On the index of fractions with square-free denominators in arithmetic progressions

We prove asymptotic formulas for the first and second moments of the index of fractions with square-free denominators of order Q streaming in a given arithmetic progression as Q→∞. A. Zaharescu was supported by NSF grant number DMS-0456615. This research was also partially supported by the CERES Program 4-147/2004 of the Romanian Ministry of Education and Research.     

A continuous review replenishment–disposal policy for an inventory system with autonomous supply and fixed disposal costs

In this study, we analyze an inventory system facing stochastic external demands and an autonomous supply (independent return flow) in the presence of fixed disposal costs and positive lead times under a continuous review replenishment–disposal policy. We derive the analytical expressions of the operating characteristics of the system; and, construct the objective function to minimize the total expected costs of ordering, holding, purchasing and disposal per unit time subject to a fill rate constraint. An extensive numerical analysis is conducted to study the sensitivity of the policy parameters and the benefit of employing a policy which allows for disposal of excess stock in this setting. We model the net demand process as the superposition of normally distributed external demand and inflows, which is expressed as a Brownian motion process. Our findings indicate that the disposal option results in considerable savings even (i) in the presence of non-zero fixed disposal costs, (ii) large actual demand rates with high return ratios (resulting in small net demands) and (iii) for moderate return ratios with high demand variability.     

Arithmetic mean of differences of Dedekind sums

Recently, Girstmair and Schoissengeier studied the asymptotic behavior of the arithmetic mean of Dedekind sums \frac1j(N) ? _{0 ￡ m < Ngcd(m,N)=1} |S(m,N)|\frac{1}{\varphi(N)} \sum_{\mathop{\mathop{ 0 \le m< N}}\limits_{\gcd(m,N)=1}} \vert S(m,N)\vert , as N → ∞. In this paper we consider the arithmetic mean of weighted differences of Dedekind sums in the form A_h(Q)=\frac1?_{\fracaq ? FQ}h(\fracaq) ×?_{\fracaq ? FQ}h(\fracaq) |s(a^￠,q^￠)-s(a,q)|A_{h}(Q)=\frac{1}{\sum_{\frac{a}{q} \in {\cal F}_{Q}}h\left(\frac{a}{q}\right)} \times \sum_{\frac{a}{q} \in {\cal F}_{\!Q}}h\left(\frac{a}{q}\right) \vert s(a^{\prime},q^{\prime})-s(a,q)\vert , where h:[0,1] ? \Bbb Ch:[0,1] \rightarrow {\Bbb C} is a continuous function with ò₀¹ h(t)  d t 1 0\int_0^1 h(t) \, {\rm d} t \ne 0 , \fracaq{\frac{a}{q}} runs over F_Q{\cal F}_{\!Q} , the set of Farey fractions of order Q in the unit interval [0,1] and \fracaq < \fraca^￠q^￠{\frac{a}{q}}<\frac{a^{\prime}}{q^{\prime}} are consecutive elements of F_Q{\cal F}_{\!Q} . We show that the limit lim _Q→∞ A _h (Q) exists and is independent of h.     

Oxygen deficiency effects on recombination lifetime and photoluminescence characteristics of ZnO thin films; correlation with crystal structure

The photoluminescence (PL), recombination lifetime (RL), and X-ray diffraction (XRD) spectra of the samples grown at various O₂ fractions of 0.290 (Zn-rich), 0.585 (moderate), and 0.836 (O-rich) over the total pressures in the growth chamber were investigated. XRD measurements revealed that all the films show highly preferred (0002) orientation. The PL measurements exhibit different dominant emissions in the ultraviolet (UV), violet and blue regions for Zn-rich, moderate and O-rich samples, respectively. Well-known green emission and high intensity of free exciton (FX) transition has been observed in Zn-rich sample after the sample is annealed at vacuum probably due to the oxygen deficiencies. Annealing the moderate sample gives rise to the UV emission at energy of 3.263 eV similar to the observed PL emission spectrum for the Zn-rich thin film. O-rich thin film exhibits a 338 meV acceptor level above the valance band maximum, most probably related to zinc vacancy (V_Zn). Free exciton RL measurements result in 568.23, 397.65, and 797.46 ps for Zn-rich, moderate, and O-rich thin films, respectively. A good correlation was found between crystallite size and the lifetime values.     

Novel thiazole‐piperazine derivatives as potential cholinesterase inhibitors

Dementia is a cognitive disorder mostly associated with Alzheimer's disease (AD) in addition to being seen in many other diseases of the central nervous system (CNS). The limited number of drugs is not sufficient to provide adequate improvement to increase the quality of life of patients suffering from this symptom; therefore, all treatment options should be evaluated in detail. In this study, new molecules, [2‐(4‐(2/3/4‐substituted phenyl)piperazin‐1‐yl)‐4‐phenylthiazol‐5‐yl][3/4‐substituted phenyl]methanone derivatives ( 1‐44 ), were obtained and analyzed in terms of their anticholinesterase activities. Kinetic mode and molecular interactions were also evaluated. An enzyme inhibition study was undertaken on acetylcholinesterase (AChE) and butyrylcholinesterase (BChE) using the Ellman method. Maestro program was used in molecular modeling studies. Forty‐four compounds were evaluated on AChE and BChE enzymes at 10^?3 and 10^?4 concentrations. The inhibition concentrations were calculated as 0.268μM to 2.104μM for six compounds ( 4 , 5 , 16 , 27 , 37 , and 38 ) on AChE. Compound 5 including the 4‐methoxy substituent (IC₅₀: 0.268μM) and compound 38 containing the 4‐methoxy and 3‐methyl substituents (IC₅₀: 0.286μM) showed the highest AChE inhibitory activity. They were further examined in terms of hydrogen bonding with Arg296 and Ar‐Ar interaction with Trp286. The activity of compound 5 was also assessed in mixed‐type kinetic mode.     

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Generalized model matching and (F,G)-invariant submodules for linear systems over rings

A generalized version of the exact model matching problem (GEMMP) is considered for linear multivariable systems over an arbitrary commutative ring K with identity. Reduced forms of this problem are introduced, and a characterization of all solutions and minimal order solutions is given, both with and without the properness constraint on the solutions, in terms of linear equations over K and K-modules. An approach to the characterization of all stable solutions is presented which, under a certain Bezout condition and a freeness condition, provides a parametrization of all stable solutions. The results provide an explicit parametrization of all solutions and all stable solutions in case K is a field, without the Bezout condition. This is achieved through a very simple characterization and a generalization to an arbitrary field K of the “fixed poles” of the model matching problem in terms of invariant factors of a certain polynomial matrix. The results also show that whenever the GEMMP has a solution, there exist solutions whose poles can be chosen arbitrarily as far as they contain the “fixed poles” with the right multiplicities (in the algebraic closure of K). Implications of these results in regard to inverse systems are shown. Equivalent simpler forms (in state space form) of the problem are shown to be obtainable. A theory of finitely generated (F,G)-invariant submodules for linear systems over rings is developed, and the geometric equivalent of the model matching problem—the dynamic cover problem—is formulated, to which the results of the previous sections provide a solution in the reduced case.     

Comparative studies of photophysical and electrochemical properties of sulfur-containing substituted metal-free and metallophthalocyanines

The synthesis, photophysical and electrochemical properties of soluble sulfur-containing 4-(methylthio) benzenethiol substituted, non-peripherally metal-free and metallo (Zn, Ga, Co, and Mn) phthalocyanine complexes (2–6) are reported for the first time. The new phthalocyanines have been characterized by FT-IR spectroscopy, ¹H-NMR, ¹³C-NMR, mass and UV–Vis spectroscopy techniques. Spectroscopic properties of these compounds were investigated in different solvents. Spectral and photophysical (fluorescence quantum yield) properties of metal free (2), zinc(II) (3), and gallium(III) phthalocyanines (4) were reported in different solvents toluene, tetrahydrofuran, and dimethyl sulfoxide. These results suggest that the solvents play role on the fluorescence quantum yields Φ _F of the synthesized complexes (2–4). The electrochemical studies exhibit that while complexes (3) and (4) give only Pc ring-based redox processes, complexes (5) and (6) give both metal and ring-based redox reactions due to the energy level of metal in the Pc core lie between the HOMO and the LUMO of the ring.     

Axially phenoxy-derivative disubstituted phthalocyanine: synthesis,characterization and photophysical properties

Research on Chemical Intermediates - We report the highly soluble axially disubstituted 4-(methoxymethyl) phenol silicon (IV) (1) phthalocyanine which was synthesized by the reaction of silicon...     

Thiochalcone substituted phthalocyanines for dye-sensitized solar cells: Relation of optical and electrochemical properties for cell performance

The new peripherally tetra-substituted metallophthalocyanines (MPcs, M=Zn, Co, Ni) bearing the chalcone, (E)-3-(4-hydroxyphenyl)-1-(thiophen-2-yl)prop-2-en-1-one, for dye-sensitized solar cells (DSSCs) were synthesized. FT-IR, ¹H NMR, ¹³C NMR, and UV–Vis spectroscopy techniques were utilized for characterization of all the MPcs. Electrochemical, optical and photovoltaic properties of all the MPcs as sensitizers were examined. Electrochemical studies reveal that while the ZnPc (4) and NiPc (6) give only Pc ring-based redox reactions, the CoPc (5) shows redox reactions based on both the central metal and the ring due to the metal 3d orbitals lying between the Pc HOMO and LUMO. The DSSC based on 5 gave the lowest power conversion efficiency (0.51%), perhaps due to the presence of a redox active central metal ion in the core of the complex, which results in a decrease electron transfer in the device. However, cells based on the other complexes including redox inactive central metal ions, which cannot reduce electron transfer, showed reasonable power conversion efficiencies of 1.27% and 1.11% for 4 and 6, respectively. The slight difference between the efficiencies can be attributed to higher molar extinction coefficient and narrower band gap of 4 than 6, which ensure a higher photocurrent and broader light absorption in the visible region.