Search

Consider edge colorings of digraphs where edges $v_1 v_2$ and $v_2 v_3$ have different colors. This coloring induces a vertex coloring by sets of edge colors, in which edge $v_1 v_2$ in the graph implies that the set color of $v_1$ contains an element not in the set color of $v_2$, and conversely. We generalize to colorings of $k$(vertex)-walks, defined so two walks have different colors if one is the prefix $c_1$ and the other is the suffix $c_2$ of a common $(k+1)$-walk. Further, the colors can belong to a poset $P$ where $c_1$, $c_2$ must satisfy $c_1 \not\leq c_2$. This set construction generalizes the lower order ideal in $P$ from a set of $k$-walk colors; these order ideals are partially ordered by containment. We conclude that a $P$ coloring of $k$-walks exists iff there is a vertex coloring by $A$ iterated $k-1$ times on $P$, where Birkhoff's $A$ maps a poset to its poset of lower order ideals. Thus the directed chromatic index problem is generalized and reduced to poset coloring of vertices. This work uses ideas, results and motivations due to Cole and Vishkin on deterministic coin tossing and Becker and Simon on vertex covers for subsets of $(n-2)$-cubes., Comment: written in 1994, 17 pages

In Parts I-III we showed that the number of ways to place $q$ nonattacking queens or similar chess pieces on an $n\times n$ chessboard is a quasipolynomial function of $n$ whose coefficients are essentially polynomials in $q$. In this part we focus on the periods of those quasipolynomials. We calculate denominators of vertices of the inside-out polytope, since the period is bounded by, and conjecturally equal to, their least common denominator. We find an exact formula for that denominator of every piece with one move and of two-move pieces having a horizontal move. For pieces with three or more moves, we produce geometrical constructions related to the Fibonacci numbers that show the denominator grows at least exponentially with $q$., Comment: 21 pp., 11 figures. Revised version of part of arXiv:1609.00853, which will be replaced with a different part of the split

Parts I-IV showed that the number of ways to place $q$ nonattacking queens or similar chess pieces on an $n\times n$ chessboard is a quasipolynomial function of $n$ whose coefficients are essentially polynomials in $q$. For partial queens, which have a subset of the queen's moves, we proved complete formulas for these counting quasipolynomials for small numbers of pieces and other formulas for high-order coefficients of the general counting quasipolynomials. We found some upper and lower bounds for the periods of those quasipolynomials by calculating explicit denominators of vertices of the inside-out polytope. Here we discover more about the counting quasipolynomials for partial queens, both familiar and strange, and the nightrider and its subpieces, and we compare our results to the empirical formulas found by Kot\v{e}\v{s}ovec. We prove some of Kot\v{e}\v{s}ovec's formulas and conjectures about the quasipolynomials and their high-order coefficients, and in some instances go beyond them., Comment: 39 pp., many figures; ancillary Mathematica programs that were used to compute examples. v2: update terminology and reference URL's. v3: 23 pp. This is expanded second half of v2; first half expanded became 1807.04741. v4: 23 pp., minor updates per published version. v5: small corrections

The number of ways to place $q$ nonattacking queens, bishops, or similar chess pieces on an $n\times n$ square chessboard is essentially a quasipolynomial function of $n$ (by Part I of this series). The period of the quasipolynomial is difficult to settle. Here we prove that the empirically observed period 2 for three to ten bishops is the exact period for every number of bishops greater than 2. The proof depends on signed graphs and the Ehrhart theory of inside-out polytopes., Comment: 15 pp.; 12 pp. without white space. v2: Updated citations, rearranged authors. 12 pp. v3: Updated citations. Rm unnec special commands. v4: 13 pp. Rev title. Corrected error; rev abstract; minor rev

To enable future human exploration missions, the lunar surface will serve as a crucial training ground and technology demonstration test site where NASA will prepare for future human missions to Mars and other destinations. Key enablers in this exploration are rover systems intended to operate in both the lunar and Mars surface environments. This study focused primarily on the Mars surface environment because, compared to the near-continuous illumination near the lunar poles, the day/night cycles and the reduced solar illumination on Mars make it much more difficult to use solar power. However, solar power is not the only power generation technology available. Top-level energy studies have shown that a radioisotope power system (RPS) has promise for supplying both power and thermal energy for crewed rovers on both the Moon and Mars. This design study investigates how an RPS could potentially meet the power and thermal needs of a pressurized rover with applications for both destinations. The design focus is on what service an RPS can provide and how it would be integrated into a rover (power and thermal interfaces, placement, radiation shielding, fairing installation on the pad, etc.).

We apply to the $n\times n$ chessboard the counting theory from Part I for nonattacking placements of chess pieces with unbounded straight-line moves, such as the queen. Part I showed that the number of ways to place $q$ identical nonattacking pieces is given by a quasipolynomial function of $n$ of degree $2q$, whose coefficients are (essentially) polynomials in $q$ that depend cyclically on $n$. Here we study the periods of the quasipolynomial and its coefficients, which are bounded by functions, not well understood, of the piece's move directions, and we develop exact formulas for the very highest coefficients. The coefficients of the three highest powers of $n$ do not vary with $n$. On the other hand, we present simple pieces for which the fourth coefficient varies periodically. We develop detailed properties of counting quasipolynomials that will be applied in sequels to partial queens, whose moves are subsets of those of the queen, and the nightrider, whose moves are extended knight's moves. We conclude with the first, though strange, formula for the classical $n$-Queens Problem and with several conjectures and open problems., Comment: 23 pp., 1 figure, submitted. This = second half of 1303.1879v1 with great improvements. V2 has a new proposition, better definitions, and corrected conjectures. V3 has results et al. renumbered to correspond with published version, and expands dictionary's cryptic abbreviations

We apply our geometrical theory for counting placements of $q$ nonattacking on an $n\times n$ chessboard, from Parts~I and II, to partial queens: that is, chess pieces with any combination of horizontal, vertical, and $45^\circ$-diagonal moves. Parts~I and II showed that for any rider (a piece with moves of unlimited length) the answer will be a quasipolynomial function of $n$ in which the coefficients are essentially polynomials in $q$. Those general results gave the three highest-order coefficients of the counting quasipolynomial and formulas for counting placements of two nonattacking pieces and the combinatorially distinct types of such placements. By contrast, the unified framework we present here for partial queens allows us to explicitly compute the four highest-order coefficients of the counting quasipolynomial, show that the five highest-order coefficients are constant (independent of $n$), and find the period of the next coefficient (which depends upon the exact set of moves). Furthermore, for three nonattacking partial queens we are able to prove formulas for the total number of nonattacking placements and for the number of their combinatorially distinct types. The method of proof, as in the previous parts, is by detailed analysis of the lattice of subspaces of an inside-out polytope., Comment: 25 pp., 5 figures. v2: Rearranged content. Better tables. Updated citations. Reordered authors. v3: Updated citations and terminology. v4: Significant improvements

By means of the Ehrhart theory of inside-out polytopes we establish a general counting theory for nonattacking placements of chess pieces with unbounded straight-line moves, such as the queen, on a polygonal convex board. The number of ways to place $q$ identical nonattacking pieces on a board of variable size $n$ but fixed shape is given by a quasipolynomial function of $n$, of degree $2q$, whose coefficients are polynomials in $q$. The number of combinatorially distinct types of nonattacking configuration is the evaluation of our quasipolynomial at $n=-1$. The quasipolynomial has an exact formula that depends on a matroid of weighted graphs, which is in turn determined by incidence properties of lines in the real affine plane. We study the highest-degree coefficients and also the period of the quasipolynomial, which is needed if the quasipolynomial is to be interpolated from data, and which is bounded by some function, not well understood, of the board and the piece's move directions. In subsequent parts we specialize to the square board and then to subsets of the queen's moves, and we prove exact formulas (most but not all already known empirically) for small numbers of queens, bishops, and nightriders. Each part concludes with open questions, both specialized and broad., Comment: 28 pp., no figures, submitted. v2 = first half of v1 with much stronger results

The function that counts the number of ways to place nonattacking identical chess or fairy chess pieces in a rectangular strip of fixed height and variable width, as a function of the width, is a piecewise polynomial which is eventually a polynomial and whose behavior can be described in some detail. We deduce this by converting the problem to one of counting lattice points outside an affinographic hyperplane arrangement, which Forge and Zaslavsky solved by means of weighted integral gain graphs. We extend their work by developing both generating functions and a detailed analysis of deletion and contraction for weighted integral gain graphs. For chess pieces we find the asymptotic probability that a random configuration is nonattacking, and we obtain exact counts of nonattacking configurations of small numbers of queens, bishops, knights, and nightriders., Comment: 21 pages, 3 figures, preprint of published version copyright Springer Basel AG 2011, Published online February 15, 2011, submitted March 15, 2007

This document reports on findings that describe the characteristics of delinquent males and the resources available to them for social skills building in the Washington, D.C. area. The study looked at the types of delinquent behavior found among boys living in the three most violent neighborhoods in Washington, and the role played by families, schools, churches, and youth-serving organizations in helping them. It also identified where and when violence involving children was taking. Finding few differences between violent juvenile offenders in D.C. and those in other cities, the study concluded that its findings could be used as a catalyst for new initiatives to reduce juvenile violence in the District. Since the research was completed the District has reorganized its funding delivery system for juvenile justice and delinquency prevention efforts. As a result, partnerships have been formed and enhanced resources are available for youth-serving agencies to develop new policies, practices, and service delivery programs. (Contains 3 tables, 4 figures, and 35 references.) (Author/JDM)

The Tutte equations are ported (or set-pointed) when the equations F(N) = g_e F(N/e) + r_e F(N\e) are omitted for elements e in a distinguished set called ports. Solutions F can distinguish different orientations of the same matroid. A ported extensor with ground set is a decomposible element in the exterior algebra over a vector space with a given basis, called the ground set, containing a distinguished subset called ports. These can represent ported matroids and have analogous dualization, deletion and contraction operations. A ported extensor function is defined using dualization, port element renaming, exterior multiplication, and contraction of non-ports. We prove that this function satisfies a sign-corrected variant of the Tutte equations over exterior algebra. For non-ported unimodular, i.e., regular matroids, our function reduces to the basis generating function and for graphs the Laplacian (or Kirchhoff) determinant. In general, the function value, as an extensor, signifies the space of solutions to Kirchhoff's and Ohm's electricity equations after projection to the variables associated to the ports. Combinatorial interpretation of various determinants (the Plucker coordinates) generalize the matrix tree theorem and forest enumeration expressions for electrical resistance. The corank-nullity polynomial, basis expansions with activities, and a geometric lattice expansion generalize to ported Tutte functions of oriented matroids. The ported Tutte functions are parametrized, which raises the problem of how to generalize known characterizations of parameterized non-ported Tutte functions., Comment: 46 pages, 3 figures corrections, additions and improved discussions of references. Editorial and spelling corrections; more acknowledgements

This report is designed to help law enforcement administrators and officers understand and institute a strategy to help prevent violence through community oriented policing services carried out in collaboration with youth-serving organizations. Descriptions of programs are based on a study that involved a survey of 579 affiliates of 7 national youth-serving organizations: (1) Boys and Girls Clubs of America; (2) Boy Scouts of America; (3) Girls Incorporated; (4) Girl Scouts of America; (5) National Association of Police Athletic Leagues; (6) National 4-H Council and USDA 4-H; and (7) YMCA of the USA. Findings suggest that popular approaches such as boot camps or curfews provide only temporary supervision. Effective prevention approaches are integral to national youth organizations. Partnerships between police and youth-serving organizations that are effective can take many forms. A review of some exemplary programs shows that they have some features in common: they are based on needs assessment and address multiple factors; and they use existing organizations and services, rather than developing new ones. Their experience suggests courses of action for police administrators, directors of youth agencies, and community leaders. An appendix lists some resource organizations. (SLD)

Structural and electronic properties of amorphous and single-phase polycrystalline films of gamma- and kappa-In2Se3 have been measured. The stable gamma phase nucleates homogeneously in the film bulk and has a high resistivity, while the metastable kappa phase nucleates at the film surface and has a moderate resistivity. The microstructures of hot-deposited and post-annealed cold-deposited gamma films are quite different but the electronic properties are similar. The increase in the resistivity of amorphous In2Se3 films upon annealing is interpreted in terms of the replacement of In-In bonds with In-Se bonds during crystallization. Great care must be taken in the preparation of In2Se3 films for electrical measurements as the presence of excess chalcogen or surface oxidation may greatly affect the film properties., Comment: 23 pages and 12 figures