Discrete mathematics is the study of mathematical structures that are fundamentally discrete rather than continuous.

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32
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740 views

On the number of complete and gap-free compositions

This is a longish post about something that has been haunting me for a while about a kind of restricted composition, namely gap-free and complete compositions. First, I will define the terms that are ...
9
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288 views

Minimal time gossip problem

The gossip problem (telephone problem) is an information dissemination problem where each of $n$ nodes of a communication network has a unique piece of information that must be transmitted to all the ...
8
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75 views

A group acting on functions of functions of functions

Given a group acting on a set $X$, there is a standard way to define an action of the group on the set of functions of $X$. This can be extended to the set of functions of functions of $X$ as I show ...
8
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495 views

A constrained topological sort?

Suppose that one has a directed, acyclic graph G, and each vertex $v$ contains a (positive) value $a_v$. Additionally, let $r$ be a constant. For my purposes, $r>1$, but this might not matter. ...
7
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188 views

Citation for subset complement result

Let $S = \{s_1, \ldots, s_n\} \subset \{1, \ldots, 2n\}$. Consider two operations on $S$, unfortunately both called complement in different setting: let $A(S) = \{1, \ldots, 2n\} \setminus S$ (set ...
7
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155 views

A variation on the Look and Say Sequence and some questions about it.

For information on the sequence mentioned in the title, see http://en.wikipedia.org/wiki/Look-and-say_sequence. This is an original problem. Suppose instead of "describing" the numbers in a string in ...
5
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57 views

How can I better solve proofs requiring the introduction of algebraic assumptions?

Today I decided to binge on discrete mathematics after a three year hiatus. I tackled three proofs, and all of them required the introduction of assumptions that seemed to not be found in the givens ...
5
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62 views

Maximal hamming distance

Here is a combinatorial problem : let $\Sigma=\{\alpha_1,\ldots,\alpha_n\}$ be an alphabet and we consider any words over $\Sigma$ of length $n$. We also define over the set of such words the Hamming ...
5
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52 views

Decrease of entropy when iterating a random discrete function

Let $m$ be a positive integer. Let $S$ be the set of non-negative integers $x$ less than $m$, with $|S|=m$. Let $X_0$ be the discrete uniform distribution over $S$, with $P(x)=\begin{cases} 1/m & ...
4
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41 views

How to solve a non-homogeneous second-order linear difference equation with both a forward and a backward difference?

Quite a long title for this: I'm looking for the general solution of the following difference equation: $$ax_{t+1} -bx_t + x_{t-1} = c + u_t$$ where $a,b,c$ are real constants and $u_t$ is a bounded ...
4
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29 views

Symmetry groups of discrete functions

I'm looking for basic information about symmetry groups of discrete functions. It is difficult to search for such information, because searching for "symmetry group" gives results that refer almost ...
4
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32 views

Subsets of cyclic group with distinct pairwise differences

Given a number $m\in\mathbb N$, let $\mathbb Z_m=\{0,1,\dots,m-1\}$ denote the ring of integers modulo $m$ (although we won't need multiplication, so any cyclic group of order $m$ will do). Given a ...
4
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70 views

How to get maximum sums from groups of numbers without exceeding a total?

This is an abstract computer code application that I'm trying to solve, but I think the following real life example kind of helps to illustrate the type of problem I'm working with. In this example, ...
4
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51 views

Nimber of selective compound games

Background/Definitions. Let $\alpha,\beta$ ordinal numbers. The Hessenberg sum $\alpha \# \beta$ is defined recursively as the smallest ordinal which is $>\alpha' \# \beta$ and $> \alpha \# ...
4
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232 views

$n$ players of paper scissor rock

Suppose there are $n$ players $(3\leq{n})$ showing Paper, Scissor or Rock simultaneously. If there is no winner then there is no payoff to any player. If there are winners and losers (e.g. $k$ ...
4
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0answers
7k views

Relationship Between The Z-Transform And The Laplace Transform

Below I've quoted Wikipedia's entry that relates the Z-Transform to the Laplace Transform. The part I don't understand is $z \ \stackrel{\mathrm{def}}{=}\ e^{s T}$; I thought $z$ was actually an ...
4
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0answers
157 views

Convex hull of balls

The convex hull is defined as the smallest convex set containing a set of points. Now we want to generalize it to a set of balls. If these balls have the same radius, then it can be shown that a ball ...
4
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194 views

Starting my nephew out on the journey to higher mathematics.

My nephew is 8 years old and shows great promise as a student. Sadly, as most of you know most programs in secondary education don't offer any foundational courses for higher mathematics. What ...
3
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29 views

A probability of a monochromatic cycle on a randomly colored lattice graph.

Let $G$ be an undirected $6 \times 6$ lattice graph. The $36$ vertices of $G$ are each randomly colored with one of $5$ colors with equal probability. Such a coloring is called "successful" if and ...
3
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35 views

Is it possible to find plaintext from ciphertext if (n) and (a) are known?

I have a couple of questions pertaining to a RSA problem. I need to decipher some ciphertext and find out what the original plaintext was. n = 2537 and a (or the exponent) = 11. Encrypting function: ...
3
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49 views

Show that $\sum_{\{a_1, a_2, \dots, a_k\}\subseteq\{1, 2, \dots, n\}}\frac{1}{a_1*a_2*\dots*a_k} = n$

Question: Show that $$\sum_{\{a_1, a_2, \dots, a_k\}\subseteq\{1, 2, \dots, n\}}\frac{1}{a_1*a_2*\dots*a_k} = n$$ (Here the sum is over all non-empty subsets of the set of the $n$ smallest positive ...
3
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32 views

Does there exist Latin square critical sets for which deleting any entry results in arbitrarily many completions?

For those familiar with Latin squares terminology, I'll get straight to the point: Q: For all $N \geq 2$, does there exists a critical set $C$ (for a Latin square of any finite order) such that ...
3
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30 views

question in product

can any expert just check my solution You bought a car for $\$2500$ down and made payments of $\$299.50$ each month for $36$ months. (a) Find the amount of the payments over the $36$ months. (b) Find ...
3
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25 views

(Counting problem) more challenging Modular N algebraic eqs - for combinatorics-permutation experts

Experts in algebra please help - Part II after Part I: we would like to know the number of solutions for this set of six of modular N algebraic equations: $$ x_1 y_2 = x_2 y_1 \pmod N \qquad (1) \\ ...
3
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38 views

How to show that $\sum\limits_{k=0}^{\lfloor0.999n\rfloor}\binom{2n}{k} < \binom{2n}{n} $ holds for large n

It seems logical to me since $\binom{2n}{n}$ is in the middle of the row in pascal triangle; therefore, the largest, and for large n the sum adds only the small ones on the left. But I do not have any ...
3
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40 views

Modular arithmetic - Suggestions to begin

I've always wanted to start studying modular arithmetic to try to solve problems like: $$\text{find } n \in \mathbb{N} : 4n^2 \equiv 1 ~(\text{mod }{10^4})$$ I have a good basis in mathematical ...
3
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61 views

Find generating function For sequences

Can anyone out here help? The exercise says: "Find the generating function for each of the sequences below (the general term is given)" Now, the question is how do you find one for those: a) $U_n = ...
3
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213 views

Maximum size of a bipartite subgraph on a random graph

Show that almost every $G \in \mathscr{G}(n,\frac{1}{2})$ contains no bipartite subgraph with more than $\frac{n^2}{8} + n^{\frac{3}{2}}$ edges. Tried using Markov's inequality by setting a = ...
3
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78 views

Product of Summations for All Subsets

We have a set $X$ of $n$ integers $\{$$x_1$, $x_2$, .. , $x_n$$\}$, for which there are $2^n$ total subsets. The summation $s$ of a subset $X'$ is simply the sum of all integers present in $X'$, ...
3
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116 views

to find the graphs having vertices with same eccentricity

I was reading a paper http://www.discuss.wmie.uz.zgora.pl/php/discuss3.php?ip=&url=plik&nIdA=11134&sTyp=HTML&nIdSesji=-1 There is a formula to calculate eccentricity in the section ...
3
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135 views

Number of non-decreasing functions of n boolean variables

If f(a) is greater than or equal to f(b) for two domain elements a,b in X^n {0,1} whenever the number of 1's in a is greater than or equal to the number of 1's in b, that function is called a ...
3
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80 views

Counting number of distinct systems

This is an enumeration problem in conjonction with some lottery problems. Given an integer $N \ge 5$. Let a ticket be a set of 5 distinct integers between $1$ and $N$. Given an integer $T$ between ...
3
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133 views

Difference between two sets of data points

I'm making a simple calibration of a z-stage, by measuring a number of points in one direction with a constant $\Delta$Z between each sample. Then I reverse the direction and measure the same number ...
3
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66 views

Define composition of small cyles and making a big graph

I am having following sub graphs and wish to compose all and make a one bigger graph (say G). After that, I want to select the closed path where it is passing along the outer vertices of that ...
3
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63 views

Upper bounds on rate of q-ary codes

Among the many upper bounds for families of codes in $\mathbb F _2 ^n$, the best known bound is the one by McEliece, Rodemich, Rumsey and Welch (MRRW) which states that the rate $R(\delta)$ ...
3
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209 views

Closed-form expression for sum of Vandermonde matrix elements

Given the Vandermonde matrix: $$\begin{pmatrix}1^0 & 1^1 & 1^2 & ... & 1^n \\ 2^0 & 2^1 & 2^2 & ... & 2^n \\ \vdots & \vdots & \vdots & \ddots & ...
3
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173 views

Find the number of specific permutations

Find the number of all permutations of the set $\left\{ 1,2,...,2n \right\}$ such that does not have compact subsequence: $\langle i, \ i+1\rangle$ or $\langle i+1, \ i\rangle$ for all $1\le i\le ...
3
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0answers
202 views

A closed-form expression for a sequence of integers

Let $(a_n)$ be the sequence of minimum values of the expression (depending on $\ell$): \begin{equation*} a_n=\arg\min \bigg\lbrace ((\ell+1)j-n)!\,(n-\ell j)!\quad \text{for}\; ...
3
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0answers
140 views

Really simple combinatorics question - Composition of n number strings

I'm doing a bit of revision and want to make sure I'm doing these two questions correctly. They are as follows: Two n-digit (leading zeros allowed) numbers are considered to be equivalent if one ...
3
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315 views

Sylvester's Theorem and Schur Theorem

I'll probably end up asking more programming questions on StackExchange forums than math questions, but I'll lead off with a math question. In my Number Theory class this past semester, I worked on a ...
3
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0answers
188 views

Expected value of the minimum of a random set of distances

You are given a rectangular $n_1\times n_2$ grid with one light bulb $b_i$ at every node. Each bulb is on or off with probability $p$ and $1-p$, respectively, and furthermore you know that exactly $m$ ...
3
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0answers
194 views

How to find $\beta$ and $\alpha$?

$\mathbb{P}$ is the prime numbers set. $p \in \mathbb{P}$ $a,b,c \in \mathbb{N}$ $n=a p^b+c$ where $c= n\bmod p$ $b$ is the highest power of $p$ who divides $n-c$ How to find $\beta$ where ...
2
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92 views

“Remainder” operation in mod 2^32

I debated posting this here, in the cryptography SE, or the programming SE. Obviously, I chose here, but I'm not confident in my choice... I'm attempting to "undo" a function, but I've hit a slight ...
2
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0answers
27 views

what does “modular” mean?

I find some similarity of the concept "modular set functions" to the cardinality function. But I don't see the cardinality function is also called "modular" or something else. I wonder what "modular" ...
2
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0answers
19 views

what is the significance of Eigen values of autocorrelation matrix?

I am trying to find auto correlation matrix of an image to get Harris corners.Paper I am referring suggest that if eigen values of auto correlation matrix are large the point will be corner point.so ...
2
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0answers
31 views

Identification of (Centers of) Cycles in a Discrete Time Dynamical System

I am studying dynamics on nonlinear Discrete Time Dynamical System of the form $$ \vec{X}_{t+1} = D(\vec{X}_t), $$ where D is some nonlinear function. I was looking for a (relatively) quick ...
2
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0answers
32 views

Inequality in matroid theory

Working on a proof in matroid theory I found there is a smooth map from an open set of $(\mathbb{C}^{\ast})^{(d−1)(n−d−1)}$ to a disjoint union of tori $(S^{1})^{\binom{n}{d}-n}.$ As a direct ...
2
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0answers
45 views

Which of the following 5 statements are true?

I am having trouble finding which of the following statements are true: Which of the following statements are true? [a] Pizza does NOT have mushrooms [b] Pizza does have mushrooms AND ...
2
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31 views

Computation of a 3-dimensional game matrix

For natural numbers $n_1 \leq n_2 \leq n_3$ we define $\beta(n_1,n_2,n_3)$ recursively to be the smallest natural number which is not among the numbers $\beta(m_1,m_2,m_3)$, where $m_1 \leq n_1 \leq ...
2
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0answers
37 views

an “alternate derivation” of Poisson summation formula and discrete Fourier transformation

Inspired by this post, I am trying to do a derivation of a Poisson summation formula. My starting point is this: $$ \frac{1}{2\pi} \int^{\infty}_{-\infty} e^{i k x} dx=\delta(k) $$ I simply wish ...