Generalized Bosbach states: part I

States have been introduced on commutative and non-commutative algebras of fuzzy logics as functions defined on these algebras with values in [0,1]. Starting from the observation that in the definition of Bosbach states there intervenes the standard MV-algebra structure of [0,1], in this paper we introduce Bosbach states defined on residuated lattices with values in residuated lattices. We are led to two types of generalized Bosbach states, with distinct behaviours. Properties of generalized states are useful for the development of an algebraic theory of probabilistic models for non-commutative fuzzy logics.     

He's homotopy perturbation method for solving the shock wave equation

In this article, we discuss the analytic solution of the fully developed shock waves. The homotopy perturbation method is used to solve the shock wave equation, which describes the flow of gases. Unlike the various numerical techniques, which are usually valid for short period of time, the solution of the presented equation is analytic for 0 < t < ∞. The results presented converge very rapidly, indicating that the method is reliable and accurate.     

Associating divisors with quadratic differentials on Klein surfaces

We extend the notion and basic properties of quadratic differential divisors to a Klein surface, using the corresponding form’s divisors from the double cover of the Klein surface.     

Optical Fiber Excitation of Fano Resonances in a Silicon Microsphere

In this article, Fano lineshape whispering gallery modes were observed in the light scattering spectrum of a silicon microsphere in near-infrared telecommunication wavelengths. A simple model is presented to explain the transition from Lorentzian lineshape to the Fano lineshape resonances with the coupled-mode theory of multiple whispering gallery modes. Polar mode spacing of 0.23 nm is observed in the spectra, which correlates well with the calculated value. The quality factor of the Lorentzian and Fano resonances are on the order of 10⁵. By using an appropriate interface design for the microsphere coupling geometries, Fano lineshape optical resonances herald novel device applications for silicon volumetric lightwave circuits.     

On Dedekind Modules

In this article the authors give the relation between a finitely-generated torsionfree Dedekind module M over a domain R and prime submodules of the 𝒪(M)-module M and the ring 𝒪(M). They also prove that M is a finitely-generated torsionfree Dedekind module over a domain R if and only if every semi-maximal submodule of R-module M is invertible.     

Iterative Approaches to Solving Equilibrium Problems and Fixed Point Problems of Infinitely Many Nonexpansive Mappings

Recently, O’Hara, Pillay and Xu (Nonlinear Anal. 54, 1417–1426, 2003) considered an iterative approach to finding a nearest common fixed point of infinitely many nonexpansive mappings in a Hilbert space. Very recently, Takahashi and Takahashi (J. Math. Anal. Appl. 331, 506–515, 2007) introduced an iterative scheme by the viscosity approximation method for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of a nonexpansive mapping in a Hilbert space. In this paper, motivated by these authors’ iterative schemes, we introduce a new iterative approach to finding a common element of the set of solutions of an equilibrium problem and the set of common fixed points of infinitely many nonexpansive mappings in a Hilbert space. The main result of this work is a strong convergence theorem which improves and extends results from the above mentioned papers.     

Edge searching weighted graphs

In traditional edge searching one tries to clean all of the edges in a graph employing the least number of searchers. It is assumed that each edge of the graph initially has a weight equal to one. In this paper we modify the problem and introduce the Weighted Edge Searching Problem by considering graphs with arbitrary positive integer weights assigned to its edges. We give bounds on the weighted search number in terms of related graph parameters including pathwidth. We characterize the graphs for which two searchers are sufficient to clear all edges. We show that for every weighted graph the minimum number of searchers needed for a not-necessarily-monotonic weighted edge search strategy is enough for a monotonic weighted edge search strategy, where each edge is cleaned only once. This result proves the NP-completeness of the problem.     

Comparing Singlet Oxygen Generation of Schiff Base Substituted Novel Silicon Phthalocyanines by Sonophotochemical and Photochemical Applications

Although the sonophotodynamic method has an effective therapeutic outcome for anticancer treatment compared with the photodynamic method, there are not enough related studies in the literature and this study aims to contribute to the development of sonophotodynamic studies. For this purpose, the Schiff base substituted silicon phthalocyanines were designed and synthesized as effective sensitizer candidates and the photophysicochemical and sonophotochemical features of the phthalocyanines were examined to increase singlet oxygen efficiency. The calculated Φ_Δ values indicate that the contribution of substituent groups improved the production of singlet oxygen compared with silicon (IV) phthalocyanine dichloride (SiPcCI₂) and also the sonophotochemical applications increased the singlet oxygen yields. The Φ_Δ values (Φ_Δ = 0.76 for axially bis-{4-[(E)-(pyridin-3-ylimino)methyl]phenol} substituted silicon (IV) phthalocyanine ( 2a ), 0.68 for axially bis-4-[(E)-{[(pyridin-3-yl)methyl]imino}methyl]phenol substituted silicon (IV) phthalocyanine ( 2b ) in photochemical study) reached to Φ_Δ = 0.98 for 2a , 0.94 for 2b in sonophotochemical study. This article will enrich the literature on increasing singlet oxygen yield.     

Swelling of <Emphasis Type="Italic">N</Emphasis>-vinylcaprolactam-dodecyl methacrylate gel in {heptane + toluene} mixtures

A copolymer gel has been synthesized from N-vinylcaprolactam and dodecyl methacrylate in ethanol using the free radical cross-linking polymerization method. Characterizations of the gel were performed by scanning electron microscopy (SEM), Fourier transform infrared spectroscopy (FTIR), and thermal gravimetric analysis (TGA) techniques. Swelling behavior of the gel was investigated in heptane, toluene, and their binary mixtures with different compositions. Swelling value in toluene is higher than that in heptane and swelling value increases with the increasing initial content of toluene in the binary mixture with heptane. The swelling values are correlated by the first- and second-order differential equations, and the best model correlating the experimental results is a second-order one. Diffusion coefficients have also been calculated for heptane and toluene at each concentration by power-law and first-order equations. While the diffusion mechanism of the gel in heptane is a Fickian one, the gel swelled in toluene exhibits a non-Fickian character. Diffusion mechanisms of the gels in binary mixtures are much more complicated. Because of the higher swelling degree in toluene compared to that in heptane, selectivity of the gel in different {heptane + toluene} mixtures with selectivity close to 1 has also been taken into consideration.