Can spells with different types of components be cast in the same action?

Would someone who is casting something with a Somatic and Material component, say Minor Illusion, also be able to cast something with only a Verbal component, say Thaumaturgy, in the same action? I’m making a character who will be casting these cantrips on a regular basis for cosmetic changes, and want to know if there is an official ruling on this.

When creating a race in Pathfinder, can you give them two racial types instead of one? [duplicate]

I noticed in the Advance race guide for Pathfinder, that most of the races have a single type except for hybrid races like half-orcs which have two for the Humanoid type.

So when your creating a race in Pathfinder, are you able to combine two racial types or is it just one type only?

Proving injectivity for an algorithm computing a function between sets of different types of partitions [closed]

I am attempting to solve the following problem:

Let $ A$ be the set of partitions of $ n$ with elements $ (a_1, \dots, a_s)$ such that $ a_i > a_{i+1}+a_{i+2}$ for all $ i < s,$ taking $ a_{s+1} = 0.$ Define $ g_n = F_{n+2}-1$ and let $ B$ be the set of partitions of $ n$ as $ b_1 \ge \dots \ge b_s$ such that $ b_i \in \{g_n\}$ for all $ i,$ and if $ b_1 = g_k$ for some $ k,$ then $ g_1, \dots, g_k$ all appear as some $ b_i.$ Prove $ |A|=|B|.$

Attempt: Let $ e_i$ be the vector with $ 1$ at position $ i$ and $ 0$ elsewhere. If $ b_1 = g_k,$ let $ c=(c_k, \dots, c_1)$ count how many times $ g_i$ appears. We calculate $ f: B \to A$ as follows:

Let $ c=(c_k,\dots,c_1), a=(0,\dots,0).$ While $ c \ne 0,$ let $ d_1 > \dots > d_k$ be the indices such that $ c_{d_i} \ne 0.$ Replace $ c, a$ with $ c-(e_{d_1}+\dots+e_{d_k}), a+(g_{d_1} e_1 + \dots + g_{d_k} e_k)$ respectively. After the while loop ends, let $ f(b)=a.$

Let $ \sum a, \sum b, \sum c$ be the sum of the components of $ a, b, c$ respectively. Since $ \sum c$ decreases after every loop, the algorithm terminates and $ f(b)$ is well-defined. Since $ c_k g_k + \dots + c_1 g_1 + \sum a$ does not change after every iteration, is $ \sum b$ at the start and $ \sum a$ at the end, we have $ \sum f(b) = \sum b = n,$ so $ f(b)$ is also a partition of $ n.$ Now $ a = (g_k, \dots, g_1)$ after the first loop, which satisfies the condition $ g_i > g_{i-1}+g_{i-2}$ since $ g_i = F_{n+2}-1 = (F_{n+1}-1)+(F_n-1)+1 > g_{i-1}+g_{i-2}.$ Furthermore, after every iteration of the loop, the difference $ a_i – (a_{i-1}+a_{i-2})$ changes by $ 0, g_{d_j}-g_{d_{j-1}} > 0,$ or $ g_{d_j}-(g_{d_{j-1}}+g_{d_{j-2}}) > 0,$ so we have $ a_i > a_{i-1} + a_{i-2}$ at the end and hence $ f(b) \in A.$ Thus, $ f: B \to A$ is well-defined.

In order to prove the injectivity of $ f,$ it suffices to prove each loop iteration as a mapping $ (c,a) \to (c’,a’)$ is injective, which would imply the mapping $ (c,0) \to (0,a)$ that the while loop creates is injective. Indeed, if $ f(b_1) = f(b_2) = a$ with $ (c_1, 0), (c_2, 0)$ being sent to $ (0, f(b_1)) = (0,a), (0, f(b_2)) = (0,a)$ respectively, then we have $ (c_1, 0) = (c_2, 0) \Rightarrow c_1 = c_2 \Rightarrow b_1 = b_2.$

Suppose $ d_1 > \dots > d_i, f_1 > \dots > f_j$ are the non-zero indices of $ c_1, c_2$ respectively and $ c_1 – (e_{d_1}+\dots+e_{d_i}) = c_2 – (e_{f_1}+\dots+e_{f_j}), a_1+g_{d_1}e_1 + \dots+ g_{d_i} e_i = a_2 + g_{f_1} e_1 + \dots + g_{f_j} e_j.$ If $ x \ge 2$ is an entry of $ c_1,$ it decreases by $ 1,$ so the corresponding entry in $ c_2$ after $ c_2$ is modified is also $ x-1,$ which means it must’ve been $ (x-1)+1 = x$ before since $ x-1>0.$ Thus, if the values of two positions of $ c_1, c_2$ differ, one is $ 1$ and the other is $ 0.$ However, if $ c_1 = (1,0), a_1 = (3,1), c_2 = (0,1), a_2 = (4,1),$ then $ (a_1, c_1), (a_2, c_2)$ both get sent to $ ((5,1), (0,0)).$ I can rule out this specific example by arguing that one of the pairs is illegal and could not have come from any choice of initial $ c,$ but I have no idea on how to do this in general.

What should I do next in order to show $ f$ is injective? Furthermore, since the problem I’m trying to prove is correct, injectivity would imply $ f$ is secretly a bijection. But I have no clue on how to even start on the surjectivity of $ f,$ so I just constructed a similar algorithm for $ g: A \to B$ in the hopes of proving $ g$ is injective too. If I can show $ f$ is injective I will probably know how to show $ g$ is.

Here is an example of $ f, g$ in practice:

Let $ n = 41, b = (12, 7, 7, 4, 4, 2, 2, 2, 1) \Rightarrow c = (1, 2, 2, 3, 1).$

$ $ ((1, 2, 2, 3, 1), (0,0,0,0,0)) \to ((0, 1, 1, 2, 0), (12, 7, 4, 2, 1)) \to ((0, 0, 0, 1, 0), (19,11,6,2,1)) \to ((21,11,6,2,1),(0,0,0,0,0)),$ $ so $ f(b) = (21,11,6,2,1).$

Let $ a = (21, 11, 6, 2, 1).$

$ $ ((21,11,6,2,1),(0,0,0,0,0)) \to ((9,4,2,0,0), (1,1,1,1,1)) \to ((2,0,0,0,0),(1,2,2,2,1)) \to ((0,0,0,0,0),(1,2,2,3,1)),$ $ so $ g(a) = (12, 7, 7, 4, 4, 2, 2, 2, 1).$

What are the most and least-resisted damage types?

Inspired by these related questions, as well as my previous question on the matter, I’m curious what the most and least resisted damage types are in DnD 5e? Ideally I am looking for a list similar to the one found in this answer to the first linked question above, but in 5e instead of 3.5.

Which damage types are the most resisted, and which are the least resisted in DnD 5e?

For simplicity’s sake, I would limit this to monsters from official WotC sources, and would classify immunity as a form of resistance (though if an answer chooses to distinguish between resistance and immunity that would be fine).

While I initially only had magical damage types in mind (Acid, Fire, Cold, Poison, Necrotic, Radiant, Lightning, Psychic, Thunder and Force), bonus points if an answer also addresses physical damage types (Slashing, Piercing, and Bludgeoning).

pkcs12 generation for EC key types using javascript

I’m generating p12 for EC keytypes using Forge library.

This library already providing p12 for RSA key types, Now, I’m trying to enhance to ECC keytypes.

  1. Generating pem format Privatekey, certs into asn1 format private key. Here I’ve written ECPrivateValidator to validate and convert into asn1 format private key.
  class: asn1.Class.UNIVERSAL,   tag: asn1.Type.SEQUENCE,   capture: 'privateKeyInfo',   value: [{     name: 'PrivateKeyInfo.Version',     class: asn1.Class.UNIVERSAL,     tag: asn1.Type.INTEGER,     capture: 'privateKeyVersion'   }] }``` Can you guide me if you have any thoughts on it? 

custom post types, custom fields and normalization

I have decided to redesign a plug I developed to use custom post types and custom fields instead of custom tables in the database.

I have some questions about abstracting data into new post types. These questions are based on the assumption that I should essentially be trying to replicate a relational database.

It’s a dictionary feature for a language learning blog. Here are the tables from the original. Simplified for readability.

CREATE TABLE words (     ID int(),     word varchar(),     parent_id int() );  CREATE TABLE nouns (     word_id int(),     plural varchar(),     [...] );  CREATE TABLE verbs (     word_id int(),     pres_part varchar(),     past_part varchar(),     [...] );  CREATE TABLE definitions (     ID int(),     word_id int(),     definition varchar(),     [...] );  CREATE TABLE examples (     ID int(),     defintion_id int(),     example varchar(),     [...] );  

Words will obviously be the first custom post type.

I’ve decided to use custom fields for the noun and verb information as they are 1:1 relationships and only existed to reduce null entries in the words table. I believe WordPress will only create an entry in the postmeta table if the fields are actually used. So that would solve that problem.

Now the main question is; should I store definitions and examples as custom post types as I would in a relational database or just use ACF repeaters? These are both genuine 1-many relationships. Although a single definition will probably only ever have a maximum of 3 examples.

The reader might wish to search for keywords within the definitions and/or examples but they would never want to view a single definition as page.

Definitions and examples might however be used outside the context of their respective word in the future. With a flashcard app for example. Is it easy to uniquely identify and request a single entry from a repeater?

What types of string properties are verifiable in polynomial time?

When given the string and the property in question as a potential certificate. Is there any classification theorem that says something along the lines of: all properties (of strings) that have this property (as a sub property) are verifiable in polynomial time?

Are there any collections of types of patterns in strings that are verifiable in poly time?

A trivial property is that a collection of strings with these properties belong to a language in NP (belonging to NP being the sub property).

I’m looking for something more concrete.

I’m looking for the common thread between string properties like these that makes these properties verifiable in poly time for any string.

i.e. is there a way to pick properties of strings out of a hat in such a way that the properties you pick are guaranteed to be verifiable in poly time in any string.

Maybe there is a way to do this with implicit complexity–where the only properties you can build (in some restricted language) are the ones that are verifiable in poly time?

Does Type:Type lead to inconsistency without general inductive types?

In e.g. Agda , it is possible to derive an element of the empty type by enabling the “type in type” option.

Every proof I have seen (and come up with) involves making a special inductive type definition. Can a contradiction be derived using type in type but with e.g. only using standard library types?