The paper introduces canonical parameters for marginally trapped surfaces in Minkowski space.
problem Determining marginally trapped surfaces in Minkowski space.
method Introducing canonical parameters and proving existence and uniqueness theorems.
result Every marginally trapped surface is determined by three smooth functions.
Short note explores conditions for Finsler functions to be uniquely metrizable.
problem Conditions for Finsler functions to be uniquely metrizable.
method Exploration of sufficient conditions for affinely rigidity.
result Discussion of open problems in Finsler function metrizability.
Computes canonical heights for arithmetic log surfaces using Hurwitz zeta function.
problem Computing canonical heights for arithmetic log surfaces.
method Introduces a canonical height and uses limits of periods to compute it.
result Explicit formulas for canonical heights of arithmetic log surfaces, including (P_1,D).
Paper introduces a new cosmological volume function and its properties.
problem Introducing a new cosmological volume function.
method Introduces and analyzes the cosmological volume function τ_V, showing it's continuously differentiable.
result τ_V leads to a canonical splitting of the metric tensor and a canonical Wick-rotated Riemannian metric.
The paper provides Weierstrass representations for minimal space-like surfaces in Minkowski space-time.
problem Minimal space-like surfaces in Minkowski space-time.
method Obtained canonical Weierstrass representations via two holomorphic functions.
result Established geometric correspondence between minimal space-like surfaces and pairs of holomorphic functions.
New method for analyzing multiple longitudinal data processes.
problem Exploring associations between multiple random processes observed jointly.
method Functional Generalized Canonical Correlation Analysis (FGCCA) based on multiblock Regularized Generalized Canonical Correlation Analysis (RGCCA).
result FGCCA framework is robust to sparsely and irregularly observed data.
Billiard motion in ellipses analyzed with canonical coordinates.
problem Understanding billiard motion in ellipses.
method Canonical coordinates and kinematic analysis.
result Explicit parametrization of billiard motions using Jacobian elliptic functions.
New stability criterion for Fano manifolds linked to anti-canonically balanced metrics.
problem Stability of anti-canonically balanced metrics on Fano manifolds.
method Analyzing the asymptotic behavior of quantized Ding functionals and introducing a new algebro-geometric stability criterion.
result Existence of anti-canonically balanced metrics implies the new stability criterion.
Paper proposes an alternative to MLE for GLMs with non-canonical link functions.
problem Challenges in MLE for GLMs with non-canonical link functions.
method Variational Inequality (VI) estimation framework.
result Established finite-sample error bounds and asymptotic normality for VI estimator.
The paper introduces various canonical parameterizations for 2D-curved shapes.
problem Comparing unparameterized simple curves in the plane.
method Proposes diverse canonical parameterizations, including arc-length and curvature-based.
result Natural parameterizations correspond to physical movements and are geometric invariants.
Quantizes canonical bases for cluster varieties of type A.
problem Deforming algebra of functions on cluster varieties.
method Natural q-deformation of Fock and Goncharov's canonical basis. result Extension to quantum symplectic double.
The paper calculates the second variation of energy functions for families of canonically polarized manifolds.
problem Computing the second variation of energy functions for families of canonically polarized manifolds.
method Analyzing the Dirichlet energy of maps between fibers and using harmonic maps.
result The energy function is plurisubharmonic under certain curvature conditions.
Radial Basis Functions Neural Networks (RBFNNs) are tools widely used in regression problems. One of their principal drawbacks is that the formulation corresponding to the training with the supervision of both the centers and the weights is a highly non-convex optimization problem, which leads to some fundamentally dif…
Study log canonical thresholds on curved metrics on group compactifications.
problem Computing log canonical thresholds for curved metrics on group compactifications.
method Associated convex functions to metrics, using asymptotic behavior to determine thresholds.
result Formula for alpha invariant in terms of polytope associated to group compactification.
The paper defines and studies canonical parameters on surfaces in 4D space.
problem Understanding surfaces in 4D space without minimal points.
method Defining and proving existence of canonical principal parameters.
result Surfaces in 4D space are uniquely determined by four functions satisfying partial differential equations.
In many Lagrangian field theories one has a Poisson bracket defined on the space of local functionals. We find necessary and sufficient conditions for a transformation on the space of local functionals to be canonical in three different cases. These three cases depend on the specific dimensions of the vector bundle of …
Canonical coordinates defined for minimal time-like surfaces in n-dimensional Minkowski space.
problem Characterizing canonical coordinates on minimal time-like surfaces.
method Introducing canonical coordinates and proving their existence and uniqueness; using analysis over the algebra of double numbers.
result Canonical coordinates on minimal time-like surfaces are characterized by a natural condition for a complex function over the algebra of double numbers.
We prove that any minimal (maximal) strongly regular surface in the three-dimensional Minkowski space locally admits canonical principal parameters. Using this result, we find a canonical representation of minimal strongly regular time-like surfaces, which makes more precise the Weierstrass representation and shows mor…
Solves natural PDEs for minimal Lorentz surfaces in 4D spacetime.
problem Natural PDEs for minimal Lorentz surfaces in R24. method Weierstrass type representations and canonical coordinates.
result Explicit solution of the system of natural PDEs.
On a manifold with a projective connection we canonically assign a second order differential operator acting on the algebra of all densities to any tensor density Sij of fixed weight λ. In particular, this implies that on any projectively connected manifold, a `bracket' (symmetric biderivation) on the algebra of…
The paper explores new Riemannian structures and their properties.
problem Understanding new classes of Riemannian manifolds.
method Study of new potential functions on Riemannian manifolds, including classical structures.
result Complete classification for rigid cases, rigidity and obstruction results for others.
Given a solution of the (backwards) Ricci flow one can construct a so called canonical soliton metric on space-time, introduced by E. Cabezas-Rivas and P. Topping. We observe that for a mean curvature flow within a (backwards) Ricci flow background, the space-time track of the mean curvature flow yields a canonical sol…
The paper simplifies complex 2D functions near their critical points.
problem Simplifying smooth functions on 2-manifolds near critical points.
method Explicit construction of coordinate changes to canonical form.
result Estimates the radius of required neighbourhoods for specific singularity types.
Develops Kähler geometry on new varieties for canonical metrics.
problem No specific problem stated; focuses on new varieties.
method Introduces new varieties, develops Kähler geometry, associates convex functions with metrics.
result Provides expression for Mabuchi functional and combinatorial sufficient condition of properness.
The author studies regions foliated by 1D families of functions and their applications.
problem Understanding regions represented as foliated forms and natural smooth maps onto them.
method Investigates natural smooth maps respecting canonical projections and moment maps, focusing on foliated regions.
result Discusses the 1st derivative of functions and critical sets in foliated regions.
Geometrically proves Zabrodin-Wiegmann conjecture for integer QH states.
problem Proving a geometric version of Zabrodin-Wiegmann conjecture for integer Quantum Hall states.
method Using Riemann surfaces, canonical sections, and asymptotic expansions, the authors construct a canonical element in cohomology and relate its norm to the partition function.
result The constant term of the asymptotic expansion of the partition function matches a geometric version of Zabrodin-Wiegmann's prediction.
Using Laurent expansions of the Kontsevich-Vishik canonical trace of holomorphic families of classical pseudodifferential operators, we define functionals on the space of Riemannian metrics and investigate their conformal properties, thereby giving a unified description of several conformal invariants and anomalies.
New method estimates sparse canonical vectors efficiently.
problem Sparse canonical vectors estimation in CCA.
method Quasi-Bayesian estimation via Rayleigh quotient function.
result Achieves minimax rate with low computational cost.
A new distortion measure optimizes function approximations in vector quantization.
problem Measuring the quality of vector quantization points for natural signals.
method A canonical distortion measure (CDM) is introduced, induced by an environment of functions on input space.
result Optimizing reconstruction error with respect to CDM yields optimal piecewise constant approximations.
Study shows Calabi-Yau metrics converge to a specific form under certain conditions.
problem Degeneration of Calabi-Yau metrics and their limits.
method Optimal transport problem and minimisation of Kontorovich functional.
result Limit data of Calabi-Yau metrics can be encoded into a unique minimiser.
Sharp inequalities for weighted log canonical thresholds derived.
problem Understanding weighted log canonical thresholds in complex analysis.
method Combining integrability estimates, complex line restrictions, and pluripotential theory.
result Uniform control of difference quotients and explicit lower bounds derived.
Study minimal timelike surfaces in 3D Lorentz-Minkowski space using holomorphic functions.
problem Characterize minimal timelike surfaces in R13. method Use a Weierstrass-type formula with holomorphic functions in split-complex numbers to find canonical parameters and corresponding holomorphic functions.
result Enneper surfaces are the only minimal timelike surfaces with polynomial parametrization of degree 3 in isothermal parameters.
Constructs currents and heights on K3 surfaces.
problem Understanding the geometry and arithmetic of K3 surfaces.
method Constructs canonical positive currents and heights on K3 surfaces, equivariant for automorphism group.
result Continuous family of currents and heights defined over an enlarged boundary of the ample cone.
This is an account of some aspects of the geometry of Kähler affine metrics based on considering them as smooth metric measure spaces and applying the comparison geometry of Bakry-Emery Ricci tensors. Such techniques yield a version for Kähler affine metrics of Yau's Schwarz lemma for volume forms. By a theorem of Chen…
Holomorphic vector fields and anti-canonical divisors on complex manifolds are studied.
problem Existence of non-trivial holomorphic vector fields on compact complex manifolds.
method Vanishing result for measure preserving holomorphic vector fields, Gibbs stability, and log terminal singularities.
result No non-trivial holomorphic vector fields on compact complex manifolds with big anti-canonical line bundle.
Uniformly observable systems can be transformed into a triangular form with non-Lipschitz functions.
problem Uniformly observable and differentially observable systems with higher order than state dimension.
method Established triangular canonical form with non-Lipschitz functions.
result Characterization of points where non-Lipschitzness occurs and its relation to uniform infinitesimal observability.
Two new methods for analyzing repeated measures data using embeddings into Reproducing Kernel Hilbert Spaces.
problem Analyzing complex data structures with multiple features over time.
method Two generalizations of canonical correlation analysis for repeated measures data using embeddings into Reproducing Kernel Hilbert Spaces.
result Consistency rates for transformation and correlation estimators, relaxing common assumptions.
Study on plane curves with special connections and curvatures.
problem Characterizing plane curves with specific geometric properties.
method Introducing tangential and geodesic curvatures, proving existence and uniqueness theorems.
result Existence and uniqueness of curves with prescribed curvatures.
Constructs independent bases for cubic curve families using Hessian structures.
problem Finding independent bases for cubic curve families.
method Uses a Hessian structure to define a cost function for constructing bases.
result Constructs valuatively independent bases for H0(X,Lk). Zeta functions of SL2-character varieties match Dedekind zeta functions of trace fields.
problem Characterizing zeta functions of SL2-character varieties of hyperbolic 3-manifolds.
method Proving equality of zeta functions and expressing special values.
result Zeta functions of SL2-character varieties are equal to Dedekind zeta functions of trace fields.
A generalized bridge is the law of a stochastic process that is conditioned on N linear functionals of its path. We consider two types of representations of such bridges: orthogonal and canonical. The orthogonal representation is constructed from the entire path of the underlying process. Thus, future knowledge of the …
The paper uses tensor decompositions to improve neural network models for tree data.
problem Encoding structural knowledge from tree-structured data efficiently.
method Introduces new aggregation functions using Canonical and Tensor-Train decompositions.
result Proposed models outperform traditional methods on tree classification tasks.
The study defines invariants for time-like surfaces with real asymptotic lines.
problem Characterizing time-like surfaces with real asymptotic lines.
method Fundamental theorem of Bonnet-type, canonical parameters, invariant functions, PDEs.
result Time-like surfaces are determined by four invariant functions, two of which can be Gauss and mean curvature.
Solves complex Hessian equations in unstable cases, proving unique canonical solutions with singularities.
problem Existence of smooth solutions to complex Hessian equations in unstable cases.
method Parabolic flows and moment-map energy functionals, focusing on J-equation and deformed Hermitian Yang-Mills equation.
result Proves existence of unique canonical solutions with singularities on Kahler surfaces.
In many Lagrangian field theories, there is a Poisson bracket on the space of local functionals. One may identify the fields of such theories as sections of a vector bundle. It is known that the Poisson bracket induces an sh-Lie structure on the graded space of horizontal forms on the jet bundle of the relevant vector …
We prove a conjecture about log canonical thresholds and volumes of klt singularities.
problem Understanding the log canonical thresholds and volumes of klt singularities.
method We use quasi-monomial valuations and techniques from klt singularities to prove the conjectures.
result We confirm Chi Li's conjecture and show that the volume of klt singularities is a constructible function.
Study characterizes hypersurfaces in product spaces with a canonical direction.
problem Characterizing hypersurfaces with a specific gradient direction in product spaces.
method Based on R. Tojeiro's work, using product manifold metrics and gradient properties.
result Characterizes hypersurfaces with the gradient as a principal direction.
New embeddings for manifolds using heat kernels.
problem Constructing canonical conformal embeddings for manifolds.
method Employing heat kernel embedding from Bérard-Besson-Gallot'94 to find canonical conformal embeddings.
result Intrinsic construction of canonical conformal embeddings with dimensions growing exponentially with t.