Exact Perturbation Research Articles

Methods for exact perturbation analysis

1. The dominant eigenvalue of the population projection matrix provides the asymptotic growth rate of a population. Perturbation analysis examines how changes in vital rates affect this rate. The standard approach to evaluating the effect of a perturbation uses sensitivities and elasticities to provide a linear approximation, which is often inappropriate. 2. A transfer function approach provides the exact relationship between growth rate and perturbation. An alternative approach derives the exact solution using symbolic algebra by calculating the matrix characteristic equation in terms of the perturbation parameters and the symptotic growth rate. 3. This provides integrated sensitivities and plots of the exact relationship. The same method may be used for any perturbation structure, however complicated. 4. The simplicity of the new method is illustrated through two examples - the killer whale and the lizard orchid.

Solution of the Davey–Stewardson equation using homotopuy analysis method

In this paper, the homotopy analysis method (HAM) proposed by Liao is adopted for solving Davey–Stewartson (DS) equations which arise as higher dimensional generalizations of the nonlinear Schrödinger (NLS) equation. The results obtained by HAM have been compared with the exact solutions and homotopy perturbation method (HPM) to show the accuracy of the method. Comparisons indicate that there is a very good agreement between the HAM solutions and the exact solutions in terms of accuracy.

Correlated band theory of spin and orbital contributions to Dzyaloshinskii-Moriya interactions

A new approach for calculations of Dzyaloshinskii-Moriya interactions in molecules and crystals is proposed. It is based on the exact perturbation expansion of total energy of weak ferromagnets in the canting angle with the only assumption of local Hubbard-type interactions. This scheme leads to a simple and transparent analytical expression for Dzyaloshinskii-Moriya vector with a natural separation into spin and orbital contributions. The main problem was transferred to calculations of effective tight-binding parameters in the properly chosen basis including spin-orbit coupling. Test calculations for La$_2$CuO$_4$ give the value of canting angle in a good agreement with experimental data.

Modeling of optimal perturbations in flat plate boundary layer using global modes: benefits and limits

The ability of capturing the noise amplifier dynamics of a boundary layer on a flat plate subjected to an adverse pressure gradient by means of a global modes strategy is numerically investigated. After a brief description of the mechanism regarding the exact optimal perturbation fields, the system is rewritten into an input–output framework. In order to achieve this, the input and output are derived from a direct-optimal growth computation through the perturbation fields at times t = 0 and t = topt where the wave packet reaches a maximum energy growth. By expanding the flow disturbance variables as a summation of global modes, their observability, controllability, and truncation error are assessed. It is shown that the truncation error decreases by taking into account highly temporally damped global modes. Therefore, we propose the ability of a reduced-order model based on global modes to capture the full dynamics of the stable subspace, under the consideration of a large number of modes.

THE KERR–SCHILD ANSATZ REVISED

Kerr–Schild metrics have been introduced as a linear superposition of the flat spacetime metric and a squared null-vector field, say k, multiplied by some scalar function, say H. The basic assumption which led to Kerr solution was that k be both geodesic and shearfree. This condition is relaxed here and Kerr–Schild Ansatz is revised by treating Kerr–Schild metrics as exact linear perturbations of Minkowski spacetime. The scalar function H is taken as the perturbing function, so that Einstein's field equations are solved order-by-order in powers of H. It turns out that the congruence must be geodesic and shearfree as a consequence of third- and second-order equations, leading to an alternative derivation of Kerr solution.

Theory of minimum spanning trees. II. Exact graphical methods and perturbation expansion at the percolation threshold

Continuing the program begun by the authors in a previous paper, we develop an exact low-density expansion for the random minimum spanning tree (MST) on a finite graph and use it to develop a continuum perturbation expansion for the MST on critical percolation clusters in space dimension d . The perturbation expansion is proved to be renormalizable in d=6 dimensions. We consider the fractal dimension D(p) of paths on the latter MST; our previous results lead us to predict that D(p)=2 for d>d(c)=6 . Using a renormalization-group approach, we confirm the result for d>6 and calculate D(p) to first order in epsilon=6-d for d<6 using the connection with critical percolation, with the result D(p)=2-epsilon/7+O(epsilon(2)) .

Uncertainty analysis for optimum plane extraction from noisy 3D range-sensor point-clouds

We utilize a more accurate range noise model for 3D sensors to derive from scratch the expressions for the optimum plane fitting a set of noisy points and for the combined covariance matrix of the plane’s parameters, viz. its normal and its distance to the origin. The range error model used by us is a quadratic function of the true range and also the incidence angle. Closed-form expressions for the Cramer–Rao uncertainty bound are derived and utilized for analyzing four methods of covariance computation: exact maximum likelihood, renormalization, approximate least-squares, and eigenvector perturbation. The effect of the simplifying assumptions inherent in these methods are compared with respect to accuracy, speed, and ease of interpretation of terms. The approximate least-squares covariance matrix is shown to possess a number of desirable properties, e.g., the optimal solution forms its null-space and its components are functions of easily understood terms like the planar-patch’s weighted centroid and scatter. It is also fast to compute and accurate enough in practice. Its experimental application to real-time range-image registration and plane fusion is shown by using a commercially available 3D range sensor.

The Uncanny Precision of the Spectral Action

Noncommutative geometry has been slowly emerging as a new paradigm of geometry which starts from quantum mechanics. One of its key features is that the new geometry is spectral in agreement with the physical way of measuring distances. In this paper we present a detailed introduction with an overview on the study of the quantum nature of space-time using the tools of noncommutative geometry. In particular we examine the suitability of using the spectral action as action functional for the theory. To demonstrate how the spectral action encodes the dynamics of gravity we examine the accuracy of the approximation of the spectral action by its asymptotic expansion in the case of the round three sphere. We find that the two terms corresponding to the cosmological constant and the scalar curvature term already give the full result with remarkable accuracy. This is then applied to the physically relevant case of the product of the three sphere by a circle where we show that the spectral action in this case is also given, for any test function, by the sum of two terms up to an astronomically small correction, and in particular all higher order terms vanish. This result is confirmed by evaluating the spectral action using the heat kernel expansion where we check that the higher order terms a4 and a6 both vanish due to remarkable cancelations. We also show that the Higgs potential appears as an exact perturbation when the test function used is a smooth cutoff function.

Immune response deviation and enhanced expression of chemokine receptor CCR4 in TBI patients due to unknown serum factors

Background Severe brain trauma leads to an activation of the immune system. To this date, neither the exact perturbation of the specific immune reaction induced by the traumatic brain injury (TBI), nor the interactions leading to the infiltration of peripheral immune cells into the brain are fully understood. Patients and methods Serum was collected from 17 patients with TBI and a long bone fracture, 24 patients with an isolated long bone fracture and from healthy individuals. The effect of the serum on normal human monocytes and T-lymphocytes was tested in vitro by assessing proliferation and expression of surface markers, chemokine receptors and cytokines. Results Serum collected from patients with a TBI and a long bone fracture increased the expression of the chemokine receptor CCR4 in monocytes when compared to patients with an isolated long bone fracture. Extending this comparison to T-lymphocytes, the serum from TBI patients induced lower proliferation rates and decreased expression of the pro-inflammatory cytokine TNF-α, while simultaneously increasing the secretion of immune-modulatory cytokines (IL-4, IL-10 and TGF-β) ( p < 0.05). Conclusion Patients with a TBI release currently unknown soluble factors into the circulating blood that up regulate expression of chemokine receptor CCR4 in peripheral blood monocytes whilst concurrently inducing expression of immunosuppressive cytokines by activated T-lymphocytes.

Study on Characteristics of Void Reactivity Coefficients for High-Conversion-Type Core of FLWR for MA Recycling

The investigation on the characteristics of void reactivity coefficients for the high-conversion-type core of the FLWR (HC-FLWR) concept for MA recycling has been performed. Void reactivity coefficients are the major restrictions for the core design of HC-FLWR for MA recycling, because the loaded MA makes void reactivity coefficients worse. Therefore, it is important to investigate the characteristics of void reactivity coefficients as a mechanism of reactor physics. Thus, in this study, the investigation of void reactivity coefficients has been performed using the exact perturbation calculations. In the exact perturbation theory, the reactivity is related to the variation in the cross section, and divided into scattering, leakage, absorption, and fission terms. Then, the worsening of the void reactivity coefficient caused by the MA loading mainly via the scattering term is found. Moreover, the void reactivity coefficient becomes better via the scattering term for the smaller fuel rod diameter, and via the leakage term for the lower core height. In addition, the 100% void reactivity coefficient, which is the restriction for the core design of HCFLWR for MA recycling, cannot be negative only by using the effect of the scattering term by reducing the fuel rod diameter. Therefore, the mechanism of achieving the negative 100% void reactivity coefficient by using the effect of the leakage term through the core height reduction is quantitatively verified.

Three types of superpotentials for perturbations in the Einstein–Gauss–Bonnet gravity

Superpotentials (antisymmetric tensor densities) in the Einstein–Gauss–Bonnet (EGB) gravity for arbitrary types of perturbations on arbitrary curved backgrounds are constructed. As a basis, the generalized conservation laws in the framework of an arbitrary D-dimensional metric theory, where conserved currents are expressed through divergences of superpotentials, are used. Such a derivation is exact (perturbations are not infinitesimal) and is approached when a solution (dynamical) is considered as a perturbed system with respect to another solution (background). Three known prescriptions are elaborated: they are the canonical Nœther theorem, the Belinfante symmetrization rule and the field-theoretical derivation. All three approaches are presented in a unique way convenient for comparisons and development. Exact expressions for the 01-component of the three types of the superpotentials are derived in the case when an arbitrary static Schwarzschild-like solution in the EGB gravity is considered as a perturbed system with respect to a background of the same type. These formulae are used for calculating the mass of the Schwarzschild–anti-de Sitter black hole in the EGB gravity. As a background, both the anti-de Sitter spacetime in arbitrary dimensions and a ‘mass gap’ vacuum, which has no maximal set of symmetries, in five dimensions are considered. Problems and perspectives for future development, including the Lovelock gravity, are discussed.

Quasilocal variables in spherical symmetry: Numerical applications to dark matter and dark energy sources

A numerical approach is considered for spherically symmetric spacetimes that generalize Lemaitre-Tolman-Bondi dust solutions to nonzero pressure ("LTB spacetimes"). We introduce quasi-local (QL) variables that are covariant LTB objects satisfying evolution equations of Friedman-Lemaitre-Robertson-Walker (FLRW) cosmologies. We prove rigorously that relative deviations of the local covariant scalars from the QL scalars are non-linear, gauge invariant and covariant perturbations on a FLRW formal "background" given by the QL scalars. The dynamics of LTB spacetimes is completely determined by the QL scalars and these exact perturbations. Since LTB spacetimes are compatible with a wide variety of "equations of state", either single fluids or mixtures, a large number of known solutions with dark matter and dark energy sources in a FLRW framework (or with linear perturbations) can be readily examined under idealized but non-trivial inhomogeneous conditions. Coordinate choices and initial conditions are derived for a numerical treatment of the perturbation equations, allowing us to study non-linear effects in a variety of phenomena, such as gravitational collapse, non-local effects, void formation, dark matter and dark energy couplings and particle creation. In particular, the embedding of inhomogeneous regions can be performed by a smooth matching with a suitable FLRW solution, thus generalizing the Newtonian "top hat" models that are widely used in astrophysical literature. As examples of the application of the formalism, we examine numerically the formation of a black hole in an expanding Chaplygin gas FLRW universe, as well as the evolution of density clumps and voids in an interactive mixture of cold dark matter and dark energy.

Study on Characteristics of Void Reactivity Coefficients for High-Conversion-Type Core of FLWR for MA Recycling

The investigation on the characteristics of void reactivity coefficients for the high-conversion-type core of the FLWR (HC-FLWR) concept for MA recycling has been performed. Void reactivity coefficients are the major restrictions for the core design of HC-FLWR for MA recycling, because the loaded MA makes void reactivity coefficients worse. Therefore, it is important to investigate the characteristics of void reactivity coefficients as a mechanism of reactor physics. Thus, in this study, the investigation of void reactivity coefficients has been performed using the exact perturbation calculations. In the exact perturbation theory, the reactivity is related to the variation in the cross section, and divided into scattering, leakage, absorption, and fission terms. Then, the worsening of the void reactivity coefficient caused by the MA loading mainly via the scattering term is found. Moreover, the void reactivity coefficient becomes better via the scattering term for the smaller fuel rod diameter, and via the leakage term for the lower core height. In addition, the 100% void reactivity coefficient, which is the restriction for the core design of HCFLWR for MA recycling, cannot be negative only by using the effect of the scattering term by reducing the fuel rod diameter. Therefore, the mechanism of achieving the negative 100% void reactivity coefficient by using the effect of the leakage term through the core height reduction is quantitatively verified.

Exact asymptotic analysis for metapopulation dynamics on correlated dynamic landscapes

We compute the mean patch occupancy for a stochastic, spatially explicit patch-occupancy metapopulation model on a dynamic, correlated landscape, using a mathematically exact perturbation expansion about a mean-field limit that applies when dispersal range is large. Stochasticity in the metapopulation and landscape dynamics gives negative contributions to patch occupancy, the former being more important at high occupancy and the latter at low occupancy. Positive landscape correlations always benefit the metapopulation, but are only significant when the correlation length is comparable to, or smaller than, the dispersal range. Our analytical results allow us to consider the importance of spatial kernels in all generality. We find that the shape of the landscape correlation function is typically unimportant, and that the variance is overwhelmingly the most important property of the colonisation kernel. However, short-range singularities in either the colonisation kernel or landscape correlations can give rise to qualitatively different behaviour.

Spin dynamics in a double quantum dot: Exact diagonalization study

We study spin dynamics and singlet-triplet decoherence due to the hyperfine interaction in a parabolic double quantum dot. We use exact diagonalization and perturbation theory for calculating the time evolution of the wave function. The probabilities for the singlet state exhibit damped oscillations and converge toward a value, which depends only on the strength of the hyperfine field and the exchange energy. We derive expressions for the asymptotic singlet probability and its variance using a $2\ifmmode\times\else\texttimes\fi{}2$ effective Hamiltonian matrix. The asymptotic values and variances for singlet probabilities are in agreement with previous experimental data.

On crustal corrections in surface wave tomography

SUMMARY Mantle models from surface waves rely on good crustal corrections. We investigated how far ray theoretical and finite frequency approximations can predict crustal corrections for fundamental mode surface waves. Using a spectral element method, we calculated synthetic seismograms in transversely isotropic PREM and in the 3-D crustal model Crust2.0 on top of PREM, and measured the corresponding time-shifts as a function of period. We then applied phase corrections to the PREM seismograms using ray theory and finite frequency theory with exact local phase velocity perturbations from Crust2.0 and looked at the residual time-shifts. After crustal corrections, residuals fall within the uncertainty of measured phase velocities for periods longer than 60 and 80 s for Rayleigh and Love waves, respectively. Rayleigh and Love waves are affected in a highly non-linear way by the crustal type. Oceanic crust affects Love waves stronger, while Rayleigh waves change most in continental crust. As a consequence, we find that the imperfect crustal corrections could have a large impact on our inferences of radial anisotropy. If we want to map anisotropy correctly, we should invert simultaneously for mantle and crust. The latter can only be achieved by using perturbation theory from a good 3-D starting model, or implementing full non-linearity from a 1-D starting model.

Dynamical Mean Field within Iterative Perturbation Theory

The problem of strong correlations belongs to the central problem of the condensed matter theory and has intensively been studied by many sophisticated methods. One of them is the dynamical mean field theory (DMFT) [1], which was successfully used to study, e.g. the metal–insulator (Mott) [2] transitions in strongly correlated electron systems. The simplest but nontrivial model that is used to study electron correlations is the Hubbard model, which in the limit of the infinite dimensions, is mapped onto a single impurity Anderson model. However, the Anderson-like models have to be solved together with appropriate self-consistent equations. In the limit of infinite dimension all spatial degrees of freedom are frozen and the self-energy is local (independent of momentum). To solve the mapped single impurity model, one can adopt several computational schemes such as the quantum Monte Carlo method, the non-crossing approximation, the exact diagonalization method or iterative perturbation theory (IPT). The last scheme (i.e. IPT), introduced for the Hubbard model at half-filling [3, 4] and in [5] for arbitrary doping, allows us to solve the single impurity Anderson model in a very efficient and relatively simple way. The objective of this paper is to discuss the numerical implementations of IPT for orbitally degenerated models.

Electromagnetoelastic behaviors of functionally graded piezoelectric solid cylinder and sphere

Analytical studies on electromagnetoelastic behaviors are presented for the functionally graded piezoelectric material (FGPM) solid cylinder and sphere placed in a uniform magnetic field and subjected to the external pressure and electric loading. When the mechanical, electric and magnetic properties of the material obey an identical power law in the radial direction, the exact displacements, stresses, electric potentials and perturbations of magnetic field vector in the FGPM solid cylinder and sphere are obtained by using the infinitesimal theory of electromagnetoelasticity. Numerical examples also show the significant influence of material inhomogeneity. It is interesting to note that selecting a specific value of inhomogeneity parameter β can optimize the electromagnetoelastic responses, which will be of particular importance in modern engineering designs.

Verification of Homogenization in Fast Critical Assembly Analyses

In the present paper, homogenization procedures for fast critical assembly analyses are investigated. Errors caused by homogenizations are evaluated by the exact perturbation theory. In order to obtain reference solutions, three-dimensional plate-wise transport calculations are performed. It is found that the angular neutron flux along plate boundaries has a significant peak in the fission source energy range. To treat this angular dependence accurately, the double-Gaussian Chebyshev angular quadrature set with S24 is applied. It is shown that the difference between the heterogeneous leakage theory and the homogeneous theory is negligible, and that transport cross sections homogenized with neutron flux significantly underestimate neutron leakage. The error in criticality caused by a homogenization is estimated at about 0.1%Δ/k/kk' in a small fast critical assembly. In addition, the neutron leakage is overestimated by both leakage theories when sodium plates in fuel lattices are voided.

On the equilibrium of a charged massive particle in the field of a Reissner–Nordström black hole

The multiyear problem of a two-body system consisting of a Reissner–Nordström black hole and a charged massive particle at rest is here solved by an exact perturbative solution of the full Einstein–Maxwell system of equations. The expressions of the metric and of the electromagnetic field, including the effects of the electromagnetically induced gravitational perturbation and of the gravitationally induced electromagnetic perturbation, are presented in closed analytic formulas.