## In general – crafting requires a formula, but not a formula book

Let’s start more generally than the alchemist. Crafting an item requires that you have the item’s formula (Core Rulebook, pg. 244):

Craft: … To Craft an item, you must meet the following requirements: …

You have the formula for the item;

You don’t have to have it in a formula book or any particular form. You merely need the formula in some fashion. That being said, formulas are typically in a written form (Core Rulebook, pg. 293):

A purchased formula is typically a schematic on rolled-up parchment of light Bulk.

The description of the item formula book on page 290 clarifies that no formula book is required to craft something. It merely holds formulas:

… there’s no need for you to copy (formulas) into a specific book as long as you can keep them on hand to reference them.

This is the core answer to your question: in general, crafting requires that the crafter have an item’s formula. It need not be in a formula book, but it must be “on hand to reference”.

## Some alchemist features require a formula book.

There are two reasons the alchemist is worth mentioning specifically: it has a class feature called formula book, and it has other class features which reference formulas.

The formula book class feature changes nothing about how to craft items. It only provides a bunch of formulas at level 1 and more as you level up.

The Advanced Alchemy class feature allows alchemists to create items as part of the morning preparations. Because of your comparison to wizards, I believe this is the feature you are referring to. Advanced Alchemy does not require a Craft roll or Crafting task, and therefore skips the previously mentioned requirements for crafting. However, it explicitly requires that a formula be in your formula book, not merely available (pg. 72):

… choose an alchemical item of your advanced alchemy level or lower that’s in your formula book, and make a batch of two of that item.

There is no mention of having to consult the formula book while creating these items, but the formula must be in there.

Quick Alchemy is different, in that it’s requirements explicitly mention that you must have the formula for the alchemical item available. It’s not enough to have the formula written in your book, you must have it present.

## algebra precalculus – Shrinkage as a percentage, \$100%\$ or \$%\$ in this formula?

I have a question regarding calculating percentage in the following:

This shows the shrinkage as a percentage:
$$text{Saving percentage} = frac{text{SizeBeforeCompresseion} – text{SizeAfterCompression}}{text{SizeBeforeCompresseion}} % tag 1$$
Example:

$$text{SizeBeforeCompresseion } =text{65 536 bytes}$$ and $$text{SizeAfterCompression} = text{16 384 bytes}$$.

The saving percentage is
$$text{Saving percentage} = frac{text{65 536} – text{16 384}}{text{65 536}} % = 0,75 %$$

Question:

$$0,75 %$$ isn’t correct (I guess?), we actually have to multiply with $$100$$ to find the right percentage, i.e. $$0,75cdot 100 %=75 %$$. So isn’t it more correct to write $$(1)$$ as
$$text{Saving percentage} = frac{text{SizeBeforeCompresseion} – text{SizeAfterCompression}}{text{SizeBeforeCompresseion}} cdot 100 % tag 2$$
?

## google sheets – How to use an expression or formula on values from IMPORTRANGE

I am trying to use an expression on the values that are imported from another spreadsheet. The IMPORTRANGE copies the raw data, and with QUERY I can then specify which columns to use, so I can skip some of them.

However, I want to process the imported values by some expression. The QUERY doesn’t seem to support different string functions like LEN, etc. But if I use an expression over the QUERY, like “=LEN(QUERY(…”, then I get only one record as a result, not a list of rows, as I would from IMPORTRANGE and QUERY.

Specifically, I have a source spreadsheet, that is filled automatically by one of my Google Forms, and there is a column with a list of personal names. This source spreadsheet is restricted to only a few users. But I have create another spreadsheet, for the public use. In this copy, I want to display all the non-sensitive columns, but for the personal names column, I want to display only a number of the names, which are comma-separated. The expression to get the number of the names is otherwise trivial:

`=IF(LEN(TRIM(A1)) > 0; COUNTA(SPLIT(A1; ",")); "")`

I could use the expression in the source spreadsheet, but it’s overwritten by Google Forms. So I need to use the expression in the copied spreadsheet.

What are the possible solutions? I haven’t found any questions similar to mine.

## evaluation – Formula: setting parameters vs replacement, different results?

I have a question that may sound stupid and have an easy answer – but I currently do not see it.
I have defined a function of four variables, $$f[a,b,c,d]=…$$, in what I think is correct Mathematica syntax. I need to evaluate this function at specific values $$a*,b*,c*,d*$$ for $$a, b, c, d$$. When I do so, I obtain $$Indeterminate$$ as a result. However, when I evaluate $$f$$ at the specific values for $$a, c, d$$, and only replace $$b$$ for its value afterwards, $$f[a*,b,c*,d*]’ /. {b rightarrow b*}$$, I do obtain a reasonable result. Any explanation for this?

PS: I can share the snippet, that shouldn’t be a problem. However, perhaps there is a general explanation and hence no need for it…? I have seen Setting the parameters in a defined function vs replacement rule in the formula, which basically asks the same question, but I don’t think the answer is satisfactory. In any case, thanks in advance.

## How to turn this recurrence formula into a convergent series?

D2n=(2-2(1-(Dn)^2 / 4)^1/2)^1/2

Pi=lim n¬∞ 2n*D2n

Is there any direct way to turn this into a series, one of the series that converges to pi.

Without any geometrical or coordinate considerations.

## pr.probability – Asymptotic inversion formula in additive number theory (sums of two sets of positive integers)

Let $$S$$ be an infinite set of positive integers, and $$T=S+S={x+y, mbox{ with } x,yin S}$$.We definte the following functions:

• $$N_S(z)$$ is asymptotic continuous version of the function counting the number of elements in $$S$$ less or equal to $$z$$.
• $$N’_S(z)$$, the derivative of $$N_S(z)$$, is the “probability” for $$z$$
(an integer) to belong to $$S$$
• $$r(z)$$ is the asymptotic continuous version of the function counting
the number of solutions $$x+y leq z$$ with $$x,yin S$$.
• $$r'(z)$$ is the derivative of $$r(z)$$.

We will work with
$$N_S(z) sim frac{az^b}{(log z)^c}.$$

Here $$frac{1}{2}< b leq 1$$ and $$a>0, cgeq 0$$. The case $$b=1, c=0$$ should be excluded. This covers a vast array of sets: sums of primes, sums of super-primes etc. The following is a known result (see here):

$$r(z) sim frac{a^2 z^{2b}}{(log z)^{2c}}cdot frac{Gamma^2(b+1)}{Gamma(2b+1)}$$
$$r'(z) sim frac{a^2 z^{2b-1}}{(log z)^{2c}}cdot frac{Gamma^2(b+1)}{Gamma(2b)}$$

More generally (see here):

$$r(z) sim zint_0^{1} N_S(z(1-v))N’_S(zv) dv.$$
$$r'(z) sim zint_0^{1} N’_S(z(1-v))N’_S(zv) dv .$$

Since $$b>frac{1}{2}$$, we have $$r'(z) rightarrow infty$$ as $$zrightarrow infty$$. This guarantees (it’s a conjecture) that barring congruence restrictions, $$T = S + S$$ contains all the positive integers except a finite number of them. The inversion formula is as follows:

Inversion formula

$$N_T(z) = z-w(z), mbox{ with } w(z) sim int_0^z expBig(-frac{1}{2} r'(u)Big)du.$$

Since $$r’$$ is a function of $$N’$$ and thus, a function of $$N$$, we have a formula linking $$N_T$$ to $$N_S$$. So if you know $$N_T$$, by inversion (it involves solving an integral equation, though we are only interested in the asymptotic value of the solution) technically, you can retrieve $$N_S$$, assuming the solution is unique (chances are that the solution is far from unique.)

Note that $$w(z)$$ represents the number of positive integers, less or equal to $$z$$, that do not belong to $$T=S+S$$. These integers are called exceptions. We also have $$w(infty) = C_S$$: this constant depends on $$S$$, but it is finite and it represents an estimate of the total number of exceptions. I tried to assess the validity of the inversion formula using some test sets $$S$$, and empirical evidence suggests that it is correct. Essentially, it is based on the following simple probabilistic argument. Let $$u(z)$$ be the probability that $$z$$ (an integer) is an exception. Then, if $$r'(z)rightarrowinfty$$ as $$zrightarrowinfty$$ and $$S$$ is free of congruence restrictions and other sources of non-randomness, then
$$u(z) sim expBig(-frac{1}{2}r'(z)Big).$$

Testing the formula on an example

I created 100 test sets $$S$$, with $$a=1, b=frac{2}{3}, c=0$$, as follows: an integer $$k$$ belongs to $$S$$ if and only if $$U_k, where the $$U_k$$‘s are independent uniform deviates on $$(0, 1)$$. I computed various statistics, but I will mention only one here. The theoretical value for $$w(infty)$$ is
$$w(infty) approx int_0^infty expBig(-frac{lambda}{2}u^{1/3}Big)du approx 63.76, mbox{ with } lambda = frac{Gamma^2(frac{5}{3})}{Gamma(frac{4}{3})}.$$

Note that the above integral can be computed explicitly. I then conjectured the value $$w(infty)$$ for each of the 100 test sets. It ranged from $$13$$ to $$199$$, with an average value of $$65.88$$. Again, $$w(infty)$$ is an estimate of the number of exceptions, that is, positive integers that can not be represented as $$x+y$$ with $$x, y in S$$. So the approximate theoretical value is in agreement with the average value inferred from my experiment.

My question

Is this inversion formula well known? Can it be of any practical use? Can it be further refined, maybe generalized to sums of three sets or made more accurate with bounds on the error term?

## sharepoint designer – Creating a workflow which automatically runs a formula to assign customized number

I have created a list which uses a calculated column to assign a customized change control number. However, whenever a new item is created/saved, the formula is not applied.

It is only when I access through list settings and open the calc column and then save, does the desired outcome display.

I am new to workflows and need help getting started.

## complexity theory – Configurations and CNF formula for neighboring configuration

A configuration of a Turing machine $$M$$ which runs in space $$S(n)$$ contains the state, the head positions, and the content of non-blank cells of all the tapes. For $$M$$ and an input $$x$$, we define its configuration graph $$G_{M,x}$$ as a directed graph whose nodes correspond to all the possible configurations and there is an edge from a configuration $$C$$ to $$C’$$ if $$C’$$ can be reached from $$C$$ in one step.

First question:
In Arora-Barak (snapshot below), they say that these nodes can be encoded in a binary string of length $$O(S(n))$$. Such encoding contains the state, all symbols under the head, and non-blank content of work tapes with special marking to denote the location of the head. I think this is not correct since such an encoding doesn’t contain the input head position. If we don’t store the input head position, which requires O(log n) bits, then two nodes can map to the same encoding, which seems wrong. Am I right? Although storing the input head position will not increase the length of the encoding since we are assuming in the book that $$S(n) > log n$$.

Second question: Next Arora-Barak says that we can construct a CNF formula $$phi_{M,x}$$ such that for any two strings $$C$$ and $$C’$$, $$phi_{M,x}(C,C’) = 1$$ iff $$C$$ and $$C’$$ encode two neighboring configuration in $$G_{M,x}$$. I am not able to figure out the proof of this claim with the kind of encoding that I have described above i.e. encoding in which we also store the index of input head. Suppose a configuration C has $$q_1, q_2, dots, q_c$$ bits for state, $$h_1,h_2, dots, h_{O(log n)}$$ bits for input head positions, $$w_1, w_2, dots, w_{S(n)}$$ bits for work tape content and $$wh_1, wh_2, dots, wh_{S(n)}$$ bits for work tape head position, where $$wh_i$$ is equal to 1 if the head is on the $$i$$th cell, else it is equal to 0. Then how with such an encoding we can construct a CNF formula as described at the beginning of this question?

## Asymptotic formula for number of topologies on the finite point-set \${1,2,dots,n}\$

Let $$n$$ be a finite positive integer.

• Is there an asymptotic formula for (or good lower / upper bounds) for the number of topologies on the point-set $${1,2,ldots,n}$$ ?
• What if we restrict to only counting topologies which are "truely distinct" ?

## real analysis – Generalization of the Jacobi theta transformation formula

Let $$thetaleft(zright):=sum_{ngeq1}e^{-n^{2}z},,mathrm{Re}left(zright)>0.$$

It is well known that $$thetaleft(zright)=left(frac{pi}{z}right)^{1/2}thetaleft(frac{pi^{2}}{z}right).$$

I would like to know if it is possible to extend this formula to the function $$theta_{ell}left(zright):=sum_{ngeq1}e^{-n^{2ell}z},,mathrm{Re}left(zright)>0,,ellinmathbb{N}^{+}.$$

I tried to work with the Poisson summation formula but I get nothing.

Thank you.