arXiv subjects

Isaac Martin

Publications and source records attributed to Isaac Martin.

3 recordsLinked to original sources

arXiv2021

Nanocircuit transmitting microwave and terahertz harmonics generated by a mode-locked laser

Earlier we focused a mode-locked laser on the tip-sample tunneling junction of a scanning tunneling microscope. This superimposed currents at hundreds of harmonics of the laser pulse-repetition rate on the DC tunneling current. The power at each harmonic is inversely proportional to its frequency. However, analysis suggests that within the tunneling junction the harmonics only decay above 45 THz as the exponent of minus the square of the product of the harmonic number and the laser duty cycle. Now this effect is applied in a nanocircuit with an optical antenna to receive the laser radiation, metal-insulator-metal tunneling diodes to generate the harmonics, and filters to select the microwave or terahertz harmonics that are transmitted by a second antenna.

physics.optics↗

arXiv2020

The number of torsion divisors in a strongly $F$-regular ring is bounded by the reciprocal of $F$-signature

Polstra showed that the cardinality of the torsion subgroup of the divisor class group of a local strongly $F$-regular ring is finite. We expand upon this result and prove that the reciprocal of the $F$-signature of a local strongly $F$-regular ring $R$ bounds the cardinality of the torsion subgroup of the divisor class group of $R$.

math.AC↗

arXiv2020

Almost Abelian Lie groups, subgroups and quotients

An almost Abelian Lie group is a non-Abelian Lie group with a codimension 1 Abelian normal subgroup. The majority of 3-dimensional real Lie groups are almost Abelian, and they appear in all parts of physics that deal with anisotropic media - cosmology, crystallography etc. In theoretical physics and differential geometry, almost Abelian Lie groups and their homogeneous spaces provide some of the simplest solvmanifolds on which a variety of geometric structures such as symplectic, K\"ahler, spin etc., are currently studied in explicit terms. Recently, almost Abelian Lie algebras were classified and studied in details. However, a systematic investigation of almost Abelian Lie groups has not been carried out yet, and the present paper is devoted to an explicit description of properties of this wide and diverse class of groups. The subject of investigation are real almost Abelian Lie groups with their Lie group theoretical aspects, such as the exponential map, faithful matrix representations, discrete and connected subgroups, quotients and automorphisms. The emphasis is put on explicit description of all technical details.

math.GR↗