13 research outputs found
On the Statistical Origin of Topological Symmetries
We investigate a quantum system possessing a parasupersymmetry of order 2, an
orthosupersymmetry of order , a fractional supersymmetry of order , and
topological symmetries of type and . 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 . We give a realization of
parafermions of order~2 using orthofermions of arbitrary order , discuss a
parasupersymmetry between 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 and the fractional supersymmetry of order .Comment: 12 page
Topological Symmetries
We introduce the notion of a topological symmetry as a quantum mechanical
symmetry involving a certain topological invariant. We obtain the underlying
algebraic structure of the Z_2-graded uniform topological symmetries of type
(1,1) and (2,1). This leads to a novel derivation of the algebras of
supersymmetry and parasupersummetry.Comment: Plain LaTeX Ref: Mod. Phys. Lett. A 15, 175-184 (2000
On the Representation Theory of Orthofermions and Orthosupersymmetric Realization of Parasupersymmetry and Fractional Supersymmetry
We construct a canonical irreducible representation for the orthofermion
algebra of arbitrary order, and show that every representation decomposes into
irreducible representations that are isomorphic to either the canonical
representation or the trivial representation. We use these results to show that
every orthosupersymmetric system of order has a parasupersymmetry of order
and a fractional supersymmetry of order .Comment: 13 pages, to appear in J. Phys. A: Math. Ge
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
Stability of synchronous state in networks of chaotic maps by matrix measure approach
Stability of synchronous state is a fundamental problem in synchronization. We study Matrix Measure as an approach for investigating of stability of synchronous states of chaotic maps on complex networks. Matrix Measure is a measure which depends on network structure. Using this measure and comparing with synchronization threshold which depends on the function of the map, show us how the synchronous state can be stabilized. We use these methods for networks with different parameters and topologies. Our numerical calculation shows that synchronous states on more dense networks are more stable. Network’s size is another effective parameter that order of value and extent of stability interval is determined by network’s size. Our results also show that among dense networks, Random and Scale-Free networks have larger stability interval of coupling strength. Finally, we use Error Function to test a prediction of Matrix Measure approach