2022-01-21T17:23:24Zhttps://researchspace.auckland.ac.nz/dspace-oai/requestoai:researchspace.auckland.ac.nz:2292/49732009-08-28T12:35:47Zcom_2292_122col_2292_4963
Spontaneous magnetisation in the plane
Nicholls, Geoff
The Arak process is a solvable stochastic process which generates coloured patterns in the plane. Patterns are made up of a variable number of random non-intersecting polygons. We show that the distribution of Arak process states is the Gibbs distribution of its states in thermodynamic equilibrium in the grand canonical ensemble. The sequence of Gibbs distributions form a new model parameterised by temperature. We prove that there is a phase transition in this model, for some non-zero temperature. We illustrate this conclusion with simulation results. We measure the critical exponents of this off-lattice model and find they are consistent with those of the Ising model in two dimensions.
2009-08-28T03:20:32Z
2009-08-28T03:20:32Z
2000
Technical Report
Department of Mathematics - Research Reports-455 (2000)
1173-0889
http://hdl.handle.net/2292/4973
Research Reports - Department of Mathematics
https://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htm
The author(s)
Department of Mathematics, The University of Auckland, New Zealand
http://www.math.auckland.ac.nz/Research/Reports/view.php?id=455
oai:researchspace.auckland.ac.nz:2292/50902009-08-28T12:40:29Zcom_2292_122col_2292_4963
Probabilistic Solutions to Merchant Problems: Locating of the False Stack
Oleinik, V.L.
Pavlov, B.
In [1] a probabilistic solution to the Infinite Merchant's Problem, an undecidable problem equivalent to the Halting Problem, was proposed. The solution uses a real Hilbert space and is based on the estimation of the exponential growth of an unbounded semigroup. In [2] was offered an alternative solution in terms of scattering processes on quantum dots. The authors reduced the problem to a special scattering problem and testify the "halting phenomenon" based on the quantum measurement of results of scattering with random input data. The a-posteriori probability of halting, subject to the negative results of multiple independent tests, was estimated. The possibility of location the number of the bag with false coins in finite-dimensional case was noticed in [1] and proved in [2]. The aim of this paper is to offer a solution of the latter problem in the infinite case. [1] C.S. Calude, B.S. Pavlov. Coins, quantum measurements, and Turing's barrier, Quantum Information Processing, 1, 1--2 (2002), 107--127. [2] V.A. Adamyan, C.S. Calude, B.S. Pavlov, A quantum scattering approach to undecidable problem. In: Quantum information and complexity, Proceedings of the Meijo Winter School 2003, Meijo University, Nagoya, Japan, 6 - 10 January 2003, edited by T Hida, K SaitÃ´ (Meijo University, Japan) and Si Si (Aichi Prefectural University, Japan).
2009-08-28T03:22:18Z
2009-08-28T03:22:18Z
2004-12
Technical Report
Department of Mathematics - Research Reports-533 (2004)
1173-0889
http://hdl.handle.net/2292/5090
Research Reports - Department of Mathematics
https://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htm
The author(s)
Department of Mathematics, The University of Auckland, New Zealand
http://www.math.auckland.ac.nz/Research/Reports/view.php?id=533
oai:researchspace.auckland.ac.nz:2292/49682009-08-28T12:30:43Zcom_2292_122col_2292_4963
On consistent social choice functions
Koray, Semih
Slinko, Arkadii
In this paper we introduce the concept of F-consistency of a social choice function relative to the given class F of social choice functions. This refines the concept of consistency (self-selectivity), introduced by the first author, and allows to discover a number of classes F for which there exist F-consistent social choice functions which are neither dictatorial nor antidictatorial. Furthermore, under certain mild conditions on F all F-consistent social choice functions are described.
2009-08-28T03:20:27Z
2009-08-28T03:20:27Z
2001-04
Technical Report
Department of Mathematics - Research Reports-461 (2001)
1173-0889
http://hdl.handle.net/2292/4968
Research Reports - Department of Mathematics
https://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htm
The author(s)
Department of Mathematics, The University of Auckland, New Zealand
http://www.math.auckland.ac.nz/Research/Reports/view.php?id=461
oai:researchspace.auckland.ac.nz:2292/51102009-11-19T00:45:30Zcom_2292_122col_2292_4963
Fitting Parameters for a Solvable Model of a Quantum Network (Maths)
Harmer, M.
A solvable model corresponding to a given quantum network is described in cite{MPP} without an explicit description of how to fit the parameters of the solvable model. Here we give a procedure to fit these parameters so that the solvable model reproduces the important features, viz. the scattering matrix for the physically relevant energies, of the quantum network, subject to the non-vanishing of a determinant.
2009-08-28T03:22:35Z
2009-08-28T03:22:35Z
2004-04
Technical Report
Department of Mathematics - Research Reports-514 (2004)
1173-0889
http://hdl.handle.net/2292/5110
Research Reports - Department of Mathematics
https://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htm
The author(s)
Department of Mathematics, The University of Auckland, New Zealand
http://www.math.auckland.ac.nz/Research/Reports/view.php?id=514
oai:researchspace.auckland.ac.nz:2292/49902009-08-28T12:38:54Zcom_2292_122col_2292_4963
Scattering on graphs and one-dimensional approximation of $N-$dimensional Schr"odinger operators
Melnikov, Y.
Pavlov, B.
In actual paper we develop the spectral analysis of Schr"odinger operators on lattice type graphs. For basic example of qubic periodic graph the problem is reduced to the spectral analysis of the regular differential operators on a fundamental star-like subgraph with a selfadjoint condition at the central node and quasiperiodic conditions at the boundary vertices. Using an explicite expression for resolvent of lattice-type operator we develop in the second sections the Lippmann- Schwinger techniques for the perturbed periodic operator and construct the corresponding scattering matrix. It serves as a base for the approximation of the multy-dimensional Schr"odinger operator by the onedimansional operator on graph : in the third section of the paper for given $N$-dimensional Schr"odinger operators with rapidly decreasing potential we construct a lattice-type operator on cubic graph embedded into ${bf R}^N$ and show that the original $N$-dimensional scattering problem can be approximated in proper sense by the corresponding scattering problem for the perturbed lattice operator.
2009-08-28T03:20:48Z
2009-08-28T03:20:48Z
2000
Technical Report
Department of Mathematics - Research Reports-439 (2000)
1173-0889
http://hdl.handle.net/2292/4990
Research Reports - Department of Mathematics
https://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htm
The author(s)
Department of Mathematics, The University of Auckland, New Zealand
http://www.math.auckland.ac.nz/Research/Reports/view.php?id=439
oai:researchspace.auckland.ac.nz:2292/49852009-08-28T12:36:39Zcom_2292_122col_2292_4963
Hermitian symplectic geometry and the Schr"{o}dinger operator on the graph
Harmer, M.
The theory of self-adjoint extensions is closely related to the theory of hermitian symplectic geometry cite{Pav,Kost:Sch,Nov3}. Here we develop this idea, showing that it may also be used to consider symmetric extensions of a symmetric operator. Furthermore we find an explicit parameterisation of the Lagrange Grassmannian in terms of the unitary matrices $U (n)$. This allows us to explicitly describe all self-adjoint boundary conditions for the Schr"{o}dinger operator on the graph in terms of a unitary matrix. We show that the asymptotics of the scattering matrix can be simply expressed in terms of this unitary matrix. \ Using the construction of the asymptotic hermitian symplectic space cite{Nov1,Nov3} we derive a formula for the scattering matrix of a graph in terms of the scattering matrices of its subgraphs. This also provides a characterisation of the discrete eigenvalues embedded in the continuous spectrum.
2009-08-28T03:20:44Z
2009-08-28T03:20:44Z
2000
Technical Report
Department of Mathematics - Research Reports-444 (2000)
1173-0889
http://hdl.handle.net/2292/4985
Research Reports - Department of Mathematics
https://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htm
The author(s)
Department of Mathematics, The University of Auckland, New Zealand
http://www.math.auckland.ac.nz/Research/Reports/view.php?id=444
oai:researchspace.auckland.ac.nz:2292/51482009-08-28T12:38:33Zcom_2292_122col_2292_4963
On modeling of amphibious population evolution
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.
2009-08-28T03:23:10Z
2009-08-28T03:23:10Z
2002-03
Technical Report
Department of Mathematics - Research Reports-480 (2002)
1173-0889
http://hdl.handle.net/2292/5148
Research Reports - Department of Mathematics
https://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htm
The author(s)
Department of Mathematics, The University of Auckland, New Zealand
http://www.math.auckland.ac.nz/Research/Reports/view.php?id=480
oai:researchspace.auckland.ac.nz:2292/51352009-08-28T12:38:14Zcom_2292_122col_2292_4963
Obstructions to directed embeddings of Eulerian digraphs in the plane
Bonnington, C. Paul
Hartsfield, Nora
Siran, Jozef
A 2-cell embedding of an Eulerian digraph in a closed surface is said to be directed if the boundary of each face is a directed closed walk in $G$. We prove Kuratowski-type theorems about obstructions to directed embeddings of Eulerian digraphs in the plane.
2009-08-28T03:22:58Z
2009-08-28T03:22:58Z
2003-01
Technical Report
Department of Mathematics - Research Reports-492 (2003)
1173-0889
http://hdl.handle.net/2292/5135
Research Reports - Department of Mathematics
https://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htm
The author(s)
Department of Mathematics, The University of Auckland, New Zealand
http://www.math.auckland.ac.nz/Research/Reports/view.php?id=492
oai:researchspace.auckland.ac.nz:2292/50022009-08-28T12:39:03Zcom_2292_122col_2292_4963
Metrizability of Manifolds by Diagonal Properties
Gartside, P.M.
Mohamad, Abdul M.
This paper investigates metrization theory of manifolds. We show that diagonal properties play a central role in developing metrizability of manifolds.
2009-08-28T03:20:59Z
2009-08-28T03:20:59Z
1999-09
Technical Report
Department of Mathematics - Research Reports-428 (1999)
1173-0889
http://hdl.handle.net/2292/5002
Research Reports - Department of Mathematics
https://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htm
The author(s)
Department of Mathematics, The University of Auckland, New Zealand
http://www.math.auckland.ac.nz/Research/Reports/view.php?id=428
oai:researchspace.auckland.ac.nz:2292/50942009-08-28T12:40:35Zcom_2292_122col_2292_4963
On the Existence of Extremal Cones and Comparative Probability Orderings
Marshall, Simon
We study the recently discovered phenomenon of existence of comparative probability orderings on finite sets that violate Fishburn hypothesis - we call such orderings and the discrete cones associated with them extremal. Conder and Slinko constructed an extremal discrete cone on the set of n=7 elements and showed that no extremal cones exist on the set of n< 7 elements. In this paper we construct an extremal cone on a finite set of prime cardinality p if p satisfies a certain number theoretical condition. This condition has been computationally checked to hold for 1,725 of the 1,842 primes between 132 and 16,000, hence for all these primes extremal cones exist.
2009-08-28T03:22:21Z
2009-08-28T03:22:21Z
2004-10
Technical Report
Department of Mathematics - Research Reports-529 (2004)
1173-0889
http://hdl.handle.net/2292/5094
Research Reports - Department of Mathematics
https://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htm
The author(s)
Department of Mathematics, The University of Auckland, New Zealand
http://www.math.auckland.ac.nz/Research/Reports/view.php?id=529
oai:researchspace.auckland.ac.nz:2292/49932009-08-28T12:38:56Zcom_2292_122col_2292_4963
Order 5 symplectic explicit Runge-Kutta Nystrom methods
Chou, Lin-yi
Sharp, P.W.
Order five symplectic ERKN methods of five stages are known to exist. However, these methods do not have free parameters with which to minimise the error coefficients. By adding one derivative evaluation per step, to give either a six-stage non-FSAL family or a seven-stage FSAL family of methods, two free parameters become available for the minimisation. This raises the possibility of improving the efficiency of order five methods despite the extra cost of taking a step. We perform the minimisation of the two families to obtain an optimal method and then compare its performance with some published methods on the two-body problem for a range of eccentricities. These comparisons along with those based on the error coefficients show the new method is significantly more efficient than the five-stage methods. The numerical comparisons also suggest the new methods can be more efficient than some existing methods of other orders.
2009-08-28T03:20:51Z
2009-08-28T03:20:51Z
1999-12
Technical Report
Department of Mathematics - Research Reports-436 (1999)
1173-0889
http://hdl.handle.net/2292/4993
Research Reports - Department of Mathematics
https://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htm
The author(s)
Department of Mathematics, The University of Auckland, New Zealand
http://www.math.auckland.ac.nz/Research/Reports/view.php?id=436
oai:researchspace.auckland.ac.nz:2292/49722009-08-28T12:35:45Zcom_2292_122col_2292_4963
Bridge estimation of the probability density at a point
Mira, Antonietta
Nicholls, Geoff
Bridge estimation, as described by Meng and Wong in 1996, is used to estimate the value taken by a probability density at a point in the state space. When the normalisation of the prior density is known, this value may be used to estimate a Bayes factor. It is shown that the multi-block Metropolis-Hastings estimators of citeN{chib01} are bridge sampling estimators. This identification leads to estimators for the quantity of interest which may be substantially more efficient. This report was submitted in July 2000. A revised version of this report was submitted in September 2003. The version below is the revised version. Print and electronic copies of the original version are available on request.
2009-08-28T03:20:31Z
2009-08-28T03:20:31Z
2003-09
Technical Report
Department of Mathematics - Research Reports-456 (2003)
1173-0889
http://hdl.handle.net/2292/4972
Research Reports - Department of Mathematics
https://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htm
The author(s)
Department of Mathematics, The University of Auckland, New Zealand
http://www.math.auckland.ac.nz/Research/Reports/view.php?id=456
oai:researchspace.auckland.ac.nz:2292/51392009-08-28T12:38:22Zcom_2292_122col_2292_4963
Deformations of Surfaces in 4-space
Yashiro, Tsukasa
In this paper we describe geometric properties of embedded liftabilities of immersed 3--manifolds in 4--space into 5--space. It is known that a regular homotopy class of an immersed orientable surface in 3--space is constructed by a pair of an embedded surface and an immersed circle on it. We found an isotopy between embedded lifts of these constructed immersions in 4--space, which covers a regular homotopy of their projections in 3--space. Also we construct non-liftable immersed 3--spheres without quadruple points.
2009-08-28T03:23:02Z
2009-08-28T03:23:02Z
2002-08
Technical Report
Department of Mathematics - Research Reports-488 (2002)
1173-0889
http://hdl.handle.net/2292/5139
Research Reports - Department of Mathematics
https://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htm
The author(s)
Department of Mathematics, The University of Auckland, New Zealand
http://www.math.auckland.ac.nz/Research/Reports/view.php?id=488
oai:researchspace.auckland.ac.nz:2292/51282009-08-28T12:38:07Zcom_2292_122col_2292_4963
Barycentric Coordinates on the Hyperbolic Plane
Harmer, Mark
We describe a `natural' set of coordinates for fundamental domains in the hyperbolic plane in the case when the fundamental domain is triangular. The metric, the measure and the Laplace-Beltrami operator are calculated in this new coordinate system. As a byproduct we give a hyperbolic analogue of the Euclidean expression of the area of a triangle in terms of its base and height.
2009-08-28T03:22:51Z
2009-08-28T03:22:51Z
2003-06
Technical Report
Department of Mathematics - Research Reports-498 (2003)
1173-0889
http://hdl.handle.net/2292/5128
Research Reports - Department of Mathematics
https://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htm
The author(s)
Department of Mathematics, The University of Auckland, New Zealand
http://www.math.auckland.ac.nz/Research/Reports/view.php?id=498
oai:researchspace.auckland.ac.nz:2292/50122009-08-28T12:39:12Zcom_2292_122col_2292_4963
FEW-BODY KREIN'S FORMULA
Kurasov, Pavel
Pavlov, B.
Selfadjoint extensions of symmetric operators with infinite deficiency indices are discussed. In particular the operators describing the system of several quantum particles are investigated in detail and a few-body analog of Krein's formula for generalized resolvents is proven. The conditions for the semiboundedness of the simplest $M$-body quantum Hamiltonian with point interactionsin in the three-dimensional space are derived
2009-08-28T03:21:09Z
2009-08-28T03:21:09Z
1999-06
Technical Report
Department of Mathematics - Research Reports-418 (1999)
1173-0889
http://hdl.handle.net/2292/5012
Research Reports - Department of Mathematics
https://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htm
The author(s)
Department of Mathematics, The University of Auckland, New Zealand
http://www.math.auckland.ac.nz/Research/Reports/view.php?id=418
oai:researchspace.auckland.ac.nz:2292/49692009-08-28T12:35:31Zcom_2292_122col_2292_4963
Reservoir offset models for Radiocarbon calibration
Jones, Martin
Nicholls, Geoff
The purpose of a reservoir offset is to enable the application of calibration data ($mu(theta)$, emph{e.g.} shortciteNP{stuiver:98}) developed for one reservoir (primary reservoir) to CRA's from another (secondary reservoir). The usual approach has been to define the activity of the secondary reservoir as some form of constant offset (with error) from the primary reservoir (emph{e.g.} citeNP{stuiver93:_model_bc}). In this case CRA's from a secondary reservoir are not independent. However, the standard procedure for incorporating offset error into calibrated distributions assumes that the CRA's from secondary reservoirs are independent ({it e.g.} citeNP{stuiver93b}), accordingly the calibrated distributions are incorrect. In many cases this calculation error will be insignificant, however the calculation error will be significant in some situations and approaches such as sample based Bayesian inference need to be adopted if a non independent reservoir offset is applied.
2009-08-28T03:20:28Z
2009-08-28T03:20:28Z
2001
Technical Report
Department of Mathematics - Research Reports-459 (2001)
1173-0889
http://hdl.handle.net/2292/4969
Research Reports - Department of Mathematics
https://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htm
The author(s)
Department of Mathematics, The University of Auckland, New Zealand
http://www.math.auckland.ac.nz/Research/Reports/view.php?id=459
oai:researchspace.auckland.ac.nz:2292/50262009-08-28T12:39:28Zcom_2292_122col_2292_4963
Isochrones and Brachistochrones
Tee, Garry J.
Christiaan Huygens proved in 1659 that a particle sliding smoothly (under uniform gravity) on a cycloid with axis vertically down reaches the base in a period independent of the starting point. He built very accurate pendulum clocks with cycloidal pendulums. Mark Denny has constructed another curve purported to give descent to the base in a period independent of the starting point: but the cycloid is the only smooth plane curve with that property. Johann Bernoulli 1st proved in 1696 that, for any pair of fixed points, the brachistochrone (the curve of quickest descent) under uniform gravity is an arc of a cycloid. In 1976, Ian Stewart asked, what is the brachistochrone for central gravity under the inverse square law? The solution is found explicitly, in terms of elliptic integrals.
2009-08-28T03:21:21Z
2009-08-28T03:21:21Z
1998-11
Technical Report
Department of Mathematics - Research Reports-405 (1998)
1173-0889
http://hdl.handle.net/2292/5026
Research Reports - Department of Mathematics
https://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htm
The author(s)
Department of Mathematics, The University of Auckland, New Zealand
http://www.math.auckland.ac.nz/Research/Reports/view.php?id=405
oai:researchspace.auckland.ac.nz:2292/50652019-12-19T03:30:19Zcom_2292_122col_2292_4963
Dade's invariant conjecture for the Chevalley groups of type G_2 in the defining characteristic
Huang, Shih-chang
This paper is part of a program to study the conjecture of E.C. Dade on counting characters in blocks for several finite groups. In this paper, we verify Dade's invariant conjecture for the Chevalley groups G_2(q) in the defining characteristic when q is not a power of 2 or 3. This implies Dade's final conjecture when q is not a power of 2 or 3.
2009-08-28T03:21:55Z
2009-08-28T03:21:55Z
2005-03
Technical Report
Department of Mathematics - Research Reports-544 (2005)
1173-0889
http://hdl.handle.net/2292/5065
Research Reports - Department of Mathematics
2041
16440
7362
7344
https://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htm
The author(s)
Department of Mathematics, The University of Auckland, New Zealand
http://www.math.auckland.ac.nz/Research/Reports/view.php?id=544
oai:researchspace.auckland.ac.nz:2292/51302009-08-28T12:38:10Zcom_2292_122col_2292_4963
Normal subgroups of low index in the modular group and other Hecke groups
Conder, Marston
Dobcsanyi, Peter
An account is given of the determination of all normal subgroups of index up to $1500$ in the modular group PSL$(2,Z) cong C_2 * C_3$, and all normal subgroups of index up to $500$ in the Hecke groups $H_q = C_2 * C_q$ for $4 le q le 12$, using an adaptation of the low index subgroups algorithm.
2009-08-28T03:22:53Z
2009-08-28T03:22:53Z
2003-06
Technical Report
Department of Mathematics - Research Reports-496 (2003)
1173-0889
http://hdl.handle.net/2292/5130
Research Reports - Department of Mathematics
https://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htm
The author(s)
Department of Mathematics, The University of Auckland, New Zealand
http://www.math.auckland.ac.nz/Research/Reports/view.php?id=496
oai:researchspace.auckland.ac.nz:2292/50882009-08-28T12:40:28Zcom_2292_122col_2292_4963
On Complexity of Lobbying in Multiple Referenda
Cristian, Robin
Fellows, Mike
Rosamond, Frances
Slinko, Arkadii
In this paper we show that lobbying in conditions of "direct democracy" is virtually impossible, even in conditions of complete information about voters preferences, since it would require solving a very computationally hard problem. We use the apparatus of parametrized complexity for this purpose.
2009-08-28T03:22:16Z
2009-08-28T03:22:16Z
2005-02
Technical Report
Department of Mathematics - Research Reports-534 (2005)
1173-0889
http://hdl.handle.net/2292/5088
Research Reports - Department of Mathematics
https://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htm
The author(s)
Department of Mathematics, The University of Auckland, New Zealand
http://www.math.auckland.ac.nz/Research/Reports/view.php?id=534
oai:researchspace.auckland.ac.nz:2292/50492009-08-28T12:39:51Zcom_2292_122col_2292_4963
Spectral Theory of Wiener-Hopf Operators and Functional Model
MacCormick, J.P.
Pavlov, B.
Using the Nagy-Foias functional model for contractions we reduce the spectral problem for Wiener-Hopf Operators with rational symbols to the spectral problem for finite matrices. In particular we suggest a simple approach to calculation of Wiener-Hopf determinants for analytic symbols.
2009-08-28T03:21:41Z
2009-08-28T03:21:41Z
1997-07
Technical Report
Department of Mathematics - Research Reports-384 (1997)
1173-0889
http://hdl.handle.net/2292/5049
Research Reports - Department of Mathematics
https://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htm
The author(s)
Department of Mathematics, The University of Auckland, New Zealand
http://www.math.auckland.ac.nz/Research/Reports/view.php?id=384
oai:researchspace.auckland.ac.nz:2292/50042009-08-28T12:39:05Zcom_2292_122col_2292_4963
Dade's Invariant Conjecture for General Linear and Unitary Groups in Non-defining Characteristics
An, Jianbei
This paper is part of a program to study the conjecture of E. C. Dade on counting characters in blocks for several finite groups. The invariant conjecture of Dade is proved for general linear and unitary groups when the characteristic of the modular representation is distinct from the defining characteristic of the groups.
2009-08-28T03:21:01Z
2009-08-28T03:21:01Z
1999-09
Technical Report
Department of Mathematics - Research Reports-426 (1999)
1173-0889
http://hdl.handle.net/2292/5004
Research Reports - Department of Mathematics
https://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htm
The author(s)
Department of Mathematics, The University of Auckland, New Zealand
http://www.math.auckland.ac.nz/Research/Reports/view.php?id=426
oai:researchspace.auckland.ac.nz:2292/51032009-08-28T12:37:38Zcom_2292_122col_2292_4963
Quasicontinuous selections of upper semicontinuous set-valued mappings
Cao, Jiling
Moors, Warren B.
In this paper, we extend a theorem of Matejdes on quasicontinuous selections of upper Baire continuous set-valued mappings from compact metric range spaces to regular T_1 range spaces. In addition, we also prove a quasicontinuous selection theorem for a special class of upper semicontinuous set-valued mappings.
2009-08-28T03:22:29Z
2009-08-28T03:22:29Z
2004-08
Technical Report
Department of Mathematics - Research Reports-521 (2004)
1173-0889
http://hdl.handle.net/2292/5103
Research Reports - Department of Mathematics
https://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htm
The author(s)
Department of Mathematics, The University of Auckland, New Zealand
http://www.math.auckland.ac.nz/Research/Reports/view.php?id=521
oai:researchspace.auckland.ac.nz:2292/49842009-08-28T12:36:37Zcom_2292_122col_2292_4963
Scattering on the annulus
Harmer, M.
We generalise the asymptotic formula for the scattering matrix in cite{BMPY} to the case of non-simple spectrum. This asymptotic formula is used to identify a simple family of switches and investigate the properties of a member of the family using numerical techniques.
2009-08-28T03:20:42Z
2009-08-28T03:20:42Z
2000
Technical Report
Department of Mathematics - Research Reports-445 (2000)
1173-0889
http://hdl.handle.net/2292/4984
Research Reports - Department of Mathematics
https://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htm
The author(s)
Department of Mathematics, The University of Auckland, New Zealand
http://www.math.auckland.ac.nz/Research/Reports/view.php?id=445
oai:researchspace.auckland.ac.nz:2292/50422009-08-28T12:39:44Zcom_2292_122col_2292_4963
CONTINUOUS BRANCHES OF INVERSES OF THE 12 JACOBI ELLIPTIC FUNCTIONS FOR REAL ARGUMENT
Tee, Garry J.
[no abstract available]
2009-08-28T03:21:35Z
2009-08-28T03:21:35Z
1998-03
Technical Report
Department of Mathematics - Research Reports-390 (1998)
1173-0889
http://hdl.handle.net/2292/5042
Research Reports - Department of Mathematics
https://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htm
The author(s)
Department of Mathematics, The University of Auckland, New Zealand
http://www.math.auckland.ac.nz/Research/Reports/view.php?id=390
oai:researchspace.auckland.ac.nz:2292/51132009-08-28T12:37:48Zcom_2292_122col_2292_4963
Exploratory analysis of similarities between common voting rules
McCabe-Dansted, John C.
Slinko, Arkadii
Nurmi (1987) investigated the relationship between voting rules by determining the frequency that two rules pick the same winner. We use statistical techniques such as clustering and multidimensional scaling to further understand the relationships between rules. We also investigate how the relationships change when elections with Condorcet winners are excluded from the frequency data, and when the homogeneity of the voting population is increased.
2009-08-28T03:22:38Z
2009-08-28T03:22:38Z
2004-03
Technical Report
Department of Mathematics - Research Reports-512 (2004)
1173-0889
http://hdl.handle.net/2292/5113
Research Reports - Department of Mathematics
https://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htm
The author(s)
Department of Mathematics, The University of Auckland, New Zealand
http://www.math.auckland.ac.nz/Research/Reports/view.php?id=512
oai:researchspace.auckland.ac.nz:2292/49702009-08-28T12:35:44Zcom_2292_122col_2292_4963
On Asymptotic Strategy-Proofness of Classical Social Choice Rules
Slinko, Arkadii
We show that, when the number of voters $n$ tends to infinity, all classical social choice rules are asymptotically strategy-proof with the proportion of manipulable profiles being of order $O(1/sqrt{n})$.
2009-08-28T03:20:29Z
2009-08-28T03:20:29Z
2000-10
Technical Report
Department of Mathematics - Research Reports-458 (2000)
1173-0889
http://hdl.handle.net/2292/4970
Research Reports - Department of Mathematics
https://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htm
The author(s)
Department of Mathematics, The University of Auckland, New Zealand
http://www.math.auckland.ac.nz/Research/Reports/view.php?id=458
oai:researchspace.auckland.ac.nz:2292/51042009-08-28T12:37:39Zcom_2292_122col_2292_4963
A New Interpretation of the Selberg Trace Formula
Cartier, P.
Voros, A.
We extend the Selberg trace formula for a hyperbolic compact Riemann surface to some new test functions, i.e., holomorphic and decreasing at infinity in a sector instead of a horizontal strip (and no longer even). As applications: 1) we interpret the trace formula as a Poisson summation formula involving the eigenvalue spectrum of the hyperbolic Laplacian on one side, and the lengths of all (real and complex) periodic geodesics of the surface on the other side; 2) we obtain a closed meromorphic continuation formula for a spectral zeta function of the hyperbolic Laplacian.
2009-08-28T03:22:30Z
2009-08-28T03:22:30Z
2004
Technical Report
Department of Mathematics - Research Reports-520 (2004)
1173-0889
http://hdl.handle.net/2292/5104
Research Reports - Department of Mathematics
https://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htm
The author(s)
Department of Mathematics, The University of Auckland, New Zealand
http://www.math.auckland.ac.nz/Research/Reports/view.php?id=520
oai:researchspace.auckland.ac.nz:2292/49742009-08-28T12:35:48Zcom_2292_122col_2292_4963
A Result on $aleph_1$-Compact Spaces
Mohamad, A.M.
In this note, we prove that every countably compact space with quasi--$S_1$--diagonal is compact. However, it is shown that it need not be metrizable.
2009-08-28T03:20:33Z
2009-08-28T03:20:33Z
2000
Technical Report
Department of Mathematics - Research Reports-454 (2000)
1173-0889
http://hdl.handle.net/2292/4974
Research Reports - Department of Mathematics
https://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htm
The author(s)
Department of Mathematics, The University of Auckland, New Zealand
http://www.math.auckland.ac.nz/Research/Reports/view.php?id=454
oai:researchspace.auckland.ac.nz:2292/50182009-08-28T12:39:18Zcom_2292_122col_2292_4963
Boundedly Metacompact or Finitistic Spaces and the Star Order of Covers
Deo, Satya
Gauld, David
After showing that the topological notion of boundedly metacompact (first named finitistic) is equivalent to metrisability for a topological manifold we then study related notions. In particular we study the star order of covers of a space. This leads us to propose a definition of dimension which we call star covering dimension.
2009-08-28T03:21:14Z
2009-08-28T03:21:14Z
1999-03
Technical Report
Department of Mathematics - Research Reports-412 (1999)
1173-0889
http://hdl.handle.net/2292/5018
Research Reports - Department of Mathematics
https://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htm
The author(s)
Department of Mathematics, The University of Auckland, New Zealand
http://www.math.auckland.ac.nz/Research/Reports/view.php?id=412
oai:researchspace.auckland.ac.nz:2292/50052009-08-28T12:39:06Zcom_2292_122col_2292_4963
The p-local Rank of a Block
An, Jianbei
Eaton, Charles W.
We generalize the p-local rank of a finite group, introduced by G. Robinson, to a p-block of a finite group and show that this has analagous properties.
2009-08-28T03:21:02Z
2009-08-28T03:21:02Z
1999-09
Technical Report
Department of Mathematics - Research Reports-425 (1999)
1173-0889
http://hdl.handle.net/2292/5005
Research Reports - Department of Mathematics
https://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htm
The author(s)
Department of Mathematics, The University of Auckland, New Zealand
http://www.math.auckland.ac.nz/Research/Reports/view.php?id=425
oai:researchspace.auckland.ac.nz:2292/49912009-08-28T12:38:54Zcom_2292_122col_2292_4963
A Generalization of Komlos Theorem on Random Matrices
Slinko, Arkadii
{bf Abstract:} In this paper we prove that, for any subset $Deltasubset Z$, the probability, that a random $Delta_{ntimes n}$ matrix is singular, is of order $Oleft(1/sqrt{n}$
2009-08-28T03:20:50Z
2009-08-28T03:20:50Z
2000-03
Technical Report
Department of Mathematics - Research Reports-438 (2000)
1173-0889
http://hdl.handle.net/2292/4991
Research Reports - Department of Mathematics
https://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htm
The author(s)
Department of Mathematics, The University of Auckland, New Zealand
http://www.math.auckland.ac.nz/Research/Reports/view.php?id=438
oai:researchspace.auckland.ac.nz:2292/50832009-08-28T12:40:21Zcom_2292_122col_2292_4963
Surface Area of Ellipsoid Segment
Tee, Garry J.
The surface area of a general segment of a 3-dimensional ellipsoid is computed.
2009-08-28T03:22:11Z
2009-08-28T03:22:11Z
2005-07
Technical Report
Department of Mathematics - Research Reports-539 (2005)
1173-0889
http://hdl.handle.net/2292/5083
Research Reports - Department of Mathematics
https://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htm
The author(s)
Department of Mathematics, The University of Auckland, New Zealand
http://www.math.auckland.ac.nz/Research/Reports/view.php?id=539
oai:researchspace.auckland.ac.nz:2292/51142009-08-28T12:37:49Zcom_2292_122col_2292_4963
Direct and Inverse Boundary Uniqueness Theorems for Analytic Functions Smooth in the Closed Unit Disc
Lee, Jeong Eun
Pavlov, B.
[no abstract available]
2009-08-28T03:22:39Z
2009-08-28T03:22:39Z
2004-04
Technical Report
Department of Mathematics - Research Reports-511 (2004)
1173-0889
http://hdl.handle.net/2292/5114
Research Reports - Department of Mathematics
https://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htm
The author(s)
Department of Mathematics, The University of Auckland, New Zealand
http://www.math.auckland.ac.nz/Research/Reports/view.php?id=511
oai:researchspace.auckland.ac.nz:2292/50862009-08-28T12:40:26Zcom_2292_122col_2292_4963
Fitting of the solvable model for scattering by Helmholtz resonator
Pavlov, B.
The analytic perturbation procedure is divergent on the continuous spectrum near the threshold of creation of resonances. For scattering on Helmholtz resonator with a small opening we suggest a modified procedure which is convergent and permits to observe the creation of the resonance. The role of the first step ("jump-start") in that perturbation procedure is played by a solvable model of the resonator, which is completely fitted based on the resonance parameters
2009-08-28T03:22:14Z
2009-08-28T03:22:14Z
2005-03
Technical Report
Department of Mathematics - Research Reports-536 (2005)
1173-0889
http://hdl.handle.net/2292/5086
Research Reports - Department of Mathematics
https://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htm
The author(s)
Department of Mathematics, The University of Auckland, New Zealand
http://www.math.auckland.ac.nz/Research/Reports/view.php?id=536
oai:researchspace.auckland.ac.nz:2292/50382009-08-28T12:39:41Zcom_2292_122col_2292_4963
Generalized Elliptic Integrals and Modular Equations
Anderson, G.D.
Qiu, S.-L.
Vamanamurthy, M.K.
Vuorinen, M.
from the Schwarz-Christoffel transformation of the upper half-plane onto a parallelogram and are naturally related to Gaussian hypergeometric functions. Certain combinations of these integrals also occur in analytic number theory in the study of Ramanujan's modular equations and approximations to $pi$. The authors study the monotoneity and convexity properties of these quantities and obtain sharp inequalities for them.
2009-08-28T03:21:31Z
2009-08-28T03:21:31Z
1998-04
Technical Report
Department of Mathematics - Research Reports-394 (1998)
1173-0889
http://hdl.handle.net/2292/5038
Research Reports - Department of Mathematics
https://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htm
The author(s)
Department of Mathematics, The University of Auckland, New Zealand
http://www.math.auckland.ac.nz/Research/Reports/view.php?id=394
oai:researchspace.auckland.ac.nz:2292/50412009-08-28T12:39:43Zcom_2292_122col_2292_4963
Separable determination of integrability and minimality of the Clarke subdifferential mapping
Borwein, Jonathan M.
Moors, Warren B.
$D$-representability of Lipschitz functions defined on arbitrary Banach spaces reduces to the study of these properties on separable Banach spaces.
2009-08-28T03:21:34Z
2009-08-28T03:21:34Z
1998-03
Technical Report
Department of Mathematics - Research Reports-391 (1998)
1173-0889
http://hdl.handle.net/2292/5041
Research Reports - Department of Mathematics
https://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htm
The author(s)
Department of Mathematics, The University of Auckland, New Zealand
http://www.math.auckland.ac.nz/Research/Reports/view.php?id=391
oai:researchspace.auckland.ac.nz:2292/49672009-08-28T12:30:42Zcom_2292_122col_2292_4963
Signed frames and Hadamard products of Gram Matrices
Peng, Irine
Waldron, Shayne
This paper concerns (redundant) representations in a Hilbert space $H$ of the form $$ f = sum_j c_jinpro{f,phi_j}phi_j, qquad forall fin H. $$ These are more general than those obtained from a tight frame, and we develop a general theory based on what are called signed frames. We are particularly interested in the cases where the scaling factors $c_j$ are unique and the geometric interpretation of negative $c_j$. This is related to results about the invertibility of certain Hadamard products of Gram matrices which are of independent interest, e.g., we show for almost every $v_1,ldots,v_ninCC^d$ $$ rank([inpro{v_i,v_j}^roverline{inpro{v_i,v_j}}^s]) = min{{r+d-1choose d-1}{s+d-1choose d-1},n}, qquad r,sge0. $$ Applications include the construction of tight frames of bivariate Jacobi polynomials on a triangle which preserve symmetries, and numerical results and conjectures about the class of tight frames in a finite dimensional space.
2009-08-28T03:20:25Z
2009-08-28T03:20:25Z
2001-04
Technical Report
Department of Mathematics - Research Reports-462 (2001)
1173-0889
http://hdl.handle.net/2292/4967
Research Reports - Department of Mathematics
https://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htm
The author(s)
Department of Mathematics, The University of Auckland, New Zealand
http://www.math.auckland.ac.nz/Research/Reports/view.php?id=462
oai:researchspace.auckland.ac.nz:2292/51562009-08-28T12:38:40Zcom_2292_122col_2292_4963
Graphs Embedded in the Plane with Finitely Many Accumulation Points
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.
2009-08-28T03:23:18Z
2009-08-28T03:23:18Z
2001-08
Technical Report
Department of Mathematics - Research Reports-472 (2001)
1173-0889
http://hdl.handle.net/2292/5156
Research Reports - Department of Mathematics
https://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htm
The author(s)
Department of Mathematics, The University of Auckland, New Zealand
http://www.math.auckland.ac.nz/Research/Reports/view.php?id=472
oai:researchspace.auckland.ac.nz:2292/51202009-08-28T12:37:58Zcom_2292_122col_2292_4963
Analytic perturbation techniques for the Friedrichs model: Intermediate Operator
Pavlov, B.
Antoniou, I.
A techniques of an intermediate operator is developed for the Friedrichs model obtained as a finite-dimensional perturbation [ {cal P} longrightarrow {cal P}_{_{varepsilon}} = {cal P} + varepsilon A ] of the momentum operator ${cal P} = frac{1}{i},,frac{d}{dx}$, defined by the corresponding operator-extension procedure. This technique permits to observe a creation of the resonance at the given point $k_{_{0}}$ via presenting the Scattering matrix for the above pair as a product of the non-analytic at $left(varepsilon,,kright) = left(0,,k_{_0}right)$ factor $S^{^{varepsilon}}_{_{0}} left(k right) $ which is the Scattering matrix to the pair $ left{{cal P}^{^{varepsilon}}_{_{0}},,{cal P}right}$ of the momentum with a local intermediate operator $ {cal P}^{^{varepsilon}}_{_{0}}$, and an analytic factor $ Sleft({cal P} + varepsilon A,,{cal P}^{^{varepsilon}}_{_{ 0}} right) $ of both variables $left(varepsilon,, kright)$ near the point $left(0,, k_{_0}right)$ which is the Scattering matrix of the pair $left({cal P}_{_{varepsilon}},{cal P}^{^{varepsilon}}_{_{ 0}}right)$. The corresponding representation is valid also for eigenfunctions of the perturbed operator.} vskip
2009-08-28T03:22:44Z
2009-08-28T03:22:44Z
2003-12
Technical Report
Department of Mathematics - Research Reports-505 (2003)
1173-0889
http://hdl.handle.net/2292/5120
Research Reports - Department of Mathematics
https://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htm
The author(s)
Department of Mathematics, The University of Auckland, New Zealand
http://www.math.auckland.ac.nz/Research/Reports/view.php?id=505
oai:researchspace.auckland.ac.nz:2292/50112009-08-28T12:39:11Zcom_2292_122col_2292_4963
Scattering on a Compact Domain with few Semiinfinite wires attached: resonance case
Mikhailova, A.
Pavlov, B.
Popov, I.
Rudakova, T.
Yafyasov, A.M.
Scattering problem for Neumann Laplacean with a continuous potential on a domain with a smooth boundary and few semiinfinite wires attached to it is studied. In resonance case when the Fermi level inthe wires coincides with some {it resonance} energy level in the domain the approximate formula for the transmission coefficient from one wire to another is derived: inthe case of weak interaction between the domain and the wires the transmission coefficient is proportional to the product of values of the corresponding resonance eigenfunction of inner problem at the points of contact.
2009-08-28T03:21:08Z
2009-08-28T03:21:08Z
1999-08
Technical Report
Department of Mathematics - Research Reports-420 (1999)
1173-0889
http://hdl.handle.net/2292/5011
Research Reports - Department of Mathematics
https://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htm
The author(s)
Department of Mathematics, The University of Auckland, New Zealand
http://www.math.auckland.ac.nz/Research/Reports/view.php?id=420
oai:researchspace.auckland.ac.nz:2292/50392009-08-28T12:39:42Zcom_2292_122col_2292_4963
1-factorizations of Cayley graphs on solvable groups
Alspach, Brian
Morton, Margaret
Qin, Yusheng
[no abstract available]
2009-08-28T03:21:32Z
2009-08-28T03:21:32Z
1998-03
Technical Report
Department of Mathematics - Research Reports-393 (1998)
1173-0889
http://hdl.handle.net/2292/5039
Research Reports - Department of Mathematics
https://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htm
The author(s)
Department of Mathematics, The University of Auckland, New Zealand
http://www.math.auckland.ac.nz/Research/Reports/view.php?id=393
oai:researchspace.auckland.ac.nz:2292/51342009-08-28T03:22:57Zcom_2292_122col_2292_4963
A generalised beta integral and the limit of the Bernstein--Durrmeyer operator with Jacobi weights
Waldron, Shayne
We give a generalisation of the multivariate beta integral. This is used to show that the (multivariate) Bernstein--Durrmeyer operator for a Jacobi weight has a limit as the weight becomes singular. The limit is an operator previously studied by Goodman and Sharma. From the elementary proof given, it follows that this operator inherits many properties of the Bernstein--Durrmeyer operator in a natural way. In particular, we determine its eigenstructure and give a differentiation formula for it.
2009-08-28T03:22:57Z
2009-08-28T03:22:57Z
2003-02
Technical Report
Department of Mathematics - Research Reports-493 (2003)
1173-0889
http://hdl.handle.net/2292/5134
Research Reports - Department of Mathematics
https://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htm
The author(s)
Department of Mathematics, The University of Auckland, New Zealand
http://www.math.auckland.ac.nz/Research/Reports/view.php?id=493
oai:researchspace.auckland.ac.nz:2292/50242009-08-28T12:39:25Zcom_2292_122col_2292_4963
Radiocarbon dating with temporal order constraints
Nicholls, Geoff
Jones, Martin
A Bayesian method has been proposed for analysing radiocarbon dates. The method takes into account stratigraphic constraints on recovered calendar dates. We find that the non-informative priors in use in the literature apply a bias towards wider date ranges which is not in general supported by substantial prior knowledge. We recommend using a prior which has a uniform marginal date range. We show how such priors are derived from a model of the deposition and observation process. We apply the method to relatively large data sets, examining the effect that various priors have on the reconstructed dates.
2009-08-28T03:21:19Z
2009-08-28T03:21:19Z
1999-02
Technical Report
Department of Mathematics - Research Reports-407 (1999)
1173-0889
http://hdl.handle.net/2292/5024
Research Reports - Department of Mathematics
https://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htm
The author(s)
Department of Mathematics, The University of Auckland, New Zealand
http://www.math.auckland.ac.nz/Research/Reports/view.php?id=407
oai:researchspace.auckland.ac.nz:2292/49802009-08-28T12:36:34Zcom_2292_122col_2292_4963
Comparisons of high order Stormer and explicit Runge-Kutta Nystrom methods for N-body simulations of the solar system
Sharp, P.W.
We compare the efficiency, stability properties, overhead, round-off error propagation and storage requirements of fixed-stepsize high order Stormer and explicit Runge-Kutta Nystrom methods for N-body simulations of the solar system. The comparisons of the round-off error propagation and efficiency are made using realistic problems, one of which requires over 500 million integration steps. We find high order ERKN methods have better stability properties and smaller overhead than Stormer methods. Our numerical tests suggest ERKN methods are more efficient than Stormer methods for shorter simulations such as one that simulates ten million years of the jovian planets. However, the superior round-off error propagation of the Stormer method Comment: a) The Math Reviews classification above is for the 2000 MSC system, b) A pdf version of the file is available with the postscript and dvi versions
2009-08-28T03:20:39Z
2009-08-28T03:20:39Z
2000-06
Technical Report
Department of Mathematics - Research Reports-449 (2000)
1173-0889
http://hdl.handle.net/2292/4980
Research Reports - Department of Mathematics
https://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htm
The author(s)
Department of Mathematics, The University of Auckland, New Zealand
http://www.math.auckland.ac.nz/Research/Reports/view.php?id=449
oai:researchspace.auckland.ac.nz:2292/50462009-08-28T12:39:46Zcom_2292_122col_2292_4963
Growth of Infinite Planar Graphs
Brand, Neal
Morton, Margaret
Vertigan, Dirk
those graphs which are also concentric a recurrence relation is given which determines the growth rate. In the more general case lower bounds on the growth rate are given. In both the concentric and the general cases, the formulae involve the local condition of excess at a vertex.
2009-08-28T03:21:38Z
2009-08-28T03:21:38Z
1998-03
Technical Report
Department of Mathematics - Research Reports-387 (1998)
1173-0889
http://hdl.handle.net/2292/5046
Research Reports - Department of Mathematics
https://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htm
The author(s)
Department of Mathematics, The University of Auckland, New Zealand
http://www.math.auckland.ac.nz/Research/Reports/view.php?id=387
oai:researchspace.auckland.ac.nz:2292/51052009-08-28T12:37:39Zcom_2292_122col_2292_4963
How to Exhibit Toroidal Maps in Space
Archdeacon, Dan
Bonnington, C. Paul
Ellis-Monaghan, Jo
Steinitz's Theorem states that a graph is the 1-skeleton of a convex polyhedron if and only if it is 3-connected and planar. The polyhedron is called a geometric realization of the embedded graph. Its faces are bounded by convex polygons whose points are coplanar. A map on the torus does not necessarily have such a geometric realization. In this paper, we relax the condition that faces are the convex hull of coplanar points. We require instead that the convex hull of the p oints on a face can be projected onto a plane so that the boundary of the convex hull of the projected points is the image of the boundary of the face. We also require that the interiors of the convex hulls of different faces do not intersect. Call this an exhibition of the map. A map is polyhedral if the intersection of any two closed faces is simply connected. Our main result is that every polyhedral toroidal map can be exhibited. As a corollary, every toroidal triangulation has a geometric realization.
2009-08-28T03:22:31Z
2009-08-28T03:22:31Z
2004-07
Technical Report
Department of Mathematics - Research Reports-519 (2004)
1173-0889
http://hdl.handle.net/2292/5105
Research Reports - Department of Mathematics
https://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htm
The author(s)
Department of Mathematics, The University of Auckland, New Zealand
http://www.math.auckland.ac.nz/Research/Reports/view.php?id=519
oai:researchspace.auckland.ac.nz:2292/51532009-08-28T12:38:37Zcom_2292_122col_2292_4963
Resonance Quantum Switch and Quantum Gate
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.
2009-08-28T03:23:15Z
2009-08-28T03:23:15Z
2002
Technical Report
Department of Mathematics - Research Reports-475 (2002)
1173-0889
http://hdl.handle.net/2292/5153
Research Reports - Department of Mathematics
https://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htm
The author(s)
Department of Mathematics, The University of Auckland, New Zealand
http://www.math.auckland.ac.nz/Research/Reports/view.php?id=475
oai:researchspace.auckland.ac.nz:2292/50512009-08-28T12:39:52Zcom_2292_122col_2292_4963
Mean value interpolation for points in general position
Waldron, Shayne
[no abstract available]
2009-08-28T03:21:43Z
2009-08-28T03:21:43Z
1997-05
Technical Report
Department of Mathematics - Research Reports-382 (1997)
1173-0889
http://hdl.handle.net/2292/5051
Research Reports - Department of Mathematics
https://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htm
The author(s)
Department of Mathematics, The University of Auckland, New Zealand
http://www.math.auckland.ac.nz/Research/Reports/view.php?id=382
oai:researchspace.auckland.ac.nz:2292/49792009-08-28T12:36:32Zcom_2292_122col_2292_4963
Weak bases and Metrizability
Mohamad, A.M.
In this paper we investigate weak bases. We give a characterization of weakly developable spaces and metrization theorems. The metrization results are: a space $X$ is metrizable if and only if $X$ has a $CWBC$--map $g$ satisfying the following conditions: begin {enumerate} item $g$ is a pseudo--strongly--quasi--N--map; item for any $A subseteq X, overline {A} subseteq bigcup {g(n,x) : x in A }$; end {enumerate} a space $X$ is metrizable if and only if $X$ has a $CWBC$--map $g$ satisfying the following conditions: begin {enumerate} item if $x in g(n,y_n)$, $y_n in g(n,x_n)$, $x_n in g(n,y_n)$ and $y_n in g(n,x)$ for all $n in N$, then $x_n$ converges to $x$; item for any $A subseteq X, overline {A} subseteq bigcup {g(n,x) : x in A }$. end {enumerate}
2009-08-28T03:20:38Z
2009-08-28T03:20:38Z
2000
Technical Report
Department of Mathematics - Research Reports-450 (2000)
1173-0889
http://hdl.handle.net/2292/4979
Research Reports - Department of Mathematics
https://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htm
The author(s)
Department of Mathematics, The University of Auckland, New Zealand
http://www.math.auckland.ac.nz/Research/Reports/view.php?id=450
oai:researchspace.auckland.ac.nz:2292/50992009-08-28T12:37:35Zcom_2292_122col_2292_4963
Symplectic operator-extension techniques and zero-range quantum models
Pavlov, B.
Kruglov, Vladimir I.
F. Berezin and L. Faddeev interpreted Fermi zero-range model as a self-adjoint extension of the Laplacian. Various modifications of this model in conventional Hilbert space possess rich spectral properties, but unavoidably have the negative effective radius and contain numerous parameters which do not have a direct physical meaning. We suggest, for spherically-symmetric scattering, a generalization of the Fermi zero-range model supplied with an indefinite metric in the inner space and a Hamiltonian of the inner degrees of freedom. Effective radius of this model may be both positive or negative. We propose also a general {it principle of analyticity} formulated in terms of $k{rm cot}delta (k)$ as a function of the scattering phase shift $delta(k)$ depending on the wave-number $k$. This principle allows us to evaluate all parameters of the model, including the indefinite metric tensor of the inner space, once the basic parameters of the model: the spectrum $sigma_{_{p}}$ of the inner Hamiltonian, the scattering length and the effective radius, are fixed, such that the sign of the effective radius is connected with the spectrum $sigma_{_{p}}$ by an appropriate consistency condition. The absolutely-continuous part of the extension plays a role of the quantum Hamiltonian of the special model.
2009-08-28T03:22:26Z
2009-08-28T03:22:26Z
2004-08
Technical Report
Department of Mathematics - Research Reports-524 (2004)
1173-0889
http://hdl.handle.net/2292/5099
Research Reports - Department of Mathematics
https://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htm
The author(s)
Department of Mathematics, The University of Auckland, New Zealand
http://www.math.auckland.ac.nz/Research/Reports/view.php?id=524
oai:researchspace.auckland.ac.nz:2292/50632009-08-28T12:40:05Zcom_2292_122col_2292_4963
Basic intervals between metrizable topologies]{Basic intervals in the partial order of metrizable topologies
McIntyre, D.W.
Watson, W.S.
This paper surveys a number of aspects of numerical methods for ordinary differential equations. The discussion includes the method of Euler and introduces Runge-Kutta methods and linear multistep methods as generalizations of Euler. Stability considerations arising from stiffness lead to a discussion of implicit methods and implementation issues. To the extent possible within this short survey, numerical methods are looked at in the context of problems arising in practical applications.
2009-08-28T03:21:53Z
2009-08-28T03:21:53Z
1997-03
Technical Report
Department of Mathematics - Research Reports-371 (1997)
1173-0889
http://hdl.handle.net/2292/5063
Research Reports - Department of Mathematics
https://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htm
The author(s)
Department of Mathematics, The University of Auckland, New Zealand
http://www.math.auckland.ac.nz/Research/Reports/view.php?id=371
oai:researchspace.auckland.ac.nz:2292/50092009-08-28T12:39:09Zcom_2292_122col_2292_4963
On Preclosed Sets and Their Generalizations
Cao, Jiling
Ganster, Maximilian
Konstadilaki, Chariklia
Reilly, Ivan
This paper continues the study of preclosed sets and of generalized preclosed sets in a topological space. Our main objective is to establish results about the relationships between the various types of generalized closed sets. As a by-product, we are able to provide characterizations of certain known classes of topological spaces by using preclosed sets and their generalizations.
2009-08-28T03:21:06Z
2009-08-28T03:21:06Z
1999-06
Technical Report
Department of Mathematics - Research Reports-421 (1999)
1173-0889
http://hdl.handle.net/2292/5009
Research Reports - Department of Mathematics
https://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htm
The author(s)
Department of Mathematics, The University of Auckland, New Zealand
http://www.math.auckland.ac.nz/Research/Reports/view.php?id=421
oai:researchspace.auckland.ac.nz:2292/49892009-08-28T12:38:53Zcom_2292_122col_2292_4963
A Lattice-valued Banach-Stone Theorem
Cao, Jiling
Reilly, Ivan
Xiong, Hongyun
Let $X$ and $Y$ be two compact Hausdorff spaces, and $E$ be a Banach lattice. We show that if there is a non-vanishing preserving Riesz isomorphism $Phi: C(X, E) to C(Y)$, then $X$ is homeomorphic to $Y$ and $E$ is Riesz isomorphic to $mathbb R$.
2009-08-28T03:20:48Z
2009-08-28T03:20:48Z
2000-03
Technical Report
Department of Mathematics - Research Reports-441 (2000)
1173-0889
http://hdl.handle.net/2292/4989
Research Reports - Department of Mathematics
https://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htm
The author(s)
Department of Mathematics, The University of Auckland, New Zealand
http://www.math.auckland.ac.nz/Research/Reports/view.php?id=441
oai:researchspace.auckland.ac.nz:2292/49882009-08-28T12:36:41Zcom_2292_122col_2292_4963
Approximability of Dodgson's rule
McCabe-Dansted, John C.
Pritchard, Geoffrey
Slinko, Arkadii
It is known that Dodgson's rule is computationally very demanding. Tideman (1987) suggested an approximation to it but did not investigate how often his approximation selects the Dodgson winner. We show that under the Impartial Culture assumption the probability that that the another approximation - we call it Dodgson Quick - for which thisconvergence of this probability to 1 is slow. We suggest convergence is exponentially fast. Also we show that Simpson and Dodgson rules are asymptotically different. We formulate, and heavily use in construction of examples, the generalization of McGarvey's theorem (1953) for weighted majority relations.
2009-08-28T03:20:47Z
2009-08-28T03:20:47Z
2006-06
Technical Report
Department of Mathematics - Research Reports-551 (2006)
1173-0889
http://hdl.handle.net/2292/4988
Research Reports - Department of Mathematics
https://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htm
The author(s)
Department of Mathematics, The University of Auckland, New Zealand
http://www.math.auckland.ac.nz/Research/Reports/view.php?id=551
oai:researchspace.auckland.ac.nz:2292/49862009-08-28T12:36:40Zcom_2292_122col_2292_4963
Sewing Ribbons on Graphs in Space
Archdeacon, Dan
Bonnington, Paul
Richter, Bruce
Siran, Jozef
An {em open ribbon} is a square with one side called the {em seam}. A {em closed ribbon} is a cylinder with one boundary component called the {em seam}. We {em sew} an open (resp.~closed) ribbon onto a graph by identifying the seam with an open (resp.~closed) walk in the graph. A {em ribbon complex} is a graph with a finite number of ribbons sewn on. We investigate when a ribbon complex embeds in 3-dimensional Euclidean space. We give several characterizations of such {em spatial} complexes which lead to algorithms. We examine special cases where: 1) each edge of the graph is incident with at most three ribbons, and 2) every ribbon is closed together with a connectivity condition.
2009-08-28T03:20:45Z
2009-08-28T03:20:45Z
2000
Technical Report
Department of Mathematics - Research Reports-443 (2000)
1173-0889
http://hdl.handle.net/2292/4986
Research Reports - Department of Mathematics
https://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htm
The author(s)
Department of Mathematics, The University of Auckland, New Zealand
http://www.math.auckland.ac.nz/Research/Reports/view.php?id=443
oai:researchspace.auckland.ac.nz:2292/50922009-08-28T12:40:34Zcom_2292_122col_2292_4963
A new proof of the girth-chromatic number theorem
Marshall, Simon
We give a new proof of the classical Erdos theorem on the existence of graphs with arbitrarily high chromatic number and girth. Rather than considering random graphs where the edges are chosen with some carefully adjusted probability, we use a simple counting argument on a set of graphs with bounded degree.
2009-08-28T03:22:20Z
2009-08-28T03:22:20Z
2004-12
Technical Report
Department of Mathematics - Research Reports-531 (2004)
1173-0889
http://hdl.handle.net/2292/5092
Research Reports - Department of Mathematics
https://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htm
The author(s)
Department of Mathematics, The University of Auckland, New Zealand
http://www.math.auckland.ac.nz/Research/Reports/view.php?id=531
oai:researchspace.auckland.ac.nz:2292/51502009-08-28T12:38:35Zcom_2292_122col_2292_4963
Maximal Embeddings of Directed Multi-Cycles
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.
2009-08-28T03:23:12Z
2009-08-28T03:23:12Z
2002-02
Technical Report
Department of Mathematics - Research Reports-478 (2002)
1173-0889
http://hdl.handle.net/2292/5150
Research Reports - Department of Mathematics
https://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htm
The author(s)
Department of Mathematics, The University of Auckland, New Zealand
http://www.math.auckland.ac.nz/Research/Reports/view.php?id=478
oai:researchspace.auckland.ac.nz:2292/50592009-08-28T12:40:01Zcom_2292_122col_2292_4963
On a projection from one co-invariant subspace onto another in character-automorphic Hardy space on a multiply connected domain
Fedorov, Sergei
The research was supported by Marsden Fund grant 96-UOA-MIS-0098
2009-08-28T03:21:50Z
2009-08-28T03:21:50Z
1997-04
Technical Report
Department of Mathematics - Research Reports-375 (1997)
1173-0889
http://hdl.handle.net/2292/5059
Research Reports - Department of Mathematics
https://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htm
The author(s)
Department of Mathematics, The University of Auckland, New Zealand
http://www.math.auckland.ac.nz/Research/Reports/view.php?id=375
oai:researchspace.auckland.ac.nz:2292/51612009-08-28T12:38:47Zcom_2292_122col_2292_4963
Metrisability of Manifolds in Terms of Function Spaces
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.
2009-08-28T03:23:23Z
2009-08-28T03:23:23Z
2001-07
Technical Report
Department of Mathematics - Research Reports-467 (2001)
1173-0889
http://hdl.handle.net/2292/5161
Research Reports - Department of Mathematics
https://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htm
The author(s)
Department of Mathematics, The University of Auckland, New Zealand
http://www.math.auckland.ac.nz/Research/Reports/view.php?id=467
oai:researchspace.auckland.ac.nz:2292/50912009-08-28T12:40:34Zcom_2292_122col_2292_4963
STRIP TEMPERATURE IN A METAL COATING LINE ANNEALING FURNACE
McGuinness, Mark
Taylor, Stephen
We discuss the work done at MISG 2004 on the mathematical modelling of a long, electric radiant furnace used to anneal strips of steel. The annealing process involves heating the steel, which is passed continuously through the furnace, to certain temperatures and then cooling it, resulting in a change in the crystalline structure of the steel. The furnace settings are often changed to cater for products with different metallurgical properties and varying dimensions. The mathematical model is desired to optimise the running of the furnace, especially during periods of change in furnace operation.
2009-08-28T03:22:19Z
2009-08-28T03:22:19Z
2004-09
Technical Report
Department of Mathematics - Research Reports-532 (2004)
1173-0889
http://hdl.handle.net/2292/5091
Research Reports - Department of Mathematics
https://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htm
The author(s)
Department of Mathematics, The University of Auckland, New Zealand
http://www.math.auckland.ac.nz/Research/Reports/view.php?id=532
oai:researchspace.auckland.ac.nz:2292/50872009-08-28T12:40:27Zcom_2292_122col_2292_4963
The product of a Baire space with a hereditarily Baire metric space is Baire
Moors, Warren B.
In this paper we prove that the product of a Baire space with a metrizable hereditarily Baire space is again Baire. This answers a recent question of J. Chaber and R. Pol.
2009-08-28T03:22:14Z
2009-08-28T03:22:14Z
2005-02
Technical Report
Department of Mathematics - Research Reports-535 (2005)
1173-0889
http://hdl.handle.net/2292/5087
Research Reports - Department of Mathematics
https://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htm
The author(s)
Department of Mathematics, The University of Auckland, New Zealand
http://www.math.auckland.ac.nz/Research/Reports/view.php?id=535
oai:researchspace.auckland.ac.nz:2292/50202009-08-28T12:39:19Zcom_2292_122col_2292_4963
Brachistochrones For Attractive Logarithmic Potential
Tee, Garry J.
Brachistochrones are constructed for attractive central force, with logarithmic potential. Each pair of points (except those with the centre between them) are connected by an unique brachistochrone.
2009-08-28T03:21:16Z
2009-08-28T03:21:16Z
1999-01
Technical Report
Department of Mathematics - Research Reports-410 (1999)
1173-0889
http://hdl.handle.net/2292/5020
Research Reports - Department of Mathematics
https://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htm
The author(s)
Department of Mathematics, The University of Auckland, New Zealand
http://www.math.auckland.ac.nz/Research/Reports/view.php?id=410
oai:researchspace.auckland.ac.nz:2292/50522009-08-28T12:39:53Zcom_2292_122col_2292_4963
On Bernstein's comparison theorem, Peano kernels of constant sign and near-minimax approximation
Waldron, Shayne
Some basic properties of what are called `B(ernstein)-monotone' seminorms are investigated. These lie between the classes of monotone and sign-monotone seminorms. It is seen that these seminorms arise naturally in Bernstein's comparison theorem, the description of Peano kernels of constant sign, and in near-minimax approximations. A number of new results are obtained including some sufficient conditions for a projection to be near-minimax which are easily seen to be satisfied by all the known examples, and a characterisation of the Peano kernels of constant sign where derivatives are replaced by divided differences
2009-08-28T03:21:44Z
2009-08-28T03:21:44Z
1997-05
Technical Report
Department of Mathematics - Research Reports-381 (1997)
1173-0889
http://hdl.handle.net/2292/5052
Research Reports - Department of Mathematics
https://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htm
The author(s)
Department of Mathematics, The University of Auckland, New Zealand
http://www.math.auckland.ac.nz/Research/Reports/view.php?id=381
oai:researchspace.auckland.ac.nz:2292/50322009-08-28T12:39:34Zcom_2292_122col_2292_4963
On the manipulability of proportional representation
Slinko, Arkadii
White, Shaun
This paper presents a new model of voter behaviour under methods of proportional representation (PR). We assume that voters are concerned, first and foremost, with the distribution of power in the post-election parliament. We abstract away from rounding, and assume that a party securing k percent of the vote wins exactly k percent of the available seats. We show that, irrespective of which positional scoring rule is adopted, there will always exist circumstances where a voter would have an incentive to vote insincerely. We demonstrate that a voter's attitude toward uncertainty can influence his or her incentives to make an insincere vote. Finally, we show that the introduction of a threshold - a rule that a party must secure at least a certain percentage of the vote in order to reach parliament - creates new opportunities for strategic voting. We use the model to explain voter behaviour at the most recent New Zealand general election.
2009-08-28T03:21:26Z
2009-08-28T03:21:26Z
2006-03
Technical Report
Department of Mathematics - Research Reports-547 (2006)
1173-0889
http://hdl.handle.net/2292/5032
Research Reports - Department of Mathematics
https://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htm
The author(s)
Department of Mathematics, The University of Auckland, New Zealand
http://www.math.auckland.ac.nz/Research/Reports/view.php?id=547
oai:researchspace.auckland.ac.nz:2292/50172009-08-28T12:39:17Zcom_2292_122col_2292_4963
About Scattering on the Ring
Bogevolnov, V.B.
Mikhailova, A.B.
Pavlov, B.
Yafyasov, A.M.
The mathematical model of a simplest quasi-one-dimensional quantum network constructed of relatively narrow waveguides (the width of the waveguide is less than the de Broghlie wavelength of the electron in the material) is developed. This model allows to reduce the problem of calculating the current through the quantum network to the construction of scattered waves for some Schr"{o}dinger equation on the corresponding one-dimensional graph. We consider a graph consisting of a compact part and few semiinfinite rays attached to it via some boundary condition depending on a parameter $beta$ (analog of the inverse exponential hight $e^{-bH}$ of a potential barrier $H$ separating the rays from the compact part). This parameter regulates the connection between the rays and the compact part. Spectral properties of the Schr"{o}dinger operator on this graph are described with a special emphasis on the resonance case when the Fermi level in the rays coincides with one of eigenvalues of the nonperturbed Schr"{o}dinger operator on the ring. An explicit expression is obtained for the scattering matrix in the resonance case for weakening connection between the rays and the compact part.
2009-08-28T03:21:13Z
2009-08-28T03:21:13Z
1999-03
Technical Report
Department of Mathematics - Research Reports-413 (1999)
1173-0889
http://hdl.handle.net/2292/5017
Research Reports - Department of Mathematics
https://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htm
The author(s)
Department of Mathematics, The University of Auckland, New Zealand
http://www.math.auckland.ac.nz/Research/Reports/view.php?id=413
oai:researchspace.auckland.ac.nz:2292/50302019-12-19T03:30:19Zcom_2292_122col_2292_4963
Dade's Conjecture for Steinberg Triality Groups $^3D_4(q)$ in Non-defining Characteristics
An, Jianbei
[no abstract available]
2009-08-28T03:21:24Z
2009-08-28T03:21:24Z
1998-08
Technical Report
Department of Mathematics - Research Reports-401 (1998)
1173-0889
http://hdl.handle.net/2292/5030
Research Reports - Department of Mathematics
2041
16440
7362
7379
https://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htm
The author(s)
Department of Mathematics, The University of Auckland, New Zealand
http://www.math.auckland.ac.nz/Research/Reports/view.php?id=401
oai:researchspace.auckland.ac.nz:2292/51062009-08-28T12:37:41Zcom_2292_122col_2292_4963
Dirichlet-to-Neumann map machinery for resonance gaps and bands of periodic Networks
Fox, C.
Oleinik, V.
Pavlov, B.
Usually spectral structure of the ordinary periodic Schr"{o}dinger operator is revealed based on analysis of the corresponding transfer-matrix. In this approach the quasi-momentum exponentials appear as eigenvalues of the transfer-matrix which correspond to quasi-periodic solutions of the homogeneous Schr"{o}dinger equation, and the corresponding Weyl functions are obtained as coordinates of the appropriate eigenvectors. This approach, though effective for tight-binding analysis of one-dimensional periodic Schr"{o}dinger operators, is inconvenient for spectral analysis on realistic periodic quantum networks with multi-dimensional period, where several leads are attached to each vertex, and can't be extended to partial Schr"{o}dinger equation. We propose an alternative approach where the Dirichlet-to-Neumann map is used instead of the transfer matrix. We apply this approach to obtain, for realistic quantum networks, conditions of existence of resonance gaps or bands.
2009-08-28T03:22:32Z
2009-08-28T03:22:32Z
2004-06
Technical Report
Department of Mathematics - Research Reports-518 (2004)
1173-0889
http://hdl.handle.net/2292/5106
Research Reports - Department of Mathematics
https://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htm
The author(s)
Department of Mathematics, The University of Auckland, New Zealand
http://www.math.auckland.ac.nz/Research/Reports/view.php?id=518
oai:researchspace.auckland.ac.nz:2292/50752009-09-25T03:38:53Zcom_2292_122col_2292_4963
Geometry of Pseudospheres II.
Marshall, T.H.
We investigate finite sequences of hyperplanes in a pseudosphere. To each such sequence we associate a square symmetric matrix, the Gram matrix, which gives information about angle and incidence properties of the hyperplanes. We find when a given matrix is the Gram matrix of some sequence of hyperplanes, and when a sequence is determined up to isometry by its Gram matrix. We also consider subspaces of pseudospheres and projections onto them. This leads to an n-dimensional cosine rule for spherical and hyperbolic simplices.
2009-08-28T03:22:04Z
2009-08-28T03:22:04Z
1997-03
Technical Report
Department of Mathematics - Research Reports-353 (1997)
1173-0889
http://hdl.handle.net/2292/5075
Research Reports - Department of Mathematics
https://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htm
The author(s)
Department of Mathematics, The University of Auckland, New Zealand
http://www.math.auckland.ac.nz/Research/Reports/view.php?id=353
oai:researchspace.auckland.ac.nz:2292/51632009-08-28T12:38:48Zcom_2292_122col_2292_4963
Trading crossings for handles and crosscaps
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).
2009-08-28T03:23:25Z
2009-08-28T03:23:25Z
2000-11
Technical Report
Department of Mathematics - Research Reports-465 (2000)
1173-0889
http://hdl.handle.net/2292/5163
Research Reports - Department of Mathematics
https://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htm
The author(s)
Department of Mathematics, The University of Auckland, New Zealand
http://www.math.auckland.ac.nz/Research/Reports/view.php?id=465
oai:researchspace.auckland.ac.nz:2292/50312009-08-28T12:39:33Zcom_2292_122col_2292_4963
The infinitude of 7-arc-transitive graphs
Conder, Marston
Walker, C.
In this paper it is shown that for all but finitely many positive integers $n$, there is a finite connected 7-arc-transitive quartic graph with the alternating group $A_n$ acting transitively on its 7-arcs, and another with the symmetric group $S_n$ acting transitively on its 7-arcs. The proof uses a construction involving permutation representations to obtain finite graphs with the desired property.
2009-08-28T03:21:25Z
2009-08-28T03:21:25Z
1998-06
Technical Report
Department of Mathematics - Research Reports-400 (1998)
1173-0889
http://hdl.handle.net/2292/5031
Research Reports - Department of Mathematics
https://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htm
The author(s)
Department of Mathematics, The University of Auckland, New Zealand
http://www.math.auckland.ac.nz/Research/Reports/view.php?id=400
oai:researchspace.auckland.ac.nz:2292/50972009-08-28T12:40:03Zcom_2292_122col_2292_4963
Condition for the Discreteness of the Laplacean on a Manifold
Harmer, Mark
Pavlov, B.
Here we propose a simple condition for the compactness of the resolvent of the Laplace-Beltrami operator on a class of smooth Riemannian manifolds.
2009-08-28T03:22:24Z
2009-08-28T03:22:24Z
2004
Technical Report
Department of Mathematics - Research Reports-526 (2004)
1173-0889
http://hdl.handle.net/2292/5097
Research Reports - Department of Mathematics
https://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htm
The author(s)
Department of Mathematics, The University of Auckland, New Zealand
http://www.math.auckland.ac.nz/Research/Reports/view.php?id=526
oai:researchspace.auckland.ac.nz:2292/51272009-08-28T12:38:07Zcom_2292_122col_2292_4963
Conformal Mappings from the Upper Half Plane to Fundamental Domains on the Hyperbolic Plane
Harmer, Mark
Martin, Gaven
We use the classical theory of Schwarz-Christoffel mappings to find conformal maps from the upper half plane to triangular regions in the hyperbolic plane. We then find the pullback of the (hyperbolic) Laplace-Beltrami operator to the upper half plane.
2009-08-28T03:22:50Z
2009-08-28T03:22:50Z
2003-06
Technical Report
Department of Mathematics - Research Reports-499 (2003)
1173-0889
http://hdl.handle.net/2292/5127
Research Reports - Department of Mathematics
https://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htm
The author(s)
Department of Mathematics, The University of Auckland, New Zealand
http://www.math.auckland.ac.nz/Research/Reports/view.php?id=499
oai:researchspace.auckland.ac.nz:2292/49772009-08-28T12:35:51Zcom_2292_122col_2292_4963
On algebraic torsion forms and their spin holonomy algebras
Bernhardt, Niels
Nagy, Paul-Andi
We study holonomy algebras generated by an algebraic element of the Clifford algebra, or equivalently,the holonomy algebras of certain spin connections in flat space. We provide some series of examples in arbitrary dimensions and prove some general properties of the holonomy algebras under some mild conditions on the generating element. We show that the first non-standard situation to look at appears in dimension $8$ and concerns $4$-forms. In this case complete structure results are obtained when moreover assuming the $4$-form to be self-dual.
2009-08-28T03:20:36Z
2009-08-28T03:20:36Z
2006-08
Technical Report
Department of Mathematics - Research Reports-552 (2006)
1173-0889
http://hdl.handle.net/2292/4977
Research Reports - Department of Mathematics
https://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htm
The author(s)
Department of Mathematics, The University of Auckland, New Zealand
http://www.math.auckland.ac.nz/Research/Reports/view.php?id=552
oai:researchspace.auckland.ac.nz:2292/51452009-08-28T12:38:28Zcom_2292_122col_2292_4963
The Majoritarian Compromise in Large Societies
Slinko, Arkadii
First, we dwell on the definition of the Majoritarian Compromise in the case of an odd number of alternatives. Then, assuming the Impartial Culture hypothesis we calculate the average maximum welfare achievable by the Majoritarian Compromise procedure and show that this social choice rule is asymptotically stable with the proportion of the number of unstable profiles to the total number of profiles being in the order of $Oleft(1/sqrt{n}right)$, where $n$ is the total number of agents.
2009-08-28T03:23:07Z
2009-08-28T03:23:07Z
2002-06
Technical Report
Department of Mathematics - Research Reports-483 (2002)
1173-0889
http://hdl.handle.net/2292/5145
Research Reports - Department of Mathematics
https://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htm
The author(s)
Department of Mathematics, The University of Auckland, New Zealand
http://www.math.auckland.ac.nz/Research/Reports/view.php?id=483
oai:researchspace.auckland.ac.nz:2292/50272009-08-28T12:39:29Zcom_2292_122col_2292_4963
On the subspaces of analytic and antianalytic functions
Fedorov, Sergei
The problem of description of those positive weights on the boundary $Gamma$ of a finitely connected domain $Omega$ for which the angle in a weighted $L_2$ space on $Gamma$ between the linear space ${cal R}(Omega)$ of all rational functions on $bar{bf {C}}$ with poles outside of $Clos Omega$ and the linear space ${cal R}(Omega)_-={bar{f}vert fin {cal R}(Omega)}$ of antianalytic rational functions, is a natural analog of the problem solved in a famous Helson-Szeg"o theorem. In this paper we solve more general problem and give a complete description (in terms of necessary and sufficient conditions) of those positive weights $w$ on $Gamma$ for which the sum of the closures in $L_2(Gamma, w)$ of the subspaces ${cal R}(Omega)$ and ${cal R}(Omega)_-$ is closed and their intersection is finite dimensional. The given description is similar to that one in the Helson-Sarason Theorem, i.e. the "modified" weight should satisfy the Muckenhoupt condition.
2009-08-28T03:21:22Z
2009-08-28T03:21:22Z
1998-09
Technical Report
Department of Mathematics - Research Reports-404 (1998)
1173-0889
http://hdl.handle.net/2292/5027
Research Reports - Department of Mathematics
https://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htm
The author(s)
Department of Mathematics, The University of Auckland, New Zealand
http://www.math.auckland.ac.nz/Research/Reports/view.php?id=404
oai:researchspace.auckland.ac.nz:2292/51402009-08-28T12:38:23Zcom_2292_122col_2292_4963
A note on immersed 3-spheres in 4-space
Yashiro, Tsukasa
There are two special immersed 3--spheres in 4--space, which generate regular homotopy classes of immersed 3--spheres in 4--space. They are constructed with a special sphere eversion called $FM$--eversion. In this note we use geometric approaches to show an immersed 3--sphere in 4--space constructed with $FM$--eversions are not liftable into 5--space.
2009-08-28T03:23:02Z
2009-08-28T03:23:02Z
2002-08
Technical Report
Department of Mathematics - Research Reports-487 (2002)
1173-0889
http://hdl.handle.net/2292/5140
Research Reports - Department of Mathematics
https://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htm
The author(s)
Department of Mathematics, The University of Auckland, New Zealand
http://www.math.auckland.ac.nz/Research/Reports/view.php?id=487
oai:researchspace.auckland.ac.nz:2292/50252009-08-28T12:39:26Zcom_2292_122col_2292_4963
Embedding Digraphs on Orientable Surfaces
Bonnington, Paul
Conder, Marston
McKenna, Patricia
Morton, Margaret
We consider a notion of embedding digraphs on orientable surfaces, applicable to digraphs in which the indegree equals the outdegree for every vertex, i.e., Eulerian digraphs. This idea has been considered before in the context of "compatible Euler tours" or "orthogonal A-trails" by Andsersen at al [1] and by Bouchet [4]. This prior work has mostly been limited to embeddings of Eulerian digraphs on predetermined surfaces, and to digraphs with underlying graphs of maximum degree at most 4. In this paper, a foundation is laid for the study of all Eulerian digraph embeddings. Results are proved which are analogous to those fundamental to the theory of undirected graph embeddings, such as Duke's Theorem [5], and an infinite family of digraphs which demonstrates that the genus range for an embeddable digraph can be any nonnegative integer is given. We show that it is possible to have genus range equal to one, with arbitrarily large minimum genus, unlike in the undirected case. The difference between the minimum genera of a digraph and its underlying graph is considered, as is the difference between the maximum genera. We say that a digraph is upper-embeddable if it can be embedded with 2 or 3 regions, and prove that every regular tournament is upper-embeddable.
2009-08-28T03:21:20Z
2009-08-28T03:21:20Z
1998-12
Technical Report
Department of Mathematics - Research Reports-406 (1998)
1173-0889
http://hdl.handle.net/2292/5025
Research Reports - Department of Mathematics
https://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htm
The author(s)
Department of Mathematics, The University of Auckland, New Zealand
http://www.math.auckland.ac.nz/Research/Reports/view.php?id=406
oai:researchspace.auckland.ac.nz:2292/51232009-08-28T12:38:02Zcom_2292_122col_2292_4963
Analysis of the Dispersion Equation for the Schr"odinger Operator on Periodic Metric Graphs
Oleinik, V.L.
Pavlov, B.
Sibirev, N.V.
The spectral analysis of the Schr"odinger operator on cubic lattice type graphs is developed. Similarly to the quantum mechanical tight-binding approximation, using the well known conception of the Dirichlet-to-Neumann map asymptotic formulae for localized negative spectral bands of the Schr"odinger operator on a periodic metric graph are established. The results are illustrated by numerical calculations.
2009-08-28T03:22:47Z
2009-08-28T03:22:47Z
2003-12
Technical Report
Department of Mathematics - Research Reports-503 (2003)
1173-0889
http://hdl.handle.net/2292/5123
Research Reports - Department of Mathematics
https://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htm
The author(s)
Department of Mathematics, The University of Auckland, New Zealand
http://www.math.auckland.ac.nz/Research/Reports/view.php?id=503
oai:researchspace.auckland.ac.nz:2292/50932009-08-28T12:40:35Zcom_2292_122col_2292_4963
The ideal generated by $sigma$-nowhere dense sets
Cao, Jiling
Greenwood, Sina
In this paper, we consider the ideal $mathscr I_sigma$ generated by all $sigma$-nowhere dense sets in a topological space. Properties of this ideal and its relations with the Volterra property are explored. We show that $mathscr I_sigma$ is compatible with the topology for any given space, an analogue to the Banach category theorem. Some applications of this result and the Banach category theorem are also given.
2009-08-28T03:22:20Z
2009-08-28T03:22:20Z
2004-11
Technical Report
Department of Mathematics - Research Reports-530 (2004)
1173-0889
http://hdl.handle.net/2292/5093
Research Reports - Department of Mathematics
https://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htm
The author(s)
Department of Mathematics, The University of Auckland, New Zealand
http://www.math.auckland.ac.nz/Research/Reports/view.php?id=530
oai:researchspace.auckland.ac.nz:2292/51642009-08-28T12:38:49Zcom_2292_122col_2292_4963
Halin's Theorem for Cubic Graphs on an Annulus
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.
2009-08-28T03:23:26Z
2009-08-28T03:23:26Z
2001-06
Technical Report
Department of Mathematics - Research Reports-464 (2001)
1173-0889
http://hdl.handle.net/2292/5164
Research Reports - Department of Mathematics
https://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htm
The author(s)
Department of Mathematics, The University of Auckland, New Zealand
http://www.math.auckland.ac.nz/Research/Reports/view.php?id=464
oai:researchspace.auckland.ac.nz:2292/50712009-08-28T12:40:11Zcom_2292_122col_2292_4963
On the Discreteness of the Free Product of Finite Cyclic groups
Gehring, F.W.
Maclachlan, C.
Martin, G.J.
For $p, q ge 2$ and max${p, q} ge 3$ we denote by $c(p, q)$ the smallest number with the following property. If $f$ and $g$ are elliptic M"{o}bius transformations of orders $p$ and $q$ and if the hyperbolic distance $delta(f, g)$ between their axes is at least $c(p, q)$, then the group $langle f, grangle$ is discrete, nonelementary andisomorphic to the free product $Z_p * Z_q$. We prove here that [ cos h (c(p, q)) = {cos(pi/p) cos (pi/q)+1 over sin (pi/p)sin (pi/q)}. ] This value is attained in the $(p, q, infty)4-triangle group. We give an application concerning the commutator parameter of the free product of cyclic groups.
2009-08-28T03:22:01Z
2009-08-28T03:22:01Z
1997-04
Technical Report
Department of Mathematics - Research Reports-358 (1997)
1173-0889
http://hdl.handle.net/2292/5071
Research Reports - Department of Mathematics
https://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htm
The author(s)
Department of Mathematics, The University of Auckland, New Zealand
http://www.math.auckland.ac.nz/Research/Reports/view.php?id=358
oai:researchspace.auckland.ac.nz:2292/51312009-08-28T12:38:11Zcom_2292_122col_2292_4963
Volterra Spaces are re-visited
Cao, Jiling
Gauld, David
In this paper, we investigate weakly Volterra spaces and relevant topological properties. New characterizations of weakly Volterra spaces are provided. An analogy of the well-known Banach category theorem in terms of Volterra properties is achieved. It is shown that every weakly Volterra homogeneous space is Volterra, and there exists a metrizable Baire space whose hyperspace of nonempty compact subsets endowed with the Vietoris topology is not weakly Volterra.
2009-08-28T03:22:54Z
2009-08-28T03:22:54Z
2003-05
Technical Report
Department of Mathematics - Research Reports-495 (2003)
1173-0889
http://hdl.handle.net/2292/5131
Research Reports - Department of Mathematics
https://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htm
The author(s)
Department of Mathematics, The University of Auckland, New Zealand
http://www.math.auckland.ac.nz/Research/Reports/view.php?id=495
oai:researchspace.auckland.ac.nz:2292/50702009-08-28T12:40:10Zcom_2292_122col_2292_4963
Attractors in quasiregular semigroups
Hinkkanen, A.
Martin, G.J.
Suppose that $f$ generates a $K$-quasimeromorphic semigroup in a domain $D$ of $overline{{R}^n}$, where $n ge 2$. Suppose that $U$ is a topological ball with $overline{f(U)} subset U$ and $overline{U} subset D$, and that $f|U$ is a homeomorphism. We prove that then $U$ contains a unique fixed point $w$ of $f$ (so that $f(w) = w)$, and there is a topological ball neighbourhood $V$ of $w$ with $overline{V} subset U$ and a quasiconformal homeomorphism $g$ of $overline{{R}^n}$ onto itself with $g(w)=0$ such that $(g circ f circ g^{-1})(x)=z/2$ for all $xin g(V)$. this allows us to classify the attracting and repelling fixed points of elements of uniformly quasimeromorphic semigroups such that the element is quasiconformally conjugate to a dilation in a neighbourhood of such a point.
2009-08-28T03:22:00Z
2009-08-28T03:22:00Z
1997-04
Technical Report
Department of Mathematics - Research Reports-359 (1997)
1173-0889
http://hdl.handle.net/2292/5070
Research Reports - Department of Mathematics
https://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htm
The author(s)
Department of Mathematics, The University of Auckland, New Zealand
http://www.math.auckland.ac.nz/Research/Reports/view.php?id=359
oai:researchspace.auckland.ac.nz:2292/51412009-08-28T12:38:24Zcom_2292_122col_2292_4963
DDAE: an integrator for ODEs, DAEs and DDEs, part I
Sharp, P.W.
Krogh, F.
DDAE is a variable order, variable stepsize Adams and BDF Fortran integrator for solving initial-value ordinary differential equations (ODEs), initial-value differential-algebraic equations (DAEs) of index 0 and 1, and delay differential equations (DDEs). The differential equations can be mixed order, and the DDEs can have both state-dependent and multiple delays and can include DAEs. DDAE has a large number of optional inputs. Options permit the user to perform a wide range of tasks and to take advantage of features of a problem to improve efficiency. The options include those for varying the interpolation, saving the solution, controlling the stepsize and order selection, and solving for g-stops. DDAE also has reverse communication (returning to the driver calling DDAE for function evaluations) as an option. This makes it easy for the user to call DDAE from other software and to use special software for solving linear equations. A distinctive feature of DDAE is the ability to group the equations and use different options for different groups. This can lead to a marked reduction in the CPU time. For example, the equations could be divided into non-stiff and stiff equations, and Adams and BDF methods used for the two groups respectively. This report summarises the features of DDAE with an emphasis on the options.
2009-08-28T03:23:03Z
2009-08-28T03:23:03Z
2002-07
Technical Report
Department of Mathematics - Research Reports-486 (2002)
1173-0889
http://hdl.handle.net/2292/5141
Research Reports - Department of Mathematics
https://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htm
The author(s)
Department of Mathematics, The University of Auckland, New Zealand
http://www.math.auckland.ac.nz/Research/Reports/view.php?id=486
oai:researchspace.auckland.ac.nz:2292/50642009-08-28T12:40:05Zcom_2292_122col_2292_4963
Numerical Methods for Differential Equations and Applications
Butcher, J.C.
This paper surveys a number of aspects of numerical methods for ordinary differential equations. The discussion includes the method of Euler and introduces Runge-Kutta methods and linear multistep methods as generalizations of Euler. Stability considerations arising from stiffness lead to a discussion of implicit methods and implementation issues. To the extent possible within this short survey, numerical methods are looked at in the context of problems arising in practical applications.
2009-08-28T03:21:54Z
2009-08-28T03:21:54Z
1997-03
Technical Report
Department of Mathematics - Research Reports-370 (1997)
1173-0889
http://hdl.handle.net/2292/5064
Research Reports - Department of Mathematics
https://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htm
The author(s)
Department of Mathematics, The University of Auckland, New Zealand
http://www.math.auckland.ac.nz/Research/Reports/view.php?id=370
oai:researchspace.auckland.ac.nz:2292/49952009-08-28T12:38:58Zcom_2292_122col_2292_4963
Experiences of Doctoral Students in Mathematics in New Zealand
Morton, Margaret
Thornley, Gillian
This study investigates factors affecting doctoral study in mathematics and mathematics education in New Zealand universities. In particular, it gives insight into the problems faced by students and provides comprehensive information for the mathematical community. A questionnaire to students gathered information including their financial support, initial motivation to pursue a doctorate, the level of satisfaction they were experiencing from their studies, their perceptions of the supervisory process, their experiences as research students, and their hopes for the future.
2009-08-28T03:20:53Z
2009-08-28T03:20:53Z
1999-12
Technical Report
Department of Mathematics - Research Reports-434 (1999)
1173-0889
http://hdl.handle.net/2292/4995
Research Reports - Department of Mathematics
https://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htm
The author(s)
Department of Mathematics, The University of Auckland, New Zealand
http://www.math.auckland.ac.nz/Research/Reports/view.php?id=434
oai:researchspace.auckland.ac.nz:2292/51262009-08-28T12:38:06Zcom_2292_122col_2292_4963
Cohomology of real Lie algebras
Josef, Silhan
We show how to describe the cohomology of a nilpotent part of some parabolic subalgebra of a semisimple Lie algebra with values in its irreducible representation. The situation in the complex case is well--known, the Kostant's result (see below) gives an explicit description of a representation of a proper reductive subalgebra on the space of the complex cohomology. The aim of this work is to read the structure of the real cohomology from the structure of the complex one. We will use the notation of Dynkin and Satake diagrams for the description of semisimple and parabolic real and complex Lie algebras and their representations.
2009-08-28T03:22:50Z
2009-08-28T03:22:50Z
2003
Technical Report
Department of Mathematics - Research Reports-500 (2003)
1173-0889
http://hdl.handle.net/2292/5126
Research Reports - Department of Mathematics
https://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htm
The author(s)
Department of Mathematics, The University of Auckland, New Zealand
http://www.math.auckland.ac.nz/Research/Reports/view.php?id=500
oai:researchspace.auckland.ac.nz:2292/51072009-08-28T12:37:42Zcom_2292_122col_2292_4963
Simulation of ancestral graphs for Monte Carlo integration
Cloete, Nicoleen
Nicholls, Geoff
Scott, David J.
An ancestral selection graph is a realization of an genealogy-process model which incorporates natural selection. The space of ancestral graphs is a countable union of spaces of unequal dimensions. We give a Markov Chain Monte Carlo algorithm simulating ancestral selection graphs. Output can be used to estimate expectations for functions defined on the space of ancestral graphs.
2009-08-28T03:22:33Z
2009-08-28T03:22:33Z
2003-08
Technical Report
Department of Mathematics - Research Reports-517 (2003)
1173-0889
http://hdl.handle.net/2292/5107
Research Reports - Department of Mathematics
https://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htm
The author(s)
Department of Mathematics, The University of Auckland, New Zealand
http://www.math.auckland.ac.nz/Research/Reports/view.php?id=517
oai:researchspace.auckland.ac.nz:2292/50692009-08-28T12:40:10Zcom_2292_122col_2292_4963
Some extended explicit Bel'tyukov pairs for Volterra integral equations of the second kind
Sharp, P.W.
We derive and investigate a family of extended explicit Bel'tyukov (EBVRK) pairs for Volterra integral equations of the second kind. The pairs use six stages, and consist of an order 3 formula completely embedded in an order 4 formula. As part of the derivation, we show that at least 6 stages are needed to form such pairs. We also examine some aspects of the structure of EBVRK pairs.
2009-08-28T03:21:59Z
2009-08-28T03:21:59Z
1997-04
Technical Report
Department of Mathematics - Research Reports-362 (1997)
1173-0889
http://hdl.handle.net/2292/5069
Research Reports - Department of Mathematics
https://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htm
The author(s)
Department of Mathematics, The University of Auckland, New Zealand
http://www.math.auckland.ac.nz/Research/Reports/view.php?id=362
oai:researchspace.auckland.ac.nz:2292/50842009-08-28T12:40:24Zcom_2292_122col_2292_4963
Monotonicity of Some Functions in Calculus
Anderson, G.D.
Vamanamurthy, M.K.
Vuorinen, M.
In the note we study a monotone analogue of L'Hospital Rule for limits of indeterminate forms. This technique greatly simplifies proofs of monotonicity of such forms.
2009-08-28T03:22:12Z
2009-08-28T03:22:12Z
2005
Technical Report
Department of Mathematics - Research Reports-538 (2005)
1173-0889
http://hdl.handle.net/2292/5084
Research Reports - Department of Mathematics
https://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htm
The author(s)
Department of Mathematics, The University of Auckland, New Zealand
http://www.math.auckland.ac.nz/Research/Reports/view.php?id=538
oai:researchspace.auckland.ac.nz:2292/50802009-08-28T12:40:17Zcom_2292_122col_2292_4963
Discrete wave scattering on a star-graph
Fedorov, Sergei
Pavlov, B.
We consider the spectral problem for the adjacency matrix of a graph composed of a compact part with a few semi-infinite periodic leads attached. Based on the spectral properties of the adjacency matrix we develop Lax-Phillips scattering theory for the corresponding discrete wave equation.
2009-08-28T03:22:08Z
2009-08-28T03:22:08Z
2005-12
Technical Report
Department of Mathematics - Research Reports-542 (2005)
1173-0889
http://hdl.handle.net/2292/5080
Research Reports - Department of Mathematics
https://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htm
The author(s)
Department of Mathematics, The University of Auckland, New Zealand
http://www.math.auckland.ac.nz/Research/Reports/view.php?id=542
oai:researchspace.auckland.ac.nz:2292/51652009-08-28T12:38:50Zcom_2292_122col_2292_4963
LEMNISCATES AND THE SPECTRUM OF THE PERTURBED SHIFT
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.
2009-08-28T03:23:27Z
2009-08-28T03:23:27Z
2001-05
Technical Report
Department of Mathematics - Research Reports-463 (2001)
1173-0889
http://hdl.handle.net/2292/5165
Research Reports - Department of Mathematics
https://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htm
The author(s)
Department of Mathematics, The University of Auckland, New Zealand
http://www.math.auckland.ac.nz/Research/Reports/view.php?id=463
oai:researchspace.auckland.ac.nz:2292/51472009-08-28T12:38:32Zcom_2292_122col_2292_4963
Russian Peasant Multiplication and Egyptian Division in Zeckendorf Arithmetic
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.
2009-08-28T03:23:09Z
2009-08-28T03:23:09Z
2002-03
Technical Report
Department of Mathematics - Research Reports-481 (2002)
1173-0889
http://hdl.handle.net/2292/5147
Research Reports - Department of Mathematics
https://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htm
The author(s)
Department of Mathematics, The University of Auckland, New Zealand
http://www.math.auckland.ac.nz/Research/Reports/view.php?id=481
oai:researchspace.auckland.ac.nz:2292/50682009-09-25T03:38:08Zcom_2292_122col_2292_4963
Dual precision software for checking explicit Runge-Kutta methods
Sharp, P.W.
Verner, J.H.
Coefficients of an explicit Runge-Kutta method which might be used for
initial-value ordinary differential equations, delay differential equations or
Volterra integral equations, often require many digits for their representation.
This can make manual checking of the coefficients unreliable.
We present a dual precision Fortran 77 package for checking the coefficients of an explicit Runge-Kutta method consisting of k formulae. In some
instances when a coefficient is wrong, the output from the package can be
used to deduce which coefficient is likely to be wrong
2009-08-28T03:21:58Z
2009-08-28T03:21:58Z
1997-03
Technical Report
Department of Mathematics - Research Reports-363 (1997)
1173-0889
http://hdl.handle.net/2292/5068
Research Reports - Department of Mathematics
https://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htm
The author(s)
Department of Mathematics, The University of Auckland, New Zealand
http://www.math.auckland.ac.nz/Research/Reports/view.php?id=363
oai:researchspace.auckland.ac.nz:2292/51432009-08-28T12:38:27Zcom_2292_122col_2292_4963
On Asymptotic Coalitional Strategy-Proofness of Social Choice Rules under the IAC Assumptoin
Slinko, Arkadii
We study the class of scoring rules and multistage elimination rules under the Impartial Anonymous Culture assumption. We show that, when the number of participating agents n tends to infinity, the proportion of the number of voting situations manipulable by a coalition of k voters to all voting situations is smaller than Ck/n, where C is the constant which depends only on the number of alternatives but not on k and n.
2009-08-28T03:23:05Z
2009-08-28T03:23:05Z
2002-06
Technical Report
Department of Mathematics - Research Reports-484 (2002)
1173-0889
http://hdl.handle.net/2292/5143
Research Reports - Department of Mathematics
https://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htm
The author(s)
Department of Mathematics, The University of Auckland, New Zealand
http://www.math.auckland.ac.nz/Research/Reports/view.php?id=484
oai:researchspace.auckland.ac.nz:2292/50082009-08-28T12:39:08Zcom_2292_122col_2292_4963
Domination Conditions for Tournaments
McKenna, Patricia
Morton, Margaret
Sneddon, Jamie
The definition for the domination graph of a tournament states that it has the same vertices as the tournament with an edge between two vertices if every other vertex is beaten by at least one of them. In this paper two new types of domination graphs are defined by using different relaxations of the adjacency definition. The first type is formed by reducing the number of vertices which must be dominated by a pair of vertices and the second by increasing the number of steps allowable for domination. Properties of these new types of domination graphs are presented with comparison between them where appropriate. In particular a full characterisation of each type is given for rotational tournements.
2009-08-28T03:21:05Z
2009-08-28T03:21:05Z
1999-08
Technical Report
Department of Mathematics - Research Reports-422 (1999)
1173-0889
http://hdl.handle.net/2292/5008
Research Reports - Department of Mathematics
https://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htm
The author(s)
Department of Mathematics, The University of Auckland, New Zealand
http://www.math.auckland.ac.nz/Research/Reports/view.php?id=422
oai:researchspace.auckland.ac.nz:2292/51442009-08-28T12:38:28Zcom_2292_122col_2292_4963
Four dimensional conformal C-spaces
Gover , A. Rod
Nagy, Paul-Andi
We investigate the structure of conformal C-spaces, a class of Riemannian manifolds which naturally arises as a conformal generalisation of the Einstein condition. A basic question is when such a structure is closed, or equivalently locally conformally Cotton. In dimension 4 we obtain a full answer to this question and also investigate the incidence of the Bach condition on this class of metrics. This is related to earlier results obtained in the Einstein-Weyl context.
2009-08-28T03:23:06Z
2009-08-28T03:23:06Z
2007-01
Technical Report
Department of Mathematics - Research Reports-555 (2007)
1173-0889
http://hdl.handle.net/2292/5144
Research Reports - Department of Mathematics
https://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htm
The author(s)
Department of Mathematics, The University of Auckland, New Zealand
http://www.math.auckland.ac.nz/Research/Reports/view.php?id=555
oai:researchspace.auckland.ac.nz:2292/51092009-08-28T12:37:43Zcom_2292_122col_2292_4963
Rankings of Multisets and Discrete Cones
Conder, Marston
Marshall, Simon
Slinko, Arkadii
We study additive representability of orders on multisets (of size $k$ drawn from a set of size $n$) which satisfy the condition of Independence of Equal Submultisets (IES) introduced by Sertel and Slinko (2002). Here we take a geometric view of those orders, and relate them to certain combinatorial objects which we call discrete cones. Following Fishburn (1996) and Conder and Slinko (2003), we define functions $f(n,k)$ and $g(n,k)$ which measure the maximal possible deviation of an arbitrary order satisfying the IES and an arbitrary almost representable order satisfying the IES, respectively, from a representable order. We prove that $g(n,k)=n-1$ whenever $nge 3$ and $(n,k)ne (5,2)$. In the exceptional case, $g(5,2)=3$. We also prove that $g(n,k)le f(n,k)le n$ and establish that for small $n$ and $k$ the functions $g(n,k)$ and $f(n,k)$ coincide.
2009-08-28T03:22:34Z
2009-08-28T03:22:34Z
2004-04
Technical Report
Department of Mathematics - Research Reports-515 (2004)
1173-0889
http://hdl.handle.net/2292/5109
Research Reports - Department of Mathematics
https://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htm
The author(s)
Department of Mathematics, The University of Auckland, New Zealand
http://www.math.auckland.ac.nz/Research/Reports/view.php?id=515
oai:researchspace.auckland.ac.nz:2292/50072009-08-28T12:39:07Zcom_2292_122col_2292_4963
Homeomorphism Groups of Manifolds*
Gartside, P.M.
Mohamad, A.M.
In this paper we present the homeomorphism groups of manifolds, explaining why non-metrizable manifolds are better behaved, with regard to their homeomorphism groups, than metrizable manifolds. A proof that the natural topology on the homeomorphism group for a one dimensional metrizable manifold is the minimum group topology but the homeomorphism group does not admit a minimum group topology for a more than one dimensional metrizable manifold will be given. Likewise, examples demonstrating how badly behaved are the homeomorphism groups of continua, in comparison with homeomorphism groups of manifolds is also given.
2009-08-28T03:21:04Z
2009-08-28T03:21:04Z
1999-09
Technical Report
Department of Mathematics - Research Reports-423 (1999)
1173-0889
http://hdl.handle.net/2292/5007
Research Reports - Department of Mathematics
https://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htm
The author(s)
Department of Mathematics, The University of Auckland, New Zealand
http://www.math.auckland.ac.nz/Research/Reports/view.php?id=423
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