The paper examines properties of self-affine Sierpiński sponges using metric invariants.
problem Investigating properties of self-affine Sierpiński sponges using metric invariants.
method Examined through maximal power law property and perfectly disconnectedness.
result Characterized self-affine Sierpiński sponges by their metric properties.
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…
Optimal ski rental strategies with machine learning predictions.
problem Minimizing ski rental costs with uncertain future days.
method Derive optimal randomized algorithms using machine learning predictions.
result Class of optimal algorithms with minimized competitive ratio.
We rephrase Gromov's definition of Markov compacta, introduce a subclass of Markov compacta defined by one building block and study cohomological dimensions of these compacta. We show that for a Markov compactum X, $\dim_{\Z_{(p)}}X=\dim_{\Q}X$ for all but finitely many primes p where Z(p) is the localization…
A new GP inference method using simplices for high-dimensional data.
problem Scalable Gaussian Processes in high dimensions.
method Developed a Simplex-GP method using a sparse simplicial grid to accelerate MVMs.
result Significantly faster GP inference in high dimensions compared to SKI.
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…
The study examines compact spaces resolvable by p-adic actions.
problem Resolving compact spaces by p-adic actions.
method Free p-adic actions on compact spaces of lower dimension.
result Compact spaces with cohomological dimension 1 under Z[1/p].
SKI accelerates GP inference with sparse grids to handle higher dimensions.
problem SKI scales poorly in high dimensions due to dense grid size.
method Sparse grids within SKI framework, novel matrix-vector multiplication algorithm.
result SKI can be scaled to higher dimensions while maintaining accuracy.
This paper analyzes error in SKI for Gaussian Processes, providing conditions for linear time inference.
problem Lack of rigorous theoretical error analysis for SKI.
method Proved error bounds for SKI Gram matrix, examined error effects, provided practical guidelines.
result Identified two dimensionality regimes for SKI's scalability-accuracy trade-offs.
Study examines diversification of mid-mountain ski tourism.
problem Understanding transformations in ski mid-mountain territories.
method Applied regional diversification theory to French ski areas.
result Identified three steps in tourism diversification paths.
The study describes how topological properties of Markov compacta can be inferred from their diagrammatic structures.
problem Detecting topological properties of Markov compacta using combinatorial diagrams.
method Developed a formalism to describe Markov compacta with finite sets of diagrams, linking topological properties to combinatorial structures.
result Topological properties of Markov compacta can be inferred from the combinatorial properties of their diagrams.
The paper improves a result about 2D ANR spaces by proving full-valuedness for certain metric compacta.
problem Proving full-valuedness for compact metric spaces, especially two-dimensional ones.
method Introducing and analyzing the homological dimension dG for metric compacta. result Two-dimensional lc^2 metric compacta are dimensionally full-valued.
Grey-box model combines GP with motion data to analyze skiing forces.
problem Analyzing forces in cross-country skiing races.
method Combines motion model formulae with Gaussian process regression.
result Grey-box approach reduces predictive uncertainty by 30-40%.
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.
problem Computing large-scale magnetic field maps in indoor environments.
method Structured kernel interpolation (SKI) with derivatives (D-SKI) for Gaussian process regression.
result Achieves better accuracy and faster computation than state-of-the-art methods.
Fine shape of local compacta represented by ordinary maps.
problem Representing fine shape of local compacta.
method Constructing a space ∣X∣ for each local compactum X such that fine shape classes correspond to homotopy classes of maps to ∣X∣. result Fine shape classes from any locally compact metrizable space Y to X bijectively correspond to homotopy classes of maps from Y to ∣X∣. Project uses machine learning to identify skiers' techniques from power meter data.
problem Identifying skiers' techniques from power meter data.
method Machine learning, specifically LSTM neural networks, applied to time-series data.
result 95% accuracy in classifying skiers' techniques with a subset of data.
Defines finite type Multivalued Shape using hyperspaces.
problem No specific problem stated; focuses on definition.
method Construction over metric compacta using hyperspaces.
result Defines finite type Multivalued Shape.
An important "stability" theorem in shape theory, due to D.A. Edwards and R. Geoghegan, characterizes those compacta having the same shape as a finite CW complex. In this note we present straightforward and self-contained proof of that theorem.
New embeddings show answer to Baker-Laidacker question can be yes or no.
problem Answer to Baker-Laidacker question about disjoint compacta in R^N.
method Use of specific wild Cantor sets and Antoine's methods.
result Answer to Baker-Laidacker question can be twofold.
SoftKI combines SKI and variational methods for scalable GP regression.
problem Scalable Gaussian Process regression on high-dimensional datasets.
method SoftKI approximates kernel via softmax interpolation from a smaller number of learned points.
result SoftKI is competitive with other approximated GP methods for modest data dimensions.
Finite approximations help reconstruct countable metric and ultrametric spaces.
problem Reconstructing countable metric and ultrametric spaces.
method Topological reconstruction using inverse limits of finite T0 spaces. result Countable metric and ultrametric spaces can be reconstructed as finite approximations.
New compacta with unique embedding properties found.
problem Embedding properties of compacta in high dimensions.
method Using sticky Cantor sets and sequences of compacta.
result Construction of compacta with specific embedding properties.
We present some results on n-dimensional compacta lying in n-dimensional products of compacta, in particular, in products of n 1-dimensional compacta. Most of our basic results are proven under the assumption that the compacta X admit essential maps into the n-sphere. The results of the present paper may be viewed as a…
The paper proposes calibration to improve algorithm performance using machine learning predictions.
problem Improving real-world performance of online algorithms with machine learning predictions.
method Calibration as a tool to bridge the gap between prediction uncertainty and algorithm design.
result Calibrated advice leads to more effective guidance in high-variance settings and significant performance improvements in real-world data.
SKI speeds up Toeplitz Neural Networks by avoiding explicit decay bias and using frequency response.
problem Efficiently compute and update Toeplitz matrices in neural networks.
method Sparse plus low-rank decomposition, asymmetric SKI, frequency response modeling.
result Achieved significant speedup with minimal performance loss.
New technique speeds up Gaussian process inference for large datasets.
problem Efficient inference for Gaussian processes in high dimensions.
method Product Kernel Interpolation (SKI) with iterative methods exploiting matrix-vector multiplications.
result Linear runtime with dimension for SKI, state-of-the-art asymptotic complexity for multi-task GPs.
New Coxeter groups yield n-dimensional Sierpiński boundaries.
problem Creating Coxeter groups with specific boundary shapes.
method Defined a class of right-angled Coxeter systems and provided conditions for their boundaries.
result Coxeter groups produce boundaries homeomorphic to n-dimensional Sierpiński compacta.
Spaces are classified as almost homology n-manifolds if their homology groups are trivial for all but the last dimension.
problem Classifying spaces as almost homology n-manifolds.
method Providing a necessary and sufficient condition for locally compact homogeneous ANR-spaces or strongly locally homogeneous ANR-spaces to be almost homology n-manifolds.
result Spaces are classified as almost homology n-manifolds based on their homology groups.
The paper confirms properties of homogeneous ANR compacta and their homological similarities.
problem Properties of homogeneous ANR compacta and their homological similarities.
method Analyzes local homological properties and cyclicity of homogeneous ANR compacta.
result Homogeneous ANR compacta have similar local homological properties to closed balls and their boundaries.
The paper explores homological dimensions and dimensional full-valuedness in metric compacta.
problem Investigating homological dimensions and dimensional full-valuedness in metric compacta.
method Comparing Čech and Steenrod homology dimensions, proving dimensional full-valuedness for specific conditions.
result Two-dimensional metric compacta with specific properties satisfy dimensionally full-valuedness.
We construct an isotopy of a planar compactum that is not the restriction of an isotopy of any planar continuum.
Applying a theorem due to Belopol'ski and Birman, we show that the Laplace-Beltrami operator on 1-forms on Rn endowed with an asymptotically Euclidean metric has absolutely continuous spectrum equal to [0,+∞).
Compacta X and Y are said to admit a stable intersection in R^n if there are maps f : X -> R^n and g : Y -> R^n such that for every sufficiently close continuous approximations f' : X -> R^n and g' : Y -> R^n of f and g we have f'(X)\cap g'(Y)\neq\emptyset. The well-known conjecture asserting that X and Y do not admit …
Classifies compact spaces by shape, finite spaces by weak homotopy.
problem Classifying compact Hausdorff spaces and finite topological spaces.
method Constructs a category that classifies spaces by shape and weak homotopy.
result Classifies compact spaces by shape, finite spaces by weak homotopy.
Let L be a countable and locally finite CW complex. Suppose that the class of all metrizable compacta of extension dimension not greater than L contains a universal element which is an absolute extensor in dimension L. Our main result shows that L is quasi-finite.
Extends Palais' theory to locally compact groups, proving slice properties.
problem Extending compact group action results to locally compact groups.
method Proves slice properties for proper actions of locally compact groups.
result Action map and slicing map properties for locally compact groups.
Cencelj and Dranishnikov showed that for certain nilpotent groups G, K(Gab,1)∈AE(X) is equivalent to K(G,1)∈AE(X) for any compacta X (here Gab is the abelianization of G). We examine the same problem for solvable groups. We also give an elementary proof of this fact for any nilpo…
We give a short answer to the question in the title: {\em dendrits}. Precisely we show that the C∗-algebra C(X) of all complex-valued continuous functions on a compactum X is projective in the category C1 of all (not necessarily commutative) unital C∗-algebras if and only if X is a…
Paper reconstructs compact metric spaces using finite approximations and inverse persistence.
problem Reconstructing the homotopy type of compact metric spaces.
method Using inverse limit of finite approximations and inverse persistence.
result Definition of inverse persistence as a new persistence process.
Study local properties of homogeneous ANR-spaces, proving dimension full-valuedness.
problem Understand local structure of homogeneous ANR-spaces.
method Describe local structure and use it to prove dimension full-valuedness.
result Every finite-dimensional homogeneous metric ANR-compactum is dimensionally full-valued.
Optimal hashing embeddings reduce linear least squares solving time.
problem Efficiently solving large-scale linear least squares problems.
method Optimal hashing sketching matrices for linear least squares.
result Ski-LLS outperforms state-of-the-art solvers on various problem types.
Fine shape theory extends strong shape to noncompact metrizable spaces.
problem Computational complexity in extending strong shape to noncompact spaces.
method Introducing FDR-embeddings and mapping cylinders to extend SSDR-maps to noncompact spaces.
result Fine shape category can be represented as a left fraction localization.
Smooth knots can be embedded into a specific Menger continuum.
problem Embedding smooth knots into a specific type of continuum.
method Explicit construction using cubical models and self-similarity of the Menger continuum.
result Every smooth knot can be isotoped into the Menger continuum.
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…
It is an open question (Pawlikowski) whether every finitely generated group can be realized as a fundamental group of a compact metric space. In this paper we prove that any countable group can be realized as the fundamental group of a compact subspace of four dimensional Euclidean space. According to theorems of Shela…
The paper uses polyhedral expansions to capture the shape of compact metric spaces.
problem Capturing the shape of compact metric spaces using finite approximations.
method Inverse sequences of polyhedra based on finite approximations of a compact metric space.
result Proves the General Principle and computes inverse persistent homology groups.
Steenrod homotopy theory is a framework for doing algebraic topology on general spaces in terms of algebraic topology of polyhedra; from another viewpoint, it studies the topology of the lim^1 functor (for inverse sequences of groups). This paper is primarily concerned with the case of compacta, in which Steenrod homot…