Special Metrics Research Articles

Concerning the existence of Einstein and Ricci soliton metrics on solvable Lie groups

In this work we investigate solvable and nilpotent Lie groups with special metrics. The metrics of interest are left-invariant Einstein and algebraic Ricci soliton metrics. Our main result shows that the existence of a such a metric is intrinsic to the underlying Lie algebra. More precisely, we show how one may determine the existence of such a metric by analyzing algebraic properties of the Lie algebra in question and infinitesimal deformations of any initial metric. Our second main result concerns the isometry groups of such distinguished metrics. Among the completely solvable unimodular Lie groups (this includes nilpotent groups), if the Lie group admits such a metric, we show that the isometry group of this special metric is maximal among all isometry groups of left-invariant metrics. We finish with a similar result for locally left-invariant metrics on compact nilmanifolds.

The ASSL approach to specifying self‐managing embedded systems

SUMMARYThe increasing complexity of contemporary embedded computing systems requires the use of self‐management in order to handle unforeseen changes in both hardware and control software. The idea behind computer systems capable of self‐management is a complex concept compound by many aspects related to both artificial intelligence and awareness. Here, the biggest challenge is still the question how to properly develop and verify such systems. In this paper, we present a formal approach to specifying embedded systems capable of self‐management. In our approach, we use the ASSL (autonomic system specification language) framework as a development environment, where self‐management features of embedded systems are specified and an implementation is automatically generated. ASSL exposes a rich set of specification constructs that help developers specify event‐driven embedded systems. Hardware is sensed via special metrics intended to drive events and self‐management policies that help the system handle critical situations in an autonomous reactive manner. We present this approach along with a simulation case study where ASSL is used to develop control software for the wide‐angle camera carried on board NASA's Voyager II Spacecraft. Copyright © 2011 John Wiley & Sons, Ltd.

IFRS Consequences on Accounting Conservatism Within Europe: The Role of Big 4 Auditors

We investigate the way auditor characteristics (i.e., reputation and industry specialization) interact on the consequences of mandatory IFRS adoption in Europe in terms of accounting conservatism. Indeed, a mandatory adoption setting may control for firm-level reporting incentives when gauging the effects of a change in accounting standards. However, companies still have some flexibility in their choice of auditor, and the later has a significant say on the accounting policy. We use a dataset of European firms adopting IFRS in 2005 in response to the IAS Regulation (2,973 unique firms) and observed between 2001 and 2008. The main findings are that: (1) the decrease in conditional conservatism (as proxied by the asymmetric timeliness of bad vs. good news), also evidenced in other studies (Ahmed et al., 2013; Andre et al., 2013) is attributable to Big 4-audited companies; (2) correlatively, unconditional conservatism is higher under IFRS in the presence of a Big 4 auditor; and (3) none of these moderating effects occur when considering auditor industry specialization metrics. We conclude that Big 4 auditors placed more emphasis on auditor risk incentives in the IFRS adoption context, by influencing overly conservative accounting practices in response to the new and uncertain accounting environment, with detrimental effects on earnings quality.

Hydration of Zwitterionic Glycine Betaine and Analogues Through Molecular Simulation

We use molecular dynamics simulation to characterize the hydration of 3 molecules important to protein stability: zwitterionic glycine, zwitterionic N,N-dimethylglycine, N,N,N-trimethylglycine (glycine betaine). Both structuring and dynamics of bulk and bound water were examined using a variety of properties and at multiple concentrations. Metrics such as radial distribution functions and residence times were used to characterize hydration. Also, we used more specialized metrics that can discriminate between subtle differences in hydration such as condensed phase order parameters, Voronoi tessellations, and multidimensional pair-pair correlation functions. Trimethylglycine was found to have a unique hydration shell that extends across the entire molecule and has no specific interactions between solute molecules. Glycine was found to aggregate and have a more disjoint hydration shell. Lastly, trimethylglycine is disperse in solution even at very high concentrations and water rapidly moves between trimethylglycine amine groups. The differences in hydration structure between the three zwitterions can be seen in Figure 1, which shows the radial distribution function between the amine and waters. This work has meaningful implications for protein stability where trimethylglycine is known to prevent protein aggregation and interfaces where trimethylglycine prevents protein adsorption.View Large Image | View Hi-Res Image | Download PowerPoint Slide

Local and Bulk Hydration of Zwitterionic Glycine and its Analogues through Molecular Simulations

Molecular dynamics simulations were used to characterize the differences in hydration between glycine and two of its zwitterionic analogues: N,N-dimethylglycine and N,N,N-trimethylglycine (glycine betaine). The hydration of dodecane and oligo(ethylene glycol) was studied for reference. Both structuring and dynamics of bulk and bound water were examined using a variety of properties and at multiple concentrations. Metrics, such as radial distribution functions and residence times, were used to characterize hydration. Also, we used more specialized metrics that can discriminate between subtle differences in hydration, such as condensed phase order parameters, Voronoi tessellations, and multidimensional pair-pair correlation functions. Trimethylglycine was found to have a unique hydration shell that extends across the entire molecule and has no specific interactions between solute molecules. Also, we found that dimethylglycine has a similar hydration structure to that of trimethylglycine despite its hydrogen-bond donor. Glycine was found to aggregate and have a more disjoint hydration shell. All three zwitterions were found to structurally affect water within 1.5-2.0 coordination shells. Lastly, trimethylglycine is disperse in solution even at very high concentrations, and water rapidly moves between trimethylglycine amine groups. This work has meaningful implications for protein stability where trimethylglycine is known to prevent protein aggregation and nonfouling interfaces where trimethylglycine prevents nonspecific protein adsorption.

Structure in space of hyperplane element with special metrics

The present work analyses intrinsic antiquaternionic structures of metric hyperplane elements. Spaces of metric hyperplane elements is Cartan space generalisation. The given spaces are cotangent bundle, where metrics is described by way of symmetric tensor. The present work analyses situation when metric tensor has specific form, as it done by work [5]. It has been proved that in such metric space treeparameter intrinsic antiquaternionic structure families exist, associated affine connections with structures have been found.

The energy of a Kähler class on admissible manifolds

On a compact complex manifold of Kahler type, the energy E(Ω) of a Kahler class Ω is given by the squared L 2-norm of the projection onto the space of holomorphic potentials of the scalar curvature of any Kahler metric representing the said class, and any one such metric whose scalar curvature has squared L 2-norm equal to E(Ω) must be an extremal representative of Ω. A strongly extremal metric is an extremal metric representing a critical point of E(Ω) when restricted to the set of Kahler classes of fixed positive top cup product. We study the existence of strongly extremal metrics and critical points of E(Ω) on certain admissible manifolds, producing a number of nontrivial examples of manifolds that carry this type of metrics, and where in many of the cases, the class that they represent is one other than the first Chern class, and some examples of manifolds where these special metrics and classes do not exist. We also provide a detailed analysis of the gradient flow of E(Ω) on admissible ruled surfaces, show that this dynamical system can be extended to one beyond the Kahler cone, and analyze the convergence of solution paths at infinity in terms of conditions on the initial data, in particular proving that for any initial data in the Kahler cone, the corresponding path is defined for all t, and converges to a unique critical class of E(Ω) as time approaches infinity.

MINIMALITY, HARMONICITY AND CR GEOMETRY FOR REEB VECTOR FIELDS

Let (M, g) be a Riemannian manifold and T1 M its unit tangent sphere bundle. Minimality and harmonicity of unit vector fields have been extensively studied by considering on T1M the Sasaki metric [Formula: see text]. This metric, and other well-known Riemannian metrics on T1 M, are particular examples of Riemannian natural metrics. In this paper we equip T1 M with a Riemannian natural metric [Formula: see text] and in particular with a natural contact metric structure. Then, we study the minimality for Reeb vector fields of contact metric manifolds and of quasi-umbilical hypersurfaces of a Kähler manifold. Several explicit examples are given. In particular, the Reeb vector field ξ of a K-contact manifold is minimal for any [Formula: see text] that belongs to a family depending on two parameters of metrics of the Kaluza–Klein type. Next, we show that the Reeb vector field ξ of a K-contact manifold defines a harmonic map [Formula: see text] for any Riemannian natural metric [Formula: see text]. Besides this, if the Reeb vector ξ of an almost contact metric manifold is a CR map then the induced almost CR structure on M is strictly pseudoconvex and ξ is a pseudo-Hermitian map; if in addition ξ is geodesic then [Formula: see text] is a harmonic map. Moreover, the Reeb vector field ξ of a contact metric manifold is a CR map iff ξ is Killing and [Formula: see text] is a special metric of the Kaluza–Klein type. Finally, in the final section, we obtain that there is a family of strictly pseudoconvex CR structures on T1S2n+1 depending on one parameter, for which a Hopf vector field ξ determines a pseudo-harmonic map (in the sense of Barletta–Dragomir–Urakawa [8]) from S2n+1 to T1S2n+1.

On a construction of majorizing measures on subsets of Rnwith special metrics

We consider processes $X_t$ with values in $L_p({\mit\Omega},\mathcal{F},P)$ and “time” index $t$ in a subset $A$ of the unit cube. A natural condition of boundedness of increments is assumed. We give a full characterization of

Sound mode hydrodynamics from bulk scalar fields

We study the hydrodynamic sound mode using gauge/gravity correspondence by examining a generic black brane background's response to perturbations. We assume that the background is generated by a single scalar field, and then generalize to the case of multiple scalar fields. The relevant differential equations obeyed by the gauge invariant variables are presented in both cases. Finally, we present an analytical solution to these equations in a special case; this solution allows us to determine the speed of sound and bulk viscosity for certain special metrics. These results may be useful in determining sound mode transport coefficients in phenomenologically motivated holographic models of strongly coupled systems.

An Automatic User-Recognition Approach Within a Reputation System Using a Nonlinear Hausdorff-Derived Metric

In this paper, we provide an automatic unsupervised recognition technique for Web community user reputations that uses a special nonlinear metric. First we describe the general framework for reputation systems. Then, we propose a feature extraction approach for the reputation system users. The resulting feature vectors (reputations) are clustered with an unsupervised classification algorithm using a nonlinear distance, derived from the Hausdorff metric for sets.

Special metrics and triality

We investigate a new 8-dimensional Riemannian geometry defined by a generic closed and coclosed 3-form with stabiliser PSU ( 3 ) , and which arises as a critical point of Hitchin's variational principle. We give a Riemannian characterisation of this structure in terms of invariant spinor-valued 1-forms, which are harmonic with respect to the twisted Dirac operator Đ on Δ ⊗ Λ 1 . We establish various obstructions to the existence of topological reductions to PSU ( 3 ) . For compact manifolds, we also give sufficient conditions for topological PSU ( 3 ) -structures that can be lifted to topological SU ( 3 ) -structures. We also construct the first known compact example of an integrable non-symmetric PSU ( 3 ) -structure. In the same vein, we give a new Riemannian characterisation for topological quaternionic Kähler structures which are defined by an Sp ( 1 ) ⋅ Sp ( 2 ) -invariant self-dual 4-form. Again, we show that this form is closed if and only if the corresponding spinor-valued 1-form is harmonic for Đ and that these equivalent conditions produce constraints on the Ricci tensor.

Lifting [formula omitted]-structures to nearly parallel [formula omitted]-structures

Hitchin shows in [N. Hitchin, The geometry of three-forms in six and seven dimensions, J. Differ. Geom. 55 (2000) 547–576. math.DG/0010054v1; N. Hitchin, Stable forms and special metrics, Global Differential Geometry: The Mathematical Legacy of Alfred Gray, in: Contemp. Math., volume 288, American Math. Soc., 2001, pp. 70–89. math.DG/0107101v1] that half-flat S U ( 3 ) -structures on a six-dimensional manifold M can be lifted to parallel G 2 -structure on the product M × R . We show that Hitchin’s approach can also be used to construct nearly parallel G 2 -structures by lifting so-called nearly half-flat structures. These S U ( 3 ) -structures are described by pairs ( ω , φ ) of stable 2- and 3-forms with d φ = λ ω 2 , for some λ ∈ R ∖ { 0 } .

On extracting physical content from asymptotically flat spacetime metrics

A major issue in general relativity, from its earliest days to the present, is how to extract physical information from any solution or class of solutions to the Einstein equations. Though certain information can be obtained for arbitrary solutions, e.g., via geodesic deviation, in general, because of the coordinate freedom, it is often hard or impossible to do. Most of the time information is found from special conditions, e.g. degenerate principle null vectors, weak fields close to Minkowski space (using coordinates close to Minkowski coordinates), or from solutions that have symmetries or approximate symmetries. In the present work, we will be concerned with asymptotically flat spacetimes where the approximate symmetry is the Bondi–Metzner–Sachs group. For these spaces the Bondi 4-momentum vector and its evolution, found from the Weyl tensor at infinity, describes the total energy–momentum of the interior source and the energy–momentum radiated. By generalizing the structures (shear-free null geodesic congruences) associated with the algebraically special metrics to asymptotically shear-free null geodesic congruences, which are available in all asymptotically flat spacetimes, we give kinematic meaning to the Bondi 4-momentum. In other words, we describe the Bondi vector and its evolution in terms of a center of mass position vector, its velocity and a spin vector, all having clear geometric meaning. Among other items, from dynamic arguments, we define a unique (at our level of approximation) total angular momentum and extract its evolution equation in the form of a conservation law with an angular momentum flux.

Distortion lower bounds for line embeddings

In this paper, we show how we can derive lower bounds and also compute the exact distortion for the line embeddings of some special metrics, especially trees and graphs with certain structure. Using linear programming to formulate a simpler version of the problem gives an interesting intuition and direction concerning the computation of general lower bounds for distortion into the line. We also show that our lower bounds on special cases of metrics are a lot better than previous lower bounds.

Analysis of set-up time models: A metric perspective

We consider model based estimates for set-up time. The general setting we are interested in is the following: given a disk and a sequence of read/write requests to certain locations, we would like to know the total time of transitions (set-up time) when these requests are served in an orderly fashion. The problem becomes nontrivial when we have, as is typically the case, only the counts of requests to each location rather then the whole input, and we can only hope to estimate the required time. Models that estimate set-up time have been suggested and heavily used as far back as the sixties. However, not much theory exists to enable a qualitative understanding of such models. To this end we introduce several properties such as (i) super-additivity which means that the set-up time estimate decreases as the input data is refined (ii) monotonicity which means that more activity produces more set-up time, (iii) Dominance which means that one model always produces higher estimates than a second model and (iv) approximation guarantees for the estimate with respect to the worst possible time, by which we can study different models. We provide criteria for super-additivity and monotonicity to hold for popular models such as the Partial Markov model (PMM). The criteria show that the estimate produced by these models will be monotone for any reasonable system. We also show that the independent reference model (IRM) based estimate functions as a worst case estimate in the sense that the estimate is guaranteed to be at least half of the actual set-up time. We also show that it dominates the PMM based estimates. Using our criteria we prove that PMM based estimates are always super additive when applied to the special metrics that correspond to seek times of disk drives. To establish our theoretical results we use the theory of finite metric spaces, and en route show a result of independent interest in that theory, which is a strengthening of a theorem of J.B. Kelly [J.B. Kelly, Hypermetric spaces and metric transforms, in: O. Shisha (Ed.), Inequalities III, 1972, pp. 149–158] about the properties of metrics that are formed by concave functions on the line.

Manifold splines with a single extraordinary point

This paper develops a novel computational technique to define and construct manifold splines with only one singular point by employing the rigorous mathematical theory of Ricci flow. The central idea and new computational paradigm of manifold splines are to systematically extend the algorithmic pipeline of spline surface construction from any planar domain to an arbitrary topology. As a result, manifold splines can unify planar spline representations as their special cases. Despite its earlier success, the existing manifold spline framework is plagued by the topology-dependent, large number of singular points (i.e., | 2 g − 2 | for any genus- g surface), where the analysis of surface behaviors such as continuity remains extremely difficult. The unique theoretical contribution of this paper is that we devise new mathematical tools so that manifold splines can now be constructed with only one singular point, reaching their theoretic lower bound of singularity for real-world applications. Our new algorithm is founded upon the concept of discrete Ricci flow and associated techniques. First, Ricci flow is employed to compute a special metric of any manifold domain (serving as a parametric domain for manifold splines), such that the metric becomes flat everywhere except at one point. Then, the metric naturally induces an affine atlas covering the entire manifold except this singular point. Finally, manifold splines are defined over this affine atlas. The Ricci flow method is theoretically sound, and practically simple and efficient. We conduct various shape experiments and our new theoretical and algorithmic results alleviate the modeling difficulty of manifold splines, and hence, promote the widespread use of manifold splines in surface and solid modeling, geometric design, and reverse engineering.

Blind carrier phase recovery for general quadrature amplitude modulation constellations

A new blind carrier phase reconstruction method that can be applied to both square- and cross-suppressed-carrier L-ary quadrature amplitude modulations (QAMs) is proposed. This method is based on a special phase metric that performs an absolute minimum for the carrier phase offset. In particular, it does not require any modification of the modulation structure. As a result, the non-coherent detection process is easy even for modulations with large QAM constellations. Simulation results will demonstrate the good performance of the estimator described, which is superior to that of conventional blind phase estimation methods.

The universal cut function and type II metrics

In analogy with classical electromagnetic theory, where one determines the total charge and both electric and magnetic multipole moments of a source from certain surface integrals of the asymptotic (or far) fields, it has been known for many years—from the work of Hermann Bondi—that the energy and momentum of gravitational sources could be determined by similar integrals of the asymptotic Weyl tensor. Recently, we observed that there were certain overlooked structures, defined at future null infinity, that allowed one to determine (or define) further properties of both electromagnetic and gravitating sources. These structures, families of complex ‘slices’ or ‘cuts’ of Penrose's , are referred to as universal cut functions. In particular, one can define from these structures a (complex) centre of mass (and centre of charge) and its equations of motion—with rather surprising consequences. It appears as if these asymptotic structures contain, in their imaginary part, a well-defined total spin-angular momentum of the source. We apply these ideas to the type II algebraically special metrics, both twisting and twist free.

Projectively flat arctangent Finsler metric

In this work, we study a class of special Finsler metrics F called arctangent Finsler metric, which is a special (α,β)-metric, where α is a Riemannian metric and β is a 1-form. We obtain a sufficient and necessary condition that F is locally projectively flat if and only if α and β satisfy two special equations. Furthermore we give the non-trivial solutions for F to be locally projectively flat. Moreover, we prove that such projectively flat Finsler metrics with constant flag curvature must be locally Minkowskian.