New Kelvin transform for anisotropic elliptic problems.
problem Semilinear and quasilinear anisotropic elliptic problems.
method Introducing a new Kelvin-type transform in the anisotropic setting.
result New insights into anisotropic elliptic problems.
Formulae for Bäcklund transformations of hyperbolic and elliptic sine-Gordon/sinh-Gordon equations.
problem Finding solutions for specific types of equations.
method Providing superposition formulae for Bäcklund transformations.
result Algebraically obtain infinitely many solutions after first integration.
Proposes elliptical transformation for improving machine learning performance in perturbation models.
problem Performance degradation of machine learning techniques in transform domain due to feature pattern differences.
method Introduces a nonlinear parametric perturbation model that transforms input feature patterns to elliptical patterns, and applies flexible block-wise dimensionality reduction.
result The proposed method outperforms PCA in classification performance and data privacy protection.
New solutions found for elliptic sinh-Gordon and sine-Gordon equations.
problem Elliptic sinh-Gordon and sine-Gordon equations on the real plane.
method Backlund transformation connecting the equations.
result New families of solutions introduced.
Study constructs transverse metrics using transformations commuting with elliptic operators.
problem Existence of transverse metrics in foliation theory.
method Applying the Average Method to construct a transverse metric.
result Pseudogroup of local transformations equicontinuous and quasi-analytic.
Paper classifies minimal graph transformations into new families of surfaces.
problem Classifying minimal graph transformations into new families of surfaces.
method Formulated and solved a coupled system of partial differential equations, reduced to solving an ordinary differential equation.
result Established rigorous equivalence to a modified problem for a harmonic function, yielding new families of minimal surfaces.
The study shows conditions for elliptic surfaces without 1-handles.
problem Finding conditions for elliptic surfaces without 1-handles.
method Analyzing knot surgeries and log-transformations of elliptic surfaces.
result Conditions for elliptic surfaces without 1-handles are established.
Invariants of knot complements remain unchanged under a specific transformation.
problem Invariance of knot complements under a special transformation.
method Bordered Heegaard Floer invariant and elliptic involution on knot complements.
result The invariant remains unchanged under the elliptic involution.
Sharp mapping properties and regularization for X-ray transform on disks of constant curvature.
problem Sharp mapping properties and regularization of X-ray transform.
method Derive functional relations and mapping properties using elliptic differential operators.
result Theoretical possibility of regularized inversions for X-ray transform.
We extend the coherent state transform (CST) of Hall to the context of the moduli spaces of semistable holomorphic vector bundles with fixed determinant over elliptic curves. We show that by applying the CST to appropriate distributions, we obtain the space of level k, rank n and genus one non-abelian theta functions w…
Elliptical Attention improves transformer performance by focusing on contextually relevant features.
problem Transformer models suffer from representation collapse and are vulnerable to contaminated samples.
method Uses Mahalanobis distance to define hyper-ellipsoidal neighborhoods for attention weights.
result Elliptical Attention reduces representation collapse and enhances model robustness.
We define a Fourier-Mukai transform for a triple consisting of two holomorphic vector bundles over an elliptic curve and a homomorphism between them. We prove that in some cases the transform preserves the natural stability condition for a triple. We also define a Nahm transform for solutions to natural gauge-theoretic…
Guillarmou extends X-ray transform to magnetic and thermostat flows.
problem Stability of magnetic X-ray transforms.
method Generalizes normal operator to thermostat and magnetic flows, proving ellipticity.
result Elliptic pseudodifferential operators of order -1 for generalized normal operators.
Quantum heat traces study new invariants from elliptic operators.
problem New invariants of elliptic operators on Riemannian manifolds.
method Relativistic and quantum heat traces, integral transforms, asymptotic expansion.
result Coefficients of asymptotic expansion determined by local and global invariants.
Study normal operators of double fibration transforms with conjugate points.
problem Normal operators of double fibration transforms with conjugate points.
method Stable conditions on the distribution of conjugate points, splitting into elliptic and Fourier integral operators.
result Normal operator splits into an elliptic pseudodifferential operator and Fourier integral operators.
For a compact Riemannian surface with boundary we study attenuated geodesic transform of functions and differential forms. We generalize several known results on uniqueness and stability of this transform dropping condition of absence of conjugate points.
We find a remarkable subalgebra of higher symmetries of the elliptic Euler-Darboux equation. To this aim we map such equation into its hyperbolic analogue already studied by Shemarulin. Taking into consideration how symmetries and recursion operators transform by this complex contact transformation, we explicitly give …
This study explores the index theory of Heisenberg elliptic and transversally Heisenberg elliptic operators using KK-theory.
problem Analyzing the index theory of Heisenberg elliptic and transversally Heisenberg elliptic operators.
method Applying Kasparov's methodology and examining specific conditions using Fourier transform of the nilpotent group C∗-algebra. result Demonstrated enhanced methods for analyzing hypoellipticity and defined transversal Heisenberg ellipticity in a KK-theoretic context. We construct a certain cross product of two copies of the braided dual H~ of a quasitriangular Hopf algebra H, which we call the elliptic double EH, and which we use to construct representations of the punctured elliptic braid group extending the well-known representations of the planar braid group attache…
We present the Nahm transform of the doubly-periodic instantons introduced in math.DG/9909069, converting them into certain meromorphic solutions of Hitchin's equations over an elliptic curve.
Geodesics in positive Lagrangian spaces map to special Lagrangian cylinders.
problem Understanding geodesics in spaces of positive Lagrangian submanifolds.
method Cylindrical transform of geodesics, solving elliptic PDEs.
result Geodesics in positive Lagrangian spaces correspond to one-parameter families of special Lagrangian cylinders.
New framework uses elliptic operators to study projective maps.
problem Understanding projective structures on Riemannian manifolds.
method Develops two elliptic operators of second and fourth order.
result Establishes a natural correspondence between analytical and geometric properties.
Authors develop a new theory to smooth spacetime connections and remove singularities in GR shock waves.
problem Singularities in General Relativity shock waves and optimal regularity of spacetime connections.
method Established a general multi-dimensional existence theory for Reintjes-Temple equations using elliptic regularity in Lp spaces. result Regularities of GR shock waves can always be removed by coordinate transformations, extending Uhlenbeck compactness to Lorentzian geometry.
Using the Weil-Brezin-Zak transform of solid state physics, we describe line bundles over elliptic curves in terms of Weyl operators. We then discuss the connection with finitely-generated projective modules over the algebra Aθ of the noncommutative torus. We show that such Aθ-modules have a natural interpretatio…
Compact sequences of manifolds with boundary converge smoothly under conformal transformations.
problem Compactness of sequences of manifolds with boundary under conformal transformations.
method Use of stable elliptic estimates and the 'flatzoomer' method.
result Smooth Cheeger-Gromov convergence of sequences of manifolds with boundary under conformal transformations.
A functional ansatz is developed which gives certain elliptic solutions of the Witten-Dijkgraaf-Verlinde-Verlinde (or WDVV) equation. This is based on the elliptic trilogarithm function introduced by Beilinson and Levin. For this to be a solution results in a number of purely algebraic conditions on the set of vectors …
Geodesic algorithms extended to arbitrary ellipsoids.
problem Computing geodesics on ellipsoids of varying eccentricity.
method Implementation of geodesic algorithms using elliptic integrals and discrete sine transform.
result Achieved high accuracy (close to machine precision) for geodesic computations.
We introduce a new elliptic operator on null hypersurfaces of four-dimensional Lorentzian manifolds. This operator depends on the first and second fundamental forms of the sections of a foliation of the null hypersurface and its novelty originates from its covariant transformation under change of foliation. It thus pro…
We show that an oriented elliptic 3-manifold admits a universally tight positive contact structure iff the corresponding group of deck transformations on S3 preserves a standard contact structure pointwise. We also relate univerally tight contact structures on 3-manifolds covered by S3 to the exceptional isomorph…
In this paper we consider the local X-ray transform for general flows. We extend the results on the local and global invertibility of the geodesic ray transform proved by Uhlmann and Vasy \cite{UV} to the X-ray transform for a general flow. The key improvement is that our argument for the ellipticity of the conjugated …
We study the transversal wave equation on a compact Riemannian foliated manifold. As applications, we get an Egorov's type theorem for transversally elliptic operators, state a relationship between the singularities of the Fourier transform of the spectrum distribution function of a transversally elliptic operator and …
This is the fourth article of our series. Here, we study weighted norm inequalities for the Riesz transform of the Laplace-Beltrami operator on Riemannian manifolds and of subelliptic sum of squares on Lie groups, under the doubling volume property and Gaussian upper bounds.
Study connects weighted isoperimetric problems to nonlocal elliptic operator extensions.
problem Sharp inequalities for weighted Poisson integrals and their extremizers.
method Formulates variational problem on conformal metric measure space.
result Sharp inequalities are linked to variational problem on CCE manifolds.
Transforms hyperbolic to flat data, deriving geometric inequalities.
problem Deriving geometric inequalities in asymptotically AdS hyperbolic spacetimes.
method Constructs transformations preserving physical quantities to relate hyperbolic to flat spacetimes.
result Derives geometric inequalities from flat counterparts.
Study light ray transform in pseudo-Euclidean space, derive inversion formula, and prove stability.
problem Analyzing light ray transform in pseudo-Euclidean space.
method Investigate normal operator, derive inversion formula, analyze as Fourier Integral Operator.
result Derive an inversion formula and prove stability estimates.
The paper studies self-Bäcklund curves in centroaffine geometry using elliptic functions.
problem Understanding self-Bäcklund curves in centroaffine geometry.
method Description of general properties and detailed analysis using elliptic functions.
result Provides a detailed description of self-Bäcklund centroaffine curves in terms of elliptic functions.
In this article, we consider a generalized Radon transform that comes up in ultrasound reflection tomography. In our model, the ultrasound emitter and receiver move at a constant distance apart along a circle. We analyze the microlocal properties of the transform R that arises from this model. As a consequence, we sh…
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.
We find that for any n-dimensional, compact, convex subset K of R^{n+1} there is an affinely-spherical hypersurface M in R^{n+1} with center at the relative interior of K, such that the disjoint union of M and K is the boundary of an (n+1)-dimensional, compact, convex set. This so-called affine hemisphere M is uniquely…
Proves X-ray transform injectivity on specific manifolds.
problem Injectivity of X-ray transform on closed Anosov manifolds and spherical boundary.
method Perturbative argument of the 0-eigenvalue of elliptic operators via microlocal analysis.
result Generically injective X-ray transform on specified manifolds.
Researchers found a Calabi-Yau structure and constructed a Bargmann type transformation on the Cayley projective plane.
problem Existence of Calabi-Yau structure and construction of Bargmann type transformation.
method Pairing of polarizations, natural Lagrangian foliation, and Kähler structure.
result Quantization of geodesic flow through elliptic Fourier integral operators.
We establish Evans-Krylov estimates for certain nonconvex fully nonlinear elliptic and parabolic equations by exploiting partial Legendre transformations. The equations under consideration arise in part from the study of the "pluriclosed flow" introduced by the first author and Tian
Paper proposes a method to efficiently cluster stretched mixtures.
problem Clustering stretched elliptical mixtures using standard methods like PCA and k-means fails.
method Proposes a non-convex program to transform data into a one-dimensional point cloud.
result Efficient first-order algorithm achieves near-optimal statistical precision.
New method of symmetrization applied to PDEs on spheres.
problem Estimating solutions of quasilinear elliptic PDEs with singular data.
method Symmetrization method applied to mappings on the sphere, using conformal transformations.
result Estimates solutions of PDEs with Dirac measures on spheres.
Study geometric relations between submanifolds of arbitrary codimension.
problem Understanding the geometric configuration of submanifolds of arbitrary codimension.
method Obtained relations between the geometry of submanifolds and their boundaries.
result Ellipticity of generalized Newton transformations implies tranversality and total geodesy.
We consider an application involving a financial quadratic portfolio of options, when the joint underlying log-returns changes with multivariate elliptic distribution. This motivates the needs for methods for the approximation of multiple integrals over hyperboloids. A transformation is used to reduce the hyperboloid i…
Injectivity of X-ray transform proven for non-smooth metrics.
problem Injectivity of X-ray transform on non-smooth metrics.
method Microlocal analysis of the normal operator, establishing ellipticity and smoothing properties.
result Injectivity of X-ray transform on L2 for metrics with finitely differentiable tensor. Existence of Ricci flat metric on Kummer K3 surface proven.
problem Proving existence of Ricci flat metric on Kummer K3 surface.
method General strategy of Donaldson's gluing construction, compact elliptic theory on usual Hölder and Sobolev spaces, explicit isometry to Gibbons-Hawking ansatz.
result Existence of a Ricci flat metric on the Kummer K3 surface.