Unified proof of Nielsen-Thurston classification via Teichmüller's theorem.
problem Classifying mapping class groups and rational maps.
method Unified proof following Bers' approach.
result Unified proof of Nielsen-Thurston classification.
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48 results for “Nielsen-Thurston classification”
For each fixed n>=2 we show how the Nielsen-Thurston classification of mapping classes for a closed surface of genus g>=2 is determined by the sequence of quantum SU(n) representations, when one considers all levels. That this is the case is a consequence of our asymptotic faithfulness property. We here provide explici…
Study of homeomorphisms on infinite type surfaces with a classification theorem.
problem Classifying homeomorphisms on surfaces of infinite type.
method Introduce tame homeomorphisms and prove a Nielsen-Thurston type classification theorem.
result For tame homeomorphisms, surfaces decompose into invariant subsurfaces with canonical decompositions.
We classify elements of a cluster modular group into three types. We characterize them in terms of fixed point property of the action on the tropical compactifications associated with the corresponding cluster ensemble. The characterization gives an analogue of the Nielsen-Thurston classification theory on the mapping …
We show that the Nielsen-Thurston classification of mapping classes of the sphere with four marked points is determined by the quantum SU(n)-representations, for any fixed integer . In the Pseudo-Anosov case we also show that the stretching factor is a limit of eigenvalues of (non-unitary) SU(2)-TQFT represen…
In this article, we propose two algorithms for determining the Nielsen-Thurston classification of a mapping class on a surface . We start with a finite generating set for the mapping class group and a word in . We show that if represents a reducible mapping class in $\Mod(S)$ then …
We study the action of the mapping class group on the real homology of finite covers of a topological surface. We use the homological representation of the mapping class to construct a faithful infinite-dimensional representation of the mapping class group. We show that this representation detects the Nielsen-Thurston …
Study shows certain diffeomorphisms cannot be dynamically coherent.
problem Dynamically coherent behavior in partially hyperbolic diffeomorphisms.
method Analyzes pseudo-Anosov components and Nielsen-Thurston classification.
result Extends previous work to larger class of diffeomorphisms.
Algorithm classifies surface homeomorphisms with polynomial time complexity.
problem Classifying surface homeomorphisms with polynomial time complexity.
method Algorithm to compute curve distances and decide Nielsen-Thurston types.
result Polynomial time classification of surface homeomorphisms.
We prove a conjecture due to Makanin: if a and b are elements of the Artin braid group B_n such that a^k=b^k for some nonzero integer k, then a and b are conjugate. The proof involves the Nielsen-Thurston classification of braids.
We give a proof of the Neilsen-Thurston classification theorem of a homeomorphism f of a standard surface of finite type as either periodic, pseudo-Anosov, or reducible. In the periodic case, we show that there exists an integer n>0 such that f is isotopic to h with h^n isotopic to the identity. This is the weaker vers…
The paper compactifies stability conditions on curves, akin to Teichmüller theory.
problem Compactifying spaces of stability conditions on curves.
method Constructing a Thurston-like compactification.
result Obtained a compactification of stability conditions space, analogous to Teichmüller theory.
The paper counts mapping classes by Nielsen-Thurston type, finding growth rates for different subsets.
problem Counting mapping classes in Teichmüller space with different subsets.
method Introduced complexity length to measure negative curvature of curve complexes.
result Growth rates for finite-order, reducible, and multitwists subsets.
The relationships between braid ordering and the geometry of its closure is studied. We prove that if an essential closed surface in the complements of closed braid has relatively small genus with respect to the Dehornoy floor of the braid, is circular-foliated in a sense of Birman-Menasco's Braid foliation the…
We develop an analogy between right-angled Artin groups and mapping class groups through the geometry of their actions on the extension graph and the curve graph respectively. The central result in this paper is the fact that each right-angled Artin group acts acylindrically on its extension graph. From this result we …
We describe a polynomial-time algorithm to compute a (tight) geodesic between two curves in the curve graph. As well as enabling us to compute the distance between a pair of curves, this has several applications to mapping classes. For example, we can use these geodesics to compute the asymptotic translation length, Ni…
New periodic solutions found in 2n-body problem, braids of pseudo-Anosov type with stretch factors as metallic ratios.
problem Periodic solutions of the 2n-body problem and their braid types.
method Analyzing braid types and stretch factors associated with pseudo-Anosov braids.
result Braids from new periodic solutions are of pseudo-Anosov type with stretch factors as metallic ratios.
We introduce an essential open book foliation, a refinement of the open book foliation, and develop technical estimates of the fractional Dehn twist coefficient (FDTC) of monodromies and the FDTC for closed braids, which we introduce as well. As applications, we quantitatively study the `gap' of overtwisted contact str…
The study connects periodic surface homeomorphisms to contact structures using rational open books.
problem Understanding the properties of contact structures associated with periodic surface homeomorphisms.
method Associate rational open books to marked data sets, study contact structures, and prove Stein fillability conditions.
result A class of data sets gives rise to Stein fillable contact structures under certain combinatorial conditions.
Let be a finite index subgroup of the mapping class group of a closed orientable surface , possibly with punctures. We give a precise condition (in terms of the Nielsen-Thurston decomposition) when an element has positive stable commutator length. In addition, we show that in these situations th…
Benardete, Gutierrez and Nitecki showed an important result which relates the geometrical properties of a braid, as a homeomorphism of the punctured disk, to its algebraic Garside-theoretical properties. Namely, they showed that if a braid sends a curve to another curve, then the image of this curve after each factor o…
Fibered commensurability on math.GT
We define and discuss a notion called fibered commensurability of outer automorphisms of free groups. This notion lets us study symmetry of outer automorphisms. The notion of fibered commensurability is first defined by Calegari-Sun-Wang on mapping class groups. The Nielsen-Thurston type of mapping classes is a commens…
We show that reducible braids which are, in a Garside-theoretical sense, as simple as possible within their conjugacy class, are also as simple as possible in a geometric sense. More precisely, if a braid belongs to a certain subset of its conjugacy class which we call the stabilized set of sliding circuits, and if it …
New 4D homeomorphism shows surprising similarities to 2D.
problem Finding a 4D pseudo-Anosov homeomorphism.
method Building a specific example of a 5-manifold fibered over the circle.
result Compact 4-manifold with unique immersed and embedded surfaces.
Let be one of the Artin groups of finite type , and affine type and . In this paper, we show that if and are elements of such that for some nonzero integer , then and are conjugate in . For the Artin …
Study growth rates of automorphisms of special groups.
problem Understanding the growth rates of automorphisms of special groups.
method Analyzing outer automorphisms of virtually special groups, showing polynomial or exponential growth, and constructing Nielsen-Thurston decompositions.
result Outer automorphism groups of virtually special groups are boundary amenable, have finite virtual cohomological dimension, and satisfy the Tits alternative.
Maximizes mixing efficiency in surface braids.
problem Finding the maximum mixing efficiency in surface braids.
method Introduced an efficient algorithm to compute topological entropy and TEPO for surface braids.
result Conjectured a novel candidate braid to have maximal mixing efficiency.
We analyze the mapping class group of extendible automorphisms of the exterior boundary W of a compression body of dimension 3 or 4, which extend over the compression body (Q,V), where V is the interior boundary. Those that extend as automorphisms of (Q,V) rel V are called discrepant automorphisms, forming the mapping …
Characterizes braid group actions on R and mapping class group actions on S1.
problem Understanding the rigidity of braid group actions on R and mapping class group actions on S1.
method Using the space of left orderings of B_n, isolated points, and conjugacy action of B_n.
result Characterizes actions of B_n on R that produce translation numbers agreeing with the standard action.
The thesis shows how automorphisms of hyperbolic groups can be represented by train track maps.
problem Representing automorphisms of hyperbolic groups using train track maps.
method Using graphs of groups and Bestvina-Handel's irreducible train track maps, the thesis constructs relative train track maps.
result Outer automorphisms of finitely-generated word hyperbolic groups satisfy a dynamical trichotomy.
Study of quadratic form associated with surface automorphisms and its applications to singularity theory.
problem Understanding the properties of quadratic forms associated with surface automorphisms and their applications to singularity theory.
method Using techniques from mapping class group theory, the authors associate a quadratic form and prove its properties using the twist formula.
result The form is positive definite under certain conditions and even in others, providing numerical invariants to distinguish different topological types of singularities.
Dual-stage sEMG classification improves gesture recognition accuracy.
problem Improving accuracy in hand gesture recognition from sEMG signals.
method Dual-stage classification approach: first stage groups similar activities, second stage classifies within groups.
result Dual-stage classification yields significantly higher accuracy than single-stage approach.
A novel method for classification with rejection using ensemble of cost-sensitive classifiers.
problem Avoid risky misclassification in error-critical applications.
method Learning an ensemble of cost-sensitive classifiers.
result Improved classification accuracy and flexibility in loss selection.
Multi-borders classificationstat.ML
The number of possible methods of generalizing binary classification to multi-class classification increases exponentially with the number of class labels. Often, the best method of doing so will be highly problem dependent. Here we present classification software in which the partitioning of multi-class classification…
Sequence classification is an important data mining task in many real world applications. Over the past few decades, many sequence classification methods have been proposed from different aspects. In particular, the pattern-based method is one of the most important and widely studied sequence classification methods in …
New NHCAs improve multi-category classification efficiency.
problem Efficient multi-category classification for real-world problems.
method Twin SVM (TWSVM), Generalized eigenvalue proximal SVM (GEPSVM), Regularized GEPSVM (RegGEPSVM), and Improved GEPSVM (IGEPSVM) with OAA, BT, and TDS approaches.
result TDS-TWSVM outperforms other methods in classification accuracy.
Classification outperforms regression in portfolio construction, yielding higher Sharpe ratios.
problem Determining which machine learning approach (classification vs. regression) is more effective for portfolio construction.
method Used stacking ensemble of gradient boosted tree, random forest, and neural network models.
result Classification yields higher Sharpe ratios and economically significant alphas compared to regression.
C-HMCNN(h) improves HMC classification by leveraging class hierarchy.
problem Hierarchical multi-label classification with class hierarchy constraints.
method Exploits class hierarchy to produce coherent predictions for multi-label classification.
result C-HMCNN(h) outperforms state-of-the-art models in HMC classification.
Advances few-shot classification by treating it as supervised learning and proposing new training techniques.
problem Formulating the ability of humans to learn from limited data in machine learning.
method Formulated few-shot classification as a supervised learning problem and introduced multi-episode and cross-way training techniques.
result Proposed training strategies accelerate the training process without accuracy loss.
We study realizations of Lie algebras by vector fields. A correspondence between classification of transitive local realizations and classification of subalgebras is generalized to the case of regular local realizations. A reasonable classification problem for general realizations is rigorously formulated and an algori…
New approach improves classification guarantees by focusing on direction rather than regression risk.
problem Improving classification guarantees in binary classification problems.
method Establishing a geometric distinction between classification and regression, leveraging scale invariance.
result Improved guarantees for classification risk compared to regression risk.
Study selective classification with halfspaces, achieving error bounds under Gaussian distributions.
problem Modeling relationships in subsets of data defined by selection rules.
method Sparse linear classifiers for subsets defined by halfspaces, focusing on Gaussian feature distributions.
result First PAC-learning algorithm for homogeneous halfspace selectors with error guarantee $\bigO*{\sqrt{\mathrm{opt}}}$.
A new network-based high-level data classification method using betweenness centrality.
problem Traditional data classification techniques focus on physical features, while high-level classification considers semantic meaning.
method Proposes a network-based high-level classification technique using betweenness centrality.
result Competent classification performance in nine real datasets compared to traditional models.
This thesis evaluates text-based vs audio-based classification of mental health interviews.
problem Classifying psychiatric illness using text-based methods.
method Design and evaluate a text classification network on mental health interviews, using belabBERT.
result Text-based classification is a strong alternative to audio-based methods.
Study on error probability for classification of heavy-tailed renewal processes.
problem Error probability in classification of heavy-tailed renewal processes.
method Asymptotic expressions for Bhattacharyya bound on misclassification error probabilities.
result Obtained asymptotic expressions for misclassification error probabilities.
Directly compute classification by learning features with class scores.
problem Classification efficiency and accuracy on various datasets.
method PCA for feature encoding, supervised learning model with encoder-decoder structure.
result Effective classification performance on multiple datasets.
This review explores resampling techniques for imbalanced binary classification.
problem Imbalanced classes lead to poor prediction results in classification.
method Classical, cost-sensitive, and Neyman-Pearson paradigms with resampling techniques and classification methods.
result Complex dynamics among resampling techniques, base methods, metrics, and imbalance ratios.
Conventional techniques for supervised classification constrain the classification rules considered and use surrogate losses for classification 0-1 loss. Favored families of classification rules are those that enjoy parametric representations suitable for surrogate loss minimization, and low complexity properties suita…