Improved asymptotic predictions for the effective slip over a corrugated topography

Accurate analytical prediction of the effective slip boundary condition in shear-driven Stokes flows directed longitudinally and transversely to a one-dimensional sinusoidal no-slip topography is explored. First, the domain perturbation technique is extended through spectral analysis and symbolic computations to obtain polynomial approximations (Taylor polynomials) of arbitrary specifiable order for the effective slip length. However, when assessed for numerical accuracy against fully-resolved numerical simulations using the finite-element-method, higher order Taylor polynomials give progressively inferior predictions in comparison to lower-order ones, unless the product of amplitude and wave-number is restricted below unity. From Domb–Sykes plots, the reason for the poor accuracy of higher order Taylor polynomials is assessed to be the finite convergence radii, approximately equaling unity, of the asymptotic power series for both longitudinal and transverse flows. For either of the flows, application of Euler transformation to the expansion parameter provide polynomial-form approximations that are accurate for amplitude values exceeding the convergence radius. The slow convergence of the Euler-transformed series can be remedied through Shanks transformation, at the cost of losing the convenience of closed forms. Finally, Padé approximants are shown to provide even more accurate but still closed-form alternatives to polynomials that work accurately at amplitudes much exceeding the above-identified convergence radii.     

Surrogate optimization of deep neural networks for groundwater predictions

Sustainable management of groundwater resources under changing climatic conditions require an application of reliable and accurate predictions of groundwater levels. Mechanistic multi-scale, multi-physics simulation models are often too hard to use for this purpose, especially for groundwater managers who do not have access to the complex compute resources and data. Therefore, we analyzed the applicability and performance of four modern deep learning computational models for predictions of groundwater levels. We compare three methods for optimizing the models’ hyperparameters, including two surrogate model-based algorithms and a random sampling method. The models were tested using predictions of the groundwater level in Butte County, California, USA, taking into account the temporal variability of streamflow, precipitation, and ambient temperature. Our numerical study shows that the optimization of the hyperparameters can lead to reasonably accurate performance of all models (root mean squared errors of groundwater predictions of 2 meters or less), but the “simplest” network, namely a multilayer perceptron (MLP) performs overall better for learning and predicting groundwater data than the more advanced long short-term memory or convolutional neural networks in terms of prediction accuracy and time-to-solution, making the MLP a suitable candidate for groundwater prediction.

     

Newtonian and special-relativistic predictions for the trajectory of a slow-moving dissipative dynamical system

We show that the trajectories predicted by Newtonian mechanics and special relativistic mechanics from the same parameters and initial conditions for a slow-moving dissipative dynamical system will rapidly disagree completely if the trajectories are chaotic or transiently chaotic. There is no breakdown of agreement if the trajectories are non-chaotic, in contrast to the slow breakdown of agreement between non-chaotic Newtonian and relativistic trajectories for a slow-moving non-dissipative dynamical system studied previously. We argue that, once the two trajectory predictions are completely different for a slow-moving dissipative dynamical system, special relativistic mechanics must be used, instead of the standard practice of using Newtonian mechanics, to correctly study its trajectory.     

Virtual test facilities for building acoustic predictions using FEM

The most important characteristic in building acoustics is the transmission loss which describes the insulation performance of a separating component. This quantity is defined by the ratio of inclined sound power to transmitted power. The transmission loss is usually measured in costly experimental set-ups of either real size facilities or down scaled test models. In order to reduce the effort and to gain further insight into the complex behavior of wave propagation in coupled structure-fluid systems, virtual test facilities using numerical methods (FEM) are established. An appropriate determination of the reverberation time is essential for obtaining good quality simulations. The results of the simulated transmission loss are presented and compared with common analytical prediction methods based on the transfer matrix method. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Regularity for the evolution of p-harmonic maps

This paper presents our study of regularity for p-harmonic map heat flows. We devise a monotonicity-type formula of scaled energy and establish a criterion for a uniform regularity estimate for regular p-harmonic map heat flows. As application we show the small data global in the time existence of regular p-harmonic map heat flow.     

Unbounded perturbation of the controllability for evolution equations

In this paper we prove that the controllability for evolution equations in Banach spaces is not destroyed, if we perturb the equation by “small” unbounded linear operator. This is done by employing a perturbation principle from linear operator theory and a characterization of surjective operators in Banach spaces. Finally, we apply these to a control system governed by partial integro-differential equations.     

Unresolved computation and optimal predictions

We present methods for predicting the solution of time‐dependent partial differential equations when that solution is so complex that it cannot be properly resolved numerically, but when prior statistical information can be found. The sparse numerical data are viewed as constraints on the solution, and the gist of our proposal is a set of methods for advancing the constraints in time so that regression methods can be used to reconstruct the mean future. For linear equations we offer general recipes for advancing the constraints; the methods are generalized to certain classes of nonlinear problems, and the conditions under which strongly nonlinear problems and partial statistical information can be handled are briefly discussed. Our methods are related to certain data acquisition schemes in oceanography and meteorology. © John Wiley & Sons, Inc.     

Fractal analysis for studying the evolution of forests

Deforestation is an important phenomenon that may create major imbalances in ecosystems. In this study we propose a new mathematical analysis of the forest area dynamic, enabling qualitative as well as quantitative statements and results. Fractal dimensions of the area and the perimeter of a forest were determined using digital images. The difference between fractal dimensions of the area and the perimeter images turned out to be a crucial quantitative parameter. Accordingly, we propose a new fractal fragmentation index, FFI, which is based on this difference and which highlights the degree of compaction or non-compaction of the forest area in order to interpret geographic features. Particularly, this method was applied to forests, where large areas have been legally or illegally deforested. However, these methods can easily be used for other ecological or geographical investigations based on digital images, including deforestation of rainforests.     

On the evolution equation for magnetic geodesics

Orbits of charged particles under the effect of a magnetic field are mathematically described by magnetic geodesics. They appear as solutions to a system of (nonlinear) ordinary differential equations of second order. But we are only interested in periodic solutions. To this end, we study the corresponding system of (nonlinear) parabolic equations for closed magnetic geodesics and, as a main result, eventually prove the existence of long time solutions. As generalization one can consider a system of elliptic nonlinear partial differential equations (PDEs) whose solutions describe the orbits of closed p-branes under the effect of a “generalized physical force”. For the corresponding evolution equation, which is a system of parabolic nonlinear PDEs associated to the elliptic PDE, we can establish existence of short time solutions.     

On the Rolewicz theorem for evolution operators

We give a short proof of a generalization of the Rolewicz theorem based on the uniform boundedness principle.

     

Models for network evolution

This paper describes mathematical models for network evolution when ties (edges) are directed and the node set is fixed. Each of these models implies a specific type of departure from the standard null binomial model. We provide statistical tests that, in keeping with these models, are sensitive to particular types of departures from the null. Each model (and associated test) discussed follows directly from one or more socio‐cognitive theories about how individuals alter the colleagues with whom they are likely to interact. The models include triad completion models, degree variance models, polarization and balkanization models, the Holland‐Leinhardt models, metric models, and the constructural model. We find that many of these models, in their basic form, tend asymptotically towards an equilibrium distribution centered at the completely connected network (i.e., all individuals are equally likely to interact with all other individuals); a fact that can inhibit the development of satisfactory tests.     

Describing critical statistical literacy habits of mind

Given the vast number of data representations that people encounter daily, it is imperative that people become critical consumers of the representations they will face. This requires the development of particular habits of mind. The purpose of this paper is to share the critical statistical literacy habits of mind framework. We articulate how we drew on the literature related to statistical literacy, critical mathematics, and critical statistical literacy to identify the habits of mind needed to enact critical statistical literacy in the context of consuming data representations. We describe the refinement process using qualitative interview data. To illustrate what each of the critical statistical literacy habits of mind looks like when enacted, we share examples from statistics teachers making sense of a data representation.     

Representations for the solutions and coefficients of evolution equations

We give new representations for the solutions and coefficients of evolution equations in the linear case. The obtained formulas contain some functional arbitrariness that can be used in identification problems. We also give classes of hyperbolic equations that admit the generalized functionally invariant solutions.     

A transport model for the evolution of urban systems

When comparing an urban system to an elasto-plastic lattice, an analogy to the solid state of matter can be exploited using the concepts of the band theory similarly. Thereafter, the population dynamics – in a region of certain stability in the state space and within appropriate energy bands – can be described in terms of Cellular Automata, with two mobile agents or pseudo particles: the inhabitant (representative of an average individual) and the recurson (representative of its multidimensional resources). As in the solid state, transition rules take the form of two coupled transport equations, comprising the terms equivalent to the generation-recombination and circulation processes. The first process can be compared to a predator–prey growth model, typical of Ecology; whereas the circulation process – composed of a drift component and a diffusion component – should be compared to the concentration-sprawl demographic balance seen in urban occupation and dynamics. Thus, it needs to be defined and determined an urban potential function, an equivalent population charge, mobility and diffusion parameters, as well as net growth factors. This analogy, discussed within the context of a case study for Great Mendoza, plausibly explains the varied growth rates of the political departments, as well as the principal urban trends for spatial occupation.     

Resolvent family for the evolution process with memory

In this paper, we investigate a class of the linear evolution process with memory in Banach space by a different approach. Suppose that the linear evolution process is well posed, we introduce a family pair of bounded linear operators, $\{(G(t),F(t)),t\ge 0\}$, that is, called the resolvent family for the linear evolution process with memory, the $F(t)$ is called the memory effect family. In this paper, we prove that the families $G(t)$ and $F(t)$ are exponentially bounded, and the family $(G(t),F(t))$ associate with an operator pair $(A,L)$ that is called generator of the resolvent family. Using $(A,L)$, we derive associated differential equation with memory and representation of $F(t)$ via L. These results give necessary conditions of the well-posed linear evolution process with memory. To apply the resolvent family to differential equation with memory, we present a generation theorem of the resolvent family under some restrictions on $(A,L)$. The obtained results can be directly applied to linear delay differential equation, integro-differential equation and functional differential equations.     

On the evolution operator for abstract parabolic equations

We find a new construction of the evolution operatorG(t, s) associated to a family {A(t), 0≦t≦T} of generators of analytic semigroups in a Banach spaceX. We study the dependence ofG (t, s) ont ands, and we give regularity results for the solution of the i.v.p.u′(t)=A(t)u(t)+f(t),u(0)=x.