Solvability of convolution equations in $K^{'}_{M}$

Abstract

Net S be a convolution operator in the space K_{M}' , M is an increasing continuous function, of distributions in R^{n} growing not faster than e^{M(kx)} for some positive integer k. We give necessary and sufficient condition on S to have S*K_{M}'=K_{M}' , then we use this condition to get necessary and sufficient conditions for solvability of systems of convolution equations in K_{M}' .

Synthesis of W‐doped TiO2 by low‐temperature hydrolysis: Effects of annealing temperature and doping content on the surface microstructure and photocatalytic activity

“…More specifically, the anatase modes (A 1g + B 1g + 3E g ) appear at 143, 196, 396, 514, and 638 cm −1 , [32] while the rutile modes (A 1g + E g ) are observed at 446 and 608 cm −1 . [30] By increasing the annealing temperature of pure TiO 2 and W-TiO 2 , we observed the shift of the peak E g (1) (144 cm −1 ) toward lower wavenumbers and the reduction of its intensity. While there is a broadening and shifting toward higher wavenumbers by increasing the percentage of doping.…”

“…This is in good agreement with other studies that reported the absence of WO 3 phase below 20 wt% of tungsten. [30,31]…”

Kirigami‐Inspired Self‐Assembly of 3D Structures

“…Finite Element Analysis: The commercially available finite element analysis package ABAQUS [71] for understanding the mechanics of the stimulus-responsive shape transformation behavior of Kirigami-cut bilayers was used. Following an already established protocol, [40,43,[72][73][74] the mismatch strain-driven bilayer morphing was modeled as an equivalent thermal expansion problem where one layer expands with respect to the other in response to a hypothetical temperature. In our computations, 4-node general purpose shell elements with throughthickness variation in material properties (the element size was determined from separate mesh refinement studies) and appropriate boundary conditions were used.…”

Synthesis and Antimicrobial Evaluation of some Functionalized Heterocycles Derived from Novel Quinolinyl Chalcone

“…Both chalcones and their heterocyclic derivatives had proven their potential in numerous demanding pharmacological activities such as antimicrobial [1,2], antimalrial [3], anti-inflammatory [4][5][6], antitumor [7,8], antifungal [9], and antioxidant activity [10][11][12][13]. On the other hand, many researchers had reported chalcones containing sulfur either as a part of heteroaryl ring (thiophene) or as a side chain (thiomethyl group) for further enhancement of their biological activities [14][15][16]. Furthermore, it had been reported that both quinolin-2ones and fused pyrimidine heterocycles represent active classes of heterocyclic compounds.…”