Mathematics

Publication  Date: June14 , 2017                        Number of Pages: 26
In this paper , we study and present solutions to 6 Mathematical Mayhem
Problems published in the February , May , and June 2006 issues of the
journal Crux Mathematicorum. Specifically :
Problems M234 , M235 , and M236; are published in the February 2006
issue; issue No2 , Vol.32 , on pages74 and 75.
Problem M255 is published in the May 2006 issue; issue No5 , Vol.32 , page265.
Problems M261 and M262 , are published in the June 2006 issue;
issue No6 , Vol.32 , page362.
We present solutions to the 6 problems , listed as Problems1-6 , in
Sections2-7 of this paper. Below , we state  three among the six problems.
  Problem1(ProblemM234): Proposed by K.R.S. Sastry , Bangalore , India.
  Let J be a two-digit number in which the two digits share no common
    divisor other than 1. Reversing the digits of J produces I , a number that is p% greater than J. Find all possible values of p , if p is a natural
number less than 100.
Our solution to this problem can be found in Section2 , pages4-8.
We show that there are only two such pairs (J , I) :
(J , I) = (12 , 21) , with the corresponding value of p being 75; p = 75.
And , (J , I) = (45 , 54) , with p = 20.
So , altogether , there are two possible values of p with that property:
p = 20 and p = 75.
Problem3( Problem M236): Proposed by K.R.S. Sastry , Bangalore , India.
Let ABC be an isosceles triangle with AB = AC , and let the lengths of the
sides be integers with no common divisor other than 1. The incenter I
divides the internal angle bisector AD such that  (AI)/(ID) = 25/24.
Determine the radius of the in-circle of triangle ABC.
We present a solution to Problem3 , in Section4 , pages12-17.
We show that , length AB = AC = 25 and base length BC = 48 ;  and the
radius of the in-circle , must  equal  24/7 .
Problem5( ProblemM261) : Proposed by Bruce Sawyer , Memorial
University of Newfoundland , St.John's , NL .
Rectangle ABCD  has  AB = (1/2)BC . On the outside of the rectangle ,
draw triangle DCF , where  angle DFC = 30 degrees and  ADF is a straight
line segment. Let E be the midpoint of AD. Determine the measure of
angle  EBF .
We give a solution to this problem in Section6 , pages 20-24.
In our method of solution , we make use of both the Law of Cosines and
the Law of Sines. We show tha the measure of the angle EBF  is
30 degrees.

The acceptance of Newton's ideas and Newtonianism in the early German Enlightenment is usually described as hesitant and slow. Two reasons help to explain this phenomenon. One is that those who might have adopted Newtonian arguments were critics of Wolffianism. ...

We studied the time course of volume and protein reabsorption of a 2-ml hydrothorax using whole (WP) or diluted (DP) homologous plasma injected into the right pleural cavity in anesthetized spontaneously breathing supine rabbits. Animals were killed at 5 (WP, n = 4; DP, n = 3), 36 (WP, n = 3; DP, n = 4), 55 (WP, n = 4), 90 (WP, n = 8; DP, n = 4), and 150 (WP, n = 4; DP, n = 5) min after the injection. The volume and protein content of the pleural liquid in control conditions (n = 12) amounted to 0.35 +/- 0.015 (SE) ml/kg and 1.8 +/- 0.27 g/100 ml, respectively, which are not significantly different at 90 min (n = 7). Pleural liquid volume decreased at a similar rate during WP or DP reabsorption according to the equation V = 0.84 +/- 0.05 X e-0.02t, with net reabsorptive flow expressed as dV/dt. The globulin quantity (Q) of the pleural liquid for WP and DP, respectively, decreased according to the equations Qwp = 1 + 1.5 X e-0.04t and Qdp = 0.7 + 0.6 X e-0.03t. Assuming a major lymphatic globulin clearance and no filtration into the cavity, we obtained lymph flow using the equation VL = dQ/dt X l/C where dQ/dt is calculated from the equations for Qwp and Qdp and C represents globulin concentration. The Starling flow (Vs) was then calculated by the equation Vs = dV/dt-VL. With increasing time, lymph flow was found to decrease progressively and was not significantly different from net flow with DP, which implied a Starling flow value of zero. During WP reabsorption, lymph flow initially exceeded the net flow, with the difference disappearing at approximately 60 min; accordingly, Starling filtration flow decreased progressively, becoming zero at the same time.

This paper proposes a generalization of the Ziegler-Nichol's (ZN) method which allows parametric identification of a linear plant modeled with one of the simplified standard models of the transfer function. The proposed method, like the standard ZN method, requires only a standard PI or PID controller, without any extra equipment. Moreover, the proposed method allows the identification even if the sustained undamped oscillations are not allowed in the plant. However, more experiments are required in this case.

Few equations for calculating body-fat percentage (BF%) from field methods have been developed in South-Asian children. The objective of this study was to assess agreement between BF% derived from primary reference methods and that from skinfold equations and bio-impedance analysis (BIA) in Indian children. We measured BF% in two groups of Indian children. In Pune, 570 rural children aged 6-8 years underwent dual-energy X-ray absorptiometry (DXA) scans. In Mysore (18)O in doubly labeled water was administered to 59 urban children aged 7-9 years. We conducted BIA at 50 kHz and anthropometry, including sub-scapular and triceps skinfold thicknesses. We used the published equations of Wickramasinghe, Shaikh, Slaughter and Dezenburg to calculate BF% from anthropometric data and the manufacturer's equation for BIA measurements. We assessed agreement with values derived from DXA and doubly labeled water using Bland-Altman analysis. Children were light and thin on average compared with international standards. There was poor agreement between the reference BF% values and those from all equations. Assumptions for Bland-Altman analysis were not met for Wickramasinghe, Shaikh and Slaughter equations. The Dezenberg equations under-predicted BF% for most children (mean difference in Pune -13.4, LOA -22.7, -4.0 and in Mysore -7.9, LOA (-13.7 and -2.2). The mean bias for the BIA equation in Pune was +5.0% and in Mysore +1.95%, and the limits of agreement were wide; -5.0, 15.0 and -7.8, 11.7 respectively. Currently available skinfold equations do not accurately predict BF% in Indian children. We recommend development of BIA equations in this population using a four-compartment model.

Spontaneous breathing consists of substantial correlated variability: Parameters characterizing a breath are correlated with parameters characterizing previous and future breaths. On the basis of dynamic system theory, negative emotion states are predicted to reduce correlated variability whereas sustained attention is expected to reduce total respiratory variability. Both are predicted to evoke sighing. To test this, respiratory variability and sighing were assessed during a baseline, stressful mental arithmetic task, nonstressful sustained attention task, and recovery in between tasks. For respiration rate (excluding sighs), reduced total variability was found during the attention task, whereas correlated variation was reduced during mental load. Sigh rate increased during mental load and during recovery from the attention task. It is concluded that mental load and task-related attention show specific patterns in respiratory variability and sigh rate.