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henceforth
henceforth, adv. (ˈhɛnsfɔəθ, hɛnsˈfɔəθ) [f. hence adv. + forth adv.] From this time forth; from now onwards.c 1350 Will. Palerne 1050 Ȝe may mete eft dernli hennes⁓forþ eche day. c 1386 Chaucer Sqr.'s T. 650 But hennes forth I wol my proces holde. 1590 Spenser F.Q. ii. i. 17 Or why should ever I hen... Oxford English Dictionary
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Henceforth - Definition, Meaning & Synonyms | Vocabulary.com
5 days agohenceforth: 1 adv from this time forth; from now on " henceforth she will be known as Mrs. Smith" Synonyms: henceforward
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henceforth
henceforth/ˌhensˈfɔ:θ; ˌhɛns`fɔrθ/ (also henceforward / ˌhensˈfɔ:wəd; ˌhɛns`fɔrwɚd/) adv (fml 文) from this time on; in future 从今以後; 今後 Henceforth I expect you to be punctual for meetings. 我希望你今後准时到会. 牛津英汉双解词典
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Doubt on Roots of a polynomial with rational coefficients I understand how $$1=0.999...= \sum_{i=1}^\infty \frac {9}{10^i}$$ $\quad \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$ (henceforth represented...
> does that mean that $x^2+x+1=0$ has the same roots as $0.\overline 9x^2+0.\overline 9x+0.\overline 9=0$ Yes, because that's two ways of representing exactly the same polynomial. > If so, then does a given n-tuple of roots satisfy $2^{(n+1)}$ nth degree rational coefficient polynomials equated to z...
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Finding the injective hull Let's suppose that I have an element $e$ of order $p$ in the group of complex numbers whose elements all have order $p^n$ for some $n\in\mathbb{N}$ (henceforth called $K$), and the module ge...
_Hint:_ The abelian group of complex numbers that are $p^n$-th roots of unity for some $n$ is isomorphic to ${\mathbb Z}\left[\frac{1}{p^n}\right]/{\mathbb Z}$, with the submodule generated by the $p$-th roots of unity corresponding to ${\mathbb Z}\frac{1}{p}$. Now the strategy is exactly the same a...
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The number of $7×7$ matrices with specific conditions I am asked to find the number of $7×7$ matrices with entries $0,1,2,...,9$ whose determinant is not a multiple of $10$. Expanding the determinant along rows (or ...
Let $M_n(m)$ be the set of _all_ $n\times n$ matrices over $\mathbb{Z}/m\mathbb{Z}$, and let $$Z_n(m)=\\{A\in M_n(m) : \det A=0\\}.$$ Chinese remainder theorem says that, for $m_1,m_2$ coprime, the map $$A\mapsto(A\bmod m_1,A\bmod m_2)$$ is a bijection between $M_n(m_1 m_2)$ and $M_n(m_1)\times M_n(...
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Speed of convergence of an integral (whose complete version gives the Mascheroni constant) Consider the following integral: $$ I(k):= \int_k^{+\infty} \frac{e^{-1/x} \log(x)}{x^2} dx\\\ =-\int_0^{1/k} e^{-t} \log t \...
If $k$ is large, $t$ is near zero, so $I(k)$ is roughly (to $0$th order) equal to $$-\int_0^{1/k} \log t \,dt = -\frac{1}{k}\left(\log\left(\frac{1}{k}\right) - 1\right)=\frac{1}{k}(\log k + 1)$$ That is, $I(k)$ is $\Theta\left(\frac{\log k}{k}\right)$.
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哪一首现代诗成了你长久的精神食粮？
Henceforth I ask not good-fortune, I myself am good-fortune, Henceforth I whimper no more, postpone no more, need nothing, Done with indoor complaints, zhihu
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What would you call a finite collection of unordered objects that are not necessarily distinct? I Just want to know the name for this if there is one because I don't think it satisifies any of the formal definitions f...
If you're looking for something like a set which may have repeated elements, standard terms are **multiset** or **bag**. See multiset on wikipedia.
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Is $\vDash \exists x ( Q x \to \forall x Qx)$ a valid sentence? Is $\vDash \exists x ( Q x \to \forall x Qx)$ a valid sentence? $Q$ is a unitary relation. I suppose that $\vDash Q x \to \forall x Qx$ , which is equi...
Whether it's even well-formed depends on the low-level details of how you define syntax. But even if it is well-formed in the syntax you use, using a variable $x$ as a dummy variable in a context where $x$ already has meaning is usually a bad idea. That said, typically in syntax that allows such a t...
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Independence of variables implies no functional representation? My question's pretty simple, I just thought the title phrases it pretty well.. Anyway, the Doob-Dynkin lemma says that $X$ is $\sigma (Y)$-measurable if...
Well, I've found a solution (I didn't think of it myself) so I'm posting it. I'm not sure what the protocol is for answering your own question. Under the assumption $X,Y$ are not a.s constant, the first answer to this question proves the contrapositive.
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The substructure generated by a subset I am given the following definition of a substructure generated by a set: "For any subset A ⊆ X of an n-ary structure (X, µ), the family of subsets X' ⊆ X containing A and close...
Consider the semigroup $\Bbb Z^+$ under the binary operation of addition. What is the smallest sub-semigroup containing the set $A = \\{2\\}$?
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Show that zero sequences satisfy the following equation I am working on the following problem and got puzzled: $\\\$ Show that every zero sequence $(a_n), a_n \neq 0$ satisfies the equation: $$ \lim_{n \rightarrow \in...
$$\frac{\sqrt {1 + a_n} - 1}{a_n} =\frac{\sqrt {1 + a_n} - 1}{a_n}\cdot\frac{\sqrt {1 + a_n} +1}{\sqrt {1 + a_n} +1} =\frac1{\sqrt {1 + a_n} + 1}$$
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Mean distance between N equidistributed points in a circle I would like to calculate the mean distance depending on circle shape points, This is a mean for calculating **all possible distances between any two points*...
The average distances among all points must be equal that the average distances from a given point. By geometry: we have that the distance is $d=2 \sin(\theta/2)$, so: $$\bar d = \frac{2}{N-1} \sum_{k=1}^{N-1} \sin\left(\frac{\pi k}{N}\right)$$ On the limit, $N\to \infty$, you replace the sum by an ...
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Trace of squared non-square matrix In reading a paper I came across this expression which I don't quite understand: $$ \frac{\lambda_1}{2N}\operatorname{tr}\left((\mathbf{H}^M\mathbf{H}^M)^T\right) $$ For context, $\l...
It looks like a typo. In the paper that you have linked, the equation(12) on page(4) rewrites $$ arg.min .J = ... -\frac{\lambda_1}{2}\Bigl(tr\bigl(\frac{1}{N}H^M(H^M)^T\bigr)+\alpha tr(\Sigma_B-\Sigma_W)\Bigr) + ... $$ with the transpose correctly placed.
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