514 research outputs found
Optical waveguide arrays: quantum effects and PT symmetry breaking
Over the last two decades, advances in fabrication have led to significant
progress in creating patterned heterostructures that support either carriers,
such as electrons or holes, with specific band structure or electromagnetic
waves with a given mode structure and dispersion. In this article, we review
the properties of light in coupled optical waveguides that support specific
energy spectra, with or without the effects of disorder, that are
well-described by a Hermitian tight-binding model. We show that with a
judicious choice of the initial wave packet, this system displays the
characteristics of a quantum particle, including transverse photonic transport
and localization, and that of a classical particle. We extend the analysis to
non-Hermitian, parity and time-reversal () symmetric Hamiltonians
which physically represent waveguide arrays with spatially separated, balanced
absorption or amplification. We show that coupled waveguides are an ideal
candidate to simulate -symmetric Hamiltonians and the transition
from a purely real energy spectrum to a spectrum with complex conjugate
eigenvalues that occurs in them.Comment: 16 pages, 12 figures, Invited Review for European Physics Journal -
Applied Physic
Tunable waveguide lattices with non-uniform parity-symmetric tunneling
We investigate the single-particle time evolution and two-particle quantum
correlations in a one-dimensional -site lattice with a site-dependent
nearest neighbor tunneling function . Since
the bandwidth and the energy levels spacings for such a lattice both depend
upon , we show that the observable properties of a wavepacket, such as
its spread and the relative phases of its constitutents, vary dramatically as
is varied from positive to negative values. We also find that the
quantum correlations are exquisitely sensitive to the form of the tunneling
function. Our results suggest that arrays of waveguides with position-dependent
evanascent couplings will show rich dynamics with no counterpart in
present-day, traditional systems.Comment: 5 pages, 4 figure
The ⊛-composition of fuzzy implications: Closures with respect to properties, powers and families
Recently, Vemuri and Jayaram proposed a novel method of generating fuzzy implications from a given pair of fuzzy implications. Viewing this as a binary operation ⊛ on the set II of fuzzy implications they obtained, for the first time, a monoid structure (I,⊛)(I,⊛) on the set II. Some algebraic aspects of (I,⊛)(I,⊛) had already been explored and hitherto unknown representation results for the Yager's families of fuzzy implications were obtained in [53] (N.R. Vemuri and B. Jayaram, Representations through a monoid on the set of fuzzy implications, fuzzy sets and systems, 247 (2014) 51–67). However, the properties of fuzzy implications generated or obtained using the ⊛-composition have not been explored. In this work, the preservation of the basic properties like neutrality, ordering and exchange principles , the functional equations that the obtained fuzzy implications satisfy, the powers w.r.t. ⊛ and their convergence, and the closures of some families of fuzzy implications w.r.t. the operation ⊛, specifically the families of (S,N)(S,N)-, R-, f- and g-implications, are studied. This study shows that the ⊛-composition carries over many of the desirable properties of the original fuzzy implications to the generated fuzzy implications and further, due to the associativity of the ⊛-composition one can obtain, often, infinitely many new fuzzy implications from a single fuzzy implication through self-composition w.r.t. the ⊛-composition
Homomorphisms on the monoid of fuzzy implications and the iterative functional equation I(x,I(x,y))=I(x,y)
Recently, Vemuri and Jayaram proposed a novel method of generating fuzzy implications, called the ⊛⊛-composition, from a given pair of fuzzy implications [Representations through a Monoid on the set of Fuzzy Implications, Fuzzy Sets and Systems, 247, 51-67]. However, as with any generation process, the ⊛⊛-composition does not always generate new fuzzy implications. In this work, we study the generative power of the ⊛⊛-composition. Towards this end, we study some specific functional equations all of which lead to the solutions of the iterative functional equation I(x,I(x,y))=I(x,y)I(x,I(x,y))=I(x,y) involving fuzzy implications which has been studied extensively for different families of fuzzy implications in this very journal, see [Information Sciences 177, 2954–2970 (2007); 180, 2487–2497 (2010); 186, 209–221 (2012)]. In this work, unlike in other existing works, we do not restrict the solutions to a particular family of fuzzy implications. Thus we take an algebraic approach towards solving these functional equations. Viewing the ⊛⊛-composition as a binary operation ⊛⊛ on the set II of all fuzzy implications one obtains a monoid structure (I,⊛)(I,⊛) on the set II. From the Cayley’s theorem for monoids, we know that any monoid is isomorphic to the set of all right translations. We determine the complete set KK of fuzzy implications w.r.t. which the right translations also become semigroup homomorphisms on the monoid (I,⊛I,⊛) and show that KK not only answers our questions regarding the generative power of the ⊛⊛-composition but also contains many as yet unknown solutions of the iterative functional equation I(x,I(x,y))=I(x,y)I(x,I(x,y))=I(x,y)
Conjugacy Relations via Group Action on the set of Fuzzy Implications
Let denote the set of all increasing bijections on [0
;
1] and
I
the set of fuzzy implications. In
[1], the authors proposed a new way of generating fuzzy implications from fuzzy....
Representations through a monoid on the set of fuzzy implications
Fuzzy implications are one of the most important fuzzy logic connectives. In this work, we conduct a systematic algebraic study on the set II of all fuzzy implications. To this end, we propose a binary operation, denoted by ⊛, which makes (I,⊛I,⊛) a non-idempotent monoid. While this operation does not give a group structure, we determine the largest subgroup SS of this monoid and using its representation define a group action of SS that partitions II into equivalence classes. Based on these equivalence classes, we obtain a hitherto unknown representations of the two main families of fuzzy implications, viz., the f- and g-implications
Lattice operations on fuzzy implications and the preservation of the exchange principle
In this work, we solve an open problem related to the preservation of the exchange principle (EP) of fuzzy implications under lattice operations ([3], Problem 3.1.). We show that generalizations of the commutativity of antecedents (CA) to a pair of fuzzy implications (I,J)(I,J), viz., the generalized exchange principle and the mutual exchangeability are sufficient conditions for the solution of the problem. Further, we determine conditions under which these become necessary too. Finally, we investigate the pairs of fuzzy implications from different families such that (EP) is preserved by the join and meet operations
Bijective transformations of fuzzy implications – An algebraic perspective
Bijective transformations play an important role in generating fuzzy implications from fuzzy implications. In [Representations through a Monoid on the set of Fuzzy Implications, Fuzzy Sets and Systems, 247, 51–67], Vemuri and Jayaram proposed a monoid structure on the set of fuzzy implications, which is denoted by II, and using the largest subgroup SS of this monoid discussed some group actions on the set II. In this context, they obtained a bijective transformation which ultimately led to hitherto unknown representations of the Yager's families of fuzzy implications, viz., f-, g -implications. This motivates us to consider whether the bijective transformations proposed by Baczyński & Drewniak and Jayaram & Mesiar, in different but purely analytic contexts, also possess any algebraic connotations. In this work, we show that these two bijective transformations can also be seen as being obtained from some group actions of SS on II. Further, we consider the most general bijective transformation that generates fuzzy implications from fuzzy implications and show that it can also be obtained as a composition of group actions of SS on II. Thus this work tries to position such bijective transformations from an algebraic perspective
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