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17

We usually denote $n$-tuples in the form $(a_1,\ldots,a_n)$, so for example $(x,y,z)$ is a triple, $(x,y)$ is a pair, somewhat redundantly $(x)$ could be called a one-tuple and $()$ a (in fact: the) zero-tuple.


13

Generally, no. But you could say: Let us denote something with (). ... and start using (), if you think this would convey your point. There is no need for academic reference validating this, it is the matter of basic author freedom. Let's say you want to wear a violet tie with blue dots. You don't ask if there is a law permitting you to do this. ...


11

We use parentheses to indicate the order of operations. To refer to your example: the operator $+$ takes two arguments, in the form of $a+b$. You can think of $+$ as a function that takes two variables. In this example, the $+$ is missing the second argument: there's nothing there. And $15 + $ isn't a valid mathematical statement -- it's not equal to ...


5

In an elementary context, $\Bbb W$ means the set of whole numbers. Some books have it as $$\Bbb W=\{0,1,2,\ldots\}$$ while others have it as $$\Bbb W=\{1,2,\ldots\}$$ Because of the ambiguity, I recommend that you avoid the use of $\Bbb W$. For the second meaning use $\Bbb Z^+$. There still is no perfect abbreviation for the first. Either meaning is also ...


4

It's certainly not conventional, and it's hard to think of any occasion when one might want to use such notation explicitly. That said, it would not be illogical to define () as zero, just as the empty set is sometimes written $\{\}$. As a precedent, an empty sum, such as $\sum_{k=1}^0x_k$, is defined to be zero by a standard convention.


4

I'd argue that using $2k-1$ is the elegant way to do it: $$\large\sum_{i=1}^\infty\sum_{k=1}^{\lceil i/2\rceil}(\text{expression using $2k-1$})$$ What you wrote in your post is one alternative, though it can be written a bit more cleanly as: $$\large\sum_{i=1}^\infty\sum_{\substack{k=1\\k\text{ odd}}}^i(\text{expression using $k$})$$


3

Anything is valid if you define it. Mathematicians tend to not only use different notation for the same thing, but often the same notation to denote different things. That's all right as long as they define up front what a given notation is supposed to mean. In many cases, the meaning is so commonplace that all mathematicians agree upon it. You'd find no ...


3

Just to provide a more or less authoritative reference as to what $\mathbb{W}$ denotes, the following is from page 2 of the book A Transition to Abstract Mathematics by Randall Maddox: As you can see, $\mathbb{W}$ denotes the set of whole numbers, but this notation is often avoided in favor of $\mathbb{N}$, and even $\mathbb{N}$ itself is often clarified ...


3

The parentheses are not necessary. If you insist on using them, they should not enclose the differential in this case of a single argument. For your typographical problem, like Michael Hardy already commented, often the $\exp$ function is used $$ \exp\left(\int f(x) dx\right) $$ instead of the $e^x$ notation. In physics some authors prefer a more ...


2

Is a function that takes an element from V and gives you an element from W.


2

You are just lacking a way to express the number of elements in the sequence, it seems. You can use $\#A$ or $|A|$ or state that the sequence has length $N$. Then append the suffix $^{th}$. The Python programming language uses the following indexing convention: elements are numbered $0$ to $N-1$, but negative indexes can be used. The element $A_{-1}$ is the ...


2

Note that $$\sum_{j=0}^{n}j=\sum_{j=1}^{n}j=\frac{n(n+1)}{2}.$$ Now setting $n=i^2-1$ gives you $$\sum_{j=0}^{i^2-1}j=\frac{(i^2-1)((i^2-1)+1)}{2}=\frac{(i^2-1)i^2}{2}=\frac{i^4-i^2}{2}.$$


2

You could make one with a 4x4 matrix. Call it S or something. \begin{array}{l} \left( {\begin{array}{*{20}{c}} {{S_{00}}}&{{S_{01}}}&{{S_{02}}}&{{S_{03}}}\\ {{S_{10}}}&{{S_{11}}}&{{S_{12}}}&{{S_{13}}}\\ {{S_{20}}}&{{S_{21}}}&{{S_{22}}}&{{S_{23}}}\\ {{S_{30}}}&{{S_{31}}}&{{S_{32}}}&{{S_{33}}} \end{array}} ...


1

Your formula being $$T(n)=\sum_{i=0}^{n-1}\sum_{j=0}^{i^2-1}\sum_{k=0}^{j-1}c$$ Let us start with the most inner loop $$\sum_{k=0}^{j-1}c=cj$$ So, $$T(n)=c\sum_{i=0}^{n-1}\sum_{j=0}^{i^2-1}j$$ For the remaining inner loop,as said (Faulhaber formula) $$\sum_{j=0}^m j=\frac{1}{2} m (m+1)$$ which makes (replacing $m$ by $i^2-1$) $$\sum_{j=0}^n ...


1

In $\int f(x)\,\mathrm dx$ you may view $f(x)\,\mathrm dx$ as if it is a multiplication. Therefore parentheses are required (not including the $\mathrm dx$) only if you integrate a sum, as in $$\int(x^2+7x)\,\mathrm dx $$ In all other cases, I think parentheses do not add clarity. If $\int$ and $f$ look too similar that may be a problem with non-printing ...


1

Some common symbols used to express the notion of "given" or "such that" are the colon ":" and the vertical bar "|". I guess your statement could then be rephrased as: $x \in \{ f(n),\ n \in \mathbb{N}\ |\ f(n) = f(n-1) + f(n-2),\ f(0) = 0,\ f(1) = 1 \} $


1

$$\forall n\Big((n\in\Bbb N\wedge n\ge 2)\to f(n)=f(n-1)+f(n-2)\Big) \wedge f(0)=0 \wedge f(1)=1$$


1

Note: It's perfectly valid to treat finite sums in the same way as for-loops in programs. You can multiple sums reshuffle in the same way as you may reorganise nested for-loops. Two hints: You mention that certain elements are equal. This is not obvious. If an element is $$M_{\alpha \beta}(\Delta x ^ \alpha)(\Delta x ^ \beta),\qquad 0\leq ...


1

The natural numbers, $\mathbb N$, are sometimes called the whole numbers, $\mathbb W$. It's ambiguous, because $-1$ is also a whole number, since it has no fractional parts.


1

Cartesian product, $$ (\mathbb{Z}^{+}) \times (\mathbb{Z}^{+}) \times (\mathbb{Z}^{+}) \times (\mathbb{Z}^{+}), $$ normally abbreviated as $(\mathbb{Z}^{+})^4$.


1

The notation $(\mathbb Z^+)^4$ is just the set of tuples of $4$ positive integers. So, it says that $(a,b,c,n)$ is a tuple of $4$ positive integers. It would be equally clear to write $a\in\mathbb Z^+,b\in\mathbb Z^+,c\in\mathbb Z^+,n\in\mathbb Z^+$ or even to write $a,b,c,n\in\mathbb Z^+$ - all of which are equivalent. The author is likely looking to ...


1

The set of all $4$-tuples of positive integers. A single element is a $1$-tuple, an ordered pair is a $2$-tuple, and an ordered foursome is a four-tuple. So it's the set: $$\left({\mathbb Z^+}\right)^4 = \left \{{(a,b,c,d): a, b, c, d \in \mathbb Z^+}\right\}$$ Where $(a,b,c,d) = (e,f,g,h)$ if and only if $a = e, b = f, c = g, d = h$. You can also think ...


1

Notation is based on common acknowledgement. Sure you can invent any random thing, for example you can even draw a monkey instead of parenthesis. But this makes people confused about the notation. If here you mean a variable, usually people put a letter here. When the letters are not used up (or not for a specific purpose, eg. Greek letter for angle, an ...


1

That's probably an Iverson bracket.


1

Maybe not as rigorous as the previous comments, but probably a bit nicer in notation: You could define an ensemble of functions $\delta_\tau(t):= \delta(t-\tau)$ and then write the convolution as $h*\delta_\tau$.



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