Dynamic processes and fixed points of set-valued nonlinear contractions in cone metric spaces

Investigations concerning the existence of dynamic processes convergent to fixed points of set-valued nonlinear contractions in cone metric spaces are initiated. The conditions guaranteeing the existence and uniqueness of fixed points of such contractions are established. Our theorems generalize recent results obtained by Huang and Zhang [L.-G. Huang, X. Zhang, Cone metric spaces and fixed point theorems of contractive maps, J. Math. Anal. Appl. 332 (2007) 1467–1475] for cone metric spaces and by Klim and Wardowski [D. Klim, D. Wardowski, Fixed point theorems for set-valued contractions in complete metric spaces, J. Math. Anal. Appl. 334 (1) (2007) 132–139] for metric spaces. The examples and remarks provided show an essential difference between our results and those mentioned above.     

Correction of stiffness and mass matrices utilizing simulated measured modal data

Measured and analytical data are unlikely to be equal due to measured noise, model inadequacies and structural damage, etc. It is necessary to update the physical parameters of analytical models for proper simulation and design studies. Starting from simulated measured modal data such as natural frequencies and their corresponding mode shapes, this study presents the equations to update the physical parameters of stiffness and mass matrices simultaneously for analytical modelling by minimizing a cost function in the satisfaction of the dynamic constraints of orthogonality requirement and eigenvalue function. The proposed equations are straightforwardly derived by Moore–Penrose inverse matrix without using any multipliers. The cost function is expressed by the sum of the quadratic forms of both the difference between analytical and updated mass, and stiffness matrices. The results are compared with the updated mass matrix to consider the orthogonality requirement only and the updated stiffness matrix to consider the eigenvalue function only, respectively. Also, they are compared with Wei’s method which updates the mass and stiffness matrices simultaneously. The validity of the proposed method is illustrated in an application to correct the mass and stiffness matrices due to section loss of some members in a simple truss structure.     

Exploring greedy criteria for the dynamic traveling purchaser problem

Given a set of products and a set of markets, the traveling purchaser problem looks for a tour visiting a subset of the markets to satisfy products demand at the minimum purchasing and traveling costs. In this paper, we analyze the dynamic variant of the problem (D-TPP) where the quantity made available in each market for each product may decrease over time. We introduce and compare several greedy strategies and test their impact on the solution in terms of feasibility and costs. In particular, we study an incremental approach where an initial naive strategy is improved and refined by a number of variants. Some of the proposed heuristics take into account either one of the two objective costs, while others are based on both traveling and purchasing costs. Extensive computational results are also provided on randomly generated instances.     

Structures and stereochemical non-rigidity of Si-substituted N-(dimethylsilylmethyl)- and N-(methylphenylsilylmethyl)amides and -lactams

Eleven new silicon-substituted N-(dimethylsilylmethyl)- and N-(methylphenylsilylmethyl)amides and -lactams bearing a chiral carbon in the amide or lactam fragment, and containing the OSiC₃X (X = Hal, OTf) coordination fragment have been synthesized and their structures determined in solution by spectroscopic means. These structures are consistent with the hypervalency model. Quantum chemical calculations adequately reflect correlations between the type of monodentate ligand X and the geometric parameters of the N-C-O-Si-X fragments.The activation parameters for enantiomerization and diastereomerization in these new compounds and the other related compounds were determined by the dynamic NMR (DNMR) method using full line-shape analysis. The free activation energy values in the absence of external nucleophiles vary from 9 to 27 kcal mol⁻¹. The entropies of activation (ΔS^#) are negative (−20 to −50 cal mol⁻¹ K⁻¹) in all cases except for the chloride derivatives of 4-phenyl-2-pyrrolidone and 4-oxazolidinone that have weaker intramolecular O → Si coordination. Irregular mechanisms of permutational isomerization were proposed on the basis of the DNMR data and the results of quantum-chemical calculations carried out by ab initio (HF) and DFT (PBE, B3PW91, 6-311++G(d,p)). Depending on the coordination environment at silicon, the mechanisms proposed involve either the dissociation of the Si-X bond followed by the Berry pseudorotation or similar in the intermediate or the cleavage of intramolecular O-Si bond with subsequent inversion at the silicon atom. The apparently simple pseudorotation mechanism involving only the pentacoordinate structures 1-21 does not appear to be favoured in any of the examples studied.     

Mass transport in micellar surfactant solutions: 2. Theoretical modeling of adsorption at a quiescent interface

Here, we apply the detailed theoretical model of micellar kinetics from part 1 of this study to the case of surfactant adsorption at a quiescent interface, i.e., to the relaxation of surface tension and adsorption after a small initial perturbation. Our goal is to understand why for some surfactant solutions the surface tension relaxes as inverse-square-root of time, 1/t(1/2), but two different expressions for the characteristic relaxation time are applicable to different cases. In addition, our aim is to clarify why for other surfactant solutions the surface tension relaxes exponentially. For this goal, we carried out a computer modeling of the adsorption process, based on the general system of equations derived in part 1. This analysis reveals the existence of four different consecutive relaxation regimes (stages) for a given micellar solution: two exponential regimes and two inverse-square-root regimes, following one after another in alternating order. Experimentally, depending on the specific surfactant and method, one usually registers only one of these regimes. Therefore, to interpret properly the data, one has to identify which of these four kinetic regimes is observed in the given experiment. Our numerical results for the relaxation of the surface tension, micelle concentration and aggregation number are presented in the form of kinetic diagrams, which reveal the stages of the relaxation process. At low micelle concentrations, "rudimentary" kinetic diagrams could be observed, which are characterized by merging of some stages. Thus, the theoretical modeling reveals a general and physically rich picture of the adsorption process. To facilitate the interpretation of experimental data, we have derived convenient theoretical expressions for the time dependence of surface tension and adsorption in each of the four regimes.     

High-capacity purification of hen egg-white proteins by ion-exchange electrochromatography with an oscillatory transverse electric field

The ion-exchange electrochromatography with an oscillatory electric field perpendicular to the mobile-phase flow driven by pressure (pIEEC) was used to separate hen egg-white (HEW) proteins. The results were compared with those of normal ion-exchange chromatography (IEC). The column was designed as three-compartment rectangular column of 2-mL with dimensions (length x width x depth) of 40 x 10 x 5 mm(3) and the electric field was applied across the direction of column width. Q Sepharose FF was packed into the central compartment as the chromatographic bed. It was confirmed that the dynamic binding capacity (DBC) of different proteins (ovotransferrin and ovalbumin) in the HEW solution increased 2.3 times when an oscillatory electric current of 30 mA at 1/20 Hz was applied in the transverse column direction. Then, the HEW proteins were separated by the pIEEC at loading amounts 2.3-fold higher than those by the IEC. When the feedstock of about one-third of the DBC was applied to the columns (i.e., 7 mL for the pIEEC and 3 mL for the IEC), similar separation efficiencies of the two chromatographic modes were achieved. Both the recovery yield and purity reached 73% to over 90%. The results indicate that the pIEEC is promising for high-capacity purification of proteins.     

pH effect on dynamic coating for capillary electrophoresis of DNA

A buffer consisting of tris(hydroxymethyl)aminomethane, 2-(N-moropholino)ethanesulfonic acid (Mes) and EDTA with constant ion strength was used to investigate the effect of buffer pH on the dynamic coating behavior of poly(N-isopropylacrylamide) (PNIPAM) for DNA separation. The atomic force microscopy (AFM) image illustrated that PNIPAM in lower-pH buffer was much more efficient in covering a silica wafer than that in higher-pH buffer. The coating performance of PNIPAM was also quantitatively analyzed by Fourier transform IR attenuated total reflectance spectroscopy and by measuring the electroosmotic flow (EOF). These results indicated that the stability of the dynamic coating was dependent on the pH of the sieving matrix and was improved by reducing the pH to the weak-acid range. The lower pH of the sieving buffer may induce the polymer more efficiently to adsorb on the capillary wall to suppress EOF and DNA–capillary wall interaction for DNA separation. The enhanced dynamic coating capacity of PNIPAM in lower-pH buffer may be attributed to the hydrogen bonds between the hydroxyl groups of the silica surface and the oxygen atom of the carbonyl groups of PNIPAM.     

A dynamic frictional contact problem for piezoelectric materials

We consider a mathematical model which describes the frictional contact between a piezoelectric body and an electrically conductive foundation. The process is dynamic, the material's behavior is modeled with an electro-viscoelastic constitutive law and the contact is described by subdifferential boundary conditions. We derive the variational formulation of the problem which is in the form of a system involving a second order evolutionary hemivariational inequality for the displacement field coupled with a time-dependent hemivariational inequality for the electric potential field. Then we prove the existence of a unique weak solution to the model. The proof is based on arguments of abstract second order evolutionary inclusions with monotone operators.     

Dynamic programming and planarity: Improved tree-decomposition based algorithms

We study some structural properties for tree-decompositions of 2-connected planar graphs that we use to improve upon the runtime of tree-decomposition based dynamic programming approaches for several NP-hard planar graph problems. E.g., we derive the fastest algorithm for Planar Dominating Set of runtime 3^tw⋅n^O(1), when we take the width tw of a given tree-decomposition as the measure for the exponential worst case behavior. We also introduce a tree-decomposition based approach to solve non-local problems efficiently, such as Planar Hamiltonian Cycle in runtime 6^tw⋅n^O(1). From any input tree-decomposition of a 2-connected planar graph, one computes in time O(nm) a tree-decomposition with geometric properties, which decomposes the plane into disks, and where the graph separators form Jordan curves in the plane.     

On exact solutions to epidemic dynamic models

In this study, we address an SIR (susceptible-infected-recovered) model that is given as a system of first order differential equations and propose the SIR model on time scales which unifies and extends continuous and discrete models. More precisely, we derive the exact solution to the SIR model and discuss the asymptotic behavior of the number of susceptibles and infectives. Next, we introduce an SIS (susceptible-infected-susceptible) model on time scales and find the exact solution. We solve the models by using the Bernoulli equation on time scales which provides an alternative method to the existing methods. Having the models on time scales also leads to new discrete models. We illustrate our results with examples where the number of infectives in the population is obtained on different time scales.