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Abstract: We consider a nonlinear parabolic model that forces solutions to stay on a $$L^2$$-sphere through a nonlocal term in the equation. We study the local and global well-posedness on a bounded domain and the whole Euclidean space in the energy space. Then, we consider the solutions’ asymptotic behavior. We prove strong convergence to a stationary state and asymptotic convergence to the ground state in bounded domains when the initial condition is positive. PubDate: 2025-04-10
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Abstract: We will address the questions and conjectures left by the recent papers by Ferreira et al. (Bollettino dell’Unione Matematica Italiana, 2023, https://doi.org/10.1007/s40574-023-00402-7), (J Algebra 638:488–505, 2024). We also present an example showing the necessity of the conditions of the results that answer the conjectures. It leads us to the generalized Vukman equation: $$\begin{aligned} f(x)+x^ng(x^{-1})=0, \end{aligned}$$ f ( x ) + x n g ( x - 1 ) = 0 , for every invertible x, where n is a nonnegative integer and f, g are additive maps on an alternative ring D. We will study this equation for the split octonion algebras, the alternative division rings, and the field $${\mathbb {Z}}_2(t)$$ of rational functions over the field $${\mathbb {Z}}_2$$ with two elements. PubDate: 2025-04-06
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Abstract: In this article, we collect main results obtained from our collaboration with Edoardo Ballico. The topics cover various aspects on the theory of vector bundles and stable sheaves related to Castelnuovo–Mumford regularity, cohomological splitting criteria, globally generated vector bundles, arithmetically Cohen–Macaulay sheaves, logarithmic vector bundles, stable sheaves of rank zero and co-Higgs sheaves. PubDate: 2025-04-01
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Abstract: The effectiveness of non-pharmaceutical interventions (NPIs) during a pandemic is challenging to assess due to the multifaceted interactions between interventions and population dynamics. Significant difficulty arises from the overlapping effects of various NPIs applied to different subgroups within a population. To address this, we propose a new mathematical model that incorporates various intervention strategies, including total and partial lockdowns, school closures, and reduced interactions among specific subgroups, such as the elderly. Our model extends previous work by explicitly accounting for the quadratic nature of control costs and the interplay between overlapping controls targeting the same population segments. Using optimal control theory, we identify intervention policies that effectively mitigate disease transmission while balancing economic and societal costs. To demonstrate the utility of our approach, we apply the model to real-world data from the COVID-19 pandemic in the State of New Jersey. Our results provide insights into the trade-offs and synergies of different NPIs and the importance of accurately capturing the relationship between a policy and the population affected. PubDate: 2025-03-19
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Abstract: Recent advances in epidemiological modelling have increasingly emphasised the role of sociological factors in describing epidemic dynamics. In this paper, we first provide an overview of the main results regarding the formulation of minimal ordinary differential equations (ODEs) models with information-dependent contact patterns. We then discuss how the minimal ODE models may be extended to integral models through the formulation of specific constitutive equations. We recall the results concerning the integral model that generalises the ODE models with prevalence-dependent contact patterns. Furthermore, we provide a follow-up by also considering the integral model with incidence-dependent contact patterns. For this new model, we study the asymptotic properties of the solutions and obtain sufficient conditions for the stability of the steady states, expressed in terms of the memory kernel and the infectivity function. We numerically show how the memory kernel affects the dynamical outcomes of the model when specific infectivity functions are considered. PubDate: 2025-03-15
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Abstract: This paper investigates the theory of normal families in the setting of several complex variables, focusing on the total derivatives of holomorphic functions. We extend classical results, such as Zalcman’s Lemma and the Zalcman-Pang Lemma, to the case of several complex variables, providing new normality criteria for families of holomorphic functions in $${\mathbb {C}}^n$$. As applications, we derive two normality theorems that highlight the relevance of our extended criteria. PubDate: 2025-03-11
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Abstract: In this article, we present a personal story for a small research domain where Edoardo Ballico brought very significant contributions. This nice small domain is the theory of uniform algebraic vector bundles on projective spaces (here below we simply write bundle for algebraic vector bundle on a projective space). This theory was essentially developed between 1971 and 1983 by a handful of colleagues and accordingly, our story dedicates one section to each of them. Concerning this period, we will also highlight a trend in the community of complex geometers, namely the shift from analytic to algebraic geometry, which we call the GAGA shift. PubDate: 2025-03-06
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Abstract: We give an overview on the landscape of polynomial interpolation theory. We will describe first the geometric approach, based on the base locus analysis of linear systems of hypersurfaces of $${\mathbb {P}}^n$$ with given degree and assigned multiplicity at a set of points. Secondly, we will consider the algebraic counterpart, with a discussion on the good postulation of fat point schemes of $${\mathbb {P}}^n$$ and their regularity index. In both cases we report on some complete, or partial, results and conjectures. PubDate: 2025-03-02
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Abstract: In this article, we briefly survey some of the recent results in the geometry of tensors with a focus on bounds for various important notions of ranks. This is written in honor of Edoardo Ballico, who made important contributions to this field, on the occasion of his 70th birthday. PubDate: 2025-02-28
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Abstract: Herd immunity is a critical concept in epidemiology, describing a threshold at which a sufficient proportion of a population is immune, either through infection or vaccination, thereby preventing sustained transmission of a pathogen. In the classic susceptible-infectious-recovered (SIR) model, which has been widely used to study infectious disease dynamics, the achievement of herd immunity depends on key parameters, including the transmission rate ($$\beta $$) and the recovery rate ($$\gamma $$), where $$\gamma $$ represents the inverse of the mean infectious period. While the transmission rate has received substantial attention, recent studies have underscored the significant role of $$\gamma $$ in determining the timing and sustainability of herd immunity. Additionally, it is becoming increasingly evident that assuming $$\gamma $$ as a constant parameter might oversimplify the dynamics, as variations in recovery times can reflect diverse biological, social, and healthcare-related factors. In this paper, we investigate how heterogeneity in the recovery rate affects herd immunity. We show empirically that the mean of the recovery rate is not a reliable metric for determining the achievement of herd immunity. Furthermore, we provide a theoretical result demonstrating that it is instead the mean recovery time, which is the mean of the inverse $$1/\gamma $$ of the recovery rate that is critical in deciding whether herd immunity is achievable within the SIR framework. A similar result is proved for the SEIR model. These insights have significant implications for public health interventions and theoretical modeling of epidemic dynamics. PubDate: 2025-02-27
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Abstract: We present a general method for characterizing the supports of small-weight codewords of algebraic geometry codes. We apply the method to investigate the weight distributions of error-correcting codes constructed from the Hermitian curve and from norm-trace curves, providing closed formulas for several parameter sets. The results presented in this paper draw on E. Ballico’s contributions to the connection between coding theory and algebraic geometry. PubDate: 2025-02-25
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Abstract: We denote by $$\mathcal {H}_{d,g,r}$$ the Hilbert scheme of smooth curves, which is the union of components whose general point corresponds to a smooth irreducible and non-degenerate curve of degree d and genus g in $$\mathbb {P}^r$$. In this article, we study $$\mathcal {H}_{16,g,5}$$ for almost every possible genus g and chasing after its irreducibility. We also study the natural moduli map $$\mathcal {H}_{d,g,5}{\mathop {\xrightarrow {\phantom{a} }}\limits ^{\mu }}\mathcal {M}_g$$ and several key properties such as gonality of a general element as well as characterizing smooth elements in each component. PubDate: 2025-02-20
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Abstract: Following the suggestions contained in [5], I will discuss constructions that generalizes the interpolation problems for divisors to the case of varieties of higher codimension, with emphasis on the case of curves arising as 0-loci of vector bundles of rank 2 in $$\mathbb {P}^3$$. PubDate: 2025-02-19
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Abstract: This survey explores the interplay between twistor geometry and projective geometry, focusing on their applications to algebraic surfaces. We explore two main topics: the inclusion of twistor fibers and lines in these surfaces, and the behavior of twistor discriminant loci, with a particular focus on degree-2 surfaces. The study highlights contributions from Ballico and collaborators, comparing the twistor spaces of $${\mathbb{C}\mathbb{P}}^3$$ and the flag threefold $${\mathbb {F}},$$ which reveal fascinating contrasts and parallels. Key findings include a detailed analysis of surfaces in $${\mathbb{C}\mathbb{P}}^3$$ and $${\mathbb {F}}$$ that either contain finite or infinite twistor fibers. The survey also touches on cubic surfaces in $${\mathbb{C}\mathbb{P}}^3$$ and their counterparts in $${\mathbb {F}},$$ where the configurations of twistor fibers lead to intriguing results. Special attention is given to surfaces of twistor degree 2, including how their geometry and singularities interact with twistor projections. In particular, we discuss smooth and singular surfaces in $${\mathbb {F}}$$ of bidegree (1, 1) and (0, 2), as well as their discriminant loci. PubDate: 2025-02-18
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Abstract: We consider a sequence of minimum problems for integral and more general functionals on sets of functions defined by bilateral constraints in variable domains. Under some conditions on the involved domains, functionals, and constraints, we prove that the sequence of minimizers of the considered problems is approximated in $$W^{1,p}$$-norms by a special $$\varGamma $$-realizing sequence for the minimizer of the $$\varGamma $$-limit functional on a limit set. This $$\varGamma $$-realizing sequence depends on the given constraints and each its element belongs to the corresponding constraint set. The crucial role in obtaining our approximation result is played by the assumption that the considered sequence of functionals satisfies the uniform convexity condition. PubDate: 2025-02-15
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