Discrete Approximations of Determinantal Point Processes on Continuous Spaces: Tree Representations and Tail Triviality

We prove tail triviality of determinantal point processes \( \mu \) on continuous spaces. Tail triviality has been proved for such processes only on discrete spaces, and hence we have generalized the result to continuous spaces. To do this, we construct tree representations, that is, discrete approximations of determinantal point processes enjoying a determinantal structure. There are many interesting examples of determinantal point processes on continuous spaces such as zero points of the hyperbolic Gaussian analytic function with Bergman kernel, and the thermodynamic limit of eigenvalues of Gaussian random matrices for \(\hbox {Sine}_2 , \hbox {Airy}_2 , \hbox {Bessel}_2 \), and Ginibre point processes. Our main theorem proves all these point processes are tail trivial.     

Statistical Analysis of the First Passage Path Ensemble of Jump Processes

The transition mechanism of jump processes between two different subsets in state space reveals important dynamical information of the processes and therefore has attracted considerable attention in the past years. In this paper, we study the first passage path ensemble of both discrete-time and continuous-time jump processes on a finite state space. The main approach is to divide each first passage path into nonreactive and reactive segments and to study them separately. The analysis can be applied to jump processes which are non-ergodic, as well as continuous-time jump processes where the waiting time distributions are non-exponential. In the particular case that the jump processes are both Markovian and ergodic, our analysis elucidates the relations between the study of the first passage paths and the study of the transition paths in transition path theory. We provide algorithms to numerically compute statistics of the first passage path ensemble. The computational complexity of these algorithms scales with the complexity of solving a linear system, for which efficient methods are available. Several examples demonstrate the wide applicability of the derived results across research areas.     

Synthesis of some new triamide derivatives via Ugi five-component reaction in aqueous solution

Triamide derivatives have been synthesized in good yields in a novel, one-pot, five-component, and efficient process by the reaction of Z-oxazolone, water, primary amines, aldehydes, isocyanides, in the presence of catalytic amount of KAl(SO₄)₂·12H₂O (alum) as a non-toxic, reusable, inexpensive, and easily available reagent via Ugi reaction in aqueous solution.     

Systematic parametric investigation on the CVD process of polysiloxane nano- and microstructures

Amorphous polysiloxane nano- and microstructures with different shapes can be synthesized from trifunctional organosilane precursors. In the present study, various polysiloxane nano- and microstructures have been produced via a chemical vapor deposition process using ethyltrichlorosilane as precursor. The structure formation and shape are the result of a delicate interplay between temperature, absolute amount of water, and relative humidity. The impact of these reaction parameters during a chemical vapor deposition process has been examined. Experiments have been performed to find a correlation between the reaction conditions and the final shape. Scanning electron microscopy data show that different structures like polysiloxane microrings, microrods, sprouts, nanofilaments, and mixtures of them can be synthesized depending on the reaction conditions. Furthermore, the in-depth comparison of the nanofilament diameters illustrates the dominating influence of relative humidity on structure formation. There is a general trend that at a higher value of relative humidity, structures with a larger diameter are formed independent from the temperature. Here, we clearly differentiate between relative humidity as major and absolute amount of water and temperature as minor important adjusting screws defining the thickness and shape of the resulting nano- and microstructures. Based on these observations, we proof the mechanism of the initial step of structure formation. It is shown that nano- and micro-sized water droplets formed on the substrate surface are likely to act as starting points for structure formation. All results described here strongly confirm the recently published droplet assisted growth and shaping mechanism.     

Auto-Bäcklund transformations and solitary wave solutions for the nonlinear evolution equation

In the present work, according to the concept of extended homogeneous balance method and with help of Maple, we get auto-Bäcklund transformations for a (2 + 1)-dimensional nonlinear evolution equation. Subsequently, by using these auto-Bäcklund transformation, exact explicit solutions of this equation are obtained.     

Optical solitons for a family of nonlinear ($$$$)-dimensional time-space fractional Schrödinger models

In this paper, the sine–cosine method is employed to construct exact solutions of the space-time fractional (\(1+1\))-dimensional nonlinear Schrödinger models. Many new families of exact traveling wave solutions of these models are successfully obtained. It is shown that the proposed method provides a more powerful mathematical tool for solving nonlinear space-time fractional evolution equations in mathematical physics.     

On some new analytical solutions for the nonlinear long–short wave interaction system

This paper deals with exact soliton solutions of the nonlinear long–short wave interaction system, utilizing two analytical methods. The system of coupled long–short wave interaction equations is investigated with the help of two analytical methods, namely, the generalized \(\tan (\phi /2)\)-expansion method and He’s semi-inverse variational method. Moreover, in this paper we generalize two aforementioned methods which give new soliton wave solutions. As a consequence, solutions are including solitons, kink, periodic and rational solutions. Moreover, dark, bright and singular solition solutions of the coupled long–short wave interaction equations have been found. All solutions have been verified back into its corresponding equation with the aid of maple package program. We depicted the physical explanation of the extracted solutions with the free choice of the different parameters by plotting some 3D and 2D illustrations. Finally, we believe that the executed methods are robust and efficient than other methods and the obtained solutions in this paper can help us to understand the soliton waves in the fields of physics and mechanics.     

High-reflectivity non-periodic sub-wavelength gratings with small-angle beam-steering ability and its application in Fabry–Perot cavity

We report a high-reflectivity non-periodic sub-wavelength gratings (SWGs) mirror with small-angle beam-steering ability for reflect light. It presents a distinctive characteristic of flexibly controlling the width of oscillation optical field for the improved Fabry–Perot (F–P) cavity. We propose a detailed principle analysis of the improved cavity. By finding out a set of SWGs with the designed structural parameters, both high reflectivity (>?93%) and beam steering (1°) can be implemented. By setting beam-steering angle and cavity length, we can control the width of oscillation optical field in the improved cavity. Beam steering ability and property of controlling the oscillation width are numerically investigated by finite element method. Simulation results prove that cavity length and steering angle can effectively control the main width of oscillation optical field, and the width is linearly associated with the cavity length.     

Optimization of cubic GaN/AlGaN quantum cascade structures for negative refraction in the THz spectral range

In this work we theoretically investigate a possibility to use cubic nitride based multi-layer periodic nanostructure as a semiconductor metamaterial. The structure design is based on an active region of a quantum cascade laser optimized to achieve optical gain in the Terahertz (THz) spectral range. In particular, we test the GaN/AlGaN quantum well configurations, which should exhibit important advantages compared to GaAs-based structures, namely room temperature operation without the assistance of magnetic field and lower doping densities. Our numerical rate-equations model is solved self-consistently and it takes into account electron-longitudinal optical phonon scattering between all the relevant states among the adjacent periods of the structure. A global optimization routine, specifically genetic algorithm is then used to generate new gain-optimized structures. This work confirms the advantages of cubic GaN designs over GaAs ones, namely feasibility of negative refraction at room temperature without the assistance of magnetic field while keeping the doping densities of the same order of magnitude.     

Multiple magnetic transitions in single crystals of Ce<Subscript>12</Subscript>Fe<Subscript>57.5</Subscript>As<Subscript>41</Subscript> and La<Subscript>12</Subscript>Fe<Subscript>57.5</Subscript>As<Subscript>41</Subscript>

Measurements of magnetic and transport properties were performed on needle-shaped single crystals of Ce₁₂Fe_57.5As₄₁ and La₁₂Fe_57.5As₄₁. The availability of a complete set of data enabled a side-by-side comparison between these two rare earth compounds. Both compounds exhibited multiple magnetic orders within 2–300 K and metamagnetic transitions at various fields. Ferromagnetic transitions with Curie temperatures of 100 and 125 K were found for Ce₁₂Fe_57.5As₄₁ and La₁₂Fe_57.5As₄₁, respectively, followed by antiferromagnetic type spin reorientations near Curie temperatures. The magnetic properties underwent complex evolution in the magnetic field for both compounds. An antiferromagnetic phase transition at about 60 K and 0.2 T was observed merely for Ce₁₂Fe_57.5As₄₁. The field-induced magnetic phase transition occurred from antiferromagnetic to ferromagnetic structure. A strong magnetocrystalline anisotropy was evident from magnetization measurements of Ce₁₂Fe_57.5As₄₁. A temperature-field phase diagram was present for these two rare earth systems. In addition, a logarithmic temperature dependence of electrical resistivity was observed in the two compounds within a large temperature range of 150–300 K, which is rarely found in 3D-based compounds. It may be related to Kondo scattering described by independent localized Fe 3d moments interacting with conduction electrons.