Estimating the turning point location in shifted exponential model of time series

We consider the distribution of the turning point location of time series modeled as the sum of deterministic trend plus random noise. If the variables are modeled by shifted exponentials, whose location parameters define the trend, we provide a formula for computing the distribution of the turning point location and consequently to estimate a confidence interval for the location. We test this formula in simulated data series having a trend with asymmetric minimum, investigating the coverage rate as a function of a bandwidth parameter. The method is applied to estimate the confidence interval of the minimum location of the time series of RT intervals extracted from the electrocardiogram recorded during the exercise test. We discuss the connection with stochastic ordering

Locally convex quasi C*-algebras and noncommutative integration

In this paper we continue the analysis undertaken in a series of previous papers on structures arising as completions of C*-algebras under topologies coarser that their norm and we focus our attention on the so-called {\em locally convex quasi C*-algebras}. We show, in particular, that any strongly *-semisimple locally convex quasi C*-algebra (\X,\Ao), can be represented in a class of noncommutative local $L^2$-spaces.Comment: 12 page

Riesz-like bases in rigged Hilbert spaces

The notions of Bessel sequence, Riesz-Fischer sequence and Riesz basis are generalized to a rigged Hilbert space \D[t] \subset \H \subset \D^\times[t^\times]. A Riesz-like basis, in particular, is obtained by considering a sequence \{\xi_n\}\subset \D which is mapped by a one-to-one continuous operator T:\D[t]\to\H[\|\cdot\|] into an orthonormal basis of the central Hilbert space \H of the triplet. The operator $T$ is, in general, an unbounded operator in \H. If $T$ has a bounded inverse then the rigged Hilbert space is shown to be equivalent to a triplet of Hilbert spaces

Trend extraction in functional data of R and T waves amplitudes of exercise electrocardiogram

The R and T waves amplitudes of the electrocardiogram recorded during the exercise test undergo strong modifications in response to stress. We analyze the time series of these amplitudes in a group of normal subjects in the framework of functional data, performing reduction of dimensionality, smoothing and principal component analysis. These methods show that the R and T amplitudes have opposite responses to stress, consisting respectively in a bump and a dip at the early recovery stage. We test these features computing a confidence band for the trend of the population mean and analyzing the zero crossing of its derivative. Our findings support the existence of a relationship between R and T wave amplitudes and respectively diastolic and systolic ventricular volumes

Fractional Sobolev Regularity for the Brouwer Degree

We prove that if $\Omega\subset \mathbb R^n$ is a bounded open set and $n\alpha> {\rm dim}_b (\partial \Omega) = d$, then the Brouwer degree deg$(v,\Omega,\cdot)$ of any H\"older function $v\in C^{0,\alpha}\left (\Omega, \mathbb R^{n}\right)$ belongs to the Sobolev space $W^{\beta, p} (\mathbb R^n)$ for every $0\leq \beta < \frac{n}{p} - \frac{d}{\alpha}$. This extends a summability result of Olbermann and in fact we get, as a byproduct, a more elementary proof of it. Moreover we show the optimality of the range of exponents in the following sense: for every $\beta\geq 0$ and $p\geq 1$ with $\beta > \frac{n}{p} - \frac{n-1}{\alpha}$ there is a vector field $v\in C^{0, \alpha} (B_1, \mathbb R^n)$ with \mbox{deg}\, (v, \Omega, \cdot)\notin W^{\beta, p}, where $B_1 \subset \mathbb R^n$ is the unit ball.Comment: 12 pages, 1 figur