A D’Alembert Formula for Flat Surfaces in the 3-Sphere

We find all the flat surfaces in the unit 3-sphere $\mathbb{S}^{3}We find all the flat surfaces in the unit 3-sphere that pass through a given regular curve of with a prescribed tangent plane distribution along this curve. The formula that solves this problem may be seen as a geometric analogue of the classical D’Alembert formula that solves the Cauchy problem for the homogeneous wave equation. We also provide several applications of this geometric D’Alembert formula, including a classification of the flat M?bius strips of  .      

Analysis of optodynamic sources generated during laser-induced breakdown in water

We propose a method for evaluating the size of the laser-induced breakdown region in water based on the detection and analysis of optodynamic waves. The breakdown region is an optodynamic source of pressure waves that propagate into the surrounding liquid as an ultrasonic pulse. In the experiment the optical breakdown was generated by a standard ophthalmic Nd:YAG laser with a pulse duration of 10 ns and a maximum energy per pulse of 10 mJ. The pulses were detected inside the liquid with a laser-beam deflection probe. The waveforms were captured in the far-field and analyzed. The analysis provides information about the apparent size of the optodynamic source, which is directly related to the size of the breakdown region. The proposed method can be adapted for online monitoring.     

ESI‐MS/MS of expanded porphyrins: a look into their structure and aromaticity

Electrospray mass spectrometry/mass spectrometry was used to investigate the gas‐phase properties of protonated expanded porphyrins, in order to correlate those with their structure and conformation. We have selected five expanded meso‐pentafluorophenyl porphyrins, respectively, a pair of oxidized/reduced fused pentaphyrins (22 and 24 π electrons), a pair of oxidized/reduced regular hexaphyrins (26 and 28 π electrons) and a regular doubly N‐fused hexaphyrin (28 π electrons). The gas‐phase behavior of the protonated species of oxidized and reduced expanded porphyrins is different. The oxidized species (aromatic Hückel systems) fragment more extensively, mainly by the loss of two HF molecules. The reduced species (Möbius aromatic or Möbius‐like aromatic systems) fragment less than their oxidized counterparts because of their increased flexibility. The protonated regular doubly fused hexaphyrin (non‐aromatic Hückel system) shows the least fragmentation even at higher collision energies. In general, cyclization through losses of HF molecules decreases from the aromatic Hückel systems to Möbius aromatic or Möbius‐like aromatic systems to non‐aromatic Hückel systems and is related to an increase in conformational distortion. Copyright © 2016 John Wiley & Sons, Ltd.     

Local structure and water cleaning ability of iron oxide nanoparticles prepared by hydro-thermal reaction

Nanoparticles (NPs) of Fe₃O₄ and γFe₂O₃ synthesized by hydrothermal reaction were characterized by X-ray diffractometry (XRD), ⁵⁷Fe-Mössbauer spectroscopy and field emission scanning electron microscopy (FE-SEM). A decrease in concentration of methylene blue (MB) aqueous solution due to bulk Fe⁰-NP γFe₂O₃ mixture with the mass ratio of 3:7 was measured by ultraviolet-visible light absorption spectroscopy (UV-Vis). The Mössbauer spectrum of NP Fe₃O₄ prepared from hydrothermal reaction was composed of two sextets with absorption area (A), isomer shift (δ) and internal magnetic field (H _int) of 56.3 %, 0.34_±0.03 mm s^???1 and 49.0_±0.30 T for tetrahedral (T _d) Fe^III, and 43.7 %, 0.66_±0.11 mm s^???1 and 44.0_±0.71 T for octahedral (O _h) Fe^II?+?III. The Fe^II/Fe^III ratio was determined to be 0.280 for NP Fe₃O₄, giving ‘x’ of 0.124 in Fe_3???xO_4. These results show that NP Fe₃O₄ prepared by hydrothermal reaction was not regular but nonstoichiometric Fe₃O₄. Consistent results were observed for XRD patterns of NP Fe_3???xO₄ indicating sharp intense peaks at 2Θ of 30.2, 35.7 and 43.3° with a large linewidth of 0.44°, yielding the crystallite size of 29–37 nm from the Scherrer’s equation. Iso-thermal annealing of NP Fe_3???xO₄ at 250 °C for 30 min resulted in the precipitation of NP γFe₂O₃ with δ of 0.33_±0.03 mm s^???1 and H _intof 46.4_±0.27 T due to magnetic tetrahedral Fe^III. The Debye temperature of NP Fe_3???xO₄ was respectively estimated to be 267_±5.45 K for Fe $^{\mathrm{III}}(T_{\mathrm{d}})$ and 282_±7.17 K for Fe $^{\mathrm{II+III}}(O_{\mathrm{h}})$ , both of which were smaller than that obtained for bulk Fe₃O₄ of 280_±4.15 K and 307_±5.70 K, indicating that the chemical environment of iron of NPs is less rigid than that of the bulk compounds. A leaching test using methylene blue (MB) and mixture of bulk Fe⁰-NP γFe₂O₃ (3:7) showed a remarkable decrease in MB concentration from 1.90 × 10^???2 to 9.49 × 10^???4 mM for 24 h with the first order rate constant (k _MB) of 2.1 × 10^???3 min^???1. This result verifies that MB decomposing ability is enhanced by using NP γFe₂O₃ compared with the k _MB of 1.1 × 10^???4 min^???1 previously obtained from the leaching test using MB and bulk mixture of Fe⁰???γFe₂O₃ (3:7).     

Drag Reduction in Wall Bounded Flows: The Effect of Polymers on the Dynamics of Turbulence

Experiments in a refractive index matched pipe flow facility were conducted using state-of-the-art laser-Doppler anemometry to study turbulent drag reduction by dilute addition of high polymers. The results were analyzed employing the invariant theory of turbulence. It was thus possible to confirm the major conclusion of preceding theoretical work, namely that the mechanism of drag reduction by long-chain polymers is associated with an increase in anisotropy of turbulence at the wall. Furthermore, theoretical considerations based on the elastic behavior of a polymer and spatial intermittency of turbulence at small scales enabled quantitative estimates to be made for the relaxation time of a polymer and its concentration that ensures maximum drag reduction. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A bias reducing technique in kernel distribution function estimation

In this paper we suggest a bias reducing technique in kerneldistribution function estimation. In fact, it uses a convex combination of three kernel estimators, and it turned out that the bias has been reduced to the fourth power of the bandwidth, while the bias of the kernel distribution function estimator has the second power of the bandwidth. Also, the variance of the proposed estimator remains at the same order as the kernel distribution function estimator. Numerical results based on simulation studies show this phenomenon, too.     

Is U(1)H a good family symmetry?

We analyze U(1)_H as a horizontal symmetry and its possibilities to explain the known elementary-fermion masses. We find that only two candidates, in the context of SU(3)_c ? SU(2)_L ? U(1)_Y ? U(1)_H nonsuper-symmetric, are able to fit the experimental result m_b«m_t. identity, but it is a common prejudice to assume that the appropriate family symmetry may explain this fact as a consequence of (i) and (ii). In what follows we will enlarge the SM gauge group with an extra U(1)_H horizontal local gauge symmetry (the simplest multi-family continuous symmetry we can think of). We then show that the structure SU(3)_c ? SU(2)_L ? U(1)_Y ? U(1)_H by itself is able to explain (ii), and that the simplest supersymmetric (SUSY) extension of this model without a μ-term can not cope with (ii).     

A note on maximally repeated sub-patterns of a point set

We answer a question raised by Brass on the number of maximally repeated sub-patterns in a set of n points in R^d. We show that this number, which was conjectured to be polynomial, is in fact Θ(2^n/2) in the worst case, regardless of the dimension d.