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14 views

Recursion relation for the coefficients of the series solution around x=0 [duplicate]

ODE: the equation is the following; $$y''' + x^2y' + xy = 0$$ y=y0x0+y1x1+...+ynxn I have written equations for y, y', y'' and y''' there are: y = SUM y_k * x^k y' = SUM y_k * k * x^(k-1) y'' = ...
2
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1answer
70 views

Primitive Recursion Functions (Programs)

The set $F_{n}$ of primitive recursive function symbols of arty $n$ can be defined inductively as \begin{array}[lr] & Z, \text{Succ} \in F_{1} & \\ \pi_{j}^{n} \in F_{n} \quad \text{for each} ...
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1answer
40 views

Combinatorics recursion question. [closed]

How many binary vectors of length n do not contain a sequence '001' ? Solve by recursion and explain your solution.
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1answer
47 views

Prove that for any $m, n \in\omega$ that (i) $m + 0 = m$, (ii) $m + n^+ = (m + n)^+$

I am getting repeatedly lost trying to approach this question: Prove that for any $m, n \in\omega$ that (i) $m + 0 = m$, (ii) $m + n^+ = (m + n)^+$ I can fairly well grasp the idea that the ...
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1answer
31 views

A question on combinatorics

Bob and Laura have bought an apartment, and are going to carpet the floor in the kitchen. The kitchen has size $2 \times n$, where $n$ is a positive integer. In how many ways can Bob and Laura ...
3
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0answers
64 views

Induction on Primitive Recursive Function

The set $F_{n}$ of primitive recursive function symbols of arity $n$ can be defined inductively as: \begin{align} & Z, \text{Succ} \in F_{1} & \\ &\pi_{j}^{n} \in F_{n} \quad \text{for ...
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1answer
40 views

Two indexes $c$ and $b$, such that $Dom(\varphi_a)=\{b\}$ and $Dom(\varphi_b)=\{c\}$

Problem: Assume $\{\varphi_i\}_{i=1}^\infty$ is an effective enumeration of the computable functions. Find two indexes $c$ and $b$, such that $Dom(\varphi_c)=\{b\}$ and $Dom(\varphi_b)=\{c\}$. ...
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2answers
51 views

Discrete Mathematics - Recursion

Given the following question by my professor: Recursively define the set of natural numbers divisible by 3. My answer: ...
4
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3answers
69 views

Recursion, multiplication and exponential

The set $F_{n}$ of primitive recursive function symbols of arty $n$ can be defined inductively as \begin{array}[lr] & Z, \text{Succ} \in F_{1} & \\ \pi_{j}^{n} \in F_{n} \quad \text{for each} ...
3
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2answers
95 views

Argument over an induction proof

My friend gave me a problem. Define a sequence $<a_n>$ by the recurrence relation :$$ a_{n+2} - 6a_{n+1} + 8a_n = 0 $$ and $a_1 = 4, a_2 = 8 $. Find the general term $a_n$ in closed form. ...
0
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1answer
45 views

Recursive function

Having difficulty with a question, was hoping someone could take a look and explain (if) where i'm going wrong. Consider the following recursive definition of a function $f:N\to N$ Base case: For ...
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0answers
63 views

Concatenation, reversal and sum are primitive recursive

Let $\mathbb N^*$ be the language of finite words with alphabet $\mathbb N$. Assume $k:\mathbb N^*\to\mathbb N$ is an effective coding. That is $k$ is a bijection whose length and member functions are ...
3
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1answer
86 views

An effective coding of $\mathbb N^*$

Problem: Assume $\Pi:\mathbb N^2\to\mathbb N\setminus\{0\}$ is a primitive recursive coding of the pairs of numbers, that is also a bijection and $(\forall (x,y)\in\mathbb N^2)(\Pi(x,y)>max(x,y))$. ...
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4answers
116 views

find $O$ and $\Omega$ bounds as tight as possible for $T(n)=n+T(\frac n 2)+T(\frac n 4)+T(\frac n 8)+…+T(\frac n {2^k})$

find $O$ and $\Omega$ bounds as tight as possible for $T(n)=n+T(\frac n 2)+T(\frac n 4)+T(\frac n 8)+...+T(\frac n {2^k})$ while k is some constant and for any $n\leq3$ $\ T(n)=c$ for k=1 ...
1
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1answer
88 views

Number of ways to arrange different poker chips. (Recursion)

Assume there are poker chips in four different colors and one of the colors is blue. In how many ways can n amount of chips be piled on top of each other without two blue ones being next to one ...
1
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1answer
150 views

Closed form for general recursive function

Does a closed form exists for general recursive functions? my guess is not, but what types can be solved or what are the constraints on a recursive function so it has a closed form, what are some ...
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0answers
67 views

An informal version of the recursion theorem

In T. Taos Analysis 1 book, on page 26, we have a proposition that tells us that recursive definitions are actually well-defined. Proposition 2.1.16: Suppose for each naturla number $n$, wh have ...
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1answer
80 views

Is definition by cases a primitive recursive function?

Primitive recursive functions can simulate every single step of a Turing machine. In order to prove this, one has to see that a function defined by state table is primitive recursive. Simply ...
0
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1answer
52 views

What will be the closed formula for the following recursive function?

What will be the closed formula for the following recursive function? F(n) = F(n/2) +1 if n is even F(n) = F(n-1) + 1 if n is odd F(1) = 0 How do we generate closed formula for such ...
0
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1answer
49 views

factoring cubic polynomail equation using Krammer's rule.

1) I have question about factoring cubic polynomials. In my note it says "Any polynomial equation with positive powers whose coefficients add to 0 will have a root of 1. Another, if sum of the ...
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0answers
37 views

How many number of ways are there for getting a special prime?

Definition of special prime : Any integer (+ve, -ve or 0) that is divisible by at least one of the single digit primes (2, 3, 5, 7) is a special prime. Thus -21, -30, 0, 5, 14 etc are special ...
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1answer
157 views

Survival Probability of a Population

A population starts with one amoeba. In each generation, each amoeba divides in two with probability $\frac{1}{2}$, or dies, with probability $\frac{1}{2}$. Let $p_n$ be the probability that the ...
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1answer
139 views

Prove that set is countable? [duplicate]

Show that the set N* of finite sequences of nonnegative integers is countable. Where do I start? I think I have to prove that there is a bijection between N* and N (set of natural numbers), but how ...
0
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0answers
62 views

Repeated function composition

I have a recursive function repeat, that composes n calls to f with a start value ...
0
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1answer
25 views

Analyse recursion $N_{t+1}=rN_t/(1+bN_t^2)$ [closed]

Given $$N_{t+1}=\frac{rN_t}{1+bN_t^2}$$ for $r>0$ and $b>0$ I need to: $1$. Find the limit of the recursion. $2$. Prove that: $$\frac{2r^2}{(4+r^2)\sqrt{b}}\le N_t \le \frac{r}{2\sqrt{b}}$$ ...
2
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2answers
2k views

Proof by Induction for a recursive sequence and a formula

So I have a homework assignment that has brought me great strain over the past 2 days. No video or online example have been able to help me with this issue either and I don't know where to turn. I’m ...
0
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1answer
65 views

How do I go about manipulating this summation equation to solve it?

In my textbook, Introduction to Algorithms, the following is shown: And I believe I understand that. However, I have a similar equation to the one on the first line, but instead of ...
2
votes
1answer
59 views

Problem solving recurrences with generating functions

I'm trying to solve this linear recurrence with generating functions, but I keep getting stuck on the last few steps. I found the generating function, but after splitting it into partial fractions and ...
1
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2answers
69 views

How does my textbook solve this summation equation for the answer?

Summations have always been my weakness in mathematics, and it's showing here as I'm very confused how my textbook, Introduction to Algorithms, goes from basically the second half of the following ...
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1answer
54 views

How does my textbook come up with this statement? I don't believe it to be true.

My textbook (Introduction to Algorithms) states the following: When polynomially comparing $n^\epsilon$ and $lgn$, it states that $n^\epsilon$ is polynomially greater for any positive $\epsilon$. ...
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3answers
77 views

Where is the value of the variable $\epsilon$ obtained in the following explanation my professor gave?

My professor gave us this explanation from the textbook Introduction to Algorithms regarding the Master Method/Theorem: As a first example, consider $$T(n)=9T(n/3)+n.$$ For this recurrence, we ...
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2answers
115 views

fill-in-the-blank induction proof

I'm stuck at homework task I'm working on. I would really appreciate some help. Here is the task: Let $f$ be a function on binary numbers defined recursively as follows. $f(0) = 1$ and ...
1
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1answer
51 views

Course-by-values recursion

I have many questions in my textbook of the kind: Assume $g$ is primitive recursive and assume $f(0)=c_0,\dots,f(n-1)=c_{n-1}$ $f(x+n)=g(<f(x),\dots,f(x+n-1)>)$ Prove that $f$ is primitive ...
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0answers
48 views

Proving a function is primitive recursive

Assume $f$, $r$, and $s$ are primitive recursive. Prove that $$h(\overline x,y) = \begin{cases} f(\overline x,y,h(\overline x,r(y))) & r(y)<y \\ s(\overline x,y) & \text{otherwise} ...
0
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1answer
139 views

“Nested” recursion preserves primitive recursive functions

Problem: Assume the functions $f$, $\pi$, and $g$ are given. They take one, two, and three arguments respectively. Prove a unique function $h$ exists such that: $$h(0,y)\cong f(y)$$ $$h(x+1,y)\cong ...
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2answers
128 views

Termination of a Fast Exponentiation problem

Here's the problem I am stuck on. There exists a fast exponentiation program like the following: Given inputs a in the set of all Real numbers, b in the set of Natural numbers, initialize ...
0
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0answers
68 views

How is the algorithm for recursively printing permutations of a set of numbers this equation our professor gave us?

I'm having a great amount of trouble understanding where my prof got $T(m, n) = n(T(m+1, n-1) + m+1 + n)$ if $n > 1$ as the recursive formula for the algorithm for recursively printing the ...
0
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1answer
25 views

Defining a recursive function $f$ on $\{a, b\}$*

I would need some help on how I can define a recursive function $f$ on $\{a, b\}$* Define a recursive function $f$ on $\{a, b\}$* which replaces any $a$ with $b$ and vice versa, for example, ...
0
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2answers
58 views

I need some help on solving a recursive function question

I'm working on a recursive function task which i'm a bit stuck at. I've tried to google it on how I can solve this task, but with no luck Here is the task: Provide a recursive function $r$ on ...
0
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1answer
73 views

How can I solve this recursion function task?

I really need help on this task. Im stuck at it and I really would appreciate your help here. Give a recursive function $r$ on $A$ that reverses a string. For instance, $r(logikk) = kkigol$ and ...
0
votes
1answer
68 views

Solving a recurrence with division

I'm having this recurrence that is giving me a lot of trouble. $$F(2^0) = 1$$ $$F(2^k) = \frac 1 2 F(2^{k-1}) + 2^k$$ I will edit my post since this does not seems to work... The initial ...
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2answers
267 views

Solving expected value with recursion

We'll start off with an example of a question of finding expected value. What is the expected number of tries to get ’6′ when rolling dice? E(x)=1/6*1 + 5/6 (1+E(x)) E(x)=6 I understand the ...
0
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1answer
25 views

Recurrence to find P(n). P(n) is the number of ways to decorate a strip of size n with tiles.

There are three kind of tile. One is of size 1. Second is of size 2 of green color. Third is of size 2 with blue color. These are the values I found but I can not figure out the formula. P1 1 p2 3 p3 ...
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1answer
62 views

Ackermann function and primitive recursiveness

If we define $b_n(m) := a(n,m)$ for all $n$ and $m \in \mathbb{N}$. For which $n$ is the function $b_n$ primitive recursive and for which $n$ it is not a primitive recursive function? Can anyone ...
3
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1answer
3k views

Is 0^infinity indeterminate?

Is a constant raised to the power of infinity indeterminate? I am just curious. Say, for instance, is 0 raised to infinity indeterminate? Or is it only 1 raised to the infinity that is?
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3answers
51 views

How does my professor go from this logarithm to the following one of a different base?

I don't understand on the second last line how my professor goes from $2^{log_3(n)} = n^{log_3(2)}$ how is that relation formed?
0
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2answers
192 views

solve $T(n)=T(n-1)+T(\frac{n}{2})+n$

Using the recursion tree i tried solving this: $T(n)=T(n-1)+T(\frac{n}{2})+n$; the tree has two parts (branches) one that of $T(n-1)$ and other branch is of $T(\frac{n}{2})$. But as the term T(n-1) ...
4
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1answer
57 views

Prove $g$ is PR. $g(n) = f(n) \text{ if n} \notin A$. $f$ is PR. $g$ is total. $A$ is a finite set.

Show that if f:N $\rightarrow$ N is primitive recursive, A $\subseteq$ N is a finite set, and g is a total function agreeing with f at every point not in A, then g is primitive recursive. My ...
2
votes
1answer
176 views

Calculate percentage with Days

I'm programming a projectscheduler and want to calculate the percentage of finishing by looping through all the subprojects and calculate their weight and how many percent is done by now and then ...
0
votes
1answer
177 views

Simultaneous recursion

I have no idea how to even start proving the following theorem: If $f_0, f_1: \mathbb{N}^r \rightarrow \mathbb{N}$ and $g_0, g_1: \mathbb{N}^{r+3} \rightarrow \mathbb{N}$ are primitive recursive, ...