Discrete mathematics is the study of mathematical structures that are fundamentally discrete rather than continuous. In contrast to real numbers that have the property of varying "smoothly", the objects studied in discrete mathematics – such as integers, graphs, and statements in logic – do not vary smoothly in this way, but have distinct, separated values. Discrete mathematics therefore excludes topics in "continuous mathematics" such as calculus or Euclidean geometry. Discrete objects can often be enumerated by integers. More formally, discrete mathematics has been characterized as the branch of mathematics dealing with countable sets (finite sets or sets with the same cardinality as the natural numbers). However, there is no exact definition of the term "discrete mathematics." Indeed, discrete mathematics is described less by what is included than by what is excluded: continuously varying quantities and related notions.
The set of objects studied in discrete mathematics can be finite or infinite. The term finite mathematics is sometimes applied to parts of the field of discrete mathematics that deals with finite sets, particularly those areas relevant to business.
Research in discrete mathematics increased in the latter half of the twentieth century partly due to the development of digital computers which operate in discrete steps and store data in discrete bits. Concepts and notations from discrete mathematics are useful in studying and describing objects and problems in branches of computer science, such as computer algorithms, programming languages, cryptography, automated theorem proving, and software development. Conversely, computer implementations are significant in applying ideas from discrete mathematics to real-world problems, such as in operations research.
Although the main objects of study in discrete mathematics are discrete objects, analytic methods from continuous mathematics are often employed as well.
In university curricula, "Discrete Mathematics" appeared in the 1980s, initially as a computer science support course; its contents were somewhat haphazard at the time. The curriculum has thereafter developed in conjunction with efforts by ACM and MAA into a course that is basically intended to develop mathematical maturity in first-year students; therefore, it is nowadays a prerequisite for mathematics majors in some universities as well. Some high-school-level discrete mathematics textbooks have appeared as well. At this level, discrete mathematics is sometimes seen as a preparatory course, not unlike precalculus in this respect.The Fulkerson Prize is awarded for outstanding papers in discrete mathematics.
Hi all,
Now naturally after completing a physics degree I am very familiar with the form and function of the Fourier Transform (FT) but never have grasped it quite conceptually. I understand that given a function f(x) I can express every functional value as a linear combination of complex...
I am simulating an ODE where the differential function is a function of sampled data points and are having trouble obtaining data in between sampled points (for the step computations of RK4). The first thing that I tried was to linearly interpolate the data, but that introduced large...
In a Hilbert-space whose dimensionality is either finite or countably infinite, we have the discrete resolution of identity
\sum_n |n\rangle \langle n| = 1
In many cases, for example to obtain the wavefunctions of the discrete states, one employs the continuous form of the resolution...
A problem i made up for some of my friends who need help with discrete distributions tables. Can you do it?
Dice Generator
Part I:
1. Construct a discrete probability distribution table for a fair six-sided dice. (Round according to example)
2. Calculate the mean, variance, and standard...
I'm trying to do some extra course work to prepare for my final next week but I'm having a lot of trouble with the book problems. They talk about a lot of things we weren't taught. Can someone help me out here?
Prove: n\niZ, n= a multiple of gcd(a,b) ⇔ n is a linear combination of a and b
This...
Hello,
I have a question regarding discrete energy levels of atoms.
If electrons must inhabit certain energy levels, when they are excited up to a higher energy level, do they instantaneously jump to that level, or do they exist for some short time 'in between'?
Furthermore, if the...
Hi everybody,
I am interested to do some DEM(Discrete Element Method) simulation
for my miniproject. So far I have seen that the most available
opensource codes require large computation resource for parallel calculation.
I don't have access to such facility and also my initial problem...
Hi guys, so I'm trying to create this thing that gets accelerometer values and integrate those values at about 25 samples a second. From what I understand if I integrate the values, that means each second I have 25 values I want to integrate. I read this research paper where someone used the...
I have a 3 rad/s sine wave that I am sampling every 2 seconds.
The slow sampling has created an alias of -0.14 rad/s.
I have plotted this over the top of the original continuous time signal, please see attachment.
My problem is that I only want to plot the first period of the discrete signal...
How do you calculate the intersection of discrete data points and an equation?
Actually I have two ways already, one is to just take the equation of the discrete points then solve it using a root-finding technique. The other would be substituting the x values of the discretized points to the...
I am currently enrolled in Calculus 1. Will be taking Calculus 2 in summer and Calculus 3 in Fall. I have already registered for these courses. One of the Math electives I have a choice in Fall Semester is either Discrete Math or Linear Algebra. Any suggestions would be greatly appreciated...
Hello all,
I've breadboarded a simple astable multivibrator to generate 38kHz (identical to http://upload.wikimedia.org/wikipedia/commons/6/6a/Transistor_Multivibrator.svg) using 2N3904s.
Upon power up, it's a solid 38Khz. After 60 seconds, it's drifted up to ~44Khz... and stabilizes to...
Homework Statement
Hi Everyone, I am trying to show why the given sum is zero. I am pretty sure it is zero.
Homework Equations
sin[8*\pi*n/5]+sin[12*\pi*n/5]
n is an integer.
The Attempt at a Solution
n----sin[8*\pi*n/5]----sin[12*\pi*n/5]
0 ----...
Hi all I need some assistance
1. Homework Statement with the attempt
How many 5-digit briefcase combinations contain
1. Two pairs of distinct digits and 1 other distinct digit. (e.g 12215)
I wasn't sure on which approach was correct.
10 * 9 * 8 (because there are three distinct...
All analitic function can be express how: f(x) = \frac{1}{0!} \frac{d^0f}{dx^0}(x_0) (x - x_0)^0 + \frac{1}{1!} \frac{d^1 f}{dx^1}(x_0) (x - x_0)^1 + \frac{1}{2!} \frac{d^2f}{dx^2}(x_0) (x - x_0)^2 + \frac{1}{3!} \frac{d^3f}{dx^3}(x_0) (x - x_0)^3 + ... that is the taylor series of the function...
I am a math major and I need to take Methods of Discrete Mathematics. What is methods of discrete mathematics? Should I take it after My calculus series( including linear/ diff. equations)? Is it easy enough to take with Calculus 2? Thanks
Hello,
I have a question I am trying to figure out how it works and I am so confused I need a break down of what is exactly going on with this problem
the Question.
"Concern three persons who each randomly choose a locker among 12 consecutive lockers"
What is the probability that no two...
Homework Statement
\sum_{x} x^k
for k ∈ Q
Homework Equations
\sum_{x} x=\frac{1}{2}x^2-\frac{1}{2}x\sum_{x} x^2=\frac{1}{3}x^3-\frac{1}{2}x^2+\frac{1}{6}x\sum_{x} x^{-1}=\Psi (x)
The Attempt at a Solution
I don't know. There isn't way to compute the antiderivative of any function and I...
Homework Statement
Suppose A={ø,1,2,{1,2},{3}}. Determine if each statement is true or false. Give a brief justification for your answers.
(I finished the majority of them except for the last two)
g) A-ø=A
h) A-ø={1,2,{1,2},{3}}
Homework Equations
The definition of difference between...
How to have better discrete math "insight"
Greetings:
I came a cross a textbook example in a discrete math book that I have been reading on my own, and I thought this example in the book was a good example of what I want to be good at:
Given integers from 0-9 arranged in a circle, is...
Hello
I am trying to solve this problem:
A coin is given with probability 1/3 for head (H) and 2/3 for tail (T).
The coin is being drawn N times, where N is a Poisson random variable with E(N)=1. The drawing of the coin and N are independent. Let X be the number of heads (H) in the N draws...
Why parity is discrete transformation?
##Px=-x##
##P\psi(x)=\psi(-x)##
when ##x## is continual variable. Could you explain me difference between discrete and continual transformation?
I'm trying to prove that the discrete form of the Fourier transform is a unitary transformation
So I used the equation for the discrete Fourier transform:
##y_k=\frac{1}{\sqrt{N}}\sum^{N-1}_{j=0}{x_je^{i2\pi\frac{jk}{N}}}##
and I put the Fourier transform into a N-1 by N-1 matrix form...
Homework Statement
is it possible to have a topological space that is neither the indiscrete nor the discrete, and very set in the topology is clopen?
Homework Equations
The Attempt at a Solution
let ##X## = {(0,1),(2,3)} with the ordinary topology on R.
(0,1) is open, but...
Hello Everyone!
Homework Statement
There exists a very large (discrete) system of N coupled masses, each of mass "m", where every pair is connected via a spring of constant "K". Assuming all motion is horizontal, find the amplitude of the oscillations of an nth mass in the system, under the...
I just want to make sure I understand the normal modes of a vibrating system of particles with discrete spacing. I have tried to drawn what I understand as the lowest and highest frequency mode of the standing waves. Is the drawing correct? Edit: actually I have drawn the maximum frequency where...
Hello!
I was told, that there are no mainstream physics theories, in which time is discrete.
My question is this:
At any present or any given future scenario, there will always exist the smallest amount of time, that a machine is able to measure. How come this variable, is not a main...
Homework Statement
There are a set number of marbles in a bag; the marbles consist of two colors. We are given the mean number of marbles of color 1 in the bag, as well as color 1's standard deviation. We are then asked to find the mean and standard deviation of color 2.Homework Equations
How...
If two fair dice are rolled 10 times, what is the probability of at least one 6 (on either die) in exactly five of these 10 rolls?
So this problem is hard to wrap my head around. I'm probably wrong on many counts, here's what I'm doing:
Two fair dice are rolled 10 times but this question only...
Hi guys,
I'm trying to solve the S.E. For a potential that's time dependent but the time variable is not continuous. Essentially the potential is a finite square well and it shifts over time but not continuously.
I.e. At time 0<t<t' it's a finite square well centered about some x'
At...
Consider a triangulated discrete manifold (a polyhedron) with known vertices (i.e. each vertex is given in terms of its $$(x,y,z)$$ coordinates ).
Assign scalar values (some kind of potentials) to each vertex (i.e. at each vertex, a $$k_t(\mathbf{v})$$ is known through its value, no...
Definition: A subset $D$ of $\mathbb R$ is said to be discrete if for every $x\in D$ there exists $\epsilon>0$ such that $(x-\epsilon,x+\epsilon)\cap D=\{x\}$.
Question: Does there exist a discrete subset $D$ of $\mathbb R$ such that the set of limit points of $D$ is an uncountable set.
___...
Corollary from book:
if d= gcd(a,b), then there exists integers x and y such that ax + by = d.
This is not an obvious statement to me. Are there any direct proofs to prove this statement? The book proves this by induction.
My proof:
Suppose d = gcd(a,b) and a and b are positive integers. a...
Hello. I am reading an introduction to induction example, and I am having the hardest time trying to determine what exactly happened in the proof. Can somebody please help? How can ##3^{k-1}## + ##3^{k-2}## + ##3^{k-3}## all of a sudden become ##3^{k-1}##+##3^{k-1}##+##3^{k-1}## and how can be...
Quantum Phys Homework:
I am given a function:
$$f(x)=\frac{1}{10}(10-x)^2\,;\,0\leq{x}\leq{10}$$
and
$$f(x)=0$$
for all other \(x\).
I need to find the average value of \(x\) where
$$\bar{x}=\frac{\int_{-\infty}^{\infty}x\,f(x)\,dx}{\int_{-\infty}^{\infty}f(x)\,dx}$$
I am not really even sure...
Homework Statement
Consider a discrete time birth and death process in which the maximal population
size is N = 6. Birth rates and death rates are directly proportional to the current
size Xt of the population at time t (t = 0; 1; 2; : : :). If the maximal population size
is reached, no more...
1. The problem statement, all variables and givenknown data
We have a system of three atoms arranged in a circular arrangements. They each have a valence electron that can tunnel to the nearest neighbor. For a tunneling rate -A/\hbar we have the Hamiltonian (shifted by an energy E_a on the...
How would I go about solving this?
We are starting to learn about venn diagrams so would creating a venn diagram be helpful?
This is what I tried so far, I created a set C consisting of all people who have taken calculus and a set D consisting of all people who have taken discrete math...
Homework Statement
Are these system specifications consistent? "Whenever the system software is being upgraded, users cannot access the file system. If users can access the file system, then they can have new files, then the system software is not being upgraded."
Homework Equations
p...
Homework Statement .
Prove that a metric space X is discrete if and only if every function from X to an arbitrary metric space is continuous.
The attempt at a solution.
I didn't have problems to prove the implication discrete metric implies continuity. Let f:(X,δ)→(Y,d) where (Y,d) is...
The N real numbers x0, ..., xN-1 are transformed into the N real numbers X0, ..., XN-1 according to one of the formula by DCT.
I would ask what is the benefit and why should we do that?
Homework Statement
Using the rules of inference, prove that if ∀x(P(x) ∨ Q(x)) and ∀x((¬P(x) ∧ Q(x)) → R(x)) are true, then ∀x(¬R(x) → P(x)) is true as well.
Homework Equations
The Attempt at a Solution
The problem arises step 5. I feel this is correct but the...
To give you a sense of strong induction and the relationship between mathematical induction and recursion (next session), let's do the pile splitting problem: Take a bunch of beads, rocks, coins, or any kind of chips. Ten is a good number. Split the pile into 2 smaller piles and multiply their...
Homework Statement
Determine the finite dimensionality of the following system:
y[n] = nx[n]
Homework Equations
y[n]= f(y[n−1], y[n−2],..., y[n−N],x[n],x[n−1],..., x[n−M],n)
Where N is how many dimensions the system has.
The Attempt at a Solution
I understand that the following system...
h is a constant in Planck's equation but I have not seen anything written saying that the frequency cannot be arbitrarily small, thus making E arbitrarily small. Is it that v is only allowed to be integral (after all, when we're measuring a frequency we're essentially counting how many times...
Hello all.
I am working on a problem and I am getting a bit confused.
Suppose we have a poisson equation that we wish to solve subject to certain boundary conditions. Let's say we are in 1D (we can later extrapolate to more dimensions).
Is it possible to impose Dirichlet boundary...