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We prove a number of results involving categories enriched over CMet, the category of complete metric spaces with possibly infinite distances. The category CPMet of path complete metric spaces is locally N1-presentable, closed monoidal, and coreflective in CMet. We also prove that the category CCMet of convex complete metric spaces is not closed monoidal and characterize the isometry-N0-generated objects in CMET, CPMET and CCMET, answering questions by Di Liberti and Rosický. Other results include the automatic completeness of a colimit of a diagram of bi-Lipschitz morphisms between complete metric spaces and a characterization of those pairs (metric space, unital C*-algebra) that have a tensor product in the CMet-enriched category of unital C*-algebras. [ABSTRACT FROM AUTHOR]

We study inequalities between the hyperbolic metric and intrinsic metrics in convex polygonal domains in the complex plane. A special attention is paid to the triangular ratio metric in rectangles. A local study leads to investigation of the relationship between the conformal radius at an arbitrary point of a planar domain and the distance of the point to the boundary. [ABSTRACT FROM AUTHOR]

We construct a Hölder continuous function on the unit interval which coincides in uncountably (in fact continuum) many points with every function of total variation smaller than 1 passing through the origin. We conclude that this function has impermeable graph—one of the key concepts introduced in this paper—and we present further examples of functions both with permeable and impermeable graphs. Moreover, we show that typical (in the sense of Baire category) continuous functions have permeable graphs. The first example function is subsequently used to construct an example of a continuous function on the plane which is intrinsically Lipschitz continuous on the complement of the graph of a Hölder continuous function with impermeable graph, but which is not Lipschitz continuous on the plane. As another main result, we construct a continuous function on the unit interval which coincides in a set of Hausdorff dimension 1 with every function of total variation smaller than 1 which passes through the origin. [ABSTRACT FROM AUTHOR]

We study a symmetric diffusion process on ℝ d , d ≥ 2, in divergence form in a stationary and ergodic random environment. The coefficients are assumed to be degenerate and unbounded but satisfy a moment condition. We derive upper off-diagonal estimates on the heat kernel of this process for general speed measure. Lower off-diagonal estimates are also shown for a natural choice of speed measure, under an additional mixing assumption on the environment. Using these estimates, a scaling limit for the Green function is proven. [ABSTRACT FROM AUTHOR]

A review of various approaches to the concept of a Finsler space based on different definitions is given. In particular, the axiomatics of a singular Finsler space is given. [ABSTRACT FROM AUTHOR]

In this paper, our main aim is to obtain two different discrete chaotic dynamical systems on the Box fractal ($B$). For this goal, we first give two composition functions (which generate Box fractal and filled-square respectively via escape time algorithm) of expanding, folding and translation mappings. In order to examine the properties of these dynamical systems more easily, we use the intrinsic metric which is defined by the code representation of the points on $B$ and express these dynamical systems on the code sets of this fractal. We then obtain that they are chaotic in the sense of Devaney and give an algorithm to compute periodic points.

We consider a class of functions defined on metric spaces which generalizes the concept of piecewise Lipschitz continuous functions on an interval or on polyhedral structures. The study of such functions requires the investigation of their exception sets where the Lipschitz property fails. The newly introduced notion of permeability describes sets which are natural exceptions for Lipschitz continuity in a well-defined sense. One of the main results states that continuous functions which are intrinsically Lipschitz continuous outside a permeable set are Lipschitz continuous on the whole domain with respect to the intrinsic metric. We provide examples of permeable sets in R d , which include Lipschitz submanifolds. [ABSTRACT FROM AUTHOR]

Avoiding singularities is a crucial task in robotics and path planning. This paper proposes a novel algorithm for detecting the closest singularity to a given pose for nine interpretations of the 3-RPR manipulator. The algorithm utilizes intrinsic metrics based on the framework's total elastic strain energy density, employing the physical concept of Green-Lagrange strain. The constrained optimization problem for detecting the closest singular configuration with respect to these metrics is solved globally using tools from numerical algebraic geometry implemented in the software package Bertini. The effectiveness of the proposed algorithm is demonstrated on a 3-RPR manipulator executing a one-parametric motion. Additionally, the obtained intrinsic singularity distances are compared with extrinsic metrics. Finally, the paper illustrates the advantage of employing a well-defined metric for identifying the closest singularities in comparison with the existing methods in the literature, highlighting its application in design optimization. [ABSTRACT FROM AUTHOR]

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We prove distance bounds for graphs possessing positive Bakry-Émery curvature apart from an exceptional set, where the curvature is allowed to be non-positive. If the set of non-positively curved vertices is finite, then the graph admits an explicit upper bound for the diameter. Otherwise, the graph is a subset of the tubular neighborhood with an explicit radius around the non-positively curved vertices. Those results seem to be the first assuming non-constant Bakry-Émery curvature assumptions on graphs.

We obtain functional inequalities for functions which are metric-preserving with respect to one of the following intrinsic metrics in a canonical plane domain: hyperbolic metric or some restrictions of the triangular ratio metric, respectively, of a Barrlund metric. The subadditivity turns out to be an essential property, being possessed by every function that is metric-preserving with respect to the hyperbolic metric and also by the composition with some specific function of every function that is metric-preserving with respect to some restriction of the triangular ratio metric or of a Barrlund metric. We partially answer an open question, proving that the hyperbolic arctangent is metric-preserving with respect to the restrictions of the triangular ratio metric on the unit disk to radial segments and to circles centered at origin.

To formulate the intrinsic metrics by using the code representations of the points on the classical fractals is an important research area since these formulas help to prove many geometrical and structural properties of these fractals. In various studies, the intrinsic metrics on the code set of the Sierpinski gasket, the Sierpinski tetrahedron, and the Vicsek (box) fractal are explicitly formulated. However, in the literature, there are not many works on the intrinsic metric that is obtained by the code representations of the points on fractals. Moreover, as seen in the studies on this subject, the contraction coefficients of the associated iterated function systems (IFSs) are the same for each fractal. In this paper, we define the intrinsic metric formula on the added Sierpinski triangle, whose IFS has different contraction factors, by using the code representations of the points of it. Finally, we give several geometrical properties of this fractal by using the intrinsic metric formula. [ABSTRACT FROM AUTHOR]

We give an explicit expression for the intrinsic metric on the Sierpinski gasket SG(3) (the mod-3 Sierpinski gasket) via code representation of its points. We also investigate the geodesics of SG(3) and determine the number of geodesics between two points. [ABSTRACT FROM AUTHOR]

In the paper weak isometries in directed groups are investigated. It is proved that for every weak isometry f in a directed group G the relation f(UL(x, y) ∩ LU(x, y)) = UL(f(x), f(y)) ∩ LU(f(x), f(y)) is valid for each x, y ∈ G. The notions of an orthogonality of two elements and of a subgroup symmetry in directed groups are introduced and it is shown that each weak isometry in a 2-isolated directed group or in an abelian directed group is a composition of a subgroup symmetry and a right translation. It is also proved that stable weak isometries in a 2-isolated abelian directed group G are directly related to subdirect decompositions of the subgroup G2 = {2x; x ∈ G} of G. [ABSTRACT FROM AUTHOR]

The box fractal is one of the typical examples of the self-similar sets obtained as the attractor of an iterated function system. In the literature, there is no work giving the intrinsic metric defined by using the code set of the box fractal. In this paper, we obtain an explicit formula for the intrinsic metric on the box fractal via the code representations of the points. [ABSTRACT FROM AUTHOR]

We prove distance bounds for graphs possessing positive Bakry-Émery curvature apart from an exceptional set, where the curvature is allowed to be non-positive. If the set of non-positively curved vertices is finite, then the graph admits an explicit upper bound for the diameter. Otherwise, the graph is a subset of the tubular neighborhood with an explicit radius around the non-positively curved vertices. Those results seem to be the first assuming non-constant Bakry-Émery curvature assumptions on graphs. [ABSTRACT FROM AUTHOR]

The Sierpinski gasket (also known as the Sierpinski triangle) is one of the fundamental models of self-similar sets. There have been many studies on different features of this set in the last decades. In this paper, initially we construct a dynamical system on the Sierpinski gasket by using expanding and folding maps. We then obtain a surprising shift map on the code set of the Sierpinski gasket, which represents the dynamical system, and we show that this dynamical system is chaotic on the code set of the Sierpinski gasket with respect to the intrinsic metric. Finally, we provide an algorithm to compute periodic points for this dynamical system. [ABSTRACT FROM AUTHOR]

The Sierpinski triangle, which is fundamantel examples of fractals, has become a field of study for various investigations with many interesting features. Intrinsic metrics, one of the important research topics of recent years, can generate interesting formulas on fractals. Thanks to the formula defined on the code set of the Sierpinski triangle, many interesting geometric features can be proven. In this paper, the relationship between the rooted binary tree and the Sierpinski triangle is addressed. [ABSTRACT FROM AUTHOR]

The computation of the distance between any two points of the Sierpinski gasket with respect to the intrinsic metric has already been investigated by several authors. However, to the best of our knowledge, in the literature there is not an explicit formula obtained by using the code set of the Sierpinski gasket. In this paper, we obtain an explicit formula for the intrinsic metric on the Sierpinski gasket via the code representations of its points. We finally give an important geometrical property of the Sierpinski gasket with regard to the intrinsic metric by using its code representation. [ABSTRACT FROM AUTHOR]

Let G be a group, ( M, d) be a metric space, $$X\subset M$$ be a compact subset and $$\varphi :G\times M\rightarrow M$$ be a left action of G on M by homeomorphisms. Denote $$gp=\varphi (g,p)$$ . The isotropy subgroup of G with respect to X is defined by $$H_X=\{g\in G; gX=X\}$$ . In this work we define the induced Hausdorff metric on $$G/H_X$$ by $$d_X(g_1H_X,g_2H_X):=d_H(g_1X,g_2X)$$ , where $$d_H$$ is the Hausdorff distance on M. Let $$\hat{d}_X$$ be the intrinsic metric induced by $$d_X$$ . In this work, we study the geometry of $$(G/H_X,d_X)$$ and $$(G/H_X,\hat{d}_X)$$ and their relationship with ( M, d). In particular, we prove that if G is a Lie group, M is a differentiable manifold endowed with a metric which is locally Lipschitz equivalent to a Finsler metric, $$X\subset M$$ is a compact subset and $$\varphi :G\times M\rightarrow M$$ is a smooth left action by isometries, then $$(G/H_X,\hat{d}_X)$$ is a $$C^0$$ -Finsler manifold. We also calculate the Finsler metric explicitly in some examples. [ABSTRACT FROM AUTHOR]

In recent years, intrinsic metrics have been described on various fractals with different formulas. The Sierpinski gasket is given as one of the fundamental models which defined the intrinsic metrics on them via the code representations of the points. In this paper, we obtain the explicit formulas of the intrinsic metrics on some self-similar sets (but not strictly self-similar), which are composed of different combinations of equilateral and right Sierpinski gaskets, respectively, by using the code representations of their points. We then express geometrical properties of these structures on their code sets and also give some illustrative examples.

We prove that the answer to the question of the title is 'as many times as you want.' More precisely, given any constant $$c>0$$ , we construct two oblique triangular bipyramids, P and Q, in Euclidean 3-space, such that P is convex, Q is nonconvex and intrinsically isometric to P, and $$\text {vol}Q>c\cdot \text {vol}P>0$$ . [ABSTRACT FROM AUTHOR]

Path integration is fundamental to human navigation. When a navigator leaves home on a complex outbound path, they are able to keep track of their approximate position and orientation and return to their starting location on a direct homebound path. However, there are several sources of error during path integration. Previous research has focused almost exclusively on encoding error-the error in registering the outbound path in memory. Here, we also consider execution error-the error in the response, such as turning and walking a homebound trajectory. In two experiments conducted in ambulatory virtual environments, we examined the contribution of execution error to the rotational component of path integration using angle reproduction tasks. In the reproduction tasks, participants rotated once and then rotated again to face the original direction, either reproducing the initial turn or turning through the supplementary angle. One outstanding difficulty in disentangling encoding and execution error during a typical angle reproduction task is that as the encoding angle increases, so does the required response angle. In Experiment 1, we dissociated these two variables by asking participants to report each encoding angle using two different responses: by turning to walk on a path parallel to the initial facing direction in the same (reproduction) or opposite (supplementary angle) direction. In Experiment 2, participants reported the encoding angle by turning both rightward and leftward onto a path parallel to the initial facing direction, over a larger range of angles. The results suggest that execution error, not encoding error, is the predominant source of error in angular path integration. These findings also imply that the path integrator uses an intrinsic (action-scaled) rather than an extrinsic (objective) metric. [ABSTRACT FROM AUTHOR]

This survey is devoted to discussing the problems of the unique determination of surfaces that are the boundaries of (generally speaking) nonconvex domains. First (in Sec. 2) we examine some results on the problem of the unique determination of domains by the relative metrics of the boundaries. Then, in Sec. 3, we study rigidity conditions for the boundaries of submanifolds in a Riemannian manifold. The final part (Sec. 4) is concernedwith the unique determination of domains by the condition of the local isometry of boundaries in the relative metrics. [ABSTRACT FROM AUTHOR]

We consider a weighted lattice Z d with conductance μ e = ∣ e ∣ − α . We show that the heat kernel of a variable speed random walk on it satisfies a two-sided Gaussian bound by using an intrinsic metric. We also show that when d = 2 and α ∈ ( − 1 , 0 ) , two independent random walks on such weighted lattice will collide infinitely many times while they are transient. [ABSTRACT FROM AUTHOR]

In this paper we discuss asymmetric length structures and asymmetric metric spaces. A length structure induces a (semi)distance function; by using the total variation formula, a (semi)distance function induces a length. In the first part we identify a topology in the set of paths that best describes when the above operations are idempotent. As a typical application, we consider the length of paths defined by a Finslerian functional in Calculus of Variations. In the second part we generalize the setting of General metric spaces of Busemann, and discuss the newly found aspects of the theory: we identify three interesting classes of paths, and compare them; we note that a geodesic segment (as defined by Busemann) is not necessarily continuous in our setting; hence we present three different notions of intrinsic metric space.

The classical Sierpinski Gasket defined on the equilateral triangle is a typical example of fractals. Sierpinski-like triangles can also be constructed on isosceles or scalene triangles. An explicit formula for the intrinsic metric on the classical Sierpinski Gasket via code representation of its points is given. The aim of this paper is to generalize this formula to the Sierpinski-like triangles. We also investigate geometrical properties of these triangles with respect to the intrinsic metric. Moreover, we describe certain properties of the classical Sierpinski gasket which are not shared by all of its analogues.

We study locally compact spaces with an intrinsic metric such that the group of metric similarities is transitive and the group of isometries is not transitive. We suggest an algebraic characterization of such spaces. [ABSTRACT FROM AUTHOR]

The notion of a homogeneous standard filtration of σ-algebras was introduced by the author in 1970. The main theorem asserted that a homogeneous filtration is standard, i.e., generated by a sequence of independent random variables (is Bernoulli), if and only if a standardness criterion is satisfied. The author has recently generalized the notion of standardness to arbitrary filtrations. In this paper we give detailed definitions and characterizations of Markov standard filtrations. The notion of standardness is essential for applications of probabilistic, combinatorial, and algebraic nature. At the end of the paper we present new notions related to nonstandard filtrations. [ABSTRACT FROM AUTHOR]

We describe the full exit boundary of random walks on homogeneous trees, in particular, on free groups. This model exhibits a phase transition; namely, the family of Markov measures under study loses ergodicity as a parameter of the random walk changes. The problem under consideration is a special case of the problem of describing the invariant (central) measures on branching graphs, which covers a number of problems in combinatorics, representation theory, and probability and was fully stated in a series of recent papers by the first author [1]-[3]. On the other hand, in the context of the theory of Markov processes, close problems were discussed as early as 1960s by E. B. Dynkin. [ABSTRACT FROM AUTHOR]