# Karl Michael SchmidtCardiff University | CU · School of Mathematics

Karl Michael Schmidt

Dr rer nat

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87

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Introduction

## Publications

Publications (87)

Introduction:
Both late-onset Alzheimer's disease (AD) and ageing have a strong genetic component. In each case, many associated variants have been discovered, but how much missing heritability remains to be discovered is debated. Variability in the estimation of SNP-based heritability could explain the differences in reported heritability.
Metho...

Sum systems are finite collections of finite component sets of non-negative integers, of prescribed cardinalities, such that their set sum generates consecutive integers without repetitions. In this present work we consider centred sum systems which generate either consecutive integers or half-integers centred around the origin, detailing some inva...

We consider square matrices arising as the sum of left and right circulant matrices and derive asymptotics of the sequence of their powers. Particular emphasis is laid on the case where the matrix has consecutive integer entries; we find explicit formulae for the eigenvalues and eigenvectors of the matrix in this case and find its Moore-Penrose pse...

INTRODUCTION: Both Alzheimer's disease (AD) and ageing have a strong genetic component. In each case, many associated variants have been discovered, but how much missing heritability remains to be discovered is debated. Variability in the estimation of SNP-based heritability could explain the differences in reported heritability.
METHODS: We comput...

Background
Alzheimer’s disease, among other neurodegenerative disorders, spans decades in individuals’ life and exhibits complex progression, symptoms and pathophysiology. Early diagnosis is essential for disease prevention and therapeutic intervention. Genetics may help identify individuals at high risk. As thousands of genetic variants may contri...

Background: Alzheimer’s disease, among other neurodegenerative disorders, spans decades in individuals’ life and exhibits complex progression, symptoms and pathophysiology. Early diagnosis is essential for disease prevention and therapeutic intervention. Genetics may help identify individuals at high risk. As thousands of genetic variants may contr...

We identify and analyse obstructions to factorisation of integer matrices into products $N^T N$ or $N^2$ of matrices with rational or integer entries. The obstructions arise as quadratic forms with integer coefficients and raise the question of the discrete range of such forms. They are obtained by considering matrix decompositions over a superalge...

We identify and analyse obstructions to factorisation of integer matrices into products NTN or N2 of matrices with rational or integer entries. The obstructions arise as quadratic forms with integer coefficients and raise the question of the discrete range of such forms. They are obtained by considering matrix decompositions over a superalgebra. We...

Background:
Genome-wide association studies (GWAS) were successful in identifying SNPs showing association with disease, but their individual effect sizes are small and require large sample sizes to achieve statistical significance. Methods of post-GWAS analysis, including gene-based, gene-set and polygenic risk scores, combine the SNP effect size...

For the Schr\"odinger equation $-d^2 u/dx^2 + q(x)u = \lambda u$ on a finite $x$-interval, there is defined an "asymmetry function" $a(\lambda;q)$, which is entire of order $1/2$ and type $1$ in $\lambda$. Our main result identifies the classes of square-integrable potentials $q(x)$ that possess a common asymmetry function. For any given $a(\lambda...

The jth divisor function \(d_j\), which counts the ordered factorisations of a positive integer into j positive integer factors, is a very well-known multiplicative arithmetic function. However, the non-multiplicative jth non-trivial divisor function \(c_j\), which counts the ordered factorisations of a positive integer into j factors each of which...

The polygenic risk scores (PRS) approach has been widely used across different traits for estimating polygenic risk, pleiotropy and disease prediction, but mostly in European populations. The predictive ability of the PRS in non-European populations is currently limited due to the lack of genetic research performed in populations of non-European an...

Divisor functions have attracted the attention of number theorists from Dirichlet to the present day. Here we consider associated divisor functions $c_j^{(r)}(n)$ which for non-negative integers $j, r$ count the number of ways of representing $n$ as an ordered product of $j+r$ factors, of which the first $j$ must be non-trivial, and their natural e...

A major controversy in psychiatric genetics is whether nonadditive genetic interaction effects contribute to the risk of highly polygenic disorders. We applied a support vector machines (SVMs) approach, which is capable of building linear and nonlinear models using kernel methods, to classify cases from controls in a large schizophrenia case–contro...

The celebrated Schoenberg theorem establishes a relation between positive definite and conditionally positive definite functions. In this paper, we consider the classes of real-valued functions P(J) and CP(J), which are positive definite and respectively, conditionally positive definite, with respect to a given class of test functions J. For suitab...

The $j$th divisor function $d_j$, which counts the ordered factorisations of a positive integer into $j$ positive integer factors, is a very well-known arithmetic function; in particular, $d_2(n)$ gives the number of divisors of $n$. However, the $j$th non-trivial divisor function $c_j$, which counts the ordered proper factorisations of a positive...

Figure S2: The LD Structure of the 115 SNPs used in Scenario B – Simulation of 115 SNPs from
Real Data, with a Proportion of Phenotypes Permuted to Maintain Effect Sizes.

Figure S1: The LD Structure of the 100 SNPs used in Scenario A – Simulation of 10 SNPs in LD
with OR=1.1 and 90 independent, unassociated SNPs.

Table SI: Comparison of the Number and Proportion of Independent Genes Below a P‐value Threshold for POLARIS, MAGMA‐PCA in GERAD data and MAGMA‐SUMMARY in IGAP data.

An equivalent representation of constant sum matrices in terms of block-structured matrices is given in this paper. This provides an easy way of constructing all constant sum matrices, including those with further symmetry properties. The block representation gives a convenient description of the dihedral equivalence of such matrices. It is also sh...

Polygenic risk scores (PRSs) are a method to summarize the additive trait variance captured by a set of SNPs, and can increase the power of set-based analyses by leveraging public genome-wide association study (GWAS) datasets. PRS aims to assess the genetic liability to some phenotype on the basis of polygenic risk for the same or different phenoty...

A sum-and-distance system is a collection of finite sets of integers such that the sums and differences formed by taking one element from each set generate a prescribed arithmetic progression. Such systems, with two component sets, arise naturally in the study of matrices with symmetry properties and consecutive integer entries. Sum systems are an...

Studying the stability of Singular Spectrum Analysis (SSA) and Multivariate Singular Spectrum Analysis (MSSA) forecasts under random perturbations of the input time series, we make the empirical observation that the reconstruction kernel of SSA as a convolution filter and the forecast recurrence vector are remarkably stable both under generated Gau...

It is known that semi-magic square matrices form a 2-graded algebra or superalgebra with the even and odd subspaces under centre-point reflection symmetry as the two components. We show that other symmetries which have been studied for square matrices give rise to similar superalgebra structures, pointing to novel symmetry types in their complement...

Using the decomposition of semimagic squares into the associated and balanced symmetry types as a motivation, we introduce an equivalent representation in terms of block-structured matrices. This block representation provides a way of constructing such matrices with further symmetries and of studying their algebraic behaviour, significantly advanci...

The radial Dirac operator with a potential tending to infinity at infinity and satisfying a mild regularity condition is known to have a purely absolutely continuous spectrum covering the whole real line. Although having two singular end-points in the limit-point case, the operator has a simple spectrum and a generalised Fourier expansion in terms...

The spectra of massless Dirac operators are of essential interest
e.g. for the electronic properties of graphene, but fundamental questions such
as the existence of spectral gaps remain open.
We show that the eigenvalues of massless Dirac operators with suitable
real-valued potentials lie inside small sets easily characterised in terms of
propertie...

The present work concerns the algebra of semi-magic square matrices. These can be decomposed into matrices of specific rotational symmetry types, where the square of a matrix of pure type always has a particular type. We examine the converse problem of categorising the square roots of such matrices, observing that roots of either type occur, but on...

We show that the absolutely continuous part of the spectral function of the one-dimensional Dirac operator on a half-line with a constant mass term and a real, square-integrable potential is strictly increasing throughout the essential spectrum (-infinity, -1] boolean OR [1, infinity). The proof is based on estimates for the transmission coefficien...

We evaluate the effect of genotyping errors on the type-I error of a general association test based on genotypes, showing that, in the presence of errors in the case and control samples, the test statistic asymptotically follows a scaled non-central distribution. We give explicit formulae for the scaling factor and non-centrality parameter for the...

The periodic Sturm-Liouville or Dirac operator on the whole real line has a purely absolutely continuous spectrum of band-gap structure; the regular end-point of the operator restricted to a half-line only introduces a single eigenvalue, if any, into each spectral gap. In applications, however, one does not always have exact periodicity of the coef...

The main purpose of this chapter is to examine the nature of the instability intervals–first their asymptotic lengths as they recede to infinity, and second the more specialised situation when all but a finite number of them are absent. To deal with the lengths we require asymptotic estimates for the eigenvalues λ
n
and μ
n
as n→∞, and the theory o...

The Floquet theory of Chapter 1 gives an overview of the global growth properties of the solutions of periodic systems. For the purposes of spectral analysis of formally symmetric systems, the oscillations or rotations of the real-valued solutions are also of similar importance. The tool for studying oscillations in Sturm-Liouville and Dirac system...

The spectra of massless Dirac operators are of essential interest e.g. for
the electronic properties of graphene, but fundamental questions such as the
existence of spectral gaps remain open. We show that the eigenvalues of
massless Dirac operators with suitable real-valued potentials lie inside small
sets easily characterised in terms of propertie...

The main result of the paper is the following characterization of the generalized arcsine density p
γ
(t) = t
γ−1(1 − t)γ−1/B(γ, γ) with \({t \in (0, 1)}\) and \({\gamma \in(0,\frac12) \cup (\frac12,1)}\) : a r.v. ξ supported on [0, 1] has the generalized arcsine density p
γ
(t) if and only if \({ {\mathbb E} |\xi- x|^{1-2 \gamma}}\) has the same v...

We propose to identify process zones in heterogeneous materials by tailored
statistical tools. The process zone is redefined as the part of the structure
where the random process cannot be correctly approximated in a low-dimensional
deterministic space. Such a low-dimensional space is obtained by a spectral
analysis performed on pre-computed soluti...

Periodic differential operators have a rich mathematical theory as well as important physical applications. They have been the subject of intensive development for over a century and remain a fertile research area. This book lays out the theoretical foundations and then moves on to give a coherent account of more recent results, relating in particu...

The solutions of periodic linear systems of differential equations are not always periodic, but their global qualitative behaviour can be analysed by studying the first period interval only. After a brief summary of the necessary concepts and results from the theory of ordinary differential equations, mainly to introduce the terminology, we establi...

In section 1.8 we studied the eigenvalues of the periodic, semi-periodic and twisted boundary-value problem on a period interval, and similarly those of a boundaryvalue problem with separated boundary conditions in section 2.3. We begin the present chapter with the observation that the eigenvalues and eigenfunctions of these boundary-value problems...

Additional information about risk genes or risk pathways for diseases can be extracted from genome-wide association studies through analyses of groups of markers. The most commonly employed approaches involve combining individual marker data by adding the test statistics, or summing the logarithms of their P-values, and then using permutation testi...

The validity of a generalised HELP inequality for a Schrödinger operator with periodic potential on a rooted homogeneous tree
is related to the quasi-stability or quasi-instability of the associated differential equation. A numerical approach to the
determination of the optimal constant in the HELP inequality is presented. Moreover, we give an exam...

We consider the Schr\"odinger operator $H$ on the half-line with a periodic
potential $p$ plus a compactly supported potential $q$. For generic $p$, its
essential spectrum has an infinite sequence of open gaps. We determine the
asymptotics of the resonance counting function and show that, for sufficiently
high energy, each non-degenerate gap contai...

It is shown that the essential spectrum of massless Dirac operators with a rotationally symmetric potential in two and three
spatial dimensions covers the whole real line. Limit values of the potential at infinity can be eigenvalues of the operator,
but outside the limit range of the potential the spectrum is purely absolutely continuous under a mi...

The following characterization of the arcsine density is established. Let [xi] be a r.v. supported on (-1,1); then [xi] has the arcsine density , -1<t<1, if and only if has the same value for almost all x[set membership, variant][-1,1].

We study the spectral properties of a class of Sturm-Liouville type operators
on the real line where the derivatives are replaced by a q-difference operator which has
been introduced in the context of orthogonal polynomials. Using the relation of this
operator to a direct integral of doubly-infinite Jacobi matrices, we construct examples
for isolat...

Although the classical Hardy inequality is valid only in the three-and higher dimensional case, Laptev and Weidl established a two-dimensional Hardy-type inequality for the magnetic gradient with an Aharonov-Bohm magnetic poten-tial. Here we consider a discrete analogue, replacing the punctured plane with a radially exponential lattice. In addition...

The interpretation of the results of large association studies encompassing much or all of the human genome faces the fundamental statistical problem that a correspondingly large number of single nucleotide polymorphisms markers will be spuriously flagged as significant. A common method of dealing with these false positives is to raise the signific...

We establish an analogue for the p -Laplacian on the half-line of an integro-differential inequality of Hardy and Littlewood and estimate the optimal constant.

We study the asymptotics of the spectral density of one-dimensional Dirac systems on the half-line with an angular momentum term and a potential tending to infinity at infinity. The problem has two singular end-points; however, as the spectrum is simple, the derivative of the spectral matrix has only one non-zero eigenvalue which we take to be the...

We study discrete, generally non-self-adjoint Hamiltonian systems, defining Weyl–Sims sets, which replace the classical Weyl circles, and a matrix-valued M-function on suitable cone-shaped domains in the complex plane. Furthermore, we characterise realisations of the corresponding differential operator and its adjoint, and construct their resolvent...

With the availability of fast genotyping methods and genomic databases, the search for statistical association of single nucleotide polymorphisms with a complex trait has become an important methodology in medical genetics. However, even fairly rare errors occurring during the genotyping process can lead to spurious association results and decrease...

In view of the linkage disequilibrium structure of the genome, the selection of maximally informative SNP markers is a fundamental issue in the design of association studies. Currently used selection methods rely on pairwise marker correlation or informativity measures for subsets of markers. Nevertheless, the selected markers do not provide a comp...

Traditionally in genetic case-control studies controls have been screened to exclude subjects with a personal history of illness. This control group has the advantage of optimal power to detect loci involved in illness, but requires more work and may incur substantial cost in recruitment. An alternative approach to screening is to use unscreened co...

It is known that one-dimensional Dirac systems with potentials q which tend to −∞ (or ∞) at infinity, such that 1/q is of bounded variation, have a purely absolutely continuous spectrum covering the whole real line. We show that, for the system on a half-line, there are no local maxima of the spectral density (points of spectral concentration) abov...

It is known that one-dimensional Dirac systems with potentials q which tend to −∞ (or ∞) at infinity, such that 1/q is of bounded variation, have a purely absolutely continuous spectrum covering the whole real line. We show that, for the system on a half-line, there are no local maxima of the spectral density (points of spectral concentration) abov...

We show that a first-order ordinary differential equation can be integrated by quadratures in the sense of Maximovič only if it arises from the linear equation by a diffeomorphic transformation of the dependent variable. In the appendix this result is applied to show that the linear second-order equation can be integrated by quadratures in a restri...

It was recently shown that the point spectrum of the separated Coulomb-Dirac operator H_0(k) is the limit of the point spectrum of the Dirac operator with anomalous magnetic moment H_a(k) as the anomaly parameter tends to 0; this spectral stability holds for all Coulomb coupling constants c for which H_0(k) has a distinguished self-adjoint extensio...

This paper reports on a new numerical procedure to count eigenvalues in spectral gaps for a class of perturbed periodic Sturm-Liouville operators. It is motivated by the desire to analyse the distribution of eigenvalues in the dense point spectrum of d-dimensional radially periodic Schrodinger operators. Our numerical results indicate that the well...

We show that the point spectrum of the standard Coulomb-Dirac operator H_0 is the limit of the point spectrum of the Dirac operator with anomalous magnetic moment H_a as the anomaly parameter tends to 0. For negative angular momentum quantum number kappa, this holds for all Coulomb coupling constants c for which H_0 has a distinguished self-adjoint...

Singular spectrum analysis (SSA) is a method of time-series analysis based on the singular value decomposition of an associated Hankel matrix. We present an approach to SSA using an effective and numerically stable high-degree polynomial approximation of a spectral projector, which also provides a means of time-series forecasting. Several numerical...

A perturbation decaying to 0 at infinity and not too irregular at 0 introduces at most a discrete set of eigenvalues into the spectral gaps of a one-dimensional Dirac operator on the half-line. We show that the number of these eigenvalues in a compact subset of a gap in the essential spectrum is given by a quasi-semiclassical asymptotic formula in...

A quick and transparent proof for Bargmann's inequality and an analogue for two–dimensional magnetic Schrödinger operators recently found by Balinsky, Evans and Lewis, is given. Furthermore, it is shown that the inequality is sharp.

This paper presents a method for the numerical investigation of the distribution of the eigenvalues introduced into a spectral gap of a periodic Dirac system by a perturbation of the type of the angular momentum term. A number of examples illustrate the effectiveness of the method and show the remarkable accuracy of the strong coupling asymptotic f...

A perturbation decaying to 0 at infinity and not too irregular at 0 introduces at most a discrete set of eigenvalues into the spectral gaps of a one-dimensional Dirac operator on the half-line. We show that the number of these eigenvalues in a compact subset of a gap in the essential spectrum is given by a quasi-semiclassical asymptotic formula in...

This paper presents a sufficient condition for a one-dimensional Dirac operator with a potential tending to infinity at infinity to have no eigenvalues. It also provides a quick proof (and suggests variations) of a related criterion given by Evans and Harris.

The approach to oscillation theory developed by Gesztesy, Simon, and Teschl produces a sharp version for the oscillation theorem for singular Sturm–Liouville operators. In the present note, an example from the stability theory of complete minimal surfaces is given in which this refinement plays a decisive role.

This paper reports on a new numerical procedure to count eigenvalues in spectral gaps for a class of perturbed periodic Sturm-Liouville operators. It is motivated by the desire to analyse the distribution of eigenvalues in the dense point spectrum of d-dimensional radially periodic Schrodinger operators. Our numerical results indicate that the well...

Under minimal hypotheses, sufficient criteria for a perturbed Dirac system to be relatively oscillatory or non-oscillatory at ∞ with respect to a reference equation with periodic coefficients are proved my means of an asymptotic analysis of generalized Prüfer angles. As illustrated by an example, they help decide whether the number of eigenvalues i...

Perturbations of asymptotic decay c/r 2 arise in the partial-wave analysis of rotationally symmetric partial differential operators. The author shows that for each end-point λ 0 of the spectral bands of a perturbed periodic Sturm-Liouville operator, there is a critical coupling constant c crit such that eigenvalues in the spectral gap accumulate at...

A potential for the one-dimensional Dirac operator is constructed such that its essential spectrum does not cover the whole real line, whereas the potential q(x) tends to ∞ as |x|→∞. Furthermore, a criterion by Hartman and Wintner for points of the essential spectrum of Sturm-Liouville operators is generalised to a purely operator-theoretical setti...

Generalizing the classical result of Kneser, the author shows that the Sturm-Liouville equation with periodic coefficients and an added perturbation term -c 2 /r 2 is oscillatory or nonoscillatory (for r→∞) at the infimum of the essential spectrum, depending on whether c 2 surpasses or stays below a critical threshold. An explicit characterization...

We give a definition of integration by quadratures of first-order ordinary differential equations, and recover a little known result by Maximovic which states that a first-order ordinary differential equation can be integrated by quadratures only if it arises from the linear equation by a diffeomorphic transformation of the dependent variable. In t...

We study the spectrum of spherically symmetric Dirac operators in three-dimensional space with potentials tending to infinity at infinity under weak regularity assumptions. We prove that purely absolutely continuous spectrum covers the whole real line if the potential dominates the mass, or scalar potential, term. In the situation where the potenti...

It is shown that the spectrum of a one-dimensional Dirac operator with a potential q tending to infinity at infinity, and such that the positive variation of 1/q is bounded, covers the whole real line and is purely absolutely continuous. An example is given to show that in general, pure absolute continuity is lost if the condition on the positive v...

For the one-dimensional Dirac operator, examples of electrostatic potentials with decay behaviour arbitrarily close to Coulomb decay are constructed for which the operator has a prescribed set of eigenvalues dense in the whole or part of its essential spectrum. A simple proof that the essential spectrum of one-dimensional Dirac operators with elect...

Using Floquet-Lyapunov theory, it is shown that for Baire-almost every periodic potential the Dirac system has all its instability intervals open. Consequently, one-dimensional Dirac operators with periodic potentials generically possess infinitely many spectral gaps. These results also hold true if only even potentials are admitted.