arXiv research

The paper examines properties of self-affine Sierpiński sponges using metric invariants.

We show that every inner metric space X is the metric quotient of a complete R-tree via a free isometric action, which we call the covering R-tree of X. The quotient mapping is a weak submetry (hence, open) and light. In the case of compact 1-dimensional geodesic space X, the free isometric action is via a subgroup of …

A carpet is a metric space which is homeomorphic to the standard Sierpiński carpet in $\mathbb{R}^2$, or equivalently, in $S^2$. A carpet is called thin if its Hausdorff dimension is $<2$. A metric space is called Q-Loewner if its $Q$-dimensional Hausdorff measure is Q-Ahlfors regular and if it satisfies a $(1,Q)$-Poin…

A carpet is a metric space homeomorphic to the Sierpinski carpet. We characterize, within a certain class of examples, non-self-similar carpets supporting curve families of nontrivial modulus and supporting Poincaré inequalities. Our results yield new examples of compact doubling metric measure spaces supporting Poinca…

It is shown that if $M$ is a strongly causal free of naked singularities space-time, then its causal structure is completely characterized by a partial order in the space of skies defined by means of a class non-negative Legendrian isotopies. It is also proved that such partial order is determined by the class of futur…

The paper shows how certain groups' boundaries relate to the Sierpiński carpet.

Paper defines topology automaton for Barański carpets and proves Hölder equivalence conditions.

A new GP inference method using simplices for high-dimensional data.

Uniform bounds found for Sierpinski carpet hyperbolic components.

We obtain a nature generalization for an affine Sierpinski carpet and Sierpinski triangle to $n$-dimensional space, by using the generations and characterizations of affinely-equivalent Sierpinski carpet. Exactly, in this paper, a Menger sponge and Sierpinski simplex in $4$-dimensional space could be drawn out clearly …

We introduce a new structured kernel interpolation (SKI) framework, which generalises and unifies inducing point methods for scalable Gaussian processes (GPs). SKI methods produce kernel approximations for fast computations through kernel interpolation. The SKI framework clarifies how the quality of an inducing point a…

SKI accelerates GP inference with sparse grids to handle higher dimensions.

We study a new class of square Sierpiński carpets $F_{n,p}$ ($5\leq n, 1\leq p<\frac{n}{2}-1$) on $\mathbb{S}^2$, which are not quasisymmetrically equivalent to the standard Sierpiński carpets. We prove that the group of quasisymmetric self-maps of each $F_{n,p}$ is the Euclidean isometry group. We also establish that …

This paper analyzes error in SKI for Gaussian Processes, providing conditions for linear time inference.

Study examines diversification of mid-mountain ski tourism.

We consider a variant of the classic Ski Rental online algorithm with applications to machine learning. In our variant, we allow the skier access to a black-box machine-learning algorithm that provides an estimate of the probability that there will be at most a threshold number of ski-days. We derive a class of optimal…

A reconstruction theorem in terms of the topology and geometrical structures on the spaces of light rays and skies of a given space-time is discussed. This result can be seen as part of Penrose and Low's programme intending to describe the causal structure of a space-time $M$ in terms of the topological and geometrical…

Efficiently maps indoor magnetic fields with SKI and D-SKI.

The paper deals with the possibly degenerate behaviour of the exterior derivative operator defined on $1$-forms on metric measure spaces. The main examples we consider are the non self-similar Sierpinski carpets recently introduced by Mackay, Tyson and Wildrick. Although topologically one-dimensional, they may have pos…

If a torsion-free hyperbolic group G has 1-dimensional boundary, then the boundary is a Menger curve or a Sierpinski carpet provided G does not split over a cyclic group. When the boundary of G is a Sierpinski carpet we show that G is a quasi-convex subgroup of a 3-dimensional hyperbolic Poincare duality group. We also…

Researchers create a Fredholm module on fractal shapes like the Cantor set.

SoftKI combines SKI and variational methods for scalable GP regression.

Power meters are becoming a widely used tool for measuring training and racing effort in cycling, and are now spreading also to other sports. This means that increasing volumes of data can be collected from athletes, with the aim of helping coaches and athletes analyse and understanding training load, racing efforts, t…

Recent work shows that inference for Gaussian processes can be performed efficiently using iterative methods that rely only on matrix-vector multiplications (MVMs). Structured Kernel Interpolation (SKI) exploits these techniques by deriving approximate kernels with very fast MVMs. Unfortunately, such strategies suffer …

The paper proposes calibration to improve algorithm performance using machine learning predictions.

We give a necessary and sufficient condition for a hyperbolic Coxeter group with planar nerve to have Sierpiński curve as its Gromov boundary.

SKI speeds up Toeplitz Neural Networks by avoiding explicit decay bias and using frequency response.

Kernel-based machine learning approaches are gaining increasing interest for exploring and modeling large dataset in recent years. Gaussian process (GP) is one example of such kernel-based approaches, which can provide very good performance for nonlinear modeling problems. In this work, we first propose a grey-box mode…

Applying a theorem due to Belopol'ski and Birman, we show that the Laplace-Beltrami operator on 1-forms on ${\bf R}^n$ endowed with an asymptotically Euclidean metric has absolutely continuous spectrum equal to $[0, +\infty)$.

Here we show existence of numerous subsets of Euclidean and metric spaces that, despite having empty interior, still support Poincaré inequalities. Most importantly, our methods do not depend on any rectilinear or self-similar structure of the underlying space. We instead employ the notion of uniform domain of Martio a…

We start by a review of the chronology of mathematical results on the Dirichlet-to-Neumann map which paved the way towards the physics of transformational acoustics. We then rederive the expression for the (anisotropic) density and bulk modulus appearing in the pressure wave equation written in the transformed coordina…

New framework links fractal complexity to separation dimension.

A generic finite presentation defines a word hyperbolic group whose boundary is homeomorphic to the Menger curve. In this article, we produce the first known examples of non-hyperbolic $CAT(0)$ groups whose visual boundary is homeomorphic to the Menger curve. The examples in question are the Coxeter groups whose nerve …

Given a probability measure on a finitely generated group, its Martin boundary is a way to compactify the group using the Green's function of the corresponding random walk. We give a complete topological characterization of the Martin boundary of finitely supported random walks on relatively hyperbolic groups with virt…

Proves convergence groups on a 2-sphere are Kleinian groups.

Optimal hashing embeddings reduce linear least squares solving time.

The set N of all null geodesics of a globally hyperbolic (d+1)-dimensional spacetime (M,g) is naturally a smooth (2d-1)-dimensional contact manifold. The sky of an event is the subset of N defined by all null geodesics through that event, and is an embedded Legendrian submanifold of N diffeomorphic to a (d-1)-dimension…

We study quasi-isometry invariants of Gromov hyperbolic spaces, focussing on the l_p-cohomology and closely related invariants such as the conformal dimension, combinatorial modulus, and the Combinatorial Loewner Property. We give new constructions of continuous l_p-cohomology, thereby obtaining information about the l…

This is a simple mathematical introduction into Feynman diagram technique, which is a standard physical tool to write perturbative expansions of path integrals near a critical point of the action. I start from a rigorous treatment of a finite dimensional case (which actually belongs more to multivariable calculus than …

The paper studies degenerations of rational maps and their limits as geometrically finite rational maps.

In this paper we provide a classification theorem for 1-dimensional boundaries of groups with isolated flats. Given a group $Γ$ acting geometrically on a $CAT(0)$ space $X$ with isolated flats and 1-dimensional boundary, we show that if $Γ$ does not split over a virtually cyclic subgroup, then $\partial X$ is homeomorp…

We define a conformal reference frame, i.e., a special projection of the six-dimensional sky bundle of a Lorentzian manifold (or the five-dimensional twistor space) to a three-dimensional manifold. We construct an example, a conformal compactification, for Minkowski space. Based on the complex structure on the skies, w…

Under Solvency II the computation of capital requirements is based on value at risk (V@R). V@R is a quantile-based risk measure and neglects extreme risks in the tail. V@R belongs to the family of distortion risk measures. A serious deficiency of V@R is that firms can hide their total downside risk in corporate network…

Let $S$ be a compact, orientable surface of hyperbolic type. Let $(k_+,k_-)$ be a pair of negative numbers and let $(g_+, g_-)$ be a pair of marked metrics over $S$ of constant curvature equal to $k_+$ and $k_-$ respectively. Using a functional introduced by Bonsante, Mondello \& Schlenker, we show that there exists a …

New approach for algorithms that learn predictors to improve performance.

Estimates nonparametric densities from mixed samples.

Researchers compute contact structures for null geodesics on specific spacetimes.

New non-rigid discrete groups found in hyperbolic spaces.