We study the relative modular classes of Lie algebroids, and we determine their relationship with the modular classes of Lie algebroids with a twisted Poisson structure.
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48 results for “relative modular structure”
Modular classes revisitedmath.DG
We present a graded-geometric approach to modular classes of Lie algebroids and their generalizations, introducing in this setting an idea of relative modular class of a Dirac structure for a certain type of Courant algebroids, called projectable. This novel approach puts several concepts related to Poisson geometry an…
We study the behavior of the modular class of a Lie algebroid under general Lie algebroid morphisms by introducing the relative modular class. We investigate the modular classes of pull-back morphisms and of base-preserving morphisms associated to Lie algebroid extensions. We also define generalized morphisms, includin…
Characterizes slopes for Markov ordering on prime pairs.
problem Investigating the Markov ordering on relatively prime integer pairs.
method Employing the stable norm on modular torus homology.
result Characterizes slopes for monotonicity of Markov ordering.
Improved algorithm for modular links provides upper volume bounds.
problem Understanding the geometry of modular links and Lorenz links.
method Bunch algorithm to study modular links and provide upper volume bounds.
result First upper volume bound independent of word exponents and quadratic in braid index.
Skew algebroid is a natural generalization of the concept of Lie algebroid. In this paper, for a skew algebroid E, its modular class mod(E) is defined in the classical as well as in the supergeometric formulation. It is proved that there is a homogeneous nowhere-vanishing 1-density on E* which is invariant with respect…
Artificial neural networks (ANNs) have achieved significant success in tackling classical and modern machine learning problems. As learning problems grow in scale and complexity, and expand into multi-disciplinary territory, a more modular approach for scaling ANNs will be needed. Modular neural networks (MNNs) are neu…
3d Modularityhep-th
We find and propose an explanation for a large variety of modularity-related symmetries in problems of 3-manifold topology and physics of 3d theories where such structures a priori are not manifest. These modular structures include: mock modular forms, Weil representations, quantum mo…
We introduce the notion of the modular class of a Lie algebroid equipped with a Nambu structure. In particular, we recover the modular class of a Nambu-Poisson manifold with its Nambu tensor as the modular class of the tangent Lie algebroid with Nambu structure We show that many known properties of th…
Machine learning classifies complex geometric patterns with high accuracy.
problem Classifying extension degree of dessins d'enfants over the rationals.
method Deep feed-forward neural network trained on machine learning.
result 0.92 accuracy in classification with 0.03 standard error.
Proves modular functors for SO(3) have integral Hodge structures.
problem Proving modular functors have integral Hodge structures.
method Based on homological models and geometric identification.
result Geometric construction of Hodge structures on SO(3) modular functors.
Modular neural networks generalize better with less data.
problem Theoretical and practical understanding of how modularity improves neural network generalization.
method Theoretical analysis of sample complexity, development of a novel learning rule.
result Modular networks require fewer samples to generalize compared to nonmodular networks, especially in high-dimensional tasks.
Constructs 3D topological field theories from a specific quantum group, linking to physics invariants.
problem Developing topological field theories from non-semisimple quantum groups.
method Using the unrolled quantum group of and a relative modular structure on weight modules.
result Establishes a connection between constructed invariants and physicists' -invariants.
Proves Hodge structures on modular functors, providing formulas for Hodge numbers.
problem Existence and uniqueness of Hodge structures on modular functors.
method Non-Abelian Hodge correspondence and Ocneanu rigidity.
result Explicit formulas for Hodge numbers in modular functors of level 2 times an odd number.
RNNs solve modular addition tasks using low rank and sparse Fourier structures.
problem Solving modular addition tasks with recurrent neural networks.
method Identified low rank structures and sparse Fourier representations in RNN weights.
result RNNs robust to removing individual frequencies but degrade with more ablation.
Classifies big mapping classes on infinite type surfaces.
problem Investigating geometric properties of hyperbolic structures on infinite type surfaces.
method Analyzing the space of convex hyperbolic structures with Fenchel-Nielsen topology and studying the dynamics of mapping classes.
result Classifies big mapping classes into three types based on their dynamics on Teichmüller subspaces.
Countable modular groups found on surfaces with infinite type.
problem Finding modular groups of infinite type surfaces.
method Proving countable modular groups for orientable infinite type surfaces.
result Every orientable infinite type surface has a countable modular group.
Origami structures are enumerated and shown to be quantum modular.
problem Counting and understanding origami structures with real structures.
method Using combinatorics of zonal polynomials and Schur polynomials, and relating to quantum modular forms and double Hurwitz numbers.
result The generating functions of certain origami structures are quantum modular forms.
In this paper, we introduce the notion of modular class of a Lie algebroid equipped with a Nambu structure satisfying some suitable hypothesis. We also introduce cohomology and homology theories for such Lie algebroids and prove that these theories are connected by a duality isomorphism when the modular class is nu…
New methods detect modular structure in neural networks, revealing surprising effects of dropout.
problem Detecting functional modules in neural networks for learning, compositionality, and generalization.
method Two families of methods: upstream and downstream, to define similarity between units.
result Dropout dramatically increased modularity, and there's little agreement between upstream and downstream methods.
Noise-driven neural networks emerge modular structures, improving robustness and generalization.
problem Artificial neural networks struggle with modular solutions, leading to poor generalization and robustness.
method Inspired by brain's modular architecture, the study uses neural noise and nonlinear responses to drive the emergence of modular solutions.
result Noise-driven modularisation improves robustness and generalization in neural networks.
Our aim is to introduce and advocate non- (non-symmetric) modular operads. While ordinary modular operads were inspired by the structure of the moduli space of stable complex curves, non- modular operads model surfaces with open strings outputs. An immediate application of our theory is a short proof that the mod…
A modular object in a symmetric monoidal bicategory is a Frobenius algebra object whose product and coproduct are biadjoint, equipped with a braided structure and a compatible twist, satisfying rigidity, ribbon, pivotality, and modularity conditions. We prove that the oriented 3-dimensional bordism bicategory of 1-, 2-…
Batalin-Vilkovisky algebra structure on Poisson manifolds with diagonalizable modular symmetrymath.DG
Study of Batalin-Vilkovisky algebra on Poisson manifolds with diagonalizable modular symmetry.
problem Exploring Batalin-Vilkovisky algebra structures on Poisson manifolds with specific symmetry conditions.
method Analysis of twisted Poincaré duality and mixed complex structure, combined with Kontsevich's deformation quantization and Koszul duality.
result Generalization of Batalin-Vilkovisky algebra structure to Poisson manifolds with diagonalizable modular symmetry.
Develops Hermitian TQFTs from quantum groups, defining new topological phases.
problem Defining Hermitian non-semisimple TQFTs.
method Categorical context and representation theory of quantum groups.
result New pseudo-Hermitian topological phases from quantum group representations.
In this paper we study the modular classes of Dirac manifolds and of Dirac maps, and we discuss their basic properties. We apply these results to explain the relationship between the modular classes of the various structures involved in the reduction of a Poisson manifold under the action by of a Poisson Lie group.
Scalable detection of statistically significant communities and hierarchies, using message-passing for modularityphysics.soc-ph
Modularity is a popular measure of community structure. However, maximizing the modularity can lead to many competing partitions, with almost the same modularity, that are poorly correlated with each other. It can also produce illusory "communities" in random graphs where none exist. We address this problem by using th…
We review the polynomial structure of the topological string partition functions as solutions to the holomorphic anomaly equations. We also explain the connection between the ring of propagators defined from special Kähler geometry and the ring of almost-holomorphic modular forms defined on modular curves.
Modular curves parametrize elliptic curves with a point of order . They can be identified with connected components of projectivized strata of meromorphic differentials. As strata of meromorphic differentials, they have a canonical walls-and-chambers structure defined by the …
Study inert and ambiguous classes in modular group using combinatorial methods.
problem Counting inert and ambiguous conjugacy classes in modular group.
method Purely combinatorial approach using word length in free product representation.
result Exact counting formulas and asymptotic growth rates for inert and ambiguous classes.
Study shows volumes of knot complements are bounded by linear functions of geodesic periods.
problem Volume calculation of knot complements associated with geodesics on modular surfaces.
method Analyzes geodesics on modular surfaces, their associated knots, and their complements' volumes.
result Volumes of knot complements are bounded linearly by the period of geodesic continued fractions.
New invariant from Viro's gl(1|1) polynomial distinguishes lens spaces.
problem Constructing a 3-manifold invariant from Viro's gl(1|1) polynomial.
method Following Costantino, Geer, and Patureau-Mirand's method in relative G-modular categories.
result The invariant can distinguish homotopy equivalent lens spaces.
Recent work extends Turaev's modular categories to non-semisimple settings.
problem Building TQFTs beyond semisimplicity.
method Generalized modular categories.
result Success in extending Turaev's construction to non-semisimple settings.
A hyperbolic conjugacy class in the modular group PSL(2,Z) corresponds to a closed geodesic in the modular orbifold. Some of these geodesics virtually bound immersed surfaces, and some do not; the distinction is related to the polyhedral structure in the unit ball of the stable commutator length norm. We prove the foll…
When market boundaries weaken: Network reconfiguration and regime-dependent cross-asset spilloversphysics.pop-ph
This paper examines cryptocurrency integration with traditional markets, showing how network structure and turbulence influence cross-asset spillovers.
problem Understanding how cryptocurrencies integrate with traditional financial markets and the impact of market stress on cross-asset spillovers.
method Combining rolling correlation networks, community structure, market-specific and system-wide Turbulence Indices, and VAR-based connectedness analysis.
result Cross-asset integration is episodic, with network structure and turbulence playing a role in transmission during stress periods.
Modular -holonomic modulesmath.GT
Introduces modular -holonomic modules to solve -difference equations.
problem Solving -difference equations in quantum invariants and Chern-Simons theory.
method Defines modular -holonomic modules with improved analyticity properties.
result Modular -holonomic modules explain structural properties of quantum invariants and Chern-Simons theory.
We study the behavior of the modular class of an orientable Poisson manifold and formulate some unimodularity criteria in the semilocal context, around a (singular) symplectic leaf. Our results generalize some known unimodularity criteria for regular Poisson manifolds related to the notion of the Reeb class. In particu…
We introduce linear holonomy on Poisson manifolds. The linear holonomy of a Poisson structure generalizes the linearized holonomy on a regular symplectic foliation. However, for singular Poisson structures the linear holonomy is defined for the lifts of tangential path to the cotangent bundle (cotangent paths). The lin…
Scaling model capacity has been vital in the success of deep learning. For a typical network, necessary compute resources and training time grow dramatically with model size. Conditional computation is a promising way to increase the number of parameters with a relatively small increase in resources. We propose a train…
A framework for modular training of robust generative models.
problem Training large generative models is resource-intensive and requires heuristic tuning.
method Modular training using a gating mechanism and a minimax game to find a robust gate.
result The modular approach can theoretically outperform monolithic baselines and is scalable.
3d Modularity Revisitedhep-th
Researchers clarify modular group representations and vertex operator algebras for 3d invariants.
problem Understanding the full set of 3d invariants and their modular properties.
method Introducing supersymmetric defects and constructing cone vertex operator algebras.
result The full vector-valued quantum modular form for \(\widetilde{
m SL}_2(\mathbb{Z})\) captures all \(\hat Z\)-invariants of a given three-manifold.
DynMSA detects market clusters for better portfolio allocation.
problem Identifying stable market clusters for effective portfolio management.
method Combining Random Matrix Theory with modularity optimization and spectral clustering.
result DynMSA outperforms baseline models in intra- and inter-cluster correlation differences.
MOCA uses modular attention to estimate causal effects from complex data.
problem Estimating causal effects from observational data with complex, non-linear, and high-dimensional treatment and outcome mechanisms.
method MOCA is a transformer-based framework that separates treatment and outcome modeling through modular design and one-way attention mechanism, with cutting-feedback to prevent outcome influence on treatment representations.
result MOCA outperforms classical estimators and machine learning approaches across various simulated and real-world scenarios.
The study examines if ReLU activation function is optimal for modularity in neural networks.
problem Finding the best activation function for modularity in neural networks.
method Comparing ReLU with other activation functions for modularity and performance.
result ReLU may not be the best choice for modularity, suggesting other functions could be more suitable.
Spin Modular Categoriesmath.GT
Modular categories are a well-known source of quantum 3-manifold invariants. In this paper we study structures on modular categories which allow to define refinements of quantum 3-manifold invariants involving cohomology classes or generalized spin and complex spin structures. A crucial role in our construction is play…
We construct modular invariants on the moduli space of quantum vacua of N=2 SYM with gauge group SU(2). We also introduce a nonchiral function K which is expressed in terms of the Seiberg-Witten and Poincare' metrics. It turns out that K has all the expected properties of the next to leading term in the Wilsonian effec…
In this paper we analyse the bipartite Colombian firms-products network, throughout a period of five years, from 2010 to 2014. Our analysis depicts a strongly modular system, with several groups of firms specializing in the export of specific categories of products. These clusters have been detected by running the bipa…
We construct families of TQFT's over the finite field Z/pZ starting from an integral TQFT obtained by Frohman and Nicas. These TQFT's are likely to describe the constant order contributions of the cyclotomic integer expansions of the Reshetikhin Turaev Ohtsuki theories. Their modular structure is intimately related to …