## Verizon Sells Media Division, Including Yahoo!, to Apollo Global Management for \$5 billion

Verizon sells its media group, including Yahoo! to Apollo Global Management for \$5 billion.

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## graphs – What is the difference between Partition and Division?

I will explain this with an example :

Consider a set $$S$$ which contains a collections of sets with a constant k such that:
$$S = { {S_1}, {S_2}, {S_3},{S_4},……….{S_k} }$$

### In Partition of Set S :

Then we can form the set again by

$$S=S_1cup S_2cup S_3cup S_4………..cup S_k$$

$$where S_icap S_j = phi$$

$$and ineq j and i,jin mathcal{N}$$

### In Division of Set S :

We just have this

$$S=S_1cup S_2cup S_3cup S_4………..cup S_k$$

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## Double lines division representation

How to make a fraction to display a numerator on top of the denominator in Google Sheet? Posted on Categories Articles

## ac.commutative algebra – Rings with terminating division chains of a given length

Let $$R$$ be an integral domain. Given $$a,bin R$$, then a division chain for $$(a,b)$$ is a sequence where we take $$r_{-1}=a$$, $$r_0=b$$, and for each $$n>0$$ we take $$r_n=r_{n-1}s_n+r_{n-2}$$ for some $$s_nin R$$. We say that the division chain terminates if $$r_n=0$$, and it terminates at length $$n$$ when $$r_ineq 0$$ for $$i.

These concepts, of course, have a lot to do with Euclidean domains, etc…

I ran across the following very interesting fact, proved by Cooke and Weinberger in 1975. Let $$K$$ be a number field, and let $$R=mathscr{O}_{K,S}$$, the ring of $$S$$-integers, where $$S$$ contains all the infinite places of $$K$$. Assume (some appropriate version of) GRH. If the unit group of $$R$$ is infinite, and $$aR+bR=R$$, then there is a terminating division chain for the pair $$(a,b)$$ of length $$5$$. Under some additional restrictions, like $$S$$ has a non-infinite place, or $$K$$ has a real embedding, the number $$5$$ can be lowered to $$4$$ or $$3$$. Moreover, they give examples showing that these numbers are best possible in some cases.

I’m interested if the following appears anywhere in the literature: For each $$n>5$$, there exists an integral domain $$R$$ such that for any $$a,bin R$$ with $$aR+bR=R$$, there is a terminating division chain for the pair $$(a,b)$$ of length $$n$$; and there is some pair of comaximal elements in $$R$$ that doesn’t have a smaller terminating length. The example will have to be somewhat complicated, since it won’t be a ring of integers over a number field.

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## python – Integer division when implementing Karatsuba’s algorithm

So im currently implementing Karatsuba’s algorithm in python and I cam across a very odd issue. I will provide two sets of code and I believe that they both theoretically do the same thing, but provide different answers (one right and one wrong).
The first set of code:

``````def decompose(x, n):
a = int(((x)//10**(n)))
b = int(x % (10**(n)))
#print("a", a, "t", "b", b)
return a, b

def digits(x):
return len(str(x))

# both x and y are n digit numbers

def multiply(x, y):
n2 = min(digits(x), digits(y))//2
if x < 10 or y < 10:
return x*y
a, b = decompose(x, n2)
# print("a", a, "t", "b", b)
c, d = decompose(y, n2)
# print("c", c, "t", "d", d)

if min(digits(a), digits(b), digits(c), digits(d)) == 1:
ac = int(a*c)
bd = int(b*d)
adbc = int((a+b)*(c+d) - a*c - b*d)
else:
ac = multiply(a, c)
bd = multiply(b, d)
adbc = multiply(a+b, c+d) - ac - bd

return (10**(2*n2))*ac + (10**(n2))*(adbc) + bd

x, y = 3141592653589793238462643383279502884197169399375105820974944592, 2718281828459045235360287471352662497757247093699959574966967627
print(multiply(x, y))
print(x*y)

``````

Second set of code:

``````def decompose(x):
n = digits(x)
a = int(((x)//10**(n//2)))
b = int(x % (10**(n//2)))
#print("a", a, "t", "b", b)
return a, b

def digits(x):
return int(len(str(x)))

# both x and y are n digit numbers

def multiply(x, y):
n2 = min(digits(x), digits(y))//2
if x < 10 or y < 10:
return x*y
a, b = decompose(x)
# print("a", a, "t", "b", b)
c, d = decompose(y)
# print("c", c, "t", "d", d)

if min(digits(a), digits(b), digits(c), digits(d)) == 1:
ac = int(a*c)
bd = int(b*d)
adbc = int((a+b)*(c+d) - a*c - b*d)
else:
ac = multiply(a, c)
bd = multiply(b, d)
adbc = multiply(a+b, c+d) - ac - bd

return (10**(2*n2))*ac + (10**(n2))*(adbc) + bd

x, y = 3141592653589793238462643383279502884197169399375105820974944592, 2718281828459045235360287471352662497757247093699959574966967627
print(multiply(x, y))
print(x*y)

``````

In the first set, I pass n2, where n2 = 32 because x and y have 64 digits. In this case I get the correct answer for the product of x and y. However, in the second set of code, I manually find the number of digits and then divide it by 2 using integer division, which should also give 32 but gives me the wrong answer. Any help would be appreciated.

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## nt.number theory – Explicit construction of division algebras of degree 3 over Q

In his book Introduction to arithmetic groups, Dave Witte Morris implicitly gives a construction of central division algebras of degree 3 over $$mathbb{Q}$$ in Proposition 6.7.4. More precisely, let $$L/mathbb{Q}$$ be a cubic Galois extension and $$sigma$$ a generator of its Galois group.If $$p in mathbb{Z}^+$$ and $$p neq tsigma(t)sigma^2(t)$$ for all $$t in L$$, then
$$D=left{ begin{pmatrix} x & y & z\ psigma(z) & sigma(x) & sigma(y)\ psigma^2(y) & psigma^2(z) & sigma^2(x) end{pmatrix} :(x,y,z)in L^3 right}$$
is a division algebra.

On page 145, just before Proposition 6.8.8, Morris claims that it is knows that every division algebra of degree 3 arises in this manner. This should follow from the fact that every central division algebra of degree 3 is cyclic. I could not find this explicit construction in my references (e.g. Pierce – Associative Algebras, though maybe I missed something) and I would like to know if there is a reference or a quick way to see that this exhausts all central division algebras of degree 3 over $$mathbb{Q}$$.

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## nt.number theory – Multiplication law in a division algebra of dimension 9 over a non-archimedean local field

Let $$k$$ be a non-archimedean local field, for example, a $$p$$-adic field (a finite extension of the filed $${Bbb Q}_p$$ of $$p$$-adic numbers).
It is well known that there is a canonical isomorphism
$${rm inv}colon {rm Br},koversetsimlongrightarrow{Bbb Q}/{Bbb Z}.$$
Let $$D$$ denote the division algebra of dimension 9 over $$k$$ with invariant $$frac13$$.

Question. How can one explicitly describe the multiplication law in $$D$$ ?

Motivation. From the multiplication law in $$D$$, I can obtain the commutation law in the 8-dimensional Lie algebra $${frak g}={frak sl}(1,D)$$.
From $$frak g$$, I can obtain an explicit trilinear alternating form on the 8-dimensional space $$frak g$$:
$$(x,y,z)mapsto ((x,y),z)quadtext{for },x,y,zin{frak g},$$
where $$(,,,)$$ denotes the Killing form.
This is a $$k$$-form of a generic alternating trilinear form on $$k^8$$.

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## División por zero en php

tengo una función que me calcula los likes y dislikes, para luego hacer un for con estrellas para el tema del rating. Pero tengo el problema de que si tiene 0 likes y 0 dislikes, luego al dividir salta un error de división por cero:

Esta es mi función:

``````function devolverRating(\$likes,\$dislikes){
return round(\$likes/(\$likes+\$dislikes)*5);
}
``````

Y el for con el que pinto las estrellas en función del rating que me devuelve, pero hay veces que a lo mejor me devuelve un 2 y me pinta las 5 estrellas o que tiene de rating 0 y me pinta 5 estrellas, sin contar lo de división por cero:

``````for(\$i=0;\$i<5;\$i++){
if(\$i<=\$rating){
echo "<i class='fas fa-star'></i>";
}
else{
echo"<i class='far fa-star'></i>";
}
}
``````

## python – Complex division for imaginary part

I seek a fast implementation of `(X / Y).imag`, where `X, Y` are complex 2D arrays (PyTorch tensors already on GPU). My approach is to move computation to a CUDA kernel via `cupy`, interfacing C and Python. Requirements:

• Faster than `(X / Y).imag`
• Using same or less memory than `(X / Y).imag`
• Output is same as `(X / Y).imag` within float precision
• `#include`s must be supported by CuPy (which is largely limited to standard CUDA libraries afaik)

My attempt:

``````extern "C" __global__
void cdiv_imag(float x(240)(240000)(2), float y(240)(240000)(2),
float out(240)(240000)) {
int i = blockIdx.x * blockDim.x + threadIdx.x;
int j = blockIdx.y * blockDim.y + threadIdx.y;
if (i >= 240 || j >= 240000)
return;

float A = x(i)(j)(0);
float B = x(i)(j)(1);
float C = y(i)(j)(0);
float D = y(i)(j)(1);

out(i)(j) = (B*C - A*D) / (C*C + D*D);
}
``````

This uses half the memory, since `X / Y` yields a temp array of reals & imaginary, but is still ultimately slower per my benchmarks on GTX 1070:

``````0.02320581099999992 sec (avg of 1000 runs)
0.01924530900000002 sec (avg of 1000 runs)
``````

Full code; Win 10 x64, Python 3.7.9, CuPy 8.3.0, PyTorch 1.8.0.

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