arXiv research

By a classical result, solutions of analytic elliptic PDEs, like the Laplace equation, are analytic. In many instances, the properties that come from being analytic are more important than analyticity itself. Many important equations are degenerate elliptic and solutions have much lower regularity. Still, one may hope …

Analyzes properties of Hopf manifolds from analytic and metric perspectives.

The book explores analytic properties of groups and their actions.

We prove that two-step analytic sub-Riemannian structures on a compact analytic manifold equipped with a smooth measure and Lipschitz Carnot groups satisfy measure contraction properties.

Examines properties of holomorphic fibrations in complex geometry.

Characterizes Bonnet surfaces using analytic conditions.

Generalizes crystallographic properties to all dimensions.

Proves Lorentzian manifold properties for analytic 3D spaces.

The paper proves properties of Finsler submanifolds and analytic maps.

We present here a result of Monomialization of real analytic two-symmetric tensor fields over regular real analytic surfaces. We apply it to the (extension of the pull-back of the) inner metric of a resolved surface of a real analytic surface singularity. Doing so we recover Hsiang & Pati property at each point of the …

Recently, Cappell and Miller extended the classical construction of the analytic torsion for de Rham complexes to coupling with an arbitrary flat bundle and the holomorphic torsion for $\bar{\partial}$-complexes to coupling with an arbitrary holomorphic bundle with compatible connection of type $(1,1)$. Cappell and Mil…

We discuss replica analytic continuation using several simple models in order to prove mathematically the validity of replica analysis, which is used in a wide range of fields related to large scale complex systems. While replica analysis consists of two analytical techniques, the replica trick (or replica analytic con…

Hybrid model speeds up galaxy simulations by incorporating baryonic properties.

In this paper we have studied the properties of covariant almost analytic vector field on Q - quasi umbilical hypersurface $M$ of a Sasakian manifold $\tilde M$ with $(φ, g, u, v, λ)-$structure and obtained the scalars $α$ and $β$ using $1 - form u, v$ covariant almost analytic for the hypersurface $M$ to be totally um…

Let (P1) be certain elliptic free-boundary problem on a Riemannian manifold (M,g). In this paper we study the restrictions on the topology and geometry of the fibres (the level sets) of the solutions f to (P1). We give a technique based on certain remarkable property of the fibres (the analytic representation property)…

We define analytic torsion of Z_2-graded elliptic complexes as an element in the graded determinant line of the cohomology of the complex, generalizing most of the variants of Ray-Singer analytic torsion in the literature. It applies to a myriad of new examples, including flat superconnection complexes, twisted analyti…

Due to the success of deep learning to solving a variety of challenging machine learning tasks, there is a rising interest in understanding loss functions for training neural networks from a theoretical aspect. Particularly, the properties of critical points and the landscape around them are of importance to determine …

GAMLA learns manifold structures with auto-encoding for global insights.

The paper studies HKKN stratifications for non-compact spaces and proves convexity properties.

New framework for calculating multivariate risk measures using Wishart process.

Analyzing static solutions in Finsler gravity, extending known results.

This paper investigates analytic properties of American option prices under the finite moment log-stable (FMLS) model. Under this model the price of American options is characterised by the free boundary problem of a fractional partial differential equation (FPDE) system. Using the technique of approximation we prove t…

The strong unique continuation property for Einstein metrics can be concluded from the well-known fact that Einstein metrics are analytic in geodesic normal coordinates. Here we give a proof of the same result that given two Einstein metrics with the same Ricci curvature on a fixed manifold, if they agree to infinite o…

Analytic surgery and gluing formula for torsion forms in fiber bundles.

We prove the automorphic property of the invariant of K3 surfaces with involution, which we obtained using equivariant analytic torsion, in the case where the dimension of the moduli space is less than or equal to 2.

The self-concordant-like property of a smooth convex function is a new analytical structure that generalizes the self-concordant notion. While a wide variety of important applications feature the self-concordant-like property, this concept has heretofore remained unexploited in convex optimization. To this end, we deve…

We provide a simple proof of a result of Rouby-Sjöstrand-Ngoc \cite{RSN} and Deleporte \cite{Deleporte}, which asserts that if the Kähler potential is real analytic then the Bergman kernel is an \textit{analytic kernel} meaning that its amplitude is an \textit{analytic symbol} and its phase is given by the polarization…

We provide a direct proof for the positivity of Chen-Nester-Tung quasi-local energy with analytic reference in spherical symmetry. A hoop-type theorem for this energy is also established. Finally, the relation between Chen-Nester-Tung and Brown-York quasi-local energies will be discussed.

We explore the efficacy of using a novel activation function in Artificial Neural Networks (ANN) in characterizing exoplanets into different classes. We call this Saha-Bora Activation Function (SBAF) as the motivation is derived from long standing understanding of using advanced calculus in modeling habitability score …

This research extends quasiplurisubharmonic functions on compact Kähler manifolds.

In this note we discuss a few properties of transnormal Finsler functions, i.e., the natural generalization of distance functions and isoparametric Finsler functions. In particular, we prove that critical level sets of an analytic transnormal function are submanifolds, and the partition of $M$ into level sets is a Fins…

In this paper we establish a strong multiplicity one type property for the length-holonomy spectrum for the three dimensional compact hyperbolic spaces. We use the analytic properties of Selberg-Gangolli-Wakayama zeta functions associated to compact hyperbolic spaces.

In this survey article, we present some panorama of groups acting on metric spaces of non-positive curvature. We introduce the main examples and their rigidity properties , we show the links between algebraic or analytic properties of the group and geometric properties of the space. Finally, we conclude with a few conj…

Constructs equivariant analytic torsion for proper actions on manifolds.

The purpose of this paper is to define semi- and subanalytic subsets and maps in the context of real analytic orbifolds and to study their basic properties. We prove results analogous to some well-known results in the manifold case. For example, we prove that if $A$ is a subanalytic subset of a real analytic quotient o…

It is a consequence of the Morse-Bott Lemma on Banach spaces that a smooth Morse-Bott function on an open neighborhood of a critical point in a Banach space obeys a Lojasiewicz gradient inequality with the optimal exponent one half. In this article we prove converses for analytic functions on Banach spaces: If the Loja…

In this paper, we specify what functions induce the bounded composition operators on a reproducing kernel Hilbert space (RKHS) associated with an analytic positive definite function defined on $\mathbf{R}^d$. We prove that only affine transforms can do so in a pretty large class of RKHS. Our result covers not only the …

We review the Reidemeister torsion, Ray-Singer's analytic torsion and the Cheeger-M"uller theorem. We describe the analytic torsion of the de Rham complex twisted by a flux form introduced by the current authors and recall its properties. We define a new twisted analytic torsion for the complex of invariant differentia…

For a smooth strictly pseudoconvex hypersurface in a complex manifold, we give a necessary and sufficient condition for being CR-diffeomorphic to a real-analytic CR manifold. Our condition amounts to a holomorphic extension property for the canonically associated function expressing $2$-jets of the formal Segre varieti…

In the paper, the author studies properties of three functions relating to the exponential function and the existence of partitions of unity, including accurate and explicit computation of their derivatives, analyticity, complete monotonicity, logarithmically complete monotonicity, absolute monotonicity, and the like.

The purpose of the paper is to introduce some conjectures regarding the analytic continuation and the arithmetic properties of quantum invariants of knotted objects. More precisely, we package the perturbative and nonperturbative invariants of knots and 3-manifolds into two power series of type P and NP, convergent in …

Let $Ω$ be a compact Riemannian manifold with smooth boundary and let $u_t$ be the solution of the heat equation on $Ω$, having constant unit initial data $u_0=1$ and Dirichlet boundary conditions ($u_t=0$ on the boundary, at all times). If at every time $t$ the normal derivative of $u_t$ is a constant function on the …

Develops theory of Anosov representations for Fuchsian groups, showing stability and analytical properties.

The abstract discusses how humans use visualizations in machine learning.

This paper is devoted to a proof of a generalized Ray-Singer conjecture for a manifold with boundary (the Dirichlet and the Neumann boundary conditions are independently given on each connected component of the boundary and the transmission boundary condition is given on the interior boundary). The Ray-Singer conjectur…

Develops Bialynicki-Birula and Morse-Bott theory for complex analytic spaces.

Holomorphic functions from knot complements link to quantum modular forms.

We propose a definition for analytic torsion of the contact complex on contact manifolds. We show it coincides with Ray-Singer torsion on any 3-dimensional CR Seifert manifold equipped with a unitary representation. In this particular case we compute it and relate it to dynamical properties of the Reeb flow. In fact th…