Generalizes HK/QK correspondence to para-geometries.
problem Tackles the HK/QK correspondence in para-geometries.
method Provides a new proof and constructs deformations.
result Shows quaternionic Kahler target manifold for para-geometries.
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48 results for “HK/QK correspondence”
We give an explicit formula for the quaternionic Kähler metrics obtained by the HK/QK correspondence. As an application, we give a new proof of the fact that the Ferrara-Sabharwal metric as well as its one-loop deformation is quaternionic Kähler. A similar explicit formula is given for the analogous (K/K) correspondenc…
A hyperkaehler manifold with a circle action fixing just one complex structure admits a natural a hyperholomorphic line bundle. This forms the basis for the construction of a corresponding quaternionic Kaehler manifold in the work of A.Haydys. We construct in this paper the corresponding holomorphic line bundle on twis…
Formulae for curvature of quaternionic Kähler manifolds derived from hyper-Kähler data.
problem Deriving curvature formulae for quaternionic Kähler manifolds.
method Using HK/QK correspondence and Levi-Civita connection, we express curvature in terms of hyper-Kähler data.
result Formulae for curvature of quaternionic Kähler manifolds, including norm of curvature tensor.
Abstract: Generalizes supergravity c-map to quaternionic manifolds.
problem Construct quaternionic manifolds from hypercomplex manifolds.
method Construct conical hypercomplex manifolds and associate quaternionic manifolds.
result Quaternionic manifolds can be associated to special complex manifolds.
When formulated in twistor space, the D-instanton corrected hypermultiplet moduli space in N=2 string vacua and the Coulomb branch of rigid N=2 gauge theories on are strikingly similar and, to a large extent, dictated by consistency with wall-crossing. We elucidate this similarity by showing that these…
Study on symmetries of quaternionic Kähler manifolds with S^1-symmetry.
problem Understanding symmetries in quaternionic Kähler manifolds.
method Associate central extensions of Lie algebras to infinitesimal automorphisms and study one-loop deformed metrics.
result One-loop deformed metrics increase cohomogeneity by at most one.
Constructing Einstein metrics on bundles over hyperKähler manifolds.
problem Finding Einstein metrics on bundles over hyperKähler manifolds.
method Constructing quaternion-Kähler and hypercomplex structures, studying infinitesimal symmetries, and performing QK reduction.
result Producing several explicit quaternion-Kähler metrics and new proofs of correspondences.
Counterexample disproves HK-conjecture for flat manifolds of dimension 9 and higher.
problem Disproving the HK-conjecture for flat manifolds of various dimensions.
method Using flat manifolds of dimension 9 and higher, constructing counterexamples.
result Minimal dimension 9 is required for counterexamples, and higher dimensions are possible.
Study on quaternionic Kähler manifolds and their integrable Hermitian structures.
problem Properties and integrability of quaternionic Kähler manifolds with Killing vector fields.
method Analysis of quaternionic Kähler manifolds with Killing vector fields, deriving conditions for integrability and conformal Kählerity.
result For a large class of quaternionic Kähler manifolds, the structure is integrable.
Defines special Joyce structures for ASK manifolds encoding real HK structures.
problem Defining Joyce structures for ASK manifolds.
method Introducing special Joyce structures and showing they encode real HK structures.
result Special Joyce structures encode real hyperkähler structures on ASK manifolds.
The main purpose of the following article is to introduce a \emph{Lie theoretical} approach to the problem of classifying pseudo quaternionic-Kähler (QK) reductions of the pseudo QK symmetric spaces, otherwise called \emph{generalized Wolf spaces}.
Locally private mechanisms' output divergence bounds derived.
problem Bounding divergence between locally private mechanisms' outputs.
method Sharp upper bounds on divergence between input and output distributions.
result Established locally private versions of estimation risk bounds.
The resolved conifold geometry is linked to a special Kähler manifold and an instanton-corrected hyperkähler manifold.
problem Understanding the geometry of the resolved conifold and its associated structures.
method Explicit description of ASK and instanton-corrected HK manifolds, relating them to twistor coordinates and solving Riemann-Hilbert problems.
result The instanton-corrected hyperkähler manifold realizes a smoothing of the semi-flat HK metric associated with the ASK geometry.
Study of quaternion-Kähler structures via harmonic flow on 8-manifolds.
problem Geometric flow of quaternion-Kähler structures on 8-manifolds.
method Formulated gradient Dirichlet flow, analyzed harmonicity, proved almost-monotonicity.
result Proved long-time existence and constructed harmonic solitons.
Quantum Kirwan maps between K-theories of G-varieties and GIT quotients.
problem Constructing maps between K-theories of G-varieties and their GIT quotients.
method Formal construction of maps in quantum K-theory, using equivariant and non-equivariant quantum K-theory.
result Presentation of quantum K-theory for smooth proper toric DM stacks.
Study decomposes uncertainty in HK-distribution parameter estimation for QUS.
problem Uncertainty in HK-distribution parameter estimation for quantitative ultrasound.
method Bayesian Neural Networks (BNNs) for parameter estimation and uncertainty decomposition.
result Decomposes total predictive uncertainty into epistemic and aleatoric components.
In the first part we survey some of the known results and conjectures on compact Hyperkaehler (HK) manifolds. In the second part we presents a program which aims to show that HK four-folds whose second cohomology (with 4-tuple cup-product) is isomorphic to that of the Hilbert square of a K3 enjoy many of the beautiful …
An explicit surjection from a set of (locally defined) unconstrained holomorphic functions on a certain submanifold of (Sp_1(C) \times C^{4n}) onto the set HK_{p,q} of local isometry classes of real analytic pseudo-hyperkähler metrics of signature (4p,4q) in dimension 4n is constructed. The holomorphic functions, calle…
We study the connections among the mapping class group of the twice punctured torus, the cyclic branched coverings of (1,1)-knots and the cyclic presentations of groups. We give the necessary and sufficient conditions for the existence and uniqueness of the n-fold strongly-cyclic branched coverings of (1,1)-knots, thro…
The study analyzes the evolution of Gaussian measures under a specific gradient flow.
problem Analyzing the evolution of Gaussian measures under a specific gradient flow.
method Derives ordinary differential equations governing the evolution of mean, covariance, and mass under the HK-Boltzmann gradient flow.
result Exponential convergence to equilibrium demonstrated through Polyak-Lojasiewicz-type inequalities.
Unified framework for analyzing gradient flows of measures with exponential decay of entropy.
problem Analyzing exponential decay of entropy functionals in gradient flows of measures.
method Characterization of global exponential decay behaviors using Hellinger-Kantorovich geometry, shape-mass decomposition, and Polyak-Łojasiewicz-type inequalities.
result Unified theoretical framework for gradient flows with complete analysis of exponential decay behaviors.
Study uses network analysis to examine Japanese overseas business networks.
problem Challenges the weak evidence supporting the existence of Japanese conglomerates (HK).
method Large dataset of 20,000 Japanese overseas subsidiaries analyzed using network techniques.
result Rejects Miwa-Ramseyer hypothesis (MRH) for global and regional datasets.
Learning to Unknotmath.GT
The UNKNOT problem solved using natural language processing and machine learning.
problem Determining if a knot is the unknot.
method Braid word representation, binary classification, Reformer and shared-QK Transformer networks, reinforcement learning, Markov moves, braid relations.
result Reformer and shared-QK Transformer networks outperform fully-connected networks in predicting the unknot.
Enhances stability ranges for Torelli and congruence subgroup homologies.
problem Improving stability ranges for specific subgroup homologies.
method Analyzes H2(Torelli subgroup of Aut(Fn)'s), H2(Torelli subgroup of mapping class groups), and Hk(congruence subgroups of GL_n(R)'s).
result Improved central stability ranges for various subgroup homologies.
Mathematical study of instanton corrected q-map spaces and their isometries.
problem Understanding the isometries of instanton corrected q-map spaces.
method Study of isometries of instanton corrected q-map spaces associated to PSR manifolds.
result Explicit example of instanton corrected q-map space with full SL(2,Z) acting by isometries.
Informative and discriminative feature descriptors play a fundamental role in deformable shape analysis. For example, they have been successfully employed in correspondence, registration, and retrieval tasks. In the recent years, significant attention has been devoted to descriptors obtained from the spectral decomposi…
Mathematical proof of S-duality and universal isometries in q-map spaces.
problem Understanding the isometries of q-map spaces and their relation to S-duality.
method Purely mathematical proof of the existence of an SL(2,R)-action on q-map spaces, describing interactions with existing isometries.
result Presentation of a (3n+6)-dimensional group of universal isometries, enlarging previous results.
Formula derived for curvature in measure spaces.
problem Deriving sectional curvature in measure spaces.
method Explicit formula derivation for sectional curvature in with metrics and .
result Curvature analysis in reveals both negative and positive components.
This work generalizes transformer attention to capture higher-order correlations efficiently.
problem Detecting triple-wise connections that were impossible for transformers.
method Developed a generalized attention scheme using Kronecker computation, showing near-linear time algorithms for bounded entries.
result A near-linear time algorithm for generalized attention computation in the bounded-entry setting.
Study compares nonorientable genus values of torus knots.
problem Comparing nonorientable genus values of torus knots.
method Examined torus knots T(p,q) with p even, q odd, and calculated differences in nonorientable three and four genus values.
result The difference between nonorientable three and four genus values on torus knots T(p,q) grows arbitrarily large for any fixed odd q, as p ranges over values of a fixed congruence class modulo q.
Sharp estimate for genus of embedded surfaces in 3-sphere.
problem Estimating the genus of embedded surfaces in the 3-sphere.
method Refined volume estimate and pinching method on the norm of traceless second fundamental form.
result Sharp pinching estimate for the genus of a surface in .
New algorithm reduces RL complexity with low switching costs.
problem Exploration-exploitation dilemma in RL with complex models.
method Monotonic Q-Learning with Upper Confidence Bound (MQL-UCB) for RL with general function approximation.
result Achieves minimax optimal regret of and near-optimal policy switching cost.
Configuration space integral for long n-knots, the Alexander polynomial and knot space cohomologymath.GT
There is a higher dimensional analogue of the perturbative Chern-Simons theory in the sense that a similar perturbative series as in 3-dimension, which is computed via configuration space integral, yields an invariant of higher dimensional knots (Bott-Cattaneo-Rossi invariant), which is constructed by Bott for degree 2…
This paper recovers input data from transformer models using attention weights.
problem Recovering input data from transformer models for security and privacy concerns.
method Introducing an algorithm to minimize the loss function between expected and actual outputs of transformers.
result The algorithm successfully recovers input data from attention weights and outputs of transformers.
Self-attention networks localize when eigenspectrum variance is small.
problem Self-attention mechanisms can lead to rank and entropy collapses, reducing model expressivity and trainability.
method Characterized attention localization using query-key eigenspectrum variance.
result Small eigenspectrum variance prevents both rank and entropy collapses, improving model performance.
Improved stock prediction model using tensor and SMC algorithm.
problem Enhance stock prediction accuracy through multi-source data fusion.
method Tensor integration for multi-sourced data, improved SMC algorithm for feature quality.
result Improved prediction accuracy demonstrated through experiments.
Chronicals boosts LLM fine-tuning speed by 3.51x over Unsloth.
problem Memory bottleneck in fine-tuning large language models.
method Four optimizations: fused kernels, cross-entropy reduction, LoRA+, and sequence packing.
result 3.51x speedup on A100-40GB with Qwen2.5-0.5B.
New algorithm reduces dynamic regret for MDPs with unknown transition and adversarial rewards.
problem Episodic linear mixture MDPs with unknown transition and adversarial rewards.
method Combines occupancy-measure-based global optimization and policy-based variance-aware value-targeted regression.
result Achieves near-optimal dynamic regret of .
New algorithms speed up attention computation for large models by limiting matrix entries.
problem Efficiently compute attention matrices for large language models.
method Developed algorithms that exploit the structure of the attention matrix to achieve subquadratic time complexity.
result There is a sharp transition at for efficient computation of attention matrices.
Weibull framework diagnoses transformer weight distributions, revealing distinct patterns across modules.
problem Diagnosing weight distributions in transformer models for better understanding of training dynamics.
method Applied Weibull distribution to diagnose weight magnitude distributions in transformer models, fitting each weight matrix independently.
result Distinct patterns of weight distributions across different transformer modules were identified.
In this paper we give a new basis, , for the Homflypt skein module of the solid torus, , which was predicted by Jozef Przytycki, using topological interpretation. The basis is different from the basis , discovered independently by Hoste--Kidwell \cite{HK} and Turaev \cite{Tu} w…
Model improves stock market prediction by integrating web news and social media sentiments.
problem Improving prediction accuracy by incorporating diverse data sources.
method Coupled matrix and tensor factorization framework to fuse heterogeneous data.
result The model predicts stock price movements more accurately by considering correlations among stocks.
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.
This work optimizes LLMs by reformulating attention networks and solving them in matrix multiplication time.
problem Optimizing the one-layer attention network objective function in LLMs.
method Provides an iterative greedy algorithm to train the loss function up to ε in time proportional to matrix multiplication.
result Solves the attention regression problem in matrix multiplication time.
Fibered Correspondencemath.DG
Base of fibered correspondence is arbitrary correspondence. Fibered correspondence is interesting when we consider relationship between different bundles. However composition of fibered correspondences may not always be defined. Reduced fibered correspondence is defined only between fibers over the same point of base. …
Generalizes uniformization to algebraic correspondences.
problem Uniformizing non-homeomorphic genus zero orbifolds.
method Constructs algebraic correspondences to simultaneously uniformize orbifolds.
result Realizes Teichmüller space of a punctured sphere in correspondences.
Study of holomorphic correspondences combining entire maps and Fuchsian groups.
problem Understanding dynamics of entire maps and their interactions with Fuchsian groups.
method Systematic study of holomorphic correspondences arising from conformal combinations of transcendental entire maps and Fuchsian groups.
result The resulting correspondence is the composition of a Möbius involution and the deleted covering correspondence of a meromorphic function with a simple pole.