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### Non-crystallographic cluster algebras

Background
Fomin and Zelevinsky have introduced cluster algebras in an influential article. To define a cluster algebra, Fomin and Zelevinsky have defined a mutation of seeds. Here, a seed ...

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### Master theorem for probabilistically inspired recurrences

Is there a general solution for multi-variable recurrences of the following form which come from Markov chain analysis?
$f(i,j,k) = p_1 f(i-1, j,k) + p_2 f(i, j-1,k) + p_3 f(i,j,k-1) + p_4 ...

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### Perturbation analysis for three term recurrences

Jacobi polynomials, denoted by $J^{(\alpha,\beta)}_n$, on $[-1,1]$ satisfy a three term recurrence
$$ J_{n+1}^{(\alpha,\beta)}(x) = (A_n+B_nx)J^{(\alpha,\beta)}_n + C_nJ_{n-1}^{(\alpha,\beta)}(x), ...

**1**

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83 views

### Bound a sum of a serie defined by a recursive integer function

I'm using a recursive function $f: \mathbb{N} \rightarrow \mathbb{N}$, that is defined as
\begin{equation}
f(n)=\lceil \log(f(n-1)) \rceil +f(n-1)
\end{equation}
where $f(1)=F\in \mathbb{N}$, and ...

**1**

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### Trying to get an idea of the maths I could use for this optimization problem

Firstly, apologies if some of the notation or terminology is odd, or if I am defining functions that have standard notation associated with them already - I am not familiar with the concepts in this ...

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149 views

### Partitions of central sets via dynamical systems

In the book "Recurrence in Ergodic Theory and Combinatorial Number Theory", 1981, Furstenberg introduced the notion of central sets.
He proved in Theorem 8.8 that in each finite partition of ...

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### Approximate closed-form solution for a recurrence

Find an (approximate) closed-form solution for $S(m, b)$.
$$S(m,b)=\sum_{i=0}^{\lfloor (e-1)/2\rfloor}{e \choose i}S(m-1, b-i) \quad +
\sum_{i=\lfloor (e-1)/2\rfloor+1}^{\min(b,e)}{e\choose ...

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374 views

### Solving a linear recurrence relation with variable coefficients.

I have the following recurrence relation:
\begin{equation}
A[n]=f_A[n-1] A[n-1] + f_B[n-1]B[n-1], \\
B[n]=g_A[n-1] A[n-1] + g_B[n-1]B[n-1],
\end{equation}
where ...

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### Recursive relation using successor function

What is the recursive relation for
H(m)=2^(m^2)
using successor function
recursive relation for multiplication:
mult(x,0)=0;
mult(x,S(y))=add(x,mult(x,y))
recursive relation for addition:
add(x,0)=x;
...

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82 views

### Recursion Formula For Mobius function

I was trying to find recursion formula for some multiplicative functions. Especially Mobius, Totient, etc. So discovered some formulas for Mobius and Totient functions, but as you see the formula is ...