Superharmonic numbers

Let τ(n) denote the number of positive divisors of a natural number n > 1 and let σ(n) denote their sum. Then n is superharmonic if σ(n) | nκτ(n) for some positive integer κ. We deduce numerous properties of superharmonic numbers and show in particular that the set of all superharmonic numbers is the first nontrivial example that has been given of an infinite set that contains all perfect numbers but for which it is difficult to determine whether there is an odd member. © 2008 American Mathematical Society

On perfect totient numbers

Let n > 2 be a positive integer and let φ denote Euler's totient function. Define φ1(n) = φ(n) and φk(n) = φ(φk-1(n)) for all integers k ≥ 2. Define the arithmetic function S by S(n) = φ(n) + φ2(n) +...+ φc(n) + 1, where φc(n) = 2. We say n is a perfect totient number if S(n) = n. We give a list of known perfect totient numbers, and we give sufficient conditions for the existence of further perfect totient numbers

Going along versus getting it right: The role of self-integrity in political conformity

People often conform to the opinions of ingroup members, even when available evidence suggests that the group is misinformed. Following insights from the social identity approach and self-affirmation theory, it was hypothesized that people conform to salient opinions in an effort to maintain global self-integrity. In a series of experiments examining Americans' approval of President Obama and his policies, approval was consistently swayed by normative information (national polling data) but not by evidentiary information (indicators of national economic health), except under theory-predicted conditions. When participants had satisfied their sense of self-integrity with a self-affirmation exercise (Democrats in Study 1, Republicans in Study 2), or when they had low levels of American identification and thus were less concerned with national norms (Democrats and Republicans in Study 3), they showed the opposite pattern and were swayed by evidence in spite of contradicting normative information. The extent to which people are influenced by norms versus evidence in political judgment is shaped by social identity, one aspect of self-integrity. The results highlight a social psychological means to attenuate and potentially reverse conformity in the face of contradicting evidence, a finding with both practical and theoretical implications

Derived sequences

We define a multiplicative arithmetic function D by assigning D(p a) = apa-1, when p is a prime and a is a positive integer, and, for n ≥ 1, we set D0(n) = n and Dk(n) = D(D k-1(n)) when k ≥ 1. We term {Dk(n)}k=0∞ the derived sequence of n. We show that all derived sequences of n < 1.5 · 1010 are bounded, and that the density of those n ∈ ℕ with bounded derived sequences exceeds 0.996, but we conjecture nonetheless the existence of unbounded sequences. Known bounded derived sequences end (effectively) in cycles of lengths only 1 to 6, and 8, yet the existence of cycles of arbitrary length is conjectured. We prove the existence of derived sequences of arbitrarily many terms without a cycle

Sorafenib dose escalation is not uniformly associated with blood pressure elevations in normotensive patients with advanced malignancies.

Hypertension after treatment with vascular endothelial growth factor (VEGF) receptor inhibitors is associated with superior treatment outcomes for advanced cancer patients. To determine whether increased sorafenib doses cause incremental increases in blood pressure (BP), we measured 12-h ambulatory BP in 41 normotensive advanced solid tumor patients in a randomized dose-escalation study. After 7 days' treatment (400 mg b.i.d.), mean diastolic BP (DBP) increased in both study groups. After dose escalation, group A (400 mg t.i.d.) had marginally significant further increase in 12-h mean DBP (P = 0.053), but group B (600 mg b.i.d.) did not achieve statistically significant increases (P = 0.25). Within groups, individuals varied in BP response to sorafenib dose escalation, but these differences did not correlate with changes in steady-state plasma sorafenib concentrations. These findings in normotensive patients suggest BP is a complex pharmacodynamic biomarker of VEGF inhibition. Patients have intrinsic differences in sensitivity to sorafenib's BP-elevating effects

Self-Affirmation Improves Problem-Solving under Stress

High levels of acute and chronic stress are known to impair problem-solving and creativity on a broad range of tasks. Despite this evidence, we know little about protective factors for mitigating the deleterious effects of stress on problem-solving. Building on previous research showing that self-affirmation can buffer stress, we tested whether an experimental manipulation of self-affirmation improves problem-solving performance in chronically stressed participants. Eighty undergraduates indicated their perceived chronic stress over the previous month and were randomly assigned to either a self-affirmation or control condition. They then completed 30 difficult remote associate problem-solving items under time pressure in front of an evaluator. Results showed that self-affirmation improved problem-solving performance in underperforming chronically stressed individuals. This research suggests a novel means for boosting problem-solving under stress and may have important implications for understanding how self-affirmation boosts academic achievement in school settings. © 2013 Creswell et al

Reading Dickens’s characters: employing psycholinguistic methods to investigate the cognitive reality of patterns in texts

This article reports the findings of an empirical study that uses eye-tracking and follow-up interviews as methods to investigate how participants read body language clusters in novels by Charles Dickens. The study builds on previous corpus stylistic work that has identified patterns of body language presentation as techniques of characterisation in Dickens (Mahlberg, 2013). The article focuses on the reading of ‘clusters’, that is, repeated sequences of words. It is set in a research context that brings together observations from both corpus linguistics and psycholinguistics on the processing of repeated patterns. The results show that the body language clusters are read significantly faster than the overall sample extracts which suggests that the clusters are stored as units in the brain. This finding is complemented by the results of the follow-up questions which indicate that readers do not seem to refer to the clusters when talking about character information, although they are able to refer to clusters when biased prompts are used to elicit information. Beyond the specific results of the study, this article makes a contribution to the development of complementary methods in literary stylistics and it points to directions for further subclassifications of clusters that could not be achieved on the basis of corpus data alone

Computing the vertices of tropical polyhedra using directed hypergraphs

We establish a characterization of the vertices of a tropical polyhedron defined as the intersection of finitely many half-spaces. We show that a point is a vertex if, and only if, a directed hypergraph, constructed from the subdifferentials of the active constraints at this point, admits a unique strongly connected component that is maximal with respect to the reachability relation (all the other strongly connected components have access to it). This property can be checked in almost linear-time. This allows us to develop a tropical analogue of the classical double description method, which computes a minimal internal representation (in terms of vertices) of a polyhedron defined externally (by half-spaces or hyperplanes). We provide theoretical worst case complexity bounds and report extensive experimental tests performed using the library TPLib, showing that this method outperforms the other existing approaches.Comment: 29 pages (A4), 10 figures, 1 table; v2: Improved algorithm in section 5 (using directed hypergraphs), detailed appendix; v3: major revision of the article (adding tropical hyperplanes, alternative method by arrangements, etc); v4: minor revisio

Stop the Top Background of the Stop Search

The main background for the supersymmetric stop direct production search comes from Standard Model ttbar events. For the single-lepton search channel, we introduce a few kinematic variables to further suppress this background by focusing on its dileptonic and semileptonic topologies. All are defined to have end points in the background, but not signal distributions. They can substantially improve the stop signal significance and mass reach when combined with traditional kinematic variables such as the total missing transverse energy. Among them, our variable M^W_T2 has the best overall performance because it uses all available kinematic information, including the on-shell mass of both W's. We see 20%-30% improvement on the discovery significance and estimate that the 8 TeV LHC run with 20 fb-1 of data would be able to reach an exclusion limit of 650-700 GeV for direct stop production, as long as the stop decays dominantly to the top quark and a light stable neutralino. Most of the mass range required for the supersymmetric solution of the naturalness problem in the standard scenario can be covered.Comment: 16 pages, 5 figure

One More Awareness Gap? The Behaviour–Impact Gap Problem

Preceding research has made hardly any attempt to measure the ecological impacts of pro-environmental behaviour in an objective way. Those impacts were rather supposed or calculated. The research described herein scrutinized the ecological impact reductions achieved through pro-environmental behaviour and raised the question how much of a reduction in carbon footprint can be achieved through voluntary action without actually affecting the socio-economic determinants of life. A survey was carried out in order to measure the difference between the ecological footprint of “green” and “brown” consumers. No significant difference was found between the ecological footprints of the two groups—suggesting that individual pro-environmental attitudes and behaviour do not always reduce the environmental impacts of consumption. This finding resulted in the formulation of a new proposition called the BIG (behaviour–impact gap) problem, which is an interesting addition to research in the field of environmental awareness gaps