Discrete Definition and 897 Threads

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.

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  1. entropy1

    B Mixing two discrete audio signals

    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...
  2. C

    I Expected Value of 2^X and 2^-X for Geometric and Poisson Distributions?

    For the following distributions find $$E[2^X]$$ and $$E[2^{-X}]$$ if finite. In each case,clearly state for what values of the parameter the expectation is finite. (a) $$X\sim Geom(p)$$ (b) $$X\sim Pois(\lambda)$$ My attempt: Using LOTUS and $$E[X]=\sum_{k=0}^{\infty}kP(X=k)=\frac{1-p}{p}$$...
  3. Haorong Wu

    I Are time and space continuous or discrete?

    In another forum, some people argue that time and space are discrete, due to Planck time and Planck length. However, I disagree with this idea. I think, the Planck time and Planck length are just some scales that we can measure, but they do not forbid continuous time and space shorter than...
  4. Math Amateur

    MHB Understand Example 3.10 (b) Karl R. Stromberg, Chapter 3: Limits & Continuity

    I am reading Karl R. Stromberg's book: "An Introduction to Classical Real Analysis". ... ... I am focused on Chapter 3: Limits and Continuity ... ... I need help in order to fully understand Example 3.10 (b) on page 95 ... ... Example 3.10 (b) reads as follows: My question is as...
  5. wirefree

    Discrete-time Signal & Periodicity condition

    Namaste I seek a clarification on the periodicity condition of discrete-time (DT) signals. As stated in Oppenheim’s Signals & Systems, for a DT signal, for example the complex exponential, to be periodic, i.e. ej*w(n+N) = ej*w*n, w/2*pi = m/N, where m/N must be a rational number. Above is...
  6. C

    MHB Problem (c) for Discrete Value Ring

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  7. S

    MHB Discrete maths problem - tracing an algorithm

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  8. C

    MHB Discrete valuation Ring which is a subring of a field K Problem

    Dear Everyone, I am stuck in the middle of a proof. Here is the background information from Dummit and Foote Abstract Algebra 2nd ed.: Let $K$ be a field. A discrete valuation on $K$ on a function $\nu$: $K^{\times} \to \Bbb{Z}$ satisfying $\nu(a\cdot b)=\nu(a)+\nu(b)$ [i.e. $\nu$ is a...
  9. A

    A Do we need stochasticity in a discrete spacetime?

    Suppose that the spacetime is discrete, with only certain positions being possible for any particle. In this case, the probability distributions of particles have nonzero values at the points on which the wavefunction is defined. Do we need randomness in the transitions of particles in such a...
  10. F

    B How Do Discrete Derivatives and Integrals Work in Calculus?

    Hello all. I've come across some math which consists of just applying the basic ideas of calculus (derivatives and integrals) onto discrete functions. (The link: http://homepages.math.uic.edu/~kauffman/DCalc.pdf ) The discrete derivative with respect to n is defined as ## \Delta_n f(n) = f(n+1)...
  11. A

    A Why is this Pilot-wave model on a discrete spacetime stochastic?

    Look at the paper in the link below: https://link.springer.com/content/pdf/10.1007%2Fs10701-016-0026-7.pdf It introduces a pilot-wave model on a discrete spacetime lattice. However, the pilot-wave model is not deterministic; the motion of quantum particles is described by a |Ψ|^2-distributed...
  12. P

    I Space is discrete or continuous?

    According to QFT or Quantum Gravitation Theory space and time are discrete or continuous?
  13. I

    I Discrete Optimization Problem?

    Consider the expression:$$A = \frac{ M! }{ r_1!\ r_2! }$$ where M = r_1 + r_2 , where r_1 = (M - 2r_2) $$A = \frac{ (r_1 + r_2)! }{ r_1!\ r_2! } \\ \ \\ \ = \frac{ ((M-2r_2) + r_2)! }{ (M-2r_2)!\ (r_2)! } \\ \ \\ \ = \frac{ (M-r_2)! }{ (M-2r_2)!\ r_2! } $$ Then, for a...
  14. Boltzman Oscillation

    How can I prove this discrete signal is periodic?

    Homework Statement Prove the discreet signal is periodic: Homework Equations for periodic funtions: x[n] = x[n + N] The Attempt at a Solution I made an equality (im going to leave the sigma out for simplicity): 2^(-abs(n-2m)) = 2^(-abs(n+N-2m)) I don't know what I need to do from...
  15. D

    I Noether's theorem for discrete symmetry

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  16. weirdoguy

    Discrete K-G eq. solutions - problem with integral

    I'm reading Klaubers QFT book and I stuck with his derivation of Hamiltonian of scalar field on page 53. To derive it one needs to deal with integrals like this: $$\int\dot{\phi}\dot{\phi}^\dagger d^3x$$ He is using discrete plane-wave solutions and after plugging them in, we end up with...
  17. A

    I Discrete to continuum Gaussian function

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  18. binbagsss

    Elliptic functions, properties of periods, discrete subgroup

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  19. JackLee

    How to read a joint discrete table?

    Homework Statement [/B] Given a group of 100 married couples, let X1 be the number of sons and X2 the number of daughters the couple has. P(X1 = 0, X2 = 2) = f(0, 2) = 8 /100 = 0.08 2. Homework Equations The Attempt at a Solution I tried to look for a similar example online, I found this...
  20. H

    MHB Discrete Mathematics - Define a relation R on S of at least four order pairs

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  21. F

    Stability of singular points in a discrete control system

    Homework Statement Give an example of a non-linear discrete-time system of the form x1(k + 1) = f1(x1(k), x2(k)) x2(k + 1) = f2(x1(k), x2(k)) With precisely four singular points, two of which are unstable, and two other singular points which are asymptotically stable. Homework Equations J =...
  22. S

    Coriolis Drift of Discrete Objects

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  23. R

    Discrete Math implications by rules of inference

    Homework Statement p→(q→r) ¬q →¬p p ----------------------- ∴r Homework EquationsThe Attempt at a Solution My book gives the following solution: (1) p - premise (2) ¬q→¬p premise (3) q, (1) and (2) and rule of detachment, (4) p and q, law of conjuctive addition . . . Can anyone explain to me...
  24. D

    B Space is Discrete? Argument Explored

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  25. 9

    I Integrating Discrete Data for Navigation: A Comprehensive Guide

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  26. T

    I Informational content in 2D discrete Fourier transform

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  27. W

    I Is Non-Commutative Geometry the Key to a Discrete Model of Spacetime?

    Physics could be fundamentally discrete. Are their any notable theories that have discrete mathematics at its core and have QM, GR and differential equations in general as emergent features?
  28. T

    Is This Approach Valid for Proving the Discrete Metric in a Metric Space?

    Homework Statement Let ##x,y\in X## such that ##X## is a metric space. Let ##d(x,y)=0## if and only if ##x=y## and ##d(x,y)=1## if and only if ##x\neq y## Homework Equations N/A The Attempt at a Solution I have already seen various approaches in proving this. Although, I just want to know if...
  29. F

    Calculating the covariance of two discrete random variables

    Homework Statement If the random variables T and U have the same joint probability function at the following five pairs of outcomes: (0, 0), (0, 2), (-1, 0), (1, 1), and (-1, 2). What is the covariance of T and U? Homework Equations σxy = E(XY) - μx⋅μy The Attempt at a Solution My issue with...
  30. S

    B An elementary confusion on discrete or continuous variable

    The question is simply posed as " identity the variables as discrete or continuous. 1) Mark of a student in an examination. 2) Family income." What I think: 1) There must be a minimum gap between two possible consecutive marks that the examiner can assign. Eg. Suppose that there are N students...
  31. S

    A Can quantum cellular automata simulate quantum continuous processes?

    Can quantum cellular automata/quantum game of life simulate quantum continuous processes in the continuous limit? At the end of this article: https://hal.archives-ouvertes.fr/hal-00542373/document it is said that: "For example, several works simulate quantum field theoretical equations in the...
  32. A

    A Rigorous transition from discrete to continuous basis

    Hi all, I'm trying to find a mathematical way of showing that given a complete set $$\left |a_i\right \rangle_{i=1}^{i=dim(H)}∈H$$ together with the usual property of $$\left |\psi\right \rangle = ∑_i \left \langle a_i\right|\left |\psi\right \rangle\left |a_i\right \rangle ∀ \left...
  33. V

    B Is space is continuous or discrete?

    I was watching a video where well known physicist Lisa Randall said that we still don't know whether space is continuous or discrete. My question is, how do we find whether space is continuous or discrete?? What type of experiments are possible? Is it being done now?? I am thinking this may be...
  34. F

    Discrete logs and non generators

    Homework Statement This is just a question that i can't seem to answer while reviewing... Is discrete log well defined when the base is not a generator? Homework EquationsThe Attempt at a Solution For example, ##2^3 \equiv 2^6 (\operatorname{mod} 7)##. Taking the discrete log of both sides...
  35. TheQuestionGuy14

    B Is Spacetime a Discrete Lattice? - Prof. Sila Beane's Experiment

    Recently I found an experiment done by Prof. Sila Beane. In this experiment he simulates a tiny porportion of the universe using a QCD Lattice where spacetime is a discrete lattice. He inputs the GZK cutoff point of a cosmic ray into the lattice and what happens is, the result is that the...
  36. Derek P

    A Discrete Models for Arguments and Continuous Variables in Quantum Mechanics

    Arguments can often be presented using a discrete model on the assumption that continuous variables can be accommodated by taking the limit as the resolution is increased. I would have thought that this would be just fine in QM where functions are continuous. But maybe mathematicians here can...
  37. Sarina3003

    Counting Sequences with Repetition Using Stars and Bars Method

    Homework Statement The question is counting how many sequence length 10 with 1,2,3 if a) increasing from left to right with repetition allowed b) increase from left to right with each number appear at least once (still with repetition allowed) Homework Equations It is the stars and bars...
  38. M

    I Probability function for discrete functions

    My textbook says that if ##X: \Omega \to \mathbb{R}## is discrete stochast (I.e., there are only countably many values that get reached), then it suffices to know the probability function ##p(x) = \mathbb{P}\{X =x\}## in order to know the distribution function ##\mathbb{P}_X: \mathcal{R} \to...
  39. N

    I Why is the Signal from a Discrete Fourier Transform considered periodic?

    https://en.wikipedia.org/wiki/Discrete_Fourier_transform Why is the signal obtained from a DFT periodic? The time signal x[n] is finite and the number of sinusoids being correlated with it is finite, yet its said the frequency spectrum obtained after the DFT is periodic. I've also read the...
  40. Van Ladmon

    Mathematica How to add legends with 6 or more discrete entries?

    I want to add legends in ArrayPlot and my code is like this: Table[ArrayPlot[ FiniteGroupData[{"CyclicGroup", i}, "MultiplicationTable"], PlotLegends -> Range[i], ColorRules -> {x_ -> ColorData["Rainbow"][(x/i)]}], {i, 1, 10}] In the first 5 graphs this work well, but after the 6th graph...
  41. qttv

    A Classical Mechanics: Continuous or Discrete universe

    Good morning. The question of the "continuous" or "discrete" nature of the universe is the subject of diatribe among the greatest physicists in the world. I would like to discuss the same topic, but asking a question about the aspect of continuum in classical mechanics. The use of mathematical...
  42. J

    Testing my Discrete Fourier Transform program

    Homework Statement I've written a program that calculates the discrete Fourier transform of a set of data in FORTRAN 90. To test it, I need to "generate a perfect sine wave of given period, calculate the DFT and write both data and DFT out to file. Plot the result- does it look like what you...
  43. CopyOfA

    A Linear regression with discrete independent variable

    Hey, I have a problem where I have a discrete independent variable (integers spanning 1 through 27) and a continuous dependent variable (50 data points for each independent variable). I am wondering about the best method of regression here. Should I just fit to the mean or median? Is there a way...
  44. B

    I Continuous Lensing Models: Discrete Data

    Hello, I am not sure if this question is better suited to the mathematics section, but I thought it would be easier to explain the problem here. In Schneider, Kochanek and Wambsganss's "Gravitational Lensing: Strong Weak and Micro" pages 279-280, they derive a relation for determining the...
  45. RJLiberator

    Understanding Discrete and Continuous PMFs: Exploring the Differences

    Homework Statement Let P_{x}(x) = \frac{x}{15}, x = 1,2,3,4,5 ; 0 elsewhere be the pmf of X. Find P(X=1 or 2), P(1/2 < X < 5/2), P(1 ≤X≤2). Homework EquationsThe Attempt at a Solution I believe what this problem is trying to show is the difference between discrete and continuous Pmf...
  46. A

    Bizarre Behavior in Discrete PMOS

    Some of my colleagues and I captured the i_D vs V_ds/V_sd curves for the CD4007 MOSFET IC (http://www.ti.com/lit/ds/symlink/cd4007ub.pdf). We did this for the NMOS and PMOS transistors. I have attached the curves to this post. The NMOS curve is as expected. However, the PMOS curve seems to be...
  47. J

    Principle of virtual work for continuous systems

    I always thought that the principle of virtual work (PVW) is valid for all structures, including continuous structures (like bars, beams, plates, etc.). However, in his book 'Fundamentals of Structural Mechanics', Hjelmstad states that the PVW is only valid for discrete systems with N particles...
  48. F

    Can Discrete Logarithms Be Added in Modular Arithmetic?

    Homework Statement Let g be a primitive root for ##\mathbb{Z}/p\mathbb{Z}## where p is a prime number. b) Prove that ##\log_g(h_1h_2) = \log_g(h_1) + \log_g(h_2)## for all ##h_1, h_2 \epsilon \mathbb{Z}/p\mathbb{Z}##. Homework Equations Let x, denoted ##\log_g(h)##, be the discrete logarithm...
  49. N

    A Averaging over the upper sum limit of a discrete function

    Hi, Let the following function: X = ∑^{L}_{k=1} f(k)/L, where f(k) is a continuous random function and L is a random discrete number. Both L and f(k) are non negative random variables. Thus, X is the average of f(k) with respect to L. Is it right to say that X equals (or approximately) to...
  50. Joppy

    MHB Discrete dynamical systems - Invertible maps

    Hi (Sleepy), I suspect this is trivial, but I couldn't find any info onlin. Consider the folowing map: $\phi_{n+1} = f(\phi_n ; \Theta, a) = (\phi_n + \Theta + a \sin \phi_n) \mod 2\pi$. I need to check if is invertible: $\phi_n = f^{-1} (\phi_{n+1}; \Theta, a)$ when a = 1/2 or 3/2...
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