Infinite Edge Research Articles

Local Spectral Multiplicity of Selfadjoint Couplings with General Interface Conditions

We consider selfadjoint operators obtained by pasting a finite number of boundary relations with one-dimensional boundary space. A typical example of such an operator is the Schrödinger operator on a star-graph with a finite number of finite or infinite edges and an interface condition at the common vertex. A wide class of “selfadjoint” interface conditions, subject to an assumption which is generically satisfied, is considered. We determine the spectral multiplicity function on the singular spectrum (continuous as well as point) in terms of the spectral data of decoupled operators.

Binaural effects and rendering of edge diffraction in geometrical acoustics

Virtual acoustic environments have many applications in entertainment, architectural planning, and hearing research. Geometrical acoustics (GA) is commonly used, offering high computational efficiency by assuming ray-like sound propagation. However, GA does not account for wave-related effects such as edge diffraction, leading to perceptually dissatisfactory results or discontinuities, e.g., when a sound source is occluded. GA can be extended by constructing edge-diffracted sound paths. Recently, we have suggested the universal diffraction filter approximation (UDFA) to model the spectral effects of diffraction from infinite and finite edges and objects composed thereof using highly efficient recursive filters. This contribution investigates how edge diffraction affects the spatial perception of a sound source in conditions where a flat plate occludes the direct sound or produces a reflection. Binaural recordings were obtained with a head and torso simulator at discrete points of a sound source trajectory in the vicinity of a flat plate, including conditions near the incident and reflection shadow boundary. In a listening test, the perceived source location was matched for conditions with and without plate. Differences in localization were observed, and an approach for binaural rendering of simulated diffraction in the GA context is suggested, replicating the localization results obtained for the dummy-head recordings.

Filter-based solutions for finite and infinite edge diffraction

Sound diffraction occurs at (building) corners, objects and openings, prominently affecting occluded and reflected sounds. Here, a physically-based, parametric filter model for diffraction at arbitrary wedges is presented. The model connects existing high-frequency asymptotic and exact solutions in geometrical acoustics using up to four half-order low-pass filters. Compact expressions for the transfer function and impulse response of the singly diffracted field are derived, with errors below 0.1 dB. A filter modification is suggested to account for the exact solution at low frequencies. The filter solution is further simplified to approximate the spectral effects of diffraction from finite wedges and objects composed thereof, referred to as universal diffraction filter approximation (UDFA). A computationally highly-efficient recursive filter implementation is presented and it is demonstrated that diffraction from flat finite objects like plates or apertures can be closely approximated. To account for effects of higher-order diffraction at the object, a heuristic filter extension is suggested. The presented filter-based diffraction solution is suited for the prediction and simulation of sound propagation including non-specular reflections in architectural acoustics. For virtual acoustics, UDFA provides an efficient and accurate approximation of edge diffraction to account for perceptually relevant frequency-dependent attenuation, extendable to arbitrary objects.

Finite element analysis of bending waves in Mindlin plates with Perfectly Matched Layers

It is important to determine the boundary conditions of walls and floors precisely when simulating the building acoustics. For a certain room, the extension of the spans can be considered as infinite edges. The Perfect Matched Layer(PML) is an artificial absorbing domain for the wave propagations and is widely used in finite element analysis to simulate the acoustical free field conditions right now. In this paper, an effective PML technique for the plate structure will be presented. The PML formulation will be derived based on the Mindlin plate theory and the implementation method will be introduced. This technique will be validated through the numerical experiments. The accuracy and limits of the presented technique will be discussed based on the numerical results compared with the analytical results. The results show that the presented PML technique is effective and accurate to simulate the plate as an infinite large plate. It is expected to implement the technique in the further research of the structure borne sound, such as floor impact sounds.

Critical Nash Value Nodes for Control Affine Embedded Graphon Mean Field Games*

Graphon Mean Field Games (GMFGs) (Caines and Huang (2021)) constitute generalizations of Mean Field Games to the case where the agents form subpopulations associated with the nodes of large graphs. The work in (Foguen-Tchuendom et al. (2021), Foguen-Tchuendom et al. (2022a)) analyzed the stationarity of equilibrium Nash values with respect to node location for large populations of non-cooperative agents with linear dynamics on large graphs embedded in Euclidean space together with their limits (termed embedded graphons). That analysis is extended in this investigation to agent systems lying in the class of control affine non-linear systems (see Isidori (1985)). Specifically, control affine GMFG systems are treated where (i) at each node αeV the drift of each generic agent system is affine in the control function, and (ii) the running costs at each node α ϵ V ⊂ Rm are exponentiated negative inverse quadratic (ENIQ) functions of the difference between a generic state and the local graphon weighted mean Zα,µG where µG:= {µβ,β ϵ V ⊂ Rm} is the globally distributed family of mean fields. The infinite cardinality node and edge limits are considered, where it is assumed that the limit embedded graphon g(α, β), (α, β) ϵ V x V, is continuously differentiable. It is shown that the equilibrium Nash value Vα is stationary with respect to the nodal location α ϵ V if and only if the corresponding mean Zα,µG is stationary with respect to nodal location.

A Universal Filter Approximation of Edge Diffraction for Geometrical Acoustics

Sound propagation in urban and indoor environments often involves diffraction at corners, finite objects and openings, resulting in perceptually relevant frequency-dependent attenuation. Geometrical acoustics (GA) have become a de-facto standard for the prediction and simulation of sound propagation and real-time virtual acoustics, including effects of edge diffraction. However, methods to account for edge diffraction often assume infinite edges, such as the uniform theory of diffraction, or are computationally involved, such as the Biot-Tolstoy-Medwin-Svensson (BTMS) method. Particularly for interactive auralization, an efficient approximation of edge diffraction is desirable. Here, an extension of GA to account for diffraction with simple-to-derive parameters and a time-domain recursive filter implementation is suggested. This universal diffraction filter approximation (UDFA) describes diffraction from infinite and finite wedges using a combination of first-order and (fractional) half-order low-pass filters to account for the frequency-dependent attenuation of the diffracted incident and reflected sound field. The physically-based UDFA exactly matches asymptotic solutions for infinite wedges and provides approximations for finite wedges. It is demonstrated that first-order diffraction from flat finite objects like a plate or an aperture can be described by combining the filters for each edge. To account for effects of higher-order diffraction, an additional heuristic filter extension is suggested.

Scattering in quantum wires and junctions of quantum wires with edge states of quantum spin Hall insulators

An integral part of scattering theory calculations in continuum quantum systems involves identifying appropriate boundary conditions in addition to writing down the correct Hamiltonian. In the simplest problem of scattering in one dimensional lattice, scattering due to an on-site potential and scattering due to an unequal bond (in otherwise translationally invariant lattice) give different results for scattering amplitudes. While the scattering problems in the continuum and the lattice models can be mapped to one another for scattering due to on-site potential, the equivalent continuum model for scattering due to an unequal bond is missing. We introduce a new parameter c in the boundary condition of the continuum model that is equivalent to scattering due to an unequal bond on the lattice. Further, we study a junction between a normal metal quantum wire and a one dimensional edge of quantum spin Hall insulator (QSHI) in continuum using the parameter c. In the case of a junction between a normal metal quantum wire and the edge of QSHI, we identify the boundary condition that permits maximum transmission. Further, we solve the scattering problem between the junction of quantum wire and QSHI using a lattice model and map it to continuum model results. The problem of transport between four channels of spinful normal metal quantum wire and two channels of QSHI edge is not well-defined. We rectify this situation by formulating the scattering problem in terms of a junction of a semi-infinite normal metal quantum wire with an infinite edge of QSHI, gapping out one semi-infinite section of the QSHI edge by a Zeeman field and applying an appropriate boundary condition at the junction. We calculate the scattering amplitudes analytically.

Using ISEF Edge Detection, Text Extraction from Natural Images of Different Languages

In this research, we presented a method for word extraction from various linguistic imagery. In computer vision research, text in images is crucial. Here, we employ edge-based text extraction with ISEF (infinite symmetrical edge filter). The best edge detector, known as ISEF, provides reliable findings for text in images. Detection, localization, tracking, and improvement are all involved in text extraction. For text extraction, numerous techniques have been proposed. Our goal is to propose a reliable method for text extraction from photos in many languages.

Finite time stabilization of the waves on an infinite network

In this paper the finite time stabilization of an infinite network of vibrating strings is studied. We consider a network with N∈N⁎,N≥2 finite edges and one infinite edge. This network is subject to homogeneous Dirichlet conditions at the endpoints of the finite edges. At the internal vertex 0, we put the continuity condition and a balanced damping condition of the form ∑j=0N(∂xuj(0,t)−∂tuj(0,t))=0. By a spectral analysis technique, we prove that when the finite edges lengths are rationally proportional, the energy of the system becomes constant within a finite time Te. We also prove that for an equilateral network we have Te=2ℓ, where ℓ is the common edge length. This is a finite time stabilization result of the system to a state with constant energy. This finite time limit energy is given in terms of the initial data and computed for a generic network. The main idea is to calculate the resolvent and to prove that it is of finite exponential type on some subspace of the energy space.

Finite index subgroups in non-large generalized Baumslag–Solitar groups

A generalized Baumslag–Solitar group (GBS-group) is a finitely generated group acting on a tree with infinite cyclic edge and vertex stabilizers. A group is large if some finite index subgroup maps onto the free group F 2. An explicit description is given for all index n subgroups in non-large GBS groups. Moreover, we give exact formulas for the numbers of subgroups, normal subgroups, and conjugacy classes of subgroups of each finite index in non-large GBS groups.

On a Model Graph with a Loop and Small Edges. Holomorphy Property of Resolvent

We consider the Schrodinger operator on a graph consisting of two infinite edges, a loop, and a glued (at the start and end points of the loop) graph obtained by e−1 times contraction of some fixed graph. The Kirchhoff conditions are imposed at interior vertices and the Dirichlet or Neumann conditions are imposed at boundary vertices of the graph. We show that the resolvent of the Schrodinger operator is holomorphic with respect to the small parameter e and write out the first three terms of the asymptotic expansion of the resolvent.

Asymptotic behaviour for local and nonlocal evolution equations on metric graphs with some edges of infinite length

We study local (the heat equation) and nonlocal (convolution-type problems with an integrable kernel) evolution problems on a metric connected finite graph in which some of the edges have infinity length. We show that the asymptotic behaviour of the solutions to both local and nonlocal problems is given by the solution of the heat equation, but on a star-shaped graph in which there are only one node and as many infinite edges as in the original graph. In this way, we obtain that the compact component that consists in all the vertices and all the edges of finite length can be reduced to a single point when looking at the asymptotic behaviour of the solutions. For this star-shaped limit problem, the asymptotic behaviour of the solutions is just given by the solution to the heat equation in a half line with a Neumann boundary condition at $$x=0$$ and initial datum $$(2 M/N ) \delta _{x=0}$$ where M is the total mass of the initial condition for our original problem and N is the number of edges of infinite length. In addition, we show that solutions to the nonlocal problem converge, when we rescale the kernel, to solutions to the heat equation (the local problem), that is, we find a relaxation limit.

Energy-efficient computation offloading strategy with tasks scheduling in edge computing

In mobile edge computing systems, the energy consumption and execution delay can be reduced dramatically by mobile edge computation offloading (MECO) . However, due to the limited computing capacity of edge cloud, an energy-efficient offloading strategy plays a significant role. In this paper, the offloading decision problem for multi-device edge computing systems based on time-division multiple access is studied. The scheduling of offloading devices at the edge cloud is considered when modelling the edge computing system. Then, the offloading decision problem is formulated as an energy consumption minimization problem with the constraint of latency tolerance. It is a mixed integer programming problem of NP-hardness. To address the problem, a Dynamic Programming-based Energy Saving Offloading (DPESO) algorithm is designed to obtain the offloading strategy including the offloading option, offloading sequence and transmission power. First, the MECO with infinite edge cloud capacity is solved by device classification and transmission power decision. Then, we sort and adjust the offloading devices to meet the latency tolerance for the MECO with finite edge cloud capacity. Finally, simulation results demonstrate that the DPESO algorithm achieves better energy efficiency than the baseline strategies and has good scalability.

Controllability of periodic bilinear quantum systems on infinite graphs

In this work, we study the controllability of the bilinear Schrödinger equation (BSE) on infinite graphs for periodic quantum states. We consider the BSE i∂tψ = −Δψ + u(t)Bψ in the Hilbert space Lp2 composed of functions defined on an infinite graph G verifying periodic boundary conditions on the infinite edges. The Laplacian −Δ is equipped with specific boundary conditions, B is a bounded symmetric operator, and u∈L2((0,T),R) with T > 0. We present the well-posedness of the BSE in suitable subspaces of D(|Δ|3/2). In such spaces, we study the global exact controllability and we provide examples involving tadpole graphs and star graphs with infinite spokes.

On knot groups acting on trees

A finitely generated group [Formula: see text] acting on a tree with infinite cyclic edge and vertex stabilizers is called a generalized Baumslag–Solitar group (GBS group). We prove that a one-knot group [Formula: see text] is a GBS group if and only if [Formula: see text] is a torus knot group, and describe all n-knot GBS groups for [Formula: see text].

$$\mathcal {F}_\pi $$-residuality of generalized Baumslag–Solitar groups

A generalized Baumslag–Solitar group is a finitely generated group that acts on a tree with infinite cyclic edge and vertex stabilizers. A group G is residually a finite $$\pi $$-group, for a set of primes $$\pi $$, if every non-trivial element of G has non-trivial image in a quotient of G that is a finite $$\pi $$-group. We provide a criterion for generalized Baumslag–Solitar groups to be residually a finite $$\pi $$-group.

Analytical and numerical methods for efficient calculation of edge diffraction by an arbitrary incident signal.

In the present study, diffraction of plane and spherically spreading signals by half-planes is considered. An existing analytical impulse response is investigated, which is exact for plane and approximate for spherical incident signals. It is shown that all its primitive functions with respect to time exist and have an explicit form involving elementary functions. The primitive functions are employed to (i) prove that the convolution of the impulse response with any bounded signal is also bounded for all times, (ii) obtain analytically the diffraction response as a combination of elementary functions for any incident signal approximated piecewise by fitting polynomials, (iii) improve the performance of the numerical convolution by orders of magnitude, and (iv) handle the convolution of very coarsely sampled incident signals. An impulse response is presented for finite-length edges, which, unlike the traditional integration formulas along the edge, is an explicit form of time. Because it is based on the impulse response for infinite edges, it inherits all aforementioned benefits associated with its primitive functions. Furthermore, it offers a substantial computational benefit compared to traditional integration formulas along the edge. Finally, the requirements for the direct application of the presented results to other impulse responses are discussed.

Infinite polytopes in Hurwitz stability region

We consider Hurwitz stability region in the parameter space of monic polynomials. Firstly, a multilinear map from the positive first octant of the cartesian coordinate system to the stability region is defined. Based on this map a necessary and sufficient condition for stability is obtained. Starting from stable points, polytopes with infinite edge lengths are defined. Stability of the edges of these polytopes is equivalent to the stability of the convex combinations of third and fourth order factor polynomials. If all edges of polytope are stable then by the edge theorem the polytope is robust stable. The obtained result is important for understanding the geometry of stability domain, and can be used for generation of convex subsets of stable polynomials and for stabilization by fixed order controllers and other related problems.

Nonlocal control of spin-spin correlations in a finite-geometry helical edge

An infinite edge of a quantum Hall system prohibits indirect exchange coupling between two spins whereas a quantum spin-Hall edge prohibits out-of-plane coupling. In this study we analyze an unexpected breakdown of this behaviors in a finite system, where the two spins can interact also via a longer path that traverses the whole perimeter of the system. We explain this using an analytical model as well as using tight binding models in real space. Based on this finding, we propose how using a lead far away from the spins can switch the coupling on and off among them non-locally.

Two-Dimensional Topological Polariton Laser.

We provide proof-of-principle illustration of lasing in a two-dimensional polariton topological insulator. Topological edge states may arise in a structured polariton microcavity under the combined action of spin-orbit coupling and Zeeman splitting in the magnetic field. Their properties and lifetime are strongly affected by gain. Thus, gain concentrated along the edge of the insulator can counteract intrinsic losses in such a selective way that the topologically protected edge states become amplified, while bulk modes remain damped. When gain is compensated by nonlinear absorption the metastable nonlinear edge states are formed. Taking a triangular structure instead of an infinite edge we observed persistent topological currents accompanied by the time-periodic oscillations of the polariton density.