts model partition functions and chromatic polynomials (the number of proper vertex colourings using Q colours) for large samples of random planar graphs with up to N=100 vertices. In the latter case, our algorithm yields a sub-exponential average running time of ~ exp(1.516 sqrt(N)), a substantial improvement over the exponential running time ~ exp(0.245 N) provided by the hitherto best known algorithm. We study the statistics of chromatic roots of random planar graphs in some detail, comparing the findings with results for finite pieces of a regular lattice.Comment: 5 pages, 3 figures. Version 2 has been substantially expanded. Version 3 shows that the worst-case running time is sub-exponential in the number of vertice

Granulicatella adiacens and Abiotrophia defectiva Native Vertebral Osteomyelitis: Three Cases and Literature Review of Clinical Characteristics and Treatment Approach

Granulicatella adiacens and Abiotrophia defectiva are an increasingly recognized cause of osteoarticular infections. We describe two cases of G. adiacens and one case of A. defectiva native vertebral osteomyelitis (NVO) and review all published cases. Nine cases of G. adiacens NVO and two cases of A. defectiva NVO were previously described. Patients were usually middle-aged men, and classical risk factors for NVO were present in half of the cases. Concomitant bacteremia was reported in 78.6% of cases, and concurrent infective endocarditis occurred in 36.4% of this sub-group of patients. Many different antibiotic schemes were recorded, with median treatment duration of 6weeks. In the most recent reports, glycopeptides represented the most frequent empirical therapy, possibly due to the increasing emergence of G. adiacens and A. defectiva penicillin-resistant strains. Stabilization surgery was rarely required (14.3% of cases), and clinical cure was generally achieved. In conclusion, Granulicatella spp. and Abiotrophia spp. NVO is rare but increasingly described. A total antibiotic course of six weeks seems to be appropriate for noncomplicated cases, and clinical outcome is generally favorable

Leukocyte Integrin Antagonists as a Novel Option to Treat Dry Age-Related Macular Degeneration

Age-related macular degeneration (AMD) is a complex multifactorial degenerative disease that leads to irreversible blindness. AMD affects the macula, the central part of the retina responsible for sharp central vision. Retinal pigment epithelium (RPE) is the main cellular type affected in dry AMD. RPE cells form a monolayer between the choroid and the neuroretina and are in close functional relationship with photoreceptors; moreover, RPE cells are part of the blood retina barrier that is disrupted in ocular diseases such as AMD. During ocular inflammation lymphocytes and macrophages are recruited, contact RPE and produce pro-inflammatory cytokines, which play an important role in AMD pathogenesis. The interaction between RPE and immune cells is mediated by leukocyte integrins, heterodimeric transmembrane receptors, and adhesion molecules, including VCAM-1 and ICAM-1. Within this frame, this study aimed to characterize RPE-leukocytes interaction and to investigate any potentially beneficial effects induced by integrin antagonists (DS-70, MN27 and SR714), developed in previous studies. ARPE-19 cells were co-cultured for different incubation times with Jurkat cells and apoptosis and necrosis levels were analyzed by flow cytometry. Moreover, we measured the mRNA levels of the pro-inflammatory cytokine IL-1\u3b2 and the expression of adhesion molecules VCAM-1 and ICAM-1. We found that RPE-lymphocyte interaction increased apoptosis and necrosis levels in RPE cells and the expression of IL-1\u3b2. This interaction was mediated by the binding of \u3b14\u3b21 and \u3b1L\u3b22 integrins to VCAM-1 and ICAM-1, respectively. The blockade of RPE-lymphocyte interaction with blocking antibodies highlighted the pivotal role played by integrins. Therefore, \u3b14\u3b21 and \u3b1L\u3b22 integrin antagonists were employed to disrupt RPE-lymphocyte crosstalk. Small molecule integrin antagonists proved to be effective in reducing RPE cell death and expression of IL-1\u3b2, demonstrating that integrin antagonists could protect RPE cells from detrimental effects induced by the interaction with immune cells recruited to the retina. Overall, the leukocyte integrin antagonists employed in the present study may represent a novel opportunity to develop new drugs to fight dry AMD