Free variables and bound variables Puns

As far as I can understand it (correct if i am wrong here as this is my own estimation, not something Iv'e read! If I am wrong then I presume it’s undefined) the above x is any natural number if n is defined as a natural number. A variable is bound by one quantifier but what happens with two? I havent been able to find the answer to this via google.

Thank you in advance guys!!

So I understand in this expression:

λxy.xu

that x and y are bound and that u is a free variable.

However, what if you have an expression like this

(λuv.v)(λx.ux)

I can tell that u and v are bound in the first expression, but in the second expression u is a free variable.

So if the question is to determine the free variables and bound variables would the answer be:

Bound: u,v,x
Free: u

EDIT: or would I be expected to alpha rename u in the first expression to something else so I could distinguish between them?

I'm having some trouble learning the restrictions on free/bound variables when dealing with rules of inferences involving quantifiers such as Universal Instantiation, Existential generalization, Universal generalization, and existential Instantiation.

I do not understand the intuition behind all of these rules as of yet, as far as I understand these are the restrictions:

Let #u be a proposition or propositional function, and #v denote replacing every free occurrence of u in #u by v. Then the restrictions are:

UI:

• Forall free occurrences of u in #u, v must be free in #v in the same place.

• The instantiating variable may already be free in u.

EG:

• v is free in #v everywhere u is free in #u

EI:

(let p be a statement derived from #v)

• v is free in #v everywhere u is free in #u

• The variables v does not occur free in p or any statement leading to p within the scope of the assumption.

UG:

• v is free in #v everywhere u is free in #u

Is this correct to the full extent?

EDIT: There is also one subtly that I am unsure about:

is it true that

(x)(exists.y)[ #xy ] iff. (exists.y)(x)[ #xy ]

if not what is the difference in meaning?

Not much more to add really. I have an outcome variable in which the scores are bound to -1 and 1, and I'd like what type of regression I can use that employs that boundary. That is, I recognize that I could just use a good old fashion OLS regression, but I don't feel it's proper for the data that I have.

I have two random variables X and Y with given (mean, variance) as (mu_x, sigma_x^2) and (mu_y, sigma_y^2), respectively. These random variables have their truncated counterparts: X* = max(0,X) and Y* = max(0,Y). We know the mean and variance of the truncated counterparts too: (mu_X*, sigma_X*^2) and (mu_Y*, sigma_Y*^2).

Can we comment anything on the covariance of the truncated random variables (i.e., E[X*Y*] - E[X*]E[Y*])?

Some conditions:

1. The distributions of the original or the truncated random variables are not available.
2. The variables may or may not be dependent because of the setup of the underlying experiment, so bounds assuming independence are not useful.

I upgraded from 4.21.2 to 4.26. My main and only widget with a bound brush variable no longer updates the Brush Texture when the bound variable is changed. I can see that the variable does receive the proper texture reference however the widget image remains white. (Top image). I was however able to update the image using the Set Brush From Texture (Bottom image).

Why is my Texture2D bound variable not updating the widget brush?

https://preview.redd.it/d9c5b8eyxoe61.png?width=1326&format=png&auto=webp&s=6d224d9ed3908f668db402a5218273ea71fee406

Asking for metaphysics purposes- Quine's idea of "to be is to be the value of a bound variable", can you be bound by a negated quantifier?

EDIT: I see I got a downvote: if this is an extremely stupid thing to be asking, please tell me! I feel I may have completely got the wrong end of the stick here. Thanks in advance. :)

Hi,

I cannot find the answer anywhere, but I have to think it exists. Perhaps I am not using the right nomenclature. I have a closure (let bound) that needs to report internal variable status (not all of it, but key ones as part of a way to get insight into the running state of a model, which this closure and it's values represent. I am having trouble finding a way to return the names (for example as keys in a hash-table and their values). For example I would like to have a list of symbols that refer to the closure lexical variables that populate both the NAME of the variable and its corresponding VALUE at that moment in time:

(loop for varname in '(active-cycles shape calculated-decay cycle-time  )
  collect (cons varname (eval varname)))

In the above example, this loop form ideally would return a list of conses with the name of the value being returned and the variable. It could also be a hash-table. It ultimately will need to be turned into JSON to send to an analysis system. If it were three or four variables this wouldn't be an issue. But since these values need to be reported out regularly from within the closure, I need some way to efficiently do this.

Thank you for any guidance...

I was playing around with equations and I found this possible interesting inequality that can be applied to any random variable, but I'm not sure if there's an easier (and possible trivial) way to obtain what I got.

Let's say we have any random variable X with mean μ and variance σ^2. And let's say we are allowed to sample it n times to get samples x_1, x_2, ..., x_n. And just like many times in statistics we try to estimate the mean from the samples, and let's say in this problem we're interested in the expected error (or an upper bound at least) of how wrong our estimate is from the mean,

As in define an error term e=|(x_1+x_2+...+x_n)/n - μ|, and we want to find an upper bound on E(e). I spent a lot of time googling and looking at different concentration inequalities in Probability theory and I got the following

P(|(x_1+x_2+...+x_n)/n - μ| > epsilon) <= min(σ^2/(n*epsilon^2), 1) [from Chebyshev inequality]

Then since the term in the expectation is always positive, using this formula:

E[|(x_1+x_2+...+x_n)/n - μ|]= \int[P(|(x_1+x_2+...+x_n)/n - μ|>x)dx] from 0 to ∞

<= \int[min(σ^2/(n*x^2), 1) dx] from 0 to ∞

<= 2σ/sqrt(n)

So if we can sample X n times then on expectation the error of the estimates mean will be at most 2σ/sqrt(n) which I think is very interesting and I can't find this fact anywhere else. And I can't figure out if it's because it's not actually interesting or if this inequality (or possibly something even tighter) can be achieved trivially using another inequality or that possibly I just didn't search hard enough.

if (isset($_POST['submit'])) {
  try {
    $connection = new PDO($dsn, $username, $password, $options);
    $user =[
      "id"           => $_POST['id'],
      "roomname"     => $_POST['roomName'],
      "carpet"       => $_POST['carpet'],
      "lengthA"      => $_POST['lengthA'],
      "lengthB"      => $_POST['lengthB'],
      "lengthC"      => $_POST['lengthC'],
      "lengthD"      => $_POST['lengthD'],
      "underlayCost" => $_POST['underlayCost'],
      "gripperCost"  => $_POST['gripperCost'],
      "roomNotes"    => $_POST['roomNotes'],
      "clientID"     => $_POST['client_fk'],
    ];

    $sql = "UPDATE rooms
            SET id = :id,
              roomname = :roomName,
              carpet = :carpet,
              lengthA = :lengthA,
              lengthB = :lengthB,
              lengthC = :lengthC,
              lengthD = :lengthD,
              underlayCost = :underlayCost,
              gripperCost = :gripperCost,
              roomNotes = :roomNotes,
              clientID = :client_fk
            WHERE id = :id";

  $statement = $connection->prepare($sql);
  $statement->execute($user);
  } catch(PDOException $error) {
      echo $sql . "<br>" . $error->getMessage();
  }

So Ive been trying to set up an updating form that updates the database, however I keep getting an error that the number of parameters binding arent equal. I can't seem to figure out what I've missed. Im new to php so I'm not the best at debugging my own code but from what ive seen online I havent missed any colons etc.

Hi everyone! Im new to kOS so I was learning on this ?official? Quick Start Tutorial. Im stuck at part where I should use kOS script to make my ship launch (hellolaunch.ks). When I run copy-pasted code, countdown will do its thing but when comes to staging, error comes (pic). I tried using just MAXTHRUST instead of SHIP:MAXTHRUST but without any luck. Also at the begging of script is thrust locked to 1, but thrust of ship is not modified.

What am I missing? Any suggentions what can be wrong here?

https://preview.redd.it/ftysio8y8ht31.png?width=380&format=png&auto=webp&s=c6d7f85fc151fe3046636bda1c6a70ae41df06bb

//hellolaunch

//First, we'll clear the terminal screen to make it look nice
CLEARSCREEN.

//Next, we'll lock our throttle to 100%.
LOCK THROTTLE TO 1.0.   // 1.0 is the max, 0.0 is idle.

//This is our countdown loop, which cycles from 10 to 0
PRINT "Counting down:".
FROM {local countdown is 10.} UNTIL countdown = 0 STEP {SET countdown to countdown - 1.} DO {
    PRINT "..." + countdown.
    WAIT 1. // pauses the script here for 1 second.
}

UNTIL SHIP:MAXTHRUST > 0 {
    WAIT 0.5. // pause half a second between stage attempts.
    PRINT "Stage activated.".
    STAGE. // same as hitting the spacebar.
}

WAIT UNTIL SHIP:ALTITUDE > 70000.

// NOTE that it is vital to not just let the script end right away
// here.  Once a kOS script just ends, it releases all the controls
// back to manual piloting so that you can fly the ship by hand again.
// If the program just ended here, then that would cause the throttle
// to turn back off again right away and nothing would happen.

So I've been thinking recently that I don't have a clear sense of how high COR(X,Y) and COR(Y,Z) can be while preserving COR(X,Z) = 0, and I feel like I generally lack intuition for why correlation distance (1-correlation) violates triangle inequality. (I found a counterexample, but that's not my point).

I feel like there's some general structure below the surface I'm standing on that I'm not quite grasping. Any thoughts on resources for inequalities in correlation?

Suppose I have a matrix that is v1 = [1, 0, 0]T v2 = 1, 0, 0]T v3 = [0, 1, 0]T v4 = [-2, -1, 0]T v5 [3, 1, 0] multiplied by a vector [x1, x2, x3, x4, x5] which equals to a zero vector [0, 0, 0]. How do I determine the free variables of the matrix? I can write it out into a system of equations but I am confused how to find the values of each x if that makes sense.

For n,m ∈ℕ find infimum of n|(n/( √2)) - m|. (It's lower than 1, Dirichlet's approximation theorem). I tried to take derivative of it in respect to n, but I'm not sure if it's even legal, and I might lose some of my answers. Any other methods?

Have a small elisp problem. Even though everywhere I read on the internet says it's possible to use variables bound by let inside of a defun in the same let like this:

(let ((something 0))
  (defun some-random-function ()
    (interactive)
    (message something)))

Doing that gives me the some-random-function: Symbol's value as variable is void: something error, as if the variable is unbound/out of scope. Am I mistaken, did something change, etc?

I'm using GNUemacs Version 24.5.1, if that matters. Would love somebody to clue me in as to what's going on here.

"Provide an example of function definition with no bound variables and at least one free variable. Is it still function definition? Justify your answer."

My initial reaction is "no". You cannot have a function definition with no bound variables. How can you? If my function definition is "Lx.x y" then "x" is bound and "y" is free, so how could you possible define a function without a bound variable?

But I'm doubting myself because of the way the question is worded. He says "Provide and example", not "Can you provide and example"... Am I way off here?

After long debate, I decided to change my properties convention from camelCase to PascalCase to conform with the C# standard.

Before, I was prefixing any private variable that had a property with an underscore. Example:

[SerializeField]
private int _someValue;
public int someValue { get { return _someValue; } }

Over time, I disliked this practice more and more. So, I bit the bullet and started. One crucial thing I forgot is the data assigned via the inspector. In one instance, there were over well over 50 occurrences of an inspector bound variable with a specific colour code.

On renaming this serialized field, instantly the values were lost. I had to revert my project version in an attempt to recover what I had lost. At this point, I nearly gave up. I considered developing some sort of editor tool to store the original value, then rename my original variable, and port the value over--but it didn't seem worth it at this stage of the project.

However, just today I discovered the solution. You can rename inspector bound variables without losing their data by using the [FormelySerializedAs] attribute in the UnityEngine.Serialization namespace.

Here is the Unity post on its usage.

I just wanted to post this here incase anyone else has encountered, or is encountering this problem.

Thank you for reading!

Every time I see a definition of "bound" vs "unbound" variable, I always see:

Bound: A bound variable is one that is within the scope of a quantifier.

Unbound: An unbound variable is one that is not within the scope of a quantifier.

Okay, I get that literal syntactic definition, but why would we ever use unbound variables? I just don't see the motivation, and the above definitions shed absolutely no light on the matter. I guess my questions are:

1. In the usual FOL you learn in an undergraduate classroom, are strings with unbounded variables even well-formed formulas?

2. Can you give an example of a meaningful sentence with an unbound variable?

3. Can you explain why you would ever need a sentence with an unbound variable? Can't you always bind your variables? I can't think of a situation where you wouldn't say about some variable x either that "there exists some x such that..." or "for all x's".

4. Where do you commonly see sentences with unbound variables? Why do I see them in encyclopedia articles that involve logic, but they're always warned against in intro to logic courses?

Thank you.