Discrete Definition and 829 Threads

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...

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...

Problem (c) for Discrete Value Ring for a unit 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...

Hello, I'm working on a discrete mathematics for computing paper and am stuck on what a symbol is trying to convey. Sorry if this seems like a stupid question (I feel stupid for not being able to work it out myself), I've just started this subject and am still getting used to it. My question...

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...

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...

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)...

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...

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

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...

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...

I am wondering if it existes some discret version of the Noether symmetry for potential with discrete symmetry (like $C_n$ ). The purpose is to describe the possible evolution of the phase space over the time without having to solve equations numerically (since even if the potential may have...

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...

I have a question regarding a paragraph in "Radiation detection and measurement" by Knoll. In the chapter about the discrete Gaussian it states that "Because the mean value of the distribution ##\bar{x}## is large , values of ##P(x)## for adjacent values of x are not greatly different from each...

Homework Statement HiI am following this proof attached and am just stuck on the bit that says: ‘since ##\Omega## is a group it follows that ##|z-\omega|<2\epsilon ## contains..’Tbh, I have little knowledge on groups , it’s not a subject I have really studied in any of my classes-so the only...

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...

Let S = {1,2,5,6 } Define a relation R on S of at least four order pairs, as (a,b)  R iff a*b is even (i.e. a multiply by b is even)

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 =...

I am writing a program aimed at 'gun nuts' designed to display Coriolis Drift of bullets (ie, 'discrete objects in free-fall', and not large fluid masses). Using the 2 equations below, I am able to calculate and display the values of Coriolis Drift (in terms of X & Y (vertical and horizontal))...

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...

I be grateful for any feedback on this argument: - First assume space is continuous - Then there is an actually infinite amount of information in a spatial volume of 10000 cubic units - There is also an actually infinite amount of information in a spatial volume of 1 cubic unit - But this is a...

I have values for the variables (C, v, g, w at all sample points) but I do not know how to evaluate the integral. This equation is supposed to be implemented on a computer as part of a larger algorithm for navigation purposes. I have a feeling that the gyroscope sensor reading and or the...

When you do a discrete Fourier transform (DFT) of a one-dimensional signal, I understand that the second half of the result is the complex conjugate of the first half. If you threw out the second half of the result, you're not actually losing any data and you would be able to recreate the entire...

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?

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

Hey guys I was just studying for finals and I came across something in my textbook that either wasn't explained properly or something I just don't get. So this page explains that the coefficient of y_n[k] is A_0, which is represented by b_0/a_0, I can see that a_0 is determined by multiplying...

Homework Statement Find ##f:R \to X##, f-continuous, where X is the discrete space. Homework EquationsThe Attempt at a Solution f is continuous at p if for any ##\epsilon > 0## there is ##\delta >0## such that ##d(f(x),f(p))<\epsilon## for all x such that ##d(x,p)<\delta##. Let ##\epsilon =...

Homework Statement I need to calculate the derivative of a function using discrete Fourier transform (DFT). Below is a simplified version of my code (just for sin function) in python Homework Equations from __future__ import division import numpy as np from pylab import * pi = np.pi def...

Let's consider a signal which is continuous in both time and amplitude. Now we consider the amplitude of this signal at specific time instants only. This is my understanding of sampling a signal in time domain. When performing a Fourier transform on a time discrete signal, we have to apply the...