Amicable pairs : a survey

In 1750, Euler [20, 21] published an extensive paper on amicable pairs, by which he added fifty-nine new amicable pairs to the three amicable pairs known thus far. In 1972, Lee and Madachy [45] published a historical survey of amicable pairs, with a list of the 1108 amicable pairs then known. In 1995, Pedersen [48] started to create and maintain an Internet site with lists of all the known amicable pairs. The current (February 2003) number of amicable pairs in these lists exceeds four million. The purpose of this paper is to update the 1972 paper of Lee and Madachy, in order to document the developments which have led to the explosion of known amicable pairs in the past thirty years. We hope that this may stimulate research in the direction of finding a proof that the number of amicable pairs is infinite

Tur\'an Graphs, Stability Number, and Fibonacci Index

The Fibonacci index of a graph is the number of its stable sets. This parameter is widely studied and has applications in chemical graph theory. In this paper, we establish tight upper bounds for the Fibonacci index in terms of the stability number and the order of general graphs and connected graphs. Tur\'an graphs frequently appear in extremal graph theory. We show that Tur\'an graphs and a connected variant of them are also extremal for these particular problems.Comment: 11 pages, 3 figure

Symptoms associated with an abnormal echocardiogram in elderly primary care hypertension patients

Background The prevalence and diagnostic value of heart failure symptoms in elderly primary care patients with hypertension is unknown. Aim To assess the prevalence, sensitivity, specificity, positive and negative predictive value of symptoms in association with an abnormal echocardiogram. Design and setting Cross-sectional screening study in five general practices in the south-east of the Netherlands. Method Between June 2010 and January 2013, 591 primary care hypertension patients aged between 60 and 85 years were included, without known heart failure and not treated by a cardiologist. All patients underwent an echocardiogram and a structured interview including assessment of heart failure symptoms: shortness of breath, fatigue, oedema, cold extremities, and restless sleep. Results and conclusion Restless sleep was reported by 25 %, cold extremities by 23%, fatigue by 19 %, shortness of breath by 17 %, and oedema by 13 %. Oedema was the only symptom significantly associated with an abnormal echocardiogram (positive predictive value was 45 %, sensitivity 20 %, and specificity 90 %, OR 2.12; 95 % CI=1.23-3.64), apart from higher age (OR 1.06; 95 % CI=1.03-1.09), previous myocardial infarction (OR 3.00; 95 % CI=1.28-7.03), and a systolic blood pressure of >160 mmHg (OR 1.62; 95 % CI= 1.08-2.41). Screening with echocardiography might be considered in patients with oedema

Spike Protein Fusion Peptide and Feline Coronavirus Virulence

Mutations can occur erratically and accompany tropism changes, resulting in unpredictable new diseases

Submergence of the Sidebands in the Photon-assisted Tunneling through a Quantum Dot Weakly Coupled to Luttinger Liquid Leads

We study theoretically the photon-assisted tunneling through a quantum dot weakly coupled to Luttinger liquids (LL) leads, and find that the zero bias dc conductance is strongly affected by the interactions in the LL leads. In comparison with the system with Fermi liquid (FL) leads, the sideband peaks of the dc conductance become blurring for 1/2<g<1, and finally merge into the central peak for g<1/2, (g is the interaction parameter in the LL leads). The sidebands are suppressed for LL leads with Coulomb interactions strong enough, and the conductance always appears as a single peak for any strength and frequency of the external time-dependent field. Furthermore, the quenching effect of the central peak for the FL case does not exist for g<1/2.Comment: 9 pages, 4 figure

The quantum structure of spacetime at the Planck scale and quantum fields

We propose uncertainty relations for the different coordinates of spacetime events, motivated by Heisenberg's principle and by Einstein's theory of classical gravity. A model of Quantum Spacetime is then discussed where the commutation relations exactly implement our uncertainty relations. We outline the definition of free fields and interactions over QST and take the first steps to adapting the usual perturbation theory. The quantum nature of the underlying spacetime replaces a local interaction by a specific nonlocal effective interaction in the ordinary Minkowski space. A detailed study of interacting QFT and of the smoothing of ultraviolet divergences is deferred to a subsequent paper. In the classical limit where the Planck length goes to zero, our Quantum Spacetime reduces to the ordinary Minkowski space times a two component space whose components are homeomorphic to the tangent bundle TS^2 of the 2-sphere. The relations with Connes' theory of the standard model will be studied elsewhere.Comment: TeX, 37 pages. Since recent and forthcoming articles (hep-th/0105251, hep-th/0201222, hep-th/0301100) are based on this paper, we thought it would be convenient for the readers to have it available on the we

Amyloid-ß and a-Synuclein Decrease the Level of Metal-Catalyzed Reactive Oxygen Species by Radical Scavenging and Redox Silencing

The formation of reactive oxygen species (ROS) is linked to the pathogenesis of neurodegenerative diseases. Here we have investigated the effect of soluble and aggregated amyloid-ß (Aß) and a-synuclein (aS), associated with Alzheimer''s and Parkinson''s diseases, respectively, on the Cu2+-catalyzed formation of ROS in vitro in the presence of a biological reductant. We find that the levels of ROS, and the rate by which ROS is generated, are significantly reduced when Cu2+ is bound to Aß or aS, particularly when they are in their oligomeric or fibrillar forms. This effect is attributed to a combination of radical scavenging and redox silencing mechanisms. Our findings suggest that the increase in ROS associated with the accumulation of aggregated Aß or aS does not result from a particularly ROS-active form of these peptides, but rather from either a local increase of Cu2+ and other ROS-active metal ions in the aggregates or as a downstream consequence of the formation of the pathological amyloid structures

Dynamics of fluctuations in a fluid below the onset of Rayleigh-B\'enard convection

We present experimental data and their theoretical interpretation for the decay rates of temperature fluctuations in a thin layer of a fluid heated from below and confined between parallel horizontal plates. The measurements were made with the mean temperature of the layer corresponding to the critical isochore of sulfur hexafluoride above but near the critical point where fluctuations are exceptionally strong. They cover a wide range of temperature gradients below the onset of Rayleigh-B\'enard convection, and span wave numbers on both sides of the critical value for this onset. The decay rates were determined from experimental shadowgraph images of the fluctuations at several camera exposure times. We present a theoretical expression for an exposure-time-dependent structure factor which is needed for the data analysis. As the onset of convection is approached, the data reveal the critical slowing-down associated with the bifurcation. Theoretical predictions for the decay rates as a function of the wave number and temperature gradient are presented and compared with the experimental data. Quantitative agreement is obtained if allowance is made for some uncertainty in the small spacing between the plates, and when an empirical estimate is employed for the influence of symmetric deviations from the Oberbeck-Boussinesq approximation which are to be expected in a fluid with its density at the mean temperature located on the critical isochore.Comment: 13 pages, 10 figures, 52 reference