On multi-term fractional differential equations with multi-point boundary conditions

In this paper, we discuss the existence and uniqueness of solutions for a new class of multi-point boundary value problems of multi-term fractional differential equations by using standard fixed point theorems. We also demonstrate the application of the obtained results with the aid of examples. The paper concludes with the study of multi-term fractional integro-differential equations supplemented with multi-point boundary conditions. Our results are new and contribute significantly to the existing literature on the topic.     

Colorimetric determination of reactive primary amino groups of macro- and microsolid supports

The several factors that could affect the sensitivity and the accuracy of the determination of solid-supported amino groups using 2-iminothiolane (Traut's reagent) and 5,5′-dithiobis-(2-nitrobenzoic acid) (Ellman's reagent) are described. The authors found that by using 0.1M phosphate buffer, pH 8.0, instead of ethanol as solvent for the reaction of the solid supports with the 2-iminothiolane, using 0.1M phosphate buffer adjusted at pH 7.27 instead of 8.0 as diluent of 5,5′-dithiobis-(2-nitrobenzoic acid), and selecting carefully the concentration of the latter reagent, it was possible to produce a very sensitive assay capable of quantitatively determining the surface amino groups of very different types of samples. The assay is well adapted for quantitative determination of amino-carrying plastic beads, permitting the determination of nanomolar quantities. In addition, the assay is well suited for microparticulated solid supports (e.g., AH-Sepharose).     

Boundary Value Problems for Riemann–Liouville Fractional Differential Inclusions with Nonlocal Hadamard Fractional Integral Conditions

In this paper, we study a new class of boundary value problems from a fractional differential inclusion of Riemann–Liouville type and nonlocal Hadamard fractional integral boundary conditions. Some new existence results for convex as well as non-convex multi-valued maps are obtained using standard fixed point theorems. The obtained results are illustrated by examples.     

On Perturbed Fractional Differential Inclusions with Nonlocal Multi-point Erdélyi–Kober Fractional Integral Boundary Conditions

In this paper, by using a nonlinear alternative for a sum of compact upper semicontinuous and contractive multivalued operators, we establish sufficient conditions for the existence of solutions for perturbed fractional differential inclusions with nonlocal multi-point Erdélyi–Kober fractional integral boundary conditions. For the applicability of the main result, we include an example.     

On a quadratic fractional Hammerstein-Volterra integral equation with linear modification of the argument

We study the existence of monotonic solutions of a quadratic fractional Hammerstein-Volterra integral equation with linear modification of the argument. The quadratic integral equation studied below contains as a special case numerous integral equations encountered in the theory of radiative transfer and in the kinetic theory of gases. We show that the quadratic fractional Hammerstein-Volterra integral equation with linear modification of the argument has a monotonic solution in the Banach space of all real functions defined and continuous on a bounded and closed interval. The concept of measure of noncompactness and a fixed point theorem due to Darbo are the main tools in carrying out our proof.     

Synthesis, structural characterization, and biological studies of new antimony(III) complexes with thiones. The influence of the solvent on the geometry of the complexes

Five new antimony(III) complexes with the heterocyclic thiones 2-mercapto-benzimidazole (MBZIM), 5-ethoxy-2-mercapto-benzimidazole (EtMBZIM), and 2-mercapto-thiazolidine (MTZD) of formulas {[SbCl(2)(MBZIM)4]+.Cl-.2H(2)O. (CH(3)OH)} (1), {[SbCl(2)(MBZIM)4]+.Cl-.3H(2)O.(CH3CN)} (2), [SbCl(3)(MBZIM)2] (3), [SbCl(3)(EtMBZIM)(2)] (4), and [SbCl(3)(MTZD)2] (5) have been synthesized and characterized by elemental analysis, FT-IR, far-FT-IR, differential thermal analysis-thermogravimetry, X-ray diffraction, and conductivity measurements. Complex {[SbCl2(tHPMT)(2)]+Cl-}, (tHPMT = 2-mercapto-3,4,5,6-tetrahydro-pyrimidine), already known, was also prepared, and its X-ray crystal structure was solved. It is shown that the complex is better described as {[SbCl3(tHPMT)(2)]} (6). Crystal structures of all other complexes (1-5) have also been determined by X-ray diffraction at ambient conditions. The crystal structure of the hydrated ligand, EtMBZIM.H2O is also reported. Compound [C(28)H(24)Cl(2)N(8)S(4)Sb.2H(2)O.Cl.(CH(3)OH)] (1) crystallizes in space group P2(1), with a = 7.7398(8) A, b = 16.724(3) A, c = 13.717(2) A, beta = 98.632(11) degrees, and Z = 2. Complex [C(28)H(24)Cl(2)N(8)S(4)S(b).Cl.3H(2)O.(CH(3)CN)] (2) corresponds to space group P2(1), with a = 7.8216(8) A, b = 16.7426(17) A, c = 13.9375(16) A, beta = 99.218(10) degrees , and Z = 2. In both 1 and 2 complexes, four sulfur atoms from thione ligands and two chloride ions form an octahedral (Oh) cationic [SbS(4)Cl(2)]+ complex ion, where chlorides lie at axial positions. A third chloride counteranion neutralizes it. Complexes 1 and 2 are the first examples of antimony(III) compounds with positively charged Oh geometries. Compound [C(14)H(12)Cl(3)N(4)S(2)S(b)] (3) crystallizes in space group P, with a = 7.3034(5) A, b = 11.2277(7) A, c = 12.0172(8) A, alpha = 76.772(5) degrees, beta = 77.101(6) degrees, gamma = 87.450(5) degrees, and Z = 2. Complex [C(18)H(20)Cl(3)N(4)O(2)S(2)S(b)] (4) crystallizes in space group P1, with a = 8.6682(6) A, b = 10.6005(7) A, c = 13.0177(9) A, alpha = 84.181(6) degrees, beta = 79.358(6) degrees, gamma = 84.882(6) degrees, and Z = 2, while complex [C(6)H(10)Cl(3)N(2)S(4)S(b)] (5) in space group P2(1)/c shows a = 8.3659(10) A, b = 14.8323(19) A, c = 12.0218(13) A, beta = 99.660(12) degrees, and Z = 4 and complex [C(8)H(16)Cl(3)N(4)S(2)S(b)] (6) in space group P1 shows a = 7.4975(6) A, b = 10.3220(7) A, c = 12.1094(11) A, alpha = 71.411(7) degrees, beta = 84.244(7) degrees, gamma = 73.588(6) degrees, and Z = 2. Crystals of complexes 3-6 grown from acetonitrile solutions adopt a square-pyramidal (SP) geometry, with two sulfur atoms from thione ligands and three chloride anions around Sb(III). The equatorial plane is formed by two sulfur and two chloride atoms in complexes 3-5, in a cis-S, cis-Cl arrangement in 3 and 5 and a trans-S, trans-Cl arrangement in 4. Finally, in the case of 6, the equatorial plane is formed by three chloride ions and one sulfur from the thione ligand while the second sulfur atom takes an axial position leading to a unique SP conformation. The complexes showed a moderate cytostatic activity against tumor cell lines.     

Study of NH stretching vibrations in small ammonia clusters by infrared spectroscopy in He droplets and ab initio calculations

Infrared spectra of the NH stretching vibrations of (NH3)n clusters (n = 2-4) have been obtained using the helium droplet isolation technique and first principles electronic structure anharmonic calculations. The measured spectra exhibit well-resolved bands, which have been assigned to the nu1, nu3, and 2nu4 modes of the ammonia fragments in the clusters. The formation of a hydrogen bond in ammonia dimers leads to an increase of the infrared intensity by about a factor of 4. In the larger clusters the infrared intensity per hydrogen bond is close to that found in dimers and approaches the value in the NH3 crystal. The intensity of the 2nu4 overtone band in the trimer and tetramer increases by a factor of 10 relative to that in the monomer and dimer, and is comparable to the intensity of the nu1 and nu3 fundamental bands in larger clusters. This indicates the onset of the strong anharmonic coupling of the 2nu4 and nu1 modes in larger clusters. The experimental assignments are compared to the ones obtained from first principles electronic structure anharmonic calculations for the dimer and trimer clusters. The anharmonic calculations were performed at the M?ller-Plesset (MP2) level of electronic structure theory and were based on a second-order perturbative evaluation of rovibrational parameters and their effects on the vibrational spectra and average structures. In general, there is excellent (<20 cm(-1)) agreement between the experimentally measured band origins for the N-H stretching frequencies and the calculated anharmonic vibrational frequencies. However, the calculations were found to overestimate the infrared intensities in clusters by about a factor of 4.     

Positive solutions for fractional differential systems with nonlocal Riemann–Liouville fractional integral boundary conditions

In this paper, we study the positive solutions of fractional differential system with coupled nonlocal Riemann–Liouville fractional integral boundary conditions. Our analysis relies on Leggett–Williams and Guo–Krasnoselskii’s fixed point theorems. Two examples are worked out to illustrate our main results.     

Waiting Time for an Almost Perfect Run and Applications in Statistical Process Control

A natural and intuitively appealing generalization of the runs principle arises if instead of looking at fixed-length strings with all their positions occupied by successes, we allow the appearance of a small number of failures. Therefore, the focus is on clusters of consecutive trials which contain large proportion of successes. Such a formation is traditionally called “scan” or alternatively, due to the high concentration of successes within it, almost perfect (success) run. In the present paper, we study in detail the waiting time distribution for random variables related to the first occurrence of an almost perfect run in a sequence of Bernoulli trials. Using an appropriate Markov chain embedding approach we present an efficient recursive scheme that permits the construction of the associated transition probability matrix in an algorithmically efficient way. It is worth mentioning that, the suggested methodology, is applicable not only in the case of almost perfect runs, but can tackle the general discrete scan case as well. Two interesting applications in statistical process control are also discussed.     

The role of hydrophobic surfaces in altering water-mediated peptide-peptide interactions in an aqueous environment

Using Born-Oppenheimer molecular dynamics within the density functional framework, we calculated the effective force acting on water-mediated peptide-peptide interaction between antiparallel β-sheets in an aqueous environment and also in the vicinity of a hydrophobic surface. From the magnitude of the effective force (corresponding to the slope of the free energy as a function of the interpeptide distance) and its sign (a negative value indicates an effective attraction, whereas a positive value indicates an effective repulsion) we can elucidate the fundamental differences of the water-mediated peptide-peptide interactions in those two environments. The computed effective forces indicate that the water-mediated interaction between peptides in an aqueous environment is attractive in the range of interpeptide distance d = 7-8 ? when hydrophobic surfaces are not nearby. Due to the stabilization of the water molecules bridging between the two β-sheets, a free energy barrier exists between the direct and indirect (water-mediated) interpeptide interactions. However, when the peptides are in the proximity of hydrophobic surfaces, this free energy barrier decreases because the hydrophobic surfaces enhance the interpeptide attraction by the destabilization and ease-to-libration of the bridging water molecules between them.