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Questions tagged [parity]

This tag is for questions relating to "Parity", a mathematical term that describes the property of an integer's inclusion in one of two categories: even or odd.

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How are there $2^n$ $n$-variable parity functions?

Source: https://www.cs.cmu.edu/~odonnell/boolean-analysis/lecture2.pdf Context: Linearity testing of boolean functions is being discussed, and they just introduced parity functions which I didn't know ...
Saksham Sethi's user avatar
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A proof that every natural number is either even or odd.

If $k \in\mathbb{Z}$ is a number, $n$ is said to be even if it is in the form $2k=n$. $2k+1$ is odd and so is $n+1$. Let $Q$ be the set of natural numbers in the form $2k$ or $2k+1$. Obviously $$1 \in ...
Edward Falls's user avatar
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What general concept, other than oriented volume, embodies all even (odd) permutations of $n$ elements?

Given an ordered orthogonal basis $v_1...v_k$ of a $k$-dimensional vector space, we can construct a $k$-vector in the geometric algebra over that space, simply by multiplying the basis vectors in the ...
Adam Herbst's user avatar
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Reference Request: Definition on parity of scalar-valued multivariate functions (even or odd)

I am studying the densities of multivariate Gaussian distributions and I am curious about the parity of those scalar-valued multivariate functions. For a function $f(\mathbf{x}): \mathbb{R}^n \...
wutai's user avatar
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Deduce $\begin{bmatrix} 1 & 0 & 1 \\ 0 & 1 & 1 \\ \end{bmatrix}$ or the simple parity-check matrix just from the final matrix-vector product equations

A linear combination like $A_{1}+ A_{2}$ serves as a backup of $A_{1}$ when $A_{2}$ is known, and serves as a backup of $A_{2}$ when $A_{1}$ is known. As a result of linearity, any two out of $A_{1}x$,...
triple_tactic's user avatar
2 votes
3 answers
112 views

Suppose we want to prove that a property $P$ is true for every integer in $ℕ_{odd}$ = $\{1,3,5,7,9,...\}$.

Suppose we want to prove that a property $P$ is true for every integer in $ℕ_{odd}$ = $\{1,3,5,7,9,...\}$. Consider the following induction mechanism: Base case: Verify the property $P(1)$ Inductive ...
Asher's user avatar
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Are there any prime solutions to $p^q - q^p =3$

Are there any prime solutions to $p^q - q^p = 3$? (For the prime numbers $p$ and $q$) If so what are they? This problem is meant to be solved with modular arithmetic. I tried proving that firstly they ...
Spinarak167's user avatar
5 votes
5 answers
1k views

Odd numbers becoming even numbers but even numbers not becoming odd numbers

When you multiply an odd number by an even number you get an even number, right? For example, $19 \cdot 2$ that is $38$ (an even number). But when you try applying that logic to even numbers it will ...
Xian Tambis's user avatar
2 votes
1 answer
75 views

How to find permutations with same order but different parity

In group (S10,o) the decomposition of a permutation p into disjoint cycles is p=(134)(2867). The order is found using the least common multiple (LCM) of cycle lengths, in this case, LCM(3,4) = 12. So, ...
Kyberchronis's user avatar
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Minimum sum between permutations of elements in a matrix

So, for the purposes of creating a heuristic for a Sokoban AI, I have to solve the following problem (if you're curious, the elements of the matrix each represent taxicab distances from boxes to ...
Math Machine's user avatar
7 votes
1 answer
273 views

Number of triangles a point lies in on a plane

On a plane , there are $2n+1$ points where no three points are co-linear. Show that for any point $P$ which is one of the points, the number of triangles the interior of which $P$ lies in is always ...
maomao's user avatar
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How to find generator matrix and parity check matrix?

The $[n, n − 1, 2]$-parity check code has messages consisting of binary strings of length $n − 1$. Messages are encoded by adding an extra symbol which is the sum (mod 2) of the previous symbols. I ...
Cooper Brian's user avatar
3 votes
1 answer
67 views

Assigning to every odd number a triangular number. Does the converse hold?

Context: A triangular number is a number of the form $n(n+1)/2$ where $n\in \mathbb N$. Question: After application of two theorems in my textbook (the first one being that "a number $n$ is ...
niobium's user avatar
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Parity in the Schrödinger equation of the hydrogen atom

Hi I'm studying quantum mechanics with Stephen Gasiorowicz's book. It says for the $n=2$ states of the hydrogen atom, the $l=0$ state has even parity and the $l=1$ state has odd parity. As far as I ...
Chan J.'s user avatar
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Prove or disprove: the average of any two odd integers is always odd

Prove or disprove the following statement: the average of any two odd integers is always odd. The proof that this statement is false is given below. Please, verify whether the one is valid or not. $\...
Vlad Mikheenko's user avatar
1 vote
1 answer
103 views

Is there a notion of a category where morphisms have parity?

In category theory, is there a name/notion for categories where the morphisms have a well-defined parity? Formally, this should mean that there is a non-constant functor from the category to the one-...
Milten's user avatar
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1 answer
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Ordered sequence of even products of nonnegative integers and possible code

I am trying to find the sequence of the even products of nonnegative integers, m and n, after sorting them. For example, $$1*6=6, 2*3=6, 6*1=6, 3*2= 6$$ What I think is the answer is the sequence is ...
Teg Louis's user avatar
2 votes
2 answers
221 views

Three grasshoppers jumping on a line

The following problem appears on pp. 86 Terrence Tao's "Solving Mathematical Problems: a personal perspective". Three grasshoppers are on a line. Each second, one (and only one) grasshopper ...
vietajumping's user avatar
1 vote
2 answers
167 views

Question in pseudovectors

I learnt that under parity transformation a vector $\vec{A}$ <---(Parity)------> $-\vec{A}$ and a pseudovector can be written as $\vec{c}=\vec{A} $ $\times$ $\vec{B}$ and since A goes negative A ...
Naveen V's user avatar
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Let α1 , …, αr be distinct even permutations, and β an odd permutation. Show then α1β, …, αr β are r distinct odd permutations.

Q: Let $\alpha_1 , …, \alpha_r$ be distinct even permutations, and $\beta$ an odd permutation. Show then $\alpha_1\beta, …, \alpha_r \beta$ are $r$ distinct odd permutations. Proof: $\alpha_i$ can be ...
martinkleins's user avatar
1 vote
1 answer
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Proving that a couple must sit together on the same motorcycle - a problem with mutually exclusive propositions

$65$ couples go on a spring outing, there are a total of $65$ motorcycles, each motorcycle carries one boy and one girl. Suppose for any two motorcycles, the following two propositions are exactly one ...
CBot's user avatar
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Ratio of odd to even $2n-$tuples

Let $o(n)$ be the number of $2n-$tuples $(a_1,a_2,\ldots,a_n,b_1,b_2,\ldots,b_n)$ such that each $a_i,b_j\in\{0,1\}$ and $a_1b_1+a_2b_2+\ldots +a_nb_n$ is odd. Similarly, let $e(n)$ be the number of $...
aqualubix's user avatar
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2 answers
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Proving If a is an odd integer then $a^2+3a+5$ is odd.

I've written a proof for which looks correct, but I wanted to know it from someone else as well. Also I feel like my proof could be made a bit shorter, so if anyone has advice for that I would ...
Csoriburi's user avatar
2 votes
0 answers
58 views

Parity of number of subsets having odd intersection with fixed number of sets

I am currently working on a combinatorial problem that reduces to the following counting problem: Let $i$ be a positive fixed integer. Given a set $S$, Is there a family of subsets $\mathcal{B}^i_S$ ...
Narek Bojikian's user avatar
0 votes
1 answer
62 views

Decrement of a Permutation vs Number of Transpositions

Given a permutation with $n$ items and $c$ independent cycles, the decrement is defined as $n-c$. Here is a simple proof showing that this number is also equal to the number of transpositions in the ...
jamman2000's user avatar
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2 answers
231 views

Switch off all the n bulbs, (n-1) at a time. A parity-based puzzle?

So, here is a puzzle-like problem: You have n light bulbs. Each one is connected to a special switch that flips the state of all the other n-1 bulbs except the bulb it is connected to. If you start ...
floatingpurr's user avatar
4 votes
2 answers
177 views

Possibility of arranging $1$ and $-1$ in a grid such that the sum of the products is $0$

Consider an $11$ x $11$ grid, where in each square, the number $1$ or $-1$ is written. One multiplies the numbers in each row and column, and then sums up these $22$ products. Is it possible for this ...
Martin Westin's user avatar
1 vote
0 answers
38 views

Maximal accuracy of logistic regression on the n-parity problem

Consider the standard logistic regression ('LR') function, $ y = \sigma(w^T \cdot x + b) $, where $\sigma$ is the logistic function ('sigmoid'). When checking the accuracy, we will consider the argmax ...
Ido4848's user avatar
  • 567
1 vote
0 answers
31 views

Checking the parity of an infinite sum

While going through a paper I stumbled on the following equality : $\begin{equation} \rho(\alpha)=\frac{1}{\pi}\sum\limits_{k\in\mathbb{Z}}\frac{e^{2ik\alpha}}{\cosh(k\zeta)}=\frac{1}{\pi}\frac{\...
Ceethemez's user avatar
1 vote
1 answer
155 views

Scalar Triple Product of Two Polar Vectors and a Pseudovector

So I have recently been trying to use methods of exterior algebra to solve quantum mechanics problems, and I still cannot seem to come to a good conclusion about the cross product of two vectors ...
Luk'yan Vilshansky's user avatar
-1 votes
1 answer
124 views

When is the 8 puzzle solvable? [duplicate]

I'm struggling to find the solution to this eight puzzle: 1 0 2 3 4 5 6 7 8 When I expressed it as a permutation with the cycle notation, I got (23456780) The signature of this permutation is (-1)^...
Marta Pleite's user avatar
0 votes
1 answer
68 views

Is there a simple way to find the parity of the remainder/quotient without performing the division?

The context is the following: I am performing some calculations on long integers in Montgomery form, and I need to know their parity without converting them back to normal form (which is slow). An ...
fjarri's user avatar
  • 191
1 vote
1 answer
52 views

The parity of permutation $(b_1, ..., b_{n-k},a_1, ..., a_k)$

Let $k$ be an integer between $1$ and $n$. Let $\sigma_1$ be the parity of $(a_1, ... ,a_k, b_1, ... ,b_{n-k})$ as a permutation of ${\{1, ... n}\}$ . So $a_1, ... ,a_k, b_1, ... ,b_{n-k}$ are all ...
stoneaa's user avatar
  • 424
1 vote
1 answer
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Proving that for an even $k$ you can't move $k$ rocks from $k$ platforms to the same platform, given a set of rules

We have $k$ rocks on $k$ different platforms placed in a circle. You can only move two different rocks at a time (not more, not less), and you can move them only one platform to the left or to the ...
bendaMan's user avatar
3 votes
0 answers
69 views

How would you detect soft locks in a tile sliding game?

I have created a tile sliding game in python (using pygame). It's a 3x3, and an online example can be found here. When my and my friends started testing this game out, comparing it to the online ...
Saksham's user avatar
  • 83
0 votes
1 answer
120 views

Parity of $p(n)$ if the parity of the self-conjugate partitions is even

Let $p(n)$ be the number of partitions of the positive integer $n$. What can we say about the parity of $p(n)$ if we know that the parity of the self-conjugate partitions is even? Actually I manually ...
user avatar
0 votes
1 answer
165 views

Probability a factor is odd

Take the above question. In the solution it says there 18 factors of 2 in the number. But how do they work this out? I see no reasoning or intuition between the line of working 10 + 5 + 2 + 1 = 18. ...
Howard Stark's user avatar
5 votes
3 answers
349 views

Parity Pattern of Triangle Numbers

I noticed that the triangle numbers given by $\frac{n(n+1)}{2}$ have a parity pattern of odd, odd, even, even, ... and I'm curious as to why. E.g: ...
Robin Andrews's user avatar
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0 answers
63 views

Can we draw a closed path made up of 9 line segments , each of which intersects exactly one of the other segments?

The solution given in Fomin's book is as follows. If such a closed path were possible, then all the line segments could be partitioned into pairs of intersecting segments. But then the number of ...
user avatar
1 vote
1 answer
402 views

Induction problem on organising a round robin tournament of n football teams

Prove that for all positive integers n, it is possible to organise a round-robin tournament of n football teams in: a) n-1 rounds if n is even b)n rounds if n is odd A round is a set of games in which ...
Garlicbreadenjoyer's user avatar
0 votes
0 answers
35 views

Showing whether a intial arrangement of numbers (odd/even) is sortable according to the constraint on sorting each time.

Let $P$ denote some arrangement of the numbers $1,2, \ldots, n$. A move on $P$ consists of exchanging the position of element 1 with the position of another element. For example, if $P=[3,1,4,2]$, ...
ProblemDestroyer's user avatar
2 votes
0 answers
38 views

Asymptotically similar functions with opposite parity, were they considered, are they useful? Case of polynomials

So, can we transform an even function into an odd function and vice versa? Let's consider this method: Transformation even->odd: Suppose $f_{even}(x)$ is a function which satisfies the following ...
Anixx's user avatar
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3 votes
1 answer
96 views

Are there an odd number of numbers in the unit interval?

Or, four questions that I think are equivalent, but I don't know if will be thought to make sense: Are there an odd number of numbers in $[0, 1]$? Are there an even number of numbers in $[0, 1)$? Are ...
Bertie Wheen's user avatar
1 vote
0 answers
79 views

Sign of permutation related to number of inversions

I define the sign of a permutation as the number of transpositions that are in any re-writing of $\sigma$ in a product of transpositions. The number of inversions of a permutation $\sigma$ of $S_n$ is ...
Miriam Del Blanco's user avatar
3 votes
1 answer
92 views

Probability that logarithm expression of $500$ values between $(0,1]$ is even

Let $a_1, a_2, a_3, \ldots, a_{500}$ be uniformly and independently chosen from the interval $(0,1]$ at random. Find the probability that $$\lceil \log_2 (a_1) \rceil + \lceil \log_4(a_2)\rceil + \...
mathisfun's user avatar
  • 466
1 vote
0 answers
60 views

What is the description of pseudo-quantities on a general manifold?

When talking about physics (on $\mathbb{R}^3$), one tends to call pseudovectors $+$-parity objects, since they transform into themselves under the parity transformation $P: (x,y,z) \mapsto (-x,-y,-z)$....
Sam's user avatar
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1 vote
1 answer
75 views

Formal expression for parity of three integers

During an X-Ray Crystal Diffraction course, we ended up with this sum : $$(-1)^{h+k} + (-1)^{k+l} + (-1)^{h+l} + 1$$ The condition of extinction for the diffraction is that the sum must be equal to 0 :...
gigipar's user avatar
  • 11
3 votes
2 answers
74 views

On the number of ways to get a sum with given parity

Fix some integers $a_1,\dots,a_n\geq 0$. Denote by $E$ and $O$ the number of ways to choose $(x_1,\dots,x_n)$ so that $0\leq x_i\leq 2a_i$ and such that $\sum_{i=1}^n x_i$ is even and odd, ...
Zuy's user avatar
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1 vote
1 answer
316 views

Why do sieves have the parity problem?

Sieve methods have the "parity problem". Terry Tao gives a "rough" statement of the problem: If A is a set whose elements are all products of an odd number of primes (or are all ...
learningmathematics's user avatar
3 votes
1 answer
120 views

Game involving operations on 1, 2, ..., 100 (Leningrad Mathematical Olympiad, 1984)

I have been reading Grade Five Competition from the Leningrad Mathematical Olympiad by Garaschuk and Liu, which is a wonderful source of interesting problems. One of the problems from the year 1984 is ...
Prism's user avatar
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