arXiv subjects
Publications and source records attributed to Paul Baird.
We study systems of {\sigma}-algebras ordered by refinement and introduce the notion of an endogenous probability measure, invariant under admissible refinement transformations. We prove existence and structural properties of such measures on finite systems and show how refinement operators induce a natural dynamical structure on the lattice of {\sigma}-algebras.
In the emerging domain of quantum algorithms, the Grover's quantum search is certainly one of the most significant. It is relatively simple, performs a useful task and more importantly, does it in an optimal way. However, due to the success of quantum walks in the field, it is logical to study quantum search variants over several kind of walks. In this paper, we propose an in-depth study of the quantum search over a hypercube layout. First, through the analysis of elementary walk operators restricted to suitable eigenspaces, we show that the acting component of the search algorithm takes place in a small subspace of the Hilbert workspace that grows linearly with the problem size. Subsequently, we exploit this property to predict the exact evolution of the probability of success of the quantum search in polynomial time.
Biconformal deformations take place in the presence of a conformal foliation, deforming by different factors tangent to and orthogonal to the foliation. Four-manifolds endowed with a conformal foliation by surfaces present a natural context to put into effect this process. We develop the tools to calculate the transformation of the Ricci curvature under such deformations and apply our method to construct Einstein $4$-manifolds. One particular family of examples have ends that collapse asymptotically to ${\mathbb R}^2$.
We explore relations between cyclic sequences determined by a quadratic difference relation, cyclotomic polynomials, Eulerian digraphs and walks in the plane. These walks correspond to closed paths for which at each step one must turn either left or right through a fixed angle. In the case when this angle is $2 π/n$, then non-symmetric phenomena occurs for $n\geq 12$. Examples arise from algebraic numbers of modulus one which are not $n$'th roots of unity.
The widespread use of generalized linear models in case-control genetic studies has helped identify many disease-associated risk factors typically defined as DNA variants, or single nucleotide polymorphisms (SNPs). Up to now, most literature has focused on selecting a unique best subset of SNPs based on some statistical perspectives. In the presence of pronounced noise, however, multiple biological paths are often found to be equally supported by a given dataset when dealing with complex genetic diseases. We address the ambiguity related to SNP selection by constructing a list of models called variable selection confidence set (VSCS), which contains the collection of all well-supported SNP combinations at a user-specified confidence level. The VSCS extends the familiar notion of confidence intervals in the variable selection setting and provides the practitioner with new tools aiding the variable selection activity beyond trusting a single model. Based on the VSCS, we consider natural graphical and numerical statistics measuring the inclusion importance of a SNP based on its frequency in the most parsimonious VSCS models. This work is motivated by available case-control genetic data on age-related macular degeneration, a widespread complex disease and leading cause of vision loss.
We prove that the problem of constructing biharmonic conformal maps on a $4$-dimensional Einstein manifold reduces to a Yamabe-type equation. This allows us to construct an infinite family of examples on the Euclidean 4-sphere. In addition, we characterize all solutions on Euclidean 4-space and show that there exists at least one non-constant proper biharmonic conformal map from any closed Einstein 4-manifold of negative Ricci curvature.
We exploit an ansatz in order to construct power series expansions for pairs of conjugate functions defined on domains of Euclidean $3$--space. Convergence properties of the resulting series are investigated. Entire solutions which are not harmonic are found as well as a $2$-parameter family of examples which contains the Hopf map.
Harmonic functions of two variables are exactly those that admit a conjugate, namely a function whose gradient has the same length and is everywhere orthogonal to the gradient of the original function. We show that there are also partial differential equations controlling the functions of three variables that admit a conjugate.
We study the soliton flow on the domain of a twistorial harmonic morphism between Riemannian manifolds of dimensions four and three. Assuming real-analyticity, we prove that, for the Gibbons-Hawking construction, any soliton flow is uniquely determined by its restriction to any local section of the corresponding harmonic morphism. For the Beltrami fields construction, we identify a contour integral whose vanishing characterises the trivial soliton flows.
We study a class of complex polynomial equations on a finite graph with a view to understanding how holistic phenomena emerge from combinatorial structure. Particular solutions arise from orthogonal projections of regular polytopes, invariant frameworks and cyclic sequences. A set of discrete parameters for which there exist non-trivial solutions leads to the construction of a polynomial invariant and the notion of a geometric spectrum. Geometry then emerges, notably dimension, distance and curvature, from purely combinatorial properties of the graph.
We use the CR geometry of the standard hyperquadric in complex projective three-space to give a detailed twistor description of conformal foliations in Euclidean three-space.
We use functions of a bicomplex variable to unify the existing constructions of harmonic morphisms from a 3-dimensional Euclidean or pseudo-Euclidean space to a Riemannian or Lorentzian surface. This is done by using the notion of complex-harmonic morphism between complex-Riemannian manifolds and showing how these are given by bicomplex-holomorphic functions when the codomain is one-bicomplex dimensional. By taking real slices, we recover well-known compactifications for the three possible real cases. On the way, we discuss some interesting conformal compactifications of complex-Riemannian manifolds by interpreting them as bicomplex manifolds.
We provide algorithms involving edge slides, for a connected simple graph to evolve in a finite number of steps to another connected simple graph in a prescribed configuration, and for the regularization of such a graph by the minimization of an appropriate energy functional.
We show how the description of a shear-free ray congruence in Minkowski space as an evolving family of semi-conformal mappings can naturally be formulated on a finite graph. For this, we introduce the notion of holomorphic function on a graph. On a regular coloured graph of degree three, we recover the space-time picture. In the spirit of twistor theory, where a light ray is the more fundamental object from which space-time points should be derived, the line graph, whose points are the edges of the original graph, should be considered as the basic object. The Penrose twistor correspondence is discussed in this context.
A shear-free ray congruence on Minkowski space is a 3-parameter family of null geodesics along which Lie transport of a complementary 2-dimensional spacelike subspace (called the screen space) is conformal. Such congruences are defined by complex analytic surfaces in the associated twistor space $\CP^3$ and are the basis of the construction of massless fields. On a more general space-time, it is unclear how to couple the massless field with the gravitational field. In this article we do this by considering the following Cauchy-type problem: given a Riemannian 3-manifold $(M^3, g_0)$ endowed with a unit vector field $U_0$ that is tangent to a conformal foliation, we require that the pair extend to a space-time $({\mathcal M}, {\mathcal G})$ endowed with a spacelike unit vector field $U_t$ in such a way that $U_t$ simultaneously generates null geodesics and is tangent to a conformal foliation on spacelike slices $t=$ const.
We prove that any (real or complex) analytic horizontally conformal submersion from a three-dimensional conformal manifold M to a two-dimensional conformal manifold N can be, locally, `extended' to a unique harmonic morphism from the heaven space of M to N.
We study $3$-dimensional Ricci solitons which project via a semi-conformal mapping to a surface. We reformulate the equations in terms of parameters of the map; this enables us to give an ansatz for constructing solitons in terms of data on the surface. A complete description of the soliton structures on all the $3$-dimensional geometries is given, in particular, non-gradient solitons are found on Nil and Sol.