Affine embeddings and intersections of Cantor sets

Let $E, F\subset \R^d$ be two self-similar sets. Under mild conditions, we show that $F$ can be $C^1$-embedded into $E$ if and only if it can be affinely embedded into $E$; furthermore if $F$ can not be affinely embedded into $E$, then the Hausdorff dimension of the intersection $E\cap f(F)$ is strictly less than that of $F$ for any $C^1$-diffeomorphism $f$ on $\R^d$. Under certain circumstances, we prove the logarithmic commensurability between the contraction ratios of $E$ and $F$ if $F$ can be affinely embedded into $E$. As an application, we show that $\dim_HE\cap f(F)<\min\{\dim_HE, \dim_HF\}$ when $E$ is any Cantor-$p$ set and $F$ any Cantor-$q$ set, where $p,q\geq 2$ are two integers with \log p/\log q\not \in \Q. This is related to a conjecture of Furtenberg about the intersections of Cantor sets.Comment: The paper will appear in J. Math. Pure. App

Dual function of Slit2 in repulsion and enhanced migration of trunk, but not vagal, neural crest cells

Neural crest precursors to the autonomic nervous system form different derivatives depending upon their axial level of origin; for example, vagal, but not trunk, neural crest cells form the enteric ganglia of the gut. Here, we show that Slit2 is expressed at the entrance of the gut, which is selectively invaded by vagal, but not trunk, neural crest. Accordingly, only trunk neural crest cells express Robo receptors. In vivo and in vitro experiments demonstrate that trunk, not vagal, crest cells avoid cells or cell membranes expressing Slit2, thereby contributing to the differential ability of neural crest populations to invade and innervate the gut. Conversely, exposure to soluble Slit2 significantly increases the distance traversed by trunk neural crest cells. These results suggest that Slit2 can act bifunctionally, both repulsing and stimulating the motility of trunk neural crest cells

The Structure and C=C Vibrational Frequencies of the all- trans Polyenes C2nH2n+2(n=2-15), C2nH2n(Me)2(n=2-13), and C2nH2n(tert-Butyl)2(n=2-5): Computational Results

Carbon-carbon bond lengths and C=C vibrational frequencies are reported for the linear, all-trans unsubstituted C2nH2n+2 (n=2-15), methyl capped C2nH2nMe2 (n=2-13), and tert-butyl capped C2nH2n(tert-butyl)2 (n=2-5) polyenes (C2h) calculated at the B3LYP/6-311++G(d,p) level. The C=C/C-C bond length alternation remains evident at this level for the unsubstituted and methyl capped polyenes as the chain length increases; the center-most difference in the length of the C-C/C=C bonds is ~0.06 Å for C30H32 and C26H26Me2. The Ag, in-phase, harmonic C=C Raman frequency for the unsubstituted polyenes decreases from 1699.2 cm-1 (n = 2) to 1528.9 cm-1 (n=15); the anharmonic frequency decreases from 1651.5 cm-1 (n = 2) to 1547.7 cm-1 (n = 8). The harmonic C=C frequency for the methyl capped polyenes decreases from 1717.9 cm-1 (n = 2) to 1539.6 cm- 1 (n= 13), and the anharmonic C=C frequency decreases from 1675.0 cm-1 (n = 2) to 1562.8 cm-1 (n = 7)

Developments of the pinned photodiode terahertz rectifier

This paper presents we presents a development of the structure of the pinned photodiode terahertz rectifier, in which the metal whisker of the antenna is separated from the semiconductor by a silane oxide layer, in order to reduce the surface defectiveness. The rectifies is the basic component of an image detection system based on the structure of actual CMOS image detectors. The structure combines a nano-antenna, fabricated on the top of a standard image sensor, the rectifier, and the readout electronics. The rectifier device proposed has vertical extension of some tenths of nanometers, can be created at the foot of the nano-whisker at the end of the terahertz antenna, above the storage well

A Note On Computing Set Overlap Classes

Let ${\cal V}$ be a finite set of $n$ elements and ${\cal F}=\{X_1,X_2, >..., X_m\}$ a family of $m$ subsets of ${\cal V}.$ Two sets $X_i$ and $X_j$ of ${\cal F}$ overlap if $X_i \cap X_j \neq \emptyset,$ $X_j \setminus X_i \neq \emptyset,$ and $X_i \setminus X_j \neq \emptyset.$ Two sets $X,Y\in {\cal F}$ are in the same overlap class if there is a series $X=X_1,X_2, ..., X_k=Y$ of sets of ${\cal F}$ in which each $X_iX_{i+1}$ overlaps. In this note, we focus on efficiently identifying all overlap classes in $O(n+\sum_{i=1}^m |X_i|)$ time. We thus revisit the clever algorithm of Dahlhaus of which we give a clear presentation and that we simplify to make it practical and implementable in its real worst case complexity. An useful variant of Dahlhaus's approach is also explained

Electronic phase separation in the rare earth manganates, (La1-xLnx)0.7Ca0.3MnO3 (Ln = Nd, Gd and Y)

All the three series of manganates showsaturation magnetization characteristic of ferromagnetism, with the ferromagnetic Tc decreasing with increasing in x up to a critical value of x, xc (xc = 0.6, 0.3, 0.2 respectively for Nd, Gd, Y). For x > xc, the magnetic moments are considerably smaller showing a small increase around TM, the value of TM decreasing slightly with increase in x or decrease in . The ferromagnetic compositions (x xc) show insulator-metal (IM) transitions, while the compositions with x > xc are insulating. The magnetic and electrical resistivity behavior of these manganates is consistent with the occurrence of phase separation in the compositions around xc, corresponding to a critical average radius of the A-site cation, , of 1.18 A. Both Tc and TIM increase linearly when < rA > > or x xc as expected of a homogenous ferromagnetic phase. Both Tc and TM decrease linearly with the A-site cation size disorder at the A-site as measured by the variance s2. Thus, an increase in s2 favors the insulating AFM state. Percolative conduction is observed in the compositions with > < rAc >. Electron transport properties in the insulating regime for x > xc conforms to the variable range hopping mechanism. More interestingly, when x > xc, the real part of dielectric constant (e') reaches a high value (104-106) at ordinary temperatures dropping to a very small (~500) value below a certain temperature, the value of which decreases with decreasing frequency.Comment: 27 pages; 11 figures, Submitted to J.Phys:Condens Matte

CMOS-Compatible Room-Temperature Rectifier Toward Terahertz Radiation Detection

In this paper, we present a new rectifying device, compatible with the technology of CMOS image sensors, suitable for implementing a direct-conversion detector operating at room temperature for operation at up to terahertz frequencies. The rectifying device can be obtained by introducing some simple modifications of the charge-storage well in conventional CMOS integrated circuits, making the proposed solution easy to integrate with the existing imaging systems. The rectifying device is combined with the different elements of the detector, composed of a 3D high-performance antenna and a charge-storage well. In particular, its position just below the edge of the 3D antenna takes maximum advantage of the high electric field concentrated by the antenna itself. In addition, the proposed structure ensures the integrity of the charge-storage well of the detector. In the structure, it is not necessary to use very scaled and costly technological nodes, since the CMOS transistor only provides the necessary integrated readout electronics. On-wafer measurements of RF characteristics of the designed junction are reported and discussed. The overall performances of the entire detector in terms of noise equivalent power (NEP) are evaluated by combining low-frequency measurements of the rectifier with numerical simulations of the 3D antenna and the semiconductor structure at 1 THz, allowing prediction of the achievable NEP