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.
Given a vector of numbers, say [exp(-a t) ] for t - [1, 2, 3, 4, 5] and choose maybe a = -2.4, how can I approximate -2.4 from using Laplace transform methods?
I know you can use regression for this, but I'd like to know the Laplace transform (or Z-transform since it is discrete) approach.
If I understand correctly, when an electron drops to a lower energy state and emits a phoTon, this is a discrete or "atomic" event in the sense that it can't be meaningfully broken down in terms of more detailed sub-processes or interactions.
Now in the case of phoNon emission, it is also...
Hello,
I recently got interested in wavelets. The main idea seems clear: we compute the inner product between the signal ##x(t)## and a chosen wavelet for different scale factors and translations of the wavelet over the signal. The inner product provides the coefficient for a wavelet with a...
Currently, as far as I know, the two main ways to express any given point on a plane is through either cartesian plane or polar coordinates. Both of which requires an ordered pair of two numbers to express a point. However, I wonder if there exists such a system that could express any given...
Is there more references for how accurate is the continuum approximation to discrete sums? Perhaps more mathematical.
What I've found:
https://lonitch.github.io/Sum-to-Int/
https://arxiv.org/pdf/2102.10941.pdf
Some examples are:
Sum to integral
$$\sum_{\mathbf{k}} \to 2 \left ( \frac{L}{2...
I have a doubt about the notation and alternative ways to represent the terms involved in sums.
Suppose that we have the following multivariable function,
$$f(x,y)=\sum^{m}_{j=0}y^{j}\sum^{j-m}_{i=0}x^{i+j}$$.
Now, let ##\psi_{j}(x)=\sum^{j-m}_{i=0}x^{i+j}##. In the light of the foregoing, is...
The following is given:
$$\displaystyle P(K = k) = \frac{1}{2}~\frac{\sqrt{2}~e^{-\frac{1}{2}~\frac{\left(k -\mu \right)^{2}}{\sigma ^{2}}}}{\sigma ~\sqrt{\pi }}$$
How can you prove that the following equalities are correct?
$$\displaystyle \sum _{k=-\infty }^{\infty }1/2\,{\frac {...
In the article A Discrete Normal Distribution of Dilip Roy in the journal COMMUNICATION IN STATISTICS Theory and methods Vol. 32, no. 10, pp. 1871-1883, 2003 one can read:
A discrete normal (##dNormal##) variate, ##dX##, can be viewed as the
discrete concentration of the normal variate ##X##...
Our current model (FLRW) is clear that the universe has a continuous temporal asymmetry. This is seen as the expansion factor grows with time, and thermodynamically with entropy.
A continuous transformation in the current model ##t \rightarrow t + dt## is not the same as ##t \rightarrow t - dt...
What was the first textbook for the modern syllabus of precaclulus which had "precalculus" in the title or subtitle?
What was the first textbook for the modern syllabus of discrete mathematics which had "discrete," "discrete mathematics" in the title or subtitle?
If you have personal...
We were discussing how much weight it would take to stop the mechanism from rotating in this thread:
https://www.physicsforums.com/threads/weight-required-to-hang-straight-down-with-known-torque.1016470/#post-6646777
I wondered if there were actually a range of weights that would stop it...
I have the following function for the normal distribution:
$$\displaystyle f \left(x \right) = \frac{1}{2}~\frac{\sqrt{2}~e^{-\frac{1}{2}~\frac{\left(x -\mu \right)^{2}}{\sigma ^{2}}}}{\sigma ~\sqrt{\pi }}$$
How can the following integrals be equal to their sums?
$$\displaystyle \int_{-\infty...
It has been 35 years since I did the math for Fourier analysis, and I have forgotten what the subtleties are. Please be kind.
So this is not a how do I calculate a DFT (though that may be my next question) but rather how do I use it, and interpret the results.
All the online and software I find...
De normal distribution has the following form:
$$\displaystyle f \left(x \right) \, = \,\frac{1}{2}~\frac{\sqrt{2}~e^{-\frac{1}{2}~\frac{\left(x -\nu \right)^{2}}{\tau ^{2}}}}{\tau ~\sqrt{\pi }}$$
and it's integral is equal to one:
$$\displaystyle \int_{-\infty }^{\infty }\!1/2\,{\frac {...
I can't understand how this approximation works ##\sum_{k=0}^m\left(\frac{k}{m}\right)^n\approx\int_0^m\left(\frac{x}{m}\right)^ndx\tag{1}##Can you please help me
Sketch of proof:
##1.## Let ##V## be open in ##Y##.
##2.## For arbitrary ##f:X\longrightarrow Y## and for arbitrary ##V##, ##f^{-1}(V)## is in ##X##.
##3.## ##f:X\longrightarrow Y## is continuous, so ##f^{-1}(V)## is open in ##X##.
##4.## Every subset ##f^{-1}(V)## of ##X## is open, so ##X##...
I was working on plotting fidelity with time for two quantum states. First I used discrete time( t= 0,1,2,3...etc) to plot my fidelity. I got constant fidelity as 1 with continuous value of time. Next I used discrete set of values ( t=0 °,30 °,60 °,90 °). Here I saw my fidelity decreases and...
Attempting this question without any guidance from my professors unfortunately as they did not teach this bit. Searched online and also there aren't many questions like this.
From what I know,
(I) Having n-1 means you should shift right by 1, which means x[0] is now equals to 0? So x[n-1] = [0 5...
I would like to seek your take on the two terms; discrete and continuous in this context,
In my understanding, when we look at height of individuals (in cms), this measure in general or in definition implies continuous data. If we are to look at specific math problem that involves height of say...
Given the commutation relation
$$\left[\phi\left(t,\vec{x}\right),\pi\left(t,\vec{x}'\right)\right]=i\delta^{n-1}\left(\vec{x}-\vec{x}'\right)$$
and define the Fourier transform as...
I am wondering if this problem has a name, and what is the most efficient way to solve it. Say you have a normalized histogram ##h(P)## (representing a pdf estimated from a large population), with ##n## bins, you want to generate a sample of points ##S## from ##h(P)## of size ##k##, such that...
Hi, please correct me if I use a wrong jargon.
If I have discrete symmetries (like for example in a crystal lattice) can I find some conserved quantity ? For example crystal momentum is conserved up to a multiple of the reciprocal lattice constant and it is linked (I think) to the periodicity...
Summary:: I'm about to take my first course in DSP. It is a one term graduate course using the 4th edition of Proakis.
I'm about to take my first course in DSP. It is a one term graduate EE course using the 4th edition of Proakis. Does anyone with experience in this have useful advice for such...
I've been struggling with the problem below for some time. It is not a homework.
A simple bubble S is a spherical surface that expands with constant speed c. A vector bubble V also expands with the same constant speed c. There is a 3d vector associated with a V.
If two S bubbles touch, they...
this is a textbook problem shared on a whattsap group by a colleague...
i have no problem in finding the value of ##k=0.08##, i have a problem with part (ii) of the problem. I have attached the solution here;
how did they arrive at the probability distribution of ##y##?
attached below is...
Hi all,
I've come across some problem where I have terms such as ##\sum_{j=1}^N \cos(2 \pi j k /N) \cos(2 \pi j k' /N)##, or ##\sum_{j=1}^N \cos(2\pi j k/ N)##, or ## \sum_{j=1}^N \cos(2\pi j k/ N) \cos(\pi j) ##. In all cases we have the extra condition that ##1 \le k,k' \le N/2-1## (and...
What is often said in Covariant LQG is that the triangulation is a truncation, and is not what is responsible for the discrete volumes one ends up with in the theory. Rather, what is responsible is the discrete spectra of the volume operator acting on the nodes of a spin network.
My confusion...
Apologies if this isn't the right forum for this. In my stats homework we have to prove that the expected value of aX and bY is aE[X]+bE[Y] where X and Y are random variables and a and b are constants. I have come across this proof but I'm a little rusty with summations. How is the jump from the...
Hi.I am looking for a book to learn about discrete mechanics (i.e. working in a 3D lattice instead of ##n## generalized coordinates).
I am particularly interested in how to derive the discrete E-L equations by extremizing the action.
I have checked Gregory and Goldstein but they do not deal...
I want to derive the discrete EL equations
$$\frac{d}{dt} \frac{\partial L}{\partial \dot \phi_a^{(i j k)}} - \frac{\partial L}{\partial \phi_a^{(i j k)}} = 0$$
We deal with a Lagrange density which only depends on the fields themselves and their first order derivatives.
We discretize space...
Hello friends,
I have a problem with a exercise sheet. Given is the impulse response of a discret element. The task is to draw the block diagram. But I think that the solution in the sheet is wrong. Because based on the difference equation (Exercise.pdf) there should be 3 delay elements. I have...
I've come across the question of continuity vs discreteness in different articles, discussions, etc. but I'm not sure that I am 100% clear on what the precise question is.
My basic interpretation of it is a question of whether the Universe is made up of lots of separate entities which all...
According to this this the Darboux transformation preserves the discrete spectrum of the Haniltonian in quantum mechanics. Is there a proof for this? My best guess is that it has to do with the fact that $$Q^{\pm}$$ are ladder operators but I'm not sure.
Hello, it's been a while since I've done any proper electrostatics, but I have a problem where I have a bunch of discrete point charges within some volume V bounded by a surface S.
I am wondering if it is possible to replace the discrete charge density in my volume V by some continuous surface...
There are 4 people that they going to split 50 gold between them. They got one extra gold that they can pay for punishment. All person makes a proposal that how can share the gold. Of the remaining players in the game, including the bidder
If more than half (half is not enough) accepts the bid...
Hello,
I understand that continuous sinusoids can have any arbitrary frequency ##f## and are always periodic with period ##T=1/f##. A continuous sinusoid looks like this: $$x(t)= sin(2\pi f t+\theta_0)$$
On the other hand, discrete-time sinusoids are not always periodic. They are periodic only...
In its flip a lid contest, a coffee chain offers prizes of 50,000 free coffees, each worth \$1.50; two new TVs, each worth \$1200; a snowmobile worth \$15 000; and sports car worth \$35 000. A total of 1 000 000 promotional coffee cups have been printed for contest. Coffee sells for \$1.50 per...
In a discrete metric space open balls are either singleton sets or the whole space ...
Is the situation the same for open sets or can there be sets of two, three ... elements ... ?
If there can be two, three ... elements ... how would we prove that they exist ... ?
Essentially, given the...
Given initial displacement ##X_0## and displacement at any time ##t## as ##x##.
Where ##x(t)=f_t(X_0)## where the functional dependence of ##x## upon ##X_0## changes with time.
For exm ##X_0=2## and ##x(t_1)=X^2_0=4,x(t_2)=X^2_0+1=5,x(t_3)=X_0^3+3=11...##and so on.
From this, is there any method...
Hello,
I'm currently working on an assignment which requires me to choose an optimal curve of power generation based on data points generated by a script I wrote (attached for reference, TideHeight1s is the source data for the script, the txt file contains the code for the .m script).
The...
I was looking at Kirchoffs Laws:
"A solid, liquid or dense gas produces a continuous spectrum".
I would expect objects to produce an emission spectrum since we would be observing the photons that come from spontaneous emission of electrons in excited states. This photons are specific to the...
I was reading introduction to quantum mechanics by DJ Griffiths and while discussing the formalism of quantum mechanics, he says that if for a hermitian operator, the eigenvalues are continuous, the eigenfunctions are non-normalizable whereas if the eigenvalues are discrete, then they can be...
I am trying to learn some topology and was looking at a problem in the back of the book asking to show that a topological space with the property that all set are closed is a discrete space which, as understand it, means that all possible subsets are in the topology and since all subsets are...
Suppose I have two audiofiles in 16 bit PCM, both recorded on a level that, except for the noise and distortion, is maximally recorded, or that the maximum recording level results in the maximum PCM level. So, the signal is recorded on the maximum level such that there is no clipping.
If we...