The function that maps a real number x to the smallest integer not less than x. (See also (floor-function).)

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3answers
85 views

Solving equations with dozens of ceil and floor functions efficiently?

I am tackling a problem which uses lots of equations in the form of: $$ q_i(x) = c_i(x) + \sum_{j=1}^N \left\lfloor \frac{q_i(x)}{P_j} \right\rfloor C_j $$ where $q_i(x)$ is the only unknown, ...
3
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2answers
57 views

Is there also an other way to show the equality?

I want to show that: $$ \left\lfloor \frac{n}{2}\right\rfloor + \left\lceil \frac{n}{2} \right\rceil=n$$ That's what I have tried: $ \left\lfloor \frac{n}{2}\right\rfloor=\max \{ m \in \mathbb{Z}: ...
3
votes
1answer
26 views

Equation with ceiling

I have to show that: $$ \lceil{ \frac{\lceil{ \frac{n}{a} \rceil }}{b}} \rceil=\lceil {\frac{n}{ab} } \rceil$$ That`s what I tried: If $n=ka, k \in \mathbb{Z}$: $ \lceil{ \frac{n}{a}} ...
0
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3answers
102 views

Proof for Concrete Mathematics 3.24

I'm reading Concrete Mathematics by Graham, Knuth, Patashnik . I found that for every integer $n$, this holds : $$n = \lceil n/m \rceil + \lceil (n-1)/m \rceil + \cdots + \lceil (n-m+1)/m \rceil$$ I ...
0
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1answer
18 views

Solving Logarithms involving ceiling function

I need to solve the equation $\lceil \log_B(M) \rceil = S$ for $B$ when $M$ and $S$ are known, $M$ and $S$ are integers, and $B < M$. Were the ceiling function not there, it would be trivial, ...
1
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1answer
46 views

Algebraic manipulation of floors and ceilings

I am trying to solve the summation $$ \sum_{n=3}^{\infty} \frac{1}{2^n} \sum_{i=3}^{n} \left\lceil\frac{i-2}{2}\right\rceil $$ I will list some of the simplifications that I've found so far, and ...
2
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1answer
59 views

Polynomial equal to the ceiling of x

For a few days, I've been looking for a polynomial who's value is equal to the ceiling function of the only variable it contains. I thought about it for while and I haven't got a clue.
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1answer
42 views

Comparing floor and ceiling fractions

Is the following true for all integers x>1: $\lfloor{\frac{2x}{3}}\rfloor \geq \lceil \frac{x}{2}\rceil$
2
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3answers
98 views

Summation with Ceilinged Logarithmic Function

According to Johann Blieberger's paper - "Discrete Loops and Worst Case Performance" (1994): $$ \sum_{i = 1}^{n}\left \lceil \log_2{(i)} \right \rceil = n\left \lceil \log_2{(n)} \right \rceil - ...
0
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0answers
45 views

Given a non-negative integer $m$ and a positive integer $n$, calculate $\lfloor \frac{m}{n} \rfloor$

Here is the problem: I have a non-negative integer $m$ and a positive integer $n$ I would like to calculate $\lfloor \frac{m}{n} \rfloor$, $\lceil \frac{m}{n} \rceil$ and $m \bmod n$ But I want to ...
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2answers
71 views

Given $\lceil x−1 \rceil$, how can I compute $\lfloor x \rfloor$ without using $modulo$?

I have $\lceil x \rceil = -\lfloor -x \rfloor$, but I can't figure out how to rely on this in order to get $\lfloor x \rfloor$ from $\lceil x−1 \rceil$. For the record, I am only interested in ...
0
votes
0answers
29 views

Inequality with nesting floors and ceilings

EDIT: I changed the inequality to a simpler more indicative format. I need to solve the following inequality for $x$ (find the minimum value of $x$) but I have trouble removing the ceiling and floors ...
2
votes
2answers
94 views

A mathematical way for defining the Floor and Ceiling functions

Given: $Floor(x)=\lfloor x \rfloor$ $Ceiling(x)=\lceil x \rceil$ Where $x$ is a real number. Is there any other (mathematical) way for defining $Floor(x)$ and $Ceiling(x)$? Restrictions: Do ...
1
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1answer
58 views

Solving equation involving the ceiling function

How can I solve the equation $$\lceil \log_{b}{1024} \rceil = n$$ where $n \in \mathbb{N}$ in terms of $b$? I have seen equations of a similar form (Solving an equation with floor function before), ...
1
vote
5answers
89 views

Floor and ceiling functions in a inequality

Given $f(x)= x/\lfloor x\rfloor$, find $f(x) \geq 3/2$. I have been stuck on this for a couple of hours. Thanks in advance.
0
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2answers
40 views

Prove that $\lfloor -a \rfloor=-\lceil a \rceil$

Let $\lfloor a \rfloor$ be the least element of {$k \in \Bbb Z : k\ge a$} and $\lceil a \rceil$ the greatest element of {$k \in \Bbb Z : k\le a$}. How can I prove that for all $a \in \Bbb R$, ...
1
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0answers
61 views

May someone tell me where I went wrong when finding a closed-form solution to a recursion?

Let $r(n)=\left\lceil\frac{n}{4}\right\rceil$, where $k \in \mathbb{N}$. Note that this can be rewritten as a recursion $R(n)=R(n-4)+1$, where, $R(0)=0, R(1)=1, R(2)=1, R(3)=1$. Since this recursion ...
2
votes
2answers
54 views

Is it possible to rewrite floor functions applied to a fraction using only the addition, multiplication, and exponentiation operators?

Let's restate this question in using mathematical notation. Let $n,k \in \mathbb{N}$. Let $f(n)=\left\lfloor{\frac{n}{k}}\right\rfloor$. Is it possible to rewrite this using the addition, ...
2
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0answers
94 views

How to solve a ceiling expression or recurrence equation?

I am trying to express the following in term of n, $$\underbrace{\hbox{$ \left \lceil \frac{3}{2}.\left \lceil { \frac{3}{2}.\left \lceil { \ldots \left \lceil \frac{3}{2}.\left \lceil { ...
0
votes
2answers
332 views

Prove that if m and n are positive integers, and x is a real number, then: ceiling((ceiling(x)+n)/m) = ceiling((x+n)/m)

I can't get eq (1) and eq (2) Question: Prove that if m and n are positive integers, and x is a real number, then: ceiling((ceiling(x)+n)/m) = ceiling((x+n)/m) Answer: Let us define the real ...
1
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1answer
84 views

sum of ceiling function inequality

I need to show the following inequality: $$\sum_{j=1}^n \left(\frac{D_j}{Q_j} \left\lceil\frac{Q_j}{T_c}\right\rceil \right) \ge \frac{\sum_{j=1}^n D_j}{\sum_{j=1}^n Q_j} ...
4
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3answers
110 views

Proving that Spec(α) and Spec(ß) partition positive integers iff α and ß are irrational and 1/α + 1/ß = 1

From Concrete Math, problem 3.13 asks: "Let α and ß be positive real numbers. Prove that Spec(α) and Spec(ß) partition positive integers if and only if α and ß are irrational and 1/α + 1/ß = 1" The ...
0
votes
1answer
53 views

Stuck solving an equation using the floor operator.

I am not entirely familiar with the equation ninja'ing involving the floor operators. Here is my problem. I need to solve for $x$. Everything is an integer, including $x$: $$ a - 1 = \lfloor {\frac{x ...
1
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0answers
37 views

Is this a valid argument ? function floor property

I'm new in that stuff in proving things. I'm always confused about when is a really valid reasoning. Then is this property, i appreciate any help. Prove, for any real x: x - 1 < ...
1
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0answers
69 views

How does a piecewise step function work?

The wiki pages aren't helping at all. What is a step function and how does it work? I've seen them used in the floor, ceil, frac and even signum functions. And although it is easy to understand what ...
1
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1answer
96 views

Solving an equation involving floor/ceiling as a summation bound

Is it possible to solve the following equation for $\alpha$? $$ M = \lfloor \alpha \rfloor + \alpha \sum_{k=\lfloor \alpha \rfloor +1}^N \frac{1}{k}$$ where $\alpha \geq 1$. Intuitively, $M$ is a ...
0
votes
3answers
272 views

How do you use floor/ceil in math, e.g. how does it work exactly?

I read the Wiki, but I am still somewhat confused. Supposedly floor returns the number lowest, and ceil the number highest. However, I am not sure I understand how this works in freehand form, or how ...
1
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0answers
565 views

Proof of floor and ceiling functions

By definition: $ \lfloor {x}\rfloor = i \Rightarrow i \le x \lt i + 1 $ (floor function) and $ \lceil {x} \rceil = j \Rightarrow j - 1 \lt x \le j $ (ceiling function) So, how is the proof that ...
0
votes
1answer
18 views

We have $T(n) \leq T(\lceil \frac{n}{5} \rceil) + T(\lceil \frac{7n}{10} \rceil)$. Show that $T(n) < c'n$.

We have $T(n) \leq T(\lceil \frac{n}{5} \rceil) + T(\lceil \frac{7n}{10} \rceil)$. Show that $T(n) < c'n$ for all $n$ and for some constant $c'$. Looks straightforward enough, but suprisingly I ...
2
votes
2answers
109 views

Derivative of $\lceil 1/x \rceil$

I'm looking for the derivative $\frac{d}{dx}\lceil 1/x \rceil$. I would like to find a real number $1<x \le y$ satisfying the minimum of $\left\lceil \frac{y}{x} \right\rceil x$, when $y$ is a ...
0
votes
1answer
49 views

Solving for variable in quotient of ceil function

Given $k=\lceil\frac m {n}\rceil$, I would like to solve for $m$ given $k$ and $n$ as positive integers. How do I solve for $m$, and is there a single solution for $m$? Perhaps one way to do this ...
1
vote
1answer
110 views

Is it possible to truncate a number just with Arithmetic operations?

Hi is it possible to truncate a number just with Arithmetic operations? For example if i have 2.5 i want to get "2" without using modulo or something other ...
0
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0answers
37 views

Least Prime Factor of a Ceiling Function that involves Primorials

Consider the following ceiling Function involving the primorial $p_k\#$: $$\left\lceil\frac{ap_{k+1} + bp_k\#}{p_{k+1}}\right\rceil + c$$ where $p_k$ is the $k$th prime, $a,b$ are positive integers ...
7
votes
7answers
544 views

How much is $\lceil\frac{1}{\infty}\rceil$?

How much is $\lceil\frac{1}{\infty}\rceil$ ? On one hand, $\frac{1}{\infty}=0$, so its ceiling is also $0$. On the other hand, for all $x\geq 1$, $\lceil\frac{1}{x}\rceil = 1$, so, when $x$ goes ...