Penrose Conditions Research Articles

Alternative to mean and least squares methods used in processing the results of scientific and technical experiments

Increasing the complexity and size of systems of various nature requires constant improvement of modeling and verification of the obtained results by experiment. It is possible to clearly conduct each experiment, objectively evaluate the summaries of the researched process, and spread the material obtained in one study to a series of other studies only if they are correctly set up and processed. On the basis of experimental data, algebraic expressions are selected, which are called empirical formulas, which are used if the analytical expression of some function is complex or does not exist at this stage of the description of the object, system or phenomenon. When selecting empirical formulas, polynomials of the form: у = А0 + А1х+ А2х2+ А3х3+…+ Аnхn are widely used, which can be used to approximate any measurement results if they are expressed as continuous functions. It is especially valuable that even if the exact expression of the solution (polynomial) is unknown, it is possible to determine the value of the coefficients An using the methods of mean and least squares. But in the method of least squares, there is a shift in estimates when the noise in the data is increased, as it is affected by the noise of the previous stages of information processing. Therefore, for real-time information processing procedures, a pseudo-reverse operation is proposed, which is performed using recurrent formulas. This procedure is a procedure of successive updating (with a shift) along the columns of the matrix of given sizes and pseudo-reversal at each step of information change. This approach is straightforward and takes advantage of the bounding method. With pseudo-inversion, it is possible to control the correctness of calculations at each step, using Penrose conditions. The need for pseudo-inversion may arise during optimization, forecasting of certain parameters and characteristics of systems of various purposes, in various problems of linear algebra, statistics, presentation of the structure of the obtained solutions, to understand the content of the incorrectness of the resulting solution, in the sense of Adomar-Tikhonov, and to see the ways of regularization of such solutions.

Power-law inflation satisfies Penrose’s Weyl curvature hypothesis

Based on entropy considerations and the arrow of time Penrose argued that the universe must have started in a special initial singularity with vanishing Weyl curvature. This is often interpreted to be at odds with inflation. Here we argue just the opposite, that Penrose's persuasions are in fact consistent with inflation. Using the example of power law inflation, we show that inflation begins with a past null singularity, where Weyl tensor vanishes when the metric is initially exactly conformally flat. This initial state precisely obeys Penrose's conditions. The initial null singularity breaks $T$-reversal spontaneously and picks the arrow of time. It can be regulated and interpreted as a creation of a universe from nothing, initially fitting in a bubble of Planckian size when it materializes. Penrose's initial conditions are favored by the initial $O(4)$ symmetry of the bubble, selected by extremality of the regulated Euclidean action. The predicted observables are marginally in tension with the data, but they can fit if small corrections to power law inflation kick in during the last 60 efolds.

On the Moore-Penrose Pseudo Inverse of the Incidence Matrix for Weighted Undirected Graph

Abstract The interaction of intelligent agents implies the existence of an environment to support it. The usual representations of this environment are graphs with certain properties. Throughput is one of the most important characteristics of such graphs. A traditional metric, such as the usual shortest paths, forms the basis of the traditional throughput index. In this case, a metric is used to synthesize the distribution of multi-coloured flows in graphs more complex than trees. To achieve better results than when using ordinary shortest paths, one can use the Euclidian metric. Working with weighted graphs requires a generalization of the explicit form of the Moore-Penrose pseudo inversed incidence matrix. The validity of the generalization is confirmed by verification of the Penrose conditions. An example of using the Euclidian metric for the distribution of computer network flows is given.

The Moore–Penrose Dual Generalized Inverse Matrix With Application to Kinematic Synthesis of Spatial Linkages

The paper initially reports about the properties of an expression of dual generalized inverse matrix currently available in the literature. It is demonstrated that such a matrix does not fulfill all the Penrose conditions. Hence, novel and computationally efficient algorithms/formulas for the computation of the Moore–Penrose dual generalized inverse (MPDGI) are herein proposed. The paper also contains a new algorithm for the singular value decomposition (SVD) of a dual matrix. The availability of these formulas allows the simultaneous solution of overdetermined systems of dual linear equations without requiring the traditional separation in primal and dual parts. This should prove useful for the solution of many kinematic problems. The algorithms/formulas herein deduced have been also tested on the kinematic synthesis of the constant transmission ratio RCCC spatial linkage.

On generalized inverses of dual matrices

Generalized inverses of real dual matrices are classified according to the set of Moore–Penrose conditions satisfied.The present paper offers theoretical and numerical insights regarding the types of dual generalized inverses that can be computed from a recently proposed formula. Moreover, the usefulness of the formula is demonstrated solving different kinematic problems. In particular, thanks to the dual matrix generalized inverse formula availability, the computation of infinite and infinitesimal screw parameters motion from redundant point and line features is obtained within a unified theoretical treatment. Numerical examples and comparison with the results from previous investigations are provided.

Localized instabilities of the Wigner equation as a model for the emergence of Rogue Waves

In this paper, we model Rogue Waves as localized instabilities emerging from homogeneous and stationary background wavefields, under NLS dynamics. This is achieved in two steps: given any background Fourier spectrum P(k), we use the Wigner transform and Penrose’s method to recover spatially periodic unstable modes, which we call unstable Penrose modes. These can be seen as generalized Benjamin–Feir modes, and their parameters are obtained by resolving the Penrose condition, a system of nonlinear equations involving P(k). Moreover, we show how the superposition of unstable Penrose modes can result in the appearance of localized unstable modes. By interpreting the appearance of an unstable mode localized in an area not larger than a reference wavelength lambda _0 as the emergence of a Rogue Wave, a criterion for the emergence of Rogue Waves is formulated. Our methodology is applied to delta spectra, where the standard Benjamin–Feir instability is recovered, and to more general spectra. In that context, we present a scheme for the numerical resolution of the Penrose condition and estimate the sharpest possible localization of unstable modes.

A fundamental solution to the CCC equations

We display a simple solution to the Penrose CCC scenario. For this solution we chose for the late stages of the previous aeon a FRW, k=0, universe with a both a cosmological constant and radiation (no mass) while for the early stages of the 'present' aeon we have again a FRW universe, k=0, with the same cosmological constant and again with radiation but with mass not yet present. The Penrose conditions force the parameters describing the radiation of the former and present aeons to be equal and the transition metric in the overlap region turns out to be flat. We further study how different rest-mass zero fields transition between the different conformally related regions. These (test) fields appears to easily allow perturbations of the geometry within the CCC scenario.

Nonlinear instability of the one-dimensional Vlasov-Yukawa system

We discuss the nonlinear instability of some class of stationary solutions to the one-dimensional Vlasov–Yukawa system with a mass parameter m. The Vlasov–Yukawa system corresponds to the short-range correction of the repulsive Vlasov–Poisson system arising from plasma physics. We show that the stationary solutions satisfying the Penrose condition are nonlinearly unstable in small mass regime. In a large mass regime, the massiveness of force carrier particles acts as stabilizer in a finite time interval. We present several numerical results to confirm our analytical results.

Criteria for the Metric Generalized Inverse and its Selections in Banach Spaces

We investigate the metric generalized inverses of linear operators in Banach spaces and their homogeneous selections, which was the research suggestion given by Nashed and Votruba (Bull. Am. Math. Soc. 80:831–835, 1974). We construct a kind of the bounded homoneneous selections for the set-valued metric generalized inverse. Criteria for the metric generalized inverses of linear operators and their homogeneous selections are given in terms of Moore–Penrose conditions.

Explicit, approximate expressions for the resolution anda posterioricovariance of massive tomographic systems

We present an approximate method to estimate the resolution, covariance and correlation matrix for linear tomographic systems Ax=b that are too large to be solved by singular value decomposition. An explicit expression for the approximate inverse matrix A− is found using one-step backprojections on the Penrose condition AA−≈I, from which we calculate the statistical properties of the solution. The computation of A− can easily be parallelized, each column being constructed independently. The method is validated on small systems for which the exact covariance can still be computed with singular value decomposition. Though A− is not accurate enough to actually compute the solution x, the qualitative agreement obtained for resolution and covariance is sufficient for many purposes, such as rough assessment of model precision or the reparametrization of the model by the grouping of correlating parameters. We present an example for the computation of the complete covariance matrix of a very large (69 043 × 9610) system with 5.9 × 106 non-zero elements in A. Computation time is proportional to the number of non-zero elements in A. If the correlation matrix is computed for the purpose of reparametrization by combining highly correlating unknowns xi, a further gain in efficiency can be obtained by neglecting the small elements in A, but a more accurate estimation of the correlation requires a full treatment of even the smaller Aij. We finally develop a formalism to compute a damped version of A−.

The construction of free–free flexibility matrices as generalized stiffness inverses

We present generalizations of the classical structural flexibility matrix. Direct or indirect computation of flexibilities as ‘influence coefficients’ has traditionally required pre-removal of rigid body modes by imposing appropriate support conditions. Here the flexibility of an individual element or substructure is directly obtained as a particular generalized inverse of the free–free stiffness matrix. This entity is called a free–free flexibility matrix. It preserves exactly the rigid body modes. The definition is element independent. It only involves access to the stiffness generated by a standard finite element program as well as a separate geometric construction of the rigid body modes. With this information, the computation of the free–free flexibility can be done by solving linear equations and does not require the solution of an eigenvalue problem or performing a singular value decomposition. Flexibility expressions for symmetric and unsymmetric free–free stiffnesses are studied. For the unsymmetric case two flexibilities, one preserving the Penrose conditions and the other the spectral properties, are examined. The two versions coalesce for symmetric matrices.

Unstable BGK Solitary Waves and Collisionless Shocks

Consider a collisionless relativistic neutral plasma. We generalize the Penrose condition for linearized instability to the relativistic case. Then we consider a general one-dimensional equilibrium (a BGK wave) that is a collisionless shock or a solitary wave. The electric potential undergoes a transition from one constant to another as $x$ runs from −∞ to +∞. We prove that if one of these constants satisfies the relativistic version of the Penrose condition, then the BGK wave is nonlinearly unstable. We also prove that the periodic relativistic BGK waves of small amplitude are nonlinearly unstable.

Boundary conditions in quantum cosmology.

The problem of boundary conditions for the cosmological wave function \ensuremath{\psi} is analyzed using a simple model as an example. The model has two degrees of freedom: the scale factor and a homogeneous scalar field. A tunneling wave function is found for this model which describes an ensemble of universes tunneling from ``nothing'' to a de Sitter space, then evolving along the lines of an inflationary scenario and eventually collapsing to a singularity. A boundary condition in superspace is proposed which selects the tunneling solution for \ensuremath{\psi}. This boundary condition can be thought of as a quantum version of the Penrose condition in classical general relativity. The probability current, the concept of time, and the possibility of an experimental test of quantum cosmology are also discussed.

Extension of the Penrose Condition

It is shown that the Penrose condition for the existence of plane wave modes that grow exponentially in time in a plasma described by the linearized Vlasov equation and Poisson's equation is necessary and sufficient for the existence of any modes that grow in time in such a plasma.