We define analogs of superconformal algebras over an arbitrary commutative associative superalgebra. In the finitely generated case the universal central extensions of these algebras are finitely presented. In particular, we show that all known superconformal algebras are finitely presented.

According to V. Kac and J. van de Leur, the superconformal algebras are the simple -graded Lie superalgebras of growth one which contains the Witt algebra. We describe an explicit classification of all cuspidal modules over the known supercuspidal algebras of rank ≥ 1, and their central extensions. Our approach reveals some unnoticed phenomena. Indeed the central charge of cuspidal modules is trivial, except for one specific central extension of the contact algebra (4). As shown in the paper, this fact also impacts the representation theory of (3), (6) and ^(2)(4). Besides these four cases, the classification relies on general methods based on highest weight theory.

We study the relationship between cyclic homology of Jordan superalgebras and second cohomologies of their Tits-Kantor-Koecher Lie superalgebras.In particular, we focus on Jordan superalgebras that are Kantor doubles of bracket algebras. The obtained results are applied to computation of second cohomologies and universal central extensions of Hamiltonian and contact type Lie superalgebras over arbitrary rings of coefficients.

Our aim is the design of an efficient decoding algorithm in group codes. The algorithm is inspired by the well known syndrome decoding algorithm for linear codes and uses the decomposition of a semisimple group algebra KG as a direct sum of two-sided ideals, each of them generated by a central idempotent of KG. When G is an abelian group the algorithm can be modified to make it very simple and efficient. Some illustrative examples are presented.

We discuss growth functions of finitely generated Jordan algebras and the related systems: Jordan pairs and Tits-Kantor-Koecher Lie algebras. In particular, we prove that the growth function of a finitely generated special Jordan algebra or a finitely generated purely exceptional Jordan algebra is asymptotically equivalent to the growth function of an associative algebra.

Let A be an algebra over a field F and let S be a generating set of A. The length of S indicates the maximal length needed to express an arbitrary element of A as a linear combination of words in the elements of S. The length of an algebra A is defined as the maximum of lengths of its generating sets. In this paper (not necessarily associative) algebras, over fields of arbitrary characteristic, having length equal to 1 are determined.

In [1], B. Klopsch proved that the Nottingham group over a finite field is verbally elliptic. We prove a similar result for fields of zero characteristic. We also prove that the Virasoro Lie algebra and some its subalgebras are polynomially elliptic.

Big data environments have become a standard solution in most public and private corporations since they allow the acquisition and processing of massive volumes of heterogeneous data, and also, act as enablers to extract useful information and insights from this data to optimize internal and external operations in these businesses. As big data tools evolve and become commodities, their democratization process helped promote new industries, business models, companies, and all sorts of new features to improve our way of life. On the other hand, the demand for flexible and powerful privacy schemes for big data has also increased, and it is now an active area of research, with different approaches to the initial problem being taken until now. In this document, we will review some of the most notorious ones, such as trying to preserve various mathematical properties in ciphertexts or using neural network-based solutions for different parts of the encryption process, allowing interesting features in the cryptographic scheme by construction. Privacy individual and social concerns of potential misuses of big data, as the primary root cause for this demand, also pose an opportunity for Cryptography to propose adaptation of standard solutions, as well as new, tailored ones for these environments. The latter should allow the proposals to tackle the specific needs of each individual big data application while addressing privacy issues in a standardized way. Finally, though it is usually considered that cryptographic schemes for big data environments are inherently resource intensive by construction, it can be seen that there are clear opportunities for efficiency improvements in current solutions for different tasks that do not require complex algorithms to be applied over encrypted space. In this document, we discuss and evaluate potential improvements in some cryptographic schemes for various tasks of different nature, considering the implications over big data setups, and deriving some open questions and possible research directions on different fields of interest.

Let F be a field of characteristic different of 2 and let M-1|1(F)((+)) denote the Jordan superalgebra of 2 x 2 matrices over the field F. The aim of this paper is to classify irreducible (unital and one-sided) Jordan bimodules over the Jordan superalgebra M-1|1(F)((+)).

The purpose of this survey is to discuss Poisson and contact brackets and related infinite dimensional superalgebras. All vector spaces are considered over the field of complex numbers $${\mathbb {C}}$$ .

We prove that an arbitrary word v of the free pro-p-group is elliptic on a pro-p-completion of a discrete finitely generated torsion residually-p group.

Let F be a finite field and let G be a finite group. We show that if C is a G-code over F with dimF⁡(C)≤3 then C is an abelian group code. Since there exist non-abelian group codes of dimension 4 when charF>2 (see the examples in [1]), we conclude that the smallest dimension of a non-abelian group code over a finite field is 4.

Probability plays a fundamental role in complexity theory, which in turn is one of the pillars of modern cryptology. However, security practitioners are not always familiar with probability theory, and thus fail to foresee the impact of (seemingly small) deviations from the theoretical description of a scheme at the implementation level. On the other hand, many cryptographic scenarios involve mutually distrusting parties, which need however to cooperate towards a joint goal. In order to attain assurance of the good behavior of one party, interactive validation methods (also known as interactive proof systems) are employed. Randomness is at the core of such methods, which most oftenwill only provide relative assurance, in the sense that they will establish correctness in a probabilistic way. In this paper we will briefly discuss the role of probability theory within modern cryptology, reviewing probabilistic proof systems as a powerful tool towards efficient protocol design, and provable security, as an invaluable framework for deriving formal security proofs.

We prove that there exist non-Abelian group codes over an arbitrary finite field.

We will present a survey on infinite dimensional graded Lie and Jordan superalgebras and their representations.

We prove that the Lie ring associated to the lower central series of a finitely generated residually-p torsion group is graded nil.

A systematic analysis of the data pertaining to the disposal method of the drugs collected by National Prescription Drug Take‐Back Day has been performed in association with United States Drug Enforcement Administration's (US‐DEA's) – Department of Justice (DOJ), DEA Public Affairs, Washington D.C. Based on the data collected from the National Prescription Drug Take‐Back Day all across United States by US DEA–DOJ, as of September 5, 2014, an estimated 4.1 million pounds (2,123 tons) of prescription medications have been removed from the circulation. As per the US DEA b– DOJ public affairs, the drugs that are collected by this program were disposed by incineration facility monitored and regulated by the United States – Environmental Protection Agency (US‐EPA). The entire disposal of the drugs that are collected by National Prescription Drug Take ‐ Back day is completed by a vendor destined by US DEA ‐DOJ. Further analysis of the disposal process reveal following pitfalls of this process: i. There are no data on file regarding pollutants generated from the incineration of Drugs from the take back programs by US DEA ‐ DOJ, 2. The practice of incineration at the rate of en mass is a potential resource for reaction intermediates, such as dioxins, fly ash, and other unknown pollutants, 3. The practice of not sorting the drugs according to the chemical toxicity prior to incineration potential harm to the environment and biota. Specific information on the class of the medication as per the chemical potencies for generating combustion based pollutants are investigated. Data analysis on the Impact of the disposal practices of drugs from take back day will be presented in Experimental Biology 2016. Support or Funding Information Professional Development Funds to S.Kannan at SWTJC for 2014–2015

In this paper we study Gelfand-Kirillov dimension in Jordan algebras. In particular we will relate Gelfand-Kirillov (GK for short) dimensions of a special Jordan algebra and its associative enveloping algebra and also the GK dimension of a Jordan algebra and the GK dimension of its universal multiplicative enveloping algebra.

It has been known some time ago that there are one-sided group codes that are not abelian codes, however the similar question for group codes was not known until we constructed an example of a non-abelian group code using the group ring F5S4. The proof needs some computational help, since we need to know the weight distribution of all abelian codes of length 24 over the prime field of 5 elements. It is natural to ask, is it really relevant that the group ring is semisimple? What happens in the case of characteristic 2 and 3? Our interest to these questions is connected also with the following open question: does the property of all group codes for the given group to be abelian depend on the choice of the base field (the similar property for left group codes does)? We have addressed this question, again with computer help, proving that there are also examples of non-abelian group codes in the non-semisimple case. The results show some interesting differences between the cases of characteristic 2 and 3. Moreover, using the group SL(2, F-3) instead of the symmetric group we can prove, without using a computer for it, that there is a code over F-2 of length 24, dimension 6 and minimal weight 10. It has greater minimum distance than any abelian group code having the same length and dimension over F-2, and moreover this code has the greatest minimum distance among all binary linear codes with the same length and dimension. The existence of such code gives a good reason to study non-abelian group codes.