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[Bruinier, Ono](http://www.aimath.org/news/partition/brunier-ono) have found an algebraic formula for partition function, which was widely reported to be a breakthrough. I am unable to understand the paper, but does it have any algorithmic consequences for fast computation of the partition function? |
CS consequences of algebraic formula for partition function? |
|algorithms|number-theory| |
Here are two hints
1. If $\bar L$ would be decidable (recursive), then so would be $L$, and you just argued that $L$ is not even recursively enumerable. Maybe you have a closer look on what $\bar L_0 \cap \bar L_1$ really is. For example
$\bar L_0 = \{ \langle M, w, 0 \rangle \mid \text{$M$ does not halts on $w$}\... |
|integers|decision-problem| |
|formal-languages|terminology|regular-expressions| |
Language member explanation |
|computational-geometry|approximation|graphics| |
|combinatorics|probability-theory|permutations| |
Intro:
I'm working with huge data set that i need to plot in browser, and since there may be up to 1M points my idea was to create different representations for different zoom levels
lets say i have 100k points, i would average two-by-two until i get 50k, then i would repeat that until i get below 500 points (my ... |
|formal-languages|turing-machines|undecidability|halting-problem| |
null |
null |
The precise mathematical skills that you'll need for your studies will vary depending on exactly which area of ML you'll be working in. In general though Bayes theorem, probability theory and statistics will be your constant companion. Getting a good grasp of these is extremely helpful when it comes to fully understand... |
null |
What is the procedure for computing the rank of a multiset after inserting an element?
For instance, lets say we have a set $S = (0,1)$ containing $n = 2$ distinct elements.
The multiset $M = (1,1)$ has rank $5$ because there are $4$ multisets less than it based on numerical ordering: $(0), (1), (0,0), (0,1)$.
... |
Computing the rank of a multiset after inserting another element |
|algorithms|combinatorics|sets|binary-arithmetic| |
was looking at details on finite fields from mathematics. finite binary fields eg $\mathbb{F_2}$ are used in CS in some places eg circuit theory.[3]
> what are some key applications of finite fields in CS?
also looking for uses of $\mathbb{F_{2^n}}$ which the mathworld link below[2] shows can be represented as b... |
what are some applications of binary finite fields in CS |
|algorithms|reference-request|discrete-mathematics|applied-theory| |
I mean is it essential? Does it have a specific application in comp science? Cause i cant seem to muster enough motivation for it |
Whats the use of studying trigonometry in computer science? |
|mathematical-analysis|mathematical-software|mathematical-programming| |
null |
was looking at details on finite fields from mathematics. finite binary fields eg $\mathbb{F_2}$ are used in CS in some places eg circuit theory.[3]
> what are some key applications of finite fields in CS?
also looking for uses of $\mathbb{F_{2}^n}$ which the mathworld link below[2] shows can be represented as b... |
I assume laziness is not an issue here and you are interested in pursuing computer science and therefore want to be laser-sharp. But even then, our basic sciences seem to crop up everywhere later in higher studies, especially mathematics. And trigonometry comes up in mathematics a little too much to be ignored. If you ... |
For one example, it shows up in the analysis of approximation algorithms for Feedback Vertex Set.
The *characteristic vector* of a simple cycle $C$ in a graph $G = (V, E)$ is a vector in $\mathbb{GF}[2]^m, m = |E|$. A vector over this field has 1s in components correspond to edges of $C$ and 0s in the other componen... |
What is the procedure for computing the rank of a [multiset][1] after inserting an element?
For instance, lets say we have a set $S = (0,1)$ containing $n = 2$ distinct elements.
The multiset $M = (1,1)$ has rank $5$ because there are $4$ multisets less than it based on numerical ordering: $(0), (1), (0,0), (0,1)... |
For one example, it shows up in the analysis of approximation algorithms for Feedback Vertex Set.
The *characteristic vector* of a simple cycle $C$ in a graph $G = (V, E)$ is a vector in $\mathbb{GF}[2]^m, m = |E|$. A vector over this field has 1s in components correspond to edges of $C$ and 0s in the other componen... |
Consider the two hash functions used to map IP addresses. $x_i$ represents a octave (or "bit field") of the address.
Hash Function 1:
$$h_a(x_1, x_2, x_3, x_4) = \sum^{4}_{i=1} a_ix_i \mod n$$
Where $n$ is a prime number closest to the total number of IP addresses being mapped. The probability of two IP addresse... |
How do the following Hash Functions compare? |
|hash-tables|hash| |
Finite fields come up in many places. Here are just a few examples:
1. The Razborov-Smolensky polynomial method.
2. Fourier analysis, as used for example in the proof of the PCP theorem, or fast integer multiplication.
3. List decoding - codes like Reed-Muller are algebraic codes.
4. Algebraization, the method us... |
* **Rotations**: that arise in Computer Graphics and Robotics , through rotation matrices, Quaternions, etc.
* <a href="http://en.wikipedia.org/wiki/CORDIC"> Cordics </a> for computing these functions on a Microprocessor / FPGA
* <a href="http://en.wikipedia.org/wiki/List_of_computer_vision_topics#Filtering.2... |
Your question is a bit confusing. These two "hashes" are incomparable.
The reason is that your second function is a **specific hash function**, while the first formulation is a **family of hash functions** (indexed by $a$).
How can you compare the two? Choose $a$ in random and compare the two instances? Compare ... |
Your question is a bit confusing. These two "hashes" are incomparable.
The reason is that your second function is a **specific hash function**, while the first formulation is a **family of hash functions** (indexed by $a$).
How can you compare the two? Choose $a$ in random and compare the two instances? Compare ... |
Your question is a bit confusing. These two "hashes" are incomparable.
The reason is that your second function is a **specific hash function**, while the first formulation is a **family of hash functions** (indexed by $a$).
How can you compare the two? Choose $a$ in random and compare the two instances? Compare ... |
Your question is a bit confusing. These two "hashes" are incomparable.
The reason is that your second function is a **specific hash function**, while the first formulation is a **family of hash functions** (indexed by $a$).
How can you compare the two? Choose $a$ in random and compare the two instances? Compare ... |
null |
A tensor product of operations, $I\otimes J$ say, acts on each subsystem separately: if $\phi$ and $\psi$ are states and $I$ and $J$ are operators then
$$(I\otimes J)(\phi\otimes \psi) = (I\phi) \otimes (J\psi)$$
In bra-ket notation the state $\phi\otimes \psi$ can be denoted $|\phi\rangle|\psi\rangle$.
In your fir... |
I can not understand the following equality $$\langle ij|(|0\rangle \langle 0|\otimes I)kl \rangle= \langle i|0\rangle \langle 0|k \rangle \langle j|I|l \rangle?$$
Also to estimate phase $\phi$ in Nielsen & Chuang book, I can not understand why
$(|0 \rangle + e^{2\pi i 2^{t-1}\phi |1 \rangle})(|0 \rangle + e^{2... |
Tensor Product in Quantum Computation |
|quantum-computing| |
What is the procedure for computing the rank of a [multiset][1] after inserting an element?
For instance, lets say we have a set $S = (0,1)$ containing $n = 2$ distinct elements.
The multiset $M = (1,1)$ has rank $5$ because there are $4$ multisets less than it based on lexicographic ordering: $(0), (1), (0,0), (... |
[Bruinier and Ono](http://www.aimath.org/news/partition/brunier-ono) have found an algebraic formula for the [partition function](http://en.wikipedia.org/wiki/Partition_function_%28number_theory%29#Partition_function), which was widely reported to be a breakthrough. I am unable to understand the paper, but does it have... |
Algorithmic consequences of algebraic formula for partition function? |
|algorithms|complexity-theory|number-theory| |
null |
null |
was looking at details on finite fields from mathematics. finite binary fields eg $\mathbb{F_2}$ are used in CS in some places eg circuit theory.[3]
> what are some key applications of finite fields in CS?
also looking for uses of $\mathbb{F_{2}^n}$ which the mathworld link below[1] shows can be represented as b... |
It is my uninformed belief that Rademacher's formula is faster in practice (and perhaps in theory also) than the Bruinier and Ono formula. While Rademacher's asymptotic expansion is an infinite sum, we know that $p(n)$ is an integer, so if we have bounds on the tails of the expansion, we can use the formula to compute ... |
I'll consider the easy $|S|=2$ case. There are $n+1$ multisets of size $n$, and so the rank of the multiset $0^{n-k} 1^k$ is
$$ r(n,k) = \sum_{m=0}^{n-1} (m+1) + k = \frac{n(n+1)}{2} + k. $$
What happens when adding an element? Adding $0$ and $1$, respectively, correspond to $r(n+1,k)$ and $r(n+1,k+1)$. We have
$$
... |
I second Raphael's remark that when shuffling cards, you don't want the deck to be "unsorted", but rather random. However, when analyzing any specific shuffle, there can be measures of randomness that can be used to *prove* that a small number of shuffles isn't enough to make the deck random enough.
As a simple exam... |
Here are two suggestions for you to try.
*Suggestion 1:* Use a linear filter. Instead of computing the average $x_{2n},x_{2n+1} \mapsto (x_{2n}+x_{2n+1})/2$, try averaging over a larger sequence, e.g. $y_n = (x_{2n-1}+2x_{2n}+2x_{2n+1}+x_{2n+2})/6$.
*Suggestion 2:* Use a conditional filter: if $x_{2n} < x_{2n+1} ... |
null |
Here are two suggestions for you to try.
*Suggestion 1:* Use a linear filter. Instead of computing the average $x_{2n},x_{2n+1} \mapsto (x_{2n}+x_{2n+1})/2$, try averaging over a larger sequence, e.g. $y_n = (x_{2n-1}+2x_{2n}+2x_{2n+1}+x_{2n+2})/6$.
*Suggestion 2:* Use a conditional filter: if $x_{2n} < x_{2n+1} ... |
IPv6 addresses in the form of `862A:7373:3386:BF1F:8D77:D3D2:220F:D7E0` are much harder to memorize or even transcribe than the 4 octets of IPv4.
There [have](http://blog.jgc.org/2011/07/pronounceable-ipv6-addresses-wpa2-psk.html?m=1) [been](http://www.halfbakery.com/idea/IPv6_20Worded_20Addresses#1260513928) atte... |
I am a bit confused with the CLRS example on this question. if someone could explain it in easier words that would be nice. thanks |
why is the randomnly built bst height O(log n) |
|binary-trees|randomness|search-trees| |
{"Voters":[{"Id":4193,"DisplayName":"user1675999"}]} |
{"Voters":[{"Id":4193,"DisplayName":"user1675999"}]} |
How do you proof that bst build randonmnly is o(logn) ] |
How do you proof that bst build randonmnly is o(logn). If you refer to chap 12.4 of the CLRS intro to algorithms. You will find the theorm with some Expected value proofs. I dont understand the equations used in it. |
Proof that a randomnly built bst height O(log n) |
Consider the two hash functions used to map IP addresses. $x_i$ represents a octave (or "bit field") of the address.
Hash Function 1:
$$h_a(x_1, x_2, x_3, x_4) = \sum^{4}_{i=1} a_ix_i \bmod n$$
Where $n$ is a prime number closest to the total number of IP addresses being mapped. The probability of two IP address... |
I was looking at details on [finite fields](http://en.wikipedia.org/wiki/Finite_field). Finite binary fields, e.g. $\mathbb{F_2}$, are used in CS in some places such as circuit theory [1].
> What are some key applications of finite fields in CS?
I am also looking for uses of $\mathbb{F_{2}^n}$ which [Mathworld](... |
What are some applications of binary finite fields in CS? |
Suppose i am making a red black search tree, and in my right subtree, i have a black node, then a red node, and it has two black children, the black children further black childrens. As such a lemma has been made that red black tree with n internal nodes has height at most 2log(n+1). Would this proof still hold for suc... |
Can you have three consecutive black nodes in red black search tree? |
|search-trees| |
|programming-languages|compilers| |
In fact, a red-black tree can have all black nodes. In that case the tree is completely balanced and has the required bound. |
i have the following pseudo code:
>Multiply(y,z)
>1. if (z==0) return 0
>2. else if (z is odd)
>3. then return (multiply(2y, floor(z/2)) + y )
>4. else return (multiply(2y, floor(z/2)))
so..the answer i have is T(z) = {0 for z=0 ; T(z/2)+1 for z>0}
why is 0 when z=0? s... |
could someone explain this recurrence equation? |
|time-complexity|algorithm-analysis|asymptotics| |
null |
The following equation is a matrix expression where B and C are $n\times n$ matrices and k is a positive integer:
$$P = \sum_{i=1}^k B_i C_i^T $$
So $P = B_1 C_1^T + B_2 C_2^T + \cdots +B_k C_k^T $
and my understanding is $B_i $ and $C_i$ are matrices themselves.
So this means P will be the sum of a b... |
How do you go about designing a vector processor architecture for the sum of matrix products? |
|computer-architecture| |
To prevent collisions, hash tables uses a methodology to chain the contents. Why cant we use another hash table allocated to each slot of the primary hash table. |
For open addressing in hash tables, why cant we create a hash table asigned to each hash table |
|hash-tables| |
How do you proof that bst build randonmnly is O(logn). If you refer to chap 12.4 of the CLRS intro to algorithms. You will find the theorm with some Expected value proofs. I dont understand the equations used in it. |
null |
The short answer is that this is, more or less, equivalent to having one hash table. Let's say you're hashing $n$ items into $m$ slots, and each slot has its own hash table of size $c$. You propose first hashing into one of the $m$ slots using some function $h$, then hashing again into one of the $c$ slots using some... |
i have the following pseudo code:
>Multiply(y,z)
>1. if (z==0) return 0
>2. else if (z is odd)
>3. then return (multiply(2y, floor(z/2)) + y )
>4. else return (multiply(2y, floor(z/2)))
so..the answer i have is
$T(z) = \begin{cases} 0 & z=0 \\ T(z/2)+1 & z>0\end{cases}$
... |
There's a bit of ambiguity in what you're looking for here. I am going to assume it's the running time of the function.
I agree that it should be 1 when $z=0$ because it generally is considered to take $O(1)$ operations to return a function value. However, this will only affect your final answer by an additive con... |
How do you prove that the expected height of a randomly built [binary search tree][1] is $O(\log n)$. There is a proof in CLRS _Introduction to Algorithms_ (chapter 12.4), but I dont understand it.
[1]: http://en.wikipedia.org/wiki/Binary_search_tree |
null |
**Comment on your proof**:
You write
"Let $c = {c_1}^2$." Strictly speaking, this doesn't make sense, since you haven't set $c_1$ to any value yet at this point. To be formally correct, you need to start with some given $c$, and then set $c_1$ to $\sqrt{c}$ and proceed from there (which is what I've done below where ... |
I am new to this topic, and in some scientific papers I've been reading about prediction in sports I encountered the term rule based reasoner. Is it this term the same as a semantic reasoner( where the two main directions are forward and backwards chaining?). If yes, can you point out the difference between a decision ... |
Difference between decision tree and rule based reasoner |
|data-mining| |
I mean is it essential? Does it have a specific application in computer science? Because I cant seem to muster enough motivation for it. |
|mathematical-analysis|applied-theory| |
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