Bush‐type Hadamard matrices and symmetric designs

Abstact: A symmetric 2‐(100, 45, 20) design is constructed that admits a tactical decomposition into 10 point and block classes of size 10 such that every point is in either 0 or 5 blocks from a given block class, and every block contains either 0 or 5 points from a given point class. This design yields a Bush‐type Hadamard matrix of order 100 that leads to two new infinite classes of symmetric designs with parameters and where m is an arbitrary positive integer. Similarly, a Bush‐type Hadamard matrix of order 36 is constructed and used for the construction of an infinite family of designs with parameters and a second infinite family of designs with parameters where m is any positive integer. © 2000 John Wiley & Sons, Inc. J Combin Designs 9: 72–78, 2001     

The Existence of a Bush-Type Hadamard Matrix of Order 324 and Two New Infinite Classes of Symmetric Designs

A symmetric 2-(324, 153, 72) design is constructed that admits a tactical decomposition into 18 point and block classes of size 18 such that every point is in either 0 or 9 blocks from a given block class, and every block contains either 0 or 9 points from a given point class. This design is self-dual and yields a symmetric Hadamard matrix of order 324 of Bush type, being the first known example of a symmetric Bush-type Hadamard matrix of order 4n ² for n > 1 odd. Equivalently, the design yields a strongly regular graph with parameters v=324, k=153, ==72 that admits a spread of cocliques of size 18. The Bush-type Hadamard matrix of order 324 leads to two new infinite classes of symmetric designs with parameters

Coherent measures and the existence of smooth Lyapunov 1-forms for flows

Let a smooth vector field V on a smooth closed manifold M be given and let Z⊂M be an isolated invariant set for the flow of V. In this situation, we give a necessary and sufficient condition for the existence of a Lyapunov 1-form for (V,Z) in terms of the relative asymptotic cycles associated with certain invariant measures of the flow.     

The One-Dimensional <Emphasis Type="Italic">Exactly 1</Emphasis> Cellular Automaton: Replication,Periodicity, and Chaos from Finite Seeds

In the Exactly 1 cellular automaton (also known as Rule 22), every site of the one-dimensional lattice is either in state 0 or in state 1, and a synchronous update rule dictates that a site is in state 1 next time if and only if it sees a single 1 in its three-site neighborhood at the current time. We analyze this rule started from finite seeds, i.e., those initial configurations that have only finitely many 1’s. Three qualitatively different types of evolution are observed: replication, periodicity, and chaos. We focus on rigorous results, assisted by algorithmic searches, for the first two behaviors. In particular, we explain why replication is observed so frequently and present a method for collecting the smallest periodic seeds. Some empirical observations about chaotic seeds are also presented.     

Capacitor-based detection of nuclear magnetization: nuclear quadrupole resonance of surfaces

We demonstrate excitation and detection of nuclear magnetization in a nuclear quadrupole resonance (NQR) experiment with a parallel plate capacitor, where the sample is located between the two capacitor plates and not in a coil as usually. While the sensitivity of this capacitor-based detection is found lower compared to an optimal coil-based detection of the same amount of sample, it becomes comparable in the case of very thin samples and even advantageous in the proximity of conducting bodies. This capacitor-based setup may find its application in acquisition of NQR signals from the surface layers on conducting bodies or in a portable tightly integrated nuclear magnetic resonance sensor.