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Questions by dd
0
votes
0
answers
21
Medium access in shared media and thought-put
Consider a scenario in which $\textbf{n}$ hosts share the same medium. The medium has bandwidth (max. data rate) $c$. Time is divided into slots of duration $T$ ... varying $p$)? Interpret your results. (c) What is the limit of maximum throughput as $n \rightarrow \infty$ ? Interpret your results.
asked
in
Computer Networks
Aug 28, 2018
209
views
computer-networks
iitb
0
votes
0
answers
22
Probability and random variable
Consider the sets $A_1, A_2, A_3 \dots A_m$ each a subset of size $k$ of $\{1,2,3, \dots , n\}$. If in a 2-colouring of $\{1,2,3, \dots , n\}$ no set $A_i$ is monochromatic, then show that $ m < 2^{k-1}$.
asked
in
Set Theory & Algebra
Apr 27, 2018
368
views
probability
random-variable
combinatorics-iitb
0
votes
1
answer
23
Set theory
Consider the sets $A_1, A_2, A_3 \dots A_m$. Prove that the number of distinct sets of the form $A_i \oplus A_j$ is at least $m$.
asked
in
Set Theory & Algebra
Apr 27, 2018
400
views
set-theory
combinatorics-iitb
0
votes
0
answers
24
Set theory and Induction
Consider the set of all subsets of a set S. A chain is a collection of subsets $P_1 \subset P_2 \subset P_3 \subset P_4 \dots \subset P_k$. A symmetric chain is one which starts at a set of size $i$ and ends at a set of size $n - i$. Prove that the poset has a decomposition into symmetric chains.
asked
in
Set Theory & Algebra
Apr 27, 2018
275
views
set-theory
combinatorics-iitb
6
votes
4
answers
25
Counting
Show that, in a grid, the number of paths from $(0,0)$ to $(n,n)$ which does not cross ( it could touch ) the line $x = y$ is \begin{align*} \frac{1}{1+n}\binom{2\cdot n}{n} = \binom{2\cdot n}{n} - \binom{2\cdot n}{n-1} \end{align*} After that, show the number of balanced paranthesis strings of length $2n$ is same as the above result.
asked
in
Set Theory & Algebra
Apr 27, 2018
2.5k
views
counting
combinatorics-iitb
0
votes
0
answers
26
Probability and set theory
Consider sets $A_1,A_2,A_3 \text{ to } A_m \text{ where }A_i \subseteq [n] \text{ and } [n] = \{1,2,3, \dots n \}$ such that there are no three distinct sets in the collection with the property $A_i \subset A_j \subset A_k.$ For even $n$, prove that \begin{align*} m ... bound on $m$ : \begin{align*} m \leq \binom{n}{\frac{n}{2}} + \binom{n}{\frac{n}{2} - 1} \end{align*}
asked
in
Set Theory & Algebra
Apr 27, 2018
292
views
probability
set-theory
combinatorics-iitb
0
votes
1
answer
27
Induction
Prove that the number of edges in a graph with exactly one triangle is at most \begin{align*} \frac{\left( n - 1 \right )^2}{4} + 2 \end{align*}
asked
in
Graph Theory
Apr 27, 2018
447
views
induction
graph-theory
combinatorics-iitb
0
votes
0
answers
28
Probability
Prove for a positive integer valued random variable, $X$ , and for positive integers $a < b$ \begin{align*} &P(X \geq b) \leq \left [ \left ( \frac{E(X)}{b} \right ) - P(a \leq X < b)\cdot \left ( \frac{a}{b} \right ) \right ] \\ \end{align*}
asked
in
Probability
Apr 27, 2018
217
views
probability
combinatorics-iitb
1
vote
1
answer
29
Set system and linear algebra
We have $m$ sets $A_1,A_2,A_3 \text{ to } A_m$. All $A_i \subseteq [n]$ where $ [n] = \{1,2,3, \dots n \}.$ Given that $|A_i| = \text{odd number}$ and $|A_i \cap A_j| = \text{even number }\forall i \neq j$. Show that $m \leq n$.
asked
in
Set Theory & Algebra
Apr 16, 2018
878
views
set-theory
linear-algebra
combinatorics-iitb
4
votes
0
answers
30
Conditional probability
A student $Y$ lives in a place $X$. In $X$ a day can be cloudy with probability $\bf0.02$ uniformly each day. $Y$'s friends may come to his house with probability $\bf0.2$ uniformly in each day irrespective of weather condition. These two events are independent. $Y$ ... on a particular day ?
asked
in
Probability
Dec 25, 2017
691
views
conditional-probability
2
votes
0
answers
31
Graph Degree sequence : Bondy and Murty : $1.1.16$
Let $d = (d_1,d_2,\dots, d_n)$ be a nonincreasing sequence of nonnegative integers, that is, $d_1 \geq d_2 \geq · · · \geq d_n \geq 0$. Show that: there is a loopless graph with degree sequence d if and only if $\sum_{i=1}^{n}d_i$ is even and $d_1 \leq \sum_{i=2}^{n}d_i$
asked
in
Graph Theory
Jul 4, 2017
430
views
graph-theory
non-gate
proof
degree-of-graph
1
vote
0
answers
32
Graph Theory : Bondy-Murty $1.1.20$
Let $S$ be a set of $n$ points in the plane, the distance between any two of which is at least one. Show that there are at most $3n$ pairs of points of S at distance exactly one. Can this be done with a unit circle and we can place at max. $6$ points on the perimeter and doing the same for other points as well ? i.e. we can get $6n/2 = 3n$ pairs at max. ?
asked
in
Graph Theory
Jul 4, 2017
306
views
graph-theory
non-gate
proof
2
votes
0
answers
33
Graphic Sequence condition
A sequence $d = (d_1,d_2,\dots , d_n)$ is graphic if there is a simple graph with degree sequence $d$ If $d = (d_1,d_2,d_3, \dots d_n)$ is graphic and $d_1 \geq d_2 \geq d_3 \geq \dots \geq d_n$ , then show that $\sum_{i=1}^{n}d_i$ is even and $\sum_{i=1}^{k}d_i \leq \left [ k(k-1) + \sum_{i=k+1}^{n} \min\{k,d_i\} \right ] \quad ,1 \leq k \leq n$.
asked
in
Graph Theory
Jul 4, 2017
401
views
non-gate
graph-theory
proof
3
votes
0
answers
34
Minimum No. of vertices required
Prove the following for graph $G$. When length of the shortest cycle in a graph is $k \geq 3$ and the minimum degree of the graph is $d$, then $G$ ... $k$.
asked
in
Graph Theory
Jul 1, 2017
387
views
graph-theory
proof
1
vote
0
answers
35
Equal coefficients
$\begin{align*} &A = \left ( p x + q \right )^{504} \text{ where p and q are +ve integers and }gcd(p,q) = 1 \\ &\text{Given } \left [ x^4 \right ] = \left [ x^5 \right ] , \text{ where } \left [ x^k \right ] \text{ denote the coefficient of }x^k \text{ in A.} \\ &\text{What is p+q ?} \end{align*}$
asked
in
Combinatory
Jun 30, 2017
199
views
discrete-mathematics
combinational-circuit
3
votes
1
answer
36
Bit-wise operation
#include <stdio.h> int main() { unsigned int m = 0; m |= 0xA38; printf("%x\n",m|(m-1)); printf("%x\n",( (m|(m-1)) + 1 ) & m ); } Find the output ?
asked
in
Programming in C
Jun 27, 2017
503
views
programming-in-c
bitwise
3
votes
1
answer
37
combinatorial argument
$\begin{align*} &\text{Prove using combinatorial argument } \\ &1) \qquad \text{For } n \geq k \geq 0 \qquad \left ( n-k \right )\cdot \binom{n}{k} = n \cdot \binom{n-1}{k} \\ &2) \qquad \text{For } n \geq 2 \qquad \quad k \cdot (k-1) \cdot \binom{n}{k} = n \cdot (n-1) \cdot \binom{n-2}{k-2} \\ \end{align*}$
asked
in
Combinatory
Jun 26, 2017
457
views
non-gate
combinatory
1
vote
0
answers
38
Integer Sequence
... $\begin{align*} |A| = |B| \end{align*}$
asked
in
Combinatory
Jun 26, 2017
210
views
non-gate
combinatory
1
vote
0
answers
39
Combinatoric properties
Prove that : $\begin{align*} &\text{For n , k , m are integers and } 0 < m \leq k < n \\ &\text{GCD}\left [ \binom{n}{m},\binom{n}{k} \right ] > 1 \\ \end{align*}$
asked
in
Combinatory
Jun 26, 2017
157
views
combinatory
non-gate
5
votes
1
answer
40
Series Summation
Series summation of $S_n$ in closed form? $\begin{align*} &S_n = \frac{1}{1.2.3.4} + \frac{1}{2.3.4.5} + \frac{1}{3.4.5.6} + \dots + \frac{1}{n.(n+1).(n+2).(n+3)} \end{align*}$
asked
in
Set Theory & Algebra
Jun 11, 2017
805
views
number-theory
summation
discrete-mathematics
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