We present a new proof of Whitney’s broken circuit theorem based on induction on the number of edges and the deletion-contraction formula.

The horizontal geoelectric field induced at the Earth’s surface during temporal variations of the geomagnetic field drives Geomagnetically Induced Currents (GIC) in groundbased conductor networks, such as electric power transmission grids. Usually studies of the geoelectric field in connection with GIC research employ the assumption of a onedimensional (1D) Earth conductivity structure. At ocea...

A theorem of Mirsky provides necessary and sufficient conditions for the existence of an N-square complex matrix with prescribed diagonal entries and prescribed eigenvalues. A simple inductive proof of this theorem is given.

e _ 2'/z 2 4 4 6 6 8 1 / 8 2 () (33) 1/4 ( 5577) which is proved as follows . For v > 2, the Pth factor is [2' '. . 2P/(2p'+ 1) . (2 1)]'áz [(2"-'-1)!!z2"!!z/2 .2v-'!!z(2v-1)!!z]'/z where n!!=n(n-2) . . .4.2 if n is even, n(n-2) . . .3 .1 if n is odd. Since 2°!!=2 z " '2° '1 and (2°-1)!!=2°! /2°!!=2°!/2z" '2°'!, this expression becomes [2z"2°-'16/2 .2 zí42'!z]I/z By induction on v, the product ...

H(⊗1 C+) denotes the Hardy space of square integrable functions analytic in each variable separately. Let P be the natural projection of L(⊗1 C+) onto H (⊗1 C+). A Hankel operator with symbol b is the linear operator from H(⊗1 C+) to H (⊗1 C+) given by Hb φ = P⊖bφ. We show that ‖Hb‖ ≃ ‖P b‖BMO(⊗n 1 C+) , where the right hand norm is S.-Y. Chang and R. Fefferman product BMO. This fact has well k...

ion identification

1 A Few Preliminaries 2 1.1 Mathematics and Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Sets and Equivalence Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 Mathematical Induction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.4 Complex and Matrix Arithmetic . . . . . . . . . ...

In this paper we present some metapredicative subsystems of analysis. We deal with reflection principles, ω-model existence axioms (limit axioms) and axioms asserting the existence of hierarchies. We show several equivalences of the introduced subsystems. In particular we prove the equivalence of Σ1 transfinite dependent choice and Π 1 2 reflection on ω-models of Σ1-DC.