Mathematical Sciences Research Collectionhttp://hdl.handle.net/10197/204720150425T19:51:40Z20150425T19:51:40ZOn the lowdimensional homology of SL_2(k[t,1/t])Hutchinson, Kevinhttp://hdl.handle.net/10197/651120150422T14:22:42Z20150301T00:00:00ZOn the lowdimensional homology of SL_2(k[t,1/t])
Hutchinson, Kevin
We prove analogues of the fundamental theorem of K theory for the second and the third homology of SL2 over an infinite field. The statements of the theorems involve Milnor–Witt K theory and refined scissors congruence groups. We use these results to calculate the lowdimensional homology of SL2 of Laurent polynomials over certain fields.
20150301T00:00:00ZTwoWeight Codes, Graphs and Orthogonal ArraysByrne, EimearSneyd, Alisonhttp://hdl.handle.net/10197/630420150119T13:09:19Z20150101T00:00:00ZTwoWeight Codes, Graphs and Orthogonal Arrays
Byrne, Eimear; Sneyd, Alison
We investigate properties of twoweight codes over finite Frobenius rings, giving constructions for the modular case. A δmodular code [15] is characterized as having a generator matrix where each column g appears with multiplicity δgR× for some δ ∈ Q. Generalizing [10] and [5], we show that the additive group of a twoweight code satisfying certain constraint equations (and in particular a modular code) has a strongly regular Cayley graph and derive existence conditions on its parameters. We provide a construction for an infinite family of modular twoweight codes arising from unions of submodules with pairwise trivial intersection. The corresponding strongly regular graphs are isomorphic to graphs
from orthogonal arrays.
20150101T00:00:00ZAngular derivatives on bounded symmetric domainsMackey, Michael C.Mellon, Paulinehttp://hdl.handle.net/10197/628620150115T12:41:12Z20030301T00:00:00ZAngular derivatives on bounded symmetric domains
Mackey, Michael C.; Mellon, Pauline
In this paper we generalise the classical JuliaWolffCarathéodory theorem to holomorphic functions defined on bounded symmetric domains.
20030301T00:00:00ZThe identity is isolated among composition operatorsChu, C.H.Hügli, R. V.Mackey, Michael C.http://hdl.handle.net/10197/628520150115T12:35:44Z20040101T00:00:00ZThe identity is isolated among composition operators
Chu, C.H.; Hügli, R. V.; Mackey, Michael C.
Let H∞(B) be the Banach algebra of bounded holomorphic functions on the open unit ball B of a Banach space. We show that the identity operator is an isolated point in the space of composition operators on H∞(B). This answers a conjecture of Aron, Galindo and Lindstrom.
20040101T00:00:00ZA Schwarz lemma and composition operatorsMackey, Michael C.Mellon, Paulinehttp://hdl.handle.net/10197/628420150113T11:06:17Z20040401T00:00:00ZA Schwarz lemma and composition operators
Mackey, Michael C.; Mellon, Pauline
We give an alternative description of the Carathéodory pseudodistance on a domain D in an arbitrary complex Banach space. This gives a Schwarz lemma for holomorphic maps of the domain.We specialise to the case of a bounded symmetric domain and obtain some applications. In particular, we give the connected components of the space of composition operators with symbol in a bounded symmetric domain. This generalises results for the space of composition operators on H∞(Δ) in [12] and for H∞(B) , B the unit ball of a Hilbert space or commutative C*algebra in [2].
20040401T00:00:00ZHomotopes of JB*triples and a Russo—Dye TheoremMackey, Michael C.http://hdl.handle.net/10197/628320150113T10:51:39Z20090101T00:00:00ZHomotopes of JB*triples and a Russo—Dye Theorem
Mackey, Michael C.
In this note, we look at homotopes of Jordan triple structures and show that, following a renorming, an isotope of a JB*triple is also a JB*triple. We also provide a proof of the Russo—Dye theorem for JBW*triples.
20090101T00:00:00ZOn the Second Parameter of an (m, p)IsometryHoffmann, PhilippMackey, Michael C.Ó Searcóid, Mícheálhttp://hdl.handle.net/10197/628220150113T10:47:27Z20110101T00:00:00ZOn the Second Parameter of an (m, p)Isometry
Hoffmann, Philipp; Mackey, Michael C.; Ó Searcóid, Mícheál
A bounded linear operator T on a Banach space X is called an (m, p)isometry if it satisfies the equation TeX , for all TeX . In this paper we study the structure which underlies the second parameter of (m, p)isometric operators. We concentrate on determining when an (m, p)isometry is a (μ, q)isometry for some pair (μ, q). We also extend the definition of (m, p)isometry, to include p = ∞ and study basic properties of these (m, ∞)isometries.
20110101T00:00:00ZLocal derivations on Jordan triplesMackey, Michael C.http://hdl.handle.net/10197/626520141222T10:03:26Z20130301T00:00:00ZLocal derivations on Jordan triples
Mackey, Michael C.
R. V. Kadison (J. Algebra 130 (1990) 494–509) defined the notion of local derivation on an algebra and proved that every continuous local derivation on a von Neumann algebra is a derivation. We provide the analogous result in the setting of Jordan triples.
20130301T00:00:00ZCompletely bounded norms of right module mapsLevene, Rupert H.Timoney, Richard M.http://hdl.handle.net/10197/615820141113T09:14:56Z20120301T00:00:00ZCompletely bounded norms of right module maps
Levene, Rupert H.; Timoney, Richard M.
It is wellknown that if T is a Dm–Dn bimodule map on the m×n complex matrices, then T is a Schur multiplier and kTkcb = kTk. If n = 2 and T is merely assumed to be a right D2module map, then we show that kTkcb = kTk. However, this property fails if m ≥ 2 and n ≥ 3. For m ≥ 2 and n = 3, 4 or n ≥ m2 we give examples of maps T attaining the supremum C(m, n) = sup{kTkcb : T a right Dnmodule map on Mm,n with kTk ≤ 1}, we show that C(m, m2) = √
m and succeed in finding sharp results for C(m, n) in certain other cases. As a consequence, if H is an infinitedimensional Hilbert space and D is a masa in B(H), then there is a bounded right Dmodule map on K(H) which is not completely bounded.
20120301T00:00:00ZSingular solutions for secondorder nondivergence type elliptic inequalities in punctured ballsGhergu, MariusLiskevich, VitaliSobol, Zeevhttp://hdl.handle.net/10197/614920141111T14:32:48Z20140724T00:00:00ZSingular solutions for secondorder nondivergence type elliptic inequalities in punctured balls
Ghergu, Marius; Liskevich, Vitali; Sobol, Zeev
We study the existence and nonexistence of positive singular solutions of secondorder nondivergence type elliptic inequalities of the form $\sum\limits_{i,j = 1}^N {a_{ij} (x)\frac{{\partial ^2 u}} {{\partial x_i \partial x_j }}} + \sum\limits_{i = 1}^N {b_i (x)\frac{{\partial u}} {{\partial x_i }} \geqslant K(x)u^p ,}  \infty < p  \infty , $ with measurable coefficients in a punctured ball B R \{0} of ℝ N , N ≥ 1. We prove the existence of a critical value p* which separates the existence region from the nonexistence region. We show that in the critical case p = p*, the existence of a singular solution depends on the rate at which the coefficients (a i j ) and (b i ) stabilize at zero, and we provide some optimal conditions in this setting.
20140724T00:00:00ZNorms of idempotent Schur multipliersLevene, Rupert H.http://hdl.handle.net/10197/613420141110T16:56:34Z20140407T00:00:00ZNorms of idempotent Schur multipliers
Levene, Rupert H.
Let D be a masa in B(H) where H is a separable Hilbert space. We find real numbers η0 < η1 < η2 < · · · < η6 so that for every bounded, normal Dbimodule map Φ on B(H), either kΦk > η6 or kΦk = ηk for some k ∈ {0, 1, 2, 3, 4, 5, 6}. When D is totally atomic, these maps are the idempotent Schur multipliers and we characterise those with norm ηk for 0 ≤ k ≤ 6. We also show that the Schur idempotents which keep only the diagonal and superdiagonal of an n × n matrix, or of an n×(n+ 1) matrix, both have norm 2 n+1 cot(π 2(n+1) ), and we consider the average norm of a random idempotent Schur multiplier as a function of dimension. Many of our arguments are framed in the combinatorial language of bipartite graphs.
20140407T00:00:00ZClustering with the multivariate normal inverse Gaussian distributionO'Hagan, AdrianMurphy, BrendanGormley, Isobel Claireet al.http://hdl.handle.net/10197/610620141022T14:02:50Z20140919T00:00:00ZClustering with the multivariate normal inverse Gaussian distribution
O'Hagan, Adrian; Murphy, Brendan; Gormley, Isobel Claire; et al.
Many modelbased clustering methods are based on a finite Gaussian mixture model. The Gaussian mixture model implies that the data scatter within each group is elliptically shaped. Hence nonelliptical groups are often modeled by more than one component, resulting in model overfitting. An alternative is to use a mean–variance mixture of multivariate normal distributions with an inverse Gaussian mixing distribution (MNIG) in place of the Gaussian distribution, to yield a more flexible family of distributions. Under this model the component distributions may be skewed and have fatter tails than the Gaussian distribution. The MNIG based approach is extended to include a broad range of eigendecomposed covariance structures. Furthermore, MNIG models where the other distributional parameters are constrained is considered. The Bayesian Information Criterion is used to identify the optimal model and number of mixture components. The method is demonstrated on three sample data sets and a novel variation on the univariate Kolmogorov–Smirnov test is used to assess goodness of fit.
20140919T00:00:00ZUniversal Taylor series, conformal mappings and boundary behaviourGardiner, Stephen J.http://hdl.handle.net/10197/564520140618T14:08:50Z20131201T00:00:00ZUniversal Taylor series, conformal mappings and boundary behaviour
Gardiner, Stephen J.
A holomorphic function f on a simply connected domain Ω is said to possess a universal Taylor series about a point in Ω if the partial sums of that series approximate arbitrary polynomials on arbitrary compacta K outside Ω (provided only that K has connected complement). This paper shows that this property is not conformally invariant, and, in the case where Ω is the unit disc, that such functions have extreme angular
boundary behaviour.
20131201T00:00:00ZBoundary Behaviour of Universal Taylor Series on Multiply Connected DomainsGardiner, Stephen J.Manolaki, Myrtohttp://hdl.handle.net/10197/564420140610T14:09:22Z20140501T00:00:00ZBoundary Behaviour of Universal Taylor Series on Multiply Connected Domains
Gardiner, Stephen J.; Manolaki, Myrto
A holomorphic function on a planar domain Ω is said to possess a universal Taylor series about a point ζ of Ω if subsequences of the partial sums of the Taylor series approximate arbitrary polynomials on arbitrary compact sets in C∖Ω that have connected complement. In the case where Ω is simply connected, such functions are known to be unbounded and to form a collection that is independent of the choice of ζ. This paper uses tools from potential theory to show that, even for domains Ω of arbitrary connectivity, such functions are unbounded whenever they exist. In the doubly connected case, a further analysis of boundary behaviour reveals that the collection of such functions can depend on the choice of ζ. This phenomenon was previously known only for domains that are at least triply connected. Related results are also established for universal Laurent series.
20140501T00:00:00ZStationary Boundary Points for a Laplacian Growth Problem in Higher DimensionsGardiner, Stephen J.Sjödin, Tomashttp://hdl.handle.net/10197/564320150423T03:00:12Z20140801T00:00:00ZStationary Boundary Points for a Laplacian Growth Problem in Higher Dimensions
Gardiner, Stephen J.; Sjödin, Tomas
It is known that corners of interior angle less than π/2 in the boundary of a plane domain are initially stationary for Hele–Shaw flow arising from an arbitrary injection point inside the domain. This paper establishes the corresponding result for Laplacian growth of domains in higher dimensions. The problem is treated in terms of evolving families of quadrature domains for subharmonic functions.
20140801T00:00:00ZPolyharmonic functions of infinite order on annular regionsKounchev, OgnyanRender, Hermannhttp://hdl.handle.net/10197/555620140423T14:06:47Z20130601T00:00:00ZPolyharmonic functions of infinite order on annular regions
Kounchev, Ognyan; Render, Hermann
Polyharmonic functions f of in nite order and type on annular regions are
systematically studied. The rst main result states that the FourierLaplace coefficients
fk;l (r) of a polyharmonic function f of in nite order and type 0 can be extended to
analytic functions on the complex plane cut along the negative semiaxis. The second
main result gives a constructive procedure via FourierLaplace series for the analytic
extension of a polyharmonic function on annular region A(r0; r1) of in nite order and
type less than 1=2r1 to the kernel of the harmonicity hull of the annular region. The
methods of proof depend on an extensive investigation of Taylor series with respect to
linear differential operators with constant coefficients.
20130601T00:00:00ZThe approximation order of polysplinesKounchev, OgnyanRender, Hermannhttp://hdl.handle.net/10197/551920140404T11:41:40Z20040701T00:00:00ZThe approximation order of polysplines
Kounchev, Ognyan; Render, Hermann
We show that the scaling spaces de ned by the polysplines of order
p provide approximation order 2p: For that purpose we re ne the re
sults on one dimensional approximation order by Lsplines obtained
in [2].
20040701T00:00:00ZPolyharmonicity and algebraic support of measuresKounchev, OgnyanRender, Hermannhttp://hdl.handle.net/10197/551120140401T08:34:23Z20070201T00:00:00ZPolyharmonicity and algebraic support of measures
Kounchev, Ognyan; Render, Hermann
Our main result states that two signed measures μ and ν with bounded
support contained in the zero set of a polynomial P(χ) are equal if they coincide on the
subspace of all polynomials of polyharmonic degree NP where the natural number NP is
explicitly computed by the properties of the polynomial P(χ). The method of proof
depends on a definition of a multivariate Markov transform which is another major
objective of the present paper. The classical notion of orthogonal polynomial of second
kind is generalized to the multivariate setting: it is a polyharmonic function which has
similar features to those in the onedimensional case.
20070201T00:00:00ZConvergence of polyharmonic splines on semiregular grids Z x aZ^n for a to 0Kounchev, OgnyanRender, Hermannhttp://hdl.handle.net/10197/551020140401T08:31:20Z20070701T00:00:00ZConvergence of polyharmonic splines on semiregular grids Z x aZ^n for a to 0
Kounchev, Ognyan; Render, Hermann
Let
p,n
∈
N
with 2
p
≥
n
+ 2
,
and let
I
a
be a polyharmonic spline of
order
p
on the grid
Z
×
a
Z
n
which satisfies the interpolating conditions
I
a
(
j,am
) =
d
j
(
am
) for
j
∈
Z
,m
∈
Z
n
where the functions
d
j
:
R
n
→
R
and the parameter
a>
0 are given. Let
B
s
(
R
n
) be the set of all integrable
functions
f
:
R
n
→
C
such that the integral
k
f
k
s
:=
Z
R
n
b
f
(
ξ
)
(1 +

ξ

s
)
dξ
is finite. The main result states that for given
σ
≥
0 there exists a
constant
c>
0 such that whenever
d
j
∈
B
2
p
(
R
n
)
∩
C
(
R
n
)
,j
∈
Z
,
satisfy
k
d
j
k
2
p
≤
D
·
(1 +

j

σ
) for all
j
∈
Z
there exists a polyspline
S
:
R
n
+1
→
C
of order
p
on strips such that

S
(
t,y
)
−
I
a
(
t,y
)
≤
a
2
p
−
1
c
·
D
·
(1 +

t

σ
)
for all
y
∈
R
n
,t
∈
R
and all 0
<a
≤
1.
20070701T00:00:00ZOn realanalytic recurrence relations for cardinal exponential BsplinesAldaz, J. M.Kounchev, OgnyanRender, Hermannhttp://hdl.handle.net/10197/550820140401T08:23:56Z20071001T00:00:00ZOn realanalytic recurrence relations for cardinal exponential Bsplines
Aldaz, J. M.; Kounchev, Ognyan; Render, Hermann
Let LN+1 be a linear differential operator of order N + 1 with constant coefficients
and real eigenvalues λ 1, ..., λ N+1, let E( N+1) be the space of all C∞solutions of
LN+1 on the real line.We show that for N 2 and n = 2, ...,N, there is a recurrence
relation from suitable subspaces εn to εn+1 involving realanalytic functions, and
with εN+1 = E(Λ N+1) if and only if contiguous eigenvalues are equally spaced.
20071001T00:00:00ZPadé approximation for a multivariate Markov transformKounchev, OgnyanRender, Hermannhttp://hdl.handle.net/10197/550320140331T09:18:05Z20081001T00:00:00ZPadé approximation for a multivariate Markov transform
Kounchev, Ognyan; Render, Hermann
Methods of Padè approximation are used to analyse a multivariate
Markov transform which has been recently introduced by the authors.
The first main result is a characterization of the rationality of the
Markov transform via Hankel determinants. The second main result
is a cubature formula for a special class of measure.
20081001T00:00:00ZOn the mixed Cauchy problem with data on singular conicsEbenfelt, PeterRender, Hermannhttp://hdl.handle.net/10197/550120140328T09:37:53Z20080801T00:00:00ZOn the mixed Cauchy problem with data on singular conics
Ebenfelt, Peter; Render, Hermann
We consider a problem of mixed Cauchy type for certain holomorphic partial differential
operators with the principal part Q2p(D) essentially being the (complex) Laplace operator to
a power, Δp. We provide inital data on a singular conic divisor given by P = 0, where P is a
homogeneous polynomial of degree 2p. We show that this problem is uniquely solvable if the
polynomial P is elliptic, in a certain sense, with respect to the principal part Q2p(D).
20080801T00:00:00ZThe Goursat problem for a generalized Helmholtz operator in the planeEbenfelt, PeterRender, Hermannhttp://hdl.handle.net/10197/550020140328T09:34:45Z20080901T00:00:00ZThe Goursat problem for a generalized Helmholtz operator in the plane
Ebenfelt, Peter; Render, Hermann
We consider the Goursat problem in the plane for partial differential operators whose principal part is the pth power of the standard Laplace operator. The data is posed on a union of 2p distinct lines through the origin. We show that the solvability of this Goursat problem depends on Diophantine properties of the geometry of lines on which the data is posed.
20080901T00:00:00ZReproducing kernels for polyharmonic polynomialsRender, Hermannhttp://hdl.handle.net/10197/549920140328T09:31:08Z20081001T00:00:00ZReproducing kernels for polyharmonic polynomials
Render, Hermann
The reproducing kernel of the space of all homogeneous polynomi
als of degree
k
and polyharmonic order
m
is computed explicitly, solving a
question of A. Fryant and M.K. Vemuri.
20081001T00:00:00ZBernstein operators for exponential polynomialsAldaz, J. M.Kounchev, OgnyanRender, Hermannhttp://hdl.handle.net/10197/549820140328T09:30:16Z20090401T00:00:00ZBernstein operators for exponential polynomials
Aldaz, J. M.; Kounchev, Ognyan; Render, Hermann
Let
L
be a linear differential operator with constant coefficients of order
n
and complex eigenvalues
λ
0
,...,λ
n
. Assume that the set
U
n
of all solutions of the
equation
Lf
= 0 is closed under complex conjugation. If the length of the interval [
a,b
]
is smaller than
π/M
n
, where
M
n
:= max
{
Im
λ
j

:
j
= 0
,...,n
}
, then there exists a basis
p
n,k
,
k
= 0
,...n
, of the space
U
n
with the property that each
p
n,k
has a zero of order
k
at
a
and a zero of order
n
−
k
at
b,
and each
p
n,k
is positive on the open interval (
a,b
)
.
Under the additional assumption that
λ
0
and
λ
1
are real and distinct, our first main
result states that there exist points
a
=
t
0
<t
1
<...<t
n
=
b
and positive numbers
α
0
,..,α
n
, such that the operator
B
n
f
:=
n
X
k
=0
α
k
f
(
t
k
)
p
n,k
(
x
)
satisfies
B
n
e
λ
j
x
=
e
λ
j
x
, for
j
= 0
,
1
.
The second main result gives a sufficient condition
guaranteeing the uniform convergence of
B
n
f
to
f
for each
f
∈
C
[
a,b
].
20090401T00:00:00ZOn the Bernstein operator of S. Morigi and M. NeamtuKounchev, OgnyanRender, Hermannhttp://hdl.handle.net/10197/549620140328T09:07:13Z20091001T00:00:00ZOn the Bernstein operator of S. Morigi and M. Neamtu
Kounchev, Ognyan; Render, Hermann
We discuss a Bernstein type operator introduced by S. Morigi and
M. Neamtu for
D
polynomials in the more general framework of exponential
polynomials
20091001T00:00:00ZShape preserving properties of generalized Bernstein operators on extended Chebyshev spacesAldaz, J. M.Kounchev, OgnyanRender, Hermannhttp://hdl.handle.net/10197/549520140328T09:05:29Z20091201T00:00:00ZShape preserving properties of generalized Bernstein operators on extended Chebyshev spaces
Aldaz, J. M.; Kounchev, Ognyan; Render, Hermann
We study the existence and shape preserving properties of a generalized
Bernstein operator
B
n
fixing a strictly positive function
f
0
, and a second function
f
1
such
that
f
1
/f
0
is strictly increasing, within the framework of extended Chebyshev spaces
U
n
.
The first main result gives an inductive criterion for existence: suppose there exists a
Bernstein operator
B
n
:
C
[
a,b
]
→
U
n
with strictly increasing nodes, fixing
f
0
,f
1
∈
U
n
.
If
U
n
⊂
U
n
+1
and
U
n
+1
has a nonnegative Bernstein basis, then there exists a Bernstein
operator
B
n
+1
:
C
[
a,b
]
→
U
n
+1
with strictly increasing nodes, fixing
f
0
and
f
1
.
In
particular, if
f
0
,f
1
,...,f
n
is a basis of
U
n
such that the linear span of
f
0
,..,f
k
is an
extended Chebyshev space over [
a,b
] for each
k
= 0
,...,n
, then there exists a Bernstein
operator
B
n
with increasing nodes fixing
f
0
and
f
1
.
The second main result says that
under the above assumptions the following inequalities hold
B
n
f
≥
B
n
+1
f
≥
f
for all (
f
0
,f
1
)convex functions
f
∈
C
[
a,b
]
.
Furthermore,
B
n
f
is (
f
0
,f
1
)convex for all
(
f
0
,f
1
)convex functions
f
∈
C
[
a,b
]
.
20091201T00:00:00ZCauchy, Goursat and Dirichlet problems for holomorphic partial differential equationsRender, Hermannhttp://hdl.handle.net/10197/549220140327T15:57:12Z20110101T00:00:00ZCauchy, Goursat and Dirichlet problems for holomorphic partial differential equations
Render, Hermann
n this paper we survey recent results about Fischer decomposi
tions of polynomials or entire functions and their applications to holomorphic
partial di erential equations. We discuss Cauchy and Goursat problems for the
polyharmonic operator. Special emphasis is given to the KhavinsonShapiro
conjecture concerning polynomial solvability of the Dirichlet problem.
20110101T00:00:00ZThe KhavinsonShapiro conjecture and polynomial decompositionsLundberg, ErikRender, Hermannhttp://hdl.handle.net/10197/549020140327T15:51:19Z20110415T00:00:00ZThe KhavinsonShapiro conjecture and polynomial decompositions
Lundberg, Erik; Render, Hermann
The main result of the paper states the following: Let ψ be a polynomial
in n variables of degree t: Suppose that there exists a constant C > 0 such
that any polynomial f has a polynomial decomposition f = ψ qf + hf with
khf = 0 and deg qf deg f + C: Then deg ψ 2k. Here ∆k is the kth
iterate of the Laplace operator ∆ : As an application, new classes of domains
in Rn are identi ed for which the KhavinsonShapiro conjecture holds.
20110415T00:00:00ZPolyharmonic Hardy spaces on the complexified annulus and error estimates of cubature formulasKounchev, OgnyanRender, Hermannhttp://hdl.handle.net/10197/548920140327T15:50:00Z20121201T00:00:00ZPolyharmonic Hardy spaces on the complexified annulus and error estimates of cubature formulas
Kounchev, Ognyan; Render, Hermann
The present paper has a twofold contribution: first, we intro
duce a new concept of Hardy spaces on a multidimensional complexified
annular domain which is closely related to the annulus of the KleinDi
rac
quadric important in Conformal Quantum Field Theory. Secondly, for
functions in these Hardy spaces, we provide error estimate for the p
oly
harmonic GaußJacobi cubature formulas, which have been introduced
in previous papers.
20121201T00:00:00ZHarmonic divisors and rationality of zeros of Jacobi polynomialsRender, Hermannhttp://hdl.handle.net/10197/548820140327T15:47:30Z20130801T00:00:00ZHarmonic divisors and rationality of zeros of Jacobi polynomials
Render, Hermann
Let
Pn
(α,β
)
(
x
)
be the Jacobi polynomial of degree
n
with parameters
αβ
The main result of the paper states the following: If
b≠
1
;
3
and
c
are nonzero rel
atively prime natural numbers then
P
(
k
+(
d
3)
=
2
;k
+(
d
3)
=
2)
n
p
b=c
6
≠ 0
for all natural
numbers
d;n
and
k
2
N
0
:
Moreover, under the above assumption, the polynomial
Q
(
x
) =
b
c
x
2
1
+
:::
+
x
2
d
1
+
b
c
1
x
2
d
is not a harmonic divisor, and the Dirichlet problem for
the cone
f
Q
(
x
)
<
0
g
has polynomial harmonic solutions for polynomial data functions.
20130801T00:00:00ZRegularity of generalized Daubechies wavelets reproducing exponential polynomials with realvalued parametersDyn, NiraKounchev, OgnyanLevin, DavidRender, Hermannhttp://hdl.handle.net/10197/548420140903T12:28:10Z20140901T00:00:00ZRegularity of generalized Daubechies wavelets reproducing exponential polynomials with realvalued parameters
Dyn, Nira; Kounchev, Ognyan; Levin, David; Render, Hermann
We investigate nonstationary orthogonal wavelets based on a nonstationary
interpolatory subdivision scheme reproducing a given set of exponentials with realvalued
parameters. The construction is analogous to the construction of Daubechies wavelets
using the subdivision scheme of DeslauriersDubuc. The main result is the existence and
smoothness of these Daubechies type wavelets.
20140901T00:00:00ZConvergence of rational Bernstein operatorsRender, Hermannhttp://hdl.handle.net/10197/547820140327T14:32:26Z20140401T00:00:00ZConvergence of rational Bernstein operators
Render, Hermann
In this paper we discuss convergence properties and error estimates of rational
Bernstein operators introduced by P. Pit¸ul and P. Sablonni`ere. It is shown that the
rational Bernstein operators converge to the identity operator if and only if the maximal
difference between two consecutive nodes is converging to zero. Further a Voronovskaja
theorem is given based on the explicit computation of higher order moments for the
rational Bernstein operator
20140401T00:00:00ZReal Bargmann spaces, Fischer decompositions and Sets of uniqueness for polyharmonic functionsRender, Hermannhttp://hdl.handle.net/10197/547420140320T15:48:50Z20080401T00:00:00ZReal Bargmann spaces, Fischer decompositions and Sets of uniqueness for polyharmonic functions
Render, Hermann
In this paper a positive answer is given to the following question of W.K.
Hayman: if a polyharmonic entire function of order k vanishes on k distinct ellipsoids
in the euclidean space Rn then it vanishes everywhere. Moreover a characterization of
ellipsoids is given in terms of an extension property of solutions of entire data functions
for the Dirichlet problem answering a question of D. Khavinson and H.S. Shapiro. These
results are consequences from a more general result in the context of direct sum decompositions
(Fischer decompositions) of polynomials or functions in the algebra A(BR)
of all realanalytic functions defined on the ball BR of radius R and center 0 whose
Taylor series of homogeneous polynomials converges compactly in BR. The main result
states that for a given elliptic polynomial P of degree 2k and sufficiently large radius
R > 0 the following decomposition holds: for each function f 2 A(BR) there exist
unique q, r 2 A(BR) such that f = Pq + r and kr = 0. Another application of this
result is the existence of polynomial solutions of the polyharmonic equation ku = 0 for
polynomial data on certain classes of algebraic hypersurfaces.
2000 Mathematical Subject Classification. Primary: 31B30. Secondary: 35A20,
14P99, 12Y05
20080401T00:00:00ZBoundary behaviour of universal Taylor seriesGardiner, Stephen J.Khavinson, Dmitryhttp://hdl.handle.net/10197/531220140129T11:55:59Z20140201T00:00:00ZBoundary behaviour of universal Taylor series
Gardiner, Stephen J.; Khavinson, Dmitry
A power series that converges on the unit disc D is called universal if its partial sums approx
imate arbitrary polynomials on arbitrary compacta in CnD that have connected complement.
This paper shows that such series grow strongly and possess a Picardtype property near
each boundary point.
20140201T00:00:00ZMetSizeR: selecting the optimal sample size for metabolomic studies using an analysis based approach.Nyamundanda, GiftGormley, Isobel ClaireFan, YueGallagher, William M.Brennan, Lorrainehttp://hdl.handle.net/10197/504320131129T09:41:08Z20131121T00:00:00ZMetSizeR: selecting the optimal sample size for metabolomic studies using an analysis based approach.
Nyamundanda, Gift; Gormley, Isobel Claire; Fan, Yue; Gallagher, William M.; Brennan, Lorraine
Background: Determining sample sizes for metabolomic experiments is important but due to the complexity of these experiments, there are currently no standard methods for sample size estimation in metabolomics. Since pilot studies are rarely done in metabolomics, currently existing sample size estimation approaches which rely on
pilot data can not be applied.
Results: In this article, an analysis based approach called MetSizeR is developed to estimate sample size for metabolomic experiments even when experimental pilot data are not available. The key motivation for MetSizeR is that it considers the type of analysis the researcher intends to use for data analysis when estimating sample
size. MetSizeR uses information about the data analysis technique and prior expert knowledge of the metabolomic experiment to simulate pilot data from a statistical model. Permutation based techniques are then applied to the simulated pilot data to estimate the required sample size.
Conclusions: The MetSizeR methodology, and a publicly available software package which implements the approach, are illustrated through real metabolomic applications. Sample size estimates, informed by the intended statistical analysis technique, and the associated uncertainty are provided.
20131121T00:00:00ZJoint SpatioTemporal Modeling of Mycobacterium bovis Infections in Badgers and Cattle  Results from the Irish Four Area ProjectKelly, Gabrielle E.http://hdl.handle.net/10197/446020140630T03:00:08Z20130601T00:00:00ZJoint SpatioTemporal Modeling of Mycobacterium bovis Infections in Badgers and Cattle  Results from the Irish Four Area Project
Kelly, Gabrielle E.
In Ireland and in the UK, bovine tuberculosis(bTB) infects cattle and wildlife badgers (Meles meleslinnaeus) and badgers contribute to the spread of the disease in cattle. Isotropic and anisotropic spatiotemporalmodels are fitted to cattle herd and badger settbTB incidence data from the Four Area Project using sequences of linear geostatistical models. An association was found between the spatial distribution of the disease in cattle and badgers in two of three areas. The limited association may be due to irregularity of sett territories,fragmentation of farms, TBtest insensitivity, temporal lags associated with transmission or nonspatial transmission. A statistical methodology is outlined whereby hypotheses related to spatial correlation structure may be tested.
20130601T00:00:00ZAnisotropic spatial clustering of TB in cattle  the implications for control policyKelly, Gabrielle E.http://hdl.handle.net/10197/428820130425T14:10:17Z20110101T00:00:00ZAnisotropic spatial clustering of TB in cattle  the implications for control policy
Kelly, Gabrielle E.
Bovine TB is a disease that affects cattle and the wildlife badger, species Meles meles, in Ireland and the UK, and badgers have been implicated in the spread of the disease in cattle. Efforts to eradicate the disease that have included localized badger culling, have not been successful. In a study to understand how the disease spreads, Kelly and More [1] determined that the disease spatially clusters in cattle herds and estimated the practical spatial ranges at which this occurs. We extend this work by examining possible anisotropy in clustering and the consequences for TB control policy.
1st Conference on Spatial Statistics 2011 – Mapping Global Change, Enschede, The Netherlands, March, 2011
20110101T00:00:00ZClustering Ordinal Data via Latent Variable ModelsMcParland, DamienGormley, Isobel Clairehttp://hdl.handle.net/10197/428420140807T13:42:43Z20110801T00:00:00ZClustering Ordinal Data via Latent Variable Models
McParland, Damien; Gormley, Isobel Claire
Item response modelling is a well established method for analysing ordinal response data. Ordinal data are typically collected as responses to a number
of questions or items. The observed data can be viewed as discrete versions of an
underlying latent Gaussian variable. Item response models assume that this latent
variable (and therefore the observed ordinal response) is a function of both respondent specific and item specific parameters. However, item response models assume
a homogeneous population in that the item specific parameters are assumed to be
the same for all respondents. Often a population is heterogeneous and clusters of
respondents exist; members of different clusters may view the items differently. A
mixture of item response models is developed to provide clustering capabilities in
the context of ordinal response data. The model is estimated within the Bayesian
paradigm and is illustrated through an application to an ordinal response data set
resulting from a clinical trial involving selfassessment of arthritis.
IFCS 2011 Symposium of the International Federation of Classification Societies (IFCS), August 30, 2011, Frankfurt
20110801T00:00:00ZA long term observational study of the impact of badger removal on herd restrictions due to bovine TB in the Irish midlands during 19892004Kelly, Gabrielle E.Condon, J.More, Simon JohnDolan, L.Higgins, I.Eves, J.http://hdl.handle.net/10197/428020140730T08:30:03Z20081001T00:00:00ZA long term observational study of the impact of badger removal on herd restrictions due to bovine TB in the Irish midlands during 19892004
Kelly, Gabrielle E.; Condon, J.; More, Simon John; Dolan, L.; Higgins, I.; Eves, J.
An observational study was carried out, using data collected from four areas in the Irish midlands, between 1989 and 2004, to critically evaluate the longterm effects of proactive badger culling and to provide insights into reactive badger culling tuberculosis (TB) prevalence in cattle. Confirmed cattle herd TB incidence is the outcome measure used throughout. Relative to reactive culling, proactive badger culling was associated with a decrease in incidence in each of the 16 years of observation, which encompassed periods of both intensive and lessintensive badger removal. By 2004, we observed a decrease of 22% [95% confidence interval (CI) 1529, P<0.001] in the entire proactive and 37% (95% CI 25–47, P<0.001), in the inner proactive removal areas. The size of the decrease increased with time (P=0.055). There was a decrease (constant over time) of at least 14% (95% CI 76–97, P=0.013) in incidence in the inner compared to the outer control area (herds ≤2 km, >2 km, from proactive removal area boundaries, respectively). Incidence in the outer proactive removal area (herds <1.6 km from the proactive removal boundary) was similar to the inner control area (P=0.890). Incidence in the outer control area and total control area, compared to a neighbouring area some distance away, increased over the course of the study. Differences with the total control area were not statistically significant but the outer control area was 11% higher than the neighbouring area by 2004 (borderline significance P=0.057).
20081001T00:00:00ZBody mass index and height over three generations: evidence from the Lifeways crossgenerational cohort studyMurrin, CelineKelly, Gabrielle E.Tremblay, Richard E.Kelleher, Cecilyhttp://hdl.handle.net/10197/427920140625T15:42:59Z20120101T00:00:00ZBody mass index and height over three generations: evidence from the Lifeways crossgenerational cohort study
Murrin, Celine; Kelly, Gabrielle E.; Tremblay, Richard E.; Kelleher, Cecily
Background: Obesity and its measure of body mass index are strongly determined by parental body size. Debate continues as to whether both parents contribute equally to offspring body mass which is key to understanding the aetiology of the disease. The aim of this study was to use cohort data from three generations of one family to
examine the relative maternal and paternal associations with offspring body mass index and how these associations compare with family height to demonstrate evidence of genetic or environmental crossgenerational transmission.
Methods: 669 of 1082 families were followed up in 2007/8 as part of the Lifeways study, a prospective observational crossgeneration linkage cohort. Height and weight were measured in 529 Irish children aged 5 to 7 years and were selfreported by parents and grandparents. All adults provided information on selfrated health, education status, and indicators of income, diet and physical activity. Associations between the weight, height, and body mass index of family members were examined with mixed models and heritability estimates computed using linear regression analysis.
Results: Selfrated health was associated with lower BMI for all family members, as was age for children. When these effects were accounted for evidence of familial associations of BMI from one generation to the next was more apparent in the maternal line. Heritability estimates were higher (h2 = 0.40) for motheroffspring pairs compared to fatheroffspring pairs (h2 = 0.22). In the previous generation, estimates were higher between mothersparents (h2 = 0.540.60) but not between fathersparents (h2 = 0.040.17). Correlations between mother and offspring across two generations remained significant when modelled with fixed variables of socioeconomic status, health, and lifestyle. A similar analysis of height showed strong familial associations from maternal and paternal lines across each generation.
Conclusions: This is the first family cohort study to report an enduring association between mother and offspring BMI over three generations. The evidence of BMI transmission over three generations through the maternal line in an observational study corroborates the findings of animal studies. A more detailed analysis of geno and
phenotypic data over three generations is warranted to understand the nature of this maternaloffspring relationship.
20120101T00:00:00ZSpatial clustering of TBinfected cattle herds prior to and following proactive badger removalKelly, Gabrielle E.More, Simon Johnhttp://hdl.handle.net/10197/427820130424T15:33:46Z20110801T00:00:00ZSpatial clustering of TBinfected cattle herds prior to and following proactive badger removal
Kelly, Gabrielle E.; More, Simon John
Bovine tuberculosis (TB) is primarily a disease of cattle. In both Ireland and the UK, badgers (Meles meles) are an important wildlife reservoir of infection. This paper examined the hypothesis that TB is spatially correlated in cattle herds, established the range of correlation and the effect, if any, of proactive badger removal on this. We also reanalysed data from the Four Area Project in Ireland, a largescale intervention study aimed at assessing the effect of proactive badger culling on bovine TB incidence in cattle herds, taking possible spatial correlation into account. We established that infected herds are spatially correlated (the scale of spatial correlation is presented), but at a scale that varies with time and in different areas. Spatial correlation persists following proactive badger removal.
20110801T00:00:00ZEstimating the extent of spatial association of Mycobacterium bovis infection in badgers in IrelandKelly, Gabrielle E.McGrath, GuyMore, Simon Johnhttp://hdl.handle.net/10197/427520130424T15:26:29Z20100201T00:00:00ZEstimating the extent of spatial association of Mycobacterium bovis infection in badgers in Ireland
Kelly, Gabrielle E.; McGrath, Guy; More, Simon John
Mycobacterium bovis infects the wildlife species badgers Meles meles who are linked with the spread of the associated disease tuberculosis (TB) in cattle. Control of livestock infections depends in part on the spatial and social structure of the wildlife host. Here we describe spatial association of M. bovis infection in a badger population using data from the first year of the Four Area Project in Ireland. Using secondorder intensity functions, we show there is strong evidence of clustering of TB cases in each the four areas, i.e. a global tendency for infected cases to occur near other infected cases. Using estimated intensity functions, we identify locations where particular strains of TB cluster. Generalized linear geostatistical models are used to assess the practical range at which spatial correlation occurs and is found to exceed 6 in all areas. The study is of relevance concerning the scale of localized badger culling in the control of the disease in cattle.
20100201T00:00:00ZSpatioTemporal Modelling of TB in Cattle HerdsKelly, Gabrielle E.http://hdl.handle.net/10197/427320130424T14:22:45Z20120801T00:00:00ZSpatioTemporal Modelling of TB in Cattle Herds
Kelly, Gabrielle E.
We examine spatial association of bovine TB in cattle herds using data from Ireland. Badgers (Meles meles), a protected species under the Wildlife Act 1976 (OAG 2012), have been implicated in the spread of the disease in cattle. Current disease control policies include reactive culling (in response to TB outbreaks) of badgers in the index and neighbouring farms. Kelly and More (2011) using generalized linear geostatistical models, established that TB clusters in cattle herds and estimated the practical spatial ranges at which this occurs. Here this work is extended by taking into account possible anisotropy. Changes in spatial association over two time periods are also examined. The results have direct implications for establishing scale and direction in reactive culling. They are also of import regarding the evaluation of vaccines for badgers and cattle.
20120801T00:00:00ZBoundary behaviour of functions which possess universal Taylor seriesGardiner, Stephen J.http://hdl.handle.net/10197/422020140429T14:14:50Z20130201T00:00:00ZBoundary behaviour of functions which possess universal Taylor series
Gardiner, Stephen J.
It is known that, for any simply connected proper subdomain Ω of the complex plane and any point ζ in Ω, there are holomorphic functions on Ω that possess ‘universal’ Taylor series expansions about ζ; that is, partial sums of the Taylor series approximate arbitrary polynomials on arbitrary compacta in ℂ\ Ω that have connected complement. This paper establishes a strong unboundedness property for such functions near every boundary point. The result is new even in the case of the disc, where it strengthens work of several authors.
20130201T00:00:00ZDetermination of a universal seriesMouze, AugustinNestoridis, VassiliPapadoperakis, IoannisTsirivas, Nikolaoshttp://hdl.handle.net/10197/403520130114T17:12:15Z20120101T00:00:00ZDetermination of a universal series
Mouze, Augustin; Nestoridis, Vassili; Papadoperakis, Ioannis; Tsirivas, Nikolaos
The known proofs for universal Taylor series do not determine a specific universal
Taylor series. In the present paper, we isolate a specific universal Taylor series by
modifying the proof in [30]. Thus we determine all Taylor coefficients of a specific
universal Taylor series on the disc or on a polygonal domain. Furthermore in non
simply connected domains, when universal Taylor series exist, we can construct a
sequence of specific rational functions converging to a universal function, provided
the boundary is good enough. The solution uses an infinite denumerable procedure
and a finite number of steps is not sufficient. However we solve a Runge's type
problem in a finite number of steps.
20120101T00:00:00ZOstrowskitype theorems for harmonic functionsManolaki, Myrtohttp://hdl.handle.net/10197/403420130114T17:07:58Z20120715T00:00:00ZOstrowskitype theorems for harmonic functions
Manolaki, Myrto
Ostrowski showed that there are intimate connections between the gap structure of a Taylor series and the behaviour of its partial sums outside the disk of convergence. This paper investigates the corresponding problem for the homogeneous polynomial expansion of a harmonic function. The results for harmonic functions display new features in the case of higher dimensions.
20120715T00:00:00ZTwophase quadrature domainsGardiner, Stephen J.Sjödin, Tomashttp://hdl.handle.net/10197/402820130114T16:24:52Z20120101T00:00:00ZTwophase quadrature domains
Gardiner, Stephen J.; Sjödin, Tomas
Recent work on twophase free boundary problems has led to the investigation of a new type of quadrature domain for harmonic functions. This paper develops a method of constructing such quadrature domains based on the technique of partial balayage, which has proved to be a useful tool in the study of onephase quadrature domains and HeleShaw flows.
20120101T00:00:00ZSentiment Analysis of Online MediaSalterTownshend, MichaelMurphy, Thomas Brendanhttp://hdl.handle.net/10197/396420140328T12:56:27Z20121218T00:00:00ZSentiment Analysis of Online Media
SalterTownshend, Michael; Murphy, Thomas Brendan
A joint model for annotation bias and document classification is presented
in the context of media sentiment analysis. We consider an Irish online media data
set comprising online news articles with user annotations of negative, positive or
irrelevant impact on the Irish economy. The joint model combines a statistical model
for user annotation bias and a Naive Bayes model for the document terms. An EM
algorithm is used to estimate the annotation bias model, the unobserved biases in the
user annotations, the classifier parameters and the sentiment of the articles. The joint
modeling of both the user biases and the classifier is demonstrated to be superior to
estimation of the bias followed by the estimation of the classifier parameters.
GfKl 2011: Joint Conference of the German Classification Society (GfKl)
and the German Association for Pattern Recognition (DAGM) August 31 to September 2, 2011 and the IFCS 2011: Symposium of the International Federation of Classification Societies (IFCS) August 30, 2011, Frankfurt am Main, Germany
20121218T00:00:00ZRecent progress on fine differentiability and fine harmonicityGardiner, Stephen J.http://hdl.handle.net/10197/396120141111T12:32:34Z20121215T00:00:00ZRecent progress on fine differentiability and fine harmonicity
Gardiner, Stephen J.
This paper describes recent results concerning the notions of differentiability and harmonicity with respect to the ne topology of classical potential theory.
Complex Analysis and Potential Theory : a conference in honour of Paul M. Gauthier and Kohur Gowrisankaran, Montreal, June 2023, 2011
20121215T00:00:00Z