New answers tagged

1

In this context, the $[\mathbf{A}|\mathbf{B}]$ notation signifies an "augmented matrix" associated with a system of linear equations. Your textbook probably has a definition of "augmented matrix". Try looking up this term in the index. The concept is simple enough: $[\mathbf{A}|\mathbf{B}]$ is just the matrix $\mathbf{A}$, but with the vector $\mathbf{B}$ ...


1

Let AX=B be a system of linear equations. If [C|D] is row equivalent to [A|B], then the system CX=D is equivalent to AX=B. $[A\mid B]$ is just a $1\times 2$ matrix consisting of two matrix blocks $A$ and $B$ as first and second column entry. So $\mid$ acts as layout separator not as division operator. You could express the same as .. If $C$ ...


1

Lipschitz functions are a special case of Holder continuous functions. The class of functions that are Holder continuous with coefficient $\alpha$ is a Banach space and is commonly denoted $C^{k,\alpha}$. Lipschitz functions are the special case with $\alpha = 1$, so you will often see them denoted $C^{k,1}$. I do not recall seeing some other specific ...


2

To expand on Tomi's answer: when considering e.g. a general non-commutative binary operation, the concepts of left- and right-composition arise naturally from the asymmetry of the mathematical structure. The names we give to those concepts are dependent on language, writing conventions, accidents of history and a mathematical culture where new things are ...


1

There are "top actions" and "bottom actions" in mathematics. Consider the following: $$\sum_{k=1}^{\infty}a_n$$ $$\lim_{h \to 0}\frac{f(x+h)-f(x)}{h}$$ There are many more examples. You can see many on this page.


1

You can find many explanations of the IEEE-754 format on the Web. In short, each 32-bit single-precision floating number consists of three parts: sign, bit 31 exponent, bits 30-23 (eight bits in total) significand, bits 22-0 (twenty three bits in total) In your case sign = 0, which means that the number is non-negative. The exponent is zero, which is a ...


1

As you say, these are entirely derived from our languages and our writing methods.


2

Writing $$f(x + yi) = f(x,y) = u(x,y) + v(x,y)i$$ then $$\frac{\partial f}{\partial x} = \frac{\partial u}{\partial x} + \frac{\partial v}{\partial x} i $$


0

Yes, there is at least one. Murray Bourne, owner of the site Interactive Mathematics, proposes writing boxed arguments: I wish to propose an alternative notation for concepts where you cannot expand in the way you do with simple algebra. It might look something like this: $\sin\boxed{x + y}$ $\log\boxed{x + y}$ $f\boxed{x}$ This would ...


4

I strongly agree with Karl's and Björn's comments regarding the Latin orgin: spatium. See: Leonhard Euler, Mechanica sive motus scientia analytice exposita, Tomus I, Petropoli, 1736 : Propositio 4 [ page 13 ] Sit spatium $AM$, sive sit linea recta sive curva, $=s$, et celeritas, quam corpus habet in M sit $c$, quae erit functio quaedam ipsius $s$. Ab ...


0

Well, I think that the answer seems to be No, that there is no such symbol in standard use in any context. More's the pity! Perhaps we can invent one.


1

Writing $\operatorname{Pr}[X \leq x]$ is an abuse of notation - nothing wrong with that, a good abuse of notation makes things much more concise and easy to read. It is a shorthand for $\operatorname{Pr}[\{ \omega \in \Omega : X(\omega) \leq x\}]$, if $X$ is a random variable with domain $\Omega$. And then $\operatorname{Pr}[X \leq x \land Y \leq y]$ would ...


1

To be rigorous, the application $Pr$ is evaluated on sets, rather than logic propositions, so the first would be more correct. Besides, I've never seen the use of $\wedge $ in these cases. the use of "$,$" is frequent instead. On the other hand this wouldn't be that bad...


4

This notion of "replacing" an element in a set $S$, while maintaining the same name $S$ for the result, is an example of what assignment statements do in imperative programming languages. You're thinking of $S$ as a variable in such a language, where at any point in time $S$ has a state, and operations mutate that state. Variables in mathematics typically ...


3

There's no widely adopted notation for this. This is not a common math operation. Technically if you "replace" a member of a set with some new member, you just get another/new set (and not the original set modified). What you're asking seems to me closer to computer science than to math. Of course you can introduce your own notation and use it, as ...


1

In my experience this might denote a sequence of numbers $x_{1}$, $x_{2}$, ..., $x_{n}$


0

[Converted from comment to answer] This looks fine to me. You could even shorten it to $$\Omega = \{x\in X \mid \operatorname{cond}(x) =\textrm{TRUE}\}$$ Addendum: Since the condition is evidently truth-valued, you could even write just $$\Omega = \{x\in X \mid \operatorname{cond}(x)\}$$


10

Number[] a = new Number[p + 1]; // range from 1 to p ... some stuff to initialize a ... Number[] ceps = new Number[Q + 1]; // range from 0 to Q ceps[0] = ln(G); for (int q = 1; q <= p; q++) { Number sum = a[q]; for (k = 1; k <= q - 1; k++) { sum += (k - q) / q * a[k] * ceps[q - k]; } ceps[q] = sum; } for (int q = p + 1; q <= Q; q++) ...


1

All the definitions given seem to agree that $0$ is always in the diagonal intersection. The sets $\{\xi\in\kappa: \xi\in\bigcap_{\alpha<\xi}X_\alpha\}$ and $\bigcap_{\alpha\in\kappa}(X_\alpha\cup\{\xi: \xi\le\alpha\})$ are the same set; in particular, $0$ is in the latter, since for every $\alpha\in\kappa$ we have $\alpha\ge0$, so $0\in\{\xi: ...


13

The $Q$ is a parameter, and $q$ is a variable ranging from $0$ to $Q$: basically, you have $Q+1$ parameters $\textrm{ceps}_0,\dots, \textrm{ceps}_Q$; or, in programming terms, you have an array $\textrm{ceps}[0\dots Q]$. Similarly, the LPC coefficients are a list of $p$ values $a_1,\dots, a_p$ (i.e., $a_q$ for $q=1\dots p$), where $p$ is another parameter. ...


7

$\{ceps_q\}_{q=0}^Q$ is the finite sequence (or array or vector in programmese) $$ceps_0,ceps_1,\ldots, ceps_Q.$$ Likewise, $\{a_q\}_{q=1}^p$ denotes $$a_1,a_2,\ldots, a_p.$$


1

I think you're right, $${x^1}^2+\dots+{x^n}^2$$ is hard to understand (although not really ambiguous once you've looked at it for long enough) but $${(x^1)}^2+\dots+{(x^n)}^2$$ is totally clear. (You could also write $x^1x^1+\dots+x^nx^n$ or $\sum_i x^ix^i$ or $\sum_i(x^i)^2$.)


1

I don't think there is any standard tensor notation convention for expressions like yours. Both your "tensor-like" proposal and the version suggested by weux082690 have their merits. In my opinion, which version to prefer depends on how often you are going to use expressions of this kind in your paper (or other document), and so how much effort you are ...


1

The notation $f|_A$ is probably best understood via a meaningful example. Before giving one (I hope it will be useful, anyway), it would probably be good to consult two decent references: 1) The Wikipedia page on the restriction of a function. 2) Abstract Algebra by Dummit and Foote (p. 3, 3rd Ed.). The relevant portion from the Wiki blurb: Let ...


0

$\langle x,y \rangle$ is a real number, so $\lvert \cdot \rvert$ is used, being the ordinary absolute value on $\mathbb{R}$. Contrast with $x,y$, which are vectors. Regarding your second question, $\langle \cdot , \cdot \rangle$ is bilinear (or sesquilinear), so it is certainly not always positive! $\langle x , -y \rangle = -\langle x,y\rangle $, and both ...


2

That's correct. Suppose we have a function $$f : Y \leftarrow X,$$ and a subset $A$ of $X$. Approach 0. Then $f \restriction_A$ is defined as the unique function $Y \leftarrow A$ that agrees with $f$ on $A$. That is: $$\mathop{\forall}_{a \in A} ((f \restriction_A)(a) = f(a))$$ However, there's a cleaner way of formalizing this. Approach 1. Write ...


3

Intuitively speaking, a function $f$ is constituted of three ingredients: a domain; a codomain; a rule (that, for each element in the domain, assigns a unique element in the codomain). If we change any of these three ingredients, we obtain a different function. In particular, if we change the domain by a subset $A$ of the original domain (keeping the ...


2

It means that I am constricting the domain of the function $f$. If $f:X\to Y$, then $g=f|_A$ means that $g:A\to Y$ where $A\subseteq X$.


1

It's true that . However, you might note that the same can be said of your ordinary integral on a line: one can easily see if $f$ is a function $[a,b]\rightarrow \mathbb R$, then it's already clear what $$\int_a^bf$$ means, and the $dx$ is just going to be tagging along. However, the advantage of writing $$\int_a^b f(x)\,dx$$ becomes more clear when one has ...


2

The designation is important. For example, we can write the scalar $I_1$ $$I_1=\int_S (\hat n\cdot \vec F) \,dS$$ the vector $I_2$ $$I_2=\int_S (\hat n\times \vec F) \,dS$$ and the dyadic (tensor, rank 2) $I_3$ $$I_3=\int_S(\hat n\,\vec F) \,dS$$ The notation for $I$, $$I=\int_S \vec F\,$$ is ambiguous without explicit designation. $$$$


0

Well, it is important to specify which are your infinitesimal variables, as you could integrate your line integral with respect to $x$, $y$, or both. For example, when you use Green's theorem: $$ \oint_C P dx +Qdy =\iint (P,Q) d\vec{S}, $$ it is important to integrate $P$ over $x$ only, and $Q$ over $y$ only. The same thing applies for surface integrals.


1

Both are correct since we assume $\{n_k\}$ is an increasing sequence of natural numbers when defining $\{x_{n_k}\}$.


2

$f(x,y)$ is just any function of the two variables; nothing else is implied. $f(x|y)$ is usually used in probability, and is used in the context of "the probability that the first variable will be of some specified value $x$, given that the second variable was $y$. An example of this may make that clearer: Say Charlie rolls 5 dice, three red and 2 blue. ...


0

Usually $f(x|\theta)$ if $x$ is r.v. and $\theta$ is another random variable (that has a distribution), with known value. $f(x,\theta)$ is the joint distribution of the two random variables $x$ and $\theta$. $$ f(x,\theta) = f(x|\theta) f(\theta)$$ Finally, $f(x;\theta)$ is when $x$ is random variable and $\theta $ is a parameter of a pdf, you can also view ...


1

In the heat equation $u(t,x)$ is a function of the two variable $t$ and $x$ that represents the temperature at time $t$ in the point $x$. So $u(0,x)=f(x)$ is the function that represents the initial temperature at point $x$, and it is a function of $x$ only. Your notation simply indicates this $f(x)$ with the symbol $u_0(x)$.


0

It's somehow correct, but it is misleading. The correct way to understand this notation is $$\mathbb Z[\sqrt{-5}]=\{a+b\mathrm i\sqrt5\,\mid a,b\in\mathbb Z\}$$


3

You are correct, but it it suffices to see that $\mathbb Z [\sqrt{-5}] = \{a_0 + a_1\sqrt{-5}: a_i \in \mathbb Z \}$


1

This is a rather old paper, so it is not a surprise to find some outdated notation. AFAIK, this $\mathbf{E}$ is not in frequent use anymore, may mean something like ensemble (French for “set”). Indeed, we have the equality $$ \underset{u}{\mathbf{E}} \, [u \in At^m \text{ and } u \leq x^{(i)}] = \{u \mid u \in At^m \text{ and } u \leq x^{(i)}\}, $$ but you ...


6

Cantor had Jewish roots, which is probably why he was familiar with the Hebrew alphabet. But it's unlikely to be the reason de jure or de facto for the choice. From Georg Cantor: His Mathematics and Philosophy of the Infinite By Joseph Warren Dauben: Not wishing to invent a new symbol himself, he chose the aleph, the first letter of the Hebrew alphabet. ...


2

According to the book "Set theory, logic and their limitations" by Moshé Machover, aleph is the first letter of the word "einsoph", which is the Hebrew word for infinity, and is also used in cabbalistic traditions as a word for God. Given Cantor's interest in the connection between the infinite and the divine, this seems like the reason for his choice of ...


3

Note that for your particular example, you could write "all of the integers in $[k,n]$ are pompous" or "all of the integers $i$ with $k \leq i \leq n$ are pompous" if you want to avoid the ellipsis. In general, though, ellipses like these are fine. An argument that is too informal and misses important details is problematic, but so is one that is ...


3

It depends on your audience. If your readers are mathematically quite immatrue, such a phrase might cause trouble. But in general, there's only one possible interpretation, so why bother spelling out the other cases?


0

First, let me say that your teacher might refuse teching you this notation because it is considered bad style in written mathematics. The symbols from formal logic (like $\forall$) should be used nearly exclusively when talking about formulas in formal logic and maybe (carefully!) as a shorthand on the blackboard. Full English sentences are just easier to ...


1

$F(x,y,t)$ is a function of three variables. So you can compute $F_x,F_y$ and $F_t$ (assuming they exist). If $x=x(t)$ and $y=y(t)$ then $F(x(t),y(t),t)$ is a function of one variable: $t.$ You can compute its derivative $\frac{d}{dt}F$ using that chain rule. As you have said $$\dfrac{d}{dt}T=F_xx'+F_yy'+F_t.$$ So, if $x$ and $y$ are constant (don't depend ...


1

Wikipedia calls this $P_p$, where the capital $P$ stands for polynomial and the lower case stands for the maximum degree. I have seen this in some textbooks as well, but it's not universal. This object is somewhat unusual; it is a vector space, but not a ring. It seems strange to have polynomials and not multiply them. It's basically $\mathbb{R}^{p+1}$ ...


3

The usual notation that I have seen extensively used is $\mathbb{R}_p[X]$.


0

From my Course: Probabilistic Methods in Finance, we denoted $R+, R++$ like so: $$R+ =\{x\in R : x\ge0\}$$ $$R++ =\{x\in R : x≫0\}.$$


0

I've just read an article which at least partly answers my questions: https://nickhigham.wordpress.com/2016/01/28/typesetting-mathematics-according-to-the-iso-standard/.


0

I see sometimes $[\![1,n]\!]$ instead of $\{1,\dots,n\}.$


0

In my document, I now use the following notations and conventions. I use $[1,w]$ for real numbered intervals and $\{1,2,...,w\}$ for integer intervals. If I reuse a specific interval several times, I introduce and name it once, e.g., as $\mathbb{N}_w=\{1,2,...,w\}$, and then refer to it using its name, e.g., $\mathbb{N}_w$. Thereby, throughout the ...



Top 50 recent answers are included