Neutrino Masses in Supersymmetry: R-Parity and Leptogenesis

In the supersymmetric standard model of particle interactions, R-parity nonconservation is often invoked to obtain nonzero neutrino masses. We point out here that such interactions of the supersymmetric particles would erase any pre-existing lepton or baryon asymmetry of the universe before the electroweak phase transition through the $B + L$ violating sphaleron processes. We then show how neutrino masses may be obtained in supersymmetry (assuming R-parity conservation) together with successful leptogenesis and predict the possible existence of new observable particles.Comment: LATEX, 12 page

Dynamical Neutrino Mass Matrix: Large (Small) Mixing Solution to Atmospheric (Solar) Neutrino Oscillations

In the wake of the SuperKamiokande announcement of atmospheric neutrino oscillations which favor the large mixing of $\nu_\mu$ with $\nu_\tau$, there is considerable theoretical activity towards a possible understanding of the solar neutrino deficit also in terms of the mixing of $\nu_e$ with the other two neutrinos. A specific dynamical model is presented here for the first time which naturally chooses the large (small) mixing solution to atmospheric (solar) neutrino oscillations.Comment: 11 pages including 4 figures, Eq.(13) corrected, conclusions unchange

Uniform Lech's inequality

Let (R,m) be a Noetherian local ring of dimension d ≥ 2. We prove that if e(Rred) > 1, then the classical Lech’s inequality can be improved uniformly for all m-primary ideals, that is, there exists ε > 0 such that e(I) ≤ d!(e(R) − ε)l(R/I) for all m-primary ideals I ⊆ R. This answers a question raised in [3]. We also obtain partial results towards improvements of Lech’s inequality when we fix the number of generators of I.La Caixa Junior Leader Postdoctoral Fellowship: LCF/BQ/PI21/1183003

Uniform Lech's inequality

Let (R,m) be a Noetherian local ring, and let M be a finitely generated R-module of dimension d. We prove that the set [Formula presented] is bounded below by 1/d!e(R‾) where R‾=R/Ann(M). Moreover, when Mˆ is equidimensional, this set is bounded above by a finite constant depending only on M. The lower bound extends a classical inequality of Lech, and the upper bound answers a question of Stückrad–Vogel in the affirmative. As an application, we obtain results on uniform behavior of the lengths of Koszul homology moduleLa Caixa Junior Leader Postdoctoral Fellowship “LCF/BQ/PI21/11830033”. PID2021-125052NA-I0

Colength, multiplicity, and ideal closure operations

In a formally unmixed Noetherian local ring, if the colength and multiplicity of an integrally closed ideal agree, then $R$ is regular. We deduce this using the relationship between multiplicity and various ideal closure operations.Comment: 7 pages, final versio

The Digital Flynn Effect: Complexity of Posts on Social Media Increases over Time

Parents and teachers often express concern about the extensive use of social media by youngsters. Some of them see emoticons, undecipherable initialisms and loose grammar typical for social media as evidence of language degradation. In this paper, we use a simple measure of text complexity to investigate how the complexity of public posts on a popular social networking site changes over time. We analyze a unique dataset that contains texts posted by 942, 336 users from a large European city across nine years. We show that the chosen complexity measure is correlated with the academic performance of users: users from high-performing schools produce more complex texts than users from low-performing schools. We also find that complexity of posts increases with age. Finally, we demonstrate that overall language complexity of posts on the social networking site is constantly increasing. We call this phenomenon the digital Flynn effect. Our results may suggest that the worries about language degradation are not warranted

Massive Neutrinos and Lepton Mixing in Unified Theories

The recent GUT (x SUSY) models which can predict the neutrino properties are reviewed.Comment: 8 pages, Talk presented at the XVI International Workshop on Weak Interactions and Neutrinos, Capri, 199