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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How exactly does minimizing the L2 norm of w in SVR affect the model's ability to fit the data?
I have a solid grasp on why we minimize the L2 norm of the weight vector ( w ) in Support Vector Machines (SVM) for classification problems — it maximizes the margin between classes and helps the ...
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What are the conditions necessary for a programming language to have no undefined behavior?
For context, yesterday I posted Does the first incompleteness theorem imply that any Turing complete programming language must have undefined behavior?. Part of what prompted me to ask that question ...
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Is "Bitwise Complement Operator" (~ tilde) distributive?
To be more precise, Is ~(a+b) = ~a + ~b? Here, "~" bitwise NOT operator.
I ran into this question while thinking about ...
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Division by zero in APL: assign return value 0 or 1
Recently I began learning a couple array based programming languages: Dyalog APL and BQN. And I cam across this peculiar manner in which division by zero is handled in Dyalog APL. Using the in-built ...
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How to represent a point cloud in the pseudocode of an algorithm?
I am writing a scientific paper in which I describe some algorithms (using pseudocode) that have point clouds as inputs. In these algorithms, I need a mathematical structure to represent a point cloud....
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How to show invariance under additive and multiplicative illumination changes
The task is to "Show that the transformation f -> ((ln fx)x, (ln fy)y) is invariant under additive and multiplicative illumination changes."
From my lecture notes, for ((ln f)x, (ln f)y), ...
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Is the root function computationally equivalent to function application?
If a function type is representable by exponentiation, does it follow that function application is represented by the right inverse, roots? It would seem that roots consume a function's input to ...
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Could Gödel’s incompleteness theorem be circumvented with a quine?
As you all probably know, Gödel’s incompleteness theorem states, that it will never be possible for mathematics to prove its own correctness.
Mainly because that proof would be part of mathematics too,...
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Reading list for mathematics and formal logic with missing truth values?
I would like a reading list of math and formal logic books which give a principled discussion of missing values.
Textbooks on mathematics and formal logic - propositional and first-order and higher-...
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How to evaluate all the binary sequences, generated from $2^{100}$ for finding all the sequeces which contain minimum $10$ zeros?
Suppose I have a set of $2^{n}$ number of binary sequences. And I have to select only those sequences which contain a minimum ${P}$ number of $0$ in it. For example, please consider the below one
Eg.
...
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Finding square root of a gram matrix over the integers [closed]
Suppose that matrix A is a symmetric positive definite matrix over the integers, i.e., $A \in Z^{n\times n}$, if B is a matrix over the real numbers, it is not difficult to find B such that $A = B \...
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Why is this mapping from binary to natural numbers surjective?
In my Lecture we came across something of importance for the Church-Turing-Thesis and i noticed one particular function, which confused me.
Function:
Let $bin(x)$ be the injective binary extension ...
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What algorithm is best for resource gathering in an RTS game in quickest time?
Though the concept can be for many RTS games but lets use the specific system of 'age of empires'. For simplicity one resource will be considered. Food.
So in the game there are villagers who gather ...
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What does x sign mean in functions
I came across this cross sign when reading a book. Does anyone know what does this mean in the context of a function?
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Translating the in-order index of a node in a complete, balanced binary tree into the level-order index
Consider the topmost part of a complete, balanced binary tree of all 64-bit numbers, exemplified here.
As highlighted by the lack of a 7*2^64/8 term it is not ...
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Are the sets in $\Sigma_{p}$ (or $\Pi_{p}$) totally ordered?
Are the sets in each stage of the arithmetic hierarchy well-ordered, with respect to : $T-$reductions, or $m-$reductions?
It is something which I have been unclear with for a while (from a ...
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Is a zero matrix nilpotent matrix?
Let a matrix $A=0$. Can I say $A$ is nilpotent? I am asking if the zero matrix is nilpotent or not.
Nilpotent means that $A^k = 0$ for some $k$ that is a non-negative integer. When $k = 1$, $A^k = A = ...
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Efficient cardinality of set overlap relation
Assume that we have a set S of sets s.
Every pair (s,s') in ...
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Is discrete math enough for computer science ? Or there other Math topics that I should also learn With it?
I want to learn computer science, SO is discrete math enough for computer science ? Or there other Math topics that I should also learn With it ?
I don’t have specific topic that I care more about ...
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How to specify mutated types mathematically?
Say I have an object which I pass to a method, and the method returns that same object, just mutated.
So it goes like this:
...
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Function notation/mapping for this type of functions?
In short, what is the proper function notation/mapping for this type of functions?
From Wikipedia regarding maps:
In the communities surrounding programming languages that treat functions as first-...
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Prove that for any finite k = 1, 2, 3 … PCPk is decidable
Let PCPk be a version of PCP where we can use at most k copies of each domino to create correspondences. Prove that for any finite k = 1, 2, 3 … PCPk is decidable.
I know this kind of PCP is called ...
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Proof the number of expected node in a binary using mathematical induction
The following algorithm constructs a binary tree.
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Papers on depth separations in neural networks
I am new to the domain of Machine learning. I have been asked to present a paper related to the mathematics behind the depth separations in Neural Networks (by Itay Safran, Ronen Eldan and Ohad Shamir)...
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A set that is not polynomial time enumerable
For a set $I\subseteq \Bbb N$, defined $s_{I}(n)=\min\{i\in I\mid i>n\}$. The set $I$ is called polynomial-time-enumerable if $s_I(n)$ is computable in $\mathsf{poly}(n)$ time.
Most of the sets I ...
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"The Annotated Turing" on listing all binary numbers between 0 and 1
In his book "The Annotated Turing" in the first sentence on page 32 Charles Petzold wrote:
These are binary numbers between 0 and 1, and (judging from the way we
created these numbers) all ...
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Monad in Haskell programming vs. Monad in category theory
I have a question about concept of monad used in Haskell programming
and category theory in math.
Recall in Haskell a monad consists of following components:
A type constructor that defines for each ...
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What are the implications of Homotopy Type Theory?
I've recently come across the topic of homotopy type theory and I'm interested to learn more. I have a very limited background in type theory.
Can anyone tell me, in functional programming terms or ...
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Which topics of mathematics should I need to learn to be a good app developer?
I'm 29 years old. I couldn't continue my studies after grade 10 due to some financial issues and I didn't have time to practice mathematics. It's been more than 11 years since I left studies. Now I ...
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What is a list of "reasonable but undecidable" theorems?
There are some theorems that go along the lines of "all reasonable properties of <math subject> are computationally undecidable." Here are two examples:
Rice's Theorem: "all ...
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A silly doubt on Computer Science Objects: Arrays, Lists, Strings, and Vectors, are they the same concept?
First of all I'm a Physics Student and I've been arguing with my brother concerning the usage and the fundamental meaning of a vector on Computer Science (CS).
I) Vectors
A vector have a axiomatic ...
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Mathematical expression for the quantity that we are maximising in the stock buying and selling problem
Problem Statement:
Say you have an array prices for which the $i^{th}$ element is the price of a given stock on day $i$.
Design an algorithm to find the maximum profit. You may complete as many ...
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What is the difference between a fraction and a float?
I understand any fraction to be a quotient of integers which isn't 0, but after coming across the term "float" in various programming languages (such as JavaScript) I misunderstand why it is ...
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How was the σ function chosen to extend the perceptron?
I am just reading about perceptrons in more depth, and now onto Sigmoid Neurons.
Some quotes:
A small change in the weights or bias of any single perceptron in the network can sometimes cause the ...
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Usefulness of Differential Geometry
I recently came across these books:
Differential Geometry and Lie Groups: A Computational Perspective
Differential Geometry and Lie Groups: A Second Course
Their subject matter really intrigues me, ...
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How do you convert bits into a different alphabet?
I have forgotten how to do this. How do I figure out what the requirements are for a 128-bit string using a certain alphabet?
That is to say, I want to generate a UUID (128-bit) value, using only the ...
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Sets in Mathematics are immutable but in Computer Science sets are mutable and called "Dynamic Sets" - truth of the statement
While reading the classic text Introduction to Algorithms by Cormen et. al. I came across the following claim:
Sets are as fundamental to computer science as they are to mathematics. Whereas ...
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A general picture of formal verification in software
I'm trying to piece together a general picture of the state of formal software verification, and I'm having a good bit of trouble. For context, I come from mostly a math background. I'm quite familiar ...
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How is expected value in entropy derived?
I was self learning about entropy and came across this equation.
$$
H = - \sum p(x) \log p(x)
$$
The equation for entropy in expected value,
$$
H(x) = \operatorname*{\mathbb{E}}_{X \sim P}[I(x)] = -\...
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Prove little o with just the definition
I have been searching for a while now but couldn't find anything about this exact pair of functions with the little $\mathcal{o}$ notation.
Given the functions $f(n) = 2^{n}$ and $g(n) = n!$ I am ...
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Where does the base come from in a subsequence count?
If sequence X has m symbols, then X has 2m possible subsequences.
Where does the base (<...
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Big O Tires Question? [duplicate]
My question is regarding the last paragraph of this excerpt from "Cracking the Coding Interview." (For some reason, my table is not formatting here.)
What's the runtime of this code?
...
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Why is addition in GF(2^8) the same as XOR?
I get the impression it has to do with either some quirk involved with limiting to 2^8 or that I'm misunderstanding what addition can be within the context of a finite field, but I'm not quite sure ...
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I don't understand formula meaning of computer vision Paper
I am master degree student at Computer Science Department.
I started master degree at this year.
Now I'm reading two papers about computer vision.
Actually before time I didn't have experience about ...
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How does the forget layer of an LSTM work?
Can someone explain the mathematical intuition behind the forget layer of an LSTM?
So as far as I understand it, the cell state is essentially long term memory embedding (correct me if I'm wrong), ...
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How are Complex Numbers and Complex Analysis used in CS? [closed]
The world is 3D and data of it is usually (as far as I know) represented and processed with real numbers. I've seen very few cases where complex numbers are used in programming and none when it comes ...
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Integer-array indexing (e.g. numpy.take) as function composition: where can I find more resources?
For context, one of Numpy's features is that an an array of integers can be passed as an index to an array, and this selects values at all of the specified positions, in any order with possible ...
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What are the other language models of computation similar to lambda calculus?
I hope this question makes sense, but I was wondering if there are other models of computation similar to lambda calculus that you can use to build up axiomatic mathematical and logical fundamentals ...
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How does a computer interpret real numbers?
I understand that the modern day digital computer works on the binary number system. I can also get, that the binary representation can be converted to rational numbers.
But I want to know how does ...
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Can we see all of mathematics as an attempt to simplify computations?
This is a rather strong claim, and therefore likely to be incorrect, but hear me out.
Firstly, when I talk of “computations”, I mean this in a broader sense than normally used, because I am including ...
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