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Securitization is a financial process where the cash flows of income-generating assets are sold to institutional investors as securities, liquidating illiquid assets. This practice presents persistent challenges due to the absence of a comprehensive mathematical framework for structuring asset-backed securities. While existing literature provides technical analysis of credit risk modeling, there remains a need for a definitive framework detailing the allocation of the inbound cash flows to the outbound positions. To fill this gap, we introduce the PEAL Method: a 10-step mathematical framework to streamline the securitization structuring across all time periods. The PEAL Method offers a rigorous and versatile approach, allowing practitioners to structure various types of securitizations, including those with complex vertical positions. By employing standardized equations, it facilitates the delineation of payment priorities and enhances risk characterization for both the asset and the liability sides throughout the securitization life cycle. In addition to its technical contributions, the PEAL Method aims to elevate industry standards by addressing longstanding challenges in securitization. By providing detailed information to investors and enabling transparent risk profile comparisons, it promotes market transparency and enables stronger regulatory oversight. In summary, the PEAL Method represents a significant advancement in securitization literature, offering a standardized framework for precision and efficiency in structuring transactions. Its adoption has the potential to drive innovation and enhance risk management practices in the securitization market.

MinRank is an NP-complete problem in linear algebra whose characteristics make it attractive to build post-quantum cryptographic primitives. Several MinRank-based digital signature schemes have been proposed. In particular, two of them, MIRA and MiRitH, have been submitted to the NIST Post-Quantum Cryptography Standardization Process. In this paper, we propose a key-generation algorithm for MinRank-based schemes that reduces the size of the public key to about 50% of the size of the public key generated by the previous best (in terms of public-key size) algorithm. Precisely, the size of the public key generated by our algorithm sits in the range of 328-676 bits for security levels of 128-256 bits. We also prove that our algorithm is as secure as the previous ones.

Social media platforms heavily changed how users consume and digest information and, thus, how the popularity of topics evolves. In this paper, we explore the interplay between the virality of controversial topics and how they may trigger heated discussions and eventually increase users' polarization. We perform a quantitative analysis on Facebook by collecting $\sim57M$ posts from $\sim2M$ pages and groups between 2018 and 2022, focusing on engaging topics involving scandals, tragedies, and social and political issues. Using logistic functions, we quantitatively assess the evolution of these topics finding similar patterns in their engagement dynamics. Finally, we show that initial burstiness may predict the rise of users' future adverse reactions regardless of the discussed topic.

Networks have always played a special role for human beings in shaping social relations, forming public opinion, and driving economic equilibria. Nowadays, online networked platforms dominate digital markets and capitalization leader-boards, while social networks drive public discussion. Despite the importance of networks in many economic and social domains (economics, sociology, anthropology, psychology,...), the knowledge about the laws that dominate their dynamics is still scarce and fragmented. Here, we analyse a wide set of online networks (those financed by advertising) by investigating their value dynamics from several perspectives: the type of service, the geographic scope, the merging between networks, and the relationship between economic and financial value. The results show that the networks are dominated by strongly nonlinear dynamics. The existence of non-linearity is often underestimated in social sciences because it involves contexts that are difficult to deal with, such as the presence of multiple equilibria -- some of which are unstable. Yet, these dynamics must be fully understood and addressed if we aim to understand the recent evolution in the economic, political and social milieus, which are precisely characterised by corner equilibria (e.g., polarization, winner-take-all solutions, increasing inequality) and nonlinear patterns.

We find new conditions that the existence of nullity of the curvature tensor of an irreducible homogeneous space $M=G/H$ imposes on the Lie algebra $\mathfrak g$ of $G$ and on the Lie algebra $\tilde{\mathfrak g}$ of the full isometry group of $M$. Namely, we prove that there exists a transvection of $M$ in the direction of any element of the nullity, possibly by enlarging the presentation group $G$. Moreover, we prove that these transvections generate an abelian ideal of $\tilde{\mathfrak g}$. These results constitute a substantial improvement on the structure theory developed in \cite{DOV}. In addition we construct examples of homogeneous Riemannian spaces with non-trivial nullity, where $G$ is a non-solvable group, answering a natural open question. Such examples admit (locally homogeneous) compact quotients. In the case of co-nullity $3$ we give an explicit description of the isometry group of any homogeneouslocally irreducible Riemannian manifold with nullity.

In 2020 Sala, Sogiorno and Taufer have been able to find the private keys of some Bitcoin addresses, thus being able to spend the cryptocurrency linked to them. This result was unexpected, since the recovery of non-trivial private keys for blockchain addresses is deemed to be an infeasible problem. In this paper we widen this analysis by mounting a similar attack to other small subsets of the set of private keys. We then apply it to other blockchains as well, examining Ethereum, Dogecoin, Litecoin, Dash, Zcash and Bitcoin Cash. In addition to the results, we also explain the techniques we have used to perform this exhaustive search for all the addresses that have ever appeared in these blockchains., Comment: 13 pages

We put in evidence and correct a mistake in the formula for the determinant 308 in Proskuryakov's linear algebra book. We apply this formula to reprove the well-known fact that the Fubini-Study metric on the complex projective space is Einstein.

We prove that the Ring Learning With Errors (RLWE) and the Polynomial Learning With Errors (PLWE) problems over the cyclotomic field $\mathbb{Q}(\zeta_n)$ are not equivalent. Precisely, we show that reducing one problem to the other increases the noise by a factor that is more than polynomial in $n$. We do so by providing a lower bound, holding for infinitely many positive integers $n$, for the condition number of the Vandermonde matrix of the $n$th cyclotomic polynomial.

Linear Feedback Shift Registers (LFRS) are tools commonly used in cryptography in many different context, for example as pseudo-random numbers generators. In this paper we characterize LFRS with certain symmetry properties. Related to this question we also classify polynomials f of degree n satisfying the property that if a is a root of f then $f(a^n)=0$. The classification heavily depends on the choice of the fields of coefficients of the polynomial; we consider the cases $K=\mathbb{F}_p$ and $K=\mathbb{Q}$., Comment: 12 pages

The policies implemented to hinder the COVID-19 outbreak represent one of the largest critical events in history. The understanding of this process is fundamental for crafting and tailoring post-disaster relief. In this work we perform a massive data analysis, through geolocalized data from 13M Facebook users, on how such a stress affected mobility patterns in France, Italy and UK. We find that the general reduction of the overall efficiency in the network of movements is accompanied by geographical fragmentation with a massive reduction of long-range connections. The impact, however, differs among nations according to their initial mobility structure. Indeed, we find that the mobility network after the lockdown is more concentrated in the case of France and UK and more distributed in Italy. Such a process can be approximated through percolation to quantify the substantial impact of the lockdown.

In response to the COVID-19 pandemic, National governments have applied lockdown restrictions to reduce the infection rate. We perform a massive analysis on near real-time Italian data provided by Facebook to investigate how lockdown strategies affect economic conditions of individuals and local governments. We model the change in mobility as an exogenous shock similar to a natural disaster. We identify two ways through which mobility restrictions affect Italian citizens. First, we find that the impact of lockdown is stronger in municipalities with higher fiscal capacity. Second, we find a segregation effect, since mobility restrictions are stronger in municipalities for which inequality is higher and where individuals have lower income per capita.

We develop a minimalist compartmental model to study the impact of mobility restrictions in Italy during the Covid-19 outbreak. We show that an early lockdown shifts the epidemic in time, while that beyond a critical value of the lockdown strength, the epidemic tend to restart after lifting the restrictions. As a consequence, specific mitigation strategies must be introduced. We characterize the relative importance of different broad strategies by accounting for two fundamental sources of heterogeneity, i.e. geography and demography. First, we consider Italian regions as separate administrative entities, in which social interactions between age classs occur. Due to the sparsity of the inter-regional mobility matrix, once started the epidemics tend to develop independently across areas, justifying the adoption of solutions specific to individual regions or to clusters of regions. Second, we show that social contacts between age classes play a fundamental role and that measures which take into account the age structure of the population can provide a significant contribution to mitigate the rebound effects. Our model is general, and while it does not analyze specific mitigation strategies, it highlights the relevance of some key parameters on non-pharmaceutical mitigation mechanisms for the epidemics.

We address the diffusion of information about the COVID-19 with a massive data analysis on Twitter, Instagram, YouTube, Reddit and Gab. We analyze engagement and interest in the COVID-19 topic and provide a differential assessment on the evolution of the discourse on a global scale for each platform and their users. We fit information spreading with epidemic models characterizing the basic reproduction numbers $R_0$ for each social media platform. Moreover, we characterize information spreading from questionable sources, finding different volumes of misinformation in each platform. However, information from both reliable and questionable sources do not present different spreading patterns. Finally, we provide platform-dependent numerical estimates of rumors' amplification.

Recently, Blanco-Chac\'on proved the equivalence between the Ring Learning With Errors and Polynomial Learning With Errors problems for some families of cyclotomic number fields by giving some upper bounds for the condition number $\operatorname{Cond}(V_n)$ of the Vandermonde matrix $V_n$ associated to the $n$th cyclotomic polynomial. We prove some results on the singular values of $V_n$ and, in particular, we determine $\operatorname{Cond}(V_n)$ for $n = 2^k p^\ell$, where $k, \ell \geq 0$ are integers and $p$ is an odd prime number.

We study new primality tests based on linear recurrent sequences of degree two exploiting a matricial approach. The classical Lucas test arises as a particular case and we see how it can be easily improved. Moreover, this approach shows clearly how the Lucas pseudoprimes are connected to the Pell equation and the Brahamagupta product. We also introduce a new specific primality test, which we will call generalized Pell test. We perform some numerical computations on the new primality tests and, for the generalized Pell test, we do not any pseudoprime up to $10^{10}$.

In this paper, we highlight that the point group structure of elliptic curves over finite or infinite fields, may be also observed on singular cubics with a quadratic component. Starting from this, we are able to introduce in a very general way a group's structure over any kind of conics. In the case of conics over finite fields, we see that the point group is cyclic and lies on the quadric; the straight line component plays a role which may be not explicitly visible in the algebraic description of point composition, but it is indispensable in the geometric description. Moreover, some applications to cryptography are described, considering convenient parametrizations of the conics. Finally, we perform an evaluation of the complexity of the operations involved in the parametric groups and consequently in the cryptographic applications.

We show a connection between the Lucas pseudoprimes and the Pell conic equipped with the Brahmagupta product introducing the Pell pseudoprimes.

We show that the extended principal bundle of a Cartan geometry of type $(A(m,\mathbb{R}),GL(m,\mathbb{R}))$, endowed with its extended connection $\hat\omega$, is isomorphic to the principal $A(m,\mathbb{R})$-bundle of affine frames endowed with the affine connection as defined in classical Kobayashi-Nomizu volume I. Then we classify the local holonomy groups of the Cartan geometry canonically associated to a Riemannian manifold. It follows that if the holonomy group of the Cartan geometry canonically associated to a Riemannian manifold is compact then the Riemannian manifold is locally a product of cones.

Despite their playful purpose social media changed the way users access information, debate, and form their opinions. Recent studies, indeed, showed that users online tend to promote their favored narratives and thus to form polarized groups around a common system of beliefs. Confirmation bias helps to account for users decisions about whether to spread content, thus creating informational cascades within identifiable communities. At the same time, aggregation of favored information within those communities reinforces selective exposure and group polarization. Along this path, through a thorough quantitative analysis we approach connectivity patterns over 1.2M of Facebook users engaged with two very conflicting narratives: scientific and conspiracy news. Analyzing such data, we quantitatively investigate the effect of two mechanisms (namely challenge avoidance and reinforcement seeking) behind confirmation bias, one of the major drivers of human behavior in social media. We find that challenge avoidance mechanism triggers the emergence of two distinct and polarized groups of users (i.e., echo chambers) who also tend to be surrounded by friends having similar systems of beliefs. Through a network based approach, we show how the reinforcement seeking mechanism limits the influence of neighbors and primarily drives the selection and diffusion of contents even among like-minded users, thus fostering the formation of highly polarized sub-clusters within the same echo chamber. Finally, we show that polarized users reinforce their preexisting beliefs by leveraging the activity of their like-minded neighbors, and this trend grows with the user engagement suggesting how peer influence acts as a support for reinforcement seeking., Comment: This version removed by arXiv administrators due to concern of copyright infringement

The efficient computation of shortest paths in complex networks is essential to face new challenges related to critical infrastructures such as a near real-time monitoring and control and the management of big size systems. In particular, using information on the minimum paths in water distribution networks (WDNs) allows to track the diffusion of contaminants and to quantify the resilience and criticality of the system. This is, ultimately, approached by considering dynamically changing path-weights that depend on the flow or on other information available at run-time. These analyses tipically include all the WDN assets but reducing the high degree of physical details with a minimum lost of key information for their performance assessment. This paper proposes a strategy to compute minimum paths that is based on a dimensionality-reduction process. Specifically, the network is partitioned into communities and suitably modified to obtain a reduced complexity representation (e.g., in terms of number of nodes and links). The paper shows how this novel, reduced representation is equivalent to the traditional network on computing the shortest paths. The proposed approach is validated considering two utility networks as case studies. The results show that the proposed method provides the exact solution for the shortest path with a computational-time reduction consistently over 50\% and up to 90\% for some cases. Furthermore, the application of the proposal on WDNs partitioning shows both hydraulic and economic advantages thanks to their monitoring and controlling at near real-time., Comment: Submitted to IJCIP

Recent studies, targeting Facebook, showed the tendency of users to interact with information adhering to their preferred narrative and to ignore dissenting information. Primarily driven by confirmation bias, users tend to join polarized clusters where they cooperate to reinforce a like-minded system of beliefs, thus facilitating fake news and misinformation cascades. To gain a deeper understanding of these phenomena, in this work we analyze the lexicons used by the communities of users emerging on Facebook around verified and unverified contents. We show how the lexical approach provides important insights about the kind of information processed by the two communities of users and about their overall sentiment. Furthermore, by focusing on comment threads, we observe a strong positive correlation between the lexical convergence of co-commenters and their number of interactions, which in turns suggests that such a trend could be a proxy for the emergence of collective identities and polarization in opinion dynamics., Comment: 11 pages, 9 figures

The social brain hypothesis fixes to 150 the number of social relationships we are able to maintain. Similar cognitive constraints emerge in several aspects of our daily life, from our mobility up to the way we communicate, and might even affect the way we consume information online. Indeed, despite the unprecedented amount of information we can access online, our attention span still remains limited. Furthermore, recent studies showed the tendency of users to ignore dissenting information but to interact with information adhering to their point of view. In this paper, we quantitatively analyze users' attention economy in news consumption on social media by analyzing 14M users interacting with 583 news outlets (pages) on Facebook over a time span of 6 years. In particular, we explore how users distribute their activity across news pages and topics. We find that, independently of their activity, users show the tendency to follow a very limited number of pages. On the other hand, users tend to interact with almost all the topics presented by their favored pages. Finally, we introduce a taxonomy accounting for users behavior to distinguish between patterns of selective exposure and interest. Our findings suggest that segregation of users in echo chambers might be an emerging effect of users' activity on social media and that selective exposure -- i.e. the tendency of users to consume information interest coherent with their preferences -- could be a major driver in their consumption patterns., Comment: PLOS One Published: March 13, 2020

Users online tend to acquire information adhering to their system of beliefs and to ignore dissenting information. Such dynamics might affect page popularity. In this paper we introduce an algorithm, that we call PopRank, to assess both the Impact of Facebook pages as well as users' Engagement on the basis of their mutual interactions. The ideas behind the PopRank are that i) high impact pages attract many users with a low engagement, which means that they receive comments from users that rarely comment, and ii) high engagement users interact with high impact pages, that is they mostly comment pages with a high popularity. The resulting ranking of pages can predict the number of comments a page will receive and the number of its posts. Pages impact turns out to be slightly dependent on pages' informative content (e.g., science vs conspiracy) but independent of users' polarization., Comment: 10 pages, 5 figures

The advent of WWW changed the way we can produce and access information. Recent studies showed that users tend to select information that is consistent with their system of beliefs, forming polarized groups of like-minded people around shared narratives where dissenting information is ignored. In this environment, users cooperate to frame and reinforce their shared narrative making any attempt at debunking inefficient. Such a configuration occurs even in the consumption of news online, and considering that 63% of users access news directly form social media, one hypothesis is that more polarization allows for further spreading of misinformation. Along this path, we focus on the polarization of users around news outlets on Facebook in different European countries (Italy, France, Spain and Germany). First, we compare the pages' posting behavior and the users' interacting patterns across countries and observe different posting, liking and commenting rates. Second, we explore the tendency of users to interact with different pages (i.e., selective exposure) and the emergence of polarized communities generated around specific pages. Then, we introduce a new metric -- i.e., polarization rank -- to measure polarization of communities for each country. We find that Italy is the most polarized country, followed by France, Germany and lastly Spain. Finally, we present a variation of the Bounded Confidence Model to simulate the emergence of these communities by considering the users' engagement and trust on the news. Our findings suggest that trust in information broadcaster plays a pivotal role against polarization of users online.

We develop a general theory for irreducible homogeneous spaces $M= G/H$, in relation to the nullity $\nu$ of their curvature tensor. We construct natural invariant (different and increasing) distributions associated with the nullity, that give a deep insight of such spaces. In particular, there must exist an order-two transvection, not in the nullity, with null Jacobi operator. This fact was very important for finding out the first homogeneous examples with non-trivial nullity, i.e. where the nullity distribution is not parallel. Moreover, we construct irreducible examples of conullity $k=3$, the smallest possible, in any dimension. None of our examples admit a quotient of finite volume. We also proved that $H$ is trivial and $G$ is solvable if $k=3$. Another of our main results is that the leaves of the nullity are closed (we used a rather delicate argument). This implies that $M$ is a Euclidean affine bundle over the quotient by the leaves of $\nu$. Moreover, we prove that $\nu ^\perp$ defines a metric connection on this bundle with transitive holonomy or, equivalently, $\nu ^\perp$ is completely non-integrable (this is not in general true for an arbitrary autoparallel and flat invariant distribution). We also found some general obstruction for the existence of non-trivial nullity: e.g., if $G$ is reductive (in particular, if $M$ is compact), or if $G$ is two-step nilpotent.

Users polarization and confirmation bias play a key role in misinformation spreading on online social media. Our aim is to use this information to determine in advance potential targets for hoaxes and fake news. In this paper, we introduce a general framework for promptly identifying polarizing content on social media and, thus, "predicting" future fake news topics. We validate the performances of the proposed methodology on a massive Italian Facebook dataset, showing that we are able to identify topics that are susceptible to misinformation with 77% accuracy. Moreover, such information may be embedded as a new feature in an additional classifier able to recognize fake news with 91% accuracy. The novelty of our approach consists in taking into account a series of characteristics related to users behavior on online social media, making a first, important step towards the smoothing of polarization and the mitigation of misinformation phenomena.

We give a proof of the existence of radial (smooth) parallel sections of vector bundles endowed with a linear connection.

Vaccine hesitancy has been recognized as a major global health threat. Having access to any type of information in social media has been suggested as a potential powerful influence factor to hesitancy. Recent studies in other fields than vaccination show that access to a wide amount of content through the Internet without intermediaries resolved into major segregation of the users in polarized groups. Users select the information adhering to theirs system of beliefs and tend to ignore dissenting information. In this paper we assess whether there is polarization in Social Media use in the field of vaccination. We perform a thorough quantitative analysis on Facebook analyzing 2.6M users interacting with 298.018 posts over a time span of seven years and 5 months. We used community detection algorithms to automatically detect the emergent communities from the users activity and to quantify the cohesiveness over time of the communities. Our findings show that content consumption about vaccines is dominated by the echo-chamber effect and that polarization increased over years. Communities emerge from the users consumption habits, i.e. the majority of users only consumes information in favor or against vaccines, not both. The existence of echo-chambers may explain why social-media campaigns providing accurate information may have limited reach, may be effective only in sub-groups and might even foment further polarization of opinions. The introduction of dissenting information into a sub-group is disregarded and can have a backfire effect, further reinforcing the existing opinions within the sub-group.

Nowadays users get informed and shape their opinion through social media. However, the disintermediated access to contents does not guarantee quality of information. Selective exposure and confirmation bias, indeed, have been shown to play a pivotal role in content consumption and information spreading. Users tend to select information adhering (and reinforcing) their worldview and to ignore dissenting information. This pattern elicits the formation of polarized groups -- i.e., echo chambers -- where the interaction with like-minded people might even reinforce polarization. In this work we address news consumption around Brexit in UK on Facebook. In particular, we perform a massive analysis on more than 1 Million users interacting with Brexit related posts from the main news providers between January and July 2016. We show that consumption patterns elicit the emergence of two distinct communities of news outlets. Furthermore, to better characterize inner group dynamics, we introduce a new technique which combines automatic topic extraction and sentiment analysis. We compare how the same topics are presented on posts and the related emotional response on comments finding significant differences in both echo chambers and that polarization influences the perception of topics. Our results provide important insights about the determinants of polarization and evolution of core narratives on online debating.