Kramer’s Sampling Theorem With Discontinuous Kernels

Kramer’s sampling theorem provides an algorithm for reconstructing a function ?, in the form $$ f(t)=\int_{a}^{b}\ F(x)K(x,t)dx,\qquad {\rm for\ some}\ F\ \in\ L^{2}(a,b), $$ from its values at a discrete set of points. In all the known examples, the kernel of the transform, K(x,t) is continuous in x and entire in t, even though the proof of the theorem shows that the continuity in x is not essential. This raises the question of whether it is possible to find an example of Kramer’s theorem with a discontinuous kernel. The aim of the paper is to answer this question in the affirmative. We show how one can construct a family of discontinuous kernels for which Kramer’s theorem holds and, in addition, each member of this family arises from a Sturm-Liouville problem, but with discontinuous initial conditions.     

Continuous catalytic Friedel-Crafts acylation in the biphasic medium of an ionic liquid and supercritical carbon dioxide

Immobilisation of catalytically-active metal salts in ionic liquids, with extraction by supercritical carbon dioxide, affords continuous Friedel-Crafts acylation, with in situ-recycling of the catalyst.     

Some oscillation criteria for second order nonlinear functional ordinary differential equations

The main objective of this article is to study the oscillatory behavior of the solutions of the following nonlinear functional differential equations (a(t)x'(t))' δ1p(t)x'(t) δ2q(t)f(x(g(t))) = 0,for 0 ≤ t0 ≤ t, where δ1 = ±1 and δ2 = ±1. The functions p,q,g : [t0, ∞) → R, f :R → R are continuous, a(t) ＞ 0, p(t) ≥ 0,q(t) ≥ 0 for t ≥ t0, limt→∞ g(t) = ∞, and q is not identically zero on any subinterval of [t0, ∞). Moreover, the functions q(t),g(t), and a(t) are continuously differentiable.     

A generalization of the prolate spheroidal wave functions

Many systems of orthogonal polynomials and functions are bases of a variety of function spaces, such as the Hermite and Laguerre functions which are orthogonal bases of and and the Jacobi polynomials which are an orthogonal basis of a weighted The associated Legendre functions, and more generally, the spheroidal wave functions are also an orthogonal basis of

The prolate spheroidal wave functions, which are a special case of the spheroidal wave functions, possess a very surprising and unique property. They are an orthogonal basis of both and a subspace of known as the Paley-Wiener space of bandlimited functions. They also satisfy a discrete orthogonality relation. No other system of classical orthogonal functions is known to possess this strange property. This raises the question of whether there are other systems possessing this property.

The aim of the article is to answer this question in the affirmative by providing an algorithm to generate such systems and then demonstrating the algorithm by a new example.

     

An Intricate Review on Nutritional and Analytical Profiling of Coconut,Flaxseed, Olive,and Sunflower Oil Blends

Vegetable oils (VOs), being our major dietary fat source, play a vital role in nourishment. Different VOs have highly contrasting fatty acid (FA) profiles and hence possess varying levels of health protectiveness. Consumption of a single VO cannot meet the recommended allowances of various FA either from saturated FA (SFA), monounsaturated FA (MUFA), polyunsaturated FA (PUFA), Ω-3 PUFAs, and medium-chain triglycerides (MCTs). Coconut oil (CO), flaxseed oil (FO), olive oil (OO), and sunflower oil (SFO) are among the top listed contrast VOs that are highly appreciated based on their rich contents of SFAs, Ω-3 PUFAs, MUFAs, and Ω-6 PUFA, respectively. Besides being protective against various disease biomarkers, these contrasting VOs are still inappropriate when consumed alone in 100% of daily fat recommendations. This review compiles the available data on blending of such contrasting VOs into single tailored blended oil (BO) with suitable FA composition to meet the recommended levels of SFA, MUFA, PUFA, MCTs, and Ω-3 to Ω-6 PUFA ratios which could ultimately serve as a cost-effective dietary intervention towards the health protectiveness and improvement of the whole population in general. The blending of any two or more VOs from CO, FO, OO, and SFO in the form of binary, ternary, or another type of blending was found to be very conclusive towards balancing FA composition; enhancing physiochemical and stability properties; and promising the therapeutic protectiveness of the resultant BOs.     

Improved visible light photocatalytic activity of TiO<Subscript>2</Subscript> nano powders with metal ions doping for glazed ceramic tiles

Self-cleaning and anti-bacterial activities of the photo-catalyst titanium dioxide make it a superior compound for use in the ceramics and glass industry. In order to achieve high self-cleaning efficiency for building products, it is important that Titania is present as anatase phase. Moreover, it is desirable that the particle sizes are in Nano-range, so that a large enough surface area is available for enhanced catalytic performance. In the present paper, Cobalt and Nickel co-doped (4%mol Ni and 4%mol Co doped TiO₂) and un-doped TiO₂ Nano powders have been prepared by sol–gel technique. They were calcined at the temperatures in the range of 475–1075 °C. Ni/Co co-doped TiO₂ postponed the anatase to rutile transformation of TiO₂ by about 200–300°C, such that before calcination at 775°C, no rutile was detected for 4 mol% Ni/Co co-doped TiO₂. A systematic decreasing on crystallite size and increasing on specific surface area of Ni/Co co-doped TiO₂ were observed. Photo-catalytic activity of anatase polymorph was measured by the decomposition rate of methylene blue under visible light. The results showed enhanced catalysis under visible light for Ni/Co co-doped TiO₂ as compared to pure TiO₂. The enhanced performance was attributed to surface chemistry change associated with a slight shift in the band gap. Depending on the temperatures ranging from 475 to 1075 °C, band gap energy of Ni and Co doped TiO₂ crystals decreased. For all samples there is a general reduction of the band gap energy from 3.00 to 2.96 eV.     

Photoluminescence properties of thenardite activated with Eu

Na₂SO₄:Eu phosphors were prepared by heating pure natural thenardite with EuF₃ at 900 °C for 20 min in air. The photoluminescence (PL) and excitation spectra of as-prepared and γ-ray-irradiated phosphors were observed at 300 K. The PL spectrum under 394 nm excitation consisted of strong narrow bands with peaks at 579, 592, 616, 652, 697 and 741 nm, assigned to the ⁵D₀→⁷F_J (J=0, 1, 2, …, 5) transitions, respectively, within Eu³⁺. The PL spectrum under 340 nm excitation consisted of a broad Eu²⁺ band with a peak at 435 nm. The excitation spectrum obtained by monitoring the violet luminescence consisted of a weak band with a peak at approximately 261 nm and a broad Eu²⁺ band with a peak at approximately 338 nm. The relative efficiency of the violet luminescence of the γ-ray-irradiated phosphor at the exposure of 46 kGy increased up to 3.0 times that of the unirradiated phosphor. The enhancement of violet luminescence by γ-ray irradiation was ascribed to the conversion of Eu³⁺ to Eu²⁺ in Na₂SO₄.