The function that maps a real number $x$ to the smallest integer greater than or equal to $x$ (which is often denoted $\lceil x\rceil$. See also (floor-function).

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13 views

Find an expression for the ones digit of a positive integer using a floor or ceiling function.

I am trying to find out how to write an expression for the ones digit for any given positive integer. For example, if n = 326, the expression should evaluate to 6. The only things I've been able to ...
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1answer
15 views

Rounding off general equation

I would like to know if there is a way to express the "roundup" function of excel or "rounding off of a number to the nearest whole number" in an equation form. e.g. in excel: roundup $(2.13,0) = 3$, ...
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1answer
32 views

Proving statements about ceiling and floor functions.

Prove or disprove the statements below. (a) For all positive real numbers x and y, $\lfloor x \cdot y\rfloor ≤ \lfloor x\rfloor \cdot \lfloor y\rfloor $. (b) For all positive real numbers x and y, ...
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1answer
48 views

Can we deduce from the equality that $\max \{n-q,q-1\} \in \left[ \frac{n}{2},n \right)$?

Suppose that we have a uniform distributed random variable in $[0,n]$. We have the following: $$\max \{n-q,q-1\}=\left\{\begin{matrix} q-1, q > \lfloor \frac{n}{2}\rfloor & \to q-1> ...
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1answer
20 views

Proving two equations containing ceiling and floor function to be equal

The problem is to show that if $n$ is a positive integer, then $$1 = C(n, 0) < C(n, 1) < ... < C(n, \left \lfloor{ n/2} \right \rfloor) = C(n, \left \lceil {n/2} \right \rceil) > ... > ...
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2answers
71 views

How to prove that the $\lceil x y \rceil \le \lceil x\rceil\lceil y\rceil$ for real numbers $x, y$?

I know intuitively that this is true, but whenever I try and break apart that intuition to see where it's coming from, I essentially end up re-writing the assumption I'm trying to prove. I've tried ...
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6answers
1k views

What is ⌊0.9 recurring ⌋? [duplicate]

For a ceiling and floor function, the number is taken to 0 decimal places. Does this process mean that 0.9 recurring inside a floor function would go to 0? Or would the mathematician take 0.9 ...
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0answers
29 views

A relation with limits

Let $t \in \mathbb{R}_+$, $\varepsilon > 0$ and $p_\varepsilon \in [0, 1]$. There is in a book the following relation: $$ \underset{\varepsilon \rightarrow 0}{lim} (1 - p_\varepsilon)^{\lceil t / ...
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1answer
26 views

Product of a prime and other number expressed in two ways are equal

Let $p \in \mathbb P$ be an odd prime and let $1\leq a \leq p-1$ be such a number that $$a p = \left\lceil \sqrt{a p}\right\rceil ^2-\left\lceil \sqrt{\left\lceil \sqrt{a p}\right\rceil ^2-a ...
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1answer
23 views

Solve an Equation with Ceiling Function

Given the following Term $$n=\left\lceil{\frac{a}{b}}\right\rceil \quad \text{with} \quad a,b\in \mathbb{R}_{>0}$$ Is there a way to get $b$ if $a$ and $n$ are given? I. e. what is the solution ...
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23 views

$R = A \times B + C$, How retrieve A and C if C is negative?

For A = 18, B = 54 and $C = {-53, 53}$ $$R = A \times B + C$$ I can retrieve A and C from R with: $$ A = \lfloor R \div B \rfloor $$ $$ C={R} \pmod {B} $$ example, A = 18, C = 53: $$ R = 18 \times ...
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1answer
17 views

Log inequality- is $(\lceil\log x\rceil - \lfloor\log m\rfloor)\cdot m+2^{\lfloor\log m\rfloor+1}\leq m\cdot(\lceil\log\frac{x}{m}\rceil+2)$?

I'm having some hard times making a tight analysis of the memory requirements for my algorithm. I want to show the following inequality, which will show my data structure can use about 2 bits per ...
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2answers
45 views

Log inequality - is $\lceil\log x\rceil - \lfloor\log y\rfloor\leq \lceil\log\frac{x}{y}\rceil+1$

Is it true that $$\forall x>y\in\mathbb N:\lceil\log_2 x\rceil - \left\lfloor\log_2 y\right\rfloor\leq \left\lceil\log_2\frac{x}{y}\right\rceil+1$$? I reached this inequality when further ...
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1answer
26 views

Rounding errors for expressions with floor/ceil funcitons

In parallel computations I frequently use the following distribution of data to processors: having a sequence of $n$ elements $x_1,\ldots,x_n$ and $P$ processors $p_1,\ldots,p_P$, processor $p_k$ ...
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1answer
33 views

Would there be no input or input does not exist?

This problem is from Discrete Mathematics and Its Applications. And the definition of inverse from the book: For part 43 (c), would the inverse not exist? For the floor function, in terms of $f(a) ...
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2answers
21 views

Matrix entries using a single function

Is it possible to write the following matrix as a single function, without any explicit if condition mathematically? For, $$1 \leq i,j\leq n,$$ $$A(i,j)=\begin{cases} 2 ~~\text{if}~~ i \neq j \\ 1 ...
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3answers
181 views

Is it possible to define $\lceil x\rceil$ in terms of $\lfloor\ldots\rfloor$?

I have tried to define the ceiling function of $x$ in terms of its floor function; I thought this might be easy, but it isn't. I can easily do this with a piecewise equation, but I need to do without ...
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2answers
39 views

Definite integral of ceiling function

I understand the basics of solving definite integrals but I have trouble with the following one because of the ceiling function. How can I solve this? $\int_0^3 [x^2] dx$
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1answer
91 views

Proof, that the floor and ceiling functions exist

I want to proof the following, with elementary properties of the integers and reals: Let $x\in \mathbb{R}$. Then there are unique $p,q\in \mathbb{Z}$, such that: $$p\leq x < p+1\text{ and ...
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2answers
21 views

Rounding half-integers to approximate product

Given two half-integers $a,b\in\mathbb Z+\frac 1 2$, the integers nearest to each is given by $N_a=\{a-0.5,\ a+0.5\}$ and $N_b=\{b-0.5,\ b+0.5\}$. Is there a general method to find, for the two given ...
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2answers
19 views

Expressing n mod m in terms of floor values?

I'm trying to prove the expression: $$\left\lceil\frac{n}m\right\rceil = \left\lfloor n+m-\frac1m\right\rfloor\;,$$ where $n,m$ are integers` So I've come across this article (PDF) which gives a ...
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2answers
51 views

Proof of a nearest-integer inequality

Let $N(y)$ be the nearest-integer function and undefined on half-integers. For all $r \in \mathbb R$ that are not half-integers, prove $$\forall{\ i \in \mathbb ...
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1answer
45 views

How to prove this apparent identity?

While solving the problem in my other question, I've come across an identity, which I've empirically found to be true, but can't seem to prove. Here I've simplified the problem a bit, so that the ...
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3answers
182 views

Proving Floor and Ceiling of a Rational Number

Suppose x,y $ \in \mathbb{Z}^+ $ Prove $\lceil x/y \rceil = \lfloor (x-1)/y \rfloor + 1$ I was considering using the definition of floor and ceiling to prove this. But this does not seem like a ...
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5answers
103 views

What is an example of a bijective function f: Z to N that isn't piecewise?

Like without using if even or odd. Like how you can define a bijection $f\colon\mathbb{N}\to\mathbb{Z}$ by is $f(n)=\lfloor n/2\rfloor\cdot(-1)^n$.
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2answers
518 views

Proving that $ \lfloor \lceil x \rceil \rfloor = \lceil x \rceil $ for all real numbers $ x $.

Let $ \mathsf{LHS} = \lfloor \lceil x \rceil \rfloor $ and $ \mathsf{RHS} = \lceil x \rceil $. Let us call $ n = \lceil x \rceil $. Case 1: $ n $ is even, i.e., there exists a $ k \in \mathbb{Z} $ ...
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2answers
99 views

Growth of $\sum_{x=1}^{n-1} \left\lceil n-\sqrt{n^{2}-x^{2} } \right\rceil$

I'm interested in the growth of $$f(n):=\sum_{x=1}^{n-1} \left\lceil n-\sqrt{n^{2}-x^{2} } \right\rceil \quad \text{for}\quad n\rightarrow\infty $$ Progress (From comments) I've got ...
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2answers
57 views

How to calculate with $\lceil \; \;\; \rceil$

I have a problem calculating with ceils. So If I have $\frac{\lceil \frac{n}{2} \rceil}{np}$, this is not the same as $\frac{\lceil \frac{1}{2} \rceil}{p}$. So do you have some rules how to ...
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0answers
23 views

Manipulating a logarithmic expression with ceilings

How can I simplify the first expression so that it satisfies the following inequality? $$ c(n+2) \log\lceil n/3\rceil - cn\log n \le 2\log\lceil n/3 \rceil-cn\log \left(\frac{3n}{n+2}\right)$$
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2answers
91 views

Inverse of a function containing the ceiling function over the natural numbers

I am wondering if there exists an inverse function for $\lceil{e^{x}}\rceil$ over the natural numbers. I don't think it is a trivial task to derive an inverse function for a function containing a ...
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3answers
541 views

Very challenging: max{floor,ceil}=?

I spotted a pattern while trying to generalize a problem. (EDIT: said problem has been removed from this post to avoid confusion. EDIT(2): Here is the problem again: ...
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1answer
28 views

Bounding a logarithmic relation

If I have the following relation $T(n) \le an\lceil \operatorname{lg} (n) \rceil - an +2bn + n$, is it possible to bound $T(n)$ such that it is in the form $T(n) \le an\operatorname{lg}(n) + bn $ for ...
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1answer
86 views

Interesting identity arising from fractional factorial design of resolution III

I am learning about statistical design of experiments, and in the process of mathematically rigorizing the concepts behind fractional factorial designs of resolution III, I derived an interesting ...
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3answers
120 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, ...
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2answers
82 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}: ...
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1answer
34 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}} ...
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3answers
124 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 ...
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1answer
25 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, ...
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1answer
109 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 ...
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1answer
74 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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3answers
121 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 - ...
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0answers
61 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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3answers
98 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 ...
3
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2answers
210 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 ...
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1answer
235 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), ...
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5answers
140 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.
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2answers
79 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 $a \in \Bbb R$, ...
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0answers
65 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
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2answers
105 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
102 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 { ...