Method counts connected 2D stratifolds with singular curves and components.
problem Counting -connected trivalent -stratifolds.
method Describes a method for counting.
result Counts -connected trivalent -stratifolds.
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48 results for “connectivity counts”
Counted essential surfaces in a knot's exterior, finding a unique pattern.
problem Counting essential surfaces in a knot's exterior.
method Counted essential surfaces by genus, using Euler totient function. Showed normal surfaces are connected by counting their components. Used Agol, Hass, and Thurston's tools to convert component counting into orbit counting.
result Found a unique pattern in the number of essential surfaces by genus.
Estimates surface count with prescribed foliations.
problem Counting square-tiled surfaces with specific foliations.
method Effective estimate with power saving error term.
result Strengthens asymptotic counting formulas.
The abstract reviews models for analyzing count data.
problem Challenges in analyzing count data with standard methods.
method Review of generalized linear models and multinomial models.
result Fundamental connections between multinomial and count models.
Researchers calculate complexity of billiard paths in regular polygons.
problem Calculating the complexity of billiard paths in regular polygons.
method Counting saddle connections on lattice surfaces, focusing on combinatorial length.
result They answered a question about billiard language complexity in regular polygons.
Given a rank 2 hermitian bundle over a 3-manifold that is non-trivial admissible in the sense of Floer, one defines its Casson invariant as half the signed count of its projectively flat connections, suitably perturbed. We show that the 2-divisibility of this integer invariant is controlled in part by a formula involvi…
A gamma process dynamic Poisson factor analysis model is proposed to factorize a dynamic count matrix, whose columns are sequentially observed count vectors. The model builds a novel Markov chain that sends the latent gamma random variables at time as the shape parameters of those at time , which are linked …
Study on counting flat connections over -orbifolds, proving moduli space compact and smooth.
problem Counting flat connections over -orbifolds.
method Analyzing moduli space of -instantons on flat bundles over torsion-free -orbifolds.
result Proves moduli space compact and smooth at the irreducible locus.
CNN estimates graphlet counts efficiently from historic graphs.
problem Difficulty in computing exact graphlet counts due to exponential growth.
method Convolutional Neural Network (CNN) framework with preprocessing techniques.
result Substantial speedup and high accuracy in estimating graphlet counts.
Better neural arithmetic logic units improve cell counting model generalization.
problem Neural networks struggle with high cell counts outside training data range.
method Introduced Neural Arithmetic Logic Units (NALU) for arithmetic operations in existing architectures.
result Improved cell counting accuracy for higher numeric ranges with better generalization.
The study counts Salem numbers linked to arithmetic hyperbolic orbifolds.
problem Bounding the proportion of Salem numbers in arithmetic lattices.
method Using results on the distribution of Salem numbers, classical methods for counting Pythagorean triples, and Gauss' lattice-counting argument.
result Improved bounds on the proportion of Salem numbers and strong exponential growth of averages.
Counting Higgs bundles for specific groups on Riemann surfaces.
problem Counting connected components of Higgs bundle moduli spaces for various groups.
method Use Cayley correspondence to count connected components of Higgs bundle moduli spaces.
result Count the number of connected components for maximal Higgs bundles over Riemann surfaces for specified groups.
Anosov groups study matrix coefficients and orbit counting in symmetric spaces.
problem Anosov groups and their matrix coefficients in symmetric spaces.
method Asymptotic analysis of matrix coefficients and higher rank measures.
result Asymptotic behavior of matrix coefficients and orbit counting results.
The study counts non-crossing permutations on surfaces of any genus.
problem Counting non-crossing permutations on surfaces of any genus.
method Polygon diagrams and arc diagrams are used to represent non-crossing permutations. The count of these diagrams exhibits interesting polynomial behavior, with leading coefficients related to intersection numbers on moduli spaces.
result The count of polygon diagrams is almost polynomial in the number of points, with leading coefficients related to intersection numbers on moduli spaces.
Counting hyperbolic multi-geodesics with individual component lengths.
problem Counting hyperbolic multi-geodesics with specific component lengths.
method Unified geometric and topological techniques, combining Mirzakhani's results and Margulis's ideas.
result Asymptotic polynomial counts of multi-geodesics in mapping class group orbits, generalizing Wolpert's conjecture.
The seemingly disjoint problems of count and mixture modeling are united under the negative binomial (NB) process. A gamma process is employed to model the rate measure of a Poisson process, whose normalization provides a random probability measure for mixture modeling and whose marginalization leads to an NB process f…
We use the Yang-Mills gradient flow on the space of connections over a closed Riemann surface to construct a Morse-Bott chain complex. The chain groups are generated by Yang-Mills connections. The boundary operator is defined by counting the elements of appropriately defined moduli spaces of Yang-Mills gradient flow li…
Counting essential surfaces in 3-manifolds yields concise formulae and detailed asymptotics.
problem Counting isotopy classes of essential surfaces in 3-manifolds.
method Normal and almost normal surfaces, Ehrhart's lattice point counting, ideal triangulations, and new essential surface testing.
result Quasi-polynomial behavior of surface counts and concise formulae for surface numbers.
How many are linear connections with prescribed Ricci tensor? How many are statistical structures? The questions are answered in the analytic case by using the Cauchy-Kowalewski theorem.
We study the counting function of topological Poincaré series associated with rational homology sphere plumbed 3-manifold with connected negative definite tree, interpreting as an alternating sum of coefficient functions associated with some Taylor expansions. It is motivated by a theorem of Szenes and Vergne which exp…
Quandle coloring detects causality in spacetime links.
problem Detecting causality in spacetime links using quandle colorings.
method Conjugation quandle of dihedral group D5 combined with Alexander-Conway polynomial.
result The conjugation quandle over D5 distinguishes causally related and unrelated events.
Many invariants of knots rely upon smoothing the knot at its crossings. To compute them, it is necessary to know how to count the number of connected components the knot diagram is broken into after the smoothing. In this paper, it is shown how to use a modification of a theorem of Zulli together with a modification of…
Mirzakhani's thesis counts geodesics on hyperbolic surfaces, finding a specific asymptotic formula.
problem Counting simple closed geodesics on hyperbolic surfaces.
method Inspired by lattice point counting, uses principles of homogeneous dynamics.
result The number of simple closed geodesics of length ≤ L is asymptotic to L^(6g-6) times a constant.
Counting geodesics on compact symmetric spaces using orbit dimensions and topological data.
problem Counting geodesics on compact symmetric spaces.
method Using orbit dimensions and topological data of the symmetric space.
result Obtained data on dimensions and connected components of focal orbits.
This is the first in a series of papers exploring the relationship between the Rohlin invariant and gauge theory. We discuss the Casson-type invariant of a 3-manifold with the integral homology of a torus, given by counting projectively flat connections. We show that its mod 2 evaluation is given by the triple cup prod…
MRCNet tackles crowd counting and density mapping in aerial imagery.
problem Accurate crowd counting and density estimation in aerial imagery.
method MRCNet is a novel encoder-decoder CNN that combines VGG-16 with FPN-inspired lateral connections.
result MRCNet outperforms state-of-the-art methods in aerial and CCTV-based crowd counting.
odeN efficiently approximates multiple temporal motifs in large networks.
problem Efficiently counting multiple temporal motifs in large temporal networks.
method odeN is a sampling-based algorithm that provides accurate probabilistic approximations of motif counts.
result odeN provides accurate approximations of motif counts in a fraction of the time needed by state-of-the-art methods.
This paper counts specific square-tiled surfaces related to hyperbolic geodesics.
problem Counting square-tiled surfaces with specific foliations.
method Conceptual and geometric methods inspired by Mirzakhani's work.
result The number of square-tiled surfaces is asymptotic to \(L^{6g-6+2n}\) times constants from Mirzakhani's count.
We extend asymptotic formulas for saddle connections on translation surfaces.
problem Counting saddle connections on translation surfaces with large genus.
method Recursive formulas and asymptotic analysis for all strata and multiplicities.
result Asymptotics for all saddle connections on translation surfaces of growing genus.
Biracks are algebraic structures related to knots and links. We define a new enhancement of the birack counting invariant for oriented classical and virtual knots and links via algebraic structures called birack dynamical cocycles. The new invariants can also be understood in terms of partitions of the set of birack la…
Unified framework for comparing clusterings from information-theoretic and pair-counting perspectives.
problem Divergent evaluations of unsupervised models due to different clustering similarity measures.
method Developed an analytical framework that unifies pair-counting and information-theoretic clustering similarity measures.
result Unified framework clarifies when and why the two regimes diverge and provides a principled basis for selecting and interpreting clustering similarity measures.
The Orbit Group of a Quandlemath.GT
We define the notion of the orbit group of a quandle via its connectivity and compute the orbit groups for some basic quandles. We also show that the orbit group counts the number of orbits of certain quandles.
The paper introduces vortex nerve complexes and new Betti numbers in CW spaces.
problem Understanding the structure and properties of CW complexes and their nerves.
method Introducing vortex nerve complexes and defining new Betti numbers for CW complexes.
result New Betti numbers (vortex , vortex nerve , shape ) are introduced and studied.
PRGDS models count tensors with sparsity and burstiness.
problem Modeling sequential count data with sparsity and burstiness.
method Poisson-randomized gamma dynamical system with alternating Poisson and gamma latent states.
result Sparse PRGDS often outperforms other models in predicting count data.
We compute the asymptotic growth rate of the number N(C, R) of closed geodesics of length less than R in a connected component C of a stratum of quadratic differentials. We prove that for any 0 < θ< 1, the number of closed geodesics of length at most R that spend at least θ-fraction of time outside of a compact subset …
Skeins on Branesmath.SG
Mathematical study connects curve counting to quantum topology.
problem Counting holomorphic curves with Lagrangian boundary in Calabi-Yau 3-folds.
method Organized 1-parameter families of holomorphic curves and matched them to HOMFLYPT skein relations.
result Holomorphic curve counts match HOMFLYPT polynomial coefficients of links in the 3-sphere.
Research connects prime numbers, graph theory, and cohomology.
problem Understanding the distribution of prime numbers using graph theory and cohomology.
method Discrete Morse-Smale complex and cohomology analysis.
result Explicit relationships between prime counting functions and cohomological properties of graphs.
New method identifies causal structure in count data using cumulants and path analysis.
problem Challenges in discovering causal structure from count data, especially due to non-identifiability.
method Poisson Branching Structural Causal Model (PB-SCM) with path analysis using high-order cumulants.
result Causal order is identifiable under specific conditions in PB-SCM using cumulant information.
In Peña (2007), MCMC sampling is applied to approximately calculate the ratio of essential graphs (EGs) to directed acyclic graphs (DAGs) for up to 20 nodes. In the present paper, we extend that work from 20 to 31 nodes. We also extend that work by computing the approximate ratio of connected EGs to connected DAGs, of …
These are notes of lectures given at the NATO Summer School, Montreal 1995. Taubes's recent spectacular work setting up a correspondence between -holomorphic curves in symplectic 4-manifolds and solutions of the Seiberg-Witten equations counts -holomorphic curves in a somewhat new way. The "standard" theory conce…
We discuss in rather general terms quantum field theories dealing with spaces of maps between Riemannian manifolds. In particular we explore the well--known connection between the renormalization group flow for non--linear sigma models and the Ricci flow.
Survey on Higgs bundle moduli spaces and their connected components.
problem Counting connected components of Higgs bundle moduli spaces.
method Analyzes moduli spaces for Higgs bundles associated with real Lie groups and closed Riemann surfaces.
result Explicit descriptions of some moduli space components are possible.
Estimates the number of connected components in a graph from a sampled subgraph.
problem Inferring the number of connected components in a larger graph from a sampled subgraph.
method A highly redundant and large-dimensional representation of the subgraph using counts of network motifs, leading to a novel estimator for the number of connected components.
result Improves upon competing algorithms for graphs with spectral gaps bounded away from zero.
Study saddle connections on hyperelliptic surfaces, finding growth rates.
problem Count saddle connections on hyperelliptic surfaces without interior intersections.
method Used horocycle renormalization to prove lower bound growth rate.
result Found saddle connections satisfy growth rate.
Unified framework counts knot representations into SU(2) and SL(2,R).
problem Counting knot representations into SL(2,R) with fixed holonomy.
method Unified framework using geometric transition and character varieties.
result SL(2,R) count determined by SU(2) count and integer h(K).
Study on moduli spaces of negatively curved metrics on surfaces.
problem Understanding the structure of moduli spaces of uniformly negatively curved metrics on surfaces.
method Construction of locally constant functionals based on geodesic string counts.
result Moduli space of metrics on is disconnected.
On the modelling of multivariate counts with Cox processes and dependent shot noise intensitiesq-fin.RM
Develops a method to model multivariate count processes with Cox processes and shot noise intensities.
problem Modeling and estimating dependent count processes using granular data.
method Multivariate Cox process with shot noise intensities, connected via Lévy copulas.
result Allows for over-dispersion, auto-correlation, and realistic features in count processes.
Study counts sub-chord diagrams to classify spherical curves.
problem Classifying spherical curves using chord diagrams.
method Counting sub-chord diagrams under specific moves.
result New invariant classifies prime reduced spherical curves.