Asymptotic Optimality of the Triangular Lattice for a Class of Optimal Location Problems

We prove an asymptotic crystallization result in two dimensions for a class of nonlocal particle systems. To be precise, we consider the best approximation with respect to the 2-Wasserstein metric of a given absolutely continuous probability measure $$f \mathrm {d}x$$ f d x by a discrete probability measure $$\sum _i m_i \delta _{z_i}$$ ∑ i m i δ z i , subject to a constraint on the particle sizes $$m_i$$ m i . The locations $$z_i$$ z i of the particles, their sizes $$m_i$$ m i , and the number of particles are all unknowns of the problem. We study a one-parameter family of constraints. This is an example of an optimal location problem (or an optimal sampling or quantization problem) and it has applications in economics, signal compression, and numerical integration. We establish the asymptotic minimum value of the (rescaled) approximation error as the number of particles goes to infinity. In particular, we show that for the constrained best approximation of the Lebesgue measure by a discrete measure, the discrete measure whose support is a triangular lattice is asymptotically optimal. In addition, we prove an analogous result for a problem where the constraint is replaced by a penalization. These results can also be viewed as the asymptotic optimality of the hexagonal tiling for an optimal partitioning problem. They generalise the crystallization result of Bourne et al. (Commun Math Phys, 329: 117–140, 2014) from a single particle system to a class of particle systems, and prove a case of a conjecture by Bouchitté et al. (J Math Pures Appl, 95:382–419, 2011). Finally, we prove a crystallization result which states that optimal configurations with energy close to that of a triangular lattice are geometrically close to a triangular lattice.

Aliasing Free For Mixed Spectra for Stable Processes

This work focuses on the symmetric alpha stable processes with continuous time frequently used in modeling the signal with indefinitely growing variance when the spectral measure is mixed: sum of a continuous meseare and discrete measure. The objective of this paper is to estimate the spectral density of the continuous part from discrete observations of the signal. For that, we propose a method based on a sample of the signal at a periodic instant. The Jackson polynomial kernel is used for construct a periodogram. We smooth this periodogram by two spectral windows taking into account the width of the interval where the spectral density is nonzero. This technique allows to circumvent the phenomenon of aliasing often encountered in the estimation from the discrete observations of a process with a continuous time.

Neural adaptations to long-term resistance training: evidence for the confounding effect of muscle size on the interpretation of surface electromyography

This study compared elbow flexor (EF; Experiment 1) and knee extensor (KE; Experiment 2) maximal compound action potential (Mmax) amplitude between long-term resistance trained (LTRT; n=15 and n=14, 6±3 and 4±1 years of training) and untrained (UT; n=14 and n=49) men; and examined the effect of normalising electromyography (EMG) during maximal voluntary torque (MVT) production to Mmax amplitude on differences between LTRT and UT. EMG was recorded from multiple sites and muscles of EF and KE, Mmax was evoked with percutaneous nerve stimulation, and muscle size was assessed with ultrasonography (thickness, EF) and magnetic resonance imaging (cross-sectional area, KE). Muscle-electrode distance (MED) was measured to account for the effect of adipose tissue on EMG and Mmax. LTRT displayed greater MVT (+66-71%, p<0.001), muscle size (+54-56%, p<0.001), and Mmax amplitudes (+29-60%, p≤0.010) even when corrected for MED (p≤0.045). Mmax was associated with the size of both muscle groups (r≥0.466, p≤0.011). Compared to UT, LTRT had higher absolute voluntary EMG amplitude for the KE (p<0.001), but not the EF (p=0.195), and these differences/similarities were maintained after correction for MED; however, Mmax normalisation resulted in no differences between LTRT and UT for any muscle and/or muscle group (p≥0.652). The positive association between Mmax and muscle size, and no differences when accounting for peripheral electrophysiological properties (EMG/Mmax), indicates the greater absolute voluntary EMG amplitude of LTRT might be confounded by muscle morphology, rather than provide a discrete measure of central neural activity. This study therefore suggests limited agonist neural adaptation after LTRT.

Families of Well Approximable Measures

We provide an algorithm to approximate a finitely supported discrete measure μ by a measure νN corresponding to a set of N points so that the total variation between μ and νN has an upper bound. As a consequence if μ is a (finite or infinitely supported) discrete probability measure on [0, 1] d with a sufficient decay rate on the weights of each point, then μ can be approximated by νN with total variation, and hence star-discrepancy, bounded above by (log N)N− 1. Our result improves, in the discrete case, recent work by Aistleitner, Bilyk, and Nikolov who show that for any normalized Borel measure μ, there exist finite sets whose star-discrepancy with respect to μ is at most ( log N ) d − 1 2 N − 1 {\left( {\log \,N} \right)^{d - {1 \over 2}}}{N^{ - 1}} . Moreover, we close a gap in the literature for discrepancy in the case d =1 showing both that Lebesgue is indeed the hardest measure to approximate by finite sets and also that all measures without discrete components have the same order of discrepancy as the Lebesgue measure.

Comparing the Slider Measure of Social Value Orientation with Its Main Alternatives

The Slider Measure of social value orientation (SVO) was introduced as an improvement from existing measures. We conduct an independent assessment of its suitability compared with the Ring Measure and the Triple Dominance Measure. Using a student sample, we assess the measures’ test-retest reliability (N = 88; using a longer time interval than previous studies) and sensitivity to random responses. Analyses pertaining to convergent validity, criterion validity, and the advantages of a continuous over a discrete measure are presented in the online appendix . Compared with alternatives, the Slider Measure has the highest test-retest reliability. However, it classifies random responses in an unbalanced way, assigning the vast majority of random responses to cooperative and individualistic, rather than altruistic and competitive, orientations. For all three measures, we propose improved ways of weeding out inconsistent responses.

Cyclic Flats of a Polymatroid

Polymatroids can be considered as “fractional matroids” where the rank function is not required to be integer valued. Many, but not every notion in matroid terminology translates naturally to polymatroids. Defining cyclic flats of a polymatroid carefully, the characterization by Bonin and de Mier of the ranked lattice of cyclic flats carries over to polymatroids. The main tool, which might be of independent interest, is a convolution-like method which creates a polymatroid from a ranked lattice and a discrete measure. Examples show the ease of using the convolution technique.

dCATCH—A Numerical Package for d-Variate near G-Optimal Tchakaloff Regression via Fast NNLS

We provide a numerical package for the computation of a d-variate near G-optimal polynomial regression design of degree m on a finite design space X ⊂ R d , by few iterations of a basic multiplicative algorithm followed by Tchakaloff-like compression of the discrete measure keeping the reached G-efficiency, via an accelerated version of the Lawson-Hanson algorithm for Non-Negative Least Squares (NNLS) problems. This package can solve on a personal computer large-scale problems where c a r d ( X ) × dim ( P 2 m d ) is up to 10 8 – 10 9 , being dim ( P 2 m d ) = 2 m + d d = 2 m + d 2 m . Several numerical tests are presented on complex shapes in d = 3 and on hypercubes in d > 3 .

Regularized Wasserstein Means for Aligning Distributional Data

We propose to align distributional data from the perspective of Wasserstein means. We raise the problem of regularizing Wasserstein means and propose several terms tailored to tackle different problems. Our formulation is based on the variational transportation to distribute a sparse discrete measure into the target domain. The resulting sparse representation well captures the desired property of the domain while reducing the mapping cost. We demonstrate the scalability and robustness of our method with examples in domain adaptation, point set registration, and skeleton layout.

The Role of Emotions and Perceived Ad Effectiveness: Evidence From the Truth FinishIt Campaign

Purpose: Examine association between emotional valence and intensity prompted by anti-tobacco advertising messages and perceived ad effectiveness among youth/young adults. Design: Online forced-exposure survey data from a nationally weighted, cross-sectional sample of youth/young adults, collected periodically over a 4-year period. Setting: National. Participants: Thirty-seven cross-sectional surveys conducted online from June 2015 to January 2018; total N = 9534. All participants, aged 15 to 21, were in the intervention; no control group. Intervention: Individuals participating in premarket testing of truth ads were forced exposed to one of 37 anti-tobacco ads. Measures: Emotional response, emotional intensity, and perceived ad effectiveness. Emotional response has been previously studied and measured. Including the discrete measure of “concerned” in positive emotions is unique to our study. It patterned with the other positive emotions when each ad was examined by each emotion. Intensity as measured in this study through the 5-point scale (“how much does this ad make you feel”) is unique in the anti-tobacco ad literature. Although several past studies ranked the degree of emotion elicited by ads, they have not incorporated the intensity of emotion as reported by the participant themselves. The scale was used to determine whether perceived ad effectiveness is similar to those used in previous studies. Analysis: Linear regressions were estimated to assess type of emotional sentiment and level of intensity in relation to perceived effectiveness of the message. Results: All 9534 participants were exposed; no control group. The βs indicate how strongly the emotion variable influences the study outcome of perceived ad effectiveness. Positive emotions (β = .76) were more highly associated with perceived ad effectiveness (β = .06). Higher intensity with positive emotional sentiment and high-intensity negative produced the highest scores for perceived ad effectiveness (β = .30). Conclusion: Eliciting a positive, high-impact emotional response from viewers can help improve perceived effectiveness, and in turn, overall ad effectiveness.