## ag.algebraic geometry – What are the main divisors of the source specified by the Galois group for an Abelian Galois map of curves?

To let $$f: X rightarrow Y$$ be an Abelian Galois coverage of non-singular complete curves over algebraic $$k$$where the order of the Galois group $$A$$ is coprime to the characteristic of $$k$$, We can display the function field $$K (X)$$ As a $$K (Y)$$ Vector space, and through Galois theory we know its structure as $$k (A)$$ Module is given by $$K (Y) (A)$$,

Given our assumptions, we see that $$K (X)$$ has a basis of eigenvectors for $$A$$, corresponding to the various one-dimensional representations of $$A$$ about $$k$$, It has an explicit basis from $$f_ lambda in K (X)$$ so that $$g.f_ lambda = lambda (g) .f$$ to the $$lambda$$ a character of $$A$$ with value in $$k$$, So the dividers shared this $$f_ lambda$$ are canonical and we have one for each character of $$A$$,

My question is, can you describe these dividers more geometrically?

I'm vaguely aware that all coverings of this form should come from Jacobian isogenies, but my knowledge of Abelian curve coverings is not great, so excuse me if this question is really simple.

## dg.differentialgeometrie – foiling of the tangent bundle from the exponential map

We first mention our motivation:
To the $$M = mathbb {R}$$ with the usual Riemann metric the Exp card $$exp: TM to M$$ is in shape$$(x, v) mapsto x + v$$
This map defines a leaf formation, the leaves of which are completely geodesic sub-diversity in relation to the Sasaki metric. You cut the zero section across. The number of intersections with the zero section is the same for all sheets. They are parallel in the sense that for every two two points $$x, y$$ in the $$M subset TM$$ and any geodetic curve connection $$x$$ to $$y$$ contained in the zero section is the tangent space of the sheet $$x$$ is transported parallel to that of $$y$$,

How can one generalize part or the whole situation in the case of any (compact) Riemannian manifold? Under what conditions on Riemannian manifold are all or part of these situations the case?

## discrete math – how many one-to-one functions are there that map from a set of cardinalities 10 to cardinalities 7, 11, 15, 20

Conceptually, this is a major problem for me. I'm pretty sure that's right, but I can't prove it to myself. Here is my attempt at an answer:

No injecting functions exist for cardinality 7 because the size of the codomain is smaller than the domain, which means that there must be an element in A that is associated with 2 or more elements in B, unless f is a subfunction.

for the cardinalities 11, 15, 20 = n I think it is P (n, 10). Which would be in the end $$n! over (n-10)!$$ right?

EDIT: I screwed up the definition of the P function: |

## gn.general topology – Holomorphic map after shrinking (Kollars lecture on the resolution of singularities)

I am reading Janos Kolloar's lecture on the resolution of singularities and have some problems understanding a detail in the proof of Thm. 1.5 on page 10:

Thm 1.5 (Riemann) To let $$F (x, y)$$ be an irreducible complex polynomial and $$C: = V (F (x, y)) subset mathbb {C} ^ 2$$ the corresponding Compex curve. Then there is one $$1$$-dimensional complex manifold $$bar {C}$$ and a real holomorphic map $$sigma: bar {C} to C$$ This is a biholomorphism, except at a finite number of points.

Proof. Since $$F$$ irreducible and $$partial F / partial y$$ only finally have a lot of common points $$Sigma subset C$$, By implicit function thoerem the first coordinate projection $$pi: C to mathbb {C}$$ is a local anylytic biholomorphisc $$C backslash Sigma$$,

We start by constructing a resolution for a small neighborhood of a point $$p in Sigma$$, To simplify the notation, it is assumed $$p = 0$$, the origin. To let $$B _ { epsilon} subset mathbb {C} ^ 2$$ denote the sphere with the radius $$epsilon$$ around the origin. chossing $$epsilon$$ small enough, we can assume that $$C cap {y = 0 } cap B _ { epsilon} = {0 }$$,

Next choose $$eta$$ small enough, we can also assume that the card is restricted

$$pi: C cap B _ { epsilon} cap pi ^ {- 1} ( Delta _ { eta}) to Delta _ { eta}$$

is right (???) and a local analytical biholomorphism other than at the origin, where $$Delta _ { eta} subset mathbb {C}$$ is the disk with the radius $$eta$$

Question: why shrink? $$eta$$ small enough suggests that the restriction of $$pi$$ to $$C cap B _ { epsilon} cap pi ^ {- 1} ( Delta _ { eta})$$ is correct map (from a topological point of view)?

In topology $$pi$$ is correct if compact for everyone $$K$$ Set the role model $$pi ^ {- 1} (K)$$ is also compact.

Some comments: I assume that from $$B _ { epsilon}$$ and $$Delta _ { eta}$$ The author means that to open Ball resp. Disc because in the case of closing the subset $$C cap B _ { epsilon}$$ would be compact and there would be no need to shrink $$eta$$ in an "appropriate" manner. I've already asked the same question in MSE without getting an exact answer that solves the problem.

## Semi-groups and monoids – prove that the given map is isomorphism

Maybe it's so easy for this site. But I couldn't get over it.

To let $$X$$ be a set and $$T_X$$ be the complete transformation semigroup on X. If $$e in T_X$$ an idiot, and if $$Y = text {image} (e)$$, then

• the map $$eT_Xe rightarrow T_Y$$. $$f to f | _ {Y}$$ is an isomorphism and $$| text {image} (f | _ {Y}) | = | text {image} (f) |$$ for all $$f in eT_Xe$$,

How can we prove that this map is isomorphism?

Actually, I try to prove after the map that it is isomorphism:

To let $$mathcal {C} _X cong mathcal {C} _n$$ be the monoid of self cards $$alpha$$ of $$X = {1 dots, n }$$ these are orderly ($$forall x, y$$. $$x le y$$ $$Rightarrow$$ $$alpha (x) le alpha (y)$$ and decreasing ($$forall x$$. $$alpha (x) le x$$).

To let $$Y = text {image} ( alpha ^ 2)$$ and $$| Y | = m$$,
Then $$alpha mathcal {C} _X alpha rightarrow mathcal {C} _Y cong mathcal {C} _m$$. $$f to f | _ {Y}$$ is an isomorphism and $$| text {image} (f | _ {Y}) | = | text {image} (f) |$$ for all $$f in alpha mathcal {C} _X alpha$$,

## Polynomial homogeneous map

Assume that $$f (x) = (p_1 (x), p_2 (x), p_2 (x))$$ is a homogeneous polynomial diffeomorphic map of $$R ^ 3 setminus {0 }$$ on to $$R ^ 3 setminus {0 }$$ i.e. $$f (tx) = t ^ n f (x)$$, to the $$t> 0$$ and some $$n in N$$, Why $$n$$ should be 1?

I am fairly new to programming and this is the second thing I did in Python.

This code creates a / Tiles directory in the folder where the script is run, and then creates an X (8) number of folders in / Tiles.

Within these 8 folders, a Y number of subfolders is created and a Z number of images per Y folder is downloaded.

Y and Z are the same number, but change depending on the X folder.

X folder 0 contains 2 ^ 0 Y folders and 2 ^ 0 Z images per Y folder.

Next X folder 2 ^ 1 etc. to 8 folders or 2 ^ 7.

Any general or python specific advice for a newbie?

I thought of a kind of loading bar because there are ~ 20,000 pictures and I looked at this https://stackoverflow.com/questions/3173320/text-progress-bar-in-the-console

But in the end I decided to just end this because I was done with the main functionality.

(..) means the code is on the same line, but I created a new code to read here without scrolling.

Thank you very much.

``````import os
import urllib.request
import ctypes

print ("The further it gets the more time it will take to fill the current folder.")
print ("Because the number of images increases exponentially.")
print ("You can check the progress by going into the latest folder.")
print ("When it is done the taskbar icon will flash and prompt you to close the window.")

# stores current directory
path = os.getcwd()
print ("The current working directory is {}.".format(path) + "n")

# adds the names of the new directory to be created
path += "/Tiles/"

# defines the access rights
access_rights = 0o755

# creates the tiles directory
try:
os.mkdir(path, access_rights)
except OSError:
print ("Creation of the tiles directory {} failed.".format(path))
else:
# for testing:
# print ("Successfully created the tiles directory {}".format(path))

# number of x subfolders
x_folders = 8
# number of y subfolders and z images - different folders have a
(..) # different number of subfolders amd images
yz_array = (2**0, 2**1, 2**2, 2**3, 2**4, 2**5, 2**6, 2**7)

# main loop
# creates the x directories
for x in range(x_folders):
try:
os.mkdir("{}/{}".format(path,x), access_rights)
except OSError:
print ("Failed to create the X directory - {}.".format(x))
break
except:
print("Something went wrong creating X folders.")
break
else:

# creates the y directories
for y in range(yz_array(x)):
try:
os.mkdir("{}/{}/{}".format(path,x,y), access_rights)
except OSError:
print ("Failed to create the Y directory - {}.".format(y))
break
except:
print("Something went wrong creating Y folders.")
break
else:
pass

for z in range(yz_array(x)):
try:
(..)/{}/{}/{}.png"
.format(x,y,z), "{}/{}/{}/{}.png".format(path,x,y,z))
except:
break
else:
pass

print("Successfully filled folder {}{}.".format(path,x) + "n")

print("Finished running.")
ctypes.windll.user32.FlashWindow(ctypes.windll.kernel32.
(..)GetConsoleWindow(),True) # flashes the taskbar icon
print("Press Enter to close")
input("n")
``````

## 8 – Track a car on a live map using the driver's phone's built-in GPS

I have a small website about my company and I want to convert it to a Drupal 8 website. For that I want to make sure that it is possible to have a live map showing the distance traveled by each driver my company (Google Map or whatever other map).

It's like having a map and a blue line that shows the current distance traveled from point A to point B by car using the driver's phone's built-in GPS.

I need something similar to this GIF image:

Tack a car on a live map example

Is it possible in Drupal 8 above? and what are the names of these modules?

Thank you very much,