In the current article we study complex cycles of higher multiplicity in a specific polynomial family of holomorphic foliations in the complex plane. The family in question is a perturbation of an exact polynomial one-form giving rise to a foliation by Riemann surfaces. In this setting, a complex cycle is defined as a …
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48 results for “cycle complexity”
The current article studies certain problems related to complex cycles of holomorphic foliations with singularities in the complex plane. We focus on the case when polynomial differential one-form gives rise to a foliation by Riemann surfaces. In this setting, a complex cycle is defined as a nontrivial element of the f…
Constructs an explicit cycle in arithmetic group cohomology.
problem Cohomology of SL at virtual cohomological dimension.
method Geometric rigidity of Voronoi tessellations and abstract framework for polyhedral tessellations.
result Explicit canonical cycle in top-dimensional homology of Voronoi complex.
Artin groups of type have special cycles and complexes with interesting properties.
problem Characterizing cycles and complexes in Artin groups of type .
method Analyzing 6-cycles and their centers/quasi-centers in the 1-skeleton of the Artin complex.
result Certain 6-cycles in the Artin complex of type have centers or quasi-centers.
Wrapping Cycles in Delaunay Complexes: Bridging Persistent Homology and Discrete Morse Theorymath.AT
New connection found between shape reconstruction methods and persistent homology.
problem Connecting shape reconstruction methods with persistent homology.
method Wrap complexes and lexicographic optimal homologous cycles.
result Lexicographically optimal homologous cycles are supported on Wrap complexes.
The paper introduces vortex cycles and nerves, inspired by Thomson's vortex atoms.
problem Understanding vortex structures and their homology.
method Introducing and analyzing non-concentric, nesting vortex cycles and nerves.
result Whitehead CW topology and Leader uniform topology outcomes of vortex cycles.
Hardness proven for embedding simplicial complexes in R^d, especially for k-dimensional ones.
problem Recognizing almost embeddability of k-dimensional complexes in R^d.
method NP-hardness proof using configuration spaces and preimage cycle properties.
result Embedding obstruction is incomplete for k-dimensional complexes in R^d.
Study 2-complexes' homology properties and torsion growth.
problem Quantitative connections between 1-cycle filling inequalities and homology complexities.
method Geometric lower bounds on first homology of finite covers.
result Geometric lower bound on first homology size of finite covers.
Characterizes real holomorphic chains on complex manifolds.
problem Representing homology classes by algebraic cycles.
method Characterization of real holomorphic chains; application to homology classes.
result Real holomorphic chains are characterized by local properties.
The paper calculates the number of closed cycles in a specific complex group.
problem Counting closed cycles in a complex group structure.
method Defined edge zeta function and used rational function formula.
result Obtained exact formula for the number of closed cycles.
Proves a conjecture about graph complexes without specific cycle lengths.
problem Graph complexes without specific cycle lengths.
method Proves stronger statements about independence complexes being contractible or homotopy equivalent to spheres.
result Independence complexes are either contractible or homotopy equivalent to spheres.
Constructs Morse homology for complex algebraic varieties.
problem Homology of vanishing cycles in complex algebraic varieties.
method Morse homology groups associated with a perturbed function on a compactified variety.
result Morse homology groups are isomorphic to the homology of vanishing cycles.
Using Kontsevich's identification of the homology of the Lie algebra l_infty with the cohomology of Out(F_r), Morita defined a sequence of 4k-dimensional classes mu_k in the unstable rational homology of Out(F_{2k+2}). He showed by a computer calculation that the first of these is non-trivial, so coincides with the uni…
Euler's theorem extended to complex structures.
problem Generalizing Euler's theorem to complex structures.
method Analyzing strongly connected, pure -dimensional regular CW-complexes.
result Evenness of cells is equivalent to generalized cycle decomposition and traversability.
Moduli of K3 families over and complex-hyperkähler metrics induced by deformed twistor cyclesmath.AG
The paper solves a geometric problem related to K3 surfaces and complex-hyperkähler metrics.
problem Geometric meaning of small deformations of twistor cycles in K3 period domain.
method Construction of a moduli space for families of marked K3 surfaces and use of Penrose's Non-linear Graviton construction.
result Small deformations of twistor cycles induce complex-hyperkähler metrics on K3 surface families.
Eigenvalues of 1-form Laplacian on hyperbolic manifolds relate to geodesic cycles.
problem Understanding the geometry of small eigenvalues on hyperbolic manifolds.
method Relating eigenvalues to cycle complexity and geodesics.
result Small eigenvalues correspond to closed geodesics with low genus surfaces.
Algorithm constructs JSJ decomposition for hyperbolic groups.
problem Constructing JSJ decompositions for hyperbolic groups.
method Combinatorial and geometric analysis of immersed cycles in CAT(0) square complexes.
result First algorithm with explicit time bound for JSJ decompositions.
Geometric model for Hodge filtered complex cobordism constructed.
problem Constructing a geometric model for Hodge filtered complex cobordism.
method Refinement of Pontryagin-Thom construction to create an explicit isomorphism.
result Explicit isomorphism between geometric and abstract models for complex manifolds.
We determine when certain state cycles represent nontrivial Khovanov homology classes by analyzing features of the state graph. Using this method, we are able to produce hyperbolic knots with arbitrarily many diagonals containing nontrivial state cycle homology classes. This gives lower bounds on the Khovanov width of …
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.
We extend the edge version of the classical Menger's Theorem for undirected graphs to -dimensional simplicial complexes with chains over the field . The classical Menger's Theorem states that two different vertices in an undirected graph can be connected by pairwise edge-disjoint paths if, and only…
We prove that tangent cones to 2-dimensional calibrated cycles are unique. Using this result we prove a rate of convergence for the mass of the blow-up of a calibrated integral 2-cycle towards the limiting density. With the same techniques, we can also prove such a rate for J-holomorphic maps between almost complex man…
Study of height jumps in Ceresa cycle using asymptotic Hodge theory.
problem Understanding height jumps in the Ceresa cycle.
method Analysis of asymptotic behavior of Hain-Reed beta-invariant in degenerating families of curves.
result Height jump of Ceresa cycle is equal to the slope of the dual graph of the curve.
A second part of detailed elementary introduction into Khovanov homologies. This part is devoted to reduced Jones superpolynomials. The story is still about a hypercube of resolutions of a link diagram. Each resolution is a collection of non-intersecting cycles, and one associates a 2-dimensional vector space with each…
Machine learning predicts US and EuroZone business cycles with high accuracy.
problem Predicting the business cycle phases in US and EuroZone.
method Three machine learning approaches were compared: Multinomial Logistic Regression (MLR) achieved the best results.
result MLR achieved 65.25% accuracy for EuroZone and 75% for US in predicting business cycle phases.
Unified framework models credit cycles and systemic risk.
problem Inadequate classical models for bubbles, crises, and credit cycles.
method Marshall-Walras price formation process and mathematical formalism.
result Unified framework reflects different economic states and systemic risk.
New linking numbers link complex cycles to Calabi-Yau 3-folds.
problem Understanding complex analytic cycles and their Massey products.
method Relating ABC Massey products to holomorphic linking numbers.
result Constructed a family of Calabi-Yau 3-folds with non-vanishing Massey products.
We develop a method to summarize causal models with cycles in cubic time.
problem Cycles in high-dimensional causal models limit applicability of existing methods.
method We relax the acyclicity assumption in LiNG models and develop a low-dimensional DAG summary.
result Our method allows recovery of a low-dimensional DAG from high-dimensional data with cycles.
We study massless deformations of generalized calibrated cycles, which describe, in the language of generalized complex geometry, supersymmetric D-branes in N=1 supersymmetric compactifications with fluxes. We find that the deformations are classified by the first cohomology group of a Lie algebroid canonically associa…
To a tropical -cycle in , we naturally associate a normal closed and -dimensional current on denoted by . Such a "tropical current" will not be an integration current along any analytic set, si…
New -manifolds created from -regular graphs with unique Eulerian cycles.
problem Creating compact -manifolds from specific graph structures.
method Defining -manifolds via compatible Eulerian cycles in -regular graphs.
result Each manifold in the class has a unique minimal ideal triangulation with tetrahedra.
Paper detects non-trivial cycles in embedding spaces using graph integrals.
problem Detecting non-trivial cycles in embedding spaces.
method Construct cycles from chord diagrams, use modified configuration space integrals, and pair arguments.
result Non-trivial cycles in embedding spaces are detected.
Study on stable translating solitons without complex cycles.
problem Understanding the topology of stable translating solitons.
method Analyzing complete -stable translating solitons to show absence of specific cycles.
result Two-dimensional complete -stable translating solitons have genus zero.
In this paper it is shown that the space of tight geodesic segments connecting any two vertices in a complex of cycles has finite, uniformly bounded dimension. The dimension is defined in terms of a discrete analogue of Jacobi fields, which are explicitly constructed and shown to give a complete description of the enti…
Paper speeds up topological signal identification and cycle matching.
problem Efficiently identifying and matching topological signals across datasets.
method Cohomological approach to persistent homology computation.
result Significantly faster performance on large-scale datasets.
We develop some theory of double fibration transforms where the cycle space is a smooth manifold and apply it to complex projective space.
In the spirit of Sullivan's paper "Cycles for the Dynamical Study of Foliated Manifolds and Complex Manifolds", existence of a contact structure on a closed manifold is shown to be equivalent to existence of an ample -invariant cone structure with no nontrivial exact structure cycles on the manifold $S^1 \time…
Study of stellar activity cycles using probabilistic methods.
problem Debate over the existence of stellar activity branches and the effects of linear trends and harmonicity assumptions.
method Application of Gaussian processes to study mean cycle periods in chromospheric activity index.
result Confirmation of two activity branches and finding only one trend in inactive population.
Study proposes a new resilience metric for stock market performance analysis.
problem Quantifying resilience cycles in stock market performance.
method Systems-oriented approach with Robustness Range and Elasticity Threshold.
result New metric quantifies non-linear resilience cycles in stock markets.
The paper introduces Lagrangian vanishing cycles to prove obstructions for symplectic foliations.
problem Lack of complexity in foliation generalizations from 3D to higher dimensions.
method Introducing strong symplectic foliations and Lagrangian vanishing cycles.
result Lagrangian vanishing cycles prevent a symplectic foliation from being strong.
The Unique Tangent Cone Property for Weakly Holomorphic Maps into Projective Algebraic Varietiesmath.DG
Paper proves unique tangent maps for complex maps into algebraic varieties.
problem Proving uniqueness of tangent maps for complex maps into algebraic varieties.
method Developed techniques to prove uniqueness of tangent maps for weakly holomorphic and locally approximable maps.
result Established unique tangent cone property for weakly holomorphic maps into projective algebraic varieties.
New method finds large curved subcomplexes, proving conjectures for specific groups.
problem Proving the K(π,1)-conjecture for Artin groups.
method Finding large non-positively curved subcomplexes in spherical Deligne complexes.
result Proves K(π,1)-conjecture for many Artin groups, except one.
Credit risk stress tests can misrepresent default probabilities due to inconsistent parameterization.
problem Misleading default probability projections in credit risk stress tests.
method Analysis of credit risk stress testing models and their parameterization.
result Current portfolios tend to align with through-the-cycle portfolios, leading to spurious default rate projections.
New method identifies causal parameters in tree-shaped linear models using cycles.
problem Identifying causal parameters from correlations in tree-shaped linear models.
method Investigates tree-shaped linear models, uses missing cycles to identify causal parameters, solves quadratic equations.
result Shows how missing cycles can be combined to obtain a unique solution for causal parameters.
We consider any pseudo holomorphic integral 2-cycle in an arbitrary almost complex manifold and perform a blow up analysis at an arbitrary point. Building upon a pseudo algebraic blow up (previously introduced by the author) we prove a geometric rate of decay for the mass ratio towards the limiting density, with an exp…
A geometric construction of Sullivan's Stiefel-Whitney homology classes of a real analytic variety is given by means of the conormal cycle of an embedding of in a smooth variety. We prove that the Stiefel-Whitney classes define additive natural transformations from certain constructible functions to homology. W…
This study optimizes cycle representatives in persistent homology using linear programming.
problem Non-uniqueness of cycle representatives in persistent homology creates ambiguity.
method Optimization of cycle representatives using linear programming methods.
result Optimization reduces the size of cycle representatives and is effective in most data sets.
This thesis explores algebraic cycles and moduli spaces over real numbers.
problem Understanding the cycle class map and its image in real algebraic geometry.
method Constructing integral Fourier transforms on Chow rings of abelian varieties over any field.
result Proof of integral Hodge conjecture for real abelian threefolds and moduli space properties.