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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. ...


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++) ...


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 $$a_1,a_2,\ldots, a_p.$$


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 ...


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 ...


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 ...


2

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


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 ...


1

The geometric language used in this case is a complete red herring, just having to do with our notation. The point is that in the various contexts you're referring to, we're dealing with binary operations, and so they provide two ways to combing any two items - either as $ab$ or $ba$. Therefore when speaking of combining $a$ with $b$, we have to specify ...


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

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


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 could do the following: "... we have $f(A)$, $f(B)$, and $f(C)$. Also we have $\kappa_A$, $\kappa_B$, and $\kappa_C$ where $f(X)$ and $\kappa_X$ are ... for all $X \in \left\{ A,B,C \right\}$ ..."


1

You can find many explanations of the IEEE-754 format on the Web. In short, each 32-bit single-precision floating-point number consists of three parts (we assume that the bits numbering begins from zero): sign, bit 31 exponent, bits 30-23 (eight bits in total) significand (or "mantissa"), bits 22-0 (twenty three bits in total) However, in your case the ...


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...


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

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

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

The bottom and top symbols $\bot,\,\top$ respectively denote contradictions and tautologies in model theory. For example, a proof by contradiction that $\sqrt{2}\notin\mathbb{Q}$ can be rewritten as a proof that $\sqrt{2}\in\mathbb{Q}\to\bot$.


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

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


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: ...


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 ...



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