Base Point Freeness, Uniqueness of Decompositions and Double Points for Veronese and Segre Varieties

We prove a base point freeness result for linear systems of forms vanishing at general double points of the projective plane. For tensors we study the uniqueness problem for the representation of a tensor as a sum of terms corresponding to points and tangent vectors of the Segre variety associated with the format of the tensor. We give complete results for unions of one point and one tangent vector.

Uncertainty analysis of 3D potential-field deterministic inversion using mixed Lp norms

The non-uniqueness problem in geophysical inversion, especially potential-field inversion, is widely recognized. It is argued that uncertainty analysis of a recovered model should be as important as finding an optimal model. However, quantifying uncertainty still remains challenging, especially for 3D inversions in both deterministic and Bayesian frameworks. Our objective is to develop an efficient method to empirically quantify the uncertainty of the physical property models recovered from 3D potential-field inversion. We worked in a deterministic framework where an objective function consisting of a data misfit term and a regularization term is minimized. We performed inversions using a mixed Lp-norm formulation where various combinations of L p (0 <= p <= 2) norms can be implemented on different components of the regularization term. Specifically, we randomly sampled the p-norm values in multiple times, and generated a large and diverse sequence of physical property models that all reproduce the observed geophysical data equally well. This suite of models offers practical insights into the uncertainty of the recovered model features. We quantified the uncertainty through calculation of standard deviations and interquartile range, as well as visualizations in box plots and histograms. The numerical results for a realistic synthetic density model created based on a ring-shaped igneous intrusive body quantitatively illustrate uncertainty reduction due to different amounts of prior information imposed on inversions. We also applied the method to a field data set over the Decorah area in the northeastern Iowa. We adopted an acceptance-rejection strategy to generate 31 equivalent models based on which the uncertainties of the inverted models as well as the volume and mass estimates are quantified.

On the Non-uniqueness Problem in Integrated Information Theory

Integrated Information Theory is currently the leading mathematical theory of conscious- ness. The core of the theory relies on the calculation of a scalar mathematical measure of consciousness, Φ, which is deduced from the phenomenological axioms of the theory. Here, we show that despite its widespread use, Φ is not a well-defined mathematical concept in the sense that the value it specifies is neither unique nor specific. This problem, occasionally referred to as “undetermined qualia”, is the result of degeneracies in the optimization routine used to calculate Φ, which leads to ambiguities in determining the consciousness of systems under study. As demonstration, we first apply the mathematical definition of Φ to a simple AND+OR logic gate system and show 83 non-unique Φ values result, spanning a substantial portion of the range of possibilities. We then introduce a Python package called PyPhi-Spectrum which, unlike currently available packages, delivers the entire spectrum of possible Φ values for a given system. We apply this to a variety of examples of recently published calculations of Φ and show how virtually all Φ values from the sampled literature are chosen arbitrarily from a set of non-unique possibilities, the full range of which often includes both conscious and unconscious predictions. Lastly, we review proposed solutions to this degeneracy problem, and find none to provide a satisfactory solution, either because they fail to specify a unique Φ value or yield Φ = 0 for systems that are clearly integrated. We conclude with a discussion of requirements moving forward for scientifically valid theories of consciousness that avoid these degeneracy issues.

A numerical approach to the non-uniqueness problem of cosmic ray two-fluid equations at shocks

ABSTRACT Cosmic rays (CRs) are frequently modelled as an additional fluid in hydrodynamic (HD) and magnetohydrodynamic (MHD) simulations of astrophysical flows. The standard CR two-fluid model is described in terms of three conservation laws (expressing conservation of mass, momentum, and total energy) and one additional equation (for the CR pressure) that cannot be cast in a satisfactory conservative form. The presence of non-conservative terms with spatial derivatives in the model equations prevents a unique weak solution behind a shock. We investigate a number of methods for the numerical solution of the two-fluid equations and find that, in the presence of shock waves, the results generally depend on the numerical details (spatial reconstruction, time stepping, the CFL number, and the adopted discretization). All methods converge to a unique result if the energy partition between the thermal and non-thermal fluids at the shock is prescribed using a subgrid prescription. This highlights the non-uniqueness problem of the two-fluid equations at shocks. From our numerical investigations, we report a robust method for which the solutions are insensitive to the numerical details even in absence of a subgrid prescription, although we recommend a subgrid closure at shocks using results from kinetic theory. The subgrid closure is crucial for a reliable post-shock solution and also its impact on large-scale flows because the shock microphysics that determines CR acceleration is not accurately captured in a fluid approximation. Critical test problems, limitations of fluid modelling, and future directions are discussed.

On Uniqueness of Dirichlet Series

We give a characterization of the ratio of two Dirichelt series convergent in a right half-plane under an analytic condition, which is applicable to a uniqueness problem for Dirichlet series admitting analytic continuation in the complex plane as meromorphic functions of finite order; uniqueness theorems are given in terms of a-points of the functions.