stochastic processes: variance of a random variable obtained from a linear transformation

Edit: I needed to review this question as suggested.

Suppose there are $ N $ Realizations of the Gaussian process denoted as vectors $ mathbf {z} _ {j} in mathbb {R} ^ {n} $ for $ j = 1, ldots, N $. Leave $ and $ be a random variable such that $ y = sum_ {j = 1} ^ {N} ( mathbf {B} mathbf {z} _ {j}) (i) $
where $ mathbf {B} $ It is a unitary matrix. What is the variance of $ y2?

Explanation: Boldface represents the vector or matrix. $ ( mathbf {B} mathbf {x}) (i) $ represents the $ i $-th vector entry $ mathbf {B} mathbf {x} $.

unit: random speed changes

I have a character that moves with a rigid body. When I jump, I seem to find, but sometimes, my character increases when he is in the air and leaves the map very quickly.
I checked it and, when it happens, it says that the speed is about 210 or around.
Why is it happening and how can i fix it? & # 39; using System.Collections;
using System.Collections.Generic;
using UnityEngine;

public class playerManager : MonoBehaviour
{
float xMovement;
float zMovement;
float xPosition;
float yPosition;
float zPosition;
public float runningSpeed;
public Animator animator;
public Camera camera;
public float cameraHeight;
public float cameraAngle;
public bool isOnGround;
public float jumpForce;
Rigidbody playerRigidbody;
public float gravityMultiplier;
Vector3 vel;
public float airSpeed;
public float groundSpeed;
public bool canJump = true;

void Start()
{
    playerRigidbody = GetComponent();
    vel = playerRigidbody.velocity;
}

void Update()
{

    xPosition = transform.position.x;
    yPosition = transform.position.y;
    zPosition = transform.position.z;
    PlayerMovements();      
    CameraManager();
}


public void CameraManager()
{
    camera.transform.position = new Vector3(xPosition+cameraAngle,cameraHeight,zPosition+cameraAngle);
}


public void PlayerMovements()
{
    //Jump
    if(Input.GetKeyDown(KeyCode.Space) & isOnGround == true & canJump == true)
    {
        runningSpeed = airSpeed;
        canJump = false;
        playerRigidbody.velocity = new Vector3(xMovement,0,zMovement);
        playerRigidbody.angularVelocity = Vector3.zero;
        playerRigidbody.constraints &= ~RigidbodyConstraints.FreezePositionY;
        playerRigidbody.AddForce(transform.up * jumpForce);
        playerRigidbody.drag = 0;

    }


    //Move the Character
    if(isOnGround == true)
    {
        xMovement = Input.GetAxis ("Horizontal");
        zMovement = Input.GetAxis ("Vertical");
        Vector3 distanceToMove = new Vector3(xMovement,0,zMovement);
        playerRigidbody.AddForce(distanceToMove * runningSpeed);
    }


    //Make the player face the direction he's going at
    transform.LookAt(new Vector3(xPosition+xMovement,yPosition,zPosition+zMovement));



    //Trigger Idle or Running Animations
    if(xMovement != 0 | zMovement != 0)
    {
        animator.SetBool("Run", true);
        animator.SetBool("Idle", false);
    }
    if((xMovement < 0.05f & xMovement > -0.05f) & (zMovement < 0.05f & zMovement > -0.05f))
    {
        animator.SetBool("Run", false);
        animator.SetBool("Idle", true);
    }           
}


public void OnCollisionEnter(Collision collidedOBJ)
{
    if(collidedOBJ.gameObject.tag=="Finish")
    {
        Debug.Log(playerRigidbody.velocity);
        Debug.Log(runningSpeed);
    }
    if(collidedOBJ.gameObject.name == "Ground")
    {
        playerRigidbody.drag = 49;
        playerRigidbody.constraints &= RigidbodyConstraints.FreezePositionY|RigidbodyConstraints.FreezeRotationX|RigidbodyConstraints.FreezeRotationY|RigidbodyConstraints.FreezeRotationZ;
        isOnGround = true;
        canJump = true;
        runningSpeed = groundSpeed;


    }
}


public void OnCollisionExit(Collision exitCollidedOBJ)
{
    if(exitCollidedOBJ.gameObject.name == "Ground")
    {
        isOnGround = false;

    }
}

}

Random QR matrices – MathOverflow

In the very good article by GW Stewart:

Stewart, G. W., The efficient generation of random orthogonal matrices with an application for condition estimators. (With mircofiche section), SIAM J. Numer. Anal. 17, 403-409 (1980). ZBL0443.65027.

The author gives the following algorithm to generate a uniform orthogonal matrix of Haar:
Leave $ M $ bean $ n times n $ matrix with Gaussian entries i.i.d. Leave $ M = QR $ be the $ QR $ decomposition of $ M $ ($ Q $ is orthogonal $ R $ it is triangular superior, with positive diagonal entries). So $ Q $ It is uniform hair.

My question is: What is the distribution of $ R? $

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C # – Making random values ​​in Java

I have a code that generates random values ​​in C #, but I want to pass it to javaHow could you do this?

uint ItemId;
                string ItemName;

                var rand = new Random().Next(1, 50);
                if (rand >= 45)
                {
                    ItemId = 7177;
                    ItemName = "rare_dragonlamp*10";
                }
                else if (rand >= 35 && rand < 45)
                {
                    ItemId = 7174;
                    ItemName = "pillar*10";
                }
                else if (rand >= 25 && rand < 35)
                {
                    ItemId = 7191;
                    ItemName = "scifirocket*10";
                }
                else if (rand >= 15 && rand < 25)
                {
                    ItemId = 7180;
                    ItemName = "rare_fountain*4";
                }
                else
                {
                    ItemId = 7180;
                    ItemName = "rare_fountain*4";

                }

graphics: combination of mesh regions of generated 3D shapes and random polygons

I am interested in generating 3D a joint BoundaryMeshRegion or a MeshRegion in different shapes and polygons. This works for RandomPolygon:

I):

r1 = BoundaryDiscretizeGraphics(
  Graphics(RandomPolygon(15, 
    DataRange -> {{0.02, 0.08}, {-0.02, 0.04}})));
r2 = BoundaryDiscretizeGraphics(
  Graphics(RandomPolygon(12, 
    DataRange -> {{0.00, 0.03}, {-0.02, 0.03}})))
bnd1 = RegionProduct(r1, MeshRegion({{0.01}, {0.05}}, Line({1, 2})));
bnd2 = RegionProduct(r2, MeshRegion({{0}, {0.02}}, Line({1, 2})));
un1 = RegionUnion(bnd2, bnd3);
Show(un1, Boxed->True, Axes->True)

Unified region of two 3D generations of RandomPolygon

I also wanted to include the shape of a country and combine it with a random polygon or other form of country in a similar way, which is possible as long as they are combined in 2D state:

ii):

r3 = BoundaryDiscretizeGraphics(CountryData("Spain", "Shape"), 
  ImageSize -> Medium, Axes -> True)
r4 = BoundaryDiscretizeGraphics(
  Graphics(RandomPolygon(15, 
    DataRange -> {{0.05, 0.3}, {-0.08, 0.08}})));
bnd3 = RegionUnion(r3, r4);
un2 = RegionProduct(bnd3, MeshRegion({{0}, {0.02}}, Line({1, 2})));
Show(un2, Boxed->True, Axes->True)

Unified region of two 3D generations of RandomPolygon and a country form respectively

However, as soon as an attempt is made to combine a generated 3D form of a country with a generated 3D form of the Polygon, the following results are obtained:

iii):

bnd31 = RegionProduct(r3, MeshRegion({{0.0}, {0.02}}, Line({1, 2})));
r5 = BoundaryDiscretizeGraphics(
  Graphics(RandomPolygon(8, 
    DataRange -> {{0.03, 0.12}, {-0.06, 0.06}})));
bnd32 = RegionProduct(r5, MeshRegion({{0.015}, {0.03}}, Line({1, 2})));
RegionUnion(bnd31, bnd32);

Iii)

In some cases, Mathematica crashes.

How would you solve this problem? My goal is to combine a 3D form of a country with a 3D form of RandomPolygon, so that both shapes have different heights from each other. However, the volumes must be connected to each other (similar to ii), since I would like to apply a RegionMemberFunction in a later step

The code for the generation of 3D shapes was found here:
How to extrude a 3D image from a binary 2D image

I am running in Mathematica 12.

In advance, thank you very much for your help!

probability: symmetry of the last stroke time of a simple random walk before a fixed time.

It is known that for the simple random walk in $ mathbb Z $ It started at $ 0 $tell $ (S_m, m in mathbb Z ^ +) $, the last batting time of $ 0 $ before $ 2n $ $$ T_ {2n} = max {m le 2n: S_m = 0 } $$
It has distribution:

$$ mathbb P (T_ {2n} = 2k) = frac {1} {2 ^ {2n}} {2k choose k} {2 (n-k) choose n-k} $$

See Durrett's book, for example. Taking the $ n to infty $ limit, one finds (after rescaling) the Arcsine distribution. This relationship is generally proven using the ballot theorem.

A striking feature of the above formula is symmetry when exchanging the role of $ k $ Y $ n-k $.

My question: is there a (simple?) Transformation in route / geometric that would help to understand this symmetry?

arrays – random URL Permanent link to Javascript

I want to add a random URL in my script section

try some random URLs with matrices

{var encryptedUrl=Base64.encode(a_to_vi);a_to_ve(i).href="https://randomurlinhere.net/page/"+ encryptedUrl+""+ User_ID;a_to_ve(i).rel="nofollow";a_to_vb++;a_to_vc+=i+":::"+ a_to_ve(i).href+"n";}}

I want to rotate the URL in the script

https://randomurlinhere.net/page/

https://randompermalink.net/page/

https://randomlinkhere.net/page/

Generate a random number in Unity without class ambiguity.

I have a problem in Unity (C #) where I would like to create a random number. I wanted to use System.Random (reference: using System) but Unity complains that it is ambiguous for them UnityEngine.Random. I cannot specify the reference (using System.Random) as random It is not a namespace. How do I specify that I want to use the random system and not the Unity?

c # – Generate unit of random numbers without class ambiguity

If you write explicitly System.Random it's too much, you can also create an alias for the type:

using Dice = System.Random;

...

Dice myDice = new Dice();

var randomNumber = myDice.Next();

Personally, I would recommend Iggy's suggestion to write the explicit type System.Random instead, there are not many additional letters and clearly express your intention.

The danger with changing the name of guys like this is that someone who reads your code (like you, six months from now, when you've forgotten that you did this) might think that Dice he is a completely new guy with unknown characteristics and wastes mental energy trying to figure out what he is doing when System.Random I would have made it immediately clear and familiar.