In the electromagnetic theory, the vacuum impedance Z0 is a universal constant, which is as important as the velocity of light c0 in vacuum. Unfortunately, however, its significance is not appreciated so well and sometimes the presence itself is ignored. It is partly because in the Gaussian system of units, which has widely been used for long time, Z0 is a dimensionless constant and of unit mag...

In this note, we present the hyperbolic Menelaus theorem in the Poincaré disc of hyperbolic geometry. 2000 Mathematical Subject Classi…cation: 30F45, 20N99, 51B10, 51M10 Keywords and phrases: hyperbolic geometry, hyperbolic triangle, gyrovector 1. Introduction Hyperbolic Geometry appeared in the …rst half of the 19 century as an attempt to understand Euclid’s axiomatic basis of Geometry. It is ...

Let f be a subhyperbolic entire transcendental function of finite order and let z0 be a repelling periodic point of f . We show that there exists at least one dynamic ray (injective curve to ∞ consisting of escaping points) that lands at z0. In fact, our result holds more generally, namely for any subhyperbolic entire function f for which each periodic address is realised by some dynamic ray in...

This document completes the paper “Smart Energy-Aware Sensors for Event-Based Control” submitted to the 51 IEEE Conference on Decision and Control by the same authors. It is not intended to be self contained; it only gives the proof of Lemma 2. I. APPENDIX We recall from [25] the following elements. The closed loop system (the system (5) with the policy (18) and the initial conditions (z0,m0)),...

The celebrated 100-year old Phragmén-Lindelöf theorem, [15, 16], is a far reaching extension of the maximum modulus theorem for holomorphic functions that in its simplest form can be stated as follows: Theorem 1.1. Let Ω ⊂ C be a simply connected domain whose boundary contains the point at infinity. If f is a bounded holomorphic function on Ω and lim supz→z0 |f(z)| ≤ M at each finite boundary p...

Here we classify finite non-Dedekindian p-groups which are not generated by their non-normal subgroups. (Theorem 1). The purpose of this paper is to classify non-Dedekindian finite p-groups which are not generated by their non-normal subgroups. It is surprising that such p-groups must be of class 2 with a cyclic commutator subgroup. We consider here only finite p-groups and our notation is stan...

Red clump giant stars can be used as distance indicators to trace the mass distribution of the Galactic bar. We use RCG stars from 44 bulge fields from the OGLE-II microlensing collaboration database to constrain analytic tri-axial models for the Galactic bar. We find the bar major axis is oriented at an angle of 24◦ – 27◦ to the Sun-Galactic centre line-of-sight. The ratio of semi-major and se...

Solutions at infinity are solutions with z0 = 0 and not all other coordinates equal to zero. We see that only z 1z 2 2 = 0 remains, or equivalently we find [0 : 1 : 0] and [0 : 0 : 1] representing two distinct solutions at infinity, but each with multiplicity two. This is an unsatisfactory result, as we expected to find two, not four solutions at infinity. The projective transformation defined ...

These lecture notes are based on Ivan Corwin’s summer 2014 MSRI summer school on SPDEs, as well as Jeffrey Kuan’s TA sessions accompanying these lectures. Please email [email protected] if you have questions or find mistakes. 1. Mild solution to the stochastic heat equation The stochastic heat equation (SHE) with multiplicative noise looks (in differential form) like { ∂tz = 1 2∂xxz + zξ z(...