Researchers find a Poisson bracket and symplectic structure for field theories.
problem Exploring the Poisson bracket and symplectic structure in the covariant canonical formalism of fields.
method Identifying the phase space as a ringed space with a graded algebra of differential forms, they found a natural Poisson bracket and symplectic structure.
result The Poisson and symplectic structures can be even or odd depending on the manifold's dimension.
Symplectic forms from two phase spaces are proven equivalent.
problem Equivalence of symplectic forms from different phase spaces.
method Proof of equivalence for theories over space-time with boundary.
result Symplectic forms derived from canonical and covariant phase spaces are equivalent.
New approach to Lagrangian systems using intrinsic geometry.
problem Developing a new framework for Lagrangian systems.
method Direct reformulation of Hamiltonian formalism, introduction of spatial equation and spatial-gauge symmetry.
result Covariant and non-covariant canonical variational principles demonstrated for Maxwell equations.
We describe a canonical form for linear differential operators that are formally self-adjoint or formally skew-adjoint.
The jet formalism for Classical Field theories is extended to the setting of Lie algebroids. We define the analog of the concept of jet of a section of a bundle and we study some of the geometric structures of the jet manifold. When a Lagrangian function is given, we find the equations of motion in terms of a Cartan fo…
Homotopy equivalence between formalities with different covariant derivatives.
problem Formality of Dolgushev depends on covariant derivative choice.
method Proved homotopy equivalence of L∞-morphisms twisted by gauge equivalent elements. result Globalized formalities with different covariant derivatives are homotopic.
A new method estimates conditional canonical correlations using random forests.
problem Estimating relationships between two sets of variables given covariates.
method Random Forest with Canonical Correlation Analysis (RFCCA)
result RFCCA provides accurate canonical correlation estimations and well-controlled Type-1 error.
When a gauge-natural invariant variational principle is assigned, to determine {\em canonical} covariant conservation laws, the vertical part of gauge-natural lifts of infinitesimal principal automorphisms -- defining infinitesimal variations of sections of gauge-natural bundles -- must satisfy generalized Jacobi equat…
We prove that the mirror map is trivial for the canonical formal families of Calabi-Yau varieties constructed by Gross and the second author. In other words, the natural coordinate in a canonical Calabi-Yau family is a canonical coordinate in the sense of Hodge theory. This implies that the higher weight periods direct…
New methods rank players using covariates and comparisons, outperforming existing algorithms.
problem Ranking players based on incomplete and noisy pairwise comparisons.
method Three spectral ranking methods incorporating player covariates.
result Proposed methods outperform existing algorithms in simulations.
Researchers create new operators from Riemannian invariants.
problem Developing new mathematical tools for Riemannian geometry.
method Introducing formally self-adjoint conformally covariant polydifferential operators.
result Found a fourth-order, conformally covariant tridifferential operator.
The homogeneous canonical formalism of Rund is applied to the second-order Lagrangian model of the self-interacting particle of Bopp. The quasi-classical free spinning particle of Mathisson appears then as a constrained subsystem of the previous system. Differential-geometric mechanisms offered in this work are formula…
Generalizes Tulczyjew triples for contact manifolds in Hamiltonian and Lagrangian formalisms.
problem Tackles the need for a geometric tool in contact manifolds.
method Introduces a generalized Tulczyjew triple for contact manifolds.
result Contact Hamiltonians and Lagrangians as sections of line bundles determine dynamics on contact phase space.
Researchers found a special basis for cycles on a K3 surface.
problem Understanding the structure of two-cycles on K3 surfaces.
method Constructed a canonical basis of two-cycles using formal sums of smooth submanifolds.
result The intersection form of the basis takes a specific canonical form.
Estimates linear model from noisy covariates and instruments using spectral regularization.
problem Estimating a linear model from many noisy covariates and instruments.
method Two-stage least squares with spectral regularization of canonical correlations.
result Upper and lower bounds on estimation error, proving optimality of the method with noisy data.
We simplify matrix computations for block matrices, especially useful for covariance and correlation matrices.
problem Complex computations for block matrices, especially for covariance and correlation matrices.
method Obtained a canonical representation for block matrices, facilitating computation of various matrix operations.
result Simplified computation of matrix operations for block matrices, particularly useful for covariance and correlation matrices.
We sketch our recent application of a non-commutative version of the Cartan `moving-frame' formalism to the quantum Euclidean space RqN, the space which is covariant under the action of the quantum group SOq(N). For each of the two covariant differential calculi over RqN based on the R-matrix formalism, we…
Compute Dolbeault and Bott-Chern cohomologies of complex solvmanifolds.
problem Compute cohomologies of complex solvmanifolds.
method Build finite-dimensional double subcomplexes and decompose them into indecomposable ones.
result Characterize the ∂∂ˉ-Lemma property and compute triple ABC-Massey products. We construct a lagrangian geometric formulation for first-order field theories using the canonical structures of first-order jet bundles, which are taken as the phase spaces of the systems in consideration. First of all, we construct all the geometric structures associated with a first-order jet bundle and, using them,…
Researchers solve Yamabe problems for specific operators, finding both uniqueness and nonuniqueness.
problem Prescribing scalar, Q-, or σ₂-curvatures in conformal classes.
method Formally self-adjoint, conformally covariant, polydifferential operators.
result Uniqueness results on the sphere, nonuniqueness in general.
We give a proof of Kontsevich's formality theorem for a general manifold using Fedosov resolutions of algebras of polydifferential operators and polyvector fields. The main advantage of our construction of the formality quasi-isomorphism is that it is based on the use of covariant tensors unlike Kontsevich's original p…
Canonical correlation analysis (CCA) is a fundamental statistical tool for exploring the correlation structure between two sets of random variables. In this paper, motivated by recent success of applying CCA to learn low dimensional representations of high dimensional objects, we propose to quantify the estimation loss…
Introduces formal frames for manifolds and their properties.
problem Understanding and generalizing frames and connections on manifolds.
method Introduces formal frames, canonical forms, and torsions.
result Equivalence of vanishing torsions to realizability of formal frames as ordinary frames.
We study generalized electric/magnetic duality in Abelian gauge theory by combining techniques from locally covariant quantum field theory and Cheeger-Simons differential cohomology on the category of globally hyperbolic Lorentzian manifolds. Our approach generalizes previous treatments using the Hamiltonian formalism …
This paper rethinks confidence calibration under covariate shifts.
problem Calibration methods struggle with covariate shifts and unstable importance weighting.
method Derives Expectation consistency condition and proposes Expectation consistency loss (ECL).
result ECL loss is compatible with various types of calibration and has the same sample complexity as ECE.
The paper explores symmetries and conservation laws in Hamiltonian systems.
problem Understanding symmetries and conservation laws in Hamiltonian systems.
method Using dynamical covariant derivative and Jacobi endomorphism, the paper finds invariant equations of symmetries and proves the canonical nonlinear connection can be determined by these symmetries.
result The canonical nonlinear connection can be determined by infinitesimal symmetries and Newtonoid vector fields.
Unified formalism for Palatini and unimodular gravity using frame bundle canonical forms.
problem Unified formalism for Palatini and unimodular gravity.
method Employing a relationship between Griffiths variational problem and Lepage-equivalent problem, using canonical forms on the first jet of frame bundle.
result Unified formalism for Palatini and unimodular gravity formulated in a geometrical fashion.
Study on estimating CCA with stochastic methods and sample complexity.
problem Estimating canonical correlation and directions from samples.
method Exact and approximate solutions using stochastic optimization and power iterations.
result Achieve optimal alignment with minimal samples and passes.
New methods test correlation between network structure and node features.
problem Assessing correlation between network structure and node-level covariates.
method Four novel methods based on linear models and canonical correlation analysis.
result Theoretical guarantees and computational efficiency for testing network dependency.
We propose a new method of valuation of portfolios and their respective investing strategies. To this end we define a canonical ensemble of portfolios that allows to use the formalism thermodynamics.
We review recent probabilistic results on covariant Schrödinger operators on vector bundles over (possibly locally infinite) weighted graphs, and explain applications like semiclassical limits. We also clarify the relationship between these results and their formal analogues on smooth (possibly noncompact) Riemannian m…
Researchers create a family of conformally covariant operators.
problem Developing a comprehensive set of conformally covariant operators.
method Constructing a family of conformally covariant tridifferential operators as tangential operators in the Fefferman--Graham ambient space.
result Symmetrization of ambient operators is formally self-adjoint.
This paper shows hypercommutative algebras on Calabi-Yau manifolds are formal.
problem Formality of hypercommutative algebras on Calabi-Yau manifolds.
method Using purity of mixed Hodge structures.
result The canonical hypercommutative algebra on compact Calabi-Yau manifolds is formal.
Robust kernel CCA method detects outliers and improves performance.
problem Kernel CO and CCO sensitivity to contaminated data.
method Proposed robust kernel CO and CCO, derived IF for CCA, robust kernel CCA method.
result Robust kernel CCA method performs better than standard kernel CCA for ideal and contaminated data.
New sparse CCA method finds interpretable associations in multi-view data.
problem Discovering interpretable associations in high-dimensional multi-view data.
method Inspired by sparse PCA, proposed a convex maximization program equivalent to non-convex sparse CCA formulation, using gradient method to reduce search space.
result Proposed two-step algorithm and new sparse CCA variants (Directed Sparse CCA, Multi-View sCCA) for multi-omic studies.
The covariant phase space of a Lagrangian field theory is the solution space of the associated Euler-Lagrange equations. It is, in principle, a nice environment for covariant quantization of a Lagrangian field theory. Indeed, it is manifestly covariant and possesses a canonical (functional) "presymplectic structure" w …
We show that for a Lie group G=Rn⋉φRm with a semisimple action φ which has a cocompact discrete subgroup Γ, the solvmanifold G/Γ admits a canonical invariant formal (i.e. all products of harmonic forms are again harmonic) metric. We show that a compact oriented aspherical manifold of dimension l…
In this paper we introduce the concept of Hamiltonian system in the canonical and Poisson settings. We will discuss the quantization of the Hamiltonian systems in the Poisson context, using formal deformation quantization and quantum group theories.
The well known conformal covariance of the Dirac operator acting on spinor fields over a semi Riemannian spin manifold does not extend to powers thereof in general. For odd powers one has to add lower order curvature correction terms in order to obtain conformal covariance. We derive an algorithmic construction in term…
We describe the cohomology of a specific type of foliation on complex manifolds.
problem Computing the basic cohomology of canonical holomorphic foliations on complex moment-angle manifolds.
method Using an Eilenberg-Moore spectral sequence and the formality of the Cartan model for the torus action.
result The basic cohomology ring of the canonical holomorphic foliation on a moment-angle manifold is similar to that of a complete simplicial toric variety.
New method improves spatial prediction validation accuracy.
problem Validation methods fail for spatial prediction tasks due to mismatch between validation and test locations.
method Proposes a new validation method that adapts existing covariate-shift ideas to spatial settings.
result Proves and demonstrates the new method's superiority in spatial prediction validation.
Solves differentiation for Lie ∞-groups using formal groupoids.
problem Differentiation of Lie ∞-groups.
method Develops homotopy theory of formal ∞-groupoids and analyzes Dold-Kan adjunction for cosimplicial algebras.
result Differentiation functor from finite-dimensional Lie ∞-groups to finite-type Lie ∞-algebras is homotopically well-behaved.
A geometric multisymplectic formulation of the classical BRST symmetry of constrained first-order classical field theories is described. To effect this we introduce graded analogues of the bundles and manifolds of the multisymplectic formulation of first-order field theories. The Lagrange-d'Alembert formalism is also d…
Anisotropic connections and parallel transport defined in Finsler spacetimes.
problem Defining and characterizing anisotropic connections and parallel transport in Finsler spacetimes.
method Introducing a new covariant derivative and parallel transport, identifying vertically trivial Finsler connections with anisotropic connections, and characterizing the Levi-Civita-Chern anisotropic connection.
result Characterization of the Levi-Civita-Chern anisotropic connection as the one preserving the length of parallely propagated vectors.
CorrCA identifies reliable dimensions in multivariate data across repetitions.
problem Finding consistent dimensions in multivariate data across trials, subjects, or raters.
method Maximizes the ratio of between-repetition to within-repetition covariance.
result CorrCA leads to repeat-reliability maximization and is equivalent to Linear Discriminant Analysis for zero-mean signals.
New method clusters stationary stochastic processes using covariance-based dissimilarity.
problem Clustering wide-sense stationary ergodic stochastic processes.
method Covariance-based dissimilarity measure with consistent algorithms for offline and online clustering.
result Asymptotically consistent algorithms for efficient clustering.
Building on the universal covering group of the general linear group, we introduce the composite spinor bundle whose subbundles are Lorentz spin structures associated with different gravitational fields. General covariant transformations of this composite spinor bundle are canonically defined.
The paper shows objective derivatives are covariant derivatives on Riemannian metrics.
problem The definition and interpretation of objective derivatives in continuum mechanics.
method Demonstrates that objective derivatives correspond to covariant derivatives on the manifold of Riemannian metrics.
result Objective derivatives are unified as covariant derivatives on the manifold of Riemannian metrics.