When are Multiples of Polygonal Numbers again Polygonal Numbers?

Euler showed that there are infinitely many triangular numbers that are three times other triangular numbers. In general, it is an easy consequence of the Pell equation that for a given square-free m > 1, the relation P=mP' is satisfied by infinitely many pairs of triangular numbers P, P'. After recalling what is known about triangular numbers, we shall study this problem for higher polygonal numbers. Whereas there are always infinitely many triangular numbers which are fixed multiples of other triangular numbers, we give an example that this is false for higher polygonal numbers. However, as we will show, if there is one such solution, there are infinitely many. We will give conditions which conjecturally assure the existence of a solution. But due to the erratic behavior of the fundamental unit in quadratic number fields, finding such a solution is exceedingly difficult. Finally, we also show in this paper that, given m > n > 1 with obvious exceptions, the system of simultaneous relations P = mP', P = nP'' has only finitely many possibilities not just for triangular numbers, but for triplets P, P', P'' of polygonal numbers, and give examples of such solutions.Comment: 17 pages, 1 figure, 2 tables. New version added a table of solutions to the second proble

Physical Exertion as a Risk Factor for Ventricular Arrhythmia: A Prospective Cohort Study

Episodes of physical exertion are associated with an immediately higher risk of cardiovascular events while physical activity over the long-term is cardioprotective. To assess the transient and long-term risk of ventricular arrhythmia (VA), we conducted a nested case-crossover study within a prospective cohort of 97 patients with implantable cardioverter-defibrillators (ICD). Within an hour of episodes of exertion, the risk of VA was 5.3 (95% CI 2.7 – 10.6) times greater compared to periods of rest. The association was higher among patients with aerobic fitness below the median (RR[relative risk]=17.5, 95% 5.2 – 58.5) than for patients with aerobic fitness above the median (RR=1.2, 95% CI 0.4 – 4.2, p-homogeneity = 0.002) and higher among patients who were sedentary (RR=52.8, 95% 10.1 – 277) compared to individuals who were not (RR=3.2, 95% 1.3 – 7.6, p-homogeneity=0.0002). We found no statistically significant difference in time-to-VA by aerobic fitness or sedentary behaviour. In this clinical cohort, there is an elevated risk of VA within an hour of exertion, particularly in patients with low aerobic fitness and sedentary behaviour

Global order status process of hi-tech companies

This paper focuses on the global order status process of high-tech companies. An effort has been made to understand how these companies approach their global order status process. Similarities and differences in their order status process are given. A brief history of tracking and tracing capabilities is also presented, highlighting FedEx and UPS. Some new trends in track and trace are also discussed

Triangular Numbers and Elliptic Curves

Some arithmetic of elliptic curves and theory of elliptic surfaces is used to find all rational solutions (r, s, t) in the function field Q(m, n) of the pair of equations r(r + 1)/2 = ms(s + 1)/2 r(r + 1)/2 = nt(t + 1)/2. It turns out that infinitely many solutions exist. Several examples will be given

Genome editing: a magical technique

This article includes different technologies associated with editing of gene. DNA can be altered using CRISPR/Cas9, TALEN and ZFN for improvement of crops and treatment of genetic disorders. This article provides details about different genes associated for rice improvement. Genome editing technique uses different methods of gene transfer for inserting a gene of interest into desired plant

Ministers and Civil Servants Relations in India: an Evaluation

In a parliamentary form of government, like in India, there exist two types of executives i.e. political or elected and permanent executive. Political executive derives its power from people and enjoys the power by virtue of constitutional position, while permanent executive or civil servant is selected on merit basis and accumulate its power due to an administrative position and technical expertise. The system of a democratic government is based on the principle of popular sovereignty wherein the supreme rests in people or their elected representative. Political executive or Minister is assisted by a civil servant. A balanced relationship between them is essential for smooth and efficient functioning of government. Minister and civil servant act as two pillars of the parliamentary form of government and weakness of any one of them will adversely affect the performance of government. Theoretically political and permanent executives perform a different role in government but in practice, their work is often overlapping and difficult to differentiate it

The last chapter of the Disquisitiones of Gauss

This exposition reviews what exactly Gauss asserted and what did he prove in the last chapter of {\sl Disquisitiones Arithmeticae} about dividing the circle into a given number of equal parts. In other words, what did Gauss claim and actually prove concerning the roots of unity and the construction of a regular polygon with a given number of sides. Some history of Gauss's solution is briefly recalled, and in particular many relevant classical references are provided which we believe deserve to be better known.Comment: 13 page

Glyphosate-Resistant Weed Control and Soybean Injury in Response to Different PPO-Inhibiting Herbicides

In Nebraska, 5-enolpyruvyl shikimate-3-phosphate synthase (EPSPS) as well as acetolactate synthase (ALS)-inhibitor-resistant weeds occur in many soybean fields where herbicides from these modes-of-action have been frequently used in the past. Currently, the protoporphyrinogen oxidase (PPO)-inhibitors are the only effective herbicides for POST control of both glyphosate- and ALS-inhibitor-resistant weeds in soybean. Greenhouse experiments were conducted in 2014 to evaluate the efficacy of PPO-inhibitors applied POST for the control of three glyphosate-resistant (GR) weeds and potential for soybean injury, when applied at two growth stages. All herbicide treatments controlled 10- and 20-cm tall GR common waterhemp ≥ 95% at 21 DAT. GR giant ragweed and kochia were controlled 86 to 99% when treated at 10-cm height and 78 to 92% at 20-cm height by 21 DAT. Herbicide treatments reduced shoot biomass in the three GR weeds 88 to 100% when treated at 10-cm height and 73 to 100% when treated at 20-cm height, at 21 DAT. Soybean injury and shoot biomass data revealed that acifluorfen and lactofen were more injurious (≥ 17%), whereas fomesafen, and fomesafen plus glyphosate were relatively safer (\u3c 10% injury). Overall, fomesafen and fomesafen plus glyphosate caused least injury to soybean and were more effective in controlling GR common waterhemp, giant ragweed, and kochia compared with acifluorfen and lactofen