The singular Hartree equation in fractional perturbed Sobolev spaces

We establish the local and global theory for the Cauchy problem of the singular Hartree equation in three dimensions, that is, the modification of the non-linear Schr\"odinger equation with Hartree non-linearity, where the linear part is now given by the Hamiltonian of point interaction. The latter is a singular, self-adjoint perturbation of the free Laplacian, modelling a contact interaction at a fixed point. The resulting non-linear equation is the typical effective equation for the dynamics of condensed Bose gases with fixed point-like impurities. We control the local solution theory in the perturbed Sobolev spaces of fractional order between the mass space and the operator domain. We then control the global solution theory both in the mass and in the energy space.Comment: Published on Journal of Nonlinear Mathematical Physics (2018

Mathematical analysis of Bose mixtures and related models: ground state theory and effective dynamics.

This PhD thesis contains new results on the mathematical study of Bose-Einstein condensation and the main part of it is devoted to mixtures of condensates, i.e., systems composed of multiple bosonic species in interaction. We prove the validity of effective ground state theories for mixtures in the Gross-Pitaevskii and mean-field regime. We show that the ground state energy asymptotics, in the large-N limit, is captured by the minimum of a suitable one-body functional. Moreover, we prove that in the ground state all species exhibit Bose-Einstein condensation onto the minimizer of that functional. For mixtures in the mean-field regime, we provide a rigorous justification of Bogoliubov\u2019s theory. This is done by computing the contribution to the ground state energy which is due to excited particles. We also prove a norm approximation for the ground state vector, in the Fock space norm. From the time-dependent viewpoint, we prove for the first time the validity of the effective equations that were previously known due to heuristic physical arguments, and that are confirmed by robust experimental evidence. Our results show that, for mixtures in the Gross-Pitaevskii and mean-field regime, the effective dynamics is governed by a system of non-linear Schr\uf6dinger equations, one for each species of the mixture. In the final part of the thesis we present additional results on problems and models related to the study on mixtures. We were able to derive the effective dynamics for spinor- and pseudo-spinor condensates. The equations that we obtain are precisely those of modern experiments with ultra-cold spin bosons. We also show that the mean-field model provide a time-dependent control of condensation that is very accurate for the typical duration times of experiments. A further result is the global well-posedness in the energy space of the singular Hartree equation. Last, we present new remarks on the adaptation of known techniques that one needs in order to prove the derivation of the magnetic Gross-Pitaevskii equation

Clinical correlates and prognostic implications of severe suicidal ideation in major depressive disorder

Suicidal ideation (SI) is a risk factor for suicidal behaviour. To ascertain the clinical correlates and prognostic impact of severe SI, we analysed 249 outpatients with major depressive disorder (MDD) and suicidal thoughts included in the COmbining Medications to Enhance Depression outcome (CO-MED) trial. Patients with severe SI (36%) were younger at disease onset (P = 0.0033), more severely depressed (P = 0.0029), had more lifetime suicidal behaviour (P < 0.0001) and psychiatric comorbidities (panic disorder: P = 0.0025; post-traumatic stress disorder: P = 0.0216), and a history of childhood maltreatment (neglect: P = 0.0054; emotional abuse: P = 0.0230; physical abuse: P = 0.0076; sexual abuse: P = 0.0016) than those experiencing low-moderate SI. After controlling for depression score, severe SI was positively correlated with lifetime suicidal behaviour (OR [95% CI]: 1.26 [1.12-1.41]), panic disorder (1.05 [1.00-1.12]), and childhood maltreatment (neglect: 1.93 [1.13-3.30]; physical abuse: 2.00 [1.11-3.69]; sexual abuse: 2.13 [1.17-3.88]), and inversely correlated with age of onset (0.97 [0.95-0.99]) and sleep-onset insomnia (0.76 [0.61-0.96]). Finally, the occurrence of serious lifetime suicidal behaviour was predicted by SI severity (2.18 [1.11-4.27]), bipolar score (1.36 [1.02-1.81]), and childhood sexual abuse (2.35 [1.09-5.05]). These results emphasise the importance of assessing childhood maltreatment and bipolar liability in MDD to estimate suicidal behaviour risk

Convergence of states for polaron models in the classical limit

We consider the quasi-classical limit of Nelson-type regularized polaron models describing a particle interacting with a quantized bosonic field. We break translation-invariance by adding an attractive external potential decaying at infinity, acting on the particle. In the strong coupling limit where the field behaves classically we prove that the model's energy quasi-minimizers strongly converge to ground states of the limiting Pekar-like non-linear model. This holds for arbitrarily small external attractive potentials, hence this binding is fully due to the interaction with the bosonic field. We use a new approach to the construction of quasi-classical measures to revisit energy convergence, and a localization method in a concentration-compactness type argument to obtain convergence of states

ZETA™ methodology and variation in the systemic risk of default: accounting for the effects of Type II (false negative) errors variation on lending

Olgiati Stefano - University of Bergamo, Department of Management, Economics and Quantitative Methods. Danovi Alessandro - University of Bergamo The loan manager - dealing with one single borrower at a time and being responsible for that single decision to lend - is exposed to the idiosyncratic risk of default of his customer just like the physician is exposed to the risk of a wrong diagnosis with our strep throat. At the same time – if we do not expect the strep throat diagnostic test kit to change - we would still expect that physician reading that test to become more careful – or update his prior beliefs – about his diagnoses when a flu epidemic is likely to kick in with a certain estimated probability (likelihood). However, this has not been the case with loan management - there is in fact some consensus that before 2007 a reduction in the standards of idiosyncratic risk assessment by lenders - prior to risks pooling - coupled with a worsening of the systemic risk scenario, is partly to blame for the well known 2007-2008 financial crisis, with some of the blame falling also on the incapacity of actuarial mathematical models (test kits) to update worst case scenarios or be calibrated continuously on the basis of variation in the likelihood of default of the underlying risks pool.The authors of this paper argue that, on the other hand, a standard Bayesian transformation of the ZETA bankruptcy prediction methodology introduced by Altman in 1968-1977 allows for a continuous a posterioriupdate of conditional Type I and II errors due to variation in the systemic likelihood of default. The Bayesian transformation can be used both to condition the loan manager's prior decision (generally based on Basel II-compliant Internal Rating Based system or Credit Agency's Rating) and to update such decision on the basis of any posterior hypothesis (based on actuarial frequentist assumptions of conditional hazard rates) regarding the creditworthiness and the probability of default of an underlying pool of securities.At the same time – under a Bayesian framework - the ZETA diagnostic test can be conditioned on the new evidence introduced by other tests to increase the total sensitivity of the default prediction models (IRB ratings, TTC ratings, logit, probit, neural) to update the commercial bank's lending decisions.A ground-state, static meta-analysis of Altman's et al. ZETA original article (1977) reveals that the odds of the commercial bank detecting a default after the ZETA score has been introduced (post-test) is 13.2 times more effective than the a priori prediction. Under the same assumptions, the odds of the commercial bank detecting a survival after (post-test) the ZETA score has been introduced is 12.2 times more effective than the a priori. Integration of the ZETA model with other default prediction models reaches a credibility interval of CI ≥ 95% when the updated likelihood of default is equal to 60%. As expected, the Efficiency Comparison Test ECZETA=.00243 is invariant under the Bayesian transformation

Credit risk management and cyclicality of bank lending to non-financial corporations in Italy during the financial crisis: 2008-2012. A modeling study

Credit to non-financial corporations in Italy is characterized, in the period June 2008-June 2012, by frequent and intense quarterly cyclical fluctuations (peak amplitude €39.2 billion). The amplitude of these fluctuations has been ascribed to the effects of Basel II accords during the financial crisis which, by imposing regulatory capital constraints on banks’ lending on the basis of credit risk estimates, induces an excessive credit reduction during economic recession and an excessive credit growth during economic expansion. In order to mitigate these cyclical effects, various techniques of buffering have been advocated. The authors have tested the opposite null hypothesis that the interaction between new credit given and defaults from outstanding loans tends to a steady state. It has been tested a quasi-linear distribution with a Cyclical Sensitivity Parameter (CSP) parameterized on variation of new credit supply in excess or defect of the rate of default of outstanding loans. It is found that, in the period June 2008-June 2012, frequent fluctuations of the total credit used by non-financial corporations are strongly related to the interaction between the default rate of outstanding loans and the growth rate of new credit supply. It’s concluded that credit risk management in Italy has been effective in parameterizing credit supply growth to outstanding credit reduction caused by defaulting loans within the Basel II regulatory framework. Basel III prospective point-in-time output buffers based on filtered Credit/GDP ratios and dynamic provisioning proposals should take into account this steady state pattern underlying frequent and intense credit cyclical fluctuations