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thereafter

thereafter, adv. (ðɛəˈrɑːftə(r), -æ-) [OE. þǽr æfter, two words, viz. þǽr, there 17 and æfter, after; ME. þer after. Cf. OS. thar after (Du. daarachter); ON. þar epter (Sw., Da., Norw. derefter).] 1. After that in time, order, or sequence; subsequently; afterwards. (Now somewhat formal.)c 897 K. ælf... Oxford English Dictionary

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Thereafter vs Therefore: How Are These Words Connected?

There are some idiomatic expressions that use "thereafter" or "therefore" in ways that do not follow the standard rules. For example, "ever thereafter" means "from that time on forever," while "and therefore" can be used to mean "so" or "consequently.". Example: "They lived happily ever thereafter.".

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thereafter

thereafter/ˌðeərˈɑ:ftə(r); ?@ -ˈæf-; ðɛr`æftɚ/ adv(fml 文) after that 此後; 其後 You will be accompanied as far as the border; thereafter you must find your own way. 有人送你到边境, 然後你就得自己走了. 牛津英汉双解词典

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sed: match pattern, and replace every occurence of a different pattern thereafter Suppose I have `aaaaabaaaaa`, and assume that `a` can be an arbitrary pattern; likewise for `b`. How do I pattern match for `aaaaaab`,...

Too complex in sed, but much easier in Perl: perl -pe '/b/g and substr($_, pos) =~ s/a/A/g' The first match `/b/g` matches the first `b` in the string which sets `pos` to its position. Then, the substring starting at that position undergoes the substitution.

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Integral solutions to expression given below. Given expression $xy=2^23^45^7(x+y)$ What are its integral solutions? I tried to solve it by converting it to $\frac{1}{2^2}\frac{1}{3^4}\frac{1}{5^7}=\frac{1}{x}+\frac{1}...

HINT: Suppose $xy=q(x+y)$ then $xy-qx-qy+q^2=q^2=(x-q)(y-q)$ That should be enough to enable you to identify the possible values.

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Tangentplane of a hyperboloid I have a hyperboloid $H$, which is $x^2+\frac{y^2}{4}-z^2=1$. I want to show that all points $P$ on $H$, in which the tangentplane to $H$ that goes through the point $(1,4,2)$, all are i...

Let $P(X,Y,Z)$. Thus, the equation of our tangent plane it's: $$2X(x-X)+\frac{Y}{2}(y-Y)-2Z(z-Z)=0$$ or $$Xx+\frac{Y}{4}y-Zz-X^2-\frac{Y^2}{4}+Z^2=0$$ or since $(X,Y,Z)$ placed on our hyperboloid, we obtain $$Xx+\frac{Y}{4}y-Zz-1=0.$$ But $(1,4,2)$ placed on this plane, which gives $$X+Y-2Z-1=0,$$ w...

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as soon as possible thereafter - Chinese translation - Linguee

is an exporting Member or an importing Member at the time of giving such. [...] notification o r as soon as possible thereafter. daccess-ods.un.org. daccess-ods.un.org. 条生效时，或如协定已经生效，则在特定日期，暂时适用. [...] 本协定。. 发出这种通知的政府应在发生此种通知之时或其后尽快 ...

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Determining eigenvectors So let us say we have the following matrix: $$ A = \begin{bmatrix} 1 & -3 \\\ 1 & 5 \\\ \end{bmatrix} $$ I can calculate that the eigenvalues are $4$ and $2$. So now it is needed to calcula...

The matrix-vector equation $$\begin{bmatrix}a&b\\\c&d\end{bmatrix}\begin{bmatrix}x\\\y\end{bmatrix}=\begin{bmatrix}A\\\B\end{bmatrix}$$ is equivalent to the equation $$\begin{bmatrix}ax+by\\\cx+dy\end{bmatrix}=\begin{bmatrix}A\\\B\end{bmatrix}$$

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Limit question concerning a triangle and its circumcircle $ABC$ is an isosceles triangle ($|AB|=|BC|$). Let $s$ be the length of the altitude from vertex $B$ to side $AC$, and let $m=|AC|$. Given that the radius of th...

With the observation that $k = sm/2$, we want $\lim \frac{2s}{m^2}$. **Hint:** Express $s$ in terms of $m$. By power of a point about the midpoint of $AC$, $ s(4-s) = (m/2) (m/2)$. Hence $ s = \frac{1}{2} \left(4 - \sqrt{16- m^2 }\right) $. **Hint:** Evaluate $\lim \frac{ 4 - \sqrt{16 - m^2 } } { m^...

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Transformation T is... "onto"? I thought you have to say a mapping is onto _something_... like, you don't say, "the book is on the top of"... Our book starts out by saying "a mapping is said to be onto R^m", but ther...

As I mentioned in my comment, the word "onto" is often used as a synonym for the word "surjective". In the same spirit, you can use "one-to-one" instead of "injective". See for example the corresponding Wikipedia article. **Edit** : I agree with the comments by Qiaochu and Jonas that "one-to-one" is...

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A question about the deduction theorem The deduction theorem states that if $T \cup \\{ \psi \\} \vdash \varphi $ and the generalisation rule is not used to prove $\varphi$ then $T \vdash \psi \rightarrow \varphi $. ...

The important thing to realise is that whatever is on the RHS of $\vdash$ has got to be an axiom (meaning: a universally true formula). On the other hand, on the LHS we can assume whatever we like. So for example if $T = \emptyset$ we can show that $\\{ \varphi (x) \\} \vdash \forall x \varphi(x)$. ...

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How do you deal with absolute values in a function when calculating volumes of solids of revolution? For example, when calculating the volume of a solid of revolution for the area between the curves `y=-abs(x-4)+4` ; ...

I've done this volume calculation using Pappus's $(2^{nd})$ Centroid Theorem: the volume of a planar area of revolution is the product of the area $A$ and the length of the path traced by its centroid R, i.e., $2πR$. The volume is simply $V=2\pi RA$. The figure below shows the area to be revolved ab...

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Piping 'false' does not give non-zero result code I understand these: true; echo "$?" # 0 false; echo "$?" # 1 true | echo "$?" # 0 But not this: false | ec...

The result of both `true | echo "$?"` and `false | echo "$?"` is misleading. The content of `"$?"` will be set _before_ piping the command `false` to the command `echo`. To execute these lines, bash sets up a pipeline of commands. The pipeline is setup, then the commands started in parallel. So in y...

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Efficient modular exponentation of powers Is there any way to efficiently compute $(((\mathrm{base}^{M_1})^{M_2})^{M_3} \dots )^{M_n}$ modulo $P$, where $P$ is prime? One way is to repeatedly do modular exponentiation...

IF $P$ divides $b,$ the remainder will be $0$ Else $(b,P)=1$ $$((b^{M_1})^{M_2})^{M_3}\cdots)^{M_n}=b^{M_1\cdot M_2\cdot M_3\cdots M_n}$$ As $P$ is prime, $a^{P-1}\equiv\pmod P$ if $(a,P)=1$ using Fermat's Little Theorem. So, we need to find $M_1\cdot M_2\cdot M_3\cdots M_n\pmod {(P-1)}$ Then use re...

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Meaning of 最も多い in this sentence > **** > Every year in Japan ??? more than 36,000 people run. I can't understand at all what does in this sentence. Literally it means 'most numerous' right? I can't understand ...

In this context, the following could safely be inferred. {}{}{}{}{} What is is the number of participants in the marathon. > "Every year, over 36,000 people, which is the largest scale in Japan, run/participate." A literal translation would be close to impossible, which is why I used "the largest sc...

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thereafter

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thereafter

Thereafter vs Therefore: How Are These Words Connected?

thereafter

sed: match pattern, and replace every occurence of a different pattern thereafter Suppose I have `aaaaabaaaaa`, and assume that `a` can be an arbitrary pattern; likewise for `b`. How do I pattern match for `aaaaaab`,...

Integral solutions to expression given below. Given expression $xy=2^23^45^7(x+y)$ What are its integral solutions? I tried to solve it by converting it to $\frac{1}{2^2}\frac{1}{3^4}\frac{1}{5^7}=\frac{1}{x}+\frac{1}...

Tangentplane of a hyperboloid I have a hyperboloid $H$, which is $x^2+\frac{y^2}{4}-z^2=1$. I want to show that all points $P$ on $H$, in which the tangentplane to $H$ that goes through the point $(1,4,2)$, all are i...

as soon as possible thereafter - Chinese translation - Linguee

Determining eigenvectors So let us say we have the following matrix: $$ A = \begin{bmatrix} 1 & -3 \\\ 1 & 5 \\\ \end{bmatrix} $$ I can calculate that the eigenvalues are $4$ and $2$. So now it is needed to calcula...

Limit question concerning a triangle and its circumcircle $ABC$ is an isosceles triangle ($|AB|=|BC|$). Let $s$ be the length of the altitude from vertex $B$ to side $AC$, and let $m=|AC|$. Given that the radius of th...

Transformation T is... "onto"? I thought you have to say a mapping is onto _something_... like, you don't say, "the book is on the top of"... Our book starts out by saying "a mapping is said to be onto R^m", but ther...

A question about the deduction theorem The deduction theorem states that if $T \cup \\{ \psi \\} \vdash \varphi $ and the generalisation rule is not used to prove $\varphi$ then $T \vdash \psi \rightarrow \varphi $. ...

How do you deal with absolute values in a function when calculating volumes of solids of revolution? For example, when calculating the volume of a solid of revolution for the area between the curves `y=-abs(x-4)+4` ; ...

Piping 'false' does not give non-zero result code I understand these: true; echo "$?" # 0 false; echo "$?" # 1 true | echo "$?" # 0 But not this: false | ec...

Efficient modular exponentation of powers Is there any way to efficiently compute $(((\mathrm{base}^{M_1})^{M_2})^{M_3} \dots )^{M_n}$ modulo $P$, where $P$ is prime? One way is to repeatedly do modular exponentiation...

Meaning of 最も多い in this sentence > **** > Every year in Japan ??? more than 36,000 people run. I can't understand at all what does in this sentence. Literally it means 'most numerous' right? I can't understand ...