Geometric Identities Research Articles

Lorentzian quantum gravity via Pachner moves: one-loop evaluation

Lorentzian quantum gravity is believed to cure the pathologies encountered in Euclidean quantum gravity, such as the conformal factor problem. We show that this is the case for the Lorentzian Regge path integral expanded around a flat background. We illustrate how a subset of local changes of the triangulation, so-called Pachner moves, allow to isolate the indefinite nature of the gravitational action at the discrete level. The latter can be accounted for by oppositely chosen deformed contours of integration. Moreover, we construct a discretization-invariant local path integral measure for 3D Lorentzian Regge calculus and point out obstructions in defining such a measure in 4D. We see the work presented here as a first step towards establishing the existence of the non-perturbative Lorentzian path integral for Regge calculus and related frameworks such as spin foams.An extensive appendix provides an overview of Lorentzian Regge calculus, using the recently established concept of the complexified Regge action, and derives useful geometric formulae and identities needed in the main text.

Spacetime Conventionalism Revisited

AbstractWe provide five rearticulations of the thesis that the structure of spacetime is conventional, rather than empirically determined, based upon variation of the structures that are empirically underdetermined and modal contexts in which this underdetermination occurs. Three of the five formulations of conventionalism will be found to fail. Two are found to open up new interesting problems for researchers in the foundations of general relativity. In all five cases, our analysis explores the interplay between geometric identities, symmetry, conformal structure, and the dynamical content of physical theories with the conventionalism dialectic deployed as a tool of explication, clarification, and exploration.

Hawking temperature as the total Gauss–Bonnet invariant of the region outside a black hole

We provide two novel ways to compute the surface gravity (kappa ) and the Hawking temperature (T_{H}) of a stationary black hole: in the first method T_{H} is given as the three-volume integral of the Gauss–Bonnet invariant (or the Kretschmann scalar for Ricci-flat metrics) in the total region outside the event horizon; in the second method it is given as the surface integral of the Riemann tensor contracted with the covariant derivative of a Killing vector on the event horizon. To arrive at these new formulas for the black hole temperature (and the related surface gravity), we first construct a new differential geometric identity using the Bianchi identity and an antisymmetric rank-2 tensor, valid for spacetimes with at least one Killing vector field. The Gauss–Bonnet tensor and the Gauss–Bonnet scalar play a particular role in this geometric identity. We calculate the surface gravity and the Hawking temperature of the Kerr and the extremal Reissner–Nordström holes as examples.

A new fluid-structure coupling model: Case of a pipe under internal pressure

This work presents the eigenfrequencies of a study on the fluid-structure interaction for a pipe with a circular section, under pressure constraints in laminar, incompressible and irrotational flow, followed by a comparison between stiffness coupling and bar coupling with different geometric ratio Radius/Length; Radius/Thickness, as a frequency geometric identity of the pipe in the industry. The realization of this study is to have better coupling, where we use the relations of mechanical behaviour of the solid on displacement-stress and for the fluid, we use the Navier-Stokes equations in cylindrical coordinates under the speed-stress form transformed into displacement by the Galerkin-Temp theory. Either the force of the fluid will be distributed on the internal surface of the cylinder by the coupling (interface) between fluid and solid. To obtain the results of the eigenfrequencies we used the hierarchical finite element method (HFEM) for both mechanical and fluid entities. For the natural frequencies, we apply the hierarchical finite elements of the Legendre polynomial which is defined in a well-known interval, thanks to this mathematical calculation we arrive at the general equations of motion matrix form of the solid, the interface and the fluid. The calculation was carried out by a MATLAB programme which determines the eigenfrequencies of the evolved system by different geometric and physical parameters mentioned. We validated this work by a table of comparison between the experimental values and the values of the programme.

Newman-Penrose formalism in quadratic gravity

The quadratic gravity constraints are reformulated in terms of the Newman-Penrose-like quantities. In such a frame language, the field equations represent a linear algebraic system for the traceless Ricci tensor components. In principle, a procedure for the combination of the Ricci components with standard geometric identities can be applied in a similar way as in the case of general relativity. These results could serve in various subsequent analyses and physical interpretations of admitted solutions to quadratic gravity. Here, we demonstrate the utility of such an approach by proving general propositions restricting the spacetime geometry under assumptions on a specific algebraic type of curvature tensors.

ОЦЕНКА ГОМОГЕННОСТИ ТОПЛИВОВОЗДУШНОЙ СМЕСИ В МАЛОЭМИССИОННОЙ КАМЕРЕ СГОРАНИЯ ГАЗОТУРБИННОГО ДВИГАТЕЛЯ С ПОМОЩЬЮ КОРРЕЛЯЦИИ ПУЛЬСАЦИЙ ДАВЛЕНИЯ И РАСХОДА ПИЛОТНОГО ТОПЛИВА

Connection between pilot fuel flow and pressure pulsations in dry low emission combustor is established on base of experimental data. Determining quantitative criteria of homogeneous or diffusion flame is formulated. The criteria give chance to evaluate quality of mixing fuel and air and combustion without emission measuring. The root-mean-square deviation of the dependence of the distribution of the air-fuel mixture concentration field over the combustion chamber cross section on the correlation coefficient between the pilot fuel ratio and the double amplitude of pressure pulsations in the flame tubes for a homogeneous (technically mixed air-fuel mixture) circuit was approximated based on the data obtained. The results obtained make it possible to significantly improve the quality of emission prediction by refining the semi-empirical mathematical model of the nitrogen oxides' generation, built on the basis of the Zeldovich thermal mechanism for low-emission operating modes of dry low emission combustor of the gas turbine engines. The account of heterogeneity of the process in the flame tubes makes it possible to move from the assumption of the geometric and gas-dynamic identity of the flame tubes to their actual heterogeneity. This conclusion allows refining the model. The developed mathematical model can be used as a virtual emission sensor in advanced tracking control systems of gas turbine engines.

Completely geometric theory I: gravitation

In order to describe motions in arbitrary physical systems, a geometric approach is proposed, in which the goal is not to find the Lagrangian, but to find the geometry of space that models the experimental space. The requirement that the observed trajectory coincides with the geodesic makes it possible to use geometric identity to find the equation for the metric. As a result, it is possible to give an interpretation of a number of observations that do not have such in the existing theories.

Geometric model of the fracture as a manifold immersed in porous media

In this work, we analyze the flow filtration process of slightly compressible fluids in porous media containing fractures with complex geometries. We model the coupled fracture-porous media system where the linear Darcy flow is considered in porous media and the nonlinear Forchheimer equation is used inside the fracture. We develop a model to examine the flow inside fractures with complex geometries and variable thickness on a Riemannian manifold. The fracture is represented as the normal variation of a surface immersed in R3. Using operators of Laplace–Beltrami type and geometric identities, we model an equation that describes the flow in the fracture. A reduced model is obtained as a low dimensional boundary value problem. We then couple the model with the porous media. Theoretical and numerical analyses have been performed to compare the solutions between the original geometric model and the reduced model in reservoirs containing fractures with complex geometries. We prove that the two solutions are close and, therefore, the reduced model can be effectively used in large scale simulators for long and thin fractures with complicated geometry.

Geometric singularity of spatial shape of urbanized territories

The article considers the theoretical foundations of the emergence of geometric singularity of the spatial form of urbanized territories in the context of translating the specific vocabulary of formation and the norms of cultural tradition. A comprehensive consideration of the problems of preserving and developing the historical appearance of the city, as well as the issues of solving the problems of forming a modern urban environment, forms the basis of the meaning and content of the article. The methodology for studying the phenomenon of geometric singularity of the spatial form of urbanized territories in the context of the historical environment of the center of the Siberian city is the basis of analysis and criticism as specific forms of activity and functioning of the urban community. The study is based on various types of ideas about the organization of the structure of external relations, understanding and designing the mechanisms for developing their own types of activities and urban environment, which ensures the normal development and regeneration of the architectural form. The main stages of the formation of urbanized territories are outlined. The problem of formation of the concept of singularity considering preservation and development of urban identity, as well as formation of modern urbanized environment is considered. The methodology for studying the mechanisms for forming unique units of the original image of the cultural and historical environment is based on the study of the geometric identity of spatial forms of co-organization of urban communities. The methods of geometric singularity of spatial form of urbanized territories considered by us in the context of translation of specific vocabulary of formation and norms of cultural traditions of historical environment of cultural and social center of the city allow carrying out described operations on various types of initial and operational data from deployment of whole spectrum of parameters to their comparative assessment and obtaining indicators of compliance of studied structure of composition with given standard. The results confirmed the existing theoretical provisions, thereby showing the effectiveness of the study.

New Magic Formula Demonstration Shows Unexpected Features of Geometrically Defined Matrices for Polyhedral Grids

Magic formulas are the geometric identities at the root of modern compatible schemes for polyhedral grids. We present rigorous yet elementary proofs of the magic formulas originating from Stokes theorem. The proofs enlighten new fundamental aspects of the mass matrices produced with the magic formulas. First, the construction of the mass matrices works for an unexpectedly broad type of mesh cells. Second, they show that dual nodes can be arbitrarily positioned thus extending the construction of the dual barycentric grid.

The Pohozaev-Schoen identity on asymptotically Euclidean manifolds: Conservation laws and their applications

The aim of this paper is to present a version of the generalized Pohozaev-Schoen identity in the context of asymptotically Euclidean manifolds. Since these kind of geometric identities have proven to be a very powerful tool when analysing different geometric problems for compact manifolds, we will present a variety of applications within this new context. Among these applications, we will show some rigidity results for asymptotically Euclidean Ricci-solitons and Codazzi-solitons. Also, we will present an almost-Schur type inequality valid in this non-compact setting which does not need restrictions on the Ricci curvature. Finally, we will show how some rigidity results related with static potentials also follow from these type of conservation principles.

Estimating marginal likelihoods from the posterior draws through a geometric identity

Abstract This article develops a new estimator of the marginal likelihood that requires only a sample of the posterior distribution as the input from the analyst. This sample may come from any sampling scheme, such as Gibbs sampling or Metropolis–Hastings sampling. The presented approach can be implemented generically in almost any application of Bayesian modeling and significantly decreases the computational burdens associated with marginal likelihood estimation compared to existing techniques. The functionality of this method is demonstrated in the context of probit and logit regressions, on two mixtures of normals models, and also on a high-dimensional random intercept probit. Simulation results show that the simple approach presented here achieves excellent stability in low-dimensional models, and also clearly outperforms existing methods when the number of coefficients in the model increases.

Geometric Hardy's inequalities with general distance functions

We establish in this paper general geometric Hardy's identities and inequalities on domains in RN in the spirit of their celebrated works by Brezis-Vázquez and Brezis-Marcus. Hardy's identities are powerful tools in establishing more precise and significantly stronger inequalities than those Hardy's inequalities in the literature. More precisely, we use the notion of Bessel pairs introduced by Ghoussoub and Moradifam to establish several Hardy identities and inequalities with the general distance functions. Our distance functions can be understood as the distance to the surfaces of codimension α∈R, and include the distance to the origin (α=N), the distance to the boundary (α=1), and even the distance to surfaces of codimension k∈N with 1≤k≤N, as special cases. Our Hardy's identities for general Bessel pairs and their special cases improve the Hardy inequalities with general distance functions d(x) by Barbatis, Filippas and Tertikas. We also establish the best constant and extremal functions for a new type of the Hardy-Sobolev-Maz'ya inequality on the domain Σ={xi>0,i=1,...,N} with the distance functiond1(x)=d(x,∂Σ)=min⁡{x1,...,xN}. Our Hardy's identities on the domain {0<d1(x)<R} with the distance function d1(x)=min⁡{x1,...,xN} are in the spirit of Brezis-Vázquez and Brezis-Marcus. More applications of our main theorems on Hardy's identities and inequalities with general distance functions are given and these results improve and sharpen many Hardy's inequalities in the literature. Interesting examples of Bessel pairs and distance functions are given as well.

GEOMETRIK AYNIYATLARNI ISBOTLASHDA KUBIK TENGLAMALAR USULI

GEOMETRIK AYNIYATLARNI ISBOTLASHDA KUBIK TENGLAMALAR USULI

Discrete symmetries in dimer diagrams

We apply dimer diagram techniques to uncover discrete global symmetries in the fields theories on D3-branes at singularities given by general orbifolds of general toric Calabi-Yau threefold singularities. The discrete symmetries are discrete Heisenberg groups, with two ZN generators A, B with commutation AB = C BA, with C a central element. This fully generalizes earlier observations in particular orbifolds of C3, the conifold and Yp,q . The solution for any orbifold of a given parent theory follows from a universal structure in the infinite dimer in R2 giving the covering space of the unit cell of the parent theory before orbifolding. The generator A is realized as a shift in the dimer diagram, associated to the orbifold quantum symmetry; the action of B is determined by equations describing a 1-form in the dimer graph in the unit cell of the parent theory with twisted boundary conditions; finally, C is an element of the (mesonic and baryonic) non-anomalous U (1) symmetries, determined by geometric identities involving the elements of the dimer graph of the parent theory. These discrete global symmetries of the quiver gauge theories are holographically dual to discrete gauge symmetries from torsion cycles in the horizon, as we also briefly discuss. Our findings allow to easily construct the discrete symmetries for infinite classes of orbifolds. We provide explicit examples by constructing the discrete symmetries for the infinite classes of general orbifolds of C3, conifold, and complex cones over the toric del Pezzo surfaces, dP1, dP2 and dP3.

From entanglement to thermodynamics and to gravity

Here we develop the connection between thermodynamics, entanglement, and gravity. By attributing thermodynamics to timeslices of a causal diamond, we show that the Clausius relation $T\Delta S_{\text{rev}}=Q$, where $\Delta S_{\text{rev}}$ is the reversible entropy change, gives rise to the non-linear gravitational equations of motion for a wide class of diffeomorphism invariant theories. We then compare the Clausius relation to the first law of causal diamond mechanics (FLCD), a geometric identity and necessary ingredient in deriving Jacobson's entanglement equilibrium proposal -- the entanglement entropy of a spherical region with a fixed volume is maximal in vacuum. Specifically we show that the condition of fixed volume can be understood as subtracting the irreversible contribution to the thermodynamic entropy. This provides a "reversible thermodynamic process" interpretation of the FLCD, and that the condition of entanglement equilibrium may be regarded as equilibrium thermodynamics for which the Clausius relation holds. Finally, we extend the entanglement equilibrium proposal to the timelike stretched horizons of future lightcones, providing an entanglement interpretation of stretched lightcone thermodynamics.

It is not in the details: Self-related shapes are rapidly classified but their features are not better remembered

Self-prioritization is a robust phenomenon whereby judgments concerning self-representational stimuli are faster than judgments toward other stimuli. The present paper examines if and how self-prioritization causes more vivid short-term memories for self-related objects by giving geometric shapes arbitrary identities (self, mother, stranger). In Experiment 1 participants were presented with an array of the three shapes and required to retain the location and color of each in memory. Participants were then probed regarding the identity of one of the shapes and subsequently asked to indicate the color of the probed shape or an unprobed shape on a color wheel. Results indicated no benefit for self-stimuli in either response time for the identification probe or for color fidelity in memory. Yet, a cuing benefit was observed such that the cued stimulus in the identity probe did have higher fidelity within memory. Experiments 2 and 3 reduced the cognitive load by only requiring that participants process the identity and color of one shape at a time. For Experiment 2, the identity probe was memory-based, whereas the stimulus was presented alongside the identity probe for Experiment 3. Results demonstrated a robust self-prioritization effect: self-related shapes were classified faster than non-self-shapes, but this self-advantage did not lead to an increase in the fidelity of memory for self-related shapes’ colors. Overall, these results suggest that self-prioritization effects may be restricted to an improvement in the ability to recognize that the self-representational stimulus is present without devoting more perceptual and short-term memory resources to such stimuli.

The generic model of general relativity

We develop a generic spacetime model in general relativity which can be used to build any gravitational model within general relativity. The generic model uses two types of assumptions: (a) geometric assumptions in addition to the inherent geometric identities of the Riemannian geometry of spacetime and (b) assumptions defining a class of observers by means of their four-velocity u^{a} which is a unit timelike vector field. The geometric assumptions as a rule concern symmetry assumptions (the so called collineations). The latter introduces the 1+3 decomposition of tensor fields in spacetime. The 1+3 decomposition results in two major results. The 1+3 decomposition of u_{a;b} defines the kinematic variables of the model (expansion, rotation, shear and four-acceleration) and defines the kinematics of the gravitational model. The 1+3 decomposition of the energy momentum tensor representing all gravitating matter introduces the dynamic variables of the model (energy density, the isotropic pressure, the momentum transfer or heat flux vector and the traceless tensor of the anisotropic pressure) as measured by the defined observers and defines the dynamics of the model. The symmetries assumed by the model act as constraints on both the kinematical and the dynamical variables of the model. As a second further development of the generic model we assume that in addition to the four-velocity of the observers u_{a} there exists a second universal vector field n_{a} in spacetime so that one has a so-called double congruence (u_{a},n_{a}) which can be used to define the 1+1+2 decomposition of tensor fields. The 1+1+2 decomposition leads to an extended kinematics concerning both fields building the double congruence and to a finer dynamics involving more physical variables. After presenting and discussing the results in their full generality we show how they are applied in practice by considering in a step by step approach the case of a string fluid in Bianchi I spacetime for the comoving observers.

Use of views and ways of construction of perspective images by students of specialization «Architecture of buildings and structures»

The section of descriptive geometry «Perspective» is one of the main subjects in the preparation of students of specialization in «Architecture of buildings and structures». The modern architect needs to know all the subtleties of urban planning, landscape architecture, bulk design, design of small architectural forms, as well as interior design for the successful realization of creative potential and aesthetic perception. A complete picture of the expressive space gives an image constructed according to the rules of the perspective. Modern techniques and methods of visualization on the plane of space-spatial compositions appeared as a result of a long period of their development and perfection. The discovery of precise laws for the construction of a perspective has allowed architects, artists, and designers to more accurately depict on the plane the forms of the visible world. In modern practice it is impossible creation of interiors, articles of way of life without visualization and using of evident image. The complete picture of expressive space (interior, landscape, design) gives an image that is built on the rules of prospect. Proper use of the types and methods of perspective images, drawings of axonometric projections provide the opportunity to apply the knowledge gained to graphically interpret the solution of various tasks in various fields of activity and in many sciences. Using the laws of perspective allows you to enhance the expressiveness of the space, identify and emphasize its depth, or, conversely, visually reduce. Using perspective views is not limited to images on the plane (painting, graphics, relief). It is found in architecture, scenography, landscape gardening. The study of the «Perspective» section allows future professionals to create visual images that carry the maximum informative. And this can be achieved by applying different ways of constructing a perspective, depending on the geometric identity of the object and showing the kind of projected object from the real point of view taking into account the features of visual perception.

A Combined SAXS and XAS Setup for Operando Electrocatalyst Degradation Studies

The noble metal nanoparticles (NPs, including oxides and/or alloys) customarily used to catalyze the electrochemical reactions at play in low temperature fuel cells and electrolyzers are prone to potential-induced dissolution1 which leads to changes in their composition and morphology. Minimizing the impact of these processes on the devices’ operation requires a better understanding of their mechanism under application-relevant operando conditions. In this regard, small angle X-ray scattering (SAXS) provides information about a sample’s NP size distribution and agglomeration, while X-ray absorption spectroscopy (XAS) is sensitive to the electronic (oxidation state) and geometric (identity and number of coordinating atoms, bond distances) structure of the element of interest. The combination of these two techniques in an operando setup would result in novel insight on the degradation mechanisms affecting noble metal nanoparticles. With this motivation, we have developed a novel setup in which the gas ionization chambers and flight tube plus 2D-detector required for XAS and SAXS acquisition respectively, are mounted in parallel on a movable table that allows to transition among techniques within a couple of minutes. This configuration allows to use both techniques in their optimal configuration, without losing signal quality while compromising on one technique while combining it with the other (as done in the past).2 To validate this approach it was used to study the operando degradation of two carbon-supported Pt-NP (Pt/C) catalysts customarily used in polymer electrolyte fuel cells, which constitute a relatively well understood system owing to the large volume of work that has been devoted to its study.3 Specifically, commercial Pt on Vulcan and Pt on graphitized Black Pearls catalysts (Pt/V and Pt/BP-g, respectively – both with 30 wt. % Pt) were processed into electrodes and electrochemically tested in an operando flow cell developed in-house.4 The degradation protocol consisted of 250 potential cycles between 0.5 and 1.5 V vs. the reversible hydrogen electrode (RHE) at 50 mV∙s-1, using 0.1 M HClO4 at 60 °C as the electrolyte. Complementing results derived from both techniques with respect to changes in particle diameter as well as oxidation state of the catalyst particles are then used to quantifiy NP-growth rates, correlation of particle size with Pt-Pt bond distance as well as NP oxide coverages. In summary, this contribution will introduce a novel, combined SAXS and XAS setup devoted to (electro)catalyst research, and exemplify its potential application based on the results derived for two Pt/C catalysts, which are fully consistent with previous literature on such materials. Cherevko, A. R. Zeradjanin, A. A. Topalov, N. Kulyk, I. Katsounaros, and K. J. J. Mayrhofer, ChemCatChem 6, 2219 (2014).Nikitenko, A. M. Beale, A. M. J. van der Eerden, S. D. M. Jacques, O. Leynaud, and M. G. O‘Brien, J. Synchrotron Rad. 15, 632 (2008).Shao-Horn, W. C. Sheng, S. Chen, P. J. Ferreira, F. Holby, and D. Morgan, Top. Catal. 46, 285 (2007).Binninger, E. Fabbri, A. Patru, M. Garganourakis, J. Han, D. Abbott, O. Sereda, R. Kötz, A. Menzel, M. Nachtegaal, and T. J. Schmidt, J. Electrochem. Soc. 163, H906 (2016).A. Gilbert, N. N. Kariuki, R. Subbaraman, A. J. Kropf, M. C. Smith, E. F. Holby, D. Morgan, and D. J. Myers, J. Am. Chem. Soc. 134, 14823 (2012).Lei, J. Jelic, L. C. Nitsche, R. Meyer, and J. Miller, Top. Catal. 54, 334 (2011).