A transformation that combines two smooth functions within a limited subset of domain?

Suppose we have two smooth functions $ f (x), g (x) $, and we would like to get another smooth function $ h (x) $ that's the closest to $ g (x) $ as possible within a bounded subset of the domain and as close as possible to $ f (x) $ as possible in the remaining part of the domain, formally,

  • $ | h (x) – g (x) | < epsilon $ for $ x in (a, b) $
  • $ | h (x) – f (x) | < delta $ for $ x in (- infty, a) cup (b, + infty) $

where $ epsilon> 0, delta> 0 $

We are looking for an analytical form of a transformation for this purpose, that is $ h (x) = mathcal {T} {(a, b)} (f (x), g (x)) $.

Is there any existing or closely related literature in the community?

Note: To describe the problem in the simplest form, we only use a one-dimensional case, but it demonstrates the same problem in more general cases (for example, scalar value function with multiple variables)

Diagonalization of linear transformation with unknown vectors based

I have been working for a few hours with this problem, but I still can't fix it. The problem says the following:

Dice $ B = {V_1, V_2, V_3 } $ Y $ B & # 39; = {V_1, V_1 + V_2, -V_1-2V_2-V_3 } $ , basis of a vector space $ V $Y $ f: V maps to V $ a linear transformation such that $ M_ (BB) & # 39; = $$ begin {pmatrix} 5 & -2 & 2 \ 0 & 1 & a \ 0 & -1 & -4 end {pmatrix} $$ $ find, if possible, $ a in Re $ such that $ 2V_2 – V_3 $ It is an autovector.

What I did: I know that for a linear transformation to be diagonalizable, then the standard matrix associated with that transformation must also be diagonalizable. However, in this case, I am completely incapable of constructing the standard matrix, because I do not know the components of the V1, V2 and V3 vects that work as the basis for V. I have reviewed my bibliography, because I feel that there should be another way of find a diagonalization of an LT without resorting to the standard matrix, but I still haven't found anything. Could someone guide me in the right way to approach this exercise?
Thank you very much in advance.

feats: when True Shapeshifter is used, does the change trigger the healing transformation?

True Shapeshifter (level 20 druid feat) says

… While under the effect of Wild Shape, you can change to any of the other forms in your Wild Shape list …

I wonder if this triggers the healing transformation

… If your next action is to cast a non-cantrip polymorph spell that only targets one creature … restore life points …

I guess that launching Wild Shape triggers Healing Transformation, because it uses the approach, but does True Shapeshifter trigger Wild Shape and, therefore, Healing Transformation?

linear algebra: consider a linear transformation L: R ^ 3 -> R ^ 3 with eigenvalues, -1,0,1 with eigenvectors respectively v1, v2, v3.

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MS Word: When transforming a photo, what causes the photo to have white transformation points versus gray transformation points?

For the photos shown, the same camera was used, but some transformation points (see photos) are white, while other photos have gray points. What causes the difference? What does the difference mean?

White transformation points allow users to select from prefabricated Correction options, where gray transformation points only allow users to change brightness directly

White transformation points

image

Gray transformation points

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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} $.

Why the "monoidal" transformation?

In Carlo Beenakker's answer to this recent MO question, it turns out that the name "monoid" was first used in mathematics by Arthur Cayley for an order surface $ 𝑛 which has a multiple order point $ 𝑛 – 1 $.

On the other hand, a (old) name for the explosion is "monoidal transformation." I suspect there should be a link that relates these two classic terminologies, but I couldn't find one. So let me ask you the following

Question. Why is the explosion known classically as "monoidal transformation"?

sprites: how to send transformation matrices to the shader, but that 6 vertices use the same matrix

I use a sprite atlas to represent a number of different sprite images on the screen, using the code below.

At the heart of all this are the mPositions Y mTextureCoordinates matrices that assign different parts of the texture to different areas of the screen.

I currently deal with translation, rotation and scaling in my own Java software, so mPositions The matrix contains points that have already been transformed.

However, now I want to transfer this responsibility to the vertex shader, so I will have to pass to the shader a transformation matrix that encodes the translation, rotation and scale operations.

It is the correct way to do it by entering another matrix mTransformations, let's say, and passing that through a GLES20.glVertexAttribPointer ?

If so, I find this a bit useless because every 6 consecutive vertices will have the same transformation matrix. Is there any way to store an array by 6 mPositions vertices and have the shader apply it to 6 consecutive vertices, then move on to the next matrix?

In addition, since I am now going to use the shader to perform the transformation, it also seems a waste to send canonical square coordinates to the shader through mPositions. In effect, I would have to fill mPositions with the same coordinates over and over again for each sprite. Despite this redundancy, is this still the right way to achieve what I am trying to achieve?

Perhaps there is some more efficient way to achieve what I want to be unaware?

    mPositions.position(0);
    mColors.position(0);
    mNormals.position(0);
    mTextureCoordinates.position(0);

    GLES20.glVertexAttribPointer(mShader.mPositionHandle, mPositionDataSize, GLES20.GL_FLOAT, false, 0, mPositions);
    GLES20.glVertexAttribPointer(mShader.mColorHandle, mColorDataSize, GLES20.GL_FLOAT, false, 0, mColors);
    GLES20.glVertexAttribPointer(mShader.mNormalHandle, mNormalDataSize, GLES20.GL_FLOAT, false, 0, mNormals);
    GLES20.glVertexAttribPointer(mShader.mTextureCoordinateHandle, mTextureCoordinateDataSize, GLES20.GL_FLOAT, false, 0, mTextureCoordinates);

    GLES20.glDrawArrays(GLES20.GL_TRIANGLES, 0, 6*numSprites);
    mPositions.position(0);
    mColors.position(0);
    mNormals.position(0);
    mTextureCoordinates.position(0);

    GLES20.glVertexAttribPointer(mShader.mPositionHandle, mPositionDataSize, GLES20.GL_FLOAT, false, 0, mPositions);
    GLES20.glVertexAttribPointer(mShader.mColorHandle, mColorDataSize, GLES20.GL_FLOAT, false, 0, mColors);
    GLES20.glVertexAttribPointer(mShader.mNormalHandle, mNormalDataSize, GLES20.GL_FLOAT, false, 0, mNormals);
    GLES20.glVertexAttribPointer(mShader.mTextureCoordinateHandle, mTextureCoordinateDataSize, GLES20.GL_FLOAT, false, 0, mTextureCoordinates);

    GLES20.glDrawArrays(GLES20.GL_TRIANGLES, 0, 6*numSprites);

geometry: how to verify if a transformation contains a mirror reflection?

In a 2D plane, a transformation that contains a mirror reflection will change an acyclic graph directed from right to left. In mathematics, this can be done with an ABC triangle, we can verify in which direction the product crosses $ vec {AB} times vec {BC} $ is pointing to, p. if the ABC plane is on $ x-y $ space sign $ vec {AB} times vec {BC} $ in $ z $ Management can serve as criteria.

How about in 3D space? Given a transformation $ tau $ that transformed the cube $ ABCDEFGH $ inside $ A & # 39; B & # 39; C & # 39; D & # 39; E & # 39; F & # 39; G & # 39; H & # 39; $or a regular tetrahedron $ ABCD $ inside $ A & # 39; B & # 39; C & # 39; D & # 39; $Is there a way to verify if it has gone through a mirror reflection?

How is the case in 4D or even in a higher space?

Transformation – How to create a custom set of coordinates based on two points in Unity?

According to your comments, I understand that you want the second point to be on the Y + axis of the local coordinate system you are building, and that the local Z + axis corresponds to the global Z + axis. We can get this like this:

Matrix4x4 GetCoordinateSystemFromPoints(
              Vector3 origin,
              Vector3 pointOnYPlusAxis)
{    
    return Matrix4x4.TRS(
             origin,
             Quaternion.LookRotation(Vector3.forward, pointOnZPlusAxis - origin),
             Vector3.One
    );
}

You can transform a point of this local space into a world space with:

var worldPoint = matrix.MultiplyPoint3x4(localPoint);

And transform yourself from world space to local space with:

Matrix4x4 inverse;
Matrix4x4.Inverse3DAffine(matrix, ref inverse);

var localPoint = inverse.MultiplyPoint3x4(localPoint); 

The disadvantage of this is that it is somewhat more expensive to invest.

Instead, you could define your own CoordinateSystem structure that includes both the direct and the inverse matrix, or an origin and quaternion pair, so that it can be transformed both inside and outside this coordinate system frequently without repeatedly calculating an inverse matrix.