Abstract:
Let $G = (V,E)$ be a graph, and for each $e \in E(G)$, let $L_e$ be a list of real numbers. Let $w:E(G) \to \cup_{e \in E(G)}L_e$ be an edge weighting function such that $w(e) \in L_e$ for each $e \in E(G)$, and let $c_w$ be the vertex colouring obtained by $c_w(v) = \sum_{e \ni v}w(e)$. We desire the smallest possible $k$ such that, for any choice of $\{L_e \,|\, e \in E(G)\}$ where $|L_e| \geq k$ for all $e \in E(G)$, there exists an edge weighting function $w$ for which $c_w$ is proper. The smallest such value of $k$ is the weight choosability number of $G$. This colouring problem, introduced by Bartnicki, Grytczuk and Niwczyk (2009), is the list variation of the now famous 1-2-3 Conjecture due to Karo\'nski, {\L}uczak, and Thomason (2004). Bartnicki et al. develop a method for approaching the problem based on the Combinatorial Nullstellensatz. Though they show that some particular classes of graphs have weight choosability number at most $3$, it was known whether their method could be extended to prove a bound which holds for all admissible graphs. In this paper, we show that this is indeed possible, showing that every graph is $(\Delta + d + 1)$-weight choosable, where $\Delta$ is the graph's maximum degree and $d$ is its degeneracy. In fact, more general results on total weight choosability are provided, where one assigns weights to edges and vertices. Improved bounds are also established for some classes of graph products.

Abstract:
The 1-2-3 Conjecture, posed in 2004 by Karonski, Luczak, and Thomason, is as follows: "If G is a graph with no connected component having exactly 2 vertices, then the edges of G may be assigned weights from the set {1,2,3} so that, for any adjacent vertices u and v, the sum of weights of edges incident to u differs from the sum of weights of edges incident to v." This survey paper presents the current state of research on the 1-2-3 Conjecture and the many variants that have been proposed in its short but active history.

Abstract:
In 1990, Hendry conjectured that every Hamiltonian chordal graph is cycle extendible; that is, the vertices of any non-Hamiltonian cycle are contained in a cycle of length one greater. We disprove this conjecture by constructing counterexamples on $n$ vertices for any $n \geq 15$. Furthermore, we show that there exist counterexamples where the ratio of the length of a non-extendible cycle to the total number of vertices can be made arbitrarily small. We then consider cycle extendibility in Hamiltonian chordal graphs where certain induced subgraphs are forbidden, notably $P_n$ and the bull.

Abstract:
Karonski, Luczak, and Thomason (2004) conjectured that, for any connected graph G on at least three vertices, there exists an edge weighting from {1,2,3} such that adjacent vertices receive different sums of incident edge weights. Bartnicki, Grytczuk, and Niwcyk (2009) made a stronger conjecture, that each edge's weight may be chosen from an arbitrary list of size 3 rather than {1,2,3}. We examine a variation of these conjectures, where each vertex is coloured with a sequence of edge weights. Such a colouring relies on an ordering of the graph's edges, and so two variations arise -- one where we may choose any ordering of the edges and one where the ordering is fixed. In the former case, we bound the list size required for any graph. In the latter, we obtain a bound on list sizes for graphs with sufficiently large minimum degree. We also extend our methods to a list variation of irregularity strength, where each vertex receives a distinct sequence of edge weights.

Abstract:
We show that the Yao graph Y4 in the L2 metric is a spanner with stretch factor 8(29+23sqrt(2)). Enroute to this, we also show that the Yao graph Y4 in the Linf metric is a planar spanner with stretch factor 8.

Abstract:
The
question whether R^{2} represents information or noise is still a fundamental
question in the study of stock price synchronicity. There are two main difficulties.
Firstly, the trait information of a company is hard to measure; Secondly, the
investors’ sophistication is ignored when we discuss the effectiveness of
market. Through the study of Chinese SSE E-interaction platform and the
measurement by textual analysis method, this article argues that the improvement
of investors’ sophistication has negative association with stock price
synchronicity. This result is more salient in companies with lower opacity or
with higher corporate governance. This paper contributes to the deepen
understanding of stock price synchronicity and new method of measuring trait
information of companies.

Abstract:
A technique that permits the calculation of the flow of agents between and within labor market states is pre-sented. A statistical agency having collected data on those flows easiest to collect and together with data on employment, unemployment and being out of the labor force, will be able to calculate the rest of the flows. The contribution of this paper is in suggesting an easy process which overcomes the difficulties statistical agencies usually have in collecting flow data.

Abstract:
In the beginning everything was explained in Biochemistry in terms of hydrogen-bonds (HB). Then, the devastating blow, known as the HB-inventory argument came; hydrogen bonding with water molecules compete with intramolecular hydrogen-bonds. As a result, the HBs paradigm fell from grace. The void created was immediately filled by Kauzmann’s idea of hydrophobic (HφO) effect which reigned supreme in biochemical literature for over 50 years (1960-2010). Cracks in the HB-inventory argument on one hand, and doubts about the adequacy of Kauzmann’s model for the HφO effect, have led to a comeback of the HBs, along with a host of new hydrophilic (HφI) effects. The HφO effects lost much of its power - which it never really had - in explaining protein folding and protein-protein association. Instead, the more powerful and richer repertoire of HφI effects took over the reins. The interactions also offered simple and straightforward answers to the problems of protein folding, and protein-protein association.

Abstract:
Three out of 125 “big questions of science”, can now be claimed to have been answered. All of these questions involve water; its structure, its role in protein folding and its role in protein-protein association.

Abstract:
In order to popularize the use of the solar-water heaters, especially in the residential and tertiary sectors with the third world, it appears to be necessary to reduce their cost while improving their performances. It is the object of this integrated storage collector thus created and tested in the south of Tunisia. It is simply made up of a tank playing the double part of solar absorber and storage tank of warm water, of a glazing to profit from the greenhouse effect and of an insulating case. Its measured energy performances, by the method of input-output proves its effectiveness to produce hot water, in spite of its simplicity of manufacture, usage and maintenance. Indeed a temperature of water exceeding 70?C is reached towards the afternoon True Solar Time, and for an efficiency of 7%. Thus, this type of collector with integrated storage is entirely satisfactory and could be available to larger mass.