Computing the Karcher mean of symmetric positive definite matrices

We present and analyze an iterative method for approximating the Karcher mean of a set of $n\times n$ positive definite matrices $A_i$, $i=1\text{,}\dots \text{,}k$, defined as the unique positive definite solution of the matrix equation ${\sum }_{i=1}^k\log (A_i^{-1}X)=0$.     

The D-optimal saturated designs of order 22

This paper attempts to prove the D-optimality of the saturated designs X${}^1$ and X${}^{11}$ of order 22, already existing in the current literature. The corresponding non-equivalent information matrices M${}^1$=(X${}^1$)${}^T$X${}^1$ and M${}^{11}$=(X${}^{11}$)${}^T$X${}^{11}$ have the maximum determinant. Within the application of a specific procedure, all symmetric and positive definite matrices M of order 22 with determinant the square of an integer and $\ge$det(M${}^1$) are constructed. This procedure has indicated that there are 26 such non-equivalent matrices M, for 24 of which the non-existence of designs X such that X${}^T$X =M is proved. The remaining two matrices M are the information matrices M${}^1$ and M${}^{11}$.     

Graphical characterization of positive definite non symmetric quasi-Cartan matrices

It is known that each positive definite quasi-Cartan matrix $A$ is $\mathbb{Z}$-equivalent to a Cartan matrix $A_{\Delta }$ called Dynkin type of $A$, the matrix $A_{\Delta }$ is uniquely determined up to conjugation by permutation matrices. However, in most of the cases, it is not possible to determine the Dynkin type of a given connected quasi-Cartan matrix by simple inspection. In this paper, we give a graph theoretical characterization of non-symmetric connected quasi-Cartan matrices. For this purpose, a special assemblage of blocks is introduced. This result complements the approach proposed by Barot (1999, 2001), for ${\mathbb{A}}_n$, ${\mathbb{D}}_n$ and ${\mathbb{E}}_m$ with $m=6,7,8$.     

Stochastic comparison of residual lifetime mixture models

In this paper, we consider stochastic comparisons of mixtures ($F^1$ and $G^1$) of residual life distributions ($F_{\theta }$ and $G_{\theta }$, $\theta >0$) arising out of different baseline distributions ($F$ and $G$) and different mixing distributions ($H_1$ and $H_2$). Under certain conditions on $F$, $G$, $H_1$ and $H_2$, we establish that the distributions $F^1$ and $G^1$ are stochastically ordered. Further, we make stochastic comparison of mixture distribution $F^1$ with the distribution of residual lifetime at random time ${\Theta }_1$.     

Thin Lehman matrices arising from finite groups

Two $n\times n$ $(0\text{,}1)$ matrices $X$, $Y$ are called thin Lehman matrices if they are solutions of the matrix equation ${\mathit{XY}}^T=J_n+I_n$, where $J_n$ is the $n\times n$ matrix of all $1$s and $I_n$ is the identity matrix. These matrices are important in the set covering problem, but few examples are known. In this paper, we will introduce the notion of $1$-overlapped factorizations of finite groups which constructs a new class of thin Lehman matrices. Moreover, we will study some structural properties of $1$-overlapped factorizations.     

Universality in the bulk holds close to given points

Let $\mu$ be a measure with compact support. Assume that $\xi$ is a Lebesgue point of $\mu$ and that ${\mu }^{\prime }$ is positive and continuous at $\xi$. Let $\left\{A_n\right\}$ be a sequence of positive numbers with limit $\infty$. We show that one can choose ${\xi }_n\in [\xi ?\frac{A_n}{n},\xi +\frac{A_n}{n}]$ such that $\underset{n\to \infty }{lim}\frac{K_n({\xi }_n,{\xi }_n+\frac{a}{{\stackrel{?}{K}}_n({\xi }_n,{\xi }_n)})}{K_n({\xi }_n,{\xi }_n)}=\frac{sin\pi a}{\pi a}\text{,}$ uniformly for $a$ in compact subsets of the plane. Here $K_n$ is the $n$th reproducing kernel for $\mu$, and ${\stackrel{?}{K}}_n$ is its normalized cousin. Thus universality in the bulk holds on a sequence close to $\xi$, without having to assume that $\mu$ is a regular measure. Similar results are established for sequences of measures.     

Fixed-parameter algorithms for vertex cover P3

Let $P_{\ell }$ denote a path in a graph $G=(V,E)$ with $\ell$ vertices. A vertex cover $P_{\ell }$ set $C$ in $G$ is a vertex subset such that every $P_{\ell }$ in $G$ has at least a vertex in $C$. The Vertex Cover$P_{\ell }$ problem is to find a vertex cover $P_{\ell }$ set of minimum cardinality in a given graph. This problem is NP-hard for any integer $\ell ⩾2$. The parameterized version of Vertex Cover$P_{\ell }$ problem called $k$-Vertex Cover$P_{\ell }$ asks whether there exists a vertex cover $P_{\ell }$ set of size at most $k$ in the input graph. In this paper, we give two fixed parameter algorithms to solve the $k$-Vertex Cover$P_3$ problem. The first algorithm runs in time $O^1\left(1.7964^k\right)$ in polynomial space and the second algorithm runs in time $O^1\left(1.7485^k\right)$ in exponential space. Both algorithms are faster than previous known fixed-parameter algorithms.     

Multivariate polynomial interpolation and sampling in Paley–Wiener spaces

In this paper, an equivalence between existence of particular exponential Riesz bases for spaces of multivariate bandlimited functions and existence of certain polynomial interpolants for functions in these spaces is given. Namely, polynomials are constructed which, in the limiting case, interpolate ${\left\{({\tau }_n,f\left({\tau }_n\right))\right\}}_n$ for certain classes of unequally spaced data nodes ${\left\{{\tau }_n\right\}}_n$ and corresponding ${?}_2$ sampled data ${\left\{f\left({\tau }_n\right)\right\}}_n$. Existence of these polynomials allows one to construct a simple sequence of approximants for an arbitrary multivariate bandlimited function $f$ which demonstrates $L_2$ and uniform convergence on ${\mathbb{R}}^d$ to $f$. A simpler computational version of this recovery formula is also given at the cost of replacing $L_2$ and uniform convergence on ${\mathbb{R}}^d$ with $L_2$ and uniform convergence on increasingly large subsets of ${\mathbb{R}}^d$. As a special case, the polynomial interpolants of given ${?}_2$ data converge in the same fashion to the multivariate bandlimited interpolant of that same data. Concrete examples of pertinent Riesz bases and unequally spaced data nodes are also given.