Pre-ordered Structures Research Articles

Statistical comparison of Path-Complete Lyapunov Functions: a Discrete-Event Systems perspective

The goal of this paper is to advertise the tight links between the theory of Discrete-Event Systems and Path-Complete Lyapunov Functions. These are algebro-combinatorial stability criteria for hybrid systems, whose meta-parameters are an automaton and a template of candidate Lyapunov functions.To do this, we analyse a phenomenon recently observed in the literature, namely the statistical ordering of Path-Complete Lyapunov Functions, which is by far not well understood yet: Path-Complete Lyapunov functions can be endowed with a preorder structure with respect to their performance. It has been recently shown that the preorder corresponding to relative worst case performance can be characterized with tools from automata theory, but the ordering corresponding to statistical relative performance has remained elusive. We advocate for a Discrete-Event Systems approach to this problem and provide preliminary results in this direction.

Development of 3D printable graphene oxide based bio-ink for cell support and tissue engineering.

Tissue engineered constructs can serve as in vitro models for research and replacement of diseased or damaged tissue. As an emerging technology, 3D bioprinting enables tissue engineering through the ability to arrange biomaterials and cells in pre-ordered structures. Hydrogels, such as alginate (Alg), can be formulated as inks for 3D bioprinting. However, Alg has limited cell affinity and lacks the functional groups needed to promote cell growth. In contrast, graphene oxide (GO) can support numerous cell types and has been purported for use in regeneration of bone, neural and cardiac tissues. Here, GO was incorporated with 2% (w/w) Alg and 3% (w/w) gelatin (Gel) to improve 3D printability for extrusion-based 3D bioprinting at room temperature (RT; 25°C) and provide a 3D cellular support platform. GO was more uniformly distributed in the ink with our developed method over a wide concentration range (0.05%-0.5%, w/w) compared to previously reported GO containing bioink. Cell support was confirmed using adipose tissue derived stem cells (ADSCs) either seeded onto 3D printed GO scaffolds or encapsulated within the GO containing ink before direct 3D printing. Added GO was shown to improve cell-affinity of bioinert biomaterials by providing more bioactive moieties on the scaffold surface. 3D cell-laden or cell-seeded constructs showed improved cell viability compared to pristine (without GO) bio-ink-based scaffolds. Our findings support the application of GO for novel bio-ink formulation, with the potential to incorporate other natural and synthetic materials such as chitosan and cellulose for advanced in situ biosensing, drug-loading and release, and with the potential for electrical stimulation of cells to further augment cell function.

The d-Majorization Polytope

We investigate geometric and topological properties of d-majorization – a generalization of classical majorization to positive weight vectors d∈Rn. In particular, we derive a new, simplified characterization of d-majorization which allows us to work out a halfspace description of the corresponding d-majorization polytopes. That is, we write the set of all vectors which are d-majorized by some given vector y∈Rn as an intersection of finitely many half spaces, i.e. as solutions to an inequality of the type Mx≤b. Here b depends on y while M can be chosen independently of y. This description lets us prove continuity of the d-majorization polytope (jointly with respect to d and y) and, furthermore, lets us fully characterize its extreme points. Interestingly, for y≥0 one of these extreme points classically majorizes every other element of the d-majorization polytope.Moreover, we show that the induced preorder structure on Rn admits minimal and maximal elements. While the former are always unique the latter are unique if and only if they correspond to the unique minimal entry of the d-vector.

Order polarities

Abstract We define an order polarity to be a polarity $(X,Y,{\operatorname{R}})$ where $X$ and $Y$ are partially ordered, and we define an extension polarity to be a triple $(e_X,e_Y,{\operatorname{R}})$ such that $e_X:P\to X$ and $e_Y:P\to Y$ are poset extensions and $(X,Y,{\operatorname{R}})$ is an order polarity. We define a hierarchy of increasingly strong coherence conditions for extension polarities, each equivalent to the existence of a preorder structure on $X\cup Y$ such that the natural embeddings, $\iota _X$ and $\iota _Y$, of $X$ and $Y$, respectively, into $X\cup Y$ preserve the order structures of $X$ and $Y$ in increasingly strict ways. We define a Galois polarity to be an extension polarity satisfying the strongest of these coherence conditions and where $e_X$ and $e_Y$ are meet- and join-extensions, respectively. We show that for such polarities the corresponding preorder on $X\cup Y$ is unique. We define morphisms for polarities, providing the class of Galois polarities with the structure of a category, and we define an adjunction between this category and the category of $\varDelta _1$-completions and appropriate homomorphisms.

Galois Connections Between a Fuzzy Preordered Structure and a General Fuzzy Structure

We continue the study of (isotone) Galois connections, also called adjunctions, in the framework of fuzzy preordered structures, which generalize fuzzy preposets by considering underlying fuzzy equivalence relations. Specifically, we present necessary and sufficient conditions so that, given a mapping $f:\mathbb {A}\rightarrow B$ from a fuzzy preordered structure $\mathbb {A}=\langle A,\approx _A,\rho _A\rangle$ into a fuzzy structure $\langle B,\approx _{B}\rangle$ , it is possible to construct a fuzzy relation $\rho _B$ that induces a suitable fuzzy preorder structure on $B$ and such that there exists a mapping $g:B\rightarrow \mathbb {A}$ such that the pair $(f,g)$ constitutes an Galois connection.

Combining X-ray scattering with dielectric and calorimetric experiments for monitoring polymer crystallization

In order to understand nucleation; crystallization and other phase transitions in polymers, polymer based composites, or in liquid crystals simultaneous experiments with a combination of different methods are useful. Due to different sample geometry, contact faces with the sample holder, and thermal conditions it is usually difficult to compare the results of several individual experiments. As an important supplement to the classical techniques for studying crystallization like X-ray scattering, or differential scanning calorimetry, measurements which test molecular mobility like dielectric or mechanical spectroscopy are of interest during isothermal and non-isothermal crystallization. From such simultaneous experiments one can learn about the existence of pre-ordered structures before formation of crystals, as detected by DSC or X-ray scattering. In this contribution we present the development of a device for simultaneous measurements of electrical properties and X-ray scattering intensities, which was extended to a microcalorimeter and allows measuring thermal properties like heat capacity and thermal conductivity additionally at the same time and at the same sample volume.

Evidence of pre-crystalline-order in super-cooled polymer melts revealed from simultaneous dielectric spectroscopy and SAXS

The existence and the formation of pre-ordered structures in super-cooled polymer melts as the initial step during the complex process of polymer crystallization are discussed. Most findings and interpretations are based on scattering and microscopic experiments despite the small contrast between the phases. Such experiments test small differences in morphology between the assumed precursors of the crystals and the surrounding melt. In contrast, relaxation experiments are probing the motions and related dynamics in the system and are therefore independent of differences in density. During the process of crystallization, material is transformed from the liquid to the solid state. Consequently, motions typical for a liquid become impossible and do not longer contribute to the measured signal. For pre-ordered structures one also expects changes in mobility due to changes in chain conformation during pre-ordering, but without additional morphological information the interpretation of relaxation experiments is often speculative. Therefore, a combination of both types of experiments seems essential. We performed simultaneously dielectric relaxation and small-angle X-ray scattering (SAXS) experiments during isothermal crystallization of polycaprolactone. A newly developed sample cell is described. Pronounced effects in the dielectric signal are observed significantly before changes in SAXS intensity or crystallinity can be detected. The observations strongly support the idea of pre-crystalline-order in the polymer melt before the formation of crystals.

Early stages of polymer crystallization—a dielectric study

The existence and the formation of pre-ordered structures as the initial step during the complex process of polymer crystallization are discussed controversially. Most of the findings and interpretations are based on scattering experiments, which test small density differences between the assumed precursors of the crystals and the surrounding melt. Because of the low contrast the interpretation of experimental results become often speculative. In contrast relaxation experiments are probing motions in the sample and are therefore independent on density contrast. During crystallization, material is transformed from the liquid to the solid state. Consequently, motions typical for a liquid become impossible and do not longer contribute to the measured signal. For pre-ordered structures we expect some changes in mobility too because of the changes in conformation on pre-ordering. We performed dielectric relaxation experiments during isothermal crystallization of PCL. Pronounced effects in ϵ′ are observed long before changes in crystallinity can be detected. The observations strongly support the idea of pre-order in the polymer melt before the formation of crystals.

Some dissenting views on the transitivity of individual preference

(1) The transitivity property is not a necessary condition for the rationality of all individual preference relations. (2) A weakened definition of the transitivity is not necessarily relevant. (3) The non-transitivity of fuzzy preference relations is not inconsistent with a fuzzy total preorder structure on the set of alternatives.