Some results on the annihilator graph of a commutative ring

Let R be a commutative ring. The annihilator graph of R, denoted by AG(R), is the undirected graph with all nonzero zero-divisors of R as vertex set, and two distinct vertices x and y are adjacent if and only if ann _R (xy) ≠ ann _R (x) ∪ ann _R (y), where for z ∈ R, ann _R (z) = {r ∈ R: rz = 0}. In this paper, we characterize all finite commutative rings R with planar or outerplanar or ring-graph annihilator graphs. We characterize all finite commutative rings R whose annihilator graphs have clique number 1, 2 or 3. Also, we investigate some properties of the annihilator graph under the extension of R to polynomial rings and rings of fractions. For instance, we show that the graphs AG(R) and AG(T(R)) are isomorphic, where T(R) is the total quotient ring of R. Moreover, we investigate some properties of the annihilator graph of the ring of integers modulo n, where n ? 1.     

Effect of electrolytes on surface tension and surface adsorption of 1-hexyl-3-methylimidazolium chloride ionic liquid in aqueous solution

Surface and bulk properties of 1-hexyl-3-methylimidazolium chloride [C₆mim][Cl] as an ionic liquid (IL) have been investigated by surface tension and electrical conductivity techniques at various temperatures. Results reveal that the ionic liquid behaves as surfactant-like and aggregates in aqueous solution. Critical aggregation concentration (cac) values obtained by conductivity and surface tension measurements are in good agreement with values found in the literature. A series of important and useful adsorption parameters including cac, surface excess concentration (Γ), and minimum surface area per molecule (A_min) at the air + water interface were estimated from surface tension in the presence and absence of different electrolytes. Obtained data show that the surface tension as well as the cac of [C₆mim][Cl] is reduced by electrolytes. Also, values of surface excess concentration (Γ) show that the IL ions in the presence of electrolyte have much larger affinity to adsorption at the surface and this affinity increased in aqueous electrolyte solution in the order of I^? > Br^? > Cl^? for counter ion of salts that was explained in terms of a larger repulsion of chloride anions from interface to the bromide and iodide anion as well as difference in their excess polarizability.     

Universality of Coproducts in Categories of Lax Algebras

Categories of lax -algebras are shown to have pullback-stable coproducts if preserves inverse images. The general result not only gives a common proof of this property in many topological categories but also shows that important topological categories, like the category of uniform spaces, are not presentable as a category of lax -algebras, with preserving inverse images. Moreover, we show that any such category of -algebras has a concrete, coproduct preserving functor into the category of topological spaces.     

Directed complete poset congruences

The study of algebraic properties of ordered structures has shown that their behavior in many cases is different from algebraic structures. For example, the analogues of the fundamental mapping theorem for sets which characterizes surjective maps as quotient sets modulo their kernel relations, is not true for order-preserving maps between posets (partially ordered sets). The main objective of this paper is to study the quotients of dcpos (directed complete partially ordered sets), and their relations with surjective dcpo maps (directed join preserving maps). The motivation of studying such infinitary ordered structures is their importance in domain theory, a theory on the borderline of mathematics and theoretical computer science.In this paper, introducing the notion of a pre-congruence on dcpos (directed complete partially ordered sets), we give a characterization of dcpo congruences. Also, it is proved that unlike natural dcpo congruences, the dcpo congruences are precisely kernels of surjective dcpo maps. Also, while it is known that the image of a dcpo map is not necessarily a subdcpo of its codomain, we find equivalent conditions on a dcpo map to satisfy this property. Moreover, we prove the Decomposition Theorem and its consequences for dcpo maps.     

Early stages of gelation in gelatin solution detected by dynamic oscillating rheology and nuclear magnetic spectroscopy

This study focuses on the early stages of the gelation of an aqueous type A (pig skin) gelatin solution. The thermo-reversible mono and triple helix formation was observed by rheology and proton NMR relaxation measurements. At high temperatures (T > 330 K), gelatin molecules form flexible random coils of small hydrodynamic radius, the elastic modulus of the solution is relatively low. On decreasing the temperature (330-320 K), mono helix formation begins, connected with an increase of the storage modulus and the hydrodynamic radius. The absence of a significant concentration dependence of this early variation of the modulus indicates the intramolecular nature of this structural change. The simultaneous decrease of the spin-spin relaxation times of the ¹H signals of certain aminoacids confirms its effect on the molecular mobility. As this affects especially the signals of arginine and lysine, we conclude that these basic aminoacids play a significant role in forming the intramolecular interactions. The formation of a three-dimensional network occurs at a point at which the viscosity begins to increase rapidly near the gel point (T < 320 K). This process is clearly dominated by intermolecular interactions, as the slope as well as the starting point of the rapid increase significantly depends on the concentration.     

Asymmetric Synthesis of Chromans Through Bifunctional Enamine-Metal Lewis Acid Catalysis

Cooperative enamine-metal Lewis acid catalysis has emerged as a powerful tool to construct carbon-carbon and carbon-heteroatom bond forming reactions. A concise synthetic method for asymmetric synthesis of chromans from cyclohexanones and salicylaldehydes has been developed to afford tricyclic chromans containing three consecutive stereogenic centers in good yields (up to 87 %) and stereoselectivity (up to 99 % ee and 11 : 1 : 1 dr). This difficult organic transformation was achieved through bifunctional enamine-metal Lewis acid catalysis. It is believed that the strong activation of the salicylaldehydes through chelating to the metal Lewis acid and the bifunctional nature of the catalyst accounts for the high yields and enantioselectivity of the reaction. The absolute configurations of the chroman products were established through X-ray crystallography. DFT calculations were conducted to understand the mechanism and stereoselectivity of this reaction.     

A collocation method for differential equations with one‐point Taylor initial conditions

This paper presents a new multistep collocation method for nth order differential equations. The interval of interest is first divided into N subintervals. By determining interval conditions in Taylor interpolation in each subinterval, Taylor polynomials are calculated with different step lengths. Then the obtained solutions in each subinterval are used as initial conditions in the next step. Optimal error is assessed using Peano lemma, which shows that the errors decay by decreasing the step length. In particular, for fixed step length h, the error is of O(m^?m), where m is the number of collocation points in each subinterval. Meanwhile, for fixed m, the error is of O(h^m). Numerical examples demonstrate the validity and high accuracy of the proposed method even for stiff problems.