Parabola - Issue 1
https://parabola.unsw.edu.au/2010-2019/volume-47-2011/issue-1
enVolume 47 Issue 1 Header
https://parabola.unsw.edu.au/content/volume-47-issue-1-header-0
<section class="field field-name-field-nav-pic-volume-issue field-type-taxonomy-term-reference field-label-above view-mode-rss"><h2 class="field-label">Volume/Issue: </h2><ul class="field-items"><li class="field-item even"><a href="/2010-2019/volume-47-2011/issue-1">Issue 1</a></li></ul></section><section class="field field-name-field-nav-pic-image field-type-image field-label-above view-mode-rss"><h2 class="field-label">Image: </h2><div class="field-items"><figure class="clearfix field-item even"><img class="image-style-volume-issue-header-image" src="https://parabola.unsw.edu.au/files/styles/volume_issue_header_image/public/promotional_images/Hammock_0.jpg?itok=v-zAFUcx" width="640" height="250" alt="" /></figure></div></section><section class="field field-name-field-nav-pic-issue-number field-type-taxonomy-term-reference field-label-above view-mode-rss"><h2 class="field-label">Issue Number: </h2><ul class="field-items"><li class="field-item even"><a href="/issue/issue-1">Issue 1</a></li></ul></section><section class="field field-name-field-nav-pic-volume-number field-type-taxonomy-term-reference field-label-above view-mode-rss"><h2 class="field-label">Volume Number: </h2><ul class="field-items"><li class="field-item even"><a href="/volume/volume-47">Volume 47</a></li></ul></section>Tue, 11 Feb 2014 04:45:43 +0000z9803847248 at https://parabola.unsw.edu.auSolutions to Problems 1341 - 1350
https://parabola.unsw.edu.au/2010-2019/volume-47-2011/issue-1/article/solutions-problems-1341-1350
<div class="field field-name-field-article-author field-type-text field-label-hidden view-mode-rss"><div class="field-items"><div class="field-item even">Various</div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden view-mode-rss"><div class="field-items"><div class="field-item even"><div><strong>Q1341</strong> A lazy weather forecaster predicts that future maximum temperatures will be the average of the preceding two days maximum temperatures. The forecaster starts his forecast by noting that yesterday's maximum temperature was $23^\circ$C and the day before it was $29^\circ$C. What temperature in $^\circ$C is the weather forecaster's long term maximum temperature forecast?</div><div><strong>ANS:</strong> The forecast temperature $T(n)$ on day $n$ satisfies the recurrence relation $$T(n)=\frac{1}{2}T(n-1)+\frac{1}{2}T(n-2)$$ with initial conditions $T(1)=29, \ T(2)=23$.</div><div> </div><div>If we seek a trial solution of the recurrence relation in the form, $T(n)=a\lambda^n$ then we find this is a valid solution for any $a$ provided that $\lambda$ is a solution of the quadratic equation $\lambda^2-\frac{1}{2}\lambda-\frac{1}{2}=0$. The solution of the quadratic equation yields $\lambda=1$ and $\lambda=-\frac{1}{2}$ thus $$T(n)=a+b\left(-\frac{1}{2}\right)^n.$$</div><div>If we use the initial conditions $T(1)=29, \ T(2)=23$ then we find that $a$ and $b$ must satisfy the simulatenous equations</div><div> </div><div>$$a-\frac{b}{2}=29$$</div><div>$$a+\frac{b}{4}=23.$$</div><div> </div><div>We then solve for $a=25$ and $b=-8$ so that</div><div>$T(n)=25-8(-\frac{1}{2})^n$ and for large $n$ we find the</div><div>long term forecast $T=25 ^\circ$C.</div></div></div></div><div class="field field-name-field-article-article-pdf field-type-file field-label-hidden view-mode-rss"><div class="field-items"><div class="field-item even"><a href="https://parabola.unsw.edu.au/files/articles/2010-2019/volume-47-2011/issue-1/vol47_no1_s.pdf">Click here to download the PDF file</a><iframe id="pdf_reader" src="https://docs.google.com/viewer?embedded=true&url=https%3A%2F%2Fparabola.unsw.edu.au%2Ffiles%2Farticles%2F2010-2019%2Fvolume-47-2011%2Fissue-1%2Fvol47_no1_s.pdf" width="600" height="780" scrolling="no" style="border: none;"></iframe></div></div></div>Tue, 20 Aug 2013 06:13:29 +0000fcuadmin78 at https://parabola.unsw.edu.auProblems Section: Problems 1351 - 1360
https://parabola.unsw.edu.au/2010-2019/volume-47-2011/issue-1/article/problems-section-problems-1351-1360
<div class="field field-name-field-article-author field-type-text field-label-hidden view-mode-rss"><div class="field-items"><div class="field-item even">Various</div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden view-mode-rss"><div class="field-items"><div class="field-item even"><p><span style="line-height: 1.5;"><strong>Q1351</strong> A city consists of a rectangular grid of roads, with $m$ roads running east-west and $n$ running north-south. Every east-west road intersects every north-south road. A construction vehicle travels around the city, visiting each intersection once (and only once) and finally returning to its starting point. As it travels it builds a fence down the middle of each road it uses: thus it constructs, in effect, a single long fence which eventually loops back on itself. How many city blocks are now inside the fence?</span></p>
<div><span style="line-height: 1.5;"><strong>Q1352 </strong>Find infinitely many triangles with integer side lengths which contain an angle of $120^\circ$.</span></div>
<div> </div>
<div><span style="line-height: 1.5;">(a) On a $4\times n$ chessboard we wish to place $2n$ knights in such a way that none attacks any other. Give three possible ways of doing this.</span></div>
<div><span style="line-height: 1.5;">(b) Prove that there is no closed knight's tour on the $4\times n$ board.</span></div>
<p> </p></div></div></div><div class="field field-name-field-article-article-pdf field-type-file field-label-hidden view-mode-rss"><div class="field-items"><div class="field-item even"><a href="https://parabola.unsw.edu.au/files/articles/2010-2019/volume-47-2011/issue-1/vol47_no1_p.pdf">Click here to download the PDF file</a><iframe id="pdf_reader" src="https://docs.google.com/viewer?embedded=true&url=https%3A%2F%2Fparabola.unsw.edu.au%2Ffiles%2Farticles%2F2010-2019%2Fvolume-47-2011%2Fissue-1%2Fvol47_no1_p.pdf" width="600" height="780" scrolling="no" style="border: none;"></iframe></div></div></div>Tue, 20 Aug 2013 06:09:43 +0000fcuadmin77 at https://parabola.unsw.edu.auIntermediate Asymptotics
https://parabola.unsw.edu.au/2010-2019/volume-47-2011/issue-1/article/intermediate-asymptotics
<div class="field field-name-field-article-author field-type-text field-label-hidden view-mode-rss"><div class="field-items"><div class="field-item even">Michael A. B. Deakin</div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden view-mode-rss"><div class="field-items"><div class="field-item even"><p>When I entered my third year of university study, I was introduced to the topic of Fluid Mechanics $-$ the mathematical analysis of the flow of liquids and gases. I found that the concept of a fluid that is analyzed in <em>that </em>context is not exactly that which applies to real fluids. The "fluids" discussed in the lectures had local properties, such as density, pressure and velocity, described by continuous functions for which it was possible to assign values at points situated in the fluid. However, we all know that real fluids are composed of atoms and/or molecules and so do not correspond to such a description. This discrepancy is addressed in the opening chapter of one of the textbooks we were set: D. E. Rutherford's<sup>2</sup> <em>Fluid Dynamics</em> (Edinburgh: Oliver & Boyd, 1959). Rutherford's careful discussion bears quoting in full.</p></div></div></div><div class="field field-name-field-article-article-pdf field-type-file field-label-hidden view-mode-rss"><div class="field-items"><div class="field-item even"><a href="https://parabola.unsw.edu.au/files/articles/2010-2019/volume-47-2011/issue-1/vol47_no1_2.pdf">Click here to download the PDF file</a><iframe id="pdf_reader" src="https://docs.google.com/viewer?embedded=true&url=https%3A%2F%2Fparabola.unsw.edu.au%2Ffiles%2Farticles%2F2010-2019%2Fvolume-47-2011%2Fissue-1%2Fvol47_no1_2.pdf" width="600" height="780" scrolling="no" style="border: none;"></iframe></div></div></div>Tue, 20 Aug 2013 06:06:43 +0000fcuadmin76 at https://parabola.unsw.edu.auThe Meanings of Equality
https://parabola.unsw.edu.au/2010-2019/volume-47-2011/issue-1/article/meanings-equality
<div class="field field-name-field-article-author field-type-text field-label-hidden view-mode-rss"><div class="field-items"><div class="field-item even">John W. Perram</div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden view-mode-rss"><div class="field-items"><div class="field-item even"><p>Using a computer algebra system (CAS) such as <em>Mathematica</em> to relate a mathematical narrative requires a more disciplined use of symbols and terms than is common in mathematical text. We distinguish 3 examples in the usage of equality, the process of naming a mathematical expression, the process of assigning a value to a variable in an expression and the conditional assignment of equality between the two sides of an equation in order to solve it. We show that <em>Mathematica</em> has 3 different constructions for describing these relationships and explain why we believe that fully integrating a CAS into mathematics education for scientists and engineers should provide a better understanding of these concepts than the traditional narrative in terms of mathematical text.</p></div></div></div><div class="field field-name-field-article-article-pdf field-type-file field-label-hidden view-mode-rss"><div class="field-items"><div class="field-item even"><a href="https://parabola.unsw.edu.au/files/articles/2010-2019/volume-47-2011/issue-1/vol47_no1_1.pdf">Click here to download the PDF file</a><iframe id="pdf_reader" src="https://docs.google.com/viewer?embedded=true&url=https%3A%2F%2Fparabola.unsw.edu.au%2Ffiles%2Farticles%2F2010-2019%2Fvolume-47-2011%2Fissue-1%2Fvol47_no1_1.pdf" width="600" height="780" scrolling="no" style="border: none;"></iframe></div></div></div>Tue, 20 Aug 2013 06:03:22 +0000fcuadmin75 at https://parabola.unsw.edu.auEditorial
https://parabola.unsw.edu.au/2010-2019/volume-47-2011/issue-1/article/editorial
<div class="field field-name-field-article-author field-type-text field-label-hidden view-mode-rss"><div class="field-items"><div class="field-item even">B. I. Henry</div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden view-mode-rss"><div class="field-items"><div class="field-item even"><p><span style="line-height: 1.5;">Welcome to the first issue of <em>Parabola Incorporating Function</em> for 2011. The first article in, by John Perram, describes different meanings of equality. For example, we might write \( y=x^2+x+1 \) to define \( y \) or to represent an equation. In the former meaning we regard \( y \) as a symbol to represent \( x^2+x+1 \). In the latter meaning we think that for a given value of \( y\) there will be values of \( x \) for which both \( y \) and \( x^2+x+1 \) have the same value. We usually know which meaning of equality to adopt based on different context. But context is a very human thing. Computer algebra systems (CAS) need to know the difference too and this can be achieved through different syntax.</span><span style="line-height: 1.5;">One possibility, is the use of \( :=\) for </span><span style="line-height: 1.5;">an assignment operator and \( =\) for equals in an equation</span><span style="line-height: 1.5;">. The meaning of \( y:=x^2+x+1\) is that \( y\) has been assigned as a name for \( x^2+x+1\) whereas \( y=x^2+x+1\) represents an equation without any assignment of the name \( y\) to \( x^2+x+1\). We could write \( E :=y=x^2+x+1\) to assign the name \( E\) to the equation \( y=x^2+x+1 \). Part of the power of CAS has come from precision in dealing with equality through syntax.</span></p></div></div></div><div class="field field-name-field-article-article-pdf field-type-file field-label-hidden view-mode-rss"><div class="field-items"><div class="field-item even"><a href="https://parabola.unsw.edu.au/files/articles/2010-2019/volume-47-2011/issue-1/vol47_no1_e.pdf">Click here to download the PDF file</a><iframe id="pdf_reader" src="https://docs.google.com/viewer?embedded=true&url=https%3A%2F%2Fparabola.unsw.edu.au%2Ffiles%2Farticles%2F2010-2019%2Fvolume-47-2011%2Fissue-1%2Fvol47_no1_e.pdf" width="600" height="780" scrolling="no" style="border: none;"></iframe></div></div></div>Tue, 20 Aug 2013 05:37:50 +0000fcuadmin74 at https://parabola.unsw.edu.au