Mathematics and Computer Studies (Peer-reviewed publications)

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Groups of singular alternating sign matrices
(International Linear Algebra Society, 2025-05-20) O'Brien, Cian; Quinlan, Rachel
We investigate multiplicative groups consisting entirely of singular alternating sign matrices (ASMs) and present several constructions of such groups. It is shown that every finite group is isomorphic to a group of singular ASMs, with a singular idempotent ASM as its identity element. The relationship between the size, the rank, and the possible multiplicative orders of singular ASMs is explored.
Weighted projections of alternating sign matrices: Latin-like squares and the ASM polytope
(Electronic Journal of Combinatorics, 2024-01-12) O'Brien, Cian
The weighted projection of an alternating sign matrix (ASM) was introduced by Brualdi and Dahl (2018) as a step towards characterising a generalisation of Latin squares they defined using alternating sign hypermatrices. Given row-vector z_n=(n,…,2,1), the weighted projection of an ASM A is equal to z_nA. Brualdi and Dahl proved that the weighted projection of an n×n ASM is majorized by the vector z_n, and conjectured that any positive integer vector majorized by z_n is the weighted projection of some ASM. The main result of this paper presents a proof of this conjecture, via monotone triangles. A relaxation of a monotone triangle, called a row-increasing triangle, is introduced. It is shown that for any row-increasing triangle T, there exists a monotone triangle M such that each entry of M occurs the same number of times as in T. A construction is also outlined for an ASM with given weighted projection. The relationship of the main result to existing results concerning the ASM polytope ASMn is examined, and a characterisation is given for the relationship between elements of ASMn corresponding to the same point in the regular n-permutohedron. Finally, the limitations of the main result for characterising alternating sign hypermatrix Latin-like squares are considered.
Alternating signed bipartite graphs and difference-1 colourings
(Elsevier, 2020-07-13) O'Brien, Cian; Jennings, Kevin; Quinlan, Rachel
We investigate a class of 2-edge coloured bipartite graphs known as alternating signed bipartite graphs (ASBGs) that encode the information in alternating sign matrices. The central question is when a given bipartite graph admits an ASBG-colouring; a 2-edge colouring such that the resulting graph is an ASBG. We introduce the concept of a difference-1 colouring, a relaxation of the concept of an ASBG-colouring, and present a set of necessary and sufficient conditions for when a graph admits a difference-1 colouring. The relationship between distinct difference-1 colourings of a particular graph is characterised, and some classes of graphs for which all difference-1 colourings are ASBG-colourings are identified. One key step is Theorem 3.4.6, which generalises Hall’s Matching Theorem by describing a necessary and sufficient condition for the existence of a subgraph H of a bipartite graph in which each vertex v of H has some prescribed degree r(v).
Alternating sign hypermatrix decompositions of Latin-like squares
(Elsevier, 2020-08-14) O'Brien, Cian
To any n × n Latin square L, we may associate a unique sequence of mutually orthogonal permutation matrices P = P_1, P_2, ..., P_n such that L = L(P ) = ∑ k_Pk . Brualdi and Dahl (2018) described a generalisation of a Latin square, called an alternating sign hypermatrix Latin-like square (ASHL), by replacing P with an alternating sign hypermatrix (ASHM). An ASHM is an n × n × n (0,1,-1)-hypermatrix in which the non-zero elements in each row, column, and vertical line alternate in sign, beginning and ending with 1. Since every sequence of n mutually orthogonal permutation matrices forms the planes of a unique n × n × n ASHM, this generalisation of Latin squares follows very naturally, with an ASHM A having corresponding ASHL L = L(A) = ∑ kA_k , where A_k is the kth plane of A. This paper addresses open problems posed in Brualdi and Dahl’s article, firstly by characterising how pairs of ASHMs with the same corresponding ASHL relate to one another and identifying the smallest dimension for which this can happen, and secondly by exploring the maximum number of times a particular integer may occur as an entry of an n × n ASHL. A construction is given for an n × n ASHL with the same entry occurring (n^2 +4n−19)/2 times, improving on the previous best of 2n.
Alternating sign matrices of finite multiplicative order
(Elsevier, 2022-07-08) O'Brien, Cian; Quinlan, Rachel
We investigate alternating sign matrices that are not permuta- tion matrices, but have finite order in a general linear group. We classify all such examples of the form P + T , where P is a permutation matrix and T has four non-zero entries, forming a square with entries 1 and −1 in each row and column. We show that the multiplicative orders of these matrices do not always coincide with those of permutation matrices of the same size. We pose the problem of identifying finite subgroups of general linear groups that are generated by alternating sign matrices. © 2022 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license
Markovianity and the Thompson Group F (Pre-published version)
(2022-10-27) Koestler, Claus; Krishnan, Arundhathi
We show that representations of the Thompson group F in the automorphisms of a noncommutative probability space yield a large class of bilateral stationary noncommutative Markov processes. As a partial converse, bilateral stationary Markov processes in tensor dilation form yield representations of F. As an application, and building on a result of Kuemmerer, we canonically associate a representation of F to a bilateral stationary Markov process in classical probability.
Pseudospectrum of an element of a Banach algebra (Pre-published version)
(Element Publishing House, 2017-03) Kulkarni, S H; Krishnan, Arundhathi
The ε -pseudospectrum Λε (a) of an element a of an arbitrary Banach algebra A is studied. Its relationships with the spectrum and numerical range of a are given. Characterizations of scalar, Hermitian and Hermitian idempotent elements by means of their pseudospectra are given. The stability of the pseudospectrum is discussed. It is shown that the pseudospectrum has no isolated points, and has a finite number of components, each containing an element of the spectrum of a. Suppose for some ε > 0 and a,b ∈ A, Λε (ax) = Λε(bx) ∀x ∈ A. It is shown that a = b if: (i) a is invertible. (ii) a is Hermitian idempotent. (iii) a is the product of a Hermitian idempotent and an invertible element. (iv) A is semisimple and a is the product of an idempotent and an invertible element. (v) A = B(X) for a Banach space X . (vi) A is a C∗-algebra. (vii) A is a commutative semisimple Banach algebra.
Pseudospectra of elements of reduced Banach algebras (Pre-published version)
(Springer, 2017) Kulkarni, S H; Krishnan, Arundhathi
Let A be a Banach algebra with identity 1 and p ∈ A be a non-trivial idempotent. Then q = 1−p is also an idempotent. The subalgebras pAp and qAq are Banach algebras, called reduced Banach algebras, with identities p and q respectively. For a ∈ A and ε > 0, we examine the relationship between the ε-pseudospectrum Λε(A, a) of a ∈ A, and ε-pseudospectra of pap ∈ pAp and qaq ∈ qAq. We also extend this study by considering a finite number of idempotents p1, · · · , pn, as well as an arbitrary family of idempotents satisfying certain conditions.
Pseudospectra of elements of reduced Banach algebras II (Pre-published version)
(Faculty of Sciences and Mathematics, University of Nis, Serbia, 2018-06) Kulkarni, S H; Krishnan, Arundhathi
Let A be a Banach algebra with identity 1 and p ∈ A be a non-trivial idempotent. Then q = 1 − p is also an idempotent. The subalgebras pAp and qAq are Banach algebras, called reduced Banach algebras, with identities p and q respectively. Let x ∈ A be such that pxp = xp, and ε > 0. We examine the relationship between the spectrum of x ∈ A, σ(A, x), and the spectra of pxp ∈ pAp, σ(pAp, pxp) and qxq ∈ qAq, σ(qAq, qxq). Similarly, we examine the relationship betweeen the ε-pseudospectrum of x ∈ A, Λε(A, x) and ε-pseudospectra of pxp ∈ pAp, Λε(pAp, pxp) and of qxq ∈ qAq, Λε(qAq, qxq).
Markovianity and the Thompson monoid F+ (Pre-published version)
(Elsevier, 2023-03-13) Koestler, Claus; Krishnan, Arundhathi; Wills, Stephen
We introduce a new distributional invariance principle, called `partial spreadability', which emerges from the representation theory of the Thompson monoid F+ in noncommutative probability spaces. We show that a partially spreadable sequence of noncommutative random variables is adapted to a local Markov filtration. Conversely we show that a large class of noncommutative stationary Markov sequences provides representations of the Thompson monoid F+. In the particular case of a classical probability space, we arrive at a de Finetti theorem for stationary Markov sequences with values in a standard Borel space.
Which graphs are rigid in lpd? (Pre-published)
(Springer, 2021-03-13) Dewar, Sean; Kitson, Derek; Nixon, Anthony
We present three results which support the conjecture that a graph is minimally rigid in d-dimensional ℓp-space, where p∈(1,∞) and p≠2, if and only if it is (d, d)-tight. Firstly, we introduce a graph bracing operation which preserves independence in the generic rigidity matroid when passing from ℓdp to ℓd+1p. We then prove that every (d, d)-sparse graph with minimum degree at most d+1 and maximum degree at most d+2 is independent in ℓdp. Finally, we prove that every triangulation of the projective plane is minimally rigid in ℓ3p. A catalogue of rigidity preserving graph moves is also provided for the more general class of strictly convex and smooth normed spaces and we show that every triangulation of the sphere is independent for 3-dimensional spaces in this class.
Constructing isostatic frameworks for the l1 and l infinity plane (Pre-published)
(Electronic Journal of Combinatorics, 2020-06-12) Clinch, Katie; Kitson, Derek
We use a new coloured multi-graph constructive method to prove that if the edge-set of a graph G = (V,E) has a partition into two spanning trees T1 and T2 then there is a map p : V → R2, p(v) = (p(v)1,p(v)2), such that |p(u)i −p(v)i| > |p(u)3−i−p(v)3−i| for every edge uv in Ti (i = 1,2). As a consequence, we solve an open problem on the realisability of minimally rigid bar-joint frameworks in the `1 or `∞-plane. We also show how to adapt this technique to incorporate symmetry and indicate several related open problems on rigidity, redundant rigidity and forced symmetric rigidity in normed spaces.
Graph rigidity for unitarily invariant matrix norms (Pre-published)
(Elsevier, 2020-11-15) Kitson, Derek; Levene, Rupert H
A rigidity theory is developed for bar-joint frameworks in linear matrix spaces endowed with a unitarily invariant matrix norm. Analogues of Maxwell's counting criteria are obtained and minimally rigid matrix frameworks are shown to belong to the matroidal class of -sparse graphs for suitable k and l. An edge-colouring technique is developed to characterise infinitesimal rigidity for product norms and then applied to show that the graph of a minimally rigid bar-joint framework in the space of 2 x 2 symmetric (respectively, hermitian) matrices with the trace norm admits an edge-disjoint packing consisting of a (Euclidean) rigid graph and a spanning tree.
Symbol functions for symmetric frameworks (Pre-published)
(Elsevier, 2021-05-15) Kitson, Derek; Kastis, Eleftherios; McCarthy, John E
We prove a variant of the well-known result that intertwiners for the bilateral shift on ℓ2(Z) are unitarily equivalent to multiplication operators on L2(T). This enables us to unify and extend fundamental aspects of rigidity theory for bar-joint frameworks with an abelian symmetry group. In particular, we formulate the symbol function for a wide class of frameworks and show how to construct generalised rigid unit modes in a variety of new contexts.
Rigidity of symmetric frameworks in normed spaces
(Elsevier, 2020-12-15) Kitson, Derek; Nixon, Anthony; Schulze, Bernd
We develop a combinatorial rigidity theory for symmetric bar-joint frameworks in a general finite dimensional normed space. In the case of rotational symmetry, matroidal Maxwell-type sparsity counts are identified for a large class of d-dimensional normed spaces (including all lp spaces with p not equal to 2). Complete combinatorial characterisations are obtained for half-turn rotation in the l1 and l-infinity plane. As a key tool, a new Henneberg-type inductive construction is developed for the matroidal class of (2,2,0)-gain-tight gain graphs.
Anomalies of the magnitude of the bias of the maximum likelihood estimator of the regression slope
(Athens Institute for Education and Research, 2015) O'Driscoll, Diarmuid; Ramirez, Donald E.
The slope of the best-fit line y h x x 0 1  ( )    from minimizing a function of the squared vertical and horizontal errors is the root of a polynomial of degree four which has exactly two real roots, one positive and one negative, with the global minimum being the root corresponding to the sign of the correlation coefficient. We solve second order and fourth order moment equations to estimate the variances of the errors in the measurement error model. Using these solutions as an estimate of the error ratio in the maximum likelihood estimator, we introduce a new estimator kap 1  . We create a function  which relates to the oblique parameter , used in the parameterization of the line from (x,h(x)) to ( ( ), ) 1 h y y  , to introduce an oblique estimator lam 1 . A Monte Carlo simulation study shows improvement in bias and mean squared error of each of these two new estimators over the ordinary least squares estimator. In O’Driscoll and Ramirez (2011), it was noted that the bias of the MLE estimator of the slope is monotone decreasing as the estimated variances error ratio approaches the true variances error ratio. However for a fixed estimated variances error ratio , it was noted that the bias is not monotone decreasing as the true error ratio κ approaches . This paper explains this anomaly by showing that as κ approaches a fixed , the bias of the MLE estimator of the slope is also dependent on the magnitude of. Other anomalies with the MLE estimator of the slope in the presence of errors in both x and y are discussed.
An investigation of the performance of five different estimators in the measurement error regression model
(Athens Institute for Education and Research, 2015) O'Driscoll, Diarmuid; Ramirez, Donald E.
In a comprehensive paper by Riggs et al.(1978) the authors analyse the performances of numerous estimators for the regression slope in the measurement error model with positive measurement error variances >0 0 for X and >0 for Y . In particular, using a Monte Carlo simulation, the authors demonstrate that the adjusted geometric mean estimator of Madansky (1959, Equation 4, p. 179), which requires knowledge of both and , performs “much worse than” the maximum likelihood estimator in the normal structural measurement error model which requires only knowledge of the ratio . The second moment estimator, coincides with the maximum likelihood estimator inthe normal structural measurement error model (Madansky (1959)). In practice κ has to be estimated by .In this paper, we show that the bias of is not only dependent on the magnitude of the difference between κ and but also on the magnitude of - . We use a fourth moment estimator to smooth the jump discontinuity in the estimator of Copas (1972) as described in ODriscoll and Ramirez (2011) and use this estimator to find estimates for each error variance and . Our Monte Carlo simulations show that the adjusted geometric mean estimator of Madansky performs much better than the ordinary least squares estimators OLS(y|x) and OLS(x|y) when the error variances are strictly positive and performs equally as well as the geometric estimator, , and the perpendicular estimator of Adcock (1878), , with κ = 1 .
A note on the computation of symmetric powers of hyperbolic forms and of trace froms on symbol algebras
(Scientific Advances Publishers, 2014) Flatley, Ronan
Let K be a field with characteristic different from 2 and let S be a symbol algebra over K. We compute the symmetric powers of hyperbolic quadratic forms over K. Also, we compute the symmetric powers of the quadratic trace form of S. In both cases, we apply a generalized form of the Vandermonde convolution in the course of the computations.
On moduli stacks of G-bundles over a curve (Pre-published version)
(Springer, 2010) Hoffmann, Norbert
Let C be a smooth projective curve over an algebraically closed eld k of arbitrary characteristic. Given a linear algebraic group G over k, let MG be the moduli stack of principal G-bundles on C. We determine the set of connected components 0(MG) for smooth connected groups G.
Moduli stacks of vector bundles on curves and the King–Schofield rationality proof (Pre-published version)
(Springer, 2010) Hoffmann, Norbert
Let C be a connected smooth projective curve of genus g ≥ 2 over an algebraically closed field k. Consider the coarse moduli scheme Bunr,d (resp. Bunr,L) of stable vector bundles on C with rank r and degree d ∈ Z (resp. determinant isomorphic to the line bundle L on C).

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