We study the problem of testing a simple null hypothesis on multivariate normal mean vector against smooth or piecewise smooth cone alternatives. We show that the mixture weights of the 2 distribution of the likelihood ratio test can be characterized as mixed volumes of the cone and its dual. The weights can be calculated by integration involving the second fundamental form on the boundary of t...

This paper investigates the existence and uniqueness of smooth positive solutions to a class of singular m-point boundary value problems of second-order ordinary differential equations. A necessary and sufficient condition for the existence and uniqueness of smooth positive solutions is given by constructing lower and upper solutions and with the maximal theorem. Our nonlinearity f t, u, v may ...

This article is the second part in the series of articles where we are developing theory of valuations on manifolds. Roughly speaking valuations could be thought as finitely additive measures on a class of nice subsets of a manifold which satisfy some additional assumptions. The goal of this article is to introduce a notion of a smooth valuation on an arbitrary smooth manifold and establish som...

In this paper, we present two results concerning the fuzzy modeling and control of nonlinear systems. The first result is on the approximation of smooth nonlinear dynamical systems using linear Takagi-Sugeno fuzzy models. The second result is on the approximation of smooth nonlinear state-feedback controllers using the the so-called parallel distributed compensation (PDC) controller. Both resul...

This paper proves two main results. First, it is shown that if Σ is a smooth manifold homeomorphic to the standard n-torus Tn = Rn/Zn and H is a real-analytically completely integrable convex hamiltonian on T Σ, then Σ is diffeomorphic to Tn. Second, it is proven that for some topological 7-manifolds, the cotangent bundle of each smooth structure admits a realanalytically completely integrable ...

We provide estimates for $s$th moments of biquadratic smooth Weyl sums, when $10\le s\le 12$, by enhancing the second author’s iterative method that delivers beyond classical convexity barrier. As a consequence, all sufficiently large intege

The max-k-sum of a set of real scalars is the maximum sum of a subset of size k, or alternatively the sum of the k largest elements. We study two extensions: First, we show how to obtain smooth approximations to functions that are pointwise max-k-sums of smooth functions. Second, we discuss how the max-k-sum can be defined on vectors in a finite-dimensional real vector space ordered by a closed...

This article is the second part in the series of articles where we are developing theory of valuations on manifolds. Roughly speaking valuations could be thought as finitely additive measures on a class of nice subsets of a manifold which satisfy some additional assumptions. The goal of this article is to introduce a notion of a smooth valuation on an arbitrary smooth manifold and establish som...

We compare a general controlled diffusion process with a deterministic system where a second controller drives the disturbance against the first controller. We show that the two models are equivalent with respect to two properties: the viability (or controlled invariance, or weak invariance) of closed smooth sets, and the existence of a smooth control Lyapunov function ensuring the stabilizabil...

We study the problem of testing a simple null hypothesis on multivariate normal mean vector against smooth or piecewise smooth cone alternatives. We show that the mixture weights of the  distribution of the likelihood ratio test can be characterized as mixed volumes of the cone and its dual. The weights can be calculated by integration involving the second fundamental form on the boundary of t...