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In this work, we investigate off-shell supersymmetric string configurations in Euclidean Ad$\mathbb{S}_5\times \mathbb{S}^5$ background. By extending the Green-Schwarz supersymmetric equations using the pure spinor formalism, we demonstrate that while the solutions to the Green-Schwarz equations are necessarily minimal surfaces, the solutions to the extended equations form infinite-dimensional families, even with fixed boundary conditions, thus preventing direct derivations of exact results via supersymmetric localization. For concreteness, we have selected a supercharge that localizes the off-shell supersymmetric configurations to an Ad$\mathbb{S}_4 \times \mathbb{S}^2$ slice and observed that the corresponding action functional reduces to an integral of a two-form., Comment: 48 pages, 1 figure

Safe and reliable disclosure of information from confidential data is a challenging statistical problem. A common approach considers the generation of synthetic data, to be disclosed instead of the original data. Efficient approaches ought to deal with the trade-off between reliability and confidentiality of the released data. Ultimately, the aim is to be able to reproduce as accurately as possible statistical analysis of the original data using the synthetic one. Bayesian networks is a model-based approach that can be used to parsimoniously estimate the underlying distribution of the original data and generate synthetic datasets. These ought to not only approximate the results of analyses with the original data but also robustly quantify the uncertainty involved in the approximation. This paper proposes a fully Bayesian approach to generate and analyze synthetic data based on the posterior predictive distribution of statistics of the synthetic data, allowing for efficient uncertainty quantification. The methodology makes use of probability properties of the model to devise a computationally efficient algorithm to obtain the target predictive distributions via Monte Carlo. Model parsimony is handled by proposing a general class of penalizing priors for Bayesian network models. Finally, the efficiency and applicability of the proposed methodology is empirically investigated through simulated and real examples.

The recently established B-RNS-GSS formalism is extended for the description of the heterotic superstring in curved backgrounds. We propose a generalized action and BRST charge defined in the small Hilbert space with the standard form of an $\mathcal{N}=(1,0)$ worldsheet superconformal theory with superconformal generator $G$ and stress tensor $T$. We show that $\big\{G,G\big\}=-2T$ implies the D=10 N=1 supergravity and super-Yang-Mills equations of motion, as well as holomorphicity of the BRST charge., Comment: 25 pages + appendices; reference arXiv number added

An algebraic structure with two constants and one ternary operation, which is not completely commutative, is put forward to accommodate ternary Boolean algebras. When the ternary operation is interpreted as Church's conditioned disjunction, Boolean algebras are characterized as a subvariety. Different interpretations for the ternary operation lead to distinct subvarieties. Rings and near-rings of characteristic 2 are used to illustrate the procedure.

We present a new approach to ternary Boolean algebras in which negation is derived from the ternary operation. The key aspect is the replacement of complete commutativity by other axioms that do not require the ternary operation to be symmetric., Comment: 8 pages

The category of mobi algebras has been introduced as a model to the unit interval of real numbers. The notion of mobi space over a mobi algebra has been proposed as a model for spaces with geodesic paths. In this paper we analyse the particular case of affine mobi spaces and show that there is an isomorphism of categories between R-modules and pointed affine mobi spaces over a mobi algebra R as soon as R is a unitary ring in which 2 is an invertible element., Comment: 19 pages. arXiv admin note: text overlap with arXiv:2001.03441, arXiv:2102.02692

In any type II superstring background, the supergravity vertex operators in the pure spinor formalism are described by a gauge superfield. In this paper, we obtain for the first time an explicit expression for this superfield in an $AdS_5 \times S^5$ background. Previously, the vertex operators were only known close to the boundary of $AdS_5$ or in the minus eight picture. Our strategy for the computation was to apply eight picture raising operators in the minus eight picture vertices. In the process, a huge number of terms are generated and we have developed numerical techniques to perform intermediary simplifications. Alternatively, the same numerical techniques can be used to compute the vertices directly in the zero picture by constructing a basis of invariants and fitting for the coefficients. One motivation for constructing the vertex operators is the computation of $AdS_5 \times S^5$ string amplitudes., Comment: References added. Minor improvements

We introduce an algebraic system which can be used as a model for spaces with geodesic paths between any two of their points. This new algebraic structure is based on the notion of mobility algebra which has recently been introduced as a model for the unit interval of real numbers. We show that there is a strong connection between modules over a ring and affine mobility spaces over a mobility algebra. However, geodesics in general fail to be affine thus giving rise to the new algebraic structure of mobility space. We show that the so called formula for spherical linear interpolation, which gives geodesics on the n-sphere, is an example of a mobility space over the unit interval mobility algebra., Comment: arXiv admin note: text overlap with arXiv:2001.03441

We introduce an algebraic system which can be used as a model for spaces with geodesic paths between any two of their points. This new algebraic structure is based on the notion of mobility algebra which has recently been introduced as a model for the unit interval of real numbers. Mobility algebras consist on a set $A$ together with three constants and a ternary operation. In the case of the closed unit interval $A=[0,1]$, the three constants are 0, 1 and 1/2 while the ternary operation is $p(x,y,z)=x-yx+yz$. A mobility space is a set $X$ together with a map $q\colon{X\times A\times X\to X}$ with the meaning that $q(x,t,y)$ indicates the position of a particle moving from point $x$ to point $y$ at the instant $t\in A$, along a geodesic path within the space $X$. A mobility space is thus defined with respect to a mobility algebra, in the same way as a module is defined over a ring. We introduce the axioms for mobility spaces, investigate the main properties and give examples. We also establish the connection between the algebraic context and the one of spaces with geodesic paths. The connection with affine spaces is briefly mentioned., Comment: 27 pages

A new procedure was recently proposed for constructing massless Type IIB vertex operators in the pure spinor formalism. Instead of expressing these closed string vertex operators as left-right products of open string vertex operators, they were instead constructed from the complex N=2 d=10 superfield whose lowest real and imaginary components are the dilaton and Ramond-Ramond axion. These Type IIB vertex operators take a simple form in the -8 picture and are related to the usual vertex operators in the zero picture by acting with picture-raising operators. In this paper, we compute explicitly this picture-raising procedure and confirm this proposal in a flat background. Work is in progress on confirming this proposal in an $AdS_5\times S^5$ background.

We show that the category of cancellative conjugation semigroups is weakly Mal'tsev and give a characterization of all admissible diagrams there. In the category of cancellative conjugation monoids we describe, for Schreier split epimorphisms with codomain B and kernel X, all morphisms h from X to B which induce a reflexive graph, an internal category or an internal groupoid. We describe Schreier split epimorphisms in terms of external actions and consider the notions of precrossed semimodule, crossed semimodule and crossed module in the context of cancellative conjugation monoids. In this category we prove that a relative version of the so-called "Smith is Huq" condition for Schreier split epimorphisms holds as well as other relative conditions., Comment: 23 pages

Recent works explore social characteristics of nodes to improve message delivery rate in Delay Tolerant Networks (DTN). This work uses machine learning techniques to create node groups organized by common interests. Messages are sent to target groups, and from there to the final destination. Simulation results using The ONE simulator show that the larger the group size the higher the message delivery rate, that reaches 100% in some cases. The paper also presents results related to the groups of interest such as message delivery rat, delivery delay and an average number of hops to deliver messages. The overall results indicate that group-based routing is a promising research filed., Comment: IV Escola Regional de Informatica (ERIN 2017), in Portuguese

We begin by introducing an algebraic structure with three constants and one ternary operation to which we call mobi algebra. This structure has been designed to capture the most relevant properties of the unit interval that are needed in the study of geodesic paths. Another algebraic structure, called involutive medial monoid (IMM), can be derived from a mobi algebra. We prove several results on the interplay between mobi algebras, IMM algebras and unitary rings. It turns out that every unitary ring with one half uniquely determines and is uniquely determined by a mobi algebra with one double. This paper is the second of a planned series of papers dedicated to the study of geodesic paths from an algebraic point of view., Comment: This paper is the continuation of "Internal monoids and groups in the category of commutative cancellative medial magmas", Portugaliae Mathematica, Vol. 73, Fasc. 3 (2016) 219-245; by the same authors

We characterise, in pointed regular categories, the ideals as the zero-classes of surjective relations. Moreover, we study a variation of the "Smith is Huq" condition: two surjective left split relations commute as soon as their zero-classes commute., Comment: 13 pages

The well known equivalence between preorders and Alexandrov spaces is extended to an equivalence between arbitrary topological spaces and spatial fibrous preorders, a new notion to be introduced., Comment: Work presented at: IX Portuguese Category Seminar, Coimbra, Portugal (1Jul2013); CT2013, Sidney, Australia (11Jul2013); Eilenberg Centenary Conference, Warsaw, Poland (23Jul2013)

For a given category B we are interested in studying internal categorical structures in B. This work is the starting point, where we consider reflexive graphs and precategories (i.e., for the purpose of this note, a simplicial object truncated at level 2). We introduce the notions of reflexive graph and precategory relative to split epimorphisms. We study the additive case, where the split epimorphisms are "coproduct projections", and the semi-additive case where split epimorphisms are "semi-direct product projections". The result is a generalization of the well known equivalence between precategories and 2-chain complexes. We also consider an abstract setting, containing, for example, strongly unital categories., Comment: 31 pages

The notion of pseudocategory, as considered in [11], is extended from the context of a 2-category to the more general one of a sesquicategory, which is considered as a category equipped with a 2-cell structure. Some particular examples of 2-cells arising form internal transformations in internal categories, conjugations in groups, derivations in crossed-modules or homotopies in abelian chain complexes are studied in this context, namely their behaviour as abstract 2-cells in a 2-cell structure. Issues such as naturality of a 2-cell structure are investigated. This article is intended as a preliminary starting work towards the study of the geometrical aspects of the 2-cell structures from an algebraic point of view., Comment: 31 pages; major re-structuring with examples added