Math Genius: $mathbb{Z}$ mod $p$ vs. $mathbb{Z}_p$

Original Source Link

What is the difference between working in $mathbb{Z}$ mod $p$ and working $mathbb{Z}_p$? I’m mainly interested in the terminology and nomenclature, I understand that the result would be the same.

This came after reading the documentation of NTL. Why do functions like SqrRootMod live in ZZ, rather than ZZ_p? In the former case one has to explicitly state that “assumes n is an odd prime”. Is it because of the word “odd”, i.e. $mathbb{Z}_p$ could also include 2?

This is quite subjective, so I apologise if I’m missing the point of the question.

When we say that $a equiv b$ (mod $p$), we are referring to an equivalence relation $equiv$ on $mathbb{Z}$ defined by $a equiv b$ if and only if $a – b$ is a multiple of $p$. The transitivity of the relation is, in itself, very useful, as are many basic theorems that allow it to be treated “like an equals sign”.

On the other hand, $mathbb{Z}_p$, refers to the set of equivalence classes of $mathbb{Z}$ with respect to this relation. For instance, when we write $0 in mathbb{Z}_p$, we are really referring to the set ${ldots,-2p, -p, 0, p, 2p, ldots}$ of all elements of $mathbb{Z}$ that are equivalent to $0$ under the relation $equiv$. The reason we might want to do this is that $mathbb{Z}_p$ is a well-defined ring with respect to the obvious addition and multiplication; we can prove this directly by defining addition and multiplication of equivalence classes, or just observe that $mathbb{Z}_p$ is the quotient of the ring $mathbb{Z}$ by its ideal $pmathbb{Z}$ (hence the common notation $mathbb{Z}/pmathbb{Z}$ instead of $mathbb{Z}_p$).

Thus, when we apply our theorems about treating $equiv$ “like an equals sign”, what we are really doing is using the fact that the quotient by the equivalence relation gives a well-defined ring, and using properties of that ring to manipulate the equivalence classes of the integers on either side of $equiv$.

But we can go further! Everything so far applies equally well to any integer $n$ in place of $p$. When (and only when) $p$ is prime, we have the result that every nonzero element of $mathbb{Z}_p$ is invertible, by which we mean that for $a in mathbb{Z}_p setminus {0}$, there exists $b in mathbb{Z}_p$ such that $ab = 1$ (which follows from Bezout’s Lemma for $mathbb{Z}$, or just from the fact that $pmathbb{Z}$ is a maximal ideal of $mathbb{Z}$). This is the final step in verifying that $mathbb{Z}_p$ is not only a ring, but also a field (often denoted $mathbb{F}_p$ for this reason). Fields have many nice properties, so we can immediately apply many general theorems. For example, the ring $mathbb{F}_p[t]$ of polynomials with coefficients in $mathbb{F}_p$ is a principal ideal domain. An understanding of this field structure is essential to proving many actual concrete results, such as Eisenstein’s Criterion for the irreducibility of an integer polynomial, and Dedekind’s Theorem on the splitting of rational prime ideals in number fields.

So, to summarise this gargantuan ramble, writing $a equiv b$ (mod $p$) is a direct statement about $a$ and $b$, whereas $mathbb{Z}_p$ is an abstract construction. However, most of the useful properties of the former rely on the ring (and field) structure of the latter.

It is only a formal difference. We just can write a congruence
$$
x^2equiv 1bmod 17
$$

as an equation $x^2=1$ in $Bbb F=Bbb Z/17Bbb Z$. The latter form is sometimes more convenient. For example, since $Bbb F$ is a field, the equation $x^2-1=(x-1)(x+1)$ has exactly two solutions, namely $x=1$ and $x=-1$. This is perhaps easier to see when we look at an equation over a field rather than a congruence.

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Math Genius: Information/references/examples on fields $mathbb R^3 to mathbb C^3$ with divergence and curl free real and imaginary parts

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In the course of some physical considerations I came across a complex vector field
$$ mathbf u = mathbf v + i mathbf w, $$ with
begin{align}
mathbf v:& mathbb R^3to mathbb R^3\
mathbf w:& mathbb R^3to mathbb R^3
end{align}

and the special propert, that it has a divergence-free imaginary and
curl free real components, that means

begin{align}
vec{nabla}times mathbf v & = mathbf 0 tag{1}\
vec{nabla}cdotmathbf w & = 0 tag{2}
end{align}

In my attempts to better understand and interpret the quantitiy described by this field, I started to wonder if:

Question 1: This property has a special name/term in complex vector analysis.

Question 2: If there are any important prominent examples of such fields.

Question 3: If there are other properties that follow as a consequence of (1) and (2).

Question 4: If anyone can hint me at literature where such fields are investigated.

I would be very grateful for any hints at this. I would hope for answers like those to this question.

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Math Genius: $mathbb{Z}$ mod $p$ vs. $mathbb{Z}_p$

Original Source Link

What is the difference between working in $mathbb{Z}$ mod $p$ and working $mathbb{Z}_p$? I’m mainly interested in the terminology and nomenclature, I understand that the result would be the same.

This came after reading the documentation of NTL. Why do functions like SqrRootMod live in ZZ, rather than ZZ_p? In the former case one has to explicitly state that “assumes n is an odd prime”. Is it because of the word “odd”, i.e. $mathbb{Z}_p$ could also include 2?

This is quite subjective, so I apologise if I’m missing the point of the question.

When we say that $a equiv b$ (mod $p$), we are referring to an equivalence relation $equiv$ on $mathbb{Z}$ defined by $a equiv b$ if and only if $a – b$ is a multiple of $p$. The transitivity of the relation is, in itself, very useful, as are many basic theorems that allow it to be treated “like an equals sign”.

On the other hand, $mathbb{Z}_p$, refers to the set of equivalence classes of $mathbb{Z}$ with respect to this relation. For instance, when we write $0 in mathbb{Z}_p$, we are really referring to the set ${ldots,-2p, -p, 0, p, 2p, ldots}$ of all elements of $mathbb{Z}$ that are equivalent to $0$ under the relation $equiv$. The reason we might want to do this is that $mathbb{Z}_p$ is a well-defined ring with respect to the obvious addition and multiplication; we can prove this directly by defining addition and multiplication of equivalence classes, or just observe that $mathbb{Z}_p$ is the quotient of the ring $mathbb{Z}$ by its ideal $pmathbb{Z}$ (hence the common notation $mathbb{Z}/pmathbb{Z}$ instead of $mathbb{Z}_p$).

Thus, when we apply our theorems about treating $equiv$ “like an equals sign”, what we are really doing is using the fact that the quotient by the equivalence relation gives a well-defined ring, and using properties of that ring to manipulate the equivalence classes of the integers on either side of $equiv$.

But we can go further! Everything so far applies equally well to any integer $n$ in place of $p$. When (and only when) $p$ is prime, we have the result that every nonzero element of $mathbb{Z}_p$ is invertible, by which we mean that for $a in mathbb{Z}_p setminus {0}$, there exists $b in mathbb{Z}_p$ such that $ab = 1$ (which follows from Bezout’s Lemma for $mathbb{Z}$, or just from the fact that $pmathbb{Z}$ is a maximal ideal of $mathbb{Z}$). This is the final step in verifying that $mathbb{Z}_p$ is not only a ring, but also a field (often denoted $mathbb{F}_p$ for this reason). Fields have many nice properties, so we can immediately apply many general theorems. For example, the ring $mathbb{F}_p[t]$ of polynomials with coefficients in $mathbb{F}_p$ is a principal ideal domain. An understanding of this field structure is essential to proving many actual concrete results, such as Eisenstein’s Criterion for the irreducibility of an integer polynomial, and Dedekind’s Theorem on the splitting of rational prime ideals in number fields.

So, to summarise this gargantuan ramble, writing $a equiv b$ (mod $p$) is a direct statement about $a$ and $b$, whereas $mathbb{Z}_p$ is an abstract construction. However, most of the useful properties of the former rely on the ring (and field) structure of the latter.

It is only a formal difference. We just can write a congruence
$$
x^2equiv 1bmod 17
$$

as an equation $x^2=1$ in $Bbb F=Bbb Z/17Bbb Z$. The latter form is sometimes more convenient. For example, since $Bbb F$ is a field, the equation $x^2-1=(x-1)(x+1)$ has exactly two solutions, namely $x=1$ and $x=-1$. This is perhaps easier to see when we look at an equation over a field rather than a congruence.

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Server Bug Fix: How do I recognise a bandit problem?

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I’m having difficulty understanding the distinction between a bandit problem and a non-bandit problem.

An example of the bandit problem is an agent playing $n$ slot machines with the goal of discovering which slot machine is the most probable to return a reward. The agent learns to find the best strategy of playing and is allowed to pull the lever of one slot machine per time step. Each slot machine obeys a distinct probability of winning.

In my interpretation of this problem, there is no notion of state. The agent potentially can utilise the slot results to determine a state-action value? For example, if a slot machine pays when three apples are displayed, this is a higher state value than a state value where three apples are not displayed.

Why is there just one state in the formulation of this bandit problem? As there is only one action (“pulling the slot machine lever” ), then there is one action. The slot machine action is to pull the lever, which starts the game.

I am taking this a step further now. An RL agent purchases $n$ shares of an asset and its not observable if the purchase will influence the price. The next state is the price of the asset after the purchase of the shares. If $n$ is sufficiently large, then the price will be affected otherwise there is a minuscule if any effect on the share price. Depending on the number of shares purchased at each time step, it’s either a bandit problem or not.

It is not a bandit problem if $n$ is large and the share price is affected? It is a bandit problem if $n$ is small and the share price is not affected?

Does it make sense to have a mix of a bandit and non-bandit states for a given RL problem? If so, then the approach to solving should be to consider the issue in its entirety as not being a bandit problem?

The bandit problem has one state, in which you are allowed to choose one lever among $n$ levers to pull.

Why is there just one state in the formulation of this bandit problem?

There is one state because the state does not change over time. Two notable consequences are that (i) pulling a lever does not change the internals of any slot machine (e.g. the distribution of rewards) and (ii) you are allowed to choose any lever without restrictions. More generally, there is no sequential aspect of the state in this problem, as future states are unaffected by past states, actions, and rewards.

It is not a bandit problem if $n$ is large and the share price is affected?

Correct! If the share price is affected, then future states would be influenced by past actions. This is because the price per share is affected, which is one aspect of the state. Thus, you would need to plan a sequential strategy for your purchases.

It is a bandit problem if $n$ is small and the share price is not affected?

It all depends on the problem: as long as the state before buying shares remains completely the same after you purchase some shares, then yes. Share price being unaffected is only one of the requirements; another example requirement is that the maximum number of shares purchased is fixed at each time step, independent of the shares purchased previously.

Does it make sense to have a mix of a bandit and non-bandit states for a given RL problem? If so, then the approach to solving should be to consider the issue in its entirety as not being a bandit problem?

It makes sense to allow the share price to either be affected or unaffected based on $n$ in the same problem. Since some actions (large $n$) change the state, then there are multiple states, and actions sequentially affect the next state. Hence it is not a bandit problem as a whole, as you correctly stated.

The agent potentially can utilise the slot results to determine a state-action value?

Correct! I suggest reading Chapter 2 of Sutton and Barto to learn some fundamental algorithms of developing such strategies.

Nice work on analyzing this problem! To help solidify your understanding and formalize the arguments above, I suggest that you rewrite the variants of this problem as MDPs and determine which variants have multiple states (non-bandit) and which variants have a single state (bandit).

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Server Bug Fix: Why do some pianists occasionally play their hands at different times?

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Piano is not my main instrument, but I have always been taught that the same rhythms should always sound together. In other words, if both hands play on the downbeat of the measure, both hands should sound simultaneously as opposed to one hand playing slightly earlier or later. The same also applies to harmonies within a single hand: you wouldn’t play your thumb and pinky together but wait a millisecond for your middle finger to join in.

And yet I’ve been finding more and more examples of pianists not doing this. Consider the following example from Chopin’s famous E-minor prelude:

enter image description here

Now consider the following recording of Cortot playing this piece. He rolls some of his left hand chords on beat 1, but pay special attention to his beat fours: the right hand more often than not articulates after the left hand.

I typically find this in older recordings, but not always; you can even sense some of it in a recent recording by Daniil Trifonov, who occasionally plays the right hand slightly before the left hand.

Is there a name to this type of playing? Is it perhaps indicative of a particular era or performance tradition (i.e., a Russian style, etc.)?

Aha, I’ve found the answer:

Asynchrony!

Asynchrony is a general term which is used to describe playing notes in a separated or not-quite-together fashion where they are written as if they should normally be played at the same time in the score, for example a chord to which an arpeggiation is applied, or a left-hand bass note and right-hand melody note both written on the same beat but actually played with one hand being placed slightly before the other.

It is apparent on many early recordings made of pianists who were born in the nineteenth century and has been the subject of detailed analysis in recent years (see Peres Da Costa’s ‘Off the Record’ cited in the bibliography). It is an area of performance practice that I find personally very interesting for its role in some of the most exquisite and in other instances most eccentric-seeming performances recorded by such artists. By incorporating it into my own playing I have found it of great effectiveness in realising the music of Chopin in particular.

Chopin as Heard: Asynchrony – An Introductory Case Study

So now here’s the resource that my impression of arpeggio rubato is more than only speculation:

Mark Arnest says in his paper

Why Couldn’t They Play With Their Hands Together?
Noncoordination Between and Within the Hands in 19th Century Piano Interpretation

The romantic pianism of a century ago differed from today’s more sober approach in nearly every respect – including attitude towards the text, tempo flexibility, agogic modifications, and even voicing. But no aspect jumps out at a listener more than the noncoordination of the hands: The older pianists don’t keep their hands consistently together.

This noncoordination was an almost universal feature of earlier pianism. A study of recordings and piano rolls by 118 pianists born between 1824 and 1880 shows that all but one engaged in the practice so some degree.

And he explains why they did it:

The purpose of noncoordination was to characterize the music, generally by heightening the expression and clarifying the rhythmic structure.

and referring to Malwine Brée, “The Groundwork of the Leschetizky Method,” Haskell House, Arnest quotes:

Neither should bass tone and melody-note always be taken precisely together, but the melody note may be struck an instant after the bass, which gives it more relief and a softer effect.

“More relief” suggests a musical accent; “a softer effect” could refer to either or both of two things: the acoustic phenomenon in which higher notes appear in the overtone sequence of lower ones, or the idea of rhythmic pulse.

and he names 5 reasons for this practice in the Romantic era like

  • Acoustics (overtones)

  • Beat versus pulse

  • Noncoordination as Accent, Separator in Contrapuntal Music and as “Orchestral” Sound.

and what they did:

Noncoordination as a Form of Tempo Rubato

for further reading:

https://www.lib.umd.edu/binaries/content/assets/public/ipam/resources-reviews-and-links/arnest-hands-together-article-pdf-5-15-12.pdf

A form of rubato. More specifically ‘playing behind the beat’. Jazz pianist Errol Gardner did something rather similar when he ‘…developed a signature style that involved his right hand playing behind the beat while his left strummed a steady rhythm and punctuation’. Though in the Chopin there’s flexibility of rhythms in both hands, the LH ‘beat-keeping’ isn’t all that strict either! Not uncommon in playing Romantic piano music.

https://en.wikipedia.org/wiki/Erroll_Garner#Playing_style

I agree with Laurence “rubato” but I would even go further and say “arpeggio rubato”. One typical feature of preludes is – if not a toccato style – the arpeggio triads.

We know the piano reduction of Bachs preludes notated in block chords.

So I could imagine that these performers are applying to the block chords of eighth notes a certain kind of arpeggio playing. Pure speculation!

Thesis: “24 Preludes Copin op. 28” (M.A. Meier)

Cosmo Buono writes here:

The overall mood of the piece is reflective and even tragic and angry. One way to infuse the piece with emotion and intrigue is to use a great deal of tempo rubato; this approach also prevents the constant eighth note rhythm of the left hand from becoming too predictable.

https://www.grandpianopassion.com/2014/06/30/chopin-prelude-e-minor-amplified/

In many genres of music, it is common to have a soloist perform rhythms rather loosely while the backing musicians follow the noted rhythms rather closely. Because the piano is a polyphonic instrument, it is common for one performer to combine the roles of both a soloist and a backing musician, and thus play some notes with freer rhythms than others.

An important thing to note if one tries to do this, however, is that successfully pulling it off generally requires more skill than playing straightforwardly. By way of analogy, some skilled bakers manage to produce “topsy-turvy” cakes where the layers are tilted at weird angles to yield a whimsical quality that is aesthetically pleasing, but when unskilled bakers try to do the same thing their cakes just look sloppy.

Harmony consist of 4 voices, the upper voice (soprano) should prevail over the others in order to project the melody, in piano technique this is accomplished by deliver extra pressure to the finger that carry the upper note. When we hear Triffonov performance it is played correctly,(together), however, in the Cortot recording you hear Rubato in the right hand in other words the upper note is played after the chord it is not my preference, why did Cortot played like that? probably he liked it and took some liberties.

As to why they are doing it, per the title of your question: compare your example to a very straightforward performance (e.g.,

) – there is very little rubato here, and almost no asynchrony. I wouldn’t say it sounds bad, but it seems to be much less complex and interesting than the recordings you brought up.

Especially for nocturnes and generally Chopin’s emotionally laden works, Cortot’s rendition seems almost superhumanly suspenseful. You could imagine it as foreground music to the most intense key scene in a Hitchcock thriller. Everytime he plays asynchronous, your mind involuntarily is reminded that something is wrong, and looking for resolution. Masterful.

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Math Genius: For a directed graph with vertex set $X$, why are the arcs members of $Xtimes X$?

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A directed graph G is defined to be pair $(X,U)$ where

a. $X$ is a set $(x_1, x_2, x_n,…, x_n)$ of elements called vertices; and

b. $U$ is a set $(u_1, u_2, u_3,…,u_n)$ of elements of the
Cartesian product $X times X$, called arcs.

I don’t understand why is it $X times X$? Does anyone here know? What is the example of that?

You can think of a digraph as being a set of points with directed edges going between them. As such, a digraph is uniquely described by $X$, its set of vertices (points) and $U$, its set of edges.

A directed edge can be described by some source $u$ and some destination $v$, where $u, v in X$. We often refer to the directed edge from $u$ to $v$ as the ordered pair $(u, v)$. Now the Cartesian product $X times X = X^2$ is the set of all ordered pairs of two elements lying in $X$. Since every edge in a digraph points from one vertex to another, it follows that the set of all possible edges is the set $X^2$ (which is all possible ordered pairs). Of course, a digraph may not have all these possible edges in it, so the edge set $U$ is a (not necessarily proper) subset of these ordered pairs.

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Math Genius: For a directed graph with vertex set $X$, why are the arcs members of $Xtimes X$?

Original Source Link

A directed graph G is defined to be pair $(X,U)$ where

a. $X$ is a set $(x_1, x_2, x_n,…, x_n)$ of elements called vertices; and

b. $U$ is a set $(u_1, u_2, u_3,…,u_n)$ of elements of the
Cartesian product $X times X$, called arcs.

I don’t understand why is it $X times X$? Does anyone here know? What is the example of that?

You can think of a digraph as being a set of points with directed edges going between them. As such, a digraph is uniquely described by $X$, its set of vertices (points) and $U$, its set of edges.

A directed edge can be described by some source $u$ and some destination $v$, where $u, v in X$. We often refer to the directed edge from $u$ to $v$ as the ordered pair $(u, v)$. Now the Cartesian product $X times X = X^2$ is the set of all ordered pairs of two elements lying in $X$. Since every edge in a digraph points from one vertex to another, it follows that the set of all possible edges is the set $X^2$ (which is all possible ordered pairs). Of course, a digraph may not have all these possible edges in it, so the edge set $U$ is a (not necessarily proper) subset of these ordered pairs.

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Math Genius: For a directed graph with vertex set $X$, why are the arcs members of $Xtimes X$?

Original Source Link

A directed graph G is defined to be pair $(X,U)$ where

a. $X$ is a set $(x_1, x_2, x_n,…, x_n)$ of elements called vertices; and

b. $U$ is a set $(u_1, u_2, u_3,…,u_n)$ of elements of the
Cartesian product $X times X$, called arcs.

I don’t understand why is it $X times X$? Does anyone here know? What is the example of that?

You can think of a digraph as being a set of points with directed edges going between them. As such, a digraph is uniquely described by $X$, its set of vertices (points) and $U$, its set of edges.

A directed edge can be described by some source $u$ and some destination $v$, where $u, v in X$. We often refer to the directed edge from $u$ to $v$ as the ordered pair $(u, v)$. Now the Cartesian product $X times X = X^2$ is the set of all ordered pairs of two elements lying in $X$. Since every edge in a digraph points from one vertex to another, it follows that the set of all possible edges is the set $X^2$ (which is all possible ordered pairs). Of course, a digraph may not have all these possible edges in it, so the edge set $U$ is a (not necessarily proper) subset of these ordered pairs.

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Math Genius: For a directed graph with vertex set $X$, why are the arcs members of $Xtimes X$?

Original Source Link

A directed graph G is defined to be pair $(X,U)$ where

a. $X$ is a set $(x_1, x_2, x_n,…, x_n)$ of elements called vertices; and

b. $U$ is a set $(u_1, u_2, u_3,…,u_n)$ of elements of the
Cartesian product $X times X$, called arcs.

I don’t understand why is it $X times X$? Does anyone here know? What is the example of that?

You can think of a digraph as being a set of points with directed edges going between them. As such, a digraph is uniquely described by $X$, its set of vertices (points) and $U$, its set of edges.

A directed edge can be described by some source $u$ and some destination $v$, where $u, v in X$. We often refer to the directed edge from $u$ to $v$ as the ordered pair $(u, v)$. Now the Cartesian product $X times X = X^2$ is the set of all ordered pairs of two elements lying in $X$. Since every edge in a digraph points from one vertex to another, it follows that the set of all possible edges is the set $X^2$ (which is all possible ordered pairs). Of course, a digraph may not have all these possible edges in it, so the edge set $U$ is a (not necessarily proper) subset of these ordered pairs.

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Linux HowTo: Term for language status

Original Source Link

I’m working on a project trying to define the relation between languages and countries. A detail on that is specifying the status of a language in a particular country. I defined four-level classification and named three of them. Probably due to being the most “neutral” of statuses, I’m having a hard time naming the third one. They are:

  • Official: The language is formally accepted and used for the functioning of the national state/government organs.
  • Recognized: Formal education and services are available in the language. Local/regional governments use it as an additional official language.
  • [Something]: The language is used in daily life and private education & services are provided in it. The state/government is mostly uninvolved.
  • Suppressed: The state or general public covertly or overtly adopts a negative stance against use of the language. Users may be derided, harassed or prosecuted.

I need a single-word name for the third status in there, in line with the others. Terms like “Neutral” seemed weak among the other three, as well as negated terms (non-x, un-x).

Another criterion is that this term should not imply the size of population using the language, so terms like minority and common don’t work well either.

Colloquial

From M-W:

1: Used in or characteristic of conversation, especially familiar and informal conversation

Quite simply, it refers to the language people actually use, regardless of whether it is officially sanctioned or not in any way.

How about accepted? From M-W:

accepted: generally approved or used. e.g., an accepted convention/practice

This seems to fit the bill nicely. It’s clearly below official and recognized and above suppressed. It pretty much calls your third category what it is.

Addendum: Another possibility is permitted. From M-W:

permit: to consent to expressly or formally

As per the comments below, I would say permitted is less positive than accepted but more positive than tolerated. Ditto re allowed, which in the context of your question means the same thing as permitted.

Of the choices I’ve suggested, I think permitted works the best:

Official: The language is formally accepted and used for the functioning of the national state/government organs.

Recognized: Formal education and services are available in the language. Local/regional governments use it as an additional official language.

Permitted: The language is used in daily life and private education & services are provided in it. The state/government is mostly uninvolved.

Suppressed: The state or general public covertly or overtly adopts a negative stance against use of the language. Users may be derided, harassed or prosecuted.

Wikipedia has an article titled List of largest languages without official status.
The OECD (Organisation for Economic Co-operation and Development) glossary of statistical terms defines Non-official language, or Unofficial language.

A language that, though relatively widely used, lacks officially sanctioned status in a particular legally constituted political entity. Example: French in Lebanon; English in Israel.

So it sounds like non-official language is an acceptable phrase.

OECD itself references a broken link from Statistics Canada, but we can see that they use the phrase non-official language in the following example: Statistics Canada 2016 census data on knowledge of languages

I would be careful how you define suppressed languages, because there is some overlap between your definition of a non-official language and a suppressed language, for instance a language that is shunned by the general public while the state has no policy toward it.

Take a look at Ethnologue

(You might take a look at https://www.ethnologue.com/ (currently behind a paywall) and use their terminology. They classify languages for a living. The OP’s grouping may be somewhat limited to political classifications.)

Heart language

Wycliffe Bible Translators calls the language that you grow up speaking and understand best a Heart Language.

Minority language speakers in Kenya experience the truth of God’s Word when they stop trying to understand Scripture in other languages and start studying it in the language they understand best.

From Kenyans Understand the Gospel in Their Heart Language

(I have worked in Burkina Faso. The children in the capital city, Ouagadougou, would speak Mòoré every day. Since the official language was French, they would learn French as a foreign language from age five or six, upon entering school. Their birth certificate and other official documents were in French. But in all other realms of daily life, it was Mòoré.)

Native / mother tongue

These terms are perhaps more neutral than heart language. A mother tongue is what you learn at your mother’s knee. Merriam-Webster defines it as:

one’s native language

Quotidian

This is just putting a fancy word on your everyday language.

Unsuppressed

Since the OP seems to be searching for political terms to classify languages, this is opposite of one of the terms.

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