POSITIVE SOLUTIONS TO(k,n-k) NONHOMOGENEOUS BOUNDARYVALUE PROBLEM

By the Schauder fixed point theory,this paper establishes the existence of positive solutions to a(k,n k) m-point boundary value problem.We show that there exists a positive constant b such that the problem has at least one positive solution when the homogeneous boundary parameter is smaller than b,and no positive solution when this parameter is greater than b.     

An object-parameter approach to predicting unknown data in incomplete fuzzy soft sets

Incomplete data in soft sets lead to uncertainty and inaccuracy in representing and handling information. This paper introduces notions of complete distance between two objects and relative dominance degree between two parameters. Based on both the notions, an object-parameter method is proposed to predict unknown data in incomplete fuzzy soft sets. The proposal makes full use of known data, including the information from the relationship between known values of all objects on a certain parameter and the information from the relationship between known values of an object on all parameters. The effectiveness of the proposal is verified by many examples under the compared investigation of classical predicted methods.     

Erratum: The average degree of an edge-chromatic critical graph II

In the article “The average degree of an edge-chromatic critical graph II” by Douglas R. Woodall (J. Graph Theory 56 (2007), 194-218), it was claimed that the average degree of an edge-chromatic critical graph with maximum degree $\mathrm{\Delta }$ is at least $\text{◂⋅▸}\frac{2}{3}\left(\mathrm{\Delta }+1\right)$ if $\mathrm{\Delta }⩾2$, at least $\text{◂+▸}\frac{2}{3}\mathrm{\Delta }+1$ if $\mathrm{\Delta }⩾8$, and at least $\text{◂⋅▸}\frac{2}{3}\left(\mathrm{\Delta }+2\right)$ if $\mathrm{\Delta }⩾15$. Unfortunately there were mistakes in the proof of the last two of these results, which are now proved only if $\mathrm{\Delta }⩾18$ and $\mathrm{\Delta }⩾30$, respectively.     

Radical polymerization in the presence of peroxide and reducing agent monomer for branched polymers

In this article, we report the radical polymerization in the presence of peroxide and commercially available or designed reducing agent monomer (RAM) for the preparation of branched poly(methyl methacrylate)s (PMMAs). The reaction behavior of the RAM was studied by NMR. Triple‐detection SEC (TD‐SEC) analysis was used to confirm the branching structure of the prepared PMMAs and to investigate the influence of peroxide concentration and RAM concentration on molecular weight and branched structure. The obtained branched PMMAs exhibited high molecular weights and relatively narrow polydispersities at high conversion of MMA. Interestingly, a significant increase in molecular weight and degree of branching of the obtained polymers are observed in higher BPO concentration, these results are quite different from that reported in the literature. The unique radical polymerization mechanism in the RAM/BPO redox‐initiated radical polymerization system resulted in branched PMMAs with high molecular weights at relatively high RAM and BPO concentrations. © 2019 Wiley Periodicals, Inc. J. Polym. Sci., Part A: Polym. Chem. 2019 , 57, 833–840     

Monitoring of the shrinkage during the photopolymerization of acrylates using hyphenated photorheometry/near‐infrared spectroscopy

This article describes a new method for the quantitative determination and time‐resolved monitoring of the polymerization shrinkage during ultraviolet (UV) photopolymerization. It is based on rheometry using a modified oscillating rheometer. Shrinkage is determined from the decrease of the gap between the rheometer plates. Moreover, near‐infrared (NIR) spectra can be recorded directly in the rheometer, which allows continuous determination of the conversion at any time of a shrinkage measurement. As both shrinkage and conversion data come from the same experiment, shrinkage can be analyzed in dependence on the current conversion achieved during UV irradiation, which enables direct investigation of correlations between both parameters. Hyphenated photorheometry/FT‐NIR spectroscopy was used for the determination of the polymerization shrinkage of pure acrylate monomers and oligomers as well as acrylate‐based formulations. Quantitative shrinkage values were found to be in excellent correlation with data that were determined by an independent method (via buoyancy measurements) and data from literature. Furthermore, the effect of ambient and irradiation conditions or the content of nanoparticles on the degree of shrinkage was studied. © 2015 Wiley Periodicals, Inc. J. Polym. Sci., Part B: Polym. Phys. 2015 , 53, 729–739     

Enhanced spin capturing polymerization: Numerical investigation of mechanism

Simulation on the kinetic scheme of enhanced spin capturing polymerizations (ESCP) were performed to reveal the influence of different parameters, e.g., initiation rate, spin capturing and side reactions on the properties (the monomer conversion, degree of polymerization, polydispersity index, and the fraction of “living” chains) of polymer obtained. The kinetic scheme of ESCP was solved numerically as well as the method of statistical moments of molecular weight distribution was applied to get some numerical expressions for analysis. The simulations showed that fast initiation rate can lead to decrease of living fraction and broadening of molecular weight distribution. Whereas high rate of spin capturing makes the polymer chains shorter but increases living character. And side reactions always decrease livingness of the polymer obtained. Finally in the case of absence of side reaction we were able to formulate criteria for formation of polymer with target molecular weight and quota of living chains via ESCP process. © 2015 Wiley Periodicals, Inc. J. Polym. Sci., Part A: Polym. Chem. 2015 , 53, 2546–2556     

Application of topological degree theory to semi-definite complementarity problem

The topological degree theory is applied to study the problem of existence of solutions to the semi-definite complementarity problem (SDCP). A notion of an exceptional family of matrices is introduced, and assertions of a non-strict alternative type are obtained. Namely, for a continuous mapping, there exists at least one of the following two items: either a solution to the SDCP, or an exceptional family of matrices. Hence, if there is no exceptional family, then at least one solution exists     

Periodic solutions to a class of biological diffusion models with hysteresis effect

This paper is concerned with a class of biological models which consists of a nonlinear diffusion equation and a hysteresis operator describing the relationship between some variables of the equations. By the viscosity approach, we show the existence of periodic solutions of the problem under consideration. More precisely, with the help of the subdifferential operator theory and Leray–Schauder theorem, we show the existence of periodic solutions to the approximation problem and then obtain the solution of the original problem by using a passage-to-limit procedure.