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# Can someone clearly explain about the lim sup and lim inf?

Can some explain the lim sup and lim inf?
In my text book the definition of these two is this.

Let $(s_n)$ be a sequence in $\mathbb{R}$. We define

and

The right side of these two equality, can I think $\sup\{s_n:n>N\}$ and $\inf\{s_n:n>N\}$ as a sequence after $n>N$?
And how these two behave as $n$ increases? My professor said that these two get smaller as $n$ increases.

Consider this example:

It alternates between something approaching $3$ from below and something approaching $5$ from above. The lim inf is $3$ and the lim sup is $5$.

The inf of the whole sequence is $3-\frac12$.

If you throw away the first term or the first two terms, the inf of what’s left is $3-\frac14$.

If you throw away all the terms up to that one and the one after it, the inf of what’s left is $3-\frac16$.

If you throw away all the terms up to that one and the one after it, the inf of what’s left is $3-\frac18$.

If you throw away all the terms up to that one and the one after it, the inf of what’s left is $3-\frac1{10}$.

. . . and so on. You see that these infs are getting bigger.

If you look at the sequence of infs, their sup is $3$.

Thus the lim inf is the sup of the sequence of infs of all tail-ends of the sequence. In mathematical notation,

Just as the lim inf is a sup of infs, so the lim sup is and inf of sups.

One can also say that $L=\liminf\limits_{n\to\infty} a_n$ precisely if for all $\varepsilon>0$, no matter how small, there exists an index $N$ so large that for all $n\ge N$, $a_n>L-\varepsilon$, and $L$ is the largest number for which this holds.


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Scalar Multiples of Matrices

# Scalar Multiples of Matrices

 Definition: A scalar is a quantity that has a magnitude (size or length) but no direction.

Unless otherwise specified, scalars in the context of Linear Algebra will be a real number $k \in \mathbb{R}$. We are now ready to define scalar multiplication on a matrix.

 Definition: If $A$ is an $m \times n$ matrix and $k \in \mathbb{R}$ a scalar, then the scalar multiple of $A$ by $k$ denoted $kA$ is an $m \times n$ matrix, all of whose entries are multiplied by $k$.

Determining a scalar multiple of a matrix is easy. For example, consider the matrix $A = A = \begin{bmatrix}1 & 2 & 3 & 4\\ 5 & 6 & 7 & 8 \end{bmatrix}$. If we wanted to figure out what the matrix $2A$ is, we would just take every entry in $A$ and multiply it by $2$ to get $2A = \begin{bmatrix}2\cdot1 & 2\cdot2 & 2\cdot3 & 2\cdot4\\ 2\cdot5 & 2\cdot6 & 2\cdot7 & 2\cdot8\end{bmatrix} = \begin{bmatrix}2 & 4 & 6& 8\\ 10 & 12 & 14 & 16\end{bmatrix}$

In general, if we have an $m \times n$ matrix $A = \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n}\\ a_{21} & a_{22} & & a_{2n}\\ \vdots & & \ddots & \vdots\\ a_{m1} & a_{m2} & \cdots & a_{mn} \end{bmatrix}$ and a scalar $k \in \mathbb{R}$ then:

(1)
\begin{align} kA = k \: \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n}\\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots\\ a_{m1} & a_{m2} & \cdots & a_{mn} \end{bmatrix} = \begin{bmatrix} ka_{11} & ka_{12} & \cdots & ka_{1n}\\ ka_{21} & ka_{22} & \cdots & ka_{2n} \\ \vdots & \vdots & \ddots & \vdots\\ ka_{m1} & ka_{m2} & \cdots & ka_{mn} \end{bmatrix} \end{align}

We will now look at some properties of scalar multiples of matrices in the following theorem.

 Theorem 1: Let $A$ and $B$ be $m \times n$ matrices, and let $k, l \in \mathbb{R}$ be scalars. Then: a) $k(A + B) = kA + kB$. b) $k(A - B) = kA - kB$. c) $(k + l)A = kA + lA$. d) $(k - l)A = kA - lA$. e) $k(lA) = (kl) A$.

We will prove (a) and (c) and leave the rest of the proofs to the reader as they follow the same format.

• Proof of (a):
(2)
\begin{align} k(A + B) = k\begin{bmatrix} a_{11} + b_{11} & a_{12} + b_{12} & \cdots & a_{1n} + b_{1n}\\ a_{21} + b_{21} & a_{22} + b_{22} & \cdots & a_{2n} + b_{2n}\\ \vdots & \vdots & \ddots & \vdots\\ a_{m1} + b_{m1} & a_{m2} + b_{m2} & \cdots& a_{mn} + b_{mn} \end{bmatrix} = \begin{bmatrix} ka_{11} + kb_{11} & ka_{12} + kb_{12} & \cdots & ka_{1n} + kb_{1n}\\ ka_{21} + kb_{21} & ka_{22} + kb_{22} & \cdots & ka_{2n} + kb_{2n}\\ \vdots & \vdots & \ddots & \vdots\\ ka_{m1} + kb_{m1} & ka_{m2} + kb_{m2} & \cdots& ka_{mn} + kb_{mn} \end{bmatrix}\\ = k\begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n}\\ a_{21} & a_{22} & \cdots & a_{2n}\\ \vdots & \vdots & \ddots & \vdots\\ a_{m1} & a_{m2} & \cdots& a_{mn} \end{bmatrix} + k\begin{bmatrix} b_{11} & b_{12} & \cdots & b_{1n}\\ b_{21} & b_{22} & & b_{2n}\\ \vdots & & \ddots & \vdots\\ b_{m1} & b_{m2} & \cdots & b_{mn} \end{bmatrix} = kA + kB \\ \blacksquare \end{align}
• Proof of (c):
(3)
\begin{align} \quad (k + l)A = \begin{bmatrix} (k+l)a_{11} & (k+l)a_{12} & \cdots & (k+l)a_{1n}\\ (k+l)a_{21} & (k+l)a_{22} & \cdots & (k+l)a_{2n}\\ \vdots & \vdots & \ddots & \vdots\\ (k+l)a_{m1} & (k+l)a_{m2} & \cdots& (k+l)a_{mn}\end{bmatrix} = \begin{bmatrix} ka_{11} + la_{11} & ka_{12} + lka_{12}& \cdots & ka_{1n} + l\\ ka_{21} + la_{21} & ka_{22} + la_{22} & \cdots & ka_{2n} + la_{2n}\\ \vdots & \vdots & \ddots & \vdots\\ ka_{m1} + la_{m1} & ka_{m2} + la_{m2} & \cdots& ka_{mn} + la_{mn} \end{bmatrix} = kA + lA \\ \blacksquare \end{align}

## Example 1

Given the matrix $A = \begin{bmatrix}2 & 4\\ 1 & 0\\ 0 & 3\\ -3 & 2\\ 1 & 3\end{bmatrix}$, find the matrix $-3A$.

To determine $-3A$, we will multiply every entry in A by -3 to obtain $-3A = \begin{bmatrix} -6 & -12\\ -3 & 0\\ 0 & -9\\ 9 & -6\\ -3 & -9 \end{bmatrix}$.

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# zbMATH — the first resource for mathematics

A note on the discriminant of a space curve. (English) Zbl 0858.32031
Let $$X\to B$$ be a deformation of a germ of an analytic space $$X_0$$ over a smooth base germ. In many situations the discriminant $$\Delta$$ is a hypersurface with remarkable properties, like being a free divisor (meaning that the module of vector fields on $$B$$, logarithmic along $$\Delta$$, is free).
The author formulates a criterion for freeness of the discriminant in terms of relative $$T^1$$s. The author applies his criterion to space curves. The proof comes about from the combination of two facts. For families $$X\to S$$ of curves with smooth general fibre the essential ingredient is that $$\omega^*_{X/S}$$, the dual of the relative dualising sheaf, is flat over $$S$$. On the other hand, this ‘$$\omega^*$$-constant’ property holds quite general for codimension two Cohen-Macaulay germs.

##### MSC:
 32S30 Deformations of complex singularities; vanishing cycles 14H50 Plane and space curves 14B12 Local deformation theory, Artin approximation, etc.
##### Keywords:
free divisor; discriminant; space curves
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##### References:
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# take -bytestring

take n, applied to a list xs, returns the prefix of xs of length n, or xs itself if n > length xs: > take 5 "Hello World!" == "Hello" > take 3 [1,2,3,4,5] == [1,2,3] > take 3 [1,2] == [1,2] > take 3 [] == [] > take (-1) [1,2] == [] > take 0 [1,2] == [] It is an instance of the more general Data.List.genericTake, in which n may be of any integral type.
O(log(min(i,n-i))). The first i elements of a sequence. If i is negative, take i s yields the empty sequence. If the sequence contains fewer than i elements, the whole sequence is returned.
O(n) take n, applied to a Text, returns the prefix of the Text of length n, or the Text itself if n is greater than the length of the Text. Subject to fusion.
O(n) take n, applied to a Text, returns the prefix of the Text of length n, or the Text itself if n is greater than the length of the Text. Subject to fusion.
takeWhile, applied to a predicate p and a list xs, returns the longest prefix (possibly empty) of xs of elements that satisfy p: > takeWhile (< 3) [1,2,3,4,1,2,3,4] == [1,2] > takeWhile (< 9) [1,2,3] == [1,2,3] > takeWhile (< 0) [1,2,3] == []
Return the contents of the MVar. If the MVar is currently empty, takeMVar will wait until it is full. After a takeMVar, the MVar is left empty. There are two further important properties of takeMVar: * takeMVar is single-wakeup. That is, if there are multiple threads blocked in takeMVar, and the MVar becomes full, only one thread will be woken up. The runtime guarantees that the woken thread completes its takeMVar operation. * When multiple threads are blocked on an MVar, they are woken up in FIFO order. This is useful for providing fairness properties of abstractions built using MVars.
Get the base name, without an extension or path. > takeBaseName "file/test.txt" == "test" > takeBaseName "dave.ext" == "dave" > takeBaseName "" == "" > takeBaseName "test" == "test" > takeBaseName (addTrailingPathSeparator x) == "" > takeBaseName "file/file.tar.gz" == "file.tar"
Get the directory name, move up one level. > takeDirectory x isPrefixOf x || takeDirectory x == "." > takeDirectory "foo" == "." > takeDirectory "/foo/bar/baz" == "/foo/bar" > takeDirectory "/foo/bar/baz/" == "/foo/bar/baz" > takeDirectory "foo/bar/baz" == "foo/bar" > Windows: takeDirectory "foo\\bar" == "foo" > Windows: takeDirectory "foo\\bar\\\\" == "foo\\bar" > Windows: takeDirectory "C:\\" == "C:\\"
Get the drive from a filepath. > takeDrive x == fst (splitDrive x)
Get the extension of a file, returns "" for no extension, .ext otherwise. > takeExtension x == snd (splitExtension x) > Valid x => takeExtension (addExtension x "ext") == ".ext" > Valid x => takeExtension (replaceExtension x "ext") == ".ext"
Get all extensions > takeExtensions "file.tar.gz" == ".tar.gz"
Get the file name. > takeFileName "test/" == "" > takeFileName x isSuffixOf x > takeFileName x == snd (splitFileName x) > Valid x => takeFileName (replaceFileName x "fred") == "fred" > Valid x => takeFileName (x </> "fred") == "fred" > Valid x => isRelative (takeFileName x)
Takes the right number of bytes from the input.
Return the contents of the TMVar. If the TMVar is currently empty, the transaction will retry. After a takeTMVar, the TMVar is left empty.
O(n) takeWhile, applied to a predicate p and a Text, returns the longest prefix (possibly empty) of elements that satisfy p. Subject to fusion.
O(i) applied to a predicate p and a sequence xs, returns the longest prefix (possibly empty) of xs of elements that satisfy p.
O(i) applied to a predicate p and a sequence xs, returns the longest suffix (possibly empty) of xs of elements that satisfy p. takeWhileR p xs is equivalent to reverse (takeWhileL p (reverse xs)).
O(1) Return the prefix of the Text of n Word16 units in length. If n would cause the Text to end inside a surrogate pair, the end of the prefix will be advanced by one additional Word16 unit to maintain its validity.
O(1) Unchecked take of k Word16s from the front of a Text.
The genericTake function is an overloaded version of take, which accepts any Integral value as the number of elements to take.
A non-blocking version of takeMVar. The tryTakeMVar function returns immediately, with Nothing if the MVar was empty, or Just a if the MVar was full with contents a. After tryTakeMVar, the MVar is left empty.
A version of takeTMVar that does not retry. The tryTakeTMVar function returns Nothing if the TMVar was empty, or Just a if the TMVar was full with contents a. After tryTakeTMVar, the TMVar is left empty.