On the Multiplicity of Zeroes of Polynomials with Quaternionic Coefficients

Regular polynomials with quaternionic coefficients admit only isolated zeroes and spherical zeroes. In this paper we prove a factorization theorem for such polynomials. Specifically, we show that every regular polynomial can be written as a product of degree one binomials and special second degree polynomials with real coefficients. The degree one binomials are determined (but not uniquely) by the knowledge of the isolated zeroes of the original polynomial, while the second degree factors are uniquely determined by the spherical zeroes. We also show that the number of zeroes of a polynomial, counted with their multiplicity as defined in this paper, equals the degree of the polynomial. While some of these results are known in the general setting of an arbitrary division ring, our proofs are based on the theory of regular functions of a quaternionic variable, and as such they are elementary in nature and offer explicit constructions in the quaternionic setting. Partially supported by G.N.S.A.G.A.of the I.N.D.A.M. and by M.I.U.R.. Lecture held by G. Gentili in the Seminario Matematico e Fisico on February 12, 2007. Received: August 2008     

Existence and stability of traveling waves for an integro-differential equation for slow erosion

We study an integro-differential equation that describes the slow erosion of granular flow. The equation is a first order nonlinear conservation law where the flux function includes an integral term. We show that there exist unique traveling wave solutions that connect profiles with equilibrium slope at ±∞. Such traveling waves take very different forms from those in standard conservation laws. Furthermore, we prove that the traveling wave profiles are locally stable, i.e., solutions with monotone initial data approach the traveling waves asymptotically as t→+∞$t\to +\infty$.     

A Cauchy kernel for slice regular functions

In this article, we show how to construct a regular, non-commutative Cauchy kernel for slice regular quaternionic functions. We prove an (algebraic) representation formula for such functions, which leads to a new Cauchy formula. We find the expression of the derivatives of a regular function in terms of the powers of the Cauchy kernel, and we present several other consequent results.     

Comparison of essential oil components and in vitro anticancer activity in wild and cultivated Salvia verbenaca

The objectives of our research were to study the chemical composition and the in vitro anticancer effect of the essential oil of Salvia verbenaca growing in natural sites in comparison with those of cultivated (Sc) plants. The oil from wild (Sw) S. verbenaca presented hexadecanoic acid (23.1%) as the main constituent, while the oil from Sc plants contained high quantities of hexahydrofarnesyl acetone (9.7%), scarce in the natural oil (0.7%). The growth-inhibitory and proapoptotic effects of the essential oils from Sw and Sc S. verbenaca were evaluated in the human melanoma cell line M14, testing cell vitality, cell membrane integrity, genomic DNA fragmentation and caspase-3 activity. Both the essential oils were able to inhibit the growth of the cancer cells examined inducing also apoptotic cell death, but the essential oil from cultivated samples exhibited the major effects.     

A Sharp Upper Bound for the¶Torsional Rigidity of Rods¶by Means of Web Functions

Using web functions, we approximate the Dirichlet integral which represents the torsional rigidity of a cylindrical rod with planar convex cross-section Ω. To this end, we use a suitably defined piercing function, which enables us to obtain bounds for both the approximate and the exact value of the torsional rigidity. When Ω varies, we show that the ratio between these two values is always larger than ¾; we prove that this is a sharp lower bound and that it is not attained. Several extremal cases are also analyzed and studied by numerical methods.     

Facile synthesis of hydantoins and thiohydantoins in aqueous solution

A series of hydantoins and thiohydantoins have been synthesized in water at room temperature from urea (or N-methylurea, or thiourea) and simple aldehydes (as glyoxal, and its simple derivatives) in the presence of phosphoric anhydride. The reaction time is 10 min using an equimolar amount of P₄O₁₀ with respect to the other reagents, but the reaction occurs also, even if with longer reaction times, with very small amounts of P₄O₁₀.In addition, this method provides a clean and ‘green’ approach to hydantoins, compounds of great interest in biological and pharmacological fields.     

First prebiotic generation of a ribonucleotide from adenine, D-ribose and trimetaphosphate

Adenosine monophosphate isomers are obtained by self-assembling of adenine, D-ribose and trimetaphosphate in aqueous solution in good yields. This generation of a ribonucleotide from its three molecular components occurs in a one-pot reaction at room temperature for about 30-40 days and with high chemio-, regio-, and stereo-selectivity. Similar results are obtained with guanine. A mechanism is also proposed.     

How does trimethylamine N-oxide counteract the denaturing activity of urea?

Trimethylamine N-oxide, TMAO, stabilizes globular proteins and is able to counteract the denaturing activity of urea. The mechanism of this counteraction has remained elusive up to now. A rationalization is proposed grounded on the same theoretical model used to clarify the origin of cold denaturation, and the denaturing activity of GdmCl versus the stabilizing one of Gdm(2)SO(4) [G. Graziano, Phys. Chem. Chem. Phys., 2010, 12, 14245-14252; G. Graziano, Phys. Chem. Chem. Phys., 2011, 13, 12008-12014]. The fundamental quantities are: (a) the difference in the solvent-excluded volume on passing from the N-state to the D-state, calculated in water and in aqueous osmolyte solution; (b) the difference in energetic attractions of the N-state and the D-state with the surrounding solvent molecules, calculated in water and in aqueous osmolyte solution. In aqueous 8 M urea + 4 M TMAO solution, the first quantity is so large and positive to counteract the second one that is large and negative due to preferential binding of urea molecules to the protein surface. This happens because aqueous 8 M urea + 4 M TMAO solution has a volume packing density markedly larger than that of water, rendering the cavity creation process much more costly. The volume packing density increase reflects the strength of the attractions of water molecules with both urea and TMAO molecules. This mechanism readily explains why TMAO counteraction is operative even though urea molecules are preferentially located on the protein surface.     

An extension of the pairing theory between divergence-measure fields and BV functions

In this paper we introduce a nonlinear version of the notion of Anzellotti's pairing between divergence-measure vector fields and functions of bounded variation, motivated by possible applications to evolutionary quasilinear problems. As a consequence of our analysis, we prove a generalized Gauss–Green formula.