We present a pumping lemma for each level of the collapsible pushdown graph hierarchy in analogy to the second author’s pumping lemma for higher-order pushdown graphs (without collapse). Using this lemma, we give the first known examples that separate the levels of the collapsible pushdown graph hierarchy and of the collapsible pushdown tree hierarchy, i.e., the hierarchy of trees generated by ...

A simplicial complex is d-collapsible if it can be reduced to an empty simplex by repeatedly removing face of dimension less than d that it is contained in a unique maximal face. These complexes have been studied since every complex that is the nerve of a family of convex sets in R d is d-collapsible. We show that the question whether a given complex is 2-collapsible can be solved in polynomial...

A conventional computer keyboard consists of as many as 101 keys. The keyboard has several sections, such as text entry section, navigation section, numeric keypad etc. each having several keys on the keyboard. The keys are arranged quite spaciously so that the human fingers can operate comfortably. The number of keys and the arrangement of keys make a keyboard quite big. Although a bigger keyb...

Thin-shell deployable reflector structures that are folded elastically in a nearly-inextensional mode have been recently realized, exploiting the recent availability of high-modulus, ultra-thin composite materials. An inherent, and significant limitation of this approach is that these structures remain “floppy” in their deployed configuration. This paper presents a general concept for increasin...

We study the first-order model checking problem on two generalisations of pushdown graphs. The first class is the class of nested pushdown trees. The other is the class of collapsible pushdown graphs. Our main results are the following. First-order logic with reachability is uniformly decidable on nested pushdown trees. Considering first-order logic without reachability, we prove decidability i...

A simplicial complex K is called d-representable if it is the nerve of a collection of convex sets in R; K is d-collapsible if it can be reduced to an empty complex by repeatedly removing a face of dimension at most d − 1 that is contained in a unique maximal face; and K is d-Leray if every induced subcomplex of K has vanishing homology of dimension d and larger. It is known that d-representabl...

Bearing capacity of single piles are occasionally predicted using the renowned theoretical methods (α and β methods). These based on laboratory tests, which can be time-consuming, but also applicable in foundation engineering practice for unsaturated soils. Full-scale pile load tests were carried out bored equipped with Expander Body Systems Federal District Brazil, known its unsaturated, colla...

Collapsible loess is a kind of soil with special properties, and widely distributed in China. When it not wetted by water, its strength generally high compressibility low. However, when collapsible water under certain pressure, the structure will be rapidly destroyed, resulting large additional subsidence. Therefore, engineering constructions on sites are carried out, appropriate foundation tre...