Spectral analysis and zeta determinant on the deformed spheres

We consider a class of singular Riemannian manifolds, the deformed spheres $S^N_k$, defined as the classical spheres with a one parameter family $g[k]$ of singular Riemannian structures, that reduces for $k=1$ to the classical metric. After giving explicit formulas for the eigenvalues and eigenfunctions of the metric Laplacian $\Delta_{S^N_k}$, we study the associated zeta functions $\zeta(s,\Delta_{S^N_k})$. We introduce a general method to deal with some classes of simple and double abstract zeta functions, generalizing the ones appearing in $\zeta(s,\Delta_{S^N_k})$. An application of this method allows to obtain the main zeta invariants for these zeta functions in all dimensions, and in particular $\zeta(0,\Delta_{S^N_k})$ and $\zeta'(0,\Delta_{S^N_k})$. We give explicit formulas for the zeta regularized determinant in the low dimensional cases, $N=2,3$, thus generalizing a result of Dowker \cite{Dow1}, and we compute the first coefficients in the expansion of these determinants in powers of the deformation parameter $k$.Comment: 1 figur

Zeta function regularization for a scalar field in a compact domain

We express the zeta function associated to the Laplacian operator on $S^1_r\times M$ in terms of the zeta function associated to the Laplacian on $M$, where $M$ is a compact connected Riemannian manifold. This gives formulas for the partition function of the associated physical model at low and high temperature for any compact domain $M$. Furthermore, we provide an exact formula for the zeta function at any value of $r$ when $M$ is a $D$-dimensional box or a $D$-dimensional torus; this allows a rigorous calculation of the zeta invariants and the analysis of the main thermodynamic functions associated to the physical models at finite temperature.Comment: 19 pages, no figures, to appear in J. Phys.

Laboratorio di Restauro architettonico

Nell'a.a. 2013-14 la didattica del Laboratorio di restauro (Scuola di Architettura Urbanistica Ingegneria delle cstruzioni del Politecnico di Milano) è stata coinvolta in una esperienza di sinergia con l'attività educativa promossa dalla Provincia a sostegno di un gruppo di studenti di scuole lecchesi, a rischio di abbandono scolastico. Tema comune di lavoro è stato la vicenda della fabbrica Faini di Lecco, che produceva materiali metallici, da decenni dismessa. Il tema del 'recupero' è stato quindi affrontato su due fronti: da una parte il gruppo lecchese di allievi delle scuole di secondo grado è stato guidato dagli insegnanti attraverso un'esperienza di laboratorio teatrale; dall'altra parte,gli studenti del Laboratorio di restauro si sono occupati, come previsto dal programma didattico, del progetto di 'recupero' degli edifici del complesso, divenuto testimonianza di archeologia industriale e in attesa di riutilizzazione. L'argomento di lavoro, affrontato da angolazioni e da soggetti evidentemente differenti - anche per età anagrafica - ha visto occasioni comuni di scambio di informazioni, di notizie storiche e di visite dell'area della ex fabbrica. Gli esiti dell'iniziativa sono stati positivi sia per gli allievi delle scuole, sia pe rgli studenti universitari. I primi hanno progettato e messo in scena uno spettacolo, rappresentato in due repliche, al termine del loro percorso di 'recupero', che ha portato anche benefici diretti sul piano scolastico. Gli studenti del Politecnico hanno svolto regolarmente le attività laboratoriali previste e ad alcuni di essi, più particolarmente interessati, è stata offerta l'occasione di seguire da vicino alcuni aspetti tecnici dell'allestimento scenografico e della regia. Inoltre, in concomitanza con le date dello spettacolo, è stata esposta una selezione di elaborati prodotti nell'ambito del Laboratorio di restauro

Ultra-high-field targeted imaging of focal cortical dysplasia: The intracortical black line sign in type IIB

BACKGROUND AND PURPOSE: Conventional MR imaging has limitations in detecting focal cortical dysplasia. We assessed the added value of 7T in patients with histologically proved focal cortical dysplasia to highlight correlations between neuropathology and ultra-high-field imaging. MATERIALS AND METHODS: Between 2013 and 2019, we performed a standardized 7T MR imaging protocol in patients with drug-resistant focal epilepsy. We focused on 12 patients in whom postsurgical histopathology revealed focal cortical dysplasia and explored the diagnostic yield of preoperative 7T versus 1.5/3T MR imaging and the correlations of imaging findings with histopathology. We also assessed the relationship between epilepsy surgery outcome and the completeness of surgical removal of the MR imaging-visible structural abnormality. RESULTS: We observed clear abnormalities in 10/12 patients using 7T versus 9/12 revealed by 1.5/3T MR imaging. In patients with focal cortical dysplasia I, 7T MR imaging did not disclose morphologic abnormalities (n = 0/2). In patients with focal cortical dysplasia II, 7T uncovered morphologic signs that were not visible on clinical imaging in 1 patient with focal cortical dysplasia IIa (n = 1/4) and in all those with focal cortical dysplasia IIb (n = 6/6). T2*WI provided the highest added value, disclosing a peculiar intracortical hypointense band (black line) in 5/6 patients with focal cortical dysplasia IIb. The complete removal of the black line was associated with good postsurgical outcome (n = 4/5), while its incomplete removal yielded unsatisfactory results (n = 1/5). CONCLUSIONS: The high sensitivity of 7T T2*-weighted images provides an additional tool in defining potential morphologic markers of high epileptogenicity within the dysplastic tissue of focal cortical dysplasia IIb and will likely help to more precisely plan epilepsy surgery and explain surgical failures

INCOME INEQUALITY AND GROWTH: PROBLEMS WITH THE ORTHODOX APPROACH

Abstract. This paper discusses the main issues about increasing inequality, whether it matters and its impact on economic activity and growth. It starts by briefly considering the empirical evidence of the share of income going to the top one percent since 1945 in the advanced countries. It then considers whether this represents an increase in the productivity of the top one percent or merely an extraction of economic rent. The empirical evidence suggests the latter is generally the case and, as a consequence, there is not likely to be a trade-off between greater income equality and efficiency (the latter being reflected in a lower economic growth rate). This is reinforced by considering the mainstream explanation of the distribution of income and by a consideration of the argument as to whether labor is paid its marginal product, which is found to be problematic. Hence, some reservations about the use of the aggregate production function are raised. The paper turns next to the question of whether or not a greater degree of inequality causes a slower economic growth, both for the advanced and the developing countries. It next considers if the increasing gap between the top one percent and the rest of the income distribution has been either responsible for, or exacerbated, the Great Recession. It concludes that the degree of inequality is an important factor in determining economic activity and one that has been ignored for too long in macroeconomics

Bilateral Cavitations of Ganglionic Eminence: A Fetal MR Imaging Sign of Halted Brain Development

SUMMARY: Ganglionic eminence is the main transitory proliferative structure of the ventral telencephalon in human fetal brain and it contributes for at least 35% to the population of cortical interneurons; however data on the human GE anomalies are scarce. We report 5 fetal MR imaging observations with bilateral symmetric cavitations in their GE regions resembling an inverted open C shape and separating the GE itself form the deeper parenchyma. Imaging, neuropathology, and follow-up features suggested a malformative origin. All cases had in common characteristics of lissencephaly with agenesis or severe hypoplasia of corpus callosum of probable different genetic basis. From our preliminary observation, it seems that GE cavitations are part of conditions which are also accompanied by severe cerebral structure derangement

Anabolic effects and inhibition of interleukin 6 production induced by neridronate on human osteoblasts

Bisphosphonates (BPs) are pharmacological compounds widely used in the treatment of a variety of bone-related diseases, particularly where the bone-turnover is skewed in favour of osteolysis. The mechanisms by which BPs reduce bone-resorption directly acting on osteoclasts (OCs) are now largely clarified even at molecular level. The researches concerning the BPs effects on osteoblasts (OBs) have instead shown variable results. Objectives: We have investigated the efficacy of neridronate (NER), an amino-BP, as anabolic agent on human OBs. Moreover, we have tried to verify if NER is able to negatively modulate the production of IL-6 on OBs stimulated or not by the pro-inflammatory cytokine Il-1b. Methods: We have tested if different concentrations of NER (from 10-11M to 10-3M), added to primary human OB cultures, could affect the cells number, the endogenous cellular alkaline phosphatase (ALP) activity, the collagen I (COLI) synthesis, the formation of mineralized nodules and the IL-6 production. Our experimental approach was performed testing a wide range of NER concentrations because, under physiological conditions, OBs seems to be exposed to variable and transient levels of the drug. Results: Our results show that NER doesn't negatively affect in vitro the viability, proliferation and cellular activity of human OBs, even after 20 days of exposure to concentrations ²10-5 M (therapeutic dose). In addition, NER seems to enhance the differentiation of cultured OBs in mature bone-forming cells. A maximum increase of COL-I synthesis (+25% after 4 days; p<0.05), ALP activity (+50% after 10 days; p<0.01) and mineralized nodules (+48% after 20 days; p<0.05) was observed in cultures treated with NER 10-8M. A maximal reduction of IL-6 secretion (-24% on IL-1b stimulated cultures and -29% on unstimulated cultures) was observed for NER 10-9 M. Conclusions: These results encourage the use of neridronate in therapy of demineralizing metabolic bone disorders