For questions on Gauss sums, a particular kind of finite sum of roots of unity.
5
votes
1answer
82 views
Gauss Sum of a Field with Four Elements
I need to calculate a couple of Gauss sums to solve a problem I'm working on, but I keep getting the wrong answer because the absolute value of what I calculate is impossible for such a sum. Can ...
5
votes
1answer
147 views
A Gauss sum like summation
I would like to calculate the following sum.
Let $\zeta$ be a primitive $n$ th root of unity for some integer $n$. Here $n$ is not necessarily prime.
The sum is
$$\sum_{j=1}^n (-1)^j ...
4
votes
1answer
141 views
Determining the Value of a Gauss Sum.
Can we evaluate the exact form of $$g\left(k,n\right)=\sum_{r=0}^{n-1}\exp\left(2\pi i\frac{r^{2}k}{n}\right) $$ for general $k$ and $n$? For $k=1$, on MathWorld we have that
...
4
votes
1answer
69 views
Prime power Gauss sums are zero
Fix an odd prime $p$. Then for a positive integer $a$, I can look at the quadratic Legendre symbol Gauss sum
$$ G_p(a) = \sum_{n \,\bmod\, p} \left( \frac{n}{p} \right) e^{2 \pi i a n / p}$$
where ...
4
votes
1answer
92 views
The Legendre symbol for an integer but not the Jacobi symbol
Let $p$ be a prime number and $\big(\frac{a}{p} \big)$ be the Legendre symbol.
Then we have the equality
$\sum_{a=1}^{p-1} \big(\frac{a}{p} \big) \zeta^a =\sum_{t=0}^{p-1} \zeta^{t^2}$, where ...
3
votes
1answer
161 views
Gauss-type sums for cube roots of primes
(Quadratic) Gauss sums express square root of any integer as a sum of roots of unity (or of cosines of rational multiples of $2\pi$, if you will) with rational coefficients.
But Kronecker-Weber ...
2
votes
1answer
55 views
Number of solutions of $N(y^{2}+x^{3}=1)=p+2ReJ(\chi,\rho)$
This is similar to a question I recently asked about. It is from Ireland's Number theory book, ch.8, ex.27 b,c. I think I can do the first part of this question, but I think there might be a trick ...
2
votes
1answer
79 views
Quadratic Gauss Sums
Let $p$ be an odd prime and $\zeta \not = 1$ be a $p^{th}$ root of unity. Let $R$ denote the set of all quadratic residues in $\mathbb{F}_p^*$.
If $\alpha=\sum_{r\in R} \zeta^r$, prove that
$$\alpha ...
2
votes
1answer
26 views
Multi-dimensional MLE Guassian
I wonder that what is the mu and sigma formula MLE(maximum likelihood estimates) for a 3 dimension guassian ? It is the same form as 1 and 2 dimension (+ 1 mu and sigma for the new vector) ?
1
vote
1answer
74 views
Fractions in limits of a summation
What if on the sum there is a fraction in the limit?
$\sum_{m=k/12}^{k}$ or $\sum_{m=0}^{k/12+1}$
thank you very much!
what type of sequence is used for summing this type of interval?
1
vote
1answer
84 views
A Trigonometric Sum Related to Gauss Sums
This is a problem given to me by fractals on Art of Problem Solving. I couldn't solve it so I'm posting it here for some thoughts on it.
Let $$S = \sum_{j = 0}^{\lfloor n/2 \rfloor} ...
1
vote
1answer
35 views
Definition: Gauss Sum - Where is the error?
In my algebraic number theory lecture we defined Gauss sums as follows. However, I am quite unsure whether this definition is correct (our lecturer is quite absentminded at times). My intuition says ...
1
vote
1answer
78 views
Gauss Newton minimization of 2D linear function
Given the input-output relation:
$
\begin{pmatrix}
y_1 \\
y_2
\end{pmatrix}
=p_1
\begin{pmatrix}
p_2 & p_3 \\
p_4 & p_4
...
0
votes
1answer
52 views
Gauss elimination with partial pivoting doubts
I have the following doubts about Gauss algorithm with partial pivoting:
Say that I sum to the second row the first row multiplied by $k$. In the $L$ matrix, should I sum to the second row the first ...
0
votes
0answers
72 views
How to prove a generalized Gauss sum formula
I read the wikipedia article on quadratic Gauss sum. link
First let me write a definition of a generalized Gauss sum.
Let $G(a, c)= \sum_{n=0}^{c-1}\exp (\frac{an^2}{c})$, where $a$ and $c$ are ...
0
votes
1answer
45 views
Is there a way to directly compute maximum of a sum of several Gaussian functions?
I have a problem which goes as follows. I am trying to predict the value of a variable $x$. I also have a set of measurements (the actual context is an image) $x^i$. I know from some training ...
