On the Statistical Origin of Topological Symmetries

We investigate a quantum system possessing a parasupersymmetry of order 2, an orthosupersymmetry of order $p$, a fractional supersymmetry of order $p+1$, and topological symmetries of type $(1,p)$ and $(1,1,...,1)$. We obtain the corresponding symmetry generators, explore their relationship, and show that they may be expressed in terms of the creation and annihilation operators for an ordinary boson and orthofermions of order $p$. We give a realization of parafermions of order~2 using orthofermions of arbitrary order $p$, discuss a $p=2$ parasupersymmetry between $p=2$ parafermions and parabosons of arbitrary order, and show that every orthosupersymmetric system possesses topological symmetries. We also reveal a correspondence between the orthosupersymmetry of order $p$ and the fractional supersymmetry of order $p+1$.Comment: 12 page

Quantum Mechanical Symmetries and Topological Invariants

We give the definition and explore the algebraic structure of a class of quantum symmetries, called topological symmetries, which are generalizations of supersymmetry in the sense that they involve topological invariants similar to the Witten index. A topological symmetry (TS) is specified by an integer n>1, which determines its grading properties, and an n-tuple of positive integers (m_1,m_2,...,m_n). We identify the algebras of supersymmetry, p=2 parasupersymmetry, and fractional supersymmetry of order n with those of the Z_2-graded TS of type (1,1), Z_2-graded TS of type (2,1), and Z_n-graded TS of type (1,1,...,1), respectively. We also comment on the mathematical interpretation of the topological invariants associated with the Z_n-graded TS of type (1,1,...,1). For n=2, the invariant is the Witten index which can be identified with the analytic index of a Fredholm operator. For n>2, there are n independent integer-valued invariants. These can be related to differences of the dimension of the kernels of various products of n operators satisfying certain conditions.Comment: Revised version, to appear in Nucl. Phys.

Artificial Intelligence and Machine Learning for Systems Analysis of the 21st Century

This paper overviews research being done at IIASA with use of machine learning (ML) methods. We elaborate on promising areas of application and advantages and challenges of using ML. These reflections are done as a part of strategic planning process going on at IIASA at the moment, which aims to come up with a new research strategy for 2021-2030, as well as a supporting research plan. It has been recognized that while applications of ML in commercial sector are numerous and become more and more powerful day to day, it is not yet so common to use ML for creating societal impact. To explore the opportunities in this context and to reflect on what IIASA’s role might be, an internal working group was initiated. This paper emerged from the internal workshop held by the working group at IIASA on June 24, 2019; the workshop invited all IIASA scientists to contribute. The workshop program can be found in Appendix A to this paper

On supersymmetric quantum mechanics

This paper constitutes a review on N=2 fractional supersymmetric Quantum Mechanics of order k. The presentation is based on the introduction of a generalized Weyl-Heisenberg algebra W_k. It is shown how a general Hamiltonian can be associated with the algebra W_k. This general Hamiltonian covers various supersymmetrical versions of dynamical systems (Morse system, Poschl-Teller system, fractional supersymmetric oscillator of order k, etc.). The case of ordinary supersymmetric Quantum Mechanics corresponds to k=2. A connection between fractional supersymmetric Quantum Mechanics and ordinary supersymmetric Quantum Mechanics is briefly described. A realization of the algebra W_k, of the N=2 supercharges and of the corresponding Hamiltonian is given in terms of deformed-bosons and k-fermions as well as in terms of differential operators.Comment: Review paper (31 pages) to be published in: Fundamental World of Quantum Chemistry, A Tribute to the Memory of Per-Olov Lowdin, Volume 3, E. Brandas and E.S. Kryachko (Eds.), Springer-Verlag, Berlin, 200