Statistics of a Family of Piecewise Linear Maps

We study statistical properties of the truncated flat spot map $f_t(x)$. In particular, we investigate whether for large $n$, the deviations $\sum_{i=0}^{n-1} \left(f_t^i(x_0)-\frac 12\right)$ upon rescaling satisfy a $Q$-Gaussian distribution if $x_0$ and $t$ are both independently and uniformly distributed on the unit circle. This was motivated by the fact that if $f_t$ is the rotation by $t$, then it has been shown that in this case the rescaled deviations are distributed as a $Q$-Gaussian with $Q=2$ (a Cauchy distribution). This is the only case where a non-trivial (i.e. $Q\neq 1$) $Q$-Gaussian has been analytically established in a conservative dynamical system. In this note, however, we prove that for the family considered here, $\lim_n S_n/n$ converges to a random variable with a curious distribution which is clearly not a $Q$-Gaussian or any other standard smooth distribution

One-Sided Derivative of Distance to a Compact Set

We give a complete and self-contained proof of a folklore theorem which says that in an Alexandrov space the distance between a point $\gamma(t)$ on a geodesic $\gamma$ and a compact set $K$ is a right-differentiable function of $t$. Moreover, the value of this right-derivative is given by the negative cosine of the minimal angle between the geodesic and any shortest path to the compact set (Theorem 4.3). Our treatment serves as a general introduction to metric geometry and relies only on the basic elements, such as comparison triangles and upper angles.Comment: 22 pages, 8 figure

Thermal Diffusion and Quench Propagation in YBCO Pancake Coils Wound with ZnO-and Mylar Insulations

The thermal diffusion properties of several different kinds of YBCO insulations and the quench properties of pancake coils made using these insulations were studied. Insulations investigated include Nomex, Kapton, and Mylar, as well as insulations based on ZnO, Zn2GeO4, and ZnO-Cu. Initially, short stacks of YBCO conductors with interlayer insulation, epoxy, and a central heater strip were made and later measured for thermal conductivity in liquid nitrogen. Subsequently, three different pancake coils were made. The first two were smaller, each using one meter total of YBCO tape present as four turns around a G-10 former. One of these smaller coils used Mylar insulation co-wound with the YBCO tape, the other used YBCO tape onto which ZnO based insulation had been deposited. One larger coil was made which used 12 total meters of ZnO-insulated tape and had 45 turns. The results for all short sample and coil thermal conductivities were ~1-3 Wm-1K-1. Finally, quench propagation velocity measurements were performed on the coils (77 K, self field) by applying a DC current and then using a heater pulse to initiate a quench. Normal zone propagation velocity (NZP) values were obtained for the coils both in the radial direction and in the azimuthal direction. Radial NZP values (0.05-0.7 mm/s) were two orders of magnitude lower than axial values (~14-17 mm/s). Nevertheless, the quenches were generally seen to propagate radially within the coils, in the sense that any given layer in the coil is driven normal by the layer underneath it.Comment: 58 pages, 5 tables, 16 fig

Statistics of a Family of Piecewise Linear Maps

We study statistical properties of a family of piecewise linear monotone circle maps ft (x) related to the Σ angle doubling map x → 2x mod 1. In particular, we investigate whether for large n, the deviations n−1 i=0 (f it (x0) − 12) upon rescaling satisfies a Q-Gaussian distribution if x0 and t are both independently and uniformly distributed on the unit circle. This was motivated by the fact that if ft is the rotation by t, then it was recently found that in this case the rescaled deviations are distributed as a Q-Gaussian with Q = 2 (a Cauchy distribution). This is the only case where a non-trivial (i.e. Q ̸= 1) Q-Gaussian has been analytically established in a conservative dynamical system. In this note, however, we prove that for the family considered here, limn Sn/n converges to a random variable with a curious distribution which is clearly not a Q-Gaussian or any other standard smooth distribution

Influence of nickel substrate grain structure on YBa2Cu3O7-x supercurrent connectivity in deformation-textured coated conductors

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