The family of translation surfaces constructed by Arnoux and Yoccoz from self-similar interval exchange maps encompasses one example from each genus greater than or equal to . We triangulate these surfaces and deduce general properties they share. The surfaces converge to a surface $(X_\i…
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48 results for “affine interval exchange maps”
The paper introduces horizon saddle connections to study dilation surfaces.
problem Understanding the geometric and dynamical properties of dilation surfaces.
method Introducing horizon saddle connections and quasi-Hopf surfaces.
result Existence of horizon saddle connections restricts the Veech group of dilation surfaces.
We study the ergodic properties of compositions of interval exchange transformations and rotations. We show that for any interval exchange transformation T, there is a full measure set of αin [0, 1) so that T composed with R_α is uniquely ergodic, where R_α is rotation by α.
A natural generalization of interval exchange maps are linear involutions, first introduced by Danthony and Nogueira. Recurrent train tracks with a single switch which we call non-classical interval exchanges, form a subclass of linear involutions without flips. They are analogs of classical interval exchanges, and are…
A natural generalization of interval exchange maps are linear involutions, first introduced by Danthony and Nogueira. Recurrent train tracks with a single switch provide a subclass of linear involutions. We call such linear involutions non-classical interval exchanges. They are related to measured foliations on orienta…
In [Mas82] and [Vee78] it was proved independently that almost every interval exchange transformation is uniquely ergodic. The Birkhoff ergodic theorem implies that these maps mainly have uniformly distributed orbits. This raises the question under which conditions the orbits yield low-discrepancy sequences. The case o…
Book explores infinite translation surfaces, challenging traditional geometry.
problem Understanding infinite-type translation surfaces.
method Detailed classification, construction, and analysis of symmetries.
result Complex dynamics in infinite-type translation flows.
Interval exchange maps are related to geodesic flows on translation surfaces; they correspond to the first return maps of the vertical flow on a transverse segment. The Rauzy-Veech induction on the space of interval exchange maps provides a powerful tool to analyze the Teichmueller geodesic flow on the moduli space of …
Study of SL(2,R) representations on a once-punctured torus, showing Cantor set spectrum.
problem Characterizing SL(2,R) representations on a once-punctured torus.
method Introduction of spectrum as a subset of projective measured laminations, analysis of dynamics of cocycles.
result Spectrum of a generic representation on a once-punctured torus is a Cantor set.
We show that there exists an interval exchange and a point so that the orbit of the point equidistributes for a measure that is not ergodic.
This paper presents quantitative shrinking target results for rotations and interval exchange transformations. To do this a quantitative version of a unique ergodicity criterion of Boshernitzan is established.
Foam cobordism groups linked to interval exchange automorphisms.
problem Understanding cobordism groups of foams.
method Using the Sah-Arnoux-Fathi invariant and K-theory.
result Identified cobordism groups with homology and K-groups.
We study relations between Rauzy classes coming from an interval exchange map and the corresponding connected components of strata of the moduli space of Abelian differentials. This gives a criterion to decide whether two permutations are in the same Rauzy class or not, without actually computing them. We prove a simil…
Constructs tail-specific prediction intervals for financial applications
problem Financial applications require strict control on the left tail
method Extends classical conformal frameworks to provide explicit tail-specific guarantees
result Improved directional calibration in skewed data
It is known since 40 years old paper by M. Keane that minimality is a generic (i.e. holding with probability one) property of an irreducible interval exchange transformation. If one puts some integral linear restrictions on the parameters of the interval exchange transformation, then minimality may become an "exotic" p…
Develops efficient time series prediction intervals.
problem Constructing reliable prediction intervals for time series data.
method Introduces exttt{EnbPI} algorithm for time series data.
result Demonstrates superior performance compared to existing methods.
An online framework optimizes efficiency in conformal prediction with a target miscoverage rate.
problem Achieving coverage and minimizing interval length in a sequential, online setting.
method Optimizes efficiency by directly optimizing the average length of intervals while maintaining coverage.
result Shows a gap between optimal performance for exchangeable and arbitrary sequences, and provides a matching algorithm for the Pareto-optimal settings.
The study examines distortion in specific homeomorphisms of Cantor sets.
problem Distortion in homeomorphisms of Cantor sets.
method Analyzes equivalence of conditions related to discontinuities and conjugacy.
result Elements are distorted if they satisfy certain conditions.
Novel method for time-series prediction with tighter confidence intervals.
problem Improving prediction intervals for time-series data.
method Kernel-based Optimally Weighted Conformal Prediction Intervals (KOWCPI) using adaptive weights.
result KOWCPI achieves narrower confidence intervals with guaranteed coverage.
Ehrenfest Model with Large Jumps in Financecond-mat.stat-mech
Changes (returns) in stock index prices and exchange rates for currencies are argued, based on empirical data, to obey a stable distribution with characteristic exponent for short sampling intervals and a Gaussian distribution for long sampling intervals. In order to explain this phenomenon, an Ehrenfest model…
Heisenberg Translation Flowsmath.DS
Study of flows on circle bundles over translation surfaces, showing decay of correlations.
problem Ergodic properties of flows on circle bundles over translation surfaces.
method Generalizing Heisenberg nilflows to more general base surfaces, showing relatively mixing.
result Showed that such flows exhibit decay of correlations in the orthogonal complement of functions constant along fibers.
We prove the existence of "half-plane differentials" with prescribed local data on any Riemann surface. These are meromorphic quadratic differentials with higher-order poles which have an associated singular flat metric isometric to a collection of euclidean half-planes glued by an interval-exchange map on their bounda…
Method constructs prediction intervals for time-varying individual treatment effects.
problem Accurately quantify uncertainty of individual treatment effects across multiple decision points.
method Conformal inference techniques for time-varying ITEs with weaker assumptions.
result Guaranteed lower bound for coverage dependent on data non-exchangeability.
We prove a computable version of de Finetti's theorem on exchangeable sequences of real random variables. As a consequence, exchangeable stochastic processes expressed in probabilistic functional programming languages can be automatically rewritten as procedures that do not modify non-local state. Along the way, we pro…
Graphs from van der Corput sequence embed into Chamanara surface.
problem Embedding graphs from van der Corput sequence into surfaces.
method Constructed -regular graphs from van der Corput sequence and Kronecker sequence, embedded into torus and Chamanara surface.
result Graphs from van der Corput sequence embed into Chamanara surface with one edge removal.
The ordinary (or classical) Birman-Wenzl-Murakami algebras were initially conceived as an algebraic framework for the Kauffman link invariant. They also appear as centralizer algebras for representations of quantum universal enveloping algebras of orthogonal or symplectic types. It was shown by Morton and Wassermann th…
A new algorithm for time series prediction intervals.
problem Non-exchangeability in time series data.
method Adaptive re-estimation of non-conformity scores.
result Significant reduction in interval width compared to existing methods.
Introduces mobility algebra for modeling geodesics on n-spheres.
problem Modeling geodesics on n-spheres using algebraic structures.
method Introduces mobility algebra and mobility spaces, showing connections to modules and affine spaces.
result Shows geodesics on n-spheres as mobility spaces over unit interval mobility algebra.
Power law for the calm-time interval of price changescond-mat.stat-mech
In this paper, we describe a newly discovered statistical property of time series data for daily price changes. We conducted quantitative investigation of the {\it calm-time intervals} of price changes for 800 companies listed in the Tokyo Stock Exchange, and for the Nikkei 225 index over a 27-year period from January …
In this paper we consider convex improper affine maps of the 3-dimensional affine space and classify their singularities. The main tool developed is a generating family with properties that closely resembles the area function for non-convex improper affine maps.
Let be a non-degenerate permutation on at least symbols. We show that the set of uniquely ergodic interval exchange transformations with permutation is path-connected.
Universal approximation for ODENet and ResNet with a single activation function.
problem Approximating complex dynamical systems with limited vector fields.
method Examined ODENet and ResNet with vector fields composed of a single activation function and affine mapping.
result ODENet and ResNet with restricted vector fields can uniformly approximate those with general vector fields.
Affine maps reveal higher rank structures in certain spaces.
problem Characterizing spaces with higher rank structures.
method Using Hadamard spaces with geometric group actions and affine maps.
result Affine maps not dilations indicate higher rank structures.
Affine Harmonic Mapsmath.DG
We introduce a class of maps from an affine flat into a Riemannian manifold that solve an elliptic system defined by the natural second order elliptic operator of the affine structure and the nonlinear Riemann geometry of the target. These maps are called affine harmonic. We show an existence result for affine harmonic…
In this paper we introduce a natural definition for the affine maps between two Finsler manifolds and and we give some geometrical properties of these affine maps. Starting from the equations of the affine maps, we construct a natural Berwald-Riemann-Lagrange geometry on the 1-jet space $J^1(TM;…
The classical Fundamental Theorem of Affine Geometry states that for , any bijection of -dimensional Euclidean space that maps lines to lines (as sets) is given by an affine map. We consider an analogous characterization of affine automorphisms for compact quotients, and establish it for tori: A bijection o…
Fluctuations in time intervals of financial data from the view point of the Gini indexphysics.data-an
We propose an approach to explain fluctuations in time intervals of financial markets data from the view point of the Gini index. We show the explicit form of the Gini index for a Weibull distribution which is a good candidate to describe the first passage time of foreign exchange rate. The analytical expression of the…
Paper proves certain closed affine manifolds without invariant lines don't exist.
problem Proving non-existence of closed affine manifolds with invariant lines.
method Developing map, holonomy, invariant line, large open subsets, modified proof.
result Developing image cannot meet invariant line if affine holonomy acts purely by translations.
Conformalized Super Learnerstat.ML
A new method combines conformal prediction with Super Learner for interval predictions.
problem Constructing reliable interval predictions for complex regression functions.
method Coupling conformal prediction with Super Learner framework.
result The conformalized SL achieves valid finite-sample coverage with competitive performance.
Two methods for quantile regression are compared and found to produce tighter intervals.
problem Comparing methods for producing prediction intervals in quantile regression.
method Two recently proposed methods combining conformal inference and quantile regression.
result Romano et al.'s method typically yields tighter prediction intervals in finite samples.
Flexible method for estimating frequencies in large datasets using sketching.
problem Estimating frequencies in very large datasets efficiently and accurately.
method Data-adaptive conformal inference method based on a smaller sketch of data.
result Proves validity of frequentist confidence intervals under data exchangeability.
Associated Derived Invariants for Geometric Mappings of Non-Symmetric Affine Connection Spacesmath.DG
New invariants found for mappings between non-symmetric affine spaces.
problem Finding new invariants for mappings between non-symmetric affine spaces.
method Obtained invariants using factored deformation tensor and novel Weyl type invariants.
result Novel Weyl type invariants for mappings between non-symmetric affine spaces.
Nonstationary Increments, Scaling Distributions, and Variable Diffusion Processes in Financial Marketsphysics.soc-ph
Arguably the most important problem in quantitative finance is to understand the nature of stochastic processes that underlie market dynamics. One aspect of the solution to this problem involves determining characteristics of the distribution of fluctuations in returns. Empirical studies conducted over the last decade …
Practical or scientific considerations often lead to selecting a subset of parameters as ``important.'' Inferences about those parameters often are based on the same data used to select them in the first place. That can make the reported uncertainties deceptively optimistic: confidence intervals that ignore selection g…
Revisits Jarrow & Turnbull model for credit and liquidity risk.
problem Modeling credit and liquidity risk in financial markets.
method Uses foreign exchange analogy and partially observable exchange rate.
result Derives tractable term structure models and explicit valuation formulae.
Modeling stock price dynamics on the Ghana Stock Exchange: A Geometric Brownian Motion approachmath.OC
Study finds GBM model accurately predicts stock prices on Ghana Stock Exchange.
problem Investigating the suitability of GBM for modeling stock price dynamics.
method Geometric Brownian Motion model applied to weekly and monthly returns of equities listed on the Ghana Stock Exchange.
result GBM model accurately forecasts stock prices with minimal deviations, as evidenced by MSE evaluations.
Solitons are special polygon midpoints under affine transformations.
problem Characterizing polygons whose midpoints under affine transformations form a new polygon.
method Analyzing midpoints polygons and their relationship to affine transformations and differential equations.
result A large class of polygons are on an orbit of a one-parameter subgroup of the affine group, and these curves are solutions to a specific differential equation.
New algorithm reveals piecewise affine structure of neural networks.
problem Lack of strong guarantees on deep neural networks' behavior in safety-critical applications.
method Developed a novel algorithm to compute the piecewise affine form of neural networks.
result Computed piecewise affine representations of neural networks with rectified linear unit activations.