Problem Of Initial Conditions Research Articles

Behaviour of the Serre equations in the presence of steep gradients revisited

We use numerical methods to study the short term behaviour of the Serre equations in the presence of steep gradients because there are no known analytical solutions for these problems. In keeping with the literature we study a class of initial condition problems that are a smooth approximation to the initial conditions of the dam-break problem. This class of initial condition problems allow us to observe the behaviour of the Serre equations with varying steepness of the initial conditions. The numerical solutions of the Serre equations are justified by demonstrating that as the resolution increases they converge to a solution with little error in conservation of mass, momentum and energy independent of the numerical method. We observe and justify four different structures of the converged numerical solutions depending on the steepness of the initial conditions. Two of these structures were observed in the literature, with the other two not being commonly found in the literature. The numerical solutions are then used to assess how well the analytical solution of the shallow water wave equations captures the mean behaviour of the solution of the Serre equations for the dam-break problem in the short term. Lastly the numerical solutions are compared to asymptotic results in the literature to approximate the depth and location of the front of an undular bore.

The initial condition problem with complete history dependency in learning models for travel choices

Learning-based models that capture travelers’ day-to-day learning processes in repeated travel choices could benefit from ubiquitous sensors such as smartphones, which provide individual-level longitudinal data to help validate and improve such models. However, the common problem of missing initial observations in longitudinal data collection can lead to inconsistent estimates of perceived value of attributes in question, and thus inconsistent parameter estimates. In this paper, the stated problem is addressed by treating the missing observations as latent variables in an instance-based learning model that is estimated via maximum simulated likelihood (MSL). The MSL method is implemented in practice using random sampling and importance sampling. Monte Carlo experimentation based on synthetic data shows that both the MSL with random sampling (MSLrs) and MSL with importance sampling (MSLis) are effective in correcting for the endogeneity problem in that the percent error and empirical coverage of the estimators are greatly improved after the correction. Compared to the MSLrs method, the MSLis method is superior in both effectiveness and computational efficiency. Furthermore, MSLis passes a formal statistical test for the recovery of the population values up to a scale with a large number of missing observations, while MSLrs systematically fails due to the curse of dimensionality. The impacts of sampling size in MSLrs and number of high probability choice sequences in MSLis on the methods’ performances are investigated the methods are applied to an experimental route-choice dataset to demonstrate their empirical application. Hausman–McFadden tests show that the estimators after correction are statistically equal to the estimators of the full dataset without missing observations, confirming that the proposed methods are practical and effective for addressing the stated problem.

An a posteriori truncation regularization method for an ill-posed problem for the heat equation with an integral boundary condition

Abstract The aim of this paper is to investigate the problem of control by the initial conditions of the heat equation with an integral boundary condition. Using the truncation method with an a posteriori parameter choice rule, we give the error estimate between the exact and the regularized solutions. A numerical implementation shows the efficiency of the proposed method.

Darkness without dark matter and energy – generalized unimodular gravity

We suggest a Lorentz non-invariant generalization of the unimodular gravity theory, which is classically equivalent to general relativity with a locally inert (devoid of local degrees of freedom) perfect fluid having an equation of state with a constant parameter w. For the range of w near −1 this dark fluid can play the role of dark energy, while for w=0 this dark dust admits spatial inhomogeneities and can be interpreted as dark matter. We discuss possible implications of this model in the cosmological initial conditions problem. In particular, this is the extension of known microcanonical density matrix predictions for the initial quantum state of the closed cosmology to the case of spatially open Universe, based on the imitation of the spatial curvature by the dark fluid density. We also briefly discuss quantization of this model necessarily involving the method of gauge systems with reducible constraints and the effect of this method on the treatment of recently! suggested mechanism of vacuum energy sequestering.

Which Derivative?

The actual state of interplay between Fractional Calculus, Signal Processing, and Applied Sciences is discussed in this paper. A framework for compatible integer and fractional derivatives/integrals in signals and systems context is described. It is shown how suitable fractional formulations are really extensions of the integer order definitions currently used in Signal Processing. The particular case of fractional linear systems is considered and the problem of initial conditions is tackled.

2D ceramic grains images manipulations: A simple geometrical characterization and grain domain recreation algorithm

The precision and accuracy of certain simulations in nanoscience, fluid dynamics and biotechnology in the analyses of boundary conditions problems with real experimental results are in general related to the characteristics of numerical approach used and subsequently to the morphological structure of lattices used along these calculations. The more the lattice used approximates to the original boundary or initial condition problem more precise the simulation would be. This work shows a simple algorithm that can be used to build huge lattices containing the main geometrical structures statistically similar to experimental 2D image data of ceramic grains by using some freeware software.

Initial conditions for inflation

Within the α-attractors framework we investigate scalar potentials with the same pole as the one featured in the kinetic term. We show that, in field space, this leads to directions without a plateau. Using this, we present a proposal, which manages to overcome the initial conditions problem of inflation with a plateau. An earlier period of proto-inflation, beginning at Planck scale, accounts for the Universe expansion and arranges the required initial conditions for inflation on the plateau to commence. We show that, if proto-inflation is power-law, it does not suffer from a sub-Planckian eternal inflationary stage, which would otherwise be a problem. A simple model realisation is constructed in the context of α-attractors, which can both generate the inflationary plateau and the exponential slopes around it, necessary for the two inflation stages. Our mechanism allows to assume chaotic initial conditions at the Planck scale for proto-inflation, it is generic and it is shown to work without fine-tuning.

Analytical Design of Consecutive Compensator for Control Systems With Delay Based on Modification of Typical Polynomial Models

The purpose of this paper is to develop an algorithm for analytical design of consecutive compensator for the control system with delay based on typical polynomial dynamical models modification. A formula relating to the characteristic frequency and the cutoff frequency of a transfer function of open loop of the desired polynomial dynamic model was derived. Using this formula a modification of polynomial models was made taking into account a value of delay element of a plant. Control of the plant with delay using the consecutive compensator has several advantages: it requires a minimum amount of measuring data and eliminates the need to introduce observer; there is no problem of non-zero initial conditions, which may arise during short-term disruption of a normal functioning of the system; a simple construction of consecutive compensator procedures for SISO and MIMO systems.

Neural AILC for Error Tracking Against Arbitrary Initial Shifts.

This paper concerns with the adaptive iterative learning control using neural networks for systems performing repetitive tasks over a finite time interval. Two standing issues of such iterative learning control processes are addressed: one is the initial condition problem and the other is that related to the approximation error. Instead of the state tracking, an error tracking approach is proposed to tackle the problem arising from arbitrary initial shifts. The desired error trajectory is prespecified at the design stage, suitable to different tracking tasks. The initial value of the desired error trajectory for each cycle is required to be the same as that of the actual error trajectory. It is just a requirement for the initial value of the desired error trajectory, but does not pose any requirement for the initial value of the actual error trajectory. It is shown that the actual error trajectory is adjustable and is able to converge to a prespecified neighborhood of the origin, while all variables of the closed-loop system are of uniform boundedness. The robustness improvement in case of nonzero approximation error is made possible due to the use of a deadzone modified Lyapunov functional. The resultant estimation for the bound of the approximation error avoids deterioration in tracking performance. The effectiveness of the designed learning controller is validated through an illustrative example.

Asymmetric dark matter and baryogenesis from pseudoscalar inflation

We show that both the baryon asymmetry of the Universe and the dark matter abundance can be explained within a single framework that makes use of maximally helical hypermagnetic fields produced during pseudoscalar inflation and the chiral anomaly in the Standard Model. We consider a minimal asymmetric dark matter model free from anomalies and constraints. We find that the observed baryon and the dark matter abundances are achieved for a wide range of inflationary parameters, and the dark matter mass ranges between 7–15 GeV . The novelty of our mechanism stems from the fact that the same source of CP violation occurring during inflation explains both baryonic and dark matter in the Universe with two inflationary parameters, hence addressing all the initial condition problems in an economical way.

The initial condition problem with complete history dependency in learning models for travel choices

Abstract: Learning-based models that capture travelers’ day-to-day learning processes in repeated travel choices could benefit from ubiquitous sensors such as smartphones, which provide individual-level longitudinal data to help validate and improve such models. However, the common problem of missing initial observations in longitudinal data collection can lead to inconsistent estimates of perceived value of attributes in question, and thus inconsistent parameter estimates. In this paper, the stated problem is addressed by treating the missing observations as latent variables. The proposed method is implemented in practice as maximum simulated likelihood (MSL) correction with two sampling methods in an instance-based learning model for travel choice, and the finite sample bias and efficiency of the estimators are investigated. Monte Carlo experimentation based on synthetic data shows that both the MSL with random sampling (MSLrs) and MSL with importance sampling (MSLis) are effective in correcting for the endogeneity problem in that the percent error and empirical coverage of the estimators are greatly improved after correction. Compared to the MSLrs method, the MSLis method is superior in both effectiveness and computational efficiency. Furthermore, MSLis passes a formal statistical test for the recovery of the population values up to a scale with a large number of missing observations, while MSLrs systematically fails due to the curse of dimensionality. The impacts of sampling size in MSLrs and number of high probability choice sequences in MSLis on the methods’ performances are investigated. The methods are applied to an experimental route-choice dataset to demonstrate their empirical application. Hausman-McFadden tests show that the estimators after correction are statistically equal to the estimators of the full dataset without missing observations, confirming that the proposed methods are practical and effective for addressing the stated problem.

Multi-phase induced inflation in theories with non-minimal coupling to gravity

In this paper we investigate the induced inflation with two flat regions: one Starobinsky-like plateau in strong coupling regime and one shorter plateau around the saddle point of the Einstein frame potential. This multi-phase inflationary scenario can be used to solve problems of classical cosmology as well as the problem of initial conditions for inflation. The inflation at the saddle-point plateau is consistent with the data and can have arbitrarily low scale. The results can be useful in the context of the Higgs-Axion relaxation and in a certain limit they are equivalent to the α-attractors.

Numerical solutions of the GEW equation using MLS collocation method

In this paper, the generalized equal width wave (GEW) equation is solved by using moving least squares collocation (MLSC) method. To test the accuracy of the method some numerical experiments are presented. The motion of single solitary waves, the interaction of two solitary waves and the Maxwellian initial condition problems are chosen as test problems. For the single solitary wave motion whose analytical solution was known [Formula: see text], [Formula: see text] error norms and pointwise rates of convergence were calculated. Also mass, energy and momentum invariants were calculated for every test problems. Obtained numerical results are compared with some earlier works. It is seen that the method is very efficient and reliable due to obtained numerical results are very satisfactorily. Stability analysis of difference equation was done by applying the moving least squares collocation method for GEW equation.

Numerical solutions of the MRLW equation using moving least square collocation method

In this paper, the Modifed Regularized Long Wave (MRLW)equation is solved by using moving least square collocation (MLSC) method.To show the accuracy of the used method several numerical test examplesare given. The motion of single solitary waves, the interaction of two solitary waves and the Maxwellian initial condition problems are chosen as test problems. For the single solitary wave motion whose analytical solution is known L2, L1 error norms are calculated. Also mass, energy and momentum in variants are calculated for every test problem. Obtained numerical resultsare compared with some earlier works. According to the obtained results, the method is very efficient and reliable

Simple cosmological solution to the Higgs field instability problem in chaotic inflation and the formation of primordial black holes

We revisit the compatibility between the chaotic inflation, which provides a natural solution to the initial condition problem, and the metastable electroweak vacuum, which is suggested by the results of LHC and the current mass measurements of top quark and Higgs boson. It is known that the chaotic inflation poses a threat to the stability of the electroweak vacuum because it easily generates large Higgs fluctuations during inflation or preheating and triggers the catastrophic vacuum decay. In this paper, we propose a simple cosmological solution in which the vacuum is stabilized during chaotic inflation, preheating and after that. This simple solution naturally predicts the formation of primordial black holes. We find interesting parameter regions where the present dark matter density is provided by them. Also, the thermal leptogenesis can be accommodated in our scenario.

Sunk Costs, Firm Heterogeneity, Export Market Entry and Exit: Evidence from India

This paper analyses the role of sunk costs and firm heterogeneity in firm decision to enter and exit export markets. Employing rich firm-level data on Indian manufacturing firms, the study points out that sunk costs in terms of previous export experience significantly explain entry and exit decisions of firms in the export market. The first set of analysis involves estimation of dynamic discrete choice model using random effects probit correcting for initial conditions problem. We find evidence that previous export experience (sunk costs) matters for export decision. However, importance of sunk costs is found to depreciate rapidly. Further, analysis across sub-sample of firms accounting for firm heterogeneity factors like size and product level information supports the hypothesis of sunk costs. Second set of analysis involving firm survival in export markets using discrete-time hazard models shows evidence of negative duration dependence. We observe that those firms which continue to export for few years are less likely to exit from export markets.

Estimation of exact initial states of fractional order systems

The initial condition problem of fractional order systems is investigated in this paper. Firstly, a named aberration phenomenon is introduced, which reveals the nature of initial condition problem. And then, in order to interpret this odd phenomenon, definitions of the Riemann–Liouville derivative and Caputo derivative are revisited. As a result, the fractional order systems described by fractional differential equations and exact state-space models are linked much more closely, and it is found that relationship between these two kinds of models basically lies in the exact initial state distribution. The results also show the inborn defects of these two derivatives. Afterward, a practical method to estimate the exact initial states is studied naturally. At last, several simulation examples carefully illustrate the effectiveness of proposed method.

Non-linearities in euro area inflation persistence

This paper investigates the nature of inflation dynamics with a special focus on inflation persistence. Using data from euro area member-states we estimate dynamic non-linear panel models addressing in detail econometric issues concerning unobserved heterogeneity, genuine state dependence, and the initial conditions problem. After controlling for observed and unobserved heterogeneity, our results suggest that the degree of inflation persistence is genuine and varies depending on whether the inflation rate is too high, within the range of ECB's target of price stability, too low or negative. This implies that policies to stabilize inflation in the short run will have longer-run effects.

Erratum to: The initial condition problem of damped quantum harmonic oscillator

Erratum to: The initial condition problem of damped quantum harmonic oscillator

Dynamical fine-tuning of initial conditions for small field inflation

Small-field inflation (SFI) is widely considered to be unnatural because an extreme fine-tuning of the initial condition is necessary for sufficiently large e-folding. In this paper, we show that the unnaturally-looking initial condition can be dynamically realised without any fine-tuning if the SFI occurs after rapid oscillations of the inflaton field and particle creations by preheating. In fact, if the inflaton field $\phi$ is coupled to another scalar field $\chi$ through the interaction $g^2 \chi^2 \phi^2$ and the vacuum energy during the small field inflation is given by $\lambda M^4$, the initial value can be dynamically set at $(\sqrt{\lambda}/g) M^2/M_{\rm pl}$, which is much smaller than the typical scale of the potential $M.$ This solves the initial condition problem in the new inflation model or some classes of the hilltop inflation models.