Isotropic Singularity Research Articles

Mechanics of reproductive differentiation in the land plants: a paradigm shift?

This article addresses the physical mechanics of gametogenesis in vascular plants. The earliest events that initiate reproductive differentiation in the land plants are not well understood. How are the few cells that initiate reproductive differentiation specified and how is that information translated into action at the cellular level? In this article I propose a physical mechanism that resolves the problem of spatial targeting without invoking dependence on diffusible morphogens or other external signals. I suggest that the initiation of archesporial differentiation can instead be attributed to the confluence of organ geometry, surficial topography, and the physical mechanics of sporangial growth, resulting in the spontaneous emergence of an isotropic singularity that locates and precipitates archesporial differentiation. In discussing the logic of single-cell target selection and the limits of stochastic molecular signaling I propose that the sporangium would be better understood as a pressurized stress-mechanical lens that focuses turgor-generated growth forces on a central location, generating a physical singularity that locates and specifies the cell or cells that become the archesporium and initiates their transition from somatic proliferation to reproductive differentiation.

Tilt in quadratic gravity

In this work, tilted source solutions in both Einstein–Hilbert General Relativity (GR) and Quadratic Gravity (QG) for the anisotropic Bianchi V model are addressed. Since the excellent CMBR match of Starobinsky’s inflation with Planck’s team measurements data, QG has acquired a prominent status in the effective sense, for sufficiently strong gravity fields. The main interest is in the numeric time evolution to the past towards the singularity and the behavior of the kinematic variables, vorticity, acceleration, and the expansion of this source substance. In QG we found that for universes with higher and smaller matter densities fall into the Kasner or isotropic singularity attractors to the past, respectively. We also found that the Kasner singularity attractor to the past has always zero vorticity, for both GR and QG theories. While for QG the isotropic singularity attractor may have divergent vorticity. For the set of assumptions and conditions supposed in this work, the isotropic singularity attractor, favors QG as compared to GR. Only in QG we were able to find a geometric singularity with divergences in all of the kinematic variables of the substance, decreasing to finite values to the future, upon time reversing. That is, we obtained an initial kinematic singularity substance, that approaches a perfect fluid source.

Stress and Geometry for Isotropic Singularities.

We develop the mathematics needed to treat the interaction of geometry and stress at any isotropic spacetime singularity. This enables us to handle the Einstein equations at the initial singularity and characterize allowed general relativistic stress-energy tensors. Their leading behaviors are dictated by an initial hypersurface conformal embedding. We also show that an isotropic big bang determines a canonical nonsingular metric on and about the initial hypersurface as well as a cosmological time. This assigns a volume and energy to the initial point singularity.

A New Anisotropic Singularity Algorithm to Characterize Geo-Chemical Anomalies in the Duolong Mineral District, Tibet, China

With the increasing exploitation of mineral resources by humans, exploring non-traditional areas for hidden resources such as deep earth and sediment-covered regions has become a significant challenge in the field of mineral exploration. Geochemical data, as a crucial information carrier of geological bodies, serves as one of the direct and effective sources for quantitative analysis of regional geological evolution and mineralization prediction studies. It plays an indispensable role in geographic information system (GIS)-based mineral exploration. Due to the neglect of spatial distribution characteristics and the variability of statistical features with spatial metrics in traditional statistical methods, this paper employs fractal/multifractal and the local singularity analysis to identify geochemical anomalies from background and characterize geochemical distributions associated with porphyry Cu-Au mineralization in the Duolong mineral district, Tibet, China. A novel algorithm for estimating the singularity index, which takes anisotropy into consideration, is proposed and practically applied to the Duolong district. By comparing with the isotropic singularity index, this new method objectively identifies anisotropic geochemical signatures and investigates non-linear behaviors of ore-forming elements, making it more practical and effective in geo-anomaly extraction. Furthermore, the current method is capable of indicating variations in geochemical distributions at different scales through directional arrows marking analytical windows. The summed-up direction of these multi-scale vectors effectively demonstrates migration trends of ore materials at each location within the study area. The new method can pinpoint the location of ore-forming element accumulation and migration directions, unlocking valuable insights from complex datasets. This promises to revolutionize our understanding of how minerals are formed and distributed within the Earth’s crust.

Cosmological models in scale-independent energy-momentum squared gravity

Scale-independent EMSG is a particular model of energy–momentum squared gravity (EMSG) in which the new terms in the Einstein field equations arising from the EMSG theory enter with the same power as the usual terms from Einstein–Hilbert part of the action. However, the model violates the local energy–momentum conservation and matter-current conservation in general and hence, permits a process of matter creation/annihilation in an expanding universe. Consequently, the scale factor dependencies of the energy densities are modified by the dimensionless model parameter α. We revisit some nostalgias such as static universes and de Sitter/steady state universes. We reproduce the original ones, moreover, present some novelties, e.g., a spatially flat static universe, de Sitter expansion by negative vacuum energy, steady state universes in the presence of arbitrary fluids with constant equation of state (EoS) parameter other than dust, etc. We also investigate the possible dynamics of dust dominated and radiation dominated universes. Depending on the value of α, dust/radiation dominated universe exhibits power-law accelerated/decelerated expansion, corresponds to a steady state model or may end in a big rip. In the framework of anisotropic cosmology, we reproduce Barrow’s quiescent universe in the presence of stiff fluid and extend it to fluids with arbitrary constant EoS parameter. We also relax the condition for isotropic initial singularity (big bang) owing to that EMSG effectively allows ultra-stiff EoS parameters.

Cosmological milestones, conformal frameworks and quiescent cosmology.

To understand the nature of the birth of our Universe and its eventual demise is a driving force in theoretical physics and astronomy and, indeed, for humanity. A zoo of definitions has appeared in the literature to catalogue different types of cosmological milestones such as 'Big Bangs', 'Big Crunches', 'Big Rips', 'Sudden Singularities', 'Bounces' and 'Turnarounds'. Quiescent cosmology is the notion that the Universe commenced in a Big Bang that was highly regular and smooth, and evolved away from this initial isotropy and homogeneity due to gravitational attraction. The quiescent cosmology concept meshes well with Penrose's ideas regarding gravitational entropy and the clumping of matter, and the associated Weyl Curvature Hypothesis. Conformal frameworks, such as the Isotropic Past Singularity (IPS), have been devised to encapsulate initial and final states for the Universe which are in accordance with these programmes. These geometric definitions are independent of models, coordinates and the equation of state of the source of the gravitational field. Much of the research on cosmological milestones has been focussed on the FRW solutions, many of which possess initial singularities which are isotropic Big Bangs. We analyse here the relationship between cosmological milestones and conformal frameworks for these solutions. We establish the general properties of FRW models which admit these conformal frameworks, including whether they satisfy various energy conditions, and are therefore physically reasonable. These results inform future development of the quiescent cosmology program. This article is part of the theme issue 'The future of mathematical cosmology, Volume 1'.

Microphysical manifestations of viscosity and consequences for anisotropies in the very early universe

It has been known that a non-perfect fluid that accounts for dissipative viscous effects can evade a highly anisotropic chaotic mixmaster approach to a singularity. Viscosity is often simply parameterised in this context, so it remains unclear whether isotropisation can really occur in physically motivated contexts. We present a few examples of microphysical manifestations of viscosity in fluids that interact either gravitationally or, for a scalar field for instance, through a self-coupling term in the potential. In each case, we derive the viscosity coefficient and comment on the applicability of the approximations involved when dealing with dissipative non-perfect fluids. Upon embedding the fluids in a cosmological context, we then show the extent to which these models allow for isotropisation of the universe in the approach to a singularity. We first do this in the context of expansion anisotropy only, i.e., in the case of a Bianchi type-I universe. We then include anisotropic 3-curvature modelled by the Bianchi type-IX metric. It is found that a self-interacting scalar field at finite temperature allows for efficient isotropisation, whether in a Bianchi type-I or type-IX spacetime, although the model is not tractable all the way to a singularity. Mixmaster chaotic behaviour, which is well known to arise in anisotropic models including anisotropic 3-curvature, is found to be suppressed in the latter case as well. We find that the only model permitting an isotropic singularity is that of a dense gas of black holes.

Mineral prospectivity modelling using singularity mapping and multifractal analysis of stream sediment geochemical data from the auriferous Hutti-Maski schist belt, S. India

The regional stream sediment geochemical survey has become an all the more important exploration technique for delineating potential mineral zones to target exploration. However, the effective appraisal of pathfinder elements has remained a fundamental challenge in exploration geochemistry. In this regard, the singularity mapping technique based on modern nonlinear theory and fractal-multifractal geometry is a new efficient method to distinguish thresholds of anomalies based on stream sediment geochemical data. The auriferous Hutti-Maski schist belt in Eastern Dharwar Craton (EDC), S. India has been taken up for delineation future targets using local singularity analysis (LSA) of stream sediment geochemical data. Ordinary Singularity Mapping (OSM) and Spatially Weighted Singularity Mapping (SWSM) are employed for mapping singularity of gold pathfinder elements (Au, As, Ag, Hg, Sb, Se, Cu, and Zn). In SWSM, a distance-based fuzzy weighting method is used to estimate the spatial weighting factor with respect to ore controlling shear zones. The local isotropic singularity index and the local anisotropic singularity index are estimated for each of the eight elements. The Student’s t-statistics based on Weight of Evidence (WOE) model is used to determine the threshold singularity index value to separate geochemical anomaly from background. Principal Component Analysis (PCA) is applied to singularity maps of multiple elements derived using OSM and SWSM techniques. The first principal component (PC-1) scores estimated by the OSM-PCA and SWSM-PCA lead to comprehensive geochemical anomaly maps. Results of two different singularity mapping techniques are compared using Geochemical Mineralization Probability Index (GMPI), Concentration-Area (C-A) fractal analysis and Prediction-Area (P-A) plots. It is observed that the SWSM method results in a smaller target with higher predictive ability compared to OSM method that delineates a larger target area.

Exponential convergence in H^1 of hp-FEM for Gevrey regularity with isotropic singularities

For functions uin H^1(Omega ) in a bounded polytope Omega subset {mathbb {R}}^dd=1,2,3 with plane sides for d=2,3 which are Gevrey regular in overline{Omega }backslash {mathscr {S}} with point singularities concentrated at a set {mathscr {S}}subset overline{Omega } consisting of a finite number of points in overline{Omega }, we prove exponential rates of convergence of hp-version continuous Galerkin finite element methods on affine families of regular, simplicial meshes in Omega . The simplicial meshes are geometrically refined towards {mathscr {S}} but are otherwise unstructured.

Anisotropic singularity: A novel way to characterize controlling effects of geological processes on mineralization

The anisotropy of geochemical signatures of ore elements caused by various geo-processes needs to be delicately analyzed. Within a fractal/multifractal context, irregular ore material accumulation caused by hydrothermal mineralization in a short spatial-temporal interval can be described by singularity theory, quantitatively and qualitatively. However, due to limits of broadly utilized singularity estimation methods, the anisotropy of geochemical anomalies has not been maturely handled, although spatial variations of geochemical anomalies can be depicted and well exposed. In this paper, a new anisotropic singularity index estimation method is proposed and further applied to characterize geochemical anomalies of ore elements in the Malipo mineral district, China. This new method takes anisotropic nature of geochemical signatures into consideration according to inspecting and comparing changing behaviors of geochemical signatures for various directions rather than conventional isotropic estimation methods. Using the direction with the most intense variations to estimate anisotropic singularity index αmax, improved patterns of geochemical behaviors with anisotropy can be achieved. Compared with other singularity estimation methods, especially the isotropic singularity by the square window-based method, the currently proposed technique objectively identifies the anisotropic geochemical features and investigates the non-linear geochemical behaviors. Moreover, the direction determined at each (sampling) location is more informative or indicative with respect to the migration paths and concentration of ore material.

Vacuum self similar anisotropic cosmologies in F(R)-gravity

The implications from the existence of a proper Homothetic Vector Field (HVF) on the dynamics of vacuum anisotropic models in $F(R)$ gravitational theory are studied. The fact that \emph{every} Spatially Homogeneous vacuum model is equivalent, formally, with a "flux" -free anisotropic fluid model in standard gravity and the induced power-law form of the functional $F(R)$ due to self-similarity enable us to close the system of equations. We found some new exact anisotropic solutions that arise as fixed points in the associated dynamical system. The non-existence of Kasner-like (Bianchi type I) solutions in proper $F(R)-$gravity (i.e. $R\neq 0$) strengthens the belief that curvature corrections will prevent the shear influence into the past thus permitting an isotropic singularity. We also discuss certain issues regarding the lack of vacuum models of type III, IV, VII$_{h}$ in comparison with the corresponding results in standard gravity.

The generic solution with isotropic Big Bang

It is shown that inclusion of the shear stresses to the cosmological evolution can stabilize the Friedmann Big Bang. This results in the existence of the generic solution with isotropic cosmological singularity.

Einstein equation at singularities

AbstractEinstein’s equation is rewritten in an equivalent form, which remains valid at the singularities in some major cases. These cases include the Schwarzschild singularity, the Friedmann-Lemaître-Robertson-Walker Big Bang singularity, isotropic singularities, and a class of warped product singularities. This equation is constructed in terms of the Ricci part of the Riemann curvature (as the Kulkarni-Nomizu product between Einstein’s equation and the metric tensor).

On the Weyl curvature hypothesis

The Weyl curvature hypothesis of Penrose attempts to explain the high homogeneity and isotropy, and the very low entropy of the early universe, by conjecturing the vanishing of the Weyl tensor at the Big-Bang singularity.In previous papers it has been proposed an equivalent form of Einstein’s equation, which extends it and remains valid at an important class of singularities (including in particular the Schwarzschild, FLRW, and isotropic singularities). Here it is shown that if the Big-Bang singularity is from this class, it also satisfies the Weyl curvature hypothesis.As an application, we study a very general example of cosmological models, which generalizes the FLRW model by dropping the isotropy and homogeneity constraints. This model also generalizes isotropic singularities, and a class of singularities occurring in Bianchi cosmologies. We show that the Big-Bang singularity of this model is of the type under consideration, and satisfies therefore the Weyl curvature hypothesis.

Fault trace-oriented singularity mapping technique to characterize anisotropic geochemical signatures in Gejiu mineral district, China

Mineralization is complex and controlled by various non-linear geo-processes. Geochemical signatures of mineralization-favored spaces inherited from multiple geo-processes are consequently anisotropic. Singularity index mapping technique efficient in identifying heterogeneity of geochemical signatures had been utilized to recognize geochemical anomalies in many cases. As an example of interdisciplinary collaboration, this paper using a newly proposed fault trace-oriented singularity index mapping technique intends to characterize hydrothermal mineralization-associated anisotropic geochemical signatures. In the case study, fault traces in the Gejiu mineral district, China are divided into segments with equal length. Centered by fault segments, a set of rectangular windows are defined to estimate singularity index. Variations of geochemical signatures along the vertical direction of fault traces are characterized. The fault trace-oriented singularity indices assigned to their corresponding fault segments term faults as positive, negative and regular fault segments to qualitatively and quantitatively explain interrelations between fault structures and hydrothermal fluids or mineralization. In comparison with frequently employed fault properties (e.g., length, density, types, etc.), the new fault attributes (i.e., positive, negative, and regular fault segment) are applicable to describe variations of physical–chemical reactions between ore-forming fluids and wall rocks along fault traces that benefit the interpretation to metallogenic mechanism. The newly proposed method can be considered as a supplement to the formerly introduced square window-based isotropic singularity index mapping technique.

ENERGY NORM A POSTERIORI ERROR ESTIMATION FOR hp-ADAPTIVE DISCONTINUOUS GALERKIN METHODS FOR ELLIPTIC PROBLEMS IN THREE DIMENSIONS

We develop the energy norm a posteriori error estimation for hp-version discontinuous Galerkin (DG) discretizations of elliptic boundary-value problems on 1-irregularly, isotropically refined affine hexahedral meshes in three dimensions. We derive a reliable and efficient indicator for the error measured in terms of the natural energy norm. The ratio of the efficiency and reliability constants is independent of the local mesh sizes and weakly depending on the polynomial degrees. In our analysis we make use of an hp-version averaging operator in three dimensions, which we explicitly construct and analyze. We use our error indicator in an hp-adaptive refinement algorithm and illustrate its practical performance in a series of numerical examples. Our numerical results indicate that exponential rates of convergence are achieved for problems with smooth solutions, as well as for problems with isotropic corner singularities.

A conformal extension theorem based on null conformal geodesics

In this article we describe the formulation of null geodesics as null conformal geodesics and their description in the tractor formalism. A conformal extension theorem through an isotropic singularity is proven by requiring the boundedness of the tractor curvature and its derivatives to sufficient order along a congruence of null conformal geodesic. This article extends earlier work by Lübbe and Tod, J. Math. Phys. 50, 112501 (2009) (“An extension theorem for conformal gauge singularities,” e-print arXiv:0710.5552).

Encoding cosmological futures with conformal structures

Quiescent cosmology and the Weyl curvature hypothesis possess a mathematical framework, namely the definition of an isotropic singularity, but only for the initial state of the universe. A complementary framework is necessary to also encode appropriate cosmological futures. In order to devise a new framework we analyse the relation between regular conformal structures and (an)isotropy, the behaviour and role of a monotonic conformal factor which is a function of cosmic time, as well as four example cosmologies for further guidance. Finally, we present our new definitions of an anisotropic future endless universe and an anisotropic future singularity which offer a promising realization for the new framework. Their irregular, degenerate conformal structures differ significantly from those of the isotropic singularity. The combination of the three definitions together could then provide the first complete formalization of the quiescent cosmology concept. For completeness we also present the new definitions of an isotropic future singularity and a future isotropic universe. The relation to other approaches, in particular to the somewhat dual dynamical systems approach, and other asymptotic scenarios is briefly discussed.

A novel spherical parallel manipulator: forward position problem, singularity analysis, and isotropy design

In this paper, a novel spherical parallel manipulator and its isotropic design is introduced. This manipulator has good accuracy and relatively a larger workspace which is free of singularities. Utilizing spherical configuration the forward position problem is solved by equivalent angle–axis representation and Bezout's method which leads to a polynomial of degree 8. Two examples are given, one for isotropic and one for nonisotrpoic design. The first case results in eight real solutions, therefore, the polynomial being minimal. Using invariant form, we study acceleration analysis, conditions for singularity and find infinite isotropic structures. Accuracy and workspace analysis are also performed and are shown to have good global conditioning index and relatively large workspace. Using isotropic design and singularity requirements, we show the workspace of isotropic design is free of singularity.

A global conformal extension theorem for perfect fluid Bianchi space-times

A global extension theorem is established for isotropic singularities in polytropic perfect fluid Bianchi space-times. When an extension is possible, the limiting behaviour of the physical space-time near the singularity is analysed.