Orientations, lattice polytopes, and group arrangements II: Modular and integral flow Polynomials of graphs

We study modular and integral flow polynomials of graphs by means of subgroup arrangements and lattice polytopes. We introduce an Eulerian equivalence relation on orientations, flow arrangements, and flow polytopes; and we apply the theory of Ehrhart polynomials to obtain properties of modular and integral flow polynomials. The emphasis is on the geometrical treatment through subgroup arrangements and Ehrhart polynomials. Such viewpoint leads to a reciprocity law on the modular flow polynomial, which gives rise to an interpretation on the values of the modular flow polynomial at negative integers and answers a question by Beck and Zaslavsky.Regal Entertainment Group (Competitive Earmarked Research Grants 600703)Regal Entertainment Group (Competitive Earmarked Research Grants 600506)Regal Entertainment Group (Competitive Earmarked Research Grants 600608

Universal relations in the finite-size correction terms of two-dimensional Ising models

Quite recently, Izmailian and Hu [Phys. Rev. Lett. 86, 5160 (2001)] studied the finite-size correction terms for the free energy per spin and the inverse correlation length of the critical two-dimensional Ising model. They obtained the universal amplitude ratio for the coefficients of two series. In this study we give a simple derivation of this universal relation; we do not use an explicit form of series expansion. Moreover, we show that the Izmailian and Hu's relation is reduced to a simple and exact relation between the free energy and the correlation length. This equation holds at any temperature and has the same form as the finite-size scaling.Comment: 4 pages, RevTeX, to appear in Phys. Rev. E, Rapid Communication

Transfer Matrices and Partition-Function Zeros for Antiferromagnetic Potts Models. V. Further Results for the Square-Lattice Chromatic Polynomial

We derive some new structural results for the transfer matrix of square-lattice Potts models with free and cylindrical boundary conditions. In particular, we obtain explicit closed-form expressions for the dominant (at large |q|) diagonal entry in the transfer matrix, for arbitrary widths m, as the solution of a special one-dimensional polymer model. We also obtain the large-q expansion of the bulk and surface (resp. corner) free energies for the zero-temperature antiferromagnet (= chromatic polynomial) through order q^{-47} (resp. q^{-46}). Finally, we compute chromatic roots for strips of widths 9 <= m <= 12 with free boundary conditions and locate roughly the limiting curves.Comment: 111 pages (LaTeX2e). Includes tex file, three sty files, and 19 Postscript figures. Also included are Mathematica files data_CYL.m and data_FREE.m. Many changes from version 1: new material on series expansions and their analysis, and several proofs of previously conjectured results. Final version to be published in J. Stat. Phy

The Random Quadratic Assignment Problem

Optimal assignment of classes to classrooms \cite{dickey}, design of DNA microarrays \cite{carvalho}, cross species gene analysis \cite{kolar}, creation of hospital layouts cite{elshafei}, and assignment of components to locations on circuit boards \cite{steinberg} are a few of the many problems which have been formulated as a quadratic assignment problem (QAP). Originally formulated in 1957, the QAP is one of the most difficult of all combinatorial optimization problems. Here, we use statistical mechanical methods to study the asymptotic behavior of problems in which the entries of at least one of the two matrices that specify the problem are chosen from a random distribution $P$. Surprisingly, this case has not been studied before using statistical methods despite the fact that the QAP was first proposed over 50 years ago \cite{Koopmans}. We find simple forms for $C_{\rm min}$ and $C_{\rm max}$, the costs of the minimal and maximum solutions respectively. Notable features of our results are the symmetry of the results for $C_{\rm min}$ and $C_{\rm max}$ and the dependence on $P$ only through its mean and standard deviation, independent of the details of $P$. After the asymptotic cost is determined for a given QAP problem, one can straightforwardly calculate the asymptotic cost of a QAP problem specified with a different random distribution $P$

Transfer matrices and partition-function zeros for antiferromagnetic Potts models. VI. Square lattice with special boundary conditions

We study, using transfer-matrix methods, the partition-function zeros of the square-lattice q-state Potts antiferromagnet at zero temperature (= square-lattice chromatic polynomial) for the special boundary conditions that are obtained from an m x n grid with free boundary conditions by adjoining one new vertex adjacent to all the sites in the leftmost column and a second new vertex adjacent to all the sites in the rightmost column. We provide numerical evidence that the partition-function zeros are becoming dense everywhere in the complex q-plane outside the limiting curve B_\infty(sq) for this model with ordinary (e.g. free or cylindrical) boundary conditions. Despite this, the infinite-volume free energy is perfectly analytic in this region.Comment: 114 pages (LaTeX2e). Includes tex file, three sty files, and 23 Postscript figures. Also included are Mathematica files data_Eq.m, data_Neq.m,and data_Diff.m. Many changes from version 1, including several proofs of previously conjectured results. Final version to be published in J. Stat. Phy

Attack or consumption of Epomophorus (Chiroptera) by Paraxerus (Rodentia) and Papio (Primates) in Tanzania

William T. Stanley, and Leah Collet

Channel Changes Choice: An Empirical Study about Omnichannel Demand Sensitivity to Fulfillment Lead Time

We examine a large transaction-level data set of an Italian omnichannel furniture retailer to study channel-specific effects of fulfillment lead time on demand. This omnichannel retailer sells the same products and has the same product fulfillment across three channels – showroom, online and catalog. A showroom channel carries no inventory but allows customers to touch and feel the products. An online channel provides a website for consumers to browse and order the products. A catalog channel sends a product catalog to all the households in Italy for them to place an order over the phone. We find that the showroom channel makes consumers less sensitive to fulfillment lead time than both online and catalog channels. In particular, a 10% increase in lead time (1.84 days from the sample mean of 18.35 days) causes a 0.85% reduction in the sales per order (~ EUR 7.6 from the sample mean of EUR 889.94) at the showroom, less than the reduction of 1.14% and 1.23% in the online and the catalog channels, respectively. This finding contradicts the common practical and theoretical assumption about homogeneous lead time sensitivity across channels. In addition, we find that niche products and experience goods accentuate the difference of lead time sensitivity between showroom and non-physical channels. We further develop a stylized model to study the implications of our empirical findings for the design of an omnichannel retailer’s facility network. Given our finding that shows the showroom wait sensitivity is smaller than online wait sensitivity, retailers should build fewer but larger showrooms than the homogeneous wait sensitivity suggests

Channel Changes Choice: An Empirical Study about Omnichannel Demand Sensitivity to Fulfillment Lead Time

We examine a large transaction-level data set of an Italian omnichannel furniture retailer to study channel-specific effects of fulfillment lead time on demand. This omnichannel retailer sells the same products and has the same product fulfillment across three channels – showroom, online and catalog. A showroom channel carries no inventory but allows customers to touch and feel the products. An online channel provides a website for consumers to browse and order the products. A catalog channel sends a product catalog to all the households in Italy for them to place an order over the phone. We find that the showroom channel makes consumers less sensitive to fulfillment lead time than both online and catalog channels. In particular, a 10% increase in lead time (1.84 days from the sample mean of 18.35 days) causes a 0.85% reduction in the sales per order (~ EUR 7.6 from the sample mean of EUR 889.94) at the showroom, less than the reduction of 1.14% and 1.23% in the online and the catalog channels, respectively. This finding contradicts the common practical and theoretical assumption about homogeneous lead time sensitivity across channels. In addition, we find that niche products and experience goods accentuate the difference of lead time sensitivity between showroom and non-physical channels. We further develop a stylized model to study the implications of our empirical findings for the design of an omnichannel retailer’s facility network. Given our finding that shows the showroom wait sensitivity is smaller than online wait sensitivity, retailers should build fewer but larger showrooms than the homogeneous wait sensitivity suggests

Additional notes on the small mammals of Malundwe Mountain, Mikumi National Park, Tanzania

Following a preliminary survey in 2004, a more intensive inventory of the small mammals of Mount Malundwe, Mikumi National Park, Tanzania, was conducted in July, 2005. Two species of insectivore, one species of bat, and five species of rodent were documented in the montane forest of this mountain. There were significantly more Praomys captured in 2005 than in 2004, but no new species of shrew or rodent were documented. One species of bat was recorded for the first time on Malundwe. The results of this study suggest that several surveys over an extended period of time may be necessary to comprehensively document the diversity and abundance of a small mammal community in an Afromontane forest.William T. Stanley, Guy Norton, Philip M. Kihaule, Leah Collett, and Kate McQuai