scholarly journals The Hilbert manifold of asymptotically flat metric extensions

2021 ◽  
Vol 53 (1) ◽  
Author(s):  
Stephen McCormick
Keyword(s):  
Phase Space ◽  
Critical Points ◽  
Boundary Data ◽  
Killing Vector ◽  
Weighted Sobolev Spaces ◽  
Stationary Solutions ◽  
Einstein Equations ◽  
Local Regularity ◽  
Static Black Hole ◽  
The Difference

AbstractIn [Commun Anal Geom 13(5):845–885, 2005], Bartnik described the phase space for the Einstein equations, modelled on weighted Sobolev spaces with local regularity $$(g,\pi )\in H^2\times H^1$$ ( g , π ) ∈ H 2 × H 1 . In particular, it was established that the space of solutions to the constraints form a Hilbert submanifold of this phase space. The motivation for this work was to study the quasi-local mass functional now bearing his name. However, the phase space considered there was over a manifold without boundary. Here we demonstrate that analogous results hold in the case where the manifold has an interior compact boundary, and the metric is prescribed on the boundary. Then, still following Bartnik’s work, we demonstrate the critical points of the mass functional over this space of extensions correspond to stationary solutions with vanishing Killing vector on the boundary. Furthermore, if this solution is smooth then it is in fact a static black hole solution. In particular, in the vacuum case, critical points only occur at exterior Schwarzschild solutions; that is, critical points of the mass over this space do not exist generically. Finally, we briefly discuss a version of the result when the boundary data is related to Bartnik’s geometric boundary data. In particular, by imposing different boundary conditions on the Killing vector, we show that stationary solutions in this case correspond to critical points of an energy resembling the difference between the ADM mass and the Brown–York mass of the boundary.

Phase space correspondence between Jordan and Einstein frames

2021 ◽  
pp. 2150087
Author(s):  
Amin Salehi
Keyword(s):  
Phase Space ◽  
Critical Point ◽  
Critical Points ◽  
Dynamical Behavior ◽  
Cosmological Parameters ◽  
Jordan Frame ◽  
Field Equations ◽  
Theories Of Gravity ◽  
The Stability ◽  
Jordan And Einstein Frames

Scalar–tensor theories of gravity can be formulated in the Einstein frame or in the Jordan frame (JF) which are related with each other by conformal transformations. Although the two frames describe the same physics and are equivalent, the stability of the field equations in the two frames is not the same. Here, we implement dynamical system and phase space approach as a robustness tool to investigate this issue. We concentrate on the Brans–Dicke theory in a Friedmann–Lemaitre–Robertson–Walker universe, but the results can easily be generalized. Our analysis shows that while there is a one-to-one correspondence between critical points in two frames and each critical point in one frame is mapped to its corresponds in another frame, however, stability of a critical point in one frame does not guarantee the stability in another frame. Hence, an unstable point in one frame may be mapped to a stable point in another frame. All trajectories between two critical points in phase space in one frame are different from their corresponding in other ones. This indicates that the dynamical behavior of variables and cosmological parameters is different in two frames. Hence, for those features of the study, which focus on observational measurements, we must use the JF where experimental data have their usual interpretation.

Implementing effective animal-based measurements for assessing animal welfare on farms and slaughter plants.

2021 ◽  
pp. 60-83
Author(s):  
Temple Grandin
Keyword(s):  
Cold Stress ◽  
Animal Welfare ◽  
Body Condition ◽  
Critical Points ◽  
Heat And Cold Stress ◽  
Standards And Guidelines ◽  
The Difference ◽  
Set Up ◽  
Animal Handling ◽  
Input Resource

Abstract This chapter describes how to write clear animal welfare standards and guidelines that will be interpreted the same way by different people; the difference between animal-based outcome measures and input resource-based standards; how to determine the most important core criteria or critical points to prevent abuse or neglect; easy-to-use measures for assessing body condition, lameness, injuries, condition of haircoat/feathers, animal handling, hygiene, heat and cold stress and the presence of abnormal behaviour and how to set up effective animal welfare auditing programmes.

COHERENT STATES WITHOUT GROUPS: QUANTIZATION ON NONHOMOGENEOUS MANIFOLDS

1993 ◽  
Vol 08 (18) ◽  
pp. 1735-1738 ◽  
Author(s):  
JOHN R. KLAUDER
Keyword(s):  
Phase Space ◽  
Coherent State ◽  
Wide Class ◽  
Coherent States ◽  
Path Integrals ◽  
Killing Vector ◽  
Single Variable ◽  
Holomorphic Representation ◽  
Representation Spaces ◽  
State Path

A wide class of single-variable holomorphic representation spaces are constructed that are associated with very general sets of coherent states defined without the use of transitively acting groups. These representations and states are used to define coherent-state path integrals involving phase-space manifolds having one Killing vector but a quite general curvature otherwise.

Superoperator methods of calculating cross sections: III. Unified description of classical and quantal scattering

1980 ◽  
Vol 58 (8) ◽  
pp. 1171-1182 ◽  
Author(s):  
R. E. Turner ◽  
R. F. Snider
Keyword(s):  
Quantum Mechanics ◽  
Phase Space ◽  
Time Dependence ◽  
Stationary State ◽  
Cross Sections ◽  
Differential Cross ◽  
Density Operator ◽  
The Difference ◽  
Unified Description ◽  
Classical And Quantum Mechanics

It is shown how differential cross sections can be obtained from the time dependence of phase space packets. This procedure is valid both for classical and quantum mechanics. Two methods are described. In one the trajectory of the packet is emphasized, while in the second the packet is appropriately spread to infinite size. Both methods are applicable to either mechanics. It is shown how the quantal results agree with those of the stationary state approach as formulated in terms of the density operator. The description is also used to elucidate the difference between the scattered flux and the generalized flux that arises naturally in the superoperator formulation.

scholarly journals Presentation of winter flower-dub lesions of peach varieties originating from different growing areas

2007 ◽  
pp. 70-73
Author(s):  
Mária Dani
Keyword(s):  
Critical Points ◽  
Frost Damage ◽  
Frost Tolerance ◽  
Great Plain ◽  
Flower Buds ◽  
Determining Factors ◽  
Tolerant Variety ◽  
The Difference

Although there are many critical points in peach production, in Hungary, winter frost damage is one of the most significant. Serious winter frost damage on the Hungarian Great Plain was the focus of our experiments. The adequate growth and the most adaptable varieties are determining factors in peach production. In our experiments, we compared three growing areas (Siófok, Sóskút, Szatymaz) and four different varieties (Suncrest, Redhaven, Meystar, Michellini). Throughout these growing areas and with varieties, we wanted to demonstrate the differences in the frost damage values of the flower buds in 2005 and 2006.In the course of the statistical trials, we found that the difference between the varieties is significant (table 2). The most tolerant variety as for frost tolerance is the ‘Michellini’ variety, and the worst is the ‘Suncrest’ variety. We also found that these data are significant. When we examined the varieties according to their growth, we got the same results (table 3). We determined that the differences between growth are significant and related to these four varieties and the two years (2005-2006), that frost damage was the highest at Szatymaz, and that it was the lowest in Sóskút.

scholarly journals AN EXACT SELF-SIMILAR SOLUTION FOR AN EXPANDING BALL OF RADIATION

2006 ◽  
Vol 15 (03) ◽  
pp. 395-404 ◽  
Author(s):  
J. PONCE DE LEON ◽  
P. S. WESSON
Keyword(s):  
Line Element ◽  
Killing Vector ◽  
Einstein Equations ◽  
Effective Density ◽  
Spherically Symmetric ◽  
Self Similar Solution ◽  
Thermodynamic Conditions ◽  
Simple Scaling ◽  
Self Similar ◽  
Boltzmann Law

We give an exact solution of the 5D Einstein equations which in 4D can be interpreted as a spherically symmetric dissipative distribution of matter, with heat flux, whose effective density and pressure are nonstatic, nonuniform, and satisfy the equation of state of radiation. The matter satisfies the usual energy and thermodynamic conditions. The energy density and temperature are related by the Stefan–Boltzmann law. The solution admits a homothetic Killing vector in 5D, which induces the existence of self-similar symmetry in 4D, where the line element as well as the dimensionless matter quantities are invariant under a simple "scaling" group.

scholarly journals CANONICAL QUANTIZATION OF THE BELINSKIĬ-ZAKHAROV ONE-SOLITON SOLUTIONS

1995 ◽  
Vol 04 (06) ◽  
pp. 749-766
Author(s):  
NENAD MANOJLOVIC ◽  
GUILLERMO A. MENA MARUGÁN
Keyword(s):  
Phase Space ◽  
Quantum Dynamics ◽  
Canonical Quantization ◽  
Soliton Solutions ◽  
Irreducible Unitary Representation ◽  
Einstein Equations ◽  
Classical Solutions ◽  
Hamiltonian Constraint ◽  
Dynamical Equations ◽  
Complete Set

We apply the algebraic quantization programme proposed by Ashtekar to the analysis of the Belinskiĭ-Zakharov classical spacetimes, obtained from the Kasner metrics by means of a generalized soliton transformation. When the solitonic parameters associated with this transformation are frozen, the resulting Belinskiĭ-Zakharov metrics provide the set of classical solutions to a gravitational minisuperspace model whose Einstein equations reduce to the dynamical equations generated by a homogeneous Hamiltonian constraint and to a couple of second-class constraints. The reduced phase space of such a model has the symplectic structure of the cotangent bundle over R+×R+. In this reduced phase space, we find a complete set of real observables which form a Lie algebra under Poisson brackets. The quantization of the gravitational model is then carried out by constructing an irreducible unitary representation of that algebra of observables. Finally, we show that the quantum theory obtained in this way is unitarily equivalent to that which describes the quantum dynamics of the Kasner model.

scholarly journals ESTIMATES OF BLOW-UP TIME FOR A NON-LOCAL REACTIVE–CONVECTIVE PROBLEM MODELLING OHMIC HEATING OF FOODS

2006 ◽  
Vol 49 (1) ◽  
pp. 215-239 ◽  
Author(s):  
C. V. Nikolopoulos ◽  
D. E. Tzanetis
Keyword(s):  
Steady State ◽  
Initial Data ◽  
Blow Up ◽  
Ohmic Heating ◽  
Steady State Solution ◽  
Critical Value ◽  
Stationary Solutions ◽  
State Solution ◽  
Non Local ◽  
The Difference

AbstractIn this work, we estimate the blow-up time for the non-local hyperbolic equation of ohmic type, $u_t+u_{x}=\lambda f(u)/(\int_{0}^1f(u)\,\mathrm{d} x)^{2}$, together with initial and boundary conditions. It is known that, for $f(s)$, $-f'(s)$ positive and $\int_0^\infty f(s)\,\mathrm{d} s\lt\infty$, there exists a critical value of the parameter $\lambda>0$, say $\lambda^\ast$, such that for $\lambda>\lambda^\ast$ there is no stationary solution and the solution $u(x,t)$ blows up globally in finite time $t^\ast$, while for $\lambda\leq\lambda^\ast$ there exist stationary solutions. Moreover, the solution $u(x,t)$ also blows up for large enough initial data and $\lambda\leq\lambda^\ast$. Thus, estimates for $t^\ast$ were found either for $\lambda$ greater than the critical value $\lambda^\ast$ and fixed initial data $u_0(x)\geq0$, or for $u_0(x)$ greater than the greatest steady-state solution (denoted by $w_2\geq w^*$) and fixed $\lambda\leq\lambda^\ast$. The estimates are obtained by comparison, by asymptotic and by numerical methods. Finally, amongst the other results, for given $\lambda$, $\lambda^*$ and $0\lt\lambda-\lambda^*\ll1$, estimates of the following form were found: upper bound $\epsilon+c_1\ln[c_2(\lambda-\lambda^*)^{-1}]$; lower bound $c_3(\lambda-\lambda^*)^{-1/2}$; asymptotic estimate $t^\ast\sim c_4(\lambda-\lambda^\ast)^{-1/2}$ for $f(s)=\mathrm{e}^{-s}$. Moreover, for $0\lt\lambda\leq\lambda^*$ and given initial data $u_0(x)$ greater than the greatest steady-state solution $w_2(x)$, we have upper estimates: either $c_5\ln(c_6A^{-1}_0+1)$ or $\epsilon+c_7\ln(c_8\zeta^{-1})$, where $A_0$, $\zeta$ measure, in some sense, the difference $u_0-w_2$ (if $u_0\to w_2+$, then $A_0,\zeta\to0+$). $c_i\gt0$ are some constants and $0\lt\epsilon\ll1$, $0\ltA_0,\zeta$. Some numerical results are also given.

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