Necessary and sufficient conditions for Hölder continuity of Gaussian processes

The continuity of Gaussian processes is an extensively studied topic and it culminates in Talagrand’s notion of majorizing measures that gives complicated necessary and sufficient conditions. In this note we study the Hölder continuity of Gaussian processes. It turns out that necessary and sufficient conditions can be stated in a simple form that is a variant of the celebrated Kolmogorov–Čentsov condition.     

Kibble–Slepian formula and generating functions for 2D polynomials

We prove a generalization of the Kibble–Slepian formula (for Hermite polynomials) and its unitary analogue involving the 2D Hermite polynomials recently proved in [16]. We derive integral representations for the 2D Hermite polynomials which are of independent interest. Several new generating functions for 2D q-Hermite polynomials will also be given.     

Calculation of the vibration–rotational transition intensities of water molecules trapped in an argon matrix: stretching O–H vibrations spectral region

The frequencies and intensities of vibration–rotational transitions of water molecules in an argon matrix were calculated for temperatures of 6 and 30 K. The rigid asymmetric top approximation was used with available literature values of the effective rotational constants in the ground and excited vibrational states. The calculations were carried out by taking into account the existence of a non-equilibrium population distribution between the rotational levels of ortho- and para-water isomers. It was assumed that the temperature relaxation of the population of rotational levels is independent of the ortho- and para-isomers. Comparison of the results of the theoretical calculations with experimental literature data shows good agreement for the majority of the rotational structure lines for symmetric and antisymmetric stretching vibrations both in the frequency values and in the values of the relative intensities.     

Liquid crystalline properties and photochromism of 4-alkyl-N-(4-alkloxysalicylidene)anilines

Eight new homologous series of 4-alkyl-N-(4-alkloxysalicylidene)anilines were synthesized. The phase transitions of these homologues were determined using differential scanning calorimetry and polarizing optical microscopy. The homologues began to exhibit liquid crystalline phases as the terminal alkyl or alkoxyl chains were lengthened. In addition, some members of these homologues series exhibited photochromism in the solid state.     

Preparation of safe water–lipid mixed electrolytes for application in ion capacitor

Aqueous electrolytes are safe, economic, and environmentally friendly. However, they have a narrow potential window. On the other hand, organic electrolytes exhibit good thermodynamic stability but are inflammable and moisture sensitive. In this study, we prepared water–PEG–lipid ternary electrolytes(TEs). To combine the advantages of water, polyethylene glycol(PEG) and propylene carbonate(PC). The nonflammable mixed electrolytes exhibited a wide potential window of about 2.8 V due to the beneficial effects of PEG and PC. Using these TEs, a lithium manganate–active carbon ion capacitor could be operated at 2.4 V with an energy density of 32 Wh/kg, based on the total active electrode material(current density of 3.3 m A/cm~2). This value was significantly higher than that achieved using an aqueous electrolyte, thereby rationalizing the higher energy density.     

Potential Applications of Cucurbit[n]urils and Their Derivatives in the Capture of Hazardous Chemicals

Cucurbit[n]urils (Q[n]s) are a relatively young family of macrocycles, consisting of glycoluril units bridged by methylene groups, and their unique structures have attracted extensive attention from chemists in recent decades. Due to the presence of a rigid hydrophobic inner cavity and two polar outer portals lined with carbonyl groups, Q[n]s not only encapsulate guest species into the cavity, but also coordinate with metal ions/clusters. Considerable achievements have been obtained in the fields of Q[n]s-based host–guest chemistry, coordination chemistry, as well as the combination of host–guest and coordination chemistry. Furthermore, the outer surface of Q[n]s has been demonstrated to be capable of interacting with definite species to generate supramolecular architectures in recent years. With more in-depth research into Q[n]s, their application studies have also emerged as a hot topic. This Minireview focuses on recent advances in the potential applications of solid-state materials based on Q[n]s and their derivatives for the capture and adsorption of hazardous chemicals from a solution or a gas mixture.     

Efficient production of nanoporous silicon carbide from rice husk at relatively low temperature

Nanoporous silicon carbide with a specific surface area of up to 186.45 m² g⁻¹ has been efficiently synthesized from waste rice husk using a magnesiothermic reduction at 950 °C as a key step. Throughout the entire process, the recovery rates of silicon, potassium and phosphorus from rice husk can reach 88.46, 91.5 and 65.5%, respectively. Turning rice husk waste into a real treasure, this promising method for producing porous SiC protects the environment and brings economic benefits.     

Synthesis of 4‐Arylthio‐3‐hydroxyphthalate by the Diels–Alder Reactionof 4‐Arylthio‐3‐hydroxy‐2‐pyrone

Treatment of 4‐arylthio‐3‐hydroxy‐2‐pyrones with acetylenedicarboxylates gave 4‐arylthio‐3‐hydroxyphthalates by the base‐catalyzed Diels–Alder reaction via a decarboxylation in good yields.     

Twist-bend nematics,liquid crystal dimers,structure–property relations

ABSTRACT

One of the current challenges in liquid crystal science is to understand the molecular factors leading to the formation of the intriguing twist-bend nematic phase (N_TB) and determine its properties. During our earlier hunt for the N_TB phase created on cooling directly from the isotropic phase and not the nematic phase, we had prepared 30 symmetric liquid crystal dimers. These had odd spacers and methylene links to the two mesogenic groups; desirable but clearly not essential features for the formation of the N_TB. Here, we report the phases that the dimers exhibit and their transition temperatures as functions of both the lengths of the spacer and the terminal chains. In addition we describe the transitional entropies, their optical textures, the X-ray scattering patterns and the ²H NMR spectra employed in characterising the phases. All of which may lead to important properties of the twist-bend nematic phase.     

Topological conjugacy of constant length substitution dynamical systems

Primitive constant length substitutions generate minimal symbolic dynamical systems. In this article we present an algorithm which can produce the list of injective substitutions of the same length that generate topologically conjugate systems. We show that each conjugacy class contains infinitely many substitutions which are not injective. As examples, the Toeplitz conjugacy class contains three injective substitutions (two on two symbols and one on three symbols), and the length two Thue–Morse conjugacy class contains twelve substitutions, among which are two on six symbols. Together, they constitute a list of all primitive substitutions of length two with infinite minimal systems which are factors of the Thue–Morse system.