Sufficient Turing instability conditions for the Schnakenberg system

A classical reaction-diffusion system, the Schnakenberg system, is under consideration in a bounded domain $\Omega\subset\mathbb{R}^m$ with Neumann boundary conditions. We study diffusion-driven instability of a stationary spatially homogeneous solution of this system, also called the Turing instability, which arises when the diffusion coefficient $d$ changes. An analytical description of the region of necessary and sufficient conditions for the Turing instability in the parameter plane is obtained by analyzing the linearized system in diffusionless and diffusion approximations. It is shown that one of the boundaries of the region of necessary conditions is an envelope of the family of curves that bound the region of sufficient conditions. Moreover, the intersection points of two consecutive curves of this family lie on a straight line whose slope depends on the eigenvalues of the Laplace operator and does not depend on the diffusion coefficient. We find an analytical expression for the critical diffusion coefficient at which the stability of the equilibrium position of the system is lost. We derive conditions under which the set of wavenumbers corresponding to neutral stability modes is countable, finite, or empty. It is shown that the semiaxis $d>1$ can be represented as a countable union of half-intervals with split points expressed in terms of the eigenvalues of the Laplace operator; each half-interval is characterized by the minimum wavenumber of loss of stability.

Counting Rational Curves on K3 Surfaces With Finite Group Actions

Göttsche gave a formula for the dimension of the cohomology of Hilbert schemes of points on a smooth projective surface $S$. When $S$ admits an action by a finite group $G$, we describe the action of $G$ on the Hodge structure. In the case that $S$ is a K3 surface, each element of $G$ gives a trace on $\sum _{n=0}^{\infty }\sum _{i=0}^{\infty }(-1)^{i}H^{i}(S^{[n]},\mathbb{C})q^{n}$. When $G$ acts faithfully and symplectically on $S$, the resulting generating function is of the form $q/f(q)$, where $f(q)$ is a cusp form. We relate the Hodge structure of Hilbert schemes of points to the Hodge structure of the compactified Jacobian of the tautological family of curves over an integral linear system on a K3 surface as $G$-representations. Finally, we give a sufficient condition for a $G$-orbit of curves with nodal singularities not to contribute to the representation.

MODELING OF THE CURING PROCESS OF A POLYMER RESIN AND CHANGES IN MICROHARDNESS IN ITS VOLUME

Curing parameters have the greatest impact on the physical and mechanical properties of FRP, therefore their optimum value is of particular importance for obtaining quality products. During curing temperature of the inner layers of the FRP can increase unevenly, which can lead to the formation of a gradient in the degree of conversion and heterogeneity of physical and mechanical properties. The article is devoted to the development of a mathematical model of the curing process of the EDT-69N resin, taking into account the kinetic parameters of curing and implementation thermophysical modeling using the finite element method. The correspondence of the family of curves for the degree of conversion along the sample cross-section and the family of microhardness curves is also shown.

Analysis of noise-reduction filter design by computer modelling

The article describes the principles of evaluating the design disadvantages of a noise-reduction filter. The characteristics of the developed equipment are influenced by many parameters, such as the inhomogeneity of the components used, the parasitic interference introduced by them, and noise induced from structural elements and external sources. At the same time, the search for and elimination of the reasons for the discrepancy between the parameters of the designed products requires more and more expenses during the development stages. That is why the problem of modeling equipment with parameters close to real ones and under conditions different from laboratory ones is urgent. To assess the influence of the above factors in the Qucs 0.0.19 program, a noise suppression filter was simulated, the introduced attenuation of which did not meet the requirements of the design documentation (this problem was identified due to the use of modern measuring equipment at the stage of serial production). A family of curves is constructed that describe the behavior of the circuit for various values of the parameters of the elements. Equivalent circuits were used to assess the influence of parasitic parameters. The effect of noise induced by structural elements was approximated using parallel connected transformers in short circuit mode. The simulation was carried out in the frequency range from 100 Hz to 1 GHz. As a result of the conducted research, it was obtained: taking into account the parasitic parameters of the elements leads to a significant change in the form of the ideal frequency response of the noise suppression filter; the value of the capacitance of the pass capacitor in the considered frequency range does not make a significant contribution to the attenuation introduced by the filter; in the low-frequency region, the magnitude of the inductance of the coils has the greatest influence on the output characteristic of the filter; in the high-frequency region, the radiating properties of structural elements begin to appear, the influence of which on the attenuation introduced by the filter is significant.

Duality of Moduli and Quasiconformal Mappings in Metric Spaces

We prove a duality relation for the moduli of the family of curves connecting two sets and the family of surfaces separating the sets, in the setting of a complete metric space equipped with a doubling measure and supporting a Poincaré inequality. Then we apply this to show that quasiconformal mappings can be characterized by the fact that they quasi-preserve the modulus of certain families of surfaces.

On the Plateau-Douglas Problem for the Willmore energy of surfaces with planar boundary curves

For a smooth closed embedded planar curve, we consider the minimization problem of the Willmore energy among immersed surfaces of a given genus greater than 1 having the given curve as boundary, without any prescription on the conormal. By general lower bound estimates, in case the curve is a circle we prove that such problem is equivalent if restricted to embedded surfaces, we prove that do not exist minimizers, and we calculate the infimum. Then we study the case in which the genus is 1 and the competitors are restricted to a suitable class of varifolds including embedded surfaces, and we prove that the non-existence of minimizers implies a lower bound on the infimum; therefore we use such criterion in order to explicitly find an infinite family of curves for which such problem does have minimizers in the corresponding class of varifolds.

Sensitivity analyses and uncertainty quantification in THM models: a benchmark study

<p>Coupled thermo-hydro-mechanical (THM) models are used for the assessment of nuclear waste disposal, reservoir engineering, and other branches of geo-environmental engineering. Model-based decision-making and design optimization in these domains require sensitivity analyses (SA) and uncertainty quantification (UQ) methods that are suitable for coupled THM problems on an engineering scale. Due to different coupling levels, non-linearities, and large spatial and temporal extents, these analyses can often be challenging both conceptually and computationally.</p><p>For an initial evaluation in a setting relevant to nuclear waste disposal we start by employing an analytical solution for thermal consolidation around a point heat source which encompasses the most relevant primary couplings and allows us to cover the entire parameter space robustly and efficiently. For uncertainty quantification, we applied an experimental design (DoE-) based history-matching approach. This approach uses DoE methods to construct a proxy model, which is used later for efficient Monte Carlo sampling and subsequent filtering of the uncertainty space of the history-match error. As a result, we obtain a family of curves that is compatible with the prior parameter set and experimental data to match, which then enables further uncertainty quantification. In our work, we demonstrate the applicability of the workflow and discuss its particular suitability to this problem class, including its (in-)sensitivity to prior parameter distribution assumptions.</p><p>For SA, we contrast the conclusions drawn via two different approaches: local one variable at a time (OVAT) and global sensitivity analysis (GSA) based on Sobol indices for different spatio-temporal settings to observe near and far-field effects as well as early- and late-stage system response. The conducted studies can serve as a benchmark for UQ and SA software designed around numerical THM simulators.</p>

On the asymptotic equivalence between the radon and the hough transforms of digital images

ABSTRACTAlthough characterized by different mathematical definitions, both the Radon and the Hough transforms ultimately take an image as input and provide, as output, functions defined on a preassigned parameter space, i.e., the so-called either Radon or Hough sinograms. The parameters in these two spaces describe a family of curves, which represent either the integration domains considered in the Radon transform, or the kind of curves to be detected by the Hough transform.It is heuristically known that the Hough sinogram converges to the corresponding Radon sinogram when the discretization step in the parameter space tends to zero. By considering generalized functions in multi-dimensional setting, in this paper we give an analytical proof of this heuristic rationale when the input grayscale digital image is described as a set of grayscale points, that is, as a sum of weighted Dirac delta functions. On these grounds, we also show that this asymptotic equivalence may have a valuable impact on the image reconstruction problem of inverting the Radon sinogram recorded by a medical imaging scanner.