Semi-local units modulo Gauss sums

For thep-th cyclotomic fieldk, Iwasawa proved thatp does not divide the class number of its maximal real subfield if and only if the odd part of the group of local units coincides with its subgroup generated by Jacobi sums related tok. We refine and give a quantitative version of this result for more general imaginary abelian fields. Our result is an analogy of the famous result on “semi-local units modulo cyclotomic units”. Partially supported by Grant-in-Aid for Scientific Research (C), Grant 09640054.     

Infrared laser radar technique using heterodyne detection for range-resolved sensing of air pollutants

An infrared, pulsed heterodyne laser radar using differential absorption technique with tunable lasers is proposed as a practical means of measuring dispersed air pollutants with range resolution. A system analysis shows that this scheme offers an adequate sensitivity for ranges up to several km.     

On the parity of the class number of an imaginary abelian field of conductor 2<Superscript><Emphasis Type="Italic">a</Emphasis></Superscript><Emphasis Type="Italic">p</Emphasis><Superscript><Emphasis Type="Italic">b</Emphasis></Superscript>

Let p be an odd prime number, and p^n₀{p^{n_0}} the highest power of p dividing 2 ^p−1 − 1. Let K_n=Q(z_pⁿ⁺¹){K_n={\bf Q}(\zeta_{p^{n+1}})} and L_n,j=K_n⁺(z_2^j+2){L_{n,j}=K_n^+(\zeta_{2^{j+2}})} for j ≥ 0. Let h_n^*{h_n^*} be the relative class number of K _n , and h _n,j the class number of L _n,j , respectively. Let n be an integer with n ≥ n ₀. We prove that if the ratio h_n^*/h_n-1^*{h_n^*/h_{n-1}^*} is odd, then h _n,j /h _n−1,j is odd for any j ≥ 0.     

Single-Mode Operation of Broad-Area Diode Laser at 670 nm by External Cavity Configuration with a Diffraction Grating

We demonstrate the single mode operation of a broad-area diode laser at 670 nm by means of an external cavity configuration using a diffraction grating. The output power of 150 mW is obtained with a spectral width of 40 MHz.     

On the parity of the class number of the 7<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>th cyclotomic field

For an odd prime number p and an integer n ≥ 0, let h _n be the class number of the p ⁿ⁺¹st cyclotomic field Q(). It is known that when p = 3 or 5, h _n is odd for all n ≥ 0. We prove that the same holds also when p = 7. The author was partially supported by Grant-in-Aid for Scientific Research (C), No. 19540005, Japan Society for the Promotion of Science.     

Optical-fibre remote sensing system of low-level propane gas using a 1·68 μm InGaAs light emitting diode

A fully optical remote detection system for propane gas was realized in real time by a near-infrared absorption method employing a 2-km long low-loss silica optical-fibre link in conjunction with an InGaAs light emitting diode at 1.68 μm. The detection sensitivity for propane gas in air, which was achieved to lower than 2 torr, i.e. nearly 12% of the lower explosion limit for propane, demonstrates the practical feasibility and reliability of this kind of fibre-optic remote gas sensor system with electrical and thermal safety.     

Semi-local units modulo Gauss sums

For the p-th cyclotomic field k, Iwasawa proved that p does not divide the class number of its maximal real subfield if and only if the odd part of the group of local units coincides with its subgroup generated by Jacobi sums related to k. We refine and give a quantitative version of this result for more general imaginary abelian fields. Our result is an analogy of the famous result on “semi-local units modulo cyclotomic units”. Received: 2 May 1997 / Revised version: 11 November 1997     

A class number formula for the p-cyclotomic field

Let p be an odd prime number and . Let be the classical Stickelberger ideal of the group ring . Iwasawa [6] proved that the index equals the relative class number of . In [2], [4] we defined for each subgroup H of G a Stickelberger ideal of , and studied some of its properties. In this note, we prove that when mod 4 and [G : H] = 2, the index equals the quotient . Received: 13 January 2006