Questions tagged [mathematical-foundations]

Questions about the relation of (subfields of) computer science to the relevant mathematical foundations and their application.

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7
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2answers
342 views

Can you operate on and draw conclusions on functions described asymptotically?

This question is homework based (not using actual problem though)! Say you have a function described as: $$f(n) \in O(2n^2) \, .$$ Can you then go on the treat this as: $$f(n) = 2n^2$$ and ...
7
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2answers
942 views

Minor Mistake in Computability, Complexity, and Languages?

In the book Computability, Complexity, and Languages (2nd edition), Martin Davis writes in chapter 1 (Preliminaries), section 2 (Functions): A partial function on a set $S$ is simply a function ...
1
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2answers
245 views

Set notation of the set of all strings

How do I present the complement using set notation? I guess it has to be shows with universal set - {aa,bb} but I do not know how to represent the universal set in ...
2
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0answers
35 views

physical significance of membership function greater than one

In fuzzy logic, when we associate an element with a set, we usually do it in terms of membership grade which suggests the "belonging" of this element to the set. Membership grade value 0 means that ...
4
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1answer
275 views

Are numbers types and what is “Number”?

I was pondering about what are the numbers. It seems like a number is data type. I mean, like Maybe in Haskell. Because, for instance, one on its own means nothing for me. However, one apple tells me ...
4
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3answers
237 views

How Types avoids Russel's Paradox?

I gone through the Russel's paradox. From Wikipedia : According to naive set theory, any definable collection is a set. Let R be the set of all sets that are not members of themselves. If R is ...
1
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1answer
181 views

Solving inhomogeneous recursion using quadratic particular solution

I am interested in solving recursions and while I was trying to solve this one: $a_n=a_{n-1}+4n-3$ I ran onto a problem, I don't see how to find an explicit solution using standard inhomogeneous ...
4
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1answer
661 views

Why is the zero-th power of the empty set {ε}?

It has been asked before why $\emptyset^\star=\{\epsilon\}$. The answer boils down to $\emptyset^\star$ being defined as $$ L^\star = \bigcup_{i=0}^\infty L^i, $$ where a word in $L^i$ is the ...
5
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2answers
2k views

Predicate Logic Notation: What does a “dot” mean?

What does a dot (.) mean in predicates? $\forall a \in A. \exists d \in D. H(a,d)$ Especially, how is the above different to $ \exists d \in D. \forall a \in A. H(a,d)$ I've never seen this used ...
2
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1answer
88 views

What kinds of problems are modeled by a recursive definition of a set of strings?

Given this definition: The set $\Sigma^*$ of strings over the alphabet $\Sigma$ is defined recursively by: BASIS STEP: $\lambda \in \Sigma^*$ (where $\lambda$ is the empty string) RECURSIVE ...
0
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1answer
148 views

Help in understanding how to apply nonlinear function in hashing

Can somebody help in brainstorming how to apply the map as a hashing function? I am aware that chaos is used in cryptography but I fail to understand how to apply it. Most popular hashing techniques ...
3
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1answer
887 views

What mathematical background is enough to go through the entire Knuth's TAOCP?

I am an undergrad. I can do mathematics fine and am a lot more interested to explore. I turned towards TAOCP as it is highly mathematically oriented so as to excel at maths and programming at the same ...
1
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1answer
51 views

What Does The Notation of Union Over K represent?

I am currently studying about polynomial time in Comp Theory. The definition for polynomial time given in my textbook as follows: $$P= \bigcup_{k}TIME(n^k)$$ I have not seen the Union over k notation ...
1
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1answer
367 views

How to read 'max min' and 'min max'?

Here is an exercise I came across while studying MaxFlow / MinCut and I'm rather stumped: Designate $G = (V, E)$ as an undirected graph that has at least two vertices. Every edge $(e)$ on this ...
4
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1answer
45 views

Reference Request: Overlaps between complexity theory and dynamical systems?

Per Wikipedia: In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in a geometrical space. Examples include the mathematical models that ...
1
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4answers
2k views

What is the role of mathematics in programming?

I am almost totally uneducated in programming and computer science in general, though I am really fascinated by these topics. I read that mathematics is important in programming and that nowadays many ...
3
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0answers
84 views

Describe data structure using equations

Good afternoon. At work I'm currently developing a system which takes user input (well structured) and then stores it in memory to do some processing. The input is basically a dataset formed by ...
0
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1answer
90 views
3
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5answers
11k views

What is the difference between $\log^2(n)$, $\log(n)^2$, $(\log n)^2$, $\log (n^2)$ and $\log \log n$?

In research articles (e.g. this one's abstract), I often see the complexity of an algorithm expressed using a somewhat ambiguous notation: $\log^2(n) = ?$ $\log(n)^2 = ?$ $(\log n)^2 = (\log(n)) \...
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0answers
88 views

What is the Necessary math to understand books?

I have many problems understanding algorithms described in the books. Well, I'm talking about the mathematical description of a problem. For example: I don't understand how the math of Unscented ...
1
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1answer
235 views

What is discrete mathematics and why am I learning about it?

I started computer science in university. And we started learning about Discrete and it's mathematics which is completely based on logic, which I understand. however, how is predicates, sets, proofs ...
2
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1answer
109 views

Equivalence relations with $\cap$, $\cup$ and $\circ$

I have the relations $E_1$ and $E_2$. A reflexive, symmetric and transitive property shows that two relations are equivalent to each other. I need to prove if this is true for the following a) $E_1 ∩...
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1answer
61 views

How to read this inductive language definition?

A language $L$ is defined recursively according to the following rules: $λ ∈ L$ If $w ∈ L$, then $bw ∈ L$ and $waa ∈ L$ I am not sure if strings from this language should mix from this definition. ...
0
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2answers
170 views

How to expand 2D graphic and functions to 3D?

So I have a program that enables the user to draw 2D-objects. To rotate them, to move them, and so on, all in 2D. I want to expand the 2D objects and functions to 3D which I don't expect to be too ...
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0answers
33 views

How much calculus do you actually use (not counting use of a graphing calculator)? [closed]

I'm wondering if it's worth it to pursue some deep study into calculus. Already, I have some familiarity with derivatives, limits, and integrals. Though, I also have a graphing calculator that's ...
3
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1answer
66 views

MAP estimation (for stationary iid gaussian environment)

This is my first post, and have been self studying Haykin's Neural Networks and Learning Machines book. I'm not sure if this is a typo or if I'm doing something wrong, but I've been stuck on a ...
4
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1answer
134 views

Right equivalent elements arising in the proof of the Schützenberger Theorem

As a part of my Bachelor thesis in computer science I should review the proof of the Schützenberger Theorem (which was given by M.P. Schützenberger himself $^{[1]}$). My question arises on page 193 in ...
9
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2answers
3k views

What are Markov chains?

I'm currently reading some papers about Markov chain lumping and I'm failing to see the difference between a Markov chain and a plain directed weighted graph. For example in the article Optimal state-...
15
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6answers
5k views

What parts of linear algebra are used in computer science?

I've been reading Linear Algebra and its Applications to help understand computer science material (mainly machine learning), but I'm concerned that a lot of the information isn't useful to CS. For ...
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4answers
98 views

What do we call a function that outputs 0 on half and 1 on the other half of all inputs?

I have a Boolean function that outputs a one on half of its inputs and outputs a zero on half of its inputs, the inputs are assumed to be coming from the uniform distribution. Another way of saying ...
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1answer
136 views

What is the significance of the following equation in computer vision?

Say I have a set of sample measurements of a one dimensional curve in the image plane $f(x)$. What is the purpose of minimizing the following functional $$E(S) = \int \left[\lambda(S''(x))^2 + (f(x) - ...
3
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2answers
359 views

Terminologies of “Process calculus” and “Process algebra” [duplicate]

In the literature, the terms of "process calculus" and "process algebra" are often interchangeable. Meanwhile, it confused me. My questions are: Are there formal, standard, and widely-accepted ...
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1answer
797 views

Applications in Computer Sciences of Partition Functions

A partition function computes the number of ways an integer $n$ can be represented as the sum of $m$ other integers. For some value $n$, we have a partition function $p(n)$. These were studied ...
-1
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1answer
104 views

Matrix usage in CS [closed]

I'm studying a major in CS. I'm interested in taking a few extra courses, specifically math to improve my knowledge as future computer scientist. Right now I'm thinking to take Matrix Fundamentals ...
-1
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1answer
47 views

Syntax checking expressed mathematically

1) Set B is the finite set of all rules for a particular programming language. 2) Set A is the infinite set of all valid strings of that language and is defined by Set B. I'm having a problem ...
2
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2answers
1k views

What is a mod b if a < b? [closed]

So when you are decrypting an encryption with modular arithmetic, I know the modulus operator (%) gives the remainder of a/b, but what if a is less than b? For example, 5%2 is 1, since the answer is ...
0
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0answers
64 views

What level of maths do I need for artificial intelligence? [duplicate]

I want to learn maths for artificial intelligence. I heard that main areas of mathematics for AI are Multivariable Calculus, Linear Algebra, Probability and Statistics and Discrete Mathematics. But ...