Growth properties of the q-Dunkl transform in the space $$L^{p}_{q,\alpha }({\mathbb {R}}_{q},|x|^{2\alpha +1}d_{q}x)$$ L q , α p ( R q , | x | 2 α + 1 d q x )

The Ramanujan Journal - Our aim in this work is to prove an analogue of Titchmarsh’s theorem [19, Theorem 84] and Younis’s theorem [20, Theorem 3.3] on the image under the q-Dunkl...     

Lipschitz conditions for the generalized discrete Fourier transform associated with the Jacobi operator on [0,π]

Our aim in this paper is to prove an analog of the classical Titchmarsh theorem on the image under the discrete Fourier–Jacobi transform of a set of functions satisfying a generalized Lipschitz condition in the space ${\mathbb{L}}_2^{(\alpha ,\beta )}$.     

Some new estimates for the Helgason Fourier transform on rank 1 symmetric spaces

New estimates are proved for the Helgason Fourier transform in the space \(L^{2}(X)\) on certain classes of functions characterized by the spherical modulus of continuity.     

Reactivity of 4‐phenylthiazoles in ruthenium catalyzed direct arylations

The reactivity of the phenyl substituent of 4‐phenylthiazoles in Ru‐catalyzed direct arylation was studied. 4‐Phenylthiazole was found to be unreactive; whereas, the introduction of an aryl unit at C5‐position of 4‐phenylthiazole enhances its reactivity, allowing the selective mono‐arylation of the phenyl unit of 4‐phenylthiazoles in moderate to high yields using 5 mol% of [Ru(p‐cymene)Cl₂]₂ catalyst precursor associated to KOPiv as base. These results reveal that the conformation and electronic properties of 4‐phenylthiazoles are crucial to allow the formation of suitable intermediates in the course of the catalytic cycle. The reaction tolerated both electron‐rich and electron‐poor aryl bromides allowing the straightforward tuning of the electronic properties of the arylated 2‐methyl‐4‐phenyl‐5‐arylthiazoles.     

Bridge‐Clamp Bis(tetrazine)s with [N]8 π‐Stacking Interactions and Azido‐s‐Aryl Tetrazines: Two Classes of Doubly Clickable Tetrazines

Click chemistry at a tetrazine core is useful for bioorthogonal labeling and crosslinking. Introduced here are two new classes of doubly clickable s‐aryl tetrazines synthesized by Cu‐catalyzed cross‐coupling. Homocoupling of o‐brominated s‐aryl tetrazines leads to bis(tetrazine)s structurally characterized by tetrazine cores arranged face‐to‐face. [N]₈ π‐stacking interactions are essential to the conformation. Upon inverse electron demand Diels–Alder (iEDDA) cycloaddition, the bis(tetrazine)s produce a unique staple structure. The o‐azidation of s‐aryl tetrazines introduces a second proximal intermolecular clickable function that leads to double click chemistry opportunities. The stepwise introduction of fluorophores and then iEDDA cycloaddition, including bioconjugation to antibodies, was achieved on this class of tetrazines. This method extends to (thio)etherification, phosphination, trifluoromethylation and the introduction of various bioactive nitrogen‐based heterocycles.     

Tetrazo[1,2-b]indazoles: Straightforward Access to Nitrogen-Rich Polyaromatics from s-Tetrazines

The straightforward access to a new class of aza-polyaromatics is reported. Starting from readily available fluorinated s-tetrazine, a cyclization process with azide leads to the formation of an unprecedented tetrazo[1,2-b]indazole or a bis-tetrazo[1,2-b]indazole (cis and trans conformers). Based on the new nitrogen core, further N-directed palladium-catalyzed ortho-C−H bond functionalization allows the introduction of halides or acetates. The physicochemical properties of these compounds were studied by a joint experimental/theoretical approach. The tetrazo[1,2-b]indazoles display solid-state π-stacking, low reduction potential, absorption in the visible range up to the near-infrared, and intense fluorescence, depending on the molecular structure.     

Biosorption of lanthanum from aqueous solutions using magnetic alginate beads

Magnetic alginate beads are potential biosorbent for sorption of lanthanum(III) from an aqueous medium. Batch experiments were carried out to study the equilibrium, kinetics, and thermodynamics of lanthanum sorption. The effects of initial solution pH, initial lanthanum concentration, and temperature on lanthanum sorption were investigated. The optimum pH value was defined to be 4. Kinetic and isotherm experiments were carried out at the optimum pH. It was enough to reach the adsorption equilibrium at 4 hours, and the maximum uptake capacity was (1.8 mmol g^?1) at 25°C. Uptake kinetics and sorption isotherms were obtained and modeled using conventional and simple equations: best results were respectively obtained with the pseudo-second-order rate equation and the Langmuir equation. The La(III) loaded magnetic alginate beads were regenerated using 0.1 M CaCl₂ without activity loss.     

Effect of Shear Flow on Nanoparticles Migration near Liquid Interfaces

The effect of shear flow on spherical nanoparticles (NPs) migration near a liquid–liquid interface is studied by numerical simulation. We have implemented a compact model through which we use the diffuse interface method for modeling the two fluids and the molecular dynamics method for the simulation of the motion of NPs. Two different cases regarding the state of the two fluids when introducing the NPs are investigated. First, we introduce the NPs randomly into the medium of the two immiscible liquids that are already separated, and the interface is formed between them. For this case, it is shown that before applying any shear flow, 30% of NPs are driven to the interface under the effect of the drag force resulting from the composition gradient between the two fluids at the interface. However, this percentage is increased to reach 66% under the effect of shear defined by a Péclet number Pe = 0.316. In this study, different shear rates are investigated in addition to different shearing times, and we show that both factors have a crucial effect regarding the migration of the NPs toward the interfacial region. In particular, a small shear rate applied for a long time will have approximately the same effect as a greater shear rate applied for a shorter time. In the second studied case, we introduce the NPs into the mixture of two fluids that are already mixed and before phase separation so that the NPs are introduced into the homogenous medium of the two fluids. For this case, we show that in the absence of shear, almost all NPs migrate to the interface during phase separation, whereas shearing has a negative result, mainly because it affects the phase separation.