Narrow Results

We prove several topological and dynamical properties of the boundary of a hierarchically hyperbolic group are independent of the specific hierarchically hyperbolic structure. This is accomplished by proving that the boundary is invariant under a “maximization” procedure introduced by the first two authors and Durham.

Network reliability plays an important role in analysis, synthesis and detection of real-world networks. In this paper, we first propose the concept of hypernetwork reliability, which generalizes the concept of network reliability. The model for hypernetwork reliability studies consists of a hypergraph with perfect reliable vertices and equal and independent hyperedge failure probability $1-p$. The measure of reliability is defined as the probability that a hypergraph is connected. Let $H$ be an $r$-uniform hypergraph with the number of vertices $n$ and the number of hyperedges $m$, where every hyperedge connects $r$ vertices. We confirm the possibility of the existence of a fixed hypergraph that is optimal or least for all hyperedges same survival possible $p$. It is simple to verify that such hypergraph exists if $m=[\frac{n-1}{r-1}]$. For a kind of 2-regular 3-uniform hypergraphs, we calculate the upper and lower bounds on the all-terminal reliability, and describe the class of hypergraphs that reach the boundary.

The trace anomaly for a conformally invariant scalar field theory on a curved manifold of positive constant curvature with boundary is considered. In the context of a perturbative evaluation of the theory's effective action explicit calculations are given for those contributions to the conformal anomaly which emerge as a result of free scalar propagation as well as from scalar self-interactions up to second order in the scalar self-coupling. The renormalization-group behavior of the theory is, subsequently, exploited in order to advance the evaluation of the conformal anomaly to third order in the scalar self-coupling. As a direct consequence the effective action is evaluated to the same order. In effect, complete contributions to the theory's conformal anomaly and effective action are evaluated up to fourth-loop order.

A finite ultraviolet cutoff near a reflecting boundary yields a stress tensor that violates the basic energy-pressure relation. Therefore, a "soft" wall described by a power-law potential, which needs no ad hoc cutoff, is being investigated by the collaboration centered at Texas A&M University and the University of Oklahoma. Progress is reported here.

We construct models where initial and boundary conditions can be found from the fundamental rules of physics, without the need to assume them, they will be derived from the action principle. Those constraints are established from physical viewpoint, and it is not in the form of Lagrange multipliers. We show some examples from the past and some new examples that can be useful, where constraint can be obtained from the action principle. Those actions represent physical models. We show that it is possible to use our rule to get those constraints directly.

In this paper, we discuss the $ℂP(N)$ model in large $N$ limit in saddle point approximation on disk and annulus with various combinations of Dirichlet and Neumann boundary conditions. We show that homogeneous condensate is not a saddle point in any of considered cases. Behavior of inhomogeneous condensate near boundary is analyzed. The condensate diverges at boundary in case of Dirichlet or Neumann boundary conditions but can be finite in case of mixed conditions.

Differential Evolution (DE) is a population-based Evolutionary Algorithm (EA) for solving optimization problems over continuous spaces. Many optimization problems are constrained and have a bounded search space from which some vectors leave when the mutation operator of DE is applied. Therefore, it is necessary the use of a boundary constraint-handling method (BCHM) in order to repair the invalid mutant vectors. This paper presents a generalized and improved version of the Centroid BCHM in order to keep the search within the valid ranges of decision variables in constrained numerical optimization problems (CNOPs), which has been tested on a robust and comprehensive set of experiments that include a variant of DE specialized in dealing with CNOPs. This new version, named Centroid$K+1$, relocates the mutant vector in the centroid formed by K random vectors and one vector taken from the population that is within or near the feasible region. The results show that this new version has a major impact on the algorithm’s performance, and it is able to promote better final results through the improvement of both, the approach to the feasible region and the ability to generate better solutions.

Due to works by Bestvina–Mess, Swarup and Bowditch, we now have complete knowledge of how splittings of a word-hyperbolic group G as a graph of groups with finite or two-ended edge groups relate to the cut point structure of its boundary. It is central in the theory that ∂G is a locally connected continuum (a Peano space). Motivated by the structure of tight circle packings, we propose to generalize this theory to cusp-uniform groups in the sense of Tukia.

A Peano space X is cut-rigid, if X has no cut point, no points of infinite valence and no cut pairs consisting of bivalent points. We prove:

Theorem. Suppose X is a cut-rigid space admitting a cusp-uniform action by an infinite group. If X contains a minimal cut triple of bivalent points, then there exists a simplicial tree T, canonically associated with X, and a canonical simplicial action of Homeo(X) on T such that any infinite cusp-uniform group G of X acts cofinitely on T, with finite edge stabilizers.

In particular, if X is such that T is locally finite, then any cusp-uniform group G of X is virtually free.

In this paper we develop some of the foundations of the theory of relatively hyperbolic groups as originally formulated by Gromov. We prove the equivalence of two definitions of this notion. One is essentially that of a group admitting a properly discontinuous geometrically finite action on a proper hyperbolic space, that is, such that every limit point is either a conical limit point or a bounded parabolic point. The other is that of a group which admits a cofinite action on a connected fine hyperbolic graph. We define a graph to be "fine" if there are only finitely many circuits a given length containing any given edge, and we develop some of the properties of this notion. We show how a relatively hyperbolic group can be assumed to act on a proper hyperbolic space of a particular geometric form. We define the boundary of a relatively hyperbolic group, and show that the limit set of any geometrically finite action of the group is equivariantly homeomorphic to this boundary. This generalizes a result of Tukia for geometrically finite kleinian groups. We also describe when the boundary is connected.

Suppose a group G is relatively hyperbolic with respect to a collection ℙ of its subgroups and also acts properly, cocompactly on a CAT(0) (or δ-hyperbolic) space X. The relatively hyperbolic structure provides a relative boundary ∂(G, ℙ). The CAT(0) structure provides a different boundary at infinity ∂X. In this paper, we examine the connection between these two spaces at infinity. In particular, we show that ∂(G, ℙ) is G-equivariantly homeomorphic to the space obtained from ∂X by identifying the peripheral limit points of the same type.

Suppose G is a finitely presented group that is hyperbolic relative to $P$ a finite collection of finitely generated proper subgroups of G. Our main theorem states that if each $P\in P$ has semistable fundamental group at $\infty$, then G has semistable fundamental group at $\infty$. The problem reduces to the case when G and the members of $P$ are all one ended and finitely presented. In that case, if the boundary $\partial (G,P)$ has no cut point, then G was already known to have semistable fundamental group at $\infty$. We consider the more general situation when $\partial (G,P)$ contains cut points.

Roger Penrose’s 2020 Nobel Prize in Physics recognizes that his identification of the concepts of “gravitational singularity” and an “incomplete, inextendible, null geodesic” is physically very important. The existence of an incomplete, inextendible, null geodesic does not say much, however, if anything, about curvature divergence, nor is it a helpful definition for performing actual calculations. Physicists have long sought for a coordinate independent method of defining where a singularity is located, given an incomplete, inextendible, null geodesic, that also allows for standard analytic techniques to be implemented. In this essay, we present a solution to this issue. It is now possible to give a concrete relationship between an incomplete, inextendible, null geodesic and a gravitational singularity, and to study any possible curvature divergence using standard techniques.

Let $0<\lambda <\mu <1$ and $\lambda +\mu >1$. In this paper, we prove that for the vast majority of such parameters the top of the planar attractor $A_{\lambda ,\mu }$ of the IFS $\{(\lambda x,\mu y),(\mu x+1-\mu ,\lambda y+1-\lambda )\}$ is the graph of a continuous, strictly increasing function. Despite this, for most parameters, $A_{\lambda ,\mu }$ has a lower box dimension strictly greater than 1, showing that the upper boundary is not representative of the complexity of the fractal. Finally, we prove that if $\lambda \mu \ge 2^{-1/6}$, then $A_{\lambda ,\mu }$ has a non-empty interior.

In this paper, we explore questions regarding the Minkowski sum of the boundaries of convex sets. Motivated by a question suggested to us by V. Milman regarding the volume of $\partial K+\partial T$ where $K$ and $T$ are convex bodies, we prove sharp volumetric lower bounds for the Minkowski average of the boundaries of sets with connected boundary, as well as some related results.

Optimal boundary control for damping the vibrations in a Mindlin-type beam is considered. Wellposedness and controllability of the system are investigated. A maximum principle is introduced and optimal control function is obtained by means of maximum principle. Also, by using maximum principle, control problem is reduced to solving a system of partial differential equations including state, adjoint variables, which are subject to initial, boundary and terminal conditions. The solution of the system is obtained by using MATLAB. Numerical results are presented in table and graphical forms.

In a recent work on the wave of advance of a beneficial technology and associated hitchhiking of cultural and biological traits, we simulated the advance of neolithic agriculture into Europe. That model embraced geographical variation of land fertility and human mobility, conversion of indigenous mesolithic hunter-gatherers to agriculture, and competition between invading farmers and indigenous converts. A key result is a sharp cultural boundary across which the agriculturalists' heritage changes from that of the invading population to that of the converts. Here we present an analytical study of the cultural boundary for some simple cases. We show that the width of the boundary is determined by human mobility and the strength of competition. Simulations for the full model give essentially the same result. The finite width facilitates irreversible gene flow between the populations, so over time genetic differences appear as gradients while e.g. linguistic barriers may remain sharp. We also examine the various assumptions of the model relating to purposeful versus. random movement of peoples and the competition between cultures, demonstrating its richness and flexibility.

A theorem due to Lichnerowicz which establishes a lower bound on the lowest nonzero eigenvalue of the Laplacian acting on functions on a compact, closed manifold is reviewed. It is shown how this theorem can be extended to the case of a manifold with nonempty boundary. Lower bounds for different boundary conditions, analogous to the empty boundary case, are formulated and some novel proofs are presented.

The problem of classifying boundary points of space-time, for example singularities, regular points and points at infinity, is an unexpectedly subtle one. Due to the fact that whether or not two boundary points are identified or even “nearby” is dependent on the way the space-time is embedded, difficulties occur when singularities are thought of as an inherently local aspect of a space-time, as an analogy with electromagnetism would imply. The completion of a manifold with respect to a pseudo-Riemannian metric can be defined intrinsically. This was done by Scott–Szekeres via an equivalence relation, formalizing which boundary sets cover other sets. This paper works through the possibilities, providing examples to show that all covering relations not immediately ruled out by the definitions are possible.

In this paper, the idea of the Ricci flow is introduced and its significance and importance to related problems in mathematics had been discussed. Several functionals are defined and their behavior is studied under Ricci flow. A unique minimizer is shown to exist for one of the functionals. This functional evaluated at the minimizer is strictly increasing. The results for the first functional considered are extended to manifold with boundary. Finally, two physically motivated examples are presented.

The issue of boundary determination in ecosystems remains problematic which is exaggerated in a dynamic, emerging innovation context, with actors joining and leaving. As part of wider research, on how firms innovate in emerging ecosystems, the boundaries of several early innovation ecosystems were explored.

Using evolutionary approaches, with extensive interviews and mapping, the wider ecosystem was initially researched. Then, five in-depth firm-focussed case studies were undertaken, and specific innovation ecosystem boundaries were mapped as they emerged and evolved.

The findings point to ‘identity’, a common early approach, being limited as the ecosystem evolves. The influence of competence and relationships play an increasing role. It is suggested that as the innovation ecosystem develops through its lifecycle, different approaches aligned to Santos and Eisenhardt’s (2005) four foci can be employed, starting with identity, then competence, then power and finally issues of efficiency.