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Computing primary solutions of equations involving primary matrix functions
No full textThe matrix equation f(X) = A, where f is an analytic function and A is a square matrix, is considered. Some results on the classification of solutions are provided. When f is rational, a numerical algorithm is proposed to compute all solutions that can be written as a polynomial of A. For real data, the algorithm yield the real solutions using only real arithmetic. Numerical experiments show that the algorithm performs in a stable fashion in finite precision arithmetic
Cuspidal Characters of Sporadic Simple Groups
No full textIn this paper the notion of an irreducible cuspidal character for fi�nite
groups of Lie type is generalized to any �finite group. All the irreducible
cuspidal characters for the �finite sporadic simple groups are then determined
A Note on Involution Centralizers in Black Box Groups
No full textHere we note a minor variation on the method in [1] which enables
calculations of CH(t) for H a subgroup of a black box group G and t an
involution of G
The inflation of viscoelastic balloons and hollow viscera
Full text linkFor the first time, the problem of the inflation of a thick-walled spherical shell is considered where the wall material is nonlinearly viscoelastic. We focus here specifically on the context where the wall has quasilinear viscoelastic constitutive behaviour. This problem is of fundamental importance in a wide range of applications but particularly in the context of biological systems such as hollow viscera including the lungs and bladder. Some canonical problems associated with the inflation and deflation of a thick walled nonlinear viscoelastic shell are described, noting that the solution of such problems, where pressures are imposed, requires the numerical solution to a nonlinear Volterra integral equation in space and time. Here a new technique to solve such equations is described. Furthermore, the limit of a thin-walled shell yields the scenario of a viscoelas-tic balloon. The associated nonlinear elastic problem of inflation of a balloon has been studied extensively but there is a paucity of studies considering the associated nonlinear viscoelastic problem. We show that, in contrast to the elastic scenario, the peak pressure associated with inflation of a neo-Hookean balloon is not independent of the shear modulus of the medium. This problem is also described in the context of intragastric balloons, which are now commonly used to treat obesity effectively
Van Dooren's Index Sum Theorem and Rational Matrices with Prescribed Structural Data
Full text linkThe structural data of any rational matrix R(\la),
i.e., the structural indices of its poles and zeros
together with the minimal indices of its left and right
nullspaces,
is known to satisfy a simple condition involving certain sums of these indices.
This fundamental constraint was first proved by Van Dooren in ;
here we refer to this result as the ``rational index sum theorem''.
An analogous result for polynomial matrices
has been independently discovered (and re-discovered)
several times in the past three decades.
In this paper we clarify the connection between these
two seemingly different index sum theorems,
describe a little bit of the history of their development,
and discuss their curious apparent unawareness of each other.
Finally, we use the connection between these results
to solve a fundamental inverse problem
for rational matrices ---
for which lists of prescribed structural data
does there exist some rational matrix R(\la)
that realizes exactly the list ?
We show that Van Dooren's condition is the \emph{only} constraint on rational realizability;
that is, a list is the structural data of some rational R(\la)
if and only if
satisfies the rational index sum condition
Bridging the gap between flat and hierarchical low-rank matrix formats: the multilevel BLR format
Full text linkMy Time at Enfield
No full textThis article centres around my time at Enfield College of
Technology. The college was part of the revolution in higher education fostered by members of both the Labour and Conservative parties, and which found expression in the Robbins Committee, set up by Harold Macmillan's government, whose report was published in 1963.
A lightly edited version of this article, without photographs, appears as Chapter 5 in the book Enfield Voices: The Birth of the People's Universities, edited by Tom Bourner and Tony Crilly and published in 2018. I am grateful to Tom and Tony, who were both colleagues at
Enfield, for the opportunity to tell something of my story.
Although certainly not unique, I was unusual in that I was a student on the Mathematics for Business, B.Sc., degree at Enfield College of Technology, and became a lecturer on the same degree the term after graduating. This article tells something of the story of my student life and working life, before, during and after Enfield
An Arbitrary Precision Scaling and Squaring Algorithm for the Matrix Exponential
Full text linkThe most popular algorithms for computing the matrix exponential are those based on the scaling and squaring technique. For optimal efficiency these are usually tuned to a particular precision of floating-point arithmetic. We design a new scaling and squaring algorithm that takes the unit roundoff of the arithmetic as input and chooses the algorithmic parameters in order to keep the forward error in the underlying Padé approximation below the unit roundoff. To do so, we derive an explicit expression for all the coefficients in an error expansion for Padé approximants to the exponential and use it to obtain a new bound for the truncation error. We also derive a new technique for selecting the internal parameters used by the algorithm, which at each step decides whether to scale or to increase the degree of the approximant. The algorithm can employ diagonal Padé approximants or Taylor approximants and can be used with a Schur decomposition or in transformation-free form. Our numerical experiments show that the new algorithm performs in a forward stable way for a wide range of precisions and that the most accurate of our implementations, the Taylor-based transformation-free variant, is superior to existing alternatives
Reduction and relative equilibria for the 2-body problem in spaces of constant curvature
Full text linkWe consider the two-body problem on surfaces of constant non-zero curvature and classify the relative equilibria and their stability. On the hyperbolic plane, for each we show there are two relative equilibria where the masses are separated by a distance . One of these is geometrically of elliptic type and the other of hyperbolic type. The hyperbolic ones are always unstable, while the elliptic ones are stable when sufficiently close, but unstable when far apart.
On the sphere of positive curvature, if the masses are different, there is a unique relative equilibrium (RE) for every angular separation except . When the angle is acute, the RE is elliptic, and when it is obtuse the RE can be either elliptic or linearly unstable. We show using a KAM argument that the acute ones are almost always nonlinearly stable. If the masses are equal there are two families of relative equilibria: one where the masses are at equal angles with the axis of rotation (`isosceles RE') and the other when the two masses subtend a right angle at the centre of the sphere. The isosceles RE are
elliptic if the angle subtended by the particles is acute and is unstable if it is obtuse. At , the two families meet and a
pitchfork bifurcation takes place. Right-angled RE are elliptic away from the bifurcation point.
In each of the two geometric settings, we use a global reduction to eliminate the group of symmetries and analyse the resulting reduced equations which live on a 5-dimensional phase space and possess one Casimir function
Integrability and dynamics of the n-dimensional symmetric Veselova top
No full textWe consider the the n-dimensional generalisation of the nonholonomic Veselova problem.
We derive the reduced equations of motion in terms of the mass tensor of the body and determine some general properties of the dynamics. In particular we give a closed formula for the invariant measure, we indicate the existence of steady rotation solutions, and obtain some results on their stability.
We then focus our attention on bodies whose mass tensor has a specific type of symmetry. We show that the phase space is foliated by invariant tori that carry quasi-periodic dynamics in the natural time variable. Our results
enlarge the known cases of integrability of the multi-dimensional Veselova top. Moreover, they show that in some previously known instances of integrability, the flow is quasi-periodic without the need of a time reparametrisation