## When a function space with compact-open topology has countable chain condition？

As in title，when a function space with compact-open topology has countable chain condition？ Are there some sufficient and necessary conditions？ Who give some references about this topic？

McCoy and Ntantu [Topological Properties of Spaces of Continuous Functions, Page 68] pointed out that Vidossich had prove that $$C_k(X)$$ has ccc if $$X$$ is submetrizable. Who can give a proof of this statement?

## Probability Density Function – Possible Values to produce a valid PDF

The question is :

f(x) = (5z/16x)t^2 + (z(x-y)/(4x))t (x,y,t belong to R)

Express z in term of x and y, given t a random continuous variable?

What are the possible values of z, t, y to produce a valid PDF?

## ASP.NET CORE validate is not a function – manual validate trigger

I have a simple asp.net core app generated from asp.net core MVC template. Scaffolded some controllers and views and I was good to go.

I have tested validation with jquery validate unobtrusive. When I use attributes in my models like [Required] everything works, the form is not submitted unless there is everything according to validation rules.

However, on a different page, I need to trigger validation manually. Both functions valid() and validate() throws

typeError validate() is ont a function

However when I go var pas = \$ ('form')[3] and call pas.jQuery331028614785259688062.unobtrusiveValidation.validate()

validation is triggered properly and displays a validation message. Problem is that this long jQuery string is every time different, therefore I cannot accept using just this string every time as this changes.

I have checked my scripts jquery validate is referenced properly, after jquery and validation scripts are initialized through @{await Html.RenderPartialAsync("_ValidationScriptsPartial");}

Any ideas?

## Moment generating function of random unit vector

Let $$X$$ be uniformly distributed on the unit sphere $$S^{n-1}$$. Is there any result concerning the calculation or bound (particularly lower bound) of $$\mathbb{E}[\exp(X^Tv)]$$ for any $$v$$?

## Custom search function with result from external url

I need to create a page in Drupal 8 where there is a text field and a search button. When the user has entered a word to search and hit the search button a list with a search result should be displayed.

The search should go against this endpoint: http://systembevakningsagenten.se/api/json/1.0/searchProduct.json?query=[SEARCHWORD] Where [SEARCHWORD] is replaced with for example Stockholm and results are shown.

If more than 5 results are shown a pager should show up.

Have search the internet for tips or solutions, but couldn’t find anything.. Is there anyone who has a clue that can point me in the right direction?

## Conditional Expectation: Intergrating indicator function multiplied by the joint denisity

I am currently reading “Measure, Integral and Probability” by Capinski, Marek (see p179). It includes some motivation for the definition of the conditional expectation. For example, given two random variables $$X,Y$$ with joint density $$f_{(X,Y)}$$ (and so the marginal and conditional densities), we want to show that for any set $$A \subset \Omega, A=X^{-1}(B), B$$ Borel, that $$\int_A\mathbb{E}(Y|X)dP= \int_A \mathbb{E}(Y)dP.$$ This is one of the defining condition of an conditional expectation. The book shows the following calculation, \begin{align} \int_A\mathbb{E}(Y|X)dP &= \int_\Omega 1_B(X)\mathbb{E}(Y|X)dP\ &= \int_\Omega 1_B(X(\omega))\left(\int_\mathbb{R}yf_{Y|X}(y|X(\omega))dy\right)dP(\omega)\ &=\int_\mathbb{R}\int_\mathbb{R}1_B(x)yf_{(Y|X)}(y|x)dy f_X(x)dx\ &=\int_\mathbb{R}\int_\mathbb{R}1_B(x)yf_{X,Y}(x,y)dxdy\ &= \int_\Omega 1_A(X)YdP\ &= \int_A YdP. \end{align} What I don’t understand is the second to last equality immediately above, i.e. $$\int_\mathbb{R} y \int_\mathbb{R}1_B(x)f_{X,Y}(x,y)dxdy = \int_\Omega 1_A(X)YdP .$$ I think it is a typo since $$X\in \mathbb{R}$$ and $$A \subset \Omega$$ — however, I cant figure the correction either!

## didBegin function doesn’t run and won’t execute

This entire code is written in Swift v5. I’m trying to make a cowboy/gunslinger game. I need a collision between the projectile (coming from the first cowboy) to hit the second cowboy. However, the didBegin function isn’t being run/called.

I’ve tried messing with the physics bodies, by using code from previous games with no prevail.

showPhysics = true.

Physics Category:

struct PhysicsCategory {     static let none              :UInt32 = 0     static let cowboy1Physics    :UInt32 = 0x1 << 1     static let cowboy2Physics    :UInt32 = 0x1 << 1     static let projectile1Physics:UInt32 = 0x1 << 0     static let projectile2Physics:UInt32 = 0x1 << 0 } 

Fire Bullet1 :

func fireProjectile1() {     let projectile1 = SKSpriteNode(imageNamed: "bullet1")     projectile1.position = cowboyNo1.position     projectile1.position.x += 5     projectile1.size = CGSize(width: projectile1.size.width / 4,           height: projectile1.size.height / 4)     self.addChild(projectile1)     projectile1.physicsBody = SKPhysicsBody(rectangleOf: projectile1.size )     projectile1.physicsBody?.isDynamic = true     projectile1.physicsBody?.categoryBitMask = PhysicsCategory.projectile1Physics     projectile1.physicsBody?.contactTestBitMask = PhysicsCategory.cowboy2Physics     projectile1.physicsBody?.collisionBitMask =  PhysicsCategory.none     projectile1.physicsBody?.usesPreciseCollisionDetection = true        projectile1.zPosition = 1     var actionArray = [SKAction]()      actionArray.append(SKAction.move(to: CGPoint(x: size.height + 1500, y: cowboyNo1.position.y), duration: TimeInterval(1.5)))     actionArray.append(SKAction.wait(forDuration: 1))     actionArray.append(SKAction.removeFromParent())     print("fire")     projectile1.run(SKAction.sequence(actionArray))   } 

Cowboy2 Physics:

func createCowboyNo2() {     cowboyNo2.position = CGPoint(x: 1800, y: 200)     cowboyNo2.zPosition = 1     cowboyNo2.size = CGSize(width: cowboyNo2.size.width / 2, height: cowboyNo2.size.height / 2)      // Cowboy2 physics     cowboyNo2.physicsBody = SKPhysicsBody(rectangleOf: cowboyNo2.size) // 1     cowboyNo2.physicsBody?.isDynamic = true // 2     cowboyNo2.physicsBody?.categoryBitMask = PhysicsCategory.cowboy2Physics // 3     cowboyNo2.physicsBody?.contactTestBitMask = 2 // 4     cowboyNo2.physicsBody?.collisionBitMask = PhysicsCategory.none     cowboyNo2.physicsBody?.usesPreciseCollisionDetection = true     physicsWorld.gravity = .zero     addChild(cowboyNo2)  } 

DidBegin:

func didBegin(_ contact: SKPhysicsContact) {     // 1     var firstBody:SKPhysicsBody     var secondBody:SKPhysicsBody     print("test")     if contact.bodyA.categoryBitMask < contact.bodyB.categoryBitMask{         firstBody = contact.bodyA         secondBody = contact.bodyB         print("contact body's")     }     else{         firstBody = contact.bodyB         secondBody = contact.bodyA         print("more contact bodys")     }     if (firstBody.categoryBitMask & PhysicsCategory.projectile1Physics) != 0 && (secondBody.categoryBitMask & PhysicsCategory.cowboy2Physics) != 0{         projectileDidCollideWithCowboy2(nodeA: firstBody.node as! SKSpriteNode, nodeB: secondBody.node as! SKSpriteNode)         print("remove child")     } 

DidCollideWithCowboy function:

func projectileDidCollideWithCowboy2(nodeA: SKSpriteNode, nodeB:     SKSpriteNode) {     print("hit")     nodeA.removeFromParent()     nodeB.removeFromParent()     score1 += 1 } 

The expected output is for the bullet to be removed and for the console to print “test”, but nothing is being printed. The projectile is firing and going straight past the cowboy. It seems as if the didBegin function isn’t being called at all in the code when it should be running.

## Fourier series representation of piecewise function

$${Expand} \; f(x)= \begin{cases} 2{A\over L}x & 0\leq x\leq {L\over 2} \ \ 2{A\over L}\left(L-x\right) & {L\over 2}\leq x\leq L \end{cases}$$

I have determined $$A_0$$ (but omitted) to be $$A_0=A$$. For $$\ A_n \$$ and $$\ B_n$$, I’m rather confused where to start. For $$A_n$$: $$A_n={2\over L}\left[2{A\over L}\int_{0}^{L\over 2}x\cos\left({2n\pi x}\over L\right)dx+2{A\over L}\left(L\int_{L\over 2}^L\cos\left({2n\pi x}\over L\right)dx-\int_{L\over 2}^Lx\cos\left({2n\pi x}\over L\right)dx\right)\right]$$ And a similar case to $$B_n$$, I’m not quite sure if I’m on the right path.

## C# convert object’s set “value” to function, if it is a function

I’m trying to execute a function when setting a certain variable to another value, for example:

myvar = new Action<string>(string s) { ///    etc.... } 

and the myvar would have a getter and setter like such:

public object myvar {    get {return null}    set {       value("hi there") // not working since "value" is technically not a function    } } 

so do I need to cast value as a function first? But what if the function is a different kind of function, like Func<string, int, string>, or a simple Action<string> ? And I only want to activate the function if it is a function, so how would I check first?