On the number of pancake stacks requiring four flips to be sorted

Using existing classification results for the 7- and 8-cycles in the pancake graph, we determine the number of permutations that require 4 pancake flips (prefix reversals) to be sorted. A similar characterization of the 8-cycles in the burnt pancake graph, due to the authors, is used to derive a formula for the number of signed permutations requiring 4 (burnt) pancake flips to be sorted. We furthermore provide an analogous characterization of the 9-cycles in the burnt pancake graph. Finally we present numerical evidence that polynomial formulae exist giving the number of signed permutations that require $k$ flips to be sorted, with $5\leq k\leq9$.Comment: We have finalized for the paper for publication in DMTCS, updated a reference to its published version, moved the abstract to its proper location, and added a thank you to the referees. The paper has 27 pages, 6 figures, and 2 table

Cycles in the burnt pancake graphs

The pancake graph $P_n$ is the Cayley graph of the symmetric group $S_n$ on $n$ elements generated by prefix reversals. $P_n$ has been shown to have properties that makes it a useful network scheme for parallel processors. For example, it is $(n-1)$-regular, vertex-transitive, and one can embed cycles in it of length $\ell$ with $6\leq\ell\leq n!$. The burnt pancake graph $BP_n$, which is the Cayley graph of the group of signed permutations $B_n$ using prefix reversals as generators, has similar properties. Indeed, $BP_n$ is $n$-regular and vertex-transitive. In this paper, we show that $BP_n$ has every cycle of length $\ell$ with $8\leq\ell\leq 2^n n!$. The proof given is a constructive one that utilizes the recursive structure of $BP_n$. We also present a complete characterization of all the $8$-cycles in $BP_n$ for $n \geq 2$, which are the smallest cycles embeddable in $BP_n$, by presenting their canonical forms as products of the prefix reversal generators.Comment: Added a reference, clarified some definitions, fixed some typos. 42 pages, 9 figures, 20 pages of appendice

Aspects of the Calendar of the Hebrew Bible and its Theological Significance

The purpose of the thesis is twofold. We endeavor to provide the exegetical base for the understanding of the calendar used by God\u27s4people in the Old Testament. There is, of course, a certain heuristic value to our topic for Old Testament studies at large. Beyond that, we find our study to be unique in its approach. Apart from the confessional position outlined above, we came to the texts without any preconceived notions about the nature, structure, and use of the Hebrew calendar. Conversely, the vast majority of works on the calendar are undertaken to prove a particular point of view. As such, these works may or may not consider evidence that is contrary to their position, and virtually none of these works will consider a position apart from their own hermeneutical stance. Our conclusions will certainly flow from our own hermeneutics, but in presenting the evidence we have sought to be as objective as possible. Rather than seeking to prove a particular calendar’s use in the Old Testament, we have tried to draw from the evidence the calendar that was most probably used. A second purpose of our topic is to discover the practical implications of Israel\u27s calendar. A calendar is a part of life so ingrained in daily affairs that one is not surprised when calendar reckoning does not receive formal treatment in a society\u27s literature. How did this intimate part of Israel\u27s life effect its religious celebrations? Further, was the calendar complementary to Israel\u27s faith, and if so, in what way? That is, did the calendar assist Israel in understanding its place before God in salvation history? Or, was the calendar simply a tool with no salvific significance at all

The area of Freedom, Security and Justice: a New Horizon for European Integration

The entry into force of the Lisbon Treaty has opened up new perspectives in the area of Justice and Home Affairs. The entry into force of the Lisbon Treaty, after a tortuous ratification process, has opened up new vistas, marked by two factors that will affect the Trio Presidency (Spain, Belgium and Hungary) with regard to Justice and Home affairs (JHA). First, the application of the Lisbon Treaty; second, the completion of the Hague Programme will give way to a new project, namely the Stockholm Programme. Another challenge to be tackled by the Trio Presidency will be to ensure the coherence and unity of the internal and external dimensions of the JHA field

Equity Commitment under Uncertainty: A Hierarchical Model of Real Option Entry Mode Choices

We develop a real option hierarchical model of entry mode choice and test predictions using a sample of US companies in Latin America and the Caribbean between 1980 and 2005. Probit results indicate that the choice between a real option non-equity mode and equity commitment is influenced by previous acquisition experience, R&D and advertising intensities, and country risk. The choice of the more flexible real option JV mode over WOEs is positively related to greater firm size and market-to-book ratio in countries with better infrastructure. In contrast, greater marketing intensity and lower country risk encourage WOEs

Equity Commitment under Uncertainty: A Hierarchical Model of Real Option Entry Mode Choices

We develop a real option hierarchical model of entry mode choice and test predictions using a sample of US companies in Latin America and the Caribbean between 1980 and 2005. Probit results indicate that the choice between a real option non-equity mode and equity commitment is influenced by previous acquisition experience, R&D and advertising intensities, and country risk. The choice of the more flexible real option JV mode over WOEs is positively related to greater firm size and market-to-book ratio in countries with better infrastructure. In contrast, greater marketing intensity and lower country risk encourage WOEs

Soil aggregation as affected by application of diverse organic materials

Application of organic materials can amend soil for improved water infiltration and reduced erodibility with effects varying with soil properties and the organic amendment type and rate. The effects of four livestock manures, three municipal biosolids, and one industrial by-product on dry and wet soil aggregate stability were evaluated at six sites in Nebraska. The amendments had similar C/N ratios but the biosolids had relatively high concentrations of lignin and cellulose. Soil organic matter (SOM) ranged from 21 to 65 g kg−1 and soils were silty clay loam, silt loam, or loamy sand. Soil was sampled for the 0- to 0.05-m depth at physiolog- ical maturity of the second corn (Zea mays L.) crop following amendment appli- cation. Aggregation was high with no amendment applied as \u3e95% of the soil was in water stable aggregates (WSA) \u3e 0.053 mm and was not affected by amend- ments with a few exceptions such as an increase in dry aggregate size and WSA 0.25–2.0 mm at one location. Dry aggregate size was much less for the loamy sand than with other soils. With SOM \u3e60 g kg−1 compared with less SOM, there was 42% more WSA \u3e2 mm and 38% less WSA \u3c2 mm diam. It cannot be concluded that organic amendment application will improve aggregation if SOM \u3e20 g kg−1 but larger effects may have occurred with: sampling sooner after amendment application; a 0- to 0.025-m sampling depth; or sampling at several months after harvest for reduced effect of the rhizosphere on aggregation

Construir Europa desde España: los nuevos desafíos de la política europea

Analysis of the political position of Spain to the EU: Europeanism, convergence, foreign policy, constitutionalism, funding cuts. Proposal of a model change