Influence of the viscosity of poly(methyl methacrylate) on the cellular structure of nanocellular materials

Three different grades of poly(methyl methacrylate) (PMMA) with different rheological properties are used for the production of nanocellular materials using gas dissolution foaming. The influences of both the viscosity of the different polymers and the processing parameters on the final cellular structure are studied using a wide range of saturation and foaming conditions. Foaming conditions affect similarly all cellular materials. It is found that an increase of the foaming temperature results in less dense nanocellular materials, with higher cell nucleation densities. In addition, it is demonstrated that a lower viscosity leads to cellular polymers with a lower relative density but larger cell sizes and smaller cell nucleation densities, these differences being more noticeable for the conditions in which low solubilities are reached. It is possible to produce nanocellular materials with relative densities of 0.24 combined with cell sizes of 75 nm and cell nucleation densities of 1015 nuclei cm−3 using the PMMA with the lowest viscosity. In contrast, minimum cell sizes of around 14 nm and maximum cell nucleation densities of 3.5 × 1016 nuclei cm−3 with relative densities of 0.4 are obtained with the most viscous one. © 2019 Society of Chemical Industr

John's ellipsoid and the integral ratio of a log-concave function

We extend the notion of John’s ellipsoid to the setting of integrable log-concave functions. This will allow us to define the integral ratio of a log-concave function, which will extend the notion of volume ratio, and we will find the log-concave function maximizing the integral ratio. A reverse functional affine isoperimetric inequality will be given, written in terms of this integral ratio. This can be viewed as a stability version of the functional affine isoperimetric inequality.Ministerio de Economía y CompetitividadFondo Europeo de Desarrollo RegionalConsejería de Industria, Turismo, Empresa e Innovación (Comunidad Autónoma de la Región de Murcia)Coordenação de aperfeiçoamento de pessoal de nivel superiorInstituto Nacional de Matemática Pura e Aplicad

Nanocellular Polymers with a Gradient Cellular Structure Based on Poly(methyl methacrylate)/Thermoplastic Polyurethane Blends Produced by Gas Dissolution Foaming

Graded structures and nanocellular polymers are two examples of advanced cellular morphologies. In this work, a methodology to obtain low-density graded nanocellular polymers based on poly(methyl methacrylate) (PMMA)/ thermoplastic polyurethane (TPU) blends produced by gas dissolution foaming is reported. A systematic study of the effect of the processing condition is presented. Results show that the melt-blending results in a solid nanostructured material formed by nanometric TPU domains. The PMMA/ TPU foamed samples show a gradient cellular structure, with a homogeneous nanocellular core. In the core, the TPU domains act as nucleating sites, enhancing nucleation compared to pure PMMA and allowing the change from a microcellular to a nanocellular structure. Nonetheless, the outer region shows a gradient of cell sizes from nano- to micron-sized cells. This gradient structure is attributed to a non-constant pressure profile in the samples due to gas desorption before foaming. The nucleation in the PMMA/ TPU increases as the saturation pressure increases. Regarding the effect of the foaming conditions, it is proved that it is necessary to have a fine control to avoid degeneration of the cellular materials. Graded nanocellular polymers with relative densities of 0.16–0.30 and cell sizes ranging 310–480 nm (in the nanocellular core) are obtained

Best approximation of functions by log-polynomials

Lasserre [La] proved that for every compact set $K\subset\mathbb R^n$ and every even number $d$ there exists a unique homogeneous polynomial $g_0$ of degree $d$ with $K\subset G_1(g_0)=\{x\in\mathbb R^n:g_0(x)\leq 1\}$ minimizing $|G_1(g)|$ among all such polynomials $g$ fulfilling the condition $K\subset G_1(g)$. This result extends the notion of the L\"owner ellipsoid, not only from convex bodies to arbitrary compact sets (which was immediate if $d=2$ by taking convex hulls), but also from ellipsoids to level sets of homogeneous polynomial of an arbitrary even degree. In this paper we extend this result for the class of non-negative log-concave functions in two different ways. One of them is the straightforward extension of the known results, and the other one is a suitable extension with uniqueness of the solution in the corresponding problem and a characterization in terms of some 'contact points'.Comment: 26 pages, 2 figure

Volume inequalitites for the i-th convolution bodies

We obtain a new extension of Rogers–Shephard inequality providing an upper bound for the volume of the sum of two convex bodies K and L. We also give lower bounds for the volume of the k-th limiting convolution body of two convex bodies K and L. Special attention is paid to the (n - 1)-th limiting convolution body, for which a sharp inequality, which is equality only when K = -L is a simplex, is given. Since the n-th limiting convolution body of K and -K is the polar projection body of K, these inequalities can be viewed as an extension of Zhang’s inequality

An extension of Berwald's inequality and its relation to Zhang's inequality

In this note prove the following Berwald-type inequality, showing that for any integrable log-concave function f:Rn→[0, ∞)and any concave function h :L →[0, ∞), where L ={(x, t) ∈Rn×[0, ∞) :f(x) ≥e−t‖f‖∞}, then p→⎛⎝1Γ(1 +p)∫Le−tdtdx∫Lhp(x, t)e−tdtdx⎞⎠1p is decreasing in p ∈(−1, ∞), extending the range of pwhere the monotonicity is known to hold true.As an application of this extension, we will provide a new proof of a functional form of Zhang’s reverse Petty projection inequality, recently obtained in [2]

El solipsismo en wittgenstein: apreciación critica desde un punto de vista fenomenológico

Nuestra crítica a los argumentos solipsistas de Wittgenstein esta formulada desde un punto de vista fenomenológico, porque consideramos que el método fenomeloloqico ha introducido en el filosofar una nueva dimensión, dentro de la cual se puede entrar a resolver la problemática del "alter ego". La línea de este método es rnuy clara. Una vez superado el dualismo tradicional suieto-obieto, pensamiento-realidad, por medio de la idea del carácter intencional de la conciencia, "la actividad filosófica fundamental correcta, consiste en observar y aprender las cosas tal como ellas mismas se presentan (como se dan inmediatamente a la conciencia) y explicarlas luego con respecto a aquellas determinaciones y relaciones presentes y aprehendidas de igual manera