Related papers: Sparse Block-Jacobi Matrices with Exact Hausdorff …

We consider Jacobi matrices with zero diagonal and off-diagonals given by elements of the hull of the Fibonacci sequence and show that the spectrum has zero Lebesgue measure and all spectral measures are purely singular continuous. In…

We consider a class of Jacobi matrices with periodically modulated diagonal in a critical hyperbolic ("double root") situation. For the model with "non-smooth" matrix entries we obtain the asymptotics of generalized eigenvectors and analyze…

An identity is proven that evaluates the determinant of a block tridiagonal matrix with (or without) corners as the determinant of the associated transfer matrix (or a submatrix of it).

In this paper we show weak convergence of the empirical eigenvalue distribution and of the weighted spectral measure of the Jacobi ensemble, when one or both parameters grow faster than the dimension $n$. In these cases the limit measure is…

Let $K$ be the attractor of a linear iterated function system (IFS) $S_j(x)=\rho_jx+b_j,$ $j=1,\cdots,m$, on the real line satisfying the generalized finite type condition (whose invariant open set $\mathcal{O}$ is an interval) with an…

First, we consider a nonnegative homogeneous block tri-diagonal matrix and obtain its convergence parameter, where some results in the field of matrix analytic method are extended to the case where block matrices have countably infinite…

We study fine properties of the so-called stable trees, which are the scaling limits of critical Galton-Watson trees conditioned to be large. In particular we derive the exact Hausdorff measure function for Aldous' continuum random tree and…

Follow-up comment by the author: Theorem 2.2 in this paper is a special case of Theorems 1.1 and 4.1 in the article "Weighted thermodynamic formalism on subshifts and applications", Asian J. Math. 16 (2012), by J. Barral and D. J. Feng. In…

It is well known due to Jarnik that the set Bad of badly approximable numbers is of Hausdorff-dimension one. If Bad(c) denotes the subset of x in Bad for which the approximation constant c > c(x), then Jarnik was in fact more precise and…

It is well known that a set of non-defect matrices can be simultaneously diagonalized if and only if the matrices commute. In the case of non-commuting matrices, the best that can be achieved is simultaneous block diagonalization. Here we…

We consider high-dimensional quadratic classifiers in non-sparse settings. The target of classification rules is not Bayes error rates in the context. The classifier based on the Mahalanobis distance does not always give a preferable…

In this short note, we show that, in any given metric space, every Lipschitz open-map image of every subset of a given metric space whose boundary is Hausdorff-null is Hausdorff-measurable with respect to the same dimension. The main…

The dimensions of sets of matrices of various types, with specified eigenvalue multiplicities, are determined. The dimensions of the sets of matrices with given Jordan form and with given singular value multiplicities are also found. Each…

We propose methodology for estimation of sparse precision matrices and statistical inference for their low-dimensional parameters in a high-dimensional setting where the number of parameters $p$ can be much larger than the sample size. We…

In this paper we show that, given a planar Reifenberg flat domain with small constant and a divergence form operator associated to a real (not necessarily symmetric) uniformly elliptic matrix with Lipschitz coefficients, the Hausdorff…

We establish variational principles for the Hausdorff and packing dimensions of a class of statistically self-affine sponges, including in particular fractal percolation sets obtained from Bara\'nski and Gatzouras-Lalley carpets and…

The paper provides an elementary proof establishing a sharp universal bound on the $(d-1)$-Hausdorff measure of the zeros of any nontrivial multivariable polynomial $p:\mathbb{R}^d\to\mathbb{R}$ within a $d$-dimensional cube of size $r$.…

We show that set of points nondense under the $\times n$-map on the circle and dense for the geodesic flow under the induced map on the circle corresponding to the expanding horospherical subgroup has full Haudorff dimension. We also show…

Let $X= \{X(t), t \in \R^N\}$ be a Gaussian random field with values in $\R^d$ defined by \[ X(t) = \big(X_1(t),..., X_d(t)\big),\qquad t \in \R^N, \] where $X_1, ..., X_d$ are independent copies of a real-valued, centered, anisotropic…

We show that there exists a dense set of frequencies with positive Hausdorff dimension for which the Hausdorff dimension of the spectrum of the critical almost Mathieu operator is positive.