The pfaffian is a classical tool which can be regarded as a generalization of the determinant. The hyperpfaffian, which was introduced by Barvinok, generalizes the pfaffian to higher dimension. This was further developed by Luque, Thibon and Abdesselam. There are several non-equivalent definitions for the hyperpfaffian, which are discussed in the introduction of this thesis. Following this we examine the extension of the Matrix-Tree theorem to the Hyperpfaffian-Cactus theorem by Abdesselam, proving it without the use of the Grassman-Berezin Calculus and with the new terminology of the non-uniform hyperpfaffian. Next we look at the extension of pfaffian orientations for counting matchings on graphs to hyperpfaffian orientations for counting matchings on hypergraphs. Finally pfaffian rings and ideal s are extended to hyperpfaffian rings and ideals, but we show that under reason able assumptions the algebra with straightening law structure of these rings cannot be extended.

The pfaffian is a classical tool which can be regarded as a generalization of the determinant. The hyperpfaffian, which was introduced by Barvinok, generalizes the pfaffian to higher dimension. This was further developed by Luque, Thibon and Abdesselam. There are several non-equivalent definitions for the hyperpfaffian, which are discussed in the introduction of this thesis. Following this we examine the extension of the Matrix-Tree theorem to the Hyperpfaffian-Cactus theorem by Abdesselam, proving it without the use of the Grassman-Berezin Calculus and with the new terminology of the non-uniform hyperpfaffian. Next we look at the extension of pfaffian orientations for counting matchings on graphs to hyperpfaffian orientations for counting matchings on hypergraphs. Finally pfaffian rings and ideal s are extended to hyperpfaffian rings and ideals, but we show that under reason able assumptions the algebra with straightening law structure of these rings cannot be extended.

Proefschrift--Leyden. / On t.p., "S̲" is in Gothic script. "Stellingen": [3] p. inserted. Bibliography: p. [55].

Thesis (doctoral)--Universität Greifswald, 1914. / Vita.

The first result of this thesis is a generation theorem for bricks. A brick is a 3-connected graph such that the graph obtained from it by deleting any two distinct vertices has a perfect matching. The importance of bricks stems from the fact that they are building blocks of a decomposition procedure of Kotzig, and Lovasz and Plummer. We prove that every brick except for the Petersen graph can be generated from K_4 or the prism by repeatedly applying certain operations in such a way that all the intermediate graphs are bricks. We use this theorem to prove an exact upper bound on the number of edges in a minimal brick with given number of vertices and to prove that every minimal brick has at least three vertices of degree three. The second half of the thesis is devoted to an investigation of graphs that admit Pfaffian orientations. We prove that a graph admits a Pfaffian orientation if and only if it can be drawn in the plane in such a way that every perfect matching crosses itself even number of times. Using similar techniques, we give a new proof of a theorem of Kleitman on the parity of crossings and develop a new approach to Turan's problem of estimating crossing number of complete bipartite graphs. We further extend our methods to study k-Pfaffian graphs and generalize a theorem by Gallucio, Loebl and Tessler. Finally, we relate Pfaffian orientations and signs of edge-colorings and prove a conjecture of Goddyn that every k-edge-colorable k-regular Pfaffian graph is k-list-edge-colorable. This generalizes a theorem of Ellingham and Goddyn for planar graphs.

The first result of this thesis is a partial result in the direction of Steinberg's Conjecture. Steinberg's Conjecture states that any planar graph without cycles of length four or five is three colorable. Borodin, Glebov, Montassier, and Raspaud showed that planar graphs without cycles of length four, five, or seven are three colorable and Borodin and Glebov showed that planar graphs without five cycles or triangles at distance at most two apart are three colorable. We prove a statement that implies the first of these theorems and is incomparable with the second: that any planar graph with no cycles of length four through six or cycles of length seven with incident triangles distance exactly two apart are three colorable. The third and fourth chapters of this thesis are concerned with the study of Pfaffian orientations. A theorem proved by William McCuaig and, independently, Neil Robertson, Paul Seymour, and Robin Thomas provides a good characterization for whether or not a bipartite graph has a Pfaffian orientation as well as a polynomial time algorithm for that problem. We reprove this characterization and provide a new algorithm for this problem. In Chapter 3, we generalize a preliminary result needed to reprove this theorem. Specifically, we show that any internally 4-connected, non-planar bipartite graph contains a subdivision of K3,3 in which each path has odd length. In Chapter 4, we make use of this result to provide a much shorter proof using elementary methods of this characterization. In the fourth and fifth chapters we investigate flat embeddings. A piecewise-linear embedding of a graph in 3-space is flat if every cycle of the graph bounds a disk disjoint from the rest of the graph. We provide a structural theorem for flat embeddings that indicates how to build them from small pieces in Chapter 5. In Chapter 6, we present a class of flat graphs that are highly non-planar in the sense that, for any fixed k, there are an infinite number of members of the class such that deleting k vertices leaves the graph non-planar.

The aim of this work is to use the Pfaffian technique, along with the Hirota bilinear method to construct different classes of exact solutions to various of generalized integrable nonlinear partial differential equations. Solitons are among the most beneficial solutions for science and technology, from ocean waves to transmission of information through optical fibers or energy transport along protein molecules. The existence of multi-solitons, especially three-soliton solutions, is essential for information technology: it makes possible undisturbed simultaneous propagation of many pulses in both directions. The derivation and solutions of integrable nonlinear partial differential equations in two spatial dimensions have been the holy grail in the field of nonlinear science since the late 1960s. The prestigious Korteweg-de Vries (KdV) and nonlinear Schrödinger (NLS) equations, as well as the ,Kadomtsev-Petviashvili (KP) and Davey-Stewartson (DS) equations, are prototypical examples of integrable nonlinear partial differential equations in (1+1) and (2+1) dimensions, respectively. Do there exist Pfaffian and soliton solutions to generalized integrable nonlinear partial differential equations in (3+1) dimensions? In this dissertation, I obtained a set of explicit exact Wronskian, Grammian, Pfaffian and N-soliton solutions to the (3+1)-dimensional generalized integrable nonlinear partial differential equations, including a generalized KP equation, a generalized B-type KP equation, a generalized modified B-type KP equation, soliton equations of Jimbo-Miwa type, the nonlinear Ma-Fan equation, and the Jimbo-Miwa equation. A set of sufficient conditions consisting of systems of linear partial differential equations involving free parameters and continuous functions is generated to guarantee that the Wronskian determinant or the Pfaffian solves these generalized equations. On the other hand, as part of this dissertation, bilinear Bäcklund transformations are formally derived for the (3+1)-dimensional generalized integrable nonlinear partial differential equations: a generalized B-type KP equation, the nonlinear Ma-Fan equation, and the Jimbo-Miwa equation. As an application of the obtained Bäcklund transformations, a few classes of traveling wave solutions, rational solutions and Pfaffian solutions to the corresponding equations are explicitly computed. Also, as part of this dissertation, I would like to apply the Pfaffianization mechanism of Hirota and Ohta to extend the (3+1)-dimensional variable-coefficient soliton equation of Jimbo-Miwa type to coupled systems of nonlinear soliton equations, called Pfaffianized systems. Examples of the Wronskian, Grammian, Pfaffian and soliton solutions are explicitly computed. The numerical simulations of the obtained solutions are illustrated and plotted for different parameters involved in the solutions.

This paper considers the statistical mechanics occupation of the edge of a single helix of DNA by simple polymers. Using Fisher's exact closed form solution for dimers on a two-dimensional lattice, a one-dimensional lattice is created mathematically that is occupied by dimers, monomers, and holes. The free energy, entropy, average occupation, and total charge on the lattice are found through the usual statistical methods. The results demonstrate the charge inversion required for a DNA helix to undergo DNA condensation.

A theorem of Edmonds characterizes when a pair of matroids has a common basis. Enumerating the common bases of a pair of matroid is a much harder problem, and includes the #P-complete problem of counting the number of perfect matchings in a bipartite graph. We focus on the problem of counting the common bases in pairs of regular matroids, and describe a class called <i>Pfaffian matroid pairs</i> for which this enumeration problem can be solved. We prove that when a pair of regular matroids is non-Pfaffian, there is a set of common bases which certifies this, and that the number of bases in the certificate is linear in the size of the ground set of the matroids. When both matroids in a pair are series-parallel, we prove that determining if the pair is Pfaffian is equivalent to finding an edge signing in an associated graph, and in the case that the pair is non-Pfaffian, we obtain a characterization of this associated graph. Pfaffian bipartite graphs are a class of graphs for which the number of perfect matchings can be determined; we show that the class of series-parallel Pfaffian matroid pairs is an extension of the class of Pfaffian bipartite graphs. Edmonds proved that the polytope generated by the common bases of a pair of matroids is equal to the intersection of the polytopes generated by the bases for each matroid in the pair. We consider when a similar property holds for the binary space, and give an excluded minor characterization of when the binary space generated by the common bases of two matroids can not be determined from the binary spaces for the individual matroids. As a result towards a description of the lattice of common bases for a pair of matroids, we show that the lattices for the individual matroids determine when all common bases of a pair of matroids intersect a subset of the ground set with fixed cardinality.

A theorem of Edmonds characterizes when a pair of matroids has a common basis. Enumerating the common bases of a pair of matroid is a much harder problem, and includes the #P-complete problem of counting the number of perfect matchings in a bipartite graph. We focus on the problem of counting the common bases in pairs of regular matroids, and describe a class called <i>Pfaffian matroid pairs</i> for which this enumeration problem can be solved. We prove that when a pair of regular matroids is non-Pfaffian, there is a set of common bases which certifies this, and that the number of bases in the certificate is linear in the size of the ground set of the matroids. When both matroids in a pair are series-parallel, we prove that determining if the pair is Pfaffian is equivalent to finding an edge signing in an associated graph, and in the case that the pair is non-Pfaffian, we obtain a characterization of this associated graph. Pfaffian bipartite graphs are a class of graphs for which the number of perfect matchings can be determined; we show that the class of series-parallel Pfaffian matroid pairs is an extension of the class of Pfaffian bipartite graphs. Edmonds proved that the polytope generated by the common bases of a pair of matroids is equal to the intersection of the polytopes generated by the bases for each matroid in the pair. We consider when a similar property holds for the binary space, and give an excluded minor characterization of when the binary space generated by the common bases of two matroids can not be determined from the binary spaces for the individual matroids. As a result towards a description of the lattice of common bases for a pair of matroids, we show that the lattices for the individual matroids determine when all common bases of a pair of matroids intersect a subset of the ground set with fixed cardinality.