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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27 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
20 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 ...
10
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3answers
139 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
28 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
69 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
20 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
13 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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0answers
52 views

Proof of nearest integer equality

Let $N(n)$ be the nearest-integer function undefined on half-integers. There are many valid ways to define $N(n)$, I like to choose $N(n) =\arg \min_{z \in \mathbb{Z}} |n-z|$. Consider the function ...
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2answers
46 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
44 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 ...
2
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3answers
104 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
81 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
451 views

⎣⎡x⎤⎦=⎡x⎤ for all real numbers x

Let LHS = ⎣⎡x⎤⎦ and RHS = ⎡x⎤ Let us call n=⎡x⎤ Case 1: n is even, there exists k in Z such that n=2k I'm not sure how to go on or if I'm setting this up right.
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2answers
93 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
53 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
20 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
30 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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0answers
55 views

Limit value of ceiling function in a trigonometric function.

What is the Left Hand Limit(LHL) and Right Hand Limit(RHL) of the $\lim_{x->0}({\sin[x]}/{[x]})$? where $[x]$ is the greatest integer function(step function) I got LHL as $\sin(1)$. Can anybody ...
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3answers
521 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
20 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
84 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
104 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
75 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
32 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
122 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
22 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
84 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
69 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
67 views

Comparing floor and ceiling fractions

Is the following true for all integers x>1: $\lfloor{\frac{2x}{3}}\rfloor \geq \lceil \frac{x}{2}\rceil$
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3answers
110 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
59 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
96 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 ...
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0answers
37 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 ...
3
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2answers
163 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
166 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
126 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
75 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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63 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 ...
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2answers
91 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
101 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 { ...
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2answers
568 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 ...
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1answer
140 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} ...
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4answers
141 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 ...
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1answer
79 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 ...
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0answers
39 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 < ...
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80 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 ...
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1answer
156 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 ...
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3answers
487 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
951 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
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1answer
20 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 ...