arXiv research

Paper simplifies and proves Bahri-Xu conjecture for various cases.

Introduces a reduction system for Artin-Tits groups, improving algorithms and proving periodicity results.

K-Nearest neighbor classifier (k-NNC) is simple to use and has little design time like finding k values in k-nearest neighbor classifier, hence these are suitable to work with dynamically varying data-sets. There exists some fundamental improvements over the basic k-NNC, like weighted k-nearest neighbors classifier (wh…

The purpose of this note is to exhibit some simple and basic constructions for smooth compact transformation groups, and some of their most immediate applications to geometry.

The paper studies invariant Einstein metrics on Lie supergroups.

Study of symplectic and Poisson reduction, proposing Poisson implosion.

In this paper we define coeffective de Rham cohomology for basic forms on a $K$--contact or Sasakian manifold $M$ and we discuss its relation with usually basic cohomology of $M$. When $M$ is of finite type (for instance it is compact) several inequalities relating some basic coeffective numbers to classical basic Bett…

We show how to reduce the general formulation of the mass-angular momentum-charge inequality, for axisymmetric initial data of the Einstein-Maxwell equations, to the known maximal case whenever a geometrically motivated system of equations admits a solution. It is also shown that the same reduction argument applies to …

We consider the fundamental problem in non-convex optimization of efficiently reaching a stationary point. In contrast to the convex case, in the long history of this basic problem, the only known theoretical results on first-order non-convex optimization remain to be full gradient descent that converges in $O(1/\varep…

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…

We first recall some basic definitions and facts about Jacobi manifolds, generalized Lie bialgebroids, generalized Courant algebroids and Dirac structures. We establish an one-one correspondence between reducible Dirac structures of the generalized Lie bialgebroid of a Jacobi manifold $(M,Λ,E)$ for which 1 is an admiss…

The paper studies Lie algebroid and groupoid quotients with forms, applying to Poisson and Dirac structures.

We define a new version of the exterior derivative on the basic forms of a Riemannian foliation to obtain a new form of basic cohomology that satisfies Poincaré duality in the transversally orientable case. We use this twisted basic cohomology to show relationships between curvature, tautness, and vanishing of the basi…

These letters, written in 1998-2000, contain various basic results about Courant algebroids (CAs), such as classification of exact and transitive CAs, reduction of CAs, description in terms of symplectic dg manifolds, a canonical generating Dirac operator, and a relation with Poisson-Lie T-duality.

Can one reduce the size of a graph without significantly altering its basic properties? The graph reduction problem is hereby approached from the perspective of restricted spectral approximation, a modification of the spectral similarity measure used for graph sparsification. This choice is motivated by the observation…

We propose a Poisson-Lie analog of the symplectic induction procedure, using an appropriate Poisson generalization of the reduction of symplectic manifolds with symmetry. Having as basic tools the equivariant momentum maps of Poisson actions, the double group of a Poisson-Lie group and the reduction of Poisson manifold…

Introduces nonlinear splittings on fibre bundles for generalizing connections.

We reframe linear dimensionality reduction as a problem of Bayesian inference on matrix manifolds. This natural paradigm extends the Bayesian framework to dimensionality reduction tasks in higher dimensions with simpler models at greater speeds. Here an orthogonal basis is treated as a single point on a manifold and is…

In this paper we study the modular classes of Dirac manifolds and of Dirac maps, and we discuss their basic properties. We apply these results to explain the relationship between the modular classes of the various structures involved in the reduction of a Poisson manifold under the action by of a Poisson Lie group.

This monograph describes a Riemannian geometric reduction approach to the three-body problem. The fundamental theorems are presented in the introductory part, whereas their proofs are provided in later chapters where specific topics are analyzed in more detail. The basic idea is to reduce the kinematic and dynamics of …

Reduction principles for proper actions on smooth manifolds.

The basic Dolbeault cohomology groups of a Sasakian manifold M are invariants of its characteristic foliation F (the orbit foliation of the Reeb flow). We show some fundamental properties of this cohomology, which are useful for its computation. In the first part of the article, we show that the basic Hodge numbers, th…

The basic cohomology of a Riemannian foliation on a complete manifold with all leaves closed is the cohomology of the leaf space. In this paper we introduce various methods to compute the basic cohomology in the presence of both closed and non-closed leaves in the simply-connected case (or more generally for Killing fo…

In [4], we gave a sharp lower bound for the first eigenvalue of the basic Laplacian acting on basic $1$-forms defined on a compact manifold whose boundary is endowed with a Riemannian flow. In this paper, we extend this result to the case of the first eigenvalue on basic $p$-forms for $p>1$. As in [4], the limiting cas…

Study on cohomology of singular foliations with localization results.

We review the basic elements of the geometrical formalism for description of gauge fields and the theory of invariant connections, and their applications to the coset space dimensional reduction of Yang-Mills theories. We also discuss the problem of classification of principal fibre bundles, which is important for the …

Scientists in many fields have the common and basic need of dimensionality reduction: visualizing the underlying structure of the massive multivariate data in a low-dimensional space. However, many dimensionality reduction methods confront the so-called "crowding problem" that clusters tend to overlap with each other i…

The paper studies Hodge structures on contact manifolds and their cohomology.

When a complex semisimple group $G$ acts holomorphically on a Kähler manifold $(X,ω)$ such that a maximal compact subgroup $K\subset G$ preserves the symplectic form $ω$, a basic result of symplectic geometry says that the corresponding categorical quotient $X/G$ can be identified with quotient of the zero-set of the m…

Leveraging the intrinsic symmetries in data for clear and efficient analysis is an important theme in signal processing and other data-driven sciences. A basic example of this is the ubiquity of the discrete Fourier transform which arises from translational symmetry (i.e. time-delay/phase-shift). Particularly important…

In this paper we prove a formula for the analytic index of a basic Dirac-type operator on a Riemannian foliation, solving a problem that has been open for many years. We also consider more general indices given by twisting the basic Dirac operator by a representation of the orthogonal group. The formula is a sum of int…

This paper tackles the reduction of redundant repeating generation that is often observed in RNN-based encoder-decoder models. Our basic idea is to jointly estimate the upper-bound frequency of each target vocabulary in the encoder and control the output words based on the estimation in the decoder. Our method shows si…

This is the pdf -version of the author's Ph.D. thesis (1995, ULB, Belgium). The notion of symeplectic symmertic space is introduced and studied via Lie theoretical and symplectic geoemetrical methods. The first chapter concerns basic poperties, however, an explicit formula for the Loos connection in the symplectic fram…

New deep network derived from rate reduction principles, explaining features and efficiency.

In this paper, we give an optimal lower bound for the eigenvalues of the basic Dirac operator on a quaternion-Kahler foliation. The limiting case is characterized by the existence of quaternion-Kahler Killing spinors. We end this paper by giving some examples.

In this paper we describe Routhian reduction as a special case of standard symplectic reduction, also called Marsden-Weinstein reduction. We use this correspondence to present a generalization of Routhian reduction for quasi-invariant Lagrangians, i.e. Lagrangians that are invariant up to a total time derivative. We sh…

This paper, in which we develop ideas introduced in \cite{MR}, focuses on \emph{reduction methods} (basically, group actions or, more generally, simmetries) for the bienergy. This type of techniques enable us to produce examples of critical points of the bienergy by reducing the study of the relevant fourth order PDE's…

In this paper, the equations governing the unsteady flow of a perfect polytropic gas in three space dimensions are considered. The basic similarity reductions for this system are performed. Reduced equations and exact solutions associated with the symmetries are obtained. This results is used to give the invariance of …

The ``classical BRST construction'' as developed by Batalin-Fradkin-Vilkovisky is a homological construction for the reduction of the Poisson algebra $P = C^\infty (W)$ of smooth functions on a Poisson manifold $W$ by the ideal $I$ of functions which vanish on a constraint locus. This ideal is called first class if $I$…

This expository paper presents elementary proofs of four basic results concerning derivatives of quasi-convex functions. They are combined into a fifth theorem which is simple to apply and adequate in many cases. Along the way we establish the equivalence of the basic lemmas of Jensen and Slodkowski.

Introduces $b$-Lie groups and studies their symplectic structures and reductions.

The study provides a formula for the volume of leaf spaces of certain foliations.

We present an explicit construction of the basic bundle gerbes with connection over all connected compact simple Lie groups. These are geometric objects that appear naturally in the Lagrangian approach to the WZW conformal field theories. Our work extends the recent construction of E. Meinrenken \cite{Meinr} restricted…

We consider the construction of the basic bundle gerbe on SU(n) introduced by Meinrenken and show that it extends to a range of groups with unitary actions on a Hilbert space including U(n), diagonal tori and the Banach Lie group of unitary operators differing from the identity by an element of a Schatten ideal. In all…

In this paper, we consider a compact Riemannian manifold whose boundary is endowed with a Riemannian flow. Under a suitable curvature assumption depending on the O'Neill tensor of the flow, we prove that any solution of the basic Dirac equation is the restriction of a parallel spinor field defined on the whole manifold…

We provide a comparative study of several widely used off-policy estimators (Empirical Average, Basic Importance Sampling and Normalized Importance Sampling), detailing the different regimes where they are individually suboptimal. We then exhibit properties optimal estimators should possess. In the case where examples …

We investigate the computational complexity of several basic linear algebra primitives, including largest eigenvector computation and linear regression, in the computational model that allows access to the data via a matrix-vector product oracle. We show that for polynomial accuracy, $Θ(d)$ calls to the oracle are nece…

Mathematical analysis of SNE and t-SNE for dimension reduction.