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Questions tagged [mathematical-analysis]

Questions related to mathematical analysis (often called analysis by mathematicians)

Mathematical analysis is a branch of mathematics.

126 questions
111
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9 answers

How/when is calculus used in Computer Science?

Many computer science programs require two or three calculus classes. I'm wondering, how and when is calculus used in computer science? The CS content of a degree in computer science tends to focus on algorithms, operating systems, data structures,…
Victor
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4 answers

What is the meaning of $O(m+n)$?

This is a basic question, but I'm thinking that $O(m+n)$ is the same as $O(\max(m,n))$, since the larger term should dominate as we go to infinity? Also, that would be different from $O(\min(m,n))$. Is that right? I keep seeing this notation,…
Zeus
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24
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2 answers

Changing variables in recurrence relations

Currently, I am self-studying Intro to Algorithms (CLRS) and there is one particular method they outline in the book to solve recurrence relations. The following method can be illustrated with this example. Suppose we have the recurrence $$T(n) =…
erickg
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19
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3 answers

Why is there the regularity condition in the master theorem?

I have been reading Introduction to Algorithms by Cormen et al. and I'm reading the statement of the Master theorem starting on page 73. In case 3 there is also a regularity condition that needs to be satisfied to use the theorem: ... 3. If $\qquad…
GrowinMan
19
votes
1 answer

Proving the (in)tractability of this Nth prime recurrence

As follows from my previous question, I've been playing with the Riemann hypothesis as a matter of recreational mathematics. In the process, I've come to a rather interesting recurrence, and I'm curious as to its name, its reductions, and its…
MrGomez
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18
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5 answers

Are the functions always asymptotically comparable?

When we compare the complexity of two algorithms, it is usually the case that either $f(n) = O(g(n))$ or $g(n) = O(f(n))$ (possibly both), where $f$ and $g$ are the running times (for example) of the two algorithms. Is this always the case? That is,…
matrixx
15
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6 answers

n*log n and n/log n against polynomial running time

I understand that $\Theta(n)$ is faster than $\Theta(n\log n)$ and slower than $\Theta(n/\log n)$. What is difficult for me to understand is how to actually compare $\Theta(n \log n)$ and $\Theta(n/\log n)$ with $\Theta(n^f)$ where $0 < f < 1$. For…
mihsathe
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11
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2 answers

How to prove that $n(\log_3(n))^5 = O(n^{1.2})$?

This a homework question from Udi Manber's book. Any hint would be nice :) I must show that: $n(\log_3(n))^5 = O(n^{1.2})$ I tried using Theorem 3.1 of book: $f(n)^c = O(a^{f(n)})$ (for $c > 0$, $a > 1$) Substituing: $(\log_3(n))^5 =…
Andre Resende
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11
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2 answers

Decidability of checking an antiderivative?

Let's suppose I have two functions $F$ and $G$ and I'm interested in determining whether $$F(x) = \int G(x)dx.$$ Let's suppose that my functions are composed of elementary functions (polynomials, exponentials, logs, and trigonometric functions), but…
templatetypedef
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11
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2 answers

Is $O$ contained in $\Theta$?

So I have this question to prove a statement: $O(n)\subset\Theta(n)$... I don't need to know how to prove it, just that in my mind this makes no sense and I think it should rather be that $\Theta(n)\subset O(n)$. My understanding is that $O(n)$ is…
Rawb
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9
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2 answers

Solving $T(n)= 3T(\frac{n}{4}) + n\cdot \lg(n)$ using the master theorem

Introduction to Algorithms, 3rd edition (p.95) has an example of how to solve the recurrence $$\displaystyle T(n)= 3T\left(\frac{n}{4}\right) + n\cdot \log(n)$$ by applying the Master Theorem. I am very confused by how it is done. So, $a=3, b=4,…
newprint
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8
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1 answer

Given a fast and a slow computer, at what sizes does the fast computer running a slow algorithm beat the slow computer running a fast algorithm?

The source of this question comes from an undergraduate course I am taking, which covers an introduction to the analysis of algorithms. This is not for homework, but rather a question asked in CLRS. You have a slow machine running at $x$ MIPS, and a…
DoggoDougal
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8
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Complexity of computer algebra for systems of trigonometric equations

As discussed in this question, I drafted a spec algorithm that hinges on finding a specific root of a system of trigonometric equations satisfying the following recurrence: $\qquad f_{p_0} = 0\\ \qquad p_0 = 2\\ \qquad \displaystyle f_{p_n}(x) =…
MrGomez
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8
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1 answer

High maths for game theory

I am a starting Ph.D. student in computer science, and I am trying to understand some classic game-theory papers, such as those by Nash, Kalai and Smorodinsky. But I find it hard to understand the mathematical parts. It seems that these papers…
Erel Segal-Halevi
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8
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2 answers

$\log^*(n)$ runtime analysis

So I know that $\log^*$ means iterated logarithm, so $\log^*(3)$ = $(\log\log\log\log...)$ until $n \leq 1$. I'm trying to solve the following: is $\log^*(2^{2^n})$ little $o$, little $\omega$, or $\Theta$ of ${\log^*(n)}^2$ In terms of the…
gfppaste
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