dnd 5e – How does a scrolling spell work when the spell has a special relationship with its hardware components?

According to the rules of Spell Scrolls, launching a scroll requires no other material component than the parchment itself. Similarly, the rules in Xanathar's Guide to Writing a Spell Scroll indicate that you must supply the hardware components at the time of creation.

Indeed. But there are some spells that have an unusual relationship with their material components. For example:

  • change of plane requires a tuning fork specific to the target plane
  • The instant convocation of Drawmij enchants the sapphire component to be summoned by crushing.
  • Secret chest of Leomund requires a trunk and its replica, one of which is sent to the ethereal plane, and the other serves to recall it or send it back.

How do these (and other) spells work when they are written in a scroll of spells?

I can guess that one change of plane scroll can be inherently linked to a single target plane according to the tuning fork used at the time of writing. So you do not have airplane offset scroll as much as one plane shift roller to the elementary plane of the air (or something like that); but in other cases, the components are not destroyed by the process of creation of the parchment; so it looks like you need scrolling. and the component (s) used in its creation to be able to use the parchment, which is rather contrary to the very notion of scrolling of spells. Is it just the DM to find something?

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Complexity Theory – Longest Single Circuit and Relationship P = NP

Given the following function: $$ : f left (G, v right) : = : size : of : the : the longest : simple : circuit : in : a : directed : graph : G : that : contains : v $$

Output: The function returns a natural number or 0, which is the largest single circuit of the directed graph G.

However, I do not understand the following statement:

it is possible to calculate the function g (G, v) in polynomial time and it is guaranteed that: $ f (G, v) -5 ≤ g (G, v) ≤ f (G, v) + $ 5 then $ P = NP $.

I do not understand why. As far as I know, the longest single circuit can be proved NP-complete using LongestSimplePath (by adding the remaining vertices to a new G graph and two directed edges, then running it) or to HamCycle (by creating an instance of longestsimplecycle (H, | V |) based on hamcycle (H) (can be done in polynomial time) and checking if the longest simple cycle equals | V | (contains the cycle of 39, Hamiltonian) as a basis for reductions, so if we know that it is possible to compute in polynomial time g (G, v), I do not understand how the fact that it is guaranteed that $ f (G, v) -5 ≤ g (G, v) ≤ f (G, v) + $ 5 help to deduce that P = NP.

There is probably a trick I do not see and who would appreciate your help. How is it possible to determine that?

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theoretical measure – Relationship between Baire sets and Baire functions

The Halmos measurement theory has a problem (51.6) which reads as follows:
The term "Baire set" is suggested by the term "Baire function" used in the analysis. Yes $ mathscr {B} $ is the smallest class of functions containing all continuous functions and containing the limit of each sequence of convergent point-to-point functions (but not necessarily uniform), the functions of $ mathscr {B} $ are called the "Baire Functions" on $ X $. A necessary and sufficient condition for a set to be a set of Baire is that it is a set of Borel and that its characteristic function is a function of Baire.

As part of Halmos, $ X $ is a locally compact Hausdorff space, and works with (generated) $ sigma $rings rather than $ sigma $-algebras. He defines the sets of Baire as the $ sigma $ring generated by the compact $ G_ delta $ sets and sets Borel as $ sigma $ring generated by the compact sets.

I can prove a direction (Baire is Borel and its characteristic function is a function of Baire), but I was not lucky with the other direction. My questions are:
1) How can I prove the reverse implication? 2) What would be the most modern equivalent statement of the problem using $ sigma $-algebras? 3) Would the proof of this statement be substantially different from the proof of the original problem?

I have tried all kinds of constructions in vain. Initially, I wanted to avoid using an equivalent transfinite definition of $ mathscr {B} $, but I relaxed this restriction that I imposed and played with these types of construction too, but without success. Any help would be greatly appreciated.

normal distribution – Relationship between noisy observations

assume $ mathbf X_1 ^ n $ is a size vector $ n $ whose elements are either $ + 1 $ or -1 $. Then we define
$$ mathbf Y ^ n = mathbf X_1 ^ n + mathbf N ^ n $$
or $ mathbf N ^ n $ is Gaussian additive noise with distribution $ mathbf N ^ n sim mathcal {N} ( mathbf 0, sigma ^ 2 mathbf I) $that is, the noise elements are distributed identically and identically (i.i.d) with variance $ sigma ^ 2 $.

Now, consider $ mathbf X_2 ^ n $ which is also a vector of $ + 1 $& # 39; sand -1 $and is identical to $ mathbf X_1 ^ n $ with the exception of $ k <n $ positions in which its values ​​are opposed to the $ mathbf X_1 ^ n $. For example for $ n = $ 5 and $ k = $ 2, we could have
$$ mathbf X_1 ^ 5 =[-1,-1,+1,+1,-1]; \
mathbf X_2 ^ 5 =[+1,-1,+1,+1,+1]; \
$$

where two vectors differ in the first and last positions.
My question is: can we get a relationship between $ || mathbf Y ^ n- mathbf X_1 ^ n || ^ 2_2 $ and $ || mathbf Y ^ n- mathbf X_2 ^ n || ^ 2_2 $. Digitally, I saw this as $ k becomes big $ || mathbf Y ^ n- mathbf X_2 ^ n || ^ 2_2 $ becomes bigger than $ || mathbf Y ^ n- mathbf X_1 ^ n || ^ 2_2 $. I'm looking for a mathematical expression to connect the two.

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postgresql – How to correctly model the parent-child relationship in a Postgres multi-tenant, multi-schema database

I have a SaaS application for multiple tenants, where each tenant gets his own scheme. When a user creates an account, I create a new schema and run a set of migrations to create tables under that schema. Then, when the user wants to authenticate, he provides information about his client (he can act from his subdomain, for example, mycompany.coolapp.com or from a Unique account ID), and the application routes the following requests to the schema of this client.

However, some tenant data will be public. For example, the user will need to create invoices and e-mail their links (containing a UUID) to their customers. When the customer clicks on the link, the application must be able to determine which tenant belongs the invoice, according to the UUID.

To do this, I created a table "public.invoices", with two columns: uuid and tenant. The latter contains the tenant's schema name, which I can then use to perform the following queries (that is, the actual invoice data) on the appropriate schema.

Question 1: Is this design sound? Is there another, more standard way to route unauthenticated / pre-authenticated requests to specific schemas in situations where each tenant gets its own schema?

Question 2: How can I tell Postgres that when the renter deletes the invoice (that of his schema), the public record (under "public.invoices") must also be deleted? I try to avoid using two queries, if an option or feature provided by the database exists. I've reviewed CASCADE, but it seems like it's only about parents and, as far as I know, there's no way to define a foreign key relationship in this scenario where the tenant's bill is the parent record.

distributed systems – What is the relationship between coherence models and consensus algorithms?

I know it:

Each replication technique must follow a consistency model. A consistency model defines a set of rules for the order of write and read operations. In some replication techniques (such as state machine replication), to execute each write operation, the replicas must reach consensus during the execution or rejection of the operation.

In my opinion, a consensus algorithm is a subset of the rules related to write operations and defined by a consistency model. Is it correct?

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