Self-similar solutions for active scalar equations in Fourier–Besov–Morrey spaces

We are concerned with a family of dissipative active scalar equation with velocity fields coupled via multiplier operators that can be of positive-order. We consider sub-critical values for the fractional diffusion and prove global well-posedness of solutions with small initial data belonging to a framework based on Fourier transform, namely Fourier–Besov–Morrey spaces. Since the smallness condition is with respect to the weak norm of this space, some initial data with large \(L^{2}\) -norm can be considered. Self-similar solutions are obtained depending on the homogeneity of the initial data and couplings. Also, we show that solutions are asymptotically self-similar at infinity. Our results can be applied in a unified way for a number of active scalar PDEs like 1D models on dislocation dynamics in crystals, Burgers’ equation, 2D vorticity equation, 2D generalized SQG, 3D magneto-geostrophic equations, among others.     

Clustering of spectra and fractals of regular graphs

We exhibit a characteristic structure of the class of all regular graphs of degree d that stems from the spectra of their adjacency matrices. The structure has a fractal threadlike appearance. Points with coordinates given by the mean and variance of the exponentials of graph eigenvalues cluster around a line segment that we call a filar. Zooming-in reveals that this cluster splits into smaller segments (filars) labeled by the number of triangles in graphs. Further zooming-in shows that the smaller filars split into subfilars labeled by the number of quadrangles in graphs, etc. We call this fractal structure, discovered in a numerical experiment, a multifilar structure. We also provide a mathematical explanation of this phenomenon based on the Ihara-Selberg trace formula, and compute the coordinates and slopes of all filars in terms of Bessel functions of the first kind.     

Self-similar solutions and large time asymptotics for the dissipative quasi-geostrophic equation

We analyze the well-posedness of the initial value problem for the dissipative quasi-geostrophic equations in the subcritical case. Mild solutions are obtained in several spaces with the right homogeneity to allow the existence of self-similar solutions. While the only small self-similar solution in the strong L^p{\cal L}^{p} space is the null solution, infinitely many self-similar solutions do exist in weak- L^p{\cal L}^{p} spaces and in a recently introduced [7] space of tempered distributions. The asymptotic stability of solutions is obtained in both spaces, and as a consequence, a criterion of self-similarity persistence at large times is obtained.     

Electron resonance studies of the renner effect

The gas phase electron resonance spectra of NCO in its ground ²Π_3/2 vibronic state and in two excited vibronic states are described. Theoretical analysis of the spectra yields effective g values for the three states. In additon the ¹⁴N magnetic hyperfine and electric quadrupole coupling constants and the electric dipole moment are determined. The theory of the Renner coupling of electronic and vibrational motion is extended, and shown to account for anomalous contributions to the g values. The theory also shows that these contributions are closely related to the Renner coupling constant.     

Reflection anisotropy spectroscopy: A new probe for the solid-liquid interface

Introducing reflection anisotropy spectroscopy (RAS) as a new probe for solid-liquid interfaces, we present results for the Au(110)/electrolyte interface which serves as a model system. We demonstrate that RAS is sensitive to surface phase transitions, step morphology, and electronic surface states. Using an empirical approach, the RA spectra are reproduced and features are identified which reflect the known character of the bias voltage driven (2x1) to (1x1) phase transition. RAS is established as an experimental technique to probe the electronic structure of solid-liquid interfaces in real time to study a wide range of interface properties.     

Directly detected Mn MRI: Application to phantoms for human hyperpolarized C MRI development

In this work we demonstrate for the first time directly detected manganese-55 (⁵⁵Mn) magnetic resonance imaging (MRI) using a clinical 3 T MRI scanner designed for human hyperpolarized ¹³C clinical studies with no additional hardware modifications. Due to the similar frequency of the ⁵⁵Mn and ¹³C resonances, the use of aqueous permanganate for large, signal-dense, and cost-effective “¹³C” MRI phantoms was investigated, addressing the clear need for new phantoms for these studies. Due to 100% natural abundance, higher intrinsic sensitivity, and favorable relaxation properties, ⁵⁵Mn MRI of aqueous permanganate demonstrates dramatically increased sensitivity over typical ¹³C phantom MRI, at greatly reduced cost as compared with large ¹³C-enriched phantoms. A large sensitivity advantage (22-fold) was demonstrated. A cylindrical phantom (d = 8 cm) containing concentrated aqueous sodium permanganate (2.7 M) was scanned rapidly by ⁵⁵Mn MRI in a human head coil tuned for ¹³C, using a balanced steady state free precession acquisition. The requisite penetration of radiofrequency magnetic fields into concentrated permanganate was investigated by experiments and high frequency electromagnetic simulations, and found to be sufficient for ⁵⁵Mn MRI with reasonably sized phantoms. A sub-second slice-selective acquisition yielded mean image signal-to-noise ratio of ~ 60 at 0.5 cm³ spatial resolution, distributed with minimum central signal ~ 40% of the maximum edge signal. We anticipate that permanganate phantoms will be very useful for testing HP ¹³C coils and methods designed for human studies.     

Data analysis method to achieve sub‐10 pm spatial resolution using extended X‐ray absorption fine‐structure spectroscopy

Obtaining sub‐10 pm spatial resolution by extended X‐ray absorption fine structure (EXAFS) spectroscopy is required in many important fields of research, such as lattice distortion studies in colossal magnetic resistance materials, high‐temperature superconductivity materials etc. However, based on the existing EXAFS data analysis methods, EXAFS has a spatial resolution limit of π/2Δk which is larger than 0.1 Å. In this paper a new data analysis method which can easily achieve sub‐10 pm resolution is introduced. Theoretically, the resolution limit of the method is three times better than that normally available. The method is examined by numerical simulation and experimental data. As a demonstration, the LaFe_1–xCr_xO₃ system (x = 0, 1/3, 2/3) is studied and the structural information of FeO₆ octahedral distortion as a function of Cr doping is resolved directly from EXAFS, where a resolution better than 0.074 Å is achieved.     

Environmentally friendly sonocatalysis promoted preparation of 1-thiocarbamoyl-3,5-diaryl-4,5-dihydro-1H-pyrazoles

An efficient and green synthesis of thiocarbamoyl-3,5-diaryl-4,5-dihydro-1H-pyrazoles via the condensation of chalcones with thiosemicarbazide in ethanol and KOH under ultrasound irradiation is reported. The products were isolated in good yields after short reaction times.