Balanced Truncation of k-Positive Systems Grussler, Christian; Damm, Tobias; Sepulchre, Rodolphe
IEEE transactions on automatic control,
2022-Jan., 2022-1-00, Volume:
67, Issue:
1
Journal Article
Peer reviewed
Open access
This article considers balanced truncation of discrete-time Hankel <inline-formula><tex-math notation="LaTeX">k</tex-math></inline-formula>-positive systems, characterized by Hankel matrices whose ...minors up to order <inline-formula><tex-math notation="LaTeX">k</tex-math></inline-formula> are nonnegative. Our main result shows that if the truncated system has order <inline-formula><tex-math notation="LaTeX">k</tex-math></inline-formula> or less, then it is Hankel totally positive (<inline-formula><tex-math notation="LaTeX">\infty</tex-math></inline-formula>-positive), meaning that it is a sum of first-order lags. This result can be understood as a bridge between two known results: the property that the first-order truncation of a positive system is positive (<inline-formula><tex-math notation="LaTeX">k=1</tex-math></inline-formula>), and the property that balanced truncation preserves state-space symmetry. It provides a broad class of systems where balanced truncation is guaranteed to result in a minimal internally positive system.
There is abundant evidence of a self-bias in cognition, with prioritised processing of information that is self-relevant. There is also abundant evidence of a positivity-bias in cognition, with ...prioritised processing of information that is positively valenced (e.g., positive emotional expressions, rewards). While the effects of self-bias and positivity-bias have been well documented in isolation, they have seldom been examined in parallel, so it is unclear whether one or other of these stimulus classes is prioritised or whether they interact. Addressing this gap, the current research aimed to establish the relative primacy of self-bias and positivity-bias using a classification task that paired self-relevant information with emotional expressions (i.e., Expt. 1) or reward information (i.e., Expt. 2). When the self was paired with relatively more positive information (i.e., smiling faces or high reward) we found evidence of a self-bias but no evidence of a positivity-bias. Whereas when the self was paired with relatively less positive information (i.e., neutral faces or low reward) we found evidence of a positivity-bias but no evidence of a self-bias. These results suggest the relative primacy of prioritised processing is flexible, context dependent and might be caused by a drive towards self-enhancement and the self-positivity bias.
The main aim of this study was to examine the norms of expressing emotions on social media. Specifically, the perceived appropriateness (i.e. injunctive norms) of expressing six discrete emotions ...(i.e. sadness, anger, disappointment, worry, joy, and pride) was investigated across four different social media platforms. Drawing on data collected in March 2016 among 1201 young Dutch users (15–25 years), we found that positive expressions were generally perceived as more appropriate than negative expressions across all platforms. In line with the objective of the study, some platform differences were found. The expression of negative emotions was rated as most appropriate for WhatsApp, followed by Facebook, Twitter, and Instagram. For positive emotion expression, perceived appropriateness was highest for WhatsApp, followed by Instagram, Facebook, and Twitter. Additionally, some gender differences were found, while age showed little variations. Overall, the results contribute to a more informed understanding of emotion expression online.
The paper focuses on the Lp-Positivity Preservation property (Lp-PP for short) on a Riemannian manifold (M,g). It states that any Lp function u with 1<p<+∞, which solves (−Δ+1)u≥0 on M in the sense ...of distributions must be non-negative. Our main result is that the Lp-PP holds if (the possibly incomplete) M has a finite number of ends with respect to some compact domain, each of which is q-parabolic for some, possibly different, values 2p/(p−1)<q≤+∞. When p=2, since ∞-parabolicity coincides with geodesic completeness, our result settles in the affirmative a conjecture by M. Braverman, O. Milatovic and M. Shubin in 2002. On the other hand, we also show that the Lp-PP is stable by removing from a complete manifold a possibly singular set with Hausdorff co-dimension strictly larger than 2p/(p−1) or with a uniform Minkowski-type upper estimate of order 2p/(p−1). The threshold value 2p/(p−1) is sharp as we show that when the Hausdorff co-dimension of the removed set is strictly smaller, then the Lp-PP fails. This gives a rather complete picture. The tools developed to carry out our investigations include smooth monotonic approximation and consequent regularity results for subharmonic distributions, a manifold version of the Brezis–Kato inequality, Liouville-type theorems in low regularity, removable singularities results for Lp-subharmonic distributions and a Frostman-type lemma. Since the seminal works by T. Kato, the Lp-PP has been linked to the spectral theory of Schrödinger operators with singular potentials Δ−V. Here we present some applications of the main results of this paper to the case where V∈Llocp, addressing the essential self-adjointness of the operator when p=2 and whether or not Cc∞(M) is an operator core for Δ−V in Lp.
•Lp-Positivity preservation property.•Removable singularities in the Lp property, and relation to completeness of the manifold.•Applications to spectral theory.
In contrast to long-held axioms of old age as a time of "doom and gloom," mounting evidence indicates an age-related positivity effect in attention and memory. However, several studies report ...inconsistent findings that raise critical questions about the effect's reliability, robustness, and potential moderators. To address these questions, we conducted a systematic meta-analysis of 100 empirical studies of the positivity effect (N = 7,129). Results indicate that the positivity effect is reliable and moderated by theoretically implicated methodological and sample characteristics. The positivity effect is larger in studies that do not constrain (vs. constrain) cognitive processing-reflecting older adults' natural information processing preferences-and in studies incorporating wider (vs. narrower) age comparisons. Analyses indicated that older adults show a significant information processing bias toward positive versus negative information, whereas younger adults show the opposite pattern. We discuss implications of these findings for theoretical perspectives on emotion-cognition interactions across the adult life span and suggest future research directions.
Compared with younger adults, older adults tend to favor positive information more than negative information in their attention and memory. This "positivity effect" has been observed in various ...paradigms, but at which stage it impacts cognitive processing and how it influences processing other stimuli appearing around the same time remains unclear. Across 4 experiments, we examined how older adults prioritize emotional information in early attention. Both younger and older adults demonstrated emotion-induced blindness-identifying targets in a rapid serial display of pictures with less accuracy after emotional compared with neutral distractors-but older adults demonstrated a positivity bias at this early attentional level. Moreover, the bias toward positive but not negative information in older adults was reduced when they had a working memory load. These results suggest that a selective bias toward positive, but not negative, information occurs early in visual processing, and the bias relies on cognitive control resources.
Chromatic quasisymmetric functions Shareshian, John; Wachs, Michelle L.
Advances in mathematics (New York. 1965),
06/2016, Volume:
295
Journal Article
Peer reviewed
Open access
We introduce a quasisymmetric refinement of Stanley's chromatic symmetric function. We derive refinements of both Gasharov's Schur-basis expansion of the chromatic symmetric function and Chow's ...expansion in Gessel's basis of fundamental quasisymmetric functions. We present a conjectural refinement of Stanley's power sum basis expansion, which we prove in special cases. We describe connections between the chromatic quasisymmetric function and both the q-Eulerian polynomials introduced in our earlier work and, conjecturally, representations of symmetric groups on cohomology of regular semisimple Hessenberg varieties, which have been studied by Tymoczko and others. We discuss an approach, using the results and conjectures herein, to the e-positivity conjecture of Stanley and Stembridge for incomparability graphs of (3+1)-free posets.
For an arbitrary-rank vector bundle over a projective manifold, J.-P. Demailly proposed several systems of equations of Hermitian-Yang-Mills type for the curvature tensor to settle a conjecture of ...Griffiths on the equivalence of Hartshorne ampleness and Griffiths positivity. In this article, we have studied two proposed systems and proved that these equations have smooth solutions for the Vortex bundle using the continuity method.
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