Cyclopentadienyl-functionalised polyhedral silsesquioxanes as building blocks for new nanostructured materials

This work is to design new organosilicon precursors, which contain both siloxane and cyclopentadienyl functionalities and use these compounds as building blocks for new nanomaterials. Polyhedral silsesquioxanes, (RSiO_1.5)_n (n = 8 and 10) with cyclopentadienyl (R = -C₅H₅ or -(CH₂)₃-C₅H₅) functionalities have been prepared by hydrolytic condensation of the correspondent silicon organic precursors RSiCl₃ or RSi(OC₂H₅)₃. The products with cyclopentadienyl groups have been used to prepare new cross-linked 3D polymeric materials by Diels-Alder reaction. The compounds have been characterised by multinuclear (¹H and ²⁹Si) NMR, ESI MS and IR spectroscopy, UV/Vis spectrometry, gel permeation chromatography and scanning electron microscopy. The approach above seems to be very promising for the preparation of new nanostructured materials.     

On minimal Sturmian partial words

Partial words, which are sequences that may have some undefined positions called holes, can be viewed as sequences over an extended alphabet A_?=A∪{?}, where ? stands for a hole and matches (or is compatible with) every letter in A. The subword complexity of a partial word w, denoted by p_w(n), is the number of distinct full words (those without holes) over the alphabet that are compatible with factors of length n of w. A function f:N→N is (k,h)-feasible if for each integer N≥1, there exists a k-ary partial word w with h holes such that p_w(n)=f(n) for all n such that 1≤n≤N. We show that when dealing with feasibility in the context of finite binary partial words, the only affine functions that need investigation are f(n)=n+1 and f(n)=2n. It turns out that both are (2,h)-feasible for all non-negative integers h. We classify all minimal partial words with h holes of order N with respect to f(n)=n+1, called Sturmian, computing their lengths as well as their numbers, except when h=0 in which case we describe an algorithm that generates all minimal Sturmian full words. We show that up to reversal and complement, any minimal Sturmian partial word with one hole is of the form aⁱ?a^jba^l, where i,j,l are integers satisfying some restrictions, that all minimal Sturmian partial words with two holes are one-periodic, and that up to complement, ?(a^N−1?)^h−1 is the only minimal Sturmian partial word with h≥3 holes. Finally, we give upper bounds on the lengths of minimal partial words with respect to f(n)=2n, showing them tight for h=0,1 or 2.     

LFC bumps on separable Banach spaces

In this note we construct a C^∞-smooth, LFC (Locally depending on Finitely many Coordinates) bump function, in every separable Banach space admitting a continuous, LFC bump function.     

On a connection between facility location and perfect graphs

We characterize the graphs for which a linear relaxation of a facility location problem defines a polytope with all integral extreme points. We use a transformation to a stable set problem in perfect graphs. Based on this transformation, these graphs can be recognized in polynomial time.     

A model of students’ combinatorial thinking

Combinatorial topics have become increasingly prevalent in K-12 and undergraduate curricula, yet research on combinatorics education indicates that students face difficulties when solving counting problems. The research community has not yet addressed students’ ways of thinking at a level that facilitates deeper understanding of how students conceptualize counting problems. To this end, a model of students’ combinatorial thinking was empirically and theoretically developed; it represents a conceptual analysis of students’ thinking related to counting and has been refined through analyzing students’ counting activity. In this paper, the model is presented, and relationships between formulas/expressions, counting processes, and sets of outcomes are elaborated. Additionally, the usefulness and potential explanatory power of the model are demonstrated through examining data both from a study the author conducted, and from existing literature on combinatorics education.     

k-Abelian pattern matching

Two words are called k-abelian equivalent, if they share the same multiplicities for all factors of length at most k. We present an optimal linear time algorithm for identifying all occurrences of factors in a text that are k-abelian equivalent to some pattern P. Moreover, an optimal algorithm for finding the largest k for which two words are k-abelian equivalent is given. Solutions for online versions of the k-abelian pattern matching problem are also proposed.     

Mechanical and thermal properties and flame retardancy of phosphorus-containing polyhedral oligomeric silsesquioxane (DOPO-POSS)/polycarbonate composites

A novel polyhedral oligomeric silsesquioxane containing 9,10-dihydro-9-oxa-10-phosphaphenanthrene-10-oxide (DOPO-POSS) has been incorporated into polycarbonate (PC) composites in order to study its effect on mechanical and thermal properties and flame retardancy. The mechanical and thermal properties of the DOPO-POSS/PC composites have been investigated by tensile and flexural testing, DSC, and DMA. Slight enhancements of yield stress, and flexural strength and modulus, and obvious decreases of fracture strength and strain of the DOPO-POSS/PC composites were observed with an increase in DOPO-POSS loading. The glass transition temperatures (T_g) of the composites were reduced with increasing DOPO-POSS loading. The morphology of the PC composites was evaluated by SEM, which indicated that the DOPO-POSS was dispersed with a particle size of 100-250 nm in the PC matrix. The thermal degradation behaviour and flame retardancies of PC composites with different DOPO-POSS loadings were investigated by TGA, LOI, UL-94 standard, and cone calorimetry. The composite had an LOI value of 30.5 and a UL-94 rating V-0 when the content of DOPO-POSS was 4%.     

Connected and alternating vectors: Polyhedra and algorithms

Given a graphG = (V, E), leta ^S, S L, be the edge set incidence vectors of its nontrivial connected subgraphs.The extreme points of = {x R ^E: a^sx |V(S)| - |S|, S L} are shown to be integer 0/± 1 and characterized. They are the alternating vectorsb ^k, k K, ofG. WhenG is a tree, the extreme points ofB 0,b ^kx 1,k K} are shown to be the connected vectors ofG together with the origin. For the four LP's associated with andA, good algorithms are given and total dual integrality of andA proven.On leave from Swiss Federal Institute of Technology, Zurich.     

Investigating undergraduate students’ proof schemes and perspectives about combinatorial proof

Combinatorics is an area of mathematics with accessible, rich problems and applications in a variety of fields. Combinatorial proof is an important topic within combinatorics that has received relatively little attention within the mathematics education community, and there is much to investigate about how students reason about and engage with combinatorial proof. In this paper, we use Harel and Sowder’s (1998) proof schemes to investigate ways that students may characterize combinatorial proofs as different from other types of proof. We gave five upper-division mathematics students combinatorial-proof tasks and asked them to reflect on their activity and combinatorial proof more generally. We found that the students used several of Harel and Sowder’s proof schemes to characterize combinatorial proof, and we discuss whether and how other proof schemes may emerge for students engaging in combinatorial proof. We conclude by discussing implications and avenues for future research.