arXiv research

Geometrically studies Moore-Penrose inverse and polar decomposition continuity.

Study submanifolds in hyperbolic space, focusing on their boundary and Laplace operator.

This paper is a continuation of arXiv:0809.1158, dealing with a general, not-necessarily torsion-free, connection. It characterizes all possible systems of generators for vector-field valued operators that depend naturally on a set of vector fields and a linear connection, describes the size of the space of such operat…

We apply the graph complex method to vector fields depending naturally on a set of vector fields and a linear symmetric connection. We characterize all possible systems of generators for such vector-field valued operators including the classical ones given by normal tensors and covariant derivatives. We also describe t…

We extend the study of the de Rham operator with ideal boundary conditions from the case of isolated conic singularities, as analyzed by Cheeger, to the case of arbitrary stratified pseudomanifolds. We introduce a class of ideal boundary operators and the notion of mezzoperversity, which intermediates between the stand…

New hierarchies derived from KP hierarchy using non-formal operators and Yang-Mills action.

Inspired by recent breakthroughs in predictive modeling, practitioners in both industry and government have turned to machine learning with hopes of operationalizing predictions to drive automated decisions. Unfortunately, many social desiderata concerning consequential decisions, such as justice or fairness, have no n…

We construct in projective differential geometry of the real dimension $2$ higher symmetry algebra of the symplectic Dirac operator ${D}\kern-0.5em\raise0.22ex\hbox{/}_s$ acting on symplectic spinors. The higher symmetry differential operators correspond to the solution space of a class of projectively invariant overde…

Motivated by the study of Hörmander's sums-of-squares operators and their generalizations, we define the convolution algebra of transverse distributions associated to a singular foliation. We prove that this algebra is represented as continuous linear operators on the spaces of smooth functions and generalized function…

Let I be a symmetrically-normed ideal of the space of bounded operators acting on a Hilbert space H. Let ${p_i}_1 ^w$ $(1\leq w \leq \infty)$ be a family of mutually orthogonal projections on H. The pinching operator associated with the former family of projections is given by P: I --> I, P(x)=\sum_{i=1}^{w} p_i x p_i.…

Study of unitary and groupoid orbits of normal operators, focusing on manifold structures and spectral conditions.

A C*algebra A generated by a class of zero-order classical pseudodifferential operator on a cylinder RxB, where B is a compact riemannian manifold, containing operators with periodic symbols, is considered. A description of the K-theory index map associated to the continuous extension to A of the principal-symbol map i…

Let U be an open subset of R^n. Let L^2=L^2(U,dx) and H^1_0=H^1_0(U) be the standard Lebesgue and Sobolev spaces of complex-valued functions. The aim of this paper is to study the group G of invertible operators on H^1_0 which preserve the L^2-inner product. When U is bounded and the border $\partial U$ is smooth, this…

Motivated by the interesting and yet scattered developments in representation theory of Banach-Lie groups, we discuss several functional analytic issues which should underlie the notion of infinite-dimensional reductive Lie group: norm ideals, triangular integrals, operator factorizations, and amenability.

Building on previous results on the quadratic helicity in magnetohydrodynamics (MHD) we investigate particular minimum helicity states. Those are eigenfunctions of the curl operator and are shown to constitute solutions of the quasi-stationary incompressible ideal MHD equations. We then show that these states have inde…

Approximate vanishing ideal is a concept from computer algebra that studies the algebraic varieties behind perturbed data points. To capture the nonlinear structure of perturbed points, the introduction of approximation to exact vanishing ideals plays a critical role. However, such an approximation also gives rise to a…

In arXiv:0707.2151 the authors introduced the theory of local representations of the quantum Teichmüller space $\mathcal{T}^q_S$ ($q$ being a fixed primitive $N$-th root of $(-1)^{N + 1}$) and they studied the behaviour of the intertwining operators in this theory. One of the main results [Theorem 20, arXiv:0707.2151] …

Characterizes Fredholm conditions for group-invariant pseudodifferential operators.

The first purpose of this paper is to point out a curious result announced by Macaulay on the Hilbert function of a differential module in his famous book The Algebraic Theory of Modular Systems published in 1916. Indeed, on page 78/79 of this book, Macaulay is saying the following: " A polynomial ideal $\mathfrak{a} \…

Resistive crossbars have attracted significant interest in the design of Deep Neural Network (DNN) accelerators due to their ability to natively execute massively parallel vector-matrix multiplications within dense memory arrays. However, crossbar-based computations face a major challenge due to a variety of device and…

Let $L^2$ be the Lebesgue space of square-integrable functions on the unit circle. We show that the injectivity problem for Toeplitz operators is linked to the existence of geodesics in the Grassmann manifold of $L^2$. We also investigate this connection in the context of restricted Grassmann manifolds associated to $p…

Study Nijenhuis operators on homogeneous spaces related to C*-algebras.

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…

Quantum trace map defines invariants for knots and links, confirming a length conjecture.

The Hochschild and cyclic homology groups are computed for the algebra of `cusp' pseudodifferential operators on any compact manifold with boundary. The index functional for this algebra is interpreted as a Hochschild 1-cocycle and evaluated in terms of extensions of the trace functionals on the two natural ideals, cor…

Spinor representations of surfaces immersed into 4-dimensional pseudo-riemannian manifolds are defined in terms of minimal left ideals and tensor decompositions of Clifford algebras. The classification of spinor fields and Dirac operators on the immersed surfaces is given. The Dirac-Hestenes spinor field on surfaces im…

Regression learns Mori-Zwanzig operators for dynamical systems.

The study proves poor ideal three-edge triangulations are minimal for certain 3-manifolds.

We give a simple method to find ideal points of the character variety of a 3-manifold from an ideal triangulation.

In this Note we introduce and study dynamical systems related to the Ricci operator on the space of Kahler metrics as discretizations of certain geometric flows. We pose a conjecture on their convergence towards canonical Kahler metrics and study the case where the first Chern class is negative, zero or positive. This …

Short proof for ideal polygons with near optimal orthogeodesic decomposition.

A method uses CG to create efficient channels for ideal observers.

We prove a Hardy inequality for uniformly elliptic operators subject to Dirichlet or mixed boundary conditions on domains $Ω$ with piecewiese smooth boundary in arbitrary Riemannian Manifolds (M, g). Employing an approach of E.B. Davies for the euclidean case, we show that it implies a sufficient geometric criterion un…

The goal of this work is to study the ideals of the Goldman Lie algebra $S$. To do so, we construct an algebra homomorphism from $S$ to a simpler algebraic structure, and focus on finding ideals of this new structure instead. The structure $S$ can be regarded as either a $\mathbb{Q}$-module or a $\mathbb{Q}$-module gen…

Financial exchange operators cater to the needs of their users while simultaneously ensuring compliance with the financial regulations. In this work, we focus on the operators' commitment for fair treatment of all competing participants. We first discuss unbounded temporal fairness and then investigate its implementati…

We investigate the rigidity of hyperbolic cone metrics on $3$-manifolds which are isometric gluing of ideal and hyper-ideal tetrahedra in hyperbolic spaces. These metrics will be called ideal and hyper-ideal hyperbolic polyhedral metrics. It is shown that a hyper-ideal hyperbolic polyhedral metric is determined up to i…

Defines timelike ideal boundary for non-positively curved Lorentzian spaces.

A taut ideal triangulation of a 3-manifold is a topological ideal triangulation with extra combinatorial structure: a choice of transverse orientation on each ideal 2-simplex, satisfying two simple conditions. The aim of this paper is to demonstrate that taut ideal triangulations are very common, and that their behavio…

The paper studies dynamical properties in semigroups modulo ideals.

We define the ideal simplicial volume for compact manifolds with boundary. Roughly speaking, the ideal simplicial volume of a manifold $M$ measures the minimal size of possibly ideal triangulations of $M$ "with real coefficients", thus providing a variation of the ordinary simplicial volume defined by Gromov in 1982, t…

The notion of ideal immersions was introduced by the author in 1990s. Roughly speaking, an ideal immersion of a Riemannian manifold into a real space form is a nice isometric immersion which produces the least possible amount of tension from the ambient space at each point. In this paper, we classify all ideal hypersur…

New formula calculates volumes of ideal hyperbolic drums.

Study of combinatorial Calabi flow on ideal circle patterns.

We construct from first principles the operator 'A-hat' that annihilates the partition functions (or wavefunctions) of three-dimensional Chern-Simons theory with gauge groups SU(2), SL(2,R), or SL(2,C) on a knot complement M. The operator 'A-hat' is a quantization of the knot complement's classical A-polynomial A(l,m).…

A biconservative submanifold of a Riemannian manifold is a sub- manifold with divergence free stress-energy tensor with respect to bienergy. These are generalizations of biharamonic submanifolds. In 2013, B. Y. Chen and M.I. Munteanu proved that $δ(2)$-ideal and $δ(3)$-ideal biharmonic hypersurfaces in Euclidean space …

The paper studies deformations of Lie ideals in Lie algebras.

The paper tackles domain generalization using functional regression.

Paper solves long-standing problem of infinite ideal polyhedra in hyperbolic space.