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ABSTRACT:The aqueous methylcellulose (MC) gels are characterized with a dynamic mechanical analyzer (DMA) under dynamic vertical compression. During the frequency sweeps of MC gels at different temperatures, the storage modulus is observed to be higher than loss modulus at lower frequencies. Both of the storage and loss modulus increases with frequency, but the rate of increase is higher for loss modulus. This leads to the first crossover between E 0 and E 00 during the frequency scan. For the frequency scan at a high temperature (80 C), a higher rate of increase is observed in storage modulus beyond the first crossover frequency. This leads to the second crossover between storage and loss modulus. Optical microscopy results indicate the presence of core-shell microstructure in aqueous MC gels. The first crossover is possibly due to the shell-sol transition, whereas the second crossover is due to the sol-shell-core transition. The validity of scaling laws at and around the first-crossover point (shell-sol transition) is checked. The scaling law is valid at the first-crossover point, but it is invalid around it. Alternate scaling equations based on reduced parameters are also used to check the universality. Irrespective of temperature, scaling laws are valid for reduced parameters.

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A graph H is p-edge colorable if there is a coloring ψ : E(H) → {1, 2, . . . , p}, such that for distinct uv, vw ∈ E(H), we have ψ(uv) = ψ(vw). The Maximum Edge-Colorable Subgraph problem takes as input a graph G and integers l and p, and the objective is to find a subgraph H of G and a p-edge-coloring of H, such that |E(H)| ≥ l. We study the above problem from the viewpoint of Parameterized Complexity. We obtain FPT algorithms when parameterized by: (1) the vertex cover number of G, by using Integer Linear Programming, and (2) l, a randomized algorithm via a reduction to Rainbow Matching, and a deterministic algorithm by using color coding, and divide and color. With respect to the parameters p + k, where k is one of the following: (1) the solution size, l, (2) the vertex cover number of G, and (3) l − mm(G), where mm(G) is the size of a maximum matching in G; we show that the (decision version of the) problem admits a kernel with O(k • p) vertices. Furthermore, we show that there is no kernel of size O(k 1− • f (p)), for any > 0 and computable function f , unless NP ⊆ coNP/poly.