The naturalness in the BLMSSM and B-LSSM

In order to interpret the Higgs boson mass and its decays naturally, we hope to examine the BLMSSM and B-LSSM. In the both models, the right-handed neutrino superfields are introduced to better explain the neutrino mass problems. In this paper, we introduce the fine-tuning to acquire the physical Higgs boson mass. Besides, the method of $$\chi ^2$$ χ 2 analyses will be adopted in the BLMSSM and B-LSSM to fit the experimental data. Therefore, we can obtain the reasonable theoretical values of the Higgs decays and muon $$g-2$$ g - 2 that are in accordance with the experimental results respectively in the BLMSSM and B-LSSM.

Mass problems and density

Recall that [Formula: see text] is the lattice of Muchnik degrees of nonempty effectively compact sets in Euclidean space. We solve a long-standing open problem by proving that [Formula: see text] is dense, i.e. satisfies [Formula: see text]. Our proof combines an oracle construction with hyperarithmetical theory.

DEEP CLASSES

A set of infinite binary sequences ${\cal C} \subseteq 2$ℕ is negligible if there is no partial probabilistic algorithm that produces an element of this set with positive probability. The study of negligibility is of particular interest in the context of ${\rm{\Pi }}_1^0 $ classes. In this paper, we introduce the notion of depth for ${\rm{\Pi }}_1^0 $ classes, which is a stronger form of negligibility. Whereas a negligible ${\rm{\Pi }}_1^0 $ class ${\cal C}$ has the property that one cannot probabilistically compute a member of ${\cal C}$ with positive probability, a deep ${\rm{\Pi }}_1^0 $ class ${\cal C}$ has the property that one cannot probabilistically compute an initial segment of a member of ${\cal C}$ with high probability. That is, the probability of computing a length n initial segment of a deep ${\rm{\Pi }}_1^0 $ class converges to 0 effectively in n.We prove a number of basic results about depth, negligibility, and a variant of negligibility that we call tt-negligibility. We provide a number of examples of deep ${\rm{\Pi }}_1^0 $ classes that occur naturally in computability theory and algorithmic randomness. We also study deep classes in the context of mass problems, examine the relationship between deep classes and certain lowness notions in algorithmic randomness, and establish a relationship between members of deep classes and the amount of mutual information with Chaitin’s Ω.