900 research outputs found
Homology of spaces of non-resultant polynomial systems in R^2 and C^2
The resultant veriety in the space of systems of homogeneous polynomials of
given degrees consists of such systems having non-trivial solutions. We
calculate the integer cohomology groups of all spaces of non-resultant systems
of polynomials , and also the rational cohomology groups of similar
systems in
Cohomology of spaces of complex knots
We develop a technique for calculating the cohomology groups of spaces of
complex parametric knots in , , and carry out these
calculations to obtain these groups of low dimensions
Cohomology of spaces of Hopf equivariant maps of spheres
For any natural , the rational cohomology ring of the space of
continuous maps (respectively, )
equivariant under the Hopf action of the circle (respectively, of the group
of unit quaternions) is naturally isomorphic to that of the Stiefel
manifold (respectively, )
Complements of caustics of simple real function singularities
All local connected components of complements of caustics of simple function
singularities are listed, and generators of their homology groups are realized
Simplified numerical form of universal finite type invariant of Gauss words
In the present paper, we study the finite type invariants of Gauss words. In
the Polyak algebra techniques, we reduce the determination of the group
structure to transformation of a matrix into its Smith normal form and we give
the simplified form of a universal finite type invariant by means of the
isomorphism of this transformation. The advantage of this process is that we
can implement it as a computer program. We obtain the universal finite type
invariant of degree 4, 5, and 6 explicitly. Moreover, as an application, we
give the complete classification of Gauss words of rank 4 and the partial
classification of Gauss words of rank 5 where the distinction of only one pair
remains.Comment: 12 pages, 3 table
Whispering gallery mode resonator based ultra-narrow linewidth external cavity semiconductor laser
We demonstrate a miniature self-injection locked DFB laser using resonant
optical feedback from a high-Q crystalline whispering gallery mode resonator.
The linewidth reduction factor is greater than 10,000, with resultant
instantaneous linewidth less than 200 Hz. The minimal value of the Allan
deviation for the laser frequency stability is 3x10^(-12) at the integration
time of 20 us. The laser possesses excellent spectral purity and good long term
stability.Comment: To be published in Optics Letter
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