2020-10-26T21:10:25Zhttps://researchspace.auckland.ac.nz/dspace-oai/requestoai:researchspace.auckland.ac.nz:2292/49652009-08-28T12:30:37Zcom_2292_122col_2292_4963Is It Ever Safe to Vote Strategically?Slinko, ArkadiiWhite, ShaunWe extend the Gibbard-Satterthwaite theorem in the following way. We prove that an onto, non-dictatorial social choice rule which is employed to choose one of at least three alternatives is safely manipulable. This means that on occasion a voter will have an incentive to make a strategic vote and know that he will not be worse off regardless of how other voters with similar preference orders would vote, sincerely or not.2009-08-282009-08-282008-07Technical ReportDepartment of Mathematics - Research Reports-563 (2008)1173-0889http://hdl.handle.net/2292/4965Research Reports - Department of Mathematicshttps://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htmThe author(s)Department of Mathematics, The University of Auckland, New Zealandoai:researchspace.auckland.ac.nz:2292/49662009-08-28T12:30:40Zcom_2292_122col_2292_4963Some results on Linearly Lindelof spacesGuo, HongfengJiang, ShouliSome new results about linearly Lindelof spaces are given here. It is proved that if X is a space of countable spread and X=Y U Z, where Y and Z are meta-Lindelof spaces, then X is linearly Lindelof. Moreover, we give a positive answer to a problem raised by A.V. Arhangel'skii and R.Z. Buzyakova.2009-08-282009-08-282006-11Technical ReportDepartment of Mathematics - Research Reports-553 (2006)1173-0889http://hdl.handle.net/2292/4966Research Reports - Department of Mathematicshttps://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htmThe author(s)Department of Mathematics, The University of Auckland, New Zealandoai:researchspace.auckland.ac.nz:2292/49672009-08-28T12:30:42Zcom_2292_122col_2292_4963Signed frames and Hadamard products of Gram MatricesPeng, IrineWaldron, ShayneThis 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.2009-08-282009-08-282001-04Technical ReportDepartment of Mathematics - Research Reports-462 (2001)1173-0889http://hdl.handle.net/2292/4967Research Reports - Department of Mathematicshttps://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htmThe author(s)Department of Mathematics, The University of Auckland, New Zealandoai:researchspace.auckland.ac.nz:2292/49682009-08-28T12:30:43Zcom_2292_122col_2292_4963On consistent social choice functionsKoray, SemihSlinko, ArkadiiIn 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.2009-08-282009-08-282001-04Technical ReportDepartment of Mathematics - Research Reports-461 (2001)1173-0889http://hdl.handle.net/2292/4968Research Reports - Department of Mathematicshttps://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htmThe author(s)Department of Mathematics, The University of Auckland, New Zealandoai:researchspace.auckland.ac.nz:2292/49692009-08-28T12:35:31Zcom_2292_122col_2292_4963Reservoir offset models for Radiocarbon calibrationJones, MartinNicholls, GeoffThe 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.2009-08-282009-08-282001Technical ReportDepartment of Mathematics - Research Reports-459 (2001)1173-0889http://hdl.handle.net/2292/4969Research Reports - Department of Mathematicshttps://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htmThe author(s)Department of Mathematics, The University of Auckland, New Zealandoai:researchspace.auckland.ac.nz:2292/49702009-08-28T12:35:44Zcom_2292_122col_2292_4963On Asymptotic Strategy-Proofness of Classical Social Choice RulesSlinko, ArkadiiWe 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 $O(1/sqrt{n})$.2009-08-282009-08-282000-10Technical ReportDepartment of Mathematics - Research Reports-458 (2000)1173-0889http://hdl.handle.net/2292/4970Research Reports - Department of Mathematicshttps://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htmThe author(s)Department of Mathematics, The University of Auckland, New Zealandoai:researchspace.auckland.ac.nz:2292/49712009-08-28T12:35:44Zcom_2292_122col_2292_4963Random-walk radiocarbon calibrationNicholls, GeoffChristen, J AndresIn this note we check the interesting recent results of Gomez Portugal Aguilar {it et al} cite{aguilar00}. These authors fit a simple random walk with Gaussian increments to the radiocarbon calibration curve. They suggest that the posterior standard deviation of the calibration curve is smaller between years where observations were made than it is at those years. We find that in contrast to their result the posterior standard deviations of the calibration curve bow out slightly between observation points in our estimates, for a model of the kind they describe. In earlier work Christen~cite{christen94} conditions the random walk to visit the measured calibration values. Following cite{aguilar00}, we consider the unconditioned posterior distribution for calibration curves, and thereby improve slightly on the reconstructions of cite{christen94}.2009-08-282009-08-282000Technical ReportDepartment of Mathematics - Research Reports-457 (2000)1173-0889http://hdl.handle.net/2292/4971Research Reports - Department of Mathematicshttps://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htmThe author(s)Department of Mathematics, The University of Auckland, New Zealandoai:researchspace.auckland.ac.nz:2292/49722009-08-28T12:35:45Zcom_2292_122col_2292_4963Bridge estimation of the probability density at a pointMira, AntoniettaNicholls, GeoffBridge 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.2009-08-282009-08-282003-09Technical ReportDepartment of Mathematics - Research Reports-456 (2003)1173-0889http://hdl.handle.net/2292/4972Research Reports - Department of Mathematicshttps://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htmThe author(s)Department of Mathematics, The University of Auckland, New Zealandoai:researchspace.auckland.ac.nz:2292/49732009-08-28T12:35:47Zcom_2292_122col_2292_4963Spontaneous magnetisation in the planeNicholls, GeoffThe 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.2009-08-282009-08-282000Technical ReportDepartment of Mathematics - Research Reports-455 (2000)1173-0889http://hdl.handle.net/2292/4973Research Reports - Department of Mathematicshttps://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htmThe author(s)Department of Mathematics, The University of Auckland, New Zealandoai:researchspace.auckland.ac.nz:2292/49742009-08-28T12:35:48Zcom_2292_122col_2292_4963A Result on $aleph_1$-Compact SpacesMohamad, 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.2009-08-282009-08-282000Technical ReportDepartment of Mathematics - Research Reports-454 (2000)1173-0889http://hdl.handle.net/2292/4974Research Reports - Department of Mathematicshttps://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htmThe author(s)Department of Mathematics, The University of Auckland, New Zealandoai:researchspace.auckland.ac.nz:2292/49752009-08-28T12:35:49Zcom_2292_122col_2292_4963Some Results on Quasi--$sigma$ and $theta$ SpacesMohamad, A.M.In this paper we show that a quasi--$G^*_{delta}$--diagonal plays a central role in metrizability. We prove that: if $X$ is a first--countable $GO$--space, then $X$ is metrizable if and only if $X$ is quasi--$sigma$--space; a $wtheta$--space is metrizable if and only if it is a quasi--Nagata space with a quasi--$G^*_{delta}(2)$--diagonal; a linearly ordered space $X$ with a quasi--$G^*_{delta}(2)$--diagonal is a $Theta$--space; a space $X$ is developable if and only if it is a $wtheta$, $beta$--space with a quasi--$G^*_{delta}(2)$--diagonal.2009-08-282009-08-282000Technical ReportDepartment of Mathematics - Research Reports-453 (2000)1173-0889http://hdl.handle.net/2292/4975Research Reports - Department of Mathematicshttps://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htmThe author(s)Department of Mathematics, The University of Auckland, New Zealandoai:researchspace.auckland.ac.nz:2292/49762009-08-28T12:35:49Zcom_2292_122col_2292_4963On Spaces with Quasi-Regular-$G_{delta}$-DiagonalsMohamad, A.M.This paper studies spaces with quasi--regular--$G_{delta}$--diagonal. It is shown that if $X$ is a normal space, then the following are equivalent: begin{enumerate} item $X$ admits a development satisfying the $3$--link property. item $X$ is a $wDelta$ with quasi--regular--$G_{delta}$--diagonal. item $X$ is a $wDelta$ with regular--$G_{delta}$--diagonal. item $X$ is $K$--semimetrizable via a semimetric satisfying $(AN)$. item There is a semimetric $d$ on $X$ such that: begin{enumerate} item [a.] if $langle x_n rangle$ and $langle y_n rangle$ are sequences both converging to the same point, then lim $d(x_n,y_n) = 0$, and item [b.] if $x$ and $y$ are distinct points of $X$ and $langle x_n rangle$ and $langle y_n rangle$ are sequences converging to $x$ and $y$, respectively, then there are integers $L$ and $M$ such that if $n > L$, then $d(x_n,y_n) > frac {1}{M}$. end {enumerate} end {enumerate}2009-08-282009-08-282000Technical ReportDepartment of Mathematics - Research Reports-452 (2000)1173-0889http://hdl.handle.net/2292/4976Research Reports - Department of Mathematicshttps://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htmThe author(s)Department of Mathematics, The University of Auckland, New Zealandoai:researchspace.auckland.ac.nz:2292/49772009-08-28T12:35:51Zcom_2292_122col_2292_4963On algebraic torsion forms and their spin holonomy algebrasBernhardt, NielsNagy, Paul-AndiWe 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.2009-08-282009-08-282006-08Technical ReportDepartment of Mathematics - Research Reports-552 (2006)1173-0889http://hdl.handle.net/2292/4977Research Reports - Department of Mathematicshttps://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htmThe author(s)Department of Mathematics, The University of Auckland, New Zealandoai:researchspace.auckland.ac.nz:2292/49782009-08-28T12:35:51Zcom_2292_122col_2292_4963Characterizations of Moore and Semi-Stratifiable SpacesMohamad, A.M.In this paper, we study Moore and semi--stratifiable spaces. We give characterizations of developable and semi--stratifiable spaces. We prove that: a regular space $X$ is semi--stratifiable if and only if it is a $beta$, quasi--semi--stratifiable and the following are equivalent for a regular $wDelta$--space $X$: begin{enumerate} item[(a)] $X$ is a Moore space; item[(b)] $X$ is a hereditarily weakly $theta$--refinable space with a quasi--${G}_delta$--diagonal; item[(c)] $X$ is a quasi--${G}^{*}_delta$--diagonal; item[(d)] $X$ is a quasi--semi--stratifiable space; item[(e)] $X$ is a quasi--$alpha$--space. end{enumerate}2009-08-282009-08-282000Technical ReportDepartment of Mathematics - Research Reports-451 (2000)1173-0889http://hdl.handle.net/2292/4978Research Reports - Department of Mathematicshttps://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htmThe author(s)Department of Mathematics, The University of Auckland, New Zealandoai:researchspace.auckland.ac.nz:2292/49792009-08-28T12:36:32Zcom_2292_122col_2292_4963Weak bases and MetrizabilityMohamad, 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}2009-08-282009-08-282000Technical ReportDepartment of Mathematics - Research Reports-450 (2000)1173-0889http://hdl.handle.net/2292/4979Research Reports - Department of Mathematicshttps://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htmThe author(s)Department of Mathematics, The University of Auckland, New Zealandoai:researchspace.auckland.ac.nz:2292/49802009-08-28T12:36:34Zcom_2292_122col_2292_4963Comparisons of high order Stormer and explicit Runge-Kutta Nystrom methods for N-body simulations of the solar systemSharp, 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 versions2009-08-282009-08-282000-06Technical ReportDepartment of Mathematics - Research Reports-449 (2000)1173-0889http://hdl.handle.net/2292/4980Research Reports - Department of Mathematicshttps://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htmThe author(s)Department of Mathematics, The University of Auckland, New Zealandoai:researchspace.auckland.ac.nz:2292/49812009-08-28T12:36:35Zcom_2292_122col_2292_4963Acousto-optic scattering in a single-mode optic fiberBadanin, A.V.Pavlov, B.Pokrovski, A.A.Prokhorov, L.V.The analysis of acousto-optic scattering in a single-mode fiber in terms of the effective equation with perturbation caused by variation of the speed of light is done. Using an ansatz based on the Lorentz transform we reduce the corresponding equation to Mathieu equation with a non-canonical small (acousto-optic) parameter. In the lowest order of perturbation theory we calculate the positions and widths of spectral lacunae for the case when the elastic wave is infinite. This result is applied for the estimation of the reflection coefficient $R$ in the lacunae using methods suggested earlier by authors for investigation of periodic nanostructures. We calculate explicitly the reflection coefficient for scattering by a segment of elastic wave of length $L$ and derive the relation $|R_{max}|^2=hbox{rm tanh}^2,(pifrac{L}{c}Deltanu)$ for the maximal reflection coefficient ($c$ stands for the phase velocity of light and $Deltanu$ denotes the width of the reflection band in $Hz$).2009-08-282009-08-282000Technical ReportDepartment of Mathematics - Research Reports-448 (2000)1173-0889http://hdl.handle.net/2292/4981Research Reports - Department of Mathematicshttps://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htmThe author(s)Department of Mathematics, The University of Auckland, New Zealandoai:researchspace.auckland.ac.nz:2292/49822009-08-28T12:36:35Zcom_2292_122col_2292_4963Requirements of an N-body integrator for astronomySharp, P.W.N-body simulations in astronomy form a challenging set of initial value problems for numerical integrators. The challenge comes from the variety of problems and their size - one recent simulation required 300 million second order equations, another 90 thousand million integration steps. A number of integrators for specific types of simulations are available. We investigate what is required of an integrator intended to efficiently perform a wide range of N-body simulations.2009-08-282009-08-282000-05Technical ReportDepartment of Mathematics - Research Reports-447 (2000)1173-0889http://hdl.handle.net/2292/4982Research Reports - Department of Mathematicshttps://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htmThe author(s)Department of Mathematics, The University of Auckland, New Zealandoai:researchspace.auckland.ac.nz:2292/49832009-08-28T12:36:36Zcom_2292_122col_2292_4963A relation between the spectrum of the Laplacean and the geometry of a compact graphHarmer, M.For the Laplacean on a compact graph with edges of commensurate length and flux-conserved boundary conditions we provide a description of the spectrum in terms of the geometry of the graph.2009-08-282009-08-282000Technical ReportDepartment of Mathematics - Research Reports-446 (2000)1173-0889http://hdl.handle.net/2292/4983Research Reports - Department of Mathematicshttps://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htmThe author(s)Department of Mathematics, The University of Auckland, New Zealandoai:researchspace.auckland.ac.nz:2292/49842009-08-28T12:36:37Zcom_2292_122col_2292_4963Scattering on the annulusHarmer, 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.2009-08-282009-08-282000Technical ReportDepartment of Mathematics - Research Reports-445 (2000)1173-0889http://hdl.handle.net/2292/4984Research Reports - Department of Mathematicshttps://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htmThe author(s)Department of Mathematics, The University of Auckland, New Zealandoai:researchspace.auckland.ac.nz:2292/49852009-08-28T12:36:39Zcom_2292_122col_2292_4963Hermitian symplectic geometry and the Schr"{o}dinger operator on the graphHarmer, 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.2009-08-282009-08-282000Technical ReportDepartment of Mathematics - Research Reports-444 (2000)1173-0889http://hdl.handle.net/2292/4985Research Reports - Department of Mathematicshttps://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htmThe author(s)Department of Mathematics, The University of Auckland, New Zealandoai:researchspace.auckland.ac.nz:2292/49862009-08-28T12:36:40Zcom_2292_122col_2292_4963Sewing Ribbons on Graphs in SpaceArchdeacon, DanBonnington, PaulRichter, BruceSiran, JozefAn {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.2009-08-282009-08-282000Technical ReportDepartment of Mathematics - Research Reports-443 (2000)1173-0889http://hdl.handle.net/2292/4986Research Reports - Department of Mathematicshttps://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htmThe author(s)Department of Mathematics, The University of Auckland, New Zealandoai:researchspace.auckland.ac.nz:2292/49872009-08-28T12:36:40Zcom_2292_122col_2292_4963Infinite Games Associated with Asymmetric Topology and ApplicationsCao, JilingIn this paper, we study two types of topological games, ${cal G}(x)$-games and ${cal G}({cal F})$-games, and topological spaces defined by them, namely $cal G$-spaces and game-compact spaces. It is shown these games are associated with $kappa$-semi-stratifiabilty, which is the duality of quasi-metrizability. Finally, we apply these games and relevant properties to study multi-valued maps. Consequently, the Choquet-Dolecki theorem on multi-valued maps is deduced. Main results of Hansell et al in cite{Ha} are generalized.2009-08-282009-08-282000Technical ReportDepartment of Mathematics - Research Reports-442 (2000)1173-0889http://hdl.handle.net/2292/4987Research Reports - Department of Mathematicshttps://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htmThe author(s)Department of Mathematics, The University of Auckland, New Zealandoai:researchspace.auckland.ac.nz:2292/49882009-08-28T12:36:41Zcom_2292_122col_2292_4963Approximability of Dodgson's ruleMcCabe-Dansted, John C.Pritchard, GeoffreySlinko, ArkadiiIt 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.2009-08-282009-08-282006-06Technical ReportDepartment of Mathematics - Research Reports-551 (2006)1173-0889http://hdl.handle.net/2292/4988Research Reports - Department of Mathematicshttps://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htmThe author(s)Department of Mathematics, The University of Auckland, New Zealandoai:researchspace.auckland.ac.nz:2292/49892009-08-28T12:38:53Zcom_2292_122col_2292_4963A Lattice-valued Banach-Stone TheoremCao, JilingReilly, IvanXiong, HongyunLet $X$ 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 $mathbb R$.2009-08-282009-08-282000-03Technical ReportDepartment of Mathematics - Research Reports-441 (2000)1173-0889http://hdl.handle.net/2292/4989Research Reports - Department of Mathematicshttps://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htmThe author(s)Department of Mathematics, The University of Auckland, New Zealandoai:researchspace.auckland.ac.nz:2292/49902009-08-28T12:38:54Zcom_2292_122col_2292_4963Scattering on graphs and one-dimensional approximation of $N-$dimensional Schr"odinger operatorsMelnikov, 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.2009-08-282009-08-282000Technical ReportDepartment of Mathematics - Research Reports-439 (2000)1173-0889http://hdl.handle.net/2292/4990Research Reports - Department of Mathematicshttps://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htmThe author(s)Department of Mathematics, The University of Auckland, New Zealandoai:researchspace.auckland.ac.nz:2292/49912009-08-28T12:38:54Zcom_2292_122col_2292_4963A Generalization of Komlos Theorem on Random MatricesSlinko, 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}$2009-08-282009-08-282000-03Technical ReportDepartment of Mathematics - Research Reports-438 (2000)1173-0889http://hdl.handle.net/2292/4991Research Reports - Department of Mathematicshttps://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htmThe author(s)Department of Mathematics, The University of Auckland, New Zealandoai:researchspace.auckland.ac.nz:2292/49922009-11-19T00:46:40Zcom_2292_122col_2292_4963High order explicit Runge-Kutta pairs for ephemerides of the solar system and the Moon. (1999)Sharp, P.W.Numerically integrated ephemerides of the solar system and the Moon require very accurate integrations of systems of second order ordinary differential equations. We present a new family of 8-9 pairs and assess the performance of two new 8-9 pairs on the equations used to create the ephemeris DE102. Part of this work is the introduction of these equations as a test problem for integrators of initial value ordinary differential equations.2009-08-282009-08-281999-12Technical ReportDepartment of Mathematics - Research Reports-437 (1999)1173-0889http://hdl.handle.net/2292/4992Research Reports - Department of Mathematicshttps://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htmThe author(s)Department of Mathematics, The University of Auckland, New Zealandoai:researchspace.auckland.ac.nz:2292/49932009-08-28T12:38:56Zcom_2292_122col_2292_4963Order 5 symplectic explicit Runge-Kutta Nystrom methodsChou, Lin-yiSharp, 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.2009-08-282009-08-281999-12Technical ReportDepartment of Mathematics - Research Reports-436 (1999)1173-0889http://hdl.handle.net/2292/4993Research Reports - Department of Mathematicshttps://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htmThe author(s)Department of Mathematics, The University of Auckland, New Zealandoai:researchspace.auckland.ac.nz:2292/49942009-08-28T12:38:57Zcom_2292_122col_2292_4963Valuation of bonds and options under floating interest rateAdamjan, V.Pavlov, B.The evolution operators, generators of which contain a numerical parameter forming a Markov process, are considered in connection with problems of financial mathematics. Under certain conditions the exact and explicit expressions for the values of the evolution operators averaged over trajectories of the process and for the corresponding variances are derived.Obtained results are applied for valuation of some financial products with account of floating interest rates.2009-08-282009-08-281999-11Technical ReportDepartment of Mathematics - Research Reports-435 (1999)1173-0889http://hdl.handle.net/2292/4994Research Reports - Department of Mathematicshttps://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htmThe author(s)Department of Mathematics, The University of Auckland, New Zealandoai:researchspace.auckland.ac.nz:2292/49952009-08-28T12:38:58Zcom_2292_122col_2292_4963Experiences of Doctoral Students in Mathematics in New ZealandMorton, MargaretThornley, GillianThis 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.2009-08-282009-08-281999-12Technical ReportDepartment of Mathematics - Research Reports-434 (1999)1173-0889http://hdl.handle.net/2292/4995Research Reports - Department of Mathematicshttps://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htmThe author(s)Department of Mathematics, The University of Auckland, New Zealandoai:researchspace.auckland.ac.nz:2292/49962009-08-28T12:38:58Zcom_2292_122col_2292_4963The Majoritarian Compromise is Asymptotically Strategy-ProofSlinko, ArkadiiIn this paper we investigate the social choice rule known as majoritarian compromise. We prove that it is asymptotically strategy-proof for $m be 3$ alternatives and that the ratio of the number of all manipulable profiles upon the total number of profiles is in the order of $O(1/sqrt{n})$.2009-08-282009-08-281999-12Technical ReportDepartment of Mathematics - Research Reports-433 (1999)1173-0889http://hdl.handle.net/2292/4996Research Reports - Department of Mathematicshttps://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htmThe author(s)Department of Mathematics, The University of Auckland, New Zealandoai:researchspace.auckland.ac.nz:2292/49972009-08-28T12:38:59Zcom_2292_122col_2292_4963Asymptotic Strategy Proofness of the Plurality and the Run-off RulesSlinko, ArkadiiIn this paper we prove that the plurality rule and the run-off procedure are asymptotically strategy-proof for any number of alternatives and that the ratio of the number of all manipulable profiles upon the total number of profiles in both cases is it the order of $O(1/sqrt{n})$.2009-08-282009-08-281999-11Technical ReportDepartment of Mathematics - Research Reports-432 (1999)1173-0889http://hdl.handle.net/2292/4997Research Reports - Department of Mathematicshttps://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htmThe author(s)Department of Mathematics, The University of Auckland, New Zealandoai:researchspace.auckland.ac.nz:2292/49982009-08-28T12:39:00Zcom_2292_122col_2292_4963Covering Properties and Metrisation of Manifolds*Gauld, DavidVamanamurthy, M.K.There are many conditions equivalent to metrisability for a topological manifold which are not equivalent to metrisability for topological spaces in general. What are the weakest such? We show that a number of weak covering properties which are equivalent to metrisability for a manifold, for example metaLindel"{o}f, may be further weakened by considering only covers of cardinality the first uncountable ordinal. Extensions to higher cardinals are discussed,2009-08-282009-08-281999-10Technical ReportDepartment of Mathematics - Research Reports-431 (1999)1173-0889http://hdl.handle.net/2292/4998Research Reports - Department of Mathematicshttps://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htmThe author(s)Department of Mathematics, The University of Auckland, New Zealandoai:researchspace.auckland.ac.nz:2292/49992009-08-28T12:39:00Zcom_2292_122col_2292_4963Generalized Convexity and InequalitiesAnderson, G.D.Vamanamurthy, M.K.Vuorinen, M.Several convexity properties are studied, with applications to power series, in particular to hypergeometric and related functions.2009-08-282009-08-282006-05Technical ReportDepartment of Mathematics - Research Reports-550 (2006)1173-0889http://hdl.handle.net/2292/4999Research Reports - Department of Mathematicshttps://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htmThe author(s)Department of Mathematics, The University of Auckland, New Zealandoai:researchspace.auckland.ac.nz:2292/50002009-08-28T12:39:01Zcom_2292_122col_2292_4963The eigenstructure of the Bernstein operatorCooper, ShaunWaldron, ShayneThe Bernstein operator $B_n$ reproduces the linear polynomials, which are therefore eigenfunctions corresponding to the eigenvalue $1$. We determine the rest of the eigenstructure of $B_n$. Its eigenvalues are $$lambda_k^{(n)}:={n!over(n-k)!}{1over n^k}, qquad k=0,1,ldots,n,$$ and the corresponding monic eigenfunctions $p_k^{(n)}$ are polynomials of degree $k$, % (with interlacing zeros) which have $k$ simple zeros in $[0,1]$. By using an explicit formula, it is shown that $p_k^{(n)}$ converges as $ntoinfty$ to a polynomial related to a Jacobi polynomial. Similarly, %for fixed $k$, the dual functionals to $p_k^{(n)}$ converge as $ntoinfty$ to measures that we identify. This diagonal form of the Bernstein operator and its limit, the identity (Weierstrass density theorem), is applied to a number of questions. These include the convergence of iterates of the Bernstein operator, and why Lagrange interpolation (at $n+1$ equally spaced points) fails to converge for all continuous functions whilst the Bernstein approximants do. We also give the eigenstructure of the Kantorovich operator. Previously, the only member of the Bernstein family for which the eigenfunctions were known explicitly was the Bernstein--Durrmeyer operator, which is self adjoint.2009-08-282009-08-281999-10Technical ReportDepartment of Mathematics - Research Reports-430 (1999)1173-0889http://hdl.handle.net/2292/5000Research Reports - Department of Mathematicshttps://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htmThe author(s)Department of Mathematics, The University of Auckland, New Zealandoai:researchspace.auckland.ac.nz:2292/50012009-08-28T12:39:02Zcom_2292_122col_2292_4963Developable Spaces and Problems of Fletcher and LindgrenMohamad, A.M.In this paper, we answer two questions of P. Fletcher and W. Lindgren. We prove that a space $X$ is $wDelta$ and has a quasi--$G^*_{delta}$--diagonal if and only if it is developable, a space $X$ is $beta$--space with a quasi--$G^{*}_{delta}$--diagonal if and only if it is semi--stratifiable, a space $X$ is $beta$, quasi--$gamma$--space and has a quasi--$G^*_{delta}$--diagonal if and only if $X$ is developable and a space $X$ is metrizable if and only if it is paracompact $beta$--space with a quasi--$G_{delta}$--diagonal.2009-08-282009-08-281999-09Technical ReportDepartment of Mathematics - Research Reports-429 (1999)1173-0889http://hdl.handle.net/2292/5001Research Reports - Department of Mathematicshttps://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htmThe author(s)Department of Mathematics, The University of Auckland, New Zealandoai:researchspace.auckland.ac.nz:2292/50022009-08-28T12:39:03Zcom_2292_122col_2292_4963Metrizability of Manifolds by Diagonal PropertiesGartside, 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.2009-08-282009-08-281999-09Technical ReportDepartment of Mathematics - Research Reports-428 (1999)1173-0889http://hdl.handle.net/2292/5002Research Reports - Department of Mathematicshttps://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htmThe author(s)Department of Mathematics, The University of Auckland, New Zealandoai:researchspace.auckland.ac.nz:2292/50032009-08-28T12:39:04Zcom_2292_122col_2292_4963The Characterization of Finite Simple Groups with no Elements of Order Six by Their Element OrdersAn, JianbeiShi, WujieWe show that all but $A_6$ non-abelian finite simple groups with no elements of order 6 are characterized by their element orders.2009-08-282009-08-281999-09Technical ReportDepartment of Mathematics - Research Reports-427 (1999)1173-0889http://hdl.handle.net/2292/5003Research Reports - Department of Mathematicshttps://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htmThe author(s)Department of Mathematics, The University of Auckland, New Zealandoai:researchspace.auckland.ac.nz:2292/50042009-08-28T12:39:05Zcom_2292_122col_2292_4963Dade's Invariant Conjecture for General Linear and Unitary Groups in Non-defining CharacteristicsAn, JianbeiThis 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.2009-08-282009-08-281999-09Technical ReportDepartment of Mathematics - Research Reports-426 (1999)1173-0889http://hdl.handle.net/2292/5004Research Reports - Department of Mathematicshttps://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htmThe author(s)Department of Mathematics, The University of Auckland, New Zealandoai:researchspace.auckland.ac.nz:2292/50052009-08-28T12:39:06Zcom_2292_122col_2292_4963The p-local Rank of a BlockAn, JianbeiEaton, 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.2009-08-282009-08-281999-09Technical ReportDepartment of Mathematics - Research Reports-425 (1999)1173-0889http://hdl.handle.net/2292/5005Research Reports - Department of Mathematicshttps://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htmThe author(s)Department of Mathematics, The University of Auckland, New Zealandoai:researchspace.auckland.ac.nz:2292/50062009-08-28T12:39:07Zcom_2292_122col_2292_4963Conditions Which Imply Metrizability in some Generalized Metric Spaces*Mohamad, A.M.In this paper we show that two important generalized metric properties are generalizations of first countability. We give some conditions on these generalized metric properties which imply metrizability. We prove that, a space $X$ is metrizable if and only if $X$ is a strongly quasi-N-space, quasi$-gamma-$space; a quasi$-gamma$ space is metrizable if and only if it is a pseudo $wN-$ space or quasi$-$Nagata$-$space with quasi $G^*_gamma-$diagonal; a space $X$ is a metrizable space if and only if $X$ has a $CWBC-$map $g$ satisfying the following conditions: 1. $g$ is a pseudo-strongly-quasi-N-map; 2. for any $A subseteq X, overline{A} subseteq cup {g(n, x) : x in A}$.2009-08-282009-08-281999-09Technical ReportDepartment of Mathematics - Research Reports-424 (1999)1173-0889http://hdl.handle.net/2292/5006Research Reports - Department of Mathematicshttps://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htmThe author(s)Department of Mathematics, The University of Auckland, New Zealandoai:researchspace.auckland.ac.nz:2292/50072009-08-28T12:39:07Zcom_2292_122col_2292_4963Homeomorphism 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.2009-08-282009-08-281999-09Technical ReportDepartment of Mathematics - Research Reports-423 (1999)1173-0889http://hdl.handle.net/2292/5007Research Reports - Department of Mathematicshttps://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htmThe author(s)Department of Mathematics, The University of Auckland, New Zealandoai:researchspace.auckland.ac.nz:2292/50082009-08-28T12:39:08Zcom_2292_122col_2292_4963Domination Conditions for TournamentsMcKenna, PatriciaMorton, MargaretSneddon, JamieThe 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.2009-08-282009-08-281999-08Technical ReportDepartment of Mathematics - Research Reports-422 (1999)1173-0889http://hdl.handle.net/2292/5008Research Reports - Department of Mathematicshttps://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htmThe author(s)Department of Mathematics, The University of Auckland, New Zealandoai:researchspace.auckland.ac.nz:2292/50092009-08-28T12:39:09Zcom_2292_122col_2292_4963On Preclosed Sets and Their GeneralizationsCao, JilingGanster, MaximilianKonstadilaki, CharikliaReilly, IvanThis 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.2009-08-282009-08-281999-06Technical ReportDepartment of Mathematics - Research Reports-421 (1999)1173-0889http://hdl.handle.net/2292/5009Research Reports - Department of Mathematicshttps://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htmThe author(s)Department of Mathematics, The University of Auckland, New Zealandoai:researchspace.auckland.ac.nz:2292/50102009-08-28T12:39:10Zcom_2292_122col_2292_4963Holomorphic forms of geometrically formal Kaehler manifoldsGrosjean, Jean-FrancoisNagy, Paul-AndiWe investigate harmonic forms of geometrically formal metrics, which are defined as those having the exterior product of any two harmonic forms still harmonic. We prove that a formal Sasakian metric can exist only on a real cohomology sphere and that holomorphic forms of a formal K"ahler metric are parallel w.r.t. the Levi-Civita connection. In the general Riemannian case a formal metric with maximal second Betti number is shown to be flat . Finally we prove that a six-dimensional manifold with $b_1 neq 1, b_2 ge 3$ and not having the cohomology algebra of $mathbb{T}^3 times S^3$ carries a symplectic structure as soon as it admits a formal metric.2009-08-282009-08-282006-05Technical ReportDepartment of Mathematics - Research Reports-549 (2006)1173-0889http://hdl.handle.net/2292/5010Research Reports - Department of Mathematicshttps://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htmThe author(s)Department of Mathematics, The University of Auckland, New Zealandoai:researchspace.auckland.ac.nz:2292/50112009-08-28T12:39:11Zcom_2292_122col_2292_4963Scattering on a Compact Domain with few Semiinfinite wires attached: resonance caseMikhailova, 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.2009-08-282009-08-281999-08Technical ReportDepartment of Mathematics - Research Reports-420 (1999)1173-0889http://hdl.handle.net/2292/5011Research Reports - Department of Mathematicshttps://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htmThe author(s)Department of Mathematics, The University of Auckland, New Zealandoai:researchspace.auckland.ac.nz:2292/50122009-08-28T12:39:12Zcom_2292_122col_2292_4963FEW-BODY KREIN'S FORMULAKurasov, PavelPavlov, 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 derived2009-08-282009-08-281999-06Technical ReportDepartment of Mathematics - Research Reports-418 (1999)1173-0889http://hdl.handle.net/2292/5012Research Reports - Department of Mathematicshttps://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htmThe author(s)Department of Mathematics, The University of Auckland, New Zealandoai:researchspace.auckland.ac.nz:2292/50132009-08-28T12:39:12Zcom_2292_122col_2292_4963HILBERT THEOREM ON LEMNISCATE AND THE SPECTRUM OF THE PERTURBED SHIFTOleinik, V.L.Pavlov, B.The spectrum of the perturbed shift operator $T: f(n)to al f(n+1)+a(n)f(n)$ in $l^2(Z)$ is considered for periodic $a(n)$ and fixed constant $al>0$. It is proven that the spectrum is continuous and fills a lemniscate. Some isospectral deformations of the sequence $a(n)$ are described. Similar facts for the perturbed shift in the spaces of sequences of some hypercomplex numbers is derived.2009-08-282009-08-281999Technical ReportDepartment of Mathematics - Research Reports-417 (1999)1173-0889http://hdl.handle.net/2292/5013Research Reports - Department of Mathematicshttps://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htmThe author(s)Department of Mathematics, The University of Auckland, New Zealandoai:researchspace.auckland.ac.nz:2292/50142009-08-28T12:39:13Zcom_2292_122col_2292_4963On the Interaction between a Group of Unitary Operators and a ProjectionTaylor, S.W.Littman, W.We consider the following objects: ${cal S}$ is a closed subspace of a Hilbert Space ${cal H}$, ${cal P}$ is the projection operator for ${cal S}$, and ${cal U}(t)$ is a strongly continuous group of unitary operators on ${cal H}$ with infinitesimal generator ${cal A}$. We let $U={cal U}(T)$, where $T>0$ is fixed. The questions that we ask are begin{itemize} item Under what conditions is $mathcal{P}Umathcal{P}$ a contraction? item Under what conditions can we steer $g in mathcal{S}$ to $hin mathcal{S}$ in the sense that we can find $fin mathcal{H}$ such that ${cal P}f=g$ and ${cal P}Uf=h$? end{itemize}2009-08-282009-08-281999-05Technical ReportDepartment of Mathematics - Research Reports-416 (1999)1173-0889http://hdl.handle.net/2292/5014Research Reports - Department of Mathematicshttps://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htmThe author(s)Department of Mathematics, The University of Auckland, New Zealandoai:researchspace.auckland.ac.nz:2292/50152009-08-28T12:39:13Zcom_2292_122col_2292_4963Boundary Feedback Stabilization of a Vibrating String with an Interior Point MassTaylor, S.W.Littman, W.We study the boundary feedback stabilization for a one-dimensional wave equation with an interior point mass. We show that if the initial data belong to a certain invariant subspace of the semigroup of operators that generates the solution of the system, then the energy will decay like $C/$time. This improves a result of Hansen and Zuazua cite{hansen} who consider decay of solutions belonging to the domain of a power of the infinitesimal generator of the semigroup.2009-08-282009-08-281999-03Technical ReportDepartment of Mathematics - Research Reports-415 (1999)1173-0889http://hdl.handle.net/2292/5015Research Reports - Department of Mathematicshttps://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htmThe author(s)Department of Mathematics, The University of Auckland, New Zealandoai:researchspace.auckland.ac.nz:2292/50162009-08-28T12:39:15Zcom_2292_122col_2292_4963Brachistochrones for Repulsive Logarithmic PotentialTee, Garry J.Brachistochrones are constructed for repulsive central force, with logarithmic potential. Each pair of points is connected by infinitely many brachistochrones, on each of which the passage time is a local minimum, and the global minimum passage time is attained on one (or on infinitely many) of those curves. The bounded brachistochrones starting at a fixed point are separated from the unbounded brachistochrones by a Critical Brachistochrone.2009-08-282009-08-281999-03Technical ReportDepartment of Mathematics - Research Reports-414 (1999)1173-0889http://hdl.handle.net/2292/5016Research Reports - Department of Mathematicshttps://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htmThe author(s)Department of Mathematics, The University of Auckland, New Zealandoai:researchspace.auckland.ac.nz:2292/50172009-08-28T12:39:17Zcom_2292_122col_2292_4963About Scattering on the RingBogevolnov, 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.2009-08-282009-08-281999-03Technical ReportDepartment of Mathematics - Research Reports-413 (1999)1173-0889http://hdl.handle.net/2292/5017Research Reports - Department of Mathematicshttps://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htmThe author(s)Department of Mathematics, The University of Auckland, New Zealandoai:researchspace.auckland.ac.nz:2292/50182009-08-28T12:39:18Zcom_2292_122col_2292_4963Boundedly Metacompact or Finitistic Spaces and the Star Order of CoversDeo, SatyaGauld, DavidAfter 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.2009-08-282009-08-281999-03Technical ReportDepartment of Mathematics - Research Reports-412 (1999)1173-0889http://hdl.handle.net/2292/5018Research Reports - Department of Mathematicshttps://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htmThe author(s)Department of Mathematics, The University of Auckland, New Zealandoai:researchspace.auckland.ac.nz:2292/50192009-08-28T12:39:18Zcom_2292_122col_2292_4963The Torsion of the Group of Homeomorphisms of Powers of the Long LineDeo, SatyaGauld, DavidBy blending techniques from Set Theory and Algebraic Topology we investigate the order of any homeomorphism of the $n$th power of the long ray or long line $L$ having finite order, finding all possible orders when $n=1, 2, 3$ or 4 in the first case and when $n=1$ or 2 in the second. We also show that all finite powers of $L$ are acyclic with respect to Alexander-Spanier cohomology.2009-08-282009-08-281999-03Technical ReportDepartment of Mathematics - Research Reports-411 (1999)1173-0889http://hdl.handle.net/2292/5019Research Reports - Department of Mathematicshttps://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htmThe author(s)Department of Mathematics, The University of Auckland, New Zealandoai:researchspace.auckland.ac.nz:2292/50202009-08-28T12:39:19Zcom_2292_122col_2292_4963Brachistochrones For Attractive Logarithmic PotentialTee, 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.2009-08-282009-08-281999-01Technical ReportDepartment of Mathematics - Research Reports-410 (1999)1173-0889http://hdl.handle.net/2292/5020Research Reports - Department of Mathematicshttps://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htmThe author(s)Department of Mathematics, The University of Auckland, New Zealandoai:researchspace.auckland.ac.nz:2292/50212009-08-28T12:39:20Zcom_2292_122col_2292_4963Boundary condition at the junctionHarmer, M.Pavlov, B.Yafyasov, A.A two-dimensional junction is modelled by a quantum graph with a resonance hode. The boundary condition at the node is calculated from the frirst principles . In low temperators limit for simplest T-junction it coincides with the the boundary condition suggested by Datta. The free parameter contained in Datta's condition is interpreted in spectral terms of an intermediate Hamiltonian. The derived boundary condition is applicable for any junction of straight semi-infinite quantum wires attached to the quantum dot, if the junction is thin : the ratio r of the diameter of the quantum dot to the width of the wires is sufficintly large $r approx 4 - 10 $.2009-08-282009-08-282006-05Technical ReportDepartment of Mathematics - Research Reports-548 (2006)1173-0889http://hdl.handle.net/2292/5021Research Reports - Department of Mathematicshttps://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htmThe author(s)Department of Mathematics, The University of Auckland, New Zealandoai:researchspace.auckland.ac.nz:2292/50222009-08-28T12:39:22Zcom_2292_122col_2292_4963Brachistochrones for Repulsive Inverse Square ForceTee, Garry J.Brachistochrones for inverse square repulsion are expressed in terms of elliptic integrals, as for inverse square attraction. But, the set of brachistochrones is much more complicated than for inverse square attraction. Each pair of points is connected by infinitely many brachistochrones, on each of which the passage time is a local minimum, and the global minimum passage time is attained on one (or on infinitely many) of those curves. The bounded brachistochrones starting at a fixed point are separated from the unbounded brachistochrones by a Critical Brachistochrone, which is expressed in terms of elementary functions.2009-08-282009-08-281999-01Technical ReportDepartment of Mathematics - Research Reports-409 (1999)1173-0889http://hdl.handle.net/2292/5022Research Reports - Department of Mathematicshttps://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htmThe author(s)Department of Mathematics, The University of Auckland, New Zealandoai:researchspace.auckland.ac.nz:2292/50232009-08-28T12:39:23Zcom_2292_122col_2292_4963Green's Function for Forcing of a Thin Floating PlateFox, ColinChung, HyuckThe Green's function for harmonic downward forcing of an infinite thin floating plate is derived. The Green's function models the response of a uniform sheet of fast ice when locally loaded at rates at which the ice may be taken to be elastic. A closed-form expression is given for the potential throughout the water and detailed expressions are given for the vertical displacement of the ice sheet. The displacement is graphed for various typical thickness of the ice sheet and for a range of frequencies of forcing.2009-08-282009-08-281999-03Technical ReportDepartment of Mathematics - Research Reports-408 (1999)1173-0889http://hdl.handle.net/2292/5023Research Reports - Department of Mathematicshttps://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htmThe author(s)Department of Mathematics, The University of Auckland, New Zealandoai:researchspace.auckland.ac.nz:2292/50242009-08-28T12:39:25Zcom_2292_122col_2292_4963Radiocarbon dating with temporal order constraintsNicholls, GeoffJones, MartinA 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.2009-08-282009-08-281999-02Technical ReportDepartment of Mathematics - Research Reports-407 (1999)1173-0889http://hdl.handle.net/2292/5024Research Reports - Department of Mathematicshttps://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htmThe author(s)Department of Mathematics, The University of Auckland, New Zealandoai:researchspace.auckland.ac.nz:2292/50252009-08-28T12:39:26Zcom_2292_122col_2292_4963Embedding Digraphs on Orientable SurfacesBonnington, PaulConder, MarstonMcKenna, PatriciaMorton, MargaretWe 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.2009-08-282009-08-281998-12Technical ReportDepartment of Mathematics - Research Reports-406 (1998)1173-0889http://hdl.handle.net/2292/5025Research Reports - Department of Mathematicshttps://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htmThe author(s)Department of Mathematics, The University of Auckland, New Zealandoai:researchspace.auckland.ac.nz:2292/50262009-08-28T12:39:28Zcom_2292_122col_2292_4963Isochrones and BrachistochronesTee, 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.2009-08-282009-08-281998-11Technical ReportDepartment of Mathematics - Research Reports-405 (1998)1173-0889http://hdl.handle.net/2292/5026Research Reports - Department of Mathematicshttps://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htmThe author(s)Department of Mathematics, The University of Auckland, New Zealandoai:researchspace.auckland.ac.nz:2292/50272009-08-28T12:39:29Zcom_2292_122col_2292_4963On the subspaces of analytic and antianalytic functionsFedorov, SergeiThe 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.2009-08-282009-08-281998-09Technical ReportDepartment of Mathematics - Research Reports-404 (1998)1173-0889http://hdl.handle.net/2292/5027Research Reports - Department of Mathematicshttps://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htmThe author(s)Department of Mathematics, The University of Auckland, New Zealandoai:researchspace.auckland.ac.nz:2292/50282009-08-28T12:39:30Zcom_2292_122col_2292_4963Planar Universal GraphsBrand, NealMorton, MargaretA graph is g-universal if it satisfies two conditions. First it must contain a subdivision of every proper planar graph of degree at most three as a subgraph. Second, the function g puts a restriction on the subdivision. In particular, for a planar graph H of degree at most three, a fixed vertex $w_0$ of H, and an arbitrary vertex w of H, the images of the vertices $w_0$ and w in the universal graph are no more than $g(d(w_0, w))$ apart. We show that a large class of planar graphs are $O(n^3)4-universal.2009-08-282009-08-281998Technical ReportDepartment of Mathematics - Research Reports-403 (1998)1173-0889http://hdl.handle.net/2292/5028Research Reports - Department of Mathematicshttps://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htmThe author(s)Department of Mathematics, The University of Auckland, New Zealandoai:researchspace.auckland.ac.nz:2292/50292009-08-28T12:39:31Zcom_2292_122col_2292_4963Computable Rings, Groups and Their IsomorphismsKhoussainov, B.Slinko, A.We investigate computable isomorphism types of groups. Our main result states that for any $ninomegacup{omega}$ there exists a computably categorical nilpotent of class $2$ group $G$ which being expanded by a finite number of constants has exactly $n$ computable isomorphism types. This result is based on the similar result for computable nonassociative rings.2009-08-282009-08-281998-08Technical ReportDepartment of Mathematics - Research Reports-402 (1998)1173-0889http://hdl.handle.net/2292/5029Research Reports - Department of Mathematicshttps://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htmThe author(s)Department of Mathematics, The University of Auckland, New Zealandoai:researchspace.auckland.ac.nz:2292/50302019-12-19T03:30:19Zcom_2292_122col_2292_4963Dade's Conjecture for Steinberg Triality Groups $^3D_4(q)$ in Non-defining CharacteristicsAn, Jianbei[no abstract available]2009-08-282009-08-281998-08Technical ReportDepartment of Mathematics - Research Reports-401 (1998)1173-0889http://hdl.handle.net/2292/5030Research Reports - Department of Mathematics20411644073627379https://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htmThe author(s)Department of Mathematics, The University of Auckland, New Zealandoai:researchspace.auckland.ac.nz:2292/50312009-08-28T12:39:33Zcom_2292_122col_2292_4963The infinitude of 7-arc-transitive graphsConder, MarstonWalker, 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.2009-08-282009-08-281998-06Technical ReportDepartment of Mathematics - Research Reports-400 (1998)1173-0889http://hdl.handle.net/2292/5031Research Reports - Department of Mathematicshttps://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htmThe author(s)Department of Mathematics, The University of Auckland, New Zealandoai:researchspace.auckland.ac.nz:2292/50322009-08-28T12:39:34Zcom_2292_122col_2292_4963On the manipulability of proportional representationSlinko, ArkadiiWhite, ShaunThis 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.2009-08-282009-08-282006-03Technical ReportDepartment of Mathematics - Research Reports-547 (2006)1173-0889http://hdl.handle.net/2292/5032Research Reports - Department of Mathematicshttps://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htmThe author(s)Department of Mathematics, The University of Auckland, New Zealandoai:researchspace.auckland.ac.nz:2292/50332009-08-28T12:39:36Zcom_2292_122col_2292_4963A Novel Numerical Solution Method for the Linear Boltzmann EquationMeylan, Michael H.We present here a novel numerical solution to the linear Boltzmann equation. The method is based on reducing the linear Boltzmann equation to a matrix partial differential equation rather than a partial integro-differential equation. A method for calculating the evolution using a complex generalised eigenfunction method is present for two simple example cases. The generalisation of this method follows straight forwardly.2009-08-282009-08-281998-05Technical ReportDepartment of Mathematics - Research Reports-399 (1998)1173-0889http://hdl.handle.net/2292/5033Research Reports - Department of Mathematicshttps://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htmThe author(s)Department of Mathematics, The University of Auckland, New Zealandoai:researchspace.auckland.ac.nz:2292/50342009-08-28T12:39:37Zcom_2292_122col_2292_4963On one recent result on the intersection of weighted hardy spacesFedorov, Sergei[no abstract available]2009-08-282009-08-281998-05Technical ReportDepartment of Mathematics - Research Reports-398 (1998)1173-0889http://hdl.handle.net/2292/5034Research Reports - Department of Mathematicshttps://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htmThe author(s)Department of Mathematics, The University of Auckland, New Zealandoai:researchspace.auckland.ac.nz:2292/50352009-08-28T12:39:37Zcom_2292_122col_2292_4963Certain cyclically presented groups are isomorphicJohnson, D.L.Kim, A.C.O'Brien, E.A.and prove that the corresponding groups in each pair are isomorphic.2009-08-282009-08-281998-04Technical ReportDepartment of Mathematics - Research Reports-397 (1998)1173-0889http://hdl.handle.net/2292/5035Research Reports - Department of Mathematicshttps://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htmThe author(s)Department of Mathematics, The University of Auckland, New Zealandoai:researchspace.auckland.ac.nz:2292/50362009-08-28T12:39:38Zcom_2292_122col_2292_4963On the Choquet-Dolecki TheoremCao, JilingMoors, Warren B.Reilly, Ivan[no abstract available]2009-08-282009-08-281998-04Technical ReportDepartment of Mathematics - Research Reports-396 (1998)1173-0889http://hdl.handle.net/2292/5036Research Reports - Department of Mathematicshttps://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htmThe author(s)Department of Mathematics, The University of Auckland, New Zealandoai:researchspace.auckland.ac.nz:2292/50372009-08-28T12:39:39Zcom_2292_122col_2292_4963The Alperin and Dade conjectures for the Fischer simple group $Fi_{23}$An, JianbeiO'Brien, E.A.use it to classify the radical subgroups and chains of the Fischer simple group $Fi_{23}$. We verify the Alperin weight conjecture and the Dade final conjectures for this group.2009-08-282009-08-281998-03Technical ReportDepartment of Mathematics - Research Reports-395 (1998)1173-0889http://hdl.handle.net/2292/5037Research Reports - Department of Mathematicshttps://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htmThe author(s)Department of Mathematics, The University of Auckland, New Zealandoai:researchspace.auckland.ac.nz:2292/50382009-08-28T12:39:41Zcom_2292_122col_2292_4963Generalized Elliptic Integrals and Modular EquationsAnderson, 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.2009-08-282009-08-281998-04Technical ReportDepartment of Mathematics - Research Reports-394 (1998)1173-0889http://hdl.handle.net/2292/5038Research Reports - Department of Mathematicshttps://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htmThe author(s)Department of Mathematics, The University of Auckland, New Zealandoai:researchspace.auckland.ac.nz:2292/50392009-08-28T12:39:42Zcom_2292_122col_2292_49631-factorizations of Cayley graphs on solvable groupsAlspach, BrianMorton, MargaretQin, Yusheng[no abstract available]2009-08-282009-08-281998-03Technical ReportDepartment of Mathematics - Research Reports-393 (1998)1173-0889http://hdl.handle.net/2292/5039Research Reports - Department of Mathematicshttps://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htmThe author(s)Department of Mathematics, The University of Auckland, New Zealandoai:researchspace.auckland.ac.nz:2292/50402009-08-28T12:39:43Zcom_2292_122col_2292_4963The Alperin and Dade Conjectures for the Conway Simple Group $Co_2$An, JianbeiO'Brien, E.A.[no abstract available]2009-08-282009-08-281998-03Technical ReportDepartment of Mathematics - Research Reports-392 (1998)1173-0889http://hdl.handle.net/2292/5040Research Reports - Department of Mathematicshttps://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htmThe author(s)Department of Mathematics, The University of Auckland, New Zealandoai:researchspace.auckland.ac.nz:2292/50412009-08-28T12:39:43Zcom_2292_122col_2292_4963Separable determination of integrability and minimality of the Clarke subdifferential mappingBorwein, 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.2009-08-282009-08-281998-03Technical ReportDepartment of Mathematics - Research Reports-391 (1998)1173-0889http://hdl.handle.net/2292/5041Research Reports - Department of Mathematicshttps://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htmThe author(s)Department of Mathematics, The University of Auckland, New Zealandoai:researchspace.auckland.ac.nz:2292/50422009-08-28T12:39:44Zcom_2292_122col_2292_4963CONTINUOUS BRANCHES OF INVERSES OF THE 12 JACOBI ELLIPTIC FUNCTIONS FOR REAL ARGUMENTTee, Garry J.[no abstract available]2009-08-282009-08-281998-03Technical ReportDepartment of Mathematics - Research Reports-390 (1998)1173-0889http://hdl.handle.net/2292/5042Research Reports - Department of Mathematicshttps://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htmThe author(s)Department of Mathematics, The University of Auckland, New Zealandoai:researchspace.auckland.ac.nz:2292/50432009-08-28T12:39:45Zcom_2292_122col_2292_4963On the Order Hereditary Closure Preserving Sum TheoremGong, JianhuaReilly, IvanThe main purpose of this paper is to prove the following two theorems, an order hereditary closure preserving sum theorem and an hereditary theorem: (1) If a topological property $mathcal{P}$ satisfies $(sum')$ and is closed hereditary, and if $mathcal{V}$ is an order hereditary closure preserving open cover of $X$ and each $V inmathcal{V}$ is elementary and possesses $mathcal{P}$, then $X$ possesses $mathcal{P}$. (2) Let a topological property $mathcal{P}$ satisfy $(sum')$ and $(beta),$ and be closed hereditary. Let $X$ be a topological space which possesses $mathcal{P}$. If every open subset $G$ of $X$ can be written as an order hereditary closure preserving (in $G$) collection of elementary sets, then every subset of $X$ possesses $mathcal{P}$.2009-08-282009-08-282006-03Technical ReportDepartment of Mathematics - Research Reports-546 (2006)1173-0889http://hdl.handle.net/2292/5043Research Reports - Department of Mathematicshttps://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htmThe author(s)Department of Mathematics, The University of Auckland, New Zealandoai:researchspace.auckland.ac.nz:2292/50442009-08-28T12:39:45Zcom_2292_122col_2292_4963LANDAUER FORMULA AND FORMING OF SPECTRAL BANDSPavlov, B.Roach, G.Yafyasov, A.barrier is compared with one corresponding to a finite periodic chain of $N$ potential barriers or wells. It is proved, that even for small periodic potentials the exponential decreasing of the transmission coefficient for growing $N$ takes place in lacunas of the corresponding periodic operator on the whole real line. Using the Landauer formula we express the conductivity of the corresponding onedimensional conductor in terms of the transmission coefficient.2009-08-282009-08-281998-03Technical ReportDepartment of Mathematics - Research Reports-389 (1998)1173-0889http://hdl.handle.net/2292/5044Research Reports - Department of Mathematicshttps://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htmThe author(s)Department of Mathematics, The University of Auckland, New Zealandoai:researchspace.auckland.ac.nz:2292/50452009-08-28T12:39:46Zcom_2292_122col_2292_4963A Note on Arc-transitive CirculantsMorton, Margaret[no abstract available]2009-08-282009-08-281998-03Technical ReportDepartment of Mathematics - Research Reports-388 (1998)1173-0889http://hdl.handle.net/2292/5045Research Reports - Department of Mathematicshttps://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htmThe author(s)Department of Mathematics, The University of Auckland, New Zealandoai:researchspace.auckland.ac.nz:2292/50462009-08-28T12:39:46Zcom_2292_122col_2292_4963Growth of Infinite Planar GraphsBrand, NealMorton, MargaretVertigan, Dirkthose 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.2009-08-282009-08-281998-03Technical ReportDepartment of Mathematics - Research Reports-387 (1998)1173-0889http://hdl.handle.net/2292/5046Research Reports - Department of Mathematicshttps://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htmThe author(s)Department of Mathematics, The University of Auckland, New Zealandoai:researchspace.auckland.ac.nz:2292/50472009-11-19T00:50:08Zcom_2292_122col_2292_4963Nonassociative Computable Rings and Their Isomorphisms. (1998)Khoussainov, B.Slinko, A.We prove that for any $ninomegacup{omega}$ there exists a ring with exactly $n$ computable isomorphism types. We also investigate the relationship between the number of computable isomorphism types of a ring and the number of computable isomorphism types of its expansion by a finite number of constants.2009-08-282009-08-281998-03Technical ReportDepartment of Mathematics - Research Reports-386 (1998)1173-0889http://hdl.handle.net/2292/5047Research Reports - Department of Mathematicshttps://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htmThe author(s)Department of Mathematics, The University of Auckland, New Zealandoai:researchspace.auckland.ac.nz:2292/50482019-12-19T03:30:19Zcom_2292_122col_2292_4963DADE'S INVARIANT CONJECTURE FOR CHEVALLEY GROUPS $G_2(q)$ IN NON-DEFINING CHARACTERISTICSAn, Jianbei[no abstract available]2009-08-282009-08-281997-07Technical ReportDepartment of Mathematics - Research Reports-385 (1997)1173-0889http://hdl.handle.net/2292/5048Research Reports - Department of Mathematics20411644073797344https://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htmThe author(s)Department of Mathematics, The University of Auckland, New Zealandoai:researchspace.auckland.ac.nz:2292/50492009-08-28T12:39:51Zcom_2292_122col_2292_4963Spectral Theory of Wiener-Hopf Operators and Functional ModelMacCormick, 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.2009-08-282009-08-281997-07Technical ReportDepartment of Mathematics - Research Reports-384 (1997)1173-0889http://hdl.handle.net/2292/5049Research Reports - Department of Mathematicshttps://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htmThe author(s)Department of Mathematics, The University of Auckland, New Zealandoai:researchspace.auckland.ac.nz:2292/50502009-08-28T12:39:51Zcom_2292_122col_2292_4963Mathematical modelling and self-consistent calculation of the charge density of two-dimensional electrons systemIvankiv, I.M.Yafyasov, A.M.Bogevolnov, V.B.Pavlov, B.Rudakova, T.V.The mathematical model of the 2D-system of electrons in the subsurface space of the homogeneous narrow-gap semiconductor was developed for accumulation layers. The calculation of the 2D-systems parameters was carried out by numerical self-consistent integration of the Schr"odinger and Poisson equations by using the Fermi and quasi-classical (WKB) descriptions of the eigenfunctions of the continuous spectrum - the states of electrons "in continuum". par It is shown that the quasi-classical approximation is preferable in comparison with the Fermi one for the description of the continuum for 2D-systems. The parameters of the two-dimensional gas were computed in wide range of temperatures (200---300,K) and potentials (0---0.2,V). There exists possibility of experimental observation of quantum subbands in accumulation layers in the subsurface space of the narrow-gap semiconductor at room temperature.2009-08-282009-08-281997-07Technical ReportDepartment of Mathematics - Research Reports-383 (1997)1173-0889http://hdl.handle.net/2292/5050Research Reports - Department of Mathematicshttps://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htmThe author(s)Department of Mathematics, The University of Auckland, New Zealandoai:researchspace.auckland.ac.nz:2292/50512009-08-28T12:39:52Zcom_2292_122col_2292_4963Mean value interpolation for points in general positionWaldron, Shayne[no abstract available]2009-08-282009-08-281997-05Technical ReportDepartment of Mathematics - Research Reports-382 (1997)1173-0889http://hdl.handle.net/2292/5051Research Reports - Department of Mathematicshttps://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htmThe author(s)Department of Mathematics, The University of Auckland, New Zealandoai:researchspace.auckland.ac.nz:2292/50522009-08-28T12:39:53Zcom_2292_122col_2292_4963On Bernstein's comparison theorem, Peano kernels of constant sign and near-minimax approximationWaldron, ShayneSome 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 differences2009-08-282009-08-281997-05Technical ReportDepartment of Mathematics - Research Reports-381 (1997)1173-0889http://hdl.handle.net/2292/5052Research Reports - Department of Mathematicshttps://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htmThe author(s)Department of Mathematics, The University of Auckland, New Zealandoai:researchspace.auckland.ac.nz:2292/50532009-08-28T12:39:53Zcom_2292_122col_2292_4963Inverse and direct theorems for best uniform approximation by polynomialsWaldron, ShayneIt is shown that the standard method of obtaining direct (Jackson) theorems for the order of best uniform approximation by algebraic polynomials from those for trigonometric polynomials also provides inverse (Bernstein) theorems.2009-08-282009-08-281997-05Technical ReportDepartment of Mathematics - Research Reports-380 (1997)1173-0889http://hdl.handle.net/2292/5053Research Reports - Department of Mathematicshttps://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htmThe author(s)Department of Mathematics, The University of Auckland, New Zealandoai:researchspace.auckland.ac.nz:2292/50542009-08-28T12:39:55Zcom_2292_122col_2292_4963Character Tables of Parabolic Subgroups of the Chevalley Groups of Type G_2An, JianbeiHuang, Shih-changThe main aim of this paper is to construct the character tables of the parabolic subgroups of the Chevalley groups G_2(q), where q is a power of a prime p > 3.2009-08-282009-08-282005-08Technical ReportDepartment of Mathematics - Research Reports-545 (2005)1173-0889http://hdl.handle.net/2292/5054Research Reports - Department of Mathematicshttps://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htmThe author(s)Department of Mathematics, The University of Auckland, New Zealandoai:researchspace.auckland.ac.nz:2292/50552009-08-28T12:39:55Zcom_2292_122col_2292_4963Sharp error estimates for multivariate positive linear operators which reproduce the linear polynomialsWaldron, ShayneA <em>sharp</em> pointwise error estimate is given for multivariate positive linear operators which reproduce the linear polynomials. This quantitative Korovkin--type theorem generalises a known univariate result. It is applied to a number of operators including the multivariate Bernstein operators, and the recently introduced Bernstein--Schoenberg type operators of Dahmen, Micchelli and Seidel.2009-08-282009-08-281997-04Technical ReportDepartment of Mathematics - Research Reports-379 (1997)1173-0889http://hdl.handle.net/2292/5055Research Reports - Department of Mathematicshttps://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htmThe author(s)Department of Mathematics, The University of Auckland, New Zealandoai:researchspace.auckland.ac.nz:2292/50562009-08-28T12:39:57Zcom_2292_122col_2292_4963The Implementation of SIRK Methods for Differential Algebraic EquationsNilsen, E.H.The aim of this report is to survey some aspects of implict differential equations, differential algebraic equations, and their numerical solution. We will formulate an algorithm to solve implicitly defined differential systems by singly implicit Runge-Kutta methods, and see how estimates of the local error can be computed. Then we will give some convergence results for Runge-Kutta methods applied to differential algebraic systems. As a part of the project we have implemented an experimental version of the stiff ODE solver STRIDE by J.C. Butcher, K. Burrage and F.H. Chipman. The new code is capable of solving systems in implicit form and some differential algebraic systems. We have used the code to verify the convergence results and the local error estimates for SIRK methods. Furthermore we have tested the experimental code on some real applications.2009-08-282009-08-281997-04Technical ReportDepartment of Mathematics - Research Reports-378 (1997)1173-0889http://hdl.handle.net/2292/5056Research Reports - Department of Mathematicshttps://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htmThe author(s)Department of Mathematics, The University of Auckland, New Zealandoai:researchspace.auckland.ac.nz:2292/50572009-08-28T12:39:59Zcom_2292_122col_2292_4963Numerical integration of systems arising from the method of linesMageroy, EinarThis report surveys the method of lines as a method for solving partial differential equations. This method involves discretising of all independent variables except one, and integration of the resulting system of ordinary differential equations in the remaining variable. An introduction to space discretisation is given, and different Runge--Kutta methods and linear multi step methods are considered with emphasis on their linear stability properties.2009-08-282009-08-281997-04Technical ReportDepartment of Mathematics - Research Reports-377 (1997)1173-0889http://hdl.handle.net/2292/5057Research Reports - Department of Mathematicshttps://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htmThe author(s)Department of Mathematics, The University of Auckland, New Zealandoai:researchspace.auckland.ac.nz:2292/50582009-08-28T12:40:00Zcom_2292_122col_2292_4963Some conditions which imply quasi-developabilityMohamad, A.M.In this paper quasi-developable spaces, quasi-WDelta-spaces, quasi-semi-stratifiable spaces and spaces with quasi-${G}^{*}_delta$-diagonal are studied. It is shown that every quasi-WDelta, quasi-semi-stratifiable space is a quasi-developable space. A regular space is quasi-semi-stratifiable if and only if it is a quasi-$beta$-space with
quasi-${G}^{*}_delta$-diagonal. A regular space is quasi-semi-stratifiable if and only if it is a quasi-$alpha$ quasi-$beta$-space. A regular quasi-$beta$-space is a quasi-Moore space if and only if it is a quasi-$gamma$-space. A quasi-first-countable quasi-semi-stratifiable space is quasi-developable. A regular quasi-$q$-space is a quasi-Moore space if and only if it is a quasi-semi-stratifiable space.2009-08-282009-08-281997-04Technical ReportDepartment of Mathematics - Research Reports-376 (1997)1173-0889http://hdl.handle.net/2292/5058Research Reports - Department of Mathematicshttps://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htmThe author(s)Department of Mathematics, The University of Auckland, New Zealandoai:researchspace.auckland.ac.nz:2292/50592009-08-28T12:40:01Zcom_2292_122col_2292_4963On a projection from one co-invariant subspace onto another in character-automorphic Hardy space on a multiply connected domainFedorov, SergeiThe research was supported by Marsden Fund grant 96-UOA-MIS-00982009-08-282009-08-281997-04Technical ReportDepartment of Mathematics - Research Reports-375 (1997)1173-0889http://hdl.handle.net/2292/5059Research Reports - Department of Mathematicshttps://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htmThe author(s)Department of Mathematics, The University of Auckland, New Zealandoai:researchspace.auckland.ac.nz:2292/50602009-08-28T12:40:02Zcom_2292_122col_2292_4963Metrization and semimetrization theorems with applications to manifoldsMohamad, A.M.This paper is a study of conditions under which a space with $S_2$ is metrizable, o-semimetrizable or semimetrizable. It is shown that: a $wMN$, $wgamma$-space is metrizable if and only if it has $S_2$, a quasi-$gamma$-space is metrizable if and only if it is a pseudo $wN$-space with $S_2$, a separable manifold is metrizable if and only if it has $S_2$ with property $(*)$, a perfectly normal manifold with quasi-${G}^{*}_delta$-diagonal is metrizable and a separable manifold is a hereditarily separable metrizable if and only if it has $theta$-${alpha}_2$.2009-08-282009-08-281997-04Technical ReportDepartment of Mathematics - Research Reports-374 (1997)1173-0889http://hdl.handle.net/2292/5060Research Reports - Department of Mathematicshttps://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htmThe author(s)Department of Mathematics, The University of Auckland, New Zealandoai:researchspace.auckland.ac.nz:2292/50612009-08-28T12:40:03Zcom_2292_122col_2292_4963Generalization of $G^{*}_delta$-diagonals and wDelta-spacesMohamad, A.M.In this paper we introduce the concepts of a quasi-$G^{*}_delta$-diagonal and quasi-wDelta-space as generalizations of the concepts of $G^{*}_delta$-diagonal and wDelta-space respectively. It is shown that a quasi-Moore space may be characterised in terms of these concepts. As a consequence we obtain the following metrization theorems: every paracompact wDelta-space with quasi-$G_delta$-diagonal is metrizable and every collectionwise normal $sigma$ quasi-wDelta-space is metrizable.2009-08-282009-08-281997-04Technical ReportDepartment of Mathematics - Research Reports-373 (1997)1173-0889http://hdl.handle.net/2292/5061Research Reports - Department of Mathematicshttps://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htmThe author(s)Department of Mathematics, The University of Auckland, New Zealandoai:researchspace.auckland.ac.nz:2292/50622009-08-28T12:40:04Zcom_2292_122col_2292_4963Finite Intervals in the Lattice of TopologiesMcIntyre, D.W.We prove two basic facts about finite intervals in the lattice of topologies on a set. One result states that a finite lattice is isomorphic to an interval of topologies if and only if it is isomorphic to an interval of topologies on a finite set, the other that not every finite lattice is an interval of topologies, although every finite lattice may be embedded into the lattice of topologies on a finite set.2009-08-282009-08-281997-04Technical ReportDepartment of Mathematics - Research Reports-372 (1997)1173-0889http://hdl.handle.net/2292/5062Research Reports - Department of Mathematicshttps://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htmThe author(s)Department of Mathematics, The University of Auckland, New Zealandoai:researchspace.auckland.ac.nz:2292/50632009-08-28T12:40:05Zcom_2292_122col_2292_4963Basic intervals between metrizable topologies]{Basic intervals in the partial order of metrizable topologiesMcIntyre, 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.2009-08-282009-08-281997-03Technical ReportDepartment of Mathematics - Research Reports-371 (1997)1173-0889http://hdl.handle.net/2292/5063Research Reports - Department of Mathematicshttps://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htmThe author(s)Department of Mathematics, The University of Auckland, New Zealandoai:researchspace.auckland.ac.nz:2292/50642009-08-28T12:40:05Zcom_2292_122col_2292_4963Numerical Methods for Differential Equations and ApplicationsButcher, 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.2009-08-282009-08-281997-03Technical ReportDepartment of Mathematics - Research Reports-370 (1997)1173-0889http://hdl.handle.net/2292/5064Research Reports - Department of Mathematicshttps://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htmThe author(s)Department of Mathematics, The University of Auckland, New Zealandetdms///col_2292_4963/100