Improved bounds for the number of forests and acyclic orientations in the square lattice

In a recent paper Merino and Welsh (1999) studied several counting problems on the square lattice $L_n$. The authors gave the following bounds for the asymptotics of $f(n)$, the number of forests of $L_n$, and $\alpha(n)$, the number of acyclic orientations of $L_n$: $3.209912 \leq \lim_{n\rightarrow\infty} f(n)^{1/n^2} \leq 3.84161$ and $22/7 \leq \lim_{n\rightarrow\infty} \alpha(n) \leq 3.70925$. In this paper we improve these bounds as follows: $3.64497 \leq \lim_{n\rightarrow\infty} f(n)^{1/n^2} \leq 3.74101$ and $3.41358 \leq \lim_{n\rightarrow\infty} \alpha(n) \leq 3.55449$. We obtain this by developing a method for computing the Tutte polynomial of the square lattice and other related graphs based on transfer matrices

Factors of sums and alternating sums involving binomial coefficients and powers of integers

We study divisibility properties of certain sums and alternating sums involving binomial coefficients and powers of integers. For example, we prove that for all positive integers $n_1,..., n_m$, $n_{m+1}=n_1$, and any nonnegative integer $r$, there holds {align*} \sum_{k=0}^{n_1}\epsilon^k (2k+1)^{2r+1}\prod_{i=1}^{m} {n_i+n_{i+1}+1\choose n_i-k} \equiv 0 \mod (n_1+n_m+1){n_1+n_m\choose n_1}, {align*} and conjecture that for any nonnegative integer $r$ and positive integer $s$ such that $r+s$ is odd, $$\sum_{k=0}^{n}\epsilon ^k (2k+1)^{r}({2n\choose n-k}-{2n\choose n-k-1})^{s} \equiv 0 \mod{{2n\choose n}},$$ where $\epsilon=\pm 1$.Comment: 14 pages, to appear in Int. J. Number Theor

Charge distribution in two-dimensional electrostatics

We examine the stability of ringlike configurations of N charges on a plane interacting through the potential $V(z_1,...,z_N)=\sum_i |z_i|^2-\sum_{i<j} ln|z_i-z_j|^2$. We interpret the equilibrium distributions in terms of a shell model and compare predictions of the model with the results of numerical simulations for systems with up to 100 particles.Comment: LaTe

Holocene Glaciation of the Arrigetch Peaks, Brooks Range, Alaska

Eleven cirque glaciers and associated deposits within the granitic Arrigetch Peaks of the west-central Brooks Range face north, minimizing insolation. Shading by surrounding mountainous terrain decreases insolation on these landforms even more significantly, favoring the formation of glacier-cored moraines. Comparison of glacier photographs taken in 1911, 1962, and 1979 reveals a record of decelerating recession. Geomorphic and lichenometric mapping suggests at least three to possibly eight phases of Holocene glacial expansion. These date between ~5000 and 300 yr B.P., based on the application of a central Brooks Range Rhizocarpon geographicum growth curve

Distribution of Oligonychus (Oligonychus) ununguis (Jacobi) (Acari: Tetranychidae) and predator mite species (Acair: Phytoseiidae) on field-grown Douglas-fir (Pseudotsuga menziesii [Mirb] Franco) Christmas trees

The shake and wash technique (samples placed in a jar with alcohol added and shaken to remove the mites) was effective in removing 100% of the predator mites, and adult spruce spider mites from Douglas-fir foliage and 98% of the spruce spider mite nymphs. Eighty-eight percent of the spruce spider mite eggs was removed. This technique was considered efficient for removal of spruce spider mite and its predators from Douglas-fir foliage. Sodium hypochlorite (0.84%) added to the alcohol did not increase the number of spruce spider mites or phytoseiid mites removed from the foliage. The intracanopy distribution of spruce spider mite and its predator mites was studied on Douglas-fir Christmas trees in the Willamette Valley, Oregon. When overwintering spruce spider mite egg densities populations were low (<5 eggs/19 cm of stem), significantly more eggs were found on the current season's growth. No significant differences were found between top and bottom halves of the tree or between compass directions. Differences between current and previous season's growth were not found when egg densities were high (>40 eggs/19 cm of stem), but significant differences were found between levels for current season's growth with more eggs found in the upper portion of the canopy. Quadri-directional differences did not exist with either low or high mite populations. Sampling tip or basal stem-halves with low overwintering egg populations did not bias population estimates. Heavy spring rainfall appeared to reduce mite populations as has been reported elsewhere by washing them off the tree and causing increased mortality. Spruce spider mite disperse to the current season's growth shortly after budbreak. Population density rapidly increased in late May and then abruptly declined in mid-July

Proof of two conjectures of Z.-W. Sun on congruences for Franel numbers

For all nonnegative integers n, the Franel numbers are defined as $$f_n=\sum_{k=0}^n {n\choose k}^3.$$ We confirm two conjectures of Z.-W. Sun on congruences for Franel numbers: \sum_{k=0}^{n-1}(3k+2)(-1)^k f_k &\equiv 0 \pmod{2n^2}, \sum_{k=0}^{p-1}(3k+2)(-1)^k f_k &\equiv 2p^2 (2^p-1)^2 \pmod{p^5}, where n is a positive integer and p>3 is a prime.Comment: 8 pages, minor changes, to appear in Integral Transforms Spec. Func

Linear Momentum Density in Quasistatic Electromagnetic Systems

We discuss a couple of simple quasistatic electromagnetic systems in which the density of electromagnetic linear momentum can be easily computed. The examples are also used to illustrate how the total electromagnetic linear momentum, which may also be calculated by using the vector potential, can be understood as a consequence of the violation of the action-reaction principle, because a non-null external force is required to maintain constant the mechanical linear momentum. We show how one can avoid the divergence in the interaction linear electromagnetic momentum of a system composed by an idealization often used in textbooks (an infinite straight current) and a point charge.Comment: 22 pages, 5 figures, to appear in Eur. J. Phy

Annotating Relationships between Multiple Mixed-media Digital Objects by Extending Annotea

Annotea provides an annotation protocol to support collaborative Semantic Web-based annotation of digital resources accessible through the Web. It provides a model whereby a user may attach supplementary information to a resource or part of a resource in the form of: either a simple textual comment; a hyperlink to another web page; a local file; or a semantic tag extracted from a formal ontology and controlled vocabulary. Hence, annotations can be used to attach subjective notes, comments, rankings, queries or tags to enable semantic reasoning across web resources. More recently tabbed Browsers and specific annotation tools, allow users to view several resources (e.g., images, video, audio, text, HTML, PDF) simultaneously in order to carry out side-by-side comparisons. In such scenarios, users frequently want to be able to create and annotate a link or relationship between two or more objects or between segments within those objects. For example, a user might want to create a link between a scene in an original film and the corresponding scene in a remake and attach an annotation to that link. Based on past experiences gained from implementing Annotea within different communities in order to enable knowledge capture, this paper describes and compares alternative ways in which the Annotea Schema may be extended for the purpose of annotating links between multiple resources (or segments of resources). It concludes by identifying and recommending an optimum approach which will enhance the power, flexibility and applicability of Annotea in many domains