Axiomatic Holonomy Maps and Generalized Yang-Mills Moduli Space

This article is a follow-up of ``Holonomy and Path Structures in General Relativity and Yang-Mills Theory" by Barrett, J. W. (Int.J.Theor.Phys., vol.30, No.9, 1991). Its main goal is to provide an alternative proof of this part of the reconstruction theorem which concerns the existence of a connection. A construction of connection 1-form is presented. The formula expressing the local coefficients of connection in terms of the holonomy map is obtained as an immediate consequence of that construction. Thus the derived formula coincides with that used in "On Loop Space Formulation of Gauge Theories" by Chan, H.-M., Scharbach, P. and Tsou S.T. (Ann.Phys., vol.167, 454-472, 1986). The reconstruction and representation theorems form a generalization of the fact that the pointed configuration space of the classical Yang-Mills theory is equivalent to the set of all holonomy maps. The point of this generalization is that there is a one-to-one correspondence not only between the holonomy maps and the orbits in the space of connections, but also between all maps from the loop space on $M$ to group $G$ fulfilling some axioms and all possible equivalence classes of $P(M,G)$ bundles with connection, where the equivalence relation is defined by bundle isomorphism in a natural way.Comment: amslatex, 7 pages, no figure

New Angle on the Strong CP and Chiral Symmetry Problems from a Rotating Mass Matrix

It is shown that when the mass matrix changes in orientation (rotates) in generation space for changing energy scale, then the masses of the lower generations are not given just by its eigenvalues. In particular, these masses need not be zero even when the eigenvalues are zero. In that case, the strong CP problem can be avoided by removing the unwanted $\theta$ term by a chiral transformation in no contradiction with the nonvanishing quark masses experimentally observed. Similarly, a rotating mass matrix may shed new light on the problem of chiral symmetry breaking. That the fermion mass matrix may so rotate with scale has been suggested before as a possible explanation for up-down fermion mixing and fermion mass hierarchy, giving results in good agreement with experiment.Comment: 14 page

ENO-wavelet transforms for piecewise smooth functions

We have designed an adaptive essentially nonoscillatory (ENO)-wavelet transform for approximating discontinuous functions without oscillations near the discontinuities. Our approach is to apply the main idea from ENO schemes for numerical shock capturing to standard wavelet transforms. The crucial point is that the wavelet coefficients are computed without differencing function values across jumps. However, we accomplish this in a different way than in the standard ENO schemes. Whereas in the standard ENO schemes the stencils are adaptively chosen, in the ENO-wavelet transforms we adaptively change the function and use the same uniform stencils. The ENO-wavelet transform retains the essential properties and advantages of standard wavelet transforms such as concentrating the energy to the low frequencies, obtaining maximum accuracy, maintained up to the discontinuities, and having a multiresolution framework and fast algorithms, all without any edge artifacts. We have obtained a rigorous approximation error bound which shows that the error in the ENO-wavelet approximation depends only on the size of the derivative of the function away from the discontinuities. We will show some numerical examples to illustrate this error estimate

Management of invasive Allee species

In this study, we use a discrete, two-patch population model of an Allee species to examine different methods in managing invasions. We first analytically examine the model to show the presence of the strong Allee effect, and then we numerically explore the model to test the effectiveness of different management strategies. As expected invasion is facilitated by lower Allee thresholds, greater carrying capacities and greater proportions of dispersers. These effects are interacting, however, and moderated by population growth rate. Using the gypsy moth as an example species, we demonstrate that the effectiveness of different invasion management strategies is context-dependent, combining complementary methods may be preferable, and the preferred strategy may differ geographically. Specifically, we find methods for restricting movement to be more effective in areas of contiguous habitat and high Allee thresholds, where methods involving mating disruptions and raising Allee thresholds are more effective in areas of high habitat fragmentation

Higher-order Representation and Reasoning for Automated Ontology Evolution

Abstract: The GALILEO system aims at realising automated ontology evolution. This is necessary to enable intelligent agents to manipulate their own knowledge autonomously and thus reason and communicate effectively in open, dynamic digital environments characterised by the heterogeneity of data and of representation languages. Our approach is based on patterns of diagnosis of faults detected across multiple ontologies. Such patterns allow to identify the type of repair required when conflicting ontologies yield erroneous inferences. We assume that each ontology is locally consistent, i.e. inconsistency arises only across ontologies when they are merged together. Local consistency avoids the derivation of uninteresting theorems, so the formula for diagnosis can essentially be seen as an open theorem over the ontologies. The system’s application domain is physics; we have adopted a modular formalisation of physics, structured by means of locales in Isabelle, to perform modular higher-order reasoning, and visualised by means of development graphs.

A Consistent Histogram Estimator for Exchangeable Graph Models

Exchangeable graph models (ExGM) subsume a number of popular network models. The mathematical object that characterizes an ExGM is termed a graphon. Finding scalable estimators of graphons, provably consistent, remains an open issue. In this paper, we propose a histogram estimator of a graphon that is provably consistent and numerically efficient. The proposed estimator is based on a sorting-and-smoothing (SAS) algorithm, which first sorts the empirical degree of a graph, then smooths the sorted graph using total variation minimization. The consistency of the SAS algorithm is proved by leveraging sparsity concepts from compressed sensing.Comment: 28 pages, 5 figure