dnd 3.5e – How can you overcome the distance limit of a dedicated wright?

In D & D 3.5 there are a number of types of homunculi that can be crafted by characters with the Craft Construct talent. Among these are dedicated wrights, which are explicitly described in their description as atypical types of homunculi who "do not go on missions or accompany their master on adventures. Instead, he stays at home and works during his master adventure (Eberron campaign) setting, P. 285). "However, they are specifically mentioned as a subtype of the standard homunculus template contained in the Monster Manual, which means that they have an innate telepathic bond with their Creator that is up to 1,500 feet away. In addition: “A homunculus never moves voluntarily beyond this area, although it can be removed by force. In this case, the creature does everything in its power to get back in touch with its master (Eberron Campaign Setting, p. 285). "

Dedicated wrights are intelligent constructs and are great for passively making money with crafting skills, especially if they are equipped with equipment to improve their skill tests. My goal is to put a group of them together and use their tireless nature to keep a smithy going around the clock while my character is not on an adventure. However, the 1,500 foot limit seems to make this impossible. After unpacking, they do not appear to be able to achieve their stated purpose. Is there an exception to this rule that I missed somewhere?

Assuming there is no solution I can travel with without my workforce stopping with every campaign. I see three options to deal with this problem. A) Replace or improve telepathic attachment, B) "falsify" telepathic attachment so that the homunculus believes that it will still work if it is not, or C) otherwise convince the homunculus that it is acceptable not to have a telepathic bond, such as an enchantment spell.

If possible, I would prefer a solution that costs 5000 gp or less. In addition, my character will take the lead at level 6 and be moderately invested in charisma, so using followers with obscure classes is perfectly acceptable for the solution.

Additional qualifiers:

  • While it is possible to improve the innate telepathic bond in each Wright using the "Improved Homunculus" feat, this triples its cost and only extends the range to 1 mile per character level. Since I doubt that every campaign is within 4 to 20 miles, this is not enough.
  • As smart as the idea may be, I would prefer not to play a dvati (I'd rather have a solution that works for every race).
  • I would prefer something cheaper than paying a wizard to do Telepathic Bond and Permanency on everyone. It would also be desirable to avoid spells with a "permanent" duration so that my workforce does not stop when my character enters an anti-magic field.

algebraic geometry – Hardy and Wright, section 5.8 Clarification (construction of regular 17-gon).

This question requires some understanding of the classic book referred to in Hardy and Wright's Introduction to the Theory of Numbers. I have the sixth edition. I try to provide most of the information needed to answer my question, but realistically, the book probably needs to be on hand. I hope the book is well known so that someone can respond to the information I give here.

Equation (5.8.2) gives the cyclotomic polynomial
$$ frac {x ^ {17} – 1} {x – 1} = x ^ {16} + x ^ {15} + ldots + 1 = 0 $$

whose roots are $ epsilon_ {k} = { it e} ( frac {k} {17}) = { text cos} k alpha + { it i} { text sin} k alpha $With $ alpha = frac {2 pi} {17} $and the notation $ { it e} ( tau) = _ { text def} e ^ {2 pi i tau} $, and with $ k = 1, l points, $ 16 Here.

Bottom of the same page, variables $ x_1 $ and $ x_2 $ are each as sums of $ epsilon $So it's immediately obvious that their sum is $ Sigma_ {1} ^ {16} epsilon_ {k} $which (?) should be the same $ -1 $ directly from (5.8.2). This is,
$$ x_1 = epsilon_1 + epsilon_9 + epsilon_ {13} + epsilon_ {15} + epsilon_ {16} + epsilon_ {8} + epsilon_ {4} + epsilon_2 $$ and $ x_2 $ is the sum of the remaining ones $ epsilon $& # 39; see Fig.

In the book of this simple step ($ x_1 + x_2 = -1 $) is reached only after a series of algebraic steps involving trigonometric identities have been performed, e.g. $ epsilon_ {k} + epsilon_ {17-k} = 2 { text cos} k alpha $, $$ x_1 = 2 ({ text cos} alpha + { text cos} 8 alpha + { text cos} 4 alpha + { text cos} 2 alpha) $$, and
$$ x_2 = 2 ({ text cos} 3 alpha + { text cos} 7 alpha + { text cos} 5 alpha + { text cos} 6 alpha) $$,

Then comes the equation that my question has to deal with. As written in my book, it reads
$$ x_1 + x_2 = 2 Sigma_ {1} ^ {8} { text cos} k alpha = 2 Sigma_ {1} ^ {16} epsilon_k = -1 $$,

I actually have 2 specific questions:

First, why is the multiplier? # 2 & # 39; present at the second summation? $ Sigma_ {1} ^ {16} epsilon_k $?

Second, why are all these efforts undertaken since then? $ x_1 + x_2 = -1 $ follows directly from their definitions together with equation (5.8.2)?

Probability – Concrete Hanson Wright Inequality?

I'm working on a paper that needs to be bound

$$ Pr left[|vec x^top Q vec y| >= tright]$$ from where $ Q $ is a matrix (random symmetric) and $ vec x, vec y $ are real mean zero subgauss random vectors. As far as I understand it, the Hanson-Wright inequality ties this as $$ 2 exp left (-c cdot min left ( frac {t ^ 2} {k ^ 4 || Q || _F ^ 2}, frac {t} {k ^ 2 || Q | | _2} right) right) $$ from where $ k $ is the subgaussian parameter and $ c $ is an absolute constant.

But I'm trying to derive an actual numerical limit. Is there a paper that gives an actual value of $ c $preferably a close? It may be different for the two sides of min (). Many Thanks.