Anxiety, emotional processing and depression in people with multiple sclerosis.

BACKGROUND: Despite the high comorbidity of anxiety and depression in people with multiple sclerosis (MS), little is known about their inter-relationships. Both involve emotional perturbations and the way in which emotions are processed is likely central to both. The aim of the current study was to explore relationships between the domains of mood, emotional processing and coping and to analyse how anxiety affects coping, emotional processing, emotional balance and depression in people with MS. METHODS: A cross-sectional questionnaire study involving 189 people with MS with a confirmed diagnosis of MS recruited from three French hospitals. Study participants completed a battery of questionnaires encompassing the following domains: i. anxiety and depression (Hospital Anxiety and Depression Scale (HADS)); ii. emotional processing (Emotional Processing Scale (EPS-25)); iii. positive and negative emotions (Positive and Negative Emotionality Scale (EPN-31)); iv. alexithymia (Bermond-Vorst Alexithymia Questionnaire) and v. coping (Coping with Health Injuries and Problems-Neuro (CHIP-Neuro) questionnaire. Relationships between these domains were explored using path analysis. RESULTS: Anxiety was a strong predictor of depression, in both a direct and indirect way, and our model explained 48% of the variance of depression. Gender and functional status (measured by the Expanded Disability Status Scale) played a modest role. Non-depressed people with MS reported high levels of negative emotions and low levels of positive emotions. Anxiety also had an indirect impact on depression via one of the subscales of the Emotional Processing Scale ("Unregulated Emotion") and via negative emotions (EPN-31). CONCLUSIONS: This research confirms that anxiety is a vulnerability factor for depression via both direct and indirect pathways. Anxiety symptoms should therefore be assessed systematically and treated in order to lessen the likelihood of depression symptoms

The universal Glivenko-Cantelli property

Let F be a separable uniformly bounded family of measurable functions on a standard measurable space, and let N_{[]}(F,\epsilon,\mu) be the smallest number of \epsilon-brackets in L^1(\mu) needed to cover F. The following are equivalent: 1. F is a universal Glivenko-Cantelli class. 2. N_{[]}(F,\epsilon,\mu)0 and every probability measure \mu. 3. F is totally bounded in L^1(\mu) for every probability measure \mu. 4. F does not contain a Boolean \sigma-independent sequence. It follows that universal Glivenko-Cantelli classes are uniformity classes for general sequences of almost surely convergent random measures.Comment: 26 page

The Traveling Salesman Problem: Low-Dimensionality Implies a Polynomial Time Approximation Scheme

The Traveling Salesman Problem (TSP) is among the most famous NP-hard optimization problems. We design for this problem a randomized polynomial-time algorithm that computes a (1+eps)-approximation to the optimal tour, for any fixed eps>0, in TSP instances that form an arbitrary metric space with bounded intrinsic dimension. The celebrated results of Arora (A-98) and Mitchell (M-99) prove that the above result holds in the special case of TSP in a fixed-dimensional Euclidean space. Thus, our algorithm demonstrates that the algorithmic tractability of metric TSP depends on the dimensionality of the space and not on its specific geometry. This result resolves a problem that has been open since the quasi-polynomial time algorithm of Talwar (T-04)

Efficient Density Estimation via Piecewise Polynomial Approximation

We give a highly efficient "semi-agnostic" algorithm for learning univariate probability distributions that are well approximated by piecewise polynomial density functions. Let $p$ be an arbitrary distribution over an interval $I$ which is $\tau$-close (in total variation distance) to an unknown probability distribution $q$ that is defined by an unknown partition of $I$ into $t$ intervals and $t$ unknown degree-$d$ polynomials specifying $q$ over each of the intervals. We give an algorithm that draws \tilde{O}(t\new{(d+1)}/\eps^2) samples from $p$, runs in time \poly(t,d,1/\eps), and with high probability outputs a piecewise polynomial hypothesis distribution $h$ that is (O(\tau)+\eps)-close (in total variation distance) to $p$. This sample complexity is essentially optimal; we show that even for $\tau=0$, any algorithm that learns an unknown $t$-piecewise degree-$d$ probability distribution over $I$ to accuracy \eps must use \Omega({\frac {t(d+1)} {\poly(1 + \log(d+1))}} \cdot {\frac 1 {\eps^2}}) samples from the distribution, regardless of its running time. Our algorithm combines tools from approximation theory, uniform convergence, linear programming, and dynamic programming. We apply this general algorithm to obtain a wide range of results for many natural problems in density estimation over both continuous and discrete domains. These include state-of-the-art results for learning mixtures of log-concave distributions; mixtures of $t$-modal distributions; mixtures of Monotone Hazard Rate distributions; mixtures of Poisson Binomial Distributions; mixtures of Gaussians; and mixtures of $k$-monotone densities. Our general technique yields computationally efficient algorithms for all these problems, in many cases with provably optimal sample complexities (up to logarithmic factors) in all parameters

Keck Spectra of Brown Dwarf Candidates and a Precise Determination of the Lithium Depletion Boundary in the Alpha Persei Open Cluster

We have identified twenty-seven candidate very low mass members of the relatively young Alpha Persei open cluster from a six square degree CCD imaging survey. Based on their I magnitudes and the nominal age and distance to the cluster, these objects should have masses less than 0.1 Msunif they are cluster members. We have subsequently obtained intermediate resolution spectra of seventeen of these objects using the Keck II telescope and LRIS spectrograph. We have also obtained near-IR photometry for many of the stars. Our primary goal was to determine the location of the "lithium depletion boundary" and hence to derive a precise age for the cluster. We detect lithium with equivalent widths greater than or equal to 0.4 \AA in five of the program objects. We have constructed a color-magnitude diagram for the faint end of the Alpha Persei main sequence. These data allow us to accurately determine the Alpha Persei single-star lithium depletion boundary at M(I$_C$) = 11.47, M(Bol) = 11.42, (R-I)$_{C0}$ = 2.12, spectral type M6.0. By reference to theoretical evolutionary models, this converts fairly directly into an age for the Alpha Persei cluster of 90 $\pm$ 10 Myr. At this age, the two faintest of our spectroscopically confirmed members should be sub-stellar (i.e., brown dwarfs) according to theoretical models.Comment: Accepted Ap

Distance sets, orthogonal projections, and passing to weak tangents

The author is supported by a Leverhulme Trust Research Fellowship (RF-2016-500).We consider the Assouad dimension analogues of two important problems in geometric measure theory. These problems are tied together by the common theme of ‘passing to weak tangents’. First, we solve the analogue of Falconer’s distance set problem for Assouad dimension in the plane: if a planar set has Assouad dimension greater than 1, then its distance set has Assouad dimension 1. We also obtain partial results in higher dimensions. Second, we consider how Assouad dimension behaves under orthogonal projection. We extend the planar projection theorem of Fraser and Orponen to higher dimensions, provide estimates on the (Hausdorff) dimension of the exceptional set of projections, and provide a recipe for obtaining results about restricted families of projections. We provide several illustrative examples throughout.PostprintPeer reviewe