The derivative expansion in asymptotically safe quantum gravity: general setup and quartic order

We present a general framework to systematically study the derivative expansion of asymptotically safe quantum gravity. It is based on an exact decoupling and cancellation of different modes in the Landau limit, and implements a correct mode count as well as a regularisation based on geometrical considerations. It is applicable independent of the truncation order. To illustrate the power of the framework, we discuss the quartic order of the derivative expansion and its fixed point structure as well as physical implications.

On the accuracy and applicability of a new implicit Taylor method and the high-order spectral method on steady nonlinear waves

This paper presents an investigation and discussion of the accuracy and applicability of an implicit Taylor (IT) method versus the classical higher-order spectral (HOS) method when used to simulate two-dimensional regular waves. This comparison is relevant, because the HOS method is in fact an explicit perturbation solution of the IT formulation. First, we consider the Dirichlet–Neumann problem of determining the vertical velocity at the free surface given the surface elevation and the surface potential. For this problem, we conclude that the IT method is significantly more accurate than the HOS method when using the same truncation order, M , and spatial resolution, N , and is capable of dealing with steeper waves than the HOS method. Second, we focus on the problem of integrating the two methods in time. In this connection, it turns out that the IT method is less robust than the HOS method for similar truncation orders. We conclude that the IT method should be restricted to M = 4, while the HOS method can be used with M ≤ 8. We systematically compare these two options and finally establish the best achievable accuracy of the two methods as a function of the wave steepness and the water depth.

COMPARISON BETWEEN SINGLE AND DOUBLE INTEGRAL TRANSFORMATION SOLUTIONS OF HEAT CONDUCTION IN SOLID-STATE ELECTRONICS

The design of modern electronic devices has been dealing with challenges on thermal control. In this work, it is proposed two different ways of modeling the temperature field in Solid State Electronics (SSE) using integral transforms, with several heat generations in the domain of the microchip and external convection. Two proposed approaches solve the heat conduction formulation on the SSE using the Classical Integral Transform Technique (CITT): One performing a single transformation (CITT-ST) and the other performing a double transformation (CITT-DT). Both methodologies are compared and achieved similar results. The simpler analytical solution by CITT-DT contrasts with a complex and cumbersome analytical manipulation of CITT-ST. The results show that CITT-ST is more efficient to obtain the solution, requiring a lower truncation order, for the problem of heat conduction in Solid State Electronics even though it has a more complex formulation.

A New Radial Basis Function Approach Based on Hermite Expansion with Respect to the Shape Parameter

Owing to its high accuracy, the radial basis function (RBF) is gaining popularity in function interpolation and for solving partial differential equations (PDEs). The implementation of RBF methods is independent of the locations of the points and the dimensionality of the problems. However, the stability and accuracy of RBF methods depend significantly on the shape parameter, which is mainly affected by the basis function and the node distribution. If the shape parameter has a small value, then the RBF becomes accurate but unstable. Several approaches have been proposed in the literature to overcome the instability issue. Changing or expanding the radial basis function is one of the most commonly used approaches because it addresses the stability problem directly. However, the main issue with most of those approaches is that they require the optimization of additional parameters, such as the truncation order of the expansion, to obtain the desired accuracy. In this work, the Hermite polynomial is used to expand the RBF with respect to the shape parameter to determine a stable basis, even when the shape parameter approaches zero, and the approach does not require the optimization of any parameters. Furthermore, the Hermite polynomial properties enable the RBF to be evaluated stably even when the shape parameter equals zero. The proposed approach was benchmarked to test its reliability, and the obtained results indicate that the accuracy is independent of or weakly dependent on the shape parameter. However, the convergence depends on the order of the truncation of the expansion. Additionally, it is observed that the new approach improves accuracy and yields the accurate interpolation, derivative approximation, and PDE solution.

On Some Sufficiency-Type Global Stability Results for Time-Varying Dynamic Systems with State-Dependent Parameterizations

This paper formulates sufficiency-type global stability and asymptotic stability results for, in general, nonlinear time-varying dynamic systems with state-trajectory solution-dependent parameterizations. The stability proofs are based on obtaining sufficiency-type conditions which guarantee that either the norms of the solution trajectory or alternative interval-type integrals of the matrix of dynamics of the higher-order than linear terms do not grow faster than their available supremum on the preceding time intervals. Some extensions are also given based on the use of a truncated Taylor series expansion of chosen truncation order with multiargument integral remainder for the dynamics of the differential system.

Irregular parameter dependence of numerical results in tensor renormalization group analysis

We study the parameter dependence of numerical results obtained by the tensor renormalization group. We often observe irregular behavior as the parameters are varied with the method. Using the two-dimensional Ising model we explicitly show that the sharp cutoff used in the truncated singular value decomposition causes this unwanted behavior when the level crossing happens between singular values below and above the truncation order as the parameters are varied. We also test a smooth cutoff, instead of the sharp one, as a truncation scheme and discuss its effects.

An exact and explicit implied volatility inversion formula

In this paper, we develop an exact and explicit (model-independent) Taylor series representation of the implied volatility based on the novel applications of an extended Faà di Bruno formula under the operator calculus setting, and the Lagrange inversion theorem. We rigorously establish that our formula converges to the true implied volatility as the truncation order increases. Numerical examples illustrate the remarkable accuracy and efficiency of the formula. The formula distinguishes from previous literature as it converges to the true exact implied volatility, is a closed-form formula whose coefficients are explicitly determined and do not involve numerical iterations.

Explicit High Accuracy Maximum Resolution Dispersion Relation Preserving Schemes for Computational Aeroacoustics

A set of explicit finite difference schemes with large stencil was optimized to obtain maximum resolution characteristics for various spatial truncation orders. The effect of integral interval range of the objective function on the optimized schemes’ performance is discussed. An algorithm is developed for the automatic determination of this integral interval. Three types of objective functions in the optimization procedure are compared in detail, which show that Tam’s objective function gets the best resolution in explicit centered finite difference scheme. Actual performances of the proposed optimized schemes are demonstrated by numerical simulation of three CAA benchmark problems. The effective accuracy, strengths, and weakness of these proposed schemes are then discussed. At the end, general conclusion on how to choose optimization objective function and optimization ranges is drawn. The results provide clear understanding of the relative effective accuracy of the various truncation orders, especially the trade-off when using large stencil with a relatively high truncation order.

Effect of Mesh Size and Mode Truncation on Reconstruction of Force Characteristics for Non Punctual Impacts on Composite Elastic Beams

dentifying characteristics of a force generated by non punctual impact is performed to better monitor the health of the impacted structure. This can be achieved through using an implemented structural model. For composite beams, the model can be constructed by means of the finite element method. In this work, the impact is assumed to be a uniform distributed pressure and the impact location is known. Reconstructing the force signal is performed by using regularized deconvolution techniques of the Toeplitz like equation giving the answer in terms of strains as function of the input force. Here, the generalized singular value decomposition based method is used in conjunction with truncation filtering. Quality of the reconstructed force is discussed as function of the mesh size and the mode truncation order.