Theory of geometric measurement – Is the rectifiability of currents independent of the choice of Riemannian metrics?

I apologize if this is a trivial question – GMT is not my area of ​​expertise but I am working on a proof which uses GMT extensively and I could not find answer to my question in the usual books.

assume $ (M ^ n, g) $ is a Riemannian variety, because $ j ≥ 0 $ denote by $ ℋ ^ j $ l & # 39; induces $ j $-Dimensional Hausdorff measurement. Let $ T ∈ D_j (M) $ designate a rectifiable $ j $-current $ (M, g) $, that is to say for all $ j $Platforms $ ω ∈ D ^ j (M) $:

T (ω) = ∫_A ⟨ω, vec {T}⟩ , θ , dℋ ^ j ; ;,

or $ A ⊂ M $ is a $ j $– rectifiable assembly (i.e. up to a $ ℋ ^ j $-nullset it is contained in a countable union of $ j $-dimensional sub-varieties of $ M $) $ vec {T}: M → Λ ^ j (TM) $ is a $ ℋ ^ j $– measurable section (a simple $ j $-vector composed of orthonormal vectors which orient the approximate tangent space $ T_x A $ of $ A $ at a.e. $ x $) and $ θ: A → ℕ $ is locally $ ℋ ^ j $integrable.

Suppose I chose a different Riemannian metrig $ g & # 39; $ sure $ M $. East $ T $ then always a rectifiable current w.r.t. $ (M, g & # 39;) $ and the Hausdorff measurement induced $ ℋ & # 39; ^ j $? Heuristically speaking, when the metric changes, both $ vec {T} $ and $ dℋ ^ j $ change and those changes should cancel. This is at least the case for the currents given by smooth sub-collectors. Is this more generally true, however?

nt.number theory – A homogeneous polynomial with zeros in a geometric progression

Let $ a_1, …, a_n in overline { mathbb {Q}} $ be algebraic numbers, such as $ frac {a_i} {a_j} $ is not a root of unity for $ i neq j $. Also, let $ m in mathbb {N} $.

Question: Is there a homogeneous polynomial $ P in overline { mathbb {Q}} (X_1, …, X_n) $ in $ n $ degree variables $ k $, such as $ P (a_1 ^ {m ^ k},…, a_n ^ {m ^ k}) = 0 $ for everyone $ k in mathbb {N} _0 $?

opengl – Geometric rendering problem

I am currently working on a 3D game engine with OpenGL 4 and C ++. The problem is that, I don't know why, my geometry is not well rendered, except for the primitives.


On the right, you can see a cube, it is rendered as expected.
In the center you can see a mesh that I created in Blender, it is not rendered as expected, on the left you can see my mesh rendered as expected BUT I have adjusted the mesh ; Z scale (and I shouldn't)

So, to be short: my stitches are not well proportioned.

I checked the coordinates of each vertex in blender and my project: they are the same.

I don't know if there is a problem with my dies, the only thing I know is that this problem only appears on the Z axis (Z is in place ).

Everything makes me think that somewhere a number is rounded but I do not see where

I leave some code here, which could be useful:

Mesh rendering code:

void RD_Mesh::render(RenderMode rndrMode) {
    if (rndrMode == RenderMode::Filled) {
        glPolygonMode(GL_FRONT_AND_BACK, GL_FILL);
    else {
        glPolygonMode(GL_FRONT_AND_BACK, GL_LINE);


    glm::mat4 mdl = glm::mat4(1.0f); //Declaring Model Matrix

    glm::mat4 translate = glm::mat4(1.0f);
    glm::mat4 scale = glm::mat4(1.0f);
    glm::mat4 rotation = glm::mat4(1.0f);

    translate = glm::translate(translate, glm::vec3(m_position.getX(), m_position.getY(), m_position.getZ()));

    scale = glm::scale(scale, glm::vec3(m_scale.getX(), m_scale.getY(), m_scale.getZ()));

    rotation = glm::rotate(rotation, glm::radians(m_rotation.getX()), glm::vec3(1.0f, 0.0f, 0.0f));
    rotation = glm::rotate(rotation, glm::radians(m_rotation.getY()), glm::vec3(0.0f, 1.0f, 0.0f));
    rotation = glm::rotate(rotation, glm::radians(m_rotation.getZ()), glm::vec3(0.0f, 0.0f, 1.0f));

    mdl = translate * rotation * scale;

    m_shader->SetMatrix("model", mdl);

    glDrawElements(GL_TRIANGLES, RAWindices.size(), GL_UNSIGNED_INT, 0);

Camera code:

void RD_Camera::SetupCamera() {
    projection = glm::perspective(glm::radians(FOV), (float)m_rndr->getWindowWidth() / m_rndr->getWindowHeigh(), m_near, m_far); //Projection matrix

    view = glm::lookAt(glm::vec3(m_pos.getX(), m_pos.getY(), m_pos.getZ()), glm::vec3(m_subject.getX(), m_subject.getY(), m_subject.getZ()), glm::vec3(0.0f, 0.0f, 1.0f)); //View matrix

    m_rndr->GetCurrentShader()->SetMatrix("projection", projection);
    m_rndr->GetCurrentShader()->SetMatrix("view", view);
    m_rndr->GetCurrentShader()->SetVec3("CamPos", m_pos);

My Vertex Shader:

#version 450 core

layout (location = 0) in vec3 aPos;
layout (location = 1) in vec3 aNormal;

out vec3 Normal;
out vec3 FragPos;

uniform mat4 projection;
uniform mat4 view;
uniform mat4 model;

void main()
    gl_Position = projection * view * model * vec4(aPos, 1.0);

    Normal = normalize(mat3(transpose(inverse(model))) * aNormal);
    FragPos = vec3(model * vec4(aPos, 1.0));

Ps: I am sorry for my English, it is not my mother tongue.

multivariable computation – What is the geometric interpretation of the integration of a multivariable function with respect to a variable?

Suppose we have a continuous function with several variables $ f (x, y) $. What would be the geometric interpretation of $ int f (x, y) dx $ or $ int f (x, y) dy $? Right now, I think that's the area under the projection curve on the xy plane, but I can't convince myself that it's true. Also, I remember it would be similar to a line integral if the function had a vector value and was parameterized, but I'm not sure that this idea still supports this one. Any thoughts or advice?

solving equations – why can't this geometric scene be drawn

I want to calculate the angle FEB in the figure below, but I can't draw this figure with the & # 39; GeometricScene & # 39; function, let alone solve it.

enter description of image here

GeometricScene[{"A", "B", "C", "E", "F"}, {Triangle[{"A", "B", "C"}], 
  PlanarAngle[{"B", "A", "C"}] == 20 °, 
  PlanarAngle[{"A", "B", "C"}] == 80 °, 
  PlanarAngle[{"A", "C", "B"}] == 80 °, Line[{"A", "F", "B"}],
   Line[{"A", "E", "C"}], 
  PlanarAngle[{"E", "B", "F"}] == 20 ° , 
  PlanarAngle[{"E", "C", "F"}] == 20 °}]

What can I do to find the answers to these questions in general?

Is there a relationship between the geometric mean and the arithmetic mean?

Is there a relationship between the geometric mean of a set and the arithmetic mean of a set? If I knew one and knew the number of terms in the set, how would I calculate the other?

Signature card of $ p $ – Geometric approximate paths at $ T ( mathbb {R}) $

Let $ f: (0, T) rightarrow mathbb {R} ^ d $ to be a rough p-geometric path and leave $ mathcal {G} _p ^ d $ to be the collection of all these paths. Does the Lyon signature card define a continuous bijection between
$ mathcal {G} _p ^ d $ and $ T ( mathbb {R} ^ d) $?

geometric topology – Hyperbolic fillings of lengths between 6 and 2 $ pi $

What is the longest slope $ gamma $ in addition to the Dehn surgery space of a cusped hyperbolic 3-collector $ M $? Here the Dehn surgery space is the space of the fillings so that the hyperbolic structure on the filling $ M ( gamma) $ can be achieved as a distortion of the original $ M $.

This question is related to Ken Baker's question:

Exceptional hyperbolic obturations of 3 cusped hyperbolic collectors

However, Ken's question is concerned with the total number of tracks in this add-in. This question focuses on the longest slope of this type, where the length is measured by displacement at the limit of a horobille trim (as in the configuration for the $ 6-the theorem or $ 2 pi $-Theorem).

Of course, it is possible that this question, as noted, has no workable answer, as there is no longest slope.

Here is a more carefully stated version:

What is the greatest $ L $ so that there is a family of tracks $ gamma_i $ in collectors $ M_i $ such as $$ lim_ {i to infty} length ( gamma_i) = L $$
and each $ M_i ( gamma_i) $ is a hyperbolic variety such that the hyperbolic structure cannot be realized as a deformation of the hyperbolic structure of $ M_i $?

Of course, the fact that the 6-theorem is clear implies that $ L geq 6 $. Also $ 2 pi $ Theorem says $ L leq 2 pi $.

Number of triples Geometric progression given a ratio [closed]

For the problem statement here
I can't finish test case # 10
The code is here

Anyway to speed up this approach?

geometry – Geometric 2 medians in one dimension

Given a set of points in a dimension, how can we find a geometric 2-median? We can easily find the median 1, which is equivalent to finding the median of a set of real numbers. In two or more dimensions, the complexity of the geometric problem with 2 medians is unknown.