Department of Mathematics - Research Reports Archive (1996-2008)

Permanent URI for this collectionhttps://hdl.handle.net/2292/4963

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LEMNISCATES AND THE SPECTRUM OF THE PERTURBED SHIFT
(Department of Mathematics, The University of Auckland, New Zealand, 2001-05) Oleinik, V.L.; Kalupin, A.P.
The spectrum of the perturbed shift operator $T$, $T: f(n)to f(n+1)+ a(n)f(n)$, in $ell^(bf Z)$ is considered for $a(n)$ taking a finite set of values. It is proven that if all values of the function $a(n)$ have uniform frequencies on $bf Z$ then the essential part of the spectrum is continuous and fills a lemniscate.
Halin's Theorem for Cubic Graphs on an Annulus
(Department of Mathematics, The University of Auckland, New Zealand, 2001-06) Archdeacon, Dan; Bonnington, C. Paul; Siran, Jozef
Halin's Theorem characterizes those locally-finite, infinite graphs that embed in the plane without accumulation points by giving a set of six topologically excluded subgraphs. We prove the analogous theorem for cubic graphs that embed in an annulus without accumulation points, finding the complete set of 29 excluded subgraphs.
Trading crossings for handles and crosscaps
(Department of Mathematics, The University of Auckland, New Zealand, 2000-11) Archdeacon, Dan; Bonnington, C. Paul; Siran, Jozef
Let $c_k = cr_k(G)$ denote the minimum number of edge crossings when a graph $G$ is drawn on an orientable surface of genus $k$. The (orientable) {em crossing sequence} $c_0,c_1,c_2,dots$ encodes the trade-off between adding handles and decreasing crossings. We focus on sequences of the type $c_0 > c_1 > c_2 = 0$; equivalently, we study the planar and toroidal crossing number of doubly-toroidal graphs. For every $epsilon > 0$ we construct graphs whose orientable crossing sequence satisfies $c_1/c_0 > 5/6-epsilon$. In other words, we construct graphs where the addition of one handle can save roughly 1/6th of the crossings, but the addition of a second handle can save 5 times more crossings. We similarly define the {em non-orientable crossing sequence} $tilde c_0, tilde c_1, tilde c_2,dots$ for drawings on non-orientable surfaces. We show that for every $tilde c_0 > tilde c_1 > 0$ there exists a graph with non-orientable crossing sequence $tilde c_0, tilde c_1, 0$. We conjecture that every strictly-decreasing sequence of non-negative integers can be both an orientable crossing sequence and a non-orientable crossing sequence (with different graphs).
The Journey of the Four Colour Theorem Through Time
(Department of Mathematics, The University of Auckland, New Zealand, 2001-06) Calude, Andreea S.
This is a historical survey of the Four Colour Theorem and a discussion of the philosophical implications of its proof. The problem, first stated as far back as 1850s, still causes controversy today. Its computer-aided proof has forced mathematicians to question the notions of proofs and mathematical truth.
Metrisability of Manifolds in Terms of Function Spaces
(Department of Mathematics, The University of Auckland, New Zealand, 2001-07) Gauld, David; Mynard, Frederic
We present conditions on the space of continuous real-valued functions on a topological manifold, either with the compact-open or pointwise convergence topology, which are equivalent to metrisability of the manifold. In addition we display some covering and related properties which are also equivalent to metrisability for a manifold.
Long initial value test problems from simulations of the Solar System
(Department of Mathematics, The University of Auckland, New Zealand, 2001-07) Sharp, P.W.
N-body simulations of the Solar System are a rich source of initial value problems for ordinary differential equations. The problems range from the simulation of two bodies requiring a few milliseconds of CPU time to large simulations of a hundred thousand bodies or more requiring CPU time counted in months. We describe five N-body models and use them with long intervals of integration to compare the integrators DIVA, STEP, RKNINT and RKSUITE.
The error growth of some symplectic explicit Runge-Kutta Nystrom methods on long N-body simulations
(Department of Mathematics, The University of Auckland, New Zealand, 2001-08) Sharp, P.W.; Vaillancourt, R.
At one extreme, the global error for symplectic explicit Runge-Kutta Nystrom (SERKN) methods consists entirely of truncation error and grows as t. At the other extreme, the global error consists entirely of random round-off error and grows stochastically as t^1.5. We use numerical testing to investigate how the global error grows for stepsizes between these two extremes. The testing is of representative SERKN methods of orders four to seven on three long N-body simulations of the Solar System. The work also provides an opportunity to introduce two new test problems for symplectic methods and to present comparisons of the efficiency of SERKN methods.
Obstructions for Embedding Cubic Graphs on the Spindle Surface
(Department of Mathematics, The University of Auckland, New Zealand, 2001-09) Archdeacon, Dan; Bonnington, C. Paul
The {em spindle surface} $S$ is the pinched surface formed by identifying two points on the sphere. In this paper we examine cubic graphs that minimally do not embed on the spindle surface. We give the complete list of 21 cubic graphs that form the topological obstruction set in the cubic order for graphs that embed on $S$. A graph $G$ is {em nearly-planar} if there exists an edge $e$ such that $G - e $ is planar. All planar graphs are nearly-planar. A cubic obstruction for near-planarity is the same as an obstruction for embedding on the spindle surface. Hence we also give the topological obstruction set for cubic nearly-planar graphs.
Graphs Embedded in the Plane with Finitely Many Accumulation Points
(Department of Mathematics, The University of Auckland, New Zealand, 2001-08) Bonnington, C. Paul; Richter, R. Bruce
Halin's Theorem characterizes those infinite connected graphs that have an embedding in the plane with no accumulation points, by exhibiting the list of excluded subgraphs. We generalize this by obtaining a similar characterization of which infinite connected graphs have an embedding in the plane (and other surfaces) with at most $k$ accumulation points. Thomassen [7] provided a different characterization of those infinite connected graphs that have an embedding in the plane with no accumulation points as those for which the ${bf Z}_2$-vector space generated by the cycles has a basis for which every edge is in at most two members. Adopting the definition that the cycle space is the set of all edge-sets of subgraphs in which every vertex has even degree (and allowing restricted infinite sums), we prove a general analogue of Thomassen's result, obtaining a cycle space characterization of a graph having an embedding in the sphere with $k$ accumulation points.
On the orientable genus of the cartesian product of a complete regular tripartite graph with a even cycle
(Department of Mathematics, The University of Auckland, New Zealand, 2001-09) Bonnington, C. Paul; Pisanski, Tomaz
We apply the technique of patchwork embeddings to find orientable genus embeddings of the Cartesian product of a complete regular tripartite graph with a even cycle. In particular, the orientable genus of $kc$ is determined for $m ge 1$ and for all $n ge 3$ and $n = 1$. For $n= 2 both lower and upper bounds are given. we see that the resulting embeddings may have a mixture of triangular and quadrilateral faces, in contrast to previous applications of patchwork method.
Arnol'd's analytic perturbation for the polar equation
(Department of Mathematics, The University of Auckland, New Zealand, 2006-09) Dillon, Samuel
Based on the paper of Arnol'd [1] on the problems of Mathieu type, we suggest a modified Feynmann diagram technique to assist in solving problems of the form [see pdf for equation]. We then use this method to calculate eigenvalues and eigenfunctions to the polar problem, [see pdf for equation] with the quasiperiodic boundary condition on [0, 2p].
Halin's Theorem for the M"obius Strip
(Department of Mathematics, The University of Auckland, New Zealand, 2001-08) Archdeacon, Dan; Bonnington, C. Paul; Debowsky, Marisa; Prestidge, Michael
Halin's Theorem characterizes those locally finite infinite graphs that embed in the plane without accumulation points by giving a set of six topologically-excluded subgraphs. We prove the analogous theorem for graphs that embed in an open M"obius strip without accumulation points. There are 153 such obstructions under the ray ordering defined herein. There are 350 obstructions under the minor ordering. There are 1225 obstructions under the topological ordering. The relationship between these graphs and the obstructions to embedding in the projective plane is similar to the relationship between Halin's graphs and ${ K_5 , K_{3,3} }$.
Resonance Quantum Switch and Quantum Gate
(Department of Mathematics, The University of Auckland, New Zealand, 2002) Bagraev, N.; Mikhailova, A.B.; Pavlov, B.; Prokhorov, L.V.
Solvable models for two- and three-terminal Quantum Switches and Quantum Gates are suggested in form of a quantum ring witha few one-dimensional quantum wires attached to it. In resonance case when the Fermi level in the wires coincides with the resonance energy level on the ring , the magnitude of the governing electric field may be specified such that the quantum current through the switch from up-leading wire to the outgoing wires may be controlled via rotation of the orthogonal projection of the field onto the plane of the device.The working parameters of the switches and gates are defined in dependence of the desired working temperature, the Fermi level and the effective mass of the electron in the wires.
Isometric tight frames
(Department of Mathematics, The University of Auckland, New Zealand, 2002-05) Reams, Robert; Waldron, Shayne
We construct a $dtimes n$ matrix, $nge d$, whose columns have equal length and whose rows are orthonormal. This is equivalent to finding an isometric tight frame of $n$ vectors in $Rd$ (or $Cd$), or writing the $dtimes d$ identity matrix $I={dover n}sum_{i=1}^n P_i$, where the $P_i$ are rank $1$ orthogonal projections. %where the $P_i$ are orthogonal projections onto $1-$dimensional subspaces.
QUASI-RELATIVISM, NARROW-GAP PROPERTY AND FORCED DYNAMICS OF ELECTRONS IN SOLIDS: FEW SOLVABLE MODELS
(Department of Mathematics, The University of Auckland, New Zealand, 2002) Pavlov, B.; Pokrovskij, A.; Strepetov, A.
A mathematical model is constructed for a super-lattice which has a small effective mass at the resonance energy. A procedure of manipulation of the resonance current through the super-lattice via the time- periodic excitation is suggested.
Maximal Embeddings of Directed Multi-Cycles
(Department of Mathematics, The University of Auckland, New Zealand, 2002-02) Ma'u, Sikimeti
We consider embeddings of Eulerian digraphs that have in-arcs alternating with out-arcs in the rotation schemes at each vertex. We define the multicycle $C^{l,m}_n$ to be the digraph on the vertex set ${v_1,v_2,ldots,v_n}$, with arcs comprising $l$ copies of the cycle $(v_1,v_2,ldots,v_n)$ and $m$ copies of the cycle $(v_n,v_{n-1}, ldots, v_1)$. We consider maximal embeddings of multicycles and show that all except the bracelet digraphs $C^{1,1}_n$ are upper-embeddable. We find that some multicycles have the maximum possible genus range, being both upper-embeddable and planar, and some multicycles have a genus range of zero.
Eigenvectors of Compound-Circulant and Alternating Circulant Matrices
(Department of Mathematics, The University of Auckland, New Zealand, 2002-02) Tee, Garry J.
A circulant matrix whose elements are square submatrices is called a compound-circulant matrix. The eigenvectors and eigenvalues had been found for symmetric compound-circulant matrices, and that method is extended to general compound-circulant matrices. That analysis is applied to Stephen J. Watson's alternating circulant matrices, which reduce to compound-circulant matrices with square submatrices of order 2.
On modeling of amphibious population evolution
(Department of Mathematics, The University of Auckland, New Zealand, 2002-03) Golubyatnikov, Vladimir P.; Likhoshvai, Vitalii A.
We consider an evolution model of population of frogs on the aqueous stage of their development. Here we study the problem of determination of the parameters of the proposed model from the observation data, in particular, from the average times of attainment of different biological ages and from the survivability function. Our model gives possibility to estimate the number of morphologically indistinguishable ages which is particularly interesting in the case of incomplete experimental data.
Russian Peasant Multiplication and Egyptian Division in Zeckendorf Arithmetic
(Department of Mathematics, The University of Auckland, New Zealand, 2002-03) Tee, Garry J.
Edouard Zeckendorf shewed that every positive integer can be represented uniquely as a sum of distinct non-consecutive Fibonacci numbers, with $ F_2 $ (but not $ F_1 $) being used for 1. Arithmetic on integers represented in Zeckendorf form is more complicated than for integers represented in binary form. But, integer multiplication can readily be performed by adapting the Russian Peasant method, and integer division can readily be performed by adapting an Ancient Egyptian method.
Ranking Committees, Words or Multisets
(Department of Mathematics, The University of Auckland, New Zealand, 2002-06) Sertel, Murat; Slinko, Arkadii
We investigate the ways in which a linear order on a finite set $A$ can be consistently extended to a linear order on a set $P_k(A)$ of multisets on $A$ of cardinality $k$. We show that, when $card(A)=3$, all linear orders on $P_k(A)$ are additive and classify them by means of Farey fractions. For $card(A)ge 4$ we show that there are non-additive consistent linear orders on $P_k(A)$, we prove that they cannot be extended to a consistent linear order on $P_K(A)$ for sufficiently large $K$. We give the lower bounds for the number of consistent linear orders on $P_2(A)$ and for the total number of consistent linear orders on $P_2(A)$.

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