An integral representation and boundary behavior of functions defined in a domain with a peak

We establish an invertible characteristic of boundary behavior of functions in Sobolev spaces defined in a space domain with a vertex of a peak on the boundary.     

Peripheral splittings of groups

We define the notion of a ``peripheral splitting' of a group. This is essentially a representation of the group as the fundamental group of a bipartite graph of groups, where all the vertex groups of one colour are held fixed--the ``peripheral subgroups'. We develop the theory of such splittings and prove an accessibility result. The theory mainly applies to relatively hyperbolic groups with connected boundary, where the peripheral subgroups are precisely the maximal parabolic subgroups. We show that if such a group admits a non-trivial peripheral splitting, then its boundary has a global cut point. Moreover, the non-peripheral vertex groups of such a splitting are themselves relatively hyperbolic. These results, together with results from elsewhere, show that under modest constraints on the peripheral subgroups, the boundary of a relatively hyperbolic group is locally connected if it is connected. In retrospect, one further deduces that the set of global cut points in such a boundary has a simplicial treelike structure.

     

Geometry of the uniform infinite half‐planar quadrangulation

We give a new construction of the uniform infinite half‐planar quadrangulation with a general boundary (or UIHPQ), analogous to the construction of the UIPQ presented by Chassaing and Durhuus, which allows us to perform a detailed study of its geometry. We show that the process of distances to the root vertex read along the boundary contour of the UIHPQ evolves as a particularly simple Markov chain and converges to a pair of independent Bessel processes of dimension 5 in the scaling limit. We study the “pencil” of infinite geodesics issued from the root vertex as reported by Curien, Ménard, and Miermont and prove that it induces a decomposition of the UIHPQ into 3 independent submaps. We are also able to prove that balls of large radius around the root are on average 7/9 times as large as those in the UIPQ, both in the UIHPQ and in the UIHPQ with a simple boundary; this fact we use in a companion paper to study self‐avoiding walks on large quadrangulations.     

Decision problems in the space of Dehn fillings

Normal surface theory is used to study Dehn fillings of a knot-manifold. We use that any triangulation of a knot-manifold may be modified to a triangulation having just one vertex in the boundary. In this situation, it is shown that there is a finite computable set of slopes on the boundary of the knot-manifold, which come from boundary slopes of normal or almost normal surfaces. This is combined with existence theorems for normal and almost normal surfaces to construct algorithms to determine precisely those manifolds obtained by Dehn filling of a given knot-manifold that: (1) are reducible, (2) contain two-sided incompressible surfaces, (3) are Haken, (4) fiber over S¹, (5) are the 3-sphere, and (6) are a lens space. Each of these algorithms is a finite computation.Moreover, in the case of essential surfaces, we show that the topology of each filled manifold is strongly reflected in the combinatorial properties of a triangulation of the knot-manifold with just one vertex in the boundary. If a filled manifold contains an essential surface then the knot-manifold contains an essential normal vertex solution which caps off to an essential surface of the same type in the filled manifold. (Normal vertex solutions are the premier class of normal surface and are computable.)     

Backgrounds in boundary string field theory

We study the role of closed string backgrounds in boundary string field theory. Background independence requires introducing dual boundary fields, which are reminiscent of the doubled field formalism. We find a correspondence between closed string backgrounds and collective excitations of open strings described by vertex operators involving the dual fields. We discuss the renormalization group flow, solutions, and stability in an example.     

Relative hyperbolicity,trees of spaces and Cannon-Thurston maps

We prove the existence of continuous boundary extensions (Cannon-Thurston maps) for the inclusion of a vertex space into a tree of (strongly) relatively hyperbolic spaces satisfying the qi-embedded condition. This implies the same result for inclusion of vertex (or edge) subgroups in finite graphs of (strongly) relatively hyperbolic groups. This generalizes a result of Bowditch for punctured surfaces in 3 manifolds and a result of Mitra for trees of hyperbolic metric spaces.     

Optimal elliptic Sobolev regularity near three-dimensional multi-material Neumann vertices

We study the optimal elliptic regularity (within the scale of Sobolev spaces) of anisotropic div-grad operators in three dimensions at a multi-material vertex on the Neumann part of the boundary of a 3D polyhedral domain. The gradient of any solution of the corresponding elliptic partial differential equation (in a neighborhood of the vertex) is p-integrable with p > 3.     

Graphs with small boundary

For a pair of vertices x and y in a graph G, we denote by d_G(x,y) the distance between x and y in G. We call x a boundary vertex of y if x and y belong to the same component and d_G(y,v)?d_G(y,x) for each neighbor v of x in G. A boundary vertex of some vertex is simply called a boundary vertex, and the set of boundary vertices in G is called the boundary of G, and is denoted by B(G).In this paper, we investigate graphs with a small boundary. Since a pair of farthest vertices are boundary vertices, |B(G)|?2 for every connected graph G of order at least two. We characterize the graphs with boundary of order at most three. We cannot give a characterization of graphs with exactly four boundary vertices, but we prove that such graphs have minimum degree at most six. Finally, we give an upper bound to the minimum degree of a connected graph G in terms of |B(G)|.     

ENUMERATION OF ROOTED VERTEX NON-SEPARABLE PLANAR MAPS

In a rooted planar map, the rooted vertex is said to be non-separable if the vertex onthe boundary of the outer face as an induced graph is not a cut-vertex. In this paper, the author derives a functional equation satisfied by the enumeratingfuuction of rooted vertex non-separable planar maps dependent on the edge number and thenumber of the edges on the outer face boundary, finds a parametric expression of itssolution, and obtains an explicit formula for the function. Particularly, the number of rooted vertex non-separable maps only replying on theedge number and that of rooted vertex non-separable tree-like maps defined in [4] accordingto the two indices, the edge number and the number of the edges on the outer face boundary,or only one index, the edge number, are also determined.     

The local structure of nodal set of solutions of Schrödinger equation on Riemannian 3-manifolds

Let (M, g) be a 3-dimensional Riemannian manifold without boundary. Consider the solution of Schrödinger equation onM. We show that locally there exists an injective Lipschitz continuous map from the nodal set of the solution away from a finite union of some small solid cones, which only intersect at the common vertex, into itself and the image set stays on a finite union of some 2-dimensional cones which have a common vertex. Moreover, the singular set of the solution is contained in the union of the solid cones.     

A Refined Green’s Function Estimate of the Time Measurable Parabolic Operators with Conic Domains

We present a refined Green’s function estimate of the time measurable parabolic operators on conic domains that involves mixed weights consisting of appropriate powers of the distance to the vertex and of the distance to the boundary.

     

Compact stable hypersurfaces with free boundary in convex solid cones with homogeneous densities

We consider a smooth Euclidean solid cone endowed with a smooth homogeneous density function used to weight Euclidean volume and hypersurface area. By assuming convexity of the cone and a curvature-dimension condition, we prove that the unique compact, orientable, second order minima of the weighted area under variations preserving the weighted volume and with free boundary in the boundary of the cone are intersections with the cone of round spheres centered at the vertex.     

Three-dimensional singularities of elastic fields near vertices

For the computation of the singular behavior of an elastic field near a three-dimensional vertex subject to displacement boundary conditions we use a boundary integral equation of the first kind whose unknown is the boundary stress. Localization at the vertex and Mellin transformation yield a one-dimensional integral equation on a piecewise circular curve γ in IR³ depending holomorphically on the complex Mellin parameter. The corresponding spectral points and packets of generalized eigenvectors characterize the desired stress field and are computed by a spline-Galerkin method with graded meshes at the corner points of the curve γ. © 1993 John Wiley & Sons, Inc.     

Maximal Controllability for Boundary Control Problems

We develop a semigroup approach to abstract boundary control problems which allows to characterize the space of all approximately reachable states. We then introduce the “maximal reachability space” giving an upper bound for this space. The abstract results are applied to the flow in a network controlled in a single vertex.     

Dirichlet–Neumann inverse spectral problem for a star graph of Stieltjes strings

We solve two inverse spectral problems for star graphs of Stieltjes strings with Dirichlet and Neumann boundary conditions, respectively, at a selected vertex called root. The root is either the central vertex or, in the more challenging problem, a pendant vertex of the star graph. At all other pendant vertices Dirichlet conditions are imposed; at the central vertex, at which a mass may be placed, continuity and Kirchhoff conditions are assumed. We derive conditions on two sets of real numbers to be the spectra of the above Dirichlet and Neumann problems. Our solution for the inverse problems is constructive: we establish algorithms to recover the mass distribution on the star graph (i.e. the point masses and lengths of subintervals between them) from these two spectra and from the lengths of the separate strings. If the root is a pendant vertex, the two spectra uniquely determine the parameters on the main string (i.e. the string incident to the root) if the length of the main string is known. The mass distribution on the other edges need not be unique; the reason for this is the non-uniqueness caused by the non-strict interlacing of the given data in the case when the root is the central vertex. Finally, we relate of our results to tree-patterned matrix inverse problems.     

Circle boundaries of planar graphs

Letg be an infinite, connected, planar graph with bounded vertex degree, which obeys a strong isoperimetric inequality and which can be embedded in the plane so that each cycle surrounds only finitely many vertices. We investigate a certain class of compactifications ofg; one of which has boundary homemorophic to a circle. We shall show that ifg is a tree or, more generally, ifg is hyperbolic, then this circle boundary supports an integral representation of any given bounded harmonic function. We further show that in the specific case of a triangulation of the plane, the graph is hyperbolic and therefore the Martin boundary is a circle.     

Magnetization formula in the bounded XYZ spin chain

The bounded XYZ spin chain is studied. We construct the vacuum states by the vertex operators of the level one modules of the elliptic algebra, and compact them through a geometric symmetry of the model called the turning symmetry. From these simplified expressions, the “magnetization formula” for magnetizations at a boundary in the bounded chain and in the half-infinite chains is derived. Applying this formula we calculate the spontaneous magnetization at a boundary in the bounded XYZ spin chain.     

Spectral Multiplicity of Selfadjoint Schrödinger Operators on Star-Graphs with Standard Interface Conditions

We analyze the singular spectrum of selfadjoint operators which arise from pasting a finite number of boundary relations with a standard interface condition. A model example for this situation is a Schrödinger operator on a star-shaped graph with continuity and Kirchhoff conditions at the interior vertex. We compute the multiplicity of the singular spectrum in terms of the spectral measures of the Weyl functions associated with the single (independently considered) boundary relations. This result is a generalization and refinement of a Theorem of I.S.Kac.     

Simple agents learn to find their way: An introduction on mapping polygons

This paper gives an introduction to the problem of mapping simple polygons with autonomous agents. We focus on minimalistic agents that move from vertex to vertex along straight lines inside a polygon, using their sensors to gather local data at each vertex. Our attention revolves around the question whether a given configuration of sensors and movement capabilities of the agents allows them to capture enough data in order to draw conclusions regarding the global layout of the polygon.In particular, we study the problem of reconstructing the visibility graph of a simple polygon by an agent moving either inside or on the boundary of the polygon. Our aim is to provide insight about the algorithmic challenges faced by an agent trying to map a polygon. We present an overview of techniques for solving this problem with agents that are equipped with simple sensorial capabilities. We illustrate these techniques on examples with sensors that measure angles between lines of sight or identify the previous location. We also give an overview over related problems in combinatorial geometry as well as graph exploration.     

The Behaviour of Elastic Fields and Boundary Integral Mellin Techniques Near Conical Points

For the computation of the local singular behaviour of an homogeneous anisotropic clastic field near the three-dimensional vertex subjected to displacement boundary conditions, one can use a boundary integral equation of the first kind whose unkown is the boundary stress. Mellin transformation yields a one - dimensional integral equation on the intersection curve 7 of the cone with the unit sphere. The Mellin transformed operator defines the singular exponents and Jordan chains, which provide via inverse Mellin transformation a local expansion of the solution near the vertex. Based on Kondratiev's technique which yields a holomorphic operator pencil of elliptic boundary value problems on the cross - sectional interior and exterior intersection of the unit sphere with the conical interior and exterior original cones, respectively, and using results by Maz'ya and Kozlov, it can be shown how the Jordan chains of the one-dimensional boundary integral equation are related to the corresponding Jordan chains of the operator pencil and their jumps across γ. This allows a new and detailed analysis of the asymptotic behaviour of the boundary integral equation solutions near the vertex of the cone. In particular, the integral equation method developed by Schmitz, Volk and Wendland for the special case of the elastic Dirichlet problem in isotropic homogeneous materials could be completed and generalized to the anisotropic case.