Non-trivial Domains Research Articles

Topological pumping in an inhomogeneous Aubry-André model

The Aubry-André model is a fundamental theoretical model that exhibits interesting topological features. In this paper, we examine topologically protected boundary states in the inhomogeneous off-diagonal Aubry-André model. In contrast to the homogeneous case, the inhomogeneity triggers boundary states at phase boundaries that separate two distinct non-trivial topological domains. Remarkably, the topological character of the boundary states is predicted through topological pumping, where a boundary state is transferred from one boundary across the bulk region to the other by adiabatically tuning the pump parameter. Moreover, the role of the off-diagonal modulation strength (λ) on the transfer efficiency of the topological pumping is addressed. To support our results, we investigate the time evolution of a continuous-time quantum walk and show that its spread rate and λ are inversely related. Our work provides a new avenue to harness topological features of the Aubry-André model, where topological pumping can be used for robust quantum transport.

Overdetermined elliptic problems in nontrivial contractible domains of the sphere

In this paper, we prove the existence of nontrivial contractible domains Ω⊂Sd, d≥2, such that the overdetermined elliptic problem{−εΔgu+u−up=0in Ω, u>0in Ω, u=0on ∂Ω, ∂νu=constanton ∂Ω, admits a positive solution. Here Δg is the Laplace-Beltrami operator in the unit sphere Sd with respect to the canonical round metric g, ε>0 is a small real parameter and 1<p<d+2d−2 (p>1 if d=2). These domains are perturbations of Sd∖D, where D is a small geodesic ball. This shows in particular that Serrin's theorem for overdetermined problems in the Euclidean space cannot be generalized to the sphere even for contractible domains.

Solving the Dirichlet problem for the Monge–Ampère equation using neural networks

The Monge–Ampère equation is a full y nonlinear partial differential equation (PDE) of fundamental importance in analysis, geometry and in the applied sciences. In this paper we solve the Dirichlet problem associated with the Monge–Ampère equation using neural networks and we show that an ansatz using deep input convex neural networks can be used to find the unique convex solution. As part of our analysis we study the effect of singularities, discontinuities and noise in the source function, we consider nontrivial domains, and we investigate how the method performs in higher dimensions. We investigate the convergence numerically and present error estimates based on a stability result. We also compare this method to an alternative approach in which standard feed-forward networks are used together with a loss function which penalizes lack of convexity.

Visualization of rainbow trapping effect in higher-order topological insulators

The rainbow trapping effect, which enables field amplification of multiple wavelengths in topological insulators, offers unprecedented opportunities for integrated ultrasonic sensors and energy harvesting. However, until now, the realization of the rainbow trapping effect has been limited to a one-dimensional (1D) or 2D interface in the electromagnetic and acoustic systems. Here, we propose a method to realize rainbow trapping based on higher-order topological corner modes (HOTCMs), where the elastic wave is localized at 0D corners rather than at the interface. We strategically design the two-dimensional elastic locally resonant metamaterial plate and construct multiple HOTCMs at the interfacial corners formed by trivial and nontrivial domains. We further investigate the interplay between nearest-neighbor couplings and next-nearest-neighbor couplings. Remarkably, the HOTCMs localized at corners with different geometric configurations are shown to be frequency dependent, providing a platform for trapping elastic waves at different corners. Moreover, we demonstrate rainbow trapping based on the HOTCMs at the corners and show the agreement between the experimental results and theoretical modeling. Our results provide a mechanism to realize the trapping of multiple frequencies and extend the use of elastic integrated devices toward higher order topological insulators.

Overdetermined elliptic problems in onduloid-type domains with general nonlinearities

In this paper, we prove the existence of nontrivial unbounded domains Ω⊂Rn+1,n≥1, bifurcating from the straight cylinder B×R (where B is the unit ball of Rn), such that the overdetermined elliptic problem{Δu+f(u)=0in Ω, u=0on ∂Ω, ∂νu=constanton ∂Ω, has a positive bounded solution. We will prove such result for a very general class of functions f:[0,+∞)→R. Roughly speaking, we only ask that the Dirichlet problem in B admits a nondegenerate solution. The proof uses a local bifurcation argument.

Boundary Tracing for Laplace's Equation with Conformal Mapping

The conformal mapping technique has long been used to obtain exact solutions to Laplace's equation in two-dimensional domains with awkward geometries. However, a major limitation of the technique is that it is only directly compatible with Dirichlet and zero-flux Neumann boundary conditions. It would be useful to have a means of adapting the technique to handle more general boundary conditions, for example, Robin or nonlinear flux conditions. Boundary tracing is an unconventional method for tackling boundary value problems with generic flux boundary conditions, where one takes a known solution to the field equation and seeks new boundaries satisfying the prescribed boundary condition. In this paper, we adapt boundary tracing for compatibility with conformal mapping to produce a new prescription for studying Laplace's equation coupled with general flux boundary conditions. We illustrate the procedure via two simple examples involving heat transfer. In both cases, we demonstrate how to construct infinite families of nontrivial domains in which the solution to the chosen flux boundary value problem is exactly equal to a selected harmonic function.

Historic wandering domains near cycles

We explain how to obtain non-trivial historic contractive wandering domains for a dense set of diffeomorphisms in certain type of C r -Newhouse domains of homoclinic tangencies in dimension d ⩾ 3 and r ⩾ 1. In particular, this gives for the first time a contribution to Takens’ last problem in the C 1 topology and in dimension d > 2. We show how these Newhouse domains can be obtained arbitrarily close to diffeomorphisms exhibiting heterodimensional cycles (in dimension d = 3) or non-transverse equidimensional cycles (in any dimension d ⩾ 3) associated with periodic points with non-real complex leading multipliers.

Symbolic Plan Recognition in Interactive Narrative Environments

Interactive narratives suffer from the narrative paradox: the tension that exists between providing a coherent narrative experience and allowing a player free reign over what she can manipulate in the environment. Knowing what actions a player in such an environment intends to carry out would help in managing the narrative paradox, since it would allow us to anticipate potential threats to the intended narrative experience and potentially mediate or eliminate them. The process of observing player actions and attempting to come up with an explanation for those actions (i.e. the plan that the player is trying to carry out) is the problem of plan recognition. We adopt the framing of narratives as plans and leverage recent advances that cast plan recognition as planning to develop a symbolic plan recognition system as a proof-of-concept model of a player's reasoning in an interactive narrative environment. In this paper we outline the system architecture, report on performance metrics that demonstrate adequate performance for non-trivial domains, and discuss the implications of treating players as plan recognizers.

Formation mechanism and energy interaction of spontaneous skyrmion in nanodisks

Topological spin textures, such as magnetic skyrmion, have great promises in data storage applications due to their inherent stability. Contrary to the general idea, simulation results here indicate that spontaneous skyrmion equilibrium states can exist generally in certain materials without the assistance of Dzyaloshinskii-Moriya interaction, external fields or defects. The spin configurations are closely related to the variations of the magnetocrystalline anisotropy K1, exchange interaction A, and saturation magnetization MS. The system with low anisotropy is conducive to the formation of vortex, while a single domain can be formed in the large range of K1 and A in the system with small MS. Landau-Lifshitz-Gilbert equation is used to investigate the variations of different interaction energies in the formation of topological domains. In the range where skyrmions are formed, the anisotropic energy, exchange energy and demagnetization energy account for 40 ~ 80%, 10 ~ 20%, 10 ~ 40% of the total energy. Meanwhile, a phase diagram of equilibrium magnetic moment distribution under different intrinsic properties is obtained. The K1, A and MS values of skyrmions range from 1.6 ~ 2.5 × 105 J/m3, 1.6 ~ 6.8 × 10-12 J/m, and 7.1 ~ 8.8 × 105 A/m in this simulation. By adjusting the material parameters, various competing interactions in magnetic nanostructures can be controlled to form skyrmion textures, which provide important insights for understanding the formation mechanism of nontrivial topological domains.

Determination of the Homotopy Type of a Morse – Smale Diffeomorphism on a 2-torus by Heteroclinic Intersection

According to the Nielsen – Thurston classification, the set of homotopy classes of orientation-preserving homeomorphisms of orientable surfaces is split into four disjoint subsets. Each subset consists of homotopy classes of homeomorphisms of one of the following types: $T_{1}$) periodic homeomorphism; $T_{2}$) reducible non-periodic homeomorphism of algebraically finite order; $T_{3}$) a reducible homeomorphism that is not a homeomorphism of algebraically finite order; $T_{4}$) pseudo-Anosov homeomorphism. It is known that the homotopic types of homeomorphisms of torus are $T_{1}$, $T_{2}$, $T_{4}$ only. Moreover, all representatives of the class $T_{4}$ have chaotic dynamics, while in each homotopy class of types $T_{1}$ and $T_{2}$ there are regular diffeomorphisms, in particular, Morse – Smale diffeomorphisms with a finite number of heteroclinic orbits. The author has found a criterion that allows one to uniquely determine the homotopy type of a Morse – Smale diffeomorphism with a finite number of heteroclinic orbits on a two-dimensional torus. For this, all heteroclinic domains of such a diffeomorphism are divided into trivial (contained in the disk) and non-trivial. It is proved that if the heteroclinic points of a Morse – Smale diffeomorphism are contained only in the trivial domains then such diffeomorphism has the homotopic type $T_{1}$. The orbit space of non-trivial heteroclinic domains consists of a finite number of two-dimensional tori, where the saddle separatrices participating in heteroclinic intersections are projected as transversally intersecting knots. That whether the Morse – Smale diffeomorphisms belong to types $T_{1}$ or $T_{2}$ is uniquely determined by the total intersection index of such knots.

Unconventional magnetization textures and domain-wall pinning in Sm\u2013Co magnets

Some of the best-performing high-temperature magnets are Sm–Co-based alloys with a microstructure that comprises an hbox {Sm}_2hbox {Co}_{17} matrix and magnetically hard hbox {SmCo}_5 cell walls. This generates a dense domain-wall-pinning network that endows the material with remarkable magnetic hardness. A precise understanding of the coupling between magnetism and microstructure is essential for enhancing the performance of Sm–Co magnets, but experiments and theory have not yet converged to a unified model. Here, transmission electron microscopy, atom probe tomography, and nanometer-resolution off-axis electron holography have been combined with micromagnetic simulations to reveal that the magnetization state in Sm–Co magnets results from curling instabilities and domain-wall pinning effects at the intersections of phases with different magnetic hardness. Additionally, this study has found that topologically non-trivial magnetic domains separated by a complex network of domain walls play a key role in the magnetic state by acting as nucleation sites for magnetization reversal. These findings reveal previously hidden aspects of magnetism in Sm–Co magnets and, by identifying weak points in the microstructure, provide guidelines for improving these high-performance magnetic materials.

An efficient solver for space–time isogeometric Galerkin methods for parabolic problems

In this work we focus on the preconditioning of a Galerkin space–time isogeometric discretization of the heat equation. Exploiting the tensor product structure of the basis functions in the parametric domain, we propose a preconditioner that is the sum of Kronecker products of matrices and that can be efficiently applied thanks to an extension of the classical Fast Diagonalization method. The preconditioner is robust w.r.t. the polynomial degree of the spline space and the time required for the application is almost proportional to the number of degrees-of-freedom, for a serial execution. By incorporating some information on the geometry parametrization and on the equation coefficients, we keep high efficiency with non-trivial domains and variable thermal conductivity and heat capacity coefficients.

Loops and Meshes Formulations for 3-D Eddy-Current Computation in Topologically Non-Trivial Domains With Volume Integral Equations

The volume integral equation method based on the current vector potential approximated by means edge element basis function is a well-established approach for 3-D eddy currents computation. The application of the method is straightforward when simply connected geometries and no connection with external circuits are involved. In this case, in fact, the solving system is easily obtained based on tree-cotree decomposition of the primal graph. However, when multiply connected geometries or external generators are considered, “additional” degrees of freedom, not related to interior cotree edges of the primal graph, need to be identified and involved to assure the consistency of the numerical solution. In this article, the link between the volume integral equation method and the circuit is investigated in detail, and the circuit view is used as a guide for systematically finding the additional degrees of freedom arising in case of multiply connected geometries and/or external circuits. In particular, the dual graph is introduced as the support of the circuit and it is shown that this is the natural frame for taking the topology into account. For multiply connected geometries, the additional degrees of freedom are related to loop currents crossing one only time (or an odd number of times) any cutting surface making the domain simply connected, and are found by applying a minimum path algorithm on the dual graph forming the circuit. In case of conducting domain connected to an external generator, an extended dual graph is introduced for finding the further additional degrees of freedom. This article also researches into the possibility to replace the usual current density of the elements, obtained via facet-element shape functions and exactly matching the current of the faces, with a uniform current density obtained by means of a minimum error procedure and approximately matching the current of the faces. The use of this uniform current density, besides improving the calculation time and the accuracy of the coupling coefficients, also allows the extension of the volume integral equation method to discretizations of the problem domain made of arbitrarily shaped polyhedral elements.

Nonexistence of solutions for elliptic equations with supercritical nonlinearity in nearly nontrivial domains

We deal with nonlinear elliptic Dirichlet problems of the form \mathrm {div}(|D u|^{p-2}D u )+f(u)=0\quad\text{ in }\Omega,\qquad u=0\ \text{ on }\partial\Omega where \Omega is a bounded domain in \mathbb R^n , n\ge 2 , p &gt; 1 and f has supercritical growth from the viewpoint of Sobolev embedding. Our aim is to show that there exist bounded contractible non star-shaped domains \Omega , arbitrarily close to domains with nontrivial topology, such that the problem does not have nontrivial solutions. For example, we prove that if n=2 , 1 &lt; p &lt; 2 , f(u)=|u|^{q-2}u with q &gt; {2p\over 2-p} and \Omega=\{(\rho\cos\theta,\rho\sin\theta)\ :\|\theta|&lt;\alpha,\ |\rho -1| &lt; s\} with 0 &lt; \alpha &lt; \pi and 0 &lt; s &lt; 1 , then for all q &gt; {2p\over 2-p} there exists \bar s &gt; 0 such that the problem has only the trivial solution u\equiv 0 for all \alpha\in (0,\pi) and s\in (0,\bar s) .

APPROXIMATING SMOOTH, MULTIVARIATE FUNCTIONS ON IRREGULAR DOMAINS

In this paper, we introduce a method known aspolynomial frame approximationfor approximating smooth, multivariate functions defined on irregular domains in$d$dimensions, where$d$can be arbitrary. This method is simple, and relies only on orthogonal polynomials on a bounding tensor-product domain. In particular, the domain of the function need not be known in advance. When restricted to a subdomain, an orthonormal basis is no longer a basis, but a frame. Numerical computations with frames present potential difficulties, due to the near-linear dependence of the truncated approximation system. Nevertheless, well-conditioned approximations can be obtained via regularization, for instance, truncated singular value decompositions. We comprehensively analyze such approximations in this paper, providing error estimates for functions with both classical and mixed Sobolev regularity, with the latter being particularly suitable for higher-dimensional problems. We also analyze the sample complexity of the approximation for sample points chosen randomly according to a probability measure, providing estimates in terms of the correspondingNikolskii inequalityfor the domain. In particular, we show that the sample complexity for points drawn from the uniform measure is quadratic (up to a log factor) in the dimension of the polynomial space, independently of $d$, for a large class of nontrivial domains. This extends a well-known result for polynomial approximation in hypercubes.

Solutions to overdetermined elliptic problems in nontrivial exterior domains

In this paper we construct nontrivial exterior domains \Omega \subset \mathbb R^N , for all N\geq 2 , such that the problem \left\{\begin{array} {ll} -\Delta u +u -u^p=0,\ u &gt;0 &amp; \text{in }\; \Omega, \\ \ u= 0 &amp; \text{on }\; \partial \Omega, \\ \ \frac{\partial u}{\partial \nu} = \text{cte} &amp; \text{on }\; \partial \Omega, \end{array}\right. admits a positive bounded solution. This result gives a negative answer to the Berestycki–Caffarelli–Nirenberg conjecture on overdetermined elliptic problems in dimension 2, the only dimension in which the conjecture was still open. For higher dimensions, different counterexamples have been found in the literature; however, our example is the first one in the form of an exterior domain.

Evolution of topological superconductivity by orbital-selective confinement in oxide nanowires

We determine the optimal conditions to achieve topological superconducting phases having spin-singlet pairing for a planar nanowire with finite lateral width in the presence of an in-plane external magnetic field. We employ a microscopic description that is based on a three-band electronic model including both the atomic spin-orbit coupling and the inversion asymmetric potential at the interface between oxide band-gap insulators. We consider amplitudes of the pairing gap, spin-orbit interactions and electronic parameters that are directly applicable to nanowires of LaAlO$_3$-SrTiO$_3$. The lateral confinement introduces a splitting of the $d$-orbitals that alters the orbital energy hierarchy and significantly affects the electron filling dependence of the topological phase diagram. Due to the orbital directionality of the $t_{2g}$-states, we find that in the regime of strong confinement the onset of topological phases is pinned at electron filling where the quasi flat heavy bands start to get populated. The increase of the nanowire thickness leads to a changeover from sparse-to-dense distribution of topologically non-trivial domains which occurs at the cross-over associated to the orbital population inversion. These findings are corroborated by a detailed analysis of the most favorable topological superconducting phases in the electron doping-magnetic field plane highlighting the role of orbital selective confinement.

Novel multilevel techniques for convergence acceleration in the solution of systems of equations arising from RBF-FD meshless discretizations

The present paper develops two new techniques, namely additive correction multicloud (ACMC) and smoothed restriction multicloud (SRMC), for the efficient solution of systems of equations arising from Radial Basis Function-generated Finite Difference (RBF-FD) meshless discretizations of partial differential equations (PDEs). RBF-FD meshless methods employ arbitrary distributed nodes, without the need to generate a mesh, for the numerical solution of PDEs. The proposed techniques are specifically designed for the RBF-FD data structure and employ simple restriction and interpolation strategies in order to obtain a hierarchy of coarse-level node distributions and the corresponding correction equations.Both techniques are kept as simple as possible in terms of code implementation, which is an important feature of meshless methods. The techniques are verified on 2D and 3D Poisson equations, defined on non-trivial domains , showing very high benefits in terms of both time consumption and work to convergence when comparing the present techniques to the most common solver approaches. These benefits make the RBF-FD approach competitive with standard grid-based approaches when the number of nodes is very high, allowing large size problems to be tackled by the RBF-FD method.

The Ohm–Rush content function II. Noetherian rings, valuation domains, and base change

The notion of an Ohm–Rush algebra, and its associated content map, has connections with prime characteristic algebra, polynomial extensions, and the Ananyan–Hochster proof of Stillman’s conjecture. As further restrictions are placed (creating the increasingly more specialized notions of weak content, semicontent, content, and Gaussian algebras), the construction becomes more powerful. Here we settle the question in the affirmative over a Noetherian ring from [N. Epstein and J. Shapiro, The Ohm-Rush content function, J. Algebra Appl. 15(1) (2016) 1650009, 14 pp.] of whether a faithfully flat weak content algebra is semicontent (and over an Artinian ring of whether such an algebra is content), though both questions remain open in general. We show that in content algebra maps over Prüfer domains, heights are preserved and a dimension formula is satisfied. We show that an inclusion of nontrivial valuation domains is a content algebra if and only if the induced map on value groups is an isomorphism, and that such a map induces a homeomorphism on prime spectra. Examples are given throughout, including results that show the subtle role played by properties of transcendental field extensions.

Domain decomposition methods with fundamental solutions for Helmholtz problems with discontinuous source terms

The direct application of the classical method of fundamental solutions (MFS) is restricted to homogeneous linear partial differential equations (PDEs). The use of fundamental solutions with different frequencies allowed the extension of the MFS to non-homogeneous PDEs, in particular, for Poisson or Helmholtz equations and for elastostatic or elastodynamic problems. This method has been called method of fundamental solutions for domains (MFS-D), but it faces an approximation problem when the non-homogeneous term presents discontinuities, because the fundamental solutions are analytic functions outside the source point set. In this paper we analyze two domain decomposition techniques for overcoming this approximation problem. The problem is set in the context of the modified Helmholtz equation, and we also establish the missing density results that justify both the MFS and the MFS-D approximations. Numerical results are presented comparing a direct and an iterative domain decomposition technique, with simulations in non-trivial domains.