Related papers: Variance Calculations and the Bessel Kernel

We consider n-by-n matrices whose (i, j)-th entry is f(X_i^T X_j), where X_1, ...,X_n are i.i.d. standard Gaussian random vectors in R^p, and f is a real-valued function. The eigenvalue distribution of these random kernel matrices is…

Classification of curves up to affine transformation in a finite dimensional space was studied by some different methods. In this paper, we achieve the exact formulas of affine invariants via the equivalence problem and in the view of…

Let $F \in \mathbf{Z}[\boldsymbol{x}]$ be a diagonal, non-singular quadratic form in $4$ variables. Let $\lambda(n)$ be the normalised Fourier coefficients of a holomorphic Hecke form of full level. We give an upper bound for the problem of…

We study integrals of the form $\int_{\Omega}f\left( d\omega\right)$, where $1\leq k\leq n$, $f:\Lambda^{k}\rightarrow\mathbb{R}$ is continuous and $\omega$ is a $\left(k-1\right)$-form. We introduce the appropriate notions of convexity,…

In a variety of applications it is important to extract information from a probability measure $\mu$ on an infinite dimensional space. Examples include the Bayesian approach to inverse problems and possibly conditioned) continuous time…

Many information sources are not just sequences of distinguishable symbols but rather have invariances governed by alternative counting paradigms such as permutations, combinations, and partitions. We consider an entire classification of…

Assumptions about invariances or symmetries in data can significantly increase the predictive power of statistical models. Many commonly used models in machine learning are constraint to respect certain symmetries in the data, such as…

We give a new probabilistic construction of solutions to real Monge-Amp\`ere equations in R^n satisfying the second boundary value problem with respect to a given target convex body P) which fits naturally into the theory of optimal…

This article deals with the adaptive and approximative computation of the Lam\'e equations. The equations of linear elasticity are considered as boundary integral equations and solved in the setting of the boundary element method (BEM).…

In this paper we obtain a description of the Hermitian operators acting on the Hilbert space $\C^n$, description which gives a complete solution to the over parameterization problem. More precisely we provide an explicit parameterization of…

We associate to each Boolean function a polynomial whose evaluations represents the distances from all possible Boolean affine functions. Both determining the coefficients of this polynomial from the truth table of the Boolean function and…

The minimum achievable statistical uncertainty in the estimation of physical parameters is determined by the quantum Fisher information. Its computation for noisy systems is still a challenging problem. Using a variational approach, we…

Learning physical properties of a quantum system is essential for the developments of quantum technologies. However, Heisenberg's uncertainty principle constrains the potential knowledge one can simultaneously have about a system in quantum…

We fix a maximal order $\mathcal O$ in $\F=\R,\C$ or $\mathbb{H}$, and an $\F$-hermitian form $Q$ of signature $(n,1)$ with coefficients in $\mathcal O$. Let $k\in\N$. By applying a lattice point theorem on the $\F$-hyperbolic space, we…

We minimize elastic energies on framed curves which penalize both curvature and torsion. We also discuss critical points using the infinite dimensional version of the Lagrange multipliers' method. Finally, some examples arising from the…

We compute the limit distribution of partial transposes (when both the number and the size of blocks tends to infinity) for a large class of ensembles of unitarily invariant random matrices. Furthermore, it is shown the asymptotic freeness…

For a square matrix, the range of its Rayleigh quotients is known as the numerical range, which is a compact and convex set by the Toeplitz-Hausdorff theorem. The largest value and the smallest boundary value (in magnitude) of this convex…

For any arbitrary algebraic curve, we define an infinite sequence of invariants. We study their properties, in particular their variation under a variation of the curve, and their modular properties. We also study their limits when the…

We answer the following question posed by Lechuga: Given a simply-connected space $X$ with both $H_*(X,\qq)$ and $\pi_*(X)\otimes \qq$ being finite-dimensional, what is the computational complexity of an algorithm computing the cup-length…

One of the great miracles of random matrix theory is that, in the $N \to \infty$ limit, many otherwise intractable matrix problems with horrendously complicated finite-$N$ expressions admit remarkably simple and elegant asymptotic…