Using restrictions to accept or reject solutions of radical equations

The standard technique for solving equations with radicals is to square both sides of the equation as many times as necessary to eliminate all radicals. Because the procedure violates logical equivalence, it results in extraneous solutions that do not satisfy the original equation, making it necessary to check all solutions against the original equation. We propose alternative solution procedures that are rigorous and simple to execute where the extraneous solutions can be identified without verification against the original equation. In this article, we review previous literature, establish and illustrate rigorous solution procedures for radical equations of depth 1 (i.e. equations where all radicals can be eliminated in one step), and deal with an ambiguity concerning the definition of real-valued solutions to radical equations. An application to defining the inverse function, resulting in a parametric radical equation, is also explained.     

Self-replication vs. reactive binary complexes--manipulating recognition-mediated cycloadditions by simple structural modifications

The rate of reaction and the selectivity of a Diels-Alder cycloaddition between a furan and a maleimide can be enhanced by the introduction of complementary recognition sites on the reactant species. Subtle manipulation of other structural elements allows the generation of the observed rate enhancements and selectivities through either self-replication or formation of a pre-reactive binary complex.     

Mechanical strength of amorphous CaCO3 colloidal spheres

Amorphous glassy CaCO3 colloidal spheres of monomodal size distribution were studied by high-resolution Brillouin light scattering. The Young modulus of 37 GPa and shear modulus of 14 GPa of glassy CaCO3 at a density of 1.9 g/cm3 were extracted from the particle vibration frequencies by employing acoustic wave scattering cross-section calculations. The line shape of the low-frequency modes is a sensitive index of the particle polydispersity.     

A simple synthesis of compounds with the benz[h]imidazo[1,2-c]quinazoline ring system

The ureas obtained from 1-amino-3(or 4)-methyl-2-naphthalenecarbonitrile and 2-chloroethyl isocyanate, or ethyl isocyanatoacetate undergo a double cyclization upon thermal decomposition, or treatment with base to yield compounds with the benz[h]imidazo[1,2-c]quinazoline ring system.     

Induced cosmology on a regularized brane in six-dimensional flux compactification

We consider a six-dimensional Einstein–Maxwell system compactified in an axisymmetric two-dimensional space with one capped regularized conical brane of codimension one. We study the cosmological evolution which is induced on the regularized brane as it moves in between known static   bulk and cap solutions. Looking at the resulting Friedmann equation, we see that the brane cosmology at high energies is dominated by a five-dimensional ρ²${\rho }^2$ energy density term. At low energies, we obtain a Friedmann equation with a term linear to the energy density with, however, negative coefficient in the small four-brane radius limit (i.e., with negative effective Newton's constant). We discuss ways out of this problem.     

No decreasing sequence of cardinals

In set theory without the Axiom of Choice (AC), we investigate the set-theoretic strength of the principle NDS which states that there is no function f on the set ω of natural numbers such that for everyn ∈ ω, f (n + 1) ? f (n), where for sets x and y, x ? y means that there is a one-to-one map g : x → y, but no one-to-one map h : y → x. It is a long standing open problem whether NDS implies AC. In this paper, among other results, we show that NDS is a strong axiom by establishing that AC^LO (AC restricted to linearly ordered sets of non-empty sets, and also equivalent to AC in ZF, the Zermelo–Fraenkel set theory minus AC) ? NDS in ZFA set theory (ZF with the Axiom of Extensionality weakened in order to allow the existence of atoms). The latter result provides a strongly negative answer to the question of whether “every Dedekind-finite set is finite” implies NDS addressed in G. H. Moore “Zermelo’s Axiom of Choice. Its Origins, Development, and Influence” and in P. Howard–J. E. Rubin “Consequences of the Axiom of Choice”. We also prove that AC^WO (AC restricted to well-ordered sets of non-empty sets) ? NDS in ZF (hence, “every Dedekind-finite set is finite” ? NDS in ZF, either) and that “for all infinite cardinals m, m + m = m” ? NDS in ZFA.     

Development of a photosystem II-based optical microfluidic sensor for herbicide detection

Herbicides are highly toxic for both human and animal health. The increased application of herbicides in agriculture during the last decades has resulted in the contamination of both soil and water. Herbicides, under illumination, can inhibit photosystem II electron transfer. Photosynthetic membranes isolated from higher plants and photosynthetic micro-organisms, immobilized and stabilized, can serve as a biorecognition element for a biosensor. The inhibition of photosystem II causes a reduced photoinduced production of hydrogen peroxide, which can be measured by a chemiluminescence reaction with luminol and the enzyme horseradish peroxidase. In the present work, a compact and portable sensing device that combines the production and detection of hydrogen peroxide in a single flow assay is proposed for herbicide detection.     

Local Lower Norm Estimates for Dyadic Maximal Operators and Related Bellman Functions

We provide lower \(L^{q}\) and weak \(L^{q}\)-bounds for the localized dyadic maximal operator on \(\mathbb {R}^{n}\), when the local \(L^{1}\) and the local \(L^{p}\) norms of the function are given. We actually do that in the more general context of homogeneous trees in probability spaces.     

Sharp Lorentz Estimates for Dyadic-like Maximal Operators and Related Bellman Functions

We precisely evaluate Bellman-type functions for the dyadic maximal operator on \(\mathbb {R}^{n}\) and of maximal operators on martingales related to local Lorentz-type estimates. Using a type of symmetrization principle, introduced for the dyadic maximal operator in earlier works of the authors, we precisely evaluate the supremum of the Lorentz quasinorm of the maximal operator on a function \(\phi \) when the integral of \(\phi \) is fixed and also the same Lorentz quasinorm of \(\phi \) is fixed. Also we find the corresponding supremum when the integral of \(\phi \) is fixed and several weak type conditions are given.     

On the denesting of nested square roots

We present the basic theory of denesting nested square roots, from an elementary point of view, suitable for lower level coursework. Necessary and sufficient conditions are given for direct denesting, where the nested expression is rewritten as a sum of square roots of rational numbers, and for indirect denesting, where the nested expression is rewritten as a sum of fourth-order roots of rational numbers. The theory is illustrated with several solved examples.