Quantum q-series identities

As analytic statements, classical $q$-series identities are equalities between power series for $|q|<1$. This paper concerns a different kind of identity, which we call a quantum $q$-series identity. By a quantum $q$-series identity we mean an identity which does not hold as an equality between power series inside the unit disk in the classical sense, but does hold on a dense subset of the boundary -- namely, at roots of unity. Prototypical examples were given over thirty years ago by Cohen and more recently by Bryson-Ono-Pitman-Rhoades and Folsom-Ki-Vu-Yang. We show how these and numerous other quantum $q$-series identities can all be easily deduced from one simple classical $q$-series transformation. We then use other results from the theory of $q$-hypergeometric series to find many more such identities. Some of these involve Ramanujan's false theta functions and/or mock theta functions.

SCMFTS: Scalable and Distributed Complexity Measures and Features for Univariate and Multivariate Time Series in Big Data Environments

Time series data are becoming increasingly important due to the interconnectedness of the world. Classical problems, which are getting bigger and bigger, require more and more resources for their processing, and Big Data technologies offer many solutions. Although the principal algorithms for traditional vector-based problems are available in Big Data environments, the lack of tools for time series processing in these environments needs to be addressed. In this work, we propose a scalable and distributed time series transformation for Big Data environments based on well-known time series features (SCMFTS), which allows practitioners to apply traditional vector-based algorithms to time series problems. The proposed transformation, along with the algorithms available in Spark, improved the best results in the state-of-the-art on the Wearable Stress and Affect Detection dataset, which is the biggest publicly available multivariate time series dataset in the University of California Irvine (UCI) Machine Learning Repository. In addition, SCMFTS showed a linear relationship between its runtime and the number of processed time series, demonstrating a linear scalable behavior, which is mandatory in Big Data environments. SCMFTS has been implemented in the Scala programming language for the Apache Spark framework, and the code is publicly available.

Convergence of Generalized Collatz Problem in k-Adic Field

In this article, we define a new series transformation called transformation and probe into its fixed point and periodicity. We extend the number field of the transform period problem to a wider field. Different constraints are imposed on then different periodic columns are formed after finite transformations. We obtain that their periodic sequences are and respectively after derivation. As an application, it can provide a reference for C problems in more complex algebraic systems.

Some q-Supercongruences from Transformation Formulas for Basic Hypergeometric Series

Several new q-supercongruences are obtained using transformation formulas for basic hypergeometric series, together with various techniques such as suitably combining terms, and creative microscoping, a method recently developed by the first author in collaboration with Zudilin. More concretely, the results in this paper include q-analogues of supercongruences (referring to p-adic identities remaining valid for some higher power of p) established by Long, by Long and Ramakrishna, and several other q-supercongruences. The six basic hypergeometric transformation formulas which are made use of are Watson’s transformation, a quadratic transformation of Rahman, a cubic transformation of Gasper and Rahman, a quartic transformation of Gasper and Rahman, a double series transformation of Ismail, Rahman and Suslov, and a new transformation formula for a nonterminating very-well-poised $${}_{12}\phi _{11}$$ 12 ϕ 11 series. Also, the nonterminating q-Dixon summation formula is used. A special case of the new $${}_{12}\phi _{11}$$ 12 ϕ 11 transformation formula is further utilized to obtain a generalization of Rogers’ linearization formula for the continuous q-ultraspherical polynomials.

Emerging investigator series: transformation of common antibiotics during water disinfection with chlorine and formation of antibacterially active products

This study determined that at chlorination conditions representative of wastewater treatment, fluoroquinolones, one of the common antibiotic classes, form antibacterially active products.