Divergent Design of Mechanical Metamaterials Clan Deducted from Arc-serpentine Curve

The exotic properties of mechanical metamaterials are determined by their unit-cells' structure and spatial arrangement, in analogy with the atoms of conventional materials. Companioned with the mechanism of structural or cellular materials1–5, the ancient wisdom of origami6–11 and kirigami12–16 and the involvement of multiphysics interaction2,17,18 enrich the programable mechanical behaviors of metamaterials, including shape-morphing8,12,14,16,19, compliance4,5,8,17,20, texture2,18,21, and topology11,18,22−25. However, typical design strategies are mainly convergent, which transfers various structures into one family of metamaterials that are relatively incompatible with the others and do not fully bring combinatorial principles3,10,26 into play. Here, we report a divergent strategy that designs a clan of mechanical metamaterials with diverse properties derived from a symmetric curve consisting of serpentines and arcs. We derived this composite curve into planar and cubic unit-cells and modularized them by attaching magnetics. Moreover, stacking each of them yields two- and three-dimensional auxetic metamaterials, respectively. Assembling with both modules, we achieved three thick plate-like metamaterials separately with flexibility, in-plane buckling, and foldability. Furthermore, we demonstrated that the hybrid of paradox properties is possible by combining two of the above assembles. We anticipate that this divergent strategy paves the path of building a hierarchical library of diverse combinable mechanical metamaterials and making conventional convergent strategies more efficient to various requests. Main

Teaching Combinatorial Principles Using Relations through the Placemat Method

The presented paper is devoted to an innovative way of teaching mathematics, specifically the subject combinatorics in high schools. This is because combinatorics is closely connected with the beginnings of informatics and several other scientific disciplines such as graph theory and complexity theory. It is important in solving many practical tasks that require the compilation of an object with certain properties, proves the existence or non-existence of some properties, or specifies the number of objects of certain properties. This paper examines the basic combinatorial structures and presents their use and learning using relations through the Placemat method in teaching process. The effectiveness of the presented innovative way of teaching combinatorics was also verified experimentally at a selected high school in the Slovak Republic. Our experiment has confirmed that teaching combinatorics through relationships among talented children in mathematics is more effective than teaching by a standard algorithmic approach.

Adding sticks to a bundle: government-imposed modifications to property

Legal economists analyze property rights in two distinct, but related enterprises. The ‘content’-enterprise treats property rights as collections of entitlements and seeks to explain the combinatorial principles at work in combining them. The ‘enforcement’-enterprise takes the existing entitlement structures as a given and focuses instead on whether and how the individual entitlements should be protected against appropriation. This article argues that the differences in how these two enterprises resolve property disputes carry policy implications: if ‘self-help’ measures parties take to appropriate or protect entitlements would lead to rent-seeking, the government should pre-emptively modify entitlement structures so that entitlements are assigned to the party that values them most.

Possible Patterns

“There are no gaps in logical space,” writes Lewis (1986), giving voice to sentiment shared by many philosophers. But different natural ways of trying to make this sentiment precise turn out to conflict with one another. One is a pattern idea: “Any pattern of instantiation is metaphysically possible.” Another is a cut and paste idea: “For any objects in any worlds, there exists a world that contains any number of duplicates of all of those objects.” Jumping off from discussions from Forrest and Armstrong (1984) and Nolan (1996), the authors use resources from model theory to show the inconsistency of certain packages of combinatorial principles and the consistency of others.

SQUARE WITH BUILT-IN DIAMOND-PLUS

We formulate combinatorial principles that combine the square principle with various strong forms of the diamond principle, and prove that the strongest amongst them holds inLfor every infinite cardinal.As an application, we prove that the following two hold inL:1.For every infinite regular cardinalλ, there exists a special λ+-Aronszajn tree whose projection is almost Souslin;2.For every infinite cardinalλ, there exists arespectingλ+-Kurepa tree; Roughly speaking, this means that this λ+-Kurepa tree looks very much like the λ+-Souslin trees that Jensen constructed inL.