At the time wolves were federally protected in the mid-1970’s, Minnesota contained the only known reproducing wolf population in the lower 48 states, except for that on Isle Royale. Over the years, much attention has been focused on studying and monitoring Minnesota’s wolves. Research efforts began in the mid-1930’s (Olson 1938) and with few lapses continue to this day. Efforts to delineate wol...

In this paper we give recursive formulas for the number of colorful tilings of small rectangular arrays. We enumerate the tilings of a 2 × n board with painted squares, dominoes, and I-trominoes. We also provide a recursion formula for the number of tilings of a 3 × n board with colorful squares and dominoes. Finally, we describe a general method for calculating the number of colorful tilings o...

Ever since the advent of the first LASER (acronym for Light Amplification by Stimulation Emission of Radiation) in 1960, there has been a steady increase in the application of lasers. Applications have kept on becoming more and more diverse as the capability of the lasers have increased. In this chapter we will enumerate and classify many of the applications of lasers and then go on to discuss ...

For each s ∈ N define the constant s with the following properties: if an entire function g(z) of type t (g)< s satisfies g (z) ∈ Z for = 0, 1, . . . , s − 1 and z= 0, 1, 2, . . . , then g is a polynomial; conversely, for any > 0 there exists an entire transcendental function g(z) satisfying the display conditin and t (g)< s + . The result 1 = log 2 is known due to Hardy and Pólya. We provide t...

We describe the couplings of six-dimensional supergravity, which contain a self-dual tensor multiplet, to n T anti-self-dual tensor matter multiplets, n V vector multiplets and n H hypermultiplets. The scalar fields of the tensor multiplets form a coset SO(nT , 1)/SO(nT ), while the scalars in the hypermultiplets form quaternionic Kähler symmetric spaces, the generic example being Sp(nH , 1)/Sp...

We prove that for fixed n there are only finitely many embeddings of Qfactorial toric varieties X into P that are induced by a complete linear system. The proof is based on a combinatorial result that implies that for fixed nonnegative integers d and n, there are only finitely many smooth d-polytopes with n lattice points. We also enumerate all smooth 3-polytopes with ≤ 12 lattice points.

This paper contains an extensive combinatorial analysis of the single-peaked domain restriction and investigates the likelihood that an election is single-peaked. We provide a very general upper bound result for domain restrictions that can be defined by certain forbidden configurations. This upper bound implies that many domain restrictions (including the single-peaked restriction) are very un...

We present a method, based on functional equations, to enumerate paths on the square lattice that avoid a horizontal half-line. The corresponding generating functions are algebraic, and sometimes remarkably simple: for instance, the number of paths of length 2n + 1 going from (0; 0) to (1; 0) and avoiding the nonpositive horizontal axis (except at their starting point) is C2n+1 , the (2n + 1)th...

Let a finite presentation be given for an associative, in general non-commulative algebra E, with identity, over a field. We study an algorithm for the construction, from this presentation, of linear, i.e, matrix, representations of this algebra. A set of vector constraints which is given as part of the initial data determines which particular representation of E is produced. This construction ...

We enumerate weighted graphs with a certain upper bound condition. We also compute the generating function of the numbers of these graphs, and prove that it is a rational function. In particular, we show that if the given graph is a bipartite graph, then its generating function is of the form p(x) (1−x)m+1 , where m is the number of vertices of the graph and p(x) is a polynomial of degree at mo...