arXiv research

Rolling two hyperboloid surfaces is described using a Monge normal form.

We find a normal form for two-input flat discrete-time systems.

The paper defines normal forms for rational 3-tangles and shows a sequence of moves to transform one form to another.

In this article, we first describe a normal form of real-analytic, Levi-nondegenerate submanifolds of $C^N$ of codimension d $\ge$ 1 under the action of formal biholomorphisms, that is, of perturbations of Levi-nondegenerate hyperquadrics. We give a sufficient condition on the formal normal form that ensures that the n…

Method studies equivalence of second order ODEs under specific transformations.

We give a complete list of normal forms for the 2-dimensional metrics that admit a transitive Lie pseudogroup of geodesic-preserving transformations and we show that these normal forms are mutually non-isometric. This solves a problem posed by Sophus Lie.

Normal forms and invariants for nondegenerate hypersurfaces in C^2.

We consider free and proper cotangent-lifted symmetries of Hamiltonian systems. For the special case of G = SO(3), we construct symplectic slice coordinates around an arbitrary point. We thus obtain a parametrisation of the phase space suitable for the study of dynamics near relative equilibria, in particular for the B…

We construct a form of swallowtail singularity in R^3 which uses coordinate transformation on the source and isometry on the target. As an application, we classify configurations of asymptotic curves and characteristic curves near swallowtail.

New form of $D_4^-$-singularities for fronts in 3D space.

We show that there are a finite number of possible pictures for a surface in a tetrahedron with local index $n$. Combined with previous results, this establishes that any topologically minimal surface can be transformed into one with a particular normal form with respect to any triangulation.

Unique continuation for X-ray transforms of one-forms with partial data.

We reduce boundary determination of an unknown function and its normal derivatives from the (possibly weighted and attenuated) broken ray data to the injectivity of certain geodesic ray transforms on the boundary. For determination of the values of the function itself we obtain the usual geodesic ray transform, but for…

We outline an approach to the inverse problem of Calderón that highlights the role of microlocal normal forms and propagation of singularities and extends a number of earlier results also in the anisotropic case. The main result states that from the boundary measurements it is possible to recover integrals of the unkno…

The study simplifies complex functions on surfaces using a special transformation.

Flow-based deep generative models learn data distributions by transforming a simple base distribution into a complex distribution via a set of invertible transformations. Due to the invertibility, such models can score unseen data samples by computing their exact likelihood under the learned distribution. This makes fl…

In this paper we form relations for the determination of the elements of the Eötvös matrix of the Earth's normal gravity field. In addition a relation between the Gauss curvature of the normal equipotential surface and the Gauss curvature of the actual equipotential surface both passing through the point P is presented…

Improves data normality with robust transformations.

We consider the moduli space MN of flat unitary connections on an open Kaehler manifold U (complement of a divisor with normal crossings) with restrictions on their monodromy transformations. Using intersection and L2 cohomologies with degenerating coefficients we construct a natural symplectic form F on MN. When U is …

Unified methodology for statistical inference in least squares and PCA via randomized sketching.

Transformer improves parameter estimation without needing closed-form solutions.

We show that in any triangulated 3-manifold, every index n topologically minimal surface can be transformed to a surface which has local indices (as computed in each tetrahedron) that sum to at most n. This generalizes classical theorems of Kneser and Haken, and more recent theorems of Rubinstein and Stocking, and is t…

In the present paper, we study the finite type invariants of Gauss words. In the Polyak algebra techniques, we reduce the determination of the group structure to transformation of a matrix into its Smith normal form and we give the simplified form of a universal finite type invariant by means of the isomorphism of this…

The study proves conditions for constant curvature submanifolds in space forms.

Firm size data usually do not show the normality that is often assumed in statistical analysis such as regression analysis. In this study we focus on two firm size data: the number of employees and sale. Those data deviate considerably from a normal distribution. To improve the normality of those data we transform them…

Twelve numerical methods for Poisson geometry concepts.

We exhibit several transformations of surfaces in R^4. First, one that takes a flat surface and gets a surface with flat normal bundle; then, one that takes a surface with flat normal bundle and gets a flat surface; finally, a one-parameter family of transformations on a flat surface with flat normal bundle and gives a…

We find a closed-form determinant for a specific sparse covariance matrix model.

Study normal operators of double fibration transforms with conjugate points.

Normality equations describe Newtonian dynamical systems admitting normal shift of hypersurfaces. They were first derived in Euclidean geometry, then in Riemannian geometry. Recently they were rederived in more general case, when geometry of manifold is given by generalized Legendre transformation. As appears, in this …

Study of singularities in mean curvature flow with focus on $\mathbb{S}^3 imes\mathbb{R}$.

SurVAE Flows combine VAEs and flows using surjective transformations.

A framework for transformer attention layers derived from SVR.

Guillarmou extends X-ray transform to magnetic and thermostat flows.

We simplify $D_4^+$-front singularities and apply to geometric invariants.

Let $K$ be a link of Conway's normal form $C(m)$, $m \geq 0$, or $C(m,n)$ with $mn\textgreater{}0$, and let $D$ be a trigonal diagram of $K.$ We show that it is possible to transform $D$ into an alternating trigonal diagram, so that all intermediate diagrams remain trigonal, and the number of crossings never increases.

Normality equations describe Newtonian dynamical systems admitting normal shift of hypersurfaces. These equations were first derived in Euclidean geometry. Then very soon they were rederived in Riemannian and in Finslerian geometry. Recently I have found that normality equations can be derived in geometry given by clas…

We give reconstruction formulas inverting the geodesic X-ray transform over functions (call it $I_0$) and solenoidal vector fields on surfaces with negative curvature and strictly convex boundary. These formulas generalize the Pestov-Uhlmann formulas in [Pestov-Uhlmann, IMRN '04] (established for simple surfaces) to ca…

New method prevents entropy collapse in Transformer training, leading to more stable and robust models.

Novel power transform unifies various mathematical functions.

The minimal standardizer of a curve system on a punctured disk is the minimal braid that transforms it into a system formed only by round curves. We give an algorithm to compute it in a geometrical way. Then, we generalize this problem algebraically to parabolic subgroups of Artin-Tits groups of spherical type and we s…

In this paper we develop the vectorial Ribaucour transformation for Euclidean submanifolds. We prove a general decomposition theorem showing that under {appropriate} conditions the composition of two or more vectorial Ribaucour transformations is again a vectorial Ribaucour transformation. An immediate consequence of t…

The paper uses transformed ANOVA to identify important fire detection variables.

The paper establishes a Poisson Poincaré-Dulac theorem for Poisson-flat connections.

Study shows stability in X-ray transform on specific hyperbolic manifolds.

Motivated by applications in computational anatomy, we consider a second-order problem in the calculus of variations on object manifolds that are acted upon by Lie groups of smooth invertible transformations. This problem leads to solution curves known as Riemannian cubics on object manifolds that are endowed with norm…

EMFs combine deep learning and probabilistic models for better density estimation.

The paper studies the distribution of random degeneracy sets on complex manifolds.