Clustering and classification via cluster-weighted factor analyzers

In model-based clustering and classification, the cluster-weighted model is a convenient approach when the random vector of interest is constituted by a response variable $Y$ and by a vector ${\varvec{X}}$ of $p$ covariates. However, its applicability may be limited when $p$ is high. To overcome this problem, this paper assumes a latent factor structure for ${\varvec{X}}$ in each mixture component, under Gaussian assumptions. This leads to the cluster-weighted factor analyzers (CWFA) model. By imposing constraints on the variance of $Y$ and the covariance matrix of ${\varvec{X}}$ , a novel family of sixteen CWFA models is introduced for model-based clustering and classification. The alternating expectation-conditional maximization algorithm, for maximum likelihood estimation of the parameters of all models in the family, is described; to initialize the algorithm, a 5-step hierarchical procedure is proposed, which uses the nested structures of the models within the family and thus guarantees the natural ranking among the sixteen likelihoods. Artificial and real data show that these models have very good clustering and classification performance and that the algorithm is able to recover the parameters very well.     

Comparing exponential location parameters with several controls under heteroscedasticity

Suppose that random samples are taken from \(k\) treatment groups and \(l\) control groups, where the observations in each group have a two-parameter exponential distribution. We consider the problem of constructing simultaneous confidence intervals for the differences between location parameters of the treatment groups and the control groups when the scale parameters may be unequal. Using the parametric bootstrap approach, we develop a new multiple comparisons procedure when the scale parameters and sample sizes are possibly unequal. We then present a simulation study in which we compare the performance of our proposed procedure with two other procedures. The results of our simulations indicate that our proposed procedure performs better than other procedures. The usefulness of our proposed procedure is illustrated with an example.     

Subsets of full measure in a generic submanifold in \mathbb C ^n are non-plurithin

In this paper we prove that if $I\subset M $ is a subset of measure $0$ in a $C^2$ -smooth generic submanifold $M \subset \mathbb C ^n$ , then $M \setminus I$ is non-plurithin at each point of $M$ in $\mathbb C ^n$ . This result improves a previous result of A. Edigarian and J. Wiegerinck who considered the case where $I$ is pluripolar set contained in a $C^1$ -smooth generic submanifold $M \subset \mathbb C ^n$ (Edigarian and Wiegernick in Math. Z. 266(2):393–398, 2010). The proof of our result is essentially different.     

The large rank of a finite semigroup using prime subsets

The large rank of a finite semigroup \(\Gamma \) , denoted by \(r_5(\Gamma )\) , is the least number \(n\) such that every subset of \(\Gamma \) with \(n\) elements generates \(\Gamma \) . Howie and Ribeiro showed that \(r_5(\Gamma ) = |V| + 1\) , where \(V\) is a largest proper subsemigroup of \(\Gamma \) . This work considers the complementary concept of subsemigroups, called prime subsets, and gives an alternative approach to find the large rank of a finite semigroup. In this connection, the paper provides a shorter proof of Howie and Ribeiro’s result about the large rank of Brandt semigroups. Further, this work obtains the large rank of the semigroup of order-preserving singular selfmaps.     

Isomorphic Steiner symmetrization of p-convex sets

In this paper we show that given a \(p\) -convex set \(K \subset \mathbb{R }^n\) , there exist \(5n\) Steiner symmetrizations that transform it into an isomorphic Euclidean ball. That is, if \(|K| = |D_n| = \kappa _n\) , we may symmetrize it, using \(5n\) Steiner symmetrizations, into a set \(K'\) such that \(c_p D_n \subset K' \subset C_p D_n\) , where \(c_p\) and \(C_p\) are constants dependent on \(p\) only.     

On the compactness of the set of invariant Einstein metrics

Let $M = G/H$ be a connected simply connected homogeneous manifold of a compact, not necessarily connected Lie group $G$ . We will assume that the isotropy $H$ -module $\mathfrak{g/h }$ has a simple spectrum, i.e. irreducible submodules are mutually non-equivalent. There exists a convex Newton polytope $N=N(G,H)$ , which was used for the estimation of the number of isolated complex solutions of the algebraic Einstein equation for invariant metrics on $G/H$ (up to scaling). Using the moment map, we identify the space $\mathcal{M }_1$ of invariant Riemannian metrics of volume 1 on $G/H$ with the interior of this polytope $N$ . We associate with a point ${x \in \partial N}$ of the boundary a homogeneous Riemannian space (in general, only local) and we extend the Einstein equation to $\partial N$ . As an application of the Alekseevsksky–Kimel’fel’d theorem, we prove that all solutions of the Einstein equation associated with points of the boundary are locally Euclidean. We describe explicitly the set $T\subset \partial N$ of solutions at the boundary together with its natural triangulation. Investigating the compactification ${\overline{\mathcal{M }}}_{1} = N$ of $\mathcal{M }_1$ , we get an algebraic proof of the deep result by Böhm, Wang and Ziller about the compactness of the set $\mathcal{E }_1 \subset \mathcal{M }_1$ of Einstein metrics. The original proof by Böhm, Wang and Ziller was based on a different approach and did not use the simplicity of the spectrum. In Appendix, we consider the non-symmetric flag manifolds $G/H$ with the second Betti number $b_2=1$ . We calculate the normalized volumes $2,6,20,82,344$ of the corresponding Newton polytopes and discuss the number of complex solutions of the algebraic Einstein equation and the finiteness problem.     

Finite groups with \mathbb{P }-subnormal subgroups

A subgroup $H$ of a group $G$ is called $\mathbb{P }$ -subnormal in $G$ whenever either $H=G$ or there is a chain of subgroups $H=H_0\subset H_1\subset \cdots \subset H_n=G$ such that $|H_i:H_{i-1}|$  is a prime for all $i$ . In this paper we study groups with $\mathbb{P }$ -subnormal 2-maximal subgroups, and groups with $\mathbb{P }$ -subnormal primary cyclic subgroups.     

On Wilking’s criterion for the Ricci flow

Wilking has recently shown that one can associate a Ricci flow invariant cone of curvature operators $C(S)$ , which are nonnegative in a suitable sense, to every $Ad_{SO(n,\mathbb{C })}$ invariant subset $S \subset \mathbf{so}(n,\mathbb{C })$ . In this article we show that if $S$ is an $Ad_{SO(n,\mathbb{C })}$ invariant subset of $\mathbf{so}(n,\mathbb{C })$ such that $S\cup \{0\}$ is closed and $C_+(S)\subset C(S)$ denotes the cone of curvature operators which are positive in the appropriate sense then one of the two possibilities holds: (a) The connected sum of any two Riemannian manifolds with curvature operators in $C_+(S)$ also admits a metric with curvature operator in $C_+(S)$ (b) The normalized Ricci flow on any compact Riemannian manifold $M$ with curvature operator in $C_+(S)$ converges to a metric of constant positive sectional curvature. We also point out that if $S$ is an arbitrary $Ad_{SO(n,\mathbb{C })}$ subset, then $C(S)$ is contained in the cone of curvature operators with nonnegative isotropic curvature.     

Inductive Moving Frames

An inductive implementation of the equivariant moving frame method is introduced for both finite-dimensional Lie group actions and infinite-dimensional Lie pseudo-groups. Given two Lie (pseudo-)groups ${\mathcal{G}}$ and ${\mathcal{H}}$ with ${\mathcal{G} \subset \mathcal{H}}$ , the inductive method streamlines the construction of a moving frame for ${\mathcal{H}}$ using the already constructed moving frame for ${\mathcal{G}}$ . As a by-product, a systematic procedure for expressing ${\mathcal{H}}$ -invariant quantities in terms of their ${\mathcal{G}}$ -invariant counterparts is obtained.     

Reproducing Subgroups of Sp(2,\mathbb{R}). Part I: Algebraic Classification

We classify the connected Lie subgroups of the symplectic group $Sp(2,{\mathbb{R}})$ whose elements are matrices in block lower triangular form. The classification is up to conjugation within $Sp(2,{\mathbb{R}})$ . Their study is motivated by the need of a unified approach to continuous 2D signal analyses, as those provided by wavelets and shearlets.     

Varieties connected by chains of lines

In this paper we give for any integer $l\geqslant 2$ a numerical criterion ensuring the existence of a chain of length $l$ of lines through two general points of an irreducible variety $X \subset \mathbb P ^{N}$ , involving the degrees and the number of homogeneous polynomials defining $X$ . We show that our criterion is sharp.     

Variance asymptotics for random polytopes in smooth convex bodies

Let $K \subset \mathbb R ^d$ be a smooth convex set and let $\mathcal{P }_{\lambda }$ be a Poisson point process on $\mathbb R ^d$ of intensity ${\lambda }$ . The convex hull of $\mathcal{P }_{\lambda }\cap K$ is a random convex polytope $K_{\lambda }$ . As ${\lambda }\rightarrow \infty $ , we show that the variance of the number of $k$ -dimensional faces of $K_{\lambda }$ , when properly scaled, converges to a scalar multiple of the affine surface area of $K$ . Similar asymptotics hold for the variance of the number of $k$ -dimensional faces for the convex hull of a binomial process in $K$ .     

A Probabilistic Symbolic Algorithm to Find the Minimum of a Polynomial Function on a Basic Closed Semialgebraic Set

We consider the problem of computing the minimum of a polynomial function \(g\) on a basic closed semialgebraic set \(E\subset \mathbb {R}^n\) . We present a probabilistic symbolic algorithm to find a finite set of sample points of the subset \(E^{\min }\) of \(E\) where the minimum of \(g\) is attained, provided that \(E^{\min }\) is non-empty and has at least one compact connected component.     

On a Property of Functions on the Sphere

According to the Knaster conjecture, for any continuous function $f:S^{n - 1} \to \mathbb{R}$ and any $n$ -point subset of the sphere $S^{n - 1}$ , there exists a rotation mapping all the points of this subset to a level surface of the function $f$ . In the present paper, this conjecture is proved for the case in which ${n = p^1 }$ for an odd prime $p$ and the points lie on a circle and divide it into equal parts.     

Ricci flow of homogeneous manifolds

We present in this paper a general approach to study the Ricci flow on homogeneous manifolds. Our main tool is a dynamical system defined on a subset $\mathcal H _{q,n}$ of the variety of $(q+n)$ -dimensional Lie algebras, parameterizing the space of all simply connected homogeneous spaces of dimension $n$ with a $q$ -dimensional isotropy, which is proved to be equivalent in a precise sense to the Ricci flow. The approach is useful to better visualize the possible (nonflat) pointed limits of Ricci flow solutions, under diverse rescalings, as well as to determine the type of the possible singularities. Ancient solutions arise naturally from the qualitative analysis of the evolution equation. We develop two examples in detail: a $2$ -parameter subspace of $\mathcal H _{1,3}$ reaching most of $3$ -dimensional geometries, and a $2$ -parameter family in $\mathcal H _{0,n}$ of left-invariant metrics on $n$ -dimensional compact and non-compact semisimple Lie groups.     

Projective curves with maximal regularity and applications to syzygies and surfaces

We first show that the union of a projective curve with one of its extremal secant lines satisfies the linear general position principle for hyperplane sections. We use this to give an improved approximation of the Betti numbers of curves ${{\mathcal C}\subset \mathbb P^r_K}$ of maximal regularity with ${{\rm deg}\, {\mathcal C}\leq 2r -3}$ . In particular we specify the number and degrees of generators of the vanishing ideal of such curves. We apply these results to study surfaces ${X \subset \mathbb P^r_K}$ whose generic hyperplane section is a curve of maximal regularity. We first give a criterion for ??an early descent of the Hartshorne-Rao function?? of such surfaces. We use this criterion to give a lower bound on the degree for a class of these surfaces. Then, we study surfaces ${X \subset\mathbb P^r_K}$ for which ${h^1(\mathbb P^r_K, {\mathcal I}_X(1))}$ takes a value close to the possible maximum deg X ? r +?1. We give a lower bound on the degree of such surfaces. We illustrate our results by a number of examples, computed by means of Singular, which show a rich variety of occuring phenomena.     

Boundary partial regularity for a class of biharmonic maps

We consider the Dirichlet problem for biharmonic maps u from a bounded, smooth domain ${\Omega\subset\mathbb R^n (n\ge 5)}$ to a compact, smooth Riemannian manifold ${N\subset{\mathbb {R}}^l}$ without boundary. For any smooth boundary data, we show that if u is a stationary biharmonic map that satisfies a certain boundary monotonicity inequality, then there exists a closed subset ${\Sigma\subset\overline{\Omega}}$ , with ${H^{n-4}(\Sigma)=0}$ , such that ${\displaystyle u\in C^\infty(\overline\Omega\setminus\Sigma, N)}$ .     

Generalized Strong Boundary Points and Boundaries of Families of Continuous Functions

If ${\mathcal{A}}$ is a family of continuous functions on a locally compact Hausdorff space X, a boundary for ${\mathcal{A}}$ is a subset ${B \subset X}$ such that every ${f \in \mathcal{A}}$ attains its maximum modulus on B. In this work we generalize the concept of strong boundary points for families of functions and show that the collection of these generalized strong boundary points is always a boundary for ${\mathcal{A}}$ . We give conditions under which all boundaries for ${\mathcal{A}}$ consist of generalized strong boundary points and under which these points coincide with classical strong boundary points. When ${\mathcal{A}}$ has sufficient algebraic structure it is proven that this construction provides a unique boundary for ${\mathcal{A}}$ consisting of boundary points, and we conclude by demonstrating how this approach provides an alternate technique for proving the existence of the Choquet and Shilov boundaries in certain function algebras.