Behavior of solutions of semilinear elliptic equations in domains with complicated boundary

In the paper, boundary value problems in a bounded domain for semilinear elliptic equations are considered. It is claimed that, for small exponent of nonlinearity, the support of a bounded solution is concentrated near the part of the boundary of the domain where the boundary condition is inhomogeneous. Estimates for the size of a neighborhood containing the support of the solution are established. For a supercritical exponent of nonlinearity, the convergence of solutions to a limit solution in the unperforated domain is established for some family of perforated domains. The rate of convergence is polynomial, and it depends on the exponent of the nonlinearity and on the rate at which the sizes of the cavities decrease simultaneously with the growth of the number of cavities and does not depend on the boundary conditions at the boundaries of the cavities. No restrictions on the displacement of the cavities are imposed.     

A robust SVD-based image watermarking using a multi-objective particle swarm optimization

The major objective in developing a robust digital watermarking algorithm is to obtain the highest possible robustness without losing the visual imperceptibility. To achieve this objective, we proposed in this paper an optimal image watermarking scheme using multi-objective particle swarm optimization (MOPSO) and singular value decomposition (SVD) in wavelet domain. Having decomposed the original image into ten sub-bands, singular value decomposition is applied to a chosen detail sub-band. Then, the singular values of the chosen sub-band are modified by multiple scaling factors (MSF) to embed the singular values of watermark image. Various combinations of multiple scaling factors are possible, and it is difficult to obtain optimal solutions. Thus, in order to achieve the highest possible robustness and imperceptibility, multi-objective optimization of the multiple scaling factors is necessary. This work employs particle swarm optimization to obtain optimum multiple scaling factors. Experimental results of the proposed approach show both the significant improvement in term of imperceptibility and robustness under various attacks.     

Comparison of ferroelectric domain delineation in triglycine sulphate by the etching technique and the pyroelectric laser technique. Application to the study of polarization reversal

It is shown that the well-known etch technique and the newly developed pyroelectric laser technique can reveal similar domains. However the latter technique is so fast that it appears as the best one to study problems such as nucleation and domain wall motion.

In the case of an alanine doped TGS crystal it is shown that domain wall motions started at a given nucleation field E₁ stop at some characteristic boundary lines in the crystal. To carry on the polarization reversal, we have to increase the electric field up to some higher value E₂ which is the minimum value of the nucleation field in the next area. Then a new domain is growing very rapidly by wall motions until the characteristic boundary lines of this area are reached. These lines are probably dislocation lines. It is shown that a direct side observation of domains is possible. Conical and cylindrical domains are clearly visible.     

Nd2Fe14B单晶磁畴结构随厚度的变化

观察了Nd₂Fe₁₄B单晶磁畴结构随试样厚度的变化。发现试样厚度存在某个临界值L₀,小于此值时,在垂直于e轴表面上的畴纹呈迷宫形,大于此值时,迷宫形条状畴在加宽的同时,其表面出现按一定间隔分布的斑点,此即与基畴反向的锥形畴底部,随试样厚度的进一步增加,基畴宽度和锥形反向畴尺寸也随之增大,后者的数目也增多,甚至出现多重锥形畴结构,本文着重讨论了锥形畴的形成机制,并估算了锥形畸形成所对应的临界厚度L₀。此外,根据实验结果,计算了Nd₂Fe₁₄B相的畴壁能。 关键词：     

Collision-number statistics for transport processes

Physical observables are often represented as walkers performing random displacements. When the number of collisions before leaving the explored domain is small, the diffusion approximation leads to incongruous results. In this Letter, we explicitly derive an explicit formula for the moments of the number of particle collisions in an arbitrary volume, for a broad class of transport processes. This approach is shown to generalize the celebrated Kac formula for the moments of residence times. Some applications are illustrated for bounded, unbounded and absorbing domains.     

High-temperature memory in multiferroic cobalt–bromine boracite

The observation of the domain dynamics of the ferroelectric–ferroelastic domains near the paraelectric–ferroelectric phase transition is reported. The study was performed by means of high-temperature polarized light microscopy. Close to the phase transition an increase in the number of domain walls was observed. After several heating–cooling cycles around the transition temperature, a memory domain microstructure at high temperature was observed. The possible physical mechanisms responsible for the high-temperature domain dynamics are discussed. Using a complex systems approach the complexity of the domain behavior is characterized.     

Efficient discretization of Laplace boundary integral equations on polygonal domains

We describe a numerical procedure for the construction of quadrature formulae suitable for the efficient discretization of boundary integral equations over very general curve segments. While the procedure has applications to the solution of boundary value problems on a wide class of complicated domains, we concentrate in this paper on a particularly simple case: the rapid solution of boundary value problems for Laplace’s equation on two-dimensional polygonal domains. We view this work as the first step toward the efficient solution of boundary value problems on very general singular domains in both two and three dimensions. The performance of the method is illustrated with several numerical examples.     

Modelling of a closed two-dimensional reactor bounded by a wavy wall

In this paper, we propose a model for a two-dimensional closed reactor bounded by a wavy wall. The left, right and top walls of the reactor are assumed to be flat surfaces while the bottom wall is a wavy surface. In order to formulate a model for such a reactor, we introduce a coordinate transformation into the dimensionless equations of a rectangular closed domain. Then the resulting equations illustrate the phenomena for a closed reactor bounded by a wavy wall. We solve these equations using the finite difference method. The astonishing results are that the intensity of streamlines and the maximum temperature within the reactor significantly increase with an increase of the number of waves in the bottom wall, the amplitude of waves and the Frank-Kamenetskii number. Converse characteristics are observed for higher values of the enlargement of a wave. Moreover, larger Rayleigh number induces stronger vortices in the flow field and reduces the maximum temperature. The Nusselt number at the bottom wavy wall is found to increase for higher values of the Frank-Kamenetskii number and the amplitude of a wave. A transition from the steady-state to the oscillatory convection is identified for a certain value of the Frank-Kamenetskii number. However, for a low value of the Rayleigh number, there occurs a transition from the steady-state to an explosion for increasing value of the Frank-Kamenetskii number. Results also demonstrate that the critical value of the Frank-Kamenetskii number, for which a transition from the steady-state to the oscillatory convection occurs, is higher for increasing values of the number of waves, the enlargement of a wave and the amplitude of a wave.     

The electromagnetic initial boundary value problem in a domain with a chiral filling

The problem of the excitation of electromagnetic oscillations by given charge and current distributions in a domain with a nonhomogeneous chiral filling is investigated. The domain where the problem is considered may be both a finite one bounded by an ideally conducting surface and an infinite supplement to an ideally conducting bounded object. The initial boundary value problem is shown to arise, for which the generalized formulation in a special functional space is given. The existence of a unique weak solution to the problem is proven using the Galerkin method.     

Why Does the Geometric Product Simplify the Equations of Physics?

In the last decades it was observed that Clifford algebras and geometric product provide a model for different physical phenomena. We propose an explanation of this observation based on the theory of bounded symmetric domains and the algebraic structure associated with them. The invariance of physical laws is a result of symmetry of the physical world that is often expressed by the symmetry of the state space for the system implying that this state space is a symmetric domain. For example, the ball of all possible velocities is a bounded symmetric domain. The symmetry on this ball follow from the symmetry of the space-time transformations between two inertial systems, which fixes the so-called symmetric velocity between them. The Lorenz transformations acts on the ball Sof symmetric velocities by conformal transformations. The ball Sis a spin ball (type IV in Cartan's classification). The Lie algebra of this ball is defined a triple product that is closely related to geometric product. The relativistic dynamic equations in mechanics and for the Lorenz force is described by this Lie algebra and the triple product.     

EFFECTS OF MODE LOCALIZATION ON INPUT-OUTPUT DIRECTIONAL PROPERTIES OF STRUCTURES

The singular-value decomposition (SVD) is employed to study the effects of localization phenomena on input-output relationships, and power and energy transmission ratios of structures. For weakly and strongly coupled systems, existence of strong localization of singular vectors and abrupt veering of singular value loci are shown. Occurrence of strong localization causes abrupt changes in input-output directional properties. In contrast to eigenvalues, singular values do not veer away by introducing a disorder in weakly coupled systems. In particular, the use of singular values and vectors is computationally advantageous in considering multiple load cases. While eigenvalue-based analyses give information about the resonance frequencies and vibration modes of a structure, singular values of the structure are related to the forced response characteristics and give the dynamic behavior in the frequency domain. Representative examples in structural dynamics are presented.     

Unified Generative Adversarial Networks for Multidomain Fingerprint Presentation Attack Detection

With the rapid growth of fingerprint-based biometric systems, it is essential to ensure the security and reliability of the deployed algorithms. Indeed, the security vulnerability of these systems has been widely recognized. Thus, it is critical to enhance the generalization ability of fingerprint presentation attack detection (PAD) cross-sensor and cross-material settings. In this work, we propose a novel solution for addressing the case of a single source domain (sensor) with large labeled real/fake fingerprint images and multiple target domains (sensors) with only few real images obtained from different sensors. Our aim is to build a model that leverages the limited sample issues in all target domains by transferring knowledge from the source domain. To this end, we train a unified generative adversarial network (UGAN) for multidomain conversion to learn several mappings between all domains. This allows us to generate additional synthetic images for the target domains from the source domain to reduce the distribution shift between fingerprint representations. Then, we train a scale compound network (EfficientNetV2) coupled with multiple head classifiers (one classifier for each domain) using the source domain and the translated images. The outputs of these classifiers are then aggregated using an additional fusion layer with learnable weights. In the experiments, we validate the proposed methodology on the public LivDet2015 dataset. The experimental results show that the proposed method improves the average classification accuracy over twelve classification scenarios from 67.80 to 80.44% after adaptation.     

FORMATION OF MULTI-BRANCHED DOMAINS IN GARNET BUBBLE FILMS BY LOW-BIAS-PULSE METHOD

The formation of multi-branched domains( MBDs) at low static bias field was first studied. With multiple branches, MBDs contain more or less same-signed vertical-Bloch-lines (VBLs) in their walls, Experiments. on four samples have shown that under certain conditions, the amount and hardness of the MBDs formed can be easily controlled, and so-called "low-bias-field method" is another effective way to form VBL chains. It was found that MBDs are formed due to the expansion of domain segments after compressed by a rectangular bias pulse. The number of VBLs nucleated in domain walls is affected by the space where the domains expand rapidly.     

Multiple domain formation induced by modulation instability in two-component Bose-Einstein condensates

The dynamics of multiple domain formation caused by the modulation instability of two-component Bose-Einstein condensates in an axially symmetric trap are studied by numerically integrating the coupled Gross-Pitaevskii equations. The modulation instability induced by the intercomponent mean-field coupling occurs in the out-of-phase fluctuation of the wave function and leads to the formation of multiple domains that alternate from one domain to another, where the phase of one component jumps across the density dips where the domains of the other exist. This behavior is analogous to a soliton train, which explains the origin of the long lifetime of the spin domains observed by Miesner et al. [Phys. Rev. Lett. 82, 2228 (1999)]].     

Description of waves in inhomogeneous domains using Heun’s equation

There are a number of model equations describing electromagnetic, acoustic or quantum waves in inhomogeneous domains and some of them are of the same type from the mathematical point of view. This isomorphism enables us to use a unified approach to solving the corresponding equations. In this paper, the inhomogeneity is represented by a trigonometric spatial distribution of a parameter determining the properties of an inhomogeneous domain. From the point of view of modeling, this trigonometric parameter function can be smoothly connected to neighboring constant-parameter regions. For this type of distribution, exact local solutions of the model equations are represented by the local Heun functions. As the interval for which the solution is sought includes two regular singular points. For this reason, a method is proposed which resolves this problem only based on the local Heun functions. Further, the transfer matrix for the considered inhomogeneous domain is determined by means of the proposed method. As an example of the applicability of the presented solutions the transmission coefficient is calculated for the locally periodic structure which is given by an array of asymmetric barriers.     

On smoothness of solutions of some elliptic functional-differential equations with degeneration

We consider second-order differential-difference equations in bounded domains in the case where several degenerate difference operators enter the equation. The degeneration leads to the fact that the multiplicity of the zero eigenvalue for the corresponding differential-difference operator becomes infinite. Regularity of generalized solutions for such equations is known to fail in the interior of the domain. However, we prove that the projections of solutions onto the orthogonal complement to the kernel of the “leading” difference operator remain regular in certain subdomains which form a decomposition of the original domain.     

Shapes of lipid monolayer domains: Solutions using elliptic functions

Solid lipid monolayer domains surrounded by a fluid phase at an air-water interface exhibit complex shapes. These intriguing shapes can be understood in terms of a competition between line tension and long-range dipole-dipole interaction. The dipolar energy has recently been relevant to a negative line tension and a positive curvature energy at the boundary, and a corresponding shape equation was derived by the variation of the approximated domain energy (Phys. Rev. Lett. 93, 206101 (2004)). Here we further incorporate surface pressure into the shape equation and show that the equation can be analytically solved: the curvature of the domain boundary is exactly obtained as an elliptic function of arc-length. We find that a circular domain can grow into bean-and peach-like domains with pressure, i.e., dipping and cuspidal transitions of circle by compression. The comparison with the experimental observation shows nice agreement.     

Stochastic dynamical analysis of a kind of vibro-impact system under multiple harmonic and random excitations

There have been many contributions concerned with non-smooth dynamics. The purpose of this study is focused on the global stochastic dynamics of a kind of vibro-impact oscillator under the multiple harmonic and bounded noisy excitations. The well-known cell-to-cell mapping method is firstly developed to investigate the incursive fractal boundaries between the attracting domains of different random attractors, and a specific Poincaré map is then set up to explore the noise-contaminated dynamical transitions in the system. Lastly, the leading Lyapunov exponents and the surrogate tests are used to identify the noise-contaminated dynamics. It is shown that several random attractors will coexist in the phase space of the randomly driven system by adjusting the parameters’ values, and fractal boundaries may also arise between the attracting domains of different random attractors. Under the joint action of the harmonic excitation and the weak bounded noise excitation, the noisy period-doubling process, similar to a deterministic one, can appear in the Poincaré’s global cross-section by increasing the strength of the bounded noisy excitation. Moreover, the noisy periodic, the noisy chaotic, and the random-dominant dynamics are also distinguished from the noise-contaminated signals.     

Dynamic stability of a spiral domain in an ac magnetic field

The dependence of the stability of a magnetic spiral domain in a film on the parameters of the film and its domain structure and on an external magnetic field is considered within a phenomenological model. The model allows one to explain a number of experimentally observed properties of dynamic spiral domains resulting from the process of self-organization of domains and domain walls in an iron-garnet film placed in an external ac magnetic field.