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whereupon

whereupon, adv. Now arch. or formal exc. in sense 4. (hwɛərəˈpɒn) [f. where 15 + upon.] I. 1. interrog. Upon what? = whereon 1; † in early use = at what? about or concerning what? upon what ground, wherefore?13.. Cursor M. 18774 (Gött.) God men of galile, Quar-apon sua wonder ȝe? 1535 Coverdale Job ... Oxford English Dictionary

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whereupon

whereuponconj after which; and then 在这以後; 然後; 於是 She laughed at him, whereupon he walked out. 她嘲笑他, 他随之离去. 牛津英汉双解词典

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Question about the global dimension of End$_A(M)$, whereupon $M$ is a generator-cogenerator for $A$ Let $A$ be a finite-dimensional Algebra over a fixed field $k$. Let $M$ be a generator-cogenerator for $A$, that mean...

The global dimension of a noetherian ring with finite global dimension is equal to the supremum of the projective dimensions of its simple modules. This is proved in most textbooks dealing with the subject. For example, this is proved in McConnell and Robson's _Noncommutative Noetherian rings_ (Coro...

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What does 春夏冬 mean? So, I was watching an anime, whereupon I found the following text (cf. screencap): > > > This text appears in the context of the character trying to find work. I'm having difficulty fig...

_The following answer is based on information taken from an earlier answer written by Tokyo Nagoya, which has been deleted by the author._ As you point out, {} is a reference to the four seasons, and in the case of , has cleverly been left off (and a added). Your intuition about the part meaning tha...

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Prove That A+B is Singular if A and B are Singular The following problem is presented: _Prove that, if A and B are singular $n \times n$ matrices, that $A+B$ is also singular._ I have the following solution, whereup...

It's not true ! Take $$A=\begin{pmatrix}1&0\\\0&0\end{pmatrix}\quad \text{and}\quad B=\begin{pmatrix}0&0\\\0&1\end{pmatrix}.$$ Notice that if $A$ and $B$ are not singular, you can also have $A+B$ singular, for example $$A=\begin{pmatrix}1&0\\\0&1\end{pmatrix}\quad \text{and}\quad B=\begin{pmatrix}-1...

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Integration - finding an explicit formula The question in my textbook asks: > If $f$ is a continuous function such that $$\int\limits_0^x{f(t)dt}=xe^{2x}+\int\limits_0^x{e^{-t}f(t)dt}$$ for all $x$, find an explicit ...

Notice that if you differentiate the equation: $$f(x) = e^{2x} + 2xe^{2x} + e^{-x}*f(x)$$ $$f(x)(1 - e^{-x}) = e^{2x} + 2xe^{2x}$$ $$f(x) = \frac{e^{2x} + 2xe^{2x}}{1 - e^{-x}}$$

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于是

【于是】yúshì[as a result;therefore;whereupon] 紧接上事之后并由于上事而出现某种结果新华字典

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Open mappings and continuous functions? Are these interrelated? In Urysohn's Metrization Theorem, at some point we define a function $F : X \rightarrow H$ from the space X into Hilbert space $H$. Whereupon we need to ...

A map is continuous if _preimage of open sets are open sets_ and it is open if _direct image of open set are open sets_. More precisely, $f:X\to Y$ is **continuous** if $f^{-1}(V)\subset X$ is open for any open set $V\subset Y$. It is open if $f(U)\subset Y$ is **open** for any open set $U\subset X$...

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Hard Lefschetz Thereom and the Cohomology of Flag Varieties Let $G$ be a compact connected connected Lie group $G$, and $B$ a Borel subgroup containing a maximal torus. Moreover, let $F = G/B$ be the associated flag m...

There is something wrong already with your statement of Borel--Weil--Bott. E.g. the simplest example of a flag variety is $\mathbb P^1$ (which is the flag variety for $SL_2$), and its Hodge numbers are $h^{0,0} = h^{1,1} = 1$, and all other $h^{a,b} = 0.$ More generally, as you write, it is true, fo...

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If PQis a focal chord, show that the interval RU is parallel to the axis of the parabola. ![enter image description here]( For part (c) of question thirteen am I only required to find the gradient of RU and prove tha...

hint...The focal chord property is $pq=-1$, whereupon you can calculate the gradient of $RU$...

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Does the creation of memory involve mRNAs crossing the synaptic gap? There is a diagram from a book titled "Teaching with the brain in mind". The diagram shows ![How memories are formed according to "teaching with the...

I have never heard of this pathway. Memory is usually assoiciated with synaptic plasticity by ‘Long-term potentiation’ (LTP), which has glutamate as a neurotransmitter. Neuroscience Exploring the brain (Bear, et al,. 2007), has a pretty good explanation of this process, if you're interrested. Motor ...

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Meaning and use of ところも I am a bit puzzled by the use of and its association with in the following sentence > **** > The attitude and behavior of the boy had caught on with some of the passengers and they conv...

, which literally just means "place", can be used to describe a quality or aspect of something. This is a metaphorical extension of 's literal meaning as a location in space/time. > **** **** Everyone has both good **qualities** and bad **qualities**. > He can be a bit of a chicken at times. (lit. H...

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Do properties of algebraic structures sometimes not carry over when their direct products are taken? I recently had a homework problem asking to prove that the direct product of rings (or rings with identity) are stil...

A direct product of integral domains is never an integral domain since $(1,0)\cdot(0,1)=0$. You can also consider PIDs; $\Bbb Z$ is a PID while $\Bbb Z\times\Bbb Z$ is no. The direct product of fields is not a field, say $\Bbb Q$ and $\Bbb Q\times\Bbb Q$. There is also a problem of checking that if ...

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Using FTOC to solve a definite integral with a function as its upper bound Let $h = e^x$ $$\frac{d}{dx} \int_2^h ln(e^{2t} + 1) dt$$ Okay, so I believe that I can assume $ln(e^{2t} + 1) = f(t)$ And then $$\int_2^h f...

As N74 mentions, we should actually have $$\int_2^h f(t) dt =F(h)-F(2)=F(e^x)-F(2)$$ Whereupon the derivative would be taken with chain rule.

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Property of angle bisectors in a triangle Let $ABC$ be a triangle having circumcenter $O$. Suppose $AH$ is the altitude from vertex $A$ and $AT$ bisects angle $A$. I would like a simple geometric proof that $AT$ also ...

Therefore, $\angle BAH$ and $\angle OAD$ are congruent complements of $\angle B$, whereupon $$\angle HAT = \angle BAT - \angle BAH = \angle CAT - \angle Therefore, $\angle BAH$ and $\angle OAD$ are congruent complements of $\angle AOD$, whereupon $$\angle HAT = \angle BAT + \angle BAH= \angle CAT + \angle

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whereupon

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whereupon

whereupon

Question about the global dimension of End$_A(M)$, whereupon $M$ is a generator-cogenerator for $A$ Let $A$ be a finite-dimensional Algebra over a fixed field $k$. Let $M$ be a generator-cogenerator for $A$, that mean...

What does 春夏冬 mean? So, I was watching an anime, whereupon I found the following text (cf. screencap): > > > This text appears in the context of the character trying to find work. I'm having difficulty fig...

Prove That A+B is Singular if A and B are Singular The following problem is presented: _Prove that, if A and B are singular $n \times n$ matrices, that $A+B$ is also singular._ I have the following solution, whereup...

Integration - finding an explicit formula The question in my textbook asks: > If $f$ is a continuous function such that $$\int\limits_0^x{f(t)dt}=xe^{2x}+\int\limits_0^x{e^{-t}f(t)dt}$$ for all $x$, find an explicit ...

于是

Open mappings and continuous functions? Are these interrelated? In Urysohn's Metrization Theorem, at some point we define a function $F : X \rightarrow H$ from the space X into Hilbert space $H$. Whereupon we need to ...

Hard Lefschetz Thereom and the Cohomology of Flag Varieties Let $G$ be a compact connected connected Lie group $G$, and $B$ a Borel subgroup containing a maximal torus. Moreover, let $F = G/B$ be the associated flag m...

If PQis a focal chord, show that the interval RU is parallel to the axis of the parabola. ![enter image description here]( For part (c) of question thirteen am I only required to find the gradient of RU and prove tha...

Does the creation of memory involve mRNAs crossing the synaptic gap? There is a diagram from a book titled "Teaching with the brain in mind". The diagram shows ![How memories are formed according to "teaching with the...

Meaning and use of ところも I am a bit puzzled by the use of and its association with in the following sentence > **** > The attitude and behavior of the boy had caught on with some of the passengers and they conv...

Do properties of algebraic structures sometimes not carry over when their direct products are taken? I recently had a homework problem asking to prove that the direct product of rings (or rings with identity) are stil...

Using FTOC to solve a definite integral with a function as its upper bound Let $h = e^x$ $$\frac{d}{dx} \int_2^h ln(e^{2t} + 1) dt$$ Okay, so I believe that I can assume $ln(e^{2t} + 1) = f(t)$ And then $$\int_2^h f...

Property of angle bisectors in a triangle Let $ABC$ be a triangle having circumcenter $O$. Suppose $AH$ is the altitude from vertex $A$ and $AT$ bisects angle $A$. I would like a simple geometric proof that $AT$ also ...