We define an order relation among oriented -complexes. We show that with respect to this relation, two -complexes over the same complex are homotopy equivalent if and only if there is an isometry between the second homology groups. We also consider minimal objects of this relation.
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48 results for “complex relations”
In this paper we introduce in study the projectively related complex Finsler metrics. We prove the complex versions of the Rapcsák's theorem and characterize the weakly Kähler and generalized Berwald projectively related complex Finsler metrics. The complex version of Hilbert's Fourth Problem is also pointed out. As an…
Proves divisibility relations for symplectic curve polynomials.
problem Divisibility relations for symplectic curve polynomials.
method New proofs of divisibility relations for Oka and Alexander polynomials of symplectic curves.
result Proves Libgober's divisibility relations for symplectic curves.
Categorifies a skein relation for links colored by one-column Young diagrams.
problem Categorification of a skein relation for links.
method Homotopy equivalence between two complexes of singular Soergel bimodules.
result Categorification of the colored skein relation.
Advances combinatorial complexes for better modeling of hierarchical and set-type relations.
problem Lack of effective modeling for complex hierarchical and set-type relations in high-dimensional data.
method Introduces combinatorial complexes as a bridge between cell complexes and hypergraphs, emphasizing their different types of relations.
result Combining set-type and hierarchical relations in a single model can be advantageous in learning tasks.
We describe a series of complexes that relate to the braid groups as the matching complexes relate to the symmetric groups. A modified construction applies as well to other complexes based on edge sets in graphs. We show that our constructions will yield Cohen-Macauley complexes provided the underlying complexes are Co…
Defines relations between Dirac structures and spinors using Courant algebroid relations.
problem Defines relations between Dirac structures and spinors using Courant algebroid relations.
method Uses Courant algebroid relations to define relations between Dirac structures and spinors.
result Proves existence results for T-dual structures and demonstrates compatibility with Type II supergravity equations.
We consider the variational complex on infinite jet space and the complex of variational derivatives for Lagrangians of multidimensional paths and study relations between them. The discussion of the variational (bi)complex is set up in terms of a flat connection in the jet bundle. We extend it to supercase using a part…
We survey results about computational complexity of the word problem in groups, Dehn functions of groups and related problems.
The geometry of relationsmath.CO
There is a canonical way to associate two simplicial complexes K, L to any relation . Moreover, the geometric realizations of K and L are homotopy equivalent. This was studied in the fifties by C.H. Dowker. In this article we prove a Galois-type correspondence for relations when…
Random walks on cell complexes link to Laplacians and Novikov-Shubin invariants.
problem Computing Novikov-Shubin invariants for complex cell structures.
method Construct random walks on cell complexes, relate to Laplacians, and use return probabilities.
result Novikov-Shubin invariants can be recovered from random walk return probabilities.
Studies amenable category's monotonicity and its relation to topological complexity.
problem Monotonicity of amenable category for degree-one maps.
method Uses amenable covers and compares with topological complexity.
result Establishes a relation between amenable category and topological complexity.
Complex - symbols relate to hyperbolic tetrahedron volumes and determinants.
problem Analyzing asymptotics of complex - symbols.
method Relating asymptotics to hyperbolic tetrahedron volumes and determinants.
result Complex - symbols' asymptotics linked to tetrahedron volumes and determinants.
The study defines and explores properties of complex Sasakian manifolds.
problem Understanding the geometric structure of complex Sasakian manifolds.
method Definition and analysis of properties, curvature relations, and flatness conditions.
result Obtained useful curvature relations and examined flatness conditions for general curvature tensor B.
Calculates cobordism ring of stably almost complex C_p-manifolds.
problem Calculating the cobordism ring of stably almost complex C_p-manifolds.
method Complete calculation using generators and relations.
result Generators compared with geometrically-defined ones by Kosniowski.
The paper extends local h-principles to complex structures on Stein manifolds.
problem Existence of local h-principles for complex structures on Stein manifolds.
method Introducing realifications of partial holomorphic relations and proving h-principles for them.
result Local h-principles can be extended to complex structures on Stein manifolds.
Hyperconnected Relator Spaces. CW Complexes and Continuous Function Paths that are Hyperconnectedmath.GT
This article introduces proximal cell complexes in a hyperconnected space. Hyperconnectedness encodes how collections of path-connected sub-complexes in a Alexandroff-Hopf-Whitehead CW space are near to or far from each other. Several main results are given, namely, a hyper-connectedness form of CW (Closure Finite Weak…
We describe how generalized complex geometry, which interpolates between complex and symplectic geometry, is compatible with T-duality, a relation between quantum field theories discovered by physicists. T-duality relates topologically distinct torus bundles, and prescribes a method for transporting geometrical structu…
MTHetGNN models complex relations in multivariate time series forecasting.
problem Complex relations among variables in multivariate time series forecasting.
method Designs a relation embedding module and a temporal embedding module, using graph neural networks and CNNs.
result Achieves state-of-the-art results in multivariate time series forecasting.
We calculate relations on characteristic classes which are obstructions preventing closed Kähler manifolds from carrying holomorphic Cartan geometries. We apply these relations to give global constraints on the phase spaces of complex analytic determined and underdetermined systems of differential equations.
Relations between allometric scalings and fluctuations in complex systems: The case of Japanese firmsphysics.soc-ph
To elucidate allometric scaling in complex systems, we investigated the underlying scaling relationships between typical three-scale indicators for approximately 500,000 Japanese firms; namely, annual sales, number of employees, and number of business partners. First, new scaling relations including the distributions o…
We show that the theory of stable complex -cobordisms, for a torus , is embedded into the theory of stable complex -cobordisms of not necessarily compact manifolds equipped with proper abstract moment maps. Thus the introduction of such non-compact cobordisms in the stable complex -cobordism theory does not…
Study provides explicit formula for complex 2D Kähler manifold quantization.
problem Quantization of complex 2D locally symmetric Kähler manifolds.
method Deformation quantization with separation of variables, solving recurrence relations.
result Explicit formula for star product on complex 2D locally symmetric Kähler manifolds.
Two related constructions are studied: (1) The diagonal complex and its barycentric subdivision related to a \textit{punctured} oriented surface equipped with a number of labeled marked points. (2) The symmetric diagonal complex and its barycentric subdivision $\math…
In statistical relational learning, the link prediction problem is key to automatically understand the structure of large knowledge bases. As in previous studies, we propose to solve this problem through latent factorization. However, here we make use of complex valued embeddings. The composition of complex embeddings …
We prove an effective version of a theorem relating curve complex distance to electric distance in hyperbolic 3-manifolds, up to errors that are polynomial in the complexity of the underlying surface. We use this to give an effective proof of a result regarding maps between curve complexes of surfaces induced by finite…
Researchers prove no unexpected relations between complex manifold numbers.
problem Proving no unexpected universal linear relations between Hodge, Betti, and Chern numbers of compact complex manifolds.
method Developed a framework to tackle more general questions involving all cohomological invariants, solved specific construction problems.
result Obtained full answers to general questions about universal relations and bimeromorphic invariants in low dimensions.
New tools analyze the complexity of left-ordering equivalence relations in groups and 3-manifolds.
problem Analyzing the complexity of conjugacy equivalence relations in left-orderable groups and 3-manifolds.
method Developed new tools to analyze the complexity of the conjugacy equivalence relation for left-orderable groups . Used these tools to demonstrate non-smoothness and initiate a systematic analysis of for 3-manifolds.
result Proved that if is not prime, then is a universal countable Borel equivalence relation, and showed that in certain cases the complexity of is bounded below by the complexity of the conjugacy equivalence relation arising from the fundamental group of each of the JSJ pieces of . Also proved that if is the complement of a nontrivial knot in , then is not smooth, and showed how determining smoothness of for all knot manifolds is related to the L-space conjecture.
Almost Hermitian Identitiesmath.DG
Study local commutation relation on almost complex manifolds.
problem Local commutation relation between Lefschetz operator and exterior differential.
method Analyzing almost complex manifolds with compatible metrics.
result Generalizes local Kähler identities to almost Hermitian manifolds.
Study isotropic curves on complex quadric with geometric relations.
problem Characterize isotropic curves on the complex quadric.
method Analyze geometric properties and relations of isotropic curves.
result Discovers relations between isotropic curves and surfaces in spaceforms.
CT improves neural network performance on cell complex data.
problem Improving predictive performance of neural networks on complex data.
method Introducing the Cellular Transformer (CT) that generalizes graph-based transformers to cell complexes.
result CT achieves state-of-the-art performance on cell complex datasets without complex enhancements.
New finding links hyperbolic manifold systolic volume to triangulation complexity.
problem Understanding the relationship between systolic volume and triangulation complexity in hyperbolic manifolds.
method Proof based on Jørgensen and Thurston's theorem of hyperbolic volume.
result Systolic volume of hyperbolic manifolds is related to triangulation complexity.
Proves universal pairing result for 2-complexes, showing lack of positivity.
problem Detecting equivalence relations in 2-complexes.
method Analogous to Freedman et al. for manifolds, but for 2-complexes.
result Universal pairing does not detect simple homotopy equivalence vs 3-deformations for 2-complexes.
Scharlemann and Thompson define the width of a 3-manifold M as a notion of complexity based on the topology of M. Their original definition had the property that the adjacency relation on handles gave a linear order on handles, but here we consider a more general definition due to Saito, Scharlemann and Schultens, in w…
We introduce the notion of a bicollapsible 2-complex. This allows us to generalize the hyperbolicity of one-relator groups with torsion to a broader class of groups with presentations whose relators are proper powers. We also prove that many such groups act properly and cocompactly on a CAT(0) cube complex.
Recent developments in Seiberg-Witten theory and relations with Complex Geometry.
New method computes first Vassiliev derivative of Khovanov homology.
problem Computing Vassiliev derivatives of Khovanov homology.
method Developed a crux complex to compute the first derivative.
result Direct computation of the first derivative of Khovanov homology.
We prove that ``almost generically'' for a one-relator group Delzant's -invariant (which measures the smallest size of a finite presentation for a group) is comparable in magnitude with the length of the defining relator. The proof relies on our previous results regarding isomorphism rigidity of generic one-relator …
A new deep learning framework for topological data.
problem Developing models for data on complex topological domains.
method Introducing combinatorial complexes and developing attention-based CCNNs.
result CCNNs outperform existing models in tasks involving mesh shape analysis and graph learning.
In this paper we prove a tertiary index theorem which relates a spectral geometric and a homotopy theoretic invariant of an almost complex manifold with framed boundary. It is derived from the index theoretic and homotopy theoretic versions of a complex elliptic genus and interestingly related with the structure of the…
Survey on manifold complexities and motion planning in robotics.
problem Understanding topological complexities of manifolds in robotic motion planning.
method Overview of topological complexities, geodesic motion planning, and connections to critical point theory.
result Estimation of motion planning complexity using Riemannian geometry and critical point theory.
Embedding-based methods for knowledge base completion (KBC) learn representations of entities and relations in a vector space, along with the scoring function to estimate the likelihood of relations between entities. The learnable class of scoring functions is designed to be expressive enough to cover a variety of real…
Unified Morse-Bott-Smale chain complex, resolves well-definedness issue.
problem Well-definedness of Morse-Bott-Smale chain complex.
method Unified five degeneracy relations into a single condition.
result Quasi-isomorphic to Morse-Smale-Witten chain complex, alternative proof of Morse Homology Theorem.
Getzler-Jones-Petrack introduced structures on the equivariant complex for manifold with smooth action, motivated by geometry of loop spaces. Applying Witten's deformation by Morse functions followed by homological perturbation we obtained a new set of structures. We extend and …
In this paper, we first provide an updated survey of the geometry of complex Cartan spaces. New characterizations for some particular classes of complex Cartan spaces are pointed out, e.g. Landsberg-Cartan, strongly Berwald-Cartan and others. We introduce the Cartan-Randers spaces which offer examples of Berwald-Cartan…
Identifying Complexity by Means of Matricescond-mat.soft
Complexity is an interdisciplinary concept which, first of all, addresses the question of how order emerges out of randomness. For many reasons matrices provide a very practical and powerful tool in approaching and quantifying the related characteristics. Based on several natural complex dynamical systems, like the str…
In this paper we define and study pseudoholomorphic vector bundles structures, particular cases of which are tangent and normal bundle almost complex structures. These are intrinsically related to the Gromov D-operator. As an application we deduce normal forms of 1-jets of almost complex structures along a submanifold.…
In this work, we move beyond the traditional complex-valued representations, introducing more expressive hypercomplex representations to model entities and relations for knowledge graph embeddings. More specifically, quaternion embeddings, hypercomplex-valued embeddings with three imaginary components, are utilized to …