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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35 views

Floor and Ceiling Series (I) [on hold]

Prove or disprove: 1) $$\sum_{n=2}^{\infty}{\frac{1}{\lfloor n^2/2 \rfloor}} = \frac{1+\zeta(2)}{2}$$ 2) $$\sum_{n=1}^{\infty}{\frac{1}{\lceil\ n^2/2 \rceil}} = \frac{\zeta(2)}{2} + \frac{\pi}{2} ...
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0answers
18 views

How to pigeonhole the primes between $p_n$ and $p_{n+1}^2$ for twin prime conjecture?

For any full list of the primes up to the $n$th prime: $P = \{2, 3,5,\dots, p_n\}$, any natural number $q$ such that $ p_n \lt q \lt p_{n+1}^2$ that is not sieved by a prime in $P$ is also a prime. ...
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3answers
242 views

How to prove this identity of ceiling function?

My book writes down this identity of least integer function: $$\lceil x\rceil +\left\lceil x + \frac{1}{n}\right \rceil + \left\lceil x + \frac{2}{n}\right \rceil + \cdots +\left\lceil x + \frac{n ...
3
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1answer
51 views

Solving recurrence equation with floor and ceil functions

I have a recurrence equation that would be very easy to solve (without ceil and floor functions) but I can't solve them exactly including floor and ceil. \begin{align} k (1) &= 0\\ k(n) &= ...
3
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2answers
31 views

Prove the following floor function identity

The identity is this: $\lfloor 2x \rfloor + \lfloor 2y \rfloor \geq \lfloor x \rfloor + \lfloor y \rfloor + \lfloor x+y \rfloor$ I am truly stumped. Help is appreciated. Thank you!
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0answers
31 views

Concrete Mathematics Floor Sum

In Concrete Mathematics, after a short introduction of the floor and ceiling functions, the problem of the "Concrete Math Club Casino" is proposed. Essentially, how many numbers in $[1,1000]$ are ...
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0answers
27 views

Factorization by multiplying and representation as difference of two squares

Definition 1.$$R: \mathbb N \to \mathbb N: \ R(n) = \lceil\sqrt{n}\rceil^2-n.$$ This is the distance from $n$ to the smallest square greater or equal to $n$. Definition 2. Let $a$ be as positive ...
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4answers
86 views

Prove that $\left\lceil \frac{n}{m} \right\rceil =\left \lfloor \frac{n+m-1}{m} \right\rfloor$

On a discrete mathematics past paper, I must prove that $$\left\lceil \frac{n}{m} \right\rceil = \left\lfloor \frac{n+m-1}{m} \right\rfloor$$ for all integers $n$ and all positive integers $m$. ...
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2answers
72 views

Least integer function and Greatest Integer Function Without using ceil() and Floor()

I was wondering if there is any mathematical way to calculate Least Integer and Greatest integer without using predefined Ceil() and Floor() Function of Programming Language.
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2answers
50 views

Ceil () and Floor()

I already know the basic rules for the both functions: $$\text{Ceil}(2.5)=3\\ \text{Floor}(2.5)=2$$ But I could not understand the following these: $$\text{Ceil}(2.6, 0.25)=2.75\\ \text{Floor}(2.6, ...
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1answer
26 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 ...
0
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1answer
27 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
84 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
54 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
46 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
82 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 ...
2
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0answers
34 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
27 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
28 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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1answer
27 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
22 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
52 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
36 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
23 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
206 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
72 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
127 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
24 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
38 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
58 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
46 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
256 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
128 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
554 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
103 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 ...
0
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2answers
66 views

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

I have a problem calculating with ceils. If I have $\frac{\lceil \frac{n}{2} \rceil}{np}$, this is not the same as $\frac{\lceil \frac{1}{2} \rceil}{p}$. Are there some rules on how to calculate ...
2
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2answers
127 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
570 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
36 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 ...
3
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1answer
89 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
145 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
87 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
135 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
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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2answers
158 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
votes
1answer
81 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.
2
votes
3answers
123 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 - ...