## 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);
}

m_mat->BindMaterial();

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);

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

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

//Rotation
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;

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

}
``````

``````#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.

``````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 °}]
RandomInstance[%]
``````

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.