arXiv research

Survey on a nonlinear d'Alembertian from general relativity.

New splitting theorem for weighted Finsler spacetimes without Berwald condition.

Researchers prove a 30-year-old cosmological conjecture about spacetime.

New calculus on spacetimes for nonlinear differential equations.

New proof of Lorentzian splitting theorems using elliptic operators.

Refines d'Alembertian for signed Lorentz distance functions in metric measure spacetimes.

New approach to gravity theory sacrifices smoothness for ellipticity.

The paper presents a probabilistic framework for SPD matrices in machine learning.

We compute the bi-Hamiltonian cohomology of an arbitrary dispersionless Poisson pencil in a single dependent variable using a spectral sequence method. As in the KdV case, we obtain that $BH^p_d(\hat{F}, d_1,d_2)$ is isomorphic to $\mathbb{R}$ for $(p,d)=(0,0)$, to $C^\infty (\mathbb{R})$ for $(p,d)=(1,1)$, $(2,1)$, $(…

Paper tackles unsupervised learning under latent label shift across domains.

Paper revisits DP-SCO in Euclidean and $\ell_p^d$ spaces, focusing on constrained and bounded sets.

We construct a family of PL triangulations of the $d$-dimensional real projective space $\mathbb{R}P^d$ on $Θ((\frac{1+\sqrt{5}}{2})^{d+1})$ vertices for every $d\geq 1$. This improves a construction due to Kühnel on $2^{d+1}-1$ vertices.

In this paper, we study the odd solution of the linearlized Einstein equation on the Schwarzschild background and in the harmonic gauge. With the aid of Regge-Wheeler quantities, we are able to estimate the odd part of Lichnerowicz d'Alembertian equation. In particular, we prove the solution decays at rate $τ^{-1+δ}$ t…

The paper proves properties of Lipschitz spacetimes with bounded Ricci curvature.

We consider discrete nets in Grassmannians $\mathbb{G}^d_r$ which generalize Q-nets (maps $\mathbb{Z}^N\to\mathbb{P}^d$ with planar elementary quadrilaterals) and Darboux nets ($\mathbb{P}^d$-valued maps defined on the edges of $\mathbb{Z}^N$ such that quadruples of points corresponding to elementary squares are all co…

The paper shows how MMD metrizes weak convergence for certain kernels.

Predicts cryptocurrency pump probability using sequence-based neural networks.

Let $S$ be an elliptic surface over a smooth curve $C$ with a section $O$. We denote its generic fiber by $E_S$. For a divisor $D$ on $S$, we canonically associate a $C(C)$-rational point $P_D$. In this note, we give a description of $P_D$ of $E_S$, when the rank of the group of $C(C)$-rational points is one. We apply …

Study cone structures on contact manifolds to understand their geometric properties.

The CAP slope is Bayes' theorem in cumulative coordinates, unlocking the weight of evidence, Somers' D, and Gini coefficient.

This is a short survey of Cheeger and Kleiner's nonembeddability theorem for Heisenberg group into $L^1$.

The moduli space of lattices of $\mathbb{C}$ is a Riemann surface of finite hyperbolic area with the square lattice as an origin. We select a lattice from the induced uniform distribution and calculate the statistics of the Teichmüller distance to the origin. This in turn identifies distribution of the distance in Teic…

Fractionally integrated generalized autoregressive conditional heteroskedasticity (FIGARCH) arises in modeling of financial time series. FIGARCH is essentially governed by a system of nonlinear stochastic difference equations ${u_t}$ = ${z_t}$ $(1-\sum\limits_{j=1}^q β_j L^j)σ_{t}^2 = ω+(1-\sum\limits_{j=1}^q β_j L^j -…

The paper analyzes convergence rates of Gaussian process approximations for scalable regression.

We show that all knots up to $6$ crossings can be represented by polynomial knots of degree at most $7$, among which except for $5_2, 5_2^*, 6_1, 6_1^*, 6_2, 6_2^*$ and $6_3$ all are in their minimal degree representation. We provide concrete polynomial representation of all these knots. Durfee and O'Shea had asked a q…

For a bounded domain $D$ and a real number $p>0$, we denote by $A^p(D)$ the space of $L^p$ integrable holomorphic functions on $D$, equipped with the $L^p$- pseudonorm. We prove that two bounded hyperconvex domains $D_1\subset \mc^n$ and $D_2\subset \mc^m$ are biholomorphic (in particular $n=m$) if there is a linear is…

We give different proofs and prove new results on the non complete solvability of some systems of complex first order p.d.e.'s, especially related to the analysis on CR manifolds.

In this paper we investigate the computational complexity of learning the graph structure underlying a discrete undirected graphical model from i.i.d. samples. We first observe that the notoriously difficult problem of learning parities with noise can be captured as a special case of learning graphical models. This lea…

In this paper we study the topology of three different kinds of spaces associated to polynomial knots of degree at most $d$, for $d\geq2$. We denote these spaces by $\mathcal{O}_d$, $\mathcal{P}_d$ and $\mathcal{Q}_d$. For $d\geq3$, we show that the spaces $\mathcal{O}_d$ and $\mathcal{P}_d$ are path connected and the …

We study the least squares regression problem \begin{align*} \min_{Θ\in \mathcal{S}_{\odot D,R}} \|AΘ-b\|_2, \end{align*} where $\mathcal{S}_{\odot D,R}$ is the set of $Θ$ for which $Θ= \sum_{r=1}^{R} θ_1^{(r)} \circ \cdots \circ θ_D^{(r)}$ for vectors $θ_d^{(r)} \in \mathbb{R}^{p_d}$ for all $r \in [R]$ and $d \in [D]…

We define local Hardy spaces of differential forms $h^p_{\mathcal D}(\wedge T^*M)$ for all $p\in[1,\infty]$ that are adapted to a class of first order differential operators $\mathcal D$ on a complete Riemannian manifold $M$ with at most exponential volume growth. In particular, if $D$ is the Hodge--Dirac operator on $…

The study examines growth of quadratic forms under Anosov subgroups.

Long and short memory in economic processes is usually described by the so-called discrete fractional differencing and fractional integration. We prove that the discrete fractional differencing and integration are the Grunwald-Letnikov fractional differences of non-integer order d. Equations of ARIMA(p,d,q) and ARFIMA(…

Study improves detection of cryptocurrency pump-and-dump schemes.

New construction shows VMRTs of unbendable curves can be Legendrian.

This econophysics work studies the long-range Ising model of a finite system with $N$ spins and the exchange interaction $\frac{J}{N}$ and the external field $H$ as a modely for homogeneous credit portfolio of assets with default probability $P_{d}$ and default correlation $ρ_{d}$. Based on the discussion on the $(J,H)…

Study of embeddings avoiding certain tangent patterns using polynomial spaces.

Let $X$ be a compact manifold, $D$ a real elliptic operator on $X$, $G$ a Lie group, $P\to X$ a principal $G$-bundle, and ${\mathcal B}_P$ the infinite-dimensional moduli space of all connections $\nabla_P$ on $P$ modulo gauge, as a topological stack. For each $[\nabla_P]\in{\mathcal B}_P$, we can consider the twisted …

Optimistic covariance-adaptive algorithms improve combinatorial semi-bandits regret.

In this note, we consider the Dirac operator $D$ on a Riemannian symmetric space $M$ of noncompact type. Using representation theory we show that $D$ has point spectrum iff the $\hat A$-genus of its compact dual does not vanish. In this case, if $M$ is irreducible then $M = U(p,q)/U(p) \times U(q)$ with $p+q$ odd, and …

We extend the model-free formula of [Fukasawa 2012] for $\mathbb E[Ψ(X_T)]$, where $X_T=\log S_T/F$ is the log-price of an asset, to functions $Ψ$ of exponential growth. The resulting integral representation is written in terms of normalized implied volatilities. Just as Fukasawa's work provides rigourous ground for Ch…

In this paper we give a new proof of a theorem by Alexandrov on the Gauss curvature prescription of Euclidean convex sets. This proof is based on the duality theory of convex sets and on optimal mass transport. A noteworthy property of this proof is that it does not rely neither on the theory of convex polyhedra nor on…

Set-valued risk measures on $L^p_d$ with $0 \leq p \leq \infty$ for conical market models are defined, primal and dual representation results are given. The collection of initial endowments which allow to super-hedge a multivariate claim are shown to form the values of a set-valued sublinear (coherent) risk measure. Sc…

The paper analyzes online learning of smooth functions in both single and multi-variable settings.

The paper proves a Calabi-Yau structure on complexifications of rank two symmetric spaces.

We study the interplay between the minimal representations of the orthogonal Lie algebra $\mathfrak{g}=\mathfrak{so}(n+2,\mathbb{C})$ and the \emph{algebra of symmetries} $\mathscr{S}(\Box^r)$ of powers of the Laplacian $\Box$ on $\mathbb{C}^{n}$. The connection is made through the construction of highest weight repres…

This paper is the third in a series that researches the Morse Theory, gradient flows, concavity and complexity on smooth compact manifolds with boundary. Employing the local analytic models from \cite{K2}, for \emph{traversally generic flows} on $(n+1)$-manifolds $X$, we embark on a detailed and somewhat tedious study …

We consider the Fractionally Integrated Exponential Generalized Autoregressive Conditional Heteroskedasticity process, denoted by FIEGARCH(p,d,q), introduced by Bollerslev and Mikkelsen (1996). We present a simulated study regarding the estimation of the risk measure $VaR_p$ on FIEGARCH processes. We consider the distr…