dg.differential geometry – Dirichlet to Neumann operator and the Riesz transform

Consider the manifold $M := mathbb{R}^3 setminus B_1$ where $B_1$ is the unit ball. Equip $M$ with an asymptotically flat metric $g$ of high order. Let $gamma$ be the induced metric on $partial M$.

The Riesz transform is the bounded linear map $R: L^2(partial M) to L^2 (Lambda^1 (T^* partial M))$ defined by

$$R (f) = d (- Delta_{gamma})^{-frac{1}{2}}$$

Define another operator $R^* : L^2(partial M) to L^2 (Lambda^1 (T^* partial M))$ by

$$R^*(f) = d N_g^{-1} (f)$$

Where $N_g: H^1(partial M) to L^2(partial M)$ is the Dirichlet to Neumann map, which is defined by the following: if $h = N_g(f)$, then $h = nu cdot nabla u $ where $nu$ is the unit normal vector field on $partial M$ and $u$ is the unique function on $M$ that goes to $0$ at infinity and satisfies $Delta_gu=0$ and $left. u right|_{partial M} = f$.

It is well known that $N_g$ is a pesudo differential operator of order 1 with principal part $- (-Delta_{gamma})^{frac{1}{2}}$. Also, $N_g$ is invertible (that’s not true on bounded domains).

My question will not be very specific. I want to understand $R^*$ more as well as the relationship between $R$ and $R^*$.

Is $R^*$ also a bounded operator? If so how does $Vert R^*Vert$ depend on $g$? What do we know about $R-R^*$? I would imagine this is a “small” operator in some sense because $N_g$ is “close” to $- (-Delta_{gamma})^{frac{1}{2}}$. Maybe $R-R^*$ is a smoothing operator (which is a psuedo differential operator of order $-infty$.)? Is there anything in the literature about $R^*$?

You can assume $g$ is the euclidean metric and so $gamma$ is the round metric on the sphere if that makes the question easier. In that case, I think $N_g = – sqrt{Delta_{gamma} + frac{1}{4}} + frac{3}{2}$

Any help or references will really be appreciated.

unity – How can I use rigidbody to addforce to transform movement and to move the transform to a target?

using UnityEngine;
using System.Collections;

public class Move : MonoBehaviour
{
    public Transform target;
    public float force = 1f;
    public Rigidbody rb;

    private void Start()
    {
        rb = GetComponent<Rigidbody>();
    }

    private void FixedUpdate()
    {
        var distance = Vector3.Distance(transform.position, target.position);

        rb.AddForce(transform.forward * force);
    }
}

In the screenshot the components on the transform. I added a Rigidbody and freezed the position on X and Y. Now the transform is moving straight getting force slowly.

The problem is if I’m not locking the X and Y on the Constraints the transform will fall down.
In the other hand locking the X and Y will prevent it to rotate or move to specific directions isn’t it ?

transform components settings

Now I want to add few stuff to it :

  • To make the transform to move to a target not just to move nonostop.

  • To make that when the transform reach a specific speed then stay at this speed for example speed 7.

  • To make that when the transform for example is in distance 10 from the target start removed force slowly until the transform is reaching the target then the force should be 0 so the transform will stop slowly smooth at the target.

  • To add a rotation to the transform when the transform start moving to the target he also will rotate facing the target.

unity – How can I increase/decrease the transform movement speed according to the distance from the target with rotation?

using System.Collections;
using System.Collections.Generic;
using UnityEngine;

public class Move : MonoBehaviour
{
    public Transform target;
    public float speed;
    public float rotationSpeed;

    private Vector3 originalPosition;
    private Quaternion originalRotation;

    // Start is called before the first frame update
    void Start()
    {
        originalPosition = transform.position;
        originalRotation = transform.rotation;
    }

    // Update is called once per frame
    void Update()
    {
        var distance = Vector3.Distance(transform.position, target.position);
        if(distance < 0.1f)
        {
            transform.rotation = Quaternion.LookRotation(originalPosition);
        }

        transform.rotation = Quaternion.Lerp(transform.rotation, target.rotation, Time.deltaTime * rotationSpeed);
        transform.position = Vector3.MoveTowards(transform.position, target.position, Time.deltaTime * speed);
    }
}

So now it’s start moving and the same time rotating smooth to facing the target.
but I want that the speed will not be static but for example the speed start at 3 so instead I want the speed to start at let’s say 1.5 and to slowly increase until to 3 so it will looks like power effect like the transform take power the speed increasing and then I want that when it’s getting close to the target to smooth decrease the speed so when it’s getting to the target for example distance 0.1 from the target then the speed should be get already to 0.

Same logic the back way when the transform reached the target rotate slowly back to the original position/rotation and again increase the move speed to 3 and when getting close to the original position start slowly decrease the speed until 0.

One of the problems I always getting messed with is how to calculate the needed rotation speed according to the movement speed. If the transform move at speed 10 the rotation speed should be higher then 3 but how much higher ? And same of the transform is slower speed of movement then what the rotation speed should be ? In some cases the transform lost for some time the track because he couldn’t finish the rotation fast enough according to the movement speed.

Where should transform method to convert from one class object to another class object should live in Java?

I have 2 POJO classes, DatabaseRecord and MyCustomRecord. In my service logic, I wanted to convert the object of type DatabaseRecord to object of type MyCustomRecord.

The code for converting should be as simple as below:

public MyCustomRecord toMyCustomRecord(DatabaseRecord databaseRecord) {
  MyCustomRecord customRecord = new MyCustomRecord();
  customRecord.setA(databaseRecord.getA());
  customRecord.setB(databaseRecord.getB());
  ...
  ...
  return customRecord;
}

Where should this method should live? These are possible places I see:

  1. DatabaseRecord POJO class
  2. MyCustomRecord POJO class
  3. The class which is doing this transformation. Can be a private class.
  4. Separate class using pattern like Adapter.

What are the reasons to choose/not to choose these locations for doing this transformation?

algorithms – Cube 3×3: Using just face rotations, transform the cube from configuration A to B

I’m looking for some tips on how to best implement the algorithm to accomplish the following:

  • Imagine a 3×3 cube with a rubik’s cube mechanism, i.e. you can freely rotate any of the 6 faces.
  • You don’t care about colors.
  • Instead you index the component 26 cubes A–Z and ignore the middle.

Illustration image (component cubes): https://paste.pics/AOYUS

Now, I would like to generate a list of face rotations that would get me from this configuration to some other configuration. E. g. this: https://paste.pics/AOYWM

Note: I don’t care which component cubes end up in positions labeled as *

Ideally the algorithm produces the shortest list of transformations necessary, but it’s not a firm condition. Any suggestions welcome, thanks!

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