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diametrically
diaˈmetrically, adv. [f. prec. + -ly2.] 1. In the manner or direction of a diameter; along the diameter; straight through.1695 Woodward Nat. Hist. Earth iii. i. (1723) 137 The Vapour..cannot penetrate the Stratum diametricaly. 1794 T. Taylor Pausanias III. 95 Its breadth, measured diametrically, may...
Oxford English Dictionary
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diametrically
diametrically/ˌdaɪəˈmetrɪklɪ; ˌdaɪə`mɛtrɪklɪ/ advcompletely; entirely 完全地; 全然地 diametrically opposed/opposite 完全相反的.
牛津英汉双解词典
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Mapping the unit circle in C to R with two diametrically opposite points having the same image. I'm stuck on this problem, along with many others, from one of my Analysis problem sheets. I'd be very grateful if someon...
Consider the function $g(x) = f(x)-f(x+\pi)$. What can you say about $f(0)$ and $f(\pi)$? Then remember the Intermediate Value Theorem.
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diameterly
† diˈameterly, adv. Obs. rare. [f. prec. + -ly2.] = diametrically 2 b.1603 Florio Montaigne iii. ix. (1632) 560 Libertie and idlenesse..are qualities diameterly contrary to that mysterie. 1633 Ames Agst. Cerem. ii. 518 So diameterly contrary to it.
Oxford English Dictionary
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Choose $3n$ points on a circle, show that there are two diametrically opposite point > On a circle of length $6n$, we choose $3n$ points such that they split the circle into $n$ arcs of length $1$, $n$ arcs of length ...
If we want to avoid having opposite points, then obviously every 1-arc must be positioned diametrically opposite to the middle of some 3-arc.
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Finding the Distance Between Two Diametrically Opposite Points on a Cone base circle The question is as follows: > Both the slant height and the base diameter of a cone are 12 inches. What is distace between two oppo...
The answer is best seen on a development. Semi-vertcal angle is $\sin^{-1}\frac12 = 30^{\circ}$ On development angle subtended at cone apex is $$ \frac12* 360^{\circ}=180^{\circ}$$ If $l=2r = 12 inches, \, $say, cone develpment is a semi-circle of r= 6 units radius. Minimum (geodesic) distance is sh...
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反其道而行之 - Wiktionary, the free dictionary
Chinese: ·to do just the opposite; to act in a diametrically opposite way
en.wiktionary.org
Explain the following combination question in deep details Suppose $32$ objects are placed along a circle at equal distances. In how many ways can $3$ objects be chosen from among them so that no two of the three chos...
There are $32$ ways to place the first object. After that there are $28$ places for the second object. That is, there are $32\cdot28=896$ ways to place the first two objects. If the first two objects are $2$ places apart, then there are $25$ places for the third; if the first two places are $15$ apa...
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Heights of two points on a circle Given a circle of arbitrary radius centred on C, let each point on the circle have an associated height which varies continuously and smoothly as you go around the circle. How can I...
Consider any two diametrically opposite points on $C$, $x$ and $x+\pi$, and their associated heights $f(x)$ and $f(x+\pi)$. Since $f(x)$ is continuous, so is $h(x)$; since $h(0)=-h(\pi)$, by the intermediate value theorem there is some $0\le x<\pi$ where $h(x)=0$, i.e. the two diametrically
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Perimeter of inscribed hexagon > In a circle of radius 3 the equilateral triangle ABC is inscribed, and the points X, Y and Z are diametrically opposite to A, B and C (respect) . Find the perimeter of the hexagon AZBX...
**Solution (you have to draw a diagram to follow along):**
Since X is diametrically opposite to A, and ABC is equilateral (which means AB=AC), we can
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Differential form on $\mathbb R\mathbb P^2$ Let $f:S^2\to\mathbb R\mathbb P^2$ be a mapping that takes the point of the unit sphere into a straight line passing through this point and the center of the sphere. Let $r:...
The idea is to push the form $\omega$ forward by the local diffeomorphism $f$. By "pushing forward" I really mean pulling back by the inverse. In other words, around any point in the projective plane you can find a small open set $U$ such that there exist two distinct sections $s_1,s_2:U \to S^2$ of...
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Show that the area of the face of the coin is $\frac{a^2}{2}(\pi-7\tan\frac{\pi}{14})$ The diagram shows a British 50 pence coin. }=\frac{r}{\sin\left(\frac{\pi}{14}\right)}$$ $$\Rightarrow r=\frac{a}{2\c...
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Proof of $+\infty=-\infty$ (Maybe) I guess we can agree that $+0 = -0$. Now, after that, I was simply looking at some graphs. The graph of $\tan x$ shows asymptotes at x = $n\pi + \pi/2$. I got to thinking, what if th...
In general (for grade 11), remember that $\infty$ is not a real number. To say that two elements are equal, they need to be equal in some set. That is, they need first be elements in some set. And $\infty$ is not an element in the set of real numbers. For example, when we say that a limit (of a func...
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Problem about continuous functions and the intermediate value theorem > Let $S^{1} := \lbrace(\cos\alpha, \sin\alpha) \subset \mathbb{R}^{2} | \alpha \in \mathbb{R}\rbrace$ be the circumference of radius $1$ and $f: S...
The idea is good, but the intermediate value theorem applies to functions mapping a closed interval $I \subset \Bbb R$ to $\Bbb R$. It would be possible to formulate a similar statement for functions $\phi: S^1 \to \Bbb R$, but it is simpler to consider $$ \phi: [0, \pi] \to \Bbb R, \quad \phi(\alph...
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What does it mean to say that a pair of points are antipodal in a topological sphere? A pair of points are antipodal if they are diametrically opposite to each other. This definition makes perfect sense when one think...
The meaning of "antipodal" depends on a choice of homeomorphism between an ellipsoid (or any other subset of $\mathbb{R}^n$ homeomorphic to a sphere) and the usual unit sphere. The Borsuk-Ulam theorem is true regardless of this choice.
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