Geodesic 3-webs on surfaces are characterized by integrable flow equations.
problem Characterizing surfaces with hexagonal geodesic 3-webs.
method Integrable flow equations and generalized hodograph transform method.
result Geodesic flow on surfaces admits cubic first integrals for hexagonal 3-webs.
Characterizes mappings preserving Pythagorean-hodograph curves.
problem Preserving Pythagorean-hodograph curves in various dimensions.
method Proves conformal functions with square rational dilation are PH-preserving.
result Conformal functions with square rational dilation are the only PH-preserving mappings.
Paper develops a framework for hyperbolic Monge-Ampère equation on strips, proving well-posedness and stability.
problem Addressing the rigidity-flexibility dichotomy for wrinkled patterns in thin elastic sheets.
method Develops hodograph transformation and parametrix-corrector decomposition to handle corner singularities and prove well-posedness.
result Proves existence and uniqueness of hodograph weak solutions and derives energy estimates for stability.
The study finds the best elliptical trajectory for planets using a variation of the hodograph theorem.
problem Finding the best elliptical trajectory for planets.
method Using a variation of the circular hodograph theorem, the study finds the best fitting ellipse for planetary trajectories by minimizing the sum of square distances from the points to the plane.
result The study finds that the best fitting ellipse for planetary trajectories minimizes the sum of square distances from the points to the plane.
In this paper we obtain the general solution to the minimal surface equation, namely its local Weierstrass-Enneper representation, by using a system of hodographic coordinates. This is done by using the method of solving the Born-Infeld equations by Whitham. We directly compute conformal coordinates on the minimal surf…
Study integrable geodesic flows on 2-surfaces with high-degree polynomial first integrals.
problem Integrable geodesic flows on 2-surfaces with high-degree polynomial first integrals.
method Semi-Hamiltonian systems of PDEs and generalized hodograph method.
result Construction of many local explicit and implicit integrable examples with polynomial first integrals of degrees 3, 4, 5.
Extends method for solving certain hydrodynamic systems.
problem Solving non-diagonalisable integrable systems of hydrodynamic type.
method Generalised hodograph method applied to F-manifolds with compatible connections.
result Provides general solution under certain assumptions.
We obtain the Weierstrass-Enneper representation for maximal graphs(whose Gauss map is one-one) in Lorentz-Minkowski space. For this we use the method of Barbishov and Chernikov, which they have used to find the solutions of Born-Infeld equation in hodographic coordinates. We could use their method in our case, because…
We present two constructions of new solutions to the dispersionless KP (dKP) equation arising from the first two Painlevé transcendents. The first construction is a hodograph transformation based on Einstein--Weyl geometry, the generalised Nahm's equation and the isomonodromy problem. The second construction, motivated…
The paper classifies affine minimal translation surfaces and finds their properties.
problem Classifying and understanding affine minimal translation surfaces.
method Using Weierstrass-Enneper formula and hodographic coordinate system.
result Classification and properties of affine minimal translation surfaces.
Given a compact manifold with boundary with unknown Riemannian metric. The problem is to reconstruct the metric in a class of conformal metrics from knowledge of lengths of all closed geodesics (kinematic data). An integral inequality is stated which implies uniqueness and stability for this problem. If the conformal c…
The paper studies magnetic geodesic flows on 2-surfaces with integrable structures.
problem Analyzing magnetic geodesic flows on 2-surfaces with additional integrals.
method Constructing exact solutions to semi-Hamiltonian systems of PDEs using generalized hodograph method and Legendre transformation.
result Exact solutions constructed for semi-Hamiltonian systems of PDEs.
This paper is focused on geometric aspects of two particular types of finite-variable reductions in the dispersionless Toda hierarchy. The reductions are formulated in terms of "Landau-Ginzburg potentials" that play the role of reduced Lax functions. One of them is a generalization of Dubrovin and Zhang's trigonometric…
This paper constructs PH spline curves with prescribed arc lengths.
problem Interpolating points, tangent directions, and curvatures with prescribed arc-length.
method Local construction of G2 planar PH biarc curves of degree 7. result Prescribed arc-length can be satisfied for any data and any chosen ratio between boundary tangents.
In this paper we construct multiparametric families of two dimensional metrics with polynomial first integral. Such integrable geodesic flows are described by solutions of some semi-Hamiltonian hydrodynamic type system. We find infinitely many conservation laws and commuting flows for this system. This procedure allows…
Starting from a homogeneous polynomial in momenta of arbitrary order we extract multi-component hydrodynamic-type systems which describe 2-dimensional geodesic flows admitting the initial polynomial as integral. All these hydrodynamic-type systems are semi-Hamiltonian, thus implying that they are integrable according t…
The present paper attempts to show an alternative approach with regards to rational Pythagorean-hodograph (PH) curves and especially more natural approach for rational PH helices (i.e. rational helices). It exploits geometric features of rational helices to obtain a simpler construction of these curves and apply this t…
This work has its origins in an attempt to describe systematically the integrable geometries and gauge theories in dimensions one to four related to twistor theory. In each such dimension, there is a nondegenerate integrable geometric structure, governed by a nonlinear integrable differential equation, and each solutio…
Study extends geodesic ray transform results to orientable surfaces.
problem Characterize and stabilize mixed and transverse ray transforms on surfaces.
method Algebraic arguments applied to various geometries and ray transforms.
result Characterization of kernel and stability for mixed and transverse ray transforms on orientable surfaces.
The paper introduces models to learn generalized transformation equivariant representations.
problem Capturing intrinsic visual structures equivariant to various transformations.
method Deterministic and probabilistic AutoEncoding Transformations (AET and AVT) models trained to learn visual representations from generic groups of transformations.
result Generalized TERs (GTERs) that are equivariant to transformations in a more general fashion.
New filter bank sparsifying transforms outperform patch-based methods for image denoising.
problem Improving image denoising performance using data-adaptive sparsifying transforms.
method Proposes a new transform learning framework using undecimated perfect reconstruction filter banks, allowing independent filter length choice.
result Filter bank sparsifying transforms outperform existing patch-based methods for image denoising.
We analyze Darboux transformations in very general settings for multidimensional linear partial differential operators. We consider all known types of Darboux transformations, and present a new type. We obtain a full classification of all operators that admit Wronskian type Darboux transformations of first order and a …
Integration procedure for Lie groupoid natural transformations.
problem Infinitesimal counterpart of natural transformations in Lie groupoids.
method Integration procedure for Lie groupoid morphisms.
result Provides smooth natural transformations between Lie groupoid morphisms.
Proposes differential and integral invariants under Mobius transformation.
problem Handling non-rigid deformation in 2-D and 3-D shapes.
method Focuses on Mobius transformation, proposes differential and integral invariants.
result Proposes differential and integral invariants under Mobius transformation.
New Lehmer Transform for analyzing non-stationary signals.
problem Analyzing non-stationary signals like brain waves.
method Proposes a new Lehmer Transform to decompose statistical moments.
result Theoretical properties of the Lehmer Transform are presented.
Introduces pseudo-codecomposition of transformation groups.
problem Understanding and categorizing transformation groups.
method Introduces pseudo-codecomposition and analyzes properties of transformation groups.
result The class of pseudo-codecomposable transformation groups is a proper intermediate class.
This paper investigates efficient Transformers and finds they scale with problem size.
problem Finding suitable replacements for standard Transformers in large-scale tasks.
method Modeling efficient Transformers (Sparse and Linear) as Dynamic Programming problems and analyzing their reasoning capabilities.
result Efficient Transformers scale with problem size, but can be more efficient for certain DP problems.
The conformal geometry of spacelike surfaces in 4-dimensional Lorentzian space forms has been studied by the authors in a previous paper, where the so-called polar transform was introduced. Here it is shown that this transform preserves spacelike conformal isothermic surfaces. We relate this new transform with the know…
Paper improves tensor completion using unitary transforms.
problem Robust tensor completion for various datasets.
method Transformed tensor SVD with unitary matrices.
result Recovered images have better PSNR than traditional methods.
Transforms classical connections using pushforwards and gauge transformations.
problem Transforming classical connections in categorical settings.
method Constructing pushforwards and applying gauge transformations to decorated path spaces.
result Combines traditional gauge transformation with affine translation.
Paper introduces graph-based transforms for video compression.
problem Efficiently represent video signals for compression.
method Develops two techniques for designing graph-based transforms (GL-GBTs and EA-GBTs).
result Graph-based transforms outperform traditional KLT in video compression.
Transformers interpret as probabilistic mixtures, offering new insights.
problem Understanding Transformers from a probabilistic perspective.
method Modeling Transformers as mixtures of Gaussian models.
result Transformers can be seen as maximum posterior probability estimators.
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.
Transformer-MGK replaces redundant heads with Gaussian key mixtures, improving efficiency and performance.
problem Redundant attention heads in transformers degrade performance and efficiency.
method Transformer-MGK replaces redundant heads with a mixture of Gaussian keys.
result Transformer-MGK accelerates training and inference, reduces parameters and FLOPs, and achieves comparable or better accuracy.
Adversarial learning improves image augmentation for neural networks.
problem Improving data augmentation for neural networks with limited data.
method Adversarial learning using an encoder-decoder architecture with a spatial transformer network.
result Our approach outperforms previous generative data augmentation methods.
B-cos transformers explain Vision Transformers' decisions.
problem Lack of holistic explanations for transformer outputs.
method Formulate each component as dynamic linear, allowing a single linear transform for summarization.
result Bcos-ViTs are highly interpretable and competitive on ImageNet.
The paper examines how polarized curves behave near singular points.
problem Analyzing the behavior of polarized curves near singular points.
method Investigates the limiting behavior of Darboux and Calapso transforms of polarized curves in the conformal n-dimensional sphere.
result For a pole of first order, all transforms converge to the original curve. For a pole of second order, a generic Darboux transform converges, but a Calapso transform has a limit point or circle.
The paper analyzes transformation models in high-dimensional settings.
problem Analyzing transformation models in high-dimensional data.
method Proposed an estimator for transformation parameter and showed asymptotic normality.
result The proposed estimator works well in small samples and tests the log-wage transformation.
XR-Transformer accelerates XMC by recursively fine-tuning on multi-resolution objectives.
problem Efficiently classifying texts with large label sets.
method Recursive multi-resolution fine-tuning of transformers.
result XR-Transformer achieves 20x faster training time and 54% Precision@1 on Amazon-3M.
Novel power transform unifies various mathematical functions.
problem Normalizing and standardizing datasets.
method Presented a novel power transform.
result Unified various mathematical functions.
Algorithm finds optimal affine transformation to minimize overall distortion.
problem Minimizing distortion in affine transformations.
method Riemannian geometry approach to define and minimize distortion.
result Mean distorting transformation found for minimizing overall distortion.
We study the dynamics of the discrete bicycle (Darboux, Backlund) transformation of polygons in n-dimensional Euclidean space. This transformation is a discretization of the continuous bicycle transformation, recently studied by Foote, Levi, and Tabachnikov. We prove that the respective monodromy is a Moebius transform…
Transformers struggle to approximate smooth functions, relying on piecewise constant approximations.
problem Understanding the expressivity of Transformers for function approximation.
method Theoretical analysis and experimental validation of Transformer's ability to approximate smooth functions.
result Transformers cannot reliably approximate smooth functions, relying on piecewise constant approximations.
The Weyl transform is introduced as a rich framework for data representation. Transform coefficients are connected to the Walsh-Hadamard transform of multiscale autocorrelations, and different forms of dyadic periodicity in a signal are shown to appear as different features in its Weyl coefficients. The Weyl transform …
Defines gauge transformations for Jacobi structures and their effects on contact groupoids.
problem Understanding transformations of Jacobi structures and their implications.
method Definition and discussion of gauge transformations for Jacobi structures and their impact on contact groupoids.
result Gauge transformations affect the contact structure of contact groupoids.
We define a transformation on harmonic maps from a Riemann surface into the 2-sphere which depends on a complex parameter, the so-called mu-Darboux transformation. In the case when the harmonic map N is the Gauss map of a constant mean curvature surface f and the parameter is real, the mu-Darboux transformation of -N i…
We begin by considering several properties commonly (but not universally) possessed by Bäcklund transformations between hyperbolic Monge-Ampère equations: wavelike nature of the underlying equations, preservation of independent variables, quasilinearity of the transformation, and autonomy of the transformation. We show…
Geometric approach uses Bäcklund transformations to create integrable discrete analogs of surface nets.
problem Creating integrable discrete analogs of surface nets and conjugate nets.
method Interpreting classical differential geometry results through Bäcklund transformations and applying permutability properties.
result Integrable discrete analogs of asymptotic and conjugate nets are constructed.