For questions on exponential sums, such as $\sum \exp(2\pi ix_n)$.

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0
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2answers
25 views

Most logical thing to do with these exponents and sums?

I'm doing homework for a programming class and came across this problem. There's no directions besides what I've shown, so I don't even know what it's asking me to do. What makes the most sense for ...
22
votes
1answer
323 views
+50

Show that $ \lim\limits_{n\to\infty}\frac{1}{n}\sum\limits_{k=0}^{n-1}e^{ik^2}=0$

TL;DR : The question is how do I show that $\displaystyle \lim_{n\to\infty}\frac{1}{n}\sum_{k=0}^{n-1}e^{ik^2}=0$ ? More generaly the question would be : given an increasing sequence of integers ...
2
votes
0answers
49 views

Subtle error with a module endormorphism on $\mathbb{Z}_8 \times \mathbb{Z}_8$

Let $a,b,c$ be arbitrary integers such that $a$ is odd and $(a,b,c)=1$. Let $R = \mathbb{Z}_8$, the set of all integer residues modulo $8$. Define an $R$-module endomorphism $\phi \colon R \times R$ ...
0
votes
4answers
51 views

If $A^{2x}=4$ what is ${A^{3x}-A^{-3x}\over A^x - A^{-x}}$

If $A^{2x}=4$ and $A > 0$, what is the numerical value of $${A^{3x}-A^{-3x}\over A^x - A^{-x}}$$ Could anyone find the solution and answer? Thanks
1
vote
0answers
26 views

When is the sum of three reduced rationals equal to an integer

When is the sum of three reduced rationals equal to an integer? This may be a duplicate of Question 1550437 but even if it is, there is no answer associated with this other question. Given three ...
1
vote
0answers
52 views

Matrix power eventually reaches partial order

Given an $n$ by $n$ matrix $A$ of non-negative integers and a column matrix $x$ of non-negative integers and of length $n$, and a partial order on the set $\{1, 2, 3, 4,\ldots, n \}$ (representing the ...
0
votes
1answer
31 views

Calculating Maximum Population Size After 106 Years, Starting With 3 Breeding Pairs

I have been looking all over for a formula or program to determine a maximum possible human population size within a $106$ year span. This specifically is in response the the claim by the Creationist ...
2
votes
0answers
47 views

Does this series converge? If so, to what?

i was solving some integral equations and some of them gave series whose convergence am not very sure of. Problem, if anyone can point how or to what the series converges, i will be more than glad. Am ...
0
votes
1answer
28 views

Notation for a specific summation

I'm trying to make the sum stop before the summation has a negative exponent. For example, I would want the sum to stop at $2^0$ in $2^3+2^2+2^1+2^0+2^{-1}$ The sum I'm dealing with is $$\sum ...
0
votes
1answer
19 views

Generate exponential weights (sum of all = 1)

I have $500$ observations and I want to make exponential weighted average of them. I want the weights to be something like $w_i = 0.999^t$ when $t$ is from $1$ to $500$ (num of observations). ...
10
votes
2answers
446 views

Sum of factors of a huge number.

I recently appeared in a math olympiad and it had this one question which had me stumped. This was a few weeks back and I have been looking for a way to find its answer ever since, but with no ...
1
vote
2answers
84 views

How to simplify a sum of exponential equation?

Suppose I have three constants $a, b, c\in R$. I have a formulation as $f=e^{ab}+e^{ac}$. Can I have some result like $f'=e^{a(b+c)}$. I know $f'$ does not hold. But I just want to combine the two ...
1
vote
0answers
14 views

How to simplify a sum for the total cost of a yearly payment including compound interest

I want to simplify the below sum for the total cost over a yearly payment including compound interest over n years. An example: we have 150 euros that need to be paid every year and an interest of ...
1
vote
0answers
17 views

Relation between the sum of the values of a polynomial $f$ over a finite field, and the additive character sum with $f$ as the polynomial argument

Let $F$ be a finite field, let $f(T) \in F[T]$ and let $\psi$ be the canonical additive character of $F$. If $\sum_{x \in F}f(x) = 0$, what can we say of $\sum_{x \in F} \psi(f(x))$?
0
votes
0answers
24 views

Can every constant be written in that way?

When we are working in the ring $\mathbb{C}[e^{\lambda x} \mid \lambda \in \mathbb{C}]$ where the elements are of the form $$\sum_{i=0}^n \alpha_i e^{\lambda_i x}$$ where $\alpha_i, \lambda_i \in ...
1
vote
0answers
29 views

A Fact Stated in Davenport's Multiplicative Number Theory

In his text Multiplicative Number Theory on page 9, Davenport mentions that another means of expanding the L-function is known and then mentions the fact that, $$ \mathcal{F} \sum_{n=1,n \; odd} ...
0
votes
1answer
12 views

Summation of gaussians

Suppose we have given constants $A_i, x_i (i=1..N)$ Is it possible to approximately calculate the sum of N gaussians in less than N iterations for any x? (may be with some preprocessing) ...
0
votes
1answer
30 views

Solving for coefficients in half sum of two exponential functions

Let $$G(x) = \frac{\exp(-ax) + \exp(-bx)}{2}$$ I need to find the coefficients $a$ and $b$. Given: A table of 10 values of x and their corresponding G(x) values. I need some help in figuring out ...
0
votes
1answer
26 views

Prove theorem that concerns to sums of exponentials.

I try to solve this theorem: Theorem: Let $x_1, x_2,...$ iid with exponential distribution with rate $\lambda$. The density function of $S_n$ is given by: $$ P(S_n \le ...
1
vote
3answers
87 views

Prove $\sum_{i=0}^n {{n \choose i} \times 2^i} = 3^n $ [duplicate]

I have to prove that $$\sum_{i=0}^n {{n \choose i} \times 2^i} = 3^n $$ Such that ${n \choose i} = \frac{n!}{i!(n-i)!} $ and $n $ is some arbitrary int I proved we can expand 2^I in a way such that ...
1
vote
2answers
56 views

log of summation expression

I am curious about simplifying the following expression: $\log \left(\sum_\limits{i=0}^{n}x_i \right)$ Is there any rule to simplify a summation inside the log?
0
votes
0answers
41 views

Average of additive character sum over all polynomials of degree at most $d$ over $GF(q)$

Let $F$ be a finite field with $q$ elements, let $E$ be its degree $n \geq 2$ extension and let $\psi$ be the canonical additive character of $E$. For $d \leq n $ let $$ \mathcal{P}_{\leq d} = \{f(x) ...
0
votes
1answer
22 views

binary search tree size calculation

how do I calculate the number of elements a binary search tree can hold if have the height? For example a tree of height 3 can have 7 elements (7=1+2+4), and a tree of height 4 can have 15 elements ...
0
votes
0answers
17 views

Additive character sum over additive subgroups of finite fields, with special monomial arguments

Let $F$ be a finite field with $q = p^n$ elements, let $\psi$ be a non-trivial additive character of $F$, let $m$ be an integer coprime to $q-1$, and let $K$ be a large subspace of $F$, say $K$ is the ...
3
votes
0answers
35 views

Additive character sum over intersection of additive and multiplicative subgroups of finite fields

Let $H$ be a multiplicative subgroup of the finite field $\mathbb{F}_q$ with $q$ elements, say $H$ is the subgroup of $d$-th powers, $d \mid q-1$. Let $L$ be a subspace of $\mathbb{F}_q$ over some ...
2
votes
3answers
75 views

How to find the value of this expression?

I just saw this question in one exam. Please help me solve it. I am not able to find any clue on where to begin. (ignore that tick it might be wrong)
2
votes
1answer
36 views

A more general Kloosterman-type sum

Let $\mathbb{F}_q$ be a finite field and let $a,b \in \mathbb{F}_q$ not both zero. Let $\psi$ be the canonical additive character on $\mathbb{F}_q$. The classical Kloosterman sum is given by $$ K(a,b) ...
0
votes
1answer
56 views

How to find the solution to this summation

This was a question asked in our exam and we have to write a code for it. We have to find the summation of following series $\log(\sum_1^n (e^{x_i}))$ where $1 < n < 10^6$ and $0 < x_i < ...
3
votes
3answers
92 views

solve equation with sum $\ln x-\ln(\sqrt[2]{ x})+\ln(\sqrt[4]{ x})-\ln(\sqrt[8]{x})+…=2$

How to solve this? Any advice? $$\ln x-\ln(\sqrt[2]{ x})+\ln(\sqrt[4]{ x})-\ln(\sqrt[8]{x})+...=2$$ Next step I do this $\sum\limits_{n=0}^\mathbb{\infty}(-1)^n\ln(x^{\frac{1}{2^n}}) = 2$ But I don't ...
2
votes
2answers
34 views

sum of fractional powers

Let $z=e^{i\theta}$ for some real number $\theta$, i.e. $z$ is complex number on the unit circle. Is there a formula for $z+z^{1/2}+z^{1/2^2}+\dots+z^{1/2^n}$? Here $n$ is a positive integer. I only ...
1
vote
2answers
28 views

Is $\int z^n e^{az}dz $ a combination of exponentials and polynomials?

We have $$I(n)=\int z^n e^{az}dz=\int z^n \left (\frac{1}{a}e^{az}\right )'dz=\frac{1}{a}z^ne^{az}-\frac{1}{a}\int nz^{n-1}e^{az}dz \\ \Rightarrow I(n)=\frac{1}{a}z^ne^{az}-\frac{1}{a}nI(n-1) \ \ ...
0
votes
0answers
43 views

Quantifier elimination in the structure of exponential sums

We consider the language $L=\{+, -, ' , T, 0, 1\}$ Let $\text{Exp}(\mathbb{C})$ (the exponential sums) be the structure in that we interpret $L$. We define $\text{Exp}(\mathbb{C})$ as the set of ...
0
votes
0answers
20 views

Population Growth-fraction exponential

Annual plants produce seeds that can over-winter for several years before germinating, but the "parent" plants do not survive themselves. An invasive annual produces S seeds in a growing season. Of ...
0
votes
1answer
28 views

Is the sum of two exponential distributed random no. is also exponential random number?

I am working on statistics after long time. Struggling with the basics. Is the sum of two exponential distributed random no. is also exponential random number?
1
vote
1answer
41 views

Upper bounding a sum of exponentials

Define $$\phi(a,b,k) = e^{-a/b} + e^{-a/(b+1)} + \cdots + e^{-a/(b+k)}.$$ For example, $$\phi(10,2,3) = e^{-10/2} + e^{-10/3} + e^{-10/4} + e^{-10/5}, $$ or $$\phi(10,2,3) = e^{-5} + e^{-3.33..} + ...
1
vote
0answers
16 views

Showing that $e^{z}$ series is concentrated around indices close to $|\Re(z)|$

So, I believe (but am having trouble showing) that the power series for $e^{z}$ is concentrated around indices close to $|\Re(z)|$ (or at any rate is negligible beyond those indices). More precisely, ...
-4
votes
1answer
35 views

Help me solve this [closed]

$(1-(1-(1-n)^{-1})^{-1})^{-1}$ Please tell me how to solve the abv sum. The ans is $n$
0
votes
0answers
33 views

Computation with Exponential Generating Function

I am working a problem on generating functions but am stuck at a technical part which I think is actually supposed to be really easy, but I just don't have much experience with such things. We're ...
0
votes
0answers
33 views

Is there a way to solve this problem - sum of powers

$N_1^a + N_2^a + \ldots + N_n^a$ Where a is any number. Example $40^7+ 37^7 + 35^7 + 32^7 + ... 1^n=R$ If Iknow what $R$ is and that $ N_n$ is restricted to $0 \lt N_n \le 40$, Is there a way to ...
0
votes
1answer
28 views

Simplifying sums of exponentials

I am trying to solve a tricky little problem involving sums of exponentials. Consider the sequences $\{a_m\}$ and $\{b_m\}$. Both of these sequences are finite and $a_m>0 ~\forall m$ and ...
2
votes
1answer
25 views

Exponent Subtraction With Same Base

Can someone explain how this is valid? Thanks so much! $3\cdot 2^{2n+2} - 3\cdot2^{2n} = 3\cdot2^{2n}$
0
votes
0answers
26 views

How to find last 9 digits of a large exponent?

I was doing some programming problem where I had to find the sum of last 9 digits of very large exponents. I can't even think of calculating that big powers, but I have to find last 9 digits of the ...
1
vote
0answers
52 views

When does sum of exponentially many exponential functions go to zero?

I have $K=e^{n\beta}$ positive numbers $D_1\leq D_2\leq ...\leq D_K$. I want to find the maximum value of $\beta$ for which $e^{-nD_1}+e^{-nD_2}+...+e^{-nD_K}$ goes to zero as $n$ goes to infinity. ...
4
votes
1answer
57 views

Infinite sum involving powers and factorials

I am interested in evaluating the following infinite sum \begin{equation} \sum_{n=0}^{\infty} \frac{\alpha^{n}}{n!}n^{\beta} \end{equation} where both $\alpha$ and $\beta$ are real numbers. However, ...
0
votes
1answer
41 views

Maximizing the sum of exponentials whose exponents sum to $N$

Let $N \geq 1$ be a sufficiently large integer, let $a > 1$ be a real number, and let $n_1, \dots, n_t$ be integers between $0$ and $K$, where $K$ divides $N$. I want to determine the following: ...
0
votes
0answers
40 views

Lower bound on sum of exponentials

Let $\mathbf{K}_{1:n}$ be a vector of known positive constants and $f(\mathbf{\theta}_{1:n})=\sum_{i=1}^n K_i \exp(\theta_i)$, for $\theta\in \mathbb{R}^n$. Is it possible to find a 'tight' lower ...
2
votes
1answer
56 views

Character sum of a type of “almost linear” surjective mappings over finite fields

Let $F$ be a finite field of characteristic a prime $p$ with $q$ elements and let $E/F$ be a finite extension of degree $n > 1$ over $F$. Let $\chi$ be the additive canonical character on $F$ ...
1
vote
3answers
90 views

Solve $e^{k_1/x}+e^{k_2/x}+\cdots+e^{k_N/x}=1$ for $x$

How can I solve $e^{k_1/x}+e^{k_2/x}+\cdots+e^{k_N/x}=1$ for $x$, where $N\geq 1, k_1,\ldots,k_N \in \mathbb{R}, k_1,\ldots,k_N < 0, x\in \mathbb{R}$ and $x >0$. I looked at the basic rules of ...
4
votes
0answers
188 views

Is Every transcendental entire function $f(z) = C + \exp a + \exp b + \exp c $?

Let $z$ be a complex number and let $f(z)$ be any transcendental entire function. Is it true that $f(z) = C + \exp a + \exp b + \exp c $ where $ a,b,c $ are entire functions of $z$ and $C$ is a ...
4
votes
1answer
63 views

question on identity of sums with exponent. [duplicate]

I want to show that for $x>0$: $$\sum_{n=-\infty}^\infty e^{-n^2\pi x}= \frac{1}{\sqrt{x}}\sum_{n=-\infty}^\infty e^{-n^2\pi / x}$$ It doesn't seem that a simple change of variables will do, like ...