How is a symbol “given meaning by a family of operations indexed by symbols”?

Practical Foundation of Programming Languages by Harper says:

Chapter 31 Symbols

A symbol is an atomic datum with no internal structure. Whereas a variable is given meaning by substitution, a symbol is given meaning by a family of operations indexed by symbols. A symbol is just a name, or index, for a family of operations.

Many different interpretations may be given to symbols according to the operations we choose to consider, giving rise to concepts such as fluid binding, dynamic classification, mutable storage, and communication channels.

A type is associated to each symbol whose interpretation depends on the particular application. For example, in the case of mutable storage, the type of a symbol constrains the contents of the cell named by that symbol to values of that type.

What does “a symbol is given meaning by a family of operations indexed by symbols” mean? Is “a symbol” given meaning by a family of operations not one of the “symbols” indexing the family of operations? What is the relation between “a symbol” and “symbols”?

What does “a symbol is just a name, or index, for a family of operations” mean? Does it mean “a symbol names or indexes a family of operations”?

When a symbol is used in each of the following example cases (which I hope you could consider as many as possible, in particular the first three cases):

  • “represent a variable in symbolic representations of equations or programs” (see the quote below),
  • “represent a word in the representation of natural language sentences” (see the quote below),
  • represent an assignable (?) in mutable storage,
  • represent something (something similar to a variable?) in fluid binding,
  • represent a class (?) in dynamic classification,
  • represent something (?) in communication channels,

how does the above quote about a symbol applies, specifically:

  • is the symbol given meaning by what family of operations indexed by symbols?
  • is the symbol just a name, or index, for what family of operations?

Thanks.


The Scheme Programming Language, 4th Edition, by Dybvig, says

Section 2.2. Simple Expressions

Symbols and variables in Scheme are similar to symbols and variables in mathematical expressions and equations. When we evaluate the mathematical expression 1 – x for some value of x, we think of x as a variable. On the other hand, when we consider the algebraic equation x 2 – 1 = (x – 1)(x + 1), we think of x as a symbol (in fact, we think of the whole equation symbolically).

While symbols are commonly used to represent variables in symbolic representations of equations or programs, symbols may also be used, for example, as words in the representation of natural language sentences.

Changed URL for a page that was indexed by Googlebot. Will redirect 301 from the old URL to the new one. But what to do with my Sitemap?

I’m planning to change a url for one of my site’s page.

Example:

From: https://www.example.com/old-post-slug

To: https://www.example.com/new-post-slug

The fact is that Google has already indexed the old url: https://www.example.com/old-post-slug

And from these DOC’s, we see that to avoid lose page ranking we should respond with a 301 - Moved permanently from the old URL pointing to the new URL.

https://support.google.com/webmasters/answer/6033049?hl=en

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QUESTION

I get that I should redirect (301) from the old URL to the new one. So when Google re-crawls, it will see that change. But what should be on my Sitemap? The old URL or the new one? Or both?

I tend to think that it would be best to keep only the new url on my Sitemap. But what if Google crawls the new URL before it sees the redirect from the old one? Wouldn’t the new page URL start off as a new page (from Google’s index perspective) with zero ranking points? How does Googlebot handles that? What is the recommended practice?

How does Indexed work in terms of sparse array

I want to use indexed given that s is an element [0,500] but I am unsure how to write that without getting a format error or a tensor error.

\[CapitalDelta]t = .0001; t = .0833; \[Sigma] = .2183; \[CapitalDelta]s = 5; s = [0, 500]; \[Mu] = ((\[Sigma]^2 Indexed[s, i]^2)/     Indexed[\[CapitalDelta]s, i]^2*\[CapitalDelta]t); \[Alpha] = (Indexed[s, i]/(    2*Indexed[\[CapitalDelta]s, i]^2*\[CapitalDelta]t)); cn1[k2_, n_] =   SparseArray[{{m_, m_} ->      1/2 + 1/2*\[Mu] +       1/2*Indexed[rate, {k2, n}]*\[CapitalDelta]t, {m_, l_} /;       l - m == 1 -> -(1/4)*\[Mu] -       1/2*Indexed[rate, {k2, n}]*\[Alpha], {m_, l_} /;       m - l == 1 -> -(1/4)*\[Mu] +       1/2*Indexed[rate, {k2, n}]*\[Alpha]}, {101, 101}] 

Show that,with the array representation for sorting an n-element heap, the leaves are the nodes indexed by n⌊n/2⌋+1,⌊n/2⌋+2,…,n

The Question of the CLRS $ 6.1-7$ exercise reads as:

Show that, with the array representation for sorting an n-element heap, the leaves are the nodes indexed by $ \lfloor n / 2 \rfloor + 1, \lfloor n / 2 \rfloor + 2, \ldots, n⌊n/2⌋+1,⌊n/2⌋+2,…,n$ .

I looked for the solution here: https://walkccc.github.io/CLRS/Chap06/6.1/

The solution was provided like this:

Let’s take the left child of the node indexed by $ \lfloor n / 2 \rfloor + 1.$

\begin{aligned} \text{LEFT}(\lfloor n / 2 \rfloor + 1) & = 2(\lfloor n / 2 \rfloor + 1) \ & > 2(n / 2 – 1) + 2 \ & = n – 2 + 2 \ & = n. \end{aligned}

I can’t understand this statement: $ LEFT(⌊𝑛/2⌋+1) > 2(𝑛/2−1)+2$

Please help me out. Thank you.