The exponential-polynomials tag has no usage guidance.

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773 views

### What is the Time Complexity of the Matrix Exponential?

While trying to compute the Matrix Exponential of an $n \times n$ array I decided to take advantage of a Python function called scipy.linalg.expm().
According to ...

**9**

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**1**answer

362 views

### Number of real roots of an exponential polynomial

Let $a_1, a_2, \dots, a_n$ and $b_1, b_2, \dots, b_n$ be real numbers, and assume that $\{a_i\} \neq \{b_i\}$. Can the equation
$$ e^{a_1 x} + e^{a_2 x} + \dots + e^{a_n x} = e^{b_1 x} + e^{b_2 x} + ...

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**3**answers

165 views

### Decidability of sum of powers exponential diophantine equation

I want to ask about decidability of exponential Diophantine equation:
$z_12^{\eta_1} + \ldots + z_n2^{\eta_n} = z$, where $z_i,\eta_i,z \in \mathbb{Z}$, and $\eta_i$ - are variables.
Can we find ...

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**0**answers

59 views

### Closed form solution for DDEs?

I am solving the equation $X−A−B e^{−Xy}−C e^{−X z}=0$ where $X, A, B$ and $C$ are 2x2 matrices and $y$ and $z$ are scalars. What will be the closed form solution ...

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**1**answer

86 views

### What is exponentially fitted osculating straight line?

While reading an article about iterative methods for solving nonlinear equations I can't understand what is exponentially fitted osculating straight line. Could someone please briefly explain this ...

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68 views

### exponential and anisotropic torus

Let $F$ be a local p-adic field and $G$ a semisimple simply connected group over $F$, $\mathfrak{g}$ its Lie algebra. Let $T$ a maximal anisotropic torus of $G$, split over an etale extension of $F$ ...

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449 views

### How to solve definite integral involving exponential function

I am trying to get a closed form for the following definite integral:
$$f(\theta)= \int_\frac{\pi}{2}^\pi \frac{1}{\sqrt{1-\alpha^2 \cos^2\theta}}\exp\left(C_2\cos\theta-C_1\sqrt{1-\alpha^2 ...

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986 views

### Why are there so few zero-dimensional polynomial system solvers and is this because there is no real market for them?

My questions involve the quotes below from wikipedia regarding solving polynomial systems, which given the size of the market for Big Data & Predictive Analysis applications I find puzzling:
...

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**0**answers

84 views

### Estimating decay of certain trigonometric polynomials

For $p=0,1,2,\dots$ and $n=0,1,2,\dots,$, let $f_{n,p}(z)=\sum_{k=0}^n k^p z^k$ be a sequence of polynomials. Restricted to the unit circle, the functions $g_{n,p}(t):=f_{n,p}(e^{it})$ are ...

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61 views

### Whether r.v. with p.g.f. $\exp [\sum\limits_{i = 1}^\infty {{q_i}({z^i}} - 1)]$ is overdispersion?

When discrete r.v. $X$ is not Poisson distributed and ${\rm{Var}}X,EX < \infty $, I want to know whether r.v. $X$ with p.g.f. $\exp [\sum\limits_{i = 1}^\infty {{q_i}({z^i}} - 1)],({q_i} \in ...

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**2**answers

821 views

### Efficient algorithms to determine the roots of: $p(x) = r^x $ in the finite field $GF(q)$, where $r$ a primitive root of the field

I need to make sure that no efficient (i.e., polynomial time) algorithm exists for the following problem:
Exponentiating Polynomial Root Problem (EPRP)
Let $p(x)$ be a polynomial with $\deg(p) \geq ...

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**2**answers

371 views

### Is equivalence of functions built from nested exponentiations a decidable problem?

Let $\mathcal{E}$ be the minimal set of symbolic expressions (without any predefined meaning) such that
The symbol $x$ is in $\mathcal{E}$, and
If expressions $P,Q\in\mathcal{E}$, then the ...

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**2**answers

508 views

### Order type of the smallest set containing the identity function and closed under exponentiation

Let $E$ be the smallest set of functions $\mathbb{N}^+\to\mathbb{N}^+$ containing the identity function $n \mapsto n$ and closed under exponentiation $(f,g) \mapsto \left(n \mapsto ...

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**1**answer

206 views

### Inequality of Partial Taylor Series

Hi,
For a given $\theta < 1$, and $N$ a positive integer, I am trying to find an $x > 0$ (preferably the smallest such $x$) such that the following inequality holds:
$$\sum_{k=0}^{N} ...

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**0**answers

102 views

### maximum of the sum of polynomials

Hi
I have $n$ polynomials, each one is positive over a certain range and the maximum value each can attain is 1. Also each polynomial has atmost one peak(maximum). Is there some way that I can show ...

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**0**answers

240 views

### solving in x involving both exponential and logarithmic function

Is it possible to solve a function with both exponential and logarithm such as
$a x^2 - b.\log(x) = c $
in closed form; where $a,b,c$ are constants and $a>0$ and $b>0$?

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**2**answers

1k views

### Sum of products of exponentials and polynomials

Hi,
I am looking for a closed-form expression for the finite sum of the product of an exponential function with a polynomial function --- that is, the sum
...

**5**

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**2**answers

449 views

### Sturm chain analogue for exponential polynomials?

I'm going to define an exponential polynomial of degree $k$ as a function $f$ of the form
$f(x) = \sum_{i=1}^k c_ie^{\alpha_ix}$ ($\alpha_i$s real).
My first question is: is there an algorithm for ...

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**1**answer

377 views

### Zeros of a combination of exponentials

Is there any known result about the necessary and sufficient conditions for the existence of zeros for a function $f(x)=\sum_{n=1}^{N} a_n e^{b_n x}$, where $a_n,b_n \in \mathbb{R}\, \forall ...

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**0**answers

202 views

### Introductory text book for Linear Recurrence Sequences

What is a good introductory text for linear recurrence sequences?
What all are the necessary prerequisite for it? (My background is in Euclidean Fourier Analysis.) After browsing through several ...

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**1**answer

1k views

### Queries about the Skolem-Mahler-Lech theorem (integer zeros of exponential polynomials)

The Skolem-Mahler-Lech Theorem says that the integer zeros of an exponential polynomial are the union of complete arithmetic progressions and a finite number of exceptional zeros. ...

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411 views

### Relation between entire function of exponential type and exponential polynomials

Is it true in general that the theory of entire function of exponential type and and that of exponential polynomials (with purely imaginary exponents) are analogous ?
Can one derive results about ...

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**3**answers

501 views

### How to find the almost period of an exponential polynomial

Let $u(t) = \Sigma_{k=1}^n c_k e^{i \lambda_k t} (c_k \in \mathbb C, \lambda_k \in \mathbb R) $ be an exponential polynomial of order $n$ with purely imaginary exponents. We can assume that the ...

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**4**answers

2k views

### Approximation by exponential polynomials

Let $u(t) = \Sigma_{k=1}^n c_k e^{\lambda_k t} (c_k \in \mathbb C, \lambda_k \in \mathbb C) $ be an exponential polynomial of $\underline{order}$ $n$.
Define $E_n$ to be the collection of all ...

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**1**answer

543 views

### Salem Inequality

I have come across this inequality in the paper "Local estimates for exponential polynomials and their applications to inequalities of the uncertainty principle type" ...

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**0**answers

927 views

### estimates of exponential polynomials

Let $ p(t) = \Sigma_{k=1}^n c_k e^{i \lambda_k t}$ be an exponential polynomial.
In the paper "Local estimates for exponential polynomials and their applications to inequalities of the uncertainty ...

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2k views

### Rational exponential expressions

Consider the following extension of polynomials. The rational exponential expressions (REXes) are given by:
The leaves 1 and $x$ for $x$ drawn from a class of variables; and
Closed under the binary ...